Floquet spectroscopy of a strongly driven quantum dot charge qubit with a microwaveresonator

J. V. Koski,1 A. J. Landig,1 A. Palyi,2, 3 P. Scarlino,1 C. Reichl,1 W.

Wegscheider,1 G. Burkard,4 A. Wallraff,1 K. Ensslin,1 and T. Ihn1

1Department of Physics, ETH Zurich, CH-8093 Zurich, Switzerland2Department of Physics, Budapest University of Technology and Economics, H-1111 Budapest, Hungary

3MTA-BME Exotic Quantum Phases ”Momentum” Research Group,Budapest University of Technology and Economics, H-1111 Budapest, Hungary4Department of Physics, University of Konstanz, D-78457 Konstanz, Germany

(Dated: February 13, 2018)

We experimentally investigate a strongly driven GaAs double quantum dot charge qubit weaklycoupled to a superconducting microwave resonator. The Floquet states emerging from strong drivingare probed by tracing the qubit - resonator resonance condition. This way we probe the resonanceof a qubit that is driven in an adiabatic, a non-adiabatic, or an intermediate rate showing distinctquantum features of multi-photon processes and Landau-Zener-Stuckelberg interference pattern.Our resonant detection scheme enables the investigation of novel features when the drive frequencyis comparable to the resonator frequency. Models based on adiabatic approximation, rotating waveapproximation, and Floquet theory explain our experimental observations.

Applying a strong drive to a quantum two-level system(qubit) gives rise to intricate physics, such as ac Starkeffect [1], multi-photon effects [2], and Landau-Zener-Stuckelberg interference [3], characterized by the emer-gence of Floquet states [4]. Many of these effects havepractical benefit: if the qubit is coupled to a supercon-ducting resonator, the ac Stark effect allows to calibratethe resonator photon number [5]. The Landau-Zener-Stuckelberg interference pattern gives relevant informa-tion on qubit decoherence [6–8], and carefully consideringthe Floquet dynamics provides means to improve the fi-delity of qubit operations [9]. Strong driving dynamicshave been investigated in various systems including su-perconducting qubits [6, 10–17] and quantum dot devices[8, 18–23]. Recent experiments have shown that stronglydriving a qubit that is coupled to a resonator can enhancethe resonator transmission [15, 24–26]. Spectroscopy ofthe Floquet quasienergies of a strongly driven systemhas been demonstrated by following its time-evolution[16, 27]. An alternative proposal is to probe the drivenqubit with a weakly coupled superconducting resonator[28, 29].

Here we report our experimental implementation of theproposal, where the resonance frequency of the resonatordetermines the probed Floquet quasienergy. Due toweak coupling, the resonator does not directly influencethe qubit energetics. We perform a set of experiments,first with adiabatic driving with which we observe multi-photon processes and Landau-Zener-Stuckelberg interfer-ence pattern similar as has been reported in other exper-iments [3, 8, 19]. In a second experiment we increase thedriving rate from adiabatic to non-adiabatic, allowing usto probe the evolution of the Floquet quasienergy. In ourthird experiment, we apply a near-resonant and a near-half-resonant drive. There we observe a vanishing of theprobe signal since the avoided crossing between the drive

field photons and the qubit energy eliminates the statesat the resonator energy. In this regime, all three energyscales present in our setup, i.e. those of the qubit, theresonator, and the drive photons, are relevant.

Our qubit is formed in a double quantum dot (DQD),shown in Fig. 1(a). Au top gates define the DQD elec-trostatically in a two-dimensional electron gas hosted ina GaAs/AlGaAs heterostructure. The number of elec-trons in the DQD is controlled with plunger gate poten-tials VL and VR and monitored by a nearby quantumpoint contact (QPC). As indicated in the charge sta-bility diagram Fig. 1 (b), the qubit is operated in thethree electron regime where the relevant charge statesare |L〉 = |(2, 1)〉 and |R〉 = |(1, 2)〉, with (n,m) notationindicating n electrons on the left and m electrons on theright dot. In the |L〉 - |R〉 basis, the system Hamiltonianis H0 = ∆σx/2 + δ0σz/2 where σx and σz are Pauli ma-trices, ∆ is the interdot tunnel coupling, and δ0 is theDC detuning energy between |L〉 and |R〉. We directlycontrol ∆ and δ0 with Vt, VL, and VR shown in Fig. 1 (a).Diagonalization of H0 determines the unperturbed qubitstates with an energy separation of εq,0 =

√∆2 + δ20 il-

lustrated in Fig. 1 (c).

The DQD is connected to a superconducting half-wavelength coplanar microwave resonator by extendingthe resonator voltage antinode to the drive gate indicatedin Fig. 1 (a), similarly as in previous work [30]. Thisleads to a coupling of the charge qubit to the resonatorelectric field characterized by Hint = g0σz(b

† + b)/2,where g0 is the zero-detuning (δ0 = 0) coupling strength,and b† is the resonator photon creation operator. Ourresonator has a resonance frequency νr = 8.32 GHz anda linewidth κ/2π = 110 MHz. By applying the methodsin [30], we estimate g0/2π ≈ 30 MHz and a qubit decoher-ence of γ2/2π ≈ 400 MHz for our device. Since g0 � γ2,the resonator is weakly coupled to the qubit and allows

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-80

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VL (mV)

VR (

mV

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VL (mV)

VR (

mV

)

FIG. 1. (a) A scanning electron micrograph of the device.Two quantum dots are defined in the positions indicated bythe orange dots. A superconducting resonator is connected tothe leftmost gate, and the qubit is driven by applying a con-tinuous tone to the rightmost gate. Unused gate electrodesare greyed out. (b) Stability diagram of the double quan-tum dot measured by the QPC. The black rectangle marksthe operation regime. (c) Charge qubit and resonator pho-ton energy as a function of δ0. The solid black line depictsthe qubit energy εq,0 =

√∆2 + δ20 . The minimum qubit en-

ergy is determined by the interdot tunnel coupling ∆ (blackdashed line). The blue line shows the cavity photon energyhνr. (d) Resonator transmission T as a function of VL andVR. The δ0 axis is depicted with an arrow.

weak probing without coherently influencing its states.When δ0 satisfies the resonance condition εq,0 = hνr, thequbit can absorb photons from the resonator. This isobserved as a decrease in transmission when probing theresonator at frequency νr. If ∆ < hνr as in Fig. 1(c), twoqubit-photon resonances occur for δ0 = ±

√(hνr)2 −∆2.

By sweeping the gate potentials across the (1, 2), (2, 1)charge transition as shown in Fig. 1(d), the two reso-nances are observed.

We drive the qubit by applying a continuous mi-crowave tone to the drive gate indicated in Fig. 1(a).This gives rise to a time-dependent detuning δ0 →δ0 + Ad cos(2πνdt) and consequently a time-dependentHamiltonian H(t) = H0 + Adσz cos(2πνdt)/2. Thedrive frequency νd has a significant effect on the qubit[3, 8, 10, 11]. As such, our experimental control parame-ters are the drive frequency νd and amplitude Ad as wellas the qubit detuning δ0 and tunnel coupling ∆.

In our first experiment, we explore the low drive fre-quency regime νd � ∆/h, νr. In this limit, the dynamics

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0 / h (GHz)

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(a)

(c) (d)

(e)

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(b)

∆ / h = 6.8 GHz

νd = 0.5 GHz

∆ / h = 6.8 GHz

νd = 1.5 GHz

∆ / h = 6.3 GHz

νd = 1.0 GHz

N = 0

N = -1

N = 1

N = 2

N = 4N = 5N = 6N = 7N = 8

N = 3

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20

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Ad /

h (

GH

z)

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8

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, ad

/ h

(G

Hz)

δ0 / h (GHz)

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0

0

∆ / h = 6.8 GHz

νd = 1.0 GHz

10

20

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h (

GH

z)

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-10 10δ

0 / h (GHz)

0

(f)

∆ / h

νr

FIG. 2. (a) The effective energy of an adiabatically drivenqubit as a function of δ0 with drive amplitudes Ad/h = 0GHz, 5 GHz, and ∼7 GHz in order of increasing energy (fromblack to red). (b) Illustation of the expected resonances asa function of δ0 and Ad for νd = 1 GHz and ∆/h = 6.8GHz. The resonance positions satisfying εq,ad = hνr +Nhνdare shown as blue lines with labels displaying the number ofphotons absorbed from the drive field. The stars in panels(a) and (b) mark the corresponding resonance positions. Thelines with unit slope emerging from the zero-Ad resonancesdefine the grey regions where resonances are not visible. (c)- (f) show the measured transmission as a function of δ0 andAd for low νd. The N = −1 resonance is indicated with adashed ellipse in panel (f). Yellow dots show the theoreticallypredicted resonance positions where one of the resonance con-ditions in Eq. (2) is fulfilled. The radius of each yellow dotis proportional to the corresponding transition strengthM inEq. (3).

are approximately adiabatic and the effective qubit en-ergy εq,ad is given by the time-averaged qubit energy

εq,ad = νd

∫ 1/νd

0

dt√

∆2 + (δ0 +Ad cos(2πνdt))2. (1)

εq,ad does not depend on νd and increases monotonicallywith increasing Ad. As illustrated in Figs. 2 (a)-(b),increasing Ad will trigger the resonance condition εq,ad =hνr for a different δ0 determined by the points where theenergy of driven qubit and that of the resonator intersect.

3

We also expect multi-photon resonances [31] satisfyinghνr = εq,ad −Nhνd, where N is an integer. As sketchedin Fig. 2 (b) for νd = 1.0 GHz, this results in replica ofthe N = 0 resonance that are visible when Ad > |N |hνd.

We now measure the transmission for νd = 0.5 GHz,νd = 1 GHz, and νd = 1.5 GHz, each with ∆/h = 6.8GHz, shown in Figs. 2(c)-(f). For zero drive amplitude,the qubit is resonant with the resonator at two detun-ing values δ0/h = ±

√ν2r − (∆/h)2 ' ±4.8 GHz, see Fig.

2(a). Increasing Ad changes the resonance condition forδ0 as illustrated in Fig. 2(b). With a sufficiently highdrive frequency as in Fig. 2 (e), the full N = 0 resonancearcs are clearly discernible, while for νd = 0.5 GHz andνd = 1.0 GHz, the resonance visibility vanishes for therange Ad

>∼ 4hνd, |δ0| >∼ 2hνd. This feature is also cap-tured by the predicted qubit visibility Eq. (3), discussedlater.

Multi-photon resonances matching εq,ad = hνr +Nhνdemerge as Ad is increased. Qualitatively, these processestake place by the qubit absorbing a single photon fromthe resonator, and N photons from the drive field. Forthese N + 1 photon processes, there are N gaps in theresonance arcs, symmetrically distributed around δ0 = 0as sketched in Fig. 2 (b). This effect is reminiscent ofLandau-Zener-Stuckelberg interference [3]. Aside the in-terference pattern, the multi-photon resonances are qual-itatively similar to that of the N = 0 resonance, includingthe vanishing visibility regimes with νd = 0.5 GHz andνd = 1.0 GHz. We further perform a measurement withνd = 1.0 GHz and ∆/h = 6.3 GHz shown in Fig. 2 (f) tofind a clear signature of a resonance corresponding toN = −1. This implies a process where the resonatorphoton is absorbed by the qubit and as a single photonby the drive field.

To describe our data more precisely, we use thequantum-electrodynamics interpretation of Floquet the-ory [4]. In this language, the coupled system of the chargequbit and its driving field is described by the Hamilto-nian [31] HF = H0 + hνdm+ Adσz(m− + m+)/4, wherethe basis is |L,m〉, |R,m〉, with m corresponding to thenumber of photons in the drive field, m =

∑mm|m〉〈m|

is the drive field photon number operator, and m+ =m†− =

∑m |m+1〉〈m|. We calculate numerically two Flo-

quet eigenstates |±〉 =∑m c±L,m|L,m〉+c

±R,m|R,m〉 with

corresponding quasienergies ε± that satisfy −hνd/2 <ε− ≤ ε+ ≤ hνd/2 [31]. All other eigenstate-quasienergypairs of HF are shifted replicas of these two, satisfying|±, n〉 = (m+)

n |±〉 and ε±,n = ε± + nhνd. The drivenqubit can absorb a resonator photon if

hνr = ε±,n − ε∓, (2)

where n is a non-negative integer. We locate numericallythe resonance positions (δ0, Ad) that satisfy one of theresonance conditions in Eq. (2), and show them in thetransmission plots in Figs. 2 (c)-(f), Fig. 3, and Fig. 4.Overall, the resonance positions match accurately with

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δ0 / h (GHz)

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d = 5.0 GHz

hνr hν

d

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Ad / h (GHz)

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(f)

0 2 4 6 8 10

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εRWA

εRWA

FIG. 3. (a)-(d) Measured transmission for increasing drivefrequency. Yellow dots show the resonance conditions inEq. (2), and their radius proportional to the correspond-ing transition strength M in Eq. (3). The black lines showthe RWA resonance conditions by Eq. (5), and orange lines(in panels (a) and (b)) show the adiabatic resonance con-ditions εq,ad = hνr with Eq. (1). (e) The energy of thestates |g, 0〉, |g, 1〉, and |e, 0〉 by RWA as a function of driveamplitude for parameters in (c). Black lines correspond toδ0/h = 0 GHz, and red lines to δ0/h = 4 GHz. The split-

ting ∆ε =√

(Ad sin(θ)/2)2 + (εq,0 − hνd)2 between |g, 1〉 and

|e, 0〉 is determined by Eq. (4). The resonance condition for|g, 0〉 and |e, 0〉 with hνr (dashed blue line) is marked by stars.(f) Corresponding plot for parameters in (d) and δ0/h = 6GHz.

the transmission minima seen in the experiment. In thespirit of Fermi’s Golden Rule, the strength M of thecorresponding transition is characterized by the squaredmatrix element of the qubit-probe coupling HamiltonianHint ∝ σz between the initial and final states [28],

M = |〈±, n|σz|∓〉|2 . (3)

In our second experiment, we move to the regime whereνd is comparable to our other frequency scales ∆/h and νrand concentrate on the N = 0 resonance. Figures 3 (a)-

4

(d) show the measured transmission as a function of δ0and Ad, displaying the evolution of the resonance patternas the configuration transitions from adiabatic to non-adiabatic regime. While Fig. 3(a) shows the arc-shapedresonance characterized by the adiabatic approximationEq. (1), for a higher drive frequency shown in Figs. 3(b)-(d), such approximation is no longer sufficient. Here, thesignatures of non-adiabaticity are the non-monotonicityof the resonance condition in Ad as a function of δ0 inFigs. 3 (b)-(c), and branching of Ad in Fig. 3 (d).

To interpret the high frequency drive data, we use a ro-tating wave approximation (RWA). We change from |L〉,|R〉 basis to the qubit eigenstate basis with Ad = 0, i.e.|g〉 = − sin(θ/2)|L〉 + cos(θ/2)|R〉, |e〉 = cos(θ/2)|L〉 +sin(θ/2)|R〉, where the mixing angle θ is given bycos(θ) = δ0/εq,0. We then limit our basis to two in-teracting states close in energy, such as |e,m = 0〉 and|g,m = 1〉, for which we can write the Hamiltonian as

HRWA =εq,02σz +

Ad4

sin(θ)(σ+m− + σ−m+) + hνdm,

(4)

where σ+ = σ†− = 12 (σx + iσy) is the qubit raising oper-

ator. As indicated in Fig. 3(e), the energy of |g,m = 1〉is offset from the energy of |g,m = 0〉 by hνd and wedetermine [31] the effective qubit energy from Eq. (4)

as εRWA = hνd ±√

(Ad sin(θ)/2)2

+ (εq,0 − hνd)2. The

resonance condition εRWA = hνr for Ad is then

Ad,RWA =2

∆εq,0

√(hνr − hνd)2 − (εq,0 − hνd)2. (5)

The data shown in Figs. 3(b) and (c) lie in theregime where we find a non-monotonic resonance con-dition as predicted by Eq. (5). Qualitatively, the dipin Ad,RWA at δ0 = 0 is due to a maximum in the qubit- drive photon coupling strength, which is proportionalto Ad sin(θ) = Ad∆/

√∆2 + δ20 as in Eq. (4), leading

to faster change in εRWA with increasing Ad as shownin Fig. 3(e). Figure 3(d) shows transmission data forνd = 12.0 GHz and ∆/h = 8.1 GHz. In this regime withνd > νr, we observe that the resonance condition forAd increases with increasing δ0. This can be understoodfrom Fig. 3(f), showing that when hνr < εq,0 < hνd, in-creasing Ad lowers the qubit energy and brings the qubiton resonance with the resonator. We find that both theRWA result in Eq. (5) and the transition strength by Eq.(3) give a good prediction for the resonance locations fordatasets shown in Figs. 3(b)-(d). We also note that ineach measurement shown in Fig. 3, the probe signal iscontinuously visible.

In our third experiment, νd is near-resonant or near-half-resonant with νr. Figure 4 (a) shows the transmis-sion for νd = 8.7 GHz, which is close to the resonatorfrequency νr ' 8.32 GHz. We observe that the trans-mission signal vanishes as Ad is increased, an effect thatcan be understood from the RWA Hamiltonian Eq. (4).

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Ad /

h (

GH

z)

δ0 / h (GHz) δ

0 / h (GHz)

(b)(a)

0

∆ / h = 6.3 GHzν

d = 8.7 GHz

∆ / h = 6.3 GHzν

d = 4.5 GHz

ε /

h (

GH

z)

0

-4

4

8

-8

hνr hν

d

|g,1

|e,0

|g,0

Ad / h (GHz)

0 2 4 6 8 10

hνr

hνd

∆ε

|g,1

|e,0

|g,0

∆ε

|e,-1

Ad / h (GHz)

ε /

h (

GH

z)

4

2

0

-2

-4

0 2 4 6 8 10

∆ε2

(d)(c)

-10 -5 0 5 10 -10 -5 0 5 10

εRWA

FIG. 4. Transmission with (a) near-resonant driving νd ' νrand (b) half-resonant driving νd ' νr/2 as a function of de-tuning and drive amplitude. Yellow dots show the resonanceconditions in Eq. (2), and their radius is proportional to thetransition strength in Eq. (3). The black lines show the RWAresonance conditions by Eq. (5). (c) Energy levels withδ0/h = 4 GHz. In panel (d), the RWA results are shownas dashed lines. Second order effects [31] form an avoidedcrossing ∆ε2 ∝ A2

d cos(θ) sin(θ) between the RWA states.

As illustrated in Fig. 4(c), when εq,0 ' hνd, the qubitenergy changes rapidly from hνd with increasing Ad dueto |e, 0〉 − |g, 1〉 hybridization. Therefore, if νd = νr, thedriven qubit cannot have an energy exactly matching hνr.Figure 4 (b) shows transmission for νd = 4.5 GHz whichclose to half νr. In this regime, the RWA result is nolonger sufficient to characterize the observed resonances.The special case of half-harmonic driving is discussed in[32]: here, the qubit is influenced by second order pho-ton processes where the hybridization gap arises when2νd ' εRWA/h, as illustrated in Fig. 4(d). However,second order coupling scales with A2

d cos(θ) sin(θ) [31],which tends to zero when δ0 → 0. This gives rise to thesmall range in δ0 where the qubit remains visible.

In conclusion, we have comprehensively investigatedFloquet energy spectra of a strongly driven charge qubitwith a weakly coupled microwave resonator as a functionof qubit detuning and drive amplitude over a large rangeof drive frequencies. In contrast to earlier experimentsstudying strongly driven systems, we have explored theregime where the qubit can be brought on resonance withthe resonator either by detuning, or by increasing driveamplitude. This feature allows to extract the Floquetquasienergy spectrum of a strongly driven charge qubit.The experiment could be extended towards measuring

5

the adiabatic phases of a doubly-driven qubit [29].

We thank Benedikt Kratochwil for a critical discussion.This work was supported by the Swiss National ScienceFoundation through the National Center of Competencein Research (NCCR) Quantum Science and Technology.AP was supported by the National Research Develop-ment and Innovation Office of Hungary within the Quan-tum Technology National Excellence Program (ProjectNo. 2017-1.2.1-NKP-2017-00001) and Grants 105149 and124723, and by the New National Excellence Programof the Ministry of Human Capacities. GB ackowledgesfunding from DFG within SFB 767.

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