# Fast high-fidelity charge readout by operating the cavity-embedded Cooper
pair transistor in the Kerr bistable regime

B. Thyagarajan<EMAIL_ADDRESS>Department of Physics
and Astronomy, Dartmouth College, Hanover, New Hampshire 03755, USA  S.
Kanhirathingal  Department of Physics and Astronomy, Dartmouth College,
Hanover, New Hampshire 03755, USA  Rigetti Computing, Berkeley, California
94710, USA  B.L. Brock  Department of Physics and Astronomy, Dartmouth
College, Hanover, New Hampshire 03755, USA  Department of Applied Physics,
Yale University, New Haven, Connecticut, USA  Juliang Li  Department of
Physics and Astronomy, Dartmouth College, Hanover, New Hampshire 03755, USA
High Energy Physics Divison, Argonne National Laboratory, 9700 South Cass
Avenue, Argonne, IL 60439, USA  M.P. Blencowe  Department of Physics and
Astronomy, Dartmouth College, Hanover, New Hampshire 03755, USA  A.J. Rimberg
<EMAIL_ADDRESS>Department of Physics and Astronomy,
Dartmouth College, Hanover, New Hampshire 03755, USA

###### Abstract

Operating the cavity-embedded Cooper pair transistor (cCPT) in the Kerr
bistable regime, we demonstrate single-shot resolution between two charge
states that are $0.09e$ apart. The measurement is performed with 94$\%$
fidelity in a duration of 3 $\mu$s. The drive power at which the measurement
is performed corresponds to only 20 intracavity photons on average in the high
oscillation amplitude state of the cCPT, which is orders-of-magnitude smaller
than that in rf-SETs. We find that the limiting factor for this mode of
operation of the cCPT is the spontaneous fluctuation-induced switching between
the two metastable oscillation amplitude states. We present empirical data on
the variation of the switching dynamics with drive parameters and cCPT DC
bias.

##  I Introduction

Fast detection of charge on the order of a fraction of an electron has long
been an important task. Versatile devices such as the quantum point contact
and the single electron transistor (SET) have been used to measure electron
lifetimes in a single electron trap [1], to map electric fields with 100 nm
spatial resolution [2], to observe macroscopic charge quantization [3], and to
study quasiparticle and electron tunneling events in real-time [4, 5]. More
recently, they have been used in the search for Majorana zero modes in
nanowires [6], and could potentially be used to detect dark matter [7, 8].
Such fast, ultrasensitive electrometers are instrumental in the readout of
silicon-based spin qubits [9, 10] where the magnetic moment of a single spin
is too small to detect directly, and is instead converted to a spin state
dependent charge which can be read out. Dispersive charge sensors operating on
the supercurrent branch of the Josephson junctions based inductive-SET (L-SET)
[11] and the single Cooper-pair box [12] are not shackled by the electron shot
noise which limits the operation of the rf-SETs [13]. The cavity-embedded
Cooper pair transistor (cCPT) used in this work has previously been shown to
achieve a charge sensitivity of 14 $\mu e/\sqrt{\mathrm{Hz}}$ operating as a
dispersive sensor in the linear regime with a single intracavity photon on
average [14], close to the theoretical quantum limit for this device [15].

The cCPT is also a rich nonlinear system whose Hamiltonian contains a Kerr
nonlinearity [16], and an emergent parametric amplifier term when the flux
line of the system is driven at twice the resonance frequency. The Kerr term
opens up the possibility of more sensitive charge detection than was achieved
in the linear regime [17]. Such a Kerr cavity coupled to a mechanical
resonator was proposed [18] and demonstrated [19] to achieve an order of
magnitude better cooling of the phonon mode compared to a linear cavity. The
Kerr nonlinearity is well known to produce bifurcations in the system response
[20]. Bifurcation amplifiers [21, 22, 23] based on a large change in the
response at a bifurcation edge have been used to read out the state of
superconducting qubits [24]. Nanomechanical devices based on the bifurcation
under a parametric drive have been used to sense charges of $~{}9e$ at room
temperature [25]. Similar devices have demonstrated charge sensing of the
order of $70e$ by manipulating the topology of the bifurcation diagram [26].

Here, using the bifurcation between a bistable and a monostable region induced
by the Kerr nonlinearity of the cCPT, we demonstrate single-shot readout of
$0.09e$ of charge in 3 $\mu$s with 94$\%$ fidelity, using fewer than 25
intracavity photons. Such low power operation ensures minimal back-action on
the system being measured [27], and also aids in the integration of these cCPT
detectors with state-of-the-art first stage amplifiers such as the TWPAs [28]
without overwhelming them beyond their compression point. Such fast high
fidelity readout is comparable to the current state-of-the-art for
semiconductor spin qubits [29, 30].

In Sec. II we present a semi-classical analysis of the nonlinear cCPT and
propose a scheme for it to function as a sensitive charge state discriminator
operating in the bistable regime. In Sec. III we experimentally study the
hysteresis in the cCPT response in the bistable regime to characterize the
extent of the bistability as a function of the drive detuning and strength. We
then implement a charge sensing protocol, and observe the presence of
fluctuation-induced spontaneous transitions between the bistable states, which
we study as a function of drive parameters and cCPT DC bias. Lastly, we
characterize our charge sensing protocol and demonstrate the optimum high-
fidelity, fast charge state readout possible with this device. In Sec. IV we
conclude by discussing some possible improvements to this work. Details of the
heterodyne measurement scheme employed in this work and the microwave
circuitry used in the dilution refrigerator are in Appendix A, and some
experimental considerations for the charge sensing scheme used in Sec. III are
detailed in Appendix B.

Figure 1: (a) Schematic of the cCPT. (b) Variation of the Kerr coefficient $K$
as a function of gate, $n_{g}$, and flux, $\Phi_{\mathrm{ext}}$, over the
operational bias range of the cCPT simulated using the extracted values of
$E_{J}$ and $E_{C}$ [16]. (c) Simulated response of the cCPT for different
drive powers, $P_{\mathrm{in}}$. The red curve is for a drive strength
$P_{\mathrm{in}}\ll P_{\mathrm{in}}^{\mathrm{(c)}}$. The blue curve is for
$P_{\mathrm{in}}>P_{\mathrm{in}}^{\mathrm{(c)}}$ and the green is for
$P_{\mathrm{in}}\gg P_{\mathrm{in}}^{\mathrm{(c)}}$. Above
$P_{\mathrm{in}}^{\mathrm{(c)}}$, we see bistability across a range of
detunings indicated by the corresponding shaded region. The $\bigtriangleup$’s
represent the stable high oscillation amplitude state, the
$\bigtriangledown$’s represent the stable low oscillation amplitude states,
and the $\Circle$’s represent the unstable state. The solid lines indicate
monostability. (d) Simulated magnitude of the reflection coefficient for the
drive powers in (c). (e) Simulated phase of the reflection coefficient for the
drive powers in (c). All simulations in (c), (d) and (e) were for a cCPT DC
bias $(n_{g},\Phi_{\mathrm{ext}})=(0,0)$ with $K/2\pi=-470$ kHz, and nominal
damping rates for this bias point [16].

##  II Theoretical Description

The cCPT, schematically depicted in Fig. 1(a), consists of two parts: (i) the
cavity, which is a $\lambda/4$ superconducting coplanar waveguide (CPW) with
its end shorted to the ground plane, and (ii) a Cooper pair transistor (CPT)
across the center line and ground plane of the CPW at its voltage anti-node.
In this geometry, the cavity and CPT form a shared SQUID loop, which couples
them together. When the CPT remains in its ground state, it modifies the
effective potential of the cavity, such that the cCPT behaves as a nonlinear
oscillator whose resonant frequency can be tuned using the effective gate
charge, $n_{g}=\frac{C_{g}V_{g}}{e}$, and the magnetic flux threading the
SQUID loop, $\Phi_{\mathrm{ext}}$. Here, $V_{g}$ is the external DC voltage
applied to the CPT island through the gate capacitance $C_{g}$. Along with the
fabrication details for the cCPT device used in this work, a detailed
characterization at low drive amplitudes where the nonlinearities do not
contribute substantially to the dynamics has been carried out in Ref. [16].
Notably, the Josephson energy, $E_{J}$, and the charging energy, $E_{C}$, were
estimated to be $E_{J}/h=14.8$ GHz and $E_{C}/h=54.1$ GHz respectively.
Finally, to drive and measure the cCPT, a probe transmission line is coupled
to the CPW through a coupling capacitor $C_{c}$.

For an input drive close to resonance, under the rotating wave approximation,
the Hamiltonian for the cCPT is given by [16, 15]

$\displaystyle H$
$\displaystyle=\hbar\omega_{0}(n_{g},\Phi_{\mathrm{ext}})a^{\dagger}a+\frac{1}{2}\hbar
K(n_{g},\Phi_{\mathrm{ext}})a^{\dagger 2}a^{2},$ (1)

where $a(a^{\dagger})$ are the annihilation (creation) operators for the
cavity mode, $\omega_{0}(n_{g},\Phi_{\mathrm{ext}})$ and
$K(n_{g},\Phi_{\mathrm{ext}})$ are the resonant frequency of the linear cCPT
Hamiltonian and the Kerr coefficient respectively. The variation of
$K(n_{g},\Phi_{\mathrm{ext}})$ over the operational range of the cCPT device
used in this work is shown in Fig. 1(b). The Kerr coefficient changes sign
with flux, attaining extremum values at half-integer multiples of $\Phi_{0}$,
and passing through zero close to $\Phi_{\mathrm{ext}}=0.25\Phi_{0}$. Kerr-
free cavities have been used to increase the dynamic range of parametric
amplifiers [31, 32].

We use input-output theory [33] to model the dynamics of the cavity mode. The
quantum Langevin equation for $a$ gives

$\displaystyle\dot{a}$
$\displaystyle=\frac{1}{i\hbar}\big{[}a,H\big{]}-\big{[}a,a^{\dagger}\big{]}\left(\frac{\kappa_{\mathrm{tot}}}{2}a-\sqrt{\kappa_{\mathrm{ext}}}a_{\mathrm{in}}(t)-\sqrt{\kappa_{\mathrm{int}}}b_{\mathrm{in}}(t)\right)$
$\displaystyle=-\left(i\left(\omega_{0}+Ka^{\dagger}a\right)+\frac{\kappa_{\mathrm{tot}}}{2}\right)a+\sqrt{\kappa_{\mathrm{ext}}}a_{\mathrm{in}}(t)+\sqrt{\kappa_{\mathrm{int}}}b_{\mathrm{in}}(t),$
(2)

and a corresponding equation for $a^{\dagger}$, where $\kappa_{\mathrm{ext}}$
is the external damping rate due to the coupling of the resonator to the probe
transmission line with the input bath operator $a_{\mathrm{in}}(t)$, and
$\kappa_{\mathrm{int}}$ is the internal damping rate associated with the
coupling of the resonator to an internal loss channel with input operator
$b_{\mathrm{in}}(t)$. The total damping rate of the cavity is
$\kappa_{\mathrm{tot}}=\kappa_{\mathrm{ext}}+\kappa_{\mathrm{int}}$. When the
input tone is a pure sine wave at frequency $\omega_{d}$ of the form $\langle
a_{\mathrm{in}}\rangle=\alpha_{\mathrm{in}}e^{-i\omega_{d}t}$, the steady
state response of the cavity is at this drive frequency. For this coherent
drive, using the semi-classical approximation, we make the ansatz $\langle
a\rangle=\alpha e^{-i\omega_{d}t}$, with
$\langle\dot{a}\rangle=-i\omega_{d}\alpha e^{-i\omega_{d}t}$ and the average
intracavity occupation number $n=\absolutevalue{\alpha}^{2}=\langle
a^{\dagger}a\rangle$. Plugging this ansatz into the expectation value of
Eq.(2) we obtain

$\displaystyle\left[-i\left(\Delta-K\absolutevalue{\alpha}^{2}\right)+\frac{\kappa_{\mathrm{tot}}}{2}\right]\alpha$
$\displaystyle=\sqrt{\kappa_{\mathrm{ext}}}\alpha_{\mathrm{in}},$ (3)

where we have defined the detuning $\Delta=\omega_{d}-\omega_{0}$. Using
Eq.(3) and the input-output relation
$a_{\mathrm{out}}(t)=a_{\mathrm{in}}(t)-\sqrt{\kappa_{\mathrm{ext}}}a(t)$
[33], [34] we find the reflection coefficient $S_{11}(\Delta)$ to be

$\displaystyle S_{11}(\Delta)$
$\displaystyle=\left(\frac{\alpha_{\mathrm{out}}}{\alpha_{\mathrm{in}}}\right)^{*}$
$\displaystyle=\frac{(\Delta-K\absolutevalue{\alpha}^{2})-i(\kappa_{\mathrm{int}}-\kappa_{\mathrm{ext}})/2}{(\Delta-K\absolutevalue{\alpha}^{2})-i(\kappa_{\mathrm{int}}+\kappa_{\mathrm{ext}})/2},$
(4)

where $a_{\mathrm{out}}(t)$ is the transmission output bath operator. Also,
from Eq.(3) and the corresponding equation for $\alpha^{*}$,
$n=\absolutevalue{\alpha}^{2}$ satisfies the cubic equation

$\displaystyle K^{2}n^{3}-2K\Delta
n^{2}+\left(\Delta^{2}+\frac{\kappa_{\mathrm{tot}}^{2}}{4}\right)n=\kappa_{\mathrm{ext}}\frac{P_{\mathrm{in}}}{\hbar\omega_{d}},$
(5)

where $P_{\mathrm{in}}=n_{\mathrm{in}}\hbar\omega_{d}$ is the power of the
input drive tone incident on the cCPT, and
$n_{\mathrm{in}}=\absolutevalue{\alpha_{\mathrm{in}}}^{2}$ is the input photon
flux. As illustrated in Figs. 1(c-e), at very low drive strengths this cubic
equation has only one real root and the oscillator exhibits only monostable
behaviour across all detunings. As the drive strength is increased beyond a
critical power
$P_{\mathrm{in}}^{\mathrm{(c)}}=\frac{\sqrt{3}}{9}\frac{\kappa_{\mathrm{tot}}^{3}}{\absolutevalue{K}\kappa_{\mathrm{ext}}}\hbar\omega_{d}^{(c)}$,
the oscillator system undergoes a bifurcation, and exhibits bistability for a
range of detunings. Here, $\omega_{d}^{(c)}$ is the drive frequency
corresponding to a detuning of
$\Delta_{c}=\operatorname{sgn}({K})\frac{\sqrt{3}}{2}\kappa_{\mathrm{tot}}$
and $(\Delta_{c},P_{\mathrm{in}}^{\mathrm{(c)}})$ corresponds to a spinode
point in the parameter space of the input drive [35]. In the bistable region,
two of the three real solutions of the cubic Eq.(5) correspond to high and low
oscillation amplitude states with corresponding values of $S_{11}$ from
Eq.(4), while the third is an unstable, experimentally inaccessible state
[36].

Figure 2: (a) Experimentally measured resonant frequency of the linear cCPT
($P_{\mathrm{in}}\ll P_{\mathrm{in}}^{\mathrm{(c)}}$), $\omega_{0}$, as a
function of the gate charge on the cCPT, $n_{g}$, at a fixed flux bias
$\Phi_{\mathrm{ext}}=0.15\Phi_{0}$ (triangles). The line is the theoretically
expected resonant frequency for the junction parameters of this device. (b)
Simulations illustrating the larger separation in reflected phase, $\delta
S_{11}$ ($\delta S_{11}^{\mathrm{(Kerr)}}$) when operating at
$P_{\mathrm{in}}>P_{\mathrm{in}}^{\mathrm{(c)}}$ (solid lines) compared to
$P_{\mathrm{in}}\ll P_{\mathrm{in}}^{\mathrm{(c)}}$ (dashed lines). (c) Phase
of the reflected signal for a forward (solid line) and reverse (dashed line)
triangular ramp of the drive amplitude, $P_{\mathrm{in}}$. The input power is
ramped between -140 dBm and -109 dBm in increasingly longer times from 2
$\mu$s to 28 $\mu$s from red to blue curves. The cCPT was biased at
($n_{g},\Phi_{\mathrm{ext}}$) = (0, 0) and the detuning was $\Delta/2\pi=-9.5$
MHz.

The variation of the resonant frequency of the cCPT with the gate, $n_{g}$, as
illustrated in Fig. 2(a), forms the basis for a sensitive dispersive charge
detector. Operating in the single-photon, linear regime, this device was
demonstrated to have a charge sensitivity of 14 $\>\mu e/\sqrt{Hz}$ [14]. Fig.
2(b) uses Eq.(4) to simulate the reflected phase as a function of drive
frequency, $\omega_{d}$, for two gate values separated by $\delta n_{g}$
corresponding to a resonant frequency shift $\delta\omega_{0}$. The $S_{11}$
for the two gate values are denoted by pink and green curves respectively,
both in the linear ($P_{\mathrm{in}}\ll P_{\mathrm{in}}^{\mathrm{(c)}}$),
single photon regime (dashed lines); and with a drive power
$P_{\mathrm{in}}>P_{\mathrm{in}}^{\mathrm{(c)}}$ (solid lines). At a given
$\omega_{d}$, $n_{g}$ can hence be inferred from the measured $S_{11}$. For
$P_{\mathrm{in}}>P_{\mathrm{in}}^{\mathrm{(c)}}$, these simulations describe
what we would observe in the absence of spontaneous transitions between the
high and the low oscillation amplitude states in the bistable region. In the
absence of these transitions, while ramping the drive detuning from a large
blue-detuned value (with respect to linear resonance, $\omega_{0}$,
$\Delta>0$), to a red-detuned value ($\Delta<0$), we expect to stay in the
high oscillation amplitude state until we reach the bifurcation detuning
further from $\omega_{0}$ for the green curve in Fig. 1(c-e). We refer to this
as the lower bifurcation point, while referring to the bifurcation detuning
closer to $\omega_{0}$ as the upper bifurcation point. Upon crossing the lower
bifurcation point, an abrupt jump from the high to the low oscillation
amplitude state is expected, with a corresponding large change in $S_{11}$, as
illustrated in Fig. 2(b). For an appropriate drive frequency denoted by the
dashed black line, the same separation in gate charge, $\delta n_{g}$,
produces a larger difference in the reflected phase between the pink and green
curves, $\delta S_{11}^{\mathrm{(Kerr)}}$, than the $\delta S_{11}$ while
operating in the linear regime. Conversely, $\delta S_{11}^{\mathrm{(Kerr)}}$
continues to remain large as the green and pink curves are brought together by
reducing $\delta n_{g}$, whereas $\delta S_{11}$ undergoes substantial
reduction while doing so. The sensitivity of the charge detector is the
smallest $\delta n_{g}$ that produces a $\delta S_{11}$ which can be detected
with a signal-to-noise-ratio (SNR) of 1 [14]. Given that the noise in the
measurement is limited by the amplifier chain in the experimental setup [14],
the larger $S_{11}^{\mathrm{(Kerr)}}$ for smaller $\delta n_{g}$ promises a
lower, much improved charge sensitivity for the device operating in this
regime.

##  III Experiments

In this section, we first describe the results of a triangular input power
ramp in order to understand the extent of the bistable region with respect to
the cCPT drive parameters at a given bias point. We then outline the protocol
we use in order to perform charge sensing based on the bifurcation described
above. Contrary to the sharp jump in $S_{11}^{(\mathrm{kerr})}$ at a precise
value of the detuning described in Sec. II, we see a non-zero probability of
obtaining a value on either end of the step illustrated in Fig. 2(b) for a
range of detunings. We discuss the results of this protocol for a range of
cCPT bias points and drive parameters. From this, we glean the optimal
conditions for charge sensing and finally perform an optimized single-shot
measurement.

In order to study the extent of the bistability, we bias the cCPT at
($n_{g},\Phi_{\mathrm{ext}}$) = (0, 0), and drive it at a fixed detuning
$\Delta/2\pi=-9.5$ MHz with a triangular amplitude ramp in the forward and the
reverse direction to check for hysteresis. This is the bias point at which we
expect minimum fluctuation in the resonant frequency of the cCPT due to charge
and flux noise [16, 37]. We perform a heterodyne measurement to obtain the
phase of the reflected signal over the course of the ramp. The RF circuitry
used in the experiments described here is detailed in Appendix A. Fig. 2(c)
plots the observed hysteresis in the phase of $S_{11}$ for different ramp
rates, each averaged over 5000 repetitions of the ramp. For fast ramps, we see
that we obtain a value for the reflected phase corresponding to the low
oscillation amplitude state for the forward ramp, and a value that corresponds
to the high oscillation amplitude state during the reverse ramp. However, as
the ramp time is increased, we observe that the spacing between the observed
phase during the forward and the reverse ramps reduces, and for this cCPT bias
point, saturates to the values represented by the blue curves, corresponding
to ramp times of $\sim 25\>\mu$s. This is because, when given enough time to
do so, the oscillator system undergoes fluctuation-induced spontaneous
switching between the high and low oscillation amplitude states over the
course of a ramp. This yields a weighted average value for the phase at each
$P_{\mathrm{in}}$ value over 5000 repetitions of the pulse sequence. The
weights depend on the average lifetimes of the high and the low oscillation
amplitude states at the chosen cCPT bias and the drive parameters. We see less
variation in the shape of the forward and reverse ramp curves for the larger
ramp times in Fig. 2(c). This provides a rough estimate of $\sim 25\>\mu$s for
the average lifetimes of these bistable states. This is a sign of spontaneous
transitions between the high and low oscillation amplitude states for a range
of input drive strengths, and will be detrimental to the charge sensing scheme
described above which counts on the sharp jump from one oscillation amplitude
state to the other at precisely a bifurcation point. A similar reduction in
the area enclosed between the curves corresponding to the forward and reverse
ramps for longer ramps was recently observed for a nonlinear semiconductor
microcavity [38].

Figure 3: (a) Charge sensing protocol. Detuning of the input pulse tone used
to initiate and readout the oscillation state in the charge sensing experiment
described in the text with representative values for the durations and
detunings of the different sections. The pink area depicts the initialization
segment to initialize the oscillator in the high oscillation amplitude state.
The phase is measured and averaged during the green segment, for a time
$t_{\mathrm{acq}}$. We wait a time $t_{\mathrm{down}}=5\>\mu$s between
consecutive pulses and set $f_{\mathrm{latch}}=0$. (b) Schematic illustrating
sigmoid S-curves for two different cCPT gate biases illustrated in pink and
green, with the black arrow denoting the maximum contrast between the two. The
inset shows a representative histogram of the reflected phase, $\phi$, upon
running the above pulse sequence $N_{\mathrm{tot}}=20,000$ times. The two
Gaussian distributions correspond to the oscillator being in the high (left)
and low (right) oscillation amplitude states respectively, with the solid
lines representing a double Gaussian fit. (c) Obtained S-curves ($\Circle$’s)
for different $n_{g}$ values at $\Phi_{\mathrm{ext}}=0.06\Phi_{0}$ and an
input drive power $P_{\mathrm{in}}=-128$ dBm. The averaging time,
$t_{\mathrm{acq}}=3\>\mu$s. The solid lines are sigmoid fits to Eq. (6). The
horizontal error bars represent the standard deviation of the resonant
frequency fluctuations due to charge and flux noise at the cCPT DC bias point
[16, 39].

While performing the charge sensing measurement, we choose an input drive
strength which gives rise to a region of bistability
($P_{\mathrm{in}}>P_{\mathrm{in}}^{\mathrm{(c)}}$) for the chosen cCPT DC bias
$(n_{g},\Phi_{\mathrm{ext}})$ with a corresponding $K<0$. In order to
deterministically initialize the oscillator in the high oscillation amplitude
state, we perform a linear ramp on the detuning of the drive tone from a blue-
to a red-detuning as illustrated in Fig. 3(a). More details on the
initialization section (shaded pink) of this protocol are provided in Appendix
B. Once initialized, we measure and average the phase of the reflected signal
for a time $t_{\mathrm{acq}}$. Performing this measurement
$N_{\mathrm{tot}}=20,000$ times, we obtain a double Gaussian histogram as
illustrated in the inset of Fig. 3(b), and extract the probability of the high
oscillation amplitude state, $P(\omega_{d})$, as the ratio of the area of the
left Gaussian to the total area of the histogram. We perform this measurement
for different detunings at the end of the initialization step of the pulse in
Fig. 3(a), and plot the obtained probability of being in the high oscillation
amplitude state for each detuning, obtaining the S-curves in Fig. 3(c). We fit
sigmoids of the form

$P(\omega_{d})=\frac{1}{1+\exp{-\frac{4.3944(\omega_{d}-\Delta_{0})}{\gamma}}},$
(6)

where $\Delta_{0}$ is the center of the sigmoid, and the numerical factor in
the exponential ensures $\gamma$ is its width between $P(\omega_{d})=0.1$ and
$P(\omega_{d})=0.9$.

As described earlier in Sec. II, we ideally expect an abrupt step in
$P(\omega_{d})$ from $1\rightarrow 0$ at the lower bifurcation point for our
ramp protocol in Fig. 3(a). However, from Fig. 3(c), we clearly do not see an
abrupt step in $P(\omega_{d})$ at just the bifurcation point, but a gradual
change in its value across a range of detunings, whose behavior for different
cCPT bias points and drive parameters we will now study.

For systems where the ratio $\frac{|K|}{\kappa_{\mathrm{tot}}}<1$ [40], close
to a bifurcation, the switching between these two metastable oscillation
amplitude states is described by a quantum activation model which predicts
fluctuation-induced escape over a metapotential barrier [35, 41]. This has
been demonstrated to accurately model the switching between these states in
nanomechanical systems [42, 43], Josephson bifurcation amplifiers [23], and in
Josephson junction array devices [44, 45]. For systems with Kerr strengths
comparable to the cavity linewidth, a quantum calculation is required to
accurately model this switching [40]. We discuss some of the possible sources
of these fluctuations in Sec. IV.

From a charge sensing point of view, we want the S-curves for two cCPT gate
biases separated by a given $\delta n_{g}$ to have a large separation between
their centers, $\Delta_{0}$, while the widths of these sigmoids, $\gamma$,
should remain small. Additionally, in order to perform single-shot
measurements separating the oscillation state using a threshold phase value at
the middle of the two Gaussian peaks in the inset in Fig. 3(b), we need to
minimize the overlap between the Gaussians.

Figure 4: (a) S-curve centers, $\Delta_{0}$, vs gate charge, $n_{g}$, for
different different flux biases, $\Phi_{\mathrm{ext}}$, for
$P_{\mathrm{in}}=-128$ dBm. The error bars are smaller than the markers. (b)
S-curve widths, $\gamma$, vs $K$ at the cCPT DC bias points in (a), for
different drive strengths. The error bars are the 95$\%$ confidence intervals
to the sigmoid fits.

Fig. 4(a) shows the variation of the centers of the sigmoid, $\Delta_{0}$, vs
cCPT gate bias, $n_{g}$, for different cCPT flux biases,
$\Phi_{\mathrm{ext}}$. For each flux bias, the largest separations between the
centers of two S-curves are observed for large gate biases. Given that we work
close to $n_{g}=0.71$, the largest separation is for flux values close to
$\Phi_{\mathrm{ext}}=0$. This is related to both the large variation in the
ground state energy of the cCPT at these DC bias points, and consequently the
linear resonance frequency, $\omega_{0}$, as in Fig. 2(a) [16], as well as the
variation of the metapotential landscape in the bistable region that limits
the extent of switching between the two oscillation amplitude states at a
given set of drive parameters. The separation between the $\Delta_{0}$ for two
distinct cCPT bias points is also found to be largest at low input powers,
$P_{\mathrm{in}}$ (data not shown). Fig. 4(b) shows the variation of the width
of the sigmoids, $\gamma$, plotted against the theoretically computed value of
the Kerr coefficient, $K$, from Fig. 1(b) at different cCPT bias points
$(n_{g},\Phi_{\mathrm{ext}})$, for different $P_{\mathrm{in}}$. The cCPT DC
bias points we are interested in based on the separation of the centers of the
sigmoids in Fig. 4(a) correspond to $K/2\pi=-600$ to -800 kHz, and we see
$\gamma$ is much smaller for lower $P_{\mathrm{in}}$ at these bias points.
This is related both to the reduction in width of the bistable region with
decreasing $P_{\mathrm{in}}$, as seen from Eq.(5), and to the reduction in
barrier height of the metapotential with increasing $P_{\mathrm{in}}$ [40,
46]. This reduced barrier height enables switching between the two oscillation
amplitude states within experimental timescales over a wider range of
detunings.

Figure 5: (a) Measured separation between peaks of Gaussians as in inset of
Fig. 3(b) vs $K$ for different $P_{\mathrm{in}}$. (b) Histogram count
$N^{(0.62)}$ ($N^{(0.71)}$) of the phase of the reflected signal for optimal
charge sensing. The data is for $N_{\mathrm{tot}}=20,000$ trials each at
$(n_{g},\Phi_{\mathrm{ext}})=(0.62,0)$ (blue) and
$(n_{g},\Phi_{\mathrm{ext}})=(0.71,0)$ (red) respectively, while driving the
cCPT with an input tone at $\omega_{d}/2\pi$ = 5.8013 GHz with
$P_{\mathrm{in}}=-128$ dBm. The dashed line denotes the threshold phase,
$\phi_{\mathrm{th}}$, used to discriminate the charge state in a single-shot.
(c) Measurement fidelity as a function of averaging time $t_{\mathrm{acq}}$
for the drive conditions in (b).

The other consideration in demonstrating single-shot readout is the resolution
of the two Gaussians in the inset of Fig. 3(b). Fig. 5(a) shows the separation
between the centers of the two Gaussians for all the cCPT DC bias points in
Fig. 4(a) for a range of $P_{\mathrm{in}}$. The smallest Kerr strengths,
$|K|$, and the lowest drive strengths, $P_{\mathrm{in}}$, give us maximum
separation. We understand this as follows. The detuning corresponding to the
maximum oscillator response is either negative (spring softening) or positive
(spring hardening), based on $\operatorname{sgn}({K})$. The degree of
softening of the oscillation response curve (Fig. 1(c)) and hence the
reflection coefficient, $S_{11}$, illustrated in Fig. 1(d-e) depends on $|K|$.
The low oscillation amplitude response is close to zero at all detunings in
the bistable region, with the corresponding $S_{11}$ close to 1. Meanwhile,
the high oscillation amplitude increases from close to zero at the upper
bifurcation point to a maximum value at the lower bifurcation point, with a
slope inversely proportional to $|K|$, with similar behavior for $S_{11}$. At
detunings close to the upper bifurcation point around which we see non-zero
probability for both high and low oscillation amplitude states within
experimental lifetimes, the high amplitude oscillation response and the
corresponding Phase$(S_{11})$ at a given detuning assume a finite non-zero
value whose magnitude depends inversely on $|K|$, as seen in Fig. 5(a), such
that the difference in Phase($S_{11})$ between the high and the zero phase low
amplitude state also depends inversely on $|K|$.

To understand the variation with $P_{\mathrm{in}}$, we compare the blue and
the green curves in Figs. 1(c-e), which are both for the same $K$. We see that
the slope of the amplitude response of both bistable states is nearly
independent of $P_{\mathrm{in}}$, but close to the upper bifurcation point,
the corresponding $S_{11}$ yields smaller separation between the high and low
oscillation amplitude states for higher $P_{\mathrm{in}}$. Fig. 5(a) hence
suggests that we work at low $|K|$ and at low $P_{\mathrm{in}}$ in order to
observe maximum separation between the reflected phase of the high and the low
oscillation amplitude states. While we saw that the latter condition also
yields S-curves with the smallest widths (Fig. 4(b)), Fig. 4(a) shows that the
$\Phi_{\mathrm{ext}}$ corresponding to small $|K|$ values correspond to poor
separation between $\Delta_{0}$ for two cCPT $n_{g}$ values, which is
contradictory to our goal. The pursuit of low $|K|$ and low $P_{\mathrm{in}}$
suggests operating the cCPT on the cusp of bistability where a large gain in
the dispersive readout is expected at a certain detuning [17]. However,
operation of the device studied here is not possible in that regime as
discussed further in Sec. IV.

With these considerations, we obtain a maximum contrast of 96.61$\%$ in the
S-curves between when the cCPT is biased at
$(n_{g},\Phi_{\mathrm{ext}})=(0.62,0)$ and at $(0.71,0)$, and driven at
$\omega_{d}/2\pi=5.8013$ GHz with $P_{\mathrm{in}}=-128$ dBm. For this drive
strength, using an averaging time, $t_{\mathrm{acq}}=3\>\mu$s, Fig. 5(b) shows
the obtained histograms with counts $N^{(0.62)}(\phi)$ and $N^{(0.71)}(\phi)$
respectively. The separation between the Gaussian peak centers is 36∘ for this
bias point and $P_{\mathrm{in}}$, and the width of each Gaussian is 12∘ for
this $t_{\mathrm{acq}}$. Using a threshold value $\phi_{\mathrm{th}}$ at the
center of the two Gaussian peaks as denoted by the dashed line, we assign a
charge state to each histogram data point. Defining the fidelity
$F^{(0.62)}=1-\frac{1}{N_{\mathrm{tot}}}\sum_{\phi=\phi_{\mathrm{th}}}^{180}N^{(0.62)}(\phi)$
and
$F^{(0.71)}=1-\frac{1}{N_{\mathrm{tot}}}\sum_{\phi=-180}^{\phi_{\mathrm{th}}}N^{(0.71)}(\phi)$,
we obtain an average fidelity $F=94.59\>\%$. The similarity between the
obtained fidelity and the measured maximum contrast which is agnostic to the
overlap of the Gaussians caused by the amplifier noise shows that for this
$t_{\mathrm{acq}}$, the limiting factor of our measurement is not the signal-
to-noise ratio of the amplifier chain, but is the broadening of the S-curves
caused by fluctuation-induced switching between the metastable oscillation
states. Finally, it is worth noting that using the above DC bias and drive
parameters in Eq. (5) along with damping rates $\kappa_{\mathrm{int}}$,
$\kappa_{\mathrm{ext}}$ extracted as described in [16, 39], we find that the
intracavity photon number, $n=8.1$ at $n_{g}=0.62$, and $n=20.94$ at
$n_{g}=0.71$ respectively in the high oscillation amplitude state. At
$n_{g}=0.71$, for the optimal drive tone, the oscillator resides predominantly
in the low oscillation amplitude state with an intracavity occupation on the
order of 0.2 photons. These are orders of magnitude fewer photons than used by
devices such as the rf-SETs [13, 47].

##  IV Discussion

The cCPT operating in the Kerr bistable regime is thus sensitive to changes in
its electrostatic environment that produce a shift of $\delta n_{g}=0.09e$ and
we have demonstrated real-time single-shot high fidelity detection of this
charge difference in $3\>\mu$s. This corresponds to a charge sensitivity per
unit bandwidth $=\delta n_{g}\sqrt{t_{\mathrm{acq}}}=155.89\>\mu
e/\sqrt{\mathrm{Hz}}$. The bandwidth of this electrometer is set by
$\kappa_{\mathrm{tot}}/2\pi\approx 1.5$ MHz. This readout is performed with
only a few tens of intracavity photons, which is several orders of magnitude
smaller than in other state-of-the-art devices [13, 48]. An application of the
cCPT as a charge sensor is to detect the state of a quantum-dot spin qubit
using spin-to-charge conversion [9, 10, 29]. The backaction by the charge
sensor on the system being measured is proportional to the number of
intracavity photons [49], making such low cavity number operation desirable
[27]. Using techniques such as defining the SET in the Si substrate [50, 51],
and extending the cCPT island [5] in order to increase the $\delta n_{g}$
induced on the cCPT island in the event of a spin tunneling out of a quantum
dot, we could work at larger $P_{\mathrm{in}}$ for the same
$\Phi_{\mathrm{ext}}$ and corresponding $K$, while still achieving sufficient
contrast and comparable fidelity for much smaller $t_{\mathrm{acq}}$. For the
more relaxed $\delta n_{g}$ requirement, low power operation with smaller
$t_{\mathrm{acq}}$ without compromising fidelity would be possible at other
$\Phi_{\mathrm{ext}}$ corresponding to smaller $|K|$ and larger phase
separation between the high and the low oscillation amplitude states as in
Fig. 5(a), while still retaining a large contrast value.

The major limitation to this mode of operation of the cCPT as a charge sensor
are the spontaneous fluctuation-induced transitions between the high and low
oscillation amplitude states in the bistable regime. The metapotential
landscape governing these transitions depends on $P_{\mathrm{in}}$ and $K$
[46, 40]. The $|K|/\kappa_{\mathrm{tot}}$ range of the cCPT lies in the
interesting ‘mesoscopic’ region where quantum effects begin to become
important [40]. Mapping the metapotential and corresponding switching rates
between the high and low oscillation amplitude state for such a device could
guide understanding of single-photon Kerr devices [52] which have been
proposed as single-photon sources [53], to generate ultra-fast pulses [54] and
to be used to implement quantum non-demolition measurements [55].

For a given metapotential, the intensity of the fluctuations present in the
system is the other factor that affects the switching rates and hence the
width of the S-curves, $\gamma$. Given that thermal activation is unlikely
since $\hbar\omega_{d}>k_{B}T$, one commonly studied source of fluctuations is
the dephasing of the oscillator caused by the modulation of the resonant
frequency [36], or equivalently, of the detuning of the drive. The phase noise
of the signal generator is typically $1/f$ in nature and is quite small in the
frequency range relevant for escape dynamics such as observed in Fig. 2(c).
The resonant frequency fluctuations for systems such as the cCPT has been
studied in detail [39], and characterized [16]. The resonant frequency
fluctuations due to charge noise arising from fluctuating two-level systems
close to the cCPT island, and magnetic flux noise arising due to unpaired
surface spins are both $1/f$ in character, and should not be relevant to the
switching dynamics either. However, the frequency modulation due to white
photon shot noise [49] induced Kerr shift considered in Ref. [39] would
explain the increase in the width of the S-curves at larger $|K|$ as in Fig.
4(b), working in tandem with the reduced barrier metapotential barrier at
larger $|K|$. A careful study showing a direct correlation between the
switching rates of the cCPT and $P_{\mathrm{in}}$ would confirm this
hypothesis, since the frequency independent power spectral density of the
photon shot noise depends on the average cavity occupation, $n$.

Another avenue to increase the sensitivity of the device would be to reduce
the quasiparticle poisoning (QP) in the device. We observe substantial
switching out of the even to the odd manifold of the CPT ($n_{g}\rightarrow
1-n_{g}$) [56] for $|n_{g}|>0.71$ [16]. As illustrated in Fig. 2(a), the cCPT
resonance frequency, $\omega_{0}$, is most sensitive to $n_{g}$ close to
$|n_{g}|=1$, and employing techniques such as effective shielding from Cooper-
pair-breaking, quasiparticle generating radiation [57, 58] could greatly
enhance the performance of the cCPT.

Still using this inherent Kerr nonlinearity, but by driving the cCPT with a
$P_{\mathrm{in}}\sim P_{\mathrm{in}}^{\mathrm{(c)}}$ close to but before the
onset of bistability where $\frac{dS_{11}}{d\Delta}\rightarrow\infty$ for some
$\Delta$, we should be able to realize a large gain in the charge sensitivity
[17]. The presence of gate and flux noise induced resonance frequency
fluctuations [16, 39] make it hard to operate at the precise $\Delta$ where
this enhancement is expected, but using the resonance frequency stabilizing
feedback scheme demonstrated in [37] should enable such operation. Enhanced
cooling of a nanomechanical resonator coupled to a nonlinear cavity operating
in this regime has been shown [19].

The cCPT Hamiltonian also has other nonlinear terms such as those of a
degenerate parametric amplifier which can be driven into resonance using an
appropriate flux pump at close to $2\omega_{0}$. The amplitude of the
parametric oscillations induced [59, 60, 61] depends on the gate bias of the
cCPT [62], and can be used as a charge sensor, similar to the qubit state
detector operating on this principle [63].

###### Acknowledgements.

We thank Ethan Williams, Hui Wang, Archana Kamal, Chandrasekhar Ramanathan,
William Braasch and Hory Mohammadpour for helpful discussions and useful
feedback on the manuscript. This work was supported by the NSF under Grants
No. DMR-1807785 (S. K., B. T., B. L. B., and A. R) and DMR-1507383 (M. P. B.),
and by a Google research award (S. K.).

##  Appendix A Experimental setup

Figure 6: Microwave circuitry used in Sec. III.

Fig. 6 shows the rf circuitry used in the experiments described in this work.
The input tone from a Keysight N5183B signal generator is mixed with an
intermediate frequency tone from a Tektronix arbitrary waveform generator
whose amplitude envelope can be ramped, or whose frequency can be chirped for
the charge sensing protocol (see Fig. 3(a)). This signal passes through
various stages of attenuation in the dilution refrigerator before driving the
cCPT, which is mounted inside a magnetic shield at the mixing chamber stage of
the refrigerator. The reflected signal goes through a circulator to a
traveling wave parametric amplifier (TWPA) [28] which serves as the first
stage amplifier. The signal is then amplified by a Low Noise Factory LNF
LNC4$\textunderscore$8C high electron mobility transistor (HEMT) and a room
temperature low noise field effect transistor (FET). This signal is then mixed
down to an intermediate frequency of 21 MHz, filtered, digitally sampled, and
demodulated to extract the phase.

The TWPA has an average gain of 18 dB over the operating bandwidth of the
cCPT, ensuring that the added noise of the amplifier chain is dominated by the
noise added by the TWPA. The added noise density referred to the input of the
amplifier chain is separately measured to be $\sim$4.67 photons/Hz (noise
temperature of 1.28 K), close to the quantum limit of 1 photon/Hz [49] for the
phase insensitive TWPA.

##  Appendix B Charge sensing protocol initialization

Here, we elaborate upon the initialization section of the charge sensing
protocol illustrated in Fig. 3(a). In order to initialize the oscillator in
the high oscillation amplitude state, we start from a detuning in the
monostable region on the positive detuning side in Fig. 1(c-e), and ramp the
detuning by $f_{\mathrm{ramp}}/2\pi=-41$ MHz in a time $t_{r}=530$ ns. The
final detuning is close to a bifurcation edge, denoted by the black dashed
line in Fig. 2(b). The oscillator is driven with this constant tone for a time
$t_{\mathrm{stab}}=4.9\>\mu$s, during which time fluctuations could cause a
transition to the low oscillation amplitude state. These values for $t_{r}$
and $t_{\mathrm{stab}}$ were settled upon after performing QuTiP [64]
simulations using a master equation solver for the exact input tone in Fig.
3(a), and seeing the system through a transient evolution period to the steady
state. This value of $t_{\mathrm{stab}}$ is also close to the nominal value of
5/$(\kappa_{\mathrm{tot}}/2\pi)$ over which transients of oscillating systems
are expected to decay, even in the region where switching between high and low
oscillation amplitude states is observed, where the oscillator dynamics are
considerably slowed [40]. The detuning could then be ramped to a slightly
larger blue-detuning, $f_{\mathrm{latch}}$, to reduce the probability of a
switching event during the measurement time, hence ‘latching’ the oscillator
in the oscillation amplitude state attained at the end of the stabilization
time [23]. However, unlike the systems studied in Refs. [43, 23], the low
oscillation amplitude state is often not a long-lived state in our system,
making the latching a little less likely, and causing the switching statistics
to depend on the additional parameters $f_{\mathrm{latch}}$ and
$t_{\mathrm{acq}}$. We thus set $f_{\mathrm{latch}}=0$.

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