# Optimizing the phase sensitivity of a Michelson interferometer with a two
mode squeezed coherent input

Stav Haldar  Hearne Institute for Theoretical Physics, Department of Physics
and Astronomy, Louisiana State University, Baton Rouge, Louisiana, 70803, USA
<EMAIL_ADDRESS>Pratik J. Barge  Hearne Institute for Theoretical Physics,
Department of Physics and Astronomy, Louisiana State University, Baton Rouge,
Louisiana, 70803, USA  Xiao-Qi Xiao  Department of Communication Engineering,
Shanghai Dianji University, Shanghai 200240, China  Hwang Lee  Hearne
Institute for Theoretical Physics, Department of Physics and Astronomy,
Louisiana State University, Baton Rouge, Louisiana, 70803, USA

###### Abstract

A Michelson-type interferometer with two-mode squeezed coherent state input is
considered. Such an interferometer has a better phase sensitivity over the
shot-noise limit by a factor of $e^{2r}$, where $r$ is the squeezing parameter
[Phys. Rev. A 102,022614 (2020)]. We show that when photon loss and noise in
the two arms is asymmetric an optimal choice of the squeezing angle can allow
improvement in phase sensitivity without any increase in input or pump power.
In particular, when loss occurs only in one arm of the interferometer, we can
have improvement in phase sensitivity for photon loss up to 80%. Hence, a
significant improvement can be made in several applications such as LiDAR,
gyroscopes and measuring refractive indices of highly absorptive/reflective
materials.

##  I Introduction

Quantum enhancement of phase estimation using optical interferometers is an
active area of research. The phase sensitivity of an interferometer using an
ordinary coherent light source scales as $1/\sqrt{n}$, where $n$ is the mean
number of input photons. This scaling limit which is due to the photon
counting error is called the shot-noise limit Zwierz  _et al._ (2012). Over
the last four decades, a lot of efforts have been made to overcome this limit.
Largely, there are three distinct approaches depending on whether it uses
squeezed states, photon-number states, or some combination of both. The first
one, squeezed-state approach, is to combine the ordinary coherent light with
squeezed state at the first beam splitter. It is the scheme that was proposed
by Caves for gravitational wave detection in the early 1980’s Caves (1981).
The phase sensitivity is shown to scale as $e^{-r}/\sqrt{n}$ or higher under
certain conditions Pezzé and Smerzi (2008); Seshadreesan  _et al._ (2011);
Lawrie  _et al._ (2019) with the squeezing parameter $r$. SU(1,1)
interferometers introduced by YMK, where the usual beam splitters are replaced
by four-wave mixers Yurke  _et al._ (1986), its coherently-boosted scheme
Plick  _et al._ (2010); Marino  _et al._ (2012); Hudelist  _et al._ (2014),
and the two-mode squeezed-vacuum scheme Anisimov  _et al._ (2010) can also be
included in this category. The second one, number-state approach, is typically
to use fixed number of photons distributed to two input ports of the
interferometer. Such an approach was first proposed by Yuen Yuen (1986) and
YMK Yurke  _et al._ (1986) in the 1980’s. Many different correlations between
the two-mode number states were proposed to go beyond the shot-noise limit
Holland and Burnett (1993); Han and Kim (1998); Kuzmich and Mandel (1998);
Berry and Wiseman (2000); Campos  _et al._ (2003); Pezzé and Smerzi (2006);
Lee  _et al._ (2009). The phase sensitivity in this case typically scales as
$1/n$, dubbed as the Heisenberg limit Higgins  _et al._ (2007); Nagata  _et
al._ (2007). Dual- Fock, or twin-Fock, states Holland and Burnett (1993); Han
and Kim (1998); Kuzmich and Mandel (1998); Campos  _et al._ (2003),
intelligent states Hillery and Mlodinow (1993); Brif and Mann (1996), and N00N
states Lee  _et al._ (2002); Resch  _et al._ (2007); Afek  _et al._ (2010) are
among the named correlated Fock states. The third category is the approach
that combines the squeezed states and the number states. This approach was
first proposed by Gerry et al. Birrittella  _et al._ (2012); Carranza and
Gerry (2012). More often than not the input state to the interferometer is
prepared by an operation—such as photon addition, subtraction, or
catalysis—made onto squeezed states to achieve quantum enhancement in phase
sensitivity Zhang  _et al._ (2021); Kumar  _et al._ (2022). These approaches
might as well be differentiated as gaussian states, non-gaussian states, and
non-gaussian operation on gaussian states in more general terms. It has been
shown in a recent paper that a coherently boosted two-mode squeezed state, or
two-mode squeezed coherent state (TMSCS) can be used as the interferometer
input to achieve sub-shot-noise phase sensitivity Xiao  _et al._ (2020). The
interferometer is a generic, ordinary SU(2) type with two beam splitters and
intensity difference measurement at the output—as opposed to the seeded
SU(1,1) type. The phase sensitivity in this case is shown to be
$e^{-2r}/\sqrt{n}$. We have a doubly enhanced phase sensitivity; one with
squeezing, the other with amplification. Thus, the value of $n$ is $e^{2r}$
times larger than the number of photons in the initial coherent states.

In the present work we consider a Michelson-type interferometer with TMSCS as
the input. First in Section II, we investigate the maximum amount of loss and
noise tolerable to maintain sensitivity below the shot-noise limit. In
particular, we report that by an optimal choice of the squeezing angle or
other input phases, the phase sensitivity can be enhanced and noise beyond the
3 dB limit tolerated. Next, in Section III, we analyze the complementary
radiation-pressure error. We find that the radiation pressure noise increases
with decreasing photon counting fluctuations, so that the standard quantum
limit remains intact. In Section IV we discuss our conclusions.

##  II Phase Sensitivity of a Michelson Interferometer

Here we describe the phase sensitivity of a Michelson interferometer (MI) with
loss and noise in both arms (See Fig. 1). The relative phase $\phi$, acquired
by the photons in one arm (variable arm) compared to the other arm (reference
arm) can be measured by its modulation of the photon number difference at the
two outputs of the interferometer (D1 and D2 in Fig. 1). This can in turn be
used to estimate the path difference between the two arms. The higher the
sensitivity of the photon number difference to the changes in relative phase
the more precise the interferometer. Here, in order to model a realistic
setting we assume loss and noise in both the reference arm and variable arm of
the interferometer.

Figure 1:  TMS: two mode squeezer, BS: beam splitter, PS: phase shifter, D1,2:
detectors. We consider that the two arms have transmissivity $\eta_{a}^{2}$
and $\eta_{b}^{2}$, respectively and thermal noise with mean photon numbers
$\bar{\gamma}_{a}$ and $\bar{\gamma}_{b}$, respectively. A two mode squeezed
coherent source is used as the input to the interferometer, created by passing
two coherent sources with average photon numbers $\alpha,\beta$ and phases
$\theta_{1},\theta_{2}$ through an optical parametric amplifier with squeezing
factor $r$ and squeezing angle $\xi$. The two 50:50 BS operations shown in
this schematic both occur at the sole beam splitter of the MI. The loss
parameters $\eta_{a,b}^{2}$ instead of $\eta_{a,b}$ have been used because for
the MI photons in each mode traverse their respective arms twice before
reaching the detectors D1 and D2.

Let $\ket{\Psi_{in}}_{ab}$ be the two-mode ($a$ and $b$) input state for the
interferometer. The output state going into the two detectors D1 and D2 is
then be given by:

$\displaystyle\ket{\psi_{out}}_{ab}$ $\displaystyle=$ (1)
$\displaystyle\hat{BS}_{ab}$ $\displaystyle\hat{BS}_{b}$
$\displaystyle(\eta_{b}^{2})\hat{BS}_{a}(\eta_{a}^{2})\hat{PS}_{ab}\hat{BS}_{ab}\ket{\Psi_{in}}_{ab},$

that is, the evolution of the state $\ket{\Psi_{in}}_{ab}$ through the
interferometer is given by the the operations of a 50:50 beam splitter –
$\hat{BS}_{ab}$, a relative phase-shift between the two arms –
$\hat{PS}_{ab}$, beam splitter operators modelling the loss and noise on both
arms – $\hat{BS}_{a}(\eta_{a}^{2})$ and $\hat{BS}_{b}(\eta_{b}^{2})$. Finally,
another operation of the 50:50 beam splitter recombines two beams.

The operators mentioned above are defined in the following way, by their
actions on the two modes $\hat{a}$ and $\hat{b}$ of the interferometer.

50:50 beam-splitter transformation:

$\displaystyle\hat{BS}_{ab}^{\dagger}\begin{bmatrix}\hat{a}^{\prime}\\\
\hat{b}^{\prime}\end{bmatrix}\hat{BS}_{ab}=\frac{1}{\sqrt{2}}\begin{bmatrix}1&i\\\
i&1\end{bmatrix}\begin{bmatrix}\hat{a}\\\
\hat{b}\end{bmatrix}=\begin{bmatrix}\hat{a}+i\hat{b}\\\
i\hat{a}+\hat{b}\end{bmatrix}.$ (2)

Beam-splitter transformations for loss and noise modelling act on one mode $a$
or $b$ of the interferometer and one thermal mode $c$ or $d$. The loss is
characterised by the transmissivity of the beam-splitter $\eta_{a}^{2}$ or
$\eta_{b}^{2}$ (higher transmissivity means lower loss). $\eta_{a,b}^{2}$
instead of $\eta_{a,b}$ have been used because for the MI photons in each mode
traverse their respective arms twice before reaching the detectors D1 and D2.
The noise is characterized by the mean photon numbers $\bar{\gamma}_{a,b}$ of
the external modes $c,d$. These external modes will eventually be traced out
(not measured). These operations are given by:

$\displaystyle\hat{BS}_{a}^{\dagger}\begin{bmatrix}\hat{a}^{\prime}\\\
\hat{c}^{\prime}\end{bmatrix}\hat{BS}_{a}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}\eta_{a}^{2}&i\sqrt{1-\eta_{a}^{4}}\\\
i\sqrt{1-\eta_{a}^{4}}&\eta_{a}^{2}\end{bmatrix}\begin{bmatrix}\hat{a}\\\
\hat{c}\end{bmatrix}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}\eta_{a}^{2}\hat{a}+i\sqrt{1-\eta_{a}^{4}}\hat{c}\\\
i\sqrt{1-\eta_{a}^{4}}\hat{a}+\eta_{a}^{2}\hat{c}\end{bmatrix},$

and a similar transformation is applied to the modes $b$ and $d$.

Phase shift transformation:

$\displaystyle\hat{PS}_{ab}^{\dagger}\begin{bmatrix}\hat{a}^{\prime}\\\
\hat{b}^{\prime}\end{bmatrix}\hat{PS}_{ab}=\begin{bmatrix}e^{i\phi}&0\\\
0&1\end{bmatrix}\begin{bmatrix}\hat{a}\\\
\hat{b}\end{bmatrix}=\begin{bmatrix}e^{i\phi}\hat{a}\\\ \hat{b}\end{bmatrix}.$
(4)

###  II.1 Two mode squeezed coherent state input

We consider as our input a TMSCS. It is created by passing two coherent
sources with average photon numbers $\alpha^{2}$, $\beta^{2}$ and phases
$\theta_{1},\theta_{2}$ through a optical parametric amplifier with squeezing
factor $r$ and squeezing angle $\xi$. Thus, the input state of the
interferometer $\ket{\Psi_{in}}_{ab}$ is given by:

$\displaystyle\ket{\Psi_{in}}_{ab}=\hat{\mathbf{S}}_{a,b}(\mathbf{z})\hat{\mathbf{D}}_{b}(\mathbf{b})\hat{\mathbf{D}}_{a}(\mathbf{a})\ket{00}_{ab},$
(5)

where,
$\hat{\mathbf{D}}_{a}(\mathbf{a})=\exp(\mathbf{a}\hat{a}^{\dagger}-\mathbf{a}^{*}\hat{a})$
and
$\hat{\mathbf{D}}_{b}(\mathbf{b})=\exp(\mathbf{b}\hat{b}^{\dagger}-\mathbf{b}^{*}\hat{b})$
are the Displacement operators for modes $a$ and $b$ entering the
interferometer with $\mathbf{a}=\alpha e^{i\theta_{1}}$ and $\mathbf{b}=\beta
e^{i\theta_{2}}$.

Further, the two mode squeezing operator implemented by the parametric
amplifier is given by:

$\displaystyle\mathbf{\hat{S}}_{a,b}(\mathbf{z})$ $\displaystyle=$
$\displaystyle\exp(r(\hat{a}\hat{b}e^{-i\xi}-\hat{a}^{\dagger}\hat{b}^{\dagger}e^{i\xi})),$
(6)

where, $\mathbf{z}=re^{i\xi}$.

Average photon number for TMSCS is given by:

$\displaystyle\bar{n}=(\alpha^{2}+\beta^{2})\cosh(2r)-2\alpha\beta\sinh(2r)\cos(\Theta)+2\sinh^{2}r\rm{,}$
(7)

where, $\Theta=\theta_{1}+\theta_{2}-\xi$. Clearly $\bar{n}$ is maximized for
$\Theta=\pi$, regardless of the individual values of
$\theta_{1},\theta_{2},\xi$.

###  II.2 Phase sensitivity

The relative phase of the two arms can be measured through its modulation of
the photon number difference between the two output detectors of the
interferometer. For brevity, we now drop the mode subscripts $a,b$ for the
operators unless the context is not obvious. The photon number difference
operator is defined as:

$\displaystyle\hat{J}_{3}=\hat{a}^{\dagger}\hat{a}-\hat{b}^{\dagger}\hat{b}.$
(8)

The phase sensitivity of the interferometer is defined by the following
relation:

$\displaystyle\Delta\phi$ $\displaystyle=$
$\displaystyle\frac{\Delta\hat{J}_{3}}{|\frac{d\langle\hat{J}_{3}\rangle}{d\phi}|},$
(9)

where
$\langle\hat{J}_{3}\rangle=\langle\psi_{out}|\hat{J}_{3}|\psi_{out}\rangle$
and
$(\Delta\hat{J}_{3})^{2}=\langle\hat{J}_{3}^{2}\rangle-\langle\hat{J_{3}}\rangle^{2}$.

With the above definitions the mean and variance of $\hat{J_{3}}$ can be
evaluated as shown in Equations (10,11). Since it is known that the phase
sensitivity increases with increasing mean photon number of the source (TMSCS
in this case), we work under the condition when $\bar{n}$ is maximum, i.e.,
$\Theta=\theta_{1}+\theta_{2}-\xi=\pi$. Within this constraint two of the
three phases can be chosen independently. For simplicity, the expressions
provided below assume $\theta_{1}=0$ and therefore $\theta_{2}=\pi+\xi$, where
$\xi$ can now be varied freely. We have also assumed that the input coherent
beams are strong, i.e., $\alpha^{2},\beta^{2}\gg 1$ and that the squeezing
$e^{2r}\gg 1$. (The covariance matrix formalism is used to obtain more general
expressions without these constraints. See Appendix V for details.)

$\displaystyle\langle\hat{J_{3}}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}\eta_{a}\eta_{b}(\alpha+\beta)^{2}\cos(\xi)\sin(\phi)$
(10) $\displaystyle(\Delta\hat{J_{3}})^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\bigg{\\{}2(\alpha^{2}+\beta^{2})\eta_{a}^{2}\eta_{b}^{2}+e^{-2r}(\alpha-\beta)^{2}\\{(2\bar{\gamma_{a}}+1)\eta_{a}^{2}+(2\bar{\gamma_{b}}+1)\eta_{b}^{2}-2(\bar{\gamma_{a}}+\bar{\gamma_{b}}+1)\eta_{a}^{2}\eta_{b}^{2}\\}$
(11) $\displaystyle+$ $\displaystyle
e^{2r}(\alpha+\beta)^{2}\big{\\{}\\{(2\bar{\gamma_{a}}+1)\eta_{a}^{2}+(2\bar{\gamma_{b}}+1)\eta_{b}^{2}-2(\bar{\gamma_{a}}+\bar{\gamma_{b}}+1)\eta_{a}^{2}\eta_{b}^{2}\\}$
$\displaystyle-$
$\displaystyle\sin\xi\\{(2\bar{\gamma_{a}}+1)\eta_{a}^{2}+(2\bar{\gamma_{b}}+1)\eta_{b}^{2}-2(\bar{\gamma_{a}}+\bar{\gamma_{b}}+1)\eta_{a}^{2}\eta_{b}^{2}\\}\big{\\}}\bigg{\\}}.$

It is clear from the above expressions that although the phase sensitivity is
maximum for $\Theta=\pi$, the choice of the individual phases within that
constraint affects the phase sensitivity without changing the mean photon
number. Or in other words, by choosing the phases optimally the phase
sensitivity can be further enhanced with same number of input photons. We show
this in the following section.

###  II.3 Enhancement in phase sensitivity

The enhancement in phase sensitivity can be calculated by comparing the
scaling of $\Delta\phi$ with mean photon number. If
$(\Delta\phi)^{2}=\frac{1}{G\bar{n}}$, then $G$ is defined as the phase
sensitivity enhancement factor over the shot noise (SN) limit and
$G=\frac{(\Delta\phi)^{2}}{(\Delta\phi_{SN})^{2}}$ Xiao  _et al._ (2020). We
will now look at this enhancement factor for the TMSCS input in different
cases. Unless stated otherwise, the phase sensitivity is maximum at $\phi=0$
and the expression for $\Delta\phi$ will thus be evaluated at $\phi=0$.

  * •

Case 1: Phase sensitivity gain compared to the SN limit in the noiseless and
loss-free case. If we set $\alpha=\beta$, and
$\theta_{1}=\theta_{2}=\pi/2,\xi=0$ in Equation (7), the SN limit for the
phase sensitivity in this case is given by:

$\displaystyle(\Delta\phi_{SN})^{2}=\frac{1}{\bar{n}}=\frac{1}{2|\alpha|^{2}e^{2r}}.$
(12)

On the other hand using Equations (10) and (11), the same choice of input
phases and setting $\eta_{a}=\eta_{b}=1$, we find the phase sensitivity in
this case as:

$\displaystyle(\Delta\phi)^{2}=\frac{1}{2|\alpha|^{2}e^{4r}}=\frac{1}{G\bar{n}},$
(13)

thus, the gain factor $G=e^{2r}$.

  * •

Case 2: Loss in one arm of interferometer, i.e. $\eta_{a}=\eta$ and
$\eta_{b}=1$. Also, the arms are noise-free, i.e.,
$\bar{\gamma_{a}}=\bar{\gamma_{b}}=0$ (except the vacuum fluctuations that
appear because of $\hat{BS}_{a}(\eta_{a}^{2})$). We assume $\theta_{1}=0$ and
$\theta_{2}=\pi+\xi$, where $\xi$ can now be varied freely. Again
$\alpha=\beta$. The phase sensitivity in this case is given by:

$\displaystyle(\Delta\phi)^{2}=(\Delta\phi_{SN})^{2}\bigg{(}\frac{1}{e^{2r}\cos^{2}(\xi)}+\frac{(1-\eta^{2})}{2\eta^{2}}\frac{(1-\sin\xi)}{\cos^{2}(\xi)}\bigg{)},$
(14)

for $\xi=0$ this reduces to known result Xiao  _et al._ (2020):

$\displaystyle(\Delta\phi)^{2}=(\Delta\phi_{SN})^{2}\bigg{(}\frac{1}{e^{2r}}+\frac{(1-\eta^{2})}{2\eta^{2}}\bigg{)}.$
(15)

To beat SN limit we need $G>1$. In the latter case Eq. (15), this translates
to $\frac{(1-\eta^{2})}{2\eta^{2}}<1$, which is true for $\eta>1/3$ Xiao  _et
al._ (2020). But, for the more general expression Eq. (14), where the
squeezing angle can be chosen freely, this limit for the tolerable loss can be
further increased to $\eta>1/5$. This is because the minimum value of
$\frac{(1-\sin\xi)}{\cos^{2}(\xi)}$ is $1/2$ which happens when $\xi\to\pi/2$.
But at the same time, at $\xi=\pi/2$ the first term in Eq. (14) blows up,
killing the advantage. Nonetheless, it can be seen that for values of $\xi$
close to $\pi/2$, $\eta$ can be pushed below $1/3$ and can ultimately approach
$1/5$. Larger the value of the squeezing parameter $r$, closer the optimum
value for $\xi$ to $\pi/2$ and closer is the lower limit of $\eta$ to 1/5.

  * •

Case 3: If we add thermal noise with mean photon number
$\bar{\gamma_{a}}=\bar{\gamma}$, also in the upper arm then the phase
sensitivity becomes:

$\displaystyle(\Delta\phi)^{2}=(\Delta\phi_{SN})^{2}$ $\displaystyle\bigg{(}$
$\displaystyle\frac{1}{e^{2r}\cos^{2}(\xi)}+$ $\displaystyle($ $\displaystyle
2\bar{\gamma}+1)\frac{(1-\eta^{2})}{2\eta^{2}}\frac{1-\sin(\xi)}{\cos^{2}(\xi)}\bigg{)}\rm{,}$

again, the maximum noise and loss that can be tolerated is increased by the
choice $\xi\to\pi/2$. It is given by the condition (also see Fig. 2):

$\displaystyle(2\bar{\gamma}+1)\frac{(1-\eta^{2})}{4\eta^{2}}\leq 1.$ (17)

  * •

Case 4: Equal loss and noise in the two arms. The expression for phase
sensitivity in this case becomes (choice of phases as in the previous cases):

$\displaystyle(\Delta\phi)^{2}=\frac{(\Delta\phi_{SN})^{2}}{\cos^{2}(\xi)}\bigg{(}\frac{1}{e^{2r}}+(2\bar{\gamma}+1)\frac{(1-\eta^{2})}{\eta^{2}}\bigg{)}.$
(18)

The gain in this case is maximum for $\xi=0$ and is given by:

$\displaystyle
G=\bigg{(}\frac{1}{e^{2r}}+(2\bar{\gamma}+1)\frac{(1-\eta^{2})}{\eta^{2}}\bigg{)}^{-1},$
(19)

with, $G>1$, whenever the following condition is satisfied:

$\displaystyle(2\bar{\gamma}+1)\frac{(1-\eta^{2})}{\eta^{2}}\leq 1.$ (20)

Figure 2:  Gain in phase sensitivity of a MI over the SN limit using a TMSCS
input, assuming loss and noise in the upper arm only. Such a configuration is
relevant for positioning and ranging applications where the upper arm acts as
the target arm and the lower arm as a reference arm. In these plots, to
indicate regions of sensitivity enhancement, we take $G=0$ whenever $G<1$.
Loss is parameterized by $\eta$, where $\eta^{2}$ is the transmissivity of the
upper arm, and noise is parameterized by $\bar{\gamma}$, i.e., the mean photon
number of the thermal state used to model the noise. The G=1 line is shown in
green which is also given by Eqn. (17). For all values of $\eta$ and
$\bar{\gamma}$ below the green curve the SN limit can be broken. The gain
factor $G$ can be increased by optimally choosing the squeezing angle. Also
this can be done without changing the average photon number of the TMSCS and
thus leads to an enhanced resistance to noise and loss without changing the
input power. A larger region in the $\eta-\bar{\gamma}$ space gives $G>1$ for
$\xi=\pi/2-0.05$ (Right), than for for $\xi=0$ (Left). Figure 3:  Gain in
phase sensitivity of a MI over the SN limit using a TMSCS input, assuming same
loss and noise in both arms. Such a symmetric configuration is relevant for
applications like gyroscopes Xiao  _et al._ (2020). In these plots, to
indicate regions of sensitivity enhancement, we take $G=0$ whenever $G<1$.
Loss is parameterized by $\eta$, where $\eta^{2}$ is the transmissivity of
each arm, and noise is parameterized by $\bar{\gamma}$, i.e., the mean photon
number of the thermal state used to model the noise. The G=1 line is shown in
green which is also given by Eq. (20). For all values of $\eta$ and
$\bar{\gamma}$ below the curve the SN limit can be broken. The gain factor $G$
is maximum for choice of squeezing angle $\xi=0$.

##  III Radiation pressure error

For applications of the MI such as gravitational wave detectors, where the
input power is high, and the end mirrors by design are free to move, radiation
pressure fluctuation is a major source of deterioration in phase sensitivity.
It was shown by Caves Caves (1981), that using a single-mode squeezed vacuum
state as one of the interferometer inputs (coherent state as the other input)
can reduce the photon counting error, i.e., fluctuations in photon number
difference by a factor of $e^{r}$. This is on account of reduced uncertainty
in one of the quadratures of the squeezed state compared to vacuum and
coherent state. At the same time, there is an increase in radiation pressure
error by the same factor of $e^{r}$, due to increased uncertainty in the other
quadrature.

Here we will show that the same kind of trade-off between photon counting
error and radiation pressure error exists for two-mode entangled squeezed
coherent states. Now, consider a TMSCS entering the two input ports of an MI.
Let the two modes be represented by $\mathbf{a}$ and $\mathbf{b}$. After
interfering at the 50:50 beam-splitter once, let’s say that the output modes
are $\mathbf{a^{\prime}}$ and $\mathbf{b^{\prime}}$, which are given by Eq.
(2). The error in measurement of path difference $\Delta z$ is then given by
the difference in radiation pressures, $\Delta\mathcal{P}$, applied by these
modes on the two end mirrors of the MI. The radiation pressure is in turn
proportional to the intensity or incident photon numbers. Therefore, for a
measurement duration $\tau$, and end mirrors each having mass $m$, the error
in $z$ due to quantum mechanical fluctuations in radiation pressure is given
by:

$\displaystyle\Delta z_{rp}=\frac{\Delta\mathcal{P}\tau}{2m}$ (21)

which is equivalent to a relative phase measurement error of:

$\displaystyle\Delta\phi_{rp}=\frac{\omega}{c}\frac{\Delta\mathcal{P}\tau}{2m}$
(22)

where $\omega$ is the angular frequency of the source and $c$ is the speed of
light,

$\displaystyle\mathcal{P}=\frac{2\hbar\omega}{c}(\hat{a}^{\prime\dagger}\hat{a}^{\prime}-\hat{b}^{\prime\dagger}\hat{b}^{\prime})$
(23)

In terms of the original input modes $\hat{a}$ and $\hat{b}$ in Eq. (2), Eq.
(23), reads:

$\displaystyle\mathcal{P}=\frac{2\hbar\omega}{c}(\hat{a}^{\dagger}\hat{b}-\hat{b}^{\dagger}\hat{a})$
(24)

The error in the momentum difference due to quantum fluctuations can be
quantified by the variance in $\mathcal{P}$ and is given by:

$\displaystyle(\Delta\mathcal{P})^{2}=\langle\mathcal{P}^{2}\rangle-\langle\mathcal{P}\rangle^{2}$
(25)

where expectation values are taken over the TMSCS given by Eq. (5), therefore,

$\displaystyle(\Delta$
$\displaystyle\mathcal{P})^{2}=(\alpha^{2}+\beta^{2})\cosh(4r)-$ (26)
$\displaystyle
2\alpha\beta\sinh(4r)\cos(\theta_{1}+\theta_{2}-\xi)+\frac{\sinh^{2}(2r)}{2}$

Putting $\mathbf{a}=\mathbf{b}$ in Eq. (26) and then substituting it in Eq.
(22) leads to a simple expression for the radiation pressure error:

$\displaystyle\Delta\phi_{rp}$
$\displaystyle=\frac{\hbar\omega\tau}{mc}\Big{[}2\alpha^{2}\cosh(4r)-2\alpha^{2}\sinh(4r)\cos(2\theta-\xi)$
(27) $\displaystyle+\frac{\sinh^{2}(2r)}{2}\Big{]}^{\frac{1}{2}}$

Now, let us consider the condition for which the photon counting error is
minimum. This happens whenever the average photon number of the TMSCS source
is maximum. In terms of choice of input phases, this translates to the
condition $\Theta=\pi$, i.e., in this case $2\theta-\xi=\pi$. Clearly under
such an assumption, the radiation pressure error given by Eq. (21), is maximum
for a given value of $\alpha$. Further, the equivalent of Eq. (13) is given
by:

$\displaystyle(\Delta\phi)^{2}={2|\alpha|^{2}e^{4r}}$ (28)

Therefore, the phase error is increased, by a gain factor $e^{2r}$ compared to
the case with no squeezing, which exactly compensates the reduction in phase
error due photon counting fluctuations.

On the other hand if the radiation pressure error has to be minimized then the
average photon number must be minimized. For given $\alpha$, this leads to the
following condition on the choice of phases: $\Theta=0$, i.e, $2\theta-\xi=0$.
The phase error due to radiation pressure in this case reduces by a factor of
$e^{2r}$. At the same time, for such a choice of the phases, the photon
counting error is maximum and the corresponding negative gain factor is
$G=e^{-2r}$. Therefore, there exists a trade-off between photon counting error
and radiation pressure error for the TMSCS input for a MI, just like single
mode squeezed vacuum state inputs Caves (1981).

##  IV Conclusions

In this paper we calculated the phase sensitivity of a Michelson
interferometer for a two mode squeezed coherent state input. Squeezed states,
by utilizing reduced quantum fluctuations in one mode, can enhance the phase
sensitivity compared to coherent states. This enhancement increases with
increasing squeezing factor $r$. Of course this is at the cost of higher input
pump power. This enhancement is tolerant to some degree of loss in the
interferometer arms. It was shown in previous work that in the case with equal
loss in both arms, a loss of up to 50% (3 dB) can be tolerated while
maintaining the advantages of squeezed states. For the asymmetric case with
loss only in one arm, this limit was shown to be 66.7%. In this work we show
that this limit can be further increased to 80% by appropriately choosing the
squeezing angle $\xi$. We show that $\xi\to\pi/2$, is the optimal choice in
this case. For the symmetric case, $\xi=0$ was found to be optimal. Further,
we also include the effects of thermal noise on the phase sensitivity. We find
that sensitivity enhancement can be achieved even in the noisy case and the
choice of optimal squeezing angles remain the same. It is important to note
that phase sensitivity enhancement for a given squeezing factor $r$ is
calculated compared to a coherent source with the same number of input
photons, the so called shot noise limit. In similar spirit, we show here that
by choosing the appropriate squeezing angle, while keeping the total number of
photons going into the interferometer fixed, we can enhance the phase
sensitivity. In this way we propose an alternative mechanism to improve the
sensitivity of an interferometer, that is, by optimally choosing the input
phases like the squeezing angle or coherent source phases. Finally, we also
show that there exists a trade-off between photon-counting error usually
relevant for low power applications and the radiation pressure error relevant
for high power uses. One can be reduced at the expense of the other, with the
reduction and increase scaling like $e^{-2r}$ and $e^{2r}$ respectively for
the TMSCS input, thus maintaining the standard quantum limit.

###### Acknowledgements.

We wish to acknowledge the support of the Army Research Office and the Air
Force Office of Scientific Research. PB and HL also acknowledge the support of
the Binational Science Foundation. We are honored to dedicate this paper to
the memory of Jonathan P. Dowling. Jonathan was an early advocate of the
emergent quantum technology from its early days. He was a great mentor,
colleague, and friend. We will miss him for a long long time.

##  V Covariance matrix formalism

It is convenient to work with the covariance matrix formalism since we deal
with states and interferometer operations which are Gaussian. A N-mode
Gaussian state has Wigner function of following form Weedbrook  _et al._
(2012)

$W(x)=\frac{\exp{\left[-\frac{1}{2}(x-\left\langle\hat{x}\right\rangle)^{T}\sigma^{-1}(x-\left\langle\hat{x}\right\rangle)\right]}}{(2\pi)^{N}\sqrt{det\sigma}}$
(29)

which is completely determined by: a $2N$-dimensional mean vector
$\hat{x}=(\hat{q}_{1},\hat{p}_{1},...\hat{q}_{N},\hat{p}_{N})^{T}$ and
$2N\times 2N$ covariance matrix

$\sigma_{jk}=\frac{1}{2}\left\langle\left\\{(\hat{x}_{j}-\langle\hat{x}_{j}\rangle),(\hat{x}_{k}-\langle\hat{x}_{k}\rangle)\right\\}\right\rangle,$
(30)

where {*,*} denotes anticommutator,
$\hat{q}_{k}=\frac{1}{\sqrt{2}}(\hat{a}^{\dagger}_{k}+\hat{a}_{k})$ and
$\hat{p}_{k}=\frac{1}{\sqrt{2}i}(\hat{a}^{\dagger}_{k}-\hat{a}_{k})$ are the
quadrature operators associated with the k${th}$ mode defined via standard
creation $(\hat{a}^{\dagger}_{k})$ and annihilation $(\hat{a}_{k})$ operators.
Action of any Gaussian unitary can be represented by a symplectic matrix $S$
with the transformations

$\hat{x}\longrightarrow S\hat{x},\hskip 28.45274pt\sigma\longrightarrow
S\sigma S^{T}.$ (31)

$\hat{x}$ and $\sigma$ at input are evolved using symplectic matrices of two-
mode squeezing unitary $(S_{TMS})$, beamsplitter $(S_{BS})$ and phase shift
$(S_{PS})$ Gard (2016) to get their final form at the output ports.

For the purpose of calculating desired expectation values we construct the
characteristic function, still a Gaussian, from final $\langle\hat{x}\rangle$
and $\sigma$ matrices as

$\chi(\Lambda)=\exp\left\\{-\frac{1}{2}\Lambda^{T}\sigma\Lambda+i\Lambda^{T}\left\langle\hat{x}\right\rangle\right\\},$
(32)

with
$\Lambda=(\Lambda_{q_{1}},\Lambda_{p_{1}},\Lambda_{q_{2}},\Lambda_{p_{2}})^{T}$
for the two mode case considered in this work. Expectation values of position
and momentum quadrature operators for the k$th$ mode are calculated by using
standard expressions

$\expectationvalue{q_{k}^{m}}=\left(-i\right)^{m}\left.\frac{\partial^{m}\chi}{\partial\Lambda^{m}_{q_{k}}}\right|_{\Lambda=0}\textrm{and
}\expectationvalue{p_{k}^{m}}=\left(-i\right)^{m}\left.\frac{\partial^{m}\chi}{\partial\Lambda^{m}_{p_{k}}}\right|_{\Lambda=0},$
(33)

which are used for intensity and variance calculation as shown in Gard  _et
al._ (2017). Using the formalism discussed above we find the general
expressions for mean and variance of the photon number difference operator
$\hat{J}_{3}$. These are given as:

$\langle\hat{J}_{3}\rangle=\frac{1}{4}e^{-2r}(\alpha^{2}\eta_{b}^{2}-\alpha^{2}\eta_{a}^{2}+2\alpha\beta\eta_{b}^{2}\sin(\theta_{1}-\theta_{2})+2\alpha\beta\eta_{a}^{2}\sin(\theta_{1}-\theta_{2})+\beta^{2}\eta_{b}^{2}\sin(\xi-2\theta_{2})+\\\
\beta^{2}\eta_{b}^{2}+\beta^{2}\eta_{a}^{2}\sin(\xi-2\theta_{2})-\beta^{2}\eta_{a}^{2}+\gamma^{2}-\eta^{2}-4\eta_{b}^{2}\bar{\gamma}_{a}e^{2r}+4\bar{\gamma}_{a}e^{2r}+4\eta_{a}^{2}\bar{\gamma}_{b}e^{2r}-4\bar{\gamma}_{b}e^{2r}+\\\
\alpha^{2}(e^{4r}-1)(\gamma^{2}+\eta^{2})\sin(\xi-2\theta_{1})+\alpha^{2}\eta_{b}^{2}e^{4r}-\alpha^{2}\eta_{a}^{2}e^{4r}-2\alpha\beta(e^{4r}-1)(\gamma^{2}-\eta^{2})\cos(-\theta_{1}-\theta_{2}+\xi)+\\\
2\alpha\beta\eta_{b}^{2}e^{4r}\sin(\theta_{1}-\theta_{2})+2\alpha\beta\eta_{a}^{2}e^{4r}\sin(\theta_{1}-\theta_{2})-\beta^{2}\eta_{b}^{2}e^{4r}\sin(\xi-2\theta_{2})+\beta^{2}\eta_{b}^{2}e^{4r}-\beta^{2}\eta_{a}^{2}e^{4r}\sin(\xi-2\theta_{2})-\\\
\beta^{2}\eta_{a}^{2}e^{4r}-2\eta_{b}^{2}e^{2r}+\gamma^{2}e^{4r}+2\eta_{a}^{2}e^{2r}-\eta^{2}e^{4r})$

$(\Delta\hat{J}_{3})^{2}=\frac{1}{8}e^{-4r}(2e^{2r}\eta_{b}^{2}-4e^{4r}\eta_{b}^{2}+2e^{6r}\eta_{b}^{2}+2e^{2r}\alpha^{2}\eta_{b}^{2}+2e^{6r}\alpha^{2}\eta_{b}^{2}+4e^{2r}\bar{\gamma}_{b}\alpha^{2}\eta_{b}^{2}+4e^{6r}\bar{\gamma}_{b}\alpha^{2}\eta_{b}^{2}+2e^{2r}\beta^{2}\eta_{b}^{2}+\\\
2e^{6r}\beta^{2}\eta_{b}^{2}+4e^{2r}\bar{\gamma}_{b}\beta^{2}\eta_{b}^{2}+4e^{6r}\bar{\gamma}_{b}\beta^{2}\eta_{b}^{2}-4e^{2r}\eta_{a}^{2}\eta_{b}^{2}+6e^{4r}\eta_{a}^{2}\eta_{b}^{2}-4e^{6r}\eta_{a}^{2}\eta_{b}^{2}+e^{8r}\eta_{a}^{2}\eta_{b}^{2}-4e^{2r}\alpha^{2}\eta_{a}^{2}\eta_{b}^{2}+4e^{4r}\alpha^{2}\eta_{a}^{2}\eta_{b}^{2}-\\\
4e^{6r}\alpha^{2}\eta_{a}^{2}\eta_{b}^{2}+2e^{8r}\alpha^{2}\eta_{a}^{2}\eta_{b}^{2}-4e^{2r}\bar{\gamma}_{a}\alpha^{2}\eta_{a}^{2}\eta_{b}^{2}-4e^{6r}\bar{\gamma}_{a}\alpha^{2}\eta_{a}^{2}\eta_{b}^{2}-4e^{2r}\bar{\gamma}_{b}\alpha^{2}\eta_{a}^{2}\eta_{b}^{2}-4e^{6r}\bar{\gamma}_{b}\alpha^{2}\eta_{a}^{2}\eta_{b}^{2}+\\\
2\alpha^{2}\eta_{a}^{2}\eta_{b}^{2}-4e^{2r}\beta^{2}\eta_{a}^{2}\eta_{b}^{2}+4e^{4r}\beta^{2}\eta_{a}^{2}\eta_{b}^{2}-4e^{6r}\beta^{2}\eta_{a}^{2}\eta_{b}^{2}+2e^{8r}\beta^{2}\eta_{a}^{2}\eta_{b}^{2}-4e^{2r}\bar{\gamma}_{a}\beta^{2}\eta_{a}^{2}\eta_{b}^{2}-4e^{6r}\bar{\gamma}_{a}\beta^{2}\eta_{a}^{2}\eta_{b}^{2}-\\\
4e^{2r}\bar{\gamma}_{b}\beta^{2}\eta_{a}^{2}\eta_{b}^{2}-4e^{6r}\bar{\gamma}_{b}\beta^{2}\eta_{a}^{2}\eta_{b}^{2}+2\beta^{2}\eta_{a}^{2}\eta_{b}^{2}-4e^{2r}\bar{\gamma}_{a}\eta_{a}^{2}\eta_{b}^{2}+8e^{4r}\bar{\gamma}_{a}\eta_{a}^{2}\eta_{b}^{2}-4e^{6r}\bar{\gamma}_{a}\eta_{a}^{2}\eta_{b}^{2}-\\\
4e^{2r}\bar{\gamma}_{b}\eta_{a}^{2}\eta_{b}^{2}+8e^{4r}\bar{\gamma}_{b}\eta_{a}^{2}\eta_{b}^{2}-4e^{6r}\bar{\gamma}_{b}\eta_{a}^{2}\eta_{b}^{2}+16e^{4r}\bar{\gamma}_{a}\bar{\gamma}_{b}\eta_{a}^{2}\eta_{b}^{2}+\eta^{2}\eta_{b}^{2}-8e^{4r}\bar{\gamma}_{a}\eta_{b}^{2}+4e^{2r}\bar{\gamma}_{b}\eta_{b}^{2}-\\\
8e^{4r}\bar{\gamma}_{b}\eta_{b}^{2}+4e^{6r}\bar{\gamma}_{b}\eta_{b}^{2}-16e^{4r}\bar{\gamma}_{a}\bar{\gamma}_{b}\eta_{b}^{2}-(-1+e^{4r})^{2}(2\alpha^{2}+2\beta^{2}+1)\eta_{a}^{2}\cos(2\phi)\eta_{b}^{2}+2e^{8r}\alpha\beta\eta_{a}^{2}\cos(\xi+2\phi-\theta_{1}-\theta_{2})\eta_{b}^{2}-\\\
2\alpha\beta\eta_{a}^{2}\cos(\xi+2\phi-\theta_{1}-\theta_{2})\eta_{b}^{2}+2e^{8r}\alpha\beta\eta_{a}^{2}\cos(\xi-2\phi-\theta_{1}-\theta_{2})\eta_{b}^{2}-2\alpha\beta\eta_{a}^{2}\cos(\xi-2\phi-\theta_{1}-\theta_{2})\eta_{b}^{2}-\\\
2e^{2r}\alpha^{2}\sin(\xi-2\theta_{1})\eta_{b}^{2}+2e^{6r}\alpha^{2}\sin(\xi-2\theta_{1})\eta_{b}^{2}-4e^{2r}\bar{\gamma}_{b}\alpha^{2}\sin(\xi-2\theta_{1})\eta_{b}^{2}+4e^{6r}\bar{\gamma}_{b}\alpha^{2}\sin(\xi-2\theta_{1})\eta_{b}^{2}-\\\
4e^{2r}\bar{\gamma}_{a}\alpha^{2}\eta_{a}^{2}\sin(\xi-2\theta_{1})\eta_{b}^{2}+4e^{6r}\bar{\gamma}_{a}\alpha^{2}\eta_{a}^{2}\sin(\xi-2\theta_{1})\eta_{b}^{2}+4e^{2r}\bar{\gamma}_{b}\alpha^{2}\eta_{a}^{2}\sin(\xi-2\theta_{1})\eta_{b}^{2}-\\\
4e^{6r}\bar{\gamma}_{b}\alpha^{2}\eta_{a}^{2}\sin(\xi-2\theta_{1})\eta_{b}^{2}+2e^{2r}\alpha^{2}\eta_{a}^{2}\sin(\xi+2\phi-2\theta_{1})\eta_{b}^{2}-2e^{6r}\alpha^{2}\eta_{a}^{2}\sin(\xi+2\phi-2\theta_{1})\eta_{b}^{2}-\\\
2e^{2r}\alpha^{2}\eta_{a}^{2}\sin(\xi-2\phi-2\theta_{1})\eta_{b}^{2}+2e^{6r}\alpha^{2}\eta_{a}^{2}\sin(\xi-2\phi-2\theta_{1})\eta_{b}^{2}+8e^{2r}\bar{\gamma}_{a}\alpha\beta\eta_{a}^{2}\sin(\theta_{1}-\theta_{2})\eta_{b}^{2}+\\\
8e^{6r}\bar{\gamma}_{a}\alpha\beta\eta_{a}^{2}\sin(\theta_{1}-\theta_{2})\eta_{b}^{2}-8e^{2r}\bar{\gamma}_{b}\alpha\beta\eta_{a}^{2}\sin(\theta_{1}-\theta_{2})\eta_{b}^{2}-8e^{6r}\bar{\gamma}_{b}\alpha\beta\eta_{a}^{2}\sin(\theta_{1}-\theta_{2})\eta_{b}^{2}+\\\
4e^{2r}\alpha\beta\sin(\theta_{1}-\theta_{2})\eta_{b}^{2}+4e^{6r}\alpha\beta\sin(\theta_{1}-\theta_{2})\eta_{b}^{2}+8e^{2r}\bar{\gamma}_{b}\alpha\beta\sin(\theta_{1}-\theta_{2})\eta_{b}^{2}+\\\
8e^{6r}\bar{\gamma}_{b}\alpha\beta\sin(\theta_{1}-\theta_{2})\eta_{b}^{2}+2e^{2r}\beta^{2}\sin(\xi-2\theta_{2})\eta_{b}^{2}-2e^{6r}\beta^{2}\sin(\xi-2\theta_{2})\eta_{b}^{2}+4e^{2r}\bar{\gamma}_{b}\beta^{2}\sin(\xi-2\theta_{2})\eta_{b}^{2}-\\\
4e^{6r}\bar{\gamma}_{b}\beta^{2}\sin(\xi-2\theta_{2})\eta_{b}^{2}+4e^{2r}\bar{\gamma}_{a}\beta^{2}\eta_{a}^{2}\sin(\xi-2\theta_{2})\eta_{b}^{2}-4e^{6r}\bar{\gamma}_{a}\beta^{2}\eta_{a}^{2}\sin(\xi-2\theta_{2})\eta_{b}^{2}-\\\
4e^{2r}\bar{\gamma}_{b}\beta^{2}\eta_{a}^{2}\sin(\xi-2\theta_{2})\eta_{b}^{2}+4e^{6r}\bar{\gamma}_{b}\beta^{2}\eta_{a}^{2}\sin(\xi-2\theta_{2})\eta_{b}^{2}-2e^{2r}\beta^{2}\eta_{a}^{2}\sin(\xi+2\phi-2\theta_{2})\eta_{b}^{2}+2e^{6r}\beta^{2}\eta_{a}^{2}\sin(\xi+2\phi-2\theta_{2})\eta_{b}^{2}+\\\
2e^{2r}\beta^{2}\eta_{a}^{2}\sin(\xi-2\phi-2\theta_{2})\eta_{b}^{2}-2e^{6r}\beta^{2}\eta_{a}^{2}\sin(\xi-2\phi-2\theta_{2})\eta_{b}^{2}+2e^{2r}\eta_{a}^{2}-4e^{4r}\eta_{a}^{2}+2e^{6r}\eta_{a}^{2}+2e^{2r}\alpha^{2}\eta_{a}^{2}+2e^{6r}\alpha^{2}\eta_{a}^{2}+\\\
4e^{2r}\bar{\gamma}_{a}\alpha^{2}\eta_{a}^{2}+4e^{6r}\bar{\gamma}_{a}\alpha^{2}\eta_{a}^{2}+2e^{2r}\beta^{2}\eta_{a}^{2}+2e^{6r}\beta^{2}\eta_{a}^{2}+4e^{2r}\bar{\gamma}_{a}\beta^{2}\eta_{a}^{2}+4e^{6r}\bar{\gamma}_{a}\beta^{2}\eta_{a}^{2}+4e^{2r}\bar{\gamma}_{a}\eta_{a}^{2}-\\\
8e^{4r}\bar{\gamma}_{a}\eta_{a}^{2}+4e^{6r}\bar{\gamma}_{a}\eta_{a}^{2}-8e^{4r}\bar{\gamma}_{b}\eta_{a}^{2}-16e^{4r}\bar{\gamma}_{a}\bar{\gamma}_{b}\eta_{a}^{2}+8e^{4r}\bar{\gamma}_{a}+8e^{4r}\bar{\gamma}_{b}+16e^{4r}\bar{\gamma}_{a}\bar{\gamma}_{b}-4(-1+e^{4r})\alpha\beta(e^{4r}\eta_{b}^{2}\eta_{a}^{2}+\\\
\gamma^{2}\eta_{a}^{2}+e^{2r}((-2(\bar{\gamma}_{a}+1)\eta_{a}^{2}-2\bar{\gamma}_{b}(\eta^{2}-1)+1)\eta_{b}^{2}+(2\bar{\gamma}_{a}+1)\eta_{a}^{2}))\cos(\xi-\theta_{1}-\theta_{2})+2e^{2r}\alpha^{2}\eta_{a}^{2}\sin(\xi-2\theta_{1})-\\\
2e^{6r}\alpha^{2}\eta_{a}^{2}\sin(\xi-2\theta_{1})+4e^{2r}\bar{\gamma}_{a}\alpha^{2}\eta_{a}^{2}\sin(\xi-2\theta_{1})-4e^{6r}\bar{\gamma}_{a}\alpha^{2}\eta_{a}^{2}\sin(\xi-2\theta_{1})-4e^{2r}\alpha\beta\eta_{a}^{2}\sin(\theta_{1}-\theta_{2})-\\\
4e^{6r}\alpha\beta\eta_{a}^{2}\sin(\theta_{1}-\theta_{2})-8e^{2r}\bar{\gamma}_{a}\alpha\beta\eta_{a}^{2}\sin(\theta_{1}-\theta_{2})-8e^{6r}\bar{\gamma}_{a}\alpha\beta\eta_{a}^{2}\sin(\theta_{1}-\theta_{2})-2e^{2r}\beta^{2}\eta_{a}^{2}\sin(\xi-2\theta_{2})+\\\
2e^{6r}\beta^{2}\eta_{a}^{2}\sin(\xi-2\theta_{2})-4e^{2r}\bar{\gamma}_{a}\beta^{2}\eta_{a}^{2}\sin(\xi-2\theta_{2})+4e^{6r}\bar{\gamma}_{a}\beta^{2}\eta_{a}^{2}\sin(\xi-2\theta_{2}))$

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