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##  Appendix A Dephasing in Active Interferometers

This appendix provides the technical derivation of 36 and 37 in Section IV,
and uses the terms defined there. This derivation produces the intermediate
steps in the relation Eq. 33, starting from the right-hand-side of that
equation. From there, the $\vec{m}$ effective observation vector can be
inserted. This vector is complex, while the left-acting matrices $\mathbbl{R}$
and $\mathbbl{S}$ are both real. The final noise expression uses a vector norm
that takes the square sum of all of the real and imaginary parts of the
resulting vector. The vector norm can formally be replaced by a matrix
Frobenius norm, notated $|\cdot|_{\text{F}}$, while the complex vector
$\vec{m}$ is split into the real matrix $\mathbbl{q}$.

$\displaystyle\left|\overset{\raisebox{-2.0pt}{\text{\tiny$\bm{\rightarrow}$}}}{\mathbbl{v}}^{\dagger}\mathbbl{H}\mathbbl{R}(\phi)\mathbbl{S}(r)\right|^{2}$
$\displaystyle=\left|\begin{bmatrix}m_{q}\\\
m_{p}\end{bmatrix}^{T}\mathbbl{R}(\phi)\mathbbl{S}(r)\right|^{2}=\left|\mathbbl{q}^{T}\mathbbl{R}(\phi)\mathbbl{S}(r)\right|^{2}_{\text{F}}$
(92) $\displaystyle\mathbbl{q}$
$\displaystyle\equiv\begin{bmatrix}\Re\\{m_{q}\\}&\Im\\{m_{q}\\}\\\
\Re\\{m_{p}\\}&\Im\\{m_{p}\\}\end{bmatrix}$ (93)

The $\mathbbl{q}$ matrix can then undergo a singular value decomposition into
two rotations acting on a real diagonal matrix.

$\displaystyle\mathbbl{R}(\theta_{D})\begin{bmatrix}\Sigma_{+}&0\\\
0&\Sigma_{-}\\\ \end{bmatrix}\mathbbl{R}(\theta_{C})$
$\displaystyle\equiv\mathbbl{q}$ (94)

The rotations are labeled $\theta_{\text{D}}$ and $\theta_{\text{C}}$ for the
differential and common rotations. The common angle expresses the average
phase on both optical quadratures, physically due to transmission or cavity
delay, whereas the differential angle expresses the rotation of the principle
squeezing axis into a specific optical quadrature. $\theta_{\text{D}}$
calculated from the SVD is the exact form of Eq. 36. The decomposition may
then be inserted into Eq. 92 to create a scalar expression taking the form of
the left-hand-side of Eq. 33.

$\displaystyle\left|\mathbbl{q}^{T}\mathbbl{R}(\phi)\mathbbl{S}(r)\right|^{2}_{\text{F}}$
$\displaystyle=\left(\Sigma_{+}^{2}e^{-2r}+\Sigma_{-}^{2}e^{+2r}\right)\cos^{2}(\phi-\theta_{D})$
$\displaystyle\hskip
10.00002pt+\left(\Sigma_{+}^{2}e^{+2r}+\Sigma_{-}^{2}e^{-2r}\right)\sin^{2}(\phi-\theta_{D})$
(95)

From there, terms can be extracted to form the relations of Eqs. 7 to 10

$\displaystyle\eta\Gamma$
$\displaystyle=\left|\overset{\raisebox{-2.0pt}{\text{\tiny$\bm{\rightarrow}$}}}{\mathbbl{v}}^{\dagger}\mathbbl{H}\right|=|m_{q}|^{2}+|m_{b}|^{2}=\Sigma_{+}^{2}+\Sigma_{-}^{2}$
(96) $\displaystyle\Xi\eta\Gamma$ $\displaystyle=\Sigma_{-}^{2}\hskip
26.00009pt(1-\Xi)\eta\Gamma=\Sigma_{+}^{2}$ (97)

Finally, dividing Eq. 97 by Eq. 96 gives the dephasing parameter in terms of
the principle squeezing levels and total observed noise magnitude.

$\displaystyle\Xi$
$\displaystyle=\frac{\Sigma_{-}^{2}}{|m_{q}|^{2}+|m_{b}|^{2}}$ (98)

The specific formulas 36 and 37 follow from the analytic computation of the
SVD for 2-by-2 matrices, which generates a specific expression for the
singular values, but is too unwieldy to include for the exact angle
$\theta(\Omega)\equiv\theta_{\text{D}}$. Instead, an approximation to
$\theta_{\text{D}}$ is given in the limit of small $\Xi$.

##  Appendix B Including Phase Noise with Dephasing

The dephasing parameter $\Xi$ is derived as an intrinsic parameter due only to
the optical system; however, it enters the response Eq. 9 exactly the same as
the non-intrinsic phase noise $\phi^{2}_{\text{rms}}$ evaluated in Eq. 6.
Applying the same expectation operator on Eq. 9 as was done for Eq. 6,
generates this sequence of squeeze-level parameters $S_{0,1,2}$.

$\displaystyle S_{0\pm}$ $\displaystyle=e^{\pm 2r}$ (99) $\displaystyle
S_{1\pm}$
$\displaystyle=(1-\phi^{2}_{\text{rms}})S_{0\pm}+\phi^{2}_{\text{rms}}S_{0\mp}$
(100) $\displaystyle S_{2\pm}$ $\displaystyle=(1-\Xi)S_{1\pm}+\Xi S_{1\pm}$
(101)

Which may be expanded and then collected into the effective dephasing
$\Xi_{\text{eff}}$.

$\displaystyle\Xi_{\text{eff}}(\Omega)$
$\displaystyle=\Xi+\phi^{2}_{\text{rms}}-2\Xi\phi^{2}_{\text{rms}}$ (102)

This equation maintains the limits that $0\leq\Xi_{\text{eff}}\leq 0.5$.

##  Appendix C Derivations of Passive Transmission Response

The response metrics for passive cavities of Eqs. 51 to 53 can certainly be
derived using Section IV, but the passivity constraints provide an alternative
derivation. This derivation provides some meaningful insight as it can be done
more natively using cavity transfer functions $\mathfrak{h}(\Omega)$. This
work chooses to only represent 2-photon matrices in the quadrature basis of
$\hat{q}(\Omega)$ and $\hat{p}(\Omega)$, rather than the sideband basis used
for $\hat{a}(\Omega)$ and $\hat{a}^{\dagger}(-\Omega)$. One can transform
between the two using the $\mathbbl{A}$ matrices defined below. For a passive
system, $\mathbbl{H}$ can be calculated using only $\mathfrak{h}(\Omega)$,
basis-changing Eq. 27, as:

$\displaystyle\mathbbl{H}(\Omega)=\mathbbl{A}\begin{bmatrix}\mathfrak{h}(+\Omega)&0\\\
0&{\mathfrak{h}}^{*}(-\Omega)\end{bmatrix}\mathbbl{A}^{-1}$ (103)
$\displaystyle\mathbbl{A}$
$\displaystyle=\frac{1}{\sqrt{2}}\begin{bmatrix}1&1\\\ -i&i\end{bmatrix}$
$\displaystyle\mathbbl{A}^{-1}$
$\displaystyle=\frac{1}{\sqrt{2}}\begin{bmatrix}1&i\\\ 1&-i\end{bmatrix}$
(104)

For a passive system $\Gamma=1$, so Eq. 33 simplifies to

$\displaystyle\eta(\Omega)S(\phi,r)$
$\displaystyle=\left|\overset{\raisebox{-2.0pt}{\text{\tiny$\bm{\rightarrow}$}}}{\mathbbl{v}}^{\dagger}\mathbbl{H}\mathbbl{R}(\phi)\mathbbl{S}(r)\right|^{2}$
(105)

When $\mathfrak{h}(\Omega)$ is reduced by loss, Eq. 105 must be extended to
include $\mathbbl{T}$ terms to couple in un-squeezed vacuum. The passivity
condition Eq. 50 includes every loss source individually accounted, but they
can be collected into the complementary loss transfer function
$\mathfrak{h}_{\text{loss}}(\Omega)$.

$\displaystyle\mathbbl{H}_{\text{loss}}$
$\displaystyle=\mathbbl{A}\begin{bmatrix}\mathfrak{h}_{\text{loss}}(+\Omega)&0\\\
0&{\mathfrak{h}}^{*}_{\text{loss}}(-\Omega)\end{bmatrix}\mathbbl{A}^{-1}$
(106) $\displaystyle\mathbbl{1}$
$\displaystyle=\mathbbl{H}\mathbbl{H}^{\dagger}+\mathbbl{H}_{\text{loss}}\mathbbl{H}_{\text{loss}}^{\dagger}$
(107)

The conservation of phase space under the given assumptionsimposes the
constraint

$\displaystyle|\mathfrak{h}_{\text{loss}}(\pm\Omega)|^{2}=1-|\mathfrak{h}(\pm\Omega)|^{2}$
(108)

The total noise of Eq. 32 can then be expressed

$\displaystyle N$
$\displaystyle=\overset{\raisebox{-2.0pt}{\text{\tiny$\bm{\rightarrow}$}}}{\mathbbl{v}}^{\dagger}\left(\mathbbl{H}\mathbbl{R}(\phi)\mathbbl{S}(r)\mathbbl{S}^{\dagger}(r)\mathbbl{R}(\phi)^{\dagger}\mathbbl{H}^{\dagger}+\mathbbl{H}_{\text{loss}}\mathbbl{H}_{\text{loss}}^{\dagger}\right)\overset{\raisebox{-2.0pt}{\text{\tiny$\bm{\rightarrow}$}}}{\mathbbl{v}}$
(109)

Together, the efficiency $\eta$ is calculated

$\displaystyle(1-\eta)$
$\displaystyle=|\overset{\raisebox{-2.0pt}{\text{\tiny$\bm{\rightarrow}$}}}{\mathbbl{v}}^{\dagger}\mathbbl{H}_{\text{loss}}|^{2}=1-\frac{|\mathfrak{h}(+\Omega)|^{2}+|\mathfrak{h}(-\Omega)|^{2}}{2}$
(110)

Now, for the remaining parameters, some factorizations into magnitude and
phase components are needed.

$\displaystyle\mathfrak{h}(\pm\Omega)$
$\displaystyle=|\mathfrak{h}(\pm\Omega)|e^{i\theta_{\pm}}$ (111)
$\displaystyle\mathbbl{H}(\Omega)$
$\displaystyle=\mathbbl{A}\begin{bmatrix}|\mathfrak{h}(+\Omega)|e^{i\theta_{+}}&0\\\
0&|\mathfrak{h}(-\Omega)|e^{-i\theta_{-}}\end{bmatrix}\mathbbl{A}^{-1}$ (112)

The factorizations then enable an SVD-like decomposition into common and
differential magnitudes and phase.

$\displaystyle C(\Omega)$
$\displaystyle\equiv\frac{|\mathfrak{h}(+\Omega)|+|\mathfrak{h}(-\Omega)|}{2}$
$\displaystyle\theta_{C}(\Omega)$
$\displaystyle\equiv\frac{\theta_{+}-\theta_{-}}{2}$ (114) $\displaystyle
D(\Omega)$
$\displaystyle\equiv\frac{|\mathfrak{h}(+\Omega)|-|\mathfrak{h}(-\Omega)|}{2}$
$\displaystyle\theta_{D}(\Omega)$
$\displaystyle\equiv\frac{\theta_{+}+\theta_{-}}{2}$ (115)
$\displaystyle\mathbbl{H}(\Omega)$
$\displaystyle=\mathbbl{A}e^{i\theta_{C}}\begin{bmatrix}(C+D)e^{i\theta_{D}}&0\\\
0&(C-D)e^{-i\theta_{D}}\end{bmatrix}\mathbbl{A}^{-1}$ (116)

The rotation operator $\mathbbl{R}(\phi)$ is a result of phase in the sideband
picture, and allows the decomposition to be reduced

$\displaystyle\mathbbl{R}(\phi)$
$\displaystyle=\mathbbl{A}\begin{bmatrix}e^{i\phi}&0\\\
0&e^{-i\phi}\end{bmatrix}\mathbbl{A}^{-1}$ (117)
$\displaystyle\mathbbl{H}(\Omega)$
$\displaystyle=e^{i\theta_{C}}\left(C\mathbbl{R}(\theta_{D})+iD\mathbbl{R}(\theta_{D}-\tfrac{\pi}{2})\right)$
(118)

plugging this back into Eq. 105 gives

$\displaystyle N$ $\displaystyle=\left|\begin{bmatrix}0\\\
1\end{bmatrix}^{\dagger}(C\mathbbl{1}-D\mathbbl{\bbsigma})\mathbbl{R}(\theta_{D}+\phi-\xi)\mathbbl{S}\right|^{2}$
$\displaystyle\mathbbl{\bbsigma}$ $\displaystyle=\begin{bmatrix}0&-i\\\
i&0\end{bmatrix}$ (119)

Using $\xi{-}\phi=\theta_{D}$ for simplicity, this then gives the final phase-
quadrature power spectrum of:

$\displaystyle N(\Omega)$ $\displaystyle=C^{2}e^{-2r}+D^{2}e^{+2r}+(1-\eta)$
(120) $\displaystyle=\eta\left((1-\Xi)e^{-2r}+\Xi e^{+2r}\right)+(1-\eta)$
(121)

Where the the second line is a result of the following relations:

$\displaystyle\Xi$ $\displaystyle=D^{2}/\eta\hskip 30.00005pt\eta=C^{2}+D^{2}$
(122)

Relaxing $\xi{-}\phi=\theta_{D}$ can be done to indicate the squeezing angle
dependence, but from the above relations, Eqs. 51 to 53 follow.

##  Appendix D Double Cavity Approximations

The transfer function equations Eqs. 60 to 64 are a reduced representation of
a double cavity system designed for resonant sideband extraction. Those
equations give the reflectivity factorized into roots, zeros and poles, from
which analytical expressions can be more easily manipulated. Those roots
represent a low order approximation of the response of two cavities, each with
differing frequency response. The interaction between the cavities from the
common mirror, the arm input mirror, causes a complicated responses that is
sensitive to multiple scales of bandwidth, delay time, and resonant
enhancement. The reflectivity transfer function of a single transverse mode
can be expressed exactly, using:

$\displaystyle\mathfrak{r}_{\text{A}}(\Omega)$
$\displaystyle=r_{\text{a}}-\frac{T_{\text{a}}\sqrt{1-\Lambda_{\text{a}}}e^{-i\Omega
2L_{\text{a}}/c+i\psi_{\text{a}}}}{1-r_{\text{a}}\sqrt{1-\Lambda_{\text{a}}}e^{-i\Omega
2L_{\text{a}}/c+i\psi_{\text{a}}}}$ (123)
$\displaystyle\mathfrak{r}_{\text{S}}(\Omega)$
$\displaystyle=r_{\text{s}}-\frac{T_{\text{s}}\mathfrak{r}_{\text{A}}(\Omega)\sqrt{1-\Lambda_{\text{s}}}e^{i\Omega
2L_{\text{s}}/c+i\psi_{\text{s}}}}{1-r_{\text{s}}\mathfrak{r}_{\text{A}}(\Omega)\sqrt{1-\Lambda_{\text{s}}}e^{i\Omega
2L_{\text{s}}/c+i\psi_{\text{s}}}}$ (124)

Where $\mathfrak{r}_{\text{S}}$ is the reflectivity of the combined cavity
system off of the signal recycling mirror with reflectivity $r_{\text{s}}$.
$\mathfrak{r}_{\text{A}}$ is the reflectivity of the arm alone, ignoring the
effect of the coupling of the cavities. In LIGO’s operating regime,
$T_{\text{a}}$ is small, to allow a large build up of arm power of the carrier
field. $T_{\text{s}}$ is large to create a low finesse cavity that only
moderately widens the arm bandwidth to be above the frequencies of
astrophysical signals. The combination of low and high cavity finesses, as
well as the discrepancy in the lengths of the arm and SEC cavities, makes
Taylor expansions or Padé approximants of Eq. 124 nontrivial to construct[35].
Furthermore, approximants tend to operate only in a limited parameter regime.
To create the approximations used in this work, the following relations are
used:

$\displaystyle T_{\text{a}}$ $\displaystyle=1-\mathfrak{r}_{\text{a}}^{2}$
$\displaystyle T_{\text{s}}$ $\displaystyle=1-\mathfrak{r}_{\text{s}}^{2}$
(125) $\displaystyle\mathfrak{r}_{\text{a}}$ $\displaystyle=1-u_{\text{a}}$
$\displaystyle\mathfrak{r}_{\text{s}}$ $\displaystyle=1-u_{\text{s}}$ (126)
$\displaystyle\sqrt{1-\Lambda_{\text{a}}}$ $\displaystyle\approx
1-\Lambda_{\text{a}}/2$ $\displaystyle\sqrt{1-\Lambda_{\text{s}}}$
$\displaystyle\approx 1-\Lambda_{\text{s}}/2$ (127)

The phase shift terms $\Psi_{\\{\\}}$, which can represent either length
detunings or the Gouy phase, are set to 0 for the fundamental mode. The
exponential term for the propagation delay is then substituted for a Padé
approximant.

$\displaystyle e^{-i\Omega 2L/c}$ $\displaystyle\approx\frac{c-i\Omega
L}{c+i\Omega L}$ (128)

From there, Eq. 123 is substituted into Eq. 124 and expanded using computer
algebra software. Now, terms are progressively dropped and the transfer
function is tested again to the exact one, maintaining the magnitude and phase
response as best as possible, even when the loss terms $\Lambda_{\\{\\}}$ are
nonzero. This leads to the following second order rational form.

$\displaystyle\mathfrak{r}_{\text{S}}(\Omega)$
$\displaystyle\approx\frac{a_{2}s^{2}+a_{1}s+a_{0}}{b_{2}s^{2}+b_{1}s+b_{0}}$
$\displaystyle s$ $\displaystyle=i\Omega$ (129) $\displaystyle a_{2}$
$\displaystyle=2L_{\text{a}}L_{\text{s}}$ $\displaystyle a_{0}$
$\displaystyle=c^{2}\left(\textstyle
u_{\text{a}}-\frac{\Lambda_{\text{a}}u_{\text{s}}}{4}\right)$ (130)
$\displaystyle b_{2}$ $\displaystyle=2L_{\text{a}}L_{\text{s}}$ $\displaystyle
b_{0}$ $\displaystyle=c^{2}\left(\textstyle
u_{\text{a}}+\frac{\Lambda_{\text{a}}u_{\text{s}}}{4}\right)$ (131)
$\displaystyle a_{1}$
$\displaystyle=cL_{\text{a}}\left(\textstyle\frac{\Lambda_{\text{s}}}{2}(1-\frac{u_{\text{a}}}{2})-u_{\text{s}}\right)$
(132) $\displaystyle b_{1}$
$\displaystyle=cL_{\text{a}}\left(\textstyle\frac{\Lambda_{\text{s}}}{2}(1-u_{\text{s}}-\frac{u_{\text{a}}}{4})+u_{\text{s}}\right)$
(133)

The rational form is then factored into roots using an approximation of the
quadratic formula. Notably, the $a_{1}$ and $b_{1}$ terms have different
numbers of summed terms, leading to the poles and zeros also having different
numbers of terms. By splitting the roots into bandwidth, $\gamma$, and loss,
$\lambda$, contributions, the presence of the loss-related terms in the poles
and zeros is symmetrized.

$\displaystyle\gamma_{\text{A}}-\lambda_{\text{A}}$
$\displaystyle\approx-\frac{a_{0}}{a_{1}}$
$\displaystyle\gamma_{\text{A}}+\lambda_{\text{A}}$
$\displaystyle\approx-\frac{b_{0}}{b_{1}}$ (134)
$\displaystyle\gamma_{\text{S}}-\lambda_{\text{S}}$
$\displaystyle\approx-\frac{a_{1}}{a_{2}}+\frac{a_{0}}{a_{1}}$
$\displaystyle\gamma_{\text{S}}+\lambda_{\text{S}}$
$\displaystyle\approx-\frac{b_{1}}{b_{2}}+\frac{b_{0}}{b_{1}}$ (135)

Solving for the individual loss and bandwidth factors for each cavity then
leads to Eqs. 60 to 63, and plugging them back in to Eq. 129 leads to Eq. 64.

##  Appendix E Multiple Transverse Mode Interferometer Model

The effort of this paper is primarily to produce simplified models the
squeezing response in light of transverse mismatch and radiation pressure. To
validate those models, it is useful to compare against a more complete, though
opaque, model that includes the exact cavity response with radiation pressure,
detuning, losses, and transverse modal mismatch. With the exception of
transverse mismatch, such a model is established and widely used for noise
modeling of for LIGO-like interferometers[12, 34]. This model is succinctly
derived here in a manner that allows transverse mismatch to be incorporated.
For simplicity, and to provide a more direct comparison, this is done for a
double cavity representing a perfectly symmetric interferometer. Future work
should simulate interferometers with arm imbalances to compare against this
exact case, but that is beyond the scope chosen here.To incorporate all of the
listed elements in an exact model, a product space is necessary to maintain
the two-photon response of each optical element across multiple inter-coupled
transverse modes. Here, only two such transverse modes are used, the
fundamental Gaussian mode and a single higher order mode such as the LG1 for
beam mismatch, or the HG01 for a misalignment. The interaction between the
modes conserves the phase space and does not leak into yet higher modes.
Similarly to Eq. 71, this is not a perfectly physical choice, but a convenient
one that is valid for small mixing between the modes. The squeezing, rotation,
and mode mixing matrices in this product space are defined below in terms of
their two-photon definitions. The rotation matrix takes on two parameters, one
common rotation $\phi$, representing a phase shift of both modes, and one
$\psi$ for the rotation solely of the higher order mode (HOM).

$\displaystyle\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{S}}(r)$
$\displaystyle\equiv\begin{bmatrix}\mathbbl{S}(r)&\mathbbl{0}\\\
\mathbbl{0}&\mathbbl{1}\end{bmatrix}$
$\displaystyle\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{R}}(\phi,\psi)$
$\displaystyle\equiv\begin{bmatrix}\mathbbl{R}(\phi)&\mathbbl{0}\\\
\mathbbl{0}&\mathbbl{R}(\psi)\mathbbl{R}(\phi)\end{bmatrix}$ (136)

The mismatch loss coupling matrix
$\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{U}}$
maintains the same parameters as before Eq. 71. The HOM phase shift term must
be converted into a quadrature rotation, and the common phase $\theta$ is
omitted.

$\displaystyle\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{U}}(\Upsilon,\psi)$
$\displaystyle\equiv\begin{bmatrix}\sqrt{1-\Upsilon}\mathbbl{1}&-\sqrt{\Upsilon}\mathbbl{R}(\psi)\\\
\sqrt{\Upsilon}\mathbbl{R}(-\psi)&\sqrt{1-\Upsilon}\mathbbl{1}\end{bmatrix}$
(137)

Additionally, basis vectors in this space are defined to simplify the
expression of single-element matrices as well as create projections for
observables.

$\displaystyle\begin{bmatrix}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Rightarrow}$}}}{\mathbf{e}}_{q0}&\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Rightarrow}$}}}{\mathbf{e}}_{p0}&\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Rightarrow}$}}}{\mathbf{e}}_{q1}&\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Rightarrow}$}}}{\mathbf{e}}_{p1}\end{bmatrix}$
$\displaystyle\equiv\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{1}}$
(138)

As in the scalar transfer function case of Appendix D, the reflectivity of the
interferometer double cavity system from the signal-recycling mirror is
needed. To make analogous equations to Eq. 123 and Eq. 124, most of the same
scalar factors are needed, but now in the product space.

$\displaystyle\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{r}}_{\text{a}}$
$\displaystyle=\sqrt{1-T_{\text{a}}}\cdot\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{1}}$
$\displaystyle\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{t}}_{\text{a}}$
$\displaystyle=\sqrt{T_{\text{a}}}\cdot\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{1}}$
(139)
$\displaystyle\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{r}}_{\text{e}}$
$\displaystyle=\sqrt{1-T_{\text{e}}}\cdot\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{1}}$
$\displaystyle\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{t}}_{\text{e}}$
$\displaystyle=\sqrt{T_{\text{e}}}\cdot\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{1}}$
(140)
$\displaystyle\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{r}}_{\text{s}}$
$\displaystyle=\sqrt{1-T_{\text{s}}}\cdot\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{1}}$
$\displaystyle\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{t}}_{\text{s}}$
$\displaystyle=\sqrt{T_{\text{s}}}\cdot\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{1}}$
(141)
$\displaystyle\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{\eta}}_{\text{a}}$
$\displaystyle=\sqrt{1-\Lambda_{\text{a}}}\cdot\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{1}}$
$\displaystyle\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{\eta}}_{\text{s}}$
$\displaystyle=\sqrt{1-\Lambda_{\text{s}}}\cdot\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{1}}$
(142)
$\displaystyle\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{L}}_{\text{a}}$
$\displaystyle=e^{i\Omega
L_{\text{a}}/c}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{R}}(k\Delta
L_{\text{a}},\psi_{\text{a}})$
$\displaystyle\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{L}}_{\text{s}}$
$\displaystyle=e^{i\Omega
L_{\text{s}}/c}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{R}}(k\Delta
L_{\text{s}},\psi_{\text{s}})$ (143)

The transmission delay matrices
$\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{L}}_{\text{a}}$
and
$\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{L}}_{\text{s}}$
use the identity Eq. 117 in this larger space. For them, the HOM picks up the
single pass Gouy phase of the arm $\psi_{\text{a}}$ and of the SEC
$\psi_{\text{s}}$. The $\Delta L_{\text{a}}$ and $\Delta L_{\text{a}}$ are
microscopic detuning lengths for each cavity.The final new component of this
double cavity matrix model is the radiation pressure. This is added ad-hoc as
a modification to the reflectivity of each of the arm cavity mirrors. It
couples the amplitude and phase quadratures only in the fundamental mode, as
that is the mode that the large carrier power $P_{\text{A}}$ resonates in each
arm. The modified reflectivities are
$\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{\rho}}_{\text{a}}$
and
$\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{\rho}}_{\text{e}}$
for the input and end mirror respectively.

$\displaystyle\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{\rho}}_{\text{a}}$
$\displaystyle=\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{r}}_{\text{a}}+8k\cdot\chi(\Omega)\cdot\frac{P_{\text{A}}}{c}\cdot\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Rightarrow}$}}}{\mathbf{e}}_{p0}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Rightarrow}$}}}{\mathbf{e}}_{q0}^{\dagger}$
(144)
$\displaystyle\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{\rho}}_{\text{e}}$
$\displaystyle=\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{r}}_{\text{e}}+8k\cdot\chi(\Omega)\cdot\frac{P_{\text{A}}}{c}\cdot\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Rightarrow}$}}}{\mathbf{e}}_{p0}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Rightarrow}$}}}{\mathbf{e}}_{q0}^{\dagger}$
(145)

The mode mismatch of this model can be added not only between the external
elements of the squeezing, interferometer and readout as shown in Fig. 5, but
also within the interferometer. The input and output mismatch matrices of
Section VI are given by
$\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{U}}_{\text{I}}$
and
$\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{U}}_{\text{O}}$,
while the mismatch between the signal recycling cavity and the arm cavity is
given by
$\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{U}}_{\text{A}}$.
These matrices act as basis transformations into and out of the respective
component basis, where the mismatch loss of the fundamental is given by an
$\Upsilon_{\\{\\}}$ parameter, and there is a (generally unknown) mismatch
phasing $\psi_{\\{\\}}$.

$\displaystyle\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{U}}_{\text{A}}$
$\displaystyle=\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{U}}(\Upsilon_{A},\psi_{A})$
$\displaystyle\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{U}}_{\text{I}}$
$\displaystyle=\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{U}}(\Upsilon_{\text{I}},\psi_{\text{I}})$
$\displaystyle\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{U}}_{\text{O}}$
$\displaystyle=\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{U}}(\Upsilon_{\text{O}},\psi_{\text{O}})$
(146)

The following Eqs. 147 to 150 are the extensions of Eq. 123 and Eq. 124 into
the product space. It is solved using noncommutative Gaussian elimination
first on the arm, then on the signal recycling cavity.
$\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{F}}_{\text{A}}$
is the round-trip closed-loop propagator from the end-mirror back to itself
via the input mirror. As such, it enters at a specific point – immediately
after the end mirror reflectivity
$\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{\rho}}_{\text{e}}$
– in the round-trip propagation sequence of the arm cavity reflectivity
$\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{r}}_{\text{A}}$.
Given the placements of the
$\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{U}}_{\text{A}}$
factors, the arm cavity reflectivity is in the modal basis of the signal
recycling cavity.

$\displaystyle\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{F}}_{\text{A}}$
$\displaystyle=\left(\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{1}}-\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{\eta}}_{\text{a}}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{\rho}}_{\text{e}}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{L}}_{\text{a}}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{\rho}}_{\text{a}}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{L}}_{\text{a}}\right)^{-1}$
(147)
$\displaystyle\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{r}}_{\text{A}}$
$\displaystyle=\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{U}}_{A}^{-1}\left(\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{r}}_{\text{a}}-\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{t}}_{\text{a}}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{L}}_{\text{a}}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{F}}_{\text{A}}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{\eta}}_{\text{a}}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{\rho}}_{\text{e}}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{L}}_{\text{a}}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{t}}_{\text{a}}\right)\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{U}}_{A}$
(148)
$\displaystyle\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{F}}_{\text{S}}$
$\displaystyle=\left(\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{1}}-\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{\eta}}_{\text{s}}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{r}}_{\text{A}}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{L}}_{\text{s}}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{r}}_{\text{s}}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{L}}_{\text{s}}\right)^{-1}$
(149)
$\displaystyle\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{r}}_{\text{S}}$
$\displaystyle=\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{U}}_{\text{I}}^{-1}\left(\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{r}}_{\text{s}}-\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{t}}_{\text{s}}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{L}}_{\text{s}}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{F}}_{\text{S}}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{\eta}}_{\text{s}}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{r}}_{\text{A}}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{L}}_{\text{s}}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{t}}_{\text{s}}\right)\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{U}}_{\text{I}}$
(150)

The propagator and reflectivity of the signal recycling cavity are constructed
similarly to the arm and, like Eq. 124, use the arm cavity reflectivity
instead of the arm input mirror reflectivity. This follows from the particular
ordering chosen during Gaussian elimination. The placements of
$\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{U}}_{\text{I}}$
indicate that
$\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{r}}_{\text{S}}$
is in the basis of the squeezer input beam.Along with the reflectivity, all of
the
$\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{H}}$
propagation and
$\\{\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{T}}\\}$
loss matrices of Section IV must be constructed. For brevity, the broadband
input and output losses from $\Lambda_{\text{I}}$ and $\Lambda_{\text{O}}$ are
not included, but are simple to incorporate. Instead, the internal
interferometer losses from the arm and signal recycling cavities are
calculated using the transmission matrices

$\displaystyle\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{T}}_{\text{R,$\text{S}$}}$
$\displaystyle=\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{t}}_{\text{S}}\sqrt{\Lambda_{\text{s}}}$
$\displaystyle\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{t}}_{\text{S}}$
$\displaystyle=\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{U}}_{\text{I}}^{-1}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{t}}_{\text{s}}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{L}}_{\text{s}}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{F}}_{\text{S}}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{\eta}}_{\text{s}}$
(151)
$\displaystyle\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{T}}_{\text{R,$\text{A}$}}$
$\displaystyle=\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{t}}_{\text{A}}\sqrt{T_{\text{e}}+\Lambda_{\text{a}}}$
$\displaystyle\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{t}}_{\text{A}}$
$\displaystyle=\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{t}}_{\text{S}}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{U}}_{A}^{-1}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{t}}_{\text{a}}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{L}}_{\text{a}}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{F}}_{\text{A}}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{\eta}}_{\text{a}}$
(152)

The output mode cleaner is applied to the reflectivity to create the total
propagation of squeezing
$\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{H}}$.
Because of the output mode cleaner, the homodyne readout projects solely in
the fundamental mode, while the LO angle $\xi\approx 0$ has some freedom due
to contrast defect in the interferometer. Together, these generate the
effective observation vector $\vec{m}$, which retains only 2 elements and is
directly applicable to the formulas of Section IV.

$\displaystyle\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{H}}$
$\displaystyle=\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{U}}_{\text{O}}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{r}}_{\text{S}}$
$\displaystyle m_{q}$
$\displaystyle=\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Rightarrow}$}}}{\mathbf{v}}^{\dagger}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{H}}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Rightarrow}$}}}{\mathbf{e}}_{q0}$
(153)
$\displaystyle\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Rightarrow}$}}}{\mathbf{v}}^{\dagger}$
$\displaystyle=\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Rightarrow}$}}}{\mathbf{e}}_{p0}^{\dagger}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{R}}(\zeta,0)$
$\displaystyle m_{p}$
$\displaystyle=\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Rightarrow}$}}}{\mathbf{v}}^{\dagger}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{H}}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Rightarrow}$}}}{\mathbf{e}}_{p0}$
(154)

While $\vec{m}$ still has only 2 elements, the vectors in the norms for the
total noise gain are still in the product space, here dimension 4. This sums
the higher order mode terms as contributors to loss, decreasing the efficiency
in the computation $\eta=|\vec{m}|^{2}/\Gamma$.

$\displaystyle\Gamma(\Omega)$
$\displaystyle=\left|\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Rightarrow}$}}}{\mathbf{v}}^{\dagger}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{H}}\right|^{2}+\left|\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Rightarrow}$}}}{\mathbf{v}}^{\dagger}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{T}}_{\text{R,$\text{a}$}}\right|^{2}+\left|\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Rightarrow}$}}}{\mathbf{v}}^{\dagger}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{T}}_{\text{R,$\text{s}$}}\right|^{2}$
(155)

From the previous expressions, the noise $N(\Omega)$ can be entirely
calculated, so squeezing can be examined. With all the machinery, it is also
useful to determine the optical gain and signal sensitivity. Below,
$\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Rightarrow}$}}}{\mathbf{S}}$ is
the signal field generated by displacement modulations $x(\Omega)$.
Displacements create phase modulations in the fundamental mode at the end
mirror. The factor of $1/\sqrt{2}$ is from the presence of the beamsplitter.
The field magnitude in the transverse mode and quadrature observed by the
homodyne is given by $S(\Omega)$.

$\displaystyle S(\Omega)$
$\displaystyle=\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Rightarrow}$}}}{\mathbf{v}}^{\dagger}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Rightarrow}$}}}{\mathbf{S}},$
$\displaystyle\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Rightarrow}$}}}{\mathbf{S}}$
$\displaystyle=\frac{1}{\sqrt{2}}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{U}}_{\text{O}}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{t}}_{\text{A}}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Rightarrow}$}}}{\mathbf{e}}_{p0}\cdot
2k\sqrt{\frac{P_{\text{A}}}{\hbar\omega}}x(\Omega)$ (156)

The signal sensitivity can then be used to define the optical gain and
sensitivity in terms of spectral density as per Section II.1, Eq. 15. Since
$s$ and
$\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Rightarrow}$}}}{\mathbf{s}}$ are
in units of quanta/second, rather than unitful power, the factor of $1/2$ in
$G$ represents the half quanta of vacuum noise.

$\displaystyle g(\Omega)$
$\displaystyle=\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Rightarrow}$}}}{\mathbf{v}}^{\dagger}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{U}}_{\text{O}}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{t}}_{\text{A}}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Rightarrow}$}}}{\mathbf{e}}_{p0},$
$\displaystyle G(\Omega)$
$\displaystyle=\frac{1}{2}\left(L_{\text{A}}\frac{\text{d}S(\Omega)}{\text{d}x(\Omega)}\right)^{-2}$
(157)

##  Appendix F Radiation Pressure Region

The previous appendix E derived the interferometer squeezing and signal
response in full generality. Due to its generality, the full expressions
obscure the physics of the ideal and nearly ideal cases. This appendix
specifically investigates the interaction of external mode mismatch on the
QRPN, where the interferometer itself is lossless and perfectly on resonance.
The coherent effects of modal mismatch and the coherent effects in QRPN could
provide an alternative explanation of the variation of $\Gamma$ required to
explain the loss measurement in the LLO data.This section concludes that is
not the case, and that the LO angle $\xi\neq 0$ variation is a more valid
explanation. This derivation also indicates some limitations in using
squeezing and QRPN as a diagnostic of the arm power, as the observed
$\Gamma(\Omega)$ and $\theta(\Omega)$ do have some dependence on
$\Upsilon_{\text{I}}$ and $\Upsilon_{\text{O}}$, and their dependence mimics
lower power in measurements.To model the ideal interferometer with mismatch,
the matrices of Section IV.1 simply need to be extended into the product space
of Appendix E to incorporate additional transverse modes. The reflectivity
matrix
$\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{R}}_{\text{S}}$
is naturally in a lower diagonal form when the interferometer is on-resonance
and has no mismatch, as all of the matrices entering in Eqs. 147 to 150 are
either diagonal or lower triangular, so the matrix inverses simplify greatly,
as each diagonal in the inverse becomes a formula like Eq. 124. In the
inverses, the off diagonal term for the radiation pressure interaction picks
up the optical gain to become Eq. 12. Together, the ideal interferometer
reflectivity for the squeezing field becomes:

$\displaystyle\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{r}}_{\text{S}}(\Omega)$
$\displaystyle=\begin{bmatrix}\mathfrak{r}_{2}(\Omega)&0&0&0\\\
\mathcal{K}(\Omega)&\mathfrak{r}_{2}(\Omega)&0&0\\\ 0&0&1&0\\\
0&0&0&1\end{bmatrix}$ (158)

Mode mismatch is then added similarly to Section VI, except using this product
space representation.

$\displaystyle\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{H}}_{\text{R}}(\Omega)$
$\displaystyle=\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{U}}(\Upsilon_{\text{O}},\psi_{\text{O}})\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{G}}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{U}}^{\dagger}(\Upsilon_{\text{I}},\psi_{\text{I}})\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{r}}_{\text{S}}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{U}}(\Upsilon_{\text{I}},\psi_{\text{I}})$
(159)

The resulting expressions are complicated, as all the interactions are
coherent. Here, row of
$\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{H}}_{\text{R}}$
corresponding to the phase quadrature of the fundamental mode is presented, as
that is the only relevant output to calculate squeezing in the ideal case,
using phase quadrature readout
$\overset{\raisebox{-2.0pt}{\text{\tiny$\bm{\rightarrow}$}}}{\mathbbl{v}}^{\dagger}=\overset{\raisebox{-2.0pt}{\text{\tiny$\bm{\rightarrow}$}}}{\mathbbl{e}}_{p0}^{\dagger}$.

$\displaystyle\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Rightarrow}$}}}{\mathbf{e}}_{p0}^{\dagger}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{H}}_{\text{R}}(\Omega)\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Rightarrow}$}}}{\mathbf{e}}_{q0}$
$\displaystyle=\mathcal{K}\sqrt{1{-}\Upsilon_{\text{O}}}\left(1{-}\Upsilon_{\text{I}}+c\sqrt{\Upsilon_{\text{I}}\Upsilon_{\text{O}}}\cos(\psi)\right)$
$\displaystyle=\mathcal{K}\sqrt{1{-}\Upsilon_{\text{O}}}\left(1-\frac{\Upsilon_{\text{R}}}{4}\right)$
(160)
$\displaystyle\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Rightarrow}$}}}{\mathbf{e}}_{p0}^{\dagger}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{H}}_{\text{R}}(\Omega)\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Rightarrow}$}}}{\mathbf{e}}_{p0}$
$\displaystyle=\sqrt{1{-}\Upsilon_{\text{O}}}$ (161)
$\displaystyle\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Rightarrow}$}}}{\mathbf{e}}_{p0}^{\dagger}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{H}}_{\text{R}}(\Omega)\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Rightarrow}$}}}{\mathbf{e}}_{q1}$
$\displaystyle=-\mathcal{K}\sqrt{\Upsilon_{\text{I}}}\sqrt{1{-}\Upsilon_{\text{I}}}\sqrt{1{-}\Upsilon_{\text{O}}}$
$\displaystyle\hskip
20.00003pt-\mathcal{K}\Upsilon_{\text{I}}\sqrt{\Upsilon_{\text{O}}}\cos(\psi)-\sqrt{\Upsilon_{\text{O}}}\sin(\psi)$
(162)
$\displaystyle\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Rightarrow}$}}}{\mathbf{e}}_{p0}^{\dagger}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{H}}_{\text{R}}(\Omega)\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Rightarrow}$}}}{\mathbf{e}}_{p1}$
$\displaystyle=-\sqrt{\Upsilon_{\text{O}}}\cos(\psi)$ (163)

These 4 terms can be separated into the purely two-photon matrix form of
Section IV, appearing as:

$\displaystyle\begin{bmatrix}\mathbbl{H}_{\text{R}}&\mathbbl{\Lambda}_{\text{R}}\\\
\cdot&\cdot\end{bmatrix}$
$\displaystyle\equiv\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{H}}_{\text{R}}(\Omega)$
(164)

These can then be used to compute the squeezing response metrics:

$\displaystyle\eta(\Omega)\Gamma(\Omega)$
$\displaystyle=(1{-}\Upsilon_{\text{O}})\left(1+\mathcal{K}^{2}\left(1-\frac{\Upsilon_{\text{R}}}{2}\right)\right)$
(165) $\displaystyle\Gamma(\Omega)$
$\displaystyle=\left|\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Rightarrow}$}}}{\mathbf{e}}_{p0}^{\dagger}\overset{\raisebox{-1.0pt}{\text{\tiny$\bm{\Leftrightarrow}$}}}{\mathbf{H}}_{\text{R}}(\Omega)\right|^{2}=1+\mathcal{K}^{2}(1{-}\Upsilon_{\text{F}})$
(166) $\displaystyle\eta_{\text{R}}(\Omega)$
$\displaystyle=(1{-}\Upsilon_{\text{O}})\frac{1+\mathcal{K}^{2}\left(1-\Upsilon_{\text{R}}/2\right)}{1+\mathcal{K}^{2}(1{-}\Upsilon_{\text{F}})}$
(167) $\displaystyle\theta_{\text{R}}(\Omega)$
$\displaystyle=\arctan\left(|\mathcal{K}|\left(1-\frac{\Upsilon_{\text{R}}}{4}\right)\right)$
(168)

Notably, the transverse mismatch losses entail certain adjustments needed to
the noise gain $\Gamma(\Omega)$ and observation angle $\theta(\Omega)$. The
$\mathcal{K}$ term in $\Gamma$ is diminished by the readout losses in exactly
the same manner that the signal experiences. This corresponds to how the
quantum noise in amplitude causes force and displacement that mimics signals,
so the quantum noise gain must be enhanced or reduced to the same
degree.$\theta(\Omega)$ behaves differently, as it reacts to both interactions
which coherent modify it at once. The influence of mismatch losses bias its
estimate of $\mathcal{K}$, and thus the arm power, downwards. This happens
because the mode mismatch moves some amount of squeezing out of the
interferometer mode, preventing it from experiencing $\mathcal{K}$, then moves
it back. This effectively causes $\mathcal{K}$ to appear reduced.The
efficiency $\eta_{\text{R}}$ is also affected by the mismatch loss and has a
slightly different adjustment of its $\mathcal{K}$ terms between the numerator
and denominator. This causes the orange model curves in the middle plots of
loss in Fig. 3 to wiggle downwards at low frequencies. This wiggle doesn’t
have particular physical significance, and can be interpreted as evidence that
$\eta\Gamma$ is a more fundamentally useful metric than $\eta$ or $\Gamma$
alone. Notably, $\eta\Gamma$ has the same dependence on mismatch from
$\Upsilon_{\text{R}}$, per factor of $\mathcal{K}$, as $\theta$.