#  Stimulated emission of signal photons from dark matter waves

Ankur Agrawal<EMAIL_ADDRESS>Present address: AWS Center for
Quantum Networking, Boston, MA 02145, USA James Franck Institute, University
of Chicago, Chicago, Illinois 60637, USA  Department of Physics, University of
Chicago, Chicago, Illinois 60637, USA  Kavli Institute for Cosmological
Physics, University of Chicago, Chicago, Illinois 60637, USA  Akash V. Dixit
Present address: National Institute of Standards and Technology, Boulder, CO
80305, USA James Franck Institute, University of Chicago, Chicago, Illinois
60637, USA  Department of Physics, University of Chicago, Chicago, Illinois
60637, USA  Kavli Institute for Cosmological Physics, University of Chicago,
Chicago, Illinois 60637, USA  Tanay Roy  Present address: Superconducting
Quantum Materials and Systems Center, Fermi National Accelerator Laboratory,
Batavia, IL 60510, USA James Franck Institute, University of Chicago, Chicago,
Illinois 60637, USA  Department of Physics, University of Chicago, Chicago,
Illinois 60637, USA  Srivatsan Chakram  Present address: Department of Physics
and Astronomy, Rutgers, the State University of New Jersey, 136 Frelinghuysen
Road, Piscataway, NJ 08854 James Franck Institute, University of Chicago,
Chicago, Illinois 60637, USA  Department of Physics, University of Chicago,
Chicago, Illinois 60637, USA  Department of Physics and Astronomy, Rutgers
University, Piscataway, New Jersey 08854, USA  Kevin He  James Franck
Institute, University of Chicago, Chicago, Illinois 60637, USA  Department of
Physics, University of Chicago, Chicago, Illinois 60637, USA  Ravi K. Naik
Computational Research Division, Lawrence Berkeley National Laboratory,
Berkeley, CA, USA  David I. Schuster  James Franck Institute, University of
Chicago, Chicago, Illinois 60637, USA  Department of Physics, University of
Chicago, Chicago, Illinois 60637, USA  Pritzker School of Molecular
Engineering, University of Chicago, Chicago, Illinois 60637, USA  Department
of Applied Physics, Stanford University, Stanford, California 94305, USA
Aaron Chou  Fermi National Accelerator Laboratory, Batavia, Illinois 60510,
USA

###### Abstract

The manipulation of quantum states of light has resulted in significant
advancements in both dark matter searches and gravitational wave detectors [1,
2, 3, 4]. Current dark matter searches operating in the microwave frequency
range use nearly quantum-limited amplifiers [5, 3, 6]. Future high frequency
searches will use photon counting techniques [1] to evade the standard quantum
limit. We present a signal enhancement technique that utilizes a
superconducting qubit to prepare a superconducting microwave cavity in a non-
classical Fock state and stimulate the emission of a photon from a dark matter
wave. By initializing the cavity in an $\left|n=4\right\rangle$ Fock state, we
demonstrate a quantum enhancement technique that increases the signal photon
rate and hence also the dark matter scan rate each by a factor of 2.78. Using
this technique, we conduct a dark photon search in a band around
$\mathrm{5.965\,GHz\,(24.67\,\mu eV)}$, where the kinetic mixing angle
$\epsilon\geq 4.35\times 10^{-13}$ is excluded at the $90\%$ confidence level.

## Introduction

The existence of dark matter (DM) is one of the greatest mysteries in physics,
which has puzzled scientists for nearly a century. Despite the lack of direct
detection, there is compelling evidence for its existence, including its
estimated contribution of 27% to the universe’s energy density and its
gravitational effects on galaxy dynamics and structure formation [7, 8, 9].
Axions and dark photons have emerged as leading candidates for dark matter due
to their cosmological origins and low-energy properties, which allow them to
exist as coherent waves with macroscopic occupation numbers [10, 11, 12, 13,
14]. Dark matter haloscope experiments in the microwave frequency range use a
cavity to resonantly enhance the oscillating electric field generated by the
DM field at a frequency corresponding to the mass of a hypothetical particle
($\nu=mc^{2}/h$)[15, 14]. Since the mass of DM is unknown a priori,
experimental searches are typically conducted as radio scans in which the
resonant cavity is tuned one step $d\nu$ at a time to test different frequency
hypotheses. A key figure of merit is therefore the frequency scan speed, which
in a photon counting experiment scales as $d\nu/dt\propto
d\nu\,R_{s}^{2}/R_{b}$ where $R_{s}$ and $R_{b}$ are the signal and background
count rates respectively.

Quantum techniques have proven useful in accelerating the scan rate of axion
and wavelike dark matter searches. Superconducting parametric amplifiers,
which are quantum-limited, have reached the standard quantum limit (SQL) and
add only 1/2 photon of noise per mode, as required by the Heisenberg
uncertainty principle for phase-preserving measurements [16, 17, 18, 19, 20].
Alternatively, qubit-based single photon detection [1] does not consider the
photon phase and can in principle measure without any added detection noise by
achieving extremely low, sub-SQL background rates.

While both of these methods employ quantum technologies, their operation is in
some sense recognizable as an ideal classical amplifier and microwave photo-
multiplier. Parametric amplifiers and cavity-qubit systems can also synthesize
inherently quantum mechanical states of light such as squeezed states [21, 22,
23] in the former and Fock states [24, 25] or Schrodinger cat states [26, 27,
28, 29, 30] for the latter. Recently, squeezed state injection paired with
phase-sensitive amplification was used to improve the scan rate of the HAYSTAC
experiment [2]. In this work, we develop a new method in which we prepare an
$n$-photon Fock state which enhances the signal rate by $\eta\,(n+1)$ times
where $\eta$ is the efficiency of detecting the state. By creating a Fock
state with $\left|n=4\right\rangle$ photons in the cavity, we observe a
$2.78$-fold enhancement in the signal rate. We show that this technique is
compatible with the previously-demonstrated noise reduction from photon
counting [1].

The power delivered to the cavity by a current density generated by dark
matter $\textbf{J}_{DM}$ is given by

$P_{s}=\int dV\,\textbf{J}_{DM}(x)\cdot\textbf{E}(x)$ (1)

and is proportional to the magnitude of oscillating electric field
$\textbf{E}(x)$ in the cavity. In the conventional scenario, the cavity is
cooled to the vacuum state, and the signal electric field builds up
monotonically over the coherence time of the cavity or of the dark matter wave
in a process akin to spontaneous emission. Alternatively, we may initialize
the cavity with a non-zero $\textbf{E}(x)$ field from a coherent state sine
wave or from a Fock state to induce stimulated emission. The Fock state has
some advantages. First, unlike the homodyne or heterodyne detection using a
coherent state pump, the Fock state is free from any shot noise, making it
possible to measure small signal amplitudes far below the SQL. Second, a Fock
state is symmetric in phase, making it equally sensitive to any instantaneous
phase of the incoming DM wave, which is unknown a priori.

We can model the action of the DM wave on a Fock state as a classical drive
amplitude $\xi$ which shifts this phase-symmetric state away from the origin
in the Wigner phase space by $\alpha$ (see Supp Fig. S6). The resultant state
comprises both in-phase components which extracted excess power from the DM
wave and also out-of-phase components which delivered their power to the DM
wave. The stimulated emission process for DM converting into photons is
enhanced by a factor of $(n+1)$ while the stimulated absorption process is
enhanced by a factor of $n$. Mathematically, the stimulated emission into the
cavity state from a dark matter wave can be described as shown below,

$\begin{split}|\left\langle
n+1\right|\mathcal{\hat{D}(\alpha)}\left|n\right\rangle|^{2}&=|\left\langle
n+1\right|e^{(\alpha a^{\dagger}-\alpha^{*}a)}\left|n\right\rangle|^{2}\\\
&\sim|\left\langle n+1\right|\alpha
a^{\dagger}\left|n\right\rangle|^{2}=(n+1)\alpha^{2}\end{split}$ (2)

where $\mathcal{\hat{D}(\alpha)}$ is the displacement operator. From Eqn. (2),
we can infer that the displacement ($\alpha\ll 1$) induced by the DM wave on a
cavity prepared in $\left|n\right\rangle$ Fock state results in population of
$\left|n+1\right\rangle$ state with probability proportional to $(n+1)$. Using
number-resolving measurements, the signal can thus be observed with
$(n+1)\times$ higher probability if the cavity is prepared in a larger
$\left|n\right\rangle$ Fock state.

We note that just as in other cases of quantum-enhanced metrology, the $(n+1)$
enhancement factor in the signal transition probability can be exactly
canceled by the $1/(n+1)$ reduction in the coherence time of the probe. As a
result, there would be no net improvement in the actual signal rate $R_{s}$.
However, this consideration does not apply when the limiting coherence time is
that of the dark matter wave rather than of the Fock photon state in the
cavity. In such cases, the signal rate is not degraded by the reduction of the
probe coherence time and retains the factor $(n+1)$. To our knowledge, this is
one of the few cases where quantum metrology can provide a realizable
improvement in a real-world application. Also, since the readout rate scales
as the inverse of the probe coherence time, the background count rate may also
scale linearly with $(n+1)$, for example for backgrounds associated with
readout errors. For experiments with fixed tuning step size $d\nu$ given for
example by the dark matter linewidth, the improvement in scan speed
$d\nu/dt\propto d\nu\,R_{s}^{2}/R_{b}$ is therefore a single factor of
$\eta\,(n+1)$.

## Fock state preparation and photon number resolving detector

We couple the cavity to a non-linear element, in this case a superconducting
transmon qubit to prepare and measure the Fock states, which are otherwise
impossible to create in a linear system such as a cavity. The device used in
this work is composed of three components - a high quality factor
($Q_{s}=4.06\times 10^{7}$) 3D multimode cavity [31] to accumulate and store
the signal induced by the dark matter (storage,
$\omega_{s}/2\pi=\mathrm{5.965\,GHz}$), a superconducting transmon qubit
($\omega_{q}/2\pi=\mathrm{4.95\,GHz}$), and a 3D cavity strongly coupled to a
transmission line ($Q_{r}=9\times 10^{3}$) used to quickly read out the state
of qubit (readout, $\omega_{r}/2\pi=\mathrm{7.789\,GHz}$) (Fig. 1 (a)). We
mount the device to the base stage of a dilution refrigerator operating at
$\mathrm{10\,mK}$.

Figure 1: Fock state preparation in a cavity dispersively coupled to a
transmon qubit. (a) A schematic of the multimode flute cavity showing the
location of the storage cavity (red), readout cavity (green), and transmon
chip with a SEM image of the Josephson junction (blue) [31]. (b) Creation of
Fock states in a particular mode of the cavity using GRAPE method.
Characterization of the cavity state using qubit spectroscopy (left) and
Wigner tomography (right). Qubit spectroscopy is performed immediately after
the optimal control (OCT) pulses, a single peak in each qubit excitation
probability ($P_{e}$) distribution confirms the creation of the correct
$\left|n\right\rangle$ Fock state. Resultant probability distribution is
fitted to a Gaussian to estimate the state preparation fidelity. Grey dashed
lines correspond to the expected shift in frequency, accounted for quadratic
dispersive shift. (Right) Wigner tomography [32] is performed by coherently
displacing the resultant cavity state in 2D phase space to map the average
parity and thus, reconstruct the cavity state density matrix.

The interaction between a superconducting transmon qubit [33, 34] and the
field in a microwave cavity is described by the Jaynes-Cummings Hamiltonian
[35] in the dispersive limit (qubit-cavity coupling $\ll$ qubit-cavity
detuning) as,

$\mathcal{H}/\hbar=\omega_{s}a^{\dagger}a+\frac{1}{2}(\omega_{q}+\chi
a^{\dagger}a)\sigma_{z}$ (3)

where $a$ and $a^{\dagger}$ are the annihilation and creation operators of the
cavity mode and $\sigma_{z}$ is the Pauli $Z$ operator of the transmon. Eqn. 3
elucidates a key feature of this interaction - a photon number dependent shift
($\chi$) of the qubit transition frequency (see Fig. 1(b))[36]. Another
important feature of this Hamiltonian is the quantum non-demolition (QND)
nature of the interaction between the qubit and cavity which preserves the
cavity state upon the measurement of the qubit state and vice-versa [37, 38,
36]. By driving the qubit at the Stark shifted frequency
($\omega_{q}+n\,\chi$), one would selectively excite the qubit if and only if
there are exactly $n$-photons in the cavity.

Recent works have shown that a single transmon has the capability to prepare
any quantum state in a cavity and perform universal control on it [39, 27, 40,
41, 42, 23]. In this study, we used a GRadient Ascent Pulse Engineering
(GRAPE) based method to generate optimal control pulses (OCT) [43, 27] that
consider the full model of the time-dependent Hamiltonian and allow us to
prepare non-classical states in a cavity. As shown in Fig. 1 (b), our approach
successfully prepared cavity Fock states with pulse duration as short as
$\mathcal{O}(1/\chi)$, which did not increase for higher Fock states.

## Stimulated emission protocol

The stimulated emission protocol is divided into two parts: the first part
involves the preparation of cavity in a desired Fock state,
$\left|n\right\rangle$ and the second part involves the detection of the
cavity in the $\left|n+1\right\rangle$ Fock state as depicted in Fig. 2(a). In
order to actively suppress any false positive events such as the cavity
accidentally starting in $\left|n+1\right\rangle$ state, we conditionally
excite the qubit with a resolved $\pi$-pulse at the $(n+1)$-shifted peak three
times. If and only if the qubit fails to excite in all three attempts do we
proceed ahead with rest of the protocol. By doing this, we can suppress the
false positive rate $\leq 3\%$.

At the end of this sequence, we measure the efficiency of the state
preparation for each $n$ by measuring the qubit excitation probability $P_{n}$
with a number resolved $\pi$-pulse centered at the $\left|n\right\rangle$
peak. The measured fidelities are $P_{0}=95.2\pm 0.3\%,P_{1}=91.2\pm
0.4\%,P_{2}=87.3\pm 0.5\%,P_{3}=81.6\pm 0.6\%,P_{4}=63.6\pm 0.7\%$.

Figure 2: Stimulated emission protocol with number resolved $\pi$-pulse and
hidden Markov model analysis. (a) Pulse sequence for stimulated emission
includes cavity initialization in a Fock state, followed by three conditional
checks to ensure cavity did not accidentally start in $\left|n+1\right\rangle$
state. The next part involves a cavity displacement drive to mimic a push from
the DM and repeated conditional qubit measurements to detect the cavity in
$\left|n+1\right\rangle$ state, where the first measurement collapses the
cavity state to either $\left|n+1\right\rangle$ or not. If in
$\left|n+1\right\rangle$, the subsequent measurements are QND and via the
quantum Zeno effect, each measurement resets the clock on the decay of the
$\left|n+1\right\rangle$ state. (b) Examples showing two measured qubit
readout sequences for a cavity initialized in $\left|n\right\rangle=1$ Fock
state followed by a small displacement drive $\alpha$. The left panel
corresponds to no change in the cavity state after the DM drive as inferred by
the absence of successful flips of the qubit state which results in a very
small probability $P(n=2)$ that the cavity was in the $\left|n\right\rangle=2$
Fock state. The right panel corresponds to an emission event where the cavity
state changed from $\left|1\right\rangle\xrightarrow{}\left|2\right\rangle$,
resulting in multiple successful flips of the qubit state. The HMM analysis of
this sequence of flips then indicates a very high likelihood ratio to be in
$\left|n\right\rangle=2$ Fock state. In case of successful detection, we
observe an exponential suppression of the detector error based false positive
probability with only linear increase in the number of repeated measurements.

After the state preparation, we apply a coherent drive to the cavity mimicking
a push from the DM wave to characterize the detector. A series of repeated QND
measurements are recorded by performing conditional $\pi$-pulses centered at
the $(n+1)$-shifted peak. The time between two successive QND measurements is
$\mathrm{5\,\mu s}$, which is relatively short compared to the lifetime of
Fock states given by $T_{1}^{s}=\mathrm{1320\,\mu s}/n$ (see Table S2). This
projective measurement resets the clock on the decay of the
$\left|n+1\right\rangle$ state [44]. We then apply a hidden Markov model (HMM)
analysis to reconstruct the cavity state and compute the probability that the
cavity state had changed from
$\left|n\right\rangle\rightarrow\left|n+1\right\rangle$ and assign a
likelihood ratio $\lambda$ associated with such events (see Supplementary
section for implementation of HMM analysis).

In principle, it is possible to prepare Fock states $\left|n>4\right\rangle$
in the device. However, due to the presence of multiple cavity modes,
simulating the complete Hamiltonian to generate the OCT pulses becomes
computationally challenging and in practice, higher $(n+1)$ Fock states are
prepared with lower fidelity. Furthermore, to prevent excessive signal photon
loss, the Fock state decay rate which is also enhanced by a factor of $(n+1)$
must remain small compared to the sum of Stark shift and readout rate, which
determines the maximum rate of number resolved measurements. For this study,
we chose $\left|n=4\right\rangle$, such that the decay probability stays below
$1\%$ between successive measurements.

## Signature of Fock enhancement

To assess the performance of the detector after preparing the cavity in a
particular Fock state $\left|n\right\rangle$, we carry out a series of
experiments. We apply a small variable displacement ($\alpha\ll 1$) to the
cavity and measure the relationship between the number of injected ($n_{\rm
inj}=|\alpha|^{2}$) and detected photons. We perform 30 repetitions of the
qubit measurement and apply a likelihood threshold of
$\lambda_{\mathrm{thresh}}=10^{3}$ to distinguish positive and negative
events. This threshold is determined based on the background cavity occupation
$n_{b}^{c}=6\times 10^{-3}$, which is assumed to be caused by photon shot-
noise from a hot cavity, as measured using the photon counting method
described in [1] (refer to Fig. S15). Errors below this value are considered
to be sub-dominant. For a cavity initialized in $\left|n\right\rangle$, the
probability of finding the cavity in the Fock state $\left|l\right\rangle$ for
a complex displacement $\alpha$ is described by the analytical expression
[45]: $P_{nl}(|\alpha|^{2})=\big{|}\left\langle
l\right|\hat{\mathcal{D}}(\alpha)\left|n\right\rangle\big{|}^{2}=(n!/l!)\alpha^{2(l-n)}e^{-|\alpha|^{2}}\times\mathcal{L}{n}^{l-n}(|\alpha|^{2})$,
where $\mathcal{L}{n}^{l-n}$ is an associated Laguerre polynomial. The data
obtained from the characterization of the detector is fitted to an expression,
represented by Eqn. (4).

$n_{\rm meas}=\eta\,P_{nl}(|\alpha|^{2}=\bar{n})+\delta$ (4)

This equation takes into account the detection efficiency, $\eta$, and false
positive probability, $\delta$. In cases where the cavity displacement
$\alpha\ll 1$, the relationship between the injected photons ($n_{\rm
inj}=|\alpha|^{2}$) and measured photons can be approximated such that
$P_{nl}(|\alpha^{2}|)\approx(n+1)|\alpha^{2}|$, as shown in Eqn. (2)

Figure 3: Stimulated emission enhancement. Mean number of measured photons
(positive events) as a function of the mean number of injected photons in the
cavity. After initializing the cavity in a Fock state $\left|n\right\rangle$
and applying a variable cavity displacement (mock DM drive), 30 repeated qubit
measurements of the cavity photon state are performed and a threshold
$\lambda_{\mathrm{thresh}}$ is applied to determine the cavity population at
$\left|n+1\right\rangle$. Background events with $\alpha=0$ are subtracted to
compare between different Fock states which may have systematic errors in the
state preparation step. $\lambda_{\mathrm{thresh}}=10^{3}$ is chosen based on
the observed background cavity occupation of $n_{b}^{c}=6\times 10^{-3}$ such
that the detector based errors are still sub-dominant. The background cavity
occupation is measured using the photon counting technique described in Ref.
[1] with repeated qubit parity measurements. Detector efficiency ($\eta$) for
each Fock state is determined from the fit as reported in the legend. The
monotonic decrease in the efficiency is attributed to - higher decay
probability to reach the same false positive probability and demolition
probability (see Supp Fig. S13). Anomalous behavior in $\left|3\right\rangle$
is attributed to the state decaying to nearby modes in the band structure
formed by multiple modes of the cavity, qubit and readout, which are close in
the energy level $\left|q,s,r\right\rangle$.

Fig. 3 displays the unique characteristic of stimulated emission enhancement,
where a higher number of detected photons is observed for the cavity that was
initialized in a higher Fock state. This result aligns with expectations and
highlights the effectiveness of stimulated emission as a method for amplifying
weak signals. The figure also showcases a clear, monotonic increase in the
slope as the Fock states increase, further emphasizing the success of this
technique. The resultant enhancement between $\left|n=4\right\rangle$ and
$\left|n=0\right\rangle$ is 2.78 ($0.45\times 5/0.81\times 1$). The reduced
efficiency $\eta=0.45$ to see the full $(n+1)$ enhancement can be explained by
the enhanced decay rate of higher Fock states making it more difficult to
achieve the fixed likelihood ratio threshold used for the comparison. For
example, consider the $\left|n=5\right\rangle$ Fock state. During the course
of 30 repeated measurements, it is 1.6 times more likely to decay than the
$\left|n=1\right\rangle$ Fock state. In addition, the higher demolition
probability 0.074 for $\left|n=4\right\rangle$ makes it more difficult to
achieve high likelihood ratio since this photon state only persists for the
first 14 quasi-QND measurements. The false positive probabilities $\delta$ are
smaller than $10^{-4}$ for all Fock states, comparable to the measured
residual photon occupation in the cavity. Further advancements could be
achieved by utilizing a system with a higher Q value and reduced demolition
probability. By combining weakly coupled qubits with the Echoed Conditional
Displacement (ECD) technique [23], Fock states can be prepared with high
fidelity while also keeping the demolition probability low. Additionally, a
detector with reduced errors from factors such as thermal population and
readout fidelity would also improve the results.

We observe anomalous behavior for the $\left|n\right\rangle=3$ data which
shows no signal enhancement. There is no direct way to investigate the cause
but we suspect leakage to a nearby mode as the 3D cavity contains multiple
modes which are closely spaced [31]. We tried this experiment on a different
cavity mode and observed similar behavior but for $\left|n\right\rangle=1$. We
have identified a couple of transitions with different modes which are closer
in energy level with $\left|g,3\right\rangle$ and which could be facilitated
by the always-on interaction of the transmon with all the modes. This
frequency collision issue can be easily resolved in future designs such that
the cavity modes are spaced further apart and the transmon has negligible
overlap with the spectator modes.

## Dark photon search

In order to conduct a dark photon search, we collect independent datasets for
a cavity prepared in different Fock states and count the number of positive
events in the absence of the mock DM drive. Additionally, we vary the dwell
time ($\tau$) between the state preparation and the beginning of the
measurement sequence in order to allow the coherent buildup of cavity field
due to the dark matter. Once the measurement sequence begins, the quantum Zeno
effect prevents further build-up of the signal field. Ideally, one would like
to choose the dwell time as close to the lifetime of Fock state as possible to
maximize the accumulation of signal and thus, the scan rate. In this work, the
dwell time was varied to compare on equal footing the dependence on $n$ and
not optimized for DM sensitivity. For this study, we chose a maximum dwell
time of $\mathrm{20\,\mu s}$ to collect reasonable statistics with
$\lambda_{\mathrm{thresh}}=10^{3}$ for all Fock states. Longer integration
times comparable to or larger than the dark matter coherence time
($\mathrm{75\,\mu s}$) which are needed to realize the full benefit of Fock
enhancement will be implemented in future dedicated experiments.

Figure 4: Measured background counts for different Fock states in the cavity
as a function of dwell time. There is no clear trend in the number of observed
background counts indicating systematic effects which could be due to the
state preparation steps. The error bars are plotted as
$\sqrt{\mathrm{Counts}}$ and the $N_{\mathrm{trials}}\sim 20,000$ for each
point.

The number of measured counts shown in Fig. 4 is fit to a functional form
given by Eqn. (5) (see Supplementary material), which has contributions coming
from a coherent source (hence the ($n+1$) Fock enhancement factor), an
incoherent source, and a state preparation dependent error

$N_{\rm meas}=a_{0}\,(n+1)\,\tau\,(N_{\rm trials}\,\tau)+b\,(N_{\rm
trials}\,\tau)+\\\ c_{n}\,N_{\rm trials}$ (5)

where $a_{0}$ and $b$ and $c_{n}$’s are the fit parameters we extract from
fitting the measured counts. The first term has two factors of $\tau$: one
from the coherent buildup of signal energy in the storage cavity for
$\tau<T_{1}^{s}$ which is included in the average signal rate $dN/dt$, and a
second factor of $\tau$ for the total integration time $t_{\rm tot}=N_{\rm
trials}\tau$.

Figure 5: Exclusion of dark photon parameter space with stimulated emission.
Shaded regions in the dark photon parameter space [13, 46] of coupling
($\epsilon$) and mass ($m_{\gamma}$) are excluded. In the orange band, dark
photon is naturally produced in models of high scale cosmic inflation [14].
The exclusion set with stimulated emission based dark photon search is shown
in the blue and red curves. On resonance with the storage cavity
($m_{\gamma^{\prime}}c^{2}=\hbar\omega_{s}$), the dark photon kinetic mixing
angle is constrained to $\epsilon\leq 4.24\times 10^{-13}$ with 90%
confidence. (Inset) The horizontal extent of the excluded region is set by the
bandwidth of the number resolved qubit $\pi$-pulse which is insensitive to any
drive outside the band. The vertical limit is set by the maximum $\epsilon$
which would result in dark photon rate greater than the value which would
degrade the fidelity of Fock state preparation significantly. The region
between the blue and red curve represents the exclusion with the stimulated
emission experiment, whereas the excluded region above the red curve is mainly
due to the failure of Fock state preparation which is easily detectable in the
experiment (See Supp Fig. S16).

By performing an ordinary least square (OLS) fit to the measured counts, we
extract the fit parameters with their uncertainties tabulated in Table 1. The
values of fit parameters obtained from performing a Maximum Likelihood
Estimate (MLE) were comparable.

Fitted Parameter | $\Theta$ | $\sigma_{\Theta}$
---|---|---
$a_{0}\,(\mathrm{s^{-2}})$ | $1.9\times 10^{3}$ | $7.662\times 10^{5}$
$b\,(\mathrm{s^{-1}})$ | $-7.26$ | $4.2\times 10^{1}$
$c_{0}$ | $3.402\times 10^{-4}$ | $4.292\times 10^{-4}$
$c_{1}$ | $1.419\times 10^{-3}$ | $4.0\times 10^{-4}$
$c_{2}$ | $5.860\times 10^{-4}$ | $4.222\times 10^{-4}$
$c_{4}$ | $7.330\times 10^{-3}$ | $6.732\times 10^{-4}$
Table 1: Fitted parameters. Best fit and statistical uncertainties
corresponding to each source of background counts. The Fock state with higher
background counts have larger $c$ value indicating that the source of counts
is related to the state preparation step and thus, it is valid to perform a
background subtraction to demonstrate the enhancement technique. For
$\tau=\mathrm{20\,\mu s}$ and $N_{\mathrm{trials}}\sim 20,000$, all three
terms in Eqn. (5) contribute roughly equally to the counts observed in Fig. 4.

The large statistical uncertainties on $a_{0}$ are due to the fact that the
other two terms dominate the measured counts and fluctuations causing $a_{0}$
to swing up and down by a large amount. With the extracted value of $a_{0}$,
we can compute the kinetic mixing angle of dark photon given by
$\epsilon=\sqrt{\frac{a_{0}}{\rho_{\mathrm{DM}}m_{\mathrm{DM}}GV}}$ (see
Supplemental material for derivation). With the measured uncertainties on all
the parameters and using error propagation, we compute the $90\%$ confidence
limit on $\epsilon$ to be $\epsilon_{fit}+1.28\,\sigma_{\epsilon}$. Thus, a
dark photon candidate on resonance with the storage cavity
($m_{\gamma^{\prime}}c^{2}=\hbar\omega_{s}$), with mixing angle $\epsilon\geq
4.35\times 10^{-13}$ is excluded at the 90% confidence level. Fig. 5 shows the
regions of dark photon parameter space excluded by the stimulated emission
based search, assuming the dark photon comprises all the dark matter density
($\rho_{\mathrm{DM}}=\mathrm{0.4\,GeV/cm^{3}}$). The detector is maximally
sensitive to dark matter candidates with masses within a narrow window around
the resonance frequency of the cavity. This window is set by the lineshape of
the dark matter [47] ($Q_{\mathrm{DM}}\sim 10^{6}$). For DM wave at
$\sim\mathrm{6\,GHz}$, the FWHM linewidth is $\mathrm{6\,kHz}$, so the
-$\mathrm{3\,dB}$ point is $\mathrm{3\,kHz}$ away from the cavity resonance
and the efficiency of a finite bandwidth number resolved qubit $\pi$-pulse is
non-zero. While the stimulated emission protocol is valid for small induced
signal, the cavity state preparation using OCT pulses excludes DM wave
corresponding to $|\alpha|>0.05$ (see Fig. S16) as it would result in larger
preparation error than experimentally observed.

## Conclusions and outlook

The preparation of a cavity in quantum states of light, such as a Fock state,
has the potential to significantly enhance the DM signal rate and hence also
the DM frequency scan rate compared to previous methods. In the current work,
we have demonstrated a signal enhancement of $2.78\times$ by initializing the
cavity in $\left|n=4\right\rangle$ versus $\left|n=0\right\rangle$ Fock state.
The corresponding improvement in attainable scan speed is reduced because of
the finite amounts of time required to prepare the initial Fock state and to
read out the final state after integrating the signal, though in an optimized
experiment the duty cycle should be comparable to that of other photon
counting experiments. Despite these imperfections, even in this initial
demonstration, the dark photon search sets an unprecedented sensitivity in an
unexplored parameter space with $90\%$ confidence level. This method holds
great promise for continued improvement as advancements in cavity coherence
times [48] and state preparation methods [23] progress, and we expect the same
improvement factor or better in future dedicated dark matter search
experiments.

While this study focuses on the detection of dark matter, the quantum-enhanced
technique presented here can be applied more widely to sense ultra-weak forces
in various settings, in cases where the signal accumulation is limited by the
coherence time of the signal rather than by that of the probe. The Fock state
stimulated emission can increase the rate of processes that involve the
delivery of small amounts of energy, and the resulting signal quanta can be
detected through number-resolved counting techniques that surpass the SQL.

## Acknowledgements

We would like to acknowledge and thank Konrad Lehnert for proposing this idea.
We thank Ming Yuan, A. Oriani, and A. Eickbusch for discussions, and the
Quantum Machines customer support team for help with implementing the control
software. We gratefully acknowledge the support provided by the Heising-Simons
Foundation. This work made use of the Pritzker Nanofabrication Facility of the
Institute for Molecular Engineering at the University of Chicago, which
receives support from Soft and Hybrid Nanotechnology Experimental (SHyNE)
Resource (NSF ECCS-1542205), a node of the National Science Foundation’s
National Nanotechnology Coordinated Infrastructure. This manuscript has been
authored by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359
with the U.S. Department of Energy, Office of Science, Office of High Energy
Physics, with support from its QuantISED program. We acknowledge support from
the Samsung Advanced Institute of Technology Global Research Partnership.

Stimulated emission of signal photons from dark matter waves

Supplemental Material

Ankur Agrawal, Akash V. Dixit, Tanay Roy, Srivatsan Chakram, Kevin He, Ravi K.
Naik,

David I. Schuster, Aaron Chou

## Conceptual overview

We use QuTip [49, 50] simulation tool to study the evolution of a cavity
initialized in different Fock states under the action of a small displacement
drive. As seen in Fig. S6, the probability of observing a dark matter induced
signal is significantly enhanced for a cavity prepared in large
$\left|n\right\rangle$ Fock state.

Figure S6: Phase-space representation of the cavity state before and after the
dark matter wave push. (Left plots) Displacing the cavity initialized in
$\left|0\right\rangle$ in an arbitrary direction by a small coherent push
$\beta_{\rm coh}\ll 1$ results in a small probability
$P_{0,1}\propto|\mathcal{M}_{0,1}|^{2}$ for creating a $\left|1\right\rangle$
component by spontaneous emission of a photon from the DM wave. The direction
of displacement is determined by the instantaneous phase of the DM wave which
is randomized every DM coherence time, but since the initial cavity state is
azimuthally symmetric, a displacement in any direction gives the same
probability $P_{01}$. The red dashed line is shows as a guide to locate the
origin w.r.t to the center of the blob. (Right plots) The cavity is
initialized in a $\left|n\right\rangle=10$ Fock state which also has an
azimuthally symmetric Wigner distribution. Displacing this distribution in an
arbitrary direction shifts some part of the distribution to larger radius and
other parts to smaller radius. For example, the lower plots shows a
displacement of $\beta_{\rm coh}=0.1$ in the positive $X=Re(\beta)$ direction.
The shift to larger radius corresponds to stimulated emission to states with
larger photon number, for example $\left|11\right\rangle$ while the shift to
smaller radius corresponds to stimulated absorption to states with smaller
photon number, for example $\left|9\right\rangle$. As shown in the histograms,
the stimulated enhancement factors ($N_{Fock}+1$) and $N_{Fock}$ give
probabilities which satisfy $P_{10,11}=11\times P_{0,1}$ and
$P_{10,9}=10\times P_{0,1}$.

## Dark matter conversion and scan rate

The expected signal rate ($R_{s}$) from DM is proportional to the volume and
the quality factor of the cavity. In DM experiments operating below
$\mathrm{1\,GHz}$ ($h\nu\ll k_{B}T$), the background rate ($R_{b}$) is limited
by the thermal occupation of the cavity such that the noise added by an
amplifier in the readout process is sub-dominant. However, at frequencies in
$\mathrm{5-30\,GHz}$ range, the DM search faces major challenges as the
signal-to-noise ratio (SNR) rapidly plummets down.

The expected DM signal decreases at higher frequencies due to following
reasons: (1) the cavity volume diminishes as the cavity dimensions shrink to
maintain the resonance condition ($V\propto\nu^{-3}$) for a
$\mathrm{TM_{0n0}}$ type mode, (2) the quality factor of a normal metal cavity
degrades due to the anomalous skin effect [51, 52]. Moreover, the noise from
quantum-limited readout process increases at higher frequencies because the
power scales as $h\nu\,(\frac{\nu}{Q_{\rm DM}})$, where $Q_{\rm DM}$ is the
quality factor of a DM wave given by the escape velocity of DM from the
galaxy. The first term in the expression represents the energy of one photon,
which is determined by the standard quantum limit. This limit specifies that
the minimum amount of noise added by an amplifier is equivalent to a mean
photon number of $\bar{n}_{SQL}\geq 1$[16, 17, 18, 19, 20]. The signal
frequency scan rate $R$ (in Hz/s), a key merit of haloscope type experiment,
is proportional to $\propto R_{s}^{2}/R_{b}$ as shown in Eqn. (S6). Quantum-
enhanced techniques to improve signal rate and reduce noise are required to
improve SNR and hence accelerate the search for the extremely weak dark matter
signal.

The integration time ($\Delta t$) required for a background limited dark
matter search is given by the time required to achieve $1\sigma$ sensitivity
determined by Poisson counting statistics: $R_{s}\Delta t>\sqrt{R_{b}\Delta
t}$, where $R_{s}=\bar{n}_{\mathrm{HP}}/\Delta t$ and background rate
$R_{b}=\bar{n}_{c}^{b}/\Delta t$. The signal accumulation time for each Fock
state preparation is given by $\tau$. For an optimal dark matter scanning
experiment, we would want to start with a high-Q cavity ($Q_{\rm cav}>Q_{\rm
DM}$) in the largest possible Fock state $\left|n\right\rangle$ such that
$\tau$ (which must be smaller than the Fock state coherence time
$Q_{cav}/(\nu*n)$) can be matched with the signal coherence time $Q_{\rm
DM}/\nu$. Hence, the integration time at each frequency point scales as
$R_{b}/R_{s}^{2}$ and, the scan rate is given by,

$R=\Delta\nu/\Delta t=(\frac{\nu}{Q_{\rm DM}})\frac{1}{N\,\tau}\propto
R_{s}^{2}/R_{b},$ (S6)

such that $\Delta t$ is the sum of many iterations of duration $\tau$ and $N$
is the total number of iterations for each Fock state preparation. While the
stimulated emission technique will generally enhance $R_{s}$ by a factor
$(n+1)$, the coherence time of the prepared quantum state is also reduced by a
factor $(n+1)$, thus also restricting the accumulation time $\tau$. The
smaller coherence time then necessitates a faster readout rate which then
generally causes the background rate $R_{b}$ to also increase by a factor
$(n+1)$ for backgrounds which are associated with the readout procedure or due
to a residual non-zero mode occupation number in the cavity. The overall
improvement in $d\nu/dt$ due to Fock enhancement is then a single factor
$\eta\,(n+1)$, where $\eta$ is the detection efficiency in the experiment.

Each measurement sequence consists of state preparation and validation
($\mathrm{2\,\mu s}+3\times t_{m}=\mathrm{17\,\mu s}$), signal integration
($\mathrm{20\,\mu s}$) and 30 repeated measurements (30$\times
t_{m}=\mathrm{150\,\mu s}$). There is an additional dead time in the
experiment due to imperfect state preparation which constitutes
$35\%\times\mathrm{17\,\mu s}\approx\mathrm{6\,\mu s}$ for
$\left|n=4\right\rangle$ Fock state. The total search time at each Fock state
is roughly $20,000\times\mathrm{193\,\mu s}=\mathrm{36.6\,s}$ with a duty
cycle of $\frac{20}{193}=10\%$ ($\mathrm{3.6\,s}$ of integration). For the
present apparatus, the duty cycle could be increased to $\sim 30\%$ if longer
signal integration times are chosen at the expense of greater systematics due
to Fock state decoherence. Higher Q cavities would enable even larger
integration times and duty cycles.

## Experimental setup

The cavities and qubit are mounted to the base plate of a dilution fridge
(Bluefors LD400) operating at $\mathrm{10\,mK}$. The device is housed in two
layers of $\mu$-metal to shield from magnetic fields. Signals sent to the
device are attenuated and thermalized at each temperature stage of the
cryostat as shown in Fig. S7. The field probing the readout resonator is
injected via the weakly coupled port (shorter dipole stub antenna). Control
pulse for the qubit are inserted through the strongly coupled readout port
(longer dipole stub antenna). The storage cavity is driven through a direct
port which is weakly coupled to the external microwave chain. Both control
lines also contain an inline copper coated XMA attenuator that is threaded to
the base plate. The signal from the readout resonator reflects off a Josephson
parametric amplifier before being amplified by a cryogenic HEMT amplifier at
the $\mathrm{4\,K}$ stage. The output is mixed down to $\mathrm{100\,MHz}$ IF
signal before being digitized. Cavity and transmon fabrication details can be
found in Supplemental materials of Ref. [1, 42].

Figure S7: Wiring diagram inside the dilution refrigerator and the room
temperature measurement setup. Quantum Machines (QM) OPX controller was used
to generate the arbitrary wave forms (DACs) and digitize the incoming readout
signal (ADCs). All the tones are up-converted using IQ modulation, with local
oscillator (LO) and AWG synced to an SRS F2575 RB source. Storage cavity is
controlled via a direct port and qubit is controlled by injecting a drive into
the strongly coupled port of the readout. Readout signal is injected into the
weakly coupled port, and the signal is routed to the JPA using non reciprocal
circulator elements. The amplified signal is routed to the HEMT for further
amplification and the signal is then mixed down to $\mathrm{100\,MHz}$ IF,
further amplified, and finally digitized. All the RF lines are heavily
filtered with homemade eccosorb filters and attenuated to minimize stray
radiation from entering the device.

## Fock state characterization

We know that a Fock state has a definite photon number but no definite phase
associated with it. Therefore, a simple qubit spectroscopy reveals as much as
information as a Wigner tomogram would do. However, we use both these methods
to compute and confirm the presence of Fock states in the cavity. Fig. 1 shows
the qubit spectroscopy on the left and Wigner tomography of the resultant Fock
states on the right. By fitting a Gaussian curve to the spectrum we estimate
the fidelities to be $P_{0}=95.2\pm 0.3\%,P_{1}=91.2\pm 0.4\%,P_{2}=87.3\pm
0.5\%,P_{3}=81.6\pm 0.6\%,P_{4}=63.6\pm 0.7\%$.

Moreover, we study the evolution of the cavity under the action of OCT pulses
by tracking the cavity state as shown in Fig. S8 (b). We perform a number
resolved qubit spectroscopy at different points in time to map out the
occupation probability of different Fock levels, in agreement with the
simulated trajectory.

Figure S8: Fock state preparation in a cavity dispersively coupled to a
transmon qubit. (a) Optimized control drives to the qubit (top) and cavity
(bottom) to transfer the qubit-cavity state from
$\left|g,0\right\rangle\xrightarrow{}\left|g,3\right\rangle$. The drive
strengths are in units of $\mathrm{MHz}$. (b) Simulated trajectory of the
cavity state using QuTip under the control drive. Measured cavity occupation
probability showing all the levels explored during the evolution before
converging to the desired state $\left|g,3\right\rangle$ at the end. The
experiment involves playing a fraction of the OCT pulses and performing qubit
spectroscopy with a number resolved $\pi$-pulse. Each vertical slice
corresponds to the cavity occupation probability in different number states as
function of the duration of pulse. Note that the simulation does not include
the presence of neighboring cavity modes as well as the measurement
infidelity. Control fields for the qubit and cavity are generated by a room
temperature field-programmable gate array (FPGA) controller, which also
provides a low-latency feedback signal to actively monitor the state of the
cavity.

In several recent works, it has been demonstrated that a single transmon is
capable of preparing any quantum state in the cavity and perform universal
control on it. The methods developed include - resonant swap of a qubit
excitation to the cavity [40], SNAP gates [39], Blockade [42], and GRadient
Ascent Pulse Engineering (GRAPE) based optimal control (OCT) pulses [41, 27]
and Echoed Conditional Displacement (ECD) control pulses [23] to solve the
inversion problem of finding control fields to transfer a quantum system from
state A to B. We use a GRAPE based method to generate a set of optimal control
pulses [41, 27] to prepare non-classical states in an cavity. In the optimal
control, we consider the full model of the time dependent Hamiltonian and
generate the control pulses which maximizes the target state fidelity as has
been previously demonstrated in nuclear magnetic resonance experiments [43] as
well as superconducting circuits [27]. The main advantage of this approach is
that the duration of state preparation pulses can be as short as 1/$\chi$ and
does not necessarily increase for higher Fock states [27, 23]. We use OCT
pulses to successfully prepare cavity Fock states as shown in Fig. 1 (b).

We briefly investigated the SNAP protocol [39] to prepare Fock states but did
not pursue further as it suffers from two issues limiting the maximum
achievable fidelity. First, the number of constructed sequences scales as
$(2n+1)$, requiring large number of gates, limiting operations which are
feasible in the presence of decoherence. Second, the constructed model fails
to account for the higher order Kerr non-linearity
($\mathcal{H}_{Kerr}/\hbar=K(a^{\dagger})^{2}(a)^{2}$) in the Hamiltonian,
which is non-negligible at higher photon occupation. This results in finite
occupation probability at the other Fock states as well.

## Calibration of number resolved $\pi$ ulse

In order to successfully probe the presence of Fock states in the cavity, we
need a very well calibrated narrow bandwidth $\pi$-pulse - amplitude as well
as frequency. We perform an amplitude Rabi experiment with a $3\mu$s
(4$\sigma$) long Gaussian pulse such that its spectral spread is smaller than
the $\chi$-shift. After a coarse amplitude sweep, we perform a finer sweep
centered at the initial guess and apply a set of 12 Gaussian pulses in
succession to amplify any gate errors such as under/over rotation, frequency
drift to get a better estimate of the $\pi$-pulse amplitude. For qubit
transition frequency, we initialize the cavity in a particular Fock state
$\left|n\right\rangle$ and perform a qubit Ramsey interferometry experiment.
It consists of two $\pi/2$ pulses separated by a variable delay time. A Fast
Fourier Transform (FFT) of the resultant time oscillations gives the shift in
the qubit transition frequency which accounts for any higher order corrections
as well. The computed transition frequencies and amplitude are used in the
actual experiment to minimize the errors.

## Device calibration

By applying a weak coherent tone at the storage cavity frequency, we induce a
variable displacement $\alpha$ of the cavity state. We calibrate the number of
photons injected into the storage cavity by varying the drive amplitude and
performing qubit spectroscopy. By fitting the qubit spectrum as shown in Fig.
S9 to a Poisson distribution, we extract the cavity occupation,
$\bar{n}=\mathrm{|\alpha|^{2}}$. Similarly, the qubit drive strength is
calibrated by performing qubit Rabi experiments with varying AWG amplitude and
pulse length. The measured drive strength and the AWG amplitude mapping is
used to send the correct OCT pulses to the device. See Supp. section in [1]
and [31] for more details on device parameter calibrations.

Figure S9: Qubit spectroscopy reveals cavity displacement (a) Pulse sequence
showing the calibration of cavity photon number by performing qubit
spectroscopy. (b) The cavity is displaced using a variable weak coherent drive
for a finite period of time. The resulting population of the cavity is
determined by performing qubit spectroscopy with a resolved $\pi$-pulse. The
cavity photon number dependent shift of the qubit transition frequency reveals
the cavity population. By fitting the spectrum to a Poisson distribution we
extract the weights of the cavity number states in the prepared coherent state
and thus, the mean photon occupation.

## Hidden Markov model analysis

We adopt the Hidden Markov Model (HMM) approach presented in [1] to consider
all the possible errors that may occur during the measurement process and
alter the state of the cavity, qubit, and readout. The cavity and qubit states
are treated as hidden variables that emit readout signals. The Markov chain is
modeled by the transition matrix ($T$) (Eqn. (S8)) which describes the
evolution of the joint cavity-qubit hidden state
$s\in[\left|n^{\prime}g\right\rangle,\left|n^{\prime}e\right\rangle,\left|ng\right\rangle,\left|ne\right\rangle]$
and the emission matrix ($E$) (Eqn. (S9)) that calculates the probability of a
readout signal $R\in$ [$\mathcal{G}$, $\mathcal{E}$] given a certain hidden
state. It’s important to note that with a number-resolved $\pi$-pulse centered
at the $n$-shifted peak, we can only determine whether the cavity is in the
$\left|n\right\rangle$ Fock state or not ($\left|n^{\prime}\neq
n\right\rangle$).

The transition and emission matrix were first introduced in [53] and
implemented in [1]. The change in qubit state probability is not affected by
the cavity state. However, the repeated measurement of the cavity state with a
number selective $\pi$-pulse introduces an additional channel through which
the cavity loses its excitation. It is called demolition probability
($p_{d}$), which signifies the non-QNDness of the projective measurement of
the cavity state with the qubit. In simple terms, a single qubit measurement
is $(1-p_{D})\%$ QND. We followed the protocol described in Ref.[54] for
repeated parity measurement and replaced it with number selective $\pi$-pulse.
Table S2 and Fig. S13 show the measured values for different Fock states. The
elements of the emission matrix represent the readout fidelities for the
ground and excited states of the qubit, which are influenced by noise from the
first stage JPA. More information on the experimental protocols used can be
found in the supplemental material.

We used the backward algorithm (Eqn. (S7)) [53, 55] to calculate the
probability ($P(n_{0})$) that the cavity state was in the
$\left|n+1\right\rangle$ Fock state after the injection of a synthetic DM
signal. The algorithm takes a set of $N+1$ measured readout signals
($R_{0},R_{1},...,R_{N}$) as input, where $n_{0}=n+1$ represents the target
Fock state and $n_{0}^{\prime}$ can take on values in the range
[$0,n,n+2,...$].

$\displaystyle
P(n_{0})=\sum_{s_{0}\in[\left|n_{0},g\right\rangle,\left|n_{0},e\right\rangle]}\sum_{s_{1}}...\sum_{s_{N}}$
$\displaystyle E_{s_{0},R_{0}}T_{s_{0},s_{1}}E_{s_{1},R_{1}}$ (S7)
$\displaystyle\cdots$ $\displaystyle T_{s_{N-1},s_{N}}E_{s_{N},R_{N}}$

In the reconstruction process, all possible scenarios are taken into account.
For instance, a readout measurement of $\mathcal{G}$ followed by $\mathcal{E}$
could happen due to the successful detection of the cavity in the
$\left|n+1\right\rangle$ state with a probability of
$P_{n_{0}n_{0}}P_{gg}F_{e\mathcal{E}}/2$. On the other hand, it could also be
caused by either a qubit heating event
($P_{n_{0}^{\prime}n_{0}^{\prime}}P_{ge}F_{e\mathcal{E}}/2$) or a readout
error ($P_{n_{0}^{\prime}n_{0}^{\prime}}P_{gg}F_{g\mathcal{E}}/2$). The data
in Fig. 2(b) illustrates the results of the measurement of the readout signals
and the reconstructed initial probabilities of the cavity state. The panels on
the left show instances when there was no emission event, while the panels on
the right depict cases where a positive emission event took place. The cavity
state shifted from $\left|1\right\rangle$ to $\left|2\right\rangle$ as a
result of a variable displacement drive $\mathcal{D}(\alpha)$, and the change
was accurately reflected in the reconstructed probability.

The reconstructed cavity state probabilities undergo a likelihood ratio test
$(\lambda=\frac{P(n_{0}=n+1)}{P(n_{0}\neq n+1)})$ to determine if the cavity
state changed or not. A positive detection of a photon signal
$\left|n\right\rangle\rightarrow\left|n+1\right\rangle$ is declared when the
threshold $\lambda>\lambda_{\mathrm{thresh}}$. The false positive rate is
limited to less than $\frac{1}{\lambda_{\mathrm{thresh}}+1}$ as a result of
this procedure. A higher detection threshold can be achieved by increasing the
number of repeated measurements, but at the cost of decreased efficiency which
is linear in the number of measurements. Hence, $\lambda_{\mathrm{thresh}}$ is
chosen to strike a balance between the detection efficiency and the false
positive rate, keeping it below the observed physical photon background. This
will be discussed further in the next section.

## Elements of hidden Markov model

The hidden Markov model relies on independent measurements of the
probabilities contained in the transition and emission matrices. The elements
of these matrices depend on the parameters of the experiment and the device,
including the lifetimes of the qubit and cavity, qubit spurious population,
and readout fidelities.

### Transmission matrix elements

The transition matrix captures the possible qubit (cavity) state changes.
Qubit relaxation $\left|e\right\rangle\rightarrow\left|g\right\rangle$ occurs
with a probability $P_{eg}^{\downarrow}=1-e^{-t_{m}/T_{1}^{q}}$. The
probability of spontaneous heating
$\left|g\right\rangle\rightarrow\left|e\right\rangle$ of the qubit towards its
steady state population is given by
$P_{ge}^{\uparrow}=\bar{n}_{q}[1-e^{-t_{m}/T_{1}^{q}}]$. Unlike a two-level
system such as a qubit, the cavity state may change from
$\left|n\right\rangle\rightarrow\left|n^{\prime}\right\rangle$ via either
decay ($\left|n\right\rangle\rightarrow\left|n-1\right\rangle$) or excitation
($\left|n\right\rangle\rightarrow\left|n+1\right\rangle$) with probabilities
$P_{n,n-1}=1-e^{-t_{m}/T_{1}^{n}}$ or
$P_{n,n+1}=\bar{n}_{c}[1-e^{-t_{m}/T_{1}^{n}}]$ respectively. The lifetime of
the Fock state is modified due to the enhanced decay [40]
($T_{1}^{n}=T_{1}^{s}/n$) as compared to the bare lifetime of a coherent state
(see Supp Fig. S12). Yet another possible source of change in the cavity state
is the repeated qubit measurement itself. In most cases, we assume the
interaction between the qubit and cavity to be QND, i.e., the measurement of
cavity state by qubit does not perturb the state. However, we measured the
QNDness of repeated qubit measurement for $\left|n\right\rangle=1$ Fock state
to be $97.4\%$ which corresponds to a demolition probability $p_{d}$ of
$2.6\%$ per measurement (see Fig. S13). Hence, we add this term in the
transition matrix. All these probabilities are computed using independently
measured qubit, cavity and readout parameters described below.

$T=\blockarray{ccccc}\left|n^{\prime}g\right\rangle&\left|n^{\prime}e\right\rangle\left|ng\right\rangle\left|ne\right\rangle\\\
\block{[cccc]c}P_{n^{\prime}n^{\prime}}P_{gg}P_{n^{\prime}n^{\prime}}P_{ge}P_{n^{\prime}n}P_{ge}P_{n^{\prime}n}P_{gg}\hskip
3.0pt\left|n^{\prime}g\right\rangle\\\
P_{n^{\prime}n^{\prime}}P_{eg}P_{n^{\prime}n^{\prime}}P_{ee}P_{n^{\prime}n}P_{eg}P_{n^{\prime}n}P_{ee}\hskip
3.0pt\left|n^{\prime}e\right\rangle\\\
P_{nn^{\prime}}P_{gg}P_{nn^{\prime}}P_{ge}P_{nn}P_{ge}P_{nn}P_{gg}\hskip
3.0pt\left|ng\right\rangle\\\
P_{nn^{\prime}}P_{eg}P_{nn^{\prime}}P_{ee}P_{nn}P_{ee}P_{nn}P_{eg}\hskip
3.0pt\left|ne\right\rangle\\\ $ (S8)

The lifetime of the qubit is determined by applying a $\pi$ pulse and waiting
for a variable time before measuring the population as shown in Fig. S10. We
map out the qubit population as a function of the delay time, fit it with an
exponential characterizing the Poissonian nature of the decay process, and
obtain $T_{1}^{q}=\mathrm{115\pm 10\,\mu s}$.

The dephasing time of the qubit is measured by a Ramsey interferometry
experiment with a $\pi/2$ pulse, variable delay, and a final $\pi/2$ with its
phase advanced by $\omega_{r}t$ where $\omega_{r}$ is the Ramsey frequency.
The phase advancement is implemented in the software. During the variable
delay period, a series of $\pi$ pulses are applied to perform spin echos and
reduce sensitivity to low frequency noise. We observe a dephasing time of
$T_{2}^{q}=\mathrm{160\pm 10\,\mu s}$ which extends to
$T_{2}^{e}=\mathrm{236\pm 6\,\mu s}$ with a single echo sequence.

Figure S10: Qubit lifetime and dephasing time measurement. (Top) $T_{1}$
measurement by sending a $\pi$-pulse to excite the transmon to
$\left|e\right\rangle$ state and monitor its decay as a function of variable
delay time. By fitting an exponential function to the qubit excitation
probability $P_{e}$, we extract the $T_{1}^{q}=\mathrm{115\pm 10\,\mu s}$.
(Bottom) $T_{2}$ measurement with a Ramsey experiment. The sequence consists
of two $\pi/2$-pulse separated by a variable delay time. The envelope of the
measured oscillations informs the $T_{2}^{e}=\mathrm{236\pm 6\,\mu s}$ and its
frequency provides us the detuning between the drive and the transmon
resonance frequency. In this case, we intentionally introduced a 60 kHz
synthetic detuning.

The storage cavity lifetime is calibrated by performing a cavity $T_{1}$
experiment. We use the OCT pulse to prepare the cavity in a
$\left|n\right\rangle=1$ Fock state, wait for a variable delay time and probe
the cavity state at the end by Rabi driving the qubit with a resolved
$\pi$-pulse. The resultant cavity population is fitted to an exponential to
obtain $T_{1}^{s}=\mathrm{1.36\pm 0.02\,ms}$ as shown in Fig. S11. To measure
the cavity dephasing time, the cavity is initialized in a superposition state
$(a\left|0\right\rangle+b\left|1\right\rangle)$ by applying a weak coherent
drive ($\alpha$) such that approximately only the first two photon states
$\left|n\right\rangle$ are populated. The contrast in the signal will be
determined by the relative amplitude ($\frac{a^{2}}{a^{2}+b^{2}}$) without any
loss of information. After a variable delay time, an identical displacement
drive with its phase advanced by $\omega_{r}\,t$ is applied before probing the
$\left|n\right\rangle=0$ with a resolved $\pi$-pulse on the qubit. The
oscillations are fitted to obtain a dephasing time of
$T_{2}^{s}=\mathrm{2.39\pm 0.02\,ms}$.

Figure S11: Storage cavity lifetime and dephasing time from $\mathrm{T_{1}}$
and Ramsey measurements. The long lived storage cavity mode is ideal for
holding a signal photon induced by the dark matter while a series of repeated
photon counting measurements is performed.

The qubit spurious population is determined by measuring the relative
populations of its ground and excited states [56]. This is done by utilizing
the $f$-level of the transmon. Two Rabi experiments are conducted swapping
population between the $\left|e\right\rangle$ and $\left|f\right\rangle$
levels. First, we apply a $\pi_{ge}$ pulse to invert the qubit population
followed by the $\left|e\right\rangle-\left|f\right\rangle$ Rabi experiment.
Second, no $\pi_{ge}$ pulse is applied before the $ef$ Rabi oscillation. The
ratio of the amplitudes of the oscillations gives us the ratio of the
populations of the excited and ground state. Assuming that $P(g)+P(e)=1$ and
measuring $\frac{P(e)}{P(g)}=0.02$, corresponds to an effective qubit
temperature of $\mathrm{54\,mK}$.

Figure S12: Decay rate of Fock states. Measured lifetime of the different Fock
states prepared in the cavity using GRAPE pulses. The decay rate is inversely
proportional to the lifetime ($\Gamma=1/T_{1}$). The curve shows enhancement
in the decay rate as a function of the Fock state $\left|n\right\rangle$. The
linear fit to the data fits well as predicted by [57], where
$T^{n}_{1}=T_{1}/n$.

In a dispersive interaction, we assume that the measurement of cavity state
via a parity or number resolved $\pi$-pulse does not perturbed the state i.e.
it is a quantum non-demolition (QND) measurement or in other words, does not
induce additional relaxation in the cavity mode. However, a recent study has
shown that parity measurements, while highly QND, can induce a small amount of
additional relaxation [54]. Hence, in order to estimate the same, but, in the
context of number resolved qubit measurement, we follow the method described
in [54], where we perform a cavity $T_{1}$ experiment interleaved with varying
number of repeated number resolved qubit measurements during the delay time.
In that experiment, the total relaxation rate was modeled as a combination of
the bare storage lifetime $\tau_{s}$ and a demolition probability $p_{d}$
associated with each qubit measurement. In Fig. S13, we show the extracted
total decay time ($\tau_{\rm tot}$) and demolition probability $p_{d}=2\pm
0.02\%$ when the cavity is prepared in $\left|n\right\rangle=1$ Fock state. In
other words, a single number resolved qubit measurement is $98\%$ QND.
Unfortunately, for $\left|n\right\rangle=2$, $p_{d}=4\pm 0.04\%$ gets worse,
which is not a good sign but probably explains why the detection efficiency in
Fig. 3 falls off at higher Fock states. This acts like an additional source of
loss to the cavity mode but only when the resolved $\pi$-pulse is on resonance
with the shifted qubit frequency.

Figure S13: QNDness of storage cavity measurement. Storage cavity $T_{1}$ measurements were performed with repeated number resolved qubit measurements interleaved during the delay time with a variable repetition interval time $\tau_{\rm rep}$. The extracted total decay time was fit to a model $1/\tau_{\rm tot}=1/\tau_{s}+p_{d}/\tau_{\rm rep}$. From the fit (red line), we infer a demolition probability per readout of $p_{d}=2.6\%$ corresponding to a QNDness of 97.4%, which is a bit lower than reported for a parity protocol [54]. The natural decay time of the storage $\tau_{s}=\mathrm{1360\,\mu s}$ is indicated by a dashed grey line.  $\left|n\right\rangle$ | $\tau_{s}(\mu s)$ | $p_{d}$ | $\sigma_{p_{d}}$
---|---|---|---
$\left|1\right\rangle$ | 1360 | 0.026 | 0.002
$\left|2\right\rangle$ | 660 | 0.040 | 0.004
$\left|3\right\rangle$ | 527 | 0.10 | 0.05
$\left|4\right\rangle$ | 319 | 0.074 | 0.012
Table S2: Demolition probability. Measured lifetime of the different Fock
states and their fitted demolition probability with error bars. $p_{d}$ can be
approximated with a linear dependence on $n$.

###  Emission matrix elements

In order to characterize the emission matrix it is necessary to measure the
readout infidelity of a particular transmon state. We consider only two
possible transmon states ($\left|g\right\rangle,\left|e\right\rangle$) in this
case as the number resolved $\pi$-pulses have very narrow spectral width
($\sigma_{\nu}\ll\alpha_{q}$). Each state is prepared 20,000 times and the
resultant quadrature values are digitized to assign a voltage in the ($I-Q$)
space. The phase of the readout pulse is pre-calibrated to align the signal
along the $I$ axis. The histogram corresponding to each state is fitted with a
sum of two Gaussian functions to estimate the overlap region and calculate the
readout fidelity, $\mathcal{F}=97\%$ (Fig. S14). A discriminator value (red
dashed line) is used to assign each readout signal either $\mathcal{G}$ or
$\mathcal{E}$ in real-time.

$E=\frac{1}{2}\hskip 3.0pt\blockarray{ccc}\mathcal{G}&\mathcal{E}\\\
\block{[cc]c}F_{g\mathcal{G}}F_{g\mathcal{E}}\hskip
3.0pt\left|n^{\prime}g\right\rangle\\\ F_{e\mathcal{G}}F_{e\mathcal{E}}\hskip
3.0pt\left|n^{\prime}e\right\rangle\\\ F_{g\mathcal{G}}F_{g\mathcal{E}}\hskip
3.0pt\left|ng\right\rangle\\\ F_{e\mathcal{G}}F_{e\mathcal{E}}\hskip
3.0pt\left|ne\right\rangle\\\ $ (S9)

Figure S14: QND readout of the transmon state. (Top) shows the two quadrature
values of the down-converted readout signal for qubit prepared in
$\left|g\right\rangle$ and $\left|e\right\rangle$ state. Each state
preparation contains 20,000 single shot points. (Bottom) By fitting a sum of
two Gaussian we can estimate the overlap region and assign a single-shot
readout fidelity as $\mathcal{F}=97\%$. The red dashed line shows an optimal
value of the threshold for tagging a qubit state based on a single shot
readout.

Readout errors are due to voltage excursions from amplifier noise or spurious
qubit transitions. The emission matrix should only contain readout errors that
occur due to voltage fluctuations. Errors due to qubit transitions during the
readout window are accounted for in the transition matrix. To disentangle the
two contributions, we subtract the readout errors caused by the spontaneous
heating and decay of the qubit to obtain $F_{g\mathcal{G}}=\mathrm{97.5\pm
1\%}$ and $F_{e\mathcal{E}}=\mathrm{96.8\pm 1\%}$.

Device Parameter | Value
---|---
Qubit frequency | $\omega_{q}=2\pi\times\mathrm{4.961\,GHz}$
Qubit anharmonicity | $\alpha_{q}=-2\pi\times\mathrm{143.2\,MHz}$
Qubit decay time | $T_{1}^{q}=\mathrm{115\pm 10\,\mu s}$
Qubit dephasing time | $T_{2}^{q}=\mathrm{160\pm 10\,\mu s}$
Qubit echo time | $T_{2}^{e}=\mathrm{236\pm 6\,\mu s}$
Qubit residual occupation | $\bar{n}_{q}=\mathrm{2\pm 1}\times 10^{-2}$
Storage frequency | $\omega_{s}=2\pi\times\mathrm{5.965\,GHz}$
Storage decay time | $T_{1}^{s}=\mathrm{1360\pm 23\,\mu s}$
Storage dephasing time | $T_{2}^{s}=\mathrm{2390\pm 286\,\mu s}$
Storage-Qubit Stark shift | $\chi=-2\pi\times\mathrm{1.285\,MHz}$
Storage residual occupation | $\bar{n}_{c}=\mathrm{6.3\pm 3}\times 10^{-3}$
Readout frequency | $\omega_{r}=2\pi\times\mathrm{7.790\,GHz}$
Readout $\left|e\right\rangle$ shift | $2\chi_{r}^{e}=-2\pi\times\mathrm{1.53\,MHz}$
Readout fidelity ($\left|g\right\rangle$) | $F_{g\mathcal{G}}=\mathrm{97.5\pm 1\%}$
Readout fidelity ($\left|e\right\rangle$) | $F_{e\mathcal{E}}=\mathrm{96.8\pm 1\%}$
Table S3: Device parameters. Measured qubit, storage, and readout cavity
parameters. These independently measured values are necessary to determine for
the transition and emission matrices. This enables the hidden Markov model to
capture the behavior of the system during the measurement sequence.

## Detector characterization

To characterize the detector, the cavity population is varied by applying a
weak drive and the cavity photon number is counted using the technique
described in the main text. In order to extract the efficiency ($\eta$) and
false positive probability ($\delta$) of the detector, the relationship
between injected photon population ($\bar{n}_{\mathrm{inj}}$) and measured
photon population ($\bar{n}_{\mathrm{meas}}$) is fit to
$\bar{n}_{\mathrm{meas}}=\eta\times\bar{n}_{\mathrm{inj}}+\delta$.

### Detector efficiency

The detector efficiency and false positive probability is determined at
varying thresholds for detection $\lambda_{\mathrm{thresh}}$. As the detection
threshold is increased, more repeated number resolved qubit measurements are
required to determine the presence of a photon. This suppresses false
positives due to qubit errors but also leads to a decrease in the detector
efficiency as events with low likelihood ratio are now rejected. Also, the
maximum achievable likelihood ratio decreases with the Fock state prepared in
the cavity. Hence, we decided to keep the false positive probability same for
comparing different Fock states at the expense of detection efficiency. We
performed a single photon counting experiment [1] to determine the occupation
number of the background photons in the cavity (Fig. S15) and chose the
threshold such that
$\frac{1}{\lambda_{\mathrm{thresh}}+1}<\bar{n}_{b}^{c}\Rightarrow\lambda_{\rm
thresh}=10^{3}$. This measured background translates into a metrological gain
of $\mathrm{11.0\,dB}$ below the SQL.

Figure S15: Photon counting to measure the cavity background. Repeated parity
measurement technique followed from [1] to measure the real photon occupation
in the cavity $\bar{n}_{b}^{c}=6.23\cdot 10^{-3}$ with detector based errors
less than $10^{-8}$.

###  Cavity backgrounds

The number of events which cross $\lambda_{\mathrm{thresh}}$ for cavity
prepared in different Fock states are not same, and are subtracted from the
$\alpha\neq 0$ events to demonstrate the enhancement effect. The number of
such events are listed in the table below and are similar to the false
positive probability from the fits in Fig. 3. The number of counts are
corrected for the detection efficiency to reflect the actual counts. There is
no reason to expect the number of counts would increase with $n$ and we
include them in the systematic uncertainties while conducting a dark photon
search. In an independent dataset, we observe a similar number of counts as
reported in Fig. 4.

$\left|n\right\rangle$ | $N_{\rm trials}$ | $N_{\rm bkgd}$ | $\%$
---|---|---|---
$\left|0\right\rangle$ | 114915 | 23 | 0.020
$\left|1\right\rangle$ | 111601 | 315 | 0.282
$\left|2\right\rangle$ | 113364 | 60 | 0.053
$\left|4\right\rangle$ | 108626 | 827 | 0.761
Table S4: Background counts. Number of background counts $N_{\rm bkgd}$
reported for cavity initialized in different Fock states out of $N_{\rm
trials}$.

## Converting background counts to dark photon exclusion

As described in the main text, for a coherent source we expect the signal rate
to be proportional to the term $a_{0}$. We will derive the expression and
compute the kinetic mixing angle which corresponds to $a_{0}$. See Supp.
Section in Ref. [1] for discussion about dark matter induced signal.

### Kinetic mixing angle exclusion

For a dark matter candidate on resonance with the cavity frequency
($m_{\mathrm{DM}}c^{2}=\hbar\omega_{c}$), the rate of photons deposited in the
cavity prepared in a Fock state $\left|n\right\rangle$ by the coherent build
up of electric field in time $\tau$ ($\tau<T_{1}^{s},Q_{\rm DM}/{\nu}$) is
given by:

$\frac{dN_{\mathrm{HP}}}{dt}=\frac{U/\omega_{s}}{\tau}=\frac{1}{2}\frac{E^{2}V}{\omega_{s}}\frac{1}{\tau}=\frac{1}{2}J^{2}_{\mathrm{DM}}(n+1)\tau^{2}\frac{GV}{\omega_{s}}\frac{1}{\tau}$
(S10)

The stimulated emission factor appears via the enhancement of magnitude of the
electric field generated inside the cavity. The volume of the cavity is
$34.5\times 0.5\times\mathrm{2.5\,cm^{3}}=\mathrm{43.13\,cm^{3}}$.
$\mathcal{G}$ encompasses the total geometric factor of the particular cavity
used in the experiment. This includes a factor of $1/3$ due to the dark matter
field polarization being randomly oriented every coherence time. For the
lowest order mode of the rectangular cavity coupled to the qubit with
$\textbf{E}=\mathrm{(}\frac{\pi x}{l})\mathrm{(}\frac{\pi y}{w})\textbf{z}$
the geometric form factor is given by:

$G=\frac{1}{3}\frac{\left|\int dVE_{z}\right|^{2}}{V\int
dV\left|E_{z}\right|^{2}}=\frac{1}{3}\frac{2^{6}}{\pi^{4}}$ (S11)

The dark photon generated current is set by the density of dark matter in the
galaxy
$\rho_{\mathrm{DM}}=\mathrm{0.4\,GeV/cm^{3}}=2\pi\times\mathrm{9.67\times
10^{19}\,GHz/cm^{3}}$:

$J^{2}_{\mathrm{DM}}=2\epsilon^{2}m^{4}A^{\prime
2}=2\epsilon^{2}m^{2}\rho_{\mathrm{DM}}$ (S12)

where $\epsilon$ is the kinetic mixing angle between the dark photon and the
visible matter. Substituting Eqn. S12 into Eqn. S10 yields the signal rate of
photons deposited in the cavity by a dark photon dark matter candidate:

$\frac{dN_{\mathrm{HP}}}{dt}=(n+1)\epsilon^{2}\rho_{\mathrm{DM}}m_{\mathrm{DM}}GV\tau$
(S13)

The total number of photons we expect to be deposited is determined by the
photon rate and the integration time ($N_{\mathrm{\rm trials}}\,\tau$) for
each Fock state:

$N_{\mathrm{HP}}=\frac{dN_{\mathrm{HP}}}{dt}\times\tau\times N_{\mathrm{\rm
trials}}\\\
=(n+1)\epsilon^{2}\rho_{\mathrm{DM}}m_{\mathrm{DM}}GV\tau^{2}N_{\mathrm{\rm
trials}}$ (S14)

Comparing the first term in Eq. 5, we can rewrite the terms to obtain
$\epsilon^{2}=\frac{a_{0}}{\rho_{\mathrm{DM}}m_{\mathrm{DM}}GV}$.

###  Calculating 90% confidence limit

Expt. Parameter | $\Theta$ | $\sigma_{\Theta}$
---|---|---
$a_{m}$ | $1.9\times 10^{3}(\mathrm{s^{-2}})$ | $\sigma_{a}=9.807\times 10^{5}(\mathrm{s^{-2}})$
$\omega_{s}$ | $\mathrm{5.965\,GHz}$ | $\sigma_{\omega_{s}}=\mathrm{25\,Hz}$
$Q_{s}$ | $5.11\times 10^{7}$ | $\sigma_{Q_{s}}=1.4\times 10^{5}$
$V$ | $\mathrm{43.13\,cm^{3}}$ | $\sigma_{V}=\mathrm{1.2\,cm^{3}}$
$G$ | $0.002$ | $\sigma_{G}=0.0002$
Table S5: Stimulated emission experimental parameters. Systematic
uncertainties of physical parameters in the experiment must be incorporated in
determining the excluded dark photon mixing angle $\epsilon$. The uncertainty
in the dark photon (HP) conversion is determined in the previous section. The
storage cavity frequency uncertainty is obtained by Ramsey interferometry. The
quality factor of the cavity is given by $Q_{s}=\omega_{s}T_{1}^{s}$ so the
uncertainty is calculated as
$\sigma_{Q_{s}}^{2}=(\omega_{s}\sigma_{T_{1}^{s}})^{2}+(T_{1}^{s}\sigma_{\omega_{s}})^{2}$.
The volume uncertainty is estimated by assuming machining tolerances of 0.005
inches in each dimension. The form a factor uncertainty is estimated from
assuming $1\%$ error in the simulated structure. Of the experimental
quantities, the DP conversion has the largest systematic uncertainty.

By estimating the strength of the coherent drive in the absence of an external
drive and the measured background counts for different Fock states, we perform
a dark photon search. We determine the dark photon mixing angle $\epsilon$
that can be excluded at the 90% confidence level by using standard error
propagation formula. We determine the standard deviation on $\epsilon$ given,
we have the error estimates for all the parameters tabulated above. The
estimated value of $\epsilon_{0}=1.6\times 10^{-15}\pm 3.30\times 10^{-13}$,
dominated by the error on $a_{0}$. We can now set the $90\%$ confidence limit
on the kinetic mixing angle term as
$\epsilon^{90\%}=\epsilon_{0}+1.28\sigma_{\epsilon}=4.24\times 10^{-13}$. This
leads us to exclude, with $90\%$ confidence, dark photon with mixing angle
$\epsilon^{90\%}$ greater than $4.24\times 10^{-13}$ as shown in Fig. S17.

Figure S16: Fock state preparation with non-zero occupation in the cavity.
Measuring the Fock state preparation probability as a function of initializing
the cavity with varying occupation before the OCT pulses. Red dashed line
corresponds to the maximum tolerable injected number of photons where the
fidelity changes significantly.

## Dark photon parameter space exclusion

A dark photon candidate that could result in more detector counts than
background counts is constrained by the cavity occupation number which
degrades the fidelity of Fock state preparation in the cavity by a significant
amount. In order to estimate the same, the cavity is prepared with varied
number of mean photons before applying the the OCT pulse. The resultant state
is measured with the same procedure as the stimulated emission protocol to
compute the fidelity. We observe that the fidelity changes significantly when
the mean injected photon number goes above $\bar{n}>0.05$ shown by the red
dashed line in Fig. S16. The maximum number of photons sourced from the dark
photon which is tolerable before the state preparation is out of control. In
principle, we can exclude any value of $\epsilon$ which is above the red curve
as it will break the first step of the stimulated emission experiment.

Figure S17: Excluded $\epsilon$ with $\mathrm{m}_{\gamma\prime}$. Shaded
regions in the dark photon parameter space of coupling ($\epsilon$) and mass
($m_{\gamma}$) are excluded with $90\%$ confidence. The horizontal extent is
set by the bandwidth of the number resolved qubit $\pi$-pulse which is
insensitive to any drive outside the band. The vertical limit is set by the
minimum $\epsilon$ which would result in dark photon rate greater than the
value which would degrade the fidelity of Fock state preparation
significantly.

The above calculations assume an infinitely narrow dark matter line. To obtain
the excluded region of the dark photon kinetic mixing angle, we must account
for the lineshape of the dark matter [47]. We convolve the dark matter
lineshape, characterized by $Q_{\mathrm{DM}}\sim\mathrm{10^{6}}$, to obtain
the region shown in Fig. S17.

The storage cavity contain an infinite set of discrete resonances each with a
unique coupling to the dark matter. We focus only on the lowest order cavity
mode that has a non-zero coupling to the dark matter as well as the qubit. In
principle, the interactions between any modes and the dark matter could result
in additional sensitivity to the dark photon. This would require the mode of
interest to have a sufficiently large geometric form factor as well as a
resolvable photon number dependent qubit shift. Future dark matter searches
could employ structures with multiple resonances to enable multiple
simultaneous searches [31].

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