Emergent macroscopic bistability induced by a single superconducting qubit Riya Sett1Farid Hassani1Duc Phan1Shabir Barzanjeh1 2Andras Vukics3yand Johannes M. Fink1z 1Institute of Science and Technology Austria ISTA 3400 Klosterneuburg Austria

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Emergent macroscopic bistability induced by a single superconducting qubit

Riya Sett,1, ∗Farid Hassani,1Duc Phan,1Shabir Barzanjeh,1, 2 Andras Vukics,3, †and Johannes M. Fink1, ‡

1Institute of Science and Technology Austria (ISTA), 3400 Klosterneuburg, Austria

2Institute for Quantum Science and Technology, University of Calgary, Calgary, Alberta, Canada

3Wigner Research Centre for Physics, H-1525 Budapest, P.O. Box 49., Hungary

(Dated: October 26, 2022)

The photon blockade breakdown in a continuously driven cavity QED system has been proposed

as a prime example for a ﬁrst-order driven-dissipative quantum phase transition. But the predicted

scaling from a microscopic system - dominated by quantum ﬂuctuations - to a macroscopic one -

characterized by stable phases - and the associated exponents and phase diagram have not been

observed so far. In this work we couple a single transmon qubit with a ﬁxed coupling strength gto

an in-situ bandwidth κtuneable superconducting cavity to controllably approach this thermody-

namic limit. Even though the system remains microscopic, we observe its behavior to become more

and more macroscopic as a function of g/κ. For the highest realized g/κ ≈287 the system switches

with a characteristic dwell time as high as 6 seconds between a bright coherent state with ≈8×103

intra-cavity photons and the vacuum state with equal probability. This exceeds the microscopic time

scales by six orders of magnitude and approaches the near perfect hysteresis expected between two

macroscopic attractors in the thermodynamic limit. These ﬁndings and interpretation are qualita-

tively supported by semi-classical theory and large-scale Quantum-Jump Monte Carlo simulations.

Besides shedding more light on driven-dissipative physics in the limit of strong light-matter coupling,

this system might also ﬁnd applications in quantum sensing and metrology.

I. INTRODUCTION

Quantum phase transitions (QPT), both ﬁrst- and

second-order [1] have been at the forefront of physics re-

search for half a century. The original idea of QPTs as

abrupt shifts in the (pure) ground state of closed quan-

tum systems as a function of a control parameter applied

mostly to condensed matter physics. Dissipative phase

transitions (DPT) occurring in the (in general, mixed)

steady state of open quantum systems [2–12], however,

broadened the scope of phase transitions to encompass

mesoscopic and later even microscopic systems, where

the interaction with the environment essentially aﬀects

the system dynamics. A DPT was ﬁrst realized experi-

mentally in a Bose-Einstein condensate interacting with

a single-mode optical cavity ﬁeld [13], and DPTs are in-

creasingly relevant to today’s quantum science and tech-

nology [14–17].

In view of this success, it is remarkable that in re-

cent years yet another phase-transition paradigm could

emerge, namely, ﬁrst-order dissipative quantum phase

transitions. A ﬁrst-order phase transition means that two

phases can coexist in a certain parameter region, like wa-

ter and ice at 0 °C for a certain range of free energy. Coex-

istence of phases in the quantum steady state seems para-

doxical, since the steady-state plus normalization condi-

tions for the density operator constitute a linear system

of equations, that admits only a single solution. That is,

given the Liouvillian superoperator Lfor the Markovian

∗riya.sett@ist.ac.at

†vukics.andras@wigner.hu

‡jﬁnk@ist.ac.at

evolution of the system, there exists only a single nor-

malized density operator ρst that satisﬁes

Lρst = 0.(1)

The resolution is that a single density operator can ac-

commodate the mixture of two macroscopically distinct

phases expressed as a ratio of the two components. In the

water analogy, at 0 °C we could symbolically write

ρst =c ρwater + (1 −c)ρice,(2)

with cgrowing from 0 to 1 as the free energy is increased.

Recently, ﬁrst-order dissipative quantum phase transi-

tions have been found in various systems. One such plat-

form is the clustering of Rydberg atoms described by

Ising-type spin models [18–23] and realized experimen-

tally [24–26]. Various other systems of ultracold atoms

[27,28] and dissipative Dicke-like models [29,30] also ex-

hibit signatures of a ﬁrst-order DPT. Other platforms in-

clude (arrays of) nonlinear photonic or polaritonic modes

[7,31–38], exciton-polariton condensates [39,40] and cir-

cuit QED [16,41–43]. In this work we observe and model

the scaling and phase diagram of a ﬁrst-order DPT in

zero dimensions, i.e. for a single qubit strongly coupled

to a single cavity mode.

II. PHOTON-BLOCKADE BREAKDOWN

The Jaynes-Cummings (JC) model - one of the most

important models in quantum science - describes the in-

teraction between atoms and photons trapped in a cavity

[44]. It is expressed by the Hamiltonian (~= 1)

HJC =ωRa†a+ωAσ†σ+iga†σ−σ†a

+iηa†e−iωt −a eiωt,(3)

arXiv:2210.14182v1 [quant-ph] 25 Oct 2022

with ωRthe angular frequency of the cavity mode with

boson operator a,ωAthat of the atomic transition with

operator σ,gthe coupling strength, ηthe drive strength,

and ωthe drive frequency. This model yields the proto-

type of an anharmonic spectrum in the strong-coupling

regime, as demonstrated in cavity [45] and circuit QED

[46], and with quantum dots in semiconductor microcav-

ities [47]. Its strong anharmonicity at single photon lev-

els is the basis of the photon blockade eﬀect [48,49], in

analogy with Coulomb blockade in quantum dots or to

polariton blockade [50]. Photon blockade means that an

excitation cannot enter the JC system from a drive tuned

in resonance with the bare resonator frequency, or simi-

larly, a second excitation from a drive tuned to resonance

with one of the single-excitation levels cannot enter.

This blockade is, however, not absolute, as it can be

broken [51–54] by strong enough driving due to a com-

bination of multi-photon events and photon-number in-

creasing quantum jumps [55]. In an intermediary ηrange,

in the time domain the system stochastically alternates

between a blockaded, dim state without cavity photons

and a bright state in which the blockade is broken and

the system resides in the highly excited quasi-harmonic

part of the spectrum resulting in a large transmission of

drive photons. In phase space, this behavior results in a

bimodal steady-state distribution

ρst =c ρbright + (1 −c)ρdim,(4)

in analogy with Eq. (2), with cgrowing from 0 to 1 with

increasing η. This eﬀect has been demonstrated experi-

mentally in a circuit QED system [42].

Bistability in the time domain or bimodality in phase

space is, however, not suﬃcient evidence for a ﬁrst-order

phase transition. It is also necessary that the two con-

stituents in the mixture Eq. (4) corresponding to the two

states in the temporal bistable signal to be macroscopi-

cally distinct as is the case in Eq. (2). It has been shown

theoretically [51,55], that the photon blockade break-

down (PBB) eﬀect has such a regime, i.e. a thermody-

namic limit, where both the timescale and the amplitude

of the bistable signal goes to inﬁnity, resulting in long-

lived and macroscopic distinct dim and bright phases. Re-

markably, this thermodynamic limit is a strong-coupling

limit, deﬁned as g→ ∞, and independent of the physical

system size, i.e. the system remains the same JC system

composed of two microscopic interacting subsystems. In

this limit, the temporal bistability is replaced by hys-

teresis, where the state of the system is determined by

its initial condition, since switching to the other state

entails an inﬁnite waiting time. The passage to the ther-

modynamic limit, i.e. the indeﬁnite increase of ghas been

termed ﬁnite-size scaling [55].

In this work, we demonstrate these additional crite-

ria that clearly signify the observed physical eﬀect as

a ﬁrst-order dissipative quantum phase transition. We

demonstrate the ﬁnite-size scaling over 7 orders of mag-

nitude towards the thermodynamic limit and back out

the phase diagram of a ﬁrst-oder DPT in zero dimen-

κxed

κvary

Qubit

Piezo nano-

positioner

(a) (b)

10.4 10.425 10.45 10.475 10.5

|S21|2

0.5

0.4

0.3

0.2

0.1

0.0

1.2 MHz

2.6 MHz

4.7 MHz

7.9 MHz

12.8 MHz

29.7 MHz

t

frequency (GHz)

FIG. 1. Experimental realization. (a) Schematics of the

experimental device consisting of a superconducting trans-

mon qubit fabricated on a silicon substrate that is placed at

the antinode of the fundamental mode of a 3D copper cav-

ity. The cavity has a ﬁxed length port (red) and an in-situ

variable length pin coupler port (blue). (b) Measured cavity

transmission spectra with the qubit far detuned for diﬀerent

coupler positions (color coded) together with a ﬁt to Eq. (5)

(dashed) and the extracted κ/2π.

sions. We realize this experiment with a superconduct-

ing qubit strongly coupled to a bandwidth-tunable mi-

crowave cavity mode and ﬁnd qualitative agreement with

large-scale Quantum-Jump Monte Carlo simulations and

semi-classical calculations of the phase boundaries.

III. EXPERIMENTAL REALIZATION

Our experimental setting incorporates a transmon

qubit [56,57] placed at the anti-node of the standing

wave of a 3D copper-cavity, as shown in Fig. 1(a), that

can be ﬂux-tuned by applying a B-ﬁeld via a millimeter-

sized superconducting bias coil mounted at the out-

side cavity wall. The transmon qubit has a maximum

Josephson energy EJ,max/h ≈48 GHz, charging energy

EC/h ≈382 MHz and a resulting maximum transition

frequency between its ground and ﬁrst excited states of

ωA/2π≈12.166 GHz. When the transmon ground to ﬁrst

excited state transition is tuned in resonance with the

cavity mode at ωR/2π≈10.4725 GHz, the directly mea-

sured coupling strength between the single photon and

the qubit transition is as high as g/2π= 344 MHz, which

is only about a factor of 3 below the so-called ultrastrong

coupling regime [58]. The relatively high absolute anhar-

monicity between subsequent transmon state transitions

is α/h ≈ −418 MHz at this ﬂux bias position.

The cavity has two ports, of which the input pin cou-

pler position is ﬁxed with an external coupling strength of

κﬁxed/2π≈500 kHz. The output coupler is attached to a

cryogenic piezo nano-positioner, which allows for adjust-

ing the pin length extending into the cavity [59]. With

this tunable coupler the coupling strength can be var-

ied in situ in a wide range κvary/2π≈20 kHz −30 MHz.

The internal cavity loss at low temperature is κint/2π≈

600 kHz, which is achieved by electro-polishing of the

high conductivity copper surface before cooldown to

10 mK in a dilution refrigerator.

All four scattering parameters are measured with a vec-

tor network analyzer to calibrate the measurement setup

and the cavity properties when the qubit is far detuned

from the cavity resonance. Figure 1(b) shows transmis-

sion measurements ﬁtted with the scattering parameter

S21 derived from the Input-Output theory of an open

quantum system [60]

S21 =√κﬁxedκvary

κ/2−i(ω−ωR).(5)

From these ﬁts, we extract all loss rates that add up to

the total cavity linewidth κ=κﬁxed +κvary +κint also

indicated in Fig. 1(b).

Time-domain characterization measurements conﬁrm

that the qubit is Purcell-limited and homogeneously

broadened at the ﬂux sweet spot [61], where the mea-

sured coherence times are T1≈0.5µs and T2≈1µs.

When the qubit frequency is tuned far below the res-

onator frequency ωA/2π≈6.083 GHz by applying an ex-

ternal magnetic ﬁeld, the measured coherence times are

T1≈18.14 µs and T2≈0.496 µs, which we attribute to

a higher Purcell limit due to the larger detuning as well

as a drastically increased ﬂux noise sensitivity. On res-

onance ωA=ωR, where the following experiments were

performed, the energy relaxation is therefore fully domi-

nated by cavity losses. The measured vacuum Rabi peak

linewidth changes with and without the qubit in reso-

nance are in agreement with a small amount of ﬂux noise

induced dephasing expected at that ﬂux bias position.

IV. PHOTON BLOCKADE BREAKDOWN

MEASUREMENT

The photon blockade (and its breakdown) phenomenon

most straightforwardly occurs when the two interacting

constituents are resonant ωA=ωR. In contrast to the

ideal two-level atom limit [51,55], when driven on reso-

nance ω=ωRthis does not lead to spontaneous dressed-

state polarization [62,63] - a second-order DPT [51], in

our experimental situation with three (or more) trans-

mon levels [42] as shown in Fig. 2(a). For low input

powers corresponding to less than a single intra-cavity

photon on average we observe a vacuum Rabi-split spec-

trum in transmission, as shown in Fig. 2(a, b) (blue line).

No transmission peak is observed at the bare cavity fre-

quency ωRup to intermediate input drive strengths η.

This means that a single photon - or even hundreds of

photons at the chosen g/κ = 39.1 - are prevented from

entering the cavity due to the presence of a single artiﬁ-

cial atom.

This blockade is observed to be broken abruptly by

further increasing the applied drive strength η, which is

proportional to square-root of the applied drive power

and the corresponding drive photon number. As ηis in-

creased by only a ﬁnite amount, the transmitted out-

put power increases by three orders of magnitude at the

bare resonator frequency, as shown in the red spectrum

in Fig. 2(b). The central sharp peak in the transmission

spectrum corresponds to a time-averaged measurement

(determined by the chosen resolution bandwidth) of a

cavity that is fully transparent for most of the integra-

tion time. This PBB eﬀect can be attributed to the non-

linearity of the lower part of the JC spectrum which is

strongly anharmonic [46,64], while the higher-lying part

of the spectrum has subsets that are closely harmonic

over a certain range of excitation numbers [65] and can

hence accommodate a closely coherent state.

In the time domain, with ηin the phase coexistence

region, the PBB eﬀect results in a bistable telegraph sig-

nal, where the system output alternates between a ‘dim’

state where the qubit-resonator system remains close to

the vacuum state unable to absorb an excitation from

the externally applied drive, and a ‘bright’ state where

the system resides in an upper-lying, closely harmonic

subset of the JC spectrum, cf. Fig. 2(c). The switches be-

tween these two classical attractors are necessarily multi-

photon events that are triggered by quantum ﬂuctua-

tions. This bistability was shown to be a ﬁnite-size pre-

cursor of what would be a ﬁrst-order DPT in the ther-

modynamic limit (g/κ → ∞) [55], where the bistability

develops into perfect hysteresis: the system is stuck in the

attractor determined by the initial condition as long as

the control parameters are set in the transition domain.

In order to investigate this dynamics qualitatively, we

record the real-time single-shot data of both quadratures

of the transmitted output ﬁeld at the bare cavity fre-

quency while applying a continuous-wave (CW) drive

tone resonant with the bare cavity, over a range of applied

drive strengths. The transmitted radiation is ﬁrst am-

pliﬁed with a high electron mobility transistor (HEMT)

at 4 K followed by another room-temperature low-noise

ampliﬁer (LNA), then down-converted with an IQ mixer

with appropriate IF frequency and ﬁnally digitized with

a digitizer. Further this recorded data is digitally low-

pass ﬁltered with appropriate resolution bandwidth and

down-converted to d.c. to extract the time-dependent

quadratures in voltage units. For example, in the case

of κ/2π= 8 MHz, the recorded data is 2.88 s long and

the ﬁnal time resolution of the extracted quadratures is

2.5µs, cf. Fig. 2(c). The selection of an appropriate res-

olution bandwidth is critical for a number of reasons:

(1) to successfully resolve frequent and sudden switching

events caused by very short dwell times at high κval-

ues, (2) to maintain a signal to noise ratio that allows

to clearly discriminate single shot measurement events

without averaging, and (3) to achieve a suﬃcient total

measurement time to resolve long dwell times with the

available memory.

From the resulting histograms in phase space,

cf. Fig. 2(d-f), which represent the scaled Husimi-Q func-

tions convolved with the added ampliﬁcation chain noise

photon number namp ≈9.2, it can be deduced that for

low drive strength the photon blockade is intact (dim

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EmergentmacroscopicbistabilityinducedbyasinglesuperconductingqubitRiyaSett,1,FaridHassani,1DucPhan,1ShabirBarzanjeh,1,2AndrasVukics,3,yandJohannesM.Fink1,z1InstituteofScienceandTechnologyAustria(ISTA),3400Klosterneuburg,Austria2InstituteforQuantumScienceandTechnology,UniversityofCalgary,Calgary,Alb...

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Emergent macroscopic bistability induced by a single superconducting qubit Riya Sett1Farid Hassani1Duc Phan1Shabir Barzanjeh1 2Andras Vukics3yand Johannes M. Fink1z 1Institute of Science and Technology Austria ISTA 3400 Klosterneuburg Austria.pdf

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Emergent macroscopic bistability induced by a single superconducting qubit Riya Sett1Farid Hassani1Duc Phan1Shabir Barzanjeh1 2Andras Vukics3yand Johannes M. Fink1z 1Institute of Science and Technology Austria ISTA 3400 Klosterneuburg Austria

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