Transmon-qubit readout using in-situ bifurcation amplification in the mesoscopic regime R. Dassonneville1 2T. Ramos3V. Milchakov1C. Mori1L. Planat1F. Foroughi1

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Transmon-qubit readout using in-situ bifurcation ampliﬁcation in the mesoscopic

regime

R. Dassonneville,1, 2 T. Ramos,3V. Milchakov,1C. Mori,1L. Planat,1F. Foroughi,1

C. Naud,1W. Hasch-Guichard,1J. J. Garc´ıa-Ripoll,3N. Roch,1and O. Buisson1

1Univ. Grenoble-Alpes, CNRS, Grenoble INP, Institut N´eel, 38000 Grenoble, France

2Aix Marseille Univ., CNRS, IM2NP, Marseille, France

3Institute of Fundamental Physics, IFF-CSIC, Calle Serrano 113b, 28006 Madrid, Spain

(Dated: January 25, 2024)

We demonstrate a transmon qubit readout based on the nonlinear response to a drive of polaritonic

meters in-situ coupled to the qubit. Inside a 3D readout cavity, we place a transmon molecule

consisting of a transmon qubit and an ancilla mode interacting via non-perturbative cross-Kerr

coupling. The cavity couples strongly only to the ancilla mode, leading to hybridized lower and

upper polaritonic meters. Both polaritons are anharmonic and dissipative, as they inherit a self-

Kerr nonlinearity Ufrom the ancilla and eﬀective decay κfrom the open cavity. Via the ancilla, the

polariton meters also inherit the non-perturbative cross-Kerr coupling to the qubit. This results in

a high qubit-dependent displacement 2χ > κ, U that can be read out via the cavity without causing

Purcell decay. Moreover, the polariton meters, being nonlinear resonators, present bistability, and

bifurcation behavior when the probing power increases. In this work, we focus on the bifurcation

at low power in the few-photon regime, called the mesoscopic regime, which is accessible when the

self-Kerr and decay rates of the polariton meter are similar U∼κ. Capitalizing on a latching

mechanism by bifurcation, the readout is sensitive to transmon qubit relaxation error only in the

ﬁrst tens of nanoseconds. We thus report a single-shot ﬁdelity of 98.6 % while having an integration

time of a 500 ns and no requirement for an external quantum-limited ampliﬁer.

I. INTRODUCTION

Qubit state readout is a mandatory step in quantum

information processing. For superconducting circuits,

the dispersive readout is the standard scheme [1,2]. It

relies on the transverse interaction between an anhar-

monic mode, whose ﬁrst two levels are used as a qubit,

and another mode, usually harmonic, used as a meter

[3,4]. This transverse interaction couples the qubit po-

larization to the meter ﬁeld quadrature and hybridizes

the qubit with the meter. In perturbation theory, the re-

sulting dispersive interaction (or perturbative cross-Kerr

coupling) corresponds in ﬁrst order to an energy-energy

interaction where the qubit state shifts the meter fre-

quency and reciprocally, the number of photons in the

meter shifts the qubit frequency. When applying a coher-

ent pulse close to the meter frequency for a time smaller

than the relaxation time of the qubit T1, the qubit state

is inferred by distinguishing, in phase-space of the ac-

quired output ﬁeld, the two pointer states of the meter

corresponding to the qubit excited or ground states. Us-

ing this dispersive readout, single-shot readout with high

ﬁdelity is nowadays routinely achieved, notably thanks to

quantum-limited Josephson Parametric Ampliﬁer (JPA)

[5]. However, the dispersive interaction contains intrinsic

limitations, due to the higher order corrections in pertur-

bation theory. The qubit states are slightly dressed by

the meter states, which leads to Purcell decay [6] and

prevents from an ideal quantum non-demolition (QND)

readout [7,8]. In addition, unwanted eﬀects for the

readout such as relaxation and excitation rate of the

qubit can increase with readout photon number n[9–

13]. To overcome these limitations, a non-perturbative

cross-Kerr coupling between the qubit and the meter has

been proposed [14,15] and demonstrated thanks to the

property of a transmon molecule [16–19] achieving high

ﬁdelity and QND single shot readout of a transmon qubit

[19]. This result was realized through a polariton meter

in its linear regime, whose signal was ampliﬁed through

an external JPA.

Alternatively to JPA, superconducting qubit readout

can also be performed using a Josephson Bifurcation Am-

pliﬁer (JBA) [20–22]. The JBA is a nonlinear pumped

resonator such as the JPA, but it is pumped at diﬀerent

working point, where it presents a nonlinear ampliﬁca-

tion relationship between its input amplitude and output

amplitude, leading to two stable states of small and large

output amplitude for input signal below and above the bi-

furcation threshold, respectively. The information on the

qubit state is then encoded into those two output states.

In addition, the bifurcation presents hysteresys leading

to a latching readout. The JBA dynamics is controlled

by the detuning between the nonlinear resonator and the

pump, the resonator losses κand its anharmonicity U.

A same-chip implementation allows a direct coupling be-

tween the qubit and the JBA, with an in-situ amplifying

bifurcation, greatly increasing the quantum detection ef-

ﬁciency [20,22–27]. Here, we propose a readout based

on bifurcation ampliﬁcation of the in-situ nonlinear po-

lariton meter.

Up to now, the bifurcation readout has been realized

in the weak anharmonicity limit U≪κin which the

bistability regime is achieved when the photon number

nin the nonlinear resonator exceeds the critical number

Ncrit =κ/(3√3U)≫1 [28,29]. However, this large

photon number, needed to reach the bistability and thus

arXiv:2210.04793v2 [quant-ph] 24 Jan 2024

bifurcation, exposes the qubit to excess backaction of the

nonlinear cavity [28–32] like inducing qubit state transi-

tions and thus rendering the readout not QND. Instead

of the usual classical regime of weak anharmonicity and

large photon number (U≪κ,Ncrit ≫1), the excess

backaction on the qubit could be weakened in the meso-

scopic regime (U∼κ,Ncrit ∼1) where bistability ap-

pears with photon number close to unity n≳Ncrit ∼1.

This little explored regime is in between the classical

regime and the quantum regime (U≫κ,Ncrit ≪1,

where the system behaves as an eﬀective quantum few-

level-system) [33–35].

In this paper, we demonstrate a transmon qubit state

latching readout using an in-situ bifurcation of a polari-

ton resonator in the mesoscopic regime. The polaritons

are superpositions of the harmonic cavity mode and the

anharmonic ancillary mode of the transmon molecule.

They result from the strong coupling/hybridization be-

tween them. Both polariton modes inherit the nonlinear-

ity of the ancilla mode, so they eﬀectively behave as two

nonlinear Kerr resonators exhibiting bistability [36–39].

By adjusting the ancilla frequency through an external

magnetic ﬂux, we control the hybridization, and conse-

quently the anharmonicity and dissipation of each po-

lariton. In Section II, we discuss details of the transmon

molecule, the ancilla-cavity hybridization, the resulting

polariton modes and their tunability. Contrary to our

previous work [19] where we considered the linear regime

with n≪Ncrit and Ncrit ≫1, here we investigate the

nonlinear regime of the polaritons at large occupation

n≳Ncrit ∼1. The polaritons response to a strong drive

and its dependence on the qubit state are detailed in Sec-

tion III. The hysteretic bistability behavior of the non-

linear upper polaritonic meter is analyzed in Section IV.

Finally, in Section V, we take advantage of this bista-

bility to perform a latching readout of the qubit state

with a high single-shot ﬁdelity and without any external

quantum-limited ampliﬁer.

II. TUNING THE ANCILLA-CAVITY

HYBRIDIZATION

A. Transmon molecule in a cavity

We use the same sample as in Ref. [19]. It consists

of a 3D-cavity containing a transmon molecule which is

made by coupling inductively and capacitively two nom-

inally identical transmons [16,19] (see Fig. 1a and Ap-

pendix A). The device has three modes of interest, the

harmonic TE101 mode ˆcof the rectangular 3D Copper

cavity with frequency ωc/2π= 7.169 GHz, and the two

orthogonal modes of the transmon molecule (for more

details see Appendix Band Ref. [19]): (i) a qubit mode,

with parameters of standard transmons, protected from

interaction with the 3D cavity mode thanks to its sym-

metric proﬁle, and (ii) an ancilla mode, with weak (∼

MHz) anharmonicity and strong interaction to the cav-

a) b)

FIG. 1. a) Scheme of the setup. The cavity mode ˆcof fre-

quency ωcis strongly coupled to an anharmonic ancilla mode

of frequency ωaand self-Kerr nonlinearity Ua. The ancilla is

also coupled to the qubit via a non-perturbative cross-Kerr

coupling of rate gzz . To perform readout, we send a coher-

ent signal on the input of the cavity mode ˆcin and measure

the transmitted cavity output ﬁeld ˆcout. b) Representation of

the system in terms of cavity-ancilla polariton modes. Lower

and upper polariton modes have distinct frequencies ωland

ωu, respectively, as well as diﬀerent self-Kerr nonlinearities

Uland Uuinherited from the ancilla. Both polaritons are

independently coupled to the qubit via cross-Kerr terms χl

and χu, which allows us to use these polariton modes as di-

rect meters of the qubit states. The readout can be extracted

from the same output ﬁeld cout due to the polaritons leakage

rate κland κu. c)-e) Measurements (dots) and predictions

(lines) for lower polariton j=l(orange) and upper polari-

ton j=u(purple) as function of the hybridization angle θof

c) the non-perturbative qubit-polariton cross-Kerr χj, d) the

self-Kerr Ujj and inter-polariton cross-Kerr Uul (green), and

e) polariton decay rates κj. The predictions are calculated

from the polariton model in Eqs. (2) and (3) using initial pa-

rameters gzz,Uaand κcand κaand plotted as black lines in

c), d) and e), respectively. f) Normalized self-Kerr polariton

nonlinearity Ujj /κjversus normalized qubit-polariton cross-

Kerr coupling 2χj/κjfor lower and upper polaritons. c-f)

The working points in the present work and in Ref. [19] are

marked by a purple and orange circle, respectively.

ity gac due to its antisymmetric mode proﬁle. Indeed,

the cavity electrical ﬁeld is aligned with the ﬁeld of the

ancilla mode while being orthogonal to the ﬁeld of the

qubit mode.

The ancilla is approximated as a weakly nonlinear

mode ˆawith frequency ωatunable by magnetic ﬂux and

self-Kerr rate Ua. Moreover, approximating the multi-

level transmon as a qubit ˆσz, the total system Hamilto-

nian including cavity, ancilla, and qubit reads,

ℏ=ωq

2ˆσz+ωaˆa†ˆa+ωcˆc†ˆc

−Ua

2ˆa†2ˆa2−gzz ˆσzˆa†ˆa+gac(ˆc†ˆa+ ˆa†ˆc).(1)

The qubit ˆσzwith frequency ωq/2π= 6.283 GHz and

coherence times T2, T1≃3µs is coupled to the ancilla

mode ˆavia a non-perturbative cross-Kerr coupling with

rate gzz. The non-perturbative nature of this coupling

allows to maximize the speed, the single-shot ﬁdelity, and

the QND properties of the readout, while minimizing the

eﬀect of unwanted decay channels such as the Purcell

eﬀect [19].

B. Ancilla-cavity hybridization leading to

polaritons

To use this coupling for reading out the state of the

qubit, we strongly hybridize the ancilla and the cav-

ity by setting their detuning ∆ac =ωa−ωcto values

comparable to or smaller than their transverse coupling

gac/2π= 295 MHz. At this operation point, |∆ac|≲gac,

this hybridization leads to two new normal modes called

upper and lower polariton modes, ˆcuand ˆcl, for highest

and lowest in frequency, respectively. They are a lin-

ear combination of ancilla and cavity ﬁelds (see Fig. 1.a-

b). They are given by a rotation ˆcu= cos(θ)ˆa+ sin(θ)ˆc,

and ˆcl= cos(θ)ˆc−sin(θ)ˆa, where the cavity-ancilla hy-

bridization angle reads tan(2θ) = 2gac/∆ac. At reso-

nance (∆ac = 0, θ=π/4), the two modes are com-

pletely hybridized into equal symmetric and antisymmet-

ric superpositions while at large detuning (|∆ac| ≫ gac,

θ−→ 0), the two normal modes tend to approach the bare

ancilla and cavity modes.

In terms of these polariton modes and using the rotat-

ing wave approximation, the total Hamiltonian takes the

form

ℏ=ωq

2ˆσz−X

j=u,l

χjˆc†

jˆcjˆσz

j=u,l

(ωjˆc†

jˆcj−Ujj

2ˆc†2

jˆc2

j)−Uulˆc†

lˆclˆc†

uˆcu,(2)

where ωu= sin2(θ)ωc+ cos2(θ)ωa+ sin(2θ)gac and

ωl= cos2(θ)ωc+ sin2(θ)ωa−sin(2θ)gac are the fre-

quencies of the upper and lower polariton modes, re-

spectively. Each polariton mode is in some proportion

cavity-like and therefore can be probed in transmission

and used for readout. Similarly, each polariton is also

ancilla-like and thus inherits nonlinearities from the an-

cilla, notably the non-perturbative cross-Kerr coupling

to the qubit. The corresponding interaction strengths

read χu=gzz cos2(θ) and χl=gzz sin2(θ), for the up-

per and lower polariton, respectively. Each polariton

also inherits an anharmonicity from the ancilla given

by Ull = sin4(θ)Uaand Uuu = cos4(θ)Ua. They also

acquire a cross-anharmonicity or cross-Kerr interaction

Uul = sin2(2θ)Ua/2, a coupling similar to the dispersive

interaction that still occurs even beyond the dispersive

regime [40]. Finally, the polaritons have eﬀective decay

rates given by a combination of the bare ancilla κaand

cavity κcdecay rates as (cf. Appendix C)

κu=κcsin2(θ) + κacos2(θ),

κl=κccos2(θ) + κasin2(θ).(3)

C. Tuning hybridization

The ancilla can be tuned at discrete frequencies in-

dependently of the qubit and cavity. This is possible

because the transmon molecule possesses two supercon-

ducting loops of diﬀerent sizes with a high area ratio of

26 (see Appendix Aand B). Using one external coil, we

can thus tune with a step precision of Φ0/26 the ﬂux de-

termining the ancilla frequency while the loop deﬁning

the qubit frequency still experiences an integer value of

ﬂux quantum Φ0. This allows to tune the hybridization

conditions and thus the diﬀerent parameters in Eqs. 2

and 3. We extracted these parameters (see Figs. 1.c-f)

as function of the hybridization angle by measuring at

diﬀerent ﬂux points. The non-perturbative cross-Kerr

couplings χland χuare well-ﬁtted by a bare qubit-

ancilla cross-Kerr coupling gzz/2π= 34.5 MHz (Fig. 1.c).

The polaritons self-Kerr and cross-Kerr couplings Ull,

Uuu, and Ulu are well ﬁtted with the polariton model

with a bare ancilla anharmoncity Ua/2π= 13.5 MHz

(Fig. 1.d). The polariton decay rates are only qualita-

tively ﬁtted by the polariton model (Fig. 1.e) with bare

cavity and bare ancilla decay rates κc/2π= 12.7 MHz

and κa/2π= 5.6 MHz. Discrepancies may be explained

by the fact that the bare ancilla decay rate can vary with

its frequency due to the presence of other losses like ﬂuc-

tuating two-level systems, or that there is residual par-

asitic transverse coupling between the ancilla and cavity

with the qubit (see Ref. [19]).

Thanks to the hybridization tunability, we can set

diﬀerent regimes (Fig. 1.f) for the polariton meters to

read out the qubit. In Ref. [19], we focused on the

linear response of the lower polariton which presents

a small self-Kerr U/κ = 0.017 at the moderate drive

n≈2≪Ncrit. This linear regime is obtained at the

zero ﬂux point (Φ/Φ0= 0) where the lower polariton is

mostly cavity-like. In this work, we focus on a diﬀerent

regime where the anharmonicity is comparable to dissi-

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Transmon-qubitreadoutusingin-situbifurcationamplificationinthemesoscopicregimeR.Dassonneville,1,2T.Ramos,3V.Milchakov,1C.Mori,1L.Planat,1F.Foroughi,1C.Naud,1W.Hasch-Guichard,1J.J.Garc´ıa-Ripoll,3N.Roch,1andO.Buisson11Univ.Grenoble-Alpes,CNRS,GrenobleINP,InstitutN´eel,38000Grenoble,France2AixMarseill...

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Transmon-qubit readout using in-situ bifurcation amplification in the mesoscopic regime R. Dassonneville1 2T. Ramos3V. Milchakov1C. Mori1L. Planat1F. Foroughi1.pdf

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Transmon-qubit readout using in-situ bifurcation amplification in the mesoscopic regime R. Dassonneville1 2T. Ramos3V. Milchakov1C. Mori1L. Planat1F. Foroughi1

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