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Technische Universität MünchenFakultät für Physik

Walther-Meißner-Institut für Tieftemperaturforschung

Bachelor Thesis

Towards a compact 3D quantummemory

Daniel Repp

Garching, 25. Juli 2016

Supervisor: Prof. Dr. Rudolf Gross

Primary supervisor : Prof. Dr. Rudolf Gross

Secondary Supervisor: Prof. Dr. Alexander Holleitner

Contents

1 Introduction 1

2 Theory 22.1 3D microwave cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Transmon qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Qubit-resonator interaction . . . . . . . . . . . . . . . . . . . . . . . 4

2.3.1 Jaynes-Cummings model . . . . . . . . . . . . . . . . . . . . . 52.3.2 Strong coupling regime . . . . . . . . . . . . . . . . . . . . . . 52.3.3 Dispersive regime . . . . . . . . . . . . . . . . . . . . . . . . . 62.3.4 AC-Stark shift . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4 Qubit dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4.1 Bloch sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4.2 Rabi oscillations . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.5 Sidebands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Experimental techniques 103.1 Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Transmon qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 Transmission measurements . . . . . . . . . . . . . . . . . . . . . . . 13

3.3.1 Single-tone spectroscopy . . . . . . . . . . . . . . . . . . . . . 133.3.2 Two-tone spectroscopy . . . . . . . . . . . . . . . . . . . . . . 14

3.4 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4.1 Two-tone setup . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4.2 Time domain setup . . . . . . . . . . . . . . . . . . . . . . . . 15

3.5 Pulse schemes for Rabi oscillations . . . . . . . . . . . . . . . . . . . 16

4 Results and discussions 194.1 Qubit and cavity characterization . . . . . . . . . . . . . . . . . . . . 19

4.1.1 Resonant frequency and dispersive shift . . . . . . . . . . . . 194.1.2 Qubit levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.1.3 Coupling constant . . . . . . . . . . . . . . . . . . . . . . . . 224.1.4 AC-Stark shift . . . . . . . . . . . . . . . . . . . . . . . . . . 22

iii

Contents

4.2 Time domain measurements . . . . . . . . . . . . . . . . . . . . . . . 244.2.1 Rabi oscillations . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.3 Sidebands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.3.1 Frequency domain . . . . . . . . . . . . . . . . . . . . . . . . 274.3.2 Time domain . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5 Conclusions and outlook 34

Bibliography 35

Acknowledgements 37

iv

Chapter 1

Introduction

Quantum computing makes use of quantum principles, such as superposition andentanglement, to conduct highly parallelized computations [1]. Possible applicationsare the efficient simulation of complex systems, database search algorithms and quan-tum cryptography[2, 3].However, up to now, no quantum computer exists that can accomplish the enormousspeed-ups promised by quantum computing outside of testing environments [4].When building a quantum computer, one has the choice between natural systemsand engineered quantum systems [5]. When taking the second approach, one hasgreater flexibility of design.Nevertheless, a problem of engineered quantum systems is a loss of coherence by cou-pling to environmental degrees of freedom. The coherence time is important, sinceit limits the number of operations that can be performed on a quantum system.Superconducting devices based on Josephson junctions are promising candidates forbuilding qubits, which are the analogue to classical bits and form the basis of quan-tum computing [6, 7]. Especially transmon qubits are widely used since they are lesssubjective to 1/f-charge noise [8].The qubit is often placed in a superconducting 3D-microwave cavity, acting as awell-defined electromagnetic environment. That way, coherence times on the orderof micro- and milliseconds can be achieved [8]. Also, the cavities provide control andreadout of the qubit by electromagnetic modes [9].In this thesis, a 3D-microwave cavity and a transmon qubit are characterized in aconfiguration different from other previous works. The idea is based on the obser-vation, that the internal losses in 3D cavities are very small and the coupling to theexternal feed lines can be controlled with high precision. By simultaneously couplingthe qubit to two cavity modes, one with high and one with low external qualityfactor, we plan to realize a compact 3D quantum memory in future experiments.After introducing the relevant theoretical concepts and the experimental setup, theresults of characterization and time-domain measurements will be presented anddiscussed. The outline of this thesis makes reference to possible applications.

1

Chapter 2

Theory

This chapter provides the necessary theoretical background for the experiments. Mi-crowave cavities and transmon qubits will be briefly described as well as a sim-ple model for light-matter interaction and the behavior of qubits in the time do-main.

2.1 3D microwave cavity

A 3D microwave cavity is a region of free space with centimeter dimensions enclosedby metallic walls on all sides so that microwave light can form standing waves. Forthe particular case of superconducting walls, these resonant modes can exhibit longcoherence times. Figure 2.1 shows a schematic drawing.

Figure 2.1: Schematic drawing of a microwave cavity and antennas with dimensionsof the cavity used in this thesis.

2

2.2 Transmon qubit

Of particular importance for this thesis are modes that are polarized perpendicularlyto the beam direction. They are called transverse electromagnetic modes (TEM).The relevant mode profiles as simulated by a finite-element solver are shown inFig. 2.2.

Figure 2.2: Cross section of the cavity perpendicular to antennas as indicated inFig. 2.1. Simulation of electromagnetic field intensities inside a 3D cavity indicatedby a qualitative colorscale. The dots indicate the antenna pins. The qubit chipposition is indicated by a black bar and the reference system is identical to Fig. 2.1

The quality factor of a mode is defined as

Q =frκ

(2.1)

with fr being the resonant mode frequency and κ the full width of the half maximumat the mode frequency.

2.2 Transmon qubit

In a quantum mechanical treatment, a resonant LC-circuit can be considered aquantum harmonic oscillator, resulting in an evenly spaced energy spectrum.A transmon qubit can be considered a LC -circuit where the inductance comes from aJosephson junction as sketched in Fig. 2.3. Since L now shows a nonlinear behavior,the energy levels are no longer equidistant [10]. As a result, this system can now be

3

Chapter 2 Theory

treated as an effective two-level system.

Figure 2.3: Equivalent circuit diagram of a transmon qubit. In this experiment, thequbit is coupled capacitively to the environment.

The qubit’s behavior is dependent on the ratio EJEC

, where EC = q2

2C = 2e2

C (Ccapacitance, e elementary charge unit) is the charging energy of the capacitanceand EJ = Φ0Ic

2π (Φ0 the flux quantum and Ic the maximum Josephson current) isthe energy needed for one charge carrier to tunnel through the Josephson junction.Increasing EJ reduces charge noise and dephasing [11] and therefore enhances thequbit’s lifetime, but reduces the anharmonicity of the energy spectrum.The transition frequencies can be expressed as [12]

f01 =

√8ECEJ − EC

h(2.2)

f12 =

√8ECEJ − 2EC

h. (2.3)

The relative anharmonicity can be calculated by

Arel =f01 − f12

f01. (2.4)

2.3 Qubit-resonator interaction

This section describes the qubit-resonator interaction by introducing the Jaynes-Cummings model.

4

2.3 Qubit-resonator interaction

2.3.1 Jaynes-Cummings model

The Jaynes-Cummings model describes the interaction between an atom and amonochromatic light field [13].The atom is modeled as a two-level system with a ground state |g〉 and an excitedstate |e〉, separated by a characteristic frequency denoted as ωq.With σz = |e〉 〈e|−|g〉 〈g|, the contribution of the isolated atom to the system Hamil-tonian can be stated as

Hq =hωq

2σz. (2.5)

The monochromatic light field contains photons of energy hωr, making its contribu-tion to the system Hamiltonian:

Hc = hωra†a (2.6)

where a† (a) is the creation (annihilation) operator and a†a the photon number op-erator. Here and in the following, the vacuum energy hωr/2 is omitted, because itdoes not contribute to the qubit dynamics.The interaction between light field and atom is characterized by a coupling con-stant g, which is the rate at which excitations can be exchanged between atom andresonator. Its contribution can be described as

Hqc = hg(a†σ− + aσ+) (2.7)

where σ− = |g〉 〈e| and σ+ = |e〉 〈g| are the qubit deexcitation and excitation opera-tors, respectively.

The Jaynes-Cummings Hamiltonian is the sum of these contributions [12]:

HJC = Hq + Hc + Hqc (2.8)

=hωq

2σz + hωra

†a+ hg(a†σ− + aσ+). (2.9)

2.3.2 Strong coupling regime

Both qubit and resonator are subject to environmental noise and therefore have alimited lifetime. The rate at which qubit excitations decay into the ground state is

5

Chapter 2 Theory

denoted by Γ, the rate for decay of the field mode by κ [5]. The strong couplingregime is defined as Γ,κ g. Here, the notation Γ is used as a shorthand for bothqubit decay mechanisms, energy relaxation and dephasing.

2.3.3 Dispersive regime

If the absolute value of detuning of qubit and resonator frequencies |∆| = |ωq − ωr|is far greater than the coupling constant g, the system is in the dispersive regime.The pure eigenstates of qubit and resonator are then good approximations of theeigenstates of the composite system, since the resonant exchange of excitations issuppressed.In this regime, the Hamiltonian simplifies to

Heff = h

(ωr +

g2

∆σz

)a†a+

h

2

(ωq +

g2

∆

)σz. (2.10)

Equation 2.10 also shows that the resonator frequency is shifted depending on thequbit state, by an amount of χ = g2

∆ , which is called the dispersive shift.

2.3.4 AC-Stark shift

By rearranging terms in Eq. 2.10, one can derive that the qubit transition frequencychanges depending on the number of photons in the resonator:

ωq = ωq + 2g2

∆

(a†a+

1

2

). (2.11)

The photon number dependent shift is the so-called AC-Stark shift and can be usedfor a photon number calibration.

2.4 Qubit dynamics

In this section, the qubit evolution in time is discussed. The concept of Bloch sphereis introduced and Rabi oscillations are described.

6

2.4 Qubit dynamics

2.4.1 Bloch sphere

Qubit dynamics is described using the Bloch sphere formalism as depicted in Fig. 2.4.Manipulations of the qubit state can be viewed as rotations about certain axes ofthe Bloch sphere.

Figure 2.4: The Bloch Sphere. Every point on the surface corresponds to a coherentsuperposition of the states |g〉 and |e〉. The angle θ describes the population ofground |g〉 and |e〉, while φ is a relative phase between |g〉 and |e〉.

2.4.2 Rabi oscillations

If a resonant drive is applied to the qubit in its ground state |Φ(0)〉 = |g〉 =

(10

), it

is subjected to a time evolution operator

URabi =

(cos(Ωτ/2) − sin(Ωτ/2)sin(Ωτ/2) cos(Ωτ/2)

)(2.12)

which generates a time-dependent state.

|Φ(τ)〉 = URabi |Φ(0)〉 =

(cos (Ωτ/2)sin (Ωτ/2)

)(2.13)

7

Chapter 2 Theory

with τ the drive pulse length and Ω the Rabi frequency. The Rabi frequency showsa power dependence of the form Ω ∝

√P [8].

2.5 Sidebands

Sidebands are defined as the sum and difference of the qubit and resonator frequen-cies.

ωblue = ωq + ωr (2.14)ωred = |ωq − ωr|. (2.15)

In practice, two slightly detuned tones ωac = ωr − δ and ωs = ωq + δ drive the bluesideband transition, since their sum corresponds to ωblue = ωq + ωr [14].The operators describing these transitions are proportional to 1/δ2 [15]. Fermi’sGolden Rule states that the transition rate is proportional to | 〈f |H |i〉 |2 ∝1/δ4, meaning that sideband transitions are difficult to observe at high detuning.398.33864pt

8

2.5 Sidebands

Figure 2.5: Schematic level diagram of the coupled qubit-cavity system in the JCapproximation. The states are tensor products of qubit states |g, e〉 and photonnumber states |n〉. The states |en〉 are lifted by an energy hωq compared to |gn〉.The resonator states are evenly spaced. The blue and red arrows indicate sidebandtransitions, whereas the green arrows correspond to the detuned tones ωac and ωswith a detuning δ.

9

Chapter 3

Experimental techniques

This chapter describes the devices, measurement setups and procedures usedthroughout this thesis.

3.1 Cavity

In this thesis, a superconducting 3D cavity made of aluminum with a purity of 95%is used. The cavity consists of two halves as illustrated in Fig. 3.1.The cavity has two antennas that can be used to feed signals into the cavity andallow for transmission measurements. Since the antennas can be considered a dis-turbance of the electromagnetic field, it is expected that the modes, which haveintensity minima near the antennas, have a greater lifetime as compared to modesthat have finite field intensities at the antenna position. The antenna depth can bevaried in order to adjust the external quality factor.

10

3.2 Transmon qubit

Figure 3.1: Photograph of microwace cavity a) closed and b) open. The cavity isconnected to external feed lines with SMA connectors. The antenna pins are partiallysurrounded by Teflon and extend into the cavity volume.

3.2 Transmon qubit

The transmon qubit used for this experiment is made of aluminum evaporated ontoa silicon chip. The Josephson junction is placed between two capacitors as shown inFig. 3.2. It behaves as a two-level system and the Jaynes-Cummings Hamiltonianfrom Eq. 2.10 can be applied to the transmon qubit-cavity system.

11

Chapter 3 Experimental techniques

Figure 3.2: Transmon schematic drawing. The position of the Josephson junction isindicated by the red arrow. The capacitors couple the qubit to the environment andprovide the capacitance EC for the transmon qubit.

The capacitors’ dimensions are 355×305 µm2 and the bridge between the two plateshas a length of 50 µm. The capacitors contain a regular grid of holes to pin fluxvortexes, which otherwise could deteriorate the qubit coherence.The transmon qubit is positioned between the two separable cavity parts (Fig. 3.3).

12

3.3 Transmission measurements

Figure 3.3: Photograph of one open cavity half. The qubit and antenna positionsare indicated by the red arrows. This setup ensures that the qubit couples equallywell to the TEM101 and TEM201 modes, but not to the TEM102 modes. This isnecessary since the TEM101 mode is used for readout and the TEM201 mode shallbe used for storing information.

3.3 Transmission measurements

This section describes the transmission-based measurement techniques used in thisthesis.

3.3.1 Single-tone spectroscopy

Single-tone transmission measurements require one microwave tone to be applied toone antenna and to be received at the second one.The first term in Eq. 2.10 shows that the TEM101 frequency ωr = ωr + g2

∆ σz isshifted depending on the qubit state.At low microwave powers, the qubit is in the ground state most of the time (< σz >=−1). But with higher power, it enters a mixed state (< σz >= 0) even if the inputfrequency is far detuned from the qubit transition frequency. Therefore, the cavityresonator frequency is shifted by χ = g2

∆ [16].The influence of the antennas on the measurements can be characterized by an

13

Chapter 3 Experimental techniques

external quality factor Qext and can be determined by measuring the total modequality factors Qtot with varying antenna depths.

1

Qtot=

1

Qext+

1

Qint(3.1)

Qint can be determined to be in the order of 106. In order to have a sufficient read-out rate, the antennas are adjusted such that the resonator is limited by Qext (cf.Tab. 4.1).

3.3.2 Two-tone spectroscopy

In two-tone transmission measurements, a readout tone and a drive tone are appliedvia the same antenna. The frequency of the readout tone is set to the TEM101 modeand that of the drive tone is varied. The drive tone interacts with the qubit andshifts the resonance frequency, if the drive corresponds to the qubit frequency.

3.4 Experimental setup

This section describes the different setups used to characterize the qubit. Allexperiments are carried out in a liquid-helium-precooled dilution refrigerator at40mK.

3.4.1 Two-tone setup

Figure 3.4 shows the setup used for two-tone characterization.

14

3.4 Experimental setup

Figure 3.4: Schematic drawing of the experimental setup used to characterize thequbit in the frequency domain. The cryostate is indicated by the dotted rectangleand contains additional filters and attenuators. This setup only shows the mostimportant devices.

A Vector network analyzer (VNA)1 is used to measure the transmission through thecavity at the frequency of the readout tone. A 20dB attenuator is connected at theVNA output to protect it from reflected signals.For two-tone measurements, one additional RF source is required. In order to be ableto drive two-photon transitions with different frequencies simultaneously (cf. Sec.2.5), in this setup, two RF sources 2 are connected in order to be able to drive two-photon transitions. RF sources and the VNA are combined using power combiners3

and are fed into the cavity.The output signal is filtered, amplified and fed to the VNA.

3.4.2 Time domain setup

Figure 3.5 shows the setup used for time domain measurements.

1Rohde&Schwarz ZVA82Agilent PSG E8267D and Rohde&Schwartz SMF100A3Mini Circuits ZX10-2-183-S+

15

Chapter 3 Experimental techniques

Figure 3.5: Time domain setup. All of the signals are pulsed. The cryostate isindicated by the dotted rectangle and contains additional filters and attenuators.This setup only shows the most important devices

Similar to the frequency domain setup, RF sources are used for driving qubit andresonator. They are combined using power combiners and are fed into the cryostat.The drive and readout signals are now pulsed4. Two mixers are used to improve thesignal on/off ratio. The output signal is then down-converted and digitized.

3.5 Pulse schemes for Rabi oscillations

This section describes the pulse schemes used in Rabi measurements. The rotationsdescribed in Sec. 4.2 are achieved by applying microwave pulses with rectangularenvelopes of sufficient length and frequency. A detailed description can be found in[8].

The pulse scheme used for Rabi oscillations with a continuous readout is depicted inFig. 3.6.

4Agilent 81160A

16

3.5 Pulse schemes for Rabi oscillations

Figure 3.6: Pulse envelopes for measuring Rabi oscillations with a continuous readout(continuous Rabi). The colors indicate the different carrier frequencies.

The measurements are performed in a time interval of 2 µs with the Rabi tonedriving the qubit for 1.5 µs starting at 0.3 µs. The readout tone is set to low powerto limit measurement-induced errors, e.g., by an AC-Stark shift.

Since the readout tone may cause additional qubit dephasing, a pulsed readout asshown in Fig. 3.7 can also be used.

Figure 3.7: Pulse scheme for measuring Rabi oscillation frequencies with a pulsedreadout.

Here, the qubit state is measured directly after the Rabi drive is switched off andthe drive duration τ is varied.

As depicted in Fig. 3.8, a Rabi pulse with a fixed duration chosen such that the qubitis in the excited state (π-pulse) followed by a waiting time τ before the readout canbe used to determine the relaxation time T1.

17

Chapter 3 Experimental techniques

Figure 3.8: Pulse scheme for T1-time determination by pulsed Rabi measurements.

18

Chapter 4

Results and discussions

In this chapter, the results obtained during this thesis are presented and dis-cussed.

4.1 Qubit and cavity characterization

This section contains the results of the cavity and qubit characterization. The datais necessary for more advanced applications, which will be dealt with in the nextsection.

4.1.1 Resonant frequency and dispersive shift

The transmission measurements of the first two cavity modes can be seen in Fig. 4.1.They are obtained by measuring transmission as a function of VNA power and of ωP.The pure eigenfrequencies are taken as the center frequency at high power and thedispersive shift χ is calculated as the difference between the resonance frequenciesat high and low power.

19

Chapter 4 Results and discussions

Figure 4.1: Single-tone transmission measurement (cf. Fig. 3.4). Dispersive shift ofa) TEM101 and b) TEM201 mode. The dispersive shift of TEM102 is barely visibleand the graph is therefore not shown here. "Magnitude" denotes the magnitude ofthe readout signal (VNA).

For the TEM101 mode, the dispersive shift is positive, whereas for the TEM201mode, it is negative. This means that the qubit frequency is positively (negatively)detuned as compared to the TEM101 (TEM201) frequency. Consequently, it mustbe between the two mode frequencies.By fitting Lorentzians to the data, the decay rates κ, the resonance frequenciesand, hence, the Q factors can be calculated. The values are summarized in Table4.1.

4.1.2 Qubit levels

The qubit levels are obtained using two-tone spectroscopy (cf. Sec. 3.3.2). Thetransmission phase is measured as a function of qubit drive power and frequency,showing a phase shift when a qubit transition is driven.

20

4.1 Qubit and cavity characterization

Figure 4.2: Two-tone transmission measurement (cf. Fig. 3.4) yielding qubit-transition frequencies as phase shift. "Phase" denotes the phase of the readoutsignal.

As shown in Fig. 4.2, the qubit levels set in at different power values. The reason forthis behavior is that the number of photons involved in the corresponding processesdiffer. The transition from the ground state |g〉 to the first excited state |e〉 onlyrequires one photon with frequency ωq/2π = 7.594GHz. This requirement is alreadyfulfilled at a low drive power.The next transition at around 7.512GHz is a two-photon process from the groundstate to the second excited state of the qubit. Because a two-photon excitation is asecond-order process, the second transition sets in at higher power. The correspond-ing frequency is lower than ωq/2π due to the anharmonicity of the qubit spectrum.From the measured spectrum, one obtains f01 = 7.594GHz and f12 = f02 − f01 =7.43GHz. The charging energy EC can be obtained by Eqs. 2.2- 2.3: EC =h(f01 − f12) = 0.164GHz · h, yielding C = 47.24 pF. Also, by rearranging Eq. 2.2,one can determine EJ = (hf01+EC)2

8EC= 45.87GHz · h, yielding Ic = 0.92 nA. The

EJ/EC ratio is therefore around 280 and the relative anharmonicity, as calculatedby Eq. 2.4, is around 0.02. As a result, it is justified to treat the transmon qubit asa two-level system.

21

Chapter 4 Results and discussions

Mode ωr/2π(GHz) Q κ(MHz) ∆(GHz) χ(MHz) g(MHz)TEM 101 5.606 2000 2.8 1.988 2 63.06TEM 201 8.904 8.45 · 104 0.1 -1.310 -3 62.69TEM 102 8.944 1515 5.9 -1.350 -0.008 3.29

Table 4.1: Relevant parameters of the cavity modes discussed in this work.

4.1.3 Coupling constant

From the measurements in Fig. 4.2 , the value ωq/2π = 7.594GHz is obtained sothat the coupling constants can be calculated from the dispersive shifts. The resultsare shown in Table 4.1.

These values for g are expected, since the qubit is at a position where the electricalfield intensities of TEM101 and TEM201 should be approximately equal. However,the TEM102 has a node at or near the qubit position, resulting in a strongly reducedcoupling constant.Furthermore, ∆ is two orders of magnitude higher than g and, thus, it is justified toassume that the system is in the dispersive regime.

4.1.4 AC-Stark shift

As described in Sec. 2.3.4, the qubit frequency changes with the photon numberoperator. Figure 4.3 shows measurements of the qubit transition frequency as afunction of the VNA power.

22

4.1 Qubit and cavity characterization

Figure 4.3: Two-tone transmission measurement (cf. Fig. 3.4) with varying VNApower. AC-Stark shift of qubit transition frequency. "Phase" denotes the phase ofthe readout signal (VNA).

Apart from the qubit frequency decrease, Fig. 4.3 shows a broadening of the qubittransition at high powers which is due to fluctuations in the photon number.For a photon number calibration, the qubit transition frequencies are plotted againstprobe power on a linear scale (cf. Fig. 4.4).The photon number is calculated using the relation

n =ωq − ωq

2g2

∆

− 1

2. (4.1)

By substituting in the linear equation

ωq − ωq = m · P (4.2)

23

Chapter 4 Results and discussions

with m = (−0.181± 0.001)GHzmW−1 being the slope from Fig. 4.4, one obtains−m/(2g

2

∆ ) = (45.25± 0.25)mW−1. This result translates into 22 µW per photon1.

0.0 0.2 0.4 0.6

0 9 18 27

7.45

7.50

7.55

Photon number

Freq

uenc

y (G

Hz)

Power (mW)

Figure 4.4: AC-Stark shifted qubit transition frequency on a linear power scale.

These results can be used to determine statistical error margins for g by rearrangingterms:

g =

√0.181GHzmW−1

45.25mW−1 · 7.594GHz− 5.606GHz2

= (63.06± 0.35)MHz.

4.2 Time domain measurements

This section deals with time domain measurements, which are used to determine theRabi frequency and qubit relaxation times.

1The calibration uses data from a previous cooldown of the measurement setup

24

4.2 Time domain measurements

4.2.1 Rabi oscillations

Figure 4.5 shows Rabi oscillation of the qubit state as well as the phase trace at adrive frequency of around 7.599GHz. The continuous-readout pulse scheme depictedin Fig. 3.6 is used.

Figure 4.5: The measurements are done in the time-domain setup depicted in Fig.3.5 and with the scheme depicted in Fig. 3.6. a) Rabi oscillations of the qubit withdrive power set to 10 dBm and readout to -10 dBm at the sources. "Phase" denotesthe phase of the readout signal. b) Time trace indicated by the yellow arrow on a).

Fitting a sine to Fig. 4.5 yields a Rabi frequency Ω = (45.81± 0.09)MHz corre-sponding to a π-pulse duration of (68.58± 0.13) ns.

Using the pulsed readout scheme of Fig. 3.7 , one gets the results shown inFig. 4.6.

25

Chapter 4 Results and discussions

0.0 0.2 0.4 0.6

Pha

se (

arb.

uni

ts)

(µs)

Figure 4.6: Pulsed qubit Rabi oscillations as depicted in Fig. 3.7 (drive power 10dBm; readout power 10 dBm; both at the sources). "Phase" denotes the phase ofthe readout signal. A sine is fitted to the data indicated by the red line.

Fitting a sine to the data in Fig. 4.6 yields a Rabi frequency of (99.29± 1.47)MHzand a π-pulse duration of (31.64± 0.05) ns. These values differ by a factor of twofrom those obtained from the continuous wave Rabi scheme. The reason is that inthe continuous case, an attenuation of 6 dB is present, resulting in a change in Rabifrequency of

√0.25 because of the dependence Ω ∝

√P .

As described in Sec. 3.5, one can use Rabi oscillations to bring the qubit into theexcited state by means of a π-pulse and, subsequently, determine the relaxation time.The result of such a measurement is expected to be an exponentially decaying signal.Indeed, as shown in Fig. 4.7, such a behavior is observed. A numerical fit yields arelaxation time T1 = (2.84± 0.05) µs.

26

4.3 Sidebands

0 2 4 6 8

12.8

13.0

13.2A

mpl

iude

(m

V)

P (µs)

Figure 4.7: Qubit relaxation measured with the protocol described in Fig. 3.8. "Am-plitudes" denotes the amplitude of the readout signal.

4.3 Sidebands

This section shows measurements of sideband transitions in frequency and time do-main.

4.3.1 Frequency domain

By adding a second drive to the two-tone measurement scheme (cf. Sec. 3.5),sidebands can be observed. First, the focus will be on the sideband of the TEM101mode.

The power of the ωs/2π-tone (cf. Sec. 2.5) is adjusted as sketched in Fig. 4.8 toaccount for the cavity filter function and to keep the number of photons in the cavityconstant.

27

Chapter 4 Results and discussions

0 2 4 6 8 10 12

0,0

0,2

0,4

0,6

0,8

1,0A

mpl

itude

(ar

b. u

nits

)

Frequency (arb. units)

Amplitude Adjusted power

Figure 4.8: Schematic plot of cavity filter function and adjusted power.

Figure 4.9: Sidebands of TEM101 mode measured by two-tone transmission as de-picted in Fig. 3.4. The blue and red sidebands are indicated by arrows.

The line parallel to the x-axis at 7.59GHz in Fig. 4.9 corresponds to pure qubitexcitations whereas the diagonal ones are sidebands. They do not appear at large

28

4.3 Sidebands

detuning δ, since their rate of occurrence is suppressed in proportion to δ−4 (cf.Sec. 2.5).The blue sideband has a negative slope, since a qubit drive tone with a frequencylarger than ωq has to be compensated by a resonator drive smaller than ωr and viceversa, so that they equal ωblue = ωac + ωs = ωq + δ + ωr − δ = ωq + ωr.The fact that the red sideband can be driven by two tones with the same sign ofdetuning, e.g. by ωs = ωq − δ and ωac − δ, can be motivated by a trigonometricidentity 2.Figure 4.9 also shows higher sidebands, setting in at smaller detuning. This is dueto the fact that those processes involve more photons, making the δ dependence inthe matrix element more pronounced.Sidebands of the TEM201 mode are measured by the same procedure and are shownin Fig. 4.10.

Figure 4.10: Sidebands of TEM201 mode measured by two-tone transmission as de-picted in Fig. 3.4. The qubit is strongly Stark-shifted around the TEM201 frequencyat 8.904GHz and slightly Stark-shifted by the TEM102 mode at 8.944GHz.

The decrease of the qubit transition frequency in the range of 8.905GHz to 8.92GHzis due to the cavity filter function and the TEM102 mode at 8.944 GHz. The closer

2cos ((ωq − δ)t) + cos ((ωr − δ)t) = cos(ωq+ωr−2δ

2

)· cos

(ωq−ωr

2

)

29

Chapter 4 Results and discussions

the frequency is swept around this value, the more photons can enter the cavity and,hence, shift the qubit frequency.The red sideband is not visible in Fig. 4.10 as opposed to 4.9. This is due to thefact that the red sideband needs an excitation already present in the system, e.g.,a photon in the cavity. Since the resonance of the TEM201 mode is much sharperthan the TEM101 mode (cf. Tab. 4.1) this is only possible for very small values ofdetuning.When choosing a detuning of 100MHz, the filter function of the TEM101 mode hasa value of 2 · 10−4 (when modeled as a Lorentzian with FWHM from Tab. 4.1 ), asopposed to 2.5 · 10−7 for the TEM201 mode. Therefore, one can either increase thedrive power or decrease the detuning to observe red sideband oscillations.These considerations apply only to the red sideband, since the blue sideband can beexcited without a photon already present in the resonator.

4.3.2 Time domain

After that, an attempt is made to resolve dynamics of the sideband transition of thequbit and the TEM201 mode. Figure 4.11 is recorded by driving the resonator withdifferent detunings δ from the TEM201 mode and sweeping the qubit drive. The sys-tem’s behavior is measured by a continuous readout scheme similar to Fig. 3.6.

30

4.3 Sidebands

Figure 4.11: Two-tone transmission measurement (cf. Fig. 3.5 and Fig. 3.6) withresonator drive detuned from the resonance frequency of the TEM201-mode. QubitRabi oscillations as well as sidebands with a) small (14MHz, ωac = 8.918GHz) andb) large (44MHz, ωac = 8.948GHz) detuning can be observed.

31

Chapter 4 Results and discussions

Apart from the qubit Rabi oscillations visible at 7.6GHz, the figures show a struc-ture, which shifts its position with detuning. In Fig. 4.11 a), δ is chosen to be14MHz, in Fig. 4.11 b), δ is set to 44MHz. In both cases, the difference betweenqubit oscillations and bright structure matches these values but with an oppositesign. From this it follows that it is the blue sideband transition, since the sum ofboth drives equals ωblue ≈ 16.5GHz.However, Rabi oscillations on the sideband transition are not visible. Since the Rabifrequency depends on the input power, an increase in microwave power is necessary.In the setup depicted in Fig. 3.5, the power limiting parts are the microwave mixers,which are used for pulsing. By using the internal pulse generator of an RF source 3,the mixers can be omitted and the input power can be increased.Furthermore, driving the sideband near the qubit transition frequency results in qubitRabi oscillations that can overlay the sideband oscillations. Therefore, the sidebandis better excited by a two-photon process with ωblue/2 ≈ (8.9GHz + 7.6GHz)/2 =8.25GHz. The detuning to the qubit frequency is then around 650MHz, whichshould suppress qubit Rabi oscillations. The resulting measurement is shown inFig. 4.12.

The fit in Fig. 4.12 yields an oscillation period of (450± 8) ns. The correspondingπ-pulse duration of 225 ns is compatible with the typical coherence times on theorder of a few microseconds. We intend to optimize the setup towards higher drivepowers to further shorten the π-pulse duration in future experiments.

3Agilent 81160A

32

4.3 Sidebands

Figure 4.12: Sideband oscillations of TEM201 mode obtained by two-tone transmis-sion measurement at 20 dBm drive power. a) Drive frequency sweep around ωblue/2.The oscillations set in close to the expected frequency. The difference can be ex-plained by an AC-Stark shift of the qubit transition frequency due to high drivepower. b) Oscillating time trace at 8.233GHz (indicated by yellow arrow in a)) c)Fit of the function A · sin(ωt − φ) · exp(−t/T ) + m · t + b to the data. The lineartrend incorporates decay effects, e.g. of the |e1〉 state d) Level diagramm with theblue arrow indicating the oscillating states and the red arrows the sources of decay.

33

Chapter 5

Conclusions and outlook

In the course of this thesis, a transmon qubit coupled to two modes of a 3D-microwavecavity has been characterized. A spectroscopic analysis successfully demonstratedthat the qubit is coupled strongly to two cavity modes, one of them with a highand the other one with a low quality factor. Using the latter for qubit readout,Rabi oscillations have been measured. Relaxation measurements have demonstratedT1-times on the order of 3 µs. Although not shown in this work, T2-times are on thesame order.Most importantly, sidebands have been observed spectroscopically for both modes.For the high-Q-mode, qubit population oscillations caused by a coherent drive yielda π-pulse duration of 250 ns for the two-photon blue sideband transition. Thesevalues clearly show that our sample can be used as a compact 3D quantum memorywith an integrated readout and storage mode in the near future.Furthermore, sidebands are considered to have applications in quantum protocolsand have already been used in a transmission line resonator [17] to entangle twoqubits. This can enable scalable quantum computation architecture.

34

Bibliography

[1] M. Homeister, Quantum Computing verstehen (Springer Fachmedien, (Wies-baden, 2015)), 4th ed., ISBN 978-3-658-10455-9.

[2] P. Kaye, An Introduction to Quantum Computing (Oxford University Press,(Oxford, 2010)), 1st ed., ISBN 978-0-19-857049-3.

[3] K. Schmeh, Kryptografie (dpunkt.verlag, (Heidelberg, 2013)), 5th ed., ISBN978-3-86490-015-0.

[4] T. Simonite, “Google’s Quantum Dream Machine”, MIT Technology Review(2015).

[5] F. Grusdt, “Introduction to Cavity QED”, Lecture notes, TU Kaiserslautern(2010).

[6] A. Wallraff, D. I. Schuster, A. Blais, J. M. Gambetta, J. Schreier, L. Frunzio,M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Sideband Transitions andTwo-Tone Spectroscopy of a Superconducting Qubit Strongly Coupled to anOn-Chip Cavity”, Phys. Rev. Lett. 99, 050501 (2007).

[7] R. Gross and A. Marx, “Applied Superconductivity”, Walther Meißner Institut(2005).

[8] M. Müting, “Time domain characterization of a transmon qubit”, Master thesis,Technische Universität München (2015).

[9] H. Paik, D. I. Schuster, L. S. Bishop, G. Kirchmair, G. Catelani, A. P. Sears,B. R. Johnson, M. J. Reagor, L. Frunzio, L. I. Glazman, S. M. Girvin, M. H.Devoret, and R. J. Schoelkopf, “Observation of High Coherence in JosephsonJunction Qubits Measured in a Three-Dimensional Circuit QED Architecture”,Phys. Rev. Lett. 107, 240501 (2011).

[10] L. S. Bishop, “Circuit Quantum Electrodynamics”, Dissertation, Yale University(2010).

[11] A. A. Houck, J. Koch, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Lifeafter charge noise: recent results with transmon qubits”, Quantum InformationProcessing 8, 105 (2009).

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[12] G. Anderson, “Circuit quantum electrodynamics with a transmon qubit in a 3Dcavity”, Master thesis, Technische Universität München (2015).

[13] E. T. Jaynes and F. W. Cummings, “Comparison of quantum and semiclassicalradiation theories with application to the beam maser”, Proceedings of the IEEE51, 89 (1963).

[14] A. Wallraff, “Sideband Transistions and Two-Tone-Spectroscopy of a Super-conducting Qubit Strongly Coupled to an On-Chip-Cavity ”, Physical ReviewLetters 99 (2007).

[15] A. Blais, J. Gambetta, A. Wallraff, D. I. Schuster, S. M. Girvin, M. H. Devoret,and R. J. Schoelkopf, “Quantum-information processing with circuit quantumelectrodynamics”, Phys. Rev. A 75, 032329 (2007).

[16] D. I. Schuster, “Circuit Quantum Electrodynamics”, Ph.D. thesis, Yale Univer-sity (2009).

[17] P. J. Leek, S. Filipp, P. Maurer, M. Baur, R. Bianchetti, J. M. Fink, M. Göppl,L. Steffen, and A. Wallraff, “Using sideband transitions for two-qubit operationsin superconducting circuits”, Phys. Rev. B 79, 180511 (2009).

36

Acknowledgements

First of all, I want to thank Prof. Rudolf Gross for giving me the opportunity towrite my bachelor thesis in the qubit group of the WMI. His lectures gave valuableinsights into the topic of condensed matter physics and inspired me greatly.

Most of all, I would like to thank Edwar Xie and Frank Deppe for always listeningto my questions and their great patience throughout, but certainly not limited to,putting this thesis into writing. I learned a great deal from them.

Also, I want to thank Peter Eder and Leanh Tuan for letting Edwar and me sharetheir cryostat when ours broke down.

Of course I want to thank the whole qubit group and the WMI for being as friendlyas you were. I learned a lot during the weekly seminars.

37