# Genus Polynomials of Cubic Graphs with Non-Real Roots

MacKenzie Carr Varpreet Dhaliwal Bojan Mohar
Department of Mathematics
Simon Fraser University
Burnaby, BC V5A 1S6, Canada Supported by the Natural Sciences and Engineering
Research Council of Canada (NSERC) Canadian Graduate Scholarship 456422823.
Email<EMAIL_ADDRESS>by the NSERC Undergraduate Student
Research Award 521692. Email<EMAIL_ADDRESS>in part by the
NSERC Discovery Grant R611450 (Canada) and by the Research Project J1-2452 of
ARRS (Slovenia). On leave from IMFM, Jadranska 19, 1000 Ljubljana. Email:
<EMAIL_ADDRESS>
###### Abstract

Given a graph $G$, its genus polynomial is $\Gamma_{G}(x)=\sum_{k\geq
0}g_{k}(G)x^{k}$, where $g_{k}(G)$ is the number of 2-cell embeddings of $G$
in an orientable surface of genus $k$. The Log-Concavity Genus Distribution
(LCGD) Conjecture states that the genus polynomial of every graph is log-
concave. It was further conjectured by Stahl that the genus polynomial of
every graph has only real roots, however this was later disproved. We identify
several examples of cubic graphs whose genus polynomials, in addition to
having at least one non-real root, have a quadratic factor that is non-log-
concave when factored over the real numbers.

##  1 Introduction

Let $G$ be a connected graph. We denote by $g_{k}(G)$ the number of 2-cell
embeddings of $G$ in an orientable surface of genus $k$. Here we count 2-cell
embeddings up to combinatorial equivalence: two 2-cell embeddings are
equivalent if their facial walks are the same. Equivalently, their clockwise
rotation systems are the same. This is also equivalent to saying that there is
an orientation-preserving homeomorphism among the surfaces that induces
identity on the embedded graph. We refer to [5] for details. The _genus
polynomial_ of $G$ is defined as

$\Gamma_{G}(x)=\sum_{k\geq 0}g_{k}(G)x^{k}.$

The following conjecture was proposed in 1989 by Gross, Robbins and Tucker
[3]. It is known as the Log-Concavity Genus Distribution (LCGD) Conjecture:

######  LCGD Conjecture ([3]).

For every graph $G$, the genus polynomial is log-concave.

Let us recall that the polynomial $\sum_{k\geq 0}g_{k}(G)x^{k}$ is _log-
concave_ if the sequence $g_{0}(G),g_{1}(G),\dots$ of its coefficients is log-
concave, meaning that $(g_{k}(G))^{2}\geq g_{k-1}(G)g_{k+1}(G)$, for every
$k\in\mathbb{N}$.

The LCGD Conjecture has been confirmed for various classes of graphs and shown
to be preserved under several graph amalgamation operations. However, the
conjecture remains widely open for general graphs. In [6], Stahl conjectured
the following stronger property of genus polynomials.

######  Conjecture 1.

For every graph $G$, the genus polynomial $\Gamma_{G}(x)$ has only real roots.

Stahl went on to prove that the conjecture holds for particular classes of
graphs including bouquets $B_{n}$, dipoles $DP_{n}$, and cobblestone paths, as
well as offer examples of graph classes whose genus generating matrices are
known but whose genus polynomials hadn’t been proven to have real roots. Liu
and Wang [4] used one such example, the class of $W_{4}$-linear graphs, to
disprove Stahl’s conjecture, but Chen [1] discovered an error in the genus
generating matrix used in this counterexample. Chen showed that the argument
of Liu and Wang does not hold using the correct genus generating matrix for
$W_{4}$-linear graphs. In addition, Chen verified that the genus polynomials
of all graphs with maximum genus 2 are real-rooted.

Stahl’s conjecture remained open until 2010, at which time Chen and Liu [2]
identified two graphs, shown in Figure 1, whose genus polynomials both have
non-real roots.

Figure 1: Two graphs whose genus polynomials have non-real roots

By computing the genus polynomials for all cubic graphs with up to 16
vertices, we identified a total of 226 such cubic graphs whose genus
polynomials have non-real roots. The number of graphs of each order is
summarized in Table 1. Since there are 40301 cubic graphs of order 18 and
510489 cubic graphs of order 20, it became infeasible to compute the number of
these that have at least one non-real root.

$n$ | # of cubic graphs | total # of
---|---|---
| with a non-real root | cubic graphs
10 | 2 | 19
12 | 5 | 85
14 | 41 | 509
16 | 178 | 4060
Table 1: The number of cubic graphs of order $n$ whose genus polynomial has a
non-real root, and the total number of cubic graphs, for $n=10,12,14,16$.

##  2 Real Roots and Log-Concavity

The question of whether all genus polynomials have real roots was of great
interest to Stahl and others due to the connection to the log-concavity of the
genus polynomial.

Stahl’s conjecture on the real-rootedness of genus polynomials, if true, would
have confirmed the LCGD Conjecture of Gross, Robbins and Tucker. This is due
to the log-concavity of polynomials with positive coefficients being preserved
under multiplication. This fact is well known, with a nice proof given by
Stanley [7].

######  Theorem 1.

If $A(x)$ and $B(x)$ are log-concave polynomials with non-negative
coefficients and no internal zero coefficients, then so is $A(x)B(x)$.

It is clear that any linear function $ax+b$ with $a,b\in\mathbb{R}_{+}$, is
log-concave. If a polynomial with positive coefficients is real-rooted, then
its roots are nonpositive and it can be factored as
$P(x)=a\prod_{j=1}^{n}(x+b_{j})$, where $a>0$ and each factor $x+b_{j}$ is
log-concave. Thus, Theorem 1 implies that $P(x)$ is log-concave.

However, Theorem 1 implies log-concavity of some polynomials with complex
roots.

######  Proposition 1.

Let $P(x)$ be a polynomial with non-negative real coefficients. If every
complex root $z$ of $P$ lies in the cone
$|\Re(z)|\geq\frac{1}{\sqrt{3}}|\Im(z)|$ with $\Re(z)\leq 0$, then $P(x)$ is
log-concave.

###### Proof.

The polynomial $P(x)$ can be factored over $\mathbb{R}$ into linear and
irreducible quadratic factors. Each real root is non-positive. This implies
that each linear factor is log-concave. Each quadratic factor $x^{2}+bx+c$ has
roots

$z=\frac{-b\pm\sqrt{b^{2}-4c}}{2}$

with $b^{2}-4c<0$. Since we have that
$|\Re(z)|\geq\frac{1}{\sqrt{3}}|\Im(z)|$, it must be the case that

$|\Re(z)|=\frac{|b|}{2}\geq\frac{1}{\sqrt{3}}\frac{\sqrt{4c-b^{2}}}{2}=\frac{1}{\sqrt{3}}|\Im(z)|.$

Note that $\Re(z)\leq 0$ guarantees that $b\geq 0$.

Now, we have that

$b\geq\sqrt{\frac{4c-b^{2}}{3}},$

which is equivalent to $b^{2}\geq c$. Therefore, each quadratic factor
$x^{2}+bx+c$ is log-concave and $P(x)$ is a product of log-concave linear and
quadratic polynomials with non-negative coefficients. By Theorem 1, $P(x)$ is
log-concave. ∎

One may be tempted to conjecture that the complex roots of genus polynomials
lie in the cone indicated in Proposition 1. However, we have identified
several cubic graphs, of orders 16 to 24, whose genus polynomials have roots
that lie outside of the cone in Proposition 1. We include these graphs in
Figure 2 and the details of the calculations in Section 3.

(a) $G(8,2)$

(b) $G(10,2)$

(c) $G(12,2)$

(d) $G_{18}$

(e) $G_{20}$

(f) $G_{22}$

Figure 2: Six graphs whose genus polynomial has non-real roots satisfying
$|\Re(z)|<|\Im(z)|/\sqrt{3}$.

##  3 Examples

The graph in Figure 2(a) is the generalized Petersen graph $G(8,2)$. Its genus
polynomial is

$\Gamma_{G(8,2)}(x)=39840x^{4}+23536x^{3}+2074x^{2}+84x+2$

which is easily shown to be log-concave. It has two real roots, approximately
equal to $-0.0572570083$ and $-0.4935182253$, and one pair of non-real roots,
approximately equal to $-0.01999390944\pm 0.03710524561i$. By direct
calculation, we see that

$\frac{1}{\sqrt{3}}(0.03710524561)\approx{}0.02142272354>0.01999390944$

Factoring $\Gamma_{G(8,2)}(x)$ over the real numbers, we get the following
factors.

$\displaystyle\Gamma_{G(8,2)}(x)\approx{}$
$\displaystyle(x+0.0572570083)(x+0.4935182253)$
$\displaystyle(x^{2}+0.03998781888x+0.001776555666)$

By direct computation, we can see that

$(0.03998781888)^{2}\approx{}0.00159902565<0.001776555666$

and thus the quadratic factor of $\Gamma_{G(8,2)}(x)$ above is not log-
concave.

Similar computation shows that the same is true for the other five graphs in
Figure 2, which all have log-concave genus polynomials. Table 2 gives a
summary of these results, including the values of the non-real roots of each
genus polynomial and their corresponding non-log-concave quadratic factors,
rounded to 11 decimal places.

$G(8,2)$ | $\Gamma_{G(8,2)}(x)$ | $39840x^{4}+23536x^{3}+2074x^{2}+84x+2$
---|---|---
| Complex roots | $-0.01999390944\pm 0.03710524561i$
| $|\Im(z)|/\sqrt{3}$ | $0.02142272354$
| Quadratic factor | $x^{2}+0.03998781888x+0.001776555666$
$G_{18}$ | $\Gamma_{G_{18}}(x)$ | $54272x^{5}+165824x^{4}+39472x^{3}$
|  | $+2480x^{2}+94x+2$
| Complex roots | $-0.01496753672\pm 0.038599441i$
| $|\Im(z)|/\sqrt{3}$ | $0.022285398$
| Quadratic factor | $x^{2}+0.02993507344x+0.001713944001$
$G(10,2)$ | $\Gamma_{G(10,2)}(x)$ | $587040x^{5}+411400x^{4}+47540x^{3}$
|  | $+2494x^{2}+100x+2$
| Complex roots | $-0.00896278346\pm 0.04522812336i$
| $|\Im(z)|/\sqrt{3}$ | $0.0261124692$
| Quadratic factor | $x^{2}+0.01792556692x+0.00212591463$
$G_{20}$ | $\Gamma_{G_{20}}(x)$ | $557248x^{5}+431656x^{4}+56602x^{3}$
|  | $+2964x^{2}+104x+2$
| Complex roots | $-0.011539073495\pm 0.0389911954i$
| $|\Im(z)|/\sqrt{3}$ | $0.0225115771616$
| Quadratic factor | $x^{2}+0.023078147x+0.001653463536$
$G_{22}$ | $\Gamma_{G_{22}}(x)$ | $692224x^{6}+2570304x^{5}+851384x^{4}$
|  | $+76726x^{3}+3550x^{2}+114x+2$
| Complex roots | $-0.0085736029\pm 0.03859372887i$
| $|\Im(z)|/\sqrt{3}$ | $0.02228209975$
| Quadratic factor | $x^{2}+0.0171472058x+0.001562982575$
$G(12,2)$ | $\Gamma_{G(12,2)}(x)$ | $8309664x^{6}+7244496x^{5}+1144338x^{4}$
|  | $+75088x^{3}+3508x^{2}+120x+2$
| Complex roots | $-0.002315938876\pm 0.04585954927i$
| $|\Im(z)|/\sqrt{3}$ | $0.02647702312$
| Quadratic factor | $x^{2}+0.004631877752x+0.002108461832$
Table 2: Summary of the genus polynomials and their roots for the graphs in
Figure 2.

##  4 Conclusion

We have identified a total of six graphs whose genus polynomials have non-real
roots satisfying $|\Re(z)|<|\Im(z)|/\sqrt{3}$ and therefore have a non-log-
concave quadratic factor when factored over the real numbers. However, we are
limited in our ability to find infinite classes of such examples by the small
number of graph classes whose genus distributions are known. One interesting
feature of these graphs is that they are all planar and 3-connected, hence the
constant coefficient of their genus polynomials is equal to 2.

## References

  * [1] Y. Chen.  A note on a conjecture of S. Stahl.  Canadian Journal of Mathematics, 64:958, 2008.
  * [2] Y. Chen and Y. Liu.  On a conjecture of S. Stahl.  Canadian Journal of Mathematics, 62:1058–1059, 2010.
  * [3] J. Gross, D. Robbins, and T. Tucker.  Genus distributions for bouquets of circles.  Journal of Combinatorial Theory Series B, 47:292–306, 1989.
  * [4] L. Liu and Y. Wang.  A unified approach to polynomial sequences with only real zeros.  Advances in Applied Mathematics, 38:542–560, 2007.
  * [5] B. Mohar and C. Thomassen.  Graphs on Surfaces.  Johns Hopkins University Press, 2001.
  * [6] S. Stahl.  On the zeros of some genus polynomials.  Canadian Journal of Mathematics, 49:617–640, 1997.
  * [7] R. Stanley.  Log-concave and unimodal sequences in algebra, combinatorics, and geometry.  Annals of the N.Y. Academy of Science, 576:500–535, 1989.


# One hundred second bit-flip time in a two-photon dissipative oscillator

C. Berdou  Laboratoire de Physique de l’Ecole normale supérieure, Mines-
Paristech, Inria, ENS-PSL, Université PSL, CNRS, Sorbonne Université, Paris,
France  A. Murani  Alice & Bob, 53 Bd du Général Martial Valin, 75015 Paris,
France  U. Réglade  Alice & Bob, 53 Bd du Général Martial Valin, 75015 Paris,
France  Laboratoire de Physique de l’Ecole normale supérieure, Mines-
Paristech, Inria, ENS-PSL, Université PSL, CNRS, Sorbonne Université, Paris,
France  W. C. Smith  M. Villiers  J. Palomo  M. Rosticher  A. Denis  P. Morfin
M. Delbecq  T. Kontos  Laboratoire de Physique de l’Ecole normale supérieure,
Mines-Paristech, Inria, ENS-PSL, Université PSL, CNRS, Sorbonne Université,
Paris, France  N. Pankratova  F. Rautschke  T. Peronnin  Alice & Bob, 53 Bd du
Général Martial Valin, 75015 Paris, France  L.-A. Sellem  P. Rouchon  A.
Sarlette  M. Mirrahimi  P. Campagne-Ibarcq  Laboratoire de Physique de l’Ecole
normale supérieure, Mines-Paristech, Inria, ENS-PSL, Université PSL, CNRS,
Sorbonne Université, Paris, France  S. Jezouin  R. Lescanne  Alice & Bob, 53
Bd du Général Martial Valin, 75015 Paris, France  Z. Leghtas
<EMAIL_ADDRESS>Laboratoire de Physique de l’Ecole normale supérieure,
Mines-Paristech, Inria, ENS-PSL, Université PSL, CNRS, Sorbonne Université,
Paris, France

###### Abstract

Current implementations of quantum bits (qubits) continue to undergo too many
errors to be scaled into useful quantum machines. An emerging strategy is to
encode quantum information in the two meta-stable pointer states of an
oscillator exchanging pairs of photons with its environment, a mechanism shown
to provide stability without inducing decoherence. Adding photons in these
states increases their separation, and macroscopic bit-flip times are expected
even for a handful of photons, a range suitable to implement a qubit. However,
previous experimental realizations have saturated in the millisecond range. In
this work, we aim for the maximum bit-flip time we could achieve in a two-
photon dissipative oscillator. To this end, we design a Josephson circuit in a
regime that circumvents all suspected dynamical instabilities, and employ a
minimally invasive fluorescence detection tool, at the cost of a two-photon
exchange rate dominated by single-photon loss. We attain bit-flip times of the
order of 100 seconds for states pinned by two-photon dissipation and
containing about 40 photons. This experiment lays a solid foundation from
which the two-photon exchange rate can be gradually increased, thus gaining
access to the preparation and measurement of quantum superposition states, and
pursuing the route towards a logical qubit with built-in bit-flip protection.

##  I Introduction

Figure 1: (Top) Principle of the experiment. A cavity is endowed with a
special mechanism (dashed left mirror) that exchanges pairs of photons (blue
double waves) at variable intensity (control knob) with a cold bath. Two meta-
stable pointer states emerge, represented by the blue distributions centered
around amplitudes $\pm\alpha$. A fraction of the cavity field (blue waves)
escapes through the weakly transmissive mirror and is collected by our
heterodyne detector. By monitoring the signal over time, we track individual
trajectories undergoing bit-flips (blue time trace). (Bottom) False-color
optical micrograph of the experimental superconducting circuit in a coplanar
waveguide geometry. The cavity is a $\lambda/2$ resonator (blue) that radiates
a field $a_{\text{out}}$ through a weakly coupled $50~{}\Omega$ port. It also
couples to a two-photon exchange device composed of a buffer mode (red)
shunted to ground through an asymmetrically threaded SQUID (ATS) as emphasized
in the insets. DC currents enter through on-chip bias tees (green) and impose
phase biases $\varphi_{L,R}$. A differential pump (purple arrows) and a common
buffer drive (red arrows) are channeled through filtered transmission lines
(orange).

The quest for a physical system suitable for quantum information processing is
intensifying at the dawn of quantum computing. Microscopic entities such as
single ions [1] and electrons [2] came forward as initial promising
candidates. More recently, human-made superconducting circuits encode quantum
bits in the form of single excitations of electromagnetic modes [3]. Despite
impressive progress [4], these systems continue to be individually plagued by
too many errors, blocking their deployment into scalable quantum machines.

A qubit with computational states $\ket{0}$ and $\ket{1}$ undergoes errors
spanned by two processes: bit-flips that randomly swap $\ket{0}$ and
$\ket{1}$, and phase-flips that scramble the phase of quantum superpositions
of $\ket{0}$ and $\ket{1}$. Unlike phase-flips, bit-flips have a clear
classical analogue. Interestingly, classical bits in a typical static random
access memory have bit-flip times in the range of $10^{15}$ seconds [5], 17
orders of magnitude larger than their quantum counterpart [6]. This
observation sparked interest in a qubit whose computational states would be
stable over macroscopic time-scales.

A nonlinear dynamical system that is open to its environment through a
carefully tailored interaction may exhibit a rich dynamical phase space
hosting multiple stable steady states [7, 8]. The switching between these
states in dissipative Kerr oscillators has long been used for amplification
[9, 10] and ultra-low power classical logic [11, 12]. Their stability is
ensured by regular energy damping, that, at the same time, decoheres their
quantum superpositions. Surprisingly, there exists a mechanism, known as two-
photon dissipation, that provides stability without inducing decoherence. A
recent qubit, coined the cat-qubit [13], is embedded in the cavity field of a
superconducting resonator that exchanges pairs of photons with its environment
[14, 15]. This defies the common intuition that a qubit must be well isolated
from its environment. Instead, the subtle interplay of drive and dissipation
pins down the cavity field on two coherent steady states with complex
amplitudes denoted $\pm\alpha$ without affecting quantum superpositions of the
two.

Increasing the number of photons $\bar{n}=|\alpha|^{2}$ in these two steady
states has two opposing effects [16]. On the one hand, their
distinguishability by an inevitably coupled uncontrolled environment increases
linearly with $\bar{n}$. This results in a linear increase of the phase-flip
error rate. Therefore, for this system to be suitable for quantum information
processing, it must operate at low photon number. On the other hand, as soon
as their separation exceeds their vacuum fluctuations, that is
$|\alpha-(-\alpha)|^{2}=4{\bar{n}}>1$, their wave-function overlap rapidly
decreases, reducing random tunneling between them and hence exponentially
increasing the bit-flip time. It is remarkable that, at least in principle, it
is possible to reach macroscopic bit-flip times with computational states
pinned by two-photon dissipation containing only a handful of photonic
excitations.

Previous experiments have succeeded in implementing a two-photon exchange
mechanism to observe the squeezing of a Schrödinger cat state out of vacuum
[15], the dynamics of a quantum gate [17], the exponential suppression of bit-
flips and linear increase of phase-flips [16]. A recent insight has been to
define a cat-qubit out of the interplay of Kerr non-linearity and single mode
squeezing [18, 19]. However, in all these implementations, the bit-flip time
saturated in the millisecond range, limited by errors impinging from the cat-
qubit tomography apparatus [16], and possible dynamical instabilities [20,
21].

In this work, we aim for the maximum bit-flip time we could achieve in a two-
photon dissipative oscillator. To reach this goal, we first design a circuit
with the objective of removing all suspected sources of dynamical
instabilities and ancillary systems that could propagate uncorrectable errors.
We fabricate a two-photon exchange dipole element close to the regime where
its energy landscape exhibits a single global minimum at any operating point,
a possible requirement for stability [21, 22]. Second, we entirely remove the
tomography apparatus: our design does not contain a transmon and readout mode.
Instead, we directly measure the field radiated by the cavity through a
travelling wave parametric amplifier (TWPA) [23], thereby accessing individual
oscillator state trajectories. We measure a bit-flip time exceeding 100
seconds for computational states pinned by two-photon dissipation and
containing about 40 photons. Our design choices came at the cost of a two-
photon exchange rate dominated by single-photon loss, hence losing our ability
to prepare quantum superposition states and hence measuring the phase-flip
rate. Guided by this benchmark, future experiments can then gradually enter
the regime suitable to implement a qubit where two-photon loss is the dominant
dissipation mechanism.

##  II The two-photon dissipative oscillator

An oscillator exchanging pairs of photons with its environment in addition to
usual energy relaxation (see Fig. 1) is modeled by the following Hamiltonian
and loss operators:

$H_{2}/\hbar={i}\epsilon_{2}a^{{\dagger}2}-{i}\epsilon_{2}^{*}a^{2},\;\;L_{2}=\sqrt{\kappa_{2}}a^{2}\;,\;\;L_{1}=\sqrt{\kappa_{a}}a\;,$
(1)

where $a$ is the annihilation operator of the mode referred to as the memory,
$\epsilon_{2}$ is the two-photon injection rate, $\kappa_{2}$ the two-photon
loss rate, and $\kappa_{a}$ is the energy damping rate. When the two-photon
injection rate overcomes the damping rate, two meta-stable pointer states
emerge:

$\ket{0}_{\alpha}=\ket{+\alpha}+\mathcal{O}(e^{-2|\alpha|^{2}})\;,\;\;\ket{1}_{\alpha}=\ket{-\alpha}+\mathcal{O}(e^{-2|\alpha|^{2}})\;,$

where $\ket{\alpha}$ is a coherent state with complex amplitude $\alpha$,
verifying $\alpha^{2}=\frac{2}{\kappa_{2}}(\epsilon_{2}-\kappa_{a}/4)\text{ if
}\epsilon_{2}>\kappa_{a}/4,\text{ and }\alpha^{2}=0\text{ otherwise}.$

The two-photon dissipation mechanism is engineered by implementing a two-to-
one photon exchange interaction with a dissipative mode referred to as the
buffer [15], modeled by the Hamiltonian

${H}_{ab}/\hbar=g_{2}^{*}a^{2}{b}^{\dagger}+g_{2}a^{{\dagger}2}{b}-\epsilon_{d}b^{\dagger}-\epsilon_{d}^{*}b\;,$
(2)

where ${b}$ is the annihilation operator of the buffer, $g_{2}$ is the two-to-
one photon coupling rate and $\epsilon_{d}$ the buffer drive amplitude. In the
limit where the buffer energy decay rate $\kappa_{b}$ is larger than
$|g_{2}|$, we recover Eq. (1) with $\kappa_{2}={4|g_{2}|^{2}}/{\kappa_{b}}$
and $\epsilon_{2}=2g_{2}\epsilon_{d}/\kappa_{b}$ [15].

Figure 2: Emergence of two meta-stable pointer states from a nonlinear
dissipative phase transition. (a1-a3) Radiated energy from the memory mode in
units of circulating photon number (color) as a function of the detuning from
the frequency matching condition
$\Delta_{a}=\frac{1}{2}\left(\omega_{p}-(2\omega_{a}-\omega_{d})\right)$
(x-axis), and the detuning from the buffer resonance
$\Delta_{b}=\omega_{d}-\omega_{b}$ (y-axis). We denote $\omega_{a,b,p,d}$ the
memory, buffer, pump and drive angular frequencies respectively. Both data and
semi-classical numerical simulations are shown in different regions of each
panel corresponding to the specified drive amplitude $\epsilon_{d}$. (b)
Radiated energy from the memory in units of circulating photon number (y-axis)
at $\Delta_{a}=\Delta_{b}=0$ as a function of the drive amplitude (x-axis).
The data correspond to an integration time of $10~{}\mu s$ with single
averaging (circles) and 10000 averages (crosses). A semi-classical model
(green solid line) captures the appearance of a critical point around
${\epsilon_{d}/2\pi\approx 3~{}\text{MHz}}$ above which the vacuum state
becomes unstable (green dashed line). A full quantum model (red solid line) is
necessary to capture the curvature at the critical point, as emphasized by the
zoom in the inset panel. (c) Histogram (color) of the $I$-quadrature
integrated over 1 ms of the field radiated by the memory (y-axis) in units of
the square root of circulating photon number as a function of the drive
amplitude (x-axis). Passed the critical point, the memory field transits from
the vacuum into two meta-stable pointer states.

##  III Circuit design

Our two-photon dissipative oscillator is implemented in a circuit quantum
electrodynamics coplanar waveguide architecture (see Fig.1). The memory mode
is the fundamental mode of a $\lambda/2$ resonator. We measure coupling and
internal loss rates $\kappa_{a}^{c}/2\pi=40$ kHz and $\kappa_{a}^{i}/2\pi=18$
kHz. In order to minimize dielectric losses [24], we target the relatively low
frequency of $\omega_{a}/2\pi=4.0457$ GHz. A thermal population of about $1\%$
was measured on a twin sample using a transmon (see [25] Sec. S4.3). An
undesired side effect of coupling the memory to a lossy mode – the buffer – is
to increase the decay rate of the memory due to the Purcell effect. To prevent
this, we designed a stop-band filter centered at the memory frequency,
consisting of three $\lambda/4$ sections [26] on both routes linking the
memory to its cold bath. The buffer mode consists of a metallic plate of
charging energy $E_{C}/h=73$ MHz shunted to ground through an Asymmetrically
Threaded SQUID (ATS) [20]. The ATS is formed by two Josephson junctions in a
loop – each of Josephson energy $E_{J}/h=37$ GHz – split in its center by an
inductance made of five junctions of total inductive energy $E_{L}/h=~{}62$
GHz. This layout defines two loops that can be biased in DC flux
$\varphi_{L,R}$. We can hence independently control the common and
differential flux through the ATS:
$\varphi_{\Sigma}=\frac{1}{2}\left(\varphi_{L}+\varphi_{R}\right)$ and
$\varphi_{\Delta}=\frac{1}{2}\left(\varphi_{L}-\varphi_{R}\right)$. Radio-
frequency (RF) signals are routed to the ATS through a 180° hybrid coupler.
The buffer drive propagates in phase through both arms of the two photon
exchange apparatus. When reaching the ATS, these waves combine, inducing
currents in the inductance and thereby driving the buffer mode. On the other
hand, the pump propagates with opposite phases, inducing common flux in the
ATS.

In the process of choosing the ATS parameters, we were guided by the intuition
that dynamical instabilities would be avoided in a system with
$2E_{J}/E_{L}\lesssim 1$ [21, 22]. However, this criterion needs to be
balanced with the requirement of large $g_{2}$ (see [25], Sec. S2). In this
experiment, we favoured stability over coupling strength and chose:
$2E_{J}/E_{L}=1.2$, a factor of $3.3$ smaller than our previous implementation
[20]. Moreover, we engineered a weak hybridization between the memory and
buffer mode in order to minimize undesired nonlinear couplings such as the
Kerr effect, with a rate estimated below $1$ Hz.

Previous experiments constructed the Wigner distribution of the memory field
through a non-linear coupling to a transmon qubit and its readout resonator
[20]. However, the finite thermal occupation of the transmon was suspected to
limit the bit-flip time to the millisecond range. Instead, we monitor our
memory through a minimally invasive detection tool: a weakly coupled
transmission line connected to a TWPA. This added leakage channel slightly
decreases the total quality factor but has the advantage of not inducing any
additional non-linear couplings to a lossy ancillary system.

##  IV Experiment calibration

The ATS contributes a non-linear potential energy of the form
$U_{\varphi_{\Sigma},\varphi_{\Delta}}(\varphi)=\frac{1}{2}E_{L}\varphi^{2}-2E_{J}\cos(\varphi_{\Sigma})\cos(\varphi+\varphi_{\Delta})$,
where $\varphi$ is the phase drop across the central inductance [20]. The
buffer and memory modes hybridize through their capacitive coupling, and hence
to the lowest order of their hybridization strength $\upsilon$, we have
$\varphi=\varphi_{b}(b+b^{\dagger}+\upsilon(a+a^{\dagger}))$, where the buffer
zero-point phase fluctuations verifies
$\varphi_{b}=\left(2E_{C}/E_{L}\right)^{1/4}$ [27]. We operate the ATS at
$\varphi_{\Sigma}=-\pi/2+\varepsilon_{p}\cos(\omega_{p}t)$, and
$\varphi_{\Delta}=\pi/2$. At this operating point the buffer resonates at
$\omega_{b}/2\pi=6.1273$ GHz with an energy decay rate $\kappa_{b}/2\pi=16$
MHz. By tuning the pump frequency at $\omega_{p}=2\omega_{a}-\omega_{b}$ and
driving the buffer mode at $\omega_{d}=\omega_{b}$, we synthesize the
Hamiltonian of Eq. (2), with $\hbar
g_{2}=-\frac{1}{2}E_{J}\varepsilon_{p}\upsilon^{2}\varphi_{b}^{3}$.

We start the experiment by measuring the buffer mode frequency map as a
function of the two DC currents. Conveniently, the desired operating point
$(\varphi_{\Sigma},\varphi_{\Delta})=(-\pi/2,\pi/2)$ is easily recognisable
since it corresponds to a saddle point of this map (see [25], Sec. S2).
Consequently, the buffer and memory modes are first order insensitive to flux
noise.

The next step is to activate the RF pump and buffer drive. We pick the largest
pump power that does not deteriorate the buffer and memory modes spectra. The
modes frequencies are Stark shifted in the presence of this strong pump.
Therefore, a precise calibration of the pump and drive frequencies is required
to rigorously verify the frequency matching conditions:
$\omega_{p}=2\omega_{a}-\omega_{d}$, and $\omega_{d}=\omega_{b}$. To this end,
we acquire the memory mode fluorescence as a function of detunings from these
matching conditions. As the drive amplitude $\epsilon_{d}$ is increased, the
region over which the drive and pump combine to populate the memory expands
around the frequency matching point (see Fig. 2a). For the remaining of the
experiment, we place ourselves at the center of these regions (colloquially
referred to as diamonds).

We calibrate the number of photons in the memory by measuring
$\overline{I^{2}+Q^{2}}$ (see [25], Sec. S4) as a function of the drive
amplitude, where $I$ and $Q$ are the in-phase and out-of-phase quadratures of
the radiated field acquired over an integration time $T_{m}$. (see Fig. 2b).
Three notable features are apparent. First, in the limit of strong drives, the
radiated energy scales linearly with the drive amplitude, a signature of the
conversion of 1 buffer photon to 2 memory photons. This is in stark contrast
with the common quadratic scaling for a driven harmonic oscillator. Moreover,
the offset of this asymptote from the origin excludes the Kerr effect as the
underlying process (see [25], Sec. S2.4). Second, the output power is close to
zero up until a critical drive amplitude, reminiscent of a non-linear
dissipative phase transition [28]. This transition occurs when the two-photon
injection rate overcomes the memory losses, setting the scale for the product
$\epsilon_{d}g_{2}$ to its value at the critical point
$\epsilon_{d}g_{2}=\kappa_{a}\kappa_{b}/8$. Furthermore, we compute the
classical dependence of $\bar{n}g_{2}^{2}$ on $\epsilon_{d}g_{2}$,
demonstrating that it is invariant under a change of $g_{2}$ (see [25], Sec.
S4). Importantly, quantum fluctuations blur the transition out of vacuum
resulting in a non-scale invariant curvature, a striking deviation from the
sharp transition expected in the classical regime. A full quantum model is
necessary to capture this third notable feature, from which we extract the key
parameter $g_{2}/2\pi=39$ kHz, and deduce $\kappa_{2}/2\pi=370$ Hz and
$\bar{n}$ for every $\epsilon_{d}$. This places our experiment in the regime
where $\kappa_{a}/\kappa_{2}=150\gg 1$. In the future we will increase the
hybridization factor $\upsilon$ to enter the regime suitable for a qubit
implementation: where the two-photon exchange rate largely dominates the
cavity losses. Finally, with the photon number calibration in hand, the
measurement records $I$ and $Q$ are rescaled to respectively coincide with a
measurement of $(a+a^{\dagger})/2$ and $(a-a^{\dagger})/2i$.

Figure 3: Real-time oscillator dynamics revealed by individual trajectories.
For photon numbers $\bar{n}=11,28,43$ (top, middle, bottom), we respectively
set the integration time to $T_{m}=0.1,1,5$ ms and the total measurement
duration to $T_{tot}=10,1000,5000$ s. (Left) Histogram of the $(I,Q)$
quadratures of the radiated field. (Center) Trajectory of the $I$-quadrature
as a function of time cropped from the full data-set. (Right) Cumulative
distribution function of the stochastic time interval $\tau_{\text{jump}}$ in
between two consecutive jumps. Its average value, that defines the bit-flip
time, is printed on each panel.

In fact, the observed phase transition corresponds to a spontaneous symmetry
breaking, where the cavity field adopts two opposite phases (or any quantum
superposition of the two in the absence of single photon loss). We observe the
emergence of these two phases by continuously acquiring, for each drive
amplitude, 10000 times the $I$-quadrature of the radiated field integrated
over $T_{m}=1$ ms, for a total measurement duration of 10 s (see Fig 2c). For
the lowest drive amplitudes ($\epsilon_{d}/2\pi\lesssim 2.7$ MHz), the cavity
state remains in the vacuum, as signaled by the Gaussian distribution centered
at $I=0$. This distribution then broadens around the critical point
($2.7~{}\text{MHz}\lesssim\epsilon_{d}/2\pi\lesssim 3.5~{}\text{MHz}$), due to
the significant overlap of the distributions of states $\ket{\pm\alpha}$ at
small $\alpha$ and the multiple flips in between during the acquisition time
$T_{m}=1$ ms. For ($3.5~{}\text{MHz}\lesssim\epsilon_{d}/2\pi\lesssim
4~{}\text{MHz}$), the two states are well resolved, and their approximately
equal weights hint towards a bit-flip time larger than the acquisition time of
$1$ ms and smaller than the full experiment duration of 10 s. For
$\epsilon_{d}/2\pi\gtrsim 4$ MHz, the field stays pinned to one of the two
computational states, hinting towards bit-flip times exceeding 10 s.

##  V Trajectories and bit-flip times

We access the dynamics of the memory by tracking individual trajectories over
time (see Fig. 3). For each trajectory, we set the drive amplitude at a fixed
value $\epsilon_{d}$, and record the $I$-quadrature of the radiated field. In
order to resolve quantum jumps, we set the integration time $T_{m}$ to be
simultaneously smaller than the bit-flip time and sufficiently large to
average out the heterodyne detection noise. To capture the statistical
properties of each trajectory, we plot the cumulative distribution function of
the interval between two consecutive jumps, denoted $\tau_{\text{jump}}$. It
shows approximately an exponential law, revealing an underlying Poisson
process. The average of $\tau_{\text{jump}}$, that defines the bit-flip time,
undergoes a spectacular increase from 1 ms to 0.3 s to 206 s for an increase
of photon number $\bar{n}$ from 11 to 28 to 43. With respect to the bare
cavity lifetime of $2.7{\rm\mu}$s, this represents a $10^{8}$ increase of the
bit-flip time, and (although inaccessible with our measurement scheme) an
estimated $2\times 43=86$ fold decrease of the phase-flip time.

We quantify the scaling of the bit-flip time with the photon number by
repeating the trajectory acquisition procedure for multiple drive amplitudes.
From each trajectory we extract the bit-flip time and the corresponding photon
number, and display them in Fig. 4. We observe two distinct regimes.
Initially, the bit-flip time rises exponentially multiplying by a factor of
about $1.4$ for every added photon. In theory, in the limit where
$\kappa_{a}/\kappa_{2}\ll 1$, this factor would approach $e^{2}\sim 7.4$ [13].
In this experiment we favoured stability over coupling strength, placing
ourselves in the opposite regime $\kappa_{a}/\kappa_{2}\sim 150$, which is
expected to decrease this factor as confirmed by numerical simulations (see
[25], Sec. S5). For $\bar{n}\gtrsim 40$ photons, the bit-flip time saturates
in the 100 second range. Although the origin of this saturation is yet to be
established, its timescale is compatible with the measured rate of highly
correlated errors in a large array of qubits [29], possibly due to high energy
particle impacts [30, 31].

##  VI Conclusion

In conclusion, we have measured timescales of order 100 seconds for bit-flips
between pointer states of a two-photon dissipative oscillator containing about
40 photons. To reach these macroscopic bit-flip times with mesoscopic photon
numbers, we designed a two-photon exchange circuit in a regime expected to
circumvent dynamical instabilities, and employed a minimally invasive
detection tool that collects the oscillator’s radiated field. Our experiment
thus puts a scale on the bit-flip times attainable with two-photon
dissipation, a necessary mechanism for quantum information processing with
cat-qubits [32]. Future work could be to uncover the phenomena causing these
bit-flip events [33, 30, 31] by monitoring oscillator trajectories over
timescales of days or weeks. Also, gradually increasing the two-photon
exchange rate so that it supersedes all loss mechanisms would lead to the
regime suitable for the cat-qubit where quantum superposition states can be
prepared and measured, thereby paving the way towards fully protected chains
of cat-qubits [34, 35].

Figure 4: Exponential suppression of bit-flips. The bit-flip time (y-axis,
log-scale) is measured (open circles) as a function of the number of photons
contained in the pointer states $\ket{0/1}_{\alpha}$ (x-axis). The bit-flip
time increases exponentially, multiplying by 1.4 per photon (tilted dashed
line) before saturating at approximately 127 s (horizontal dashed line).

##  VII Acknowledgments

We thank Lincoln Labs for providing a Josephson Traveling-Wave Parametric
Amplifier. The devices were fabricated within the consortium Salle Blanche
Paris Centre. This work was supported by the QuantERA grant QuCOS, by ANR
19-QUAN-0006-04. Z.L. acknowledges support from ANR project ENDURANCE, and
EMERGENCES grant ENDURANCE of Ville de Paris. This work has been supported by
the Paris Île-de-France Region in the framework of DIM SIRTEQ. This project
has received funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (grant
agreements No. 851740 and No. 884762).

##  VIII Author contributions

C.B simulated and fabricated the device. C.B, A.M and Z.L measured the device.
C.B, A.M, U.R, R.L, S.J and Z.L analyzed the data and co-wrote the manuscript
with input from all authors. W.C.S, M.V, A.D and P.M designed the sample
holder. Support was provided by J.P, N.P and M.R for nanofabrication, M.D,
T.K, R.L and T.P for experimental tools, F.R for microwave engineering, L.A.S,
P.R, A.S and M.M for theory. Z.L, R.L, P.C and M.M conceived the experiment.

## References

  * Bruzewicz  _et al._ [2019] C. D. Bruzewicz, J. Chiaverini, R. McConnell, and J. M. Sage, Trapped-ion quantum computing: Progress and challenges, Applied Physics Reviews 6, 021314 (2019), https://doi.org/10.1063/1.5088164 .
  * Vandersypen and Eriksson [2019] L. M. K. Vandersypen and M. A. Eriksson, Quantum computing with semiconductor spins, Physics Today 72, 38 (2019), https://doi.org/10.1063/PT.3.4270 .
  * Blais  _et al._ [2021] A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff, Circuit quantum electrodynamics, Rev. Mod. Phys. 93, 025005 (2021).
  * Arute  _et al._ [2019] F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. S. L. Brandao, D. A. Buell, B. Burkett, Y. Chen, Z. Chen, B. Chiaro, R. Collins, W. Courtney, A. Dunsworth, E. Farhi, B. Foxen, A. Fowler, C. Gidney, M. Giustina, R. Graff, K. Guerin, S. Habegger, M. P. Harrigan, M. J. Hartmann, A. Ho, M. Hoffmann, T. Huang, T. S. Humble, S. V. Isakov, E. Jeffrey, Z. Jiang, D. Kafri, K. Kechedzhi, J. Kelly, P. V. Klimov, S. Knysh, A. Korotkov, F. Kostritsa, D. Landhuis, M. Lindmark, E. Lucero, D. Lyakh, S. Mandrà, J. R. McClean, M. McEwen, A. Megrant, X. Mi, K. Michielsen, M. Mohseni, J. Mutus, O. Naaman, M. Neeley, C. Neill, M. Y. Niu, E. Ostby, A. Petukhov, J. C. Platt, C. Quintana, E. G. Rieffel, P. Roushan, N. C. Rubin, D. Sank, K. J. Satzinger, V. Smelyanskiy, K. J. Sung, M. D. Trevithick, A. Vainsencher, B. Villalonga, T. White, Z. J. Yao, P. Yeh, A. Zalcman, H. Neven, and J. M. Martinis, Quantum supremacy using a programmable superconducting processor, Nature 574, 505 (2019).
  * Autran  _et al._ [2014] J. Autran, D. Munteanu, P. Roche, and G. Gasiot, Real-time soft-error rate measurements: A review, Microelectronics Reliability 54, 1455 (2014).
  * Earnest  _et al._ [2018] N. Earnest, S. Chakram, Y. Lu, N. Irons, R. K. Naik, N. Leung, L. Ocola, D. A. Czaplewski, B. Baker, J. Lawrence, J. Koch, and D. I. Schuster, Realization of a $\mathrm{\Lambda}$ system with metastable states of a capacitively shunted fluxonium, Phys. Rev. Lett. 120, 150504 (2018).
  * Dykman and Krivoglaz [1980] M. Dykman and M. Krivoglaz, Fluctuations in nonlinear systems near bifurcations corresponding to the appearance of new stable states, Physica A: Statistical Mechanics and its Applications 104, 480 (1980).
  * Guckenheimer and Holmes [1983] J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields (Springer New York, New York, NY, 1983) pp. 1–65.
  * Siddiqi  _et al._ [2004] I. Siddiqi, R. Vijay, F. Pierre, C. M. Wilson, M. Metcalfe, C. Rigetti, L. Frunzio, and M. H. Devoret, Rf-driven josephson bifurcation amplifier for quantum measurement, Phys. Rev. Lett. 93, 207002 (2004).
  * Krantz  _et al._ [2016] P. Krantz, A. Bengtsson, M. Simoen, S. Gustavsson, V. Shumeiko, W. D. Oliver, C. M. Wilson, P. Delsing, and J. Bylander, Single-shot read-out of a superconducting qubit using a josephson parametric oscillator, Nature Communications 7, 10.1038/ncomms11417 (2016).
  * Mabuchi [2011] H. Mabuchi, Nonlinear interferometry approach to photonic sequential logic, Applied Physics Letters 99, 153103 (2011).
  * Kerckhoff and Lehnert [2012] J. Kerckhoff and K. W. Lehnert, Superconducting microwave multivibrator produced by coherent feedback, Physical Review Letters 109, 10.1103/physrevlett.109.153602 (2012).
  * Mirrahimi  _et al._ [2014] M. Mirrahimi, Z. Leghtas, V. V. Albert, S. Touzard, R. J. Schoelkopf, L. Jiang, and M. H. Devoret, Dynamically protected cat-qubits: A new paradigm for universal quantum computation, New J. Phys. 16, 045014 (2014).
  * Carmichael [2008] H. J. Carmichael,  _Statistical Methods in Quantum Optics 2_ (Springer Berlin Heidelberg, 2008).
  * Leghtas  _et al._ [2015] Z. Leghtas, S. Touzard, I. M. Pop, A. Kou, B. Vlastakis, A. Petrenko, K. M. Sliwa, A. Narla, S. Shankar, M. J. Hatridge, M. Reagor, L. Frunzio, R. J. Schoelkopf, M. Mirrahimi, and M. H. Devoret, Confining the state of light to a quantum manifold by engineered two-photon loss, Science 347, 853 (2015).
  * Lescanne  _et al._ [2020] R. Lescanne, M. Villiers, T. Peronnin, A. Sarlette, M. Delbecq, B. Huard, T. Kontos, M. Mirrahimi, and Z. Leghtas, Exponential suppression of bit-flips in a qubit encoded in an oscillator, Nature Physics 16, 509 (2020).
  * Touzard  _et al._ [2018] S. Touzard, A. Grimm, Z. Leghtas, S. O. Mundhada, P. Reinhold, C. Axline, M. Reagor, K. Chou, J. Blumoff, K. M. Sliwa, S. Shankar, L. Frunzio, R. J. Schoelkopf, M. Mirrahimi, and M. H. Devoret, Coherent oscillations inside a quantum manifold stabilized by dissipation, Phys. Rev. X 8, 021005 (2018).
  * Puri  _et al._ [2017] S. Puri, S. Boutin, and A. Blais, Engineering the quantum states of light in a Kerr-nonlinear resonator by two-photon driving, npj Quantum Inf. 3, 18 (2017).
  * Grimm  _et al._ [2020] A. Grimm, N. E. Frattini, S. Puri, S. O. Mundhada, S. Touzard, M. Mirrahimi, S. M. Girvin, S. Shankar, and M. H. Devoret, Stabilization and operation of a kerr-cat qubit, Nature 584, 205 (2020).
  * Lescanne  _et al._ [2019] R. Lescanne, L. Verney, Q. Ficheux, M. H. Devoret, B. Huard, M. Mirrahimi, and Z. Leghtas, Escape of a driven quantum josephson circuit into unconfined states, Phys. Rev. Applied 11, 014030 (2019).
  * Verney  _et al._ [2019] L. Verney, R. Lescanne, M. H. Devoret, Z. Leghtas, and M. Mirrahimi, Structural instability of driven josephson circuits prevented by an inductive shunt, Phys. Rev. Applied 11, 024003 (2019).
  * [22] M. Burgelman, A. Sarlette, and M. Mirrahimi, In preparation, .
  * Macklin  _et al._ [2015] C. Macklin, K. OBrien, D. Hover, M. E. Schwartz, V. Bolkhovsky, X. Zhang, W. D. Oliver, and I. Siddiqi, A near-quantum-limited Josephson traveling-wave parametric amplifier, Science 350, 307 (2015).
  * Wang  _et al._ [2015] C. Wang, C. Axline, Y. Y. Gao, T. Brecht, Y. Chu, L. Frunzio, M. H. Devoret, and R. J. Schoelkopf, Surface participation and dielectric loss in superconducting qubits,  _Applied Physics Letters_ , Appl. Phys. Lett. 107, 162601 (2015).
  * [25] See supplementary information, .
  * Pozar [2012] D. M. Pozar,  _Microwave Engineering_, 4th ed. (Wiley, 2012).
  * Girvin [2014] S. M. Girvin, Circuit QED: Superconducting qubits coupled to microwave photons, in  _Quantum Machines: Measurement and Control of Engineered Quantum Systems (Les Houches Session XCVI)_, edited by M. Devoret, B. Huard, R. Schoelkopf, and L. F. Cugliandolo (Oxford University Press, 2014) pp. 113–256.
  * Mylnikov  _et al._ [2021] V. Y. Mylnikov, S. O. Potashin, G. S. Sokolovskii, and N. S. Averkiev, Dissipative phase transition in systems with two-photon driving and dissipation near the critical point, in  _2021 Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC)_ (IEEE, 2021).
  * McEwen  _et al._ [2021] M. McEwen, L. Faoro, K. Arya, A. Dunsworth, T. Huang, S. Kim, B. Burkett, A. Fowler, F. Arute, J. C. Bardin, A. Bengtsson, A. Bilmes, B. B. Buckley, N. Bushnell, Z. Chen, R. Collins, S. Demura, A. R. Derk, C. Erickson, M. Giustina, S. D. Harrington, S. Hong, E. Jeffrey, J. Kelly, P. V. Klimov, F. Kostritsa, P. Laptev, A. Locharla, X. Mi, K. C. Miao, S. Montazeri, J. Mutus, O. Naaman, M. Neeley, C. Neill, A. Opremcak, C. Quintana, N. Redd, P. Roushan, D. Sank, K. J. Satzinger, V. Shvarts, T. White, Z. J. Yao, P. Yeh, J. Yoo, Y. Chen, V. Smelyanskiy, J. M. Martinis, H. Neven, A. Megrant, L. Ioffe, and R. Barends, Resolving catastrophic error bursts from cosmic rays in large arrays of superconducting qubits, Nature Physics 18, 107 (2021).
  * Vepsäläinen  _et al._ [2020] A. P. Vepsäläinen, A. H. Karamlou, J. L. Orrell, A. S. Dogra, B. Loer, F. Vasconcelos, D. K. Kim, A. J. Melville, B. M. Niedzielski, J. L. Yoder, S. Gustavsson, J. A. Formaggio, B. A. VanDevender, and W. D. Oliver, Impact of ionizing radiation on superconducting qubit coherence, Nature 584, 551 (2020).
  * Cardani  _et al._ [2021] L. Cardani, F. Valenti, N. Casali, G. Catelani, T. Charpentier, M. Clemenza, I. Colantoni, A. Cruciani, G. D’Imperio, L. Gironi, L. Grünhaupt, D. Gusenkova, F. Henriques, M. Lagoin, M. Martinez, G. Pettinari, C. Rusconi, O. Sander, C. Tomei, A. V. Ustinov, M. Weber, W. Wernsdorfer, M. Vignati, S. Pirro, and I. M. Pop, Reducing the impact of radioactivity on quantum circuits in a deep-underground facility, Nature Communications 12, 10.1038/s41467-021-23032-z (2021).
  * Gautier  _et al._ [2021] R. Gautier, A. Sarlette, and M. Mirrahimi, Combined dissipative and hamiltonian confinement of cat qubits (2021), arXiv:2112.05545 .
  * Vool  _et al._ [2014] U. Vool, I. M. Pop, K. Sliwa, B. Abdo, C. Wang, T. Brecht, Y. Y. Gao, S. Shankar, M. Hatridge, G. Catelani, M. Mirrahimi, L. Frunzio, R. J. Schoelkopf, L. I. Glazman, and M. H. Devoret, Non-Poissonian quantum jumps of a fluxonium qubit due to quasiparticle excitations, Phys. Rev. Lett. 113, 247001 (2014).
  * Guillaud and Mirrahimi [2019] J. Guillaud and M. Mirrahimi, Repetition cat qubits for fault-tolerant quantum computation, Physical Review X 9, 10.1103/physrevx.9.041053 (2019).
  * Chamberland  _et al._ [2022] C. Chamberland, K. Noh, P. Arrangoiz-Arriola, E. T. Campbell, C. T. Hann, J. Iverson, H. Putterman, T. C. Bohdanowicz, S. T. Flammia, A. Keller, G. Refael, J. Preskill, L. Jiang, A. H. Safavi-Naeini, O. Painter, and F. G. Brandão, Building a fault-tolerant quantum computer using concatenated cat codes, PRX Quantum 3, 10.1103/prxquantum.3.010329 (2022).
  * Azouit  _et al._ [2016] R. Azouit, A. Sarlette, and P. Rouchon, Adiabatic elimination for open quantum systems with effective lindblad master equations, in  _2016 IEEE 55th Conference on Decision and Control (CDC)_ (IEEE, 2016).
  * Bartolo  _et al._ [2016] N. Bartolo, F. Minganti, W. Casteels, and C. Ciuti, Exact steady state of a kerr resonator with one- and two-photon driving and dissipation: Controllable wigner-function multimodality and dissipative phase transitions, Phys. Rev. A 94, 033841 (2016).
  * Barchielli and Gregoratti [2009] A. Barchielli and M. Gregoratti,  _Quantum Trajectories and Measurements in Continuous Time_ (Springer Berlin Heidelberg, 2009).
  * Tilloy [2018] A. Tilloy, Exact signal correlators in continuous quantum measurements, Phys. Rev. A 98, 010104 (2018).
  * Johansson  _et al._ [2012] J. Johansson, P. Nation, and F. Nori, Qutip: An open-source python framework for the dynamics of open quantum systems, Computer Physics Communications 183, 1760 (2012).
  * Johansson  _et al._ [2013] J. Johansson, P. Nation, and F. Nori, Qutip 2: A python framework for the dynamics of open quantum systems, Computer Physics Communications 184, 1234 (2013).

Supplementary Material

##  S1 Device fabrication and wiring

#### Wafer preparation

We sputter 120 nm of Nb on a 2-inch intrinsic silicon wafer, with a 280 µm
thickness and a resistivity larger than 10 k$\Omega$cm. We fabricate twelve
10$\times$11 mm chips on the same wafer. We dice the individual chips at the
end of the fabrication process and select the sample that is best suited for
the experiment.

#### Circuit patterning

We pattern the large features of the circuit ($>5$ µm) using laser
lithography. We spin positive resist (S1805), expose the pattern, then develop
in MF319 for 1 min and rinse in deionized (DI) water for 1 min. The wafer is
then etched in a reactive ion etching machine with a SF6 plasma and a 10 s
overetch. A 30 min lift-off step follows in a 50 °C acetone bath with
sonication. Finally, the sample is rinsed in isopropyl alcohol (IPA) for 1
min, blow dried and cleaned in an O2 plasma for 20s, thus stripping residual
organic contaminants.

#### Junction patterning

Our Josephson junctions are fabricated from Dolan bridges patterned with
electron beam (e-beam) lithography. We spin two layers of resist: first,
methacrylic acid/ methyl methacrylate (MAA EL13) baked for 3 min at 185 °C and
second, poly(methyl methacrylate) (PMMA A3) baked for 30 min at 185 °C. Once
the e-beam patterning completed, we develop in a IPA:H2O (3:1) bath at 6 °C
for 2 min, rinse for 10 s in IPA and blow dry. Finally, residual organic
contaminants below the bridges are stripped by an O2 plasma for 10 s.

#### Junction deposition

The wafer is then introduced in an e-beam evaporator. We start with a 2 min
argon milling step at an angle of $\pm$ 30° to prepare for a good electrical
contact with the Nb layer. We deposit two layers of aluminium (35 nm then 70
nm thick) at an angle of $\pm$ 30°, separated by a static oxidation in a pure
O2 atmosphere at 10 mbar for 10 min. Before venting to air, the chamber is
filled with 300 mbar of O2 for 5 min. We lift-off in a 50 °C acetone bath for
1 h, transfer the wafer to a new acetone bath for 5 min and sonicate for 10 s,
then rinse in IPA and blow dry. Images of the fabricated junctions are
displayed in Fig. S1.

Figure S1: (a) Optical micrograph of the asymmetrically threaded SQUID (ATS)
made of aluminium (light grey) deposited on the niobium circuit (grey) over a
silicon substrate (dark grey). (b-d) Scanning electron microscope images of
the small single junctions (b,d) forming the SQUID loop and the five array
junctions (c) forming the inductive shunt. The small junctions are 275 nm
$\times$ 700 nm. The array junctions, which would ideally all be equal in
area, are in fact composed of three 600nm $\times$ 3.9 µm and two 270 nm
$\times$ 3.9 µm junctions. This results in a critical current density of about
450 nA$/$µm2. For clarity, small arrows point to the location of each
junction. Figure S2: Wiring of the experiment. Measurement apparatus for the
memory (blue labels), buffer (red labels), and TWPA pump (black label) connect
to the experiment through RF lines (black lines). DC voltage sources are used
to drive flux lines (green lines). Dashed lines indicate the different
temperature stages of the dilution refrigerator. Additional information is
provided in the legend (grey background), annotations and in the text. Figure
S3: Lumped element model of the circuit. The buffer (red) with bare frequency
$\omega_{b,0}/2\pi$ is constituted with an ATS (inductive energy $E_{L}$, mean
Josephson energy $E_{J}$, asymmetry $\Delta E_{J}$), and a capacitor with
charging energy $E_{C}$. The ATS loops are threaded with fluxes $\varphi_{L}$,
$\varphi_{R}$ (green) and is connected to the memory (blue), with bare
frequency $\omega_{a,0}/2\pi$. The phase $\varphi$ is indicated with an arrow
(black).

##  S2 Circuit Analysis

###  S2.1 Asymmetrically Threaded SQUID

The Asymmetrically Threaded SQUID (ATS) is the non-linear inductive dipole
that mediates the exchange of pairs of photons between the memory and its
environment. This dipole consists of a SQUID split in its center by an
inductance at a specific DC flux bias and is represented by the circuit of
Fig. S3.

The inductive energy of the ATS writes [16]

$\begin{split}U_{\varphi_{\Sigma},\varphi_{\Delta}}(\varphi)=\frac{1}{2}E_{L}\varphi^{2}&-2E_{J}\cos(\varphi_{\Sigma})\cos(\varphi+\varphi_{\Delta})\\\
&+2\Delta
E_{J}\sin(\varphi_{\Sigma})\sin(\varphi+\varphi_{\Delta})\,.\end{split}$ (S1)

where $E_{L}$ is the inductive energy of the shunt inductance, $E_{J}\pm\Delta
E_{J}$ are the Josephson energies of the left and right Josephson junctions
respectively, $\varphi$ is the superconducting phase difference across the ATS
and $\varphi_{\Sigma,\Delta}=(\varphi_{L}\pm\varphi_{R})/2$ are related to the
common and differential flux threading the ATS with $\varphi_{L,R}$ threading
the left and right loop of the ATS respectively.

#### Symmetries

The ATS potential has the following translational symmetries

$U_{\varphi_{\Sigma},\varphi_{\Delta}}(\varphi)=U_{\varphi_{\Sigma}+\pi,\varphi_{\Delta}+\pi}(\varphi)=U_{\varphi_{\Sigma}+\pi,\varphi_{\Delta}-\pi}(\varphi)\,.$
(S2)

and an inversion symmetry center at
$(\varphi_{\Sigma},\varphi_{\Delta})=(\pi/2,\pi/2)$ such that

$U_{\pi/2+\varphi_{\Sigma},\pi/2+\varphi_{\Delta}}(\varphi)=U_{\pi/2-\varphi_{\Sigma},\pi/2-\varphi_{\Delta}}(-\varphi)$
(S3)

Combining these three symmetries gives rise to a second inversion symmetry
center located at $(\varphi_{\Sigma},\varphi_{\Delta})=(\pi/2,-\pi/2)$. Hence,
all the information about the system is contained in the region
$\varphi_{\Sigma}\in[0,\pi]$, $\varphi_{\Delta}\in[-\pi/2,\pi/2]$. Note that
provided $\Delta E_{J}=0$, we have additional symmetry axes
$\varphi_{\Sigma}=0$ and $\varphi_{\Delta}=0$ such that

$U_{\varphi_{\Sigma},\varphi_{\Delta}}(\varphi)=U_{-\varphi_{\Sigma},\varphi_{\Delta}}(\varphi)=U_{\varphi_{\Sigma},-\varphi_{\Delta}}(-\varphi)$
(S4)

#### Saddle points

Let us study the potential around the inversion symmetry points
$(\varphi_{\Sigma},\varphi_{\Delta})=(\pi/2+\epsilon,\pm\pi/2+\delta)$

$\begin{split}U(\varphi,\epsilon,\delta)=\frac{1}{2}E_{L}\varphi^{2}&\mp
2E_{J}\sin(\epsilon)\sin(\varphi+\delta)\\\ &\pm 2\Delta
E_{J}\cos(\epsilon)\cos(\varphi+\delta)\end{split}$ (S5)

For small $\epsilon$ and $\delta$

$\begin{split}U(\varphi,\epsilon,\delta)&=\frac{1}{2}E_{L}\varphi^{2}\\\
&\mp(-2\Delta E_{J}+2E_{J}\epsilon\delta+\Delta
E_{J}(\epsilon^{2}+\delta^{2}))\cos(\varphi)\\\ &\mp(2E_{J}\epsilon+2\Delta
E_{J}\delta)\sin(\varphi)\,.\end{split}$ (S6)

For $\epsilon=\delta=0$, the potential reaches its minimum at
$\varphi_{\text{min}}=0$. At $\epsilon,\delta\neq 0$, we search for a first
order perturbation of $\varphi_{\text{min}}$. Solving for
$\frac{\partial}{\partial_{\varphi}}U(\varphi_{\text{min}},\epsilon,\delta)=0$,
we get

$\begin{split}\varphi_{\text{min}}=\pm\frac{2E_{J}\epsilon+2\Delta
E_{J}\delta}{E_{L}\mp 2\Delta E_{J}}\,.\end{split}$ (S7)

Around the minimum $\varphi_{\text{min}}$, the second derivative of the
potential with respect to $\varphi$, i.e. the inductive energy of the ATS
writes

$\begin{split}\frac{\partial^{2}}{{\partial\varphi}^{2}}U(\varphi_{\text{min}},\epsilon,\delta)&=E_{L}\mp
2\Delta E_{J}+E_{L}\varphi_{\text{min}}^{2}\pm 2E_{J}\epsilon\delta\\\
&\mp\Delta E_{J}(\varphi_{\text{min}}^{2}-\epsilon^{2}-\delta^{2})\end{split}$
(S8)

The ATS inductive energy has no linear terms in $\epsilon$ or $\delta$ so the
points $(\varphi_{\Sigma},\varphi_{\Delta})=(\pi/2,\pm\pi/2)$ are critical
points of the inductive map of the ATS as a function of $\epsilon$ and
$\delta$. Its quadratic dependence around the critical point has the following
matrix representation

$\begin{split}M(E_{L},E_{J},\Delta E_{J})&=\frac{4(E_{L}\mp\Delta
E_{J})}{(E_{L}\mp 2\Delta E_{J})^{2}}\begin{bmatrix}E_{J}^{2}&E_{J}\Delta
E_{J}\\\ E_{J}\Delta E_{J}&\Delta E_{J}^{2}\end{bmatrix}\\\
&\pm\begin{bmatrix}\Delta E_{J}&E_{J}\\\ E_{J}&\Delta
E_{J}\end{bmatrix}\end{split}$ (S9)

the determinant of which writes

$\det(M)=E_{L}^{2}\frac{\Delta E_{J}^{2}-E_{J}^{2}}{(E_{L}\mp 2\Delta
E_{J})^{2}}\,.$ (S10)

The determinant is negative (provided $\Delta E_{J}<E_{J}$) hence the critical
point is a saddle point. This property is used to tune the DC working point
experimentally (see Fig. S5). When $\Delta E_{J}\neq 0$, the two points
$(\varphi_{\Sigma},\varphi_{\Delta})=(\pi/2,\pm\pi/2)$ are non equivalent
saddle points of the ATS with inductive energy $E_{L}\mp 2\Delta E_{J}$.

###  S2.2 Circuit Hamiltonian

The dynamics of the circuit displayed in Fig. 1 is well captured by a reduced
lumped element model (see Fig. S3) with the following Hamiltonian [16]:

$\begin{split}H&=\hbar\omega_{a,0}a^{\dagger}a+\hbar\omega_{b,0}b^{\dagger}b\\\
&-2E_{J}\cos(\varphi_{\Sigma})\cos(\varphi+\varphi_{\Delta})\\\ &+2\Delta
E_{J}\sin(\varphi_{\Sigma})\sin(\varphi+\varphi_{\Delta})\end{split}$ (S11)

where $a,b$ are the memory and buffer annihilation operators. The buffer’s
angular frequency verifies $\omega_{b,0}=\sqrt{8E_{L}E_{C}}/\hbar$, where
$E_{L},E_{C}$ are the energies associated to the buffer’s inductive and
capacitive shunt respectively. The angular frequency of the memory is denoted
$\omega_{a,0}$. We denote $2E_{J}$ the sum of the Josephson energies of the
single junctions composing the SQUID loop, and $2\Delta E_{J}$ their
difference. During fabrication we aim for the smallest possible junction
asymmetry, however in practice we are left with $\Delta E_{J}/E_{J}\approx
0.5\%$ (see [25], Sec. S3.1) which leads to spurious Kerr and cross-Kerr
effects. We neglect $\Delta E_{J}$ in the rest of the analysis. The ATS is
threaded with a common and differential flux
$\varphi_{\Sigma,\Delta}=\frac{1}{2}(\varphi_{L}\pm\varphi_{R})$, where
$\varphi_{L,R}$ is the flux threading the left and right loop of the ATS. In
the limit where the hybridization factor $\upsilon$ between the buffer and
memory is much smaller than 1, the phase across the ATS denoted $\varphi$,
verifies
$\varphi=\varphi_{b}\left(b+b^{\dagger}+\upsilon(a+a^{\dagger})\right)$, where
the zero point phase fluctuations for the buffer reads
$\varphi_{b}=\left(2E_{C}/E_{L}\right)^{1/4}$.

###  S2.3 First order Hamiltonian at the operating point

By flux pumping the ATS around a well chosen DC working point [16]

$\begin{split}\varphi_{\Sigma}&=\frac{\pi}{2}+\epsilon_{p}\cos(\omega_{p}t)\\\
\varphi_{\Delta}&=\frac{\pi}{2}\end{split}$ (S12)

we engineer a two-to-one photon exchange Hamiltonian between the memory and
the buffer, provided the pump frequency $\omega_{p}$ is close to the matching
condition $\omega_{p}=2\omega_{a}-\omega_{b}$. This two-to-one photon exchange
Hamiltonian converts the strong single photon losses of the buffer into an
effective two-photon loss channel for the memory.

Likewise, a microwave drive at frequency $\omega_{d}$ close to the buffer
frequency, is converted into an effective two-photon drive of the memory (or
squeezing) at frequency $(\omega_{d}+\omega_{p})/2$. By definition, this
frequency is close to the memory frequency.

For the memory, the combination of the two-photon loss and two-photon drive,
stabilizes two coherent states with frequency $(\omega_{p}+\omega_{d})/2$ of
equal amplitude and opposite phase. The heterodyne demodulation frequency
$\omega_{\text{dm}}$ for the memory is constrained accordingly

$\omega_{\text{dm}}=\frac{\omega_{p}+\omega_{d}}{2}\,.$ (S13)

By going in the frame rotating at frequency $\omega_{\text{dm}}$ for the
memory and $\omega_{d}$ for the buffer and performing first order rotating
wave approximation (RWA), the Hamiltonian (S11) writes [16]

$H/\hbar=-\Delta_{a}a^{\dagger}a-\Delta_{b}b^{\dagger}b+g_{2}^{*}a^{2}b^{\dagger}+g_{2}{a^{2}}^{\dagger}b$
(S14)

where $\Delta_{a}=\omega_{\text{dm}}-\omega_{a}$,
$\Delta_{b}=\omega_{d}-\omega_{b}$ and $\omega_{a}$ and $\omega_{b}$ are
respectively the memory and buffer frequency accounting for the AC-stark shift
due to the pump [16].

Incorporating the buffer drive and the dissipation of the two modes, the
dynamics of the system is governed by

$\begin{split}H/\hbar&=-\Delta_{a}a^{\dagger}a-\Delta_{b}b^{\dagger}b\\\
&\qquad+g_{2}^{*}a^{2}{b}^{\dagger}+g_{2}a^{{\dagger}2}{b}-\epsilon_{d}b^{\dagger}-\epsilon_{d}^{*}b\\\
L_{a}&=\sqrt{\kappa_{a}}a\\\ L_{b}&=\sqrt{\kappa_{b}}b\end{split}$ (S15)

where $\epsilon_{d}$ is the buffer drive strength and $\kappa_{a}$ and
$\kappa_{b}$ are the single photon loss rate of the memory and the buffer
respectively.

We gain further insight on the dynamics of the system by performing the
adiabatic elimination of the buffer. This is justified provided
$g_{2}\ll\kappa_{b}$. Following the method of [36], we find that the reduced
dynamics of the memory is given by

$\begin{split}H_{a}/\hbar&=-\Delta_{a}a^{\dagger}a-\epsilon_{d}\gamma
a^{{\dagger}2}-\epsilon_{d}\gamma^{*}a^{2}+\Delta_{b}|\gamma|^{2}a^{{\dagger}2}a^{2}\\\
&L_{a^{2}}=\sqrt{\kappa_{b}|\gamma|^{2}}a^{2}\\\
&L_{a}=\sqrt{\kappa_{a}}a\end{split}$ (S16)
$\text{with}\quad\gamma=\frac{g_{2}}{\Delta_{b}+i\kappa_{b}/2}\,.$ (S17)

At $\Delta_{a}=\Delta_{b}=0$, we recover eq. (1)

$\begin{split}H_{2}/\hbar&=i\epsilon_{2}a^{{\dagger}2}-i\epsilon_{2}^{*}a^{2}\\\
&L_{2}=\sqrt{\kappa_{2}}a^{2}\\\ &L_{1}=\sqrt{\kappa_{a}}a\end{split}$ (S18)

with $\epsilon_{2}=2\epsilon_{d}g_{2}/\kappa_{b}$ and
$\kappa_{2}=4|g_{2}|^{2}/\kappa_{b}$.

###  S2.4 Steady-state photon number

In this paragraph, we derive the stationary mean photon number in the memory
using a semi-classical approximation

In the interaction picture, the dynamics arising from (S15) writes

$\begin{split}\frac{da}{dt}&=\left(i\Delta_{a}-\frac{\kappa_{a}}{2}\right)a-2ig_{2}a^{\dagger}b\\\
\frac{db}{dt}&=\left(i\Delta_{b}-\frac{\kappa_{b}}{2}\right)b-ig_{2}^{*}a^{2}+i\epsilon_{d}\,.\end{split}$
(S19)

We perform a mean-field approximation on mode $a$ and $b$, and compute the
steady-state of the simplified dynamics. The operator $a$ and $b$ are replaced
by their mean value, the complex numbers $\alpha$ and $\beta$.

This system always admits a solution in which the memory is in vacuum and
corresponds to

$\begin{split}\alpha&=0\\\
\beta&=\frac{-\epsilon_{d}}{\Delta_{b}+i\kappa_{b}/2}\,.\end{split}$ (S20)

This solution is stable provided it is the only solution of eq. (S19) for a
given set parameters.

Assuming $\alpha\neq 0$, we can write

$\begin{split}i\frac{\kappa_{a}}{2}+\Delta_{a}&=2g_{2}\beta
e^{-2i\theta_{a}}\\\
\left(i\frac{\kappa_{b}}{2}+\Delta_{b}\right)\beta&=g_{2}^{*}\alpha^{2}-\epsilon_{d}\end{split}$
(S21)

where $\theta_{a}=\arg(\alpha)$.

Solving for $\beta$ in the first equation and injecting in the second one, we
get

$\begin{split}|\alpha|^{2}&=\frac{\epsilon_{d}}{g_{2}^{*}}e^{-2i\theta_{a}}+z\\\
\text{with}\quad
z&=\frac{(i\kappa_{a}/2+\Delta_{a})(i\kappa_{b}/2+\Delta_{b})}{2|g_{2}|^{2}}\,.\end{split}$
(S22)

#### Zero-detuning

When $\Delta_{a}=\Delta_{b}=0$, eq. (S22) simplifies into

$|\alpha|^{2}=\frac{\epsilon_{d}}{g_{2}^{*}}e^{-2i\theta_{a}}-\frac{\kappa_{a}\kappa_{b}}{8|g_{2}|^{2}}\,.$
(S23)

leading to

$\begin{split}|\alpha|^{2}&=\max\left[\left|\frac{\epsilon_{d}}{g_{2}^{*}}\right|-\frac{\kappa_{a}\kappa_{b}}{8|g_{2}|^{2}},0\right]\\\
&=\max\left[\left|\frac{\epsilon_{d}}{g_{2}^{*}}\right|\left(1-\frac{\kappa_{a}}{4|\epsilon_{2}|}\right),0\right]\,.\end{split}$
(S24)

We recover the critical point when the two-photon drive overcomes the cavity
dissipation.

In the absence of calibrated input or output lines, the power radiated by the
memory is defined up to a constant, in particular the quantity
$|g_{2}\alpha|^{2}$ writes

$|g_{2}\alpha|^{2}=\max\left[|\epsilon_{d}g_{2}|-\frac{\kappa_{a}\kappa_{b}}{8},0\right]$
(S25)

which, as a function of $\epsilon_{d}g_{2}$, has a slope 1 and an x-intercept
$\kappa_{a}\kappa_{b}/8\,.$

In terms of the single mode effective quantities, eq. (S24) rewrites for
$\epsilon_{2}\geq\kappa_{a}/4$

$|\alpha|^{2}=\frac{1}{2\kappa_{2}}(4\epsilon_{2}-\kappa_{a})\qquad\text{(two-
photon dissipation)}\;.$

If instead of two-photon dissipation, Kerr effect of amplitude $K$ was
limiting the amplitude of the pointer states [37, 18], a similar semi-
classical analysis predicts a mean photon number for
$\epsilon_{2}\geq\kappa_{a}/4$

$|\alpha|^{2}=\frac{1}{2K}\sqrt{(4\epsilon_{2})^{2}-(\kappa_{a})^{2}}\qquad\text{(dissipative
Kerr)}\;,$

which is qualitatively different from what is observed in this experiment.

#### General case

Since $\theta_{a}$ and $\alpha$ are on separate sides of eq. (S22), we can
geometrically solve the system in the complex plane. The right-hand side is a
circle of radius $|\epsilon_{d}/g_{2}^{*}|$ centered on $z$. The left-hand
side is the positive real axis. In this picture, there can be 0, 1, or 2
intersections between this circle and the real positive axis, giving rise to
1, 3 or 5 solutions for the system, one for vacuum plus two for each
intersection since $\pm\alpha$ are both valid solutions. Experimentally
observed solutions are the ones that give rise to the largest field in the
memory. Hence $|\alpha|^{2}$, the mean photon number in steady-state, writes:

$\\!\\!|\alpha|^{2}=\begin{cases}\max\\!\left[\text{Re}(z)\\!+\\!\sqrt{\left|\frac{\epsilon_{d}}{g_{2}^{*}}\right|^{2}\\!\\!-\text{Im}(z)^{2}},0\right]\\!\\!,&\\!\\!\\!\text{if
}\left|\frac{\epsilon_{d}}{g_{2}^{*}}\right|^{2}\\!\\!\\!>\\!\text{Im}(z)^{2}\\\
0\,,&\\!\\!\\!\text{otherwise.}\end{cases}$ (S26)

In the $\Delta_{a},\Delta_{b}$ coordinates, the region where $\alpha^{2}$ is
non-zero forms a diamond shape. Here after, we provide the equation for the
borders of this feature colloquially referred to as a diamond.

The $\Delta_{a},\Delta_{b}$-plane is divided in two domains depending on the
sign of the quantity $\Delta_{a}\Delta_{b}-\kappa_{a}\kappa_{b}/4\,.$

The top-right and bottom-left edges of the diamonds are located in the
positive domain and given by

$\kappa_{a}\Delta_{b}+\kappa_{b}\Delta_{a}=\pm 4|\epsilon_{d}g_{2}|\,.$ (S27)

Note that the slope of this edge is given by $-\kappa_{b}/\kappa_{a}$. The
bottom-right and top-left edges are located in the negative domain and given
by

$(\frac{\kappa_{a}^{2}}{4}+\Delta_{a}^{2})(\frac{\kappa_{b}^{2}}{4}+\Delta_{b}^{2})=|2\epsilon_{d}g_{2}|^{2}\,.$
(S28)

Hence the border of the diamond only depends on the product
$\epsilon_{d}g_{2}$ and does not carry information on $g_{2}$ nor
$\epsilon_{d}$ independently. Moreover, we cannot determine $g_{2}$ and
$\epsilon_{d}$ even with the full diamond information (not only the edges).
Indeed, from the measurement of the rates $\kappa_{a}$, $\kappa_{b}$ and the
knowledge of the applied detunings $\Delta_{a}$, $\Delta_{b}$, one has
independently access to the quantity

$z^{\prime}=\frac{1}{2}(i\frac{\kappa_{a}}{2}+\Delta_{a})(i\frac{\kappa_{b}}{2}+\Delta_{b})\,.$
(S29)

In the absence of photon number calibration, we only learn from eq. S26 that
the power radiated by the memory is proportional to

$\braket{I^{2}}\propto\text{Re}(z^{\prime})+\sqrt{|\epsilon_{d}g_{2}|^{2}-\text{Im}(z^{\prime})^{2}}\,.$
(S30)

Thus, we only have access to the product $\epsilon_{d}g_{2}$ but not $g_{2}$
or $\epsilon_{d}$ independently.

##  S3 Tuning the experiment

Our experiment requires DC and RF powering. For optimal operation, two DC
currents and two RF powers and frequencies need to be fine tuned. In this
section we describe the sequence of calibration experiments we perform to set
the working point.

###  S3.1 DC currents

The first calibration experiment we perform is to extract the buffer and
memory frequencies as a function of the common and differential flux in the
ATS loop (see Fig. S4). From these maps we identify the circuit parameters and
locate the ATS saddle points.

In the following we describe the measurement protocol to acquire the buffer
frequency flux map. We set a tone at frequency $f$ on the buffer port and
record its reflected amplitude and phase as a function of the DC voltages
$V_{\Sigma,\Delta}=(V_{L}\pm V_{R})/2$ (see Fig. S2). The physical controls
$V_{\Sigma,\Delta}$ are transformed to the common and differential flux basis
$\varphi_{\Sigma,\Delta}$ to match the symmetries of the circuit Hamiltonian
(S11). A variation in the reflected signal is detected at flux points
$\varphi_{\Sigma,\Delta}(f)$ where the buffer frequency enters the vicinity of
$f$. This sequence is repeated by scanning $f$ in between 5.2 GHz and 9 GHz in
steps of 100 MHz. In Fig. S4, the frequency $f$ is encoded in the color of
pixels located at $\varphi_{\Sigma,\Delta}(f)$. We repeat the same protocol on
the memory port to extract the memory frequency flux map.

The theory plots in Fig S4 are obtained for the numerical diagonalization of
the Hamiltonian in Eq. (S11). From the ATS symmetries, we know that there
exist two nonequivalent families of saddle points, those generated from
${(\varphi_{\Delta},\varphi_{\Sigma})=(-\pi/2,\pi/2)}$, and those from
${(\varphi_{\Delta},\varphi_{\Sigma})=(\pi/2,\pi/2)}$. The junction asymmetry
$\Delta E_{J}$ lifts the degeneracy of the buffer frequency between these
points.

We refine the flux and frequency sweeps around these saddle points in order to
precisely pin down their location. In Fig. S5, we directly display the
reflected amplitude on the buffer port at different frequencies $f$. A saddle
point is easily identified as the closing of the buffer frequency contour
line. Note that the saddle point at $(-\pi/2,\pi/2)$ appears at
$f_{b1}=6.00\text{ GHz}$ as shown in the top middle panel. The second one
appears at $f_{b2}=6.04\text{ GHz}$ as shown in the bottom middle panel.

The parameters entering Eq. (S11) are listed in table 1. The charging energy
$E_{C}$ is extracted from 3D finite elements electromagnetic simulations. The
inductive energy $E_{L}$ and the junction asymmetry $\Delta E_{J}$ are
computed from the buffer frequencies at the saddle points that verify, in the
weakly hybridized limit:

$f_{b1,b2}=\frac{1}{h}\sqrt{8E_{C}(E_{L}\pm 2\Delta E_{J})}\;.$

The Josephson energy $E_{J}$ is extracted from the maximum buffer frequency
$f_{b_{\text{max}}}$ measured in the buffer flux map of Fig. S4. This maximum
value $f_{b_{\text{max}}}=8.9~{}\text{GHz}$ is reached for
${(\varphi_{\Delta},\varphi_{\Sigma})=(0,0)}$ and verifies, in the weakly
hybridized limit:
${f_{b_{\text{max}}}=\frac{1}{h}\sqrt{8E_{C}(E_{L}+2E_{J})}}$.

The memory frequency $f_{a}$ is extracted from the memory frequency flux map
in Fig. S4 at the saddle points. Due to the weak hybridization with the
buffer, the memory frequency difference at the two saddle points is
negligible. Finally we numerically find the hybridization factor $\upsilon$
that produces a memory frequency flux map in agreement with the data in Fig.
S4.

$E_{C}/h$ | 72.6 MHz
---|---
$E_{L}/h$ | 62.40 GHz
$E_{J}/h$ | 37.00 GHz
$\Delta E_{J}/h$ | 0.207 GHz
$\omega_{b,0}/2\pi$ | 6.020 GHz
$\varphi_{b}$ | 0.220
$\omega_{a,0}/2\pi$ | 4.0457 GHz
$\upsilon$ | 3.6%
Table 1: Buffer and memory parameters entering the Hamiltonian (S11). $E_{c}$
and $E_{l}$ are respectively the buffer’s charging and inductive energy.
$E_{J}$ and $\Delta E_{J}$ are the ATS mean Josephson energy and asymmetry.
$\omega_{b,0}/2\pi$ and $\omega_{a,0}/2\pi$ are buffer and memory bear
frequencies. $\varphi_{b}$ is the buffer zero point fluctuations. $\upsilon$
is the hybridization strength. From these numbers, we can estimate the Kerr
non-linearity of the memory $K_{a}<1$ Hz.

$\omega_{b}/2\pi$ | 6.1273 GHz
---|---
$\omega_{a}/2\pi$ | 4.0458 GHz
$\omega_{p}/2\pi$ | 2.07 GHz

$\kappa^{i}_{a}/2\pi$ | 18 kHz
---|---
$\kappa^{c}_{a}/2\pi$ | 40 kHz
$\kappa_{b}/2\pi$ | 16 MHz

$g_{2}/2\pi$ | 39 kHz
---|---
$\kappa_{2}/2\pi$ | 370 Hz
$\overline{n}_{\text{sat}}$ | 43

Table 2: Parameters at the operating point of the experiment.
$\omega_{b}/2\pi$, $\omega_{a}/2\pi$, $\omega_{p}/2\pi$ are the buffer, memory
and pump frequencies. $\kappa_{a}^{i}$, $\kappa_{a}^{c}$ are the internal and
coupling loss rates of the memory, $\kappa_{b}$ is the loss rate of the
buffer, $g_{2}$ is the two-photon coupling rate, $\kappa_{2}/2\pi$ is the two-
photon dissipation rate and $\bar{n}_{\rm sat}$ is the average number of
photons at the bit-flip time saturation. The following parameters are given
with confidence intervals: $\kappa_{a}^{i}\in[15,22]$ kHz,
$\kappa_{a}^{c}\in[39,42]$ kHz, $\kappa_{b}^{i}\in[13,20]$ MHz,
$g_{2}/2\pi\in[30,46]$ MHz, $\kappa_{2}\in[270,410]$,
$\overline{n}_{\text{sat}}\in[43,54]$. Figure S4: Measured (top) and simulated
(bottom) frequencies (color) of the buffer (left) and memory (right) as a
function of the differential flux (x-axis) and common flux (y-axis) in the ATS
loop. The orange cross marks the flux point at which we operate the
experiment. Figure S5: Each panel displays the measured relative amplitude
(color) of the reflected signal on the buffer port at the frequency $f$
specified in each label box, as a function of the differential (x-axis) and
common (y-axis) phase biases. The dashed grey lines are guides for the eye
corresponding to $\varphi_{\Delta}=\pm\pi/2$ and $\varphi_{\Sigma}=\pi/2$.

###  S3.2 Phase-locking

In the laboratory frame, at any point in time, the phase of the pointer states
resulting from the junction mixing process is given by

$\theta_{a}=(\theta_{p}+\theta_{d})/2$ (S31)

where $\theta_{p}$ and $\theta_{d}$ are respectively the pump and drive tone
phase. The pump tone is directly generated by the microwave signal generator
and pulsed via a microwave switch whereas the drive tone is pulsed via an IQ-
mixer (Fig. S2).The resulting phases of the tones are

$\theta_{p}=\theta_{p}^{LO},\quad\theta_{d}=\theta_{d}^{LO}+\theta_{d}^{IF}$
(S32)

where $\theta_{p}^{LO}$ and $\theta_{d}^{LO}$ are the Local Oscillator (LO)
phases of the microwave generator and $\theta_{d}^{IF}$ is the Intermediate
Frequency (IF) signal phase delivered by the Arbitrary Waveform Generator
(AWG) channel to pulse the drive tone. The radiated signal from the memory is
demodulated in a frame with phase

$\theta_{dm}=\theta_{dm}^{LO}+\theta_{dm}^{IF}$ (S33)

where $\theta_{dm}^{LO}$ is the phase of the demodulation LO and
$\theta_{dm}^{IF}$ is the phase of the Analog-Digital Converter (ADC). In
order to phase-lock the pointer states with the demodulation frame, we should
ensure

$\theta_{a}-\theta_{dm}={\rm cst}$ (S34)

The three LOs are generated with a single four channel Anapico signal
generators, and the two IFs with a single Quantum Machines OPX. The accuracy
of these instruments ensure that all the LOs share the same time reference and
all the IFs share the same time reference. However, given the high frequencies
at stake, the instrument sharing the same 50 MHz clock is not sufficient for
this two time references to be considered identical. The LO time is referred
to as $t$ and the IF time as $t^{\prime}$. Hence

$\begin{split}\theta_{a}-\theta_{dm}&=(\omega_{p}^{LO}t+\omega_{d}^{LO}t+\omega_{d}^{IF}t^{\prime})/2+{\rm
cst}\\\ &\quad-(\omega_{dm}^{LO}t+\omega_{dm}^{IF}t^{\prime})+{\rm cst}\\\
&=((\omega_{p}^{LO}+\omega_{d}^{LO})/2-\omega_{dm}^{LO})t\\\
&\quad+(\omega_{d}^{IF}/2-\omega_{dm}^{IF})t^{\prime}\,.\end{split}$ (S35)

and the phase-locking condition imposes the frequency matching conditions

$\begin{split}(\omega_{p}^{LO}+\omega_{d}^{LO})/2-\omega_{dm}^{LO}=0\\\
\omega_{d}^{IF}/2-\omega_{dm}^{IF}=0\end{split}$ (S36)

###  S3.3 Pump and drive frequencies

Once the DC biases are tuned at one saddle point and the memory and buffer
resonance frequencies are determined by direct spectroscopy in reflection on
their respective ports, we proceed to tune the pump and drive tones. The pump
frequency is determined via two-tone spectroscopy: a weak drive tone is used
to perform buffer spectroscopy while sweeping the pump frequency around the
frequency matching condition $\omega_{p}=2\omega_{a}-\omega_{b}$. As this
operation is being performed, we also perform the heterodyne detection of the
field radiated by the memory (Fig. S6).

When the two-to-one photon exchange is resonant, a sharp feature is observed
within buffer resonance, referred to as a diamond, and the memory starts to
radiate power. The discrepancy between the ideal and measured diamond shape is
used as a witness for the appearance of higher order processes. The pump
amplitude is set so as to maximize $g_{2}$ while mitigating detrimental higher
order effects.

In order to minimize the amount of data collected and accurately zoom on the
diamond feature Fig. S6a is acquired in the following way :

  * •

at a fixed pump frequency (x-axis of Fig. S6) the buffer spectroscopy is done
by varying the buffer IF frequency with a fixed LO frequency. Due to the
frequency constraint of eq. (S36) the heterodyne detection of the radiated
memory field is done by varying the memory IF frequency with a fixed LO
frequency.

  * •

for the demodulation frequency to remain close to the memory frequency while
varying the pump frequency, (y-axis of Fig. S6) the pump and drive LO
frequencies are varied in opposite directions. In this way, we have
$\omega_{p}+\omega_{b}^{LO}=2\omega_{dm}^{LO}=\text{cst.}$

In these coordinates, $\Delta_{b}$ and $2\Delta_{a}$ are varied along the
x-axis and $\Delta_{b}$ is varied along the y-axis. If there was no Stark
shift, the buffer resonance would be a diagonal line of slope 1 and the memory
line (when the two-photon drive is tuned with the memory mode) a vertical
line. In practice, these two lines are distorted and we numerically fit the
buffer and memory frequencies as a function of the pump frequency to perform
the change of basis leading to the diamond of Fig. S6d in the
$\Delta_{a},\Delta_{b}$ coordinate system.

To perform this change of coordinates, we evaluate the following functions
from measurements by linear interpolation:

$\begin{split}\Delta_{b}=f(\omega^{IF}_{d},\omega_{p})&=\omega_{d}^{IF}+\omega_{d}^{LO}-\omega_{b}[\omega_{p}]\\\
\Delta_{a}=g(\omega^{IF}_{d},\omega_{p})&=\frac{\omega_{b}^{IF}}{2}+\frac{\omega_{p}+\omega_{d}^{LO}}{2}-\omega_{a}[\omega_{p}]\\\
&=\omega_{dm}^{IF}+\omega_{dm}^{LO}-\omega_{a}[\omega_{p}]\,.\end{split}$
(S37)

For each data point, we can now compute the actual value of $\Delta_{a}$ and
$\Delta_{b}$. This enables us to display radiated energy by the memory in the
basis of Hamiltonian eq. (S15): previously distorted, the diamonds recover
their shape.

By construction, diamond center should coincide with the zero detuning point.
By exploiting the diamonds inversion symmetry, we can verify that the maximum
of the auto-correlation function

$\begin{split}&(\Delta_{a}^{0},\Delta_{b}^{0})\,=\\\
&\max_{(\Delta_{a},\Delta_{b})}\left(\iint|\alpha|^{2}(\delta_{a},\delta_{b})|\alpha|^{2}(\Delta_{a}-\delta_{a},\Delta_{b}-\delta_{b})\delta_{a}\delta_{b}\right)\end{split}$
(S38)

gives back the zero detuning point, up to a slight discrepancy due to the
diamond imperfection. Experimentally, we find the zero detuning point as the
convergence point of the diamond feature at vanishingly small drive amplitude
(see Fig. S6c). All three centers (construction, auto-correlation,
experimental) are shown in Fig. S7 and lie in a small region at the center of
the diamond.

Figure S6: (a) Relative amplitude (color) of the reflected signal on the
buffer port as a function of the pump frequency $\omega_{p}$ (left y-axis),
drive local oscillator (LO) frequency $\omega_{d}^{\text{LO}}$ (right y-axis)
and drive intermediate frequency $\omega_{d}^{\text{IF}}$ (x-axis). The drive
frequency is given by
$\omega_{d}=\omega^{\text{IF}}_{d}+\omega^{\text{LO}}_{d}$. For each pump
frequency, the drive LO frequency is set to
$\omega^{\text{LO}}_{d}=(2\omega_{a}-\omega_{p})+200~{}\text{MHz}$, such that
vertical lines correspond to constant detuning from the frequency matching
condition ($\Delta_{a}=cte$). The buffer drive resonance condition
$\Delta_{b}=0$ is determined by fitting each horizontal cuts of the map
(dashed red line). In the vicinity of $\Delta_{a}=0$ and $\Delta_{b}=0$, a
sharp feature indicates that the two-to-one photon exchange transition is
resonant (dotted green line). Spurious transitions appear near the frequency
matching condition $\omega_{p}+\omega_{d}=\omega_{2a}$ (blue dashed line),
where $\omega_{2a}$ is the frequency of the second harmonic of the memory
$\lambda/2$-resonator measured independently. (b) Zoom on the two-to-one
photons exchange transition for increasing drive amplitude $\epsilon_{d}$. (c)
radiated energy from the memory in units of circulating photon number (color)
as a function the pump frequency $\omega_{p}$ (left y-axis) and two-photon
drive intermediate frequency $\omega_{dm}^{\text{IF}}$ (x-axis), for
increasing drive amplitude. On these panels, the two-photon drive LO frequency
is set to $\omega^{\text{LO}}_{dm}=\omega_{a}+100~{}\text{MHz}$. When the two-
to-one photon exchange transition is resonant, the engineered two-photon drive
populates the memory. The average occupation of the memory is determined
thanks to an undercoupled port via heterodyne detection. (d) radiated energy
from the memory in units of circulating photon number (color) as a function of
the pump and drive detuning from the frequency matching condition $\Delta_{a}$
(x-axis), and the drive detuning from the buffer $\Delta_{b}$ (y-axis). In
these coordinates, the feature takes the shape of a regular diamond. Figure
S7: radiated energy from the memory in units of circulating photon number
(color) as a function of the detuning from the frequency matching condition
$\Delta_{a}$ (x-axis), and the detuning from the buffer resonance $\Delta_{b}$
(y-axis). Left column displays data, right column displays semi-classical
simulations for the corresponding drive amplitude. Orange cross shows the
position of the maximum of the auto-correlation for the largest drive
amplitude. Red cross shows the zero detuning point given by direct fit of
memory and buffer spectroscopy. Red circle is the point at which well averaged
data were taken to perform the fit of $g_{2}$. Figure S8: Two photon coupling
calibration. (a) radiated energy from the memory (y-axis) as a function of
drive amplitude (x-axis). There are two regimes: when the drive amplitude is
small, single photon loss overcome the two photon drive, and the memory stays
in the vacuum. Passed the critical point, the memory gets populated by a
coherent state with photon number asymptotically proportional to the drive
amplitude. The axes units are chosen so that the critical point is at
$\kappa_{a}\kappa_{b}/8$ and the asymptotic slope is $1$. The data correspond
to an integration time of $10~{}\mu s$ with 10000 averages (crosses). The
semi-classical model (green solid line) captures the position of the critical
point but fails to explain the curvature of the experimental data. A full
numerical simulation is used to reproduce the data where the only fitting
parameter is $g_{2}$ (red lines). (b) Zoom on the curvature around the
critical point (gray rectangle from (a) ), emphasizing the agreement between
simulations and experimental data.

##  S4 Photon number calibration

It is of central importance that our macroscopic bit-flip times were observed
for states containing only a few tens of photons. Indeed it is only in the low
photon number regime that this system can operate as a coherent qubit. A
reliable calibration of the number of photons $\bar{n}$ is therefore key to
this work. In this section, we describe two calibration methods, and check
their consistency. We start by computing the mapping between the cavity field
properties and the measured quadratures. Then, we detail the method used in
the main text to calibrate $\bar{n}$ from the curvature at the critical point.
Finally, we describe a method relying on the measurement of the detection
efficiency $\eta$.

###  S4.1 Heterodyne detection

The heterodyne detection of the field radiated by the memory results in two
signals that are integrated over a integration time $T_{m}$ to give out
$(I,Q)$ pairs time traces

$\begin{split}I_{t}&=\sqrt{G}\int_{t}^{t+T_{m}}\left(\sqrt{2\kappa_{a}^{c}\eta}{\rm
Tr}\left(\rho_{t^{\prime}}(a+a^{\dagger})/2\right){\rm d}t^{\prime}+{\rm
d}W_{I}\right)\\\
Q_{t}&=\sqrt{G}\int_{t}^{t+T_{m}}\left(\sqrt{2\kappa_{a}^{c}\eta}{\rm
Tr}\left(\rho_{t^{\prime}}(a-a^{\dagger})/2i\right){\rm d}t^{\prime}+{\rm
d}W_{Q}\right)\,,\end{split}$ (S39)

where $G$ is the gain of the amplification chain, $\kappa_{a}^{c}$ is the
coupling rate of the memory, $\eta$ is the quantum detection efficiency,
$\rho_{t}$ is the instantaneous state and ${\rm d}W_{I}$, ${\rm d}W_{Q}$ are
the noises added to each quadrature that verify ${\rm d}W_{I}^{2}={\rm
d}W_{Q}^{2}={\rm d}t$. The statistics of the distribution of the $(I,Q)$ pairs
collected over time gives information about the memory state. In particular,
we can verify that in the general case [38, 39] and in the limit of small
$T_{m}$

$\begin{split}\overline{I^{2}+Q^{2}}&=2GT_{m}+2G\kappa_{a}^{c}\eta
T_{m}^{2}\text{Tr}(\rho_{\infty}a^{\dagger}a)\\\
&=2GT_{m}+2G\kappa_{a}^{c}\eta T_{m}^{2}\bar{n}\\\ \end{split}$ (S40)

where $\bar{n}$ is the mean photon number, $\overline{I^{2}+Q^{2}}$ is the
statistical average over the $(I,Q)$ pairs collected over time and
$\rho_{\infty}$ is the steady-state density operator of the cavity. In our
specific case, we have verified both numerically and experimentally that this
limit is practically reached for $T_{m}=10$ ${\rm\mu}$s. In eq. (S40), the
offset $GT_{m}$ can be calibrated out from the average of
$\overline{I^{2}+Q^{2}}$ when the cavity is in vacuum which results in the
average energy radiated by the cavity over a period $T_{m}$ of

$\overline{I^{2}+Q^{2}}-\left.\overline{I^{2}+Q^{2}}\right|_{\text{vac}}=2G\kappa_{a}^{c}\eta
T_{m}^{2}\bar{n}\,.$ (S41)

###  S4.2 Critical point

For various values of buffer drive amplitude $\epsilon_{d}$, we measure the
average energy radiated by the memory for a duration $T_{m}=10$ ${\rm\mu}$s
according to eq. (S40) (see Fig. S8) which is proportional to $\bar{n}$. The
following paragraphs aim at calibrating this proportionality constant. First,
we calibrate the axes such that the only unknown parameter is $g_{2}$ using
the semi-classical analysis. Then, we use the quantum fluctuation at the
critical point to determine $g_{2}$. Finally, we explain how we use this
calibration to estimate the memory photon number in a given trajectory.

#### Semi-classical

As shown by equation (S24) in the semi-classical approximation, at zero
detuning, the critical point appears when
$|\epsilon_{d}g_{2}|=\kappa_{a}\kappa_{b}/8$ and from there the mean photon
number in the memory increases linearly with the drive amplitude. Using these
properties, we calibrate the drive amplitude axis in units of
$|\epsilon_{d}g_{2}|$: the x-intercept of the linear dependence at large
photon number (semi-classical regime) is located at $\kappa_{a}\kappa_{b}/8$.
The x-axis being calibrated, we linearly stretch the y-axis such that the
asymptotic slope is 1 in the strong drive regime. According to eq. (S25), this
transformation enforces the y-axis to $|\alpha g_{2}|^{2}$ and leads to the
scaled data of Fig. S8.

This rescaling crucially depends on the values of $\kappa_{a}$ and
$\kappa_{b}$ which are determined as follows. We measure the reflection
coefficient of the memory in the presence of a pump tone slightly detuned from
the frequency matching condition. This enables to capture the shift of
parameters (frequency, internal losses, coupling losses) due to nonlinear
effects arising from the pump while disabling the two-photon losses. From this
measurement we numerically fit $\kappa_{a}^{i}/2\pi\in[15,22]$ kHz and
$\kappa_{a}^{c}/2\pi\in[39,42]$ kHz. The same protocol fails to determine
precisely $\kappa_{b}$ due to the background induced by the band-stop filters
and the strong dependence of the buffer parameters on the pump frequency.
Instead, we use the diamond property derived in eq. (S27) that the top-right
and bottom-left edge of the diamond have a slope of $-\kappa_{b}/\kappa_{a}$.
We find $\kappa_{b}/2\pi$ in the range $[13,20]$ MHz.

Figure S9: radiated energy from the memory in units of circulating photon
number (color) as a function of the detunings $\Delta_{a}$ (x-axis), and
$\Delta_{b}$ (y-axis) defined in the text, for the drive amplitude
$\epsilon_{d}/2\pi=12.1$ MHz. The red solid line displays the fitted slope of
the top-right and bottom-left edge, yielding the ratio $\kappa_{b}/\kappa_{a}$
(see (S27)). The red dashed-dotted lines and red dashed lines respectively
give the upper and lower bound on this parameter (determined by graphical
reading).

We later propagate the parameter range found for $\kappa_{a}$ and $\kappa_{b}$
on the rest of the calibration to give a robust confidence interval for
$g_{2}$ and $\bar{n}$.

We can independently check the calibration of $|\alpha g_{2}|^{2}$ by studying
the excess internal losses arising from the two-photon dissipation. When the
two-photon dissipation becomes resonant, the internal losses of the memory
measured by direct spectroscopy increase drastically and become non-linear as
a function of the probe power. The effective internal losses of the memory
write

$\kappa_{a,\text{eff}}^{i}=\kappa_{a}^{i}+2\kappa_{2}|\alpha|^{2}$ (S42)

where $\kappa_{a}^{i}$ is the bare internal losses of the cavity and
$|\alpha|^{2}$ is the average circulating photon number due to the
spectroscopy tone. The excess losses rewrite $8|\alpha g_{2}|^{2}/\kappa_{b}$
and provides an independent calibration of $|\alpha g_{2}|^{2}$ that we find
in good agreement with the previous method.

#### Quantum signature

At the critical point, the semi-classical analysis fails to capture the
curvature of the mean photon number $\bar{n}$ as a function of the drive
amplitude $\epsilon_{d}$ (Fig. 2). This curvature results from the quantum
fluctuations at the dissipative phase transition [28]. Instead, we perform a
quantum analysis and compute the average photon number in the steady state
$\rho_{\infty}$ of the Lindblad equation generated by (S16),
$\bar{n}=\text{Tr}(\rho_{\infty}a^{\dagger}a)$ using the steadystate function
imported from the QuTiP python package. Once we express $|\alpha g_{2}|^{2}$
as a function of $|\epsilon_{d}g_{2}|$ (see Fig. S8) the only fitting
parameter is $g_{2}$. Given the range of $\kappa_{a}$ and $\kappa_{b}$, we
estimate $g_{2}/2\pi\in[30,46]$ kHz.

#### Trajectory calibration

We analyse the bit-flip time scale over several orders of magnitude, hence we
increase the integration time $T_{m}$ to keep manageable amount of data in
long bit-flip traces. Thanks to the previous calibration, we can readily get
the memory photon number from the $(I,Q)$ statistics of the trace. Indeed,
from eq. S40, we have both $G$ from the value of
$\left.\overline{I^{2}+Q^{2}}\right|_{\text{vac}}$ and $2G\kappa_{a}^{c}\eta$
from the calibration of $\bar{n}$. When given a trace with different
integration time $T_{m}^{\prime}$, we determine

$\bar{n}=\frac{\overline{I^{2}+Q^{2}}-2GT_{m}^{\prime}}{2G\kappa_{a}^{c}\eta
T_{m}^{\prime 2}}\,.$ (S43)

###  S4.3 Quantum detection efficiency

A different route leading to $\bar{n}$ is to measure the detection efficiency
$\eta$. Indeed, for a coherent state containing a number of photons $\bar{n}$,
the measured mean $\overline{I}$ and standard deviation $\sigma(I)$ of the
$I$-quadrature verify

$\bar{n}=\left(\frac{\overline{I}}{\sigma(I)}\right)^{2}\frac{1}{2\eta\kappa_{a}^{c}T_{m}}\;.$
(S44)

Inversely, from the calibration of $\bar{n}$ from the previous section, we
estimate $\eta\simeq 3\%$.

In this section, we independently evaluate $\eta$ by fabricating a device
containing a memory mode coupled to a transmon that serves as an in-situ
measurement of $\bar{n}$.

Figure S10: False-color optical micrograph of the detection efficiency chip in
Nb (grey) on Si (dark blue). The memory resonator (blue) is capacitively
coupled to a transmon (green).The inset is centered on the transmon and its
Josephson junction in Al (light grey). We can address the memory and collect
the reflected signal (blue waves) via the bottom $50\Omega$ port. The left
$50\Omega$ port is dedicated to drive the transmon (green waves). This sample
was also used to measure the memory thermal population of about $1\%$.

The fabricated memory mode is identical to the one described in the main text,
and the entire two-photon exchange apparatus is replaced with a transmon qubit
(see Fig. S10). The chip was mounted in a similar sample holder and measured
with an identical wiring as the experiment described in the main text. The
characteristics of the chip are listed in Table. 3.

$T_{1}$ | $19.3\,{\rm\mu s}$
---|---
$T_{2}$ | $24.3\,{\rm\mu s}$
$\kappa_{a}^{c}/2\pi$ | $38\,{\rm kHz}$
$\kappa_{a}^{i}/2\pi$ | $17\,{\rm kHz}$
$\chi/2\pi$ | $1.75\,{\rm MHz}$
Table 3: Parameters of the device used to calibrate the quantum detection
efficiency $\eta$. The transmon qubit lifetime and coherence times are denoted
$T_{1}$ and $T_{2}$. The memory coupling and internal loss rates are denoted
$\kappa_{a}^{c}$ and $\kappa_{a}^{i}$, and $\chi$ corresponds to the
dispersive coupling rate between the transmon and the memory.

Our evaluation of $\eta$ follows three steps. First, we perform a standard
spectroscopy in reflection of the memory mode in order to emulate a
measurement signal that is directly proportional to the intra-cavity field
amplitude $\braket{a}$. Second, for a given amplitude $a_{\rm in}$, we
calibrate the cavity photon number $\bar{n}=\braket{a^{\dagger}a}$ by
resolving the photon number splitting of the qubit. Third, for each calibrated
photon number we measure the fluctuations of the outgoing field $a_{\rm out}$
and retrieve $\eta$ by inverting Eq. (S44). We detail each step of this
procedure below.

#### Memory spectroscopy

For various incoming signal amplitudes $S_{\rm in}$, we perform a spectroscopy
measurement recording the reflected signal $S_{\rm out}$. Using the results of
the resonance fit, we can then translate the data in the $(I,Q)$ plane in
order to emulate a transmission signal: $S_{\rm t}=A\braket{a}$, where $A$ is
an unknown scaling factor to be calibrated.

#### Photon number resolved qubit spectroscopy

For various resonant signal amplitudes $S_{\rm in}$ we activate a drive on the
transmon at a fixed amplitude $S_{q}$ with a varying detuning $\Delta_{q}$.
The data $S_{\rm t}(\Delta_{q},S_{\rm in},S_{q})$ are then fitted to the
result of a numerical simulation that we detail in the following. Using the
steadystate function of the QuTiP package [40, 41], we solve the following
dynamics

$\displaystyle\partial_{t}{\rho}$ $\displaystyle=$
$\displaystyle-i\left[H,\rho\right]$ $\displaystyle+$ $\displaystyle
D\left[\sqrt{\kappa_{a}}a\right]\rho+D\left[\sqrt{\kappa_{1}}q\right]\rho+D\left[\sqrt{\kappa_{\rm\phi}}q^{\dagger}q\right]\rho$
$\displaystyle H$ $\displaystyle=$ $\displaystyle\Delta_{q}q^{\dagger}q-\chi
a^{\dagger}aq^{\dagger}q+\Omega_{a}\left(a+a^{\dagger}\right)+\Omega_{q}\left(q+q^{\dagger}\right)\;,$

where $a$ (resp. $q$) is the memory (resp. qubit) mode annihilation operator,
$\kappa_{1}=\frac{1}{T_{1}}$,
$\kappa_{\phi}=\frac{1}{T_{2}}-\frac{1}{2T_{1}}$, $\Omega_{a}$ (resp.
$\Omega_{q}$) is the drive on the memory (resp. qubit) and the remaining
parameters are defined in table 3. From this simulation we extract
$\braket{a}(\Delta_{q},\Omega_{a},\Omega_{q})$, that is used to fit the
dataset $S_{\rm t}(\Delta_{q},S_{\rm in},S_{q})$, where the fit parameters are
the proportionality constants relating $S_{\rm in}$ to $\Omega_{a}$, $S_{q}$
to $\Omega_{q}$ and $S_{\rm t}$ to $\braket{a}$ (see Fig. S11).

#### Output field statistics

For every drive amplitude $S_{\rm in}$, the previous fit estimates the intra-
cavity field $\braket{a}$, and hence $\bar{n}$. By acquiring histograms of the
output field $S_{\rm t}$, we now invert Eq. (S44) and retrieve $\eta\simeq
7\%$, a factor two larger than the previously estimated value. This deviation
can be attributed to differences in the RF connections of the two samples.
These values may be explained by lossy elements (two circulators, one Eccosorb
filter and two directional couplers) between the sample and the TWPA.

Figure S11: Calibration of the detection efficiency $\eta$. (Top) Qubit
spectroscopy showing photon number splitting: data (solid lines) and fit
(dashed lines). (Bottom) Product $\bar{n}\eta$ computed using equation S44 as
a function of the square input signal $S_{\text{in}}^{2}$ in units of photon
number.

##  S5 Bit-flip time simulations

We numerically simulate the dynamics of the memory described in Eq. (1) using
the mesolve function imported from the QuTiP python package [40, 41]. We run
the simulation for three different values of $g_{2}$ (or equivalently
$\kappa_{2}$). For each of these values, we sweep $\epsilon_{2}$ in order to
vary $\bar{n}=|\alpha|^{2}$ in the range of 4 to 40 photons. We initialize the
memory in the coherent state $\ket{\alpha}$, and fit the expectation value of
the annihilation operator $a$ to an exponentially decaying function. The
extracted decay time corresponds to the bit-flip time. In Fig. S12, we display
the computed bit-flip time as a function of the product
$\bar{n}\times(g_{2}/2\pi)^{2}$, since it is a well calibrated quantity in our
experiment. The data lie in the vicinity of the simulation results for
$g_{2}/2\pi=39$ kHz, thus confirming our calibration of $g_{2}$.

Figure S12: Numerical simulations (solid lines) of the bit-flip time (y-axis)
for three values of $g_{2}$ (labels) as a function of the number of photons in
the memory $\bar{n}$ multiplied by $(g_{2}/2\pi)^{2}$ (x-axis). The data
(dots) from Fig. 4 of the main text qualitatively matches the simulations for
$g_{2}/2\pi=39$ kHz, thus confirming our calibration of $g_{2}$.