# Onset of transmon ionization in microwave single-photon detection

Yuki Nojiri<EMAIL_ADDRESS>Walther-Meißner-Institut, Bayerische
Akademie der Wissenschaften, 85748 Garching, Germany  Physics Department,
School of Natural Sciences, Technical University of Munich, 85748 Garching,
Germany  Kedar E. Honasoge  Walther-Meißner-Institut, Bayerische Akademie der
Wissenschaften, 85748 Garching, Germany  Physics Department, School of Natural
Sciences, Technical University of Munich, 85748 Garching, Germany  Achim Marx
Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, 85748
Garching, Germany  Kirill G. Fedorov<EMAIL_ADDRESS>Rudolf Gross
<EMAIL_ADDRESS>Walther-Meißner-Institut, Bayerische Akademie der
Wissenschaften, 85748 Garching, Germany  Physics Department, School of Natural
Sciences, Technical University of Munich, 85748 Garching, Germany  Munich
Center for Quantum Science and Technology (MCQST), Schellingstr. 4, 80799
Munich, Germany

###### Abstract

By strongly driving a transmon-resonator system, the transmon qubit may
eventually escape from its cosine-shaped potential. This process is called
transmon ionization (TI) and known to be detrimental to the qubit coherence
and operation. In this work, we investigate the onset of TI in an
irreversible, parametrically-driven, frequency conversion process in a system
consisting of a superconducting 3D-cavity coupled to a fixed-frequency
transmon qubit. Above a critical pump power we find a sudden increase in the
transmon population. Using Rényi entropy, Floquet modes, and Husimi Q
functions, we infer that this abrupt change can be attributed to a quantum-to-
classical phase transition. Furthermore, in the context of the single-photon
detection, we measure a TI-uncorrected detection efficiency of up to 86% and
estimate a TI-corrected value of up to $78$% by exploiting the irreversible
frequency conversion. Our numerical simulations suggest that increasing the
detuning between the pump and qubit frequencies and increasing the qubit
anharmonicity can suppress the TI impact. Our findings highlight the general
importance of the TI process when operating coupled qubit-cavity systems.

##  I Introduction

In superconducting circuits and cavity quantum electrodynamics systems,
Josephson junctions play a pivotal role. Characterized by their strong
nonlinearity, and low losses, these junctions are fundamental for various
quantum information applications. Their crucial role is evident in the
realization of parametrically driven devices, including quantum amplifiers [1,
2, 3, 4, 5, 6], frequency converters [7, 8, 9], nonclassical light generators
[10, 11], stabilizers [12, 13], and microwave single-photon detectors (SPDs)
[14, 15]. This is attributed to three- or four-wave mixing processes enabled
by a strong microwave drive, denoted as the ”pump”. Recently, SPDs have
attracted particular attention as they can be applied for many quantum
protocols, such as quantum teleportation [16], entanglement distillation [17],
entanglement swapping [18], quantum erasers [19], quantum repeaters [20],
Gaussian boson sampling [21], and quantum error correction [22].

The development of SPDs in the microwave domain is particularly challenging.
Compared to devices operating in the optical regime, this is primarily
attributed to the lower photon energy [23, 24, 25, 26, 14, 15] and the high
performance standards demanded by applications in quantum information, quantum
sensing, and quantum illumination [27, 28, 29, 30, 31, 32, 33]. Despite these
obstacles, various microwave SPD designs with quantum efficiencies exceeding
50% have been demonstrated in recent years [23, 24, 25, 26, 14, 15]. The
strategic significance of microwave SPDs becomes evident in applications such
as dark matter axion search [29], spin fluorescence detection [34, 35], and
quantum radar [36, 32]. For example recently, the application of microwave
SPDs in quantum radar systems demonstrated an experimental quantum advantage
in a target detection [32]. In the field of spin fluorescence detection, using
microwave SPDs allows one to surpass the signal-to-noise-ratio performance of
conventional electron spin resonance methods [34, 35]. Microwave SPDs are also
expected to further advance the rapidly growing fields of microwave quantum
communication and sensing [11, 37, 38, 33].

Figure 1: (a) Principle of the microwave single-photon detection. A coherent
incoming photon (green) is absorbed by a buffer mode, $\hat{a}_{b}$, and is
converted to a pair of qubit-waste excitations described with operators
$\hat{a}_{q}$ and $\hat{a}_{w}$, respectively, with the interaction strength
$g_{4}\xi_{p}$. Due to the engineered fast dissipation of the waste mode,
$\kappa_{w}\gg\left|g_{4}\xi_{p}\right|$, the inverse process
$(\hat{a}_{b}^{\dagger}\hat{a}_{q}\hat{a}_{w})$ is effectively suppressed. (b)
Schematic of the experimental setup. The transmon is mounted in one arm of the
horseshoe-shaped 3D superconducting cavity and coupled to the cavity
waste/buffer modes, represented by orange/green shaded areas depicting
corresponding electric field distributions, respectively. The buffer (green),
readout (orange), and strong pump pulses (purple) are shaped using an
arbitrary waveform generator (AWG) for the time-domain experiments. The
frequencies of the buffer, $\omega_{b}$, readout, $\omega_{w}$, and pump tone,
$\omega_{p}$, are set by external microwave generators.

A common technique for detecting single photons by microwave SPDs involves the
use of transmon qubits, the most common qubit type used in quantum computation
[25, 26, 14, 15]. The transmon ionization (TI) emerges when the qubit escapes
from the Josephson potential by strong microwave driving, transferring the
qubit into a complex regime involving many quantum levels [39, 40, 41, 42].
The phenomena occuring in this regime are known under various names, such as
the quantum-to-classical phase transition [43, 44, 45], first-order
dissipative phase transition [46, 47, 48, 49], chaotic dynamics [50, 42], and
breakdown of photon-blockade [46, 45]. Various tools and representations, such
as the Rényi entropy, Floquet theory, and Husimi Q functions, are employed to
identify potential phase transitions. Notably, within the framework of the
quantum Duffing oscillator, the Rényi entropy serves as a quantifier of purity
of the quantum system, $P=\text{Tr}\left(\hat{\rho}^{2}\right)$, while the
Floquet theory provides insights into the temporal evolution under periodic
driving. Concurrently, the Husimi Q function offers a phase space
representation, revealing significant quantum fluctuations which arise from
tunneling between two metastable states. While this bistability is known to be
useful for high-fidelity readout schemes [41, 51], its influence on the SPD
behavior is not well understood so far. Given the importance of such devices
in quantum applications, we investigate its complex dynamics, and quantify the
impact of the TI on the SPD performance.

In the present work, we focus on the onset of TI in the irreversible frequency
conversion process in strongly driven transmon-resonator systems applied for
single-photon detection. The model and the detection principle of our device
are explained in Sec. II. In Sec. III, we study the conversion of the incoming
photons to the transmon excitations and waste resonator photons as a function
of the pump power. Once the pump power eventually approaches a critical
threshold, it triggers the sudden increase in the transmon population in
response to the incoming buffer photons. We study this process by using the
Rényi entropy, the Floquet theory, and the Husimi Q function. Our results
indicate that the abrupt change of the transmon response can be attributed to
a quantum-to-classical phase transition. Subsequently in Sec. IV, we provide a
comprehensive analysis of the irreversible frequency conversion in our system,
leading to a single-photon detection efficiency of $86$%. Then, we consider
the influence of TI on the detector performance and estimate the TI-corrected
detection efficiency to be notably lower, around $78$%. Concluding our study,
we systematically evaluate strategies to enhance the SPD detection effi-ciency
by properly adjusting the system parameters. In particular, we find that
increasing the detuning between the pump and qubit frequencies and increasing
the qubit anharmonicity are the most promising steps for further increasing
the detection efficiency. Furthermore, our analysis suggests that the spectral
arrangement of the buffer and waste frequencies should be implemented based on
the sign of the qubit anharmonicity to ensure that the pump drive and multi-
photon qubit transitions do not coincide in the frequency space.

Figure 2: (a) Triple-tone pulsed spectroscopy of the coupled transmon-
resonator system as explained in the main text. An inclined feature (white
dots) indicates the four-wave mixing process. (b) Reference two-tone pulsed
spectroscopy of the coupled transmon-resonator system without the buffer
signal. (c) Phase shift of the transmon response $\Delta\varphi$ normalized to
the phase shift between the ground and first excited states $\varphi_{ge}$ as
a function of the steady state buffer photon number $n_{b}$. Different colors
correspond to specific pump power values as indicated by the color code. (d)
Transmon phase response for three selected buffer photon numbers
$n_{b}=0.00,0.16,0.63$ (red, orange, blue) as a function of the pump power.
The vertical black dashed line indicates the CPP, at which the abrupt change
of the qubit population can be observed. Solid lines are the simulation
results using Eq. (1) with $n_{b}=0.00,0.16,0.63$ (red, orange, blue). (e) The
buffer and waste resonance frequencies as a function of the pump power at
$\omega_{p}/2\pi=5.1595$ GHz. A distinct jump of the resonance frequencies can
be observed in both resonators at the CPP of $-67$ dBm.

##  II Demonstration of irreversible frequency conversion process

In our experiments, we use a horseshoe-shaped aluminum 3D cavity, which
contains a transmon qubit chip (as schematically shown in Fig. 1(b)). We
utilize two of the cavity modes: the waste mode at $\omega_{w}/2\pi=7.609$ GHz
with a decay rate of $\kappa_{w}/2\pi=16.7$ MHz and the buffer mode at
$\omega_{b}/2\pi=7.955$ GHz with a decay rate of $\kappa_{b}/2\pi=3.7$ MHz.
The qubit frequency is $\omega_{q}/2\pi=5.664$ GHz, with $T_{1}=28\
\mathrm{\mu}$s and $T_{2}=16\ \mathrm{\mu}$s, limited by the Purcell effect.
The qubit is coupled to the waste and buffer modes with coupling strengths of
$g_{w}/2\pi=30$ MHz and $g_{b}/2\pi=18$ MHz, respectively. In a driven system,
the ac-Stark effect induces shifts in all three eigenfrequencies as a function
of the pump power. These shifted frequencies are denoted by
$\omega^{\prime}_{j}$ for the $j$-th mode, where $j=\\{q,b,w\\}$. The
Hamiltonian for the irreversible frequency conversion process in the rotating
frame of
$\hat{\mathcal{H}}_{0}/\hbar=\omega^{\prime}_{q}\hat{a}_{q}^{\dagger}\hat{a}_{q}+\omega^{\prime}_{w}\hat{a}_{w}^{\dagger}\hat{a}_{w}+\omega^{\prime}_{b}\hat{a}_{b}^{\dagger}\hat{a}_{b}$
is formulated as [39]

$\displaystyle\hat{\mathcal{H}}_{4}/\hbar$
$\displaystyle\approx\sum_{k=2}^{N_{t}-1}\frac{\chi^{(k)}}{k!}\left(\hat{a}_{q}^{\dagger}\right)^{k}\hat{a}_{q}^{k}$
$\displaystyle+\chi_{qw}\hat{a}_{q}^{\dagger}\hat{a}_{q}\hat{a}_{w}^{\dagger}\hat{a}_{w}+\chi_{qb}\hat{a}_{q}^{\dagger}\hat{a}_{q}\hat{a}_{b}^{\dagger}\hat{a}_{b}+i\varepsilon_{b}\left(\hat{a}_{b}^{\dagger}-\hat{a}_{b}\right)$
$\displaystyle+g_{4}\left(\xi_{p}\hat{a}_{q}^{\dagger}\hat{a}_{w}^{\dagger}\hat{a}_{b}e^{i\Delta_{qwbp}t}+\xi_{p}^{*}\hat{a}_{q}\hat{a}_{w}\hat{a}_{b}^{\dagger}e^{-i\Delta_{qwbp}t}\right)$
$\displaystyle+2i\varepsilon_{q}\cos\left(\omega_{p}t\right)\left(\hat{a}_{q}^{\dagger}e^{i\omega^{\prime}_{q}t}-\hat{a}_{q}e^{-i\omega^{\prime}_{q}t}\right),$
(1)

where $\hat{a}_{q}$, $\hat{a}_{w}$, and $\hat{a}_{b}$ are the annihilation
operators of the qubit, waste, and buffer modes, respectively, and $N_{t}$ is
the dimension of the transmon Hilbert space. The detuning involving all four
frequencies is defined as
$\Delta_{qwbp}=\omega^{\prime}_{q}+\omega^{\prime}_{w}-\omega^{\prime}_{b}-\omega_{p}$
and the driving strength of the transmon and buffer mode as
$\varepsilon_{q,b}$, respectively. The term $\chi^{(k)}$ corrects for the
$k$-th eigenfrequency of the transmon, with $\chi^{(2)}$ being its
anharmonicity. In addition, $\chi_{qw}$ and $\chi_{qb}$ are the cross-Kerr
interaction strengths of the waste and buffer modes, respectively. Lastly,
$g_{4}$ is the strength of the four-wave mixing interaction, and $\xi_{p}$ is
the pump amplitude at the pump frequency, $\omega_{p}$. Below the regime,
where $\chi^{(2)}\gg\varepsilon_{q,b},\kappa_{b,w},1/T_{1,2}$, the bosonic
annihilation operator $\hat{a}_{q}$ can be replaced by the Pauli operator
$\hat{\sigma}$. The irreversible frequency conversion process is based on the
dissipation-engineered four-wave mixing process of a nonlinear Josephson
junction element [15]. When an incoming photon arrives at the buffer mode
frequency, the buffer photon is converted to the excited state of the qubit
and a waste photon by a strong pump pulse, corresponding to the parametrically
activated conversion process
$(\hat{a}_{b}\hat{\sigma}^{\dagger}\hat{a}_{w}^{\dagger})$. However, under the
conditions, $\kappa_{w}\gg\left|g_{4}\xi_{p}\right|$, the waste resonator
state rapidly decays to the vacuum state, making the process irreversible and
prohibiting the inverse process
$(\hat{a}_{b}^{\dagger}\hat{\sigma}\hat{a}_{w})$. As a result, information of
the incoming photon is stored in the qubit state, as shown in Fig. 1 (a).

By performing a triple-tone spectroscopy composed of the pump and buffer
pulses with identical duration of $t_{b}=20$ $\mu$s, followed by a readout
pulse at the waste mode frequency with a duration of $t_{r}=2.5$ $\mu$s (see
Fig. 1), we observe the conversion process of the buffer photons to the qubit
excited state. As shown in Fig. 2 (a), when the buffer pulse is switched on,
we observe the phase response of the transmon in the pump frequency range of
$\omega_{p}/2\pi=5.15-5.18$ GHz. This response disappears in the absence of
the buffer signal (see Fig. 2 (b)).

##  III onset of transmon ionization

Figure 3: Rényi entropy, Floquet simulation, and Husimi Q distribution using
the Hamiltonian in Eq. (1). We choose $\varepsilon_{b}=0$ for the case
”without buffer drive”. The vertical black dashed line marks the CPP at
$P_{c}=-67$ dBm. (a) Rényi entropy, $S_{2}$, obtained from the simulated
reduced transmon density matrix without (blue) and with (orange) the buffer
drive as a function of the pump power. At the CPP, $P_{c}=-67$ dBm, the Rényi
entropy remains rather smooth in the absence of buffer photons, while its
sudden increase is observed for finite buffer signal powers. (b) The overlap
between the transmon state $\left|n_{q}00\right\rangle$ and the Floquet mode
$\left|410\right\rangle_{F}$ as a function of pump power. At the CPP, we find
a drastic increase of this overlap for transmon states characterized by
different $n_{q}$. (c) Probability of finding specific Floquet modes in the
simulated density matrix as a function of the pump power. A sudden increase of
the probabilities can be observed. Insets show Husimi Q distributions of the
Floquet states $\left|101\right\rangle_{F}$, $\left|222\right\rangle_{F}$ and
$\left|410\right\rangle_{F}$ at $-66.2$ dBm (vertical dash-dotted line). (d)
Simulation of the transmon Husimi Q function in the absence (top) and presence
(bottom) of the buffer signal for various pump powers. In the absence of the
buffer signal, minimal variation is observed around the CPP. Conversely, in
the presence of the buffer signal, the Husimi Q function spreads across the
phase space, progressively converging to a double-peak distribution for higher
pump powers. The red dashed box highlights the transmon Husimi Q function at
the CPP.

In this section, we present a systematic study of the buffer photon conversion
into the transmon-waste excitations as a function of the pump power. Our
measurements reveal a strong increase in the transmon population above a
certain critical pump power (CPP) of $P_{c}=-67$ dBm, as illustrated in Fig. 2
(c,d). To maximize the qubit population for each pump power, we adjust our
pump frequencies accordingly. We further note that the phase shift of the
transmon response $\Delta\varphi$ is directly related to the transmon
population based on the dispersive readout technique [52, 53].

For the pump powers below the CPP, we observe a monotonic increase in the
transmon population with increasing the steady-state buffer photon number.
This result is in agreement with theoretical predictions, which will be
discussed in the next section. Conversely, for pump powers above the CPP, we
again observe a monotonic increase in transmon population, but with a
discontinuity in the transmon response at the CPP (see Fig. 2 (c)). In an
independent measurement using a two-tone spectroscopy approach, we also find a
step-like change of the buffer and waste resonance frequencies at a similar
pump power, where the pump power is varied at the fixed pump frequency (see
Fig. 2 (e)). This observation is consistent with previous studies [51, 39] and
numerical simulations [54, 40, 41], and suggests that here the transmon is
entering the ionization region, where it eventually ”escapes” from the
Josephson potential.

We explore the system dynamics numerically by utilizing the Hamiltonian
presented in Eq. (1). By relying on experimentally determined parameters, we
solve the Lindblad master equation given in Eq. (9) and subsequently validate
the proposed Hamiltonian by comparing our results with measurements shown in
Fig. 2 (d). During the simulation, the Hilbert space dimensions of the
transmon, waste, and buffer modes are chosen to be 9, 3, and 3, respectively.
The eigenenergy calculations of the transmon with the experimental parameters
show that the first eight transmon states are confined states, while the ninth
lies above the Josephson junction potential corresponding to the first ionized
state.

To interpret the dynamics of our system, we utilize the Rényi entropy, Floquet
theory, and Husimi Q function, as shown in Fig. 3. We begin by considering the
Rényi entropy. This quantity can offer valuable insights into the phase
transition of our tripartite system around the critical point [55, 56, 47,
57]. Mathematically, the Rényi entropy is defined as

$S_{\alpha}=-\frac{1}{1-\alpha}\log_{2}\text{Tr}\left(\hat{\rho}_{t}^{\alpha}\right),$
(2)

with the so-called order parameter $\alpha$. The reduced density matrix of the
transmon, $\hat{\rho}_{t}$, is obtained by tracing out both the waste and
buffer modes, i.e., $\hat{\rho}_{t}=\text{Tr}_{w,b}\left(\hat{\rho}\right)$.
In this study, we choose $\alpha=2$, as this entropy serves as a key metric
for quantifying the purity, and thereby, the ”classicality” of the associated
density matrix. Intuitively, a system with higher entropy is more mixed or
less pure, and consequently, is more classical in nature. This entropy allows
us to observe the quantum-to-classical phase transition. As illustrated in
Fig. 3 (a), a distinct jump in entropy is evident at the CPP, a feature that
is absent without the buffer drive. This marked change offers a clear
indication of the transition of the system from a predominantly quantum state
to a more classical one, thus, supporting the observed quantum-to-classical
phase transition [48, 40, 39, 41, 42].

Furthermore, we apply the Floquet theory, which is particularly suited for
examining strongly-driven systems, making it suitable for our quantum-to-
classical phase transition studies [50, 42]. We analyze the overlap between
the transmon and Floquet states as a function of the pump power. The Floquet
modes are designated as $\left|n_{q}n_{b}n_{w}\right\rangle_{F}$, aligning
with $\left|n_{q}n_{b}n_{w}\right\rangle$ at minimal pump levels, which are
the eigenstates of $\hat{\mathcal{H}}_{0}$. As illustrated in Fig. 3 (b), a
distinct transition in the overlap of the Floquet state,
$\left|410\right\rangle_{F}$, is evident at the CPP, $P_{c}=-67$ dBm.
Comparable trends are noted for Floquet modes $\left|101\right\rangle_{F}$ and
$\left|222\right\rangle_{F}$ (not shown). We further investigate the dynamical
properties of probabilities corresponding to these Floquet modes in the
resulting density matrix, deduced from the Lindblad master simulation. In line
with the previous findings, a sharp increase in these probabilities is
observed at the CPP, as shown in Fig. 3 (c). Henceforth, after the CPP, these
Floquet states, $\left|101\right\rangle_{F}$, $\left|222\right\rangle_{F}$,
and $\left|410\right\rangle_{F}$, can be viewed as predominant states
characterizing the system for the increased pump powers. The insets of Fig. 3
(c) present the Husimi Q distributions of these Floquet states at $-66.2$ dBm,
slightly above the CPP. They reveal extensive delocalization, where
$\left|101\right\rangle_{F}$ and $\left|222\right\rangle_{F}$ states are
further characterized by the double-peak distributions. Here, the transmon
Husimi Q function is defined as

$Q=\left\langle\alpha_{q}\right|\hat{\rho}_{t}\left|\alpha_{q}\right\rangle/\pi,$

where $\left|\alpha_{q}\right\rangle$ represents the coherent state with the
complex transmon field amplitude $\alpha_{q}$.

Lastly, we aim to affirm that this bimodal nature observed in the Husimi Q
function of the Floquet modes distinctly manifests in the phase space of the
transmon after the CPP. In Fig. 3 (d), we show the simulation results of the
Husimi Q representation of the transmon in scenarios with and without the
buffer signal. We employ the double peak of the Husimi Q distribution in the
phase space as an indicator of coexisting states. This characteristic bimodal
structure of the Q function is a recognized feature of dissipative quantum
phase transitions, typically observed in quantum Duffing oscillators [58, 46,
54, 47, 49]. Thus, the identification of such a structure within our system
suggests the occurrence of the first-order dissipative phase transition in the
transmon-resonator system. As shown in the top panel of Fig. 3 (d), the
Gaussian-like distribution of the Husimi Q function remains largely unchanged
until around $P_{p}=-61$ dBm, if the buffer drive is switched off. In other
words, the delocalization does not appear even after $P_{c}=-67$ dBm. This
localized distribution complies with the transmon population, predominantly
occupying the ground state (see Fig. 2 (d)). Conversely, with the buffer drive
switched on, the system undergoes a delocalization in the phase space around
the CPP, progressively converging to a double-peak formation at stronger pump
powers, as illustrated in the bottom panel of Fig. 3 (d) at $-57$ dBm. This
observation, along with prior simulation results, reveals that the
delocalization in phase space is influenced by the emergence of double-peak
distributions. These distributions are characterized by the Floquet modes,
$\left|101\right\rangle_{F}$, $\left|222\right\rangle_{F}$, and
$\left|410\right\rangle_{F}$. Importantly, the manifestation of this bimodal
structure is not a gradual transition but occurs abruptly at the CPP. This
phenomenon is consistent with the sudden increase in the transmon population
observed experimentally, as shown in Fig. 2 (d).

##  IV single-photon detection performance

We further investigate our device as a single-photon detector. An important
figure of merit of such a device is its detection efficiency. We extract
corresponding efficiencies for various pump frequencies and powers up to the
CPP at various calibrated buffer signal powers (see Fig. 4). The performance
of the conversion process is characterized by a separate conversion efficiency
defined as [15]

$\eta_{c}=4\frac{\kappa_{\rm nl}\kappa_{b}}{\left(\kappa_{\rm
nl}+\kappa_{b}\right)^{2}},$ (3)

with $\kappa_{\rm nl}=4\left|g_{4}\xi_{p}\right|^{2}/\kappa_{w}$. Its
relationship with the qubit population
$n_{q}=\left\langle\hat{\sigma}^{\dagger}\hat{\sigma}\right\rangle$ is

$n_{q}=n^{*}\left(1-\exp\left(\eta_{c}\left|b_{\rm
in}\right|^{2}t_{b}\right)\right),$ (4)

where $\left|b_{\rm in}\right|^{2}$ is the buffer photon flux and $t_{b}$ is
the buffer pulse length. Here, $n^{*}\leq 1$ is the saturated qubit
population, whence the detection efficiency is defined as

$\eta_{\det}:=n^{*}\eta_{c}.$ (5)

In actual experiments, reduced detection efficiencies, $\eta_{\det}<1$, are
attributed to factors such as a finite energy relaxation rate and finite
thermal waste photon numbers, which stem from heating due to the elevated pump
powers and microwave noise from the input lines [59, 60, 61]. The detection
efficiencies $\eta_{\det}$ are extracted using the Lindblad master equation
simulations of the Hamiltonian given in Eq. (1): We simulate the qubit
population $n_{q}$ for various values of $g_{4}\xi_{p}$, $\left|b_{\rm
in}\right|^{2}$, and thermal bath temperatures at a fixed buffer pulse length
$t_{b}$. We finally choose $\eta_{\rm det}$, such that the simulated
$n_{q}(n_{b})$ fits all experimentally measured curves (see Fig. 4 (b)). For
these simulations, the qubit driving term is neglected, i.e.,
$\varepsilon_{q}=0$. Additionally, the detuning parameter $\Delta_{qwbp}$ is
set to zero, and the transmon Hilbert space dimension is set to $N_{t}=3$,
i.e., we take the second excited state into account. These fits are conducted
for every pump power and frequency, as shown in Fig. 4 (a). Figure 4 (c) shows
a Lorentzian fit to obtain the optimal detection efficiencies at each pump
power. Finally, we utilize the pump power dependence of $\eta_{c}$, such that
$n^{*}$ and $\eta_{c}$ are determined by fitting the optimal detection
efficiencies as a function of the pump power using Eq. (3) and Eq. (5), as
demonstrated in Fig. 4 (d). In our studies, the maximum conversion and
detection efficiencies are $\eta_{c}=(96\pm 8)$% and $\eta_{\rm det}=(86\pm
6)$%, respectively. In other words, we have a detection infidelity of
$1-\eta_{\det}=(14\pm 6)\%$. As derived in App. B, by taking into account the
qubit energy relaxation rate, the saturated qubit population can be determined
as

$n_{q}^{*}=\frac{\eta_{c}\left|b_{\rm in}\right|^{2}}{\eta_{c}\left|b_{\rm
in}\right|^{2}+\gamma_{q}}.$ (6)

In our case, for a buffer photon flux of $\left|b_{\rm in}\right|^{2}=1.22$
MHz corresponding to the steady-state photon number of $n_{b}=0.25$ we obtain
$n_{q}^{*}=0.97$. Notably, although the buffer photon flux significantly
surpasses the decoherence rate observed in our measurements, we can reach a
detection infidelity of only a few per cent. Inserting this value into Eq.
(5), we find the detection infidelity of 7%.

Figure 4: (a) Detection efficiency as a function of the pump frequency and
power. (b) Qubit population as a function of the steady state buffer photon
number $n_{b}$. The red dots are the measured data points, while the blue line
is fitted by numerically solving the quantum Lindblad master equation with
$t_{b}=0.55$ $\mu$s (see Eq. (1)). (c) Lorentzian fit of $\eta_{\rm det}$ at
$P_{p}=-67$ dBm and $\omega_{p}/2\pi=5.156$ GHz. (d) Single-photon detection
efficiency $\eta_{\rm det}$ as a function of the pump power. The blue dots are
the optimal detection efficiency extracted from the Lorentzian fit (blue line
in (c)). The solid line is the fit using Eq. (3) and Eq. (5). The dashed line
shows the case for $n^{*}=1$. (e) The qubit population of the state
$\left|n_{q}=3\right\rangle$ as a function of $n_{b}$ for various pump powers
used in the simulation. The population is calculated as $\left\langle
3\right|\hat{a}^{\dagger}_{q}\hat{a}_{q}\hat{\rho}_{t}\left|3\right\rangle=3\times\left\langle
3\right|\hat{\rho}_{t}\left|3\right\rangle=3\times\text{P}_{q}(n_{q}=3)$,
where the probability being in the state $\left|n_{q}=3\right\rangle$ is
denoted as $\text{P}_{q}(n_{q}=3)=\left\langle
3\right|\hat{\rho}_{t}\left|3\right\rangle$. Figure 5: Detection efficiency,
qubit population contribution of the state $\left|n_{q}=3\right\rangle$, and
sensitivity as a function of the pump power for different system parameters.
TI-uncorrected efficiencies $\eta_{\det}$ and sensitivities $S$ are
represented by solid lines, while TI-corrected efficiencies
$\eta_{\det}^{\prime}$ and sensitivities $S^{\prime}$ are represented by
dashed lines. The qubit population of $\left|n_{q}=3\right\rangle$ is
calculated as $\left\langle
3\right|\hat{a}^{\dagger}_{q}\hat{a}_{q}\hat{\rho}_{t}\left|3\right\rangle=3\times\text{P}_{q}(n_{q}=3)$,
where the probability being in the state $\left|n_{q}=3\right\rangle$ is
denoted as $\text{P}_{q}(n_{q}=3)$. (a) The impact of various dissipation
rates of the waste mode $\kappa_{w}/2\pi=4.6,8.6,16.6,30$ MHz (blue, orange,
green, red line). The green dots are the experimentally extracted values for
$\kappa_{w}/2\pi=16.6$ MHz. (b) The impact of various four-wave interaction
strengths $g_{4}/2\pi=0.9,1.8,3.6,7.2,14.4$ MHz (purple, red, green, orange,
blue line). The green dots represent the experimentally extracted values for
$g_{4}/2\pi=3.6$ MHz. (c) The impact of various buffer frequencies
$\omega_{b}/2\pi=9,10,11$ GHz (green, orange, blue line), and (d)
anharmonicities $\chi^{(2)}/2\pi=-490,234,672,988$ MHz (blue, orange, green,
red line). For the simulation with the positive anharmonicities, we consider
typical flux qubit values as a reference. (a) and (b) are simulated at $T_{\rm
eff}=98$ mK, while (c) and (d) are obtained in the zero temperature limit,
$T=0$ K.

In the case of a non-zero waste photon number, the combined quantum system can
partially reverse the four-wave mixing process. Given this scenario, the
saturated qubit population is quantified as

$n_{w}^{*}=\frac{1-6n_{\text{th},w}}{1-4n_{\text{th},w}},$ (7)

under the condition $n_{\text{th},w}\ll 1$ (see App. B for more details).We
employ the dark count probability of the qubit to estimate $n_{\text{th},w}$.
To achieve this, we use the measurement data taken for $n_{b}=0$, as shown in
Fig. 2 (d), and find the dark count probability of $(5.4\pm 8.3)\%$,
corresponding to the effective temperature of $T_{\rm eff}=98$ mK. Using the
formula $n_{\text{th},w}=(\exp(\hbar\omega_{w}/k_{\rm B}T_{\rm eff})-1)^{-1}$,
we obtain $n_{w}^{*}=0.94$. This result leads to the detection infidelity of
10%. Adding up both infidelities, we obtain a total infidelity of 17%, which
aligns well with the measured detection infidelity of 14%.

The discussion so far has omitted effects of the TI process. To address this,
we consider the impact of the higher qubit level excitations, $\left|n_{q}\geq
3\right\rangle$, which may additionally lead to an overestimation of the
detection efficiency. For that purpose, we extract the occupation probability
of $\left|n_{q}\geq 3\right\rangle$ as a function of $n_{b}$ at the various
pump powers, and find that the occupation probabilities of the states
$\left|n_{q}\geq 4\right\rangle$ are negligibly small. In Fig. 4 (e), the
buffer photon number and pump power dependence of the state
$\left|n_{q}=3\right\rangle$ can be clearly observed. Crucially, the
dependence on the buffer photon number leads to an overestimation of the
detection efficiency. This effect arises because the occupation of this state
additionally contributes to the qubit population. In order to estimate the TI
influence on the detection efficiency, we remove the contribution of higher
excited states, i.e. $\sum_{n_{q}\leq 2}\left\langle
n_{q}\right|\hat{a}^{\dagger}_{q}\hat{a}_{q}\hat{\rho}_{t}\left|n_{q}\right\rangle$,
and extract its efficiency with the same method used in the experiments. Upon
this adjustment, we observe a reduction in the detection efficiency by 10%
with the optimal parameters. We assume that the correction applied to the
simulated data is also applicable to the experimental results. Under this
assumption, the original detection efficiency of $\eta_{\rm det}=86$% reduces
to the TI-corrected detection efficiency of $\eta_{\rm det}^{\prime}=78$%.

Next, we conduct an extensive analysis to find out how to enhance the
detection efficiency and sensitivity of our SPD. Furthermore, we also discuss
how to suppress the influence of the TI process. Referring to Fig. 5, we
present the TI-uncorrected and TI-corrected detection efficiencies and
sensitivities across various parameters. The sensitivity is defined as [62]

$\displaystyle S=$ $\displaystyle\frac{\hbar\omega_{b}\sqrt{r_{\rm
dc}}}{\eta_{\rm det}},$ $\displaystyle S^{\prime}=$
$\displaystyle\frac{\hbar\omega_{b}\sqrt{r_{\rm dc}}}{\eta_{\rm
det}^{\prime}},$ (8)

with the dark count rate $r_{\rm dc}$. We denote $S$ as the TI-uncorrected
sensitivity, and $S^{\prime}$ as the TI-corrected one. We observe an early
rise of the detection efficiency $\eta_{\rm det}$ ($\eta_{\rm det}^{\prime}$)
for the higher dissipation rate of the waste mode $\kappa_{w}$, as
demonstrated in Fig. 5 (a). However, the increase of $\kappa_{w}$ also leads
to an enhancement of the Purcell decay rate of the qubit. Thus,
$\varepsilon_{q}$ increases for the same pump power accelerating the onset of
the TI, which sets the upper bound of the conversion process. Consequently,
despite the early rise in detection efficiency, $\eta_{\rm det}$ ($\eta_{\rm
det}^{\prime}$) fails to achieve higher values for larger $\kappa_{w}$. The
early increase of the probability $\text{P}_{q}(n_{q}=3)$ supports the
argument of this early onset of the TI. Therefore, the optimal detection
efficiency, $\eta_{\rm det}$ ($\eta_{\rm det}^{\prime}$), along with the
sensitivity $S$ ($S^{\prime}$), is more favorably achieved at lower
$\kappa_{w}$. As shown in Fig. 5 (b), an enhancement in the coupling strength
$g_{4}$ improves both $\eta_{\rm det}$ ($\eta_{\rm det}^{\prime}$) and $S$
($S^{\prime}$). Since the CPP does not change under $g_{4}$ variation, we can
reach higher $\eta_{\rm det}$ ($\eta_{\rm det}^{\prime}$) before reaching the
CPP. Additionally, when we tune the buffer frequency $\omega_{b}$ to higher
frequencies, Fig. 5 (c) predicts that $\text{P}_{q}(n_{q}=3)$ decreases. This
is attributed to a large detuning between pump and qubit frequencies, which
reduces the TI impact. In other words, $\eta_{\rm det}$ converges to
$\eta_{\rm det}^{\prime}$. Finally, we investigate the influence of the TI for
different anharmonicities $\chi^{(2)}$. We observe that large
$\left|\chi^{(2)}\right|$ values suppress the TI process, as expected from the
previous studies [63, 64, 65]. Accordingly, $\eta_{\rm det}$ asymptotically
approaches $\eta_{\rm det}^{\prime}$. Notably, the configuration of the buffer
and waste frequencies should be chosen considering a particular value of
$\chi^{(2)}$. In general, the pump frequency can match a frequency of a
certain multi-photon process at a certain pump power due to the power-
dependent frequency shift of the qubit frequency. In the regime of negative
anharmonicity, this situation can occur if $\omega_{w}<\omega_{b}$, such that
$\omega_{p},\omega_{0k}/k<\omega_{q}^{\prime}$, where $\omega_{0k}/k$ is the
multi-photon transition frequency. If the pump frequency is close to such a
transition frequency, the dark count rate increases, which eventually degrades
the SPD performance. For example for $\chi^{(2)}/2\pi=-490$ MHz, the SPD can
reach $\eta_{\rm det}\approx 0.5$, until the pump collides with the two-photon
process in the spectral domain. In simulations with positive anharmonicity
values, we use a flux qubit configuration [65], and achieve detection
efficiencies up to $95$%. Importantly, for the flux qubit, multi-photon
processes become relevant at frequencies exceeding the qubit frequency, and
given that $\omega_{p}<\omega_{q}^{\prime}$ for $\omega_{w}<\omega_{b}$, such
processes are effectively far-detuned, resulting in a reduction of dark counts
(Fig. 5 (d)).

Lastly, we compare the sensitivity between the cases with $T_{\rm eff}=98$ mK
(Fig. 5 (a,b)) and $T=0$ K (Fig. 5 (c,d)). Although the sensitivity for an
ideal environment ($T=0$ K) generally improves, it does not reach the present
state of the art value of $S=10^{-22}\ \rm W/\sqrt{Hz}$ [62]. This is
attributed to the filtering effect of our cavity, which requires stronger pump
powers to reach the same detection efficiency, and hence, results in a higher
dark count probability due to the pump heating. In our case, the dark count
rate gives $26\pm 4$ kHz at the pump power of $P_{p}=-67$ dBm, at which we
measure the maximum $\eta_{\rm det}$ ($\eta_{\rm det}^{\prime}$). Despite of
the cavity filtering effect, the dark count rate reported here is comparable
to the values reported in the previous studies [24, 25, 26].

##  V Conclusion

In this work, we have presented an experimental and numerical study of the TI
onset and the SPD performance of a transmon qubit coupled to a multi-mode 3D
cavity. We have investigated the dependence of the transmon population on the
buffer photon number, $n_{b}$, and pump power, and observed that the transmon
is highly-sensitive to $n_{b}$ within the region close to a critical pump
powers (CPP). Through a comprehensive analysis employing the Rényi entropy,
Floquet theory, and Husimi Q function, we have elaborated an occurrence of a
quantum-to-classical phase transition around the CPP. Our numerical
simulations align well with experimental observations, emphasizing the
accuracy of our chosen methodologies.

We have also investigated the device as a single-photon detector, and
extracted its detection efficiencies for various pump frequencies and powers
up to the CPP. Our measurements show that a maximum TI-uncorrected detection
efficiency of 86% can be achieved, while we estimate the TI-corrected
detection efficiency to be 78%. The conversion efficiency $\eta_{c}$ between
the buffer and the qubit-waste subsystems is limited by the TI, while the
reduction of the saturated qubit population $n^{*}$ can be attributed to a
finite thermal waste photon number and the qubit energy relaxation rate.
Finally, increasing the frequency detuning between the pump and qubit
frequencies and the qubit anharmonicity is expected to strongly suppress the
influence of the TI process and leads to a distinct improvement of the TI-
corrected detection efficiency and sensitivity. The spectral positioning of
the buffer and waste frequencies have to be determined based on the sign of
the qubit anharmonicity to ensure that the pump and multi-photon processes do
not overlap.

In conclusion, our systematic studies reveal the specific advantages inherent
to the onset of TI, suggesting its prospective extension to high-efficiency
microwave SPDs compatible with 3D cavity architectures. Moreover, our study
emphasizes the necessity for a careful analysis of the efficiency extraction,
particularly given the impact of the TI process. These insights bear
significant implications for the development of parametric device
applications, which are essential in advancing quantum information processing
and communication with superconducting circuits.

## Acknowledgments

We thank Joan Agusti, Yvan Briard, Louis Garbe and Peter Rabl for insightful
discussions. We acknowledge support by the German Research Foundation via
Germany’s Excellence Strategy (EXC2111-390814868), the German Federal Ministry
of Education and Research via the project QUARATE (Grant No.13N15380). This
research is part of the Munich Quantum Valley, which is supported by the
Bavarian state government with funds from the Hightech Agenda Bayern Plus.

##  Appendix A Experimental techniques

Figure 6: Picture of the device. (top) Optical image of the Josephson
junction. (center) Optical image of the transmon qubit with its antenna pads.
(bottom) Photograph of two halves of the Al horseshoe cavity with the transmon
qubit chip inside.

###  A.1 Microwave single-photon detector sample

As outlined in Sec. II, our single-photon detection device is comprised of two
primary components: a horseshoe cavity and a transmon qubit chip. The cavity,
constructed from aluminum (Al) with a high purity level of $99.99$%, has been
fabricated by the workshop at the Walther-Meißner-Institut. To address the
relatively low thermal conductance of superconducting aluminum, we mount a
gold-plated copper plate, 1.5 mm in thickness, onto the cavity. This plate
significantly enhances the thermal conductance of the system, ensuring more
efficient heat removal.

The transmon qubit, central to our detection mechanism, is fabricated on a
silicon chip with spatial dimension of $3.5$ mm by $10$ mm. For the
fabrication of the transmon qubit, a standard lift-off technique is used. We
evaporate aluminum through a PMMA/CSAR resist mask written in an electron-beam
lithography step. The antenna of the qubit has a size of $890$ $\mu$m by $330$
$\mu$m, ensuring a sufficient dipole interaction strength with the cavity.
Additionally, the Josephson junctions have an area of $240\times
240\,\text{nm}^{2}$.

A visual representation of the assembly, showing the integration of the open
horseshoe cavity with the embedded transmon qubit chip, can be found in Fig.
6.

Figure 7: Schematic drawing of dry dilution refrigerator embedding Al
horseshoe cavity with the transmon qubit chip inside.

###  A.2 Cryogenic setup

The cryogenic arrangement utilized for our experimental measurements is
illustrated in Fig. 7.

For the generation of arbitrary waveforms, we employ the HDAWG from Zurich
Instruments. Additionally, the setup incorporates three microwave sources (R&S
SGS 100A). To integrate these three pulsed drives, we utilize two power
combiners. Initially, the pump and buffer pulses are merged, followed by their
subsequent combination with the readout pulse. This combined signal is then
directed to the cryostat, facilitating both the frequency conversion process
and the qubit readout (see also Fig. 1 in the main text).

The combined signals travel through a series of attenuators mounted at various
cooling stages within the cryostat, followed by a low-pass filter to suppress
high-frequency noise. Upon reaching the mixing chamber stage, the signal is
delivered to the cavity-qubit system. The Al horseshoe cavity, housing the
transmon qubit chip, is shielded from external magnetic fields by a cryoperm
shield. The single-port design of the cavity is coupled with a circulator to
separate the output signal from the incoming signal.

The output path includes a low-pass filter, followed by passage through two
isolators. Amplification is achieved via a HEMT amplifier located at the 4 K
stage, and subsequently, through a room-temperature amplifier.

##  Appendix B Saturated qubit population

In this section, we derive the influence of the qubit decoherence and residual
waste photon number on the saturated qubit population. For that purpose, we
briefly review the derivation of the nonlinear decay by tracing out the waste
modes with adiabatic elimination [15]. After canceling out the ac-Stark terms,
neglecting the driving terms and restricting the Hilbert space of the qubit
mode to a two-level system ($\hat{a}_{q}\rightarrow\hat{\sigma}$) in Eq. (1),
we arrive at

$\displaystyle\hat{\mathcal{H}}^{\prime\prime}_{4}=$
$\displaystyle\hat{\mathcal{H}}_{w}+\hat{\mathcal{H}}_{qb}$
$\displaystyle\hat{\mathcal{H}}_{w}/\hbar=$
$\displaystyle\left(\Delta_{w}-\chi_{qw}\hat{\sigma}^{\dagger}\hat{\sigma}\right)\hat{a}_{w}^{\dagger}\hat{a}_{w}+g_{4}\xi_{p}\hat{a}_{b}\hat{a}_{w}^{\dagger}\hat{\sigma}^{\dagger}+\rm
h.c.$ $\displaystyle\hat{\mathcal{H}}_{qb}/\hbar=$
$\displaystyle\chi_{qb}\hat{\sigma}^{\dagger}\hat{\sigma}{\hat{a}_{b}}^{\dagger}\hat{a}_{b},$

which dynamics is described by the Lindblad master equation

$\displaystyle\frac{\text{d}}{\text{d}t}\hat{\rho}=$
$\displaystyle-i\left[\hat{\mathcal{H}}_{w},\hat{\rho}\right]+\mathcal{L}_{qb}\left[\hat{\rho}\right]$
$\displaystyle+\kappa_{w}\left(n_{\text{th},w}+1\right)\mathcal{D}[\hat{a}_{w}]\hat{\rho}+\kappa_{w}n_{\text{th},w}\mathcal{D}[\hat{a}_{w}^{\dagger}]\hat{\rho},$
(9)

where $n_{\text{th},w}=(\exp(\hbar\omega_{w}/k_{\rm B}T_{\rm eff})-1)^{-1}$
and

$\displaystyle\mathcal{L}_{qb}\left[\hat{\rho}\right]=$
$\displaystyle-i\left[\hat{\mathcal{H}}_{qb},\hat{\rho}\right]+\kappa_{b}\mathcal{D}[\hat{a}_{b}]\hat{\rho}+\gamma_{q}\mathcal{D}[\hat{\sigma}]\hat{\rho}.$

For the adiabatic elimination of the waste mode, we assume

$\left|\frac{g_{4}\xi_{p}}{\kappa_{w}}\right|,\left|\frac{\chi_{jj^{\prime}}}{\kappa_{w}}\right|\sim\delta\ll
1,$

and $n_{\text{th},w}=0$, such that the waste mode is dominantly in the vacuum
state due to the fast decay rate $\kappa_{w}$. Hence, we can reduce the
Hilbert space of the waste mode to
$\mathcal{H}_{w}=\text{span}\left(\left|0_{w}\right\rangle,\left|1_{w}\right\rangle\right)$.
In particular, for the density matrix
$\hat{\rho}\in\mathcal{H}_{q}\otimes\mathcal{H}_{b}\otimes\mathcal{H}_{w}$,
the reduced density matrices acting on the qubit and buffer Hilbert space have
the following relations

$\displaystyle\left\langle 0_{w}\right|\hat{\rho}\left|0_{w}\right\rangle=$
$\displaystyle\hat{\rho}_{00},$ $\displaystyle\left\langle
0_{w}\right|\hat{\rho}\left|1_{w}\right\rangle=$
$\displaystyle\delta\hat{\rho}_{01},$ $\displaystyle\left\langle
1_{w}\right|\hat{\rho}\left|1_{w}\right\rangle=$
$\displaystyle\delta^{2}\hat{\rho}_{11},$ $\displaystyle\left\langle
0_{w}\right|\hat{\rho}\left|2_{w}\right\rangle=$
$\displaystyle\delta^{2}\hat{\rho}_{02},$

with $\left|j_{w}\right\rangle,\left|j^{\prime}_{w}\right\rangle$ being the
Fock basis of the waste mode. Note that
$\hat{\rho}_{00}\in\mathcal{H}_{q}\otimes\mathcal{H}_{b}$. By projecting Eq.
(9) with $\left\langle 0_{w}\right|...\left|0_{w}\right\rangle,\left\langle
0_{w}\right|...\left|1_{w}\right\rangle,\left\langle
1_{w}\right|...\left|1_{w}\right\rangle$, respectively, we obtain

$\displaystyle\frac{\text{d}}{\text{d}t}\hat{\rho}_{00}=$
$\displaystyle-i\delta\left[g_{4}\xi_{p}^{*}\hat{a}_{b}^{\dagger}\hat{\sigma}\hat{\rho}_{10}-g_{4}\xi_{p}\hat{\rho}_{01}\hat{a}_{b}\hat{\sigma}^{\dagger}\right]+\delta^{2}\kappa_{w}\hat{\rho}_{11}$
$\displaystyle+\mathcal{L}_{qb}\left[\hat{\rho}_{00}\right]+\mathcal{O}\left(\delta^{3}\right)$
(10) $\displaystyle\delta\frac{\text{d}}{\text{d}t}\hat{\rho}_{01}=$
$\displaystyle
i\hat{\rho}_{00}\left[g_{4}\xi_{p}^{*}\hat{a}_{b}^{\dagger}\hat{\sigma}\right]+i\delta\hat{\rho}_{01}\left[\Delta_{w}-\chi_{qw}\hat{\sigma}^{\dagger}\hat{\sigma}\right]$
$\displaystyle-\delta\frac{\kappa_{w}}{2}\hat{\rho}_{01}+\delta\mathcal{L}_{qb}\left[\hat{\rho}_{01}\right]+\mathcal{O}\left(\delta^{3}\right)$
(11) $\displaystyle\delta^{2}\frac{\text{d}}{\text{d}t}\hat{\rho}_{11}=$
$\displaystyle-i\delta\left[g_{4}\xi_{p}\hat{a}_{b}\hat{\sigma}^{\dagger}\hat{\rho}_{01}-g_{4}\xi_{p}^{*}\hat{\rho}_{10}\hat{a}_{b}^{\dagger}\hat{\sigma}\right]$
$\displaystyle+i\delta^{2}\left[\Delta_{bwq}-\chi_{qw}\hat{\sigma}^{\dagger}\hat{\sigma},\hat{\rho}_{11}\right]$
$\displaystyle-\delta^{2}\kappa_{w}\hat{\rho}_{11}+\delta^{2}\mathcal{L}_{qb}\left[\hat{\rho}_{11}\right]+\mathcal{O}\left(\delta^{3}\right)$
(12)

Focusing on the relevant dynamics, we find that Eqs. (11) and (12) include a
damping term of order $\delta^{0}$, while all terms in Eq. (10) are of order
$\delta^{2}$. Hence, this allows us to treat $\hat{\rho}_{01}$ and
$\hat{\rho}_{11}$ as a steady state (adiabatic approximation), which leads to

$\displaystyle\hat{\rho}_{01}\approx$
$\displaystyle\hat{\rho}_{00}\frac{ig_{4}\xi_{p}^{*}/\delta}{\frac{\kappa_{w}}{2}-i\left(\Delta_{w}-\chi_{qw}\right)}\hat{a}_{b}^{\dagger}\hat{\sigma},$
(13)

where we have used
$\hat{a}_{b}^{\dagger}\hat{\sigma}\hat{\sigma}^{\dagger}\hat{\sigma}=\hat{a}_{b}^{\dagger}\sigma$.
As for the steady state solution for $\hat{\rho}_{11}$, we get

$\displaystyle\hat{\rho}_{11}=$
$\displaystyle\frac{1}{\left(\frac{\kappa_{w}}{2}\right)^{2}+\left(\Delta_{w}-\chi_{qw}\right)^{2}}\frac{\left|g_{4}\xi_{p}\right|^{2}}{\delta^{2}}\hat{a}_{b}\hat{\sigma}^{\dagger}\hat{\rho}_{00}\hat{a}_{b}^{\dagger}\hat{\sigma}.$
(14)

Inserting the steady state solutions into Eq. (10), we obtain

$\displaystyle\frac{\text{d}}{\text{d}t}\hat{\rho}_{00}=$
$\displaystyle\left[\frac{i\left|g_{4}\xi_{p}\right|^{2}\left(\Delta_{w}-\chi_{qw}\right)}{\left(\frac{\kappa_{w}}{2}\right)^{2}+\left(\Delta_{w}-\chi_{qw}\right)^{2}}\hat{a}_{b}^{\dagger}\hat{a}_{b}\hat{\sigma}\hat{\sigma}^{\dagger},\hat{\rho}_{00}\right]$
$\displaystyle+\kappa_{\rm
nl}\mathcal{D}\left[\hat{a}_{b}\hat{\sigma}^{\dagger}\right]\hat{\rho}_{00}+\mathcal{L}_{qb}\left[\hat{\rho}_{00}\right]$
(15)

with the nonlinear decay rates $\hat{L}_{\rm nl}=\sqrt{\kappa_{\rm
nl}}\hat{a}_{b}\hat{\sigma}^{\dagger}$

$\displaystyle\kappa_{\rm nl}$
$\displaystyle\equiv\frac{\left|g_{4}\xi_{p}\right|^{2}}{\left(\frac{\kappa_{w}}{2}\right)^{2}+\left(\Delta_{w}-\chi_{qw}\right)^{2}}\kappa_{w}\stackrel{{\scriptstyle\Delta_{w}=\chi_{qw}}}{{=}}\frac{4\left|g_{4}\xi_{p}\right|^{2}}{\kappa_{w}}.$
(16)

###  B.1 Influence of qubit decoherence

Here, we delve into the impact of qubit decoherence on its dynamics. Notably,
it becomes evident that the qubit population never reaches unity. This is a
direct implication of the fact that qubit decoherence inherently reduces the
detection efficiency in single-photon detection scenarios.

The qubit decoherence term and the buffer driving term
$i\varepsilon_{b}^{\prime}\left(\hat{a}_{b}-\hat{a}_{b}^{\dagger}\right)$ are
added to the Lindblad master equation in Eq. (15), such that we obtain

$\displaystyle\frac{\text{d}}{\text{d}t}\hat{\rho}_{00}=$
$\displaystyle\kappa_{\rm
nl}\mathcal{D}\left[\hat{a}_{b}\hat{\sigma}^{\dagger}\right]\hat{\rho}_{00}+\kappa_{b}\mathcal{D}\left[\hat{a}_{b}\right]\hat{\rho}_{00}$
$\displaystyle+\varepsilon_{b}^{\prime}\left[\hat{a}_{b}-\hat{a}_{b}^{\dagger},\hat{\rho}_{00}\right]+\gamma_{q}\mathcal{D}\left[\hat{\sigma}\right]\hat{\rho}_{00}.$
(17)

Since the time evolution of the buffer mode is dependent on the qubit state,
we first consider $\hat{\rho}_{g}\equiv\left\langle
g\right|\hat{\rho}_{00}\left|g\right\rangle$ and
$\hat{\rho}_{e}\equiv\left\langle
e\right|\hat{\rho}_{00}\left|e\right\rangle$, such that

$\displaystyle\frac{\text{d}}{\text{d}t}\hat{\rho}_{g}=$
$\displaystyle-\frac{1}{2}\kappa_{\rm
nl}\left(\hat{\rho}_{g}\hat{a}_{b}\hat{a}_{b}^{\dagger}+\hat{a}_{b}\hat{a}_{b}^{\dagger}\hat{\rho}_{g}\right)+\kappa_{b}\mathcal{D}\left[\hat{a}_{b}\right]\hat{\rho}_{g}$
$\displaystyle+\varepsilon_{b}^{\prime}\left[\hat{a}_{b}-\hat{a}_{b}^{\dagger},\hat{\rho}_{g}\right]+\gamma_{q}\hat{\rho}_{e},$
(18a) $\displaystyle\frac{\text{d}}{\text{d}t}\hat{\rho}_{e}=$
$\displaystyle\kappa_{\rm
nl}\hat{a}_{b}\hat{\rho}_{g}\hat{a}_{b}^{\dagger}+\kappa_{b}\mathcal{D}\left[\hat{a}_{b}\right]\hat{\rho}_{e}$
$\displaystyle+\varepsilon_{b}^{\prime}\left[\hat{a}_{b}-\hat{a}_{b}^{\dagger},\hat{\rho}_{e}\right]-\gamma_{q}\hat{\rho}_{e}.$
(18b)

In our case, since the applied buffer tone is a coherent tone, we can safely
assume that the buffer is in a coherent state $\left|\beta\right\rangle$ with
the dimensionless amplitude $\beta=-2\varepsilon^{\prime}_{b}/(\kappa_{\rm
nl}+\kappa_{b})$. That is,
$\hat{\rho}_{g/e}\propto\left|\beta\right\rangle\left\langle\beta\right|$. We
can now calculate the qubit ground and excited state probability by tracing
them, $p_{g/e}=\text{Tr}(\hat{\rho}_{g/e})$. This leads us to formulate the
differential equation as

$\displaystyle\frac{\text{d}}{\text{d}t}\left[\begin{array}[]{c}p_{g}\\\
p_{e}\end{array}\right]=$
$\displaystyle\left[\begin{array}[]{cc}-\eta_{c}\left|b_{\rm
in}\right|^{2}&\gamma_{q}\\\ \eta_{c}\left|b_{\rm
in}\right|^{2}&-\gamma_{q}\end{array}\right]\left[\begin{array}[]{c}p_{g}\\\
p_{e}\end{array}\right],$ (25)

which solution gives

$\displaystyle\left[\begin{array}[]{c}p_{g}(t)\\\ p_{e}(t)\end{array}\right]=$
$\displaystyle\left[\begin{array}[]{cc}\frac{\eta_{c}\left|b_{\rm
in}\right|^{2}}{\eta_{c}\left|b_{\rm
in}\right|^{2}+\gamma_{q}}e^{-\eta_{c}\left|b_{\rm
in}\right|^{2}t-\gamma_{q}t}+\frac{\gamma_{q}}{\eta_{c}\left|b_{\rm
in}\right|^{2}+\gamma_{q}}&\frac{\gamma_{q}}{\eta_{c}\left|b_{\rm
in}\right|^{2}+\gamma_{q}}\left(1-e^{-\eta_{c}\left|b_{\rm
in}\right|^{2}t-\gamma_{q}t}\right)\\\ \frac{\eta_{c}\left|b_{\rm
in}\right|^{2}}{\eta_{c}\left|b_{\rm
in}\right|^{2}+\gamma_{q}}\left(1-e^{-\eta_{c}\left|b_{\rm
in}\right|^{2}t-\gamma_{q}t}\right)&\frac{\eta_{c}\left|b_{\rm
in}\right|^{2}}{\eta_{c}\left|b_{\rm
in}\right|^{2}+\gamma_{q}}+\frac{\gamma_{q}}{\eta_{c}\left|b_{\rm
in}\right|^{2}+\gamma_{q}}e^{-\eta_{c}\left|b_{\rm
in}\right|^{2}t-\gamma_{q}t}\end{array}\right]\left[\begin{array}[]{c}p_{g}(0)\\\
p_{e}(0)\end{array}\right].$ (32)

We find the excitation probability of the transmon qubit for sufficiently long
time $t$ is

$p_{e}(t\rightarrow\infty)=\frac{\eta_{c}\left|b_{\rm
in}\right|^{2}}{\eta_{c}\left|b_{\rm in}\right|^{2}+\gamma_{q}}<1.$ (33)

It is noteworthy that the saturation level of the qubit population shows a
relative dependency on the qubit energy relaxation rate $\gamma_{q}$, and the
”effective buffer photon conversion rate” $\eta_{c}\left|b_{\rm
in}\right|^{2}$. Particularly, the impact of the energy relaxation rate
becomes significant when the value of $\eta_{c}\left|b_{\rm in}\right|^{2}$ is
comparably small. This observation underscores an interplay or a steady-state
dynamics between the information gain from the incoming buffer photon and its
subsequent loss.

###  B.2 Influence of residual waste photon number

In our prior discussions, we assumed that our quantum system experienced only
vacuum noise, excluding the influence of the coherent drive. This is a
standard assumption, primarily justified by the extremely low temperatures of
dilution refrigerators, typically in the range of $10-50$ mK. Nevertheless,
when it comes to parametric devices, strong system pumping can induce an
effective temperature increase for the quantum apparatus. This is particularly
relevant in our case: the presence of a residual waste photon number can
reduce detection efficiency. This is attributed to the fact that a partial
reversal process ($\hat{a}^{\dagger}_{b}\hat{a}_{w}\hat{\sigma}$) is allowed.
Given this potential complication, it is imperative to thoroughly examine how
the residual waste photon number affects the qubit’s temporal evolution.

We model the residual waste photon number as arising from a thermal bath.
Hence, the Eqs. (10), (11) and (12) are modified to

$\displaystyle\frac{\text{d}}{\text{d}t}\hat{\rho}_{00}=$
$\displaystyle-i\delta\left[g_{4}\xi_{p}^{*}\hat{a}_{b}^{\dagger}\hat{\sigma}\hat{\rho}_{10}-g_{4}\xi_{p}\hat{\rho}_{01}\hat{a}_{b}\hat{\sigma}^{\dagger}\right]$
$\displaystyle+\delta^{2}\kappa_{w}\left(1+n_{\text{th},w}\right)\hat{\rho}_{11}-\kappa_{w}n_{\text{th},w}\hat{\rho}_{00}$
$\displaystyle+\mathcal{L}_{qb}\left[\hat{\rho}_{00}\right]+\mathcal{O}\left(\delta^{3}\right)$
(34) $\displaystyle\delta\frac{\text{d}}{\text{d}t}\hat{\rho}_{01}=$
$\displaystyle
i\hat{\rho}_{00}\left[g_{4}\xi_{p}^{*}\hat{a}_{b}^{\dagger}\hat{\sigma}\right]-\delta\frac{\kappa_{w}\left(1+4n_{\text{th},w}\right)}{2}\hat{\rho}_{01}$
$\displaystyle+\delta\mathcal{L}_{qb}\left[\hat{\rho}_{01}\right]+\mathcal{O}\left(\delta^{3}\right)$
(35) $\displaystyle\delta^{2}\frac{\text{d}}{\text{d}t}\hat{\rho}_{11}=$
$\displaystyle-i\delta\left[g_{4}\xi_{p}\hat{a}_{b}\hat{\sigma}^{\dagger}\hat{\rho}_{01}-g_{4}\xi_{p}^{*}\hat{\rho}_{10}\hat{a}_{b}^{\dagger}\hat{\sigma}\right]$
$\displaystyle-\delta^{2}\kappa_{w}\left(1+3n_{\text{th},w}\right)\hat{\rho}_{11}+n_{\text{th},w}\hat{\rho}_{00}$
$\displaystyle+\delta^{2}\mathcal{L}_{qb}\left[\hat{\rho}_{11}\right]+\mathcal{O}\left(\delta^{3}\right)$
(36)

Here, we set $\Delta_{w}=\chi_{qw}$ for simplicity. With the same argument, we
can find the steady state solutions for $\hat{\rho}_{01}$ and
$\hat{\rho}_{11}$

$\displaystyle\hat{\rho}_{01}\approx$
$\displaystyle\hat{\rho}_{00}\frac{2ig_{4}\xi_{p}^{*}}{\delta\kappa_{w}(1+4n_{\text{th},w})}\hat{a}_{b}^{\dagger}\hat{\sigma}$
$\displaystyle\hat{\rho}_{11}\approx$
$\displaystyle\frac{1}{\delta^{2}\kappa_{w}\left(1+3n_{\text{th},w}\right)}\frac{4\left|g_{4}\xi_{p}\right|^{2}}{\kappa_{w}(1+4n_{\text{th},w})}\hat{a}_{b}\hat{\sigma}^{\dagger}\hat{\rho}_{00}\hat{a}_{b}^{\dagger}\hat{\sigma}$
$\displaystyle+\frac{1}{\delta^{2}\kappa_{w}\left(1+3n_{\text{th},w}\right)}\kappa_{w}n_{\text{th},w}\hat{\rho}_{00}$

and insert them into

$\displaystyle\frac{\text{d}}{\text{d}t}\hat{\rho}_{00}\approx$
$\displaystyle\kappa_{\rm
nl}\left(1-4n_{\text{th},w}\right)\mathcal{D}[\hat{a}_{b}\hat{\sigma}^{\dagger}]\hat{\rho}_{00}$
$\displaystyle-2\kappa_{\rm
nl}n_{\text{th},w}\hat{a}_{b}\hat{\sigma}^{\dagger}\hat{\rho}_{00}\hat{a}_{b}^{\dagger}\hat{\sigma}+\mathcal{L}_{qb}\left[\hat{\rho}_{00}\right]$
$\displaystyle+\mathcal{O}\left(\delta^{3},n_{\text{th},w}^{2}\right).$ (37)

The Lindblad master equation therefore is read with the buffer driving term
$i\varepsilon_{b}^{\prime}\left(\hat{a}_{b}-\hat{a}_{b}^{\dagger}\right)$ as

$\displaystyle\frac{\text{d}}{\text{d}t}\hat{\rho}_{00}=$
$\displaystyle\kappa_{\rm
nl}\left(1-4n_{\text{th},w}\right)\mathcal{D}\left[\hat{a}_{b}\hat{\sigma}^{\dagger}\right]\hat{\rho}_{00}$
$\displaystyle-2\kappa_{\rm
nl}n_{\text{th},w}\hat{a}_{b}\hat{\sigma}^{\dagger}\hat{\rho}_{00}\hat{a}_{b}^{\dagger}\hat{\sigma}+\kappa_{b}\mathcal{D}\left[\hat{a}_{b}\right]\hat{\rho}_{00}$
$\displaystyle+\varepsilon_{b}^{\prime}\left[\hat{a}_{b}-\hat{a}_{b}^{\dagger},\hat{\rho}_{00}\right].$
(38)

After following the same procedure as outlined previously, the differential
equation is formulated as

$\displaystyle\frac{\text{d}}{\text{d}t}p_{g}=$ $\displaystyle-\kappa_{\rm
nl}\left(1-4n_{\text{th},w}\right)\left|\beta^{\prime}\right|^{2}p_{g},$ (39a)
$\displaystyle\frac{\text{d}}{\text{d}t}p_{e}=$ $\displaystyle\kappa_{\rm
nl}\left(1-6n_{\text{th},w}\right)\left|\beta^{\prime}\right|^{2}p_{g}$ (39b)

with

$\beta^{\prime}=-\frac{2\varepsilon_{b}^{\prime}}{\kappa_{\rm
nl}\left(1-4n_{\text{th},w}\right)+\kappa_{b}}.$

Finally, we obtain the qubit evolution

$p_{e}(t)=\frac{1-6n_{\text{th},w}}{1-4n_{\text{th},w}}\left(1-\exp\left(-\eta_{c}^{\prime}\left|b_{\rm
in}\right|^{2}t\right)\right)$ (40)

with the modified conversion efficiency

$\eta_{c}^{\prime}:=\frac{4\kappa_{b}\kappa_{\rm
nl}\left(1-4n_{\text{th},w}\right)}{(\kappa_{\rm
nl}\left(1-4n_{\text{th},w}\right)+\kappa_{b})^{2}}.$ (41)

We clearly see that the qubit population is limited by the waste photon number
$n_{\text{th},w}$ and can never reach unity.

## References

  * Yurke  _et al._ [1989] B. Yurke, L. R. Corruccini, P. G. Kaminsky, L. W. Rupp, A. D. Smith, A. H. Silver, R. W. Simon, and E. A. Whittaker, Observation of parametric amplification and deamplification in a Josephson parametric amplifier, Phys. Rev. A 39, 2519 (1989).
  * Castellanos-Beltran  _et al._ [2008] M. A. Castellanos-Beltran, K. D. Irwin, G. C. Hilton, L. R. Vale, and K. W. Lehnert, Amplification and squeezing of quantum noise with a tunable Josephson metamaterial, Nat. Phys. 4, 928 (2008).
  * Bergeal  _et al._ [2010] N. Bergeal, F. Schackert, M. Metcalfe, R. Vijay, V. E. Manucharyan, L. Frunzio, D. E. Prober, R. J. Schoelkopf, S. M. Girvin, and M. H. Devoret, Phase-preserving amplification near the quantum limit with a Josephson ring modulator, Nature 465, 64 (2010).
  * Abdo  _et al._ [2014] B. Abdo, K. Sliwa, L. Frunzio, and M. Devoret, Directional amplification with a Josephson circuit, Phys. Rev. X 3, 3 (2014).
  * Malnou  _et al._ [2021] M. Malnou, M. R. Vissers, J. D. Wheeler, J. Aumentado, J. Hubmayr, J. N. Ullom, and J. Gao, Three-Wave Mixing Kinetic Inductance Traveling-Wave Amplifier with Near-Quantum-Limited Noise Performance, PRX Quantum 2, 1 (2021).
  * Qiu  _et al._ [2023] J. Y. Qiu, A. Grimsmo, K. Peng, B. Kannan, B. Lienhard, Y. Sung, P. Krantz, V. Bolkhovsky, G. Calusine, D. Kim, A. Melville, B. M. Niedzielski, J. Yoder, M. E. Schwartz, T. P. Orlando, I. Siddiqi, S. Gustavsson, K. P. O. Brien, and W. D. Oliver, Broadband squeezed microwaves and amplification with a Josephson travelling-wave parametric amplifier, Nat. Phys. 19, 706 (2023).
  * Abdo  _et al._ [2011] B. Abdo, F. Schackert, M. Hatridge, C. Rigetti, and M. Devoret, Josephson amplifier for qubit readout, Appl. Phys. Lett. 99, 16 (2011).
  * Kamal  _et al._ [2014] A. Kamal, A. Roy, J. Clarke, and M. H. Devoret, Asymmetric frequency conversion in nonlinear systems driven by a biharmonic pump, Phys. Rev. Lett. 113, 247003 (2014).
  * Jiang  _et al._ [2023] Y. Jiang, E. P. Ruddy, K. O. Quinlan, M. Malnou, N. E. Frattini, and K. W. Lehnert, Accelerated Weak Signal Search Using Mode Entanglement and State Swapping, Phys. Rev. Appl. 10, 1 (2023).
  * Pechal  _et al._ [2014] M. Pechal, L. Huthmacher, C. Eichler, S. Zeytino, A. A. Abdumalikov, S. Berger, A. Wallraff, and S. Filipp, Microwave-Controlled Generation of Shaped Single Photons in Circuit Quantum Electrodynamics, Phys. Rev. X 4, 041010 (2014).
  * Flurin  _et al._ [2012] E. Flurin, N. Roch, F. Mallet, M. H. Devoret, and B. Huard, Generating entangled microwave radiation over two transmission lines, Phys. Rev. Lett. 109, 183901 (2012).
  * Murch  _et al._ [2012] K. W. Murch, U. Vool, D. Zhou, S. J. Weber, S. M. Girvin, and I. Siddiqi, Cavity-assisted quantum bath engineering, Phys. Rev. Lett. 109, 183602 (2012).
  * 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, Science 347, 853 (2015).
  * Dassonneville  _et al._ [2020] R. Dassonneville, R. Assouly, T. Peronnin, P. Rouchon, and B. Huard, Number-Resolved Photocounter for Propagating Microwave Mode, Phys. Rev. Appl. 14, 044022 (2020).
  * Lescanne  _et al._ [2020] R. Lescanne, S. Deléglise, E. Albertinale, U. Réglade, T. Capelle, E. Ivanov, T. Jacqmin, Z. Leghtas, and E. Flurin, Irreversible Qubit-Photon Coupling for the Detection of Itinerant Microwave Photons, Phys. Rev. X 10, 021038 (2020).
  * Ma  _et al._ [2012a] X. S. Ma, T. Herbst, T. Scheidl, D. Wang, S. Kropatschek, W. Naylor, B. Wittmann, A. Mech, J. Kofler, E. Anisimova, V. Makarov, T. Jennewein, R. Ursin, and A. Zeilinger, Quantum teleportation over 143 kilometres using active feed-forward, Nature 489, 269 (2012a).
  * Abdelkhalek  _et al._ [2016] D. Abdelkhalek, M. Syllwasschy, N. J. Cerf, J. Fiurášek, and R. Schnabel, Efficient entanglement distillation without quantum memory, Nat. Commun. 7, 11720 (2016).
  * Ma  _et al._ [2012b] X. S. Ma, S. Zotter, J. Kofler, R. Ursin, T. Jennewein, Č. Brukner, and A. Zeilinger, Experimental delayed-choice entanglement swapping, Nat. Phys. 8, 479 (2012b).
  * Scully and Drühl [1982] M. O. Scully and K. Drühl, Quantum eraser: A proposed photon correlation experiment concerning observation and ”delayed choice” in quantum mechanics, Phys. Rev. A 25, 2208 (1982).
  * Yuan  _et al._ [2008] Z. S. Yuan, Y. A. Chen, B. Zhao, S. Chen, J. Schmiedmayer, and J. W. Pan, Experimental demonstration of a BDCZ quantum repeater node, Nature 454, 1098 (2008).
  * Deng  _et al._ [2023] Y. H. Deng, S. Q. Gong, Y. C. Gu, Z. J. Zhang, H. L. Liu, H. Su, H. Y. Tang, J. M. Xu, M. H. Jia, M. C. Chen, H. S. Zhong, H. Wang, J. Yan, Y. Hu, J. Huang, W. J. Zhang, H. Li, X. Jiang, L. You, Z. Wang, L. Li, N. L. Liu, C. Y. Lu, and J. W. Pan, Solving Graph Problems Using Gaussian Boson Sampling, Phys. Rev. Lett. 130, 190601 (2023).
  * Jin  _et al._ [2012] X. M. Jin, Z. H. Yi, B. Yang, F. Zhou, T. Yang, and C. Z. Peng, Experimental quantum error detection, Sci. Rep. 2, 626 (2012).
  * Chen  _et al._ [2011] Y. F. Chen, D. Hover, S. Sendelbach, L. Maurer, S. T. Merkel, E. J. Pritchett, F. K. Wilhelm, and R. McDermott, Microwave photon counter based on josephson junctions, Phys. Rev. Lett. 107, 217401 (2011).
  * Inomata  _et al._ [2016] K. Inomata, Z. Lin, K. Koshino, W. D. Oliver, J.-S. Tsai, T. Yamamoto, and Y. Nakamura, Single microwave-photon detector using an artiﬁcial L-type three-level system, Nat. Commun. 7, 12303 (2016).
  * Kono  _et al._ [2018] S. Kono, K. Koshino, Y. Tabuchi, A. Noguchi, and Y. Nakamura, Quantum non-demolition detection of an itinerant microwave photon, Nat. Phys. 14, 546 (2018).
  * Besse  _et al._ [2018] J. C. Besse, S. Gasparinetti, M. C. Collodo, T. Walter, P. Kurpiers, M. Pechal, C. Eichler, and A. Wallraff, Single-Shot Quantum Nondemolition Detection of Individual Itinerant Microwave Photons, Phys. Rev. X 8, 021003 (2018).
  * Varnava  _et al._ [2008] M. Varnava, D. E. Browne, and T. Rudolph, How good must single photon sources and detectors be for efficient linear optical quantum computation?, Phys. Rev. Lett. 100, 060502 (2008).
  * Hadfield [2009] R. H. Hadfield, Single-photon detectors for optical quantum information applications, Nat. Photon. 3, 696 (2009).
  * Lamoreaux  _et al._ [2013] S. K. Lamoreaux, K. A. Van Bibber, K. W. Lehnert, and G. Carosi, Analysis of single-photon and linear amplifier detectors for microwave cavity dark matter axion searches, Phys. Rev. D 88, 035020 (2013).
  * Chunnilall  _et al._ [2014] C. J. Chunnilall, I. P. Degiovanni, S. Kück, I. Müller, and A. G. Sinclair, Metrology of single-photon sources and detectors: a review, Opt. Eng. 53, 081910 (2014).
  * Barzanjeh  _et al._ [2015] S. Barzanjeh, S. Guha, C. Weedbrook, D. Vitali, J. H. Shapiro, and S. Pirandola, Microwave quantum illumination, Phys. Rev. Lett. 114, 080503 (2015).
  * Assouly  _et al._ [2023] R. Assouly, R. Dassonneville, T. Peronnin, A. Bienfait, and B. Huard, Quantum advantage in microwave quantum radar, Nat. Phys. 19, 1418 (2023).
  * Kronowetter  _et al._ [2023] F. Kronowetter, F. Fesquet, M. Renger, K. Honasoge, Y. Nojiri, K. Inomata, Y. Nakamura, A. Marx, R. Gross, and K. G. Fedorov, Quantum Microwave Parametric Interferometer, Phys. Rev. Appl. 20, 024049 (2023).
  * Wang  _et al._ [2023] Z. Wang, L. Balembois, M. Rančić, E. Billaud, M. Le Dantec, A. Ferrier, P. Goldner, S. Bertaina, T. Chanelière, D. Esteve, D. Vion, P. Bertet, and E. Flurin, Single-electron spin resonance detection by microwave photon counting, Nature 619, 276 (2023).
  * Billaud  _et al._ [2023] E. Billaud, L. Balembois, M. Le Dantec, M. Rančić, E. Albertinale, S. Bertaina, T. Chanelière, P. Goldner, D. Estève, D. Vion, P. Bertet, and E. Flurin, Microwave Fluorescence Detection of Spin Echoes, Phys. Rev. Lett. 131, 100804 (2023).
  * Las Heras  _et al._ [2017] U. Las Heras, R. Di Candia, K. G. Fedorov, F. Deppe, M. Sanz, and E. Solano, Quantum illumination reveals phase-shift inducing cloaking, Sci. Rep. 7, 9333 (2017).
  * Fedorov  _et al._ [2021] K. G. Fedorov, M. Renger, S. Pogorzalek, R. Di Candia, Q. Chen, Y. Nojiri, K. Inomata, Y. Nakamura, M. Partanen, A. Marx, R. Gross, and F. Deppe, Experimental quantum teleportation of propagating microwaves, Sci. Adv. 7, eabk0891 (2021).
  * Renger  _et al._ [2021] M. Renger, S. Pogorzalek, Q. Chen, Y. Nojiri, K. Inomata, Y. Nakamura, M. Partanen, A. Marx, R. Gross, F. Deppe, and K. G. Fedorov, Beyond the standard quantum limit for parametric amplification of broadband signals, npj Quantum Inf. 7, 160 (2021).
  * 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. Appl. 11, 014030 (2019).
  * Verney  _et al._ [2019] L. Verney, R. Lescanne, M. H. Devoret, and Z. Leghtas, Structural Instability of Driven Josephson Circuits Prevented by an Inductive Shunt, Phys. Rev. Appl. 10, 024003 (2019).
  * Shillito  _et al._ [2022] R. Shillito, A. Petrescu, J. Cohen, J. Beall, M. Hauru, M. Ganahl, A. G. Lewis, G. Vidal, and A. Blais, Dynamics of Transmon Ionization, Phys. Rev. Appl. 18, 034031 (2022).
  * Cohen  _et al._ [2023] J. Cohen, A. Petrescu, R. Shillito, and A. Blais, Reminiscence of classical chaos in driven transmons, PRX Quantum 4, 020312 (2023).
  * Boissonneault  _et al._ [2010] M. Boissonneault, J. M. Gambetta, and A. Blais, Improved superconducting qubit readout by qubit-induced nonlinearities, Phys. Rev. Lett. 105, 100504 (2010).
  * Bishop  _et al._ [2010] L. S. Bishop, E. Ginossar, and S. M. Girvin, Response of the Strongly Driven Jaynes-Cummings Oscillator, Phys. Rev. Lett. 105, 100505 (2010).
  * Pietikäinen  _et al._ [2019] I. Pietikäinen, J. Tuorila, D. S. Golubev, and G. S. Paraoanu, Photon blockade and the quantum-to-classical transition in the driven-dissipative Josephson pendulum coupled to a resonator, Phys. Rev. A 99, 063828 (2019).
  * Carmichael [2015] H. J. Carmichael, Breakdown of photon blockade: A dissipative quantum phase transition in zero dimensions, Phys. Rev. X 5, 031028 (2015).
  * Mavrogordatos [2016] T. K. Mavrogordatos, Quantum phase transitions in the driven dissipative Jaynes-Cummings oscillator: From the dispersive regime to resonance, EPL 116, 54001 (2016).
  * Fink  _et al._ [2018] T. Fink, A. Schade, S. Höfling, C. Schneider, and A. Imamoglu, Signatures of a dissipative phase transition in photon correlation measurements, Nat. Phys. 14, 365 (2018).
  * Chen  _et al._ [2023] Q.-M. Chen, M. Fischer, Y. Nojiri, M. Renger, E. Xie, M. Partanen, S. Pogorzalek, K. G. Fedorov, A. Marx, F. Deppe, and R. Gross, Quantum behavior of a superconducting Duffing oscillator at the dissipative phase transition, Nat. Commun. 14, 2896 (2023).
  * Utermann  _et al._ [1994] R. Utermann, T. Dittrich, and P. Hänggi, Tunneling and the onset of chaos in a driven bistable system, Phys. Rev. E 49, 273 (1994).
  * Reed  _et al._ [2010] M. D. Reed, L. Dicarlo, B. R. Johnson, L. Sun, D. I. Schuster, L. Frunzio, and R. J. Schoelkopf, High-fidelity readout in circuit quantum electrodynamics using the jaynes-cummings nonlinearity, Phys. Rev. Lett. 105, 173601 (2010).
  * Gambetta  _et al._ [2008] J. Gambetta, A. Blais, M. Boissonneault, A. A. Houck, D. I. Schuster, and S. M. Girvin, Quantum trajectory approach to circuit QED: Quantum jumps and the Zeno effect, Phys. Rev. A 77, 012112 (2008).
  * Krantz  _et al._ [2019] P. Krantz, M. Kjaergaard, F. Yan, T. P. Orlando, S. Gustavsson, and W. D. Oliver, A quantum engineer’s guide to superconducting qubits, Applied Physics Reviews 6, 1 (2019).
  * Elliott and Ginossar [2016] M. Elliott and E. Ginossar, Applications of the Fokker-Planck equation in circuit quantum electrodynamics, Phys. Rev. A 94, 043840 (2016).
  * Rényi [1960] A. Rényi, On measures of information and entropy, Proceedings of the fourth Berkeley Symposium on Mathematics, Statistics and Probability 1, 547 (1960).
  * Jin and Korepin [2004] B. Q. Jin and V. E. Korepin, Quantum spin chain, Toeplitz determinants and the Fisher-Hartwig conjecture, J. Stat. Phys. 116, 79 (2004).
  * Qolibikloo and Ghodsi [2019] S. Qolibikloo and A. Ghodsi, More on phase transition and Rényi entropy, Eur. Phys. J. C 79, 406 (2019).
  * Hines  _et al._ [2005] A. P. Hines, R. H. McKenzie, and G. J. Milburn, Quantum entanglement and fixed-point bifurcations, Phys. Rev. A 71, 042303 (2005).
  * Córcoles  _et al._ [2011] A. D. Córcoles, J. M. Chow, J. M. Gambetta, C. Rigetti, J. R. Rozen, G. A. Keefe, M. Beth Rothwell, M. B. Ketchen, and M. Steffen, Protecting superconducting qubits from radiation, Appl. Phys. Lett. 99, 181906 (2011).
  * Geerlings  _et al._ [2013] K. Geerlings, Z. Leghtas, I. M. Pop, S. Shankar, L. Frunzio, R. J. Schoelkopf, M. Mirrahimi, and M. H. Devoret, Demonstrating a driven reset protocol for a superconducting qubit, Phys. Rev. Lett. 110, 120501 (2013).
  * Jin  _et al._ [2015] X. Y. Jin, A. Kamal, A. P. Sears, T. Gudmundsen, D. Hover, J. Miloshi, R. Slattery, F. Yan, J. Yoder, T. P. Orlando, S. Gustavsson, and W. D. Oliver, Thermal and Residual Excited-State Population in a 3D Transmon Qubit, Phys. Rev. Lett. 114, 240501 (2015).
  * [62] L. Balembois, J. Travesedo, L. Pallegoix, A. May, E. Billaud, M. Villiers, D. Estève, D. Vion, P. Bertet, and E. Flurin, Practical Single Microwave Photon Counter with $10^{\mathrm{-22}}$ $\mathrm{W/\sqrt{Hz}}$ sensitivity, arXiv:2307.03614 .
  * Gambetta  _et al._ [2011] J. M. Gambetta, F. Motzoi, S. T. Merkel, and F. K. Wilhelm, Analytic control methods for high-fidelity unitary operations in a weakly nonlinear oscillator, Phys. Rev. A 83, 012308 (2011).
  * Chen  _et al._ [2016] Z. Chen, J. Kelly, C. Quintana, R. Barends, B. Campbell, Y. Chen, B. Chiaro, A. Dunsworth, A. G. Fowler, E. Lucero, E. Jeffrey, A. Megrant, J. Mutus, M. Neeley, C. Neill, P. J. O’Malley, P. Roushan, D. Sank, A. Vainsencher, J. Wenner, T. C. White, A. N. Korotkov, and J. M. Martinis, Measuring and Suppressing Quantum State Leakage in a Superconducting Qubit, Phys. Rev. Lett. 116, 020501 (2016).
  * Yan  _et al._ [2016] F. Yan, S. Gustavsson, A. Kamal, J. Birenbaum, A. P. Sears, D. Hover, T. J. Gudmundsen, D. Rosenberg, G. Samach, S. Weber, J. L. Yoder, T. P. Orlando, J. Clarke, A. J. Kerman, and W. D. Oliver, The flux qubit revisited to enhance coherence and reproducibility, Nat. Commun. 7, 12964 (2016).