# Fundamental sensitivity limit of lossy cavity-enhanced interferometers with
external and internal squeezing

M. Korobko  Institut für Laserphysik und Zentrum für Optische
Quantentechnologien der Universität Hamburg,
Luruper Chaussee 149, 22761 Hamburg, Germany  J. Südbeck  Institut für
Laserphysik und Zentrum für Optische Quantentechnologien der Universität
Hamburg,
Luruper Chaussee 149, 22761 Hamburg, Germany  S. Steinlechner  Faculty of
Science and Engineering, Maastricht University, Duboisdomein 30, 6229 GT
Maastricht, The Netherlands  Nikhef, Science Park 105, 1098 XG Amsterdam, The
Netherlands  R. Schnabel  Institut für Laserphysik und Zentrum für Optische
Quantentechnologien der Universität Hamburg,
Luruper Chaussee 149, 22761 Hamburg, Germany

###### Abstract

Quantum optical sensors are ubiquitous in various fields of research, from
biological or medical sensors to large-scale experiments searching for dark
matter or gravitational waves. Gravitational-wave detectors have been very
successful in implementing cavities and quantum squeezed light for enhancing
sensitivity to signals from black hole or neutron star mergers. However, the
sensitivity to weak forces is limited by available energy and optical
decoherence in the system. Here, we derive the fundamental sensitivity limit
of cavity and squeezed-light enhanced interferometers with optical loss. This
limit is attained by the optimal use of an additional internal squeeze
operation, which allows to mitigate readout loss. We demonstrate the
application of internal squeezing to various scenarios and confirm that it
indeed allows to reach the best sensitivity in cavity and squeezed-light
enhanced linear force sensors. Our work establishes the groundwork for the
future development of optimal sensors in real-world scenarios where, up until
now, the application of squeezed light was curtailed by various sources of
decoherence.

##  I Introduction

Laser interferometers serve a multiple of applications, including biological
macromolecule sensors [1], medical sensors [2], accelerometers [3], and
extraordinary sensitive gravitational-wave detection [4]. Highest
sensitivities in laser interferometry are accomplished by injecting a quasi-
monochromatic carrier light in a significantly displaced coherent state
alongside strongly squeezed vacuum states that cover the signal band’s
sideband frequencies [5, 6]. This process is complemented by utilizing optical
cavities along the interferometer arms [7, 8], and reducing light
decoherence[9, 10]. Ideally, the power of the carrier light is enhanced up to
a threshold at which either the measured sample gets disturbed or the
interferometer’s optical hardware compromises its quality. The cavities
resonantly enhance both the power of the carrier light as well as the signal
due to constructive interference over the many cavity round trips. Preceding
and current gravitational-wave observatories employ the highest power that the
interferometer can sustain, enhancement cavities, and externally produced
squeezed vacuum states [11, 12, 13, 14, 15, 16]. Laser interferometers have
been enhanced in proof-of-principle experiments for biological sensors [17,
18], magnetic sensors [19], optomechanical sensors [20, 21], and dark matter
sensors [22, 23]. However, decoherence limits the quantum enhancement provided
by squeezed light, stemming from optical loss that incorporates imperfect
photo-electric detection efficiency and various propagation losses.
Decoherence always reduces the sensitivity that can be achieved.

The Quantum Cramer-Rao Bound (QCRB) [24, 25] defines the best possible
sensitivity of linear sensors across all signal frequencies in the absence of
decoherence for a given optical power. QCRB associates the device’s
sensitivity to the interaction between the meter and the object, and
ultimately to the energy fluctuations of a meter [26]. Attaining QCRB is only
possible with a pure quantum state upon interaction [27, 28], with quantum
decoherence in any practical setup preventing this, as was investigated in [9,
10, 29, 30]. However, not all sources of decoherence impact the sensitivity in
the same way. We recognize three distinct loss sources: injection, readout and
internal, see Fig. 1. Injection loss occurs upon injecting externally squeezed
state into the detector cavity and limits the available squeeze factor.
Readout loss impacts the light propagating from the cavity towards the
photodetector, including the photodetection process itself. Internal loss
happens during the measurement process itself and directly affects the purity
of the state upon interaction with the object.

In this work, we posit that internal loss establishes the fundamental limit
for cavity-enhanced interferometric sensing. In contrast, readout loss can be
mitigated by internal squeezing, which requires a second-order non-linear
crystal placed inside the interferometer’s cavity pumped with second harmonic
light [31, 32, 33, 34, 35]. The pump phase specifies the quadrature phase
subject to parametric amplification. We examine a general case incorporating
all three loss sources, and investigate the enhancement in sensitivity as well
as sensitivity-bandwidth product [33]. We also demonstrate the benefit of
internal squeezing compared to output amplification [5].

Our limit supports our recent experimental readout loss mitigation via
internal squeezing [36]. Furthermore, it provides a comprehensive framework
whithin which the specific design of cavity-enhanced detectors can be
optimised – for gravitational-wave detection and beyond.

Figure 1: Illustration of a future gravitational-wave detector able to operate
at the fundamental sensitivity limit. Optical losses occur inside the
detectors’s cavities formed by the signal-recycling mirror (SRM) and the arm
cavities; upon readout, including the propagation losses, e.g. through the
Faraday isolator (FI) used to inject external squeezing, as well as the loss
at the photodetectors. Injected squeezing also experiences injection loss due
to various generation and propagation inefficiencies. Internal squeezing is
crucial in achieving the fundamental sensitivity limit in this detector. All
three squeeze operation: internal, external and output amplification are
required in the general case.

##  II Decoherence-induced limit in a lossy cavity-enhanced detector with
internal squeezing

We model a force sensor as a Fabry-Perot cavity with a mirror that is
displaced under an action of an external force, which imprints phase
modulation on the reflected light field. Such a model is equivalent to the
full interferometer, as shown in Fig. 1, in terms of quantum noise [37].
Internal squeezing is produced by a non-linear medium inside the cavity, and
squeezed vacuum injected externally, see Fig. 2. In addition, we complement
the detector by an output amplifier [5], which amplifies the signal quadrature
by a factor of $\zeta$ (in power) and proportionally deamplifies the
orthogonal quadrature. In our study we focus on the shot-noise limited
sensitivity, assuming quantum back-action to be evaded by appropriate measures
[38].

Figure 2: Schematic representation of the internal squeezing cavity. An
external force moves the mirror $x$, producing the phase shift on the light
field. Signal together with noise are amplified by a factor $G$ inside a
nonlinear crystal. Outgoing field $d$ is separated from the incoming squeezed
field $v$. Optical loss $r_{\rm{i},\rm{int},\rm{det}}$ (injection, internal
and readout correspondingly) is introduced as a beam-splitter with
corresponding amplitude reflectivity. Optical loss couples in vacuum fields
$n^{\rm{c},\rm{int},\rm{i},\rm{det}}$ that destroy coherence of squeezed
state. Fields $a,b$ define counter-propagating quantum fields inside the
cavity, field $d$ is detected with a homodyne detector.

We follow the input-output formalism [39, 40, 41] for solving the equations of
motion for the light fields shown in Fig. 2. We further make a single-mode
approximation, assuming the cavity to have high enough finesse to have a well-
defined single mode within a free spectral range of the longitudinal
resonances. A signal at the Fourier frequency $\Omega$ couples only to this
mode. The complete derivation can be found in Appendix, where we compute the
simplified noise spectral density $S_{n}(\Omega)$ and the spectral shape of
the signal transfer function $\mathcal{T}(\Omega)$:

$\displaystyle S_{n}(\Omega)=$
$\displaystyle\frac{1}{\mathcal{D}(\Omega)}\left[\mathcal{D}(\Omega)-4qT_{c}(1-\epsilon_{\rm
read})-\right.$ $\displaystyle-(1-\beta^{-1})(1-\epsilon_{\rm read})\times$
$\displaystyle\left.\times\left((q-T_{c}+\epsilon_{\mathrm{int}})^{2}+16\Omega^{2}\tau^{2}\right)\right]$
(1) $\displaystyle|\mathcal{T}(\Omega)|^{2}=$ $\displaystyle\frac{8\pi
P_{c}}{\hbar\lambda c}\frac{4T_{c}(1-\epsilon_{\rm
read})}{\mathcal{D}(\Omega)}$ (2) $\displaystyle\mathcal{D}(\Omega)=$
$\displaystyle(q+T_{c}+\epsilon_{\mathrm{int}})^{2}+16\Omega^{2}\tau^{2}.,$
(3)

where $q$ is the gain in a single pass through the crystal, $T_{c}$ is the
power transmission through the front mirror, $\epsilon_{\rm int,read}$ is the
internal and readout power loss correspondingly, $\tau$ is a single-pass time
inside the cavity, $P_{c}$ is the total power inside the cavity (which we
assume to be fixed), $\lambda$ is the central wavelength, $c$ is the speed of
light, and $\beta=e^{2r_{\rm ext}}$, where $r_{\rm ext}$ is the external
squeezing rate. We assumed the output amplification and the injection loss to
be zero, and will add them later in the text. The derived expression allows to
optimize internal squeezing $q$ in order to achieve the highest sensitivity
$S_{n}(\Omega)/|\mathcal{T}(\Omega)|^{2}$ and study the limiting cases.

We first consider a low-frequency case: $\Omega\approx 0$, which allows to see
the characteristic behavior of different limiting cases better and doesn’t
affect the conceptual understanding. We make a reference to a QCRB in a
lossless case, $\epsilon_{\mathrm{int}}=0,\epsilon_{\rm read}=0$:

$S_{hh}^{\mathrm{QCRB}}=\frac{\hbar\lambda c}{8\pi
P_{c}L^{2}}\times\frac{(T_{c}-q)^{2}}{\beta T_{c}},$ (4)

where we normalized the sensitivity to the GW strain $h_{0}=x/L$. Notably, in
the lossless case the sensitivity becomes unlimited,
$S_{hh}^{\mathrm{QCRB}}=0$, at the parametric threshold:
$\gamma_{s}=\gamma_{c}$ or at infinite injected squeezing, $1/\beta=0$.
Optical losses will prevent from achieving this QCRB. It is therefore possible
to compute the new limit $S_{hh}^{\epsilon}$, induced by the losses. We start
by computing the reference sensitivity without internal squeezing, $q=0$, in
two limiting cases: when there is no externally injected squeezing and when it
is infinitely large:

$\displaystyle\left.S_{hh}^{\epsilon}\right|_{q=0,\beta=1}$
$\displaystyle=\,\frac{\hbar\lambda c}{8\pi
P_{c}L^{2}}\frac{1}{1-\epsilon_{\rm read}}\times$
$\displaystyle\,\times\left(\frac{T_{c}}{4}\epsilon_{\rm
read}+\frac{1}{2}\epsilon_{\mathrm{int}}+\frac{\epsilon_{\mathrm{int}}^{2}}{4T_{c}}\right),$
(5) $\displaystyle\left.S_{hh}^{\epsilon}\right|_{q=0}$
$\displaystyle\xrightarrow[\beta\rightarrow\infty]{}\,\frac{\hbar\lambda
c}{8\pi P_{c}L^{2}}\frac{1}{1-\epsilon_{\rm read}}\times$
$\displaystyle\,\times\left(\frac{T_{c}}{4}\epsilon_{\rm
read}+\frac{2-\epsilon_{\rm
read}}{2}\epsilon_{\mathrm{int}}+\frac{\epsilon_{\rm
read}\epsilon_{\mathrm{int}}^{2}}{4T_{c}}\right).$ (6)

Another characteristic reference is the case when internal squeezing is at its
threshold value: $q=q^{\rm th}:=T_{c}+\epsilon_{\rm int}$:

$\displaystyle\left.S_{hh}^{\epsilon}\right|_{\beta=1}=$
$\displaystyle\frac{\hbar\lambda c}{8\pi P_{c}L^{2}}\frac{1}{1-\epsilon_{\rm
read}}\left(T_{c}\epsilon_{\rm
read}+\epsilon_{\mathrm{int}}+\frac{\epsilon_{\mathrm{int}}^{2}}{4T_{c}}\right),$
(7) $\displaystyle S_{hh}^{\epsilon}\xrightarrow[\beta\rightarrow\infty]{}$
$\displaystyle\frac{\hbar\lambda c}{8\pi P_{c}L^{2}}\frac{1}{1-\epsilon_{\rm
read}}\left(T_{c}\epsilon_{\rm
read}+\epsilon_{\mathrm{int}}+\frac{\epsilon_{\mathrm{int}}^{2}}{4T_{c}}\right).$
(8)

This case has been discussed in Ref. [30] as the ultimate detection limit. It
is interesting to note that in the case without internal squeezing the limit
is 4 times lower than in the case with at-threshold internal squeezing. This
occurs due to 6 dB of signal deamplification in the case of internal squeezing
at threshold [33].

These two references allow comparison with an optimal sensitivity $S_{hh}^{\rm
opt}$, for which the optimal internal gain $q_{\mathrm{opt}}$ allows to reach
the new sensitivity limit:

$\displaystyle S_{hh}^{\mathrm{opt}}$ $\displaystyle=\frac{\hbar\lambda
c}{8\pi P_{c}L^{2}}\left(\frac{T_{c}\epsilon_{\rm read}}{1+\epsilon_{\rm
read}(\beta-1)}+\epsilon_{\mathrm{int}}\right),$ (9) $\displaystyle q^{\rm
opt}$ $\displaystyle=T_{c}\left(1-\frac{2\epsilon_{\rm
read}}{\beta(1-\epsilon_{\rm read})-\epsilon_{\rm read}}\right)-\epsilon_{\rm
int}.$ (10)

This limit constitutes the ultimate limit on the sensitivity, and overcomes
the limit in Ref. [30]. Note, that in this case the QCRB is not zero, since
the internal squeezer operates below threshold. In Fig. 3 we show the benefit
from optimal internal squeezing compared to the case without internal
squeezing for different values of losses and externally injected squeezing.
One can see that the benefit is higher for lower external squeezing, since
then internal squeezing works to generate more squeezing. For higher external
squeezing, the benefit increases with increased readout loss as it is
compensated by internal squeezing. Internal loss defines the sensitivity
limit, so the higher it is, the smaller is the benefit from internal
squeezing. There also exists a readout loss value for which it’s optimal not
to squeeze internally, since the introduced signal deamplification exactly
cancels a potential benefit.

Figure 3: Relative sensitivity improvement from optimal internal squeezing
compared to the case without internal squeezing. Top: relative improvement as
a function of readout loss for different levels of external squeezing, for
$\epsilon_{\mathrm{int}}=0.001$. Internal squeezing is almost always
beneficial, except for a single value of readout loss for each curve. For
higher readout loss, it is optimal to amplify inside the interferometer
cavity. Bottom: relative improvement as a function of readout loss for
different levels of internal loss, for 15 dB external squeezing. For high
internal loss, internal squeezing becomes beneficial when the detection loss
is high. For low internal loss, internal squeezing is almost always
beneficial, except for a single readout loss value. For losses higher than
that it becomes optimal to amplify inside the cavity. For all plots
$T_{c}=0.01,\Omega=0$.

In the absence of external squeezing we achieve:

$\left.S_{hh}^{\mathrm{opt}}\right|_{\beta=1}=\frac{\hbar\lambda c}{8\pi
P_{c}L^{2}}\left(T_{c}\epsilon_{\rm read}+\epsilon_{\mathrm{int}}\right)$ (11)

which overcomes the limit for the at-threshold operation. The optimal gain
here is below the threshold

$\left.q_{\mathrm{opt}}\right|_{\beta=1}=q_{\mathrm{th}}-2(T_{c}\epsilon_{\rm
read}+\epsilon_{\mathrm{int}}).$ (12)

In this case the purpose of internal squeezing is to generate as much
squeezing as possible, at the same time keeping the deamplification of the
signal as low as possible, taking into consideration the losses in the system.
This regime was explored by us experimentally before in Ref. [33].

Finally, we can compute the ultimate limit for infinite squeezing, which is
achieved by maximally amplifying the signal quadrature inside the cavity
$q=-q^{\rm th}$:

$S_{hh}^{\mathrm{opt}}\xrightarrow[\beta\rightarrow\infty]{}\frac{\hbar\lambda
c}{8\pi P_{c}L^{2}}\epsilon_{\mathrm{int}}.$ (13)

This is the ultimate limit to the sensitivity of a linear detector, and it can
be significantly lower than the sensitivity at threshold (8). In Fig. 4 we
show the benefit from optimal internal squeezing compared to the on-threshold
case. As expected from the equations, when the readout loss is small, the
benefit is negligible (since internal squeezing works best when it can
compensate the readout loss). However, as the readout loss and externally
injected squeezing increase, the benefit increases too, and at the same time
it requires generating less squeezing inside, quickly transitioning into the
amplification regime (negative values for $q$). For example, already for a
readout loss of around 5% (which is smaller than the current loss in
gravitational-wave detectors) it becomes optimal to amplify inside (for 15 dB
of external squeezing).

Figure 4: Relative sensitivity improvement from optimal internal squeezing
compared to the case of internal squeezing at threshold: the improvement
$S_{hh}^{\mathrm{opt}}/S_{hh}^{\epsilon}$ (solid), and the corresponding
optimal internal gain relative to the threshold
$q_{\mathrm{opt}}/q_{\mathrm{th}}$ (dashed). (Top) relative improvement as a
function of readout loss for different levels of external squeezing, for
$\epsilon_{\mathrm{int}}=0.001$; (bottom) relative improvement as a function
of internal loss for different levels of readout loss, for 15 dB external
squeezing. Internal gain is optimized to squeezing for low losses, but for
high losses switches to parametric amplification regime (values below zero).
The plots demonstrate that optimised internal squeezing is always beneficial
and leads to an improvement compared to the limit presented in Ref. [30]. For
all plots $T_{c}=0.01,\Omega=0$.

###  II.1 Sensitivity-bandwidth relation

In the previous section we investigated only the effect of internal squeezing
on the peak sensitivity. It is important to also consider the detection
bandwidth and the sensitivity-bandwidth product as a generic measure [33].
Internal squeezing reduces the effective bandwidth of the system, and the
higher the squeezing is, the smaller is the bandwidth. However, since the peak
sensitivity increases faster, the overall sensitivity-bandwidth product is
enhanced beyond the standard sensitivity-bandwidth limit (SSBL) for coherent
light. Eqs. (40),(42) allow to analyse the sensitivity-bandwidth product by
defining the bandwidth $\mathcal{B}$ as the half-width half-maximum of the
sensitivity. While the full expressions for the bandwidth are rather bulky, we
can analyse them numerically.

In Fig. 5, we show the gain in the SNR relative to the case without internal
squeezing, and the corresponding reduction in the squeezer’s bandwidth, when
no externally injected squeezing is present. As in the previous Section, here
the clear optimum point is reached for every internal gain value.

When external squeezing is injected, internal squeezing acts in different ways
depending on the loss level, as described in the previous Section. The
bandwidth is always smaller than in the case without internal squeezing, but
its dependence on the losses is nontrivial. In Fig. 6, we show the enhancement
relative to the case with external squeezing, demonstrating that the
sensitivity can be always enhanced, despite the reduction in bandwidth. We
also show the enhancement relative to the SSBL, demonstrating that although
the main contribution comes from the external squeezing, internal squeezing
helps to gain sensitivity. In this case the bandwidth can also be enhanced in
some parameter regimes, due to nontrivial dependence of the bandwidth on
losses and squeezing levels.

Figure 5: Internal squeezing operating without external squeezing. The noise
is reduced due to the produced squeezing, and the signal is deamplified at the
same time. The SNR gain (top) and the reduction in the bandwidth (bottom)
depends on the internal gain. For different values of readout loss, there
exists an optimal internal gain for which the SNR is maximized. Although the
bandwidth is reduced, the sensitivity-bandwidth product is still increased, as
we demonstrated experimentally in Ref. [33]. In both cases
$T_{c}=0.01,\epsilon_{\rm int}=0$.  Figure 6: Effect of optimal internal
squeezing on the detection bandwidth (top) and the peak sensitivity-bandwidth
product $\mathcal{S}\times\mathcal{B}$ (bottom). Left: the changes relative to
the standard interferometer without internal or external squeezing. Right: the
changes relative to the baseline interferometer without internal squeezing.
The optimal internal squeezing is different for different parameters, and
influences the bandwidth differently. The sensitivity-bandwidth product is
enhanced always except for a single specific value of readout loss. Bottom
left: the solid line is a combination of optimal internal squeezing with
external squeezing, dashed line is only external squeezing. external squeezing
provides the main contribution to the enhancement, internal squeezing helps to
suppress the readout losses. For all plots
$T_{c}=0.01,\epsilon_{\mathrm{int}}=0.0001$.

##  III Injection loss and output amplification

Finally, in order to complete the consideration, we provide the expression for
the sensitivity including the injection loss for external squeezing as well as
an additional amplifier before the detection, as it was first proposed by
Caves [5]. Such three-stage squeezing scheme (input, internal, output) would
yield the best sensitivity with any combination of losses.

In order to include the injection loss $\epsilon_{i}$, we add a loss term when
computing the spectral density, _i.e._
$S_{vv}=\beta(1-\epsilon_{i})+\epsilon_{i}$. The output amplification would be
represented as a linear gain $\zeta$ just before the detection loss term.

The resulting sensitivity with optimized internal gain (in the single-mode
approximation):

$S_{hh}^{\mathrm{opt}}(\Omega)=\frac{\hbar\lambda c}{8\pi P_{c}L^{2}}\times\\\
\times\left[\frac{T_{c}\epsilon_{\mathrm{det}}\left(1-\epsilon_{i}(1+\beta)\right)}{\beta\epsilon_{\rm
read}+\zeta(1-\epsilon_{\rm read})(1-(1-\beta)\epsilon_{i})}+\right.\\\
+\epsilon_{\mathrm{int}}+\frac{4\Omega^{2}\tau^{2}}{T_{c}\beta\zeta\left(1-\epsilon_{\mathrm{det}}\right)}\left(\beta\epsilon_{\mathrm{det}}+\right.\\\
\biggl{.}\left.+\zeta(1-\epsilon_{\rm
read})(1-(1-\beta)\epsilon_{i})\right)\biggr{]}.$ (14)

It is interesting to consider limiting cases here as well. In the limit of
infinite external squeezing (at zero frequency):

$S_{hh}^{\mathrm{opt}}(0)\xrightarrow[\beta\rightarrow\infty]{}\frac{\hbar\lambda
c}{8\pi
P_{c}L^{2}}\left[\frac{T_{c}\epsilon_{\mathrm{det}}\epsilon_{i}}{\epsilon_{\mathrm{det}}+\zeta\epsilon_{i}\left(1-\epsilon_{\mathrm{det}}\right)}+\epsilon_{\mathrm{int}}\right].$
(15)

Without output amplification ($\zeta=1$), the limit is dependent on the
injection loss. The injected state is no longer pure, and that impurity
contributes to the interaction Hamiltonian, thus preventing from achieving the
QCRB, as discussed in the introduction. Internal squeezing is unable to
compensate that effect. Injection loss effectively reduces the level of
external squeezing, and the state can be kept rather pure by optimizing the
external squeezing strength. An additional effect of destroyed correlations
between two quadratures of the field used for back-action evasion, will
ultimately limit the sensitivity in back-action-dominated region, which we do
not consider here.

Interestingly, in the limiting case of infinite output amplification, the
result is independent on the external squeezing $\zeta\rightarrow\infty$:

$S_{hh}^{\mathrm{opt}}(0)\xrightarrow[\zeta\rightarrow\infty]{}\frac{\hbar\lambda
c}{8\pi P_{c}L^{2}}\epsilon_{\rm int},\quad
q_{\mathrm{opt}}=(T_{c}-\epsilon_{\mathrm{int}}).$ (16)

Here, the readout loss is fully compensated by output amplification, and
internal squeezing can be used to create maximal squeezing (taking into
account balance between signal deamplification and internal loss).

Finally, we can compare the limits from the internal squeezing and output
amplification. In essence, they achieve the same thing: they allow to evade
the readout loss. Output amplification without internal squeezing reaches the
limit:

$\left.S_{hh}(0)\right|_{q=0}\xrightarrow[\zeta\rightarrow\infty,\beta\rightarrow\infty]{}\frac{\hbar\lambda
c}{8\pi P_{c}L^{2}}\left(\epsilon_{\rm
int}+\frac{\epsilon_{i}(T_{c}-\epsilon_{\rm int})^{2}}{4T_{c}}\right).$ (17)

Comparing this expression with Eq.(15) for $\zeta=1$ we will find that they
both reach the same limit if there’s no injection loss. However, if injection
loss is present, these expressions are different, and in different regimes
either one or the other work better. That does not mean, however, that output
amplification is sufficient to reach the ultimate limit. Here we considered a
perfect output amplification with no added losses, while internal squeezing is
considered imperfect (i.e. taking into account the losses). This added loss
would lead to an increased decoherence-induced limit, higher than that of
internal squeezing. The specific implementation of the output amplifier
requires a separate study.

##  IV Discussion and conclusion

Internal squeezing mitigates the decoherence effect of optical loss,
establishing a generally improved sensitivity limit. Uniquely, internal
squeezing enhances sensitivity in two ways: either by directly maximizing the
SNR when the loss is minimal, or by amplifying both the signal and the noise
to evade significant readout losses. In this work, we offer a detailed
exploration of various regimes for the application of internal squeezing
within realistic parameters.

We also compared internal squeezing with the ideal output amplification [5],
which was recently explored in other contexts [42, 43, 44, 45, 46]. By
highlighting the similarities and differences between internal squeezing and
this approach, we delineate the limits each can achieve independently and
present an optimal design combining both to counteract the three primary loss
sources. Our argument suggests that in the presence of input, internal, and
readout loss, the optimal configuration incorporates both output amplification
and internal squeezing. Internal squeezing usually would suffice for low
injection loss, but the optimal configuration would be determined by the full
parameter set.

In our study, we assumed that such an output amplifier would introduce no loss
– an unrealistic assumption. Practically, with the presence of an output
amplifier, the readout loss must be divided into three stages: before, inside,
and after the amplifier. In the GW detectors – the primary target for such
designs – the major source of readout loss would occur before the output
amplifier: on the Faraday isolator used to inject squeezed light, on the mode-
matching to the output mode-cleaner used to purify the optical mode leaving
the interferometer, and inside this mode-cleaner. This scenario limits the
utility of output amplification as it would only effectively compensate for
photodetection inefficiency, which is typically very low. Contrarily, internal
squeezing can compensate for all these decoherence sources.

In addition to the optical loss, other decoherence mechanisms are present in
realistic detectors: phase noise [6, 47], dephasing [48], mode mismatch [49],
and others. The impact of these sources on the performance of internal
squeezing and the fundamental limits will be the direction of future research.

Internal squeezing is an exceptionally versatile tool for enhancing the
sensitivity of various sensors. It offers manipulation of the sensitivity
[33], the bandwidth [35], or even modifies the dynamics of test masses in
optical cavities [34]. This paper introduces a new facet to the study of
internal squeezing, demonstrating its capability to achieve the fundamental
limit imposed by optical decoherence. To complement this theoretical study, we
experimentally demonstrated mitigation of decoherence and optimal internal
squeezing in Ref. [36]. Together with the current paper, it paves the way for
practical implementation of internal squeezing in quantum sensing and
underlines its advantages in various practical scenarios.

Acknowledgements We thank F.Khalili and H.Miao for insightful discussions.
This work was supported by the Deutsche Forschungsgemeinschaft (DFG) under
Germany’s Excellence Strategy EXC 2121 “Quantum Universe”-390833306.

*

##  Appendix A Appendix A: Derivation of spectral density

Using the perturbation theory, we decompose the light field into a steady-
state amplitude with amplitude $A_{0}$ and laser carrier frequency
$\omega_{0}$ and a slowly varying noise amplitude $a(t)$ (see details in
[41]):

$\displaystyle
A(t)=\sqrt{\frac{2\pi\hbar\omega_{0}}{\mathcal{A}c}}\left[A_{0}e^{-i\omega_{0}t}+a(t)e^{-i\omega_{0}t}\right]+{\rm
h.c.}$ (18) $\displaystyle
a(t)=\int_{-\infty}^{\infty}a(\omega_{0}+\Omega)e^{-i\Omega
t}\frac{d\Omega}{2\pi},$ (19)

where $\mathcal{A}$ is the laser beam cross-section area and $\hbar$ is the
reduced Plank constant. Note that we omit the hats on the operator for
brevity, although all the fields are quantised. Since we are interested in the
steady-state solutions for the fields, we consider only the noise fields in
the frequency domain. We further define the two-photon amplitude and phase
quadratures at a sideband frequency $\Omega$ correspondingly:

$\displaystyle
a_{x}(\Omega)=\frac{a(\omega_{0}+\Omega)+a^{\dagger}(\omega_{0}-\Omega)}{\sqrt{2}},$
(20) $\displaystyle
a_{y}(\Omega)=\frac{a(\omega_{0}+\Omega)-a^{\dagger}(\omega_{0}-\Omega)}{i\sqrt{2}}.$
(21)

These operators obey the commutation relation:

$\displaystyle[a_{x}(\Omega),a_{x}(\Omega^{\prime})]=[a_{y}(\Omega),a_{y}(\Omega^{\prime})]=0,$
(22)
$\displaystyle[a_{x}(\Omega),a_{y}(\Omega^{\prime})]=[a_{x}(\Omega),a_{y}(\Omega^{\prime})]=2\pi
i\delta(\Omega+\Omega^{\prime}).$ (23)

Using these two-photon quadratures, we can apply the input-output formalism
[39, 40] and find the steady-state fields in the system.

The signal $s$ appears only in the equations for the phase quadrature of the
light field. We model the optical loss by a beamsplitter reflecting some part
of the light fields to the environment and mixing in some vacuum from the
environment. The optical parametrical amplification process is not included in
the model, we treat crystal as a gain media that linearly amplifies with a
gain $G$ a certain quadrature (amplitude in our case) and deamplifies the
orthogonal to it; we call $q$ a squeezing factor in a single pass through the
crystal. We also consider the output amplification of the field before
detection [5], which amplifies the signal quadrature by a factor of $\zeta$
(in power) and proportionally deamplifies the orthogonal quadrature.

We start by deriving the full input-output relations for the fields in the
process, as shown in Fig. 2.

$\displaystyle\begin{cases}b^{x}=&a^{x}e^{2i\Omega\tau}G^{2}r_{b}t_{\rm
int}+n_{c}^{x}t_{b}t_{\rm int}Ge^{i\Omega\tau}+\\\ &+r_{\rm int}n_{\rm
int}^{x},\\\ a^{x}=&r_{c}b^{x}+t_{c}t_{i}v^{x}+t_{c}r_{i}n_{i}^{x}y,\\\
d^{x}=&\zeta^{-1/2}t_{b}(t_{c}b^{x}-r_{c}t_{i}v^{x}-r_{c}r_{i}n_{i}^{x})+r_{d}n_{d}^{x},\end{cases}$
(24) $\displaystyle\begin{cases}b^{y}=&a^{y}e^{2i\Omega\tau}G^{-2}r_{b}t_{\rm
int}+n_{c}^{y}t_{b}t_{\rm int}G^{-1}e^{i\Omega\tau}+\\\ &+r_{\rm int}n_{\rm
int}^{y}+st_{\rm int}G^{-1}e^{i\Omega\tau},\\\
a^{y}=&r_{c}b^{y}+t_{c}t_{i}v^{y}+t_{c}r_{i}n_{i}^{y},\\\
d^{y}=&\zeta^{1/2}t_{b}(t_{c}b^{y}-r_{c}t_{i}v^{y}-r_{c}r_{i}n_{i}^{y})+r_{d}n_{d}^{y},\end{cases}$
(25)

where $r_{c,b,\rm{int},i}$ are the amplitude reflectivity of the front mirror,
back mirror, internal loss and injection loss; $t=\sqrt{1-r^{2}}$ are the
corresponding transmissivities; $\tau$ is the cavity single-pass time.

Then, solving equations for the output field, we arrive at:

$\displaystyle d^{x}=$
$\displaystyle\frac{\zeta^{-1/2}t_{d}}{G^{-2}-r_{c}r_{b}t_{\rm
int}e^{2i\Omega\tau}}\left[t_{c}r_{\rm int}n_{\rm int}^{x}G^{-2}+\right.$ (26)
$\displaystyle+n_{i}^{x}\left(-r_{c}r_{i}G^{-2}+r_{b}r_{i}t_{\rm
int}e^{2i\Omega\tau}\right)+$ (27)
$\displaystyle\left.+v^{x}\left(-r_{c}t_{i}G^{-2}+r_{b}t_{i}t_{\rm
int}e^{2i\Omega\tau}\right)\right]+n_{e}^{x}r_{d},$ (28) $\displaystyle
d^{y}=$ $\displaystyle\frac{\zeta^{1/2}t_{d}}{G^{2}-r_{c}r_{b}t_{\rm
int}e^{2i\Omega\tau}}\left[st_{c}t_{\rm int}e^{q+i\Omega\tau}\right.$ (29)
$\displaystyle+t_{c}r_{\rm int}n_{\rm int}^{y}G^{2}+$ (30)
$\displaystyle\left.+n_{i}^{y}\left(-r_{c}r_{i}G^{2}+r_{b}r_{i}t_{\rm
int}e^{2i\Omega\tau}\right)+\right.$ (31)
$\displaystyle\left.+v^{y}\left(-r_{c}t_{i}G^{2}+r_{b}t_{i}t_{\rm
int}e^{2i\Omega\tau}\right)\right]+n_{e}^{y}r_{d}.$ (32)

Solving the equations for amplitude quadrature and maximizing the field
strength inside the cavity, we arrive at the threshold value:

$e^{-q^{\rm th}}=(r_{b}r_{c}\sqrt{1-\epsilon_{\rm int}}).$ (33)

We obtain the spectral shape of transfer function $\mathcal{T}(\Omega)$ for
the signal $s$, where we introduce power transmissivities:
$T_{n}=t_{n}^{2},R_{n}=r_{n}^{2}$ and losses $\epsilon_{n}=r_{n}^{2}$:

$\displaystyle|\mathcal{T}(\Omega)|^{2}=$ $\displaystyle\frac{\zeta
G^{2}T_{c}(1-\epsilon_{d})(1-\epsilon_{\rm int})}{|\mathcal{D}(\Omega)|^{2}},$
(34) $\displaystyle|\mathcal{D}(\Omega)|^{2}=$ $\displaystyle
G^{4}+R_{b}R_{c}(1-\epsilon_{\rm int})-$ $\displaystyle-$ $\displaystyle
2G^{2}\sqrt{R_{b}R_{c}(1-\epsilon_{\rm int})}\cos 2\Omega\tau.$ (35)

For computing the noise spectral density, we note that external squeezing is
injected into the cavity, with the spectral density $S_{vv}=\beta=e^{2r_{\rm
ext}}$, where $r_{\rm ext}$ is the external squeezing rate. The phase of
external squeezing is set to squeezing or anti-squeezing, since any other
angle would introduce non-trivial dynamics to the system, which we explored
before in a loss-less case in Ref. [34], while a full treatment requires a
separate investigation. This results the noise spectral density:

$\displaystyle S_{n}(\Omega)=$ $\displaystyle 1-\frac{(1-\epsilon_{\rm
read})}{|\mathcal{D}(\Omega)|^{2}}\left(S_{1}\epsilon_{\rm int}+S_{2}+\right.$
$\displaystyle\left.+S_{3}\sqrt{R_{b}R_{c}\left(1-\epsilon_{\rm
int}\right)}\cos 2\Omega\tau\right)$ (36) $\displaystyle S_{1}(\Omega)=$
$\displaystyle-R_{b}R_{c}+\zeta\left(G^{2}T_{c}(T_{b}-G^{2})+\right.$
$\displaystyle\left.+R_{b}\beta^{-1}(1-\epsilon_{i})+R_{b}\epsilon_{i}\right)$
(37) $\displaystyle S_{2}(\Omega)=$ $\displaystyle
G^{4}+R_{b}R_{c}-\zeta\left[\left(\epsilon_{i}+\beta^{-1}\left(1-\epsilon_{i}\right)\right)\times\right.$
$\displaystyle\left.\times\left(R_{b}-G^{4}R_{c}\right)+G^{2}T_{b}T_{c}\right]$
(38) $\displaystyle S_{3}(\Omega)=$ $\displaystyle
2G^{2}\left(-1+\zeta\beta^{-1}\left(1-(1-\beta)\epsilon_{i}\right)\right)$
(39)

The sensitivity is defined as a noise-to-signal ratio:
$S_{n}(\Omega)/|\mathcal{T}(\Omega)|^{2}$.

The expression for the spectral density and signal transfer function can be
simplified by making the standard single-mode approximation. The amplitude
transmissivities of the coupling and the back mirrors, as well as the internal
loss, are much smaller than unity, and we can approximate correspondingly
$R_{\mathrm{c}}\approx 1-T_{c}/2$, and set $T_{b}=0$, effectively including
into the internal loss. The single-pass gain $G$ is close to unity, so we can
approximate $G\approx 1+q/4$, where $q$ is the double-pass power gain factor;
the frequency of interest is much smaller than the FSR of the cavity
$\Omega\ll 1/2\tau$, which enables us to make a Taylor expansion:
$\cos\Omega\tau\approx 1-\Omega^{2}\tau^{2}/2$. The signal $s$ is due to the
modulation adds a phase shift on the light reflected of the movable mirror:
$e^{2ikx(\Omega)}\approx 1+2ik_{p}x(\Omega)$, where $k_{p}$ is the light wave
vector, and $x(\Omega)$ is a small mirror displacement. The signal is then
represented as $s(\Omega)=\sqrt{8\pi P_{c}(\hbar\lambda c)^{-1}}x(\Omega)$. We
further posit, that the average light power inside remains fixed and does not
get amplified by the parametric process. Finally, we first study the case
without injection loss, $\epsilon_{i}=0$, and the output amplification,
$\zeta=1$.

With these assumptions we achieve the full expression for the noise spectral
density and the signal transfer function in the single-mode approximation:

$\displaystyle S_{n}(\Omega)=$
$\displaystyle\frac{1}{\mathcal{D}(\Omega)}\left[\mathcal{D}(\Omega)-4qT_{c}(1-\epsilon_{\rm
read})-\right.$ $\displaystyle-(1-\beta^{-1})(1-\epsilon_{\rm read})\times$
$\displaystyle\left.\times\left((q-T_{c}+\epsilon_{\mathrm{int}})^{2}+16\Omega^{2}\tau^{2}\right)\right]$
(40) $\displaystyle|\mathcal{T}(\Omega)|^{2}=$ $\displaystyle\frac{8\pi
P_{c}}{\hbar\lambda c}\frac{4T_{c}(1-\epsilon_{\rm
read})}{\mathcal{D}(\Omega)}$ (41) $\displaystyle\mathcal{D}(\Omega)=$
$\displaystyle(q+T_{c}+\epsilon_{\mathrm{int}})^{2}+16\Omega^{2}\tau^{2}.$
(42)

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