Observation of Higher-Order Sideband Transitions

and First-Order Sideband Rabi Oscillations

in a Superconducting Flux Qubit Coupled to a SQUID Plasma Mode

Yoshihiro SHIMAZU�, Masaki TAKAHASHI, and Natsuki OKAMURA

Department of Physics, Yokohama National University, Yokohama 240-8501, Japan

(Received January 29, 2013; accepted May 13, 2013; published online June 13, 2013)

We report results of spectroscopic measurements and time-domain measurements of a superconducting flux qubit.The dc superconducting quantum interference device (SQUID), used for readout of the qubit, and a shunt capacitorformed an LC resonator generating a SQUID plasma mode. Higher-order red and blue sidebands were observed in asimple measurement scheme because the resonant energy of the resonator, 600MHz, was comparable to the thermalenergy. We also observed Rabi oscillations on the carrier transition and the first-order sideband transitions. Because thequbit was coupled to a single arm of the dc SQUID, the qubit-SQUID coupling was significant at zero bias current,where these phenomena were observed. The ratios between the Rabi periods for the carrier transition and the sidebandtransitions are compared with those estimated from the coupling constant, which was separately determined. The resultmay be explained by assuming initial excitation of the resonator.

KEYWORDS: flux qubit, Josephson junction, sideband, circuit QED, quantum computation

1. Introduction

Work toward the development of future quantuminformation technology, including quantum simulators1)

and quantum computers,2) has led to the extensive study ofvarious types of quantum bits (qubits) over the last decade.3)

Compared with other types of qubits, superconductingqubits4) have the advantages of scalability and stronginteraction with external fields, which may allow fast androbust control and readout. Superconducting qubits easilycouple to superconducting resonators, such as LC resona-tors,5,6) coplanar waveguide resonators,7) and three-dimen-sional (3D) cavities.8) The coupled systems, which areknown as circuit quantum electrodynamics (QED) systems,9)

have been given much attention as important hybridquantum circuits.10) The resonators in these systems can beemployed as a data bus to transfer quantum informationbetween qubits. They are also used to read out the qubitstate8) and to build quantum gates,11,12) such as controlled-NOT gates.

In this paper, we present experimental results from athree-Josephson-junction (3-JJ) flux qubit13) coupled to asuperconducting quantum interference device (SQUID)plasma mode14,15) that is associated with a dc SQUID forreadout of the qubit. This mode is equivalent to an LCresonance in which L is the Josephson inductance of the dcSQUID and C is the shunt capacitance of the SQUID. Sincethe intrinsic nonlinearity of the Josephson inductance is verysmall in typical experimental conditions, the SQUID plasmamode is well represented by a linear resonator. Quantumentanglement between a flux qubit and a resonator was firstobserved in the coupled system of a flux qubit and theSQUID plasma mode.14)

For the sample we studied, the resonant energy of theresonator was comparable to the thermal energy. Thisenabled us to observe higher-order red- and blue-sidebandtransitions, in which the photon number changed by morethan one, using a simple scheme. We note that suchtransitions in a superconducting qubit-resonator system have

only been reported,16) to the best of our knowledge, for a fluxqubit coupled with a linear LC resonator. These transitionsare similar to Raman transitions in atoms and molecules,wherein the low-energy modes involved in the transition canbe vibrational and rotational modes. We also observed Rabioscillations on the carrier transition and the first-ordersideband transitions.14) These observations were made atzero bias current. The significant qubit-resonator coupling inthis bias condition is due to our sample geometry, in whichthe qubit was coupled to a single arm of the dc SQUID, incontrast to the conventionally studied flux qubits.14,15)

We compared the ratios between the Rabi periods for thecarrier transition and the sideband transitions with thoseestimated from the qubit-resonator coupling constant g,which was separately determined. The result can beexplained by assuming a small number of excited photonsin the initial state. We show that, under appropriateconditions, the analysis of sideband Rabi oscillations wouldallow the determination of the photon distribution in theresonator, which has been examined previously by spectro-scopic means,17) and the coupling constant g. In previousstudies, g has been estimated from vacuum Rabi splitting7,18)

or vacuum Rabi oscillations5,19) in the resonant regime andfrom the dispersive shift8) of a resonator frequency in thedispersive regime. The method for estimating g on the basisof Rabi periods of the carrier transition and the sidebandtransitions should be very useful in the deep dispersiveregime, where estimation based on the vacuum Rabisplitting, vacuum Rabi oscillations, or the dispersive shiftis impossible.

2. Experimental Methods

Figures 1(a) and 1(b) show a schematic and an opticalmicrograph of the sample. The 3-JJ flux qubit is galvanicallyconnected to the readout dc SQUID. Electron-beamlithography was used for the fabrication. After fabricatingthe bottom plate of the shunt capacitors, made fromaluminum, its surface was oxidized in ambient air. Thenthe top plates of the shunt capacitors, the qubit, and the

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SQUID were fabricated. The two shunt capacitors with anarea of �4000 �m2 each were connected in series. The shuntcapacitors and the dc SQUID were linked by superconduct-ing wires with a total length of 400 �m and a width of0.3 �m. The small Josephson junctions were formed usingshadow deposition of two films of aluminum, with athickness of 25 nm each. The areas of two of the junctionsin the qubit were approximately 0.03 �m2 each, and the thirdjunction was smaller by a factor of 0.8. The areas of thejunctions of the SQUID were also approximately 0.03 �m2.The dimensions of the loop of the qubit were 25 �m by2.5 �m.

The qubit is dominantly coupled to a single arm of the dcSQUID,20) in contrast to the conventionally studied fluxqubits,14,15,21) wherein the qubit is coupled to both armsof the dc SQUID symmetrically. We will show that thedifference in the geometry has a significant effect on thequbit-SQUID coupling.

The measurement was performed at 20mK, the basetemperature of a dilution refrigerator. For spectroscopy andmeasurement of Rabi oscillations,21) a bias current pulseconsisting of a short pulse of duration 20 ns and a trailingplateau of duration 2.5 �s was applied to the SQUIDimmediately after a microwave pulse. Then the switchingprobability Psw of the SQUID was recorded, typically after5000 trials. The electrical leads to the sample were carefullyfiltered, and magnetic shielding was provided, using asuperconducting lead shield and a permalloy shield. A smallsuperconducting magnet (diameter: 20mm) was used toapply a magnetic flux to the sample.

3. Results and Discussion

The circulating current of the qubit changes its directionwhen fQ ¼ �Q=�0 crosses 0:5þN, where �Q is themagnetic flux in the qubit loop, �0 ¼ h=2e, and N is aninteger. This current reversal is exhibited by the step-likevariation in Psw shown in Fig. 2(a). In this measurement,under irradiation of a 10-GHz microwave pulse with aduration of 400 ns, the height of the bias current pulse was

changed linearly as a function of fQ. For low microwavepower, we observed resonant peaks and dips when theenergy separation between the ground state and the firstexcited state of the qubit h�Q was equal to mh� where �is the microwave frequency and m is an integer. Theappearance of resonances corresponding to m > 1 is due tothe possible multi-photon processes and the harmonicsproduced by the nonlinearity of the microwave mixers usedto produce the pulses. The presence of the harmonics wasconfirmed using a spectrum analyzer. Figure 2(a) shows theresonant peaks and dips for m ¼ 2 and 3 at � ¼ 10GHz.

Figure 2(b) shows the energy dispersion curve of thequbit, which was obtained by locating the resonant peak anddip for m ¼ 1 as a function of the applied magnetic flux.This result fits well to the theoretical energy splitting,13)

h�Q ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi"2 þ�2

p; ð1Þ

where " ¼ 2Ip�0ð fQ � 0:5Þ, Ip is the circulating current inthe qubit, and � is the minimum energy splitting (gap).From this fit, we obtained the parameter values Ip ¼ 217 nAand � ¼ 15:4GHz.

At fQ ¼ 0:4954, where �E ¼ 16:7GHz, Psw vs � curveswere obtained at various microwave powers, as shown inFig. 3. In this measurement, the SQUID bias current Ib waszero during the microwave pulses, which were 100-ns long.One remarkable observation that was made is that sidebandpeaks gradually appear with increasing microwave power, as

(a) Ib

shunt capacitor

25 μm

(b)

Fig. 1. (Color online) (a) Schematic diagram of the sample. The crosses

represent small Josephson junctions. The combination of the shunt capacitor

and the dc SQUID generates a SQUID plasma mode, which couples to the

three-Josephson-junction flux qubit. The resonator frequency can be tuned

via the bias current Ib and the magnetic flux in the SQUID loop. (b) Optical

image of the qubit galvanically connected to the SQUID.

20

30

40

50

60

70

Psw

(%

)

fQ - 0.5

-0.04 -0.02 0.00 0.02 0.04

(a)

-0.005 0.000 0.00515.0

15.5

16.0

16.5

17.0

17.5

18.0

18.5

Freq

uenc

y (G

Hz)

fQ - 0.5

(b)

Fig. 2. (Color online) (a) Switching probability Psw of the dc SQUID as a

function of fQ � 0:5 where fQ is the magnetic flux in the qubit loop divided

by �0 ¼ h=2e. A microwave pulse with frequency 10GHz is applied to

excite the qubit. Resonant peaks and dips are shown. (b) Qubit spectroscopy.

The solid curve is a fit to the theoretical formula for the qubit frequency �Q.

Y. SHIMAZU et al.J. Phys. Soc. Jpn. 82 (2013) 074710 FULL PAPERS

074710-2 #2013 The Physical Society of Japan

can be seen in the figure. These sideband peaks are attributedto red and blue sidebands in the coupled system of the qubitand resonator. The transition frequency for the sidebands isgiven by �Q þM� where M is an integer other than 0 andwhere � is the resonator frequency. The transition for thesidebands is schematically shown in Fig. 4. M represents thechange in the quantum number of the resonator, which isalso referred to as the resonator photon number. In Fig. 3,the sideband transitions for jMj ¼ 2 are clearly visible.Those for jMj ¼ 3 are barely observed at the highestmicrowave power. The apparent downward shift in resonatorfrequency with increasing power can be attributed to thenonlinearity of the resonator.22) Because � � 0:6GHzcorresponds to 30mK, which is near the cryogenictemperature of 20mK, a significant amount of photons arethermally excited, in contrast to the case of the previousstudy.14) These thermally excited photons are seeding thered-sideband generation process and leading to the well-defined appearance of the first and second red sidebands asshown in Fig. 3.

� was also spectroscopically determined when thefrequency of the microwave pulse was varied around0.6GHz. From the dependence of the resonator frequencyon the magnetic flux and Ib,

14,22) shown in Fig. 5, weconclude that the origin of the resonator mode is the SQUID

plasma mode associated with the combined system of thereadout dc SQUID and the shunt capacitance C. Theeigenfrequency of the SQUID plasma mode is given by

�pl ¼ 1

2�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðLSQ þ LsÞC

p ; ð2Þ

where LSQ is the Josephson inductance of the dc SQUID andLs is the stray inductance. For a symmetric dc SQUID, LSQ

is given by

LSQ ¼ �0

2�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2Ic cosð�fSQÞÞ2 � I2b

p ; ð3Þ

where fSQ ¼ �SQ=�0,�SQ is the magnetic flux threading theSQUID loop, and Ic is the critical current of the junction inthe SQUID. The observed resonant frequency is comparedwith the theoretical curves for �pl in Fig. 5. Good agreementbetween them is found. Using Ic ¼ 0:2 �A and the fit shownin Fig. 5, we obtain the values Ls ’ 2300 pH and C ’ 17 pF.We note that the possible asymmetry (�10%) of the SQUIDjunctions does not change these estimates significantlybecause the asymmetry has a second-order effect on LSQ.

The coupling between the flux qubit and the SQUIDplasma mode is described by the Hamiltonian15)

H ¼ h

2ð"�z þ��xÞ þ h

2� ayaþ 1

2

� �þ hfg1ðay þ aÞ þ g2ðay þ aÞ2g�z; ð4Þ

where �z and �x are Pauli matrices written in the persistentcurrent states basis and ay (a) is the photon creation(annihilation) operator; also,

g1 ¼ 1

2

d"

dIb�i0;

g2 ¼ 1

4

d"2

d2Ib�i20;

where

�i0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

h�

2ðLSQ þ LsÞ

s

represents the rms fluctuations of the current in the oscillatorground state. When the qubit is coupled to both arms of the

15 16 17 18 1935

40

45

50

55 +12 dBm +7 -3 -13

Psw

(%

)

Frequency (GHz)

Fig. 3. (Color online) Power dependence of Psw as a function of the

microwave excitation frequency, measured at fQ ¼ 0:4954 and Ib ¼ 0. The

traces are offset vertically for clarity. With increasing power, blue- and red-

sideband transitions of higher orders appear beside the carrier transition.

|g, 0>

|g, 2>|g, 1>

|g, 3>

|e, 0>|e, 1>

|e, 2>|e, 3>

Ω

νQ

νQ + 2ΩνQ − Ω

red sideband

blue sideband

|g, 4 >

|e, 4>

νQ

νQ − 2Ω

νQ + Ω

Fig. 4. (Color online) Energy-level diagram for the qubit-resonator

system. jg ni and je ni denote the qubit in the ground state and in the

excited state, respectively, accompanied by n photons. The resonator

energy, h�, is considerably smaller than the excitation energy of the qubit,

h�Q. The carrier transition from jg 2i is shown by the green solid line, while

the red- and blue-sideband transitions (of first and second order) are

represented by red dashed and blue dotted lines, respectively.

0.2 0.4 0.6 0.8 1.00.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

Res

onat

or f

requ

ency

(G

Hz)

fSQ

Ib/I

c=0

0.53 1.05

Fig. 5. (Color online) The resonator frequency, that is, the center

frequency of the resonant peak for the SQUID plasma mode, as a

function of fSQ ¼ �SQ=�0, where �SQ is the magnetic flux threading the

SQUID loop, for different values of Ib. The solid lines are fits to the

theoretical expression for the resonator frequency. See text for the

parameters used for the fits.

Y. SHIMAZU et al.J. Phys. Soc. Jpn. 82 (2013) 074710 FULL PAPERS

074710-3 #2013 The Physical Society of Japan

SQUID symmetrically, g1 becomes zero at Ib ¼ I�b ’ 0.15,23)

For the sample under investigation, wherein the qubit iscoupled to a single arm of the dc SQUID, the curve of " vs Ibis shown in Fig. 6. The data agree well with the parabolicfit.15,23) The decoupling point (g1 ¼ 0) is given by I�b ’�0:1 �A, which is �80% of the SQUID switching current.On the basis of Fig. 6, we find g1 ¼ 100� 20MHz at Ib ¼ 0

using the above expression. This large coupling, g1=� ’0:2, allowed the clear observation of the sideband transitionsat Ib ¼ 0.

We also found that � exhibited a monotonic dependenceon Ib. The reason for this is not understood at present.However, this dependence was very weak, and thus does notaffect the conclusions of the present paper.

The origin of the second-order sideband transitions(� ¼ �Q � 2�) may be due to the nonlinear term (the g2term) in Eq. (4). However, even for a flux qubit coupled to alinear LC resonator, qubit transitions involving the exchangeof up to 10 photons have been observed.16) Further study isneeded to fully understand the origin of the higher-ordersideband transitions.

We observed Rabi oscillations for the carrier transition(� ¼ �Q) and the first blue- and red-sideband transitions(� ¼ �Q ��),14) as shown in Fig. 7. We now discuss thesesideband Rabi oscillations. These data were again taken atIb ¼ 0 and fQ ¼ 0:502. These oscillations were measuredin a different experimental run from that in which thespectroscopic data shown in Fig. 3 was taken. There is aslight difference in both Ip and � between these measure-ments. The oscillations are well fitted by exponentiallydecaying sinusoids with a small exponentially changingbackground. From the fit, the Rabi periods for the carriertransition and the blue- and red-sideband transitions arefound to be �center ¼ 2:6� 0:1 ns, �blue ¼ 29� 2 ns, and�red ¼ 20� 3 ns, respectively. The decay time constants are13–20 ns.

The ratio between these periods can be compared with thecorresponding theoretical predictions. We take into accountthe excitation of photons in the resonator because the photonenergy is comparable to the thermal energy for the samplewe studied. The state vectors of the combined system of the

qubit and the resonator with n photons are denoted as jg niand je ni for the qubit in the ground state and in the excitedstate, respectively. The blue-sideband transition betweenjg ni and je nþ 1i takes place with a period �blueðnÞ, whilethe red-sideband transition between jg nþ 1i and je ni takesplace with a period �redðnÞ (n ¼ 0; 1; 2; . . .). The ratiosbetween the periods are theoretically given by

�blueðnÞ�center

¼ �ð�Q þ�Þ2

ffiffiffiffiffiffiffiffiffiffiffinþ 1

pg1�Q cos �

; ð5Þ�redðnÞ�center

¼ �ð�Q ��Þ2

ffiffiffiffiffiffiffiffiffiffiffinþ 1

pg1�Q cos �

; ð6Þ�blueðnÞ�redðnÞ ¼ �Q þ�

�Q ��; ð7Þ

where cos � ¼ "=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi"2 þ�2

p(see the Appendix).24)

The observed ratio �blue=�red ¼ 1:5� 0:3 is larger than thetheoretical value of 1.06 given by Eq. (7). We attribute thisto the frequency dependence of the microwave attenuationof the cable; higher frequency microwaves are more stronglyattenuated leading to slower Rabi oscillations.

At the operating point for the data shown in Fig. 7, weobtain cos � ¼ 0:13� 0:04; then using Eqs. (5) and (6) andthe previously estimated value of g1 ¼ 100� 20MHz, wefind

�blueðnÞ�center

’ �redðnÞ�center

¼ ð21� 8Þ � 1ffiffiffiffiffiffiffiffiffiffiffinþ 1

p :

Note that �Q � � here. Despite the large uncertainty, wecan conclude that the observed ratios

�blue�center

¼ 11� 1;�red�center

¼ 8� 2

are not consistent with the theoretical values for n ¼ 0, butare consistent with those for n ’ 3. We note that thediscrepancy between the theoretical values for n ¼ 0 and theobserved ones cannot be explained in terms of the monotonicfrequency dependence of the microwave attenuation.

The above result, implying the initial presence of a smallnumber of photons, is reasonable, considering that the

0

2

4

6ε

(GH

z)

Ib (μA)

-0.2 -0.1 0.0 0.1

Fig. 6. (Color online) Qubit energy bias " induced by Ib. The origin of

" is chosen at its minimum. The linear coupling constant g1, which is

proportional to d"=dIb, becomes zero at Ib ¼ I�b ’ �0:1 �A. The solid line

is a parabolic fit.

35

40

45

50

55

60

65

17.3 GHz

16.8 GHz

Psw

(%

)

Pulse length (ns)

16.3 GHz

0 10 20 30 40 50 60

Fig. 7. (Color online) Rabi oscillations for the carrier transition at 16.8

GHz and blue- and red-sideband transitions at 17.3 and 16.3GHz,

respectively, measured at fQ ¼ 0:502 and Ib ¼ 0. The black solid lines are

fits to exponentially decaying sinusoids. The ratios between the Rabi periods

can be discussed in terms of the qubit-resonator coupling constant.

Y. SHIMAZU et al.J. Phys. Soc. Jpn. 82 (2013) 074710 FULL PAPERS

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photon energy corresponding to 30mK is comparable tothe thermal energy of the system. However, the photondistribution cannot be determined on the basis of astraightforward comparison between the theoretical valuesand the observed ones for �blue=�center and �red=�center. Wefound the relaxation time of the qubit to be T1 ’ 100 ns atIb ¼ 0 and Ib ¼ I�b . The photon lifetime was Tphoton ’100 ns, which was derived from the width of the resonantpeak of the SQUID plasma mode. The Ramsey-interferencedecay time was T2Ramsey ’ 25 ns at Ib ¼ I�b . These timescales are comparable to the decay time of the sideband Rabioscillations shown in Fig. 7. In this case, these relaxationprocesses can combine with sideband excitation to cause thesideband transitions with different values of n to becomemixed with each other. Therefore, to determine the photondistribution, the qubit relaxation, the dephasing of the qubit,and the lifetime of the photons should be taken into account.Further theoretical analysis is needed to simulate theobserved sideband Rabi oscillations.

If the photon lifetime and qubit decoherence time areconsiderably longer than the periods of the sideband Rabioscillations, the photon distribution function can bedetermined from the analysis of the sideband Rabi oscilla-tions; after a time t corresponding to the duration of theresonant blue-sideband pulse (� ¼ �Q þ�), the expectationvalue of �z, written in the energy eigenstates basis, is givenby

h�zi ¼ �X1n¼0

pðnÞ cosð2��blue

ffiffiffiffiffiffiffiffiffiffiffinþ 1

ptÞ; ð8Þ

where pðnÞ is the probability for n photons to be excited inthe initial state and �blue ¼ �blueð0Þ�1 is the Rabi frequencyof the blue-sideband transition between jg 0i and je 1i.25,26)The red-sideband Rabi oscillations are described similarlywith �blue replaced by �red ¼ �redð0Þ�1 (the Rabi frequencyof the red-sideband transition between jg 1i and je 0i). Itshould be noted that the Rabi oscillations given by Eq. (8)have the same form as the Rabi oscillations for a Rydbergatom interacting with a resonant cavity with photons with aprobability distribution given by pðnÞ.27) If photons are in thecoherent state, collapse and revival of the Rabi oscillationsmay be observed as shown in Ref. 27. Meanwhile, if thephotons obey a thermal distribution (Bose–Einstein statis-tics), the fit to Eq. (8) allows estimation of the effectivetemperature Teff of the system. Calculation assuming athermal distribution with Teff comparable to the photonenergy has shown that the dominant frequency of blue-(red-) sideband Rabi oscillations is given by �blue (�red)irrespective of Teff. Therefore, our result cannot notexplained in terms of Bose–Einstein distribution of photons.Deviation from the Bose–Einstein statistics and the probablepeak of the photon distribution at around n ’ 3 may becaused by possible environmental modes of 1–2GHz.

When the photon energy is considerably higher than thethermal energy,14) the initial excitation of the photons isnegligible, and thus the blue-sideband transition betweenjg 0i and je 1i should be observed. In this case, we canestimate the coupling constant g from the Rabi periods forthe carrier and the blue-sideband transition using Eq. (5)with n ¼ 0, under the assumption that the photon lifetimeand qubit decoherence time are sufficiently long. The

frequency characteristics of the microwave line must betaken into account in this analysis. This method forestimating g is particularly useful in the deep dispersiveregime. It compliments the estimate of g on the basis ofvacuum Rabi splitting and vacuum Rabi oscillations in theresonant regime and that based on the dispersive shift of aresonator frequency in the dispersive regime. We note thatthe dispersive shift g2=�Q for the system under investigationis only 0.6MHz, which is too small to be observed.

Our observation of the resonator states involving manyexcitations of photons may provide a basis for futureexperiments manipulating many composite levels of thecoupled system, including demonstration of quantum gatesusing sideband transitions11,12,28) and dynamical coolingsuch as sideband cooling, which was realized in an ion-trapexperiment.29) Dynamical cooling using a tunable qubit30)

and realizing nonclassical states of a resonator31) are alsointeresting.

The important characteristics of the SQUID plasma modein comparison with resonators with a fixed resonantfrequency, such as an LC resonator or a transmission lineresonator, are the tunability of � and the intrinsicnonlinearity. Using the tunability of �, we may realizeresonance between the resonator and a 3-JJ flux qubit at thesymmetry point.5) The degree of nonlinearity of the SQUIDplasma mode can be easily tuned via a bias current andan applied magnetic flux. By enhancing the nonlinearityconsiderably, the SQUID plasma mode could be used as aqubit system that is similar to the phase qubit.

4. Conclusion

In conclusion, we observed higher-order red- and blue-sideband transitions in a coupled system of a flux qubit and aSQUID plasma mode. The resonant energy of the SQUIDplasma mode was comparable to the thermal energy, whichallowed this observation to be made in a simple scheme.Rabi oscillations for the carrier transition and the first red-and blue-sideband transitions were also observed. Thesignificant qubit-resonator coupling at Ib ¼ 0, where theseobservations were made, was due to the sample geometry;the qubit was coupled to a single arm of the dc SQUID. Wecompared the ratios between the Rabi periods of the carriertransition and the sideband transitions, �blue=�center and�red=�center, with those estimated from the qubit-resonatorcoupling constant g, which was separately determined fromthe dependence of the qubit flux bias on the bias current. Theresult may be explained by assuming the initial excitationof n ’ 3 resonator photons. If the photon lifetime andqubit decoherence time are considerably longer than thesideband Rabi periods, the photon distribution function canbe determined from the analysis of sideband Rabi oscilla-tions. In addition, if the resonator energy is considerablyhigher than the thermal energy, g can be estimated byexamining the ratio of the Rabi periods. This method forestimating g is useful, particularly in the deep dispersiveregime.

Acknowledgment

This work was supported by a Grant-in-Aid for ScientificResearch from the Japan Society for the Promotion ofScience.

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Appendix

In this Appendix, we present the effective Hamiltonian forthe flux qubit coupled to the resonator with a linear couplingconstant g.5,6,24) The flux penetrating the qubit loop isassumed to be modulated sinusoidally with a frequency �.The Hamiltonian written in the qubit energy eigenstatesbasis is given by H ¼ HQR þHD,where

HQR ¼ h

2�Q�z þ h� ayaþ 1

2

� �þ hgð�z cos � � �x sin �Þðay þ aÞ ðA�1Þ

and the driving term is

HD ¼ A cos 2��t

2ð�z cos � � �x sin �Þ: ðA�2Þ

Here, cos � ¼ "=�Q, and A is the amplitude of the energy-bias modulation. The nonlinear term shown in Eq. (4) isneglected here because it is not relevant to the carriertransition and first-order sideband transitions, which arefocused on in this Appendix. The eigenstates of HQR arereferred to as the dressed states. We assume g cos � � andg sin � j�Q ��j.

The driving term written in the dressed states basis is

H 0D ¼ A cos 2��t

2f�z cos � � ð�þ þ ��Þ sin �g

þ gA�Q sin2 � cos 2��t

�2Q ��2�zðay þ aÞ

� gA�Q sin � cos � cos 2��t

�ð�Q þ�Þ ð�þay þ ��aÞ

þ gA�Q sin � cos � cos 2��t

�ð�Q ��Þ ð�þaþ ��ayÞ ðA�3Þto the first order of g, where

�� ¼ 1

2ð�x � i�yÞ:

In the interaction picture using the unperturbedHamiltonian

H0 ¼ h

2�Q�z þ h� ayaþ 1

2

� �;

H 0D is transformed to

H 00D ¼ A cos 2��t

2f�z cos � � ð�þe2�i�Qt þ ��e�2�i�QtÞ sin �g

þ gA�Q sin2 � cos 2��t

�2Q ��2�zðaye2�i�t þ ae�2�i�tÞ

� gA�Q sin � cos � cos 2��t

�ð�Q þ�Þ� ð�þaye2�ið�Qþ�Þt þ ��ae�2�ið�Qþ�ÞtÞþ gA�Q sin � cos � cos 2��t

�ð�Q ��Þ� ð�þae2�ið�Q��Þt þ ��aye�2�ið�Q��ÞtÞ: ðA�4Þ

The ratios between the Rabi periods for the carrier transition(� ¼ �Q) and the first blue- and red-sideband transitions(� ¼ �Q ��) shown in Eqs. (5)–(7) are obtained fromEq. (A�4) using the rotating-wave approximation.

�yshimazu@ynu.ac.jp

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