Observing the escape of a driven quantum Josephson circuit into unconfined states

Raphael Lescanne,1, 2, ∗ Lucas Verney,2, 1 Quentin Ficheux,1, 3 Michel H.

Devoret,4 Benjamin Huard,3 Mazyar Mirrahimi,2, 5 and Zaki Leghtas6, 1, 2, †

1Laboratoire Pierre Aigrain, Ecole Normale Superieure,PSL Research University, CNRS, Universite Pierre et Marie Curie,

Sorbonne Universites, Universite Paris Diderot, Sorbonne Paris-Cite,24 rue Lhomond, 75231 Paris Cedex 05, France

2QUANTIC team, INRIA de Paris, 2 Rue Simone Iff, 75012 Paris, France3Laboratoire de Physique, Ecole Normale Superieure de Lyon, 46 allee d’Italie, 69364 Lyon Cedex 7, France

4Department of Applied Physics, Yale University, 15 Prospect St., New Haven, CT 06520, USA5Yale Quantum Institute, Yale University, 17 Hillhouse Av., New Haven, CT 06520, USA

6Centre Automatique et Systemes, Mines-ParisTech,PSL Research University, 60, bd Saint-Michel, 75006 Paris, France

(Dated: January 23, 2019)

Josephson circuits have been ideal systems to study complex non-linear dynamics which canlead to chaotic behavior and instabilities. More recently, Josephson circuits in the quantum regime,particularly in the presence of microwave drives, have demonstrated their ability to emulate a varietyof Hamiltonians that are useful for the processing of quantum information. In this paper we showthat these drives lead to an instability which results in the escape of the circuit mode into statesthat are not confined by the Josephson cosine potential. We observe this escape in a ubiquitouscircuit: a transmon embedded in a 3D cavity. When the transmon occupies these free-particle-likestates, the circuit behaves as though the junction had been removed, and all non-linearities are lost.This work deepens our understanding of strongly driven Josephson circuits, which is important forfundamental and application perspectives, such as the engineering of Hamiltonians by parametricpumping.

I. INTRODUCTION

Superconducting circuits are one of the leading plat-forms to implement quantum technologies. They hosthighly coherent electromagnetic modes whose parametersare engineered to fulfill a particular function. Joseph-son junctions (JJ) mediate non linear couplings betweenthe circuit modes which can be arranged in a variety oftopologies. By tailoring the type and strength of cou-pling, one can perform a multitude of tasks, such as am-plifying signals [1–3], generating non-classical light [4, 5],stabilizing single quantum states [6, 7] or manifolds [8],releasing and catching propagating modes [9, 10], andsimulating quantum systems [11]. The full in-situ con-trol of the couplings mediated by the Josephson non-linearity necessitates a key ingredient: a strong coherentmicrowave drive, referred to as a pump. Typically, thepump frequency is not resonant with any mode and in-stead satisfies a specific frequency matching condition inorder to select the desired coupling (so-called parametricpumping). In the low pump power regime, the systembehaves in a stable manner, and the coupling strengthincreases with the pump power.

However, as the pump power is increased to seekstronger couplings, this non-linear out-of-equilibriumopen quantum system can display complex dynamicsleading to the degradation of coherence times [8, 12] and

∗ raphael.lescanne@lpa.ens.fr† zaki.leghtas@mines-paristech.fr

instabilities [13]. This behavior is reminiscent of the dy-namics of RF current biased Josephson junctions, whichhave been thoroughly studied in the classical regime inthe context of the Josephson voltage standard [14]. Un-derstanding these dynamics in the quantum regime isessential, both from a fundamental and application per-spective. A particularly urgent matter is to guide us to-wards circuit designs which prevent instabilities, makingthem suitable for parametric pumping.

This work focuses on the transmon [15], a widely usedsuperconducting circuit because of its long coherencetimes. Usually, its first two energy levels are used asa qubit to encode quantum information. Actually, thetransmon has an infinite number of energy levels whichfall in two categories. First, those with energies smallerthan the Josephson energy and that are confined by theJosephson potential. Second, those which lie above thecosine potential which we refer to as unconfined states.Due to their high energies, unconfined states have alwaysbeen considered irrelevant for circuit dynamics and dis-regarded [16–21].

In this paper, we show that unconfined states play acentral role in the dynamics of strongly driven Josephsoncircuits. This time-periodic system has periodic orbitsknown as Floquet states. In the dissipative steady stateregime, the system converges to a statistical mixture ofFloquet states. For weak pump powers, the steady stateremains pure and occupies low energy confined states.Above a critical pump power, the steady state suddenlyjumps to a statistical mixture of many Floquet states,with a significant population on unconfined states. The

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dramatic change in the steady state when the pumppower is slightly increased above the threshold is a sig-nature of structural instability [13].

II. DESCRIPTION OF THE EXPERIMENT

Our device consists of a single transmon in a 3D coppercavity [22] coupled to a transmission line. The system iswell modeled by the circuit depicted in Fig. 1a, whoseHamiltonian is given by [15]

H(t) = 4ECN2 − EJ cos (θ) + ~ωaa†a+ ~gN

(a+ a†

)+ ~Ap(t)

(a+ a†

), (1)

whereN is the Cooper pair number operator and θ is thephase operator [23]. The operator cos (θ) is the trans-fer operator for Cooper pairs across the junction whilea† and a are the creation and annihilation operators forthe resonator mode. The Josephson and charging ener-gies are denoted EJ and EC , respectively. The angu-lar frequency of the resonator in the absence of the JJ(EJ taken to be 0) is denoted ωa and is known as thebare resonator frequency and g is the coupling rate be-tween the transmon and the resonator. Note that thisdefinition of g differs from the one usually used in theJaynes-Cummings Hamiltonian [24]. The pump couplescapacitively to the circuit, and is accounted for by thesecond line in (1), where Ap(t) = Ap cos(ωpt), Ap andωp being the pump amplitude and angular frequency.

Our physical picture of the dynamics of a pumpedtransmon in a cavity is the following. When the pumppower is sufficiently large, the transmon is excited aboveits bounded potential well (Fig. 1c,d). This is reminis-cent to an electron escaping from the bound states of theatomic Coulomb potential and leaving the atom ionized.The transmon can occupy highly excited states whichresemble charge states, or equivalently, plane waves inthe phase representation. A classical analogy would bea strongly driven pendulum which rotates indefinitely,making turns around its anchoring point (Fig. 1b). Whenoccupying such a state, the transmon behaves like a freeparticle which is almost not affected by the cosine po-tential. Thus the Josephson energy can be set to zero inEq. (1). The Hamiltonian is now that of a superconduct-ing island capacitively coupled to a resonator, with noJosephson junction. The island-resonator coupling termcommutes with the Hamiltonian of the island, which con-sists of the charging energy only. The coupling is nowlongitudinal and therefore the cavity frequency is fixedto the bare frequency ωa, independently of the islandstate.

The dynamics governed by Hamiltonian (1) is difficultto numerically simulate in the regime of large pump am-plitudes for two reasons. First, the simulated Hilbertspace needs to be large. Indeed, the number of trans-mon states needs to be much larger than the number ofstates in the well. For our experimental parameters we

find that about 45 states are necessary. Moreover, theresonator is also driven to large photon number states.Second, the pump amplitude is so large that we reachregimes where Ap � ωa. It is therefore excluded touse the rotating wave approximation in its usual form.These two difficulties are surmounted by the theory in-troduced in [13]. Using an adequate change of frame inwhich the resonator is close to the vacuum, we can de-crease the number of computed states to 10 for the res-onator, hence reducing the dimensionality of the Hilbertspace (Appendix VII E). Since H(t) is 2πω−1p -periodic,Floquet-Markov theory [26, 27] is well suited [28, 29] tofind the system steady state, as shown in Fig. 1c. Thistheory assumes weak coupling to the bath, but can treatpumps of arbitrary amplitude and frequency. For smallpump powers, the transmon remains close to its groundstate, and after the power is increased beyond a criticalvalue, it is driven into highly excited states, above itscosine potential (orange dashed line in Fig. 1c).

The parameters of Hamiltonian (1) are deduced frommeasurements involving a low number of excitations inthe system. In this regime, it is well approximated by theHamiltonian of two non-linearly coupled effective modes,one transmon qubit-like and one resonator-like, respec-tively denoted by subscripts q and r [30]:

Hlow(t)/~ = ωqa†qaq −

αq2

(a†q)2aq

2 − χqra†qaqa†rar

+ ωra†rar −

αr2

(a†r)2ar

2

+ Ap,r(t)(ar + a†r

), (2)

where ωq,r are the modes angular frequencies, αq,r theirKerr non-linearities, and χqr their dispersive coupling.In this basis, the pump is represented by the term inAp,r(t) = Ap,r cos(ωpt), with a renormalized amplitudeAp,r, where we neglect the linear coupling of the pumpto mode q.

We measure ωq/2π = 5.353 GHz, ωr/2π = 7.761 GHz,αq/2π = 173 MHz and χqr/2π = 5 MHz. In order tocompute EJ , EC , g and ωa, we write Hamiltonian (1)in matrix form in the charge basis for the transmon andthe Fock basis for the resonator. We calculate the energydifferences of the lowest energy levels, and fit the mea-sured values with: EC/h = 166 MHz, EJ/h = 23.3 GHz,g/2π = 179 MHz and ωa/2π = 7.739 GHz. We also inferαr/2π = 43 kHz.

III. OBSERVING THE TRANSMON ESCAPEINTO UNCONFINED STATES

We experimentally observe the transmon jump intohighly excited states by performing the spectroscopy ofthe resonator while the pump is applied (Fig. 2). Forsmall pump powers, the system is well described by thelow energy Hamiltonian (2). The pump populates theresonator with photons at the pump frequency with an

average number nr =∣∣∣ Ap,r/2i(ωr−ωp)+κr/2

∣∣∣2, and shifts the

3

FIG. 1. Principle of the experiment demonstrating the effect of a strong pump on a Josephson circuit. (a) A superconductingisland (within the dashed lines) is coupled to ground through a Josephson junction (JJ), which is shunted by a capacitor,forming a transmon mode (orange). It is capacitively coupled to an LC resonator (blue) which represents the lowest mode ofa 3D waveguide cavity. The system is pumped and probed through a transmission line (black). (b) Classical picture: Thetransmon is represented by a pendulum [15], where the deviation angle from equilibrium, denoted θ, is the phase differenceacross the JJ. The gravitational potential energy represents the Josephson energy. For small pump powers (left), the pendulumacquires a small kinetic energy and oscillates around its equilibrium. For large pump powers (right), it acquires sufficientenergy to escape from its trapping potential well, and rotates indefinitely. (c) Numerical simulation of the population (color)of the energy levels of the transmon (y-axis) as a function of the number of photons n (Appendix VII B) in the resonator dueto the pump (x-axis). For small n, the transmon is close to its ground state, except for small population bursts (see text).Above a critical n (here about 170 photons), the transmon jumps into highly excited states above its cosine potential (orangedashed line). (d) Quantum picture: cosine potential and energy levels (orange lines) for charge offset Ng = 0. Light orangeareas represent the band of levels appearing when varying Ng [25]. Large pump powers (big arrow) induce transitions from theground state |g〉 with a phase localized around θ = 0, into highly excited states with a delocalized phase.

resonance to ωr(nr) = ωr − 2αrnr [8, Supplement sec-tion 2.1]. We calibrate the photon number nr (Ap-pendix VII B) and, using the anharmonicity αr deducedfrom the model (2), we plot ωr(nr) (solid line in Fig. 2b).The data points match this prediction in the regime ofsmall photon numbers, over a wide range of pump fre-quencies. This linear dependence is disrupted by abruptfrequency jumps, as seen for a pump power of about30P1ph in Fig. 2a. This behavior suggests that the systemfrequencies, at this specific power, have shifted into res-onance with a process which excites the transmon. Thisresembles the population bursts observed for example at90 photons in the simulation of Fig. 1c, before the popu-lation jumps at around 170 photons.

Above a critical nr (of about 100 photons for the mostdetuned pump frequencies), the resonance jumps to anew frequency close to ωa/2π (arrow in Fig. 2b), whichis independent of the pump frequency and power. This is

the resonator frequency expected when the transmon ishighly excited so that the JJ behaves as an “open”. Sucha jump of the resonance has been used for qubit readout[31] and modeled in previous works with a strong drivewhich plays the role of the pump and the probe, appliedclose to resonance with the resonator. Those models in-volve two-level [17] and multi-level [18] approximationsfor the transmon. Such approximations are incompat-ible with our following measurements. In recent work[28, 29], numerical simulations suggest that the resonatorfrequency jump is accompanied by an increase in thetransmon population. In the next paragraphs, we exper-imentally measure a sudden population increase whichcorresponds to the jump of the transmon state out of itspotential well into unconfined states. In order to modelthis effect, it is crucial to consider all levels confined inthe cosine potential and a number of levels above whichdepends on the pump power.

4

7.73 7.74 7.75 7.76Probe frequency (GHz)

0

50

100

150

200

Pum

p po

wer/P

1ph

(a)

7.73 7.74 7.75 7.76Resonance frequency (GHz)0

100

200

Aver

age

phot

on n

umbe

r (nr)

Pump frequency (GHz)

7.47.7397.7458.18.3(ωr − 2nrαr)/2π

200040006000

ωa/2π(b)

0.1

0.2

0.3

0.4

0.5

0.60.70.80.91.0

Rela

tive

refle

cted

pow

er

FIG. 2. Effect of the pump on the cavity resonance frequency. (a) Relative reflected power of a weak probe as a function ofprobe frequency (x-axis), and pump power (y-axis) in units of P1ph, the power needed to populate the cavity with one photonin average. The pump frequency is fixed at 8.1 GHz, about 300 MHz above the cavity frequency. The reduced reflected poweris due to internal losses when the probe is resonant with the cavity. We indicate the fitted cavity resonance ωr/2π with whitediamonds. As the pump power increases, two regimes are distinguishable. For small powers, ωr shifts linearly with the pumppower. Above a critical power, the cavity resonance jumps to a new frequency ωa/2π = 7.739 GHz which is independentof pump power. (b) Fitted cavity resonance as a function of pump power for various pump frequencies. Pump powers areconverted into a mean number of photons nr (Appendix VII B). The y-axis of (a) and (b) slightly differ because nr takesinto account the resonator frequency shift. The general behavior is the same for all pump frequencies. The low power lineardependence is well reproduced by the AC Stark shift [8] for an independently measured Kerr αr (solid gray line).

For states well confined inside the cosine potential, thetransmon state is encoded in the resonance frequency ofthe resonator. As seen from Hamiltonian (2), each ad-ditional excitation in the transmon pulls the resonatorfrequency to ωr(nq) = ωr−nqχqr (Fig. 3a). However, forstates above the cosine potential, the resonator adopts afrequency ωa which is now independent of the particulartransmon state.

The highly excited nature of the transmon can be con-firmed by turning off the pump, and waiting for the trans-mon to decay to lower lying states which can be resolvedby probing the resonator frequency. In this experiment,the highest of these states was the sixth excited state|e6〉. As expected, after the pump is turned off (t = 0 inFig. 3b), we observe a population decay of the transmoncompatible with a highly excited state at t ≤ 0.

IV. EFFECT OF PUMP-INDUCEDQUASIPARTICLES

Strongly pumping a superconducting circuit generatesnon-equilibrium quasiparticles [32, 33], which poses thequestion: what is the influence of quasiparticles on thisdynamics? We detect whether quasiparticles have beengenerated by the pump pulse by measuring the trans-mon T1 after a time delay large enough to ensure thatall modes have decayed back to their ground states. Wefind that for the pump frequency and power used for theexperiment of Fig. 3b, the T1 is unaffected (Fig. 8). Thisindicates that the pump can expel the transmon out of itspotential well, without generating a measurable amountof quasiparticles, and therefore the simulation of Fig. 1c,which does not include quasiparticles, is relevant.

In order to observe a measurable decrease in T1, weneed to insert a much larger number of photons in theresonator. We enter this regime by applying a pump atfrequency ωa, so that the pump is resonant with the res-onator at large powers. As shown in Fig. 4b, the T1 dropsat high pump powers indicating that we have successfully

5

7.730 7.735 7.740 7.745 7.750 7.755 7.760 7.765 7.770Probe frequency (GHz)

0

10

20

30

40

50

60

Pum

p-pr

obe

del

ay ti

me

(s)

nr=560(b)

ωa/2π

0.0

0.25

0.50

0.75

Rela

tive

refle

cted

pow

er|g⟩|e1

⟩|e2

⟩|e3

⟩|e4

⟩|e5

⟩|e6

⟩

(a)

datafit

0.1

0.2

0.3

0.4

0.50.60.70.80.91.0

Rela

tive

refle

cted

pow

er

FIG. 3. Probing the transmon decay out of highly excited states populated by a strong off-resonant pump. (a) Cavityspectroscopy after the transmon is prepared in various eigenstates. Each transmon state |ek〉 (k =1 to 6) is prepared using ak-photon π-pulse (Fig. 6) and higher states could not be prepared due to charge noise (Appendix VII D). (b) Time resolvedmeasurement: first, a pump is applied for 50 µs at 8.1 GHz populating the cavity with a sufficiently large photon number toinduce a jump in ωr, here nr = 560. After a time-delay t, a weak probe is applied for 2 µs. We plot the relative reflected powerof the probe as a function of probe frequency (x-axis), and time-delay t (y-axis). For t < 0, as in Fig. 2, the cavity is probedwhile the pump is on, and the resonance frequency is ωa/2π, confirming that the system has jumped. At t > 0, the photonsin the cavity rapidly decay at a rate κr = 1/(55 ns), and the cavity can now be used, as in Fig. 3a, to read the transmonstate. The transmon is found in a highly excited state, and after t = 60 µs, it has fully decayed into its ground state which iscompatible with T1 = 14 µs.

generated quasiparticles. Furthermore, we find that thetransmon T1 recovers its nominal value after a few mil-liseconds, a typical timescale for quasiparticle recombina-tion [32, 33]. In Fig. 4a, we probe the transmon dynamicsas in Fig. 3b, in the presence of pump-induced quasi-particles. The increased quasiparticle density does notqualitatively modify the transmon dynamics, but onlyincreases the decay rate back to its ground state.

V. CONCLUSION

In conclusion, we have measured the dynamics of atransmon in a cavity in the presence of a strong off-resonant pump. Our measurements are in qualitativeagreement with our theory describing the transmon es-caping out of its potential well. The transmon then be-haves like a free particle which no longer induces anynon-linearity on the cavity. This escape, which is dueto the boundedness of the cosine potential, could be pre-vented by adding an unbounded parabolic potential, pro-vided by an inductive shunt. One could then further in-

crease the non-linear couplings by increasing the pumppower, without driving the transmon into free particlestates and losing the resource of non-linearity. Our re-search points towards inductively shunted JJs [34] as bet-ter suited than standard transmons for pump-inducednon-linear couplings [13].

VI. ACKNOWLEDGMENTS

The authors thank Pierre Rouchon, Takis Kontos,Benoit Doucot and Caglar Girit for fruitful discussionsand Matthieu Dartiailh for developing the instrumentcontrol software [35]. This work was supported by theANR grant ENDURANCE, and the EMERGENCESgrant ENDURANCE of Ville de Paris. The devices werefabricated within the consortium Salle Blanche ParisCentre. MHD acknowledges ARO support.

6

(1) (2) (3) (4) (5) (6) (7)100

101

T1 (

s)

(b)

0

20

40

60(a) nr 1

(1)

nr=35 or nr=6800

(2)

nr=52 or nr=9000

(3)

nr=11000

(4)

7.73 7.760

20

40

60nr=16000

(5)7.73 7.76

nr=22000

(6)7.73 7.76

nr=45000

(7)7.73 7.76

nr=67000

(8)

0.1

0.2

0.30.4

0.60.81.0

Rela

tive

refle

cted

pow

er

Probe frequency (GHz)

Pum

p-pr

obe

dela

y tim

e (

s)

FIG. 4. Transmon relaxation following a pump pulse of increasing power. (a) Same experiment as in Fig. 3b with increasingpump power (a1 to a8). The pump frequency is fixed at fp = ωa/2π = 7.739 GHz in order to populate the cavity with a largenumber of photons. Each pump power is converted into the mean number of photons nr in the cavity. The conversion factordepends on the observed cavity frequency at t ≤ 0, indicated by the arrows (Appendix VII B). (a1) The pump power is too lowto induce a shift or a jump in the cavity frequency. (a2-a3) At intermediate pump powers, the system is in a statistical mixtureof two configurations, as indicated by the two arrows at t < 0. First, a weakly shifted resonance due to the AC Stark shift witha small cavity population. Second, a resonance at ωa coinciding with the pump frequency and hence a large cavity population.(a4-a5) At higher pump powers, the cavity frequency always jumps to ωa and the transmon decays as in Fig. 3b. (a6-a8) Ateven higher pump powers, the transmon decay rate increases significantly. (b) Transmon T1 (y-axis) measured 50 µs after apump is applied. The pump duration and powers (x-axis) correspond to the ones used for the experiments shown in (a). Weobserve a decrease in T1 as the pump power increases, which is consistent with an increasing quasi-particle density induced bythe pump [32, 33], and explains the significantly higher decay rates in (a6-a8). For pump power (3), T1 is marginally reduced,while the system dynamics is highly affected (a3). The T1 corresponding to (a8) was too small to be accurately measured.

VII. APPENDIX

A. Sample fabrication

The sample measured in this work is the same oneas in [36]. The transmon is made of a single aluminumJosephson junction and is embedded in a copper cavityof 26.5 × 26.5 × 9.5 mm3 thermalized on the base plateof a dilution fridge at about 10 mK. More details can befound in [36, Sections 1.A and 1.B].

B. Photon number calibration

In this section, we explain how we convert a pumppower in mW, to an average number of photons atthe pump frequency inside the cavity. We exploit theAC Stark shift and measurement induced-dephasing [37],where a drive on the resonator inserts photons which de-

tune and dephase the qubit. We use a 2-level approxi-mation of the low energy Hamiltonian from Eq. (2):

H(2)low/~ = ωq

σz2

+ ωra†rar − χqr |e〉 〈e|a†rar

+ Ap,r cos(ωpt)(ar + a†r

)(3)

where σz = |e〉 〈e|−|g〉 〈g|, and |g〉 , |e〉 are the transmon’sground and first excited states. Moving into a framerotating at the pump frequency ωp for the resonator, andthe qubit frequency ωq for the qubit, and performing therotating wave approximation (RWA), we get

H(2)low/~ = ∆a†rar − χqr |e〉 〈e|a†rar

+ Ap,rar/2 +Ap,ra†r/2 ,

where ∆ = ωr − ωp.Depending on the qubit state |g〉 [resp: |e〉], the res-

onator reaches a steady state which is a coherent state of

7

0.000.250.500.751.001.251.501.752.00

ωp/2π = 7.4 GHzPp,max = 1.0x10 ¹ mW

(a)

ωp/2π = 7.739 GHzPp,max = 2.0x10 mW

(b)

30 25 20 15 10 5 00.000.250.500.751.001.251.501.752.00

ωp/2π = 7.762 GHzPp,max = 5.7x10 mW

(d)

30 25 20 15 10 5 0

ωp/2π = 7.8 GHzPp,max = 1.5x10 ³ mW

(e)

30 25 20 15 10 5 0

ωp/2π = 8.1 GHzPp,max = 1.3x10 ¹ mW

(f)

0

Pp,max

Linea

rly in

crea

sing

powe

r

data fit

ωp/2π = 7.753 GHzPp,max = 5.0x10 mW

(c)

Re(δtot) (MHz)

−Im

(δto

t) (M

Hz)

FIG. 5. Calibration to convert the pump power (mW) into an average number of photons nr, using the AC Stark shift andmeasurement-induced dephasing [37]. By performing a Ramsey experiment in the presence of a pump at frequency ωp andpower Pp, we measure the detuning between the drive and the qubit δq(ωp, Pp), and the qubit dephasing rate γφ(ωp, Pp). Weintroduce δtot = δq− iγφ. For each plot (a) to (f), the pump frequency is fixed, and γtot is plotted in the complex plane (hollowcircles), for increasing Pp (blue is 0 mW, red is Pp,max). For each pump frequency, the data points are fitted with Eq. (5), whereδm is given in Eq. (4), and the constant C is the only fitting parameter. All other parameters: ∆, κr, χqr, Pp, δ0 are measuredindependently. (a),(f) Large detuning between the pump and the cavity. The pump mostly induces a detuning on the qubit,without inducing any dephasing. (d) The pump is resonant with the cavity, and mostly induces dephasing on the qubit. (b),(c), (e) Intermediate regimes where the pump both dephases and detunes the qubit.

amplitude αg [resp: αe], where:

αg =−iAp,r/2i∆ + κr/2

αe =−iAp,r/2

i(∆− χqr) + κr/2.

The measurement induced complex detuning is [37]

δm = −χqrαeα∗g

= −χqrA2p,r/4

(i(∆− χqr) + κr/2) (−i∆ + κr/2),

Re(δm) being the AC Stark frequency shift, and -Im(δm)being the measurement induced dephasing rate. Thequantity A2

p,r is proportional to the output power of ourinstrument Pp, and hence we write it in the followingform: A2

p,r = κ2rCPp, where C is the proportionality con-stant we aim to calibrate and is expressed in a numberof photons per mW. Since our microwave input lines donot have a perfectly flat transmission over a GHz range,C can be dependent on ωp. We get:

δm(ωp, Pp) = −C κ2rχqrPp/4

(i(∆− χqr) + κr/2) (−i∆ + κr/2).

(4)For each pump frequency ωp, we vary the pump power

Pp between 0 and Pp,max, and perform a T2 Ramsey

experiment. The Ramsey fringes oscillation frequencyand exponential decay provide a complex frequency shiftδtot, such that Re(δtot) is the oscillation frequency, and-Im(δtot) is the dephasing rate. We have

δtot(ωp, Pp) = δ0 + δm(ωp, Pp) , (5)

where δ0 is due to the chosen drive-qubit detuning andto the qubit dephasing rate in the absence of the pump.Thus, we measure δtot, and the only unknown is the con-stant C, which we fit at each pump frequency. In Fig. 5,we plot for each pump frequency, δtot (open circles), andthe fitted values (crosses), where C is the only fitting pa-rameter. Once we know C, the average photon numberis given by

nr(ωp, Pp) =

∣∣∣∣ −iAp,r/2i(ωr − ωp) + κr/2

∣∣∣∣2 (6)

= CPpκ2r/4

(ωr − ωp)2 + κ2r/4, (7)

where ωr is the measured resonance frequency which it-self depends on ωp and Pp. We use Eq. (7) to calculateall the given values of nr, and for the y-axis of Fig. 2b.

8

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.5

1.0

1.5

2.0

∆f = 9.9x10 ² MHz

(b) |g⟩ |e1⟩

δ2 − δ12π

= 0.1 MHz(1)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.2

0.4

0.6

0.8

∆f = 0.20 MHz

|g⟩ |e2⟩

δ2 − δ12π

= 0.1 MHz(2)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

∆f = 0.30 MHz

|g⟩ |e3⟩

δ2 − δ12π

= 0.1 MHz(3)

1.0 1.5 2.0 2.5 3.0 3.5 4.0

0.0

0.1

0.2

0.3

0.4

∆f = 0.40 MHz

|g⟩ |e4⟩

δ2 − δ12π

= 0.1 MHz(4)

2.5 5.0 7.5 10.0 12.5 15.0

0.0

0.5

1.0

1.5

∆f = 0.51±0.01 MHz

|g⟩ |e5⟩

δ2 − δ12π

= 0.1 MHz(5)

5 10 15 20

0.0

0.1

0.2

0.3

0.4

∆f = 0.6±0.1 MHz

|g⟩ |e6⟩

δ2 − δ12π

= 0.1 MHz(6)

−π/2 0 π/2

θ

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2En

ergy

/EJ

|g⟩

|e1⟩

|e2⟩

|e3⟩

ω0k

2πk

|e4⟩

|e5⟩

|e6⟩

fq

(a)

FT frequency (MHz)

FT a

mpl

itude

FIG. 6. Characterization of k−photon transitions. (a) Bottom of the cosine potential well of the transmon (grey line) in unitsof EJ (y-axis), as a function of the phase across the junction (x-axis). The ground state |g〉 and the first six excited states|ek=1...6〉 are represented (black horizontal lines). Each excited state |ek〉 is prepared with a k-photon transition (arrows) with adrive at ωd = ω0k/k (see text). (b1)-(b6) Fourier transform (FT) of a Ramsey signal (dots) for two different drive detunings δ1and δ2 = δ1 + 2π× 0.1 MHz. The data is fitted (line) by the Fourier transform of an exponentially decaying cosine function (asum of two cosines is needed for (b6)), which provides the detuning between the k-photon drive with effective frequency kωd andthe transition frequency ω0k. We verify that the difference between these detunings ∆f scales as k (δ2 − δ1) /2π, as expectedfor a k−photon transition. Unlike (b1)-(b5), in (b6) we observe two peaks for each curve. This shows that the Ramsey signalhas two frequency components, which is a consequence of quasi-particles tunneling across the junction (see text and Fig.7).

C. multi-photon qubit drive

In the main paper, we show that at large pump pow-ers, when the resonator frequency jumps, the transmonis in a highly excited state. We observe the decay of thetransmon to its ground state (Fig. 3b), and verify thatthe measured resonances coincide with the resonator fre-quency when the transmon is prepared in excited state|ek〉 (Fig. 3a). In order to drive the transmon from itsground state into state |ek〉, we apply a strong pulse atfrequency ω0k/k where

ω0k =Ek − E0

~. (8)

Ek is the eigenenergy of eigenstate |ek〉, and E0 is theeigenenergy of the ground state |g〉. This induces a k-photon transition between |g〉 and |ek〉, as schematicallyrepresented in Fig. 6b. This k−photon transition is thenused to perform all standard qubit experiments, such asRabi oscillations between |g〉 and |ek〉, T2 Ramsey andT1 measurements. We verify that this is indeed a k-photon transition by performing a Ramsey measurement:two π/2 rotations with a drive at frequency ωd slightlydetuned from ω0k/k, separated by a varying time, andfollowed by the measurement of σz. We obtain the os-cillation frequency of the Ramsey fringes by taking theFourier transform of the measurement (Fig. 6). If we de-tune the drive frequency by δ: ωd = ω0k/k− δ, then kωd

is detuned from the resonance ω0k by kδ. We performa first Ramsey experiment with a drive detuned by δ1,which serves as a reference. A second Ramsey experimentis performed with a detuning δ2 = δ1 + 2π × 0.1 MHz.By subtracting the frequencies of the Ramsey fringes,denoted ∆f , we get ∆f = k (δ2 − δ1) /2π, as shown inFig. 6b1-b6.

D. Charge noise

Observing Fig. 6b, we see that the Ramsey signal isnot composed of a single frequency. This is reminiscentof charge noise, where a single quasiparticle is tunnel-ing across the junction in and out of the superconduct-ing island, thus changing its charge parity. This paritychange corresponds to changes between Ng and Ng±1/2,which occur on the millisecond timescale [38]. Our Ram-sey measurements take 16 seconds, hence our data showsthe average of these charge offset configurations. Notethat despite the fact that we are deep in the transmonregime: EJ/EC = 140, the higher energy levels dispersewith charge offset.

We confirm this effect by plotting the Fourier trans-form of the Ramsey signal for the |g〉 to |e6〉 transitionvs. time (Fig. 7). We see the slow symmetric drift of thetwo frequencies, due to the background charge motion[38].

9

The following energy level (|e7〉 and above) fluctuatemuch more than |e6〉 with charge offsets, and this is whywe could not prepare the transmon in states above |e6〉.

E. Numerical simulation

In this section, we describe the numerical simulationof Fig. 1c. The details of this simulation can be foundin [13], and for completeness, we explain the main prin-ciples here. We denote ρ(t) the density operator ofthe transmon-resonator system. This system’s dynam-ics are driven by the time dependent Hamiltonian H(t)of Eq. (1), and by a coupling to a bath through an op-erator X. Typically, X = a + a†, and the bath is atransmission line. For simplicity, we neglect the directcoupling of the transmon to the bath, but this can easilybe included in the theory by changing X. We recall thatH(t) is periodic with period Tp = 2π/ωp, where ωp is thepump frequency. The goal is to find ρ(t) in the infinitetime limit.

After a sufficiently long time, ρ(t) reaches a limit-cycle:a Tp-periodic trajectory denoted ρ(t). We find this limit-cycle using Floquet-Markov theory [26]. This theory as-sumes a weak coupling to the bath, but can treat pumpsof arbitrary amplitudes and frequencies.

a. Change of frame: The first step is to write theHamiltonian in a frame where the resonator is close tothe vacuum state. We do this by solving the quantumLangevin equations when EJ = 0. We will use the fol-lowing useful formulas [39, (4.12)]:

[θ,N ] = i ,

and hence for any analytic function f

[θ, f(N)] = i∂f

∂N(N) ,

and also

[a, f(a†)] =∂f

∂a†(a†) .

We denote a0,N0,θ0 the solutions of the Langevin equa-tions associated to Hamiltonian (1) with EJ = 0. Wefind:

d

dta0 =

1

i~[a0,H] = −iω0a0 − igN0 − iAp(t)

d

dtN0 =

1

i~[N0,H] = 0

d

dtθ0 =

1

i~[θ0,H] = 8EC/~N0 + g(a0 + a†0) ,

where Ap(t) = Ap cos(ωpt), and Ap ≥ 0. These equationsare linear, and are therefore easy to solve. We introducethe expectation values: 〈a0〉 = a0, 〈N0〉 = N0, 〈θ0〉 = θ0.

Using some approximations [13], we find

a0(t) =Ap/2

(ωp − ωa)e−iωpt

N0(t) = 0

θ0(t) = gAp

ωp(ωp − ωa)sin(ωpt) .

We define

n =

∣∣∣∣ Ap/2

ωp − ωa

∣∣∣∣2 . (9)

For simplicity we assume ωp > ωa. If ωp < ωa then oneneeds to multiply the following solutions by −1. We get

a0(t) =√ne−iωpt

N0(t) = 0 (10)

θ0(t) = 2g

ωp

√n sin(ωpt) .

Comparing Eq. (9), and Eq. (7), we see that the expres-sions of n and nr are very similar, and differ only by thepresence of subscripts r, and the term in κr in Eq. (7).In the limit of large detunings: |ωr − ωp| � κr, andhence the term in κr can be neglected. The pump ampli-tude Ap,r slightly differs from Ap due to the transmon-resonator mode hybridization. For our experimental pa-rameters, Ap,r and Ap differ by less than 1%. Themeasured resonator frequency ωr roughly varies betweenωr and ωa, spanning about 20 MHz. For the simu-lation of Fig. 1b, we consider a pump at frequencyωp/2π = 8.1 GHz, which is detuned from the resonatorby about 340 MHz. This detuning is much larger thanthe frequency variation of ωr. Hence n and nr differ byless than 15%.

We now move to a frame around the solutions givenin Eq. (10), introducing θ = θ − θ0, N = N − N0 anda = a−a0. This frame is chosen such that the deviationof the resonator from the vacuum is small, and hence wecan use a low truncation for the resonator mode. Thischange of frame leads to the following Hamiltonian [13]:

H = 4ECN2 − EJ cos

(θ + θ0(t)

)+ ~ωaa†a+ ~gN

(a+ a†

)= 4ECN

2 − EJ(

cos(θ) cos(θ0(t))− sin(θ) sin(θ0(t)))

+ ~ωaa†a+ ~gN(a+ a†

)We write H as a matrix in the basis spanned by

|Tk〉 ⊗ |Fl〉, where |Tk〉 are eigenstates of the transmon

Hamitlonian 4ECN2 − EJ cos

(θ)

, and |Fl〉 are Fock

states: eigenstates of the resonator Hamiltonian ~ωaa†a.We truncate this basis to M = 450 states, with 45 statesfor the transmon and 10 states for the resonator.

10

b. Floquet analysis: Using functions provided by theQuantum Toolbox in Python (Qutip), we compute the

Floquet states of H(t), which we denote |ψα(t)〉 for α =1, . . . ,M . We can then calculate the dissipation operatorX matrix elements in the Floquet basis |ψα〉 [26, (245)].

With these matrix elements, we can now calculatethe master equation followed by the density matrix [26,(251)], and compute its steady state in the Floquet ba-sis: ρ(t) =

∑α ραα |ψα(t)〉 〈ψα(t)|. We recall that ρ(t)

is Tp-periodic and we define ρ0 = ρ(0). We can expressρ0 in the basis {|Tk〉 ⊗ |Fl〉}k,l, and calculate the par-tial trace with respect to the cavity ρ0q = Trcav(ρ0) =∑l 〈Fl| ρ0 |Fl〉. The diagonal of ρ0q: diag(ρ0q) is the list

of the transmon eigenstate populations. Note that attimes tm = 0+mTp, θ0(tm) = 0, hence the unitary whichtransforms states between the displaced frame back tothe original one is the identity for the transmon degreeof freedom. In Fig. 1c, we plot diag(ρ0q) as a function of

n, where n given in Eq. (9).

F. Quasiparticle generation

In section IV, we discuss whether the strong pumpwe apply on the system generates quasiparticles. If thepump significantly increases the quasiparticle density, thetransmon T1 will decrease [32, 33]. In Fig. 8a, we repro-duce the experiment of Fig. 3b for various pump powers,which correspond to various numbers of photons nr. Forsmall nr, the transmon is not excited, while for large nr,the transmon is driven to highly excited states. Addi-tionally, after each pump pulse, we measure the trans-mon T1 after a time delay chosen so that all modes havedecayed back to their ground state. We find no decreasein T1 (Fig. 8b) which means that the pump can expel thetransmon out of its potential well without producing ameasurable amount of quasiparticles.

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12

0 20 40 60 80 100 120Time (min)

2.5

5.0

7.5

10.0

12.5

15.0

17.5

20.0

FT fr

eque

ncy

(MHz

)

FIG. 7. Fourier transform (FT) of the Ramsey signal of the |g〉 to |e6〉 transition, measured repeatedly every 32 seconds overa period of about 2 hours. Due to quasiparticle tunneling, two frequencies appear in each Ramsey signal. These frequenciesdrift over a timescale of several minutes due to background charge motion [38].

(1) (2) (3) (4) (5)100

101

T1 (

s) (b)

7.73 7.750

20

40

60

Pum

p-pr

obe

del

ay ti

me

(s)

(a)nr¿ 1

(1)7.73 7.75

nr=62

(2)7.73 7.76

nr=120 or nr=110

(3)7.73 7.76

nr=220

(4)7.73 7.76

nr=560

(5)0.1

0.2

0.30.4

0.60.81.0

Rela

tive

refle

cted

pow

er

Probe frequency (GHz)

FIG. 8. Transmon relaxation following a pump pulse of increasing power. (a) Same experiment as in Fig. 3b for increasingpump power from a1 to a5, a5 corresponding to the maximum power the microwave source could deliver. (a1-a2) The pumpinduces an AC Stark shift proportional to nr at t < 0 but does not significantly excite the transmon. (a3) At intermediate pumppowers, the system is in a statistical mixture of two configurations, as indicated by the two dips at t < 0. One corresponds tothe AC Stark shifted cavity, the other to the bare cavity frequency ω indicating that the transmon gets highly excited. (a4-a5)At higher pump powers, the cavity frequency always jumps to ω and the transmon decays as in Fig. 3b. (b) Transmon T1

(y-axis) measured 50 µs after a pump is applied. The pump duration and powers (x-axis) correspond to the ones used for theexperiments shown in (a). T1 does not decrease as the pump power increases which indicates that the pump does not generatea measurable increase in the quasiparticle density [32, 33].