Time-domain control of light-matter interaction with superconducting ...

Technische Fakultät Walther-Meißner- BayerischeUniversität für Institut für Akademie derMünchen Physik Tieftemperaturforschung Wissenschaften

Time-domain controlof

light-matter interactionwith

superconducting circuits

Diploma ThesisThomas Losinger

Advisor: Prof. Dr. Rudolf GrossGarching, 2012-11-07

Time-domain control of light-matter interaction with superconducting circuits

Contents

List of Figures iii

List of Tables v

List of Symbols and Abbreviations vii

1 Introduction and Motivation 1

2 Theoretical background 52.1 Josephson physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Josephson junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.2 RCSJ model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.3 3 Josephson junction flux qubit . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Quantum harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Jaynes-Cummings model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4 Dynamic and decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Experimental setup 233.1 Cryogenic setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.1 Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.1.1.1 Superconducting niobium resonator and antenna . . . . . . . . . 233.1.1.2 3 Josephson junction flux qubit and coupling junction . . . . . . 25

3.1.2 Cryostat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.1.2.1 Microwave input lines . . . . . . . . . . . . . . . . . . . . . . . . 273.1.2.2 Microwave output line . . . . . . . . . . . . . . . . . . . . . . . . 293.1.2.3 DC lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Room temperature setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2.1 Continuous wave spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2.1.1 Single tone continuous wave spectroscopy . . . . . . . . . . . . . 313.2.1.2 Two-tone continuous wave spectroscopy . . . . . . . . . . . . . . 32

3.2.2 Pulsed wave spectroscopy and time-domain measurements . . . . . . . . . 333.2.2.1 Pulse generation . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2.2.2 Pulse detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4 Measurement results 474.1 Continuous wave spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.1.1 Flux calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.1.2 High power continuous wave spectroscopy . . . . . . . . . . . . . . . . . . 514.1.3 Photon number calibration . . . . . . . . . . . . . . . . . . . . . . . . . . 534.1.4 Low power continuous wave spectroscopy . . . . . . . . . . . . . . . . . . 57

4.2 Time-domain measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.2.1 ACQIRIS card measurements . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.2.1.1 Pulsed two-tone spectroscopy . . . . . . . . . . . . . . . . . . . . 62

i


4.2.1.2 Rabi oscillation measurements . . . . . . . . . . . . . . . . . . . 644.2.2 FPGA board measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.2.2.1 Rabi oscillation measurements . . . . . . . . . . . . . . . . . . . 69

5 Conclusion and Outlook 77

6 Acknowledgments 79

Bibliography 81

A Digital heterodyne IQ mixer calibration 85A.1 Mathematical calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85A.2 MATLAB code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

B Photon number calibration 101

C Persönliche Erklärung 105

ii


List of Figures

2.1 Josephson junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 RCSJ model of a Josephson junction . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Tilted washboard potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 3 Josephson junction flux qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 Potential of a 3 Josephson junction flux qubit . . . . . . . . . . . . . . . . . . . . 112.6 Qubit hyperbola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.7 Energy levels of a multi-mode resonator . . . . . . . . . . . . . . . . . . . . . . . 142.8 Lumped element circuit for a superconducting resonator . . . . . . . . . . . . . . 162.9 Energy levels of the Jaynes-Cummings Hamiltonian . . . . . . . . . . . . . . . . . 202.10 The Bloch sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.11 The π-pulse pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.1 The chip design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2 Shadow evaporation techique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3 Schematic of the experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . 263.4 Photograph of the cryogenic stage . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.5 Continuous wave spectroscopy setup . . . . . . . . . . . . . . . . . . . . . . . . . 323.6 Functional principle of two-tone spectroscopy . . . . . . . . . . . . . . . . . . . . 333.7 Pulse generation setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.8 IQ detector setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.9 Functional principle of an IQ mixer . . . . . . . . . . . . . . . . . . . . . . . . . . 363.10 Effects of the digital filters on ACQIRIS card data . . . . . . . . . . . . . . . . . 373.11 Origin of the ACQIRIS card artifacts . . . . . . . . . . . . . . . . . . . . . . . . . 393.12 1 dB compression point of amplifiers in the IQ detector . . . . . . . . . . . . . . . 403.13 IQ mixer calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.14 Recorded raw data in comparison to calibrated data . . . . . . . . . . . . . . . . 443.15 Reconstructed amplitude and phase for a calibration pulse . . . . . . . . . . . . . 45

4.1 Second harmonic resonator mode . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.2 Flux calibration: Full range scan . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.3 Flux calibration: Detailed scan of the anticrossings . . . . . . . . . . . . . . . . . 504.4 Single tone continuous wave spectroscopy at high power . . . . . . . . . . . . . . 514.5 Numerical fit of the Jaynes-Cummings Hamiltonian to high power data . . . . . . 524.6 Photon number calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.7 Low power continuous wave spectroscopy with fits . . . . . . . . . . . . . . . . . 574.8 Low power two-tone spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.9 Fittet energy spectrum of the sample . . . . . . . . . . . . . . . . . . . . . . . . . 614.10 Pulsed wave two-tone spectroscopy cabling and pulse patterns . . . . . . . . . . . 624.11 Pulsed wave two-tone spectroscopy measurement result . . . . . . . . . . . . . . . 634.12 Time-domain measurements cabling and pulse patterns . . . . . . . . . . . . . . . 644.13 Recorded quadratures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.14 Amplitude and phase of a time-domain measurement with the ACQIRIS card . . 664.15 Energy relaxation times for an ACQIRIS card measurement . . . . . . . . . . . . 67

iii


4.16 ACQIRIS and FPGA measurement in comparison . . . . . . . . . . . . . . . . . . 694.17 Energy relaxation times for a FPGA board measurement . . . . . . . . . . . . . . 704.18 Heaviside window function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.19 Rabi osclillations for different drive power values . . . . . . . . . . . . . . . . . . 734.20 Fourier spectrum and power dependence of the oscillations . . . . . . . . . . . . . 744.21 Rabi oscillation measurement detuned form degeneracy point . . . . . . . . . . . 75

A.1 Points in the real projective plain RP2 . . . . . . . . . . . . . . . . . . . . . . . . 86A.2 IQ mixer calibration: The transformations step by step . . . . . . . . . . . . . . . 88A.3 IQ mixer calibration: Amplitude extraction . . . . . . . . . . . . . . . . . . . . . 89A.4 IQ mixer calibration: Signal vs. Time . . . . . . . . . . . . . . . . . . . . . . . . 93

iv


List of Tables

3.1 Maximum power dissipation per temperature stage . . . . . . . . . . . . . . . . . 273.2 Resonator’s thermal noise and dissipated power . . . . . . . . . . . . . . . . . . . 283.3 Antenna’s thermal noise and dissipated power . . . . . . . . . . . . . . . . . . . . 28

4.1 Resonator modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.2 Flux calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.3 Photon numbers on average and corresponding probe powers . . . . . . . . . . . 564.4 Fit parameters for high and low power spectroscopy in comparison . . . . . . . . 584.5 Vacuum Rabi levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

v


vi


List of Symbols and Abbreviations

a(~r,t) amplitude of a wave functiona†, a photon creation and annihilation operatorA(t) time dependent amplitude extracted from an IQ measurement~A vector potentialα relative form factor of the third Josephson junction of the qubitAC alternating currentβ relative form factor of the coupling junctionc velocity of light in vacuum c = 2.99792 · 108 m/sC capacity in the RCSJ modelCk capacity per unit length in the RCSJ modelCin capacity of the input port of the resonatorCout capacity of the output port of the resonatorCPW coplanar waveguideδΦ detuning of the flux from half a flux quantaδωi detuning of the mode i form the qubit excitation frequency ωQ

∆ energy gap of a qubitd thickness of the insulating layer in a Josephson junctionDC direct currentDTG data timing generatore elementary charge e = 1.60218 · 10−19 Cε flux dependent energy bias of a qubit

EJD potential energy of a driven Josephson junctionEJS potential energy of an undriven Josephson junctionEJ0 Josephson energy

FWHM full width half maximumΓ1 energy relaxation rate of the qubitΓ2 total dephasing rate of the qubit

ΓDS energy relaxation rate of the dressed stateEn excitation energy of n photons in a resonatorEQ qubit potentialEQE qubit excitation energygi coupling strength of the qubit to the i-th resonator mode

|g〉 , |e〉 ground and excited state of the qubitHi Hilbert space of a quantum harmonic oscillator with mode iHdisp

JC Jaynes-Cummings Hamiltonian in the dispersive limitHJC Jaynes-Cummings HamiltonianHMM multi-mode harmonic oscillator HamiltonianHMMJC Hilbert space of a multi-mode Jaynes-Cummings Hamiltonian

vii


HNLT transmission line Hamiltonian with a nonlinearityHSM single mode harmonic oscillator HamiltonianHSMJC Hilbert space of a single mode Jaynes-Cummings HamiltonianHQ qubit HamiltonianHQ Hilbert space of a qubitHZPE zero point energy of a multi-mode harmonic oscillator Hamiltonian~ reduced Planck constant ~ = h/2π = 1.05457 · 10−34 Js

HEMT high electron mobility transistorI external driving currentIC critical supercurrent of a Josephson junctionID displacement current trough the capacitively part of the RCSJ modelIP persistent current of a qubitIR current trough the resistively part of the RCSJ modelIS supercurrent of a Josephson junctionkB Boltzmann constant kB = 1.38065 · 10−23 J/Kκi loss rate of the i-th resonator model geometrical length of the resonatorLk inductance per unit length in the RCSJ modelLJ Josephson inductanceLS inductance of a Josephson junctionM mutual inductanceme electron mass me = 9.10938 · 10−31 kgm∗ effective mass of a particlenCP density of Cooper pairsni photon number operator of mode inPh number of thermal photons in the resonator|n,±〉i dressed state of mode i with n photons and the qubitωi angular frequency of the cavity mode iωIF intermediate frequency of an IQ mixerωJ Josephson frequencyωLO local oscillator frequency of an IQ mixerωP probe tone frequencyωQ qubit excitation frequencyωQ dispersively shifted qubit excitation frequencyωS spectroscopy tone frequencyωSa sampling frequency of a data acquisition cardφ gauge invariant phase difference over a Josephson junctionϕ Bloch azimuthal angleΦ magnetic flux through the qubit loopΦ0 magnetic flux quantum Φ0 = h/2e = 2.06783 · 10−15 WbΦ scalar potential in the Schrödinger equationΨ wave function of a superconductor|Ψ〉 Bloch vectorPSG Agilent E8267D PSG

viii


poa photons on averageq charge

QED quantum electrodynamicsQIP quantum information processingρ(t)) time dependent phase extracted from an IQ measurementRCSJ resistively capacitively shunted junctionRF radio frequencyσx, σz Pauli operatorsSMF Rohde & Schwarz SMF100ASNR signal to noise ratioT1 energy relaxation time of the qubitTDS energy relaxation time of the dressed stateτ pulse width in a time-domain measurementΘ Bloch polar angleϑ mixing angleR resistor in the RCSJ modelU voltage drop over a Josephson junctionUV ultravioletVNA vector network analyzerWMI Walther-Meissner-Institutξ(~r,t) phase of a superconductor’s wave functionZ0 characteristic impedance of an element in the RCSJ model

ix


x


1 Introduction and Motivation

In classical computation the information is stored in the classical states 0 and 1. Consider-ing a quantum computer where the information is processed in quantum mechanical two levelsystems with eigenstates |g〉 and |e〉 the situation is more complex, as the system is no longerdescribed by classical electrodynamics but by quantum mechanics. Already in the 1980s it wasproposed by Feynman and Deutsch to use quantum mechanical phenomena for simulation [1]and computation [2]. These two proposals have given rise to the field of quantum informationprocessing (QIP). In quantum computation one solves problems that would not be computablewith classical computers in a suitable time. A famous example is the prime factorization of largenumbers via Shor’s algorithm [3]. The fact that the time it takes to factorize a large numbergrows exponentially with the length of the key on a classical computer is a key element of cryp-tography where the key is produced from the product of two prime numbers. To decrypt theinformation without knowledge of the two prime numbers one has to factorize the key. If the keyis long enough this is not computable on a classical computer within a suitable amount of time.But if a quantum computer and Shor’s algorithm is used the time increases only polynomiallywith the length of the key. It was recently demonstrated that 143 could be factorized with aNMR system [4] and 15 was successfully factorized in a superconducting phase qubit system [5].A second application for QIP is quantum simulation [6, 7], where one quantum system whichis hard to investigate is mapped onto a different quantum system that could be well controlledand realized more straight forwardly. For example one can simulate a relativistic quantum sys-tem with a non-relativistic quantum system [8, 9]. For a functional quantum computer we needquantum bits (qubits) which fulfill the DiVincenzo criteria [10] which were proposed in 2000.These criteria demand the existence of a quantum two level system which can be used as a qubitand that it can be prepared in an initial state. Further the decoherence times have to be largeenough such that a sufficient number of operations can be performed within that time. Theseoperations are realized by quantum gates. After the operations have been applied to the systemthe readout has to be performed with high accuracy. Finally the system should be extendableto a larger number of qubits easily. There are several approaches to implement qubits which tryto fulfill all these criteria.

Promising realizations can be found in the field of cavity quantum electrodynamics (QED). Hereone uses highly reflective mirrors to confine an optical light field which interacts with trappedatoms or ions. These systems allow to investigate the interaction of the atom/ion with the lightfield. This interaction is characterized by the coupling strength which should be large in order totransfer photons efficiently from the light field to the atom such that the atom is excited. In thisfield it has been shown that one can store information in a single atom [11] and that it is possibleto entangle up to 14 qubits [12]. The main benefit of these approaches is the relatively longcoherence time of several milliseconds [13] while the increase of the coupling strength is still anissue. With this approach it is also possible to perform a quantum non demolition measurementso it is possible to read out the quantum state without destroying the quantum state. For thedevelopment of such a measurement technique where the wave function does not collapse throughthe readout Serge Haroche and David J. Wineland were awarded the Nobel Prize in 2012.

1


Another promising approach to study light-matter interaction is circuit QED. Here artificialatoms (qubits) such as flux qubits [14, 15, 16, 17] or transmons [18, 19, 20] are realized witha superconducting circuit. A superconducting resonator forms the equivalent to the highly re-flective mirrors in cavity QED. Here the confined photon field is shifted from the optical to themicrowave regime. The transition frequency of the artificial atom and the resonance frequency ofthe resonator are designed via electronic engineering which yields a large experimental ground.This approach is very favorable since the needed micro and nano fabrication technology is al-ready available. A flux qubit can be realized with a superconducting loop which is interruptedby 3 Josephson junctions [14, 15]. A flux can be trapped in the loop. This flux is used to varythe transition frequency of the artificial atom. In circuit QED the coherent transfer of photonsin a three-resonator circuit was recently demonstrated [21]. The coherence times still are notsufficiently long since they are in the order of tens of microseconds [22, 23]. The coherence timesare very important because only within this time one can perform algorithms. If one wants toread out the information after a certain time t longer than the coherence time the informationis lost.

Two important components of a quantum computer are, first quantum bits (qubits) where theinformation is processed and second a bus system such that information can be transferred fromone qubit to another. To perform operations it is essential to have gates with a very high fi-delity. A fundamental gate in QIP is the so called controlled-NOT (CNOT) gate [24]. Anotherpromising gate where a high fidelity could be achieved is the Mølmer-Sørenson gate [25] sinceit is based on second order transitions. Gates are realized by applying pulse sequences on thequbits. These pulses have to be performed within the coherence time of the system. Therefore,one can estimate the number of possible operations by the quotient of the coherence time andthe time needed to perform one gate operation. Since the time for the gate operation is more orless fixed it is essential to increase the coherence time to compute complex quantum algorithms.At the WMI Josephson junction flux qubits have been successfully investigated spectroscopicallyfor several years [17, 26, 27]. Up to now the coherence times of these qubits have not beeninvestigated in details.

At the WMI three Josephson junction flux qubits are used since they yield larger couplingstrengths than for example transmons. It was demonstrated [17] that the ultra strong couplingregime can be realized by using a coupling junction in a shared edge of the qubit and the res-onator. The size of the coupling junction influences the coupling strength. A current field ofresearch at the WMI is the determination of the maximal coupling strength of a flux qubitto a superconducting resonator. So the first issue we address in this thesis is the determina-tion of the coupling strength for an area of the coupling junction different than in ref. [17] infrequency-domain measurements. The second issue we would like to address in this thesis is themeasurement of the coherence times of a coupled qubit resonator system. Therefore, we have tobuild up, characterize and calibrate a detector for time-domain measurements.

2


In this thesis we successfully measure the energy relaxation time of qubits in a coplanar trans-mission line resonator fabricated at the WMI for the first time. Therefore, we introduce thetheory necessary to describe the artificial atom and the light field in the resonator. We use theJaynes-Cummings model [28] which was derived in the field of quantum optics but also holdsfor our artificial system. Furthermore, we introduce the physics of the measurement techniqueused in two-tone spectroscopy and the time-domain measurements. For the latter one we give ashort introduction into the loss mechanisms of the system since these determine the coherencetimes. Afterwards we proceed with a detailed description of the used measurement setup. Weintroduce the production steps of our sample, depict the configuration of the measurement setupinside the dilution refrigerator and the room temperature setup. Furthermore, we concentrate onthe detector which we built up for the time-domain measurements, here we perform a detailedcharacterization and introduce a digital heterodyne IQ mixer calibration algorithm which wedeveloped during the course of this thesis. Finally we present the experimental results of theperformed measurements. We begin with detailed continuous wave single and two-tone spec-troscopy such that we can calibrate the applied flux to the qubit and the photon number in theresonator. Afterwards we perform spectroscopy measurements with negligible photon numberin the readout resonator mode. From a fit of the Jaynes-Cummings Hamiltonian to the dataacquired we are able to determine the coupling strengths gi of the qubit to the resonator modei. Furthermore, we gain knowledge of the characteristic frequencies of our system such that wecan apply a signal of the correct frequency in the time-domain measurements. During thesemeasurements we observe an oscillatory behavior of the response of the sample with the lengthτ of the applied pulse which strongly indicates the observation of Rabi oscillations.

3


4


2 Theoretical background

In this thesis we investigate a coupled qubit resonator system. This quantum mechanical systemis theoretically described by the Hamiltonian of the Jaynes-Cummings model [28]. In this chapterwe introduce the basic ingredients needed to develop the Jaynes-Cummings Hamiltonian i.e. the3 Josephson junction flux qubit and the quantum mechanical oscillator with multiple modeexcitations. Therefore, we start with an introduction to Josephson physics of one Josephsonjunction and develop a model for a 3 Josephson junction flux qubit. We proceed with a descriptionof the single mode quantum harmonic oscillator and expand it to a multi-mode picture. In thenext step we use a fourth Josephson junction located at the shared edge of the qubit and theresonator. We treat this coupling junction as a nonlinearity in the resonator which shouldenhance the coupling. In the last part of this chapter we provide the basic theory necessary tointerpret the time-resolved measurement results of section 3.2.2.

2.1 Josephson physics

The history of superconductivity began in 1911 when Heike Kamerlingh Onnes discovered thedisappearance of the resistance of mercury at liquid helium temperature [29, 30]. In 1935,the London brothers developed a phenomenological theory of superconductivity [31] where theyintroduced a macroscopic wave function of the form

Ψ(~r,t) = a(~r,t) exp(iξ(~r,t)

). (2.1)

With a time and space dependent amplitude a(~r,t) and phase ξ(~r,t). For better readability wedrop the dependence on (~r,t) in the rest of this thesis. This ansatz is used to solve the timedependent Schrödinger equation

(i~∂

∂t− qΦ

)Ψ =

(−i~~∇− q

c~A)2

2m∗Ψ (2.2)

with the reduced Planck constant ~ = h/2π = 1.05457 · 10−34 Js [32], the charge q, the scalarpotential Φ, the velocity of light in vacuum c = 2.99792 · 108 m/s [32], the vector potential ~Aand the effective mass m∗. It took more than 20 years before Bardeen, Cooper, and Schriefferwere able to developed a microscopic theory [33, 34] to solve the time dependent Schrödingerequation (2.2) in 1957. For their BCS theory they received the Nobel Prize in 1972. They foundthat the amplitude of the wave function in equation (2.1) is given as

a =√nCP. (2.3)

Where nCP is the density of the Cooper pairs in the superconductor. Cooper pairs are quasipar-ticles formed by two electrons in momentum space. As a result the charge is given as q = 2e andthe effective mass m∗ = 2me in equation (2.2) with the elementary charge e = 1.60218 · 10−19 C[32] and the mass of the electron me = 9.10938 · 10−31 kg [32]. So the wave function

Ψ =√nCP exp (iξ) (2.4)

describes the condensate of all Cooper pairs and not the single particles. Thus we have a beautifulexample for quantum mechanics on a macroscopic scale.

5


2.1.1 Josephson junction

Already in 1962 Brian Josephson considered two superconducting materials (SC) with wavefunctions Ψ1,2 =

√n1,2 exp (iξ1,2) separated by a thin insulating layer I with thickness d [35] as

depicted in Figure 2.1. If we observe coherent Cooper pair tunneling we call such a configurationa Josephson junction.

Figure 2.1: A sketch of a Josephson junction. The two superconductors with the macroscopicwave functions Ψ1 and Ψ2 are separated by a thin insulating layer I. For a given insulatingmaterial, the thickness d determines the tunneling rate of the Cooper pairs because it determinesthe overlap of the two wave functions.

He realized that due to the macroscopic wave function and the formation of Cooper pairs in themomentum space tunneling is a coherent process over a length of several thousands of atoms(due to Heisenberg’s uncertainty relation [36]) and as a result the Cooper pairs are able to tunnelthrough the thin barrier if the thickness d is not too large such that the wave functions on bothsides overlap. For this work and the prediction of the Josephson effects via equations (2.5) and(2.6) he was awarded the Nobel Prize in 1973.

IS = IC sin (φ) (2.5)∂φ

∂t=

2π

Φ0U (2.6)

So the supercurrent IS over the junction is determined by a critical supercurrent IC and thegauge invariant [37] phase difference φ of the two wave functions of the superconductors

φ := ξ2 − ξ1 −2π

Φ0

∫ 2

1

~Ad~l. (2.7)

The relation (2.5) is known as the first Josephson equation or current phase relation. Thisphenomenon is also known as DC Josephson effect because a DC current can flow without anyexternal applied voltage. The second Josephson equation (2.6) or voltage phase relation combinesthe partial time derivative of the phase difference φ with the voltage U across the junction via theflux quantum Φ0 = 2.06783 · 10−15 Wb. If the voltage V is time-independent, we can solve thepartial differential equation (2.6) via separation of variables. With the result we enter equation(2.5) and end up with the AC Josephson effect (2.8).

IS = IC sin

(φ0 +

2πU

Φ0t

)(2.8)

Equation (2.8) is a remarkable example for a nonlinear effect in electronic circuits. Here a DCvoltage induces an alternating current with Josephson frequency ωJ = 2πU/Φ0. Since we arenow familiar with the Josephson effects we can proceed with a derivation of the potential energyEJS stored in the junction. This is done by integrating the power with respect to time from timet′ = 0 where the phase difference φ′ = 0 to a certain time t where the phase difference is φ [38].

EJS =

∫ t

0Pdt ′ =

∫ t

0ISUdt ′ =

∫ t

0

ICΦ0

2πsin(φ′) ∂φ′∂t′

dt ′ =ICΦ0

2π

∫ φ

0sin(φ′)

dφ′

= EJ0 (1− cos (φ))

(2.9)

6


Where we introduced the Josephson energy EJ0 = ICΦ0/2π. Equation (2.9) represents an energywhich is 2π periodic in φ with minima for 2πn and n ∈ Z. The fact that energy can be storedin the junction also indicates that a Josephson junction is a nonlinear device. We use relation(2.9) in sections 2.1.3 and 2.2 again. Another useful quantity we can derive from the Josephsonequations (2.5) and (2.6) is the so called Josephson inductance LJ. Therefore, we have a look atthe time derivative of the first Josephson equation (2.5)

dIS

dt= IC cos (φ)

dφ

dt= IC cos (φ) .

2π

Φ0U (2.10)

When we rearrange this equation we end up with

LS = U

(dISdt

)−1

=Φ0

2πIC cos (φ)= LJ

1

cos (φ), (2.11)

where LJ = Φ0/(2πIC) denotes the Josephson inductance. So the ideal Josephson junction actsas a nonlinear inductance where the energy EJS is stored in the inductance LS [38].

2.1.2 RCSJ model

To model a Josephson junction such that it is usable for electronic engineering it is necessaryto develop an equivalent circuit diagram (see Figure 2.2). This was achieved with the so calledresistively and capacitively shunted junction model (RCSJ model) which was developed byStewart and McCumber in 1968 [39, 40].

Figure 2.2: The Josephson junction in the RCSJ model. The current source applies a current Ito the junction (dashed box) which is modeled as a parallel circuit of a normal resistance R, anideal Josephson junction with critical current IC and a capacitor with capacity C. The voltageU drops on that equivalent circuit.

The current I generated by the source splits up into three parts, one enters the resistor such thatIR = U/R, the second flows through the junction IS = IC sin (φ) and the third is a displacementcurrent in the capacitor ID = CdU/dt. Where R denotes the normal resistance and C thecapacity of the junction. By applying Kirchhoff’s current law [41], we obtain the equation

I =∑

i=S,R,D

Ii = IC sin (φ) +U

R+ C

dU

dt. (2.12)

In the small junction limit the spatial derivative of the phase difference φ over the junction isnegligible [42] as a result the total and the partial time derivatives are identical such that wecan use the second Josephson equation (2.6) to replace the voltage U by the phase difference φor vice versa [37] and end up with

I = IC sin (φ) +Φ0

2πR

dφ

dt+CΦ0

2π

d2φ

dt2. (2.13)

7


Equation (2.13) is a nonlinear differential equation of second order without an analytical solutionexcept for the case when the capacity vanishes such that it is reduced to first order [37]. However,by multiplying equation (2.13) with the constant term Φ0/2π and neglecting the dissipation term∝ dφ/dt, we end up with the equation

C

(Φ0

2π

)2 d2φ

dt2=

Φ0

2πI − EJ0 sin (φ) . (2.14)

Equation (2.14) is of the same type as m~a = ~F = −~∇EJD and we can treat the phase differenceφ as a phase particle. It can be shown [38] that the potential EJD when the junction is drivenby a current is given as

EJD = EJ0

(1− cos (φ)− I

ICφ

). (2.15)

The potential described by equation (2.15) is called tilted washboard potential. In Figure 2.3 wedepict the potential for various values of the driving current I.

Figure 2.3: The so called tilted washboard potential for a driven Josephson junction in theRCSJ model. When the driving current I is turned off, the potential is identical to the undrivenpotential EJS given in equation (2.9) and tunneling of a phase particle in both directions ispossible with the same probability (blue line). If we turn on the current and are below thecritical current IC of the junction tunneling into one direction is more probable than in the otherdirection (red line). If the driving current is higher than the critical current the potential doesno longer exhibit stable minima (magenta line).

8


In the case that the driving current I is switched off (see blue line of Figure 2.3), we end upwith the potential of the undriven junction EJS given in equation (2.9) where the equilibriumis stable and tunneling of a phase particle in both directions appears with the same probabilitydue to the height of the potential barrier and its width. If we slowly increase the driving currenttunneling into one direction becomes more favorable than in the other (red line) because thepotential barriers have different heights. When the driving current I is equal to the criticalcurrent IC of the junction (green line) the system is in an unstable equilibrium because aninfinitesimal distortion results in an acceleration of the phase particle. If the current is furtherincreased the system is no longer stable (magenta line). A great advantage of the RCSJ modelis that it describes the dynamics of the junction quite well without a detailed description of themicroscopic current transport [37].

2.1.3 3 Josephson junction flux qubit

In this section we introduce the 3 Josephson junction flux qubit [14, 15], which serves as anartificial atom and is one of the main components of our sample of a coupled qubit resonatorsystem. Therefore, we have a look at the qubit potential and its dependencies, further on weidentify the ground |g〉 and excited state |e〉 with the direction in which the persistent current IP

is flowing. If we place 3 Josephson junctions in series and connect the two open ends, we end upwith a superconducting ring interrupted by 3 Josephson junctions. Such a device can be used asa flux qubit. We assume that two of the junctions are equal and the third is smaller by a designfactor α with respect to the area of the junction [14, 15] as depicted in Figure 2.4.

Figure 2.4: Sketch of a three Josephson junction flux qubit. A superconducting loop is intersectedby three Josephson junctions (blue blocks). Two of the junctions are designed to be equal suchthat they have the same potential energy EJ0, capacity C and inductance LJ. The third differsby a parameter α ≈ 0.7 with respect to the area of the junction. The ground |g〉 and exitedstate |e〉 are coded in clockwise (green arrow) or counter clockwise (magenta arrow) rotation ofthe persistent current IP .

The ground |g〉 and excited state |e〉 are coded in clockwise (green arrow) or counter clockwise(magenta arrow) rotation of the persistent current IP which is caused by the flux Φ trapped inthe qubit loop. If the inductance of the superconducting loop is small compared to the Josephsoninductance LJ of the junctions we can express the potential energy of the loop as the sum of thepotential energies of each junction derived in equation (2.9)

EQ = EJ0 (2 + α− cos (φ1)− cos (φ2)− α cos (φ3)) , (2.16)

where i = 1, 2 indicates the two equal junctions and i = 3 is the so called α-junction.

9


Further it is justified to use fluxoid quantization which was discovered by R. Doll and M. Näbauerat the Walther-Meissner-Institut (WMI) [43] and by B.S. Deaver and W.M. Fairbank [44] in 1961.Hence we find the relation [15]

φ1 − φ2 + φ3 = −2πΦ

Φ0. (2.17)

We use equation (2.17) to eliminate φ3 from equation (2.16)

EQ = EJ0

(2 + α− cos (φ1)− cos (φ2)− α cos

(2π

Φ

Φ0+ φ1 − φ2

)). (2.18)

Since the significant variations of the persistent current IP occur in a flux range of several mΦ0

around the degeneracy point of the qubit at (n+ 1/2) Φ0 with n ∈ Z [14, 15] it is reasonable tointroduce the flux detuning δΦ as

δΦ := minn∈Z

(Φ− 2n+ 1

2Φ0

). (2.19)

So δΦ measures the minimal distance from the flux Φ to the nearest half flux quantum. Therefore,the lower and upper bounds are given as −0.5Φ0 ≤ δΦ < 0.5Φ0. For simplicity we assume forthe rest of this thesis that Φ is located around 1/2 Φ0 such that the minimum is given at n = 0,so finally the qubit potential is of the form

EQ = EJ0

(2 + α− cos (φ1)− cos (φ2)− α cos

(π + 2π

δΦ

Φ0+ φ1 − φ2

)). (2.20)

A contour plot of the qubit potential EQ for α = 0.72 and two different values of δΦ is presentedin Figure 2.5 (a) and (b). In Figure 2.5 (a) the flux detuning is zero and the potential is periodicwith period 2π in each degree of freedom. There are two identical minima around (0, 0)T, theminima of the double well potential are located at [45]

φ1 = −φ2 = ± arccos

(1

2α

). (2.21)

where we used that the flux detuning δΦ = 0. We would like to mention that we assume that theso called intra cell minima [14] remain at the constant value given in equation (2.21) for smallperturbations of the flux. The potential energy EQ which belong to these minima are no longerdegenerated for a small detuning of δΦ = −20 mΦ0 in Figure 2.5 (b). A one dimensional cut alongthe line φ1 +φ2 = 0 where the intra cell minima are located is visualized in Figure 2.5 (c) and (d).It is worth mentioning that detuning the flux through the qubit loop causes a manipulation ofthe qubit potential. The two level system needed to encode the qubit is represented by the lowestexcitation in each minimum of the double well potential (d). We would like to point out thatwe can change the double well by tuning the flux Φ as a result we lower and higher the groundstate |g〉 and exited state |e〉 of the qubit. This also causes a change of the qubits excitationenergy EQE which is the energy difference of the ground and excited state. The amount of fluxdetuning is characterized by the value of δΦ.

10


Figure 2.5: Equipotential lines of the qubit potential for two different detuning values. The qubitpotential (2.20) for phases φ1, φ2 from −2π to 2π, δΦ = 0 and α = 0.72 is plotted in (a). It turnsout that the potential is 2π-periodic in each direction and it results in a double well potentialalong the blue line. In (b) we detuned the flux by δΦ = −20 mΦ0, the periodicity remains butthe double well potential is no longer symmetric. In (c) we present a one dimensional cut alongthe line φ1 +φ2 = 0 [blue and red lines in (a) and (b)] through the intra cell minima (2.21). Theinset of the black box is visualized in (d).

According to ref. [45] the Hamiltonian of the qubit HQ for values around δΦ = 0 can be writtenas

HQ =ε

2σz +

∆

2σx =

1

2

(ε ∆∆ −ε

). (2.22)

With σx and σz as the Pauli operators, ∆ as the energy gap of the qubit and a flux dependentenergy bias

ε := 2δΦ∂EQ

∂δΦ

∣∣∣∣(φ1,−φ1)T=min

= 2δΦIP, (2.23)

where the minimum is given in equation (2.21). According to ref. [45] we quantify the persistentcurrent IP of the qubit to

IP = IC

√1−

(1

2α

)2

. (2.24)

11


Therefore, we express the qubit exitation energy as

EQE = ~ωQ =√

∆2 + ε2, (2.25)

where ωQ denotes the qubit excitation frequency. The minimum of the qubit excitation energyEQE is given by the flux independent energy gap ∆ for a flux detuning of δΦ = 0. The resultingenergy diagram depending on the flux detuning δΦ of the qubit hyperbola is depicted in Figure2.6 (a).

Figure 2.6: The qubit hyperbola and the persistent current dependencing on the flux detuningδΦ. In (a) we depict the qubit energy diagram according to equation (2.25). It has a hyperbolicdependence on the flux detuning δΦ. With vanishing detuning the levels are separated by thequbit energy gap ∆. In (b) the slope of the hyperbola is plotted. For large detunings the valueof the slope converges to the persistent current of the qubit IP.

At detuning δΦ = 0 we need the energy gap ∆ to excite the qubit from state |g〉 to state |e〉. Thered lines indicate the case for vanishing energy gap. For large detunings the slope of the qubithyperbola converges to

limδΦ→±Φ0/2

∂EQE

∂δΦ= ±IP (2.26)

which is visualized in Figure 2.6 (b). Here the current circulating in the qubit loop is showndependencing on the flux detuning. For large detunings we can distinguish between clockwiseand counter clockwise rotating currents, but when we are close to zero detuning the two currentsoverlap. This can be understood by a degeneracy of the qubit energy levels in the double wellpotential of equation (2.20) which is visualized in Figure 2.5.

12


2.2 Quantum harmonic oscillator

The second important component of our sample is the superconducting niobium resonator, whichin good approximation is a one dimensional structure with negligible width and height terminatedby a capacitor at each end. Each mode of the resonator is modeled as a harmonic oscillator.We present different scenarios of such models depending on the modes which are present in theresonator. In an analogous way to its mechanical equivalent, a mass attached to a spring [32],the Hamiltonian of the harmonic oscillator is given by

HSM = ~ω(a†a+

1

2

), (2.27)

where ω is the eigenfrequency of the oscillator, a annihilates a photon at frequency ω and a†

creates a photon at frequency ω. In the Fock space where |n〉 denotes the state with n photonsin the resonator we find the expressions [32]

a |n〉 =√n |n− 1〉 (2.28)

a† |n〉 =√n+ 1 |n+ 1〉 (2.29)

a |0〉 = 0, (2.30)

since |n〉 is an eigenstate of the photon number operator n := a†a. These three relations holdfor any bosonic particle. Since this is an orthonormal eigenbasis we can define the Hilbert spaceof harmonic oscillator with a single mode as

HSM = |n〉 , n ∈ N0 . (2.31)

According to ref. [32] the eigenenergies of the time-independent Schrödinger equation

HSMΨ = ~ω(a†a+

1

2

)Ψ = EΨ (2.32)

are given as

En =

(n+

1

2

)~ω. (2.33)

This energy spectrum is visualized in Figure 2.7 (a).

13


Figure 2.7: The energy levels for different scenarios of a resonator. In (a) only a single mode ispresent, the Fock state |0〉 denotes the vacuum level. The higher excitations are separated bythe energy ~ω. The energy spectrum of a multi-mode resonator with equidistant mode spacingis illustrated in (b) where |n〉i denotes the state of n photons of mode i. The equidistant modespacing can be seen as a resonance condition which can be written as ωi = (i+ 1)ω0 with i ∈ N0.It turns out, that for example the states |2〉0 which corresponds to the lowest mode with respectto frequency has the same excitation energy as the state |1〉1. In (c) we illustrate the mostgeneral case of a multi-mode resonator. Here the resonance condition or the equidistant modespacing is no longer fulfilled and the energy spectrum is rich. Such a situation can be causedby a nonlinearity such as a Josephson junction embedded in the resonator such that we createa mode dependent phase drop [46]. This situation causes more than one parabola in the energyspectrum for clarity we decided to show only one. A sketch of the current density of the firstthree modes of a multi-mode resonator without a nonlinearity (d). The electrical length of theresonator is proportional to the geometric length independent of the mode and we end up withthe energy spectrum depicted in (b). In (e) a nonlinearity is placed at 1/4 of the resonatorslength. The mode dependent phase drop [46] modifies the electrical length such that modes withother wavelengths than in the undistorted case fit into the resonator. Since the nonlinearity isplaced at 1/4 of the resonators length the third harmonic ω3 would not be affected by the phasedrop since at this position the current distribution i(z) is zero. This scenario causes the energyspectrum depicted in (c).

14


Up to now we only talked about a resonator with a single mode, but for a sufficiently large amountof energy we can also excite higher harmonics. This can be expressed as the superposition of thesingle mode Hamiltonians and the Hamiltonian of the new system can be written as

H =∑i

~ωi(a†i ai +

1

2

)with i ∈ N0. (2.34)

This equation is a functional in an infinite dimensional Hilbert space formed from a tensorproduct of the Hilbert spaces Hi for every single mode i. However to make it accessible for anumerical simulation in a classical computer one has to assign a maximum number of Fock statesN which leads to a square matrix of finite dimension. Under this assumption we can subtractthe zero point energy HZPE which is given as

HZPE =

N∑i=0

~ωi2, (2.35)

such that we end up with the new multi-mode Hamiltonian

HMM = H − HZPE =∑i

~ωi(a†i ai

). (2.36)

We end up with a single vacuum mode and count the number of excitations. We would like tomention that we are still able to normalize the energy level with zero excitations to a certainvalue. For example in Figure 2.7 (b) and (c) we set the constant to ~ω0/2 to emphase the analogyto the situation depicted in (a). While in the energy level diagram of our qubit resonator systemin Figure 4.9 the vacuum level is set to zero. However in a multi-mode resonator we are able todistinguish two cases. The first case is when the different modes are resonant which means that

ωi = (i+ 1)ω0 with i ∈ N0. (2.37)

Here the mode spacing of each mode in the resonator is equidistant such that the first harmonicω1 has twice the frequency/energy of the fundamental mode ω0. An example of an energyspectrum in the resonant case with renormalized vacuum mode is visualized in Figure 2.7 (b).The most general and therefore the most interesting case is a scenario where the condition (2.37)is violated. As proposed in ref. [46] we can create such a system by adding a Josephson junctionto the resonator. This nonlinearity in the system causes a mode dependent phase drop [46].Therefore, we shift the modes such that they are no longer on resonance, which means that

ωi 6= (i+ 1)ω0 with i ∈ N. (2.38)

We would like to mention that equation (2.36) does not contain the nonlinearity explicitly.Furthermore, the mode ω1 is still called the first harmonic and so forth to keep a commonly usedterm. The interaction between the oscillator and the nonlinearity shifts the modes in a nontrivialcorrelation [46]. In the sample used in this thesis the coupling junction is placed at 1/4 of theresonators length [see Figure 3.1 (a)]. So the third harmonic ω3 is not affected by a phase dropsince the current distribution i(z) at this position is zero. Up to now we modeled the multi-moderesonator as sum of quantum harmonic oscillators to investigate the energy spectrum, but wedid not describe the resonator from an electrical engineering point of view. In network theorythe harmonic oscillator is modeled as a transmission line with lumped elements because thephysical dimensions of the circuit are comparable to the electrical wavelength of the microwave[47, 48]. As we use a superconducting niobium resonator the appropriate lumped element circuitis depicted in Figure 2.8.

15


Figure 2.8: The superconducting niobium resonator in a coplanar waveguide (CPW) architecture.In (a) the niobium (blue layer) is fabricated on the silicon substrate (gray layer) as described insection 3.1.1.1. It is capacitively coupled to the center conductor of the measurement cables viathe capacitors Cin and Cout. The length of the resonator l is defined by the distance between thecoupling capacitors Cin and Cout. The equivalent circuit diagram consisting of discrete capacitorsand inductors is shown in (b).

16


In Figure 2.8 (a) the fabricated superconducting niobium resonator is characterized by its lengthl and its input and output capacitors Cin and Cout. We use a coplanar waveguide architecture(CPW) which is the two dimensional equivalent to a coaxial cable. In (b) we show the equiv-alent circuit diagram for the modeled lossless transmission line. We divide the total length lof the resonator into n discrete elements with a length ∆zk where the transmission line has aninductance Lk and capacity Ck per unit length such that the equations

l =n∑k=1

∆zk (2.39)

L =n∑k=1

Lk (2.40)

C =

n∑k=1

Ck (2.41)

are fulfilled. We would like to mention that the distances ∆zk do not have to be equal, onlythe inductances Lk and the capacitances Ck have the same value. If we assume a homogeneouswaveguide and substrate the lengths ∆zk will become equal, too. Under that conditions we canquantify the characteristic impedance to [47, 48]

Z0 =

√LkCk. (2.42)

The microwave signal of angular frequency ω in the transmission line is then given as the solutionof the so called telegrapher equations. The solutions can be found in refs. [47, 48]. Accordingto that solutions the voltage and current in the transmission line is given as propagating wavestraveling in both directions. These propagating waves reduce to an infinite discrete set of standingwaves when we introduce the boundary condition given by the electrical length of the resonatorfixed by the capacitors Cin and Cout. As a result the angular frequencies ω are also reduced toan infinite set of the resonator modes ωi. In the case of a multi-mode resonator with equidistantmode spacing we end up with the situation depicted in Figure 2.7 (b) and (d). If we take intoaccount a nonlinearity such as a Josephson junction which causes a mode dependent phase drop[46] the situation is of more interest. According to ref. [46] that such a system can be modeledby the Hamiltonian in a lumped element formalism

HNLT =q2

2 (C + CJ)+δΦ2

2L− EJ0 cos

(2πδΦ

Φ0

). (2.43)

Where q is the charge in the resonator and δΦ is the flux detuning introduced in equation (2.23).The first term takes into account the capacity of the resonator and the junction, while the secondand the third term represent the inductance of the resonator and the junction respectively.Depending on which of the two conjugate variables (flux or charge) is used one has to verifywhich terms in equation (2.43) can be neglected. As we are using flux qubits we have to estimateif we can neglect the charge term.

17


2.3 Jaynes-Cummings model

So far we discussed the quantum behavior of a superconducting flux qubit and a superconductingresonator as two independent systems. In this thesis we couple these two systems, so the Hamil-tonian of the new system is the superposition of the two independent systems plus an additionalinteraction term. In the most general case this results in the Hamiltonian

H =~2ωQσz +

∑i

(~ωia†i ai + ~gi

(a†i + ai

)(σ+ + σ−)

). (2.44)

The first term corresponds to the qubit eigenstates |g〉 and |e〉 the series represents the multi-mode resonator modeled as a sum of quantum harmonic oscillators and the interaction of eachmode i with the qubit. The mode dependent coupling strength is denoted by gi. According toref. [45] this dependence arises from

hgi = MIPIi, (2.45)

where M is the so called mutual inductance between the qubit and the resonator, IP is thepersistent current of the qubit and Ii is the vacuum current of the i-th cavity mode [45] given as

Ii =

√~ωiL. (2.46)

So the current in the resonator Ii depends on the mode and the total inductance of the resonatorgiven in equation (2.41). If the condition ωQ + ωi > |ωQ − ωi| > gi is fulfilled, where the lattercondition is the more important one, we can neglect the fast rotating terms ωQ +ωi. This is theso called rotating wave approximation (RWA) which simplifys the Hamiltonian to [49]

HJC =~2ωQσz +

∑i

(~ωia†i ai + ~gi

(a†i σ− + aiσ+

)), (2.47)

where we neglected the so called counter rotating terms a†i σ+ and aiσ−. In 1963 this ansatzwas derived for the first time by Jaynes and Cummings [28]. Therefore, equation (2.47) is calledthe Jaynes-Cummings Hamiltonian. The operator combination a†i σ− and aiσ+ represent theprocess that a photon is absorbed by the qubit from the photon field of the resonator mode ior vice versa. By considering these processes the Jaynes-Cummings Hamiltonian preserves thenumber of excitations in the system. Due to the second diagonal entries of the σz operator theHamiltonian (2.47) has to be diagonalized in a suitable Hilbert space. In our case this Hilbertspace is given by

HMMJC = Hq ⊗i Hi (2.48)

where ⊗i denotes the tensor product. The subscript i should indicate that for every resonatormode this multiplication has to be executed and HQ is the Hilbert space of the qubit representedas

HQ =

(10

),

(01

). (2.49)

In this thesis we choose that the ground state of the qubit is be represented by |g〉 = (1, 0)T andexited state by |e〉 = (0, 1)T. The Hi correspond to the Fock states of the resonator mode ωi.So in analogy to the Hilbert space (2.31) it is given by

Hi = |n〉i , n ∈ N0 . (2.50)

18


In a multi-mode picture the vacuum mode still has to be renormalized, this is very important forthe numerical fits of the theory presented here with the experimental data discussed in Chapter4. However we now concentrate on one mode i as the Jaynes-Cummings Hamiltonian (2.47) doesnot contain a coupling of different resonator modes. We would like to mention that we do notdrop the index i. In analogy to ref. [49] we reduce the Hilbert space (2.48) to

HSMJC = HQ ⊗Hi =

|n, e〉i =

(0|n〉i

), |n, g〉i =

(|n〉i0

), n ∈ N0

. (2.51)

In this Hilbert space the eigenstates can be calculated analytically by introducing the mixingangle ϑ [45, 49, 50] (

|n,−〉i|n,+〉i

)=

(cos (ϑ) − sin (ϑ)sin (ϑ) cos (ϑ)

)(|n〉i |g〉|n− 1〉i |e〉

). (2.52)

For a detailed derivation on the mixing angle ϑ we would like to refer to ref. [49]. According toref. [45] the mixing angle is given by the expression

ϑ =1

2arctan

(2gi√n

δωi

)(2.53)

where we introduced the frequency detuning δωi := ωQ − ωi of the resonator mode i and thequbit excitation frequency ωQ given in equation (2.25). With this new parameter we are able todistinguish two different cases. In the first case we find that the qubit frequency ωQ ≈ ωi suchthat the frequency detuning δωi is closed to zero. In this case we receive [45, 49]

limδωi→0,gi>0

sin (ϑ) = limδωi→0

cos (ϑ) =1√2

(2.54)

As a result the dressed eigenstates are of the form(|n,−〉i|n,+〉i

)=

1√2

(|n〉i |g〉 − |n− 1〉i |e〉|n〉i |g〉+ |n− 1〉i |e〉

)(2.55)

and with energy levels separated by a factor

2gi√n (2.56)

as visualized in Figure 2.9 (a). In the uncoupled system (dashed lines) the energy states wherethe total number of excitations is equal are degenerate, while by introducing the mixing angle ϑthe eigenstates are separated with a photon number dependent energy splitting (solid lines). Inthe case when the frequency detuning |δωi| > gi which is the so called dispersive limit [45, 50]the interaction changes and the Hamiltonian can be approximated as [51]

HdispJC ≈ ~

(ωi +

g2i

δωiσz

)a†i ai +

~2

(ωQ +

g2i

δωi

)σz, (2.57)

such that we end up with an energy spectrum as depicted in Figure 2.9 (b). Where the firstterm represents the resonator with a shifted resonance frequency and the second corresponds tothe qubit whose excitation frequency is also shifted i.e. the resonator frequency of the coupledsystem (solid lines in 2.9 (b)) is shifted by the value ±g2

i /δωi depending on the qubit state.In both cases of the resonant and the dispersive coupled qubit resonator system the resonatoris affected by a frequency shift such that this mechanism forms the physical basis of two-tonespectroscopy introduced in section 3.2.1.2 and used in Chapter 4.

19


Figure 2.9: The energy spectrum of the Jaynes-Cummings Hamiltonian for two different cases.In (a) the excitation frequency of the qubit ωQ and the resonator mode ωi are equal. The dashedgreen lines represent the uncoupled system when the qubit is in state |g〉 while the magenta linesrepresent the excited state |e〉. For the coupled system the eigenstates split up and the levelspacing is depends on the photon number. In the dispersive case the frequency detuning δωiis large in comparison to the coupling gi of the qubit to the resonator mode ωi is depicted in(b). The resonator mode is shifted by ±g2

i /δωi depending on the qubit state, while the qubitexcitation frequency is changed by (2n+ 1)g2

i /δωi [50].

However the most interesting term in the Hamiltonian (2.57) is

~g2i

δωi

(a†i ai

)σz =

~g2i

δωiniσz (2.58)

which from this point of view causes a photon number dependent shift of the qubit frequency.Where ni is the photon number operator of mode i which counts the number of photons withmode i in the resonator. This is more obvious if we rearrange the Hamiltonian (2.57) to

HdispJC ≈ ~ωia†i ai +

~2

(ωQ +

g2i

δωi+

2g2i

δωia†i ai

)σz = ~ωia†i ai +

~2

(ωQ + (2ni + 1)

g2i

δωi

)σz.(2.59)

In the dispersive limit of the coupled system the qubit frequency ωQ is shifted by the value2ng2

i /δωi for a fixed number of photons and a vacuum Lamb shift g2i /δωi [50]. This photon

number dependence of the AC-Zeeman shift helps us to calibrate the photon number in section4.1.3. However, up to now we only talked about steady states which is experimentally accessibletrough spectroscopy. When we are interested in dynamics and the characteristic time scales wehave to consider decoherence effects. Therefore, we discuss the corresponding theory in the nextpart.

20


2.4 Dynamic and decoherence

In classical computation, information is stored in the classical states 0 and 1. When informationis stored in a quantum system with eigenstates |g〉 and |e〉 the situation is more complex sinceone has to care about superposition of states, which can be expressed as [38]

|Ψ〉 = cos

(Θ

2

)|g〉+ sin

(Θ

2

)exp (iϕ) |e〉 . (2.60)

The state |Ψ〉 is a superposition of the two pure eigenstates on the so-called Bloch sphere withthe polar angle Θ and the azimuthal angle ϕ as depicted in Figure 2.10.

Figure 2.10: The Bloch sphere representation of a qubit. The Bloch vector |Ψ〉 at a certain timeis a superposition of the ground |g〉 and excited state |e〉 characterized by the angles ϕ and Θ.

It is important to keep this picture in mind to understand the physics and experimental accessi-bility of decoherence and the characteristic time scales linked to it. The qubit or in general thetwo level system can be driven from the ground state |g〉 to the excited state |e〉 by applyinga π-pulse (see Figure 2.11) at its excitation frequency ωQ along the z-axis, so via a couplingto σz and therefore changing the value of the polar angle Θ. However, due to coupling of thetwo level system to the environment various noise sources [26, 52] perturb the system and causedecoherence. An important parameter for decoherence processes is the energy relaxation rateΓ1. We can calculate this rate via Bloch-Redfield-Theory [53, 54], according to ref. [26] in thecase of a flux qubit as the two level system the relaxation rate is given as

Γ1 = πSΓ1

Φ(ωQ), (2.61)

where SΓ1

Φ(ωQ)is the symmetrized noise spectral density [26] depending on the qubit excitation

frequency ωQ.

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Figure 2.11: An example of a Rabi or π-pulse which excites the two level system from the groundstate |g〉 to the exited state |e〉. In time-domain measurements the length τ of a π-pulse at thequbit excitation frequency ωQ is the first value which has to be specified.

The inverse of the energy relaxation rate is called the energy relaxation time T1 := (Γ1)−1

and it can be interpreted as the time which the qubit needs to relax from the exited state |e〉to the ground state |g〉. This time is experimentally accessible via so called Rabi oscillationmeasurements which are presented in Chapter 4. Despite time dependent motion of the polarangle Θ the azimuthal angel ϕ can also vary in time. This process is often called dephasing[26, 50]. According to refs. [26, 55] there are several mechanisms which cause dephasing. Thetotal dephasing is a sum of all these mechanisms and we find the expression [50]

Γ2 =Γ1

2+∑m

Γ2,m. (2.62)

For a detailed analysis of the different dephasing mechanisms we would like to refer to refs.[26, 50, 52, 55]. We would like to mention that the measured dephasing time T2 := (Γ2)−1

depends on the experimental technique one applies [26]. For all these experimental techniqueslike Spin-Echo and Ramsey fringes [26, 50] the pulse length τ of a π-pulse has to be known.The determination of that pulse width is inevitable and should be one of the first measurementsin the time-domain. All rates introduces so far influence the timescale for which a functionalquantum computer can store the information and perform algorithms accurately. In summarysince the two level system is not perfectly isolated from the environment1 the system is affectedby decoherence. To determine the decoherence rates one has to perform time-domain measure-ments. In a first time-domain measurement one has to determine the length of a π-pulse in orderto gain access to the explicit form of the pulse sequences which one has to apply in subsequentmeasurements.

In this chapter we presented the necessary theory to describe the coupled qubit resonator system.The flux qubit is described in an RCSJ model and its energy spectrum is determined by the fluxtrough the qubit loop which causes a persistent current in the loop. The resonator is modeledas a sum of harmonic oscillators where we described two scenarios. In the first scenario themode spacing is equidistant and in the second case the mode spacing is non-equidistant due to aJosephson junction which causes a mode dependent phase drop [46]. The energy spectrum of thecoupled qubit resonator system can be understood by concerning the Jaynes-Cummings Hamil-tonian [28]. In the last section we presented a short introduction to decoherence in our systemand depicted the idea how to determine the relaxation and dephasing rates in an experimentalsetup.

1On the other side if the system would be perfectly isolated, we would not be able to determine the state ofthe system by measurement.

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3 Experimental setup

As mentioned in the previous parts the aim of this work is to couple a superconducting resonatorto a flux qubit and observe the response of it in continuous wave spectroscopy, pulsed wavespectroscopy and time-domain measurements. As we intend to observe quantum mechanicalphenomena the sample has to be cooled down to very low temperatures (in our case approximately50 mK) to reduce the number of thermal photons in the resonator to a negligible value and asa result the quantum properties become dominant. We first introduce the experimental setupused in the dilution refrigerator including the sample and afterwards treat the room temperaturecomponents.

3.1 Cryogenic setup

In this section we introduce the sample used for our experiments and explain the different pro-duction steps. The sample is mounted into a sample holder which is then built into an in-housefabricated cryostat with input and output chains. These chains consist of microwave cables,attenuators, circulators and amplifiers and have an optimized signal to noise ratio under theboundary condition of a limited cooling power at the different temperature stages. To couplemagnetic flux into the qubit loop we use DC lines connected to a superconducting coil with apersistent current switch. These components are also introduced.

3.1.1 Sample

The sample we investigate is produced with the help of optical and electron beam lithographyas well as reactive-ion etching at the WMI. Here we give a short overview of several productionsteps which are necessary to fabricate a superconducting resonator and a flux qubit. At thispoint the author wants to thank M. Haeberlein for the production of the sample, because theauthor was not involved in the production of the sample and without the contribution of Mr.Haeberlein this thesis would not have been possible.

3.1.1.1 Superconducting niobium resonator and antenna

The resonator is produced via optical lithography and reactive-ion etching. The main productionsteps are as follows. At the beginning one sputters approximately 100 to 200 nm niobium ontoa Si-substrate. Afterwards one spins a resist on top of the niobium. In the next step one coversthe resist with a chromium-plated mask which contains a negative of the pattern one wants toproduce [see Figure 3.1 (a)] and applies a flood exposure of UV light on the resist which is notcovered by the mask in a mask aligner. Afterwards one applies chemical development to removethe resist which was exposed to UV light followed by a reactive-ion etching process to removethe niobium which is not covered by the resist. Finally, one does a resist stripping. Now onehas produced a chip with a niobium resonator and an antenna on it as well as the connectionpads needed to contact the sample. If one has produced more than one sample it might be ofinterest to check one of the samples in a 4 K cryostat (as the critical temperature of niobium isTNbC = 9.2 K) to gain information about the superconducting resonator for example its quality

factor.

23


Figure 3.1: The Chip design (5 by 10 mm) used in the course of this thesis with a transmissionline resonator (blue). The resonators coupling capacitors Cin and Cout located at the greenboxes fix the boundary conditions for the resonator modes since they fix the geometrical lengthto 19 mm. The shape of the resonator takes into account that the so called box modes, whicharise from the geometrical extensions are shifted to higher frequencies such that they are not inthe bandwidth of our experiment. One of the coupling capacitors is depicted in (b). During thereactive-ion etching the inner conductor of the CPW has been interrupted in order to build acoupling capacitor. The qubit is located at 1/4 of the resonator length [magenta box in (a)] suchthat it can be excited by a signal in the antenna (red). An SEM picture of the qubit is depictedin (c). The qubit is colored for clarity. The Josephson junctions of the qubit [blue boxes in (c)]and the coupling junction [red box in (c)] are clearly visible. The right edge of the loop alsoforms the shared edge of the qubit and the resonator and determines the mutual inductance M .During the course of this thesis a chip with the same fabrication parameters is used.

24


3.1.1.2 3 Josephson junction flux qubit and coupling junction

In order to fabricate the 3 Josephson junction flux qubit on a chip we have to produce Josephsonjunctions out of aluminum and aluminum oxide. There are two identical junctions and the thirdso called α-junction with a relative size α ≈ 0.7 compared to the others. This qubit is locatedat 1/4 of the length of the resonator. Via a shared edge the qubit is coupled galvanically andinductively with an geometric inductance Lgeo to the resonator [45]. We try to increase theinductive coupling with the help of a fourth junction with a relative size factor β ≈ 1 comparedto the other junctions of the qubit [see Figure 3.1 (c)]. The inductive coupling is determined bythe mutual inductance M as could be seen from equation (2.45). In our case we assume that themutual inductance is given asM = Lgeo +LJ. Since the geometric inductance Lgeo is fixed by thedesign of the qubit the inductance the coupling junction LJ could be changed easier1. However,we can tune the inductance of the coupling junction and therefore enhance the coupling. It wasreasonable to reduce the size of the fourth junction because according to ref. [45] the mutualinductance M is proportional to the Josephson inductance LJ ∝ 1/A of the coupling junction,where A denotes the area of the junction. We can enhance the coupling gi by reducing the sizeof the junction as could be seen from equation (2.45). To produce Josephson junctions we haveto produce a pattern for the shadow evaporation process via electron beam lithography [56, 57]at first. A sketch of the shadow evaporation process is depicted in Figure 3.2.

Figure 3.2: A sketch of the shadow evaporation technique. By depositing material under twodifferent angles on a substrate with the help of a mask we would deposit it on two differentlocations and we would end up with a nonfunctional sample (a). This can be solved by usinga mask structure as depicted in (b). After the evaporation and oxidation process we producedtunnel junctions but also ghost structures [56, 57].

1Both values are fixed during the fabrication process. But the production of a functional flux qubit withdifferent sizes of the coupling junction is more straight forward than a in the case of different areas.

25


In the shadow evaporation process we deposit aluminum under an angle ζ on the chip [red tracein Figure 3.2 (a)] in a first step. To produce the thin insulating tunnel barrier of the Josephsonjunction we oxidize the aluminum we evaporated on the chip in the first step. The oxidation timedetermines the thickness of the insulating barrier. With a second evaporation under an angle−ζ (blue trace) we deposit another layer of aluminum afterwards. After a lift-off we producedthe desired sample [see Figure 3.1 (c)] for this thesis. For a more detailed description of theseprocesses we would like to refer to refs. [45, 58].

3.1.2 Cryostat

After the sample is mounted in a suitable sample holder it is cooled down to a base tempera-ture of 43 mK of the 3He/4He dilution refrigerator. For a detailed introduction to the differentcomponents of a dilution refrigerator we would like to refer to refs. [45, 59]. The sample stagetemperature is stabilized at approximately 50 mK via a resistance measurement bridge and a tem-perature controller. To protect the experiment from fields that might perturb the measurementthe cryostat is covered by a cryoperm and mu-metal shield which are surrounded by a shieldingroom finally. The basic experimental setup which is used during this thesis and mounted insidethis shielding room is depicted in Figure 3.3. Other measurement equipment located outside ofthe shielding room is introduced in section 3.2.

Figure 3.3: A schematic of the experimental setup inside the shielding room. The different tem-perature stages are visualized including the devices at the specific temperature. The attenuatorsthermalize inner and outer conductor of the feed lines and are chosen such that the signal tonoise ratio (SNR) arriving at the sample is optimized, the numbers can be found in the text. Thecirculators in the output guarantee that no signal is entering the sample from higher temperaturestages but the signal from the sample can pass trough.

26


3.1.2.1 Microwave input lines

The input line of the resonator is connected to a Rohde & Schwarz ZVA24 [60] vector networkanalyzer (VNA) during continuous wave spectroscopy and thermalized at each temperature stagevia an attenuator to reduce thermal noise by thermalizing inner and outer conductor at thesame time. According to ~ω = kBT where kB = 1.38065 · 10−23 J/K represents the Boltzmannconstant a thermal photon with a frequency ω/2π of 5 GHz would correspond to a temperatureT of approximately 240 mK. If we would connect the inner and outer conductor which are atroom temperature to the resonator without thermalization we would populate the resonator withmore than 1200 thermal photons. By thermalizing the lines at different temperature stages weattenuate not only the signal but also the noise of the higher temperature stage but we add anamount of thermal photons corresponding to the actual temperature stage. The added noisequanta can be calculated with [59]

nPh =1

2+

1

exp(

~ωkBT

)− 1

. (3.1)

The term 1/2 is due to the vacuum fluctuations. The task is to optimize the signal to noiseratio (SNR) entering the sample. So obviously it would be the best to attenuate only at the lasttemperature stage. Because there one attenuates the signal and the high temperature noise bya desired value of the attenuator and adds only a small number of thermal photons. However,due to the limited cooling power this is not possible and one has to dissipate power at highertemperature stages where the cooling power is sufficient. Therefore, a boundary condition of ouroptimization problem is the limited cooling power at each temperature stage. According to ref.[61] the maximum allowed power dissipation values for each stage of our cryostat are as given inTable 3.1.

stage temperature (K) max. dissipation (µW)

1K-pot 1.2 20000still 0.700 500

step exchanger 0.100 200sample stage 0.050 50

Table 3.1: The dissipated power of each temperature stage in the cryostat we use is limited bythe values listed here.

So we calculate the power Pdiss dissipated by the attenuator Att at the different temperaturestages, via the formula [61]

Pdiss (µW) = 1000(

10Pin(dBm)

10 − 10Pin(dBm)−Att(dB)

10

). (3.2)

Here we only consider the input power Pin of the signal entering from the next higher temperaturestage via the resonator and the antenna chain because the power of the thermal photons is severalorders of magnitude lower. By using equations (3.1) and (3.2) consistently for the two feed linesin Figure 3.3 we end up with the numbers presented in Table 3.2 for the resonator feed line ata frequency of ω/2π = 7.106 GHz which corresponds to the 3λ/2 mode of the resonator. Thismode is used as probe frequency in two-tone spectroscopy measurements in section 4.1. Thedissipated power is calculated for an output of the VNA of −45 dBm2 which is one of the highestinput powers used during continuous wave spectroscopy.

2The output power of the VNA is −5 dBm which is attenuated by 50 dB at roomtemperature (see Figure 3.5)such that −45 dBm enter the resonator input line.

27


stage T (K) Att (dB) nPh Pdiss (µW)

room temperature 300 — 880 0.028

He bath 4.2 10 100 < 0.005

still 0.700 10 12.1 < 0.005

step exchanger 0.100 10 1.75 < 0.005

sample stage 0.050 20 0.519 < 0.005

Table 3.2: The dissipated power the resonator feed line for the used attenuator configurationintroduced in Figure 3.3. The numbers in the fourth column are the sums of the attenuatedphotons from the next higher temperature stage and the thermal photons added at the specifictemperature stage. Here we used a resonator frequency of ω/2π = 7.106 GHz. It turns outthat we are close to the limit given by vacuum noise. The dissipated power is calculated for aninput power of −45 dBm at the resonator feed line. The output power of 5 dBm of the VNAwas attenuated by 50 dB and applied to the resonator during the photon number calibrationmeasurements.

During continuous wave spectroscopy the on-chip-antenna is driven by a microwave source SMF100A from Rohde & Schwarz [62]. While in the time-domain measurements we use a vector signalgenerator E8267D PSG from Agilent [63] to drive the antenna. The inner and outer conductor ofthis feed line is also thermalized by some attenuators at different temperatures. In Table 3.3 wecalculate the dissipated power in the antenna feed line for a frequency of ω/2π = 4.96 GHz whichis proportional to the energy gap ∆ of the qubit. According to equation (2.25) it is reasonable touse this frequency as we want to excite the qubit via the antenna. Therefore, we have at least tospend the energy of the gap at the degeneracy point. The relative high input power of −15 dBmwhich is used to calculate the dissipated power is applied by the SMF to the antenna input portat the shielding room during the first two-tone spectroscopy experiments in this thesis.

stage T (K) Att (dB) nPh Pdiss (µW)

room temperature 300 — 1260 —He bath 4.2 20 30.3 31.3still 0.700 3 18.1 0.158

step exchanger 0.100 5 6.34 0.108sample stage 0.050 20 0.572 0.050

Table 3.3: The dissipated power the antenna feed line for the used attenuator configurationintroduced in Figure 3.3. The numbers in the fourth column are the sums of the attenuatedphotons from the next higher temperature stage and the thermal photons added at the specifictemperature stage. Here we used a frequency of ω/2π = 4.96 GHz, which is equivalent to thequbits energy gap ∆. The dissipated power is calculated for an input power of −15 dBm at theantenna feed line. So this input power can be provided from the SMF in the case of continuouswave spectroscopy or the PSG during the pulsed wave spectroscopy.

So the dominant part of the total dissipated power which is the sum of all power which aredissipated is that from the antenna feed line. However, these values are still smaller than themaximum values in Table 3.1 therefore, the cryostat is able to cool our experiment sufficiently.

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3.1.2.2 Microwave output line

The output of the resonator is connected to two circulators in series to protect it against noisefrom higher temperature stages, the first one (7) in Figure 3.4 located on the sample stage at atemperature of approximately 50 mK (see Figure 3.3 for wiring and Figure 3.4 for photos) andthe second (8) in Figure 3.4 is placed at the still stage (3) and therefore at a temperature of700 mK. A circulator is a three port device, in our case lets call the ports input, output andterminated. As depicted in Figure 3.3 the input of the first circulator is connected to the outputof the sample holder (5) and the output of the circulator is linked to a higher temperature stage.The ferrite attached to the wave guides inside the circulator causes the scattering matrix of thecirculator to become antisymmetric, which means that a signal entering from the input portis weakly damped to the output port but highly damped to the terminated port. The physicsbehind this process is as follows in the case of a phase shift circulator. The signal entering thecirculator is split up in two equal parts where the ferrite causes a decrease of the group velocityin one of the arms. So as a result on one port constructive and on the other port destructiveinterference is realized [47, 64]. In our case this means that a signal entering the input interferesconstructively at the output and destructively at the terminated port. If now a signal or noiseis entering the output port it interferes destructively at the input port and so it is not able topass trough to the sample. At the terminated port it interferes constructively but that is notrelevant because this port is terminated by 50 Ω. A circulator where one of the three ports isterminated by 50 Ω is also called an isolator. However, due to the ferrite a circulator used incombination with a flux qubit has to be shielded such that the magnetic field of the ferrite doesnot perturb our qubit. A cold high electron mobility transistor (HEMT) amplifier (9) in Figure3.4 from Low Noise Factory Model LNF-LNC4_8A is mounted at 4 K to amplify the measuredsignal with a gain of 40 dB. According to the data sheet [65] the bandwidth is limited to a rangeof 4 to 8 GHz, here the gain and the added noise are almost constant. At room temperaturethere is a third circulator and a second amplifier JS2 from Miteq with a gain of approximately27 dB and a bandwidth from 2 to 8 GHz. Finally the signal enters the detection port of the VNAin the case of continuous wave spectroscopy. In summary the bandwidth of our experiment is ofa limited range from 4 to 8 GHz.

3.1.2.3 DC lines

Up to now we introduced the microwave cables built into the cryostat which are necessary toprovide the RF signals to our experiment. However, as we intend to investigate a coupledflux qubit resonator system we have to provide external flux to the qubit loop. So we haveto apply a magnetic field with the help of a superconducting magnet coil [(6) in Figure 3.4]which is connected to DC lines. The coil is fabricated out of NbTi wire in a Cu matrix andhas an inductance of approximately 1 mH [45]. Driving a current If through the coil generatesa magnetic field and as a result a flux which can penetrate the qubit loop. When the desiredflux is trapped in the qubit loop we use a persistent current switch to disconnect the currentsource from the magnet coil by keeping the flux in the coil as well as in the qubit loop constant.This is useful for long time measurements at a constant flux value because we are not sensitiveto current fluctuations in the current source.

29


Figure 3.4: A photograph of the cryogenic stage. In (a) the temperature decreases from 4.2 Kat the 4K-flange (1) over the 1K-pot (2), the still (3) at approximately 700 mK and the mixingchamber (4) at a base temperature of approximately 43 mK. A view from the left side of theupper part of the sample stage (green box) is depicted in (b). The sample mounted in the sampleholder (5) is biased with an external flux via the superconducting magnet coil (6). The backsideof the lower part of the sample stage (red box) is illustrated in (c). After the signal passed thesample it enters a circulator (7), which is thermalized at approximately 50 mK. Then the signalgoes all the way up and enters a second circulator (8) at the still stage and is amplified by a coldHEMT (9) at 4.2 K before it is sent to the room temperature components. More details can befound in the text.

30


3.2 Room temperature setup

In addition to continuous wave spectroscopy we also want to perform pulsed wave spectroscopyand time-domain measurements. While continuous wave spectroscopy is a well established tech-nique at the WMI pulsed measurements in the context of quantum information processing arenot. In this section we explain the continuous wave setup which is in use and then introduce indetail how we generate the pulses sent to the sample inside the cryostat and how we detect theresponse of our sample via an IQ mixer and a data acquisition card.

3.2.1 Continuous wave spectroscopy

A well established technique to investigate the interplay of a qubit and a resonator coupledtogether is continuous wave spectroscopy [17, 18, 20]. This can be done by using a single probetone or by using two different tones, a so called probe tone with frequency ωP and a spectroscopytone ωS. The latter method is also a part of the basic functional principle of a Rabi oscillationmeasurement where the probe tone is no longer continuous but pulsed. However, in this partwe introduce the devices used to perform single and two-tone continuous wave spectroscopy andexplain the basic functional principle of these measurements.

3.2.1.1 Single tone continuous wave spectroscopy

Single tone continuous wave spectroscopy is a fundamental technique. We need a signal sourcewhich continuously generates a signal at a desired frequency. This signal is then sent to the samplewe investigate and the response of this signal is recorded by a spectrometer which detects thesame frequency at which the signal source is emitting the signal. If one hits a resonance frequencyof the sample it can transmit the signal to the detector, hence the measured transmission is large.If one is now off resonance all the power is reflected and the detected transmission decreases.In our case we send a continuous microwave signal and observe the response of the samplewith a vector network analyzer VNA. Therefore, we connect the two ports of our VNA ZVA24from Rohde & Schwarz to the input and output of our superconducting resonator as labeledin Figure 3.3. A VNA is able to generate a signal of desired amplitude and frequency withinits specifications with a built-in microwave generator and detect the response in transmissionor if desired even in reflection at the same time. In this thesis we concentrate on transmissionmeasurements during continuous wave spectroscopy. A VNA is even more powerful, it can detectnot only the amplitude but also the phase of the response. Therefore, the generated signal issplit up, the first part is sent to the sample and the second to a internal receiver, where it isused as a phase reference. That is why we are able to measure a complex signal (amplitude andphase) which we can interpret as a vector in R2 [66]. As mentioned, a VNA is able to measuretransmission and reflection so it contains an S-parameter test set.

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3.2.1.2 Two-tone continuous wave spectroscopy

Up to now we only talked about single tone continuous wave spectroscopy, where we could inves-tigate the behavior of the coupled qubit resonator system dependent on the applied frequencyand power. The next step and therefore a more sophisticated technique is the so called two-tonecontinuous wave spectroscopy, where one applies two signals which can differ in frequency, poweror both to the experiment. If one would only have one input line one has to use a power com-biner to send the signals to the sample as it was done in ref. [45]. However, one can also use anon-chip antenna to apply the second tone to the qubit. In this thesis this experimental approachis realized at the WMI for the first time. We use once again the ZVA24 to apply the so calledprobe tone ωP to the resonator and the second spectroscopy tone ωS is generated by the SMF100A and sent to the experiment via the antenna input line in the cryostat as depicted in Figure3.5.

Figure 3.5: The continuous wave setup used in single and two-tone experiments in this thesis. Forsingle tone continuous wave spectroscopy one does not need the SMF 100A so one can disconnectit and terminate the antenna feed line with a 50 Ω resistor. The input signal to the resonator isattenuated by 50 dB due to the limited dynamical range of the ZVA24.

All signal generators are connected to a 10 MHz Rubidium reference source from Stanford Re-search Systems Model FS 725 [67]. In our setup the measurement principle is as follows. Imaginethat your probe tone ωP is on resonance with the sample and the spectroscopy tone is off reso-nance or switched off as in Figure 3.6. So the VNA detects a high transmission of the total signalwhich enters the sample. If the spectroscopy tone remains off resonant with the sample, the lattercan not be excited and the transmission remains almost constant. If now the spectroscopy tonehits a resonance of the sample it can excite the system with microwave photons of frequencyωS . According to the mechanism described in section 2.3 this causes an AC-Zeeman shift [blueline in Figure 3.6 (a)]. Due to the linewidth of the signals the detected transmission measuredat the fixed probe frequency ωP decreases by ∆Tr. We analyze the measured transmission ofthe probe signal as a function of the spectroscopy tone ωS to detect the resonance of the qubiti.e. we are able to measure the qubit hyperbola if we would apply an additional flux sweep. Inthe time-domain measurement we use the phase change ∆Ph depicted in Figure 3.6 (b) to gaininformation about the sample. The phase shift is also caused by the AC-Zeeman shift of theprobe tone frequency.

32


Figure 3.6: The functional principle of a two-tone spectroscopy for the measured transmission(a) and phase (b). When signal generator SMF attached to the antenna is off (red line) we lookfor a transmission peak of the resonator to assign the probe tone frequency ωP. By operating thesignal generator (blue line) on a resonance frequency of the system it affected by an AC-Zeemanshift as described in section 2.3, due to the linewidth of the signals this causes also a reducedtransmission on the probe tone frequency ωP as pointed out in detail on the text. For this datawe used a driving amplitude of the antenna of −30 dBm and a frequency of ωS = 5.04 GHz at theoutput of the SMF. We measured at the degeneracy point of the qubit such that ωS = ωQ = ∆/h.

3.2.2 Pulsed wave spectroscopy and time-domain measurements

With the experimental techniques introduced so far we are able to characterize our sample verywell in the frequency-domain, but we still suffer a lack of information regarding the timescales.First of all we are interested in the relaxation time of our qubit T1 and second in the dephasingtime T2. A naive approach to access these times is that one performs a Fourier analysis to thedata received from spectroscopy experiments. In the theory of Fourier analysis one has access tothe full spectrum i.e. to all frequencies but due to the limited bandwidth of the experiment andthe fact that one would have to perform a discrete and not an analytical Fourier transformationone would end up with an approximation of the time response which is not satisfying. So one hasto perform a time based measurement. Therefore, we build up an IQ detector. Furthermore, weimplement protocols that generate and detect pulses with a length of a few ns up to µs efficiently.In this section we introduce our approach to this task.

33


3.2.2.1 Pulse generation

Pulsed measurements with more than one signal source are very sensitive to the phase positionof the different tones, so it is important to provide a reference signal which guarantees a stablephase difference between each signal source. This is very important since we average over morethan 106 traces. If the phase difference would not be stable it might happen that the informationof the experiment one is interested in cancels out. The basic hardware and its wiring is depictedin Figure 3.7.

Figure 3.7: The 10 MHz reference signal is applied to the two microwave sources (SMF andPSG) and to the data timing generator (DTG) from Tektronix. The latter is used to apply atrigger signal to the two sources as well as to the data acquisition card. With the SMF we driveour resonator. We use the PGS to apply a pulse to the qubit via the on chip antenna to excitethe qubit. The two channels of the data acquisition card are recording a signal from an IQ mixer.

We use a 10 MHz Rubidium reference source from Stanford Research Systems Model FS 725 [67].This reference signal is sent to a microwave source SMF 100 A, a vector signal generator E8267DPSG from Agilent [63] and a data timing generator from Tektronix Model DTG 5334 [68]. Fromthe latter device a trigger pulse of desired shape with respect to delay and width is applied tothe microwave sources and the data acquisition card. In this thesis we use rectangular pulseswith a width from 5 ns up to 3µs. The SMF applies a microwave signal of desired amplitudeand frequency to the input port of the resonator while the qubit is driven by the PSG. We usethe internal mixer of the PSG to generate pulses similar to that depicted in Figure 2.11 duringthe time-domain measurements in section 4.2. The output of the resonator is connected to anIQ mixer which splits up the signal into two parts. These two chains are connected to the dataacquisition card which records the amplitude and phase of the response at a certain time, thedetails on this procedure are explained in the next part.

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3.2.2.2 Pulse detection

The pulses generated as described in the previous part are sent into the cryostat. After theypassed the sample and the amplification chain depicted in Figure 3.3 the pulses are split up withan IQ mixer into two channels. Finally these two channels are detected with the help of anACQIRIS card from Agilent Model DC440 [69] with a sampling rate of 400 MSa/s3. A samplingrate of 150 MSa/s4 is used in the FPGA (f ield programmable gate array) board from InnovativeIntegration [70] to speed up the measurements. We can speed up the measurements due to thefact that the FPGA is able to average the acquired data very efficient such that we are able touse every second trigger pulse for data acquisition. This is not possible with the ACQIRIS card,where on average from 16 pulses only the first one is recorded while the other 15 pulses are lostbecause the average calculations are still taking place. As a result the measurement time for thesame experiment carried out with the FPGA card is reduced by a factor of 8 in comparison tothe ACQIRIS card measurement but with a reduced time resolution of 150 points per µs insteadof 400 points per µs. However, we first concentrate on the ACQIRIS card. Due to the fact thatthe data acquisition card is not able to work at GHz frequencies the signal has to be converteddown to several MHz by an IQ mixer with additional components such as amplifiers and filters.The used hardware is depicted in Figure 3.8.

Figure 3.8: The signal from the experiment is first filtered using a band pass and then converteddown in frequency and split up into an in phase signal (I) and a quadrature component (Q).To avoid compression in the amplifiers of the I- and Q-chain we attenuate the signal by 20 dB.Before entering the ACQIRIS card we apply a low pass filter with a bandwidth from DC up to32 MHz.

The signal from the output chain of the experiment inside the shielding room is band pass filteredby a Mini-Circuit VHF-4600+ [71] before it enters the RF port of the IQ mixer from MITEQModel IR0408LC2Q , which is connected to a second microwave source. In our case a secondSMF which provides the local oscillator frequency at an output power of the SMF of 12.5 dBm.The first SMF is used to drive the resonator. The bandwidth of the used IQ mixer is from 4 to8 GHz. The basic functional principle of an ideal IQ mixer is shown in Figure 3.9.

3the maximum sampling rate is 420MSa/s, during the experiments we use a sampling rate of 400MSa/s4the maximum sampling rate is 200MSa/s but the used logic supports a sampling rate of 150MSa/s

35


Figure 3.9: The basic idea of an ideal IQ mixer is to convert the frequency down from the RFinput ωP with help of a local oscillator ωLO to the intermediate frequency ωIF := ωP − ωLO andto split the signal into an in phase signal I and a quadrature component Q

One splits the measured RF signal into an in phase channel I and an a quadrature componentchannel Q and converts it down to an intermediate frequency ωIF := ωP−ωLO such that one canextract the amplitude A(t) and phase ρ(t) at any time of the experiment [50]. The intermediatefrequency ωIF is obtained from the local oscillator frequency ωLO and the probe tone ωP at theRF input port by using an addition theorem. In this theorem we end up with the sum andthe difference of the two frequencies. The sum is filtered out from a band pass inside the IQmixer while the difference is kept as intermediate frequency ωIF. However, due to imperfectionsin both output arms e.g. the two amplifiers (both from MITEQ model AU15-25 [72], witha typical gain of 65 dB within their bandwidth from 1 to 300 MHz) do not have exactly thesame characteristics and the phase shift of the IQ mixer is not exactly π/2 this setup has to becalibrated as described in Appendix A. Nevertheless, it is also worth to have a look at the rawdata before starting calibration. The raw data is low pass filtered with a Mini-Circuit SLP-30+[73] and recorded by the ACQIRIS card. To record the calibration pulse we operate the PSG atan output power of −100 dBm with a frequency ωP/2π = 7.109 GHz and connect it to the RFinput of the IQ mixer setup depicted in Figure 3.8 via a microwave cable directly without thesample between the PSG and the IQ mixer. The local oscillator microwave source was operatingat ωLO = 7.099 Ghz and a power of 12.5 dBm such that the intermediate frequency is ωIF =10 MHz. This intermediate frequency is identical to the one used in section 4.2. We set ωIF > 0to avoid signal spectrum inversion [64]. The internal mixer of the PSG was used to fold thecontinuous microwave generated by the PSG itself with a rectangular pulse with a period of 3µsand a delay and width of 1µs generated by the DTG (see Figure 3.7 for wiring). So if the IQmixer would work perfectly we expect a recorded signal which is zero for the first microsecond,a sine with a fixed amplitude A(t) = const. and a time independent phase ρ(t) == const. inthe next microsecond and once again zero for the rest of the pulse. The data of the recordedcalibration pulse is depicted in Figure 3.10 (a).

36


Figure 3.10: A calibration pulse recorded by the ACQIRIS card for 3µs and averaged 20 milliontimes for a output power of −100 dBm at the PSG. The raw data in (a) still seems to be noisyand it is clearly visible that channel 2 has a DC offset. In (d) we show a zoom in of the greenbox in (a) detect a spurious signal with a frequency of 200 Mhz. By applying a digital slidingwindow filter of that frequency on both channels we end up with the figure depicted in (b).Channel 2 is still perturbed by the pattern shown in (e) as a zoom in of the magenta box in (b).We would like to stress that (e) shows the pattern only once and that it has a width of 32 datapoints corresponding to a frequency of 12.5 MHz. If we compensate this pattern with a hoppingwindow filter we end up with the figure in (c).

37


As depicted in Figure 3.10 (a) the data is of desired shape, the fact that the sine does not start atexactly 1µs is due to delays caused by the internal mixer of the PSG. The intermediate frequencyseems to be 10 MHz because we observe 10 periods in one microsecond. Although the data areaveraged 20 million times they still seem to be noisy, which is not expected because for a largenumber of averages and due to the law of large numbers the signal trace should be smooth andnoise should cancel out. By having a closer look at the data it turns out that these are artifactsadded by the ACQIRIS card, which could be filtered digitally. The first feature we discover isthat there is a high frequency sinusoidal structure with a frequency of 200 MHz on the signal sas shown in Figure 3.10 (d). This is exactly half the sampling frequency of the AQCIRIS cardand it is hard to imagine that is caused by the signal because of the mounted low pass filterfrom DC to 32 MHz. This pattern could be removed easily by applying a so called digital slidingwindow filter. The idea of such a filter is that one averages over one period τ from a time tto a time t + τ and takes this new value as the value for time t. We have to choose the sizeof the window w such that the condition 2πw/ωSa = τ is fulfilled, where ωSa/2π = 400 MHzis the sampling frequency of the ACQIRIS card. In this case if we want to filter a 200 MHzartifact from the signal s we have to choose a window which consists of two data points suchthat the width w = 2. By integrating a sine or cosine function over a whole period the resultis zero because the positive and negative areas are equal. So we use numerical quadrature andimplement a modified trapezoidal rule for this task in the form of

si =1

w

w−1∑j=0

s(i+j) with i = 1, 2, . . . S − w, (3.3)

where S denotes the length of the signal (product of the sampling rate and the readout time).Due to the fact that the filter acts from the first data point to the S−w-th data point, we looseone period by applying this filter. As a result, by applying the sliding window filter one filtersout the frequency 1/τ and also higher harmonics and ends up with the average value of all otherfrequencies. A MATLAB function of this filter can be found in Appendix A.2. The result of thisoperation on every channel is visualized in Figure 3.10 (b). Channel 1 now appears smooth andas expected for 20 million averages, none the less channel 2 still has a pattern with a period of80 ns on it. According to the sampling rate ωSa of the ACQIRIS card the width of the patternis w = 32 data points. The shape of the pattern is depicted in Figure 3.10 (e). To obtain theshape of the pattern and filter it out one could do something similar to a sliding window filter.So we use what we call hopping window digital filter. The respective MATLAB function canbe found in Appendix A.2. To extract the shape of the pattern we average not over a periodas we do in the sliding window filter but now we average over the value of the signal s at acertain position of the pattern pi over a number of periods K, this operation is given by themathematical expression (3.4).

pi =1

K

K−1∑j=0

s(i+jw) with i = 1, 2, . . . w (3.4)

Where pi is the value of the pattern at position i and w is the width of the pattern/window asdefined in the case of a sliding window filter. The next step is to subtract this pattern from thesignal trace so one has to shift the window by a fixed steps of with w. In detail we subtract thatpattern p from the first 32 points of the signal s than the pattern is hopping to the next 32 datapoints where it is subtracted and so forth. There exists a pitfall if the length of the recordedsignal is not a multiple of the width w of the window. There are two possible solutions eitherone disregards the tail of the signal or one subtracts only the first part of the pattern which hasthe same length as the tail. We decided to disregard the tail of the data presented in this thesisbecause the readout times where chosen long enough such that we do not loose any information.

38


Another advantage of the hopping window filter is that it also acts as a DC filter, because onesubtracts a fixed structure from the signal and as one could see from equation (3.4) it takes intoaccount constant terms such as DC components, as could be seen in in Figure 3.10 (c). Herewe end up with a smooth signal which could now be used to calibrate the IQ mixer. However,we had a closer look at the origin of the artifacts and it turns out that they are added by theACQIRIS card itself and are linked to the usage of the external reference input port where theSRS FS 725 is linked to, so a possible reason for the presented features is an internal problem ofthe ACQIRIS card as could be seen from the three different scenarios in Figure 3.11.

Figure 3.11: Three calibration pulses recorded by the ACQIRIS card for 3µs and averaged 20million times for a power output of −100 dBm at the PSG for different reference sources. In(a) we connected the SRS to the external reference port of the ACQIRIS and the 10 MHz signalfrom the SRS is used as reference frequency. In (b) we disconnected the SRS and measured withthe internal reference frequency. Compared to (a) we got rid of the artifacts but received biggerDC offsets and the phase of the sinusoidal signal changed. The external reference was connectedbut the internal was used for the measurement in (c). Despite different DC offsets there is nodifference in comparison to (b).

39


In Figure 3.11 (a) the raw data first time presented in Figure 3.10 (a) is shown again for compar-ison, here we connected the SRS to the external reference port of the ACQIRIS and the 10 MHzsignal from the SRS is used as reference frequency, while in (b) the SRS is disconnected, theinput port is terminated and the internal reference is used during the measurement. It turnedout that we get rid of the artifacts we have in (a) but we observe larger DC offsets. The phaseof the sinusoidal signals at the beginning of the pulse also changed, which is not surprising if another phase reference is in use. In the last case depicted in (c) we connect the SRS again butstill use the internal reference source, despite different DC offsets the recorded data is the sameas in (b). So there is strong indication that the circuit of the external reference of the ACQIRIScard is damaged. However, this does not perturb our measurements because as we showed herewe implement a powerful digital filter which cancels out the artifacts. The next thing one hasto guarantee is that one does not operate the amplifiers above their compression points. So wemeasured the 1 dB compression point of the devices mounted as visualized in Figure 3.8 at roomtemperature. Therefore, the PSG was directly connected to the RF input of the IQ mixer. Thecompression point denotes the lowest input power at which the gain is reduced by more than1 dB. The results of this measurement for an uncalibrated but digitally filtered IQ mixer is shownin Figure 3.12.

Figure 3.12: The 1 dB compression point of the used amplifiers in the IQ detector depictedin Figure 3.8. We plotted the measured gain vs. the RF-input power of the IQ mixer. Thedashed horizontal lines are a fit to the part where the gain is almost constant (input power from−65 dBm to −30 dBm.) and provide a gain of 44.7 dB for Q and 44.2 dB for I. The small stepsat low powers arise from internal step attenuator from the input source. The 1 dB compressionpoint is then given by −23.1 dBm for the I-channel and −23.6 dBm for channel Q, which is moreclearly visible in the inset.

40


As we now want to concentrate on the IQ mixer itself, channel 1 of the ACQIRIS card will bedenoted by I and channel 2 with Q for the same reason. From the time trace recorded by theACQIRIS card we fitted a sine for each power value to receive a fitted amplitude u. Because thecircuit is matched to R = 50 Ω the gain could be calculated via:

Pout =U2effR

=u2

2R(3.5)

gain(dB) = 10

(1 + 2 log10

(u(V)

1 V

))− Pin(dBm) (3.6)

The steps at low power in both channels arise from the internal step attenuators of the signalsource. If there were no other loss mechanisms in the two chains one would expect a theoreticalgain of 45 dB. Here the measured values of 44.7 dB for Q and 44.2 dB for I are in the expectedrange. It turns out that the 1 dB compression point is at −23.1 dBm for the I-channel and−23.6 dBm for channel Q. The difference in power shows that a calibration of an IQ mixer isinevitable. The measurements presented in section 3.2.2 is driven at powers such that the RFinput power of the IQ mixer is approximately −100 dBm, so that we do not operate the amplifiersbeyond their compression point. After we got rid of the artifacts added by the ACQIRIS cardand we showed that we do not operate the amplifiers beyond their compression point, we cannow concentrate on calibrating the IQ mixer i.e. we apply a correction matrix which removesDC offsets, equals the amplitudes of the two channels and sets the phase shift between themto π/2. Despite the fact that there already exists a digital homodyning calibration method [50]we decided to implement our own digital heterodyning calibration method. In a homodyningcalibration method the intermediate frequency ωIF is set to zero while in a heterodyning methodit is of the order of several MHz [50]. Our heterodyning method is based on fitting an ellipse[74] in homogeneous coordinates in the real projective plain RP2 [75, 76] to the recorded datain a numerical stable way [77] and afterwards transforming the ellipse into a circle. By doingso we are able to remove DC offsets of both channels, let them be phase separated by π/2 andthe amplitudes of the two channels will become equal to the radius of the circle. A detailedderivation of the correction values from the fitted ellipse parameters can be found in AppendixA. Here we introduce the main steps of the correction algorithm and show the effects to the datapresented in Figure 3.10. We take the calibration pulse shown in Figure 3.10 (c), remember thatthere have been applied digital filters on it, and plot I vs. Q for a time from 1.2 to 2.1µs, thisis visualized in Figure 3.13 (b).

41


Figure 3.13: A calibration pulse recorded by the ACQIRIS card for 3µs and averaged 20 milliontimes for a output power of −100 dBm at the PSG. The raw data in (a) is already digitallyfiltered to remove the artifacts generated by the ACQIRIS card. For the calibration of the IQmixer only the region between 1.2µs and 2.1µs is used. Therefore, we plot I vs. Q in (b) andfit an ellipse to the data points. After the calibration algorithm was applied to the data and thefitted ellipse we end up with a circle depicted in (c). In (d) channel I and Q are plotted versustime here it is visible that the two signals have the same amplitude and are phase separated byπ/2.

It turns out, that as expected the signal is shifted by a DC offset in both channels. The fittedvalues are DCI = −34µV for I and DCQ = −52µV for Q. The amplitude of the respectivechannels is AMI = 510µV and AMQ = 543µV. The phase difference of the signal is fitted tobe δ = 4.5. We apply a correction matrix C of the form written down in equation (3.7) to therecorded data given in homogeneous coordinates.

C = Rη−δAMR−ηDC =

∗ ∗ ∗∗ ∗ ∗0 0 1

(3.7)

Here the matrix DC takes into account DC offsets in both channels so its explicit shape is

DC =

0 0 −DCI

0 0 −DCQ

0 0 1

. (3.8)

With DCI and DCQ as the fitted values for a DC offset.

42


The next three transformation matrices Rη−δAMR−η correct the amplitude and phase differenceof both channels at the same time in the eigenbasis of the ellipse. Therefore, we lost the phase ofchannel I at the beginning of the pulse, as we want to recover the phase of channel I we correctthe angle error received in the dilation AM by an additional rotation with the angle −δ. Theorigin of this angle error is due to the fact that the dilution factor in the first direction of theeigenbasis of the ellipse differs from the factor according to the second eigenvector.

Rη−δAMR−η =cos(η − δ) − sin(η − δ) 0sin(η − δ) cos(η − δ) 0

0 0 1

AMISMA 0 0

0 AMISMI 0

0 0 1

cos(η) sin(η) 0− sin(η) cos(η) 0

0 0 1

(3.9)

Where η denotes the angle between the semi-mayor axis SMA and the x-axis. We decided toset the amplitude of the calibrated signal for both channels to the analytically calculated valuefor the amplitude of channel I AMI. SMI corresponds to the length of the semi-minor axis ofthe ellipse. All the correction values presented here can be calculated from the fitted parametersa, b, . . . , f of the ellipse, which we extracted from the recorded data by using Fitzgibbon’s method[74]. The main differences to a homodyning calibration method [50] are that we do not have totransform the data in a rotating frame which results in a multiplication of a universal matrix(due to time dependency) to each signal point. Further we are in a more general case whereno small angle approximation is used to assign the correction matrix and finally we are ableto remove DC offsets. These would be converted up to the intermediate frequency ωIF in therotating frame due to the matrix multiplication in the homodyne method. After we applied thecorrection matrix C we end up with Figure 3.13 (c). Now I and Q are calibrated which meansthat they have the same amplitude and are phase separated by π/2. This is more clearly visiblein a parametric representation with respect to the time in Figure 3.13 (d). All four graphs havein common that they are affected by a drift which arises from the nearly linear increase of themeasured amplitude in I and Q over the calibration region [see Figure 3.13 (a)]. The origin of thisincrease might be a ground loop or a periodic signal with a frequency of a few kHz. This resultsin an oscillation of the amplitude A(t) and phase ρ(t) at the intermediate frequency ωIF. If wewould apply a digital sliding window filter at that frequency on the channels I and Q we wouldalso cancel out the information we want to get through the measurement. Therefore, we applya digital sliding window filter after we extracted the time dependent amplitude A(t) and phaseρ(t). This has to be clarified in future works. However, before we proceed with a reconstructionof the time dependent amplitude A(t) and phase ρ(t) which carries the information about ourexperiment we would like to compare the recorded raw data with the filtered and calibrated data,this is visualized in Figure 3.14.

43


Figure 3.14: In (a) the recorded raw data of the ACQIRIS card is shown, while in (b) we presentdata filtered by using a sliding window filter with a frequency of 200 MHz, a hopping windowfilter at a frequency of 12.5 MHz and finally calibrating the two channels of the IQ mixer byusing the digital heterodyning method introduced in this thesis.

The applied digital filters and the heterodyne calibration method improves the data quality a lot,when we compare the raw data in Figure 3.14 (a) to the calibrated and filtered data in (b). Thesliding window filter smoothed the two curves while the hopping window filter removed the DCoffset of channel Q and the periodic structure presented in Figure 3.10 (e). Finally the heterodynecalibration method causes the amplitudes of channel I and Q to become equal and let the twosignal channels be phase shifted by π/2. The so recorded and calibrated pulses can now be usedto extract the time dependent amplitude A(t) and phase ρ(t) of the experiment. In the case of acalibration pulse (which in our case is a pure sine signal with a time independent amplitude andphase) the amplitude and phase deviation should be almost constant since there is no device inthe measurement chain that disturbs the signal generated by the microwave source. However,with a sample mounted between the signal generator and the detector we get a response fromwhich we can extract for example the characteristic time scales of our coupled qubit resonatorsystem. It can be shown [50] that one can calculate the amplitude and phase out of the recordedquadratures I and Q via

A(t) =√I2

(t) +Q2(t) (3.10)

ρ(t) = arctan

(Q(t)

I(t)

)(3.11)

By applying these formulas to the data presented 3.13 (d) we end up with an amplitude andphase for every recorded time increment. The result is visualized in Figure 3.15.

44


Figure 3.15: The reconstructed amplitude and phase for a calibration pulse. In (a) the re-constructed amplitude A(t)for the data presented in Figure 3.13 (d) is plotted for each timeincrement. The blue circles depict the calibrated data before a digital sliding window filter atthe intermediate frequency ωIF was applied while the green circles show the recorded data afterthat filter was used. The red line indicates the case of perfect calibration without any source ofnoise. The same as in (a) but for the extracted phase ρ(t) is shown in (b).

In Figure 3.15 (a) the amplitude at every recorded time increment is centered around 510µVwhich is the fitted amplitude of channel I and so the desired value as we set it equation (3.9). In(b) the phase is centered around zero as expected for a time independent phase of the calibrationpulse. The blue circles visualize the data which was filtered by a digital sliding window filterof 200 MHz, a hopping window filter of 12.5 MHz and calibrated via the digital heterodyningcalibration method introduced in this thesis on detail in Appendix A. Due to the linear increaseof the test pulse in the calibration region [see Figure 3.13 (a)] we apply a digital sliding windowfilter at the intermediate frequency ωIF = 10 MHz to compensate this error such that we end upwith the green circles. We would like to mention that we lost one period in comparison to theunfiltered data (blue circles) as a result of the definition of the digital sliding window filter (3.3).By using all these filter and calibration methods presented in this chapter we improve the dataquality a lot such that we can extract the physical information of time-domain measurementspresented in section 4.2 with high accuracy. In summary we showed that we can handle thehardware required to perform pulsed measurements, split up the recorded signal with the helpof an IQ mixer and record both channels with an ACQIRIS card. Furthermore, we understandthe shape of the raw data, can identify the source of the artifacts as the reference port of theACQIRIS card and filter them out digitally. We demonstrated how to calibrate the IQ mixerused in the setup with a powerful heterodyne calibration algorithm.

45


46


4 Measurement results

As mentioned in the previous parts we study a superconducting resonator coupled to a flux qubitand observe the response for continuous wave spectroscopy and time-domain measurements. Aswe intend to observe quantum mechanical phenomena the sample is cooled down to very lowtemperatures to neglect thermal population in the resonator as described in detail in section3.1.2.1. Here we present the necessary measurements to characterize our system in the frequency-and time-domain. We begin with a determination of the resonator modes followed by a fluxcalibration. Afterwards we perform continuous wave single- and two-tone spectroscopy includinga photon number calibration. Finally we present pulsed wave spectroscopy and time-domainmeasurements of the coupled qubit resonator system which were successfully performed in thequbit group of the WMI for the first time.

4.1 Continuous wave spectroscopy

In this section we present all important measurements to characterize the coupled qubit-resonatorsystem in the frequency-domain. First we are interested in the modes of the resonator when thesample is cooled down to approximately 50 mK. Therefore, we perform a wide range frequencyscan with the vector network analyzer (VNA) and search for peaks in the transmitted amplitudein a single tone continuous wave spectroscopy (data not shown). Afterwards we perform a moredetailed scan on the peaks and fit a Lorentzian to the recorded data. In Figure 4.1 this isvisualized for the second harmonic and yields a resonance frequency of ω2/2π = 7.1057 GHz forthe resonator far detuned from the degeneracy point of the qubit. Taking into account the fullwidth half maximum (FWHM) κ2 = 831 kHz from the fit data we can estimate the quality factorof our superconducting resonator Q2 = ω2/(2πκ2) ≈ 8500. By repeating this procedure for thefundamental mode and the first harmonic we enumerate that the three experimentally accessiblemodes have resonance frequencies given in Table 4.1. It is worth mentioning that the limitedamplifier bandwidth ranging from 4 to 8 GHz causes a low signal to noise ratio (SNR) of thefundamental mode, making it challenging to record a clear spectrum such that the uncertaintiesprevail the accuracy of the fit. We would like to mention that all three experimentally accessiblemodes are affected by a mode-dependent phase drop as discussed on detail in section 2.2. Dueto the limited bandwidth we are not able to determine the mode ω3 which would not be affectedby a phase drop.

mode i frequency ωi/2π (GHz) κi (MHz) Qi (i+ 1) · ω0/2π (GHz)0 (λ/2) 2.642 — — 2.6421 (λ) 5.067 1.25 4054 5.284

2 (3λ/2) 7.106 0.831 8551 7.926

Table 4.1: The three experimentally accessible resonator modes with their center frequencies.If we compare the second and fifth column, it turns out that the coupling junctions causes anon-equidistant spacing of the modes.

47


Figure 4.1: A frequency scan of the second harmonic far detuned from the degeneracy point ofthe qubit. The Lorentzian fit yields a resonance frequency of 7.1057 GHz. The deviations of thefit from the measured data for low transmission values arise from noise.

4.1.1 Flux calibration

Now that we know the resonance frequencies of our resonator modes we can continue with a fluxsweep to determine which amount of current through the magnet coil corresponds to a flux of1 Φ0 in the qubit loop. We vary the current from −10 mA to +10 mA and use a frequency windowfrom 7.105±0.015 GHz to record the measured transmission with the VNA. We observe periodicanticrossings with a period of 1 Φ0 [14, 15]. The result of this measurement for an input powerof approximately 2 photons on average (poa) in the resonator is visualized in Figure 4.2 and itturns out that we observe three anticrossings. We express the power used for the measurementin number of photons in the resonator. These numbers are calculated in section 4.1.3. To carryout the photon number calibration we have to do a flux calibration and a determination of thecoupling strengths gi first. When this is done we can determine the average photon numberin the measurements. However, the measurement presented in Figure 4.2 gives us a roughestimation of the current to flux conversion factor. Therefore, we performed measurements witha higher resolution of each anticrossing to determine our conversion factor with high precision.By identifying distinctive points pj,i (see Figure 4.3) and taking the average of the degeneracypoint qj for each current range we are able to define a conversion factor from current to flux.The numbers for the points pj,i presented in Table 4.2 represent the current value at frequenciesof 7.1 GHz and 7.1136 GHz where the recorded transmission through the coupled qubit resonatorsystem reaches a maximum (see Figure 4.3). As a result we are able to label the axis in units offlux detuning δΦ instead of the coil current If which was applied for the measurement. Therefore,the mean value of the current q corresponding to 1 Φ0 is given by:

q =1

2

3∑j=2

(qj − qj−1) =q3 − q1

2= 6.444 mA (4.1)

48


Figure 4.2: A scan over the full range of the coil current If from −10 mA to +10 mA for an inputpower of 2 poa in the resonator mode ω2. Within this range we observe three anticrossings (bluewindows) which we use for flux calibration. The discontinuity at approximately 7 mA is causedby a temperature instability which occurred at that time as could be seen from our temperaturemonitoring (data not shown).

Measurement j pj,1 (mA) pj,2 (mA) pj,3 (mA) pj,4 (mA) qj = 1/4∑4

i=1 pj,i (mA)1 −8.257 −8.249 −8.207 −8.199 −8.228

2 −1.821 −1.813 −1.774 −1.766 −1.794

3 +4.630 +4.678 +4.678 +4.688 +4.659

Table 4.2: The flux current values of the four distinctive points taken from Figure 4.3 for eachanticrossing available in the range of the current source. In the last column we present theaverage value of the current where the corresponding flux is n/2 · Φ0

Due to the fact that this mean value q is a telescoping series we calculate the variance σq withthe formula:

σq =1

2(|q2 − q1 − q|+ |q3 − q2 − q|) = 9.5µA (4.2)

so we define the variance as the average of the absolute deviations in every single period. Finallythe applied coil current If can be converted into flux in units of Φ0

If(mA)

Φ(Φ0)= 6.444± 0.009. (4.3)

According to ref. [45] the systematic error of the used 16 bit analog digital converter which isused to set the current If to a certain value is given by 0.3 nA such that the statistical errorsprevail.

49


Figure 4.3: A more detailed scan over the three anticrossings first observed in Figure 4.2 in afrequency window of 7.106±0.010 GHz over a coil current range of 0.2 mA. The distinctive pointspj,i (black circles) are taken at a frequency of 7.1 GHz for pj,1,4 and 7.1136 GHz for pj,2,3. Inall three measurements the resonator mode is populated with approximately 2 poa.

50


4.1.2 High power continuous wave spectroscopy

In this section we provide first results of high power continuous wave spectroscopy experiments.Furthermore, we fit the Jaynes-Cummings Hamiltonian (2.47) to the data to estimate the cou-pling strengths gi, the energy gap of the qubit ∆ and its persistent current IP. This enables usto determine the photon number population in the resonator in section 4.1.3. We do this for allthree experimentally accessible modes, the fundamental mode, the first harmonic and the secondharmonic in a single tone continuous wave experiment. In the data presented in Figure 4.4 theoutput power of the VNA is set to a value such that according to the photon number calibrationin section 4.1.3 the second harmonic of the resonator is populated with 2 poa.

Figure 4.4: The three experimentally accessible modes recorded in a continuous wave spec-troscopy experiment with the setup depicted in Figure 3.5. The second harmonic is depictedin (a). We observe a clearly visible anticrossing symmetric around the degeneracy point withdetuning δΦ = 0. The discontinuities (black arrows) arise from higher order transitions [45]. In(b) and (c) we show spectroscopy data for the first harmonic and the fundamental mode. Dueto the limited bandwidth of the cold HEMT amplifier the signal to noise in the (c) is lower.

51


For the second harmonic in Figure 4.4 (a) we observe an anticrossing where the qubit excitationfrequency ωQ is on resonance with the second harmonic of the resonator ω2 at δΦ ≈ ±4 mΦ0.In this region the recorded resonator transmission is reduced due to the new eigenstates of thecoupled qubit resonator system as described in section 2.3. The discontinuities (black arrows)are caused by higher order transitions which can occur at large drive powers [45]. The recordedtransmission spectrum for the first harmonic in Figure 4.4 (b) is reduced for the same reasonaround the degeneracy point of the qubit. Finally the fundamental mode (c) is dispersivelyshifted around the degeneracy point of the qubit. We would like to mention that for a photonnumber calibration the coupling strengths gi have to be known [45]. These can be estimatedfrom a numerical fit of the multi-mode Jaynes-Cummings Hamiltonian (2.47) to the recordeddata. The result of such a fit is presented in Figure 4.5.

Figure 4.5: A numerical fit of the recorded spectra to the theoretical Hamiltonian (2.47). Thedata of the second (a) and first harmonic (b) agree well with the fit. In (c) the qubit excitationfrequency was recorded in a two-tone spectroscopy experiment. In (d) we present the fit tothe fundamental mode. The used fitting parameters are the coupling strengths to each modeg0, g1, g2, the energy gap of the qubit ∆ and the persistent current of our qubit IP. Some ofthese parameters are used to calibrate the photon number n in our system.

52


In Figure 4.5 we use the three single tone continuous wave spectroscopy measurements presentedin Figure 4.4 and a two-tone continuous wave spectrum [see Figure 4.5 (c)] for the fit. At thispoint the author wants to thank Juan José García-Ripoll from Universidad Complutense deMadrid for coding the main parts of the fitting program. The numerical fit of the multi-modeJaynes-Cummings Hamiltonian (2.47) shows that the recorded experimental data agrees verywell with the theory. The fitted coupling of the qubit to the second harmonic in Figure 4.5 (a)is approximately 76.8 MHz which results in a relative coupling strength g2/ω2 ≈ 1 %. In (b)the fit of the theory to the experimental data yields a relative coupling rate of approximately1.5 %. A two-tone spectroscopy was performed in (c). We are able to detect the qubit hyperbolayielding a qubit delta ∆/h ≈ 4.94 GHz and a persistent current IP ≈ 205 nA. The fit also showsthat the mode spacing in the resonator is non-equidistant as the first exited state of the firstharmonic |1〉1 has an eigenenergy of ω1/~ = 5.067 GHz [lowest horizontal line on the right handside of (c)] while the eigenenergy of the eigenstate |2〉0 is 2ω0/~ = 5.284 GHz (middle horizontalline). According to ref. [46] this is caused by a mode dependent phase drop over the couplingjunction. We would like to point out that this eigenstate is present in the theory as well as inthe numerical fit. But we do not observe it in the experimental recorded frequency spectrumbecause it refers to a two photon process with an energy of 2~ω0 in the fundamental mode. Thistwo-tone spectroscopy data also shows that the second harmonic ω2 (upper horizontal line) isalmost constant in the recorded region, this is important as we can use the second haromic as theread out mode in the photon number calibration. In Figure 4.5 (d) we fit the energy spectrum tothe fundamental mode ending up with a relative coupling of approximately 0.5 %. We would liketo point out that all three coupling rates are on the order of 1 % of the respective resonator modeand there is no evidence for ultra strong or even deep ultrastrong coupling as expected from adecrease in the size of the coupling junction. This relation was proposed in ref. [45]. However, wediscuss this in more detail in section 4.1.4. The next step is a calibration of the photon numbersin the resonator. Therefore, we present a two-tone spetroscopy in the next section.

4.1.3 Photon number calibration

In this section we calibrate the photon number in the resonator via a two-tone spectroscopy.We perform a frequency sweep for different probe tone powers PP from −25 to +5 dBm in stepsof 1 dBm generated by the VNA. During the measurement the flux detuning is constant at aconstant flux value close to δΦ = 0. The probe tone frequency is fixed to the resonance frequencyof the second harmonic ωP/2π = ω2/2π = 7.109 GHz. The spectroscopy tone generated by theSMF at a power of PS = −30 dBm is swept from ωS/2π = 4.5 GHz to 5.3 GHz and applied tothe qubit via the antenna. The condition PS > PP is fulfilled due to the 50 dB attenuator atroom temperature at the output port of the VNA as depicted in Figure 3.5. The experimentalresult for the power range from PP = −25 dBm to 5 dBm is depicted in Figure 4.6. The blueregion at PP = −22 dBm for a spectroscopy frequency from 5.2 GHz to 5.3 GHz is caused by acommunication error between the VNA and the measurement computer such that we could notrecord the acquire data.

53


Figure 4.6: Photon number calibration in a two-tone spectroscopy. In (a) we depict the recordedspectra for each probe power PP and the spectroscopy tone ωS. For 0 to 5 dBm we observe asingle dip which represents the dispersively shifted qubit excitation energy. For probe powervalues below −1 dBm we observe two dips. The fitted center frequencies of a Lorentzian fitto each dip are visualized in (b). We simulate the interaction of the first harmonic ω1 withthe photon number dependent dispersively shifted qubit excitation energy ωQ and diagonalizedthe Hamiltonian such that we end up with the eigenstates depicted in (c). We use numericaloptimization to find the linear relation of the average photon number n and the applied probepower PP visualized in (d). To verify that theory and the measured spectra agree we use thefitted line (red line in (d)) to plot the experimental data (circles) and the theoretical model (solidlines) in (e).54


According to equation (2.59) the dispersively shifted qubit excitation frequency is given by

ωQ = ωQ +(g2)2

δω2(2n+ 1) . (4.4)

Since there is a linear dependence between the dispersively shifted qubit excitation frequency ωQ

and the photon number n in the second harmonic resonator mode ω2 we expect to measure asingle transmission dip in a two-tone spectroscopy measurement which represents this dispersivelyshifted excitation frequency. If we consider the used probe tone frequency ωP/2π = ω2/2π =7.109 GHz and a qubit excitation frequency ωQ ≈ ∆/h = 4.94 GHz we determine the frequencydetuning to δω2/2π = −2.166 GHz for a measurement closed to vanishing flux detuning. Fora coupling strengt g2/2π = 76.8 MHz the shift per photon is expected to be 2 (g2)2 /δω2 =2π (−5.45 MHz), so the measured qubit excitation energy should decrease with increasing power.If we compare these estimations to the acquired data in Figure 4.6 (a) we find that this holdsonly for probe powers PP larger than 0 dBm. Below this power value we observe two dips. Byfitting a Lorentz distribution to each dip, we end up with the frequencies depicted in Figure 4.6(b). For the rest of this section we denote the dip at higher frequency (blue circles) with ν+ andthe one at lower frequency (green circles) is denoted by ν−. For low powers the two dips seemto remain at a constant value within the frequency window ω1 ± g1 around the first harmonicresonator mode. This gives rise to the assumption that the condition δω1 < g1 is fulfilled andthat the observed two dips are the dressed states |m,+〉1 and |m,−〉1 where m denotes thephoton number in the first harmonic of the resonator as described in section 2.3. So we describethis system with the Hamiltonian

H

~=

(ω1 g1

g1 ωQ

), (4.5)

where the first harmonic of the resonator ω1 interacts with the dispersively shifted qubit ex-citation frequency ωQ (4.4) via the coupling g1. According to the calculations carried out inAppendix B the eigenstates of the diagonalized Hamiltonian (4.5) are given as

λ± =

(ω1 + ωQ + (2n+ 1) (g2)2

δω2

)±√(

ω1 − ωQ + (2n+ 1) (g2)2

δω2

)2+ 4 (g1)2

2. (4.6)

The eigenstates λ± for a qubit excitation frequency of ωQ/2π = 5.04 GHz, coupling strengthsof g1/2π = 86.5 MHz, and g2/2π = 88.2 MHz are visualized in Figure 4.6 (c). The coupling g1

is chosen such that the level splitting for the lowest probe tone power PP is given by 2g1. Thequbit excitation energy ωQ is close to the fitted value for the qubit energy gap ∆/h = 5.02 GHzin the low power two-tone spectroscopy of Figure 4.7 (f). The small deviation of 20 MHz of theused ωQ/2π = 5.04 GHz and the expected qubit excitation frequency at the degeneracy point ofthe fit ∆/h = 5.02 GHz might be due to the accuracy of the fitting algorithm from which weobtained ∆. The used coupling strength g2 is also obtained from the low power spectroscopyfit. If we compare the measured eigenfrequencies ν± with the simulated eigenfrequencies λ± wefind that they agree very well. This gives rise to the assumption that the Hamiltonian (4.5)describes the measured data properly. In the next step we have to correlate the average photonnumber n from which we simulate the eigenstates λ± with the applied probe power PP for thetwo measured frequencies ν±. Therefore, we use numerical optimization via the ansatz

minn

∥∥∥∥(λ+

λ−

)−(ν+

ν−

)∥∥∥∥2

. (4.7)

55


So for each pair of measured frequencies ν± we determine the average photon number n forwhich the distance between the theoretical eigenfrequencies λ± and the measured frequencies ν±is minimal in the Euclidean norm ‖ ‖2. After these calculations we end up with pairs of probepowers PP and average photon numbers n for which the condition (4.7) is fulfilled. These pairsare visualized in Figure 4.6 (d). To complete the photon number calibration we fit a line troughthe origin (red line in (d)). By converting the applied probe powers PP to photon numbers onaverage n we find that the measured frequency pairs ν± agree very well (see Figure 4.6 (e))with the theoretical eigenfrequencies described by the Hamiltonian (4.5). Finally we are ableto calibrate the average photon number n of the second harmonic mode ω2 by simulating theinteraction of the dispersively shifted qubit excitation energy ωQ and the first harmonic modeω1. An excerpt of average photon numbers n and corresponding probe power PP are given inTable 4.3.

average photon number n probe power PP (dBm)0.25 -201 -142 -10.5

Table 4.3: The probe power PP in dependence of the average photon number n in the secondharmonic ω2 according to the calibration.

Here we use the dressed states of a qubit and the resonator mode ω1 to calibrate the photonnumber in the resonator mode ω2 which dispersively shifts the qubit excitation energy. Withthe performed photon number calibration, we are now able to gauge the additional losses in theresonator input line which are caused for example by cable loss and insertion loss. For n = 1 poawe send in −14 dBm from the output port of the VNA. Since we know our resonator frequencyω2/2π = 7.106 GHz and the FHWM κ2 = 831 kHz of the second harmonic from Figure 4.1 thepower in the resonator is

Prf = 10 log10

(n~ω2κ2

1 mW

)= −144 dBm (4.8)

The theoretical power which arrives in the resonator is Pth = −114 dBm. This value is calculatedfrom the output power of the VNA and the attenuator configuration of 50 dB at roomtemperature(see Figure 3.5) and 50 dB inside the shielding room (see Figure 3.3). So the difference of 30 dBis caused by the additional losses mentioned above. However, since we calibrated the photonnumber in the second harmonic of the resonator we are now able to perform continuous waveexperiments with negligible photon number on average in the resonator in the next section.

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4.1.4 Low power continuous wave spectroscopy

Based on the performed photon number calibration in the last section we are now able to repeatthe continuous wave spectroscopy experiments of section 4.1.2 with a negligible number of pho-tons in the resonator. As a result we are able to determine the characteristic frequencies in oursystem with more precision since they are no longer perturbed by the power dependent shifts dis-cussed in section 2.3. This step is very important since we want to perform pulsed measurementswith fixed frequencies in the last part of the thesis. The result of a single tone continuous wavespectroscopy measurement of the first and second harmonic as well as a two-tone continuouswave spectroscopy experiment is visualized in Figure 4.7. In the two-tone spectroscopy we use aprobe tone frequency of ωP = 7.109 GHz and a spectroscopy power of PS = −30 dBm.

Figure 4.7: Measurement results for low power spectroscopy. In (a) we observe an anticrossingwhen the second harmonic ω2 is resonant with the qubit excitation energy ωQ at a negligiblenumber of photons in the resonator in a single tone continuous wave spectroscopy. In (b) thespectroscopy data for the first harmonic is visualized. A two-tone spectroscopy experiment witha higher resolution than in Figure 4.5 was performed in (c). The pictures (d) to (f) show a fitof the Jaynes-Cummings Hamiltonian (2.47) to the experimental data. As a result we end upwith coupling strengths g1/2π ≈ 71.9 MHz and g2/2π ≈ 88.2 MHz. The fit yields a qubit deltaof ∆/h ≈ 5.02 GHz and a persistent current of IP ≈ 208 nA. Futhermore we find |δω1| < g1

around vanishing flux detuning δΦ.

57


In Figure 4.7 (a) we observe an anticrossing of the second harmonic with the qubit hyperbola.Since we measure with a negligible number of photons the higher order transitions we observein Figure 4.4 (a) are no longer present in the spectrum. The first harmonic is shifted to lowerfrequencies around the degeneracy point of the qubit at δΦ = 0. In (c) we depict the experimentalresults of a two-tone spectroscopy. This picture shows a very nice coincidence since for vanishingflux detuning δΦ the resonance detuning |δω1| < g1 such that we can investigate the dressedstates as depicted in Figure 2.9 (a). As a result we are able to observe the transition from thedispersive to the resonant interaction regime of the Jaynes-Cummings Hamiltonian describedin section 2.3 by sweeping the flux detuning δΦ from finite values to zero. In analogy to thetheoretical results described in ref. [45] we observe a reduction of the measured transmissionwhen we undergo the transition from the dispersive to the resonant interaction in the Jaynes-Cummings Hamiltonian. The pictures (d) to (f) show fits to the Jaynes-Cummings Hamiltonian.We find relative coupling rates of approximately 1 % such that the qubit is strongly coupled tothe resonator. In agreement with the fact that we are at zero frequency detuning δω1 when theflux detuning δΦ is also closed to zero the fit shows that the qubits energy gap ∆/h ≈ 5.02 GHzis close to the excitation energy of the first harmonic of the resonator ω1/2π = 5.067 GHz so thecondition

|δω1/2π| = |ωQ − ω1| ≈ |∆/h− ω1| = 47 MHz < g1/2π ≈ 71.9 MHz (4.9)

is fulfilled. For flux detuning close to zero the dressed states |1,−〉1 and |1,+〉1 are formed fromthe energy levels |0〉1,|1〉1,|g〉 and |e〉 as described by equation (2.55). We do not observe ananticrossing in which the state |2〉0 with two photons is involved since this would not preservethe number of excitations which is one fundamental assumption of the Jaynes-Cummings modelintroduced in section 2.3. If we compare the results of the fit in section 4.1.2 to that in thissection we find the numbers presented in Table 4.4.

high power spectroscopy low power spectroscopyg1/2π (MHz) 72.4 71.9g2/2π (MHz) 76.8 88.2∆/h (GHz) 4.94 5.02IP (nA) 205 208

Table 4.4: Fit parameters of the Jaynes-Cummings Hamiltonian (2.47) for the performed highpower and low power continuous wave spectroscopy measurement.

The values for the coupling g1 of the qubit to the first harmonic of the resonator and the persistentcurrent IP agree very well. The deviations of the values for the qubit energy gap ∆ and thecoupling g2 also agree within the accuracy of the fit. With the higher resolution of the performedtwo-tone spectroscopy presented in Figure 4.7 (c) in comparison to that in Figure 4.5 (c) we areable to detect a rich stucture around the degeneracy point of the qubit. Since the probe tone ofthe VNA is set to a value such that the second harmonic of the resonator is populated with anegligible number of photons but the spectroscopy tone power PS = −30 dBm is still two ordersof magnitude larger in the two-tone measurement we observe some less pronounced side arms[blue arrows in Figure 4.7 (c)]. We have a closer look on that before we proceed with time-domain measurements. A two-tone spectroscopy experiment with a higher frequency resolutionthan in Figure 4.7 (c) for a probe tone frequency of ωP = 7.109 GHz and a spectroscopy powerof PS = −30 dBm is presented in Figure 4.8 (a).

58


Figure 4.8: Zoom in around the degeneracy point of the qubit in two-tone spectroscopy is shownin (a). We present a cut for vanishing detuning δΦ = 0 in (b). It turns out that there is anadditional pattern which we analyzed for a flux detuning of δΦ = 1.37 mΦ0 in (c). We multipliedthe data in (c) by −1 and plot it on a linear scale.

In Figure 4.8 (a) we observe two side arms (black arrows). As already mentioned the condition|δω1| < g1 is fulfilled such that we observe the dressed states |1,−〉1 and |1,+〉1 for vanishing fluxdetuning δΦ. Therefore, we present a one dimensional cut through the experimental data takenat δΦ = 0 in (b). It turns out that the dip at higher frequency is sharp and the other is flattened.Since we are interested in spectroscopy frequencies at which they reach their minimum, we fitteda Lorentzian to them [red and green lines in Figure 4.8 (b)]. The two center frequencies are fittedto

ν+ = 5.132 GHz, (4.10)ν− = 4.968 GHz. (4.11)

(4.12)

As a result they are split by the value

δν = ν+ − ν− = 164 MHz. (4.13)

59


According to equation (2.56) the photon number dependent energy level splitting in the resonantcase is given by

2g1

√m = 2 · 71.9 MHz

√m = 143.8 MHz

√m (4.14)

Where m denotes the photon number corresponding to the first harmonic ω1. If we assumeδν = 2g1

√m we find that m ≈ 1.3. The deviation from m = 1 might be due to the fact that the

Lorentzian from which we extracted ν− is a superposition of more than one vacuum Rabi level ordue to the accuracy of the fitting program which determines the coupling strengths. However, thefitted and calculated energy level splitting agree well within the scope of measurement accuracy.This indicates that we are able to observe the dressed states of a resonantly coupled qubitresonator system with a dispersive readout via a second resonator mode. If we apply a largerflux detuning δΦ ≈ 1.37 mΦ0 we change the qubit excitation energy via the flux dependentenergy bias ε [see blue line in Figure 4.8 (a)]. Therefore, the measured signal and linewidth ofthe dip which corresponds to the dressed state |1,−〉1 decreases. Furthermore, the dip is shiftedto slightly higher frequencies. Its limit is given by the resonator mode ω1/2π = 5.067 GHzwhich occurs for large detuning such that we end up in the dispersive limit. This is causedby the increase of the frequency detuning δω1 since by an increase of the flux detuning δΦwe increase the flux dependent energy bias ε given in equation (2.23) and therefore the qubitexcitation frequency ωQ [see equation (2.25)]. As a result the mixing angle ϑ form equation (2.53)converges to zero for large flux detuning δΦ. So for increasing flux detuning the contribution ofthe qubit to the state |1,−〉1 decreases [see equation (2.55)] and at a certain flux detuning thedressed states are no longer present since the condition g1 > δω1 is violated. Furthermore, thepeak of the dressed state |1,+〉1 is also shifted to higher frequencies and exhibits side arms. Thisis more clearly visible in Figure 4.8 (c). Here we are able to determine the center frequencies ofthe peaks to

ν|1,+〉1 = 5.316 GHz (4.15)ν|3,+〉1 = 5.211 GHz (4.16)ν|4,+〉1 = 5.194 GHz (4.17)ν|1,−〉1 = 5.054 GHz. (4.18)

The frequencies are given as the center frequencies of a Lorentzian fit. The choice of the indicesfor this frequencies will become clear below. These side arms look very similar to the vacuumRabi resonances observed in ref. [78]. We are able to identify two side arms |3,+〉1 (magenta)and |4,+〉1 (cyan) in Figure 4.8 (c). According to ref. [78] the energy levels of the n-photonsubspaces are given as

2πEn = ω1 ±g√n, (4.19)

where g denotes the mode spacing. If we assume that the center frequency ν|1,+〉1 = 5.316 GHzcorrespondes to the state with n = 1 we are able to calculate the mode spacing to

g/2π = ν|1,+〉1 − ω1/2π = 5.316 GHz− 5.067 GHz = 240 MHz. (4.20)

With this result we enter equation (4.19) again to end up with the numbers presented in Table4.5.

60


photon number n state energy En (GHz)1 |1,+〉1 5.1362 |2,+〉1 5.2683 |3,+〉1 5.2064 |4,+〉1 5.187

Table 4.5: The excitation energy of the four lowest vacuum Rabi levels according to equation(4.19).

The black horizontal lines in Figure 4.8 represent these numbers. It is worth mentioning that thefitted center frequencies ν|3,+〉1 and ν|4,+〉1 agree well with the excitation energies of the vacuumRabi levels. The fact that we could not observe a peak where the photon number should beequal to two could be caused by the line width of the resonance peak located around 5.3 GHz.We are not able to observe the symmetric behavior of the energy level splitting in contrast to thevacuum Rabi levels observed in ref. [78] since for the minus in equation (4.19) we are perturbedby the levels which would arise from the vacuum Rabi levels for negative flux detuning. However,the results presented here seem to be consistent with vacuum Rabi transition.

Up to now we characterized the coupled qubit resonator system in the frequency-domain. Weused single tone and two-tone continuous wave spectroscopy methods and fitted the theory of theJaynes-Cummings Hamiltonian (2.47) to the data acquired with a negligible number of photonsin the readout mode of the resonator. As a result we present the theoretical energy spectrum ofour sample in Figure 4.9.

Figure 4.9: The fitted energy spectrum of the Jaynes-Cummings Hamiltonian (2.47) to the datapresented in Figure 4.7. The vacuum mode is denoted by |0〉 For a large flux detuning δΦ weare able to identify the pure resonator eigenstates |n〉i where n denotes the number of photonsin the resonator mode i.

61


Figure 4.9 visualizes the energy levels of our sample for various flux detunings from −15 mΦ0 to+15 mΦ0. The spectrum is symmetric around zero detuning. We find that the Jaynes-CummingsHamiltonian (2.47) describes our sample very well. For zero flux detuning the qubit’s excitationenergy ωQ is equal to the energy gap ∆/h ≈ 5.02 GHz such that for the first harmonic thecondition δω1 < g1 is fulfilled which leads to a rich structure around zero flux detuning.

4.2 Time-domain measurements

In this part we present time-domain measurements on the coupled qubit resonator system. Toperform these experiments we use pulses with a pulse width τ at the resonance frequencies ofour sample for various flux values. To find the correct frequency we use the energy spectrumof the Jaynes-Cummings Hamiltonian (2.47) presented in Figure 4.9. Therefore, it was essentialto perform the spectroscopy experiments in the first part of this thesis. In a first approachto this task we perform a pulsed wave two-tone spectroscopy to check if the used hardwareintroduced and calibrated in section 3.2.2 works well in the experimental setup. Since the checkof the calibration and filtering methods introduced in section 3.2.2.2 was performed under testingconditions without a mounted sample between the signal generator and the detector. Afterwardswe successfully perform a time-domain measurement of a coupled qubit resonator system withthe ACQIRIS card in the qubit group of the WMI for the first time. In the last part we repeatthis measurement with an FPGA board to speed up the measurement.

4.2.1 ACQIRIS card measurements

In this section we present the measurement results recorded with the ACQIRIS card. First weperform a pulsed wave two-tone spectroscopy experiment to check the hardware and the softwarealgorithms developed during this thesis. Afterwards we perform a time-domain measurement ofthe coupled qubit resonator system to investigate the response of the sample on Rabi pulses.

4.2.1.1 Pulsed two-tone spectroscopy

To test the IQ mixer, the digital filter methods and the heterodyne calibration algorithm weperform a pulsed wave two-tone spectroscopy. A simplified circuit diagram and the used pulsepattern are depicted in Figure 4.10.

Figure 4.10: The setup of the pulsed wave two-tone spectroscopy experiment. On the left handside we depicted a simplified circuit diagram which indicates which signal source generates thesignals used in the setup. The used pulse pattern for the readout time t of the ACQIRIS cardand the pulse width τ are depicted on the right hand side.

62


We use one SMF to apply the spectroscopy tone ωS to the qubit via the antenna and a secondone is used as local oscillator ωLO for the IQ mixer, both are driven continuously. The PSG isdriven at ωP/2π = 7.109 GHz and at an output power such that the resonator is populated with1 poa (see Table 4.3). The output power of the SMF is PS = −30 dBm. The DTG is used with aperiod of 4µs to send a trigger of 3µs to the ACQIRIS card to define the length of the readouttime t. We use a second channel of the DTG to set the pulse width τ of the probe tone signalto 1µs. In this experiment we vary the spectroscopy tone ωS from 4.8 to 5.6 GHz in steps of25 MHz and the flux is swept from δΦ ≈ −1.5 mΦ0 to approximately 2 mΦ0. For each data pointwe record over 2.3 · 106 traces for averaging. From the recorded time traces which look similarto the calibration pulses presented in section 3.2.2.2 we extract the amplitude for a certain fluxvalue and a specific spectroscopy tone from the two fitted quadrature amplitudes AMI and AMQ

via the formula

B =

√(AMI)

2 + (AMQ)2. (4.21)

We are interested in the average amplitude B while the probe tone ωP was applied to theexperiment and not in the amplitude A(t) at every time increment of the readout time t, since wewant to perform a spectroscopy experiment. The result of this experiment is depicted in Figure4.11.

Figure 4.11: The recorded transmission spectra for a pulsed wave two-tone spectroscopy exper-iment. We are able to reproduce the results of the continuous wave two-tone spectroscopy inFigure 4.7 (c). However, this measurement technique is very inefficient, due to the large amountof time which is needed for averaging the time traces in the ACQIRIS card. We are able toresolve the side arms (black arrows) first presented in Figure 4.7 (c).

63


In general we can reproduce the experimental result of a continuous wave two-tone experimentwith the pulsed measurement technique presented in Figure 4.11. The mechanism for a decreaseof the recorded amplitude while the spectoscopy tone ωS is on resonance with the system is thesame as for continuous wave two-tone spectroscopy and described in section 2.3 and depicted inFigure 3.6. Due to the fact that the averaging methods in a VNA are much more sophisticatedthan the simple method used here the measurement time is a factor of two larger and theresolution is not even a fourth in comparision to the measurement presented in Figure 4.7 (c).However, this measurement where the sample is mounted beteen the signal generators and thebuilt up and calibrated IQ detector for the first time confirms that the detector operates asexpected and is able to deliver information about the experiment. With this measurementtechnique we are also able to detect the side arms which were depicted in Figure 4.7 (c). However,in the next step we reduce the pulse width τ by two to three orders of magnitude to performtime-domain measurements.

4.2.1.2 Rabi oscillation measurements

In this section we present experimental results of time-domain measurements on a coupled qubitresonator system. A simplified picture of the experimental setup and the pulse pattern usedin the experiment is presented in Figure 4.12. For a detailed introduction of the measurementsetup used in the experiments see section 3.2.2, here we provide an overview of the experimentalparameters.

Figure 4.12: A simplified circuit diagram for the time-domain measurements on the left handside. The used pulse pattern for the readout time t of the ACQIRIS card and the pulse width τare depicted on the right hand side. The readout time t and the pulse width τ differ by two tothree orders of magnitude and that the pulse always starts at t = 1µs and ends at t = 1µs + τ .

As we intend to measure Rabi oscillations we have to excite the qubit from the ground state|g〉 to the excited state |e〉 at its excitation frequency ωQ such that we can extract the energyrelaxation time T1 of the qubit. To reduce the effect of flux fluctuations shifting the qubit’sexcitation frequency we perform this measurement around the degeneracy point of the qubitwhere the qubit hyperbola is very flat. From the spectroscopy experiments in section 4.1.4 weknow that for a flux detuning of δΦ ≈ 0 the excitation frequency of the dressed state |m,−〉where the qubit is in the ground state has a frequency of ω|m,−〉/2π = 4.96 Ghz.

64


Therefore, we operate the PSG at ωS/2π = 4.96 GHz and an output power of −20 dBm. Wekeep the indices P and S which indicated the probe tone and spectroscopy tone in a continuouswave two-tone spectroscopy since the spectroscopy is the long time limit for the time-domainmeasurements presented here. The SMF generates a weak continuous microwave signal such thatthe second harmonic of the resonator is populated with approximately 1 poa at a fixed frequencyωP/2π = 7.109 GHz, since this is the dispersively shifted resonator frequency according to thespectroscopy experiments in section 4.1.4. The second SMF provides the local oscillator frequencyωLO/2π = 7.099 GHz at a power of 12.5 dBm. The response of the sample is then converted downto an intermediate frequency ωIF/2π = 10 MHz with the IQ detector depicted in Figure 3.8. Thesignal of the channels I and Q is recorded with the ACQIRIS card for a readout time t of 3µs.The pulse width τ is swept from 5 to 38 ns and applied to the antenna with a delay of 1µs afterthe ACQIRIS card has begun to record the data. Each time trace for a certain pulse width isrecorded and averaged in the ACQIRIS card over 23 ·106 times. Two examples for these averagedtime traces for two different pulse widths τ are presented in Figure 4.13.

Figure 4.13: Recorded raw data traces for I and Q and two different pulse widths τ . In (a) wedepicted the recorded I and Q channel for a pulse width τ = 15 ns. While the delay is set to 1µsthe response is seen at 1.2µs due to the signal processing in the internal mixer of the PSG. In(b) the pulse width is 22 ns, in comparrison to (a) the decrease in the measured amplitude fromapproximately 1.3 to 1.7µs is larger. The raw data presented here has already been filtered bya sliding window and a hoppping window filter as described in section 3.2.2.2. The heterodynecalibration algorithm uses the region from 0.1 to 1.0µs to extract the correction matrix which isapplied to the complete trace.

65


In Figure 4.13 (a) we depict the recorded and filtered data for a pulse width τ = 15 ns. A decreaseof the measured amplitude is clearly visible for both channels after approximately 1.3µs. For apulse width τ = 22 ns [see Figure 4.13 (b)] the decrease is even more prominent. In both casesthe decrease of the measured amplitude can be explained with the same mechanism as for thetwo-tone spectroscopy, namely an AC-Zeeman shift. After t ≈ 2.5µs the signal has the sameamplitude as within the first microsecond of the recorded trace. This is a first hint that wecan excite the qubit with the applied pulse and that the system relaxes after a certain time.To determine the so called energy relaxation time T1 we have to extract the time dependentamplitude A(t) and phase ρ(t) from the recorded I and Q channel as described in section 3.2.2.2.Therefore, we use the recorded data for a readout time t from 0.1 to 1.0µs to apply the heterodynecalibration algorithm from Appendix A to obtain a correction matrix C for each pulse width τ .For every readout time increment we multiply the data of channel I and Q with the correctionmatrix of the respective pulse width. Afterwards we calculate the amplitude A(t) and phase ρ(t)

via the equations (3.10) and (3.11). Finally we apply a sliding window filter at the intermediatefrequency ωIF for the reasons explained in section 3.2.2.2. The result of these calculations isvisualized in Figure 4.14.

Figure 4.14: The extracted amplitude (a) and phase (b) are depicted as a function of the readouttime t for pulse widths τ = 5, 6, . . . 38 ns. We would like to mention that all filter and calibrationalgorithms as described in section 3.2.2.2 in detail have already been applied to the depicteddata. The MATLAB code to calculate the data presented here can be found in the AppendixA.2.

The amplitude A(t) in Figure 4.14 (a) for a fixed pulse width τ shows a clearly visible response.After the pulse was applied we detect a decrease in the amplitude and after a certain time itrelaxes to the initial value. The observed response varies for different pulse widths τ . In (b)we depict the phase response of the system to the performed experiment. As expected for asuccessful calibration the phase difference in the first third of the readout time t is almost zerofor every pulse width τ but for different pulse widths we observe an oscillatory behavior of thephase. For all applied pulse widths τ we find that we can excite the system. The obtainedresult of Figure 4.14 (b) looks very similar to that in ref. [19]. For the rest of this thesis weconcentrate on the phase ρ(t) to analyze the experimental results, calculate energy relaxationtimes and investigate the oscillatory behavior of the response on the pulse width τ . The resultof this analysis is depicted in Figure 4.15.

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Figure 4.15: The extracted phase ρ(t) for different pulse widths τ is depicted in (a). The fittedenergy relaxation rate on a minimum of the observed response [magenta line in (a)] is given byTDS ≈ 108 ns which is visualized in (b). In (c) we depict the same as in (b) but on a maximum[cyan line in (a)], here the energy relaxation time is TDS ≈ 110 ns. We extract the maximum ofthe phase shift (d) by applying the Chebyshev norm for readout times from 1.17 to 1.36µs ateach pulse width τ . We find that the phase shift shows an oscillatory behavior. The fitted sineyields a Rabi frequency of ωRabi/2π ≈ 78 MHz [red line in (b)].

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First we concentrate on the energy relaxation time. In Figure 4.15 (b) we present the recordedresponse for a pulse width τ = 15 ns [magenta line in (a)]. We use the measured data markedas green dots to fit an exponential decay to the data. Here we end up with a relaxation timeTDS ≈ 108 ns. In the spectroscopy experiments we found that at the degeneracy point of thequbit at which we perform the time-domain measurements presented here we also drive thedressed states, since we fulfill the condition |ωQ − ω1| < g1. Therefore, the energy relaxationtime of the qubit T1 can be calculated via a Matthiessen’s rule [79]

(TDS)−1 = ΓDS = Γ1 + κ1, (4.22)

Γ1 = (TDS)−1 − κ1 =1

108 ns− 1.25 MHz = 8.01 MHz, (4.23)

T1 = (Γ1)−1 = 125 ns. (4.24)

So the fitted relaxation rate of the dressed states ΓDS is a superposition of the resonator lossrate κ1 and the qubit energy relaxation rate Γ1. We have to regard the relaxation time TDS

with suspicion since the relaxation rate ΓDS = 9.25 MHz is close to the intermediate frequencyωIF which was filtered with a sliding window filter. So the numbers presented here only are ableto quantify the order of magnitude of the qubit relaxation time T1. If we apply the equation(4.23) to the fitted relaxation rate TDS = 110 ns for the time trace presented in Figure 4.15 (c)we can determine the qubit energy relaxation rate to T1 = 127 ns. We would like to mentionthat the calculated T1 time for a pulse width of τ = 15 ns agrees well with the calculated T1 timefor a pulse width τ = 22 ns, which indicates that the qubit is excited for every pulse width weapplied in this experiment. In the next step we investigate the oscillatory behavior of the phaseρ(t) with the pulse width τ . Therefore, we apply the Chebyshev norm1 to the data presentedin Figure 4.15 (a) for readout times from 1.17 to 1.36µs at each pulse width τ . The maximumphase shift for the different pulse widths τ is depicted in 4.15 (d). It might be possible thatthe reduced visibility for pulse widths between 6 and 14 ns is caused by the response time ofthe resonator. We use the measured data marked with green dots to fit a sine (red curve) tothe data. This fit yields a frequency of ωRabi/2π ≈ 78 MHz. The white curve in (a) is a sineof this frequency. If we could proof that this frequency scales linear with the amplitude of thespectroscopy power PS then it would be another indication that we observed Rabi oscillations.With a pulse width sweep increment of 1 ns we can not observe oscillations with a frequencygreater than (1 ns/2)−1 = 500 MHz. However, such a time-domain measurement is successfullyperformed in the qubit group of the WMI for the first time. To speed up the measurements andto confirm the results received up to now we replace the ACQIRIS card by an FPGA board fromInnovative Integration (Model: X5-RX).

4.2.2 FPGA board measurements

To speed up the measurements and to avoid the digital filtering of the raw data we replace theACQIRIS card by an FPGA board. The rest of the measurement setup depicted in Figures3.7 and the pulse sequences in 4.12 remain unchanged. In a first experiment we reproduce themeasurement results of the ACQIRIS card, afterwards we investigate the oscillatory behavior ofthe sample with the pulse width τ and its power dependence.

1The Chebyshev norm in a compact space like Rn is equivalent to the maximum norm.

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4.2.2.1 Rabi oscillation measurements

Since we replace the ACQIRIS card by the FPGA board we no longer have to apply the slidingwindow filter at 200 MHz and the hopping window filter at 12.5 MHz, since the artifacts whichmade it necessary to apply these filters are caused by internal errors of the ACQIRIS card.However, the sliding window filter at the intermediate frequency ωIF still has to be used after wecalibrated the IQ mixer since the ground loop is still not compensated (see section 3.2.2.2 for adetailed discussion). For a first check we perform a measurement with the same parameters asin ACQIRIS card measurement in section 4.2.1.2. So the SMF operates at ωP/2π = 7.109 GHzand an output power such that the second harmonic resonator mode is populated with 1 poa(see Table 4.3). The pulse at a frequency of ωS/2π = 4.96 GHz at an output power of PS =−20 dBm is generated by the PSG. The local oscillator is driven continuously at a frequencyωLO/2π = 7.099 GHz and a power of 12.5 dBm. The results of this measurement in comparisonto the ACQIRIS card measurement are presented in Figure 4.16.

Figure 4.16: We compare the measurement results where we use the ACQIRIS card for readout(a) to the measurement with the FPGA board (b) for the same experimental parameters. In(b) the signal is slightly weaker despite the fact that less digital filters are used. In general themeasurement result is reproducible.

When we compare the two subfigures in Figure 4.16 it turns out that as expected they look verysimilar. We observe a low phase response for pulse widths between 7 and 18 ns in (b) as it is alsothe case in (a) where the ACQIRIS card is used. This indicates that the reduced visibility is eithercaused by the response time of the resonator or by the used IQ mixer. However, there are twomain differences of the ACQIRIS card measurement in (a) and the FPGA board measurementin (b). First the signal to noise ratio is higher in (b), this may indicate that the digital filtersapplied to the ACQIRIS card can be optimized. The second and more important difference is thetime needed to perform the measurement. Each measurement trace is averaged 23 ·106 times butit took approximately 17 h to acquire the data with the ACQIRIS card but less than one hourto record the data with the FPGA board. The better signal to noise ratio and the reduction ofmeasurement time for achieving similar measurement results suggests to use the FPGA board infuture experiments. We now proceed with a detailed analysis of the FPGA board measurementpresented in Figure 4.17 to confirm the measurement results recorded with the ACQIRIS card.

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Figure 4.17: In agreement with the ACQIRIS measurement of section 4.2.1.2 we find that thephase shift shows an oscillatory behavior when we use the FPGA board (a). The fitted energyrelaxation rate on a maximum [magenta line in (a)] is given as TDS ≈ 111 ns which is visualizedin (b). In (c) we depicted the same as in (b) but on a minimum [cyan line in (a)], here theenergy relaxation time is found to be TDS ≈ 138 ns. The fitted sine yields a Rabi frequency ofωRabi/2π ≈ 70 MHz [red curve in (d)].

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Figure 4.17 (a) visualizes the measurement result of an experiment which is performed with thesame parameters as the measurement discussed in section 4.2.1.2 but now with the FPGA boardas recording device instead of the ACQIRIS card. Despite the problem of the applied slidingwindow filter at the intermediate frequency ωIF it is worth to analyze the energy relaxationtime in analogy to the results of section 4.2.1.2. The time trace for a pulse width τ = 25 nsis visualized in Figure 4.17 (b). We use equation (4.23) to calculate a qubit energy relaxationtime of T1 ≈ 129 ns from the fitted dressed state relaxation rate TDS ≈ 111 ns. In Figure 4.17(c) we visualize the recorded time trace for a minimum in the oscillation. Here we find a qubitrelaxation time of T1 ≈ 167 ns. This value is approximately one third larger than the valueat the maximum. However, since all qubit relaxation times presented so far are between 100and 200 ns we can assume that the real relaxation time value is of this order of magnitude.Similar to the measurements with the ACQIRIS card we observe an oscillatory behavior of therecorded time traces with the pulse width τ in Figure 4.17 (a). This is more clearly visualizedin Figure 4.17 (d) where we plot the phase shift extracted form the Chebyshev norm for readouttimes from 1.17 to 1.36µs at each pulse width τ . The fitted sine (red curve) yields a Rabifrequency of ωRabi/2π ≈ 70 MHz. We are affected by a reduced visibility for pulse width form7 to 18 ns. However, the phase shift for pulse widths of 5 and 6 ns agree with the fitted sine.This indicates that the optimization of the detector might be an issue in our setup. In the nextstep we concentrate on the oscillatory behavior of the recorded traces with the pulse width τ .It was shown in refs. [16, 19] that the Rabi oscillation frequency ωRabi depends linearly on thedriving amplitude. Therefore, we vary the spectroscopy tone power PS from −26 to −10 dBmin steps of 2 dBm. Since we measured at zero flux detuning the spectroscopy tone frequency isωS/2π = 4.96 GHz. The resonator readout mode ω2/2π = 7.109GHz is populated with 1 poaand driven continuously by a SMF. The local oscillator is driven continuously at a frequencyωLO/2π = 7.099 GHz and a power of 12.5 dBm. The pulse width is varied from 5 to 55 ns insteps of 1 ns. Each recorded time trace for the respective pulse width is averaged 20 · 106 times.For all power values PS we receive data similar to that visualized in Figure 4.17 (a). To end upwith comparable data we determine the phase shift no longer via the Chebyshev norm since itis not robust against noise. To determine the phase shift ρ in an automated data analysis scriptwe have to determine the readout time increment in which the pulse of width τ ends. Therefore,we analyzed the readout times for a pulse width of τ = 5 ns for all probe powers PP in theperformed measurement. In a histogram it turned out that the maximum of the observed phaseshift is located at the readout time t0 = 1.2533µs (data not shown). We assume that the pulsewith width τ = 5 ns ends in this time increment. To weaken this assumptions we average over7 time increments centered around t0. As the pulses are defined to start at t = 1µs and endat t = 1µs + τ we expect the maximal response of the sample is also shifted to larger readouttimes t. The sampling rate of the FPGA board is ωSa/2π = 150 MHz which yields that two timeincrements are separated by (150 MHz)−1 = 6.667 ns ≈ 7 ns. Therefore, we assume that for pulsewidths between 5 and 11 ns the respective pulse also ends at the time increment t0. But for apulse width τ = 12 ns the pulse will end at the time increment t0 + 6.6667 ns and so forth. Sofor longer pulse widths the maximum response of the observed phase shift is recorded in a timeincrement after t0. The average window has to be shifted too for larger pulse widths τ . So wedefine a Heaviside window function which averages the recorded data as

H(t,τ) =

17 −3 ≤ (t−t0)ωSa

2π −⌊τ−5

7

⌋≤ 3

0 else(4.25)

where the center time is t0 = 1.2533µs as introduced above. The b c denotes the floor operation.

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We would like to mention that the argument of the condition in equation (4.25)

(t− t0)ωSa

2π−⌊τ − 5

7

⌋(4.26)

is always an integer, such that the statement is true for exactly 7 time increments. So theHeaviside function (4.25) is built in a way such that we can compensate the shift of the responsefor larger pulse widths τ . Since for every seventh pulse width the center time t0 is shifted to thenext readout time increment (see Figure 4.18).

Figure 4.18: A visualization of the Heaviside window function (4.25) which we used for anautomated determination of the phase ρ during the data analysis. Every seven pulse widthincrements, which is in a good approximation the inverse of the sampling rate of the FPGAboard, the Heaviside window is shifted one readout time increment higher.

The weighting is set to 1/7 since we want to average the phase ρ(t) over seven readout timeincrements by using the scalar product

ρ =∑t

ρ(t)H(t,τ). (4.27)

The phases ρ calculated in this way for every pulse width τ are depicted in Figure 4.19 for fourdifferent powers PS.

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Figure 4.19: Typical data for the mean phase deviation ρ around the maximum as a function ofthe pulse width τ for different drive power values PS. We observe a linear increase of the phaseas well as an oscillation. For large power (a) and (b), the frequency seems to be nearly constant.while for low power (c) and (d) the frequency is decreasing with decreasing power.

In Figure 4.19 it turns out that for every power we observe a linear increasing trend in the phaseρ for increasing pulse widths τ . This trend is superposed by an oscillatory signal. Therefore, weuse a fit function of the form

ρ = c0 cos (ωRabiτ) + c1τ + c2 (4.28)

with fitting parameter c0, c1, c2 and ωRabi. The red lines show the result of this fitting function.For large power PS the frequency of the sinusoidal signal seems to saturate [see (a) and (b) inFigure 4.19. However, for lower powers (c) and (d) the frequency decreases for decreasing powerPS. In addition to the fit of equation (4.28) we perform a Fourier analysis of the data. Tosuppress low frequencies in the spectra we subtract the linear term c1τ + c2 from the data suchthat we end up with the spectra presented in Figure 4.20. For the five lowest power values inFigure 4.20 (a) we are able to identify a single peak for each power PS while for higher power weobserve more than one peak. This may indicate that we would have to modify our fit functionfor higher driving power since equation (4.28) is only able to model one frequency.

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Figure 4.20: The Fourier spectrum of the phase deviation for different drive powers PS is depictedin (a). We subtracted the linear term of the fit function (4.28) to end up with the spectrapresented here (green lines). We would like to mention that they are shifted for clarity. Theblue circles indicate the Rabi frequency from the fit function (4.28). They agree well with therespective maximum in the spectrum. The frequencies from the fit function and the prominentfrequencies of the Fourier spectrum for the respective drive amplitude are depicted in (b). Thered line indicates a linear increase of the frequency. We performed a power sweep and analyzedthe Rabi frequencies as a function of the drive amplitude. For low powers the Rabi frequencyωRabi increases linearly with the amplitude. For large drive powers they seem to converge to aconstant value.

The blue circles in Figure 4.20 (a) indicate the fitted frequencies of that fit function. They agreewell with the peaks in the spectra. To clarify if the observed oscillations are in agreement withRabi oscillations we have to analyze the linear dependence of the frequencies on the drivingamplitude. The result of this analysis is visualized in Figure 4.20 (b). It turns out that for lowdriving amplitudes the frequencies obtained from the fit seem to depend linearly but for higheramplitudes (blue circles) they converge to a constant value which coincides with the couplingstrengths we determined when we analyzed the level splitting of the dressed state in section4.1.3 and 4.1.4. By taking into account the prominent frequencies of the Fourier spectrum [greentriangles in Figure 4.20 (b)] we find that these frequencies agree well with that from the fitfunction (4.28). However, if we also take into account the peaks labeled with the arrows in thespectra they seem to depend linear on the driving amplitude. Such a phenomenon where theRabi frequency converges towards resonant frequencies present in the system under study wasalso observed in ref. [16] but in a frequency range which was two orders of magnitude larger thenin our case. However, the data presented so far do not disagree with Rabi oscillations. Withan improved experimental setup the origin of this oscillatory behavior should be investigated infuture experiments.

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In the next measurement we apply a flux detuning δΦ ≈ 1 mΦ0 such that the qubit excitationfrequency ωQ/2π = 5.34 GHz. As a result the dressed states of the qubit and the first harmonicω1 of the resonator should no longer be present and we expect to observe driven Rabi oscillationsof the qubit and the second harmonic ω2 of the resonator. Therefore, we set the PSG to ωS/2π =5.34 GHz and a drive power of PS = −10 dBm. The SMF which drives the resonator continuouslyat a frequency ωP = 7.109 GHz at an output power corresponding to 1 poa in the resonator (seeTable 4.3). We record time traces from 0 to 3µs with the FPGA board for pulse widths τ from5 to 55 ns. Each readout time trace is averaged 20 · 106 times. The result of this measurementis visualized in Figure 4.21 (a).

Figure 4.21: Typical data for a Rabi oscillation measurement in the case where the qubit isdetuned from its degeneracy point is depicted in (a). We observe a response of the sampleindependent of the applied pulse widths. In (b) we visualize the extracted phase shift aroundthe maximum of the observed response. We still observe a slight linear trend of the phase ρ.The fit (red line) yields a Rabi frequency of ωRabi/2π = 144.2 MHz. The Fourier spectrum of thepoints in (b) corrected by the linear trend is depicted in (c). The peak is found at a frequencyof 143.9 MHz.

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If we compare Figure 4.21 (a) to Figure 4.17 (a) we find that the oscillatory behavior is clearlyvisible even for short pulse lengths τ and that the periodicity is shorter. If we use the Heavisidewindow function (4.25) for a center readout time t0 = 1.2533µs we end up with phase shifts ρfor every pulse width τ as depicted in (b). The red curve is a fit to the function

ρ = c0 exp

(−τc4

)cos (ωRabiτ) + c1τ + c2 (4.29)

So we extended the fit function (4.28) by an exponential decay of the amplitude of the oscilla-tory term. This fit yields a Rabi frequency ωRabi/2π = 144.2 MHz. To confirm this frequencywe perform a Fourier analysis in Figure 4.21 (c). To do this, we subtract the linear trend of thefit function (4.29) from the phase ρ presented in Figure 4.21 (b). The spectrum is dominated bya single peak at a center frequency of 143.9 MHz. If we compare the fitted Rabi frequency andthe center frequency of the Fourier spectrum we find that they agree very well. If we take intoaccount the fitted coupling strength to the low power spectroscopy data presented in Figure 4.7(e) we find 2g2/2π = g2/π = 143.8 MHz. According to ref. [50] the frequency of driven Rabioscillations is given by νRabi =

√nSg2/π. If we assume nS = 1 we have evidence that the data

presented in Figure 4.21 shows driven Rabi oscillations. To clarify this we have to perform apower sweep at this flux detuning value. Since here the qubit hyperbola presented in Figure4.7 (f) is quite steep we are very sensitive to flux fluctuations which change the qubit excitationfrequency ωQ via the flux dependent energy bias ε given in equation (2.23). This difficulty isstill an issue in the experimental setup. However, for the measurement presented here we founda Rabi frequency of 143.9 MHz this corresponds to a time of approximately 7 ns which it takesto apply a full rotation on the Bloch sphere. If we now want to excite the qubit from the groundstate |g〉 to the excited state |e〉 we have to use a pulse width τ ≈ 3.5 ns.

In this Chapter we performed continuous wave spectroscopy experiments to investigate the char-acteristic frequencies of our sample and its dependence on the flux detuning δΦ. We determinedthe coupling strengths of the qubit resonator system at a negligible photon number populationin the resonator. Further we found indications for vacuum Rabi levels in our system. We suc-cessfully performed pulsed two-tone spectroscopy and time-domain measurements in the qubitgroup of the WMI for the first time. We are able to detect a reproducible oscillatory behavior ofthe phase response of the sample with two different data acquisition cards. The acquired dataindicates that these oscillations are similar to Rabi oscillations but this has to be clarified infuture experiments with an improved setup.

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5 Conclusion and Outlook

During the course of this thesis we have extended the measurement tool box available at theWMI to time-domain measurements. Therefore, we have built up a functional IQ detector and apowerful heterodyne calibration algorithm. After a detailed characterization of the detector weinvestigated a coupled qubit resonator system. In preliminary measurements we have performedspectroscopy measurements to determine the coupling strengths.

The study of a coupled qubit-resonator system is an interesting field. The usage of an arti-ficial atom which is realized in a superconducting circuit yields larger coupling strengths thanin cavity QED where natural atoms are used. These artificial atoms are coupled to a supercon-ducting resonator. At the WMI three Josephson junction flux qubits are used to realize artificialatoms. Flux qubits are favorable since they yield larger coupling strengths than for exampletransmons. At the WMI it was demonstrated that the ultra strong coupling regime can be real-ized by using a coupling junction in the shared edge of the qubit and the resonator. A currentfield of research at the WMI is the determination of the maximal coupling strength of a flux qubitto a superconducting resonator. It was proposed in ref. [45] that the coupling can be enhancedby a decrease of the area of the coupling junction. In the course of this thesis we used a coupledqubit resonator system with a relative size β = 1 of the coupling junction for two reasons. Firstwe investigate this sample in the frequency-domain to verify the relationM ∝ 1/A of the mutualinductance and the area of the coupling junction and second to determine the energy relaxationrate of the system in the time-domain.

In the frequency-domain measurements we have found coupling strengths of g1/2π ≈ 72 MHzand g2/2π ≈ 88 MHz of the qubit to the first and second harmonic of the resonator. This showsthat the relation of the mutual induction and the size of the coupling junction is more com-plex than M ∝ 1/A. For experimentalists it would be useful to have a theory which predictsthe coupling strength as a function of the size of the coupling junction, here work is already inprogress. Furthermore, we have determined the qubit energy gap to ∆/h ≈ 5.02 GHz and itspersistent current to IP ≈ 208 nA. We have calibrated the photon number of the readout modeof the resonator. With the on-chip antenna we have been able to perform two-tone spectroscopymeasurements. This has demonstrated that on-chip antennas can be implemented successfullyin the qubit group at the WMI. In the recorded two-tone spectroscopy we have found that at thedegeneracy point δΦ = 0 of the qubit the frequency detuning δω1 of the first harmonic ω1 withthe qubit excitation frequency ωQ is less than the coupling strength g1 to this mode such thatthe spectrum exhibits the dressed states of our qubit and the first harmonic ω1 of the resonator.As a result we have found that the excitation levels of the observed side arms in the recordedspectra agree well with vacuum Rabi levels.

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We have been able to build up a functional detector for time-domain measurements whose mainfunctional device is an IQ mixer. Furthermore, we have demonstrated that the developed digitalheterodyne calibration method of the IQ mixer is able to compensate the imperfections causedby the manufacturing tolerances. In the next step of this thesis we successfully have performedtime-domain measurements on a coupled qubit resonator system at the WMI for the first time.To speed up these measurements by a factor of 10 and to confirm the results recorded with theACQIRIS card we have repeated the measurement with an FPGA board. It turned out that withboth data acquisition cards we get highly reproducible results. The decrease of the measurementtime while the signal to noise ratio remains constant shows that the FPGA board should be themost favorable data acquisition card in future works for these kind of measurements. We havebeen able to determine the energy relaxation time T1 of our flux qubit of approximately 120 ns.The observed oscillatory behavior of the response as a function of the pulse width is a strongindication that we observed Rabi oscillations. In the performed power sweep we have found acorrelation of the observed frequencies and the drive amplitude which supports this statement.We have found that the dependence of the observed Rabi oscillations on the driving amplitudeat the degeneracy point of our qubit where we have been affected by the dressed states yieldsmore interesting data as in the case of driven Rabi oscillations when we detune our flux qubitfrom its degeneracy point. Nonetheless this has to be confirmed in future experiments with anoptimized pulse generation and detection setup. For first measurements without an optimizeddetector and pulse generation setup the signal to noise ratio is very good.

To improve future time-domain measurements we have to optimize the pulse generation whichmeans that the envelope of the pulse has to be modified. Furthermore, ghost pulses and the slewrate of the signal might also be an issue towards an optimization of the experimental setup. Therequired hardware for this task is an arbitrary waveform generator (AWG). Further we shouldimprove the detector by identifying the source of the ground loop and compensate it such thatwe do no longer have to use digital filters. However, since we are now able to determine thelength of a π-pulse we would be able to perform Ramsey and Spin echo experiments to accessthe dephasing time of the qubit with an improved detector. With this π-pulse we are in principleable to create a single photon source, since we excite the qubit with one photon by applying sucha pulse. A single photon source is one of the main requirements to test a CNOT gate operationwhich may pose a longterm goal in the qubit group at the WMI.

The study of a coupled qubit-resonator system is an interesting field. The usage of an arti-ficial atom which is realized in a superconducting circuit provides a larger experimental groundthan in the case of natural atoms. The performed experiments in the frequency- and time-domainprovided an inside towards a functional quantum computer which can be used for quantum com-putation and simulation.

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6 Acknowledgments

First of all I would like to thank Prof. Rudolf Gross for providing me the opportunity to do myDiploma Thesis in the qubit group at the Walther Meissner Institut. His advice and inspiringdidactic style during the interesting discussions was inestimably to solve the tasks during thiswork.

I thank Dr. Achim Marx for his advice and great experience concerning any question about thedilution refrigerator I was using during the experimental term of this thesis. His contributionshelped a lot to bring the theoretical knowledge from lecture courses and the practice of using acryostat together.

I am especially thankful to Alexander Baust for the great collaboration during the last year,the introduction to the cryostat and the measurement instruments. He always was available forquestions and his experience was essential for the progress concerning measurement techniquedevelopment at the WMI. I felt great pleasure to explore the sample together with him. Thissymbiosis was a big advantage to solve several issues.

Dr. Frank Deppe is to be thanked for the fruitful discussions and for his introduction to thephysics of superconductors and low temperature physics. He was able to awaken my demand toinvestigate coupled qubit resonator systems in the qubit group of the WMI during my advancedstudy period.

Dr. Hans Huebl’s knowledge on measurement techniques especially the technical issues combinedwith the physics behind it was very helpful to solve several tasks. His abundance of patience isreally impressive. Therefore, I want to thank him.

Special thanks go to Max Haeberlein for the production of the sample which I investigated duringthe last 12 months. Without his contribution this thesis and the results presented here wouldnever had been possible. His sophisticated knowledge on theoretical topics and production tech-nique was very helpful in several discussions.

I also would like to thank Elisabeth Hoffmann for the kind introduction into the clean roomduring the very beginning of my thesis.

Furthermore, I would like to thank Edwin Menzel and Peter Eder. Their detailed knowledge onmeasurement techniques and instruments helped a lot to implement the measurement protocolsand filtering techniques in the time based measurements in this work.

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I’m also thankfully to all members of the qubit group for the fruitful discussions and the greatatmosphere not only concerning the work at the institute but also some group activities duringfree time. This provided me a picture of the WMI not only as a place to work but also as a placeto meet friends.

The WMI workshop is to be thanked for the accurate fabrication of various parts which improvedthe measurement setup in the laboratory. Robert Müller, head of the diploma student’s work-shop, was always willing to provide a workbench when mechanical works have been necessary.For this, I would like to thank him very much.

Dr. Juan José García-Ripoll from Universidad Complutense de Madrid has to be thanked forhis theoretical work on coupled qubit resonator systems and the fitting algorithms he providedto proof the correlation between the measurement carried out at the WMI and the developedtheory. This collaboration and his will to provide a theoretical basis for the experiments is veryfruitful.

Further I would like to thank my office partners, especially Michael Schreier and Nikolaj Bittner.The interesting discussions and their ideas from other qualified fields provided a great workingatmosphere in our room and from time to time a nice chit-chat.

Finally I would like to thank my parents Brigitte and Johann Losinger for their never endinglove. Their support and encouragement was priceless during my studies at the TU Munich.

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http://mathworld.wolfram.com/Ellipse.html


A Digital heterodyne IQ mixer calibra-tion

Here we introduce a calibration method for an IQ mixer where the two channels are representedby points in the real projective plain. Therefore, we first give a short introduction to the realprojective plain RP2 . The central element of the calibration algorithm is fitting a ellipse with thehelp of Fitzgibbon’s method [74, 77]. The basic idea of any projective space is that we embed theEuclidean space which we are interested in in a space of higher dimension than the dimension ofEuclidean space itself. Furthermore we identify scalar multiples as the same object. Therefore,a point in the Euclidean plain which is equivalent to the vector space R2 is represented by anequivalence class in the real projective plain RP2 [75, 76].´

A.1 Mathematical calculations

Definition 1 (Set of points P in the real projective plain RP2 ) The set of all points inthe real projective plain RP2 is given by the quotient structure

P =R3 \

~0

R \ 0.

An illustration of one object in P is shown in Figure A.1. We introduce a disjoint copy of P andidentify it as lines.

Definition 2 (Set of lines L in the real projective plain RP2 ) The set of all lines in thereal projective plain RP2 is given by the quotient structure

L =R3 \

~0

R \ 0.

The last definition we have to assign before we are able to introduce the real projective plainRP2 is a relation which indicates the condition that a point is an element of a line, if we thinkof a line as an infinite set of points.

Definition 3 (Incidence relation I) Let [P ] ∈ P and [l] ∈ L be homogenous coordinates,then the Incidence relation I is defined by:

[P ] I [l]⇔ 〈P, l〉 =3∑i=1

Pili = 0

and we say P lies on l.

So here the Incidence relation I is defined as a scalar product. Finally we define the real projectiveplain RP2 .

Definition 4 (real projective plain RP2 ) The triple (P,L, I) is called real projective plainRP2 .

85


Figure A.1: Points in the real projective plain RP2 [76]. The blue rectangle should representRP2 , while the green line is the equivalence class of a point [P ]. The red dot is one representativeP of the point [P ] here with a z-coordinate z = 1.

Since we defined points in the real projective plain we want to link them with points in theEuclidean plain, therefore, we define two mappings.

Definition 5 (Homogenisation) The mapping

H : R2 → P(xy

)7→

xy1

.is called Homogenisation.

So we are able to transform points from the Euclidean plain to the real projective plain RP2 .P and L are powerful objects, as we are now allowed to talk about infinite far points and oneinfinite far away line. Therefore, we have to take care when we define the inverse mapping of H,since infinite far away objects do not have a representative in R2.

Definition 6 (Dehomogenisation) The mapping

D : R3 \

(x, y, 0)T | x, y ∈ R→ R2 x

yz

7→ 1

z

(xy

).

is called Dehomogenisation.

Now we are able to talk about points and lines in the real projective plain RP2 and we are able toidentify points in the Euclidean plain with points in the real projective plain RP2 , but we wouldalso like to talk about the line which connects two points and a way to calculate this specificline.

86


Definition 7 (Join ∨) For two different points [P1] , [P2] ∈ P we define the operation Join ∨to calculate the line [l] ∈ L which connects the two points via:

l := P1 ∨ P2 = P1 × P2.

As we would like to deal with ellipses in a proper way we define an ellipse as follows with thehelp of homogeneous coordinates.

Definition 8 (Ellipse) Let p1, ..., pn be a number of points in RP2 with n ≥ 5, then these pointsare on an ellipse, if and only if there exists a matrix E of the form:

E =

a b db c ed e f

with det(E) 6= 0, det

(a bb c

)> 0 and det(E)/(a+ c) < 0

such that the equation pTi Epi = 0 is fulfilled ∀ i = 0, . . . , n. We call E the descriptive matrix of

the ellipse or just ellipse.

For further details on the fascinating world of projective geometry we refer to ref. [75]. If wedrop the conditions to the determinants in Definition 8 we get a more general form i.e. a conic.However, to extract the parameters a, b, . . . , f representing the ellipse we use the well establishedFitzgibbon method [74] which includes a constraint such that always an ellipse is fitted to thepoints which gained from the experiment. We use Fitzgibbon’s method because it was shown,that it’s robust against noise [77]. Now that we know all coefficients of the Matrix E, we extractthe correction values out of the matrix. It is worth mentioning that all the calculated valuesonly use the parameters a, b, . . . , f of the ellipse and they do not take into account in whichmathematical structure the ellipse is embedded. For example the algorithm of Fitzgibbon fitsthe parameter in RP5 to a given set of points, the calculations we present here work in the realprojective plain RP2 and the figures depicted in this section are plotted in the Euclidean plain.We begin with the DC offset of both channels, this manifests itself in an ellipse not centered atO := (0, 0, 1)T as it was shown in ref. [80] the offset’s are given by

DC1 =cd− beb2 − ac

, (A.1)

DC2 =ae− bdb2 − ac

. (A.2)

and therefore, the first projective transformation one wants to apply is given by a translation ofthe form

DC =

0 0 −DC1

0 0 −DC2

0 0 1

. (A.3)

Here the power of using homogeneous coordinates becomes clearly visible for the first time, weare able to express a translation as a matrix multiplication instead of a vector addition as aresult the correction matrix presented here is able to consider and correct DC offsets as well asamplitude offsets and phase deviation errors of the channels I and Q. In comparison to the methodused in ref. [50] were one would need a 2 by 2 correction matrix and a 2 dimensional translationvector to be as powerful as the method presented here. The result of this transformation can beseen in Figure A.2 (b). The next step consists of correcting the amplitude and phase differenceof channels at the same time, therefore, the angle between the x-axis and the semi-major axishas to be known. It can be shown [80] that it is given by

η =

12 arctan

(a−c2b

)for b 6= 0 and a < c

π2 + 1

2 arctan(a−c2b

)for b 6= 0 and a > c

. (A.4)

87


Figure A.2: The transformations for the IQ mixer calibration for each step on simulated datawithout noise. The raw data with a DC offset, a mismatch in the amplitude of both channelsand a difference in phase unequal to π/2 is shown in (a). In (b) the DC offset is removed byapplying a translation matrix DC. We rotate the ellipse by the angle −η to go into the eigenbasiswhere we can easily transform the ellipse into a circle by the dilation AM and finally rotatingback by η (c) to (e). In the last step (f) we restore the phase of channel 1 to its original valuefrom the experiment by rotating with the angle −δ.

88


Figure A.3: Idea how to extract amplitudes of both channels from the fitted ellipse. The am-plitude of both channels is given by the projection of the ellipse on the respective axis. So onestarts with solving a quadratic equation (a) and looks up for the case where the discriminant inVieta’s Theorem vanishes (b).

Furthermore we need the length of the semi-axes and the amplitudes of both channels, first thetwo semi-axes lengths can be calculated via [80]

sa1 =

√√√√ 2(ae2 + cd2 + fb2 − 2bde− acf)

(b2 − ac)(√

(a− c)2 + 4b2 − (a− c)) , (A.5)

sa2 =

√√√√ 2(ae2 + cd2 + fb2 − 2bde− acf)

(b2 − ac)(−√

(a− c)2 + 4b2 − (a− c)) , (A.6)

SMA = max sa1, sa2 , (A.7)SMI = min sa1, sa2 . (A.8)

With SMA as the semi-mayor axis and SMI as the semi-minor axis. Now we can extract theamplitudes of channel 1 and 2 form matrix E, this can be done analytically by considering theidea depicted in Figure A.3.

Theorem 1 Let E be an ellipse as defined in Definition 8 and centered at O then the horizontaland vertical projection to the axes are given by

AM1 =

(cd− be)(

1±√

1− (e2−cf)(b2−ac)(dc−be)2

)b2 − ac

, (A.9)

AM2 =

(ae− bd)

(1±

√1− (d2−af)(b2−ac)

(ae−bd)2

)b2 − ac

. (A.10)

The condition that the ellipse is centered at O guarantees that |AM+i | = |AM−i | for i = 1, 2,

where i corresponds to the channel numbers, otherwise the solution is shifted by the DC offsetsmentioned above.

89


Proof 1 For every point p = (x, y, 1)T the condition of Definition 8 is fulfilled. This conditioncan be rewritten by

ax2 + 2bxy + cy2 + 2dx+ 2ey + f = 0. (A.11)

Now we set x = const. and rearrange

cy2 + 2(e+ bx)y + ax2 + 2dx+ f = 0. (A.12)

Vieta’s Theorem yields

y1/2 =−2(e+ bx)±

√4 (e+ bx)2 − 4c (ax2 + 2dx+ f)

2c. (A.13)

Thus we get the two y-values for which the line x = const. intersects the ellipse E [see FigureA.3 (a)], at the maximum value and therefore at the projection to the x-axis, the two points areidentical and therefore the discriminant vanishes [see Figure A.3 (b)], this implies:

e2 + 2bex+ b2x2 − acx2 − 2cdx− cf = 0 (A.14)(b2 − ac)x2 + 2(be− cd)x+ e2 − cf = 0 (A.15)

Now we use Vieta’s Theorem again:

AM1 =−2 (be− cd)±

√4 (be− cd)2 − 4 (e2 − cf) (b2 − ac)

2 (b2 − ac)

=(cd− be)±

√(cd− be)2 − (e2 − cf) (b2 − ac)

b2 − ac

=

(cd− be)(

1±√

1− (e2−cf)(b2−ac)(dc−be)2

)b2 − ac

(A.16)

Repeating this with y = const. to receive AM2:

ax2 + 2(d+ by)x+ cy2 + 2ey + f = 0 (A.17)

x1/2 =−2(d+ by)±

√4 (d+ by)2 − 4a (cy2 + 2ey + f)

2a(A.18)

d2 + 2bdy + b2y2 − acy2 − 2cey − af = 0 (A.19)(b2 − ac)x2 + 2(bd− ae)x+ d2 − af = 0 (A.20)

AM2 =−2 (bd− ae)±

√4 (bd− ae)2 − 4 (d2 − af) (b2 − ac)

2 (b2 − ac)

=(ae− bd)±

√(ae− bd)2 − (d2 − af) (b2 − ac)

b2 − ac

=

(ae− bd)

(1±

√1− (d2−af)(b2−ac)

(ae−bd)2

)b2 − ac

(A.21)

90


Now we have all ingredients to transform the ellipse to a circle with desired radius r, here wechoose r = AM1. One can also calibrate to the amplitude of channel 2 AM2 or to any othervalue of interest. This is now applied by a series of projective transformation of the form.

RηAMR−η =

cos(η) − sin(η) 0sin(η) cos(η) 0

0 0 1

AM1SMA 0 0

0 AM1SMI 0

0 0 1

cos(η) sin(η) 0− sin(η) cos(η) 0

0 0 1

. (A.22)

We rotate the ellipse by the angle −η such that the semi-mayor axis is equivalent with the x-axis,and the semi-minor axis with the y-axis and as a result we are in an orthonormal eigenbasis ofthe ellipse, then by applying a dilation to the ellipse such that it is now a circle and finallyrotating back by the angle η, see Figure A.2 (c) - (e) for a visualization of these steps. We nowalready have achieved that the two channels have the same amplitude and are phase shifted byπ/2, but due to the fact that we correct this in the eigenbasis of the ellipse we lost the phase ofchannel 1 and 2 at the beginning of the applied pulse. The phase of channel 2 at the beginningis lost because we want it to be phase shifted from channel 1 by π/2. However, there is a wayto recover the phase of channel 1 by correcting the angle in an orthogonal basis. We do this, sothat we can easily compare this result to an other method used by another group [50].

Theorem 2 Let p be a point on an ellipse E, then the angle error caused by the dilation AM isgiven by

δ := arctan

(SMA− SMI

SMA + SMI

). (A.23)

Proof 2 As p ∈ RP2 its coordinates are given by p = (x, y, 1)T and

AM · p =

AM1SMA 0 0

0 AM1SMI 0

0 0 1

·xy

1

=

AM1SMAxAM1SMI y

1

. (A.24)

We show that the point O is a fixpoint of the transformation AM

AM ·O =

AM1SMA 0 0

0 AM1SMI 0

0 0 1

·0

01

=

001

= O. (A.25)

We are now interested in the angle between the two lines h1, h2 defined as

h1 := p ∨O =

xy1

×0

01

=

y−x0

, (A.26)

h2 := AMp ∨O =

AM1SMAxAM1SMI y

1

×0

01

=

AM1SMI y

−AM1SMAx0

. (A.27)

Since we are only interested in measuring angles and not distances, it is sufficient to switch fromthe orthonormal basis (1, 0, 0)T, (0, 1, 0)T, (0, 0, 1)T to the orthogonal basis(y, 0, 0)T, (0, x, 0)T, (0, 0, 1)T and receive new coordinates

h1 =

1−10

, (A.28)

h2 =

AM1SMI

−AM1SMA0

. (A.29)

91


We now use Laguerre’s Formula [75], therefore we define the two complex numbers

q1 := 1− i, (A.30)

q2 :=AM1

SMI− i

AM1

SMA= u− iv. (A.31)

so the angle can be calculated:

δ =1

2iln

(q∗1q2

q1q∗2

)=

1

2i

((1 + i)(u− iv)

(1− i)(u+ iv)

)=

1

2iln

(u+ v + i(u− v)

u+ v − i(u− v)

)

=1

2iln

(q3

q∗3

)=

1

2i

ln

∣∣∣∣q3

q∗3

∣∣∣∣︸︷︷︸=1

+ i arg

(q3

q∗3

)=

1

2i

2i arctan

Im(q3q∗3

)Re(q3q∗3

)+∣∣∣ q3q∗3 ∣∣∣

= arctan

Im(q3q∗3

)Re(q3q∗3

)+ 1

.(A.32)

We have a closer look at the argument of arctan:

Im(q3q∗3

)Re(q3q∗3

)+ 1

=Im(q3q3|q3|2

)Re(q3q3|q3|2

)+ |q3|2

|q3|2=

Im (q3q3)

Re (q3q3) + |q3|2. (A.33)

So we calculate q3q3

q3q3 = (Re (q3) + i Im (q3)) (Re (q3) + i Im (q3))

= (Re (q3))2 − (Im (q3))2 + 2i Re (q3) Im (q3) .(A.34)

With these results we enter equation (A.32) again

δ = arctan

(Im (q3q3)

Re (q3q3) + |q3|2

)= arctan

(2 Re (q3) Im (q3)

(Re (q3))2 − (Im (q3))2 + (Re (q3))2 + (Im (q3))2

)= arctan

(Im (q3)

Re (q3)

)= arctan

(u− vu+ v

)= arctan

(AM1SMI −

AM1SMA

AM1SMI + AM1

SMA

)

= arctan

(SMA− SMI

SMA + SMI

).

(A.35)

It is worth mentioning that δ only depends on the value of the semiaxes so it is a global propertyof the ellipse and therefore to the imperfections off the signal and not of each individual point.A lower and upper bound of δ is given by 0 ≤ δ ≤ π/4.

92


Figure A.4: The signal of the two channels plotted versus time. The dashed lines represents theuncalibrated data from Figure A.2 (a). If one would stop calibration with the situation in FigureA.2 (e) one would end up with the dot-dashed lines. It is clearly visible that the two signals havethe same amplitude and are phase shifted by π/2. However, we can even reconstruct the phaseof channel 1 (solid lines) by rotating by −δ, as a result the sinusoidal movement of the blue solidline follows perfectly the movement of the uncalibrated channel 1 data. If channel 1 would nothave been perturbed by a DC offset the dashed and the solid blue line would be identical.

The lower bound due to SMA = SMI which is the case if the data is already calibrated andwe have a perfect circle. The upper bound can be found with the condition SMI = 0. In thatcase the ellipse is a degenerated double line which occurs if and only if the two signals are inphase or phase shifted by π. The exact mathematical condition on here is (2n− 1)π with n ∈ N.Only under this condition the method presented here does not work accurately because of thebreakdown of Fitzgibbon method. To correct this error δ we define a new rotation matrix

R−δ =

cos(δ) sin(δ) 0− sin(δ) cos(δ) 0

0 0 1

. (A.36)

The effect of this rotation is depicted in Figure A.2 (f). As it is difficult to extract the phaseinformation from Figure A.2 we present the two signals in parameter form i.e. plotted againsttime in Figure A.4. Without the rotation R−δ we would receive a calibrated signal with identicalamplitudes and a phase shift of π/2 of the two arms (dot-dashed lines in Figure A.4), but withthe extra rotation we are able to recover the phase of channel 1 (solid lines). One would expectto apply R−δ directly after AM . Since rotation matrices of the same dimension form an Abeleangroup and we do not have to care about the order how we apply it to the points.

93


Finally the correction matrix C is given by

C = R−δRηAMR−ηDC =

∗ ∗ ∗∗ ∗ ∗0 0 1

. (A.37)

By multiplying this matrix C from the left to any set of points given in homogeneous coordinateswith the measured response of channel 1 on the x-coordinate and channel 2 on the y-coordinateone is able to calibrate your measurement a posteriori. The result of this for ideal data is shownin Figure A.4. If one would like to implement this correction matrix C in a numeric program,one would use the group ability of rotation matrices and replace R−δRη by Rη−δ to reduce thenumber of operations applied and therefore speed up the program. However, here we decidedto do it step by step to show the influence of every single matrix operation. Finally we wouldlike to present the MATLAB code in the case of a time-domain measurement with the AQCIRIScard as data acquisition card.

A.2 MATLAB code

Here we present the MATLAB code to analyze a time-domain measurement, therefore we callthe script CallAuswPulsedTwotone.m.

1 %This script calculates the result of a time−domain measurement recorded2 %with the ACQIRIS card. If we use the FPGA board the applied filters on the3 %I and Q raw data do not have to be applyed and the function4 %"readPulsedTwoToneAQCIRIS.m" has to be replaced by5 %"readPulsedTwoToneFPGA.m"6 %7 % written by Thomas Losinger8

9 %clear workspace10 clear11

12 %input arguments:13 %strings14 path = ''; % string with path to raw data15 %scalars16 mixfreq = 10; % IF frequency of the IQ−Mixer in MHz17 samplingRate = 400; % sampling rate of the ACQIRIS card in MHz18 NumSamples = 1201; % total number of readout time increments19 %vectors20 power = [−20]; % probe powers in dBm21 times = [5:38]; % pulse width in ns22

23 %begin execution24 [Iraw,Qraw]=readPulsedTwotoneACQIRIS(path,NumSamples,times,power);25 % The function "readPulseTwoToneACQIRIS" reads the rawdata from a WMI intern26 % data format and writes it into the two MATLAB variables Iraw and Qraw.27 % Therefore we will not publish the code of this function. The output is of28 % the form29 %30 % dim(Iraw) = lenght(NumSamples)+1, lenght(times)+1, lenght(power)31 % Iraw(1,1,jj) = power(jj)32 % Iraw(1,2:end,:) = times33 % Iraw(2:end,1,:) = recorded readout times in us34 % Iraw(2:end,2:end,:) = recorded data in uV35

36

94


37 %apply ditital filters (these can be neglected if the FPGA board is used)38 % 200 MHz sliding window filter on both cannels39 Ifil=slidingWindow(Iraw,200,samplingRate);40 Qfil=slidingWindow(Qraw,200,samplingRate);41

42 % 12.5 MHz hopping window filter only on channel Q43 Qfilpat=hoppingWindow(Qfil,12.5,samplingRate);44 Ifilpat=Ifil(1:length(Qfilpat),:,:);45

46 %calibrate IQ mixer47 [Amplitude, Phase, CorrValues]=calibrateIQ(Ifilpat,Qfilpat, mixfreq);48

49 %apply IF sliding window filter (also with the FPGA board)50 AmplitudeFil=slidingWindow(Amplitude,mixfreq,samplingRate);51 PhaseFil=slidingWindow(Phase,mixfreq,samplingRate);52

53 %draw a nice picture54 plotresult(AmplitudeFil,path,1);55 plotresult(PhaseFil,path,0);56

57 %save workspace in subfolder Analysis (this folder must exist)58 save([path,'\Analysis\',datestr(now,'yyyymmdd'),'data']);

This script uses the functions slidingWindow.m

1 function [ filtered ] = slidingWindow( unfiltered, filterFreq, SamplingRate)2 %slidingWindow applies a digital sliding window filter to the rawdata to3 %filter the filterFreq and higher harmonics4 % input arguments5 % unfiltered contains the raw data in the form6 % dim(unfiltered) = n,m,p7 % unfiltered(1,1,jj) = probe power in dBm8 % unfiltered(1,2:end,:) = pulse widths in ns9 % unfiltered(2:end,1,:) = recorded readouttimes in us

10 % unfiltered(2:end,2:end,:) = recorded data which should be filtered11 % filterFreq contains the frequency at which the filter should operate12 % SamplingRate contains the sampling rate of the data acquisition card13 %14 % output arguments15 % filtered contains the filtered data of the same format as unfiltered16 % but with fewer readout times17 %18 % written by Thomas Losinger19

20 [n,m,p]=size(unfiltered);21 fillen=floor(SamplingRate./filterFreq)−1; % floor to receive an integer22 % initiate memory23 filtered=unfiltered(1:n−fillen,:,:);24 %apply filter25 for jj=1:p26 for ii=2:m27 for kk=2:n−fillen28 filtered(kk,ii,jj)=mean(unfiltered(kk:(kk+fillen),ii,jj));29 end30 end31 end32 end

and hoppingWindow.m

95


1 function [ filtered ] = hoppingWindow( unfiltered, filterFreq, SamplingRate)2 %hoppingWindow applies a digital hopping window filter to the rawdata to3 %filter a reqular stucture with frequency filterFreq4 % input arguments5 % unfiltered contains the raw data in the form6 % dim(unfiltered) = n,m,p7 % unfiltered(1,1,jj) = probe power in dBm8 % unfiltered(1,2:end,:) = pulse widths in ns9 % unfiltered(2:end,1,:) = recorded readouttimes in us

10 % unfiltered(2:end,2:end,:) = recorded data which should be filtered11 % filterFreq contains the frequency at which the filter should operate12 % SamplingRate contains the sampling rate of the data acquisition card13 %14 % output arguments15 % filtered contains the filtered data of the same format as unfiltered16 % but with fewer readoutimes17 %18 % written by Thomas Losinger19

20 [n,m,p]=size(unfiltered);21 fillen=floor(SamplingRate./filterFreq); % floor to receive an integer22 % initiate memory23 filtered=unfiltered(1:(floor((n−1)./fillen)*fillen+1),:,:);24 %apply filter25 for jj=1:p26 for ii=2:m27 %determine the pattern which should be filtered over 5 periods28 pattern=zeros(fillen,1);29 for k=1:530 pattern=pattern+unfiltered(1+((fillen*k−(fillen−1)):(k*fillen)),ii,jj);31 end32 pattern=pattern/5;33 % filter the pattern34 for kk=1:floor((n−1)/fillen)35 filtered(1+((fillen*kk−(fillen−1)):(32*kk)),ii,jj)= ...36 unfiltered(1+((fillen*kk−(fillen−1)):(fillen*kk)),ii,jj)−pattern;37 end38 end39 end40 end

to filter the digital artifact added by the ACQIRIS card presented in Figure 3.10 (d) and (e). Inthe next step we calibrate the IQ mixer via the function calibrateIQ.m.

1 function [ Amp, Phase, CorrVal] = calibrateIQ( Iraw, Qraw, mixfreq)2 %calibrateIQ calibrates an IQ mixer3 % input parameters4 % Iraw and Qraw contains the uncalibrated data in the form5 % dim(Iraw) = n, m, p6 % Iraw(1,1,jj) = probe power in dBm7 % Iraw(1,2:end,:) = pulse widths in ns8 % Iraw(2:end,1,:) = recorded readout times in us9 % Iraw(2:end,2:end,:) = recorded data in uV

10 % mixfreq is the IF frequency of the IQ mixer in MHz11 %12 % output parameters13 % Amp and Phase contain the extraced amplitude in uV14 % and phase in deg in the same form as the input parameters Iraw and Qraw15 % CorrVal contains the important correction values (DC offset of the16 % channels and their amplitudes in uV such as the phase deviation in deg17 % for each pulse width in ns

96


18 %19 % written by Thomas Losinger20

21 [n,m,p]=size(Iraw);22 %initialize memory for Amp, Phase and CorrVal23 Amp=Iraw;24 Phase=Iraw;25 CorrVal=zeros(6,m−1,p);26 %write pulsewidth into first row27 CorrVal(1,:,:)=Iraw(1,2:end,:);28

29 %enable user to select the calibration region therefore we plot the first30 %recorded trace31 rawdataplot=figure();32 hold on33 plot(Iraw(2:end,1,1),Iraw(2:end,2,1),'−',Qraw(2:end,1,1),Qraw(2:end,2,1),'−r');34 legend('Iraw','Qraw');35 xlabel('time (\mus)');36 ylabel('Iraw, Qraw (V)');37 title(['raw data for pulse width = ',num2str(Iraw(1,2,1)),' ns and probe power = ...

',num2str(Iraw(1,1,1)),' dBm']);38 hold off39 %enable user selected start stop for calibration:40 startpoint=input('Signaltime start in \mus: ','s');41 startpoint=str2double(startpoint);42 stoppoint=input('Signaltime stop in \mus: ','s');43 stoppoint=str2double(stoppoint);44 time=Iraw(startpoint≤Iraw(2:end,1) & Iraw(2:end,1)≤stoppoint,1);45 omegat=2.*pi.*mixfreq.*Iraw(2:end,1,1);46 for jj=1:p47 for ii=2:m48 y1=Iraw(2:length(time)+1,ii,jj);49 y2=Qraw(2:length(time)+1,ii,jj);50 Y=[y1,y2,ones(size(time))]';51 %start calibtation by using Fitzgibbon method52 ellipsePara=EllipseDirectFit([Y(1:2,:)]');53 %assign Matrix from the extracted fitparameters54 E=[ellipsePara(1),ellipsePara(2)/2,ellipsePara(4)/2;...55 ellipsePara(2)/2,ellipsePara(3),ellipsePara(5)/2;...56 ellipsePara(4)/2,ellipsePara(5)/2,ellipsePara(6)];57 %calculate correction matrix58 corrvalues=extractCorrectionValues(E);59 %write corrvalues into Corval60 CorrVal(2:end,ii−1,jj)=[corrvalues(1:2);corrvalues(6:8)];61 %build correction matrix62 RotMat1=[cos(corrvalues(5)),sin(corrvalues(5)),0;...63 −sin(corrvalues(5)),cos(corrvalues(5)),0;...64 0,0,1];65 AmpMat=[corrvalues(6)./corrvalues(3),0,0;...66 0,corrvalues(6)./corrvalues(4),0;...67 0,0,1];68 etamindel=corrvalues(5)−corrvalues(8);69 RotMat2=[cos(etamindel),−sin(etamindel),0;...70 sin(etamindel),cos(etamindel),0;...71 0,0,1];72 CorrMat=RotMat2*AmpMat*RotMat1*[1,0,−corrvalues(1);0,1,−corrvalues(2);0,0,1];73 %multiply correction matrix to all data points of current trace74 temp=CorrMat*[Iraw(2:end,ii,jj),Qraw(2:end,ii,jj),ones(n−1,1,1)]';75 %calculate Amplitude and Phase76 Amp(2:end,ii,jj)=sqrt(temp(1,:).^2+temp(2,:).^2);77 Phase(2:end,ii,jj)=unwrap(atan2(temp(2,:),temp(1,:)));78 %now subtract ideal phase from unwrapped phase

97


79 Phase(2:end,ii,jj)=(Phase(2:end,ii,jj)−(Phase(2,ii,jj)+omegat)).*180./pi;80 %and correct it by mean phase during the calibration81 meanPhase=mean(Phase(startpoint≤Iraw(2:end,1) & Iraw(2:end,1)≤stoppoint,ii,jj));82 Phase(2:end,ii,jj)=Phase(2:end,ii,jj)−meanPhase;83 end84 end85 % convert phase deviation from rad in deg86 CorrVal(6,:)=CorrVal(6,:)./pi.*180;87 end

In the first step of the calibration we use Fitzgibbon [74, 77] method to fit an ellipse with thefunction EllipseDirectFit.m.

1 function A = EllipseDirectFit(XY);2 %3 % Direct ellipse fit, proposed in article4 % A. W. Fitzgibbon, M. Pilu, R. B. Fisher5 % "Direct Least Squares Fitting of Ellipses"6 % IEEE Trans. PAMI, Vol. 21, pages 476−480 (1999)7 %8 % Our code is based on a numerically stable version9 % of this fit published by R. Halir and J. Flusser

10 %11 % Input: XY(n,2) is the array of coordinates of n points x(i)=XY(i,1), ...

y(i)=XY(i,2)12 %13 % Output: A = [a b c d e f]' is the vector of algebraic14 % parameters of the fitting ellipse:15 % ax^2 + bxy + cy^2 +dx + ey + f = 016 % the vector A is normed, so that ||A||=117 %18 % This is a fast non−iterative ellipse fit.19 %20 % It returns ellipses only, even if points are21 % better approximated by a hyperbola.22 % It is somewhat biased toward smaller ellipses.23 %24 centroid = mean(XY); % the centroid of the data set25

26 D1 = [(XY(:,1)−centroid(1)).^2, (XY(:,1)−centroid(1)).*(XY(:,2)−centroid(2)),...27 (XY(:,2)−centroid(2)).^2];28 D2 = [XY(:,1)−centroid(1), XY(:,2)−centroid(2), ones(size(XY,1),1)];29 S1 = D1'*D1;30 S2 = D1'*D2;31 S3 = D2'*D2;32 T = −inv(S3)*S2';33 M = S1 + S2*T;34 M = [M(3,:)./2; −M(2,:); M(1,:)./2];35 [evec,eval] = eig(M);36 cond = 4*evec(1,:).*evec(3,:)−evec(2,:).^2;37 A1 = evec(:,find(cond>0));38 A = [A1; T*A1];39 A4 = A(4)−2*A(1)*centroid(1)−A(2)*centroid(2);40 A5 = A(5)−2*A(3)*centroid(2)−A(2)*centroid(1);41 A6 = A(6)+A(1)*centroid(1)^2+A(3)*centroid(2)^2+...42 A(2)*centroid(1)*centroid(2)−A(4)*centroid(1)−A(5)*centroid(2);43 A(4) = A4; A(5) = A5; A(6) = A6;44 A = A/norm(A);45

46 end % EllipseDirectFit

98


Afterwards we extract the correction values from the fit parameters based on the calculations insection A.1. This is achieved by using the function extractCorrectionValues.m.

1 function [ erg ] = extractCorrectionValues( E )2 %extractCorrectionValues takes a fitted Ellipse E and extracts correction3 %values such as the DCoffset of Channel 1 and 2, the semiaxes, the angle4 %eta from the majoraxis with respect to the x−axis, the amplidute of5 % Channel 1 and 2 as well as the phase deviation ∆

6 %7 % input argument8 % A 3x3 Matrix E of the form (a,b,d;b,c,e;d,e,f)9 %

10 % output argument erg with values11 % DC offset Channel 112 % DC offset Channel 2,13 % semimajor axis SMA,14 % semiminor axis SMI,15 % angle eta between semimajor axis and x−axis,16 % amplitude Channel 1,17 % amplitude Channel 2,18 % phase deviation ∆

19 %20 % written by Thomas Losinger21

22 %initialize memory23 erg=zeros(8,1);24

25 %start calculation26 alpha= E(1,2).^2−E(1,1).*E(2,2);27 %calculate center of ellipse DC_1 and DC_228 erg(1)=(E(2,2).*E(1,3)−E(1,2).*E(2,3))./alpha;29 erg(2)=(E(1,1).*E(2,3)−E(1,2).*E(1,3))./alpha;30 %calculate angle eta between semimayor axis and −axis31 erg(5)=−0.5.*acot((E(2,2)−E(1,1))./(2.*E(1,2)));32 %calculate semiaxes SMA and SMI33 beta=2.*(E(1,1).*E(2,3).^2+E(2,2).*E(1,3).^2+E(3,3).*E(1,2).^2 ...34 −2.*E(1,2).*E(1,3).*E(2,3)−E(1,1).*E(2,2).*E(3,3));35 gamma=sqrt(1+(4.*E(1,2).^2)./(E(1,1)−E(2,2)).^2);36 a=sqrt(beta./(alpha.*((E(2,2)−E(1,1)).*gamma−(E(2,2)+E(1,1)))));37 b=sqrt(beta./(alpha.*((E(1,1)−E(2,2)).*gamma−(E(2,2)+E(1,1)))));38 erg(3)=max(a,b);39 erg(4)=min(a,b);40 if(a<b)41 erg(5)=pi./2+erg(5);42 end43 %calculate the amplitudes of the channels AM_1 and AM_244 beta=E(2,2).*E(1,3)−E(1,2).*E(2,3);45 gamma=E(1,1).*E(2,3)−E(1,2).*E(1,3);46 erg(6)=((beta+sqrt(beta.^2+alpha.*(E(2,2).*E(3,3)−E(2,3).^2)))./alpha)−erg(1);47 erg(7)=((gamma+sqrt(gamma.^2+alpha.*(E(1,1).*E(3,3)−E(1,3).^2)))./alpha)−erg(2);48 if erg(6)<049 erg(6)=((beta−sqrt(beta.^2+alpha.*(E(2,2).*E(3,3)−E(2,3).^2)))./alpha)−erg(1);50 end51 if erg(7)<052 erg(7)=((gamma−sqrt(gamma.^2+alpha.*(E(1,1).*E(3,3)−E(1,3).^2)))./alpha)−erg(2);53 end54 %calculate phase deviation ∆

55 erg(8)=atan((erg(3)−erg(4))./(erg(3)+erg(4)));56 end

99


Finally we visualize the results via plotresult.m

1 function [] = plotresult( data, path, caseselect )2 %plotresult plots the data in an imagesc plot and saves it to path3 % data contains the information to plot in the form4 % dim(data) = n, m, p5 % data(1,1,jj) = prob power in dBm6 % data(1,2:end,:) = pulse widths in ns7 % data(2:end,1,:) = recorded readouttimes in us8 % data(2:end,2:end,:) = recorded data9 % path is a string it contains the path to save the picture

10 % caseselect is 0 to plot the phase11 % is 1 to plot the amplitude12 %13 % written by Thomas Losinger14

15 p=size(data,3);16

17 for jj=1:p18 %determine if phase or amplitude should be ploted19 if caseselect == 020 zaxtit = 'phase (deg)';21 titlename = ['PhaseProbePower',num2str(data(1,1,jj)),'dbM'];22 else23 zaxtit = 'amplitude (\muV)';24 titlename = ['AmplitudeProbePower',num2str(data(1,1,jj)),'dbM'];25 end26 %draw the picture27 PlotRes=figure();28 imagesc( data(1,2:end,jj),data(2:end,1,jj), data(2:end,2:end,jj));29 colorbar;30 xlabel('pulse width (ns)');31 ylabel('readout time (\mus)');32 zaxis = zaxtit;33 cb = colorbar('vert');34 zlab = get(cb,'ylabel');35 set(zlab,'String',zaxis);36 set(gca,'YDir','normal');37 title(titlename);38 %save image to the subfolder Analysis (this folder must exist)39 file=[path,'\Analysis\',titlename,datestr(now,'yyyymmdd')];40 saveas(PlotRes,file,'jpg');41 saveas(PlotRes,file,'fig');42 end43 end

100


B Photon number calibration

In this chapter we carry out the necessary calculations on the photon number calibration ofsection 4.1.3. We assume that the Hamiltonian of the observed dressed states is of the form

H

~=

(ω1 g1

g1 ωQ

)(B.1)

where ω1 denotes the eigenfrequency of the first harmonic of the resonator, g1 is the coupling ofthe qubit and the first harmonic of the resonator. Finally ωQ denotes the dispersively shiftedqubit excitation frequency from the Hamiltonian (2.57). Since we use the second harmonic forthe photon number calibration in the experiment we find the explicit form.

ωQ = ωQ + (2n+ 1)(g2)2

δω2. (B.2)

Here ωQ denotes the pure qubit excitation frequency as defined in equation (2.25), g2 is thecoupling of the qubit to the second harmonic and δω2 = ωQ − ω2 is the frequency detuning ofthe qubit and the second harmonic. In the next step we diagonalize the Hamiltonian (B.1) tofind the eigenfrequencies λ± of the system. We use the ansatz

det(H − ~λ · Id2

)= 0 (B.3)

det

(ω1 − λ g1

g1 ωQ − λ

)= (ω1 − λ) (ωQ − λ)− (g1)2 = 0 (B.4)

λ2 − (ω1 + ωQ)λ+ ω1ωQ − (g1)2 = 0 (B.5)

According to Vieta’s Theorem the solutions of the characteristic polynom (B.5) are given as

λ± =

(ω1 + ωQ)±√

(ω1 + ωQ)2 − 4(ω1ωQ − (g1)2

)2

=(ω1 + ωQ)±

√(ω1)2 + 2ω1ωQ + (ωQ)2 − 4ω1ωQ + 4 (g1)2

2

=(ω1 + ωQ)±

√(ω1 − ωQ)2 + 4 (g1)2

2=: η ± µ.

(B.6)

If we use the explicit form of ωQ given in equation (B.2) we end up with

λ± =

(ω1 + ωQ + (2n+ 1) (g2)2

δω2

)±√(

ω1 − ωQ + (2n+ 1) (g2)2

δω2

)2+ 4 (g1)2

2. (B.7)

If ωQ = ω1 the level splitting is 2g1 which agrees well with the data depicted in Figure 2.9 (a).We would like to mention that the photon number dependency of g1 is assumed to be constantsince the spectroscopy power PS is constant.

101


However, since we now know the eigenfrequencies of the Hamiltonian (B.1) for a given photonnumber n in the second harmonic mode of the resonator and since we have some fitted eigenfre-quencies of the performed power sweep in section 4.1.3 at a given output power of the VNA, wehave to link the photon number scale and the power scale in the next step. Therefore, we usenonlinear optimization i.e. we minimize the distance between the measured1 eigenfrequencies ν±and the theoretical eigenfrequencies λ±. First we define the function

f(n) : R→ R2 n 7→(λ+

λ−

). (B.8)

Further we treat the measured frequencies ν± as a vector in R2 of the form (ν+, ν−)T withν+ > ν− such that we can formulate the optimization problem

minn

∥∥∥∥f(n)−(ν+

ν−

)∥∥∥∥2

. (B.9)

For each measured frequency pair ν± from which we know the output power of the VNA wedetermine the photon number n in the eucledian norm ‖ ‖2 for which the deviation of themeasured frequencies ν± to the theoretical eigenfrequencies λ± is minimal. As a result we endup with pairs of values for the output power of the VNA and a photon number. To these setof points we fit a line through the origin [see Figure 4.6 (d)] to calibrate the photon number.However, here we present a more detailed expression on the objective function of the optimizationproblem (B.9).∥∥∥∥f(n)−

(ν+

ν−

)∥∥∥∥2

=

∥∥∥∥(λ+

λ−

)−(ν+

ν−

)∥∥∥∥2

=

√(λ+ − ν+)2 + (λ− − ν−)2

=

√(λ+)2 + (λ−)2 − (2λ+ν+ + 2λ−ν−) + (ν+)2 + (ν−)2

(B.10)

With the two auxiliary calculations based on the notation in equation (B.6).

Auxiliary calculation 1

(λ+)2 + (λ−)2 = (η + µ)2 + (η − µ)2 = 2η2 + 2µ2

=1

2(ω1 + ωQ)2 +

1

2

[(ω1 − ωQ)2 + 4 (g1)2

]= (ω1)2 + (ωQ)2 + 2 (g1)2

(B.11)

Auxiliary calculation 2

2λ+ν+ + 2λ−ν− = 2 (η + µ) ν+ + 2 (η − µ) ν− = 2η (ν+ + ν−) + 2µ (ν+ − ν−)

= (ω1 + ωQ) (ν+ + ν−) +

√(ω1 − ωQ)2 + 4 (g1)2 (ν+ − ν−)

(B.12)

At this point we stop our analytic calculations and determine the photon number n via numericalcalculations in MATLAB. Therefore, we call the script PhotonCalScript.m.

1From the measured spectra we extract center frequencies of a Lorentzian fit, these frequencies are denotedwith ν±.

102


1 % This script calculates a photon number calibration in the case that the2 % dressed states were observed in a two−tone spectroscopy power sweep of3 % the probe tone with k power values. It needs a vector v of the form:4 %5 % dim(v) = 3,k6 % v(1,:) = probe power values in dBm7 % v(2,:) = center frequency of the state |m,+>8 % v(3,:) = center frequency of the state |m,−>9 %

10 % written by Thomas Losinger11

12 %Set parameters13

14 omegaResOne = 5.067; % first harmonic resonator mode in GHz15 omegaResTwo = 7.106; % second harmonic resonator mode in GHz16 omegaQubit = 5.04; % qubit excitation frequency in GHz17 gOne = 0.0865; % coupling of qubit and first harmonic18 gTwo = 0.0882; % coupling of qubit and second harmonic19

20 % initiate variable21 photonnumbers=[v(1,:);zeros(1,length(v))];22

23 % start calculation24 a = omegaQubit−omegaResTwo; % frequency detuning25 b = gTwo.^2./a; % dispersive frequency shift26 c = omegaResOne.^2; % first harmonic squared27 d = 2.*gOne.^2; % twice g_1 squared28 e = omegaResOne+omegaQubit; % sum of first harmonic and qubit excitation frequency29 f = omegaResOne−omegaQubit; % difference of first harmonic and qubit excitation ...

frequency30 for jj = 1:length(v)31 g = v(2,jj)+v(3,jj); % sum of the center frequencies32 h = v(2,jj)−v(3,jj); % difference of the center frequencies33 l = v(2,jj).^2; % squared center frequency of the state |m,+>34 m = v(3,jj).^2; % squared center frequency of the state |m,−>35 %build objective function36 objfunc = @(n) sqrt(c+(omegaQubit+(2*n+1)*b).^2+d−((e+(2*n+1)*b)*g+ ...37 h*sqrt((f−(2*n+1)*b).^2+2*d))+l+m);38 photonnumbers(2,jj)=fminsearch(objfunc,1);39 end40

41 % determine output power to photon number conversion factor by fitting a42 % line trough the origin on a linear−linear−scale43 y = inline('a.*x','a','x');44 slope = nlinfit(10.^(photonnumbers(1,:)./10),photonnumbers(2,:),y,1);

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C Persönliche Erklärung

Mit der Abgabe der Diplomarbeit versichere ich, dass ich die Arbeit selbständig verfasst undkeine anderen als die angegebenen Quellen und Hilfsmittel benutzt habe.

Ort, Datum, Unterschrift

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Time-domain control of light-matter interaction with superconducting ...

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