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Li, Jian; Silveri, M. P.; Kumar, K. S.; Pirkkalainen, J.-M.; Vepsäläinen, A.; Chien, W.C.;Tuorila, J.; Sillanpää, M.A.; Hakonen, P.J.; Thuneberg, E.V.; Paraoanu, G.S.Motional averaging in a superconducting qubit

Published in:Nature Communications

DOI:10.1038/ncomms2383

Published: 01/01/2013

Document VersionPublisher's PDF, also known as Version of record

Please cite the original version:Li, J., Silveri, M. P., Kumar, K. S., Pirkkalainen, J-M., Vepsäläinen, A., Chien, W. C., ... Paraoanu, G. S. (2013).Motional averaging in a superconducting qubit. Nature Communications, 4, 1-6. [1420].https://doi.org/10.1038/ncomms2383

ARTICLE

Received 13 Aug 2012 | Accepted 11 Dec 2012 | Published 29 Jan 2013

Motional averaging in a superconducting qubitJian Li1, M.P. Silveri2, K.S. Kumar1, J.-M. Pirkkalainen1, A. Vepsalainen1, W.C. Chien1, J. Tuorila2,

M.A. Sillanpaa1,3, P.J. Hakonen1, E.V. Thuneberg2 & G.S. Paraoanu1

Superconducting circuits with Josephson junctions are promising candidates for developing

future quantum technologies. Of particular interest is to use these circuits to study effects

that typically occur in complex condensed-matter systems. Here we employ a super-

conducting quantum bit—a transmon—to perform an analogue simulation of motional

averaging, a phenomenon initially observed in nuclear magnetic resonance spectroscopy. By

modulating the flux bias of a transmon with controllable pseudo-random telegraph noise we

create a stochastic jump of its energy level separation between two discrete values. When the

jumping is faster than a dynamical threshold set by the frequency displacement of the levels,

the initially separate spectral lines merge into a single, narrow, motional-averaged line. With

sinusoidal modulation a complex pattern of additional sidebands is observed. We show that

the modulated system remains quantum coherent, with modified transition frequencies, Rabi

couplings, and dephasing rates. These results represent the first steps towards more

advanced quantum simulations using artificial atoms.

DOI: 10.1038/ncomms2383

1 O. V. Lounasmaa Laboratory, Aalto University School of Science, P.O. Box 15100, FI-00076 AALTO, Espoo, Finland. 2 Department of Physics, University ofOulu, P.O. Box 3000, FI-90014, Oulu, Finland. 3 Department of Applied Physics, Aalto University School of Science, P.O. Box 11100, FI-00076, Aalto, Espoo,Finland. Correspondence and requests for materials should be addressed to M.P.S. (email: matti.silveri@oulu.fi) or to G.S.P. (email: sorin.paraoanu@aalto.fi).

NATURE COMMUNICATIONS | 4:1420 | DOI: 10.1038/ncomms2383 | www.nature.com/naturecommunications 1

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The ability to resolve energy variations DE occurring in atime interval Dt is fundamentally limited by the energy-time uncertainty relation DEDt\�h. Consider for example a

system of spin-1/2 particles filling a porous material (Fig. 1a) inan external magnetic field; then, if the pore wall is paramagnetic,the particles near it will experience a modified local magneticfield. As a result, static particles will produce two spectral peaks inthe nuclear magnetic resonance spectrum at frequencies o1 ando2 (Fig. 1b). In contrast, particles moving swiftly back and forthbetween the two regions on timescales Dt shorter than�hðDEÞ� 1¼ðo2�o1Þ� 1 are not able to discriminate betweenthe two energy values. Then, as the particles move faster, theoutcome in the spectroscopy is not simply a continuousbroadening and overlapping of the two peaks. Instead, a newpeak emerges at the average frequency o0 with a width smallerthan the energy separation DE/�h, denoted as motional averagingand narrowing1,2, respectively. In atomic ensembles andcondensed-matter systems, this occurs via fast variations of theelectronic state populations, chemical potential, molecularconformation, effective magnetic fields, lattice vibrations, micro-electric fields producing ac-Stark shifts and so on1–6. Closelyrelated phenomena are the Dyakonov-Perel effect7, the Dicke linenarrowing of Doppler spectra8, and the quantum Zeno effect9.

In this paper, we report the observation of motional averagingand narrowing in a single artificial atom, a quantum bit (qubit),with a simulated fast-fluctuating environment under directexperimental control. This should be contrasted with the typicalsituation in condensed-matter systems, where one has a largenumber of particles and the experimentalist can only indirectlyattempt to change the fluctuation rate, typically by modifying athermodynamic function of state, such as temperature orpressure.

ResultsFrequency-modulated transmon qubit. Our device is a circuitQED system10, consisting of a tunable transmon coupled to aquarter-wavelength coplanar waveguide resonator. The circuitschematic and the simplified experimental set-up are shown inFig. 1d and in Supplementary Fig. S1. The applied bias flux Fthrough the qubit loop consists of a constant part Fdc and a time-dependent part Fac(t) with an amplitude much smaller than Fdc.The Hamiltonian of the transmon qubit11 is

HðtÞ¼ �h2½o0þ xðtÞ�sz; ð1Þ

where o0¼op½cos2ðpFdc/F0Þþ d2sin2ðpFdc/F0Þ�14� EC/�h. Here

F0¼ h/2e is the magnetic flux quantum, the Josephson plasmafrequency is op¼

ffiffiffiffiffiffiffiffiffiffiffiffi8ECEJp

/�h, sx;y;z denote the Pauli matrices,the single-electron charging energy is EC¼ e2/2(CGþCB)Eh� 0.35 GHz, EJ¼ EJ1þ EJ2E24EC is the maximum Josephsonenergy, and d¼ (EJ1� EJ2)/EJE0.11 denotes the junctionasymmetry. We choose Fdc so that o0/2p¼ 2.62 GHz, which isfar detuned from the bare resonator frequency or/2p¼ 3.795 GHz, allowing the dispersive measurement of thequbit through the resonator by standard homodyne andheterodyne techniques12.

The time-dependent part �hxðtÞ of the energy splitting,determined by Fac(t), is controlled by an arbitrary waveformgenerator (AWG) via a fast flux line. The random telegraph noise(RTN) is realized by feeding pseudo-random rectangular pulses tothe on-chip flux bias coil. Ideally, the dynamics of �hxðtÞ is astationary, dichotomous Markovian process, characterized by anaverage jumping rate w and symmetrical dwellings at frequencyvalues of ±x (see Fig. 1c). The number of jumps is a Poissonprocess with the probability Pn(t)¼ (wt)ne� wt/n! for exactly njumps within a time interval t. This Poissonian process simulatesthe temporal variations causing motional averaging in atomicensembles and condensed-matter systems (Fig. 1).

Simulation of motional averaging. In order to find the effectivetransition energies and decoherence rates of the modulated sys-tem, we calculate the absorption spectrum1,2. For a qubitsubjected to RTN fluctuations as in equation (1), we exploit thequantum regression theorem13 to find the spectrum (see details inSupplementary Note S1)

SðoÞ¼ 12p

Z 1�1

eiðo�o0Þt�G2 j t j e� iR t

0xðtÞdt

D Exdt; ð2Þ

where the expectation value is taken over noise realizations andG2¼GjþG1/2 is the decoherence rate in the absence ofmodulation. For our sample G2E2p� 3 MHz (determined inan independent measurement). For the Poisson process x(t),equation (2) gives1,2,14,15

SðoÞ¼ 1p

2wx2þG2 �G22þ �o2þ x2� �

ðx2� �o2þG2 �G2Þ2þ 4ðG2þ wÞ2 �o2; ð3Þ

with �G2¼G2þ 2w and �o¼o�o0. In our measurement, wehave recorded the population of qubit’s excited state, as shown inFig. 2 and in Supplementary Fig. S2. The qubit is excited with a

�0 �2�1

�0+�

t

S(�)

Static particles

Moving particles

a b c

�

�0−�

�0

�1�2CC

VNA

Resonator

Qubitdc+modulations

Φ

d

CG

CB

EJ1 EJ2

�probe

�

Iflux

25 mKRoom temperature

Port 1

Port 2

12

S1.6

0.4

0.8

1.2

<�(t+�)� (t )>� = �2e

−2�|�|

Figure 1 | Schematics of motional averaging and simplified experimental set-up. (a) Illustration of a single pore filled with a medium consisting of

spin-1/2 particles that are either static or moving. White and grey areas denote different magnetic environments, producing the energy splittings �ho1 and

�ho2, respectively (see b,c). (b) Schematic spectrum of the motional averaging. The solid lines denote the spectrum of static particles, while the dashed line

corresponds to moving particles. (c) A sample trajectory of the energy splitting �h½o0þ xðtÞ� in equation (1). (d) Schematic diagram of the sample device

and the simplified experimental set-up. Two asymmetric Josephson junctions (EJ1 and EJ2) are capacitively shunted (CB) and coupled (CG) to a resonator.

The time-dependent transition frequency of the qubit is controlled with fast flux pulses on top of a dc bias (Iflux¼ Idcþ Ipulse), generated by an AWG.

A vector network analyser (VNA) is used for probing the resonator at oprobeEor. One microwave source (o) is used for driving the qubit. Additional details

are shown in Methods and in Supplementary Fig. S1.

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transverse drive

HdriveðtÞ¼ �hg cosðotÞsx: ð4Þ

The simulation of the occupation probability presented in Fig. 2a(see Methods and Supplementary Note S2) includes also the effectof power broadening due to strong driving amplitudes(g4G1,G2).

Two intuitively appealing limits result from equation (3) (seealso Fig. 2) at the opposite sides of a dynamical threshold definedby w¼ x. In the case of slow jumping, w � x, and resolvableenergy variations, DEDt � �hx/w � �h, the qubit absorbs energyat o0±x with the total decoherence rate G02¼G2þ w. Thecorrelation time of the displacement process �hxðtÞ is tx¼ (2w)� 1

and the linewidth (full width at half maximum 2G2) broadens bythe amount of the reduced mean life-time of a qubit excitation(exchange broadening1). In contrast, for fast jumping processes,w � x, and when the variations are not resolvable,DEDt � �hx/w � �h, the qubit absorbs energy only at thefrequency o0 with G02¼G2þ x2/2w. The increase indecoherence rate by x2/2w can be related to the excursions ofthe accumulated phase uðtÞ¼

R t0 xðtÞdt (equation (2)) for single

noise realizations, especially to the diffusion coefficient of theprocess16. Surprisingly, the averaged spectral line is well-localizedaround the mean value, since x2/2w � 2x. The effect of motionalnarrowing can be best seen in Fig. 2c as a raise in the height of thecentre peak because the additional decoherence x2/2w decreaseswith increasing jumping rate w. When w is comparable with x,there is a cross-over region where absorption is reduced and thepeak broadens due to enhanced decoherence (see Fig. 2). This isimportant for the improvement of qubit dephasing times as itimplies that a longitudinally coupled two-level system (TLS)fluctuator is the most poisonous when its internal dynamicsoccurs approximately at the same frequency as the coupling to thequbit.

A counter-intuitive aspect of motional averaging is that thesystem is at any time in either one of the states � �hx and spendsno time in-between. Yet spectroscopically it is not seen in thestates � �hx; instead it has a clear signature in the middle. Also,the spectrum (3) decays as 1/o4 far in the wings, showing a non-Lorentzian character that originates from the non-exponentialdecay of the correlator s� ðtÞsþ ð0Þh i14. This is because thefrequent sz-changes have a similar effect to a continuous

measurement of the system, slowing its dynamics in analogywith the quantum Zeno effect9,17.Sinusoidal modulation of frequency. To further explore theseeffects, we have used sinusoidal waves to modulate the qubitenergy splitting: �h½o0þ d cosðOtÞ�. In Fig. 3a,b, the experimentaldata and the corresponding numerical simulation are shown.Resolved sidebands appear at o¼o0±kO (k¼ 0,1,2,y), and areamplitude-modulated with Bessel functions Jk(d/O), where d andO denote the modulation amplitude and frequency, respectively.The Hamiltonian Hþ Hdrive (equations (1) and (4)) can betransformed to a non-uniformly rotating frame18,19 (seeMethods). The theoretical prediction for the steady-stateoccupation probability on the resolved sidebands (O4g4G2) is

Pe¼X1

k¼ �1

G22G1

gJkdO

� �� �2

G22þ o0þ kO�oð Þ2þ G2

G1gJk

dO

� �� �2 ; ð5Þ

which is compared with experimental data in Fig. 3c and inSupplementary Fig. S3. These population oscillations can bealternatively understood as a photon-assisted version of thestandard Landau–Zener–Stuckelberg interference20,21 (seeMethods and Supplementary Fig. S4). Also, when the qubitdrive is tuned close to the resonator frequency oEor and whenthe driving amplitude g is large and equal to the frequency of themodulation g¼O, the system can be seen as a realization of aquantum simulator of the ultrastrong coupling regime22,23. In ourset-up, the simulated coupling rate is estimated to reach over 10%of the effective resonator frequency (see Supplementary Note S3),comparable to earlier reports24,25 where the ultrastrong couplingwas obtained by sample design. The sinusoidal modulation allowsa different perspective on motional averaging. The RTN noisecomprises many different modulation frequencies O (Fig. 1c).Each frequency creates different sidebands (k¼ 1,2,3,y), whichoverlap and average away. At large O’s only the central bandk¼ 0 survives because the only non-vanishing Bessel function ofzero argument is J0(0)¼ 1.

Coherence of the modulated system. Finally, we show that it ispossible to use the modulated system as a new, photon-dressedqubit. Indeed, we can drive Rabi oscillations on the central bandand on the resolved sidebands (see Fig. 4 and SupplementaryFig. S5). For fast RTN modulation, w � x, and under the strongdrive in equation (4), we construct (see Supplementary Note S2) a

a b c

� (M

Hz)

–200

–150

–100 –5

0 0 50 100

150

200

–200

–150

–100 –5

0 0 50 100

150

200

50

75

100

125

150

175

200

225

250

0

0.1

0.2

0.3

0.4

0.5

� (M

Hz)

50

75

100

125

150

175

200

225

250

0

0.1

0.2

0.3

0.4

0.5

–200 –100 0 100 2000

0.2

0.4

0.6

0.8

1� = 50 MHz� = 100 MHz� = 200 MHz

(�−�0)/2� (MHz) (�−�0)/2� (MHz)(�−�0)/2� (MHz)

Pe

Figure 2 | Motional averaging. (a) Numerical simulation of the occupation probability Pe plotted in the o� w plane with parameters x/2p¼43 MHz,

g/2p¼ 8 MHz, G1/2p¼ 1 MHz, and G2/2p¼ 3 MHz. (b) Measured excited state occupation probability Pe under the RTN modulation. (c) Horizontal

cuts of the measured occupation probability in b, denoted with white lines, at three different jumping rates (solid lines). For clarity, the curves are displaced

vertically by 0.3. The dashed lines indicate the corresponding results of the numerical simulation in a. See Supplementary Fig. S2 for the zero-detuning

(o¼o0) occupation probability Pe with different x.

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms2383 ARTICLE

NATURE COMMUNICATIONS | 4:1420 | DOI: 10.1038/ncomms2383 | www.nature.com/naturecommunications 3

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master equation15 describing the Rabi-oscillation with the

detuned Rabi frequency gd¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðo�o0Þ2þ g2

qand with the

total decoherence rate G02¼G2þ x2/2w. In the case of sinusoidalmodulation, the Rabi frequencies on the sidebands are obtained

as gðkÞeff ¼ g j Jkðd/OÞ j .

DiscussionOur work could open up several research directions. For example,a key limitation in solid-state based quantum processing ofinformation is the decoherence due to fluctuating TLSs in thedielectric layers fabricated on-chip26–28. We anticipate, resting onthe motional narrowing phenomenon, that the dephasing timesof the existing superconducting qubits may be dramaticallyimproved if one is able to accelerate the dynamics of thelongitudinally coupled TLSs. The quantum coherence of theresolved sidebands and of the central band suggest that

low-frequency modulation offers an additional tool forimplementing quantum gates29, in analogy with the use ofvibrational sidebands in the ion trap computers30. Also, if the twotones used to drive and modulate the qubit satisfy certainconditions, the system can be mapped into an effective qubit-harmonic oscillator system with ultrastrong coupling, openingthis regime for experimental investigations. Finally, the recentprogress in the field of nanomechanical oscillators makes possiblethe study of frequency jumps in nanomechanical resonators16,which is predicted to be accompanied by squeezing31. Our workpaves the way for further simulations of quantum coherencephenomena using superconducting quantum circuits32,33.

MethodsMeasurement set-up. The electronic measurement set-up at room temperature isillustrated in Supplementary Fig. S1. For qubit spectroscopy with RTN modula-tions, the dc flux bias and RTN modulations are generated by an Agilent 81150AAWG. The qubit driving signal from an Agilent E8257D analogue signal generator

cba

0

0.1

0.2

0.3

0.4

0.5

0.6

Pe

10 20 30 402.56

2.58

2.6

2.62

2.64

2.66

�/2

� (G

Hz)

Rabi pulse width (ns)

10 20 30 402.56

2.58

2.6

2.62

2.64

2.66

�/2

� (G

Hz)

0 0.5 1 1.5 2 2.5 3 3.50

10

20

30

40

50

60

g eff

/2�

(MH

z)

0−band1−band

1 2 3 4 520406080

100120140

Rabi pulse amplitude (Arb)

g/2�

(M

Hz)

0−band

RTN modulation

1 2 3 4 520

40

60

80

100

120

Rabi pulse amplitude (Arb)

g eff

/2�

(MH

z)

0−band1−band

Sinusoidalmodulation

J0(�/�)

J1(�/�)

0.7

�/�

Figure 4 | Rabi oscillations with modulations. (a) Rabi oscillations versus driving frequency with RTN modulation: measurement and numerical simulation

with o0/2p¼ 2.610 GHz, x/2pE95 MHz, wE600 MHz, g/2p¼91 MHz, G1/2p¼ 1 MHz, and G2/2p¼ 3 MHz. The corresponding Rabi oscillations in the

case of sinusoidal modulation are shown in Supplementary Fig. S5. (b) The measured Rabi frequency of the random and sinusoidal modulation g and geff,

respectively, as a function of Rabi-pulse amplitude (same scaling in the both panels). The solid and dashed lines are fits to the data in the linear regime.

(c) Rabi frequencies versus amplitude of the sinusoidal modulation. The horizontal axis is the dimensionless scaled modulation amplitude d/O, which is

also the argument for the effective Rabi frequency gðkÞeff ¼ g j Jkðd/OÞ j plotted with g/2p¼60 MHz and colour coding for k: 0 (red dotted) and 1 (blue

dashed). The (red) circles and (blue) triangles indicate the measured Rabi frequencies at the centre of 0-band and 1-band, respectively.

100

200

300

400

500Ω

/2�

(MH

z)

0

0.1

0.2

0.3

0.4

0.5

100

200

300

400

500

Ω/2

� (M

Hz)

0

0.1

0.2

0.3

0.4

0.5

–300 –200 –100 0 100 200 3000

0.1

0.2

0.3

0.4

0.5

Pe

a b c

(�−�0)/2� (MHz) (�−�0)/2� (MHz)

(�−�0)/2� (MHz)–600

–400

–200 0

200

400

600

–600

–400

–200 0

200

400

600

Figure 3 | Occupation probabilities of the excited state of the qubit under sinusoidal modulations. (a) Measured excited state occupation probability Pe

with amplitude d/2p¼ 250 MHz. (b) Simulation with G1/2p¼ 1 MHz, G2/2p¼ 3 MHz, and g/2p¼ 20 MHz. c The cut along the dashed line in a,b at

O/2p¼ 210 MHz. The (black) circles are the experimental data. The (blue) line denotes the analytical occupation probability obtained from equation (5) by

taking k¼ � 1,0,1, which overlaps perfectly with the simulation in the parameter regime, O � g 4G2. For k¼0 (0-band), the linewidth is narrower than

the bare linewidth without modulation (see Methods and Supplementary Fig. S3).

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and the coplanar waveguide cavity probe signal from an Agilent N5230C PNA-Lnetwork analyser are combined together by a Mini-Circuits ZFSC-2-10G powersplitter/combiner and sent to the cavity input line of the dilution refrigerator. Thesignal from the cavity output line of the dilution refrigerator is amplified anddetected by a PNA-L network analyser. For sinusoidal modulations, the tones aregenerated by a R&S SMR27 microwave signal generator and they are added to thedc bias generated by the Agilent 81150A via a bias tee (not shown). Between theradio frequency instruments (analogue signal generator and network analyser) andtheir corresponding lines, dc blocks are used for breaking possible ground loops.All instruments are synchronized with a SRS FS725 Rubidium frequency standard(not shown in the figure).

For Rabi-oscillation measurements, high ON/OFF ratio Rabi pulses are gener-ated by mixing a continuous microwave signal from the Agilent E8257D withrectangular pulses from a SRS DG645 digital delay generator via two identicalMini-Circuits frequency mixers. The Agilent N5230C PNA-L is used as a signalgenerator. Its output signal is split into two parts. One part is used for generatingmeasurement pulses in a similar fashion to the Rabi pulses; the other part acts as alocal oscillator signal to mix the output signal from the coplanar waveguide cavitydown via a Marki IQ mixer. The IQ data is filtered, amplified by a SRS SR445A pre-amplifier, and digitized by an Agilent U1082A-001 digitiser.

Measurement controlling and data processing are done by MATLAB running ona measurement computer. The communication between the measurement com-puter and the instruments is realized through IEEE-488 GPIB buses. To generateRTN pulses, we use MATLAB’s internal Poisson random number generatorpoissrnd(l) to obtain binary RTN sequences, and load the sequences into theAgilent 81150A AWG. Each binary RTN sequence consists of 50,000 data pointsand around 5,000 random jumps in average. The mean jumping rate w (of AWG’soutput) is modulated by changing the clock frequency n of the AWG: wE5,000� n.To verify this relation between w and n, we observe the RTN sequences at differentn’s by a fast oscilloscope (10 GS per s), count the number n of edges (jumpingevents) for certain period of time t, and calculate the real mean jumping rate by itsoriginal definition, w¼ n/t. As an example, a 0.5-ms long RTN sequence withestimated mean jumping rate wE5,000� 10 KHz¼ 50 MHz is shown inSupplementary Fig. S1. Twenty-five jumps are counted during 0.5 ms, which givesw¼ 50 MHz. As long as no100 KHz, the formula wE5,000� n gives a good esti-mation of the mean jumping rate. For n4100 KHz, the real mean jumping rate issmaller than the estimated one, due to the intrinsic � 2 ns rising/falling time of theAWG.

Numerical simulations. The occupation probability of Figs 2a,c, 3b, 4a, andSupplementary Figs S2 and S4 is a result of solving numerically the master equation

drðtÞdt¼ � i

�hHðtÞþ HdriveðtÞ; rðtÞ� �

� Gj

4½sz ; ½sz ; r��

þ G1

22s� rsþ � sþ s� r� rsþ s�ð Þ;

ð6Þ

in the frame rotating at o and applying the rotating wave approximation(the pure dephasing rate Gf¼G2�G1/2). For the random modulation (Figs 2a,c,and 4a, and Supplementary Fig. S2), we apply the method of quantum trajectories13

and the imperfections of the experimentally realized wave forms (the raising/falling time � 2:0 ns and the sampling time 0.5 ns) are included in thesimulation.

Multi-photon transition process in sinusoidal modulation. The effects of sinu-soidally modulating the transition frequency of an atom is a generic problem intheoretical physics, and has attracted a lot of interest in the past, most notably inthe quantum-optics community34,35. We address now this problem in the contextof superconducting qubits. We consider a transmon qubit whose energy splitting ismodulated sinusoidally: �h½o0 þ d cosðOtÞ�. Even though O � o0, the qubit can beexcited with the low-frequency signal in the presence of an additional high-frequency drive o. This can be seen as a multi-photon transition process involvingquanta from both fields (in equation (5)), or, as a photon-assisted Landau–Zener–Stuckelberg (LZS) interference18,20,21,36 between the dressed states |k, nS and|m, n� 1S.

Let us first consider the multi-photon transition processes associated with thesinusoidal-modulated Hamiltonian, defined in equations (1) and (4),

H¼ �h2½o0 þ d cosðOtÞ�sz þ �hg cosðotÞsx : ð7Þ

The qubit is driven both in the longitudinal (sz) and in the transverse (sx)direction. We show below how to eliminate the time-dependence from thelongitudinal drive. After moving to a non-uniformly rotating frame with theunitary transformation18,19

U ¼ exp � i2

o0tþ dO

sinOt

� �sz

; ð8Þ

the effective Hamiltonian is H0 ¼ UyHU þ i�hð@t UyÞU , and by using the Jacobi-Anger relations, we get

H0 ¼ �hg2

eiot þ e� iot� �

eio0 tX1

k¼ �1Jk

dO

� �eikOt sþ þ h:c:

" #: ð9Þ

By assuming that the transverse drive is close to a resonance, that is, oEo0±kO(k¼ 0,1,2y) and that the resonances are resolvable O4g4G2, we transform backwith U ¼ exp½iðo0 þ kO�oÞtsz/2� and ignore all but the resonant terms, that is,all the fast rotating terms (rotating wave approximation, RWA). The resultingRWA Hamiltonian reads

H 0RWA ¼�h2ðo0 þ kO�oÞsz þ gJk

dO

� �sx

; ð10Þ

describing, in the Bloch spin representation, precessions around the vectorO¼ (gJk(d/O),0,o0þ kO�o) with the effective Rabi frequency

gðkÞeff ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðo0 �oþ kOÞ2 þ g2J2

k d/Oð Þq

: ð11Þ

Exactly at the multi-photon resonance o¼o0±kO (k¼ 0,1,2y), the effectiveRabi frequency is |gJk(d/O)|, and it is plotted in Fig. 4c.

In the time domain, the measured Rabi oscillations are presented inSupplementary Fig. S5 together with numerical simulations of Bloch equationsexploiting the effective Rabi frequency in equation (11). To find the steady-stateoccupation probability in the presence of relaxation with rate G1 and decoherencewith rate G2, we solve the Bloch equations analytically. In the RWA, that is whenO4g4G2 is satisfied, one is allowed to add up independent contributions from allthe resonances (the resolved sidebands). The result for the steady-state excited stateoccupation probability is equation (5). In the non-modulated case, thecorresponding expression for the occupations is simply

Pe ¼G2

2G1g2

G22 þðo0 �oÞ2 þ G2

G1g2: ð12Þ

The occupation probabilities in equations (5) and (12) are compared with theexperimental steady-state occupation probabilities in Supplementary Fig. S3. Aninteresting effect is that the linewidth of the modulated qubit on the central band isalways smaller than the linewidth on the qubit in the absence of the modulationdue to reduced power broadening.

Photon-assisted Landau–Zener–Stuckelberg interference. The spectra seen inthe experiment under sinusoidal modulation can be also interpreted as a photon-assisted LZS effect. Note that in the absence of the driving field standard LZSprocesses are not possible: indeed the qubit energy separation is one order ofmagnitude higher than the modulation frequency, therefore the standard LZSprobability is negligibly small. However, the system can still perform LZS transi-tions by absorbing a photon from the driving field. This photon-assisted LZSinterference can be seen by transforming the Hamiltonian Hþ Hdrive fromequation (7) into the frame rotating at o around the z axis (unitary transformationU ¼ expð� iosz/2Þ), where it has exactly the same form (in the RWA) as that ofLZS interference18,21,36, namely

H¼ �h2½o0 �oþ d cosðOtÞ�sz þ

�h2

gsx : ð13Þ

As illustrated in Supplementary Fig. S4, LZS transition events may occur whenthe modulation amplitude d is of the order of, or larger than, the detuning betweenthe driving frequency and the qubit splitting o–o0. The LZS transitions occurbetween the transversely dressed states |k, nS and |m, n� 1S, where n refers to thephoton number of the transverse driving field. The phase difference of the twostates gathered between the consecutive tunnelling events leads to eitherconstructive or destructive interference observed as maxima or minima in theoccupation probability of the excited state, shown in Supplementary Fig. S4. TheLZS interference seen in our system corresponds to the so-called fast passagelimit21. In this limit, the expression for the occupation probability can be calculatedanalytically21 and the result agrees exactly with our equation (5).

To prove experimentally that this picture is valid, we have scanned themodulation amplitude d and the driving frequency o at fixed O. We observe aninterference pattern of the steady-state occupation probability, which is in goodagreement with the theoretical prediction, see Supplementary Fig. S4.

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AcknowledgementsWe thank S. Girvin and J. Viljas for discussions. This work was done under the Center ofExcellence ‘Low Temperature Quantum Phenomena and Devices’ (project 250280) of theAcademy of Finland. G.S.P. acknowledges support from the Academy of Finland, pro-jects 141559, 253094, and 135135. M.P.S. was supported by the Magnus EhrnroothFoundation and together with J.-M.P. by the Finnish Academy of Science and Letters(Vilho, Yrjo ja Kalle Vaisala Foundation). J.-M.P. acknowledges also the financial supportfrom Emil Aaltonen Foundation and KAUTE Foundation. The contribution of M.A.S.was done under an ERC Starting Grant. J.L. and M.P.S. acknowledge partial support fromNGSMP.

Author contributionsJ.L., K.S.K., and G.S.P. designed and performed the experiment. M.P.S. carried out thetheoretical work. A.V., W.C.C., and M.P.S. contributed to experiments. J.-M.P. fabricatedthe sample. J.T. and E.V.T. provided theoretical support. J.L. and M.P.S. co-wrote themanuscript in co-operation with all the authors. M.A.S., P.J.H., E.V.T. and G.S.P. pro-vided support and supervised the project.

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How to cite this article: Li. J. et al. Motional averaging in a superconducting qubit.Nat. Commun. 4:1420 doi: 10.1038/ncomms2383 (2013).

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