# Lewis and Berry phases for a gravitational wave interacting with a quantum
harmonic oscillator

$\mathbf{Soham}$ $\mathbf{Sen}^{\dagger}$, $\mathbf{Manjari}$
$\mathbf{Dutta}^{*}$ and $\mathbf{Sunandan}$
$\mathbf{Gangopadhyay}^{\ddagger}$ 111
<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>Department
of Astrophysics and High Energy Physics
S.N. Bose National Centre for Basic Sciences
JD Block, Sector III, Salt Lake, Kolkata 700 106, India

###### Abstract

In this work, we compute the Lewis and Berry phases for a gravitational wave
interacting with a two dimensional quantum harmonic oscillator in the
transverse-traceless gauge. We have considered a gravitational wave consisting
of the plus polarization term only. Considering the cross polarization term to
be absent makes the Hamiltonian separable in terms of the first and the second
spatial coordinates. We then compute the Lewis phase by assuming a suitable
form of the Lewis invariant considering only quadratic order contributions
from both position and momentum variables. Next, we obtain two Lewis
invariants corresponding to each separable part of the full Hamiltonian of the
system. Using both Lewis invariants, one can obtain two Ermakov-Pinney
equations, from which we finally obtain the corresponding Lewis phase. Then
making an adiabatic approximation enables us to isolate the Berry phase for
the full system. After this we obtain some explicit expressions of the Berry
phase for a plane polarized gravitational wave with different choices of the
harmonic oscillator frequency. Finally, we consider a gravitational wave with
cross polarization only interacting with an isotropic two dimensional harmonic
oscillator. For this we obtain the Lewis phase and the total Berry phase of
the system, which is found to be dependent upon the cross polarization part of
the gravitational wave.

## 1 Introduction

In 1916, Albert Einstein predicted that gravitational radiation must exist by
linearizing the general theory of relativity. This was the birth of
gravitational waves. The detection of gravitational wave in 2015 (from two
colliding neutron stars), led to an upsurge in research related to
gravitational waves and its various aspects almost after a century had passed
from the initial theoretical prediction in 1916. From its initial detection,
several theoretical models have been proposed to investigate the effects of
gravitational waves in several quantum mechanical scenarios. Although it is
theoretically more prudent to consider the background geometry to be curved as
most of the gravitational wave detectors (LIGO and VIRGO) are ground based
detectors, the usual theoretical notion is to consider the gravitational wave
as a fluctuation over a flat Minkowski background. The idea of observing
gravitational waves directly was pioneered by Weber in the 1960s. He for the
first time introduced the so-called resonant bar detectors [1, 2], and claimed
that gravitational waves have been detected. Thereafter, there has been a
monumental effort to increase the sensitivity of these detectors. The
possibility of increasing their sensitivity to higher levels has also led to
the idea of probing quantum gravity effects in these setups [3, 4, 6, 5, 7, 8,
9, 10]. The basic model in the resonant bar detector setup can be interpreted
as a quantum mechanical forced gravitational wave harmonic oscillator system.
The incoming gravitational wave interacts with the elastic matter. The tiny
vibrations in the detector are termed phonons.

The analysis involving the interaction between a gravitational wave and a
quantum harmonic oscillator is quite well known in the recent literature. The
interaction of classical gravitational waves with matter was studied in [11]
using a quantum mechanical approach. The study of time dependent quantum
harmonic oscillators is another area of research that is of immense interest
among the scientific community as well. This started with the work of Lewis
et. al. [12]. The method involves using exact invariants to critically study
these systems [13, 14]. There have been numerous analyses involving the
damping in a one dimensional quantum harmonic oscillator [15, 16, 17, 18, 19].
There also have been few studies involving damping in a two dimensional
quantum harmonic oscillator [20] and later extended to noncommutative space
[21]. An explicit analysis of a damped harmonic oscillator with time dependent
frequency in a noncommutative space in the presence of magnetic field was done
in [22].

On the other hand in a process if a very gradual change of external parameters
occurs [23, 24], then we call it an adiabatic process with which two
characteristic time scales are involved. These two characteristic time scales
are termed as “external time” and “internal time”. The “external time”
signifies the parameters (of the system), over which the system changes
significantly and the “internal time” is related to the motion of the system.
For an adiabatic process, the “external time” is dominant over the internal
time which signifies that the Hamiltonian parameters evolve quite slowly over
time. Computing the Lewis phase and doing the adiabatic approximation, gives
an efficient way to obtain this phase.

In [25, 26], Berry used a Hamiltonian with more than one time-dependent
parameter. One of the criteria he used in his work was that the eigenfunctions
of the Hamiltonian must be complex numbers. Now if the set of parameters
changes adiabatically for such a system through a closed path (such that it
can return to its initial position), then the system acquires a geometric
phase [29] and for these systems, the time reversal symmetry of the
Hamiltonian must be broken [30]. He showed that along with the time dependent
dynamical phase, a time independent and path dependent geometric phase is also
picked up by the system. This path dependent geometric phase is also known as
the “Berry phase” which is a physically measurable quantity. As it is not as
large as compared to the dynamical phase, it is possible to pick up the
geometric phase using sophisticated experiments [27, 28]. The connection
between Berry’s phase and Lewis’s phase for the quadratic Hamiltonians was
first explored in the work [31]. Few more interesting investigations of such
phases have been performed especially in the case of quantum harmonic
oscillators [36, 32, 33, 34, 35, 37, 38]. Information theoretic viewpoint of
the computation of such phases can be found in [40, 39] .

In this study, our primary objective is to obtain Berry’s geometric phase for
the non-relativistic quantum harmonic oscillator interacting with
gravitational waves. The motivation for looking at Berry phases in this kind
of setup is that they can prove to be a very strong detection tool for
gravitational waves. Our approach to looking for the Berry phase employs the
Lewis method [12]. This technique offers an elegant way to derive the Berry
phase by applying the adiabatic approximation once the Lewis phase has been
obtained. Here, we consider a gravitational wave in the transverse-traceless
gauge interacting with a two dimensional time dependent quantum harmonic
oscillator with time dependent frequency. As a result, the system Hamiltonian
becomes time dependent. We have then followed the analysis by Lewis et. al.
[12] to obtain an invariant corresponding to the total Hamiltonian of the
system. The analysis provides a direct connection between the eigenstates of
the time-dependent invariant and the solution of the Schrödinger’s equation.
This Lewis invariant method is very elegant in the sense that it needs no
prior approximation to obtain the Lewis phase corresponding to the system and
the geometric part of the phase is obtained using a standard adiabatic
approximation which is the Berry phase of the system [31]. Apart from this
Lewis invariant approach one can resort to a path integral approach for
obtaining Berry’s geometric phase as well [41, 42]. We start our analysis by
considering the plus polarization of the gravitational wave. We find out that
the Hamiltonian can be decoupled into two individual Hamiltonians
corresponding to the two spatial directions. We then obtain the Lewis phase
and eventually the Berry phase of the two-dimensional harmonic oscillator-
gravitational wave system. The importance of looking for the Berry phase in
the quantum harmonic oscillator-gravitational wave system is evident. Finding
a non-trivial detectable Berry phase can prove to be important in the
detection of gravitational waves in resonant bar detector systems [5, 7, 8,
9]. The most important insight that we obtain via this analysis is that the
Lewis phase, which can be considered to be the total phase of the system, has
no contribution from the gravitational wave in its dynamic part after applying
the adiabatic approximation. The contribution of the gravitational wave gets
captured entirely in the geometric part of the Lewis phase. After that we have
considered a gravitational wave, carrying cross polarization only, interacting
with a two dimensional and isotropic harmonic oscillator system. We observe
that the Hamiltonian can be decoupled as the plus polarization case by
considering a simple change of coordinates which effectively rotates the
coordinate system by an angle $\frac{\pi}{4}$ in the clockwise direction.
Again by following the same analysis, we obtain the Lewis phase and eventually
the total Berry phase of the system. Finally, in an appendix, we have tried to
obtain the Lewis invariant for a two dimensional harmonic oscillator system
interacting with a gravitational wave carrying both plus and cross
polarizations. Due to the presence of some self-contradictory constraint
equations, we were unable to obtain a valid form of the Lewis invariant for
the same. Hence, we keep it as an open problem and wish to follow up in future
work.

The organization of the paper goes as follows. In section 2, we discuss the
Hamiltonian of the system. In section 3, we have discussed in brief the method
of Lewis invariant and obtained the Lewis invariant corresponding to our
system in section 4. In section 5, we have obtained the creation and
annihilation operators from the Lewis invariant obtained in section 4. In
section 6, we have obtained the Lewis and the Berry phases for our system
Hamiltonian. Next in section 7, we have obtained a few explicit Berry phases
for some distinct set of choices of the Harmonic oscillator frequencies. In
section 8, we have calculated the contribution of the boundary term towards
the total geometric phase of the system. In section 9, we have analyzed the
case of a gravitational wave with only a cross polarization interacting with
an isotropic two dimensional harmonic oscillator and obtained the
corresponding Lewis and Berry phases for the system. Finally, we have
concluded the paper in section 10.

## 2 Hamiltonian of the system

In this paper we consider a two dimensional simple harmonic oscillator
interacting with a gravitational wave in the transverse-traceless gauge. The
background metric in the linearized approximation can then be expressed as the
flat Minkowski metric plus the perturbation due to the gravitational wave

$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ (1)

where $\eta_{\mu\nu}=\text{diag}\\{-1,1,1,1\\}$ with $\mu,\nu=\\{0,1,2,3\\}$.
The form of $h_{\mu\nu}$ in the transverse traceless gauge is given in the
matrix form as

$h_{\mu\nu}=\begin{pmatrix}0&0&0&0\\\ 0&h_{+}(t)&h_{\times}(t)&0\\\
0&h_{\times}(t)&-h_{+}(t)&0\\\ 0&0&0&0\end{pmatrix}$ (2)

where $h_{+}(t)=2\chi(t)\varepsilon_{+}$ and
$h_{\times}(t)=2\chi(t)\varepsilon_{\times}$, with $\varepsilon_{+}$ and
$\varepsilon_{\times}$ denoting the plus and the cross polarization of the
gravitational waves having amplitude $\chi(t)$. If we now define the
polarization tensor in three spatial dimensions to be
$\varepsilon_{l}=(\varepsilon_{\times},0,\varepsilon_{+})$, we can write down
the fluctuation term in terms of the Pauli spin matrix elements
$\sigma^{l}_{jk}$ in the following way

$\begin{split}h_{jk}(t)&=2\chi(t)\left(\varepsilon_{+}\sigma^{3}_{jk}+\varepsilon_{\times}\sigma^{1}_{jk}\right)\\\
&=2\chi(t)\varepsilon_{l}\sigma^{l}_{jk}\end{split}$ (3)

where $l=\\{1,2,3\\}$ and $j,k=\\{1,2\\}$. To write down eq.(3) from
$h_{\mu\nu}$, it is important to note that the matrix $h_{\mu\nu}$ can be
reduced effectively to $2\times 2$ matrix spanning the $x$ and $y$ directions
only. Before explicitly making use of the gravitational wave perturbation term
in eq.(3), we shall write down the classical form of harmonic oscillator-
gravitational wave interaction Hamiltonian (with time dependent frequency) as
[3]

$H=\frac{1}{2m}p_{k}p^{k}+\Gamma^{j}_{~{}0k}x^{k}p_{j}+\frac{1}{2}m\omega^{2}(t)x_{k}x^{k}$
(4)

where up to $\mathcal{O}(h)$ we can express
$\Gamma^{j}_{~{}0k}\cong\frac{1}{2}\eta^{jm}\partial_{0}h_{mk}$ ($j,k,m\simeq
1,2$). In order to obtain the above Hamiltonian of the system, one needs to
start from the action given by

$S=\int
dt\left(\frac{1}{2}m\dot{x}_{k}\dot{x}^{k}-\frac{1}{2}mR^{j}_{~{}0k0}x_{j}x^{k}-V\right)$
(5)

where $j,k=\\{1,2\\}$ and $V$ is the harmonic oscillator potential which is
given by

$V=\frac{1}{2}m\omega^{2}x_{k}x^{k}~{}.$ (6)

In general, it is more convenient to consider $\omega$ as a constant but in
our analysis we consider the frequency of the harmonic oscillator to be time
dependent ($\omega=\omega(t)$). Dropping the boundary term, we can recast the
form of the action in eq.(5) given as

$S=\int
dt\left(\frac{1}{2}m\dot{x}_{k}\dot{x}^{k}-m\Gamma^{j}_{~{}0k}\dot{x}_{j}x^{k}-V\right)$
(7)

where $R^{j}_{~{}0k0}=-\partial_{0}\Gamma^{j}_{~{}0k}$ and the modified
Lagrangian of the system is given by

$L^{\prime}=\frac{1}{2}m\dot{x}_{k}\dot{x}^{k}-m\Gamma^{j}_{~{}0k}\dot{x}_{j}x^{k}-V~{}.$
(8)

From this modified Lagrangian, one gets back our starting Hamiltonian in
eq.(4). Before proceeding further, we make a comment about the boundary term
that we have dropped in eq.(7) to obtain the modified form of the action in
eq.(7). The boundary term has the form given as

$\mathcal{B}=\frac{m}{2}\int_{0}^{t}d\tau\frac{d}{d\tau}\left(\Gamma^{j}_{~{}0k}(\tau)x_{j}(\tau)x^{k}(\tau)\right)~{}.$
(9)

Note that the boundary term may contribute towards the overall geometric
phase. We shall calculate this contribution in section 8. Using eq.(3) in
eq.(4) and raising the phase space variables to operator status, we can
finally write down the Hamiltonian for the quantum harmonic oscillator to be

$\begin{split}\hat{H}(t)=&\frac{a(t)}{2}(\hat{p}_{1}^{2}+\hat{p}_{2}^{2})+\frac{b(t)}{2}(\hat{x}_{1}^{2}+\hat{x}_{2}^{2})+d(t)(\hat{x}_{1}\hat{p}_{1}+\hat{p}_{1}\hat{x}_{1})-d(t)(\hat{x}_{2}\hat{p}_{2}+\hat{p}_{2}\hat{x}_{2})+f(t)(\hat{x}_{1}\hat{p}_{2}+\hat{p}_{1}\hat{x}_{2})\end{split}$
(10)

where $a(t)=\frac{1}{m}$, $b(t)=m\omega^{2}(t)$,
$d(t)=\frac{1}{2}\varepsilon_{+}\dot{\chi}(t)$, and
$f(t)=\varepsilon_{\times}\dot{\chi}(t)$. It is important to note that
$d(t),f(t)\sim\mathcal{O}(h)$. Instead of progressing further with the form of
the Hamiltonian $H(t)$ in eq.(10), we simplify the form of the above
Hamiltonian by setting $f(t)=0$ in eq.(10). This can be done without any loss
of generality, since removing the $f(t)$ term, implies looking at the
gravitational wave interacting with the two dimensional harmonic oscillator
having plus polarization only. Later in section (9), we have considered the
case where the gravitational wave can be considered to have the cross
polarization only without any loss of generality222We have discussed the case
where the gravitational wave carries both polarizations, in the Appendix..
Apart from setting $f(t)=0$ (which is equivalent to setting
$\varepsilon_{\times}=0$), any alternative choice, such as $f(t)=$ constant,
would result in setting the factor $\dot{\chi}(t)$ to a constant. This in turn
would imply $d(t)$=constant. As we shall see in section 6, the Berry phase of
the system depends explicitly upon the factor $\dot{d}(t)$, which therefore
becomes zero when $f(t)$=constant.

Setting $f(t)=0$, the Hamiltonian in eq.(10) can now be recast in the
following form,

$\begin{split}\hat{H}_{+}(t)=&\frac{a}{2}\hat{p}_{1}^{2}+\frac{b(t)}{2}\hat{x}_{1}^{2}+d(t)(\hat{x}_{1}\hat{p}_{1}+\hat{p}_{1}\hat{x}_{1})+\frac{a}{2}\hat{p}_{2}^{2}+\frac{b(t)}{2}\hat{x}_{2}^{2}-d(t)(\hat{x}_{2}\hat{p}_{2}+\hat{p}_{2}\hat{x}_{2})\\\
=&\hat{H}_{1}(t)+\hat{H}_{2}(t)\end{split}$ (11)

where

$\displaystyle\hat{H}_{1}(t)$
$\displaystyle=\frac{a}{2}\hat{p}_{1}^{2}+\frac{b(t)}{2}\hat{x}_{1}^{2}+d(t)(\hat{x}_{1}\hat{p}_{1}+\hat{p}_{1}\hat{x}_{1})~{},$
(12) $\displaystyle\hat{H}_{2}(t)$
$\displaystyle=\frac{a}{2}\hat{p}_{2}^{2}+\frac{b(t)}{2}\hat{x}_{2}^{2}-d(t)(\hat{x}_{2}\hat{p}_{2}+\hat{p}_{2}\hat{x}_{2})~{}.$
(13)

Note that, we can effectively write down the Hamiltonian as a sum of two
Hamiltonians describing the dynamics of the system corresponding to each
spatial dimension. We will now try to find out the Lewis invariant
corresponding to the Hamiltonian given in eq.(11).

## 3 Basic introduction to Lewis invariant

The model Hamiltonian obtained in eq.(10) is an explicit function of time.
Following the analysis in [12], it may be possible to construct a time
dependent Hermitian operator $\hat{I}(t)$ such that

$\begin{split}\frac{d\hat{I}}{dt}=\frac{\partial\hat{I}}{\partial
t}+\frac{1}{i\hbar}[\hat{I},\hat{H}]=0~{}.\end{split}$ (14)

If a time dependent Schrödinger state vector $|\psi\rangle$ satisfies the
following relation

$i\hbar\frac{\partial|\psi_{k}\rangle}{\partial t}=\hat{H}|\psi_{k}\rangle$
(15)

then we can deduce the following relation (by making use of eq.(14))

$\begin{split}i\hbar\frac{\partial}{\partial
t}(\hat{I}|\psi_{k}\rangle)=&i\hbar\frac{\partial\hat{I}}{\partial
t}|\psi_{k}\rangle+\hat{I}\left[i\hbar\frac{\partial|\psi_{k}\rangle}{\partial
t}\right]\\\ =&\left[i\hbar\frac{\partial\hat{I}}{\partial
t}+\hat{I}\hat{H}\right]|\psi_{k}\rangle\\\
=&\hat{H}(\hat{I}|\psi_{k}\rangle)\end{split}$ (16)

where to obtain the last line of the above equation, we have made use of
eq.(14). Eq.(16) suggests that the action of $\hat{I}(t)$ on the Schrödinger
state vector creates another state vector. If one now assumes that the
invariant operator is one among a complete set of commuting observables, then
it is straightforward to find out a complete set of eigenstates for the
Hermitian invariant operator $\hat{I}(t)$. We assume that for the invariant in
our case, the eigenstate is $|\varphi_{k}\rangle$. Then we can write down the
following relation

$\hat{I}(t)|\varphi_{k}\rangle=\zeta|\varphi_{k}\rangle$ (17)

with $\zeta$ being the eigenvalue. Following the analysis in [12], one can
write down the time dependent solution of the Schrödinger equation in eq.(15)
in terms of the eigenstate $|\varphi_{k}\rangle$ as follows

$|\psi_{k}\rangle=e^{i\theta_{k}(t)}|\varphi_{k}\rangle$ (18)

where $\theta_{k}(t)$ is a real function of time. Substituting eq.(18) back in
the time dependent Schrödinger equation given in eq.(15), we obtain the
following relation

$e^{i\theta_{k}(t)}\left(i\hbar\partial_{t}-\hbar\dot{\theta}_{k}(t)\right)|\phi_{k}\rangle=e^{i\theta_{k}(t)}\hat{H}|\phi_{k}\rangle~{}.$
(19)

We can recast eq.(19) via the action of $\langle\psi_{k}|$ which gives

$\dot{\theta}_{k}(t)=\frac{1}{\hbar}\langle\phi_{k}|i\hbar\partial_{t}-\hat{H}|\phi_{k}\rangle~{}.$
(20)

The real function $\theta_{k}(t)$ is also known as the Lewis phase factor.
Later we shall make use of eq.(20) to obtain the form of the Lewis phase and
eventually the Berry phase for our model system. With the initial introduction
regarding the invariant operator, we will try to find a suitable form of
$\hat{I}(t)$ corresponding to the Hamiltonian in eq.(10).

## 4 Lewis invariant of the system

The form of the Hamiltonian in eq.(11) suggests that the most general form of
the Lewis invariant can be taken as

$\begin{split}\hat{I}(t)=&\alpha_{1}(t)\hat{p}_{1}^{2}+\alpha_{2}(t)\hat{p}_{2}^{2}+\beta_{1}(t)\hat{x}_{1}^{2}+\beta_{2}(t)\hat{x}_{2}^{2}+\delta_{1}(t)(\hat{x}_{1}\hat{p}_{1}+\hat{p}_{1}\hat{x}_{1})+\delta_{2}(t)(\hat{x}_{2}\hat{p}_{2}+\hat{p}_{2}\hat{x}_{2})\end{split}$
(21)

where $\alpha_{1},\alpha_{2},\beta_{1},\beta_{2},\delta_{1}$ and $\delta_{2}$
are all unknown time dependent parameters. The Lewis invariant method provides
differential equations for these unknown parameters which yields the form of
the invariant. In the following discussion, we shall develop the differential
equations for these unknown time-dependent parameters. To proceed, we note
that it is possible to break the invariant $\hat{I}(t)$ in eq.(21) into two
parts given by

$\hat{I}(t)=\hat{I}_{1}(t)+\hat{I}_{2}(t)$ (22)

where the forms of $\hat{I}_{1}$ and $\hat{I}_{2}(t)$ are given as follows

$\hat{I}_{k}(t)=\alpha_{k}(t)\hat{p}_{k}^{2}+\beta_{k}(t)\hat{x}_{k}^{2}+\delta_{k}(t)(\hat{x}_{k}\hat{p}_{k}+\hat{p}_{k}\hat{x}_{k})~{};~{}k=1,2.$
(23)

We shall now try to determine all the undetermined constants by inserting the
form of $\hat{I}(t)$ from eq.(21) back in eq.(14) and equating the
coefficients of all quadratic operators to zero. Computing
$i\hbar\frac{\partial\hat{I}(t)}{\partial t}$ and $[\hat{I},\hat{H}]$, we
obtain a set of ordinary differential equations given by

$\displaystyle\dot{\alpha}_{1}-4\alpha_{1}d+2\delta_{1}a=0~{},$ (24)
$\displaystyle\dot{\alpha}_{2}+4\alpha_{2}d+2\delta_{2}a=0~{},$ (25)
$\displaystyle\dot{\beta}_{1}+4\beta_{1}d-2\delta_{1}b=0~{},$ (26)
$\displaystyle\dot{\beta}_{2}-4\beta_{2}d-2\delta_{2}b=0~{},$ (27)
$\displaystyle\dot{\delta}_{1}-\alpha_{1}b+\beta_{1}a=0~{},$ (28)
$\displaystyle\dot{\delta}_{2}-\alpha_{2}b+\beta_{2}a=0~{},$ (29)

We shall now introduce two time dependent parameters $\rho_{1}(t)$ and
$\rho_{2}(t)$ such that

$\alpha_{1}(t)=\kappa_{1}\rho_{1}^{2}(t)~{},~{}\alpha_{2}(t)=\kappa_{2}\rho_{2}^{2}(t)$
(30)

with $\kappa_{1}$ and $\kappa_{2}$ being two time independent unknown
constants. Using the forms of $\alpha_{1}$ and $\alpha_{2}$, we can now obtain
the form of the other unidentified time dependent parameters up to
$\mathcal{O}(h)$ to be

$\displaystyle\beta_{1}(t)$
$\displaystyle=\frac{\kappa_{1}}{a^{2}}\left(\dot{\rho}_{1}^{2}+\rho_{1}\ddot{\rho}_{1}-2\rho_{1}^{2}\dot{d}-4\rho_{1}\dot{\rho}_{1}d+ab\rho_{1}^{2}\right)~{},$
(31) $\displaystyle\beta_{2}(t)$
$\displaystyle=\frac{\kappa_{2}}{a^{2}}\left(\dot{\rho}_{2}^{2}+\rho_{2}\ddot{\rho}_{2}+2\rho_{2}^{2}\dot{d}+4\rho_{2}\dot{\rho}_{2}d+ab\rho_{2}^{2}\right)~{},$
(32) $\displaystyle\delta_{1}(t)$
$\displaystyle=-\frac{\kappa_{1}}{a}(\rho_{1}\dot{\rho_{1}}-2\rho_{1}^{2}d)~{},$
(33) $\displaystyle\delta_{2}(t)$
$\displaystyle=-\frac{\kappa_{2}}{a}(\rho_{2}\dot{\rho_{2}}+2\rho_{2}^{2}d)~{},$
(34)

To further simplify the forms of $\beta_{1}(t)$ and $\beta_{2}(t)$ from
eq.(s)(31,32), we substitute the forms of these two parameters (as well as
$\delta_{1}(t)$ and $\delta_{2}(t)$) back in the following differential
equations

$\displaystyle\dot{\beta}_{1}+4\beta_{1}d-2\delta_{1}b=0~{},$ (35)
$\displaystyle\dot{\beta}_{2}-4\beta_{2}d-2\delta_{2}b=0~{}.$ (36)

We can finally obtain from the above two differential equations (using the
forms of the parameters), two non-linear equations involving the parameters
$\rho_{1}(t)$ and $\rho_{2}(t)$ as follows

$\displaystyle\ddot{\rho}_{1}(t)+\left(ab(t)-2\dot{d}(t)\right)\rho_{1}(t)=\frac{a^{2}\xi_{1}^{2}}{\rho_{1}^{3}(t)}~{},$
(37)
$\displaystyle\ddot{\rho}_{2}(t)+\left(ab(t)+2\dot{d}(t)\right)\rho_{2}(t)=\frac{a^{2}\xi_{2}^{2}}{\rho_{2}^{3}(t)}$
(38)

where $\xi_{1}^{2}$ and $\xi_{2}^{2}$ are integration constants. It is
important to note that $\varepsilon_{+}^{2}+\varepsilon_{\times}^{2}=1$ and
for a gravitational wave in the absence of the cross polarization term,
$\epsilon_{+}=1$ which indicates $d(t)=\frac{1}{2}\dot{\chi}(t)$.
Eq.(s)(37,38) are the Ermakov-Pinney [43, 44] equations corresponding to the
two directions presented in this analysis.

We shall now make use of the Ermakov-Pinney equations to subsequently simplify
the forms of $\beta_{1}(t)$ and $\beta_{2}(t)$ from eq.(s)(31,32) given as
follows

$\displaystyle\beta_{1}(t)$
$\displaystyle=\frac{\kappa_{1}}{a^{2}}\left(\dot{\rho}_{1}^{2}-4\rho_{1}\dot{\rho}_{1}d\right)+\frac{\kappa_{1}\xi_{1}^{2}}{\rho_{1}^{2}}~{},$
(39) $\displaystyle\beta_{2}(t)$
$\displaystyle=\frac{\kappa_{2}}{a^{2}}\left(\dot{\rho}_{2}^{2}+4\rho_{2}\dot{\rho}_{2}d\right)+\frac{\kappa_{2}\xi_{2}^{2}}{\rho_{2}^{2}}~{}.$
(40)

With the analytical forms of the parameters, we can now recast the separable
Lewis invariants from eq.(23) as follows

$\displaystyle\hat{I}_{1}(t)=$
$\displaystyle\kappa_{1}\rho_{1}^{2}\hat{p}_{1}^{2}+\left(m^{2}\kappa_{1}(\dot{\rho}_{1}^{2}-2\rho_{1}\dot{\rho_{1}}\dot{\chi}(t))+\frac{\kappa_{1}\xi_{1}^{2}}{\rho_{1}^{2}}\right)\hat{x}_{1}^{2}-m\kappa_{1}\rho_{1}(\dot{\rho}_{1}-\dot{\chi}(t)\rho_{1})(\hat{x}_{1}\hat{p}_{1}+\hat{p}_{1}\hat{x}_{1})~{},$
(41) $\displaystyle\hat{I}_{2}(t)=$
$\displaystyle\kappa_{2}\rho_{2}^{2}\hat{p}_{2}^{2}+\left(m^{2}\kappa_{2}(\dot{\rho}_{2}^{2}+2\rho_{2}\dot{\rho_{2}}\dot{\chi}(t))+\frac{\kappa_{2}\xi_{2}^{2}}{\rho_{2}^{2}}\right)\hat{x}_{2}^{2}-m\kappa_{2}\rho_{2}(\dot{\rho}_{2}+\dot{\chi}(t)\rho_{2})(\hat{x}_{2}\hat{p}_{2}+\hat{p}_{2}\hat{x}_{2})~{}.$
(42)

It is important to note that $\kappa_{1}$ and $\kappa_{2}$ being arbitrary
constants, can be set to unity for a simplified analysis. In order to proceed
further, we would try to compute the raising and lowering operators
corresponding to the new separable invariant operators $\hat{I}_{1}$ and
$\hat{I}_{2}$.

## 5 Obtaining the creation and annihilation operators

In this section, we will obtain the creation and annihilation operators
corresponding to the invariant operator $\hat{I}_{1}(t)$ using a simple
“completing the square” approach and proceed to do the same for
$\hat{I}_{2}(t)$. Our primary aim is to recast $\hat{I}_{1}(t)$ in such a way
that
$\hat{I}_{1}(t)=A_{1}^{2}(\hat{x}_{1},\hat{p}_{1})+A_{2}^{2}(\hat{x}_{1},\hat{p}_{1})+A_{3}$,
where $A_{3}$ is a constant. We can recast $\hat{I}_{1}(t)$ in the following
way

$\begin{split}\frac{\hat{I}_{1}}{\kappa_{1}}=&\frac{\xi_{1}^{2}\hat{x}_{1}^{2}}{\rho_{1}^{2}}+\rho_{1}^{2}\hat{p}_{1}^{2}+m\rho_{1}(\rho_{1}\dot{\chi}(t)-\dot{\rho}_{1})\hat{p}_{1}\hat{x}_{1}+m\rho_{1}(\rho_{1}\dot{\chi}(t)-\dot{\rho}_{1})\hat{x}_{1}\hat{p}_{1}+m^{2}(\rho_{1}\dot{\chi}(t)-\dot{\rho}_{1})^{2}\hat{x}_{1}^{2}-m^{2}(\rho_{1}\dot{\chi}(t)-\dot{\rho}_{1})^{2}\hat{x}_{1}^{2}\\\
&+m^{2}(\dot{\rho}_{1}^{2}-2\rho_{1}\dot{\rho}_{1}\dot{\chi}(t))\hat{x}_{1}^{2}~{}.\end{split}$
(43)

From the last line of the above equation, we can see that the final two terms
cancel each other up to $\mathcal{O}(h)$. Hence, we can recast eq.(43) in the
following form

$\begin{split}\frac{\hat{I}_{1}}{\kappa_{1}}=&\frac{\xi_{1}^{2}\hat{x}_{1}^{2}}{\rho_{1}^{2}}+\left(\rho_{1}\hat{p}_{1}+m(\rho_{1}\dot{\chi}(t)-\dot{\rho}_{1})\hat{x}_{1}\right)^{2}~{}.\end{split}$
(44)

From now on, we shall be working in the $\hbar=c=1$ unit. As a result, the
commutation relation between the position and momentum variables takes the
form $[\hat{x}_{1},\hat{p}_{1}]=[\hat{x}_{2},\hat{p}_{2}]=i$. We can now
recast eq.(44) in the following form

$\begin{split}\hat{I}_{1}=&\kappa_{1}\left[\frac{\xi_{1}\hat{x}_{1}}{\rho_{1}}-i\left(\rho_{1}\hat{p}_{1}+m(\rho_{1}\dot{\chi}(t)-\dot{\rho}_{1})\hat{x}_{1}\right)\right]\left[\frac{\xi_{1}\hat{x}_{1}}{\rho_{1}}+i\left(\rho_{1}\hat{p}_{1}+m(\rho_{1}\dot{\chi}(t)-\dot{\rho}_{1})\hat{x}_{1}\right)\right]+\kappa_{1}\xi_{1}~{}.\end{split}$
(45)

As $\kappa_{1}$ and $\xi_{1}$ are both arbitrary constants, we can choose
$\kappa_{1}=\frac{1}{2}$ and $\xi_{1}=1$. We can again recast eq.(45) as
follows

$\begin{split}\hat{I}_{1}(t)=\hat{a}^{\dagger}_{1}(t)\hat{a}_{1}(t)+\frac{1}{2}\end{split}$
(46)

where

$\begin{split}\hat{a}_{1}&=\frac{1}{\sqrt{2}\rho_{1}}\left(\hat{x}_{1}+i\rho_{1}^{2}\hat{p}_{1}+im\rho_{1}(\rho_{1}\dot{\chi}-\dot{\rho}_{1})\hat{x}_{1}\right)\\\
&=\frac{1}{\sqrt{2}\rho_{1}}\left(\hat{x}_{1}+i\rho_{1}^{2}\hat{p}_{1}+im\rho_{1}(\rho_{1}\Gamma^{1}_{~{}01}-\dot{\rho}_{1})\hat{x}_{1}\right)\end{split}$
(47)

where $\Gamma^{1}_{~{}01}=\dot{\chi}(t)$. It can be easily checked using the
form of eq.(47) that $[\hat{a}_{1},\hat{a}_{1}^{\dagger}]=1$. We can also
recast the invariant operator $\hat{I}_{2}$ as follows

$\hat{I}_{2}(t)=\hat{a}_{2}^{\dagger}(t)\hat{a}_{2}(t)+\frac{1}{2}$ (48)

where

$\begin{split}\hat{a}_{2}&=\frac{1}{\sqrt{2}\rho_{2}}\left(\hat{x}_{2}+i\rho_{2}^{2}\hat{p}_{2}-im\rho_{2}(\rho_{2}\dot{\chi}+\dot{\rho}_{2})\hat{x}_{2}\right)\\\
&=\frac{1}{\sqrt{2}\rho_{2}}\left(\hat{x}_{2}+i\rho_{2}^{2}\hat{p}_{2}+im\rho_{2}(\rho_{2}\Gamma^{2}_{~{}02}-\dot{\rho}_{2})\hat{x}_{2}\right)\end{split}$
(49)

where $\Gamma^{2}_{02}=-\dot{\chi}(t)$ and
$[\hat{a}_{2},\hat{a}_{2}^{\dagger}]=1$. From eq.(s)(47,49), we can write down
a generalized commutation relation involving the ladder operators given by

$[\hat{a}_{j},\hat{a}_{k}^{\dagger}]=\delta_{jk}$ (50)

for $j,k=\\{1,2\\}$. With the forms of the creation and annihilation operators
in hand, we shall now proceed to obtain the Lewis phase for the system.

## 6 Lewis and Berry phase for the system

In this section, we shall obtain the Lewis phase and eventually the Berry
phase for the system. To obtain the Lewis phase, we need to properly construct
the eigenstates corresponding to the Lewis invariant operators. It is
important to note that the total Lewis invariant of the system can be divided
into two Lewis invariants (eq.(22)) and each of the invariant operators can be
expressed as a number operator plus a constant. Hence, it is straightforward
to write down the total eigenstate corresponding to $\hat{I}(t)$ as a tensor
product of two number states corresponding to each of the two directions given
by

$|n_{1},n_{2}\rangle=|n_{1}\rangle\otimes|n_{2}\rangle$ (51)

such that

$(\hat{a}_{1}\otimes\mathbb{1}_{2})|0,n_{2}\rangle=0~{},\text{and,
}(\mathbb{1}_{1}\otimes\hat{a}_{2})|n_{1},0\rangle=0~{}.$ (52)

We can further define the action of the invariant operator on the number state
as follows

$\begin{split}\hat{I}(t)|n_{1},n_{2}\rangle&=\left(\hat{I}_{1}(t)\otimes\mathbb{1}_{2}+\mathbb{1}_{1}\otimes\hat{I}_{2}(t)\right)|n_{1},n_{2}\rangle\\\
&=\left(n_{1}+\frac{1}{2}\right)|n_{1},n_{2}\rangle+\left(n_{2}+\frac{1}{2}\right)|n_{1},n_{2}\rangle\\\
&=(n_{1}+n_{2}+1)|n_{1},n_{2}\rangle~{}.\end{split}$ (53)

It is straightforward to write down the number state as an action of creation
operators on the individual vacuum states as follows

$|n_{1},n_{2}\rangle=\frac{1}{\sqrt{n_{1}!n_{2}!}}\left((\hat{a}_{1}^{\dagger})^{n_{1}}|0\rangle\right)\otimes\left((\hat{a}_{2}^{\dagger})^{n_{2}}|0\rangle\right)~{}.$
(54)

For $\hbar=1$, we can recast eq.(20) (and for
$|\phi_{n}\rangle=|n_{1},n_{2}\rangle$) as follows

$\begin{split}\dot{\theta}(t)=&\langle
n_{1},n_{2}|i(\partial_{t})_{1}\otimes\mathbb{1}_{2}+\mathbb{1}_{1}\otimes
i(\partial_{t})_{2}-(\hat{H}_{1}(t)\otimes\mathbb{1}_{2}+\mathbb{1}_{1}\otimes\hat{H}_{2}(t))|n_{1},n_{2}\rangle\\\
=&\langle n_{1}|i(\partial_{t})_{1}-\hat{H}_{1}(t)|n_{1}\rangle\langle
n_{2}|\mathbb{1}_{2}|n_{2}\rangle+\langle
n_{1}|\mathbb{1}_{1}|n_{1}\rangle\langle
n_{2}|i(\partial_{t})_{2}-\hat{H}_{2}(t)|n_{2}\rangle\\\ =&\langle
n_{1}|i(\partial_{t})_{1}-\hat{H}_{1}(t)|n_{1}\rangle+\langle
n_{2}|i(\partial_{t})_{2}-\hat{H}_{2}(t)|n_{2}\rangle\\\
=&\dot{\theta}_{1}(t)+\dot{\theta}_{2}(t)\end{split}$ (55)

where

$\displaystyle\dot{\theta}_{1}(t)$ $\displaystyle=\langle
n_{1}|i(\partial_{t})_{1}-\hat{H}_{1}(t)|n_{1}\rangle~{},$ (56)
$\displaystyle\dot{\theta}_{2}(t)$ $\displaystyle=\langle
n_{2}|i(\partial_{t})_{2}-\hat{H}_{2}(t)|n_{2}\rangle~{}.$ (57)

We shall now make use of eq.(s)(56,57) to obtain the forms of the Lewis phases
$\theta_{1}(t)$ and $\theta_{2}(t)$. In order to obtain the form of
$\dot{\theta}_{1}(t)$, we at first need to calculate the commutator bracket
$[\hat{a}_{1},i\partial_{t}-\hat{H}_{1}(t)]$ given as follows

$\begin{split}[\hat{a}_{1},i\partial_{t}-\hat{H}_{1}]|\psi\rangle=\left(-i\dot{\hat{a}}_{1}-[\hat{a}_{1},\hat{H}_{1}]\right)|\psi\rangle~{}.\end{split}$
(58)

The analytical form of $-i\dot{a}_{1}$ is given by

$-i\dot{\hat{a}}_{1}=\frac{i\dot{\rho}_{1}}{\sqrt{2}\rho_{1}^{2}}\hat{x}_{1}+\frac{\dot{\rho}_{1}\hat{p}_{1}}{\sqrt{2}}+\frac{m}{\sqrt{2}}\left[\dot{\rho}_{1}\Gamma^{1}_{~{}01}+\rho_{1}\dot{\Gamma}^{1}_{~{}01}-\ddot{\rho}_{1}\right]\hat{x}_{1}$
(59)

and the form of $[\hat{a}_{1},\hat{H}_{1}]$ is given by

$\begin{split}[\hat{a}_{1},\hat{H}_{1}]&=\frac{i\hat{p}_{1}}{\sqrt{2}m\rho_{1}}+\frac{i\Gamma^{1}_{~{}01}\hat{x}_{1}}{\sqrt{2}\rho_{1}}+\frac{\rho_{1}m\omega_{1}^{2}\hat{x}_{1}}{\sqrt{2}}+\frac{\dot{\rho}_{1}\hat{p}_{1}}{\sqrt{2}}+\frac{m\Gamma^{1}_{~{}01}\dot{\rho}_{1}\hat{x}_{1}}{\sqrt{2}}~{}.\end{split}$
(60)

Summing eq.(59) with eq.(60) and making use of the Ermakov-Pinney equation in
eq.(37) (for $\xi_{1}=1$ and
$\dot{d}(t)=\frac{1}{2}\dot{\Gamma}^{1}_{~{}01}$), we can express the form of
the commutator bracket in eq.(58) as follows

$[\hat{a}_{1},i\partial_{t}-\hat{H}_{1}]=-\frac{1}{m\rho_{1}^{2}}\hat{a}_{1}~{}.$
(61)

Making use of the analytical form of the commutator bracket given in eq.(61),
again we can recast the right hand side of eq.(56) as

$\begin{split}\langle
n_{1}|i(\partial_{t})_{1}-\hat{H}_{1}(t)|n_{1}\rangle=-\frac{n_{1}}{m\rho_{1}^{2}}+\langle
0|i\partial_{t}-\hat{H}_{1}|0\rangle~{}.\end{split}$ (62)

Note that the second term in the right hand side of the above equation is a
constant arbitrary phase, and therefore can be chosen conveniently. A
convenient form of the zero state contribution in our current analysis can be
chosen as [12, 35, 36]333Note that the choice made here is inconsequential as
we shall see in the subsequent discussion that this does not contribute to the
Berry phase.

$\langle 0|i\partial_{t}-\hat{H}_{1}|0\rangle=-\frac{1}{2m\rho_{1}^{2}}~{}.$
(63)

Using, eq.(s)(62,63), we can write down the Lewis phase $\theta_{1}(t)$ as
follows

$\theta_{1}(t)=-\left(n_{1}+\frac{1}{2}\right)\int_{0}^{t}\frac{d\tau}{m\rho_{1}^{2}(\tau)}~{}.$
(64)

Following a similar analysis, we can obtain the other Lewis phase
$\theta_{2}(t)$ as follows

$\theta_{2}(t)=-\left(n_{2}+\frac{1}{2}\right)\int_{0}^{t}\frac{d\tau}{m\rho_{2}^{2}(\tau)}~{}.$
(65)

With the forms of $\theta_{1}(t)$ and $\theta_{2}(t)$ in hand, we can now
proceed to obtain the geometric part of the phase making use of the adiabatic
approximation. In the adiabatic approximation,
$\ddot{\rho}_{1}(t)=\ddot{\rho}_{2}(t)=0$, and the modified Ermakov-Pinney
equations corresponding to the two coordinates take the form

$\displaystyle\omega^{2}(t)-\dot{\Gamma}^{1}_{~{}01}(t)$
$\displaystyle\simeq\frac{1}{m^{2}\rho_{1}^{4}(t)}~{},$ (66)
$\displaystyle\omega^{2}(t)+\dot{\Gamma}^{1}_{~{}01}(t)$
$\displaystyle\simeq\frac{1}{m^{2}\rho_{2}^{4}(t)}~{}.$ (67)

Since $\dot{\Gamma}^{1}_{~{}01}(t)=\ddot{h}_{11}(t)/2$ is very small (as
$h_{11}(t)$ is a very small quantity), it is evident that
$\dot{\Gamma}^{1}_{~{}01}(t)<<\omega^{2}(t)$. We can therefore simplify
eq.(s)(66,67) to the following forms

$\displaystyle\frac{1}{m\rho_{1}^{2}(t)}$
$\displaystyle\simeq\omega(t)-\frac{\dot{\Gamma}^{1}_{~{}01}(t)}{2\omega(t)}~{},$
(68) $\displaystyle\frac{1}{m\rho_{2}^{2}(t)}$
$\displaystyle\simeq\omega(t)+\frac{\dot{\Gamma}^{1}_{~{}01}(t)}{2\omega(t)}~{}.$
(69)

It is straightforward to show that, corresponding to the two coordinate
directions, if one considers the harmonic oscillator frequencies to be
different in the two coordinate directions then eq.(s)(68,69) take the forms
given by

$\displaystyle\frac{1}{m\rho_{1}^{2}(t)}$
$\displaystyle=\omega_{1}(t)-\frac{\dot{\Gamma}^{1}_{~{}01}(t)}{2\omega_{1}(t)}~{},\frac{1}{m\rho_{2}^{2}(t)}=\omega_{2}(t)+\frac{\dot{\Gamma}^{1}_{~{}01}(t)}{2\omega_{2}(t)}~{}.$
(70)

Substituting eq.(70) back in eq.(s)(64,65), we obtain the following two
relations

$\displaystyle\tilde{\theta}_{1}(t)$
$\displaystyle=-\left(n_{1}+\frac{1}{2}\right)\left(\int_{0}^{t}\omega_{1}(\tau)d\tau-\int_{0}^{t}\frac{\dot{\Gamma}^{1}_{~{}01}(\tau)}{2\omega_{1}(\tau)}d\tau\right)~{},$
(71) $\displaystyle\tilde{\theta}_{2}(t)$
$\displaystyle=-\left(n_{2}+\frac{1}{2}\right)\left(\int_{0}^{t}\omega_{2}(\tau)d\tau+\int_{0}^{t}\frac{\dot{\Gamma}^{1}_{~{}01}(\tau)}{2\omega_{2}(\tau)}d\tau\right)$
(72)

where $\tilde{\theta}_{n}(t)$ (for $n=\\{1,2\\}$) denotes the Lewis phase
$\theta_{n}(t)$ in the adiabatic approximation. From eq.(s)(71,72), it is
straightforward to observe that the first integrals introduce a dynamic phase
whereas the second integrals are geometric in nature. Now we consider that the
Hamiltonian of the system completes an adiabatic cycle at $t=\mathcal{T}$ in
the parameter space and as a result it is possible to write down

$\mathcal{R}(0)=\mathcal{R}(\mathcal{T})~{};~{}~{}\mathcal{R}=(b,d)~{}.$ (73)

We can now easily write down the first order time derivative in terms of
$\mathcal{R}$ as

$\frac{d}{dt}=\frac{d\mathcal{R}}{dt}\nabla_{\mathcal{R}}~{}.$ (74)

Hence, corresponding to the two coordinate directions and making use of
eq.(74), we can write down Berry’s geometric phases to be of the following
form

$\displaystyle\Theta_{1}^{G}$
$\displaystyle=\left(n_{1}+\frac{1}{2}\right)\oint^{\mathcal{R}}\frac{1}{\omega_{1}}\vec{\nabla}_{\mathcal{R}}\left(\Gamma^{1}_{~{}01}\right).d\vec{\mathcal{R}}~{},$
(75) $\displaystyle\Theta_{2}^{G}$
$\displaystyle=-\left(n_{2}+\frac{1}{2}\right)\oint^{\mathcal{R}}\frac{1}{\omega_{2}}\vec{\nabla}_{\mathcal{R}}\left(\Gamma^{1}_{~{}01}\right).d\vec{\mathcal{R}}~{}.$
(76)

Similar results for the Berry phases were obtained using a completely
different method in [38]. The presence of the Christoffel symbol
$\Gamma^{1}_{~{}01}$ in the expressions for the Berry phases corresponding to
the two spatial directions in the above two equations indicates the
possibility that observing a non-zero Berry phase in this set up is equivalent
to detecting a gravitational wave. We would like to make a remark now. There
is a possibility that a contribution to the Berry phase may also arise from
the boundary term (eq.(9)). In section 8, we shall calculate this contribution
explicitly and show that it vanishes.

## 7 Explicit Berry phase calculation

In this section, we shall make use of eq.(s)(75,76) to compute some explicit
forms of the Berry phases corresponding to the two coordinate directions.

For a linearly polarized gravitational wave

$h_{jk}(t)=2f_{0}\cos\Omega
t\left(\varepsilon_{+}\sigma^{3}_{jk}+\varepsilon_{\times}\sigma^{1}_{jk}\right)$
(77)

where we have set $\chi(t)=f_{0}\cos\Omega t$ in eq.(3) with $f_{0}$ being the
amplitude of the gravitational wave and $\Omega$ being the frequency of the
same. It is important to note that we have considered that the gravitational
wave is carrying plus polarization only. Using eq.(77), we can write down the
first order time derivative of the Christoffel symbol
$\dot{\Gamma}^{1}_{~{}01}=\frac{\ddot{h}_{11}(t)}{2}=-f_{0}\Omega^{2}\varepsilon_{+}\cos\Omega
t$. For a linearly polarized gravitational wave, eq.(s)(75,76) give

$\displaystyle\Theta_{1}^{G}$
$\displaystyle=-f_{0}\Omega^{2}\varepsilon_{+}\left(n_{1}+\frac{1}{2}\right)\int_{0}^{\frac{2\pi}{\Omega}}d\tau\frac{\cos\Omega\tau}{2\omega_{1}(\tau)}~{},$
(78) $\displaystyle\Theta^{G}_{2}$
$\displaystyle=f_{0}\Omega^{2}\varepsilon_{+}\left(n_{2}+\frac{1}{2}\right)\int_{0}^{\frac{2\pi}{\Omega}}d\tau\frac{\cos\Omega\tau}{2\omega_{2}(\tau)}~{}.$
(79)

Now for an explicit set of choices of the frequencies
$\omega_{1}(t)=\omega_{01}\cos\Omega t$ and
$\omega_{2}(t)=\omega_{02}\cos\Omega(t)$, we can evaluate $\Theta_{1}^{G}$ and
$\Theta_{2}^{G}$ as follows

$\displaystyle\Theta_{1}^{G}$
$\displaystyle=-\frac{f_{0}\pi\Omega\varepsilon_{+}}{\omega_{01}}\left(n_{1}+\frac{1}{2}\right)~{},$
(80) $\displaystyle\Theta_{2}^{G}$
$\displaystyle=\frac{f_{0}\pi\Omega\varepsilon_{+}}{\omega_{02}}\left(n_{2}+\frac{1}{2}\right)~{}.$
(81)

Our second choice of the frequencies are
$\omega_{1}(t)=\omega_{1}+\tilde{\omega}_{1}\cos\Omega t$ and
$\omega_{2}(t)=\omega_{2}+\tilde{\omega}_{2}\cos\Omega t$, such that
$|\omega_{1}|\neq|\tilde{\omega}_{1}|$ and
$|\omega_{2}|\neq|\tilde{\omega}_{2}|$. Under this choice of frequencies, we
can evaluate $\Theta_{1}^{G}$ and $\Theta_{2}^{G}$ given by

$\displaystyle\Theta_{1}^{G}=$ $\displaystyle-\frac{\pi
f_{0}\Omega\varepsilon_{+}}{\tilde{\omega}_{1}}\left(n_{1}+\frac{1}{2}\right)~{},$
(82) $\displaystyle\Theta_{2}^{G}=$ $\displaystyle\frac{\pi
f_{0}\Omega\varepsilon_{+}}{\tilde{\omega}_{2}}\left(n_{2}+\frac{1}{2}\right)~{}.$
(83)

## 8 Geometric phase from the boundary term

In this section, we shall calculate the geometric phase from the boundary term
in eq.(9). We start by rewriting the term (eq.(9)) as

$\begin{split}\mathcal{B}&=\frac{m}{2}\int_{0}^{\mathcal{T}}d\tau\frac{d}{d\tau}\left(\Gamma^{j}_{~{}0k}(\tau)x_{j}(\tau)x^{k}(\tau)\right)\\\
&=\frac{m}{2}\oint^{\mathcal{R}^{\prime}}d\vec{\mathcal{R}}^{\prime}.\vec{\nabla}_{\mathcal{R}^{\prime}}\left(\Gamma^{j}_{~{}0k}(\tau)x_{j}(\tau)x^{k}(\tau)\right)\end{split}$
(84)

where $\mathcal{R}^{\prime}(0)=\mathcal{R}^{\prime}(\mathcal{T})$. Eq.(84)
reveals that $\mathcal{B}$ is geometric in nature. We can now raise the
phasespace position variables to operator status and after a little bit of
calculation, we can recast eq.(84) in the following form (where the
gravitational wave is considered to be carrying plus polarization only)

$\begin{split}\hat{\mathcal{B}}&=\frac{m}{2}\left\\{\Gamma^{1}_{~{}01}(\tau)\hat{x}_{1}^{2}(\tau)\otimes\hat{\mathbb{1}}_{2}+\Gamma^{2}_{~{}02}(\tau)~{}\hat{\mathbb{1}}_{1}\otimes\hat{x}_{2}^{2}(\tau)\right\\}\Bigr{\rvert}_{\tau=0}^{\tau=\mathcal{T}}\\\
&=\frac{m}{4}\dot{h}_{+}(\tau)\left(\hat{x}_{1}^{2}(\tau)\otimes\hat{\mathbb{1}}_{2}-\hat{\mathbb{1}}_{1}\otimes\hat{x}_{2}^{2}(\tau)\right)\Bigr{\rvert}_{\tau=0}^{\tau=\mathcal{T}}\\\
&=\hat{\mathcal{B}}_{1}\otimes\hat{\mathbb{1}}_{2}+\hat{\mathbb{1}}_{1}\otimes\hat{\mathcal{B}}_{2}\end{split}$
(85)

with the definitions
$\hat{\mathcal{B}}_{1}\equiv\frac{m}{4}\left(\dot{h}_{+}(\mathcal{T})\hat{x}_{1}^{2}(\mathcal{T})-\dot{h}_{+}(0)\hat{x}_{1}^{2}(0)\right)$
and
$\hat{\mathcal{B}}_{2}\equiv-\frac{m}{4}\left(\dot{h}_{+}(\mathcal{T})\hat{x}_{2}^{2}(\mathcal{T})-\dot{h}_{+}(0)\hat{x}_{2}^{2}(0)\right)$.
The total geometric phase can be obtained by calculating $\langle
n_{1}|\hat{\mathcal{B}}_{1}|n_{1}\rangle$ and $\langle
n_{2}|\hat{\mathcal{B}}_{2}|n_{2}\rangle$ separately and adding them. Before
proceeding further, we need to obtain $\hat{x}_{1}^{2}(\tau)$ and
$\hat{x}_{2}^{2}(\tau)$ in terms of the corresponding raising and lowering
operators. Using the form of the lowering operator in the first coordinate
direction from eq.(47) and making use of its complex conjugate, we obtain

$\begin{split}\hat{x}_{1}(\tau)=\frac{\rho_{1}(\tau)}{\sqrt{2}}\left(\hat{a}_{1}+\hat{a}_{1}^{\dagger}\right)\implies\hat{x}_{1}^{2}(\tau)=\frac{\rho_{1}^{2}(\tau)}{2}\left(\hat{a}_{1}^{2}+\hat{a}_{1}^{\dagger
2}+2\hat{a}_{1}^{\dagger}\hat{a}_{1}+1\right)~{}.\end{split}$ (86)

Similarly by using eq.(49) and its complex conjugate, we obtain

$\begin{split}\hat{x}_{2}(\tau)=\frac{\rho_{2}(\tau)}{\sqrt{2}}\left(\hat{a}_{2}+\hat{a}_{2}^{\dagger}\right)\implies\hat{x}_{2}^{2}(\tau)=\frac{\rho_{2}^{2}(\tau)}{2}\left(\hat{a}_{2}^{2}+\hat{a}_{2}^{\dagger
2}+2\hat{a}_{2}^{\dagger}\hat{a}_{2}+1\right)~{}.\end{split}$ (87)

We shall now compute the contribution of the boundary term towards the
geometric phase in the first coordinate direction as follows

$\begin{split}\langle
n_{1}|\hat{\mathcal{B}}_{1}|n_{1}\rangle&=\frac{1}{2}\langle
n_{1}|\frac{m}{4}\left(\dot{h}_{+}(\mathcal{T})\rho_{1}^{2}(\mathcal{T})-\dot{h}_{+}(0)\rho_{1}^{2}(0)\right)\left(\hat{a}_{1}^{2}+\hat{a}_{1}^{\dagger
2}+2\hat{a}_{1}^{\dagger}\hat{a}_{1}+1\right)|n_{1}\rangle\\\
&=\frac{m}{4}\left(n_{1}+\frac{1}{2}\right)\left(\dot{h}_{+}(\mathcal{T})\rho_{1}^{2}(\mathcal{T})-\dot{h}_{+}(0)\rho_{1}^{2}(0)\right)~{}.\end{split}$
(88)

Our next aim is to substitute the form of $\rho_{1}^{2}(\tau)$ in the above
equation. Using the Ermakov-Pinney equation (eq.(68)) under the adiabatic
approximation (with $\omega(\tau)$ replaced by $\omega_{1}(\tau)$), we have

$\begin{split}m&\rho_{1}^{2}(\tau)=\frac{1}{\omega_{1}(\tau)-\frac{\dot{\Gamma}^{1}_{~{}01}(\tau)}{2\omega_{1}(\tau)}}\\\
\implies&\rho_{1}^{2}(\tau)\simeq\frac{1}{m\omega_{1}(\tau)}+\frac{\ddot{h}_{+}(\tau)}{4m\omega_{1}^{3}(\tau)}~{}.\end{split}$
(89)

Substituting eq.(89) back in eq.(88) and keeping terms upto $\mathcal{O}(h)$,
we obtain

$\langle
n_{1}|\hat{\mathcal{B}}_{1}|n_{1}\rangle\simeq\frac{1}{4}\left(n_{1}+\frac{1}{2}\right)\left(\frac{\dot{h}_{+}(\mathcal{T})}{\omega_{1}(\mathcal{T})}-\frac{\dot{h}_{+}(0)}{\omega_{1}(0)}\right)~{}.$
(90)

Similarly, the term $\langle n_{2}|\hat{B}_{2}|n_{2}\rangle$ can be calculated
and is given by

$\langle
n_{2}|\hat{\mathcal{B}}_{2}|n_{2}\rangle\simeq-\frac{1}{4}\left(n_{2}+\frac{1}{2}\right)\left(\frac{\dot{h}_{+}(\mathcal{T})}{\omega_{2}(\mathcal{T})}-\frac{\dot{h}_{+}(0)}{\omega_{2}(0)}\right)~{}.$
(91)

Hence, the total geometric phase due the boundary term in an anisotropic
harmonic oscillator reads

$\begin{split}\langle
n_{1},n_{2}|\hat{\mathcal{B}}|n_{1},n_{2}\rangle&=\langle
n_{1}|\hat{\mathcal{B}}_{1}|n_{1}\rangle\langle
n_{2}|\hat{\mathbb{1}}_{2}|n_{2}\rangle+\langle
n_{1}|\hat{\mathbb{1}}_{1}|n_{1}\rangle\langle
n_{2}|\hat{\mathcal{B}}_{2}|n_{2}\rangle\\\ &=\langle
n_{1}|\hat{\mathcal{B}}_{1}|n_{1}\rangle+\langle
n_{2}|\hat{\mathcal{B}}_{2}|n_{2}\rangle\\\
&=\frac{1}{4}\left(n_{1}+\frac{1}{2}\right)\left(\frac{\dot{h}_{+}(\mathcal{T})}{\omega_{1}(\mathcal{T})}-\frac{\dot{h}_{+}(0)}{\omega_{1}(0)}\right)-\frac{1}{4}\left(n_{2}+\frac{1}{2}\right)\left(\frac{\dot{h}_{+}(\mathcal{T})}{\omega_{2}(\mathcal{T})}-\frac{\dot{h}_{+}(0)}{\omega_{2}(0)}\right)~{}.\end{split}$
(92)

For an isotropic harmonic oscillator with the oscillator frequency
$\omega_{1}(\tau)=\omega_{2}(\tau)=\omega(\tau)$, interacting with a
gravitational wave, we obtain

$\langle
n|\hat{B}|n\rangle=\frac{1}{4}\left(n_{1}-n_{2}\right)\left(\frac{\dot{h}_{+}(\mathcal{T})}{\omega(\mathcal{T})}-\frac{\dot{h}_{+}(0)}{\omega(0)}\right)$
(93)

where $|n\rangle=|n_{1},n_{2}\rangle=|n_{1}\rangle\otimes|n_{2}\rangle$. Both
$h(\tau)$ and $\omega(\tau)$ belongs to the parameter space
$\mathcal{R}^{\prime}$ and we therefore know, $h(\mathcal{T})=h(0)$ and
$\omega(\mathcal{T})=\omega(0)$. For example, the linearly polarized
gravitational wave from eq.(77) has the component
$h_{+}(\tau)=2f_{0}\cos\Omega\tau$. Here, $\mathcal{T}=\frac{2\pi}{\Omega}$
and $h_{+}(2\pi/\Omega)=h_{+}(0)$. Therefore, from eq.(92) (anisotropic
harmonic oscillator) and eq.(93) (isotropic harmonic oscillator), we find that
the contribution to the geometric phase coming from the boundary term (eq.(9))
vanishes.

## 9 Incident gravitational wave with cross polarization only

In this section, we shall consider a gravitational wave incident on a two
dimensional isotropic harmonic oscillator with cross polarization only
($\varepsilon_{+}=0$). The Hamiltonian operator of the system can be expressed
as

$\begin{split}\hat{H}(t)&=\frac{a(t)}{2}\left(\hat{p}_{1}^{2}+\hat{p}_{2}^{2}\right)+\frac{b(t)}{2}(\hat{x}_{1}^{2}+\hat{x}_{2}^{2})+f(t)(\hat{x}_{1}\hat{p}_{2}+\hat{p}_{1}\hat{x}_{2})\\\
&=\frac{a(t)}{2}\left(\hat{p}_{1}^{2}+\hat{p}_{2}^{2}\right)+\frac{b(t)}{2}(\hat{x}_{1}^{2}+\hat{x}_{2}^{2})+\frac{f(t)}{2}(\hat{x}_{1}\hat{p}_{2}+\hat{p}_{2}\hat{x}_{1})+\frac{f(t)}{2}(\hat{p}_{1}\hat{x}_{2}+\hat{x}_{2}\hat{p}_{1})\end{split}$
(94)

where $a(t)=\frac{1}{m}$, $b(t)=m\omega^{2}(t)$, and
$f(t)=\varepsilon_{\times}\dot{\chi}(t)$ with $\chi(t)$ denoting the amplitude
of the gravitational wave as before. To write down the last line in eq.(94),
we have made use of the commutation relation
$[\hat{x}_{i},\hat{p}_{j}]=i\hbar\delta_{ij}$ ($i,j=1,2$). Before trying to
obtain a Lewis invariant corresponding to the harmonic oscillator-
gravitational wave system, we make a change of coordinates as follows

$\hat{x}_{+}\equiv\frac{1}{\sqrt{2}}\left(\hat{x}_{1}+\hat{x}_{2}\right)~{},~{}~{}\hat{x}_{-}\equiv\frac{1}{\sqrt{2}}\left(\hat{x}_{1}-\hat{x}_{2}\right)~{}.$
(95)

Corresponding to these new coordinate directions, one can now define two new
momentum operators in the following way

$\begin{split}\hat{p}_{+}\equiv\frac{1}{\sqrt{2}}\left(\hat{p}_{1}+\hat{p}_{2}\right)~{},~{}~{}\hat{p}_{-}\equiv\frac{1}{\sqrt{2}}\left(\hat{p}_{1}-\hat{p}_{2}\right)~{}.\end{split}$
(96)

It is easy to check the following set of commutation relations among the new
position and momentum operators,

$\begin{split}[\hat{x}_{+},\hat{p}_{+}]=[\hat{x}_{-},\hat{p}_{-}]=i\hbar~{},~{}~{}[\hat{x}_{+},\hat{x}_{-}]=[\hat{p}_{+},\hat{p}_{-}]=[\hat{x}_{+},\hat{p}_{-}]=[\hat{x}_{-},\hat{p}_{+}]=0~{}.\end{split}$
(97)

We can therefore claim that $\hat{p}_{+}$ is the momentum conjugate to
$\hat{x}_{+}$ and $\hat{p}_{-}$ is the momentum conjugate to $\hat{x}_{-}$. In
terms of these newly defined operators, we can write down the old operators as

$\hat{x}_{1}=\frac{1}{\sqrt{2}}(\hat{x}_{+}+\hat{x}_{-})~{},~{}~{}\hat{x}_{2}=\frac{1}{\sqrt{2}}(\hat{x}_{+}-\hat{x}_{-})~{},~{}~{}\hat{p}_{1}=\frac{1}{\sqrt{2}}(\hat{p}_{+}+\hat{p}_{-})~{},~{}\text{and}~{}~{}\hat{p}_{2}=\frac{1}{\sqrt{2}}(\hat{p}_{+}-\hat{p}_{-})~{}.$
(98)

Making use of the above equation, one can rewrite the Hamiltonian in eq.(94)
in terms of the new “$\pm$” operators as follows

$\begin{split}\hat{H}(t)&=\frac{a}{2}\left(\hat{p}_{+}^{2}+\hat{p}_{-}^{2}\right)+\frac{b(t)}{2}\left(\hat{x}_{+}^{2}+\hat{x}_{-}^{2}\right)+\frac{f(t)}{2}(\hat{x}_{+}\hat{p}_{+}+\hat{p}_{+}\hat{x}_{+})-\frac{f(t)}{2}(\hat{x}_{-}\hat{p}_{-}+\hat{p}_{-}\hat{x}_{-})\\\
&=\frac{a}{2}\hat{p}_{+}^{2}+\frac{b(t)}{2}\hat{x}_{+}^{2}+\frac{f(t)}{2}(\hat{x}_{+}\hat{p}_{+}+\hat{p}_{+}\hat{x}_{+})+\frac{a}{2}\hat{p}_{-}^{2}+\frac{b(t)}{2}\hat{x}_{-}^{2}-\frac{f(t)}{2}(\hat{x}_{-}\hat{p}_{-}+\hat{p}_{-}\hat{x}_{-})\\\
&=\hat{H}_{+}(t)+\hat{H}_{-}(t)\end{split}$ (99)

where
$\hat{H}_{\pm}(t)=\frac{a}{2}\hat{p}_{\pm}^{2}+\frac{b(t)}{2}\hat{x}_{\pm}^{2}\pm\frac{f(t)}{2}(\hat{x}_{\pm}\hat{p}_{\pm}+\hat{p}_{\pm}\hat{x}_{\pm})$.
It is very interesting to see that by applying the coordinate changes
introduced in eq.(95) (and the conjugate momentum operators in eq.(96)), the
coupled Hamiltonian in the $\\{1,2\\}$ basis now gets decoupled. The reason
behind this decoupling can be easily understood via a simple diagram.

Figure 1: In the new $`\pm^{\prime}$ coordinate basis, the cross polarization
effectively behaves like the plus polarization and therefore it is possible to
decouple the Hamiltonian in this new coordinate system.

In Figure 1, 1(a) denotes a time slice of an incoming gravitational wave
carrying a plus polarization only in the $\\{x_{1},x_{2}\\}$ coordinate
system. Figure 1(b) denotes an incident gravitational wave with cross
polarization only in the same coordinate basis. Next, we rotate the coordinate
system by an angle $\pi/4$ in the clockwise direction and the new coordinate
axes are now annotated as $x_{+}$ and $x_{-}$. As can be seen from Figure
1(c), in this new coordinate system the cross polarization effectively behaves
as the plus polarization. This is the primary reason behind the decoupling of
the Hamiltonian as can be seen from eq.(99) after implementing the coordinate
changes implemented in eq.(95).

Our next aim is to obtain the Lewis invariant corresponding to the Hamiltonian
given in eq.(99). Due to the striking structural similarities among the
Hamiltonian in eq.(99) with that of the Hamiltonian for the plus polarization
case in eq.(11), we can proceed to obtain the Lewis phases as before. As
before, one can propose the Lewis invariants corresponding to the $x_{+}$ and
$x_{-}$ coordinate directions as

$\hat{I}_{\pm}(t)=\alpha_{\pm}(t)\hat{p}_{\pm}^{2}+\beta_{\pm}(t)\hat{x}_{\pm}^{2}+\delta_{\pm}(t)(\hat{x}_{\pm}\hat{p}_{\pm}+\hat{p}_{\pm}\hat{x}_{\pm})~{}.$
(100)

We introduce two time dependent parameters $\rho_{+}(t)$ and $\rho_{-}(t)$ as
follows

$\alpha_{\pm}(t)=\kappa_{\pm}\rho_{\pm}^{2}(t)~{}.$ (101)

Calculating $i\hbar\frac{\partial\hat{I}_{\pm}(t)}{\partial
t}+[\hat{I}_{\pm}(t),\hat{H}_{\pm}(t)]=0$ for the proposed forms of the two
invariants in eq.(100), we obtain six ordinary differential equations. Making
use of eq.(101) in these set of differential equations, we eventually obtain
two Ermakov-Pinney equations as

$\displaystyle\ddot{\rho}_{+}(t)+\left(ab(t)-\dot{f}(t)\right)\rho_{+}(t)$
$\displaystyle=\frac{a^{2}\xi_{+}^{2}}{\rho_{+}^{3}(t)}~{},$ (102)
$\displaystyle\ddot{\rho}_{-}(t)+\left(ab(t)+\dot{f}(t)\right)\rho_{-}(t)$
$\displaystyle=\frac{a^{2}\xi_{-}^{2}}{\rho_{-}^{3}(t)}$ (103)

where $\xi_{+}^{2}$ and $\xi_{-}^{2}$ are the two integration constants. Using
the above two Ermakov-Pinney equations, we can obtain the form of the
parameters as

$\displaystyle\beta_{+}=\frac{\kappa_{+}}{a^{2}}\left(\dot{\rho}_{+}^{2}-2\rho_{+}\dot{\rho}_{+}f(t)\right)+\frac{\kappa_{+}\xi_{+}^{2}}{\rho_{+}^{2}}~{}$
$\displaystyle,~{}\beta_{-}=\frac{\kappa_{-}}{a^{2}}\left(\dot{\rho}_{-}^{2}+2\rho_{-}\dot{\rho}_{-}f(t)\right)+\frac{\kappa_{-}\xi_{-}^{2}}{\rho_{-}^{2}}~{},$
(104)
$\displaystyle\delta_{+}(t)=-\frac{\kappa_{+}}{a^{2}}\left(\rho_{+}\dot{\rho}_{+}-\rho_{+}^{2}f(t)\right)~{}$
$\displaystyle,~{}\delta_{-}(t)=-\frac{\kappa_{-}}{a^{2}}\left(\rho_{-}\dot{\rho}_{-}+\rho_{-}^{2}f(t)\right)~{}.$
(105)

Using eq.(s)(101,104,105), we can rewrite the Lewis invariants as follows

$\displaystyle\hat{I}_{+}(t)$
$\displaystyle=\kappa_{+}\rho_{+}\hat{p}_{+}^{2}+\left(\kappa_{+}m^{2}\left(\dot{\rho}_{+}^{2}-2\varepsilon_{\times}\dot{\chi}(t)\rho_{+}\dot{\rho}_{+}\right)+\frac{\kappa_{+}\xi_{+}^{2}}{\rho_{+}^{2}}\right)\hat{x}_{+}^{2}-m\kappa_{+}\rho_{+}\left(\dot{\rho}_{+}-\varepsilon_{\times}\dot{\chi}(t)\rho_{+}\right)\left(\hat{x}_{+}\hat{p}_{+}+\hat{p}_{+}\hat{x}_{+}\right)~{},$
(106) $\displaystyle\hat{I}_{-}(t)$
$\displaystyle=\kappa_{-}\rho_{-}\hat{p}_{-}^{2}+\left(\kappa_{-}m^{2}\left(\dot{\rho}_{-}^{2}+2\varepsilon_{\times}\dot{\chi}(t)\rho_{-}\dot{\rho}_{-}\right)+\frac{\kappa_{-}\xi_{-}^{2}}{\rho_{-}^{2}}\right)\hat{x}_{-}^{2}-m\kappa_{-}\rho_{-}\left(\dot{\rho}_{-}+\varepsilon_{\times}\dot{\chi}(t)\rho_{-}\right)\left(\hat{x}_{-}\hat{p}_{-}+\hat{p}_{-}\hat{x}_{-}\right)~{}.$
(107)

Making use of the completing the square approach, one can now obtain the
following annihilation operators corresponding to the $``\pm"$ coordinate
directions as

$\displaystyle\hat{a}_{+}$
$\displaystyle=\frac{1}{\sqrt{2}\rho_{+}}\left(\hat{x}_{+}+i\rho_{+}^{2}\hat{p}_{+}+im\rho_{+}\left(\varepsilon_{\times}\dot{\chi}\rho_{+}-\dot{\rho}_{+}\right)\hat{x}_{+}\right)~{},$
(108) $\displaystyle\hat{a}_{-}$
$\displaystyle=\frac{1}{\sqrt{2}\rho_{-}}\left(\hat{x}_{-}+i\rho_{-}^{2}\hat{p}_{-}-im\rho_{-}\left(\varepsilon_{\times}\dot{\chi}\rho_{-}+\dot{\rho}_{-}\right)\hat{x}_{-}\right)~{}.$
(109)

Again it is straightforward to check that
$[\hat{a}_{A},\hat{a}^{\dagger}_{B}]=\delta_{AB}$ and
$[\hat{a}_{A},\hat{a}_{B}]=[\hat{a}^{\dagger}_{A},\hat{a}^{\dagger}_{B}]=0$
where $A,B=+,-$. Instead of going through the same procedure, it is possible
to write down the Lewis phases, corresponding to the two coordinate directions
as follows

$\displaystyle\theta_{+}(t)=-(n_{+}+\frac{1}{2})\int_{0}^{t}\frac{d\tau}{m\rho_{+}^{2}(\tau)}~{},$
(110)
$\displaystyle\theta_{-}(t)=-(n_{-}+\frac{1}{2})\int_{0}^{t}\frac{d\tau}{m\rho_{-}^{2}(\tau)}$
(111)

where $n_{+}$ and $n_{-}$ denotes the eigenvalues of the number operators
$\hat{N}_{+}=\hat{a}^{\dagger}_{+}\hat{a}_{+}$ and
$\hat{N}_{-}=\hat{a}^{\dagger}_{-}\hat{a}_{-}$ corresponding to the eigenstate
$|n_{+},n_{-}\rangle=|n_{+}\rangle\otimes|n_{-}\rangle$. Here the number state
of the system is expressed as a tensor product state of the individual number
states. Using the adiabatic approximation in eq.(s)(102,103), we can recast
the above Lewis phases as

$\displaystyle\theta_{+}(t)$
$\displaystyle\simeq-\left(n_{+}+\frac{1}{2}\right)\int_{0}^{t}\omega(\tau)d\tau+\left(n_{+}+\frac{1}{2}\right)\int_{0}^{t}d\tau\frac{\varepsilon_{\times}\ddot{\chi}(\tau)}{2\omega(\tau)^{2}}~{},$
(112) $\displaystyle\theta_{-}(t)$
$\displaystyle\simeq-\left(n_{-}+\frac{1}{2}\right)\int_{0}^{t}\omega(\tau)d\tau-\left(n_{-}+\frac{1}{2}\right)\int_{0}^{t}d\tau\frac{\varepsilon_{\times}\ddot{\chi}(\tau)}{2\omega(\tau)^{2}}~{}.$
(113)

The second integrals in eq.(s)(112,113) are geometric in nature. Therefore,
corresponding to the two new coordinate directions, we can write down Berry’s
geometric phases as

$\displaystyle\Theta^{G}_{+}$
$\displaystyle=\left(n_{+}+\frac{1}{2}\right)\oint^{\mathcal{R}}\frac{1}{\omega}\vec{\nabla}_{\mathcal{R}}\left(\Gamma^{1}_{~{}02}\right).d\vec{\mathcal{R}}~{},$
(114) $\displaystyle\Theta^{G}_{-}$
$\displaystyle=-\left(n_{-}+\frac{1}{2}\right)\oint^{\mathcal{R}}\frac{1}{\omega}\vec{\nabla}_{\mathcal{R}}\left(\Gamma^{1}_{~{}02}\right).d\vec{\mathcal{R}}$
(115)

where $\mathcal{R}(0)=\mathcal{R}(\mathcal{T})$ and $\mathcal{R}=(b,f)$. Using
eq.(s)(114,115), one can now write down the total Berry phase of the system as

$\Theta^{G}=\left(n_{+}-n_{-}\right)\oint^{\mathcal{R}}\frac{1}{\omega}\vec{\nabla}_{\mathcal{R}}\left(\Gamma^{1}_{~{}02}\right).d\vec{\mathcal{R}}~{}.$
(116)

It is quite easily understandable that the total Berry phase of the system is
independent of the choice of the coordinate basis. Contrary to the claim made
in [38], we do observe that for different values of $n_{+}$ and $n_{-}$, there
exists a Berry phase which is dependent on the cross polarization of the
gravitational wave. Hence, not only the plus polarization, the cross
polarization also carries signatures of a geometric phase.

## 10 Summary

In this work we have considered a gravitational wave interacting with an
isotropic two dimensional harmonic oscillator. Starting from the traditional
representation of the gravitational wave (in the transverse traceless gauge)
to be in both the plus and cross polarizations, we proceed by omitting the
cross polarization terms. The reason behind dropping the cross polarization is
to avoid unnecessary complications during the calculation. We have then used
the Lewis invariant method to obtain a suitable Lewis invariant corresponding
to the total Hamiltonian of the system. Our main goal is to obtain a Berry
phase from the Lewis phase by making an adiabatic approximation. We follow
essentially the approach in [35] in achieving this task. In order to find a
Lewis invariant, we have used an ansatz considering contributions from all
quadratic order terms in the phase space variables and finally obtained two
separable Lewis invariants. Using the “completing the square” approach, we
have then obtained the ladder operators corresponding to the two coordinate
directions from the obtained Lewis invariants. We then make use of the form of
the ladder operator to obtain a commutation relation between the ladder
operator and $i\partial_{t}-\hat{H}_{k}$ operator, $k=1,2$. Using this
commutation relation, we finally obtain the analytical form of $\langle
n_{k}|i(\partial_{t})_{k}-\hat{H}_{k}|n_{k}\rangle$ which is essentially equal
to the total time derivative of the Lewis phase $\theta_{k}$ ($k=1,2$). Our
next aim was to extract the geometric part of the phase in the Lewis phase
itself. In order to do so we have made use of the Ermakov Pinney equations
corresponding to the two spatial directions and made use of the adiabatic
approximation to obtain the geometric part of the Lewis phase. Under this
approximation, the Lewis phase breaks into two separate parts. The first part
is dynamic in nature whereas the second part, which is completely dependent on
the gravitational perturbation due to the incoming gravitational wave, is
geometrical in nature. These are the Berry phases which arise solely due to
the passing of the gravitational wave. In order to truly underpin the
underlying nature of Berry’s geometric phase we have made use of the linearly
polarized gravitational wave template and made use of explicit values of the
frequencies of the harmonic oscillator system. The Berry phase corresponding
to the two spatial directions entirely depends upon the amplitude of the
linearly polarized gravitational wave, amplitude of the plus polarization, and
frequency of the incident gravitational wave. The approach that we have taken
to obtain the Berry phase is based on finding the Lewis invariant and Lewis
phase and then making an adiabatic approximation. This approach differs from
the one in [38] where similar results for the Berry phase were obtained using
a ladder operator approach. The main objective of our study is to show that a
non-trivial Berry phase in the setup that we have considered, can prove to be
an effective tool for the detection of gravitational waves. In this regard, we
have also calculated the contribution of the boundary term towards the total
geometric phase of the system for both the isotropic and anisotropic harmonic
oscillator. Now due to the periodicity of the parameters belonging to the
parameter space, we observe that during a complete period of revolution the
boundary term picks up no geometric phase for the well known gravitational
wave templates. Finally, we consider a gravitational wave carrying cross
polarization only. For this analysis, we have considered a gravitational wave
carrying cross polarization only with an isotropic two dimensional harmonic
oscillator. In order to calculate the Lewis phases for the new harmonic
oscillator-gravitational wave system, we have made use of a rotated coordinate
system such that the cross polarization is effectively similar to the plus
polarization in this basis. Such a rotated coordinate system helps us to
decouple the Hamiltonian and progress similarly to the plus polarization case.
We then obtain the Lewis phases corresponding to the two new coordinate
directions and eventually arrive at the geometric Berry phases by applying the
adiabatic approximation. The total Berry phase of the system does not depend
upon the choice of the coordinate basis and as can be observed from eq.(116),
it is dependent upon the cross polarization of the gravitational wave. This is
quite contrary to the claim made in [38] which states that the Berry phase is
generated entirely by the plus polarization of the gravitational wave and does
not depend upon the cross polarization of the gravitational wave. Therefore,
our analysis with a gravitational wave carrying the cross polarization only is
not in accord with the claim made in [38].

## Appendix: Form of the coefficients in the Lewis invariant when the
gravitational wave consists of both the plus and cross polarization terms

In this appendix, we try to obtain a Lewis invariant considering the
Hamiltonian in eq.10 which includes the interaction of the gravitational wave
with the two dimensional harmonic oscilltaor carrying both the plus and cross
polarization. Here we intend to reveal our progress with both the
polarizations. We make use of the following ansatz of the invariant
corresponding to the Hamiltonian in eq.(10).

$\begin{split}\hat{I}(t)&=\alpha_{1}(t)\hat{p}_{1}^{2}+\alpha_{2}(t)\hat{p}_{2}^{2}+\beta_{1}(t)\hat{x}_{1}^{2}+\alpha_{2}(t)\hat{x}_{2}^{2}+\delta_{1}(t)(\hat{x}_{1}\hat{p}_{1}+\hat{p}_{1}\hat{x}_{1})+\delta_{2}(t)(\hat{x}_{2}\hat{p}_{2}+\hat{p}_{2}\hat{x}_{2})+\lambda_{1}(t)\hat{x}_{1}\hat{p}_{2}+\lambda_{2}(t)\hat{p}_{1}\hat{x}_{2}\\\
&+\lambda_{3}(t)\hat{p}_{1}\hat{p}_{2}+\lambda_{4}(t)\hat{x}_{1}\hat{x}_{2}\end{split}$
(117)

where
$\alpha_{1},\alpha_{2},\beta_{1},\beta_{2},\delta_{1},\delta_{2},\lambda_{1},\lambda_{2},\lambda_{3}$,
and $\lambda_{4}$ are all unknown time dependent parameters. Computing
$i\hbar\frac{\partial\hat{I}(t)}{\partial t}$ and $[\hat{I},\hat{H}]$, we
obtain a set of ordinary differential equations given by

$\displaystyle\dot{\alpha}_{1}-4\alpha_{1}d+2\delta_{1}a-\lambda_{3}f=0~{},$
(118)
$\displaystyle\dot{\alpha}_{2}+4\alpha_{2}d+2\delta_{2}a-\lambda_{3}f=0~{},$
(119)
$\displaystyle\dot{\beta}_{1}+4\beta_{1}d-2\delta_{1}b+\lambda_{4}f=0~{},$
(120)
$\displaystyle\dot{\beta}_{2}-4\beta_{2}d-2\delta_{2}b+\lambda_{4}f=0~{},$
(121)
$\displaystyle\dot{\delta}_{1}-\alpha_{1}b+\beta_{1}a-\lambda_{1}f=0~{},$
(122)
$\displaystyle\dot{\delta}_{1}-\alpha_{1}b+\beta_{1}a+\lambda_{2}f=0~{},$
(123)
$\displaystyle\dot{\delta}_{2}-\alpha_{2}b+\beta_{2}a+\lambda_{1}f=0~{},$
(124)
$\displaystyle\dot{\delta}_{2}-\alpha_{2}b+\beta_{2}a-\lambda_{2}f=0~{},$
(125)
$\displaystyle\dot{\lambda}_{1}-2\delta_{1}f+2\delta_{2}f+4\lambda_{1}d-\lambda_{3}b+\lambda_{4}a=0~{},$
(126)
$\displaystyle\dot{\lambda}_{2}+2\delta_{1}f-2\delta_{2}f-4\lambda_{2}d-\lambda_{3}b+\lambda_{4}a=0~{},$
(127)
$\displaystyle\dot{\lambda}_{3}-2\alpha_{1}f-2\alpha_{2}f+(\lambda_{1}+\lambda_{2})a=0~{},$
(128)
$\displaystyle\dot{\lambda}_{4}+2\beta_{1}f+2\beta_{2}f-(\lambda_{1}+\lambda_{2})b=0~{}.$
(129)

Comparing eq.(122) with eq.(123) (or eq.(124) with eq.(125)), one can easily
find out that $\lambda_{1}=-\lambda_{2}=\lambda_{0}$. Substituting
$\lambda_{1}=-\lambda_{2}=\lambda_{0}$ in eq.(s)(126,127), we obtain one
differential equation and a constraint equation given by

$\displaystyle\dot{\lambda}_{0}-2\delta_{1}f+2\delta_{2}f=0~{},$ (130)
$\displaystyle 4\lambda_{0}d-\lambda_{3}b+\lambda_{4}a=0~{}.$ (131)

Next we use the parametrization
$\alpha_{1}(t)=\kappa_{1}\rho_{1}^{2}(t)~{},~{}\alpha_{2}(t)=\kappa_{2}\rho_{2}^{2}(t)$,
where $\kappa_{1}$ and $\kappa_{2}$ are two time independent unknown
constants. Then, in terms of $\rho_{1}$ and $\rho_{2}$, we can obtain the
other time dependent coefficients up to $\mathcal{O}(h)$ as

$\displaystyle\beta_{1}(t)$
$\displaystyle=\frac{\kappa_{1}}{a^{2}}\left(\dot{\rho}_{1}^{2}+\rho_{1}\ddot{\rho}_{1}-2\rho_{1}^{2}\dot{d}-4\rho_{1}\dot{\rho}_{1}d+ab\rho_{1}^{2}\right)~{},$
(132) $\displaystyle\beta_{2}(t)$
$\displaystyle=\frac{\kappa_{2}}{a^{2}}\left(\dot{\rho}_{2}^{2}+\rho_{2}\ddot{\rho}_{2}+2\rho_{2}^{2}\dot{d}+4\rho_{2}\dot{\rho}_{2}d+ab\rho_{2}^{2}\right)~{},$
(133) $\displaystyle\delta_{1}(t)$
$\displaystyle=-\frac{\kappa_{1}}{a}(\rho_{1}\dot{\rho_{1}}-2\rho_{1}^{2}d)~{},$
(134) $\displaystyle\delta_{2}(t)$
$\displaystyle=-\frac{\kappa_{2}}{a}(\rho_{2}\dot{\rho_{2}}+2\rho_{2}^{2}d)~{},$
(135) $\displaystyle\lambda_{0}(t)$
$\displaystyle=\frac{2}{a}\int_{0}^{t}dt^{\prime}f(t^{\prime})(\kappa_{2}\rho_{2}(t^{\prime})\dot{\rho}_{2}(t^{\prime})-\kappa_{1}\rho_{1}(t^{\prime})\dot{\rho}_{1}(t^{\prime}))~{},$
(136) $\displaystyle\lambda_{3}(t)$
$\displaystyle=2\int_{0}^{t}dt^{\prime}f(t^{\prime})(\kappa_{1}\rho_{1}^{2}(t^{\prime})+\kappa_{2}\rho_{2}^{2}(t^{\prime}))~{},$
(137) $\displaystyle\lambda_{4}(t)$
$\displaystyle=\frac{2b}{a}\int_{0}^{t}dt^{\prime}f(t^{\prime})(\kappa_{1}\rho_{1}^{2}(t^{\prime})+\kappa_{2}\rho_{2}^{2}(t^{\prime}))~{}.$
(138)

These equations are difficult to solve exactly and hence we go back to the
analysis in section 4.

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