representation must qualify that photon counting is not as universally
accessible to signal analysis as the timeseries data from homodyne readout.

###  VII.4 Next Generation Gravitational Wave Detectors

For photon counting, the temporal basis must be orthonormal and must be chosen
apriori, before detection. Current technology also requires a basis of Fourier
modes passed through readout cavities, though future technology can feasibly
improve this restriction. All of this indicates major tradeoffs for matched
template searches if switching to photon counting measurement, despite
potential benefits in avoiding quantum noise.However, homodyne readout and
photon counting are not exclusive techniques and can be combined, when
temporal-mode basis not directed to a counting readout, they will instead land
on a standard readout. After the detection of an astronomical event, many
additional science tasks follow to analyze the waveform. Many of these tasks
can be framed as testing for excess residuals after subtracting the best-fit
waveform or otherwise testing for deviations from a “standard-model” waveform.
Thus, if the expected waveform can be established early during detection, then
all deviations exist in orthogonal templates. This poses photon counting as a
means to perform quantum hypothesis tests of excess residuals, with reduced
quantum noise, for gravitational wave analysis.Such potential applications for
this are: tests of general relativity; searching for deviations from numerical
relativity simulations; searching for weak, delayed gravitational wave
“echoes”; and testing for deviations between inspiral and merger portions of
waveforms. Binary neutron star signals have similar potential applications:
Searches for late-inspiral phase shifts in the signal from tidal damping in
the equation of state; as well as searches for high-frequency resonances after
the merger.Applying photon counting techniques to gravitational wave
interferometers will have profound experimental challenges, but further
development of sub-shot-noise measurement techniques and data analysis may
reveal new science possibilities. Future generations proposed gravitational
wave detectors further motivate this study for two reasons.First, they have a
larger separation between their quantum and classical noise contributions due
to the various scalings with the facility length [32, 33]. The gain-bandwidth
trade-offs[34] cause quantum noise to scale less favorably than the next most
dominant noises, primarily the coating thermal noise as well as the residual
gas noise of molecules traversing the vacuum beam pipes.Figure 6 shows the
spectra of a configuration of the proposed Cosmic Explorer observatory design.
The gap between the quantum and classical noise contribution can be orders of
magnitude in the power spectral density. The FIRSD heuristic expression then
indicates that photon counting – even without further quantum enhancement –
can outperform squeezing in the science cases where it is found to be
applicable. Furthermore, it can outperform narrow-band resonant tunings, where
the optical cavities are optimized for specific high-frequency science
objectives[35]. This is because the detector operates close to the limits of
anticipated internal losses [36].

Figure 6: The classical and quantum noise of Cosmic Explorer[37], a proposed
observatory design. Top shows the fundamental noises in a proposed upgraded
configuration using cryogenic silicon technologies with 3 mega-Watts of arm
power (per table IV of [38]). The proposed design includes 10dB of squeezing,
so the standard quantum limit noise contribution, without squeezing, is
included. The FIRSD spectrum is computed from Eq. 176. Bottom shows the
background flux from classical noises in both upper and lower sidebands using
Eq. 85. The noise curves are calculated using the scientific software pygwinc.

Second, next generation observatories will detect substantially more events
than the current generation [37, 39]. For the Cosmic Explorer design, $10^{5}$
binary neutron star coalescence events are predicted to be observed per year
of operation. More events potentially mitigates some of the lost information
of purely excess-residual searches, and indicates that the Fisher information
as an applicable figure of merit in aggregating information from many
events.The bottom of Fig. 6 shows the low count rate expected from the
fundamental classical noises. For each event detection, some number
independent waveform tests might be defined. For binary neutron star
postmerger physics, the number of independent tests can likely related to the
maximum resonance Q-factor of oscillation features and the band over which
they are expected to appear[40]. The noise flux rate times the number of
independent tests indicates the average number of background photons observed
per event. Until a photon is observed, upper bounds on the postmerger signal
power may be the primary statistical conclusion, but that thousands of events
may allow postmerger-waveform morphology to be accessible even for the
cosmologically-distant neutron star events comprising the majority of the
detections.It is worth reiterating the experimental challenges ahead for
photon counting, but also reiterate that the technique may open new potential
for future detectors. It shows promise for the observatories to even more
fully utilize the advantages of their long baseline designs as well as the
reduced classical noise from emerging physics instrumentation techniques [41].

###  VII.5 Quantum Enhancements

The classical Fisher Information and variance calculations for photon counting
(Section V) show that the information scales proportionally to optical gain,
as one would expect, yet inversely proportional to the noise background. An
implication is that for arbitrarily low-background instruments with nonzero
optical losses, the limits implied from [42] do not universally apply. As a
result, the conclusions of [22] either, do not apply to photon counting, or
must be interpreted differently.Squeezed states are the principle method of
incorporating a quantum enhancement into interferometers, to reduce quantum
noise below the shot-noise SQL in homodyne readout. Soon, advanced frequency-
dependent squeezed states will be utilized to also reduce measurement back-
action. The technique of squeezing works by continuously preparing states over
the measurement bandwidth using nonlinear optics and “injecting” them into the
interferometer from the output port. They then reflect from the Michelson
interferometer[28] towards the readout. Any signal modulations act as a
displacement operation on those states, during the interferometer
reflection.Squeezed states carry energy and carry pairs of photons, entangled
between upper and lower sideband frequencies [43, 44]. This likely limits the
utility of squeezed states for low-background photon counting, as losses
decohere those pairs, leading to background counts. Photon counting is a
nonlinear and non-Gaussian observation of the optical states in each temporal
mode. This implies that, analogously to squeezing, non-Gaussian states should
be prepared and injected into the dark port of the interferometer to enhance
photon counting readouts.Squeezed states are generated using relatively few
optical elements, yet span all of the temporal basis read in the homodyne
timeseries, changing the observed quantum contribution to the spectrum. Each
and every temporal mode in any basis of the observation band then contains a
squeezed state. Implementing this with non-Gaussian states to saturate a time-
bandwidth basis will require considerable investigation and development. Such
ideas are already being explored in microwave axion detectors [45], and may
have analogous developments for optical interferometers – if photon counting
can be proven to be experimentally viable in high power instruments. Many
experimental challenges of creating custom temporal mode quantum states
remains, but may reveal new prospects for precision optical interferometer
physics.

###  VII.6 Quantum Information

Given known experimental limits, a principle question to ask is how well
current observation and quantum enhancement techniques saturate provable
theoretical measurement limits. To that end, interferometers have been
extensively studied. When reading a timeseries, the optical gain and sensing
noise are related through the “energetic quantum limit[46]”, bounding the
achievable precision when using light as a probe for displacement from the
probe energy and its uncertainty. Extending these bounds and including the
measurement back-action333Measurement back-action, which can imply other
definitions for a “standard quantum limit”, is mentioned here for
completeness, but not analyzed in this work.  of “too precisely” probing the
location of finite mass oscillators, continuously, leads to the Quantum
Cramer-Rao Bound (QCRB)[42, 47]. Applied, this bound establishes fundamental
limits of phase-sensitive time-series measurement in lossy, realizable
detectors [48, 36, 49] with a realization that injecting Gaussian squeezed
states approach saturating channel bounds.The timeseries readout of
interferometers is extremely convenient for all manner of scientific tests, as
it linearly relates perturbations in the interferometer to modulations in the
readout power. Standard, intuitive signal analysis is then applied to
statistically define a “detection test” for: wide-bandwidth stochastic
backgrounds, coherent or incoherent narrow-bandwidth fields, or finite-
duration events. For events, a significant detection then leads to a different
statistical problem, inferring the most likely waveform given measured data.
Alternative to this waveform inference is the statistical test for significant
deviations or “features” between the measured waveform and a set of expected
event waveforms, either in a single instance or while aggregating many events,
depending on the scientific goals.While convenient, the limitations of the
timeseries readout has major implications [22]. In particular, the technique
of using squeezed light approaches saturation of quantum waveform estimation
bounds, implying that optical losses and decoherence establish the fundamental
precision limit for a given interferometer circulating power.Properly
comparing theoretical and experimental quantum-measurement limits in a well-
posed manner requires fully defining the statistical test necessary to achieve
a science result. A specific example of this is the established knowledge that
the timeseries readout and its associated QCRB does _not_ saturate more
general quantum bounds. There are two notable examples of this. The first
example is given by the Kennedy and Dolinar receivers[50, 51, 52, 53, 54],
developed for hypothesis testing to discriminate between sets of candidate
waveforms. These receivers are designed instead to saturate the Helstrom
Bound[19, 55] in detecting a weakly-displaced coherent signal, are applicable
to interferometer waveform estimation applications [56]. These _do not_ rely
on non-classical states of light yet achieve the 3db improvement over homodyne
readout from the Helstrom bound. Their statistical application generalizes as
continuous hypothesis testing[57]. The second example is the aforementioned
quantum spectrum analysis [18], which concludes that for the purposes of
estimating a weak, stochastic signal over any bandwidth, the homodyne
timeseries is inferior to using frequency-resolved photon counting. This work
generalizes that result to signal-recycled interferometers with generalized a
temporal-mode basis. In summary, optimizing for waveform estimation by
implementing by a timeseries readout, even with squeezed light, can limit the
output of instruments which either: search for incoherent signals, or frame
their science as discriminating between finite sets of candidate waveform
models carrying some total signal power.The aforementioned quantum bounds have
a common feature in that they are applied to detecting weak signals of some
power density, rather than the specific sensitivity limit of an
interferometer. This analysis does not include measurement back-action or
quantum enhancements, in which case the Michelson interferometer can be
considered a relatively simple transducer, with an associated optical gain,
that emits weak signals from new physics. This leads to, for interferometers,
the conclusion that the QCRB for waveform estimation[42] can be surpassed for
certain physics applications. If this conclusion is connected to additional
prospects for quantum enhancement using non-Gaussian states, then there is new
potential for a wider variety pf application-specific quantum information
bounds for interferometer signal science use-cases.

##  VIII Event Based Searches

Gravitational wave detection is a key application of modern high power
Michelson interferometers. Some astrophysical science signals are linear time
invariant stochastic backgrounds, but gravitational wave detections, so far,
are from astronomical events, where a signal of finite duration is emitted and
detected. The stochastic analysis developed in the earlier sections constructs
chi-square power observations and their analogous implementation using photon
counting. For event-based searches, this makes the formalism so far developed
more similar to incoherent “burst” searches.A work that extensively develops
and describes similar time-bandwidth arguments to relate excess-power burst
searches against matched-template is provided by Flanagan and Hughes (F&H)[58,
59]. In particular, the heuristic comparison for burst searches that appears
to apply is the relation eqn (2.15) of [58].

$\displaystyle\frac{\sigma_{\textsc{S/N},\text{band-
pass}}}{\sigma_{\textsc{S/N},\text{matched}}}$
$\displaystyle\approx\frac{1}{\sqrt{N_{\text{bins}}}}$ $\displaystyle
N_{\text{bins}}\approx 2\Delta T_{\text{bank}}\Delta F_{\text{bank}}$ (178)

The $(S/N)$ in [58] is analogous to the square-root of the Fisher information
or, equivalently, the inverse of the standard deviation of an unbiased
estimator. The factor $N_{\text{bins}}$ is, heuristically, the time and
bandwidth spanned by the bank of templates. This section develops comparisons
of photon counting with timeseries waveform inference, showing that the
benefits of low classical noise remain, and that the statistical penalty is
smaller than this heuristic expectation.

###  VIII.1 Template Search Analysis

Event searches are implemented by convolving444Computationally, it is
substantially more sophisticated than brute force convolution, using Fourier-
domain waveform templates over an optimized grid of candidates. a set of model
waveform templates over the homodyne timeseries while checking a search
statistic “score” against a threshold. Sufficient score then signifies a
significant detection. Over the set of candidate waveforms, this implements
matched-template search. The temporal-mode basis used in this work,
particularly for photon counting, is not compatible with convolution nor is it
compatible with search over continuous space of parameterized waveforms. This
is due to the ortho-normality property (cf. Eq. 39) of the temporal-mode
basis. Similarly, it cannot freely test a space of waveforms to perform
parameter estimation. This might suggest that photon counting is not promising
for event-based science; however, the event detection initiates follow-up
science on the waveform. Photon counting can potentially improve deep, sub-
shot noise tests for new features of waveforms.To define such tests, start
with the strain waveform broken into well-known and unknown components.

$\displaystyle h(t)$ $\displaystyle=h_{0}(t)+h_{1}(t|\bm{\beta})$ (179)

$h_{1}(t|\bm{\beta})$ represents the uncertain components of the waveform, and
the uncertainty is parameterized with a vector $\bm{\beta}$ that has some
apriori distribution $P(\bm{\beta})$. The dimensionality of the parameter
space is given by $N_{\text{param}}=\textsc{Dim}[\bm{\beta}]$.One can assume
that the majority of the signal energy for $h_{0}(t)$ is at times earlier than
$h_{1}(t)$ and typically at frequencies that are lower than $h_{1}(t)$, which
implies that $h_{0}(t)$ and $h_{1}(t|\bm{\beta})$ are highly orthogonal. The
measurement of $h_{0}$ is assumed to be performed using homodyne readout and
estimated with high SNR. This assumption doesn’t strongly affect formulas
below, but is stated to indicate that waveform inference on $h_{0}(t)$ is
needed to determine source properties and establish the prior distributions
$P(\bm{\beta})$ for the waveform models of $h_{1}(t|\bm{\beta})$.Assuming
either the raw interferometer response $g(\Omega)$, or the response
$\mathfrak{g}(\Omega)$ filtered through $\mathfrak{q}(\Omega)$, the field at
the output port of the interferometer is:

$\displaystyle\hat{d}(\Omega)$
$\displaystyle=i\delta(\Omega)\overline{l}+i\mathfrak{g}(\Omega)Lh(\Omega)$
$\displaystyle\hskip
20.00003pt+\mathfrak{q}(\Omega)\big{(}i\check{\theta}(\Omega)+\check{n}(\Omega)\big{)}+\hat{a}(\Omega)$
(180)

Where $L$ is the length of the interferometer arms that converts from
gravitational metric strain to displacement.To develop the timeseries search
statistic, first define the unit-normalized template, weighted by the
instrument response.

$\displaystyle\overline{h}_{1}(\Omega|\bm{\beta})$
$\displaystyle\equiv\frac{\overline{\mathfrak{g}}(\Omega)h_{1}(\Omega|\bm{\beta})}{\sqrt{\int_{-\infty}^{\infty}|\overline{\mathfrak{g}}(\Omega)h_{1}(\Omega|\bm{\beta})|^{2}\frac{\mathrm{d}\Omega}{2\pi}}}$
(181)

####  VIII.1.1 Homodyne Detections

First, we will derive known statistics for matched-template search using the
framework and notation of this work.To simplify, we will assume that the
classical noise and optical gain is not varying with frequency
$\mathfrak{q}(\Omega)S_{\check{\theta}}(\Omega)\rightarrow\overline{S}_{\check{\theta}}$,
and $\overline{\mathfrak{g}}(\Omega)\rightarrow\overline{\mathfrak{g}}$. This
then allows us to create a measurement operator, power operator, and then
unbiased statistic on the single template
$\kappa_{\bm{\beta}}(\Omega)=\overline{h}_{1}(\Omega|\bm{\beta})$. For this,
the statistic is built on the underlying Gaussian measurement, normalized by
the homodyne measurement gain $\overline{l}$, without creating the power
measure chi-square of Section IV.1.

$\displaystyle\braket{\hat{D}_{\bm{\beta}}/\overline{l}}$
$\displaystyle=2|\overline{\mathfrak{g}}|LH(\bm{\beta})$
$\displaystyle\textsc{Var}[\hat{D}_{\bm{\beta}}/\overline{l}]$
$\displaystyle=2\overline{S}_{\check{\theta}}+1$ (182)

These use $H(\bm{\beta})$, for the RMS strain energy of the
$h_{1}(\Omega|\bm{\beta})$ waveform, expressed:

$\displaystyle H^{2}(\bm{\beta})$
$\displaystyle=\int_{-\infty}^{\infty}\left|h_{1}(\Omega|\bm{\beta})\right|^{2}\frac{\mathrm{d}\Omega}{2\pi}$
(183)

Rescaling and shifting the measurement creates the unbiased estimators.

$\displaystyle\hat{\mathcal{N}}_{\bm{\beta}}$
$\displaystyle=\frac{\hat{D}_{\bm{\beta}}}{2|\overline{\mathfrak{g}}|LH(\bm{\beta})\overline{l}}$
$\displaystyle\textsc{Var}[\hat{\mathcal{N}}_{\bm{\beta}}]$
$\displaystyle=\frac{2\overline{S}_{\check{\theta}}+1}{\left(2|\overline{\mathfrak{g}}|LH(\bm{\beta})\right)^{2}}$
(184)

using Eq. 83 and Eq. 84 to express the strain power-spectral-density in terms
of the given noise density operators

$\displaystyle L^{2}S_{h}(\Omega)$
$\displaystyle=\textsc{Psd}[\text{Qu}(\Omega)]+\textsc{Psd}[\text{Cl}(\Omega)]$
(185)

this then gives a relation between the variance and a score threshold

$\displaystyle\overline{\rho}_{\hat{D}}^{2}$
$\displaystyle\equiv\frac{1}{\textsc{Var}[\hat{\mathcal{N}}_{\bm{\beta}}]}=\frac{2H^{2}(\bm{\beta})}{\overline{S}_{h}}=\frac{2L^{2}H^{2}(\bm{\beta})}{\textsc{Psd}[\text{Qu}]+\textsc{Psd}[\text{Cl}]}$
(186)

Which agrees with the exact, optimally-weighted calculation from eqn 1.1 (or
2.7) of F&H[58].

$\displaystyle\rho_{\hat{D}}^{2}$ $\displaystyle\equiv
4\int_{0}^{\infty}\frac{|h_{1}(\Omega)|^{2}}{S_{h}(\Omega)}\frac{\mathrm{d}\Omega}{2\pi}$
(187)
$\displaystyle=2\int_{-\infty}^{\infty}\frac{2|\overline{g}(\Omega)|^{2}|Lh_{1}(\Omega)|^{2}}{(1+2S_{\check{\theta}}(\Omega))}\frac{\mathrm{d}\Omega}{2\pi}\approx\overline{\rho}_{\hat{D}}^{2}$
(188)

The difference is that the optimal template weighting moves the strain
spectral density within the integral. The necessary weighting factors and
their optimization were not included in this derivation for simplicity.The
search score $\rho^{2}$ indicates the expected significance for a given signal
power. For a matched template search, there is some spacing of $\bm{\beta}$
over which the waveforms are orthogonal, and thus there is some total number,
$N_{\text{shapes}}$ of independent tests of the search space (cf. 2.8 of
[58]). Additionally, there is some probability threshold, $\epsilon$ that is
used to indicate a significant detection or event.From eqn 2.8 of F&H[58], the
score threshold follows the relation

$\displaystyle\text{erfc}\left(\frac{\rho_{\hat{D}}}{\sqrt{2}}\right)$
$\displaystyle=\frac{\epsilon}{N_{\text{shapes}}}$ (189)

Leading to eqn 2.9 of F&H[58]

$\displaystyle\overline{\rho}_{\hat{D}}$
$\displaystyle\approx\sqrt{2\ln(N_{\text{shapes}}/\epsilon)}$ (190)

The number of shapes is also related to the number of “sufficiently
independent” templates in the computational search space, whereby the
statistical penalty between $\ln(N_{\text{shapes}})$ and signal power loss
from under-sampling are made similar.For sub-shot noise detections, we assume
that many events will need to be “stacked” or evaluated in a aggregate
statistic. In some cases, it may be possible to re-scale and transform the
timeseries across many events, such that waveforms can be coherently added,
realizing the benefit of [60]. In that case, the average-event search
threshold becomes

$\displaystyle\text{erfc}\left(\rho_{\hat{D}}\sqrt{\frac{N_{\text{events}}}{2}}\right)$
$\displaystyle=\frac{\epsilon}{N_{\text{shapes}}}$ (191)

This will improve the average threshold strain power by
$\sqrt{N_{\text{events}}}$. Given that inspirals have many parameters, it is
perhaps more reasonable to assume that only incoherent stacking may be
possible for certain tests. these will have the scaling

$\displaystyle\text{erfc}\left(\rho_{\hat{D}}^{2}\sqrt{\frac{N_{\text{events}}}{2}}\right)$
$\displaystyle=\frac{\epsilon}{N_{\text{shapes}}}$ (192)

which leads to the detection threshold for the signal power of

$\displaystyle H^{2}_{\hat{D}}$
$\displaystyle\approx\overline{S}_{h}\sqrt{\frac{\ln(N_{\text{shapes}}/\epsilon)}{2N_{\text{events}}}}$
$\displaystyle\overline{S}_{h}\equiv\frac{2\overline{S}_{\check{\theta}}+1}{2|\overline{\mathfrak{g}}|^{2}L^{2}}$
(193)

Before analyzing photon counting, it is worth noting the general scale of the
number of shapes. From Eqn. 7.5 and eqn 7.12 F&H[59] The number of independent
waveforms scales can be expected to scale volume of the parameter space. In
which case

$\displaystyle\ln(N_{\text{shapes}})\propto N_{\text{params}}$ (194)

and thus ultimately

$\displaystyle H^{2}_{\hat{D}}$
$\displaystyle\propto\overline{S}_{h}\sqrt{\frac{N_{\text{params}}}{N_{\text{events}}}}$
(195)

The crossover for when matched template search and excess-power burst search
can then be considered as when $N_{\text{params}}\sim N_{\text{bins}}$, under
the amended heuristic

$\displaystyle\frac{\textsc{S/N}_{\text{band-
pass}}}{\textsc{S/N}_{\text{matched}}}$
$\displaystyle\propto\sqrt{\frac{N_{\text{params}}}{N_{\text{bins}}}}$ (196)

####  VIII.1.2 Photon Counting Detections

As mentioned, photon counting intrinsically appears more like a burst search,
as it is forced to use excess-power measurements. This deleterious assumption
is only partially true. Avoiding it requires us to assume that measuring the
photon power in individual template measurements becomes possible. I.e. the
ability to individually measure $\hat{E}_{i}$ rather than the summation
$\hat{E}$. In the case of quantum spectrum analysis, individual line filters
using cavities or gratings could achieve this [18]. For more general
templates, or templates with optimized spectral leakage properties, this means
using emerging technology such as quantum memories.To start on the statistical
analysis, we first must establish the required number of templates. The total
number of templates, $N_{\text{bins}}$ is more exactly determined from the Von
Neuman Entropy over a density matrix or phase-space representation,
$K(t,t^{\prime})$, of the templates, averaged over the parameter space. This
representation of the waveform uncertainty has the construction.

$\displaystyle K_{h_{1}}(t,t^{\prime})$
$\displaystyle=\int_{\bm{\beta}}\overline{h}_{1}(t|\bm{\beta})\overline{h}_{1}(t^{\prime}|\bm{\beta})P(\bm{\beta})\mathrm{d}\bm{\beta}$
(197)

The number of bins is related to the information entropy on this phase-space
density-matrix of the waveforms.

$\displaystyle N_{\text{bins}}$ $\displaystyle=e^{S_{K}},$
$\displaystyle{S}_{K}$
$\displaystyle=-\int_{-\infty}^{\infty}K_{h_{1}}(t,t)\ln\big{(}K_{h_{1}}(t,t)\big{)}\mathrm{d}t$
(198)

Where the number of bins also approximately corresponds to the time-bandwidth
product spanned by the set of templates. This number of bins establishes the
number of ortho-normal templates that will be needed to fully capture the
event signal power for any given event.Now assume a template basis set
$\kappa_{i}(t)\in\mathcal{K}$ that are unit-normalized and ortho-normal. In
the event that the local oscillator light is disabled or filtered away by
$\mathfrak{q}(\Omega)$, emerging technology can template-match to the optical
field, implementing the operators

$\displaystyle\hat{d}_{i}$
$\displaystyle\equiv\int_{-\infty}^{\infty}\hat{d}(t)\kappa_{i}(t)\mathrm{d}t\equiv\int_{-\infty}^{\infty}\hat{d}(\Omega)\kappa_{i}(-\Omega)\mathrm{d}\Omega$
(199)

A number operator can be built from these field operators, to measure how many
photons have accumulated in each template.

$\displaystyle\hat{E}_{i}$
$\displaystyle\equiv\hat{d}_{i}^{\dagger}\hat{d}_{i}$ (200)

With the intent that each of these power measurements can be individually
addressed during observation. Using an array of quantum memories, each with an
orthogonal, modulated coupling rate then implements the template set
$\mathcal{K}$. Each one can then be projectively-measured to determine the
occupation.Using the set of measurements of $\hat{E}_{i}$, an unbiased search
statistic can be build using a weighted combination of them.

$\displaystyle\hat{G}(\bm{\beta})$
$\displaystyle\equiv\frac{1}{\overline{G}(\bm{\beta})}\sum_{i=0}^{N_{\text{bins}}}(\hat{E}_{i}-\overline{S}_{i})W_{i}(\bm{\beta})$
(201) $\displaystyle\overline{G}(\bm{\beta})$
$\displaystyle\equiv\sum_{i=0}^{N_{\text{bins}}}\braket{(\hat{E}_{i}-\overline{S}_{i})W_{i}(\bm{\beta})}$
(202)

A good choice of weights, determined in a manner similar to Section A.3, is
based on the relative occupation of the templates by the true waveform.

$\displaystyle W_{i}(\bm{\beta})$
$\displaystyle=\left|\int_{-\infty}^{\infty}\overline{h}_{1}(\Omega|\bm{\beta})\kappa_{i}(-\Omega)\frac{\mathrm{d}\Omega}{2\pi}\right|^{2}$
(203)

This choice leads to the properties

$\displaystyle\sum_{i=0}^{N_{\text{bins}}}W_{i}(\bm{\beta})$ $\displaystyle=1$
$\displaystyle\sum_{i=0}^{N_{\text{bins}}}W_{i}^{2}(\bm{\beta})$
$\displaystyle\equiv\frac{1}{N_{\text{split}}(\bm{\beta})}$ (204)

This $N_{\text{split}}(\bm{\beta})$ is a measure of how many of the templates
are covered by the waveform given by the parameter $\bm{\beta}$. For good
choices of templates optimized for particularly high-probability parameters
$N_{\text{split}}\sim 1$. For low probability parameters, or poorly optimized
parameters $N_{\text{split}}\sim N_{\text{bins}}$.The calculations on the
normalization factors enter the computation for the gain-normalization factor,
$\overline{G}$, giving

$\displaystyle\overline{G}(\bm{\beta})$
$\displaystyle=\frac{\int_{-\infty}^{\infty}|\mathfrak{g}(\Omega)Lh_{1}(\Omega|\bm{\beta})|^{2}\frac{\mathrm{d}\Omega}{2\pi}}{N_{\text{split}}(\bm{\beta})}$
(205)

Now, for simplicity, we make the assumption that the classical noises are
constant $\overline{S}_{i}\rightarrow\overline{S}$ and that the optical gain
is constant $\mathfrak{g}(\Omega)\rightarrow\overline{\mathfrak{g}}$. The
variance for the unbiased estimator, $\braket{\hat{G}(\bm{\beta})}=1$, is then

$\displaystyle\textsc{Var}[\hat{G}(\bm{\beta})]$
$\displaystyle=\frac{N_{\text{split}}(\bm{\beta})\overline{S}}{\big{(}|\overline{\mathfrak{g}}|^{2}L^{2}H^{2}(\bm{\beta})\big{)}^{2}}$
(206)

over the whole parameter space, the numerical efficiency factor can be
averaged

$\displaystyle\overline{N}_{\text{split}}$
$\displaystyle=\int_{\bm{\beta}}N_{\text{split}}(\bm{\beta})P(\bm{\beta})\mathrm{d}\bm{\beta}$
(207)

so the average penalty factor should be based on $\overline{N}_{\text{split}}$
rather than $N_{\text{bins}}$.One can expect that $N_{\text{split}}$ depends
on the tiling of ortho-normal templates within the waveform parameter space,
giving that, on average, one might assume $N_{\text{split}}\sim
2^{N_{\text{param}}}$ from a lattice geometry. More efficient “sphere-packing”
of the templates within the waveform space might improve this bound, relating
the statistical efficiency of photon counting to the Hamming Bound for
waveform discrimination.With the statistic constructed, we assume that many
events are detected and incoherently stacked. This is the assumption behind
Fisher Information arguments, and it allows us to assume a Gaussian error
distribution in the limit of many events.

$\displaystyle\overline{\rho}_{G}^{2}$
$\displaystyle\equiv\frac{1}{\textsc{Var}[\hat{G}(\bm{\beta})]}\approx\frac{\big{(}L^{2}H^{2}(\bm{\beta})\big{)}^{2}}{2\overline{N}_{\text{split}}\textsc{Psd}[\text{Cl}(\Omega)]\textsc{Psd}[\text{Qu}(\Omega)]}$
(208)

After many measurements, the score threshold relates to the statistical
threshold by,

$\displaystyle\text{erfc}\left(\rho_{G}\sqrt{\frac{N_{\text{events}}}{2}}\right)$
$\displaystyle=\frac{\epsilon}{N_{\text{bins}}}$ (209)

which leads to the detection threshold on the signal energy of

$\displaystyle H^{2}_{G}$
$\displaystyle\approx\overline{S}^{\prime}_{h}\sqrt{\frac{2\overline{N}_{\text{split}}\ln(N_{\text{bins}}/\epsilon)}{N_{\text{events}}}}$
$\displaystyle\overline{S}^{\prime}_{h}\equiv\frac{\sqrt{\overline{S}_{\check{\theta}}+\overline{S}_{\check{n}}}}{|\overline{\mathfrak{g}}|^{2}L^{2}}$
(210)

####  VIII.1.3 Ratio Relation

With the detection threshold estimates above, we can now estimate a modified
heuristic for the FIRST of Eq. 176 to evaluate the scaling with parameter and
time-bandwidth bin dimensionality.

$\displaystyle\overline{S}^{\prime}_{h}$
$\displaystyle\approx\frac{\sqrt{2\textsc{Psd}[\text{Qu}]\textsc{Psd}[\text{Cl}]}}{L^{2}}$
(211) $\displaystyle\overline{S}_{h}$
$\displaystyle\approx\frac{\textsc{Psd}[\text{Qu}]+\textsc{Psd}[\text{Cl}]}{L^{2}}$
(212)

For a system that is primarily dominated by quantum noise, this gives

$\displaystyle
H^{2}_{G}\propto\sqrt{8\frac{\overline{N}_{\text{split}}}{\overline{N}_{\text{params}}}\frac{\textsc{Psd}[\text{Cl}]}{\textsc{Psd}[\text{Qu}]}}H^{2}_{\hat{D}}$
(213)

With a constant of proportionality that will depend on the specifics of the
template search and will require further study. The key point of this analysis
is to show that there is some level of classical noise at which photon
counting over multiple events is always superior over a finite set of
templates. That level does depend on information arguments related to the size
of the search parameter space, but that the new definition of
$\overline{N}_{\text{split}}$ allows us to analyze the continuum between fully
continuous “waveform inference” and fully “discrete” hypothesis testing. In
actuality, the difference in performance depends critically on the relative
dimensionality and distribution of the waveforms between the dimensionality
relations
$1\leq\overline{N}_{\text{params}}\leq\overline{N}_{\text{split}}\leq
N_{\text{bins}}$.

##  IX Conclusions

This work methodically describes the measurement process and readout of a
Michelson interferometer, starting with a description of the measurement
process in Section III. The standard homodyne readout, typically implemented
using the interferometer destructive fringe light, provides a continuous
timeseries. This gives a linear, classical recording of the passing signals.
At the quantum sensitivity limit of the interferometer, there is a background
noise on the timeseries, from the shotnoise of independently arriving photons
at the detector. In this case, the majority of the photons are provided by the
local oscillator field, rather than the signal itself. The derivation of
Section IV shows how the local oscillator provides a static field for the
signal field modulations to with. Calculating an overlap integral with any
single temporal mode, the quantum noise statistics will be Gaussian, where any
signal in the chosen temporal mode will offset the mean value. Squaring the
overlap integral then constructs a chi-square test which can indicate whether
a signal is present. By Fisher-information, this chi-square test is as
efficient as any other test for the presence of a signal, when using the
timeseries data.In short, searching for signals in a timeseries uses the
sequence: define a temporal-mode, perform an overlap-integral, then square it
to test for signal power in excess of shot-noise power. This led to an
analogous derivation for photon counting, whereby temporal modes are defined,
and the overlap integral is performed as part of the derivation, rather than
as a calculation. The act of squaring is then performed using single-photon
detectors, which directly observe signal power. The conclusion of this
derivation is that now only the signal and background classical noises
introduce shot-noise, so the Fisher Information can be arbitrarily high, and
the associated measurement variance low.This is a curious conclusion, as weak
signals may only rarely result in observed photon emissions, and thus limit
the information available. This is true and reflects that quantum limits are
still at play, but the so called “standard quantum limit” from “vacuum
fluctuations” are not a necessary limit to Michelson interferometers. When
searching for weak stochastic signals from new physics, the search
acceleration of this new technique can be dramatic, when interferometers are
designed to minimize their classical noise contributions to be well-below the
shot noise. For such searches, the derivation provided used the temporal-mode
basis only as a formal tool to build the search statistics. Physically, the
temporal-modes can be selected using optical cavities, and this is an
immediate prospect for implementing photon counting searches for
interferometers.Cavities have many drawbacks, physically and statistically,
and in time may be surpassed by a physical implementation of matched-template-
search, directly in the optical domain with high-contrast and low spectral-
leakage. This suggests interesting new experimental avenues using quantum
memory devices for fundamental physics measurements.After deriving the
statistics and benefits of photon counting, Section VI considered wider
classes of Michelson interferometers - those with signal recycling cavity
enhancements. There, the measurement bandwidth is traded for optical gain.
Both wideband and narrow band searches were considered. For wide-band
searches, where the signal has a bandwidth or coherence-time comparable to
light-crossing time of the interferometer, signal recycling does not provide
substantial benefit and photon counting maintains its advantages. A special
case for this statement is when the filter cavities used to implement photon
counting pass less bandwidth than the signal provides. In this case, signal
recycling can added to recover the full statistical benefit of photon counting
over homodyne readout. For narrow-band signals, particularly those with signal
power at an unknown frequency - such as in searches for dark-matter, photon
counting is beneficial only when the interferometer has extremely low
classical noise, such that even the resonantly-enhanced classical noise is
below the shot noise limit.The formal sections outline specific areas where
photon counting can benefit interferometer searches. The outlook, Section VII,
then speculates on further application areas. In particular, photon counting
shows that known limits to squeezing from losses do not imply as universal
limits to quantum measurement in interferometers, or at least must be adapted
or reinterpreted for photon counting. For applications that can be posed as
binary or N-ary tests, rather than continuous waveform inference, a new
statistical measure, the “Fisher-Information representative spectral density”,
is introduced to used to reason about potential future benefits of photon
counting for domain-specific applications. Applications, such as
gravitational-wave astronomy and instrumentation, that typically use power-
spectral density as the figure of merit can then project more complex
statistical measures, under some bold assumptions that eventual technology can
provide similar spectral resolution as current classical timeseries
computations.In total, the technical derivations in this work bridge recent
development in quantum measurement theory to physics applications best suited
for optical interferometers. This motivates the substantial work ahead to
demonstrate the experimental feasibility to realize “Counting Quantum Limited”
rather than “Standard Quantum Limited” performance.AcknowledgementsLM
gratefully acknowledges helpful comments and discussions from Chris Stoughton,
Jocelyn Read, Rana Adhikari, and Yanbei Chen. This research towards the GQuEST
experiment is supported by the Heising-Simons Foundation through grant
2022-3341.

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##  Appendix A Derivations and Notes

This section carries snippets of technical qualifications and derivations that
are useful but combersome to include inline in the main text

###  A.1 Band Limited Power Observations

The form Eq. 51 is ill-defined due to the singular nature of the quantum
density operators $\hat{a}(t)$ contained in $\hat{d}(t)$. More realistically,
it requires the convolution with some kernel $K(t)$ that encapsulates the
finite bandwidth of the photo detector opto-electronic properties.

$\displaystyle\hat{P}_{d}(t)=\int_{\infty}^{\infty}\hat{d}^{\dagger}(t)\hat{d}(t-t^{\prime})K(t^{\prime})\mathrm{d}t^{\prime}$
(214)

The power operator in the frequency domain for the output port of the
interferometer is then expressed as

$\displaystyle\hat{P}_{d}(\Omega)=K(\Omega)\Big{(}\hat{d}^{\dagger}(\omega+\Omega)\circledast\hat{d}(\omega+\Omega)\Big{)}$
(215)

The $K(t)$ is introduced here for correctness, but is cumbersome to carry. The
main text assumes the limit $K(t^{\prime})\rightarrow\delta(t^{\prime})$,
$K(\Omega)\rightarrow 1$.

###  A.2 Homodyne Field Observable Gaussian Characteristic Function, Equation
63

This is a short derivation of the Characteristic function of a Gaussian using
the Homodyne observable, related to Eq. 63.

$\displaystyle\textsc{Cf}\left[\tilde{d},\hat{D}_{ci}\right]$
$\displaystyle=\Braket{e^{-i\hat{D}_{ci}\tilde{d}}}=\Braket{e^{-i(2\overline{g}\check{\varphi}_{ci}+2\check{\theta}_{ci}+i\hat{a}_{ci}-i\hat{a}_{ci}^{\dagger})\overline{l}\tilde{d}}}$
(216)
$\displaystyle=e^{i\overline{l}\tilde{d}(2\overline{g}\check{\varphi}_{ci}+2\check{\theta}_{ci})}\Braket{e^{\overline{l}\tilde{d}\hat{a}_{ci}^{\dagger}-\overline{l}\tilde{d}\hat{a}_{ci}}}=e^{i\overline{l}\tilde{d}(2\overline{g}\check{\varphi}_{ci}+2\check{\theta}_{ci})}\Braket{e^{(\overline{l}\tilde{d})^{2}\frac{[\hat{a}_{ci}^{\dagger},\hat{a}_{ci}]}{2}}e^{\overline{l}\tilde{d}\hat{a}_{ci}^{\dagger}}e^{-\overline{l}\tilde{d}\hat{a}_{ci}}}$
(217)
$\displaystyle=e^{i\overline{l}\tilde{d}(2\overline{g}\check{\varphi}_{ci}+2\check{\theta}_{ci})-\frac{(\overline{l}\tilde{d})^{2}}{2}}$
(218)

Where the lines Eq. 217 utilizes the Campbell-Baker-Hausdorff formula to
normally-order the raising and lowering operators.

###  A.3 Full Variance of Homodyne Chi-Square

This section utilizes the weighting parameters of Eq. 72 to create the optimal
search statistic. The weighting is an ad-hoc factor that is included in a
manner that preserves the unbiased mean of the estimator. It represents taking
a weighted average when aggregating many independent chi-square
measurements.We start from Eq. 72 and write the full variance for the weighted
expression.

$\displaystyle\hat{\chi}^{2}_{2N}$
$\displaystyle\equiv\frac{1}{(\sum_{i=0}^{N}W_{i})}\sum_{i=0}^{N}\frac{W_{i}}{2|\overline{g}(\Omega_{i})|^{2}\Phi(\Omega_{i})}\left(\hat{V}_{+i}-\braket{\hat{V}_{+i}}\Big{|}_{S_{\varphi}=0}\right)$
(219) $\displaystyle\textsc{Var}[\hat{\chi}^{2}_{2N}]$
$\displaystyle\approx\frac{1}{(\sum_{i=0}^{N}W_{i})^{2}}\sum_{i=1}^{N}\left(\frac{W_{i}}{2|\overline{g}(\Omega_{i})|^{2}\Phi(\Omega_{i})}\right)^{2}\left(2|\overline{g}(\Omega)|^{2}S_{\check{\varphi}}(\Omega_{i})+2S_{\check{\theta}}(\Omega_{i})+1\right)$
(220)

The derivative of the variance with respect to $W_{i}$ shows that the variance
is minimized, derivative zero, when

$\displaystyle W_{i}$
$\displaystyle=\frac{\left(2|\overline{g}(\Omega_{i})|^{2}\Phi(\Omega_{i})\right)^{2}}{2|\overline{g}(\Omega)|^{2}S_{\check{\varphi}}(\Omega_{i})+2S_{\check{\theta}}(\Omega_{i})+1}\approx\frac{\left(2|\overline{g}(\Omega_{i})|^{2}\Phi(\Omega_{i})\right)^{2}}{2S_{\check{\theta}}(\Omega_{i})+1}$
(221)

Where the approximation is in the weak signal limit, but includes classical
noise contributions. Plugging in this optimal weighting recovers the usual
formulas for optimal stochastic searches such as eqn. 7.293 of Creighton and
Anderson [61]. The summation can also be converted, using Eq. 50, into an
integral form.

$\displaystyle\frac{1}{\textsc{Var}[\hat{\chi}^{2}_{2N}]}$
$\displaystyle\approx\sum_{i=1}^{N}\frac{\left(2|\overline{g}(\Omega_{i})|^{2}\Phi(\Omega_{i})\right)^{2}}{2S_{\check{\theta}}(\Omega_{i})+1}\approx\Delta
T\int_{0}^{\infty}\frac{\left(2|\overline{g}(\Omega)|^{2}\Phi(\Omega)\right)^{2}}{2S_{\check{\theta}}(\Omega)+1}\frac{\mathrm{d}\Omega}{2\pi}\approx\Delta
T\int_{0}^{\infty}\mathcal{I}[{\hat{D}_{+i}}|\alpha]\frac{\mathrm{d}\Omega_{+i}}{2\pi}$
(222)

The last integral represents a summation over all possible orthogonal
templates within the detection time $\Delta T$. The integral forms indicate
that the optimal weighting saturates the Fisher information over the whole
frequency band.

###  A.4 Properties of Gaussian distributions

When first introduced, the signal and noise processes were described as random
noise processes with two-point correlation functions of Eq. 33 that include
delta-function factors. This makes them singular, and difficult to reason
about their distribution. Once those signals are integrated with templates,
they become substantially better behaved and higher-order correlation
functions and moments can be defined.First, the characteristic function for
the Gaussian processes is given by:

$\displaystyle\braket{e^{i\tilde{e}\Re\\!\\{\check{\varphi}_{\pm i}\\}}}$
$\displaystyle=e^{-S_{\varphi}(\Omega_{i})\tilde{e}^{2}/4}$
$\displaystyle\braket{e^{i\tilde{e}\Im\\!\\{\check{\varphi}_{\pm i}\\}}}$
$\displaystyle=e^{-S_{\varphi}(\Omega_{i})\tilde{e}^{2}/4}$ (223)

Another useful higher order moment is the characteristic function of the
square of an expectation value. Those are computed using the Gaussian PDF over
the variable $\varphi_{R}$

$\displaystyle\braket{e^{i\tilde{e}\Re\\!\\{\check{\varphi}_{\pm i}\\}^{2}}}$
$\displaystyle=\sqrt{\frac{2}{\pi
S_{\varphi}}}\int_{-\infty}^{\infty}e^{ie\varphi_{R}^{2}}e^{-\frac{2\varphi_{R}^{2}}{S_{\varphi}}}\mathrm{d}\varphi_{R}=\sqrt{\frac{2}{\pi
S_{\varphi}}}\int_{-\infty}^{\infty}e^{-\varphi_{R}^{2}(\frac{2}{S_{\varphi}}+ie)}\mathrm{d}\varphi_{R}=\sqrt{\frac{2}{2+ieS_{\varphi}}}$
(224)

These characteristic functions are equivalent to those of a scaled chi-square
distribution.

###  A.5 Single-mode Energy Characteristic Function, Equation 117

This section derives the number operator for a Michelson interferometer in the
Heisenberg picture.This derivation starts from Eq. 103 or Eq. 105 and will
result in the expressions used for Eq. 117. First expressions for
$\hat{E}_{\pm i}$ are factorized with the classical-fluctuation operator
$\check{Y}_{\pm i}$

$\displaystyle\hat{E}_{\pm i}$ $\displaystyle=\big{(}\check{Y}_{\pm
i}-i\hat{a}_{\pm i}\big{)}^{\dagger}\big{(}\check{Y}_{\pm i}-i\hat{a}_{\pm
i}\big{)}$ $\displaystyle\check{Y}_{\pm i}$
$\displaystyle=\mathfrak{g}(\Omega_{\pm i})\check{\varphi}_{\pm
i}+q(\Omega_{\pm i})\check{n}_{\pm i}$ (225)

Similarly to the homodyne observation derivation of Section A.2, the
expectations can be reduced have applying a normal ordering to the
expressions. For this case, the Campbell-Baker-Hausdorff formula is not
sufficient, and a generalized normal ordering relation [62] is used:

$\displaystyle e^{X\hat{a}_{\pm i}^{\dagger}\hat{a}_{\pm i}}$
$\displaystyle=:e^{\hat{a}_{\pm i}^{\dagger}\hat{a}_{\pm i}(e^{X}-1)}:$ (226)

Here, the $::$ symbols indicate that the operators of the expression are
normally-ordered.To calculate the PMF, the normal ordering is immediately
applied to eliminate the quantum variables. The PDF of the classical signal
and noise is then used to calculate the expectations from the factor
$\overline{S}_{\pm i}$ of Eq. 106. The derivation can be simplified by
applying LABEL:eq:ChiSq_character_by_Gaussian at LABEL:eq:CF_Geom_mid to skip
to the end.

$\displaystyle\textsc{Cf}\left[\tilde{e},\hat{E}_{+i}\right]$
$\displaystyle\equiv\Braket{e^{i\hat{E}\tilde{e}}}=\braket{e^{i\tilde{e}(\check{Y}_{\pm
i}-i\hat{a}_{\pm i})^{\dagger}(\check{Y}_{\pm i}-i\hat{a}_{\pm
i})}}=\braket{:e^{(\check{Y}_{\pm i}-i\hat{a}_{\pm
i})^{\dagger}(\check{Y}_{\pm i}-i\hat{a}_{\pm
i})(e^{i\tilde{e}}-1)}:}=\braket{e{\check{Y}_{\pm
i}}{{}^{2}(e^{i\tilde{e}}-1)}}{}{eq:CF_{G}eom_{m}id}$ (227)
$\displaystyle=\frac{1}{\pi\overline{S}_{\pm
i}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(\varphi_{R}^{2}+\varphi_{I}^{2})(e^{i\tilde{e}}-1)}e^{-\frac{\varphi_{R}^{2}+\varphi_{I}^{2}}{\overline{S}_{\pm
i}}}\mathrm{d}\varphi_{R}\mathrm{d}\varphi_{I}=\frac{1}{\overline{S}_{\pm
i}}\int_{0}^{\infty}e^{-\varphi(e^{i\tilde{e}}-1+\frac{1}{\overline{S}_{\pm
i}})}\varphi\mathrm{d}\varphi$ (228) $\displaystyle=\frac{1}{\overline{S}_{\pm
i}(e^{i\tilde{e}}-1)+1}$ (229)

Which is the characteristic function of a Geometric distribution with a
Bernoulli trial probability $p=1/(1+\overline{S}_{\pm i})$. The Geometric
distribution implements Bose statistics on a single temporal mode of the
light, with a thermal occupation. This implies a mean occupation number and
resulting temperature:

$\displaystyle\braket{\hat{E}_{\pm i}}=(1-p)/p=\overline{S}_{\pm
i}=e^{\frac{\hbar\omega}{k_{\text{B}}T_{\omega}}}=e^{\frac{\hbar\Omega_{+i}}{k_{\text{B}}T_{\Omega}}}$
(230)

The two definitions of temperature are for the absolute temperature,
$T_{\omega}$, of a system that would create photons at frequency $\omega$ vs.
the sideband temperature, $T_{\Omega}$, for creating acoustic modulations.

###  A.6 Single-mode Energy Probability Mass, Equation 118

This derivation extends the previous one to generate the probability mass
function for photon counting a single signal mode Eq. 118. It starts form of
the characteristic equation which has not been entirely simplified

$\displaystyle\textsc{Pdf}\left[E,\hat{E}_{+i}\right]$
$\displaystyle=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{-iE\tilde{e}}\braket{e{\check{Y}_{\pm
i}}{{}^{2}(e^{i\tilde{e}}-1)}}{\mathrm{d}}{\tilde{e}}=\frac{1}{2\pi}\Braket{e^{-\check{Y}_{\pm
i}^{2}}\int_{-\pi}^{\pi}\sum_{n=0}^{\infty}\frac{\check{Y}_{\pm
i}^{2n}}{n!}e^{-iE\tilde{e}}e^{in\tilde{e}}\mathrm{d}\tilde{e}}=\Braket{e^{-\check{Y}_{\pm
i}^{2}}\frac{\check{Y}_{\pm i}^{2E}}{E!}}$ (231)
$\displaystyle=\frac{1}{E!}\frac{1}{\pi\overline{S}_{\pm
i}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}(\varphi_{R}^{2}+\varphi_{I}^{2})^{E}e^{-\varphi_{R}^{2}-\varphi_{I}^{2}}e^{-\frac{\varphi_{R}^{2}+\varphi_{I}^{2}}{\overline{S}_{\pm
i}}}\mathrm{d}\varphi_{R}\mathrm{d}\varphi_{I}$ (232)
$\displaystyle=\frac{1}{E!}\frac{2\pi}{\pi\overline{S}_{\pm
i}}\int_{-\infty}^{\infty}\varphi^{2E}e^{-\varphi^{2}}e^{-\frac{\varphi^{2}}{\overline{S}_{\pm
i}}}\varphi\mathrm{d}\varphi=\frac{1}{E!}\frac{2}{\overline{S}_{\pm
i}}\int_{-\infty}^{\infty}\varphi^{2E}e^{-\varphi^{2}(1+\frac{1}{\overline{S}_{\pm
i}})}\varphi\mathrm{d}\varphi$ (233)
$\displaystyle=\frac{1}{E!}\frac{1}{\overline{S}_{\pm
i}}\int_{0}^{\infty}x^{E}e^{-x(1+\frac{1}{\overline{S}_{\pm
i}})}\mathrm{d}x=\frac{1}{E!}\frac{1}{\overline{S}_{\pm
i}}\left(1+\frac{1}{\overline{S}_{\pm
i}}\right)^{-1-E}\int_{0}^{\infty}y^{E}e^{-y}\mathrm{d}y$ (234)
$\displaystyle=\frac{1}{\overline{S}_{\pm
i}}{\left(1+\frac{1}{\overline{S}_{\pm i}}\right)}^{-1-E}$ (235)

This relates the distribution fully back to the geometric distribution.

$\displaystyle p$ $\displaystyle\equiv\frac{1}{\overline{S}_{\pm i}+1}$
$\displaystyle\left(1+\frac{1}{\overline{S}_{\pm i}}\right)^{-1-E}$
$\displaystyle\equiv p{(1-p)}^{E}$ $\displaystyle\frac{1}{\overline{S}_{\pm
i}+1}\left(1-\frac{1}{\overline{S}_{\pm i}+1}\right)^{E}$ $\displaystyle\equiv
p{(1-p)}^{E}$ (236)

From above, the variance of the Geometric distribution can be utilized to
calculate the full expressions of Eq. 108 and Eq. 113.

$\displaystyle\textsc{Var}[\hat{E}]$
$\displaystyle=\frac{1-p}{p^{2}}=\overline{S}_{\pm i}(\overline{S}_{\pm i}+1)$
$\displaystyle\textsc{Var}[\hat{G}_{\alpha}]$
$\displaystyle\approx\frac{2}{N_{\hat{E}}}\left(\frac{1}{|\overline{g}|^{2}\overline{\Phi}}\right)^{2}\overline{S}_{\pm
i}(1+\overline{S}_{\pm i})$ $\displaystyle\overline{S}_{\pm i}$
$\displaystyle=|\overline{g}|^{2}\alpha\overline{\Phi}+\overline{S}_{\check{\theta}}+\overline{S}_{\check{n}}+\frac{2\overline{B}\Delta
T}{N_{\hat{E}}}$ (237)

###  A.7 Full Narrowband Search Scan

These expressions are for the full relationship for Eq. 169 without assuming
optimal bandwidths.

$\displaystyle\frac{\textsc{Var}[\hat{G}]_{\text{SR,scan}}}{\textsc{Var}[\hat{\chi}^{\prime
2}_{2N}]_{\text{SR,scan}}}$
$\displaystyle=\left(|\overline{g}|^{2}\overline{S}_{\check{c}}+\frac{\Delta
T\overline{B}}{\mathcal{N}_{q}}\right)\left(\frac{c}{8L\gamma_{\eta}}\right)\left(\frac{\Delta
Tc}{2L}\right)\frac{\mathcal{N}_{q}\gamma_{\varphi}(\gamma_{q}^{-1}+\gamma_{g}^{-1})}{\mathcal{N}_{\Phi}^{2}}$
(238)
$\displaystyle=\left(|\overline{g}|^{2}\overline{S}_{\check{c}}+\frac{\Delta
T\overline{B}}{\mathcal{N}_{q}}\right)\left(\frac{c}{8L\gamma_{\eta}}\right)\frac{\left(\gamma_{\varphi}+\gamma_{g}+\gamma_{\eta}\right)^{2}}{\eta_{g}\gamma_{\varphi}}\frac{\gamma_{q}(\gamma_{q}^{-1}+\gamma_{g}^{-1})}{\gamma_{q}+\gamma_{g}+\gamma_{\eta}}$
(239)

###  A.8 Application Example from Statistical Equations

Here, the heuristic argument developed to demonstrate the application of
photon counting towards a stochastic signal search from quantum gravity of
Section II is derived from the equations of Section IV and Section V. Both
start from the relationship between the shot-noise limited displacement
sensitivity and the optical gain of an interferometer:

$\displaystyle S_{\hat{D}}$
$\displaystyle\equiv\frac{1}{2|\overline{g}|^{2}}=\textsc{Psd}[\text{Quantum}]$
(240)

Using this definition and the optimal search from Eq. 222, the expressions for
the search time can be derived. The $1\sigma$ observing time is established
from the condition $\textsc{Var}[\hat{\chi}^{2}_{2N}]=1$. In the case where
the dominant noise is quantum noise, and classical noise is a negligible
contribution, Eq. 222 can be given.

$\displaystyle 1=\frac{1}{\textsc{Var}[\hat{\chi}^{2}_{2N}]}$
$\displaystyle\approx\Delta
T_{\hat{D}}^{1\sigma}\int_{0}^{\infty}\left(2|\overline{g}(\Omega)|^{2}\Phi(\Omega)\right)^{2}\frac{\mathrm{d}\Omega}{2\pi}\approx\Delta
T_{\hat{D}}^{1\sigma}\left(2|\overline{g}|^{2}\overline{\Phi}\right)^{2}\int_{0}^{\infty}\frac{\Phi(\Omega)^{2}}{\overline{\Phi}^{2}}\frac{\mathrm{d}\Omega}{2\pi}$
(241)

And then reduced to the following equations.

$\displaystyle\Delta T_{\hat{D}}^{1\sigma}$
$\displaystyle=\frac{N_{\hat{D}}}{\Delta F_{\hat{D}}}$ $\displaystyle
N_{\hat{D}}$ $\displaystyle\equiv\Delta F_{\hat{D}}\Delta
T_{\hat{D}}^{1\sigma}=\frac{S_{\hat{D}}^{2}}{(\alpha\overline{\Phi})^{2}}$
$\displaystyle\Delta F_{\hat{D}}$
$\displaystyle=\int_{0}^{\infty}\left(\frac{\Phi(\Omega)}{\overline{\Phi}}\right)^{2}\frac{\mathrm{d}\Omega}{2\pi}$
(242)

The application section Section II then uses a heuristic set of arguments to
determine the search time from photon counting, resulting in the following
equations

$\displaystyle E_{\varphi}$
$\displaystyle=\frac{\alpha\overline{\Phi}}{4S_{\hat{D}}}\Delta\overline{T}_{E}\Delta
F_{E}$ $\displaystyle\Delta F_{E}$
$\displaystyle=\int_{-\infty}^{\infty}\frac{\Phi(\Omega)}{\overline{\Phi}}\frac{\mathrm{d}\Omega}{2\pi}$
$\displaystyle E_{\varphi}$ $\displaystyle=\Delta
T_{E}^{1\sigma}\int_{-\infty}^{\infty}P_{\varphi}(\Omega)\frac{\mathrm{d}\Omega}{2\pi}$
$\displaystyle P_{\varphi}(\Omega)$
$\displaystyle=\frac{S_{\varphi}(\Omega)}{4S_{\hat{D}}(\Omega)}$ (243)

using $E_{\varphi}=1$ to imply the $1\sigma$ statistical time is done first
under the assumption of negligible classical noise. In the equations above,
negligible classical noise is assumed by $P_{\varphi}(\Omega)$ including only
the power contributions from the signal. The reduced time
$\overline{T}_{E}^{1\sigma}$ denotes the time to measure one signal photon,
while the full time, $T_{E}^{1\sigma}$ includes the classical noise
contributions. In the application section, these appeared ad-hoc, but can be
related to the statistics of the photon-counting section.

$\displaystyle\Delta\overline{T}_{E}^{1\sigma}$
$\displaystyle\equiv\frac{4S_{\hat{D}}}{\alpha\overline{\Phi}\Delta
F_{E}}=\frac{2}{|\overline{g}|^{2}\alpha\overline{\Phi}\Delta F_{E}}$
$\displaystyle\Delta T_{E}^{1\sigma}$
$\displaystyle=\Delta\overline{T}_{E}^{1\sigma}\left(1+\frac{\textsc{Psd}[\text{Cl}]}{\overline{\Phi}}+\overline{B}\Delta\overline{T}_{E}^{1\sigma}\right)$
(244)

The full, classical noise included, integration time can be derived from Eq.
113 (including the signal contribution term) where
$1=\textsc{Var}[\hat{G}_{\alpha}]$ indicates the $1\sigma$ observing time.
Expanding that condition leads to the expressions

$\displaystyle\Delta T_{E}^{1\sigma}\Delta F_{E}$ $\displaystyle\approx
2\left(\frac{1}{|\overline{g}|^{2}\overline{\Phi}}\right)^{2}\left(|\overline{g}|^{2}\overline{\Phi}+\overline{S}_{\check{\theta}}+\overline{S}_{\check{n}}+\frac{2\overline{B}\Delta
T}{N_{\hat{E}}}\right)=\frac{4S_{\hat{D}}}{\overline{\Phi}}\left(1+\frac{\overline{S}_{\check{\theta}}+\overline{S}_{\check{n}}}{|\overline{g}|^{2}\overline{\Phi}}+\frac{2\overline{B}}{|\overline{g}|^{2}\overline{\Phi}\Delta
F_{E}}\right)$ (245)

The right hand side is directly convertible into the expressions of Eq. 244.