# Probing Ultralight Tensor Dark Matter with the Stochastic Gravitational-Wave
Background from Advanced LIGO and Virgo’s First Three Observing Runs

Rong-Zhen Guo<EMAIL_ADDRESS>School of Fundamental Physics and
Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS, Hangzhou
310024, China  School of Physical Sciences, University of Chinese Academy of
Sciences, No. 19A Yuquan Road, Beijing 100049, China  Yang Jiang
corresponding author<EMAIL_ADDRESS>School of Physical Sciences,
University of Chinese Academy of Sciences, No. 19A Yuquan Road, Beijing
100049, China  CAS Key Laboratory of Theoretical Physics, Institute of
Theoretical Physics, Chinese Academy of Sciences,Beijing 100190, China  Qing-
Guo Huang  corresponding author<EMAIL_ADDRESS>School of Fundamental
Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study,
UCAS, Hangzhou 310024, China  School of Physical Sciences, University of
Chinese Academy of Sciences, No. 19A Yuquan Road, Beijing 100049, China  CAS
Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences,Beijing 100190, China

###### Abstract

Ultralight bosons are attractive dark-matter candidates and appear in various
scenarios beyond standard model. They can induce superradiant instabilities
around spinning black holes (BHs), extracting the energy and angular momentum
from BHs, and then dissipated through monochromatic gravitational radiation,
which become promising sources of gravitational wave detectors. In this
letter, we focus on massive tensor fields coupled to BHs and compute the
stochastic gravitational wave backgrounds emitted by these sources. We then
undertake a search for this background within the data from LIGO/Virgo
O1$\sim$ O3 runs. Our analysis reveals no discernible evidence of such
signals, allowing us to impose stringent limits on the mass range of tensor
bosons. Specifically, we exclude the existence of tensor bosons with masses
ranging from $4.0\times 10^{-14}$ to $2.0\times 10^{-12}$ eV at $95\%$
confidence level.

Introduction. The detection of gravitational waves (GWs) from the merger of
binary black holes (BHs) Abbott  _et al._ (2019a) and neutron stars Abbott
_et al._ (2021a) ushered in a new era of GW astronomy. Nowadays, GWs have been
used to test our basic understanding of different aspects of fundamental
physics Barack  _et al._ (2019); Baibhav  _et al._ (2021); Abbott  _et al._
(2019b, 2021b); Sathyaprakash  _et al._ (2019); Jiang and Huang (2023a, 2022,
b). Especially, it provides entirely new avenues for the detection and
constraints of ultralight bosonsBarack  _et al._ (2019); Baibhav  _et al._
(2021); Yuan  _et al._ (2021, 2022); Bertone  _et al._ (2020); Brito  _et al._
(2017a, b); Isi  _et al._ (2019); Tsukada  _et al._ (2019); Brito  _et al._
(2013), such as spin-0 QCD axions, axion-like particles in string axiverse and
spin-1 dark photons Arvanitaki  _et al._ (2010); Arvanitaki and Dubovsky
(2011); Weinberg (1978); Peccei and Quinn (1977); Marsh (2016); Holdom (1986),
which are important candidates of dark matter (DM).

The coupling of these fields to Standard Model particles is very weak, making
direct detection in the laboratory difficult Marsh (2016); Feng (2010).
However, if the ultralight bosonic field with mass $m_{b}$ exists around a
spinning BH, it can trigger superradiant instability if its typical
oscillation frequency $\omega_{R}=m_{b}c^{2}/\hbar$ satisfies
$0<\omega_{R}<m\Omega_{H}$, where $c$ is speed of light, $m$ is the azimuthal
quantum number of the unstable mode and $\Omega_{H}$ is horizon angular
velocity of the BH (see Brito  _et al._ (2015a) for a review). This mechanism
results in the transfer of angular momentum and energy from the BH to the
bosonic field. Ultimately, when $\omega_{R}\sim m\Omega_{H}$, superradiant
instability suppresses and corotating and non-axisymmetric boson condensate
forms. The condensate is dissipated through the emission of almost
monochromatic GWs with frequency $f_{0}\sim\omega_{R}/\pi$ Arvanitaki  _et
al._ (2015); Brito  _et al._ (2015b). Thus, it leads to a wealth of observable
astrophysical phenomena Brito  _et al._ (2015a); Dolan (2007); Brito  _et al._
(2013, 2017b); East and Pretorius (2017); Brito  _et al._ (2017a), which have
aroused much attention to probe ultralight bosons or the nature of BHs
recentlyGuo  _et al._ (2023, 2022); Jia  _et al._ (2023); Zhou  _et al._
(2023); Cardoso  _et al._ (2018); Xia  _et al._ (2023); Cannizzaro  _et al._
(2023).

Until now, there have been no positive results from direct detection of such
nearly monochromatic GW signals Palomba  _et al._ (2019); Tsukada  _et al._
(2021); Ng  _et al._ (2021); Abbott  _et al._ (2022); Sun  _et al._ (2020).
Another effective method to constrain the parameters of ultralight bosons is
to exploit the potential abundance of faint signals in the universe, that is,
to search for stochastic GW background (SGWB) they generate. From an
astronomical view, this background is generated by all the unresolved sources
in Universe. During O1$\sim$ O3 observing runs of LIGO and Virgo, the upper
intensity of SGWB has been set Abbott  _et al._ (2017a, 2019c, 2021c). The
SGWB from BH-boson system was calculated by Brito  _et al._ (2017a, b). So
far, there have been studies utilizing SGWB to constrain the parameters of
ultralight bosons Brito  _et al._ (2017a); Tsukada  _et al._ (2019, 2021);
Yuan  _et al._ (2021, 2022). The authors of Tsukada  _et al._ (2019) and
Tsukada  _et al._ (2021) provide the first search of SGWB in the first
observing run of LIGO sourced by scalar and vector boson, respectively.
However, the parameter constraints for ultralight tensor dark matterAoki and
Mukohyama (2016); Babichev  _et al._ (2016a, b); Aoki and Maeda (2018) have
not been thoroughly investigated as extensively as scalar and vector dark
matter Wu  _et al._ (2023).

In this letter, we explore the impact of ultralight tensor dark matter, which
can be described by linearized ghost-free bimetric theory, and calculate the
energy spectra of corresponding SGWB. Bayesian framework is adopted to search
for such signal in LIGO and Virgo’s observing period. From now on we use the
units $G=c=\hbar=1$.

Superradiant instability of ultralight tensor dark matter. The dynamics of
tensor dark matter can be described by spin-2 field $H_{\alpha\beta}$ on BH
background with initial mass $M_{i}$ and initial dimensionless angular
momentum $\chi_{i}=J_{i}/M_{i}^{2}$. This field can be regarded as linear
massive tensor perturbation in the ghost-free bi-metric gravity theory Brito
_et al._ (2020); Babichev  _et al._ (2016b), which equation of motion on
Ricci-flat background reads:

$\displaystyle\square
H_{\alpha\beta}+2R_{\alpha\gamma\beta\delta}H^{\gamma\delta}-m_{b}^{2}H_{\alpha\beta}$
$\displaystyle=0,$ (1) $\displaystyle\nabla^{\alpha}H_{\alpha\beta}=0,\quad
H^{\alpha}{}_{\alpha}$ $\displaystyle=0,$

where the box operator, Riemann tensor $R_{\alpha\gamma\beta\delta}$,
contractions are constructed with the background metric, and $m_{b}$ is the
mass of the spin-2 field.

For ultralight ($m_{b}M\ll 1$) tensor dark matter, there are several different
mechanisms for triggering instability. There exists unstable monopolar ($m=0$)
modes, which are also existed in the spherical BH case. Such instabilities
appear first, affecting the spacetime geometry of BHs and eventually forming
hairy BHs East and Siemonsen (2023); Brito  _et al._ (2013); Babichev and
Fabbri (2013). This instability has an extremely short characteristic
timescale ($\tau_{\textrm{mono}}\sim m_{b}^{-1}$) Brito  _et al._ (2013), and
due to its monopolarity, no GW produces during this process. Therefore, the
impact of this process can be ignored. And subsequent superradiant instability
leads to the formation of the quasi-bound state, which characteristic
frequency is complex Brito  _et al._ (2013); Dias  _et al._ (2023):

$\displaystyle\frac{\omega_{R}}{m_{b}}$ $\displaystyle\simeq
1-\frac{(m_{b}M)^{2}}{2(\ell+n+S+1)^{2}},$ (2) $\displaystyle\omega_{I}$
$\displaystyle\propto(m_{b}M)^{4\ell+5+2S}\left(\omega_{R}-m\Omega_{\mathrm{H}}\right),$

where $\ell\geq 0,n\geq 0$, and the integer $S=(0,\pm 1,\pm 2)$ are the mode
total angular momentum, overtone number, and polarization with
$m\in[-\ell,\ell]$. We use the result on Kerr background, because these modes
are significant support at $r\sim M(m_{b}M)^{-2}$ Brito  _et al._ (2013), the
deformation compared to the Kerr BH for the hairy BH is therefore negligible
under ultralight limit. It is clearly that for $m\geq 1$ cases, if
$\omega_{R}-m\Omega_{H}<0$, these states would grow exponentially until
reaching the saturation point, $\omega_{R}\sim m\Omega_{H}$ in timescale
$\tau_{\text{inst }}\equiv 1/\omega_{I}$. In this letter, we concentrate on
two dominant unstable appearing in non-relativistic approximation, a dipole
mode ($l=m=1,S=0$) and a quadrapole mode ($l=m=2,S=-2$). Although they share
the comparable timescale, it has been shown Ficarra  _et al._ (2019) that the
dipole mode can be reabsorbed by the BH, giving back almost all its mass and
spin. The loss of mass and angular momentum is thus almost entirely
attributable to the quadrapole mode itself.

According to the conservation of mass and angular momentum, we can get the
relationship between initial BH mass $M_{i}$, spin $J_{i}$ and the mass
$M_{f}$ and spin $J_{f}$ when superradiant instability saturated at
$\omega_{R}=m\Omega_{H}$:

$J_{f}=J_{i}-\frac{m}{\omega_{R}}\left(M_{i}-M_{f}\right)=J_{i}-\frac{m}{\omega_{R}}M_{T}^{\mathrm{max}},$
(3)
$M_{f}=\frac{m^{3}-\sqrt{m^{6}-16m^{2}\omega_{R}^{2}\left(mM_{i}-\omega_{R}J_{i}\right)^{2}}}{8\omega_{R}^{2}\left(mM_{i}-\omega_{R}J_{i}\right)},$
(4)

where $M_{T}^{\mathrm{max}}$ is the maximal total mass of tensor dark matter.
This quasi-bound state dissipates its energy by GWs in a typical timescale
$\tau_{\mathrm{GW}}$:

$\tau_{\mathrm{GW}}=M_{f}\left(\frac{d\tilde{E}}{dt}\frac{M_{T}^{\max}}{M_{f}}\right)^{-1},$
(5)

where $d\tilde{E}/dt$ is the reduced GW flux. In the qradrapole case, it reads
Brito  _et al._ (2020)

$\tau_{\mathrm{GW}}\sim
290\left(\frac{M_{f}}{30M_{\odot}}\right)^{2}\left(\frac{3M_{\odot}}{M_{T}^{\mathrm{max}}}\right)\left(\frac{m_{b}M_{f}}{0.2}\right)^{-10}\
\textrm{sec}.$ (6)

Since the hierarchy of timescales $\tau_{\mathrm{GW}}\gg\tau_{\text{inst }}\gg
M$, we can get the total GW energy of the whole system emitted in a duration
time $\Delta t=t_{0}-t_{f}$ by quasi-adiabatic approximation Brito  _et al._
(2017b)

$E_{\text{GW}}=\frac{M_{T}^{\mathrm{max}}\Delta t}{\Delta
t+\tau_{\mathrm{GW}}},$ (7)

where $t_{0}$ is the age of the Universe and $t_{f}$ characterizes the time
when the hairy $\mathrm{BH}$ is formed.

SGWB from boson clouds. In this section, we aim to model the SGWB from the
superradiant instability. As described in Eq. (8) Allen and Romano (1999)

$\Omega_{\text{GW}}(f)=\frac{1}{\rho_{c}}\frac{\mathrm{d}\rho_{\text{GW}}}{\mathrm{d}\ln
f},$ (8)

it is characterized by the energy density per logarithm frequency and
$\rho_{c}$ is the critical energy density required for a spatially flat
Universe. To figure out this background, we need to sum the GWs radiated by
all the possible sources. In our work, it is the superposition of all the
individual BH-boson system triggering superradiance. The general formula is

$\Omega_{\text{GW}}(f)=\frac{f}{\rho_{c}}\int\mathrm{d}z\frac{\mathrm{d}t}{\mathrm{d}z}\int\mathrm{d}\bm{\theta}\;\mathcal{R}(\bm{\theta};z)\frac{\mathrm{d}E_{s}}{\mathrm{d}f_{s}}(\bm{\theta};m_{b},f),$
(9)

where $\mathrm{d}t/\mathrm{d}z$ is the derivative of lookback time over
redshift. It is determined by the $\Lambda\text{CDM}$ cosmology model Condon
and Matthews (2018):

$\frac{\mathrm{d}t}{\mathrm{d}z}=\frac{1}{(1+z)H_{0}\sqrt{\Omega_{M}(1+z)^{3}+\Omega_{\Lambda}}}.$
(10)

$\mathcal{R}(\bm{\theta};z)$ is the formation rate of BH with parameter
$\bm{\theta}$ per comoving volume at redshift $z$.
$\mathrm{d}E_{s}/\mathrm{d}f_{s}$ is the energy spectrum of a single event in
the source frame. Since the GW radiation is nearly monochromatic, the spectrum
is well approximated by

$\frac{\mathrm{d}E_{s}}{\mathrm{d}f_{s}}=E_{\text{GW}}\delta(f_{s}-f_{0})=E_{\text{GW}}\delta[f(1+z)-f_{0}],$
(11)

where $f_{0}=\omega_{R}/\pi$ and we replace the source-frame frequency $f_{s}$
by detected frequency due to the redshift of the sources.

Two kinds of BH populations are considered in our work, which are isolated
stellar origin BHs (SBHs) and binary SBH merging remnants. For isolated BH
channel, the formation rate is

$\mathcal{R}^{\text{iso}}(\chi,M;z)=p(\chi)R^{\text{iso}}(M;z).$ (12)

By writing down Eq. (12), we have separate the relation of BH spin since its
distribution can not be derived from theory and here we adopt a uniform prior
distribution:

$p(\chi)=\begin{cases}\frac{1}{\chi_{\mathrm{u}}-\chi_{\mathrm{l}}},&\chi_{\mathrm{l}}\leq\chi\leq\chi_{\mathrm{u}}\\\
0,&\text{otherwise.}\end{cases}$ (13)

The remaining part can be determined by astrophysical model

$R^{\text{iso}}(M;z)=\psi(z_{f})\frac{\xi(M_{*})}{M_{*}}\frac{\mathrm{d}M_{*}}{\mathrm{d}M},$
(14)

where $M_{*}$ is the mass of progenitor star. Its population is the product of
star formation rate and initial mass function. We adopt the Salpeter initial
mass function Salpeter (1955), which is a power law $\xi(M_{*})\propto
M^{-1.35}$ and normalized between $0.1M_{\odot}\sim 100M_{\odot}$. For star
formation rate, we use a model proposed in Springel and Hernquist (2003)

$\psi(z)=\nu\frac{a\exp[b(z-z_{m})]}{a-b+b\exp[a(z-z_{m})]}.$ (15)

As suggested in Vangioni  _et al._ (2015), the fitting parameters are taken to
be $\nu=0.178\,M_{\odot}/\text{yr}/\text{Mpc}^{3}$, $a=2.37$, $b=1.80$ and
$z_{m}=2.00$. $z_{f}$ is the redshift at progenitor birth, so the
corresponding lookback time satisfies $t_{f}=t-\tau(M_{*})$, where
$\tau(M_{*})$ denotes the lifetime of star and we calculate this quantity
based on Schaerer (2002). Besides, to determine the derivative item in Eq.
(14), we use the relation of mass between BH and progenitor star metalliticity
in Fryer  _et al._ (2012); Belczynski  _et al._ (2016).

For merging remnant channel,

$\mathcal{R}^{\text{rem}}(m_{1},m_{2};z)=p(m_{1},m_{2})R^{\text{rem}}(m_{1},m_{2};z).$
(16)

The adopted mass distribution of binary system is a broken power law model
Abbott  _et al._ (2021d, c). That is

$\displaystyle
p(m_{1},m_{2})\propto\begin{cases}m_{1}^{-1.58},&m_{\text{mim}}\leq
m_{1}<m_{*}\\\ m_{1}^{-5.59},&m_{*}\leq m_{1}<m_{\text{max}}\\\
0,&\text{otherwise}\end{cases}$ $\displaystyle m_{\text{min}}=3.96M_{\odot},$
$\displaystyle m_{\text{max}}=87.14M_{\odot},$ $\displaystyle
m_{*}=m_{\text{min}}+0.43(m_{\text{max}}-m_{\text{min}}).$

$R^{\text{rem}}$ appeared in Eq. (16) is the binary merger rate. It is the
convolution of the binary formation rate with the distribution of time delays
between formation and merger:

$R^{\text{rem}}(m_{1},m_{2};z)=\int_{t_{\text{min}}}^{t_{\text{max}}}R_{f}(m_{1},m_{2};z_{f})p(t_{d})\,\mathrm{d}t_{d}.$
(17)

Following Abbott  _et al._ (2016, 2018, 2021c), we have assumed that
$R_{f}(m_{1},m_{2};z_{f})$ is proportional to a metallicity-weighted star
formation rate. The parameters of weighting rectification are the same as
Abbott  _et al._ (2021c) and the time distribution is taken as
$p(t_{d})\propto t_{d}^{-1}$ with $50\,\text{Myr}\leq t_{d}\leq
13.5\,\text{Gyr}$. Numerical Relativity suggests that the mass and spin of BH
remnant are Berti  _et al._ (2007); Scheel  _et al._ (2009); Barausse and
Rezzolla (2009)

$\displaystyle\begin{split}M&=m_{1}+m_{2}-(m_{1}+m_{2})\Bigg{[}\left(1-\sqrt{\frac{8}{9}}\right)\nu\\\
&\phantom{=}-4\nu^{2}\left(0.19308+\sqrt{\frac{8}{9}}-1\right)\Bigg{]},\end{split}$
$\displaystyle\chi$
$\displaystyle=\nu\left(2\sqrt{3}-3.5171\nu+2.5763\nu^{2}\right).$ (18)

The local merger rate is normalized as Abbott  _et al._ (2021c)

$\int\mathcal{R}^{\text{rem}}(\bm{m};0)\,\mathrm{d}\bm{m}=19\;\text{Gpc}^{-3}\text{yr}^{-1}.$
(19)

Figure 1: The energy spectra of SGWB from superradiant instability of spin-2
field. Solid curves are backgrounds from isolated BH channel with uniform
prior $\chi\in[0,1]$ while dashed curves represent BBH remnant contributions.
The power-law integrated sensitivity curves Thrane and Romano (2013) of LIGO
O3 run, Einstein Telescope (ET) Abbott  _et al._ (2017b) and Cosmic Explorer
(CE) Punturo  _et al._ (2010) are also shown.

Add the contributions from these two kinds of BH populations up and we work
out the actual background. Typical spectra within LIGO’s frequency band are
shown in Fig. 1. We show the spectra of these two channels separately. The
background is dominated by isolated BH population. Its amplitude is about 4
orders of magnitude larger than BBH remnant channel.

Data analysis and results. SGWB will cause a correlation in the output of
detector pairs. An estimator can be constructed with Fourier transform of the
strain data as follows Romano and Cornish (2017)

$\hat{C}_{IJ}(f)=\frac{2}{T}\frac{\text{Re}[\tilde{s}_{I}^{*}(f)\tilde{s}_{J}(f)]}{\gamma_{IJ}(f)S_{0}(f)},$
(20)

where $S_{0}(f)=(3H_{0}^{2})/(10\pi^{2}f^{3})$. $\gamma_{IJ}(f)$ is the
overlap reduction function Allen and Romano (1999) of detector pair $IJ$ and
$T$ denotes observing time. Eq. (20) has been normalized to
$\langle\hat{C}_{I}J(f)\rangle=\Omega_{\text{GW}}(f)$ and the variance is

$\sigma_{IJ}^{2}(f)=\frac{1}{2T\Delta
f}\frac{P_{I}(f)P_{J}(f)}{\gamma_{IJ}^{2}(f)S_{0}^{2}(f)}.$ (21)

$P_{I,J}(f)$ appeared in Eq. (21) are the power spectral densities of the
interferometers. To evaluate the mass of spin-2 boson, we perform Bayesian
analysis. After glitch gating and non-stationary rejection, the noise of
detectors is well approximated by uncorrelated Gaussian signal. Therefore, the
total likelihood is

$p(\bm{\hat{C}}|\bm{\theta})\propto\exp\left[-\sum_{IJ}\sum_{f}\frac{\left(\hat{C}_{IJ}(f)-\Omega(f;\bm{\theta})\right)^{2}}{2\sigma^{2}_{IJ}(f)}\right],$
(22)

where $\bm{\theta}$ denotes a sequence of parameters to be determined by
analysis and we have multiplied the likelihood from all the frequency bins and
detector pairs. The ratio of model evidence

$\mathcal{B}_{\text{\,noise}}^{\text{\,model}}=\frac{p(\bm{\hat{C}}\,|\,\text{Model
of signal})}{p(\bm{\hat{C}}\,|\,\text{Pure noise})},$ (23)

so-called Bayes factor illustrates the statistical significance of the
presence of the signal.

Figure 2: Posterior distributions of $\log m_{b}$ assuming a uniform spin
distribution for the isolated SBHs. The orange dash lines denote $95\%$
exclusion intervals.

We set a log-uniform prior between $10^{-14}\sim 10^{-11}\,\text{eV}$ to the
mass of tensor field. To figure out the influence of spin distribution, we
have assumed three different combinations of $\chi_{\mathrm{l,u}}$ and treated
their energy spectrum separately. As a result, the logarithm of Bayes factors
are $-0.29\sim-0.26$, indicating that there are no such signals in the
observing periods. The probability distributions of the mass and $95\%$
exclusion ranges by assuming a uniform distribution of spin between
$[0,\,1],\;[0,\,0.5]$ and $[0.5,\,1]$ are displayed in Fig. 2. The
corresponding excluded range of $\log(m_{b}/\text{eV})$ is around
$-13.4\sim-11.7$ and it does not a show strong connection with the spin
assumptions we adopted. Compared to scalar and vector fields, the observing
data puts a stronger constraint on the mass of tensor field. This is within
our expectations as tensor field has a shorter timescale of superradiant
instability comparing with scalar and vector fields with same mass Brito  _et
al._ (2015a); East (2017).

Discussions and conclusions.  In this work, we search for the SGWB signal
produced by ultralight tensor dark matter around both isolated SBHs and binary
SBHs remnants in LIGO/Virgo O1$\sim$ O3 runs. We find no evidence to suggest
the presence of the signal then we place constraints on the mass of tensor
bosons. Assuming a uniform distribution of spin between $[0,\,1],\;[0,\,0.5]$
and $[0.5,\,1]$ for isolated SBHs, we exclude the mass range $10^{-13.4}\sim
10^{-11.7}\text{eV}$. The relationship between the excluded mass range and the
spin distribution is not very tightly connected.

It is noteworthy that we notice recent advances in the purely numerical
spectral analysis of ultralight tensor dark matter East and Siemonsen (2023);
Dias  _et al._ (2023). These studies contain several new modes beyond the
hydrogenic approximation we used in the work. Since these modes stem from
nonseparability of the field equations, their dynamical impact becomes very
difficult to compute. This aspect remains a subject for future investigations
and warrants careful consideration in subsequent research endeavors.

Acknowledgements. We thank Yuan Chen for useful discussions. QGH is supported
by the grants from NSFC (Grant No. 12250010, 11975019, 11991052, 12047503),
Key Research Program of Frontier Sciences, CAS, Grant No. ZDBS-LY-7009, the
Key Research Program of the Chinese Academy of Sciences (Grant No. XDPB15). We
acknowledge the use of HPC Cluster of ITP-CAS.

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# Tunable partial polarization beam splitter and optomechanically induced
Faraday effect

Xuan Mao  These authors contributed equally to this work Department of
Physics, State Key Laboratory of Low-Dimensional Quantum Physics, Tsinghua
University, Beijing 100084, China  Guo-Qing Qin  These authors contributed
equally to this work Department of Physics, State Key Laboratory of Low-
Dimensional Quantum Physics, Tsinghua University, Beijing 100084, China  Hong
Yang  Zeguo Wang  Department of Physics, State Key Laboratory of Low-
Dimensional Quantum Physics, Tsinghua University, Beijing 100084, China  Min
Wang  Beijing Academy of Quantum Information Sciences, Beijing 100193, China
Gui-Qin Li  Department of Physics, State Key Laboratory of Low-Dimensional
Quantum Physics, Tsinghua University, Beijing 100084, China  Frontier Science
Center for Quantum Information, Beijing 100084, China  Peng Xue
<EMAIL_ADDRESS>Beijing Computational Science Research Center, Beijing,
China  Gui-Lu Long<EMAIL_ADDRESS>Department of Physics, State Key
Laboratory of Low-Dimensional Quantum Physics, Tsinghua University, Beijing
100084, China  Beijing Academy of Quantum Information Sciences, Beijing
100193, China  Frontier Science Center for Quantum Information, Beijing
100084, China  Beijing National Research Center for Information Science and
Technology, Beijing 100084, China  School of Information, Tsinghua University,
Beijing 100084, China

###### Abstract

Polarization beam splitter (PBS) is a crucial photonic element to separately
extract transverse-electric (TE) and transverse-magnetic (TM) polarizations
from the propagating light fields. Here, we propose a concise, continuously
tunable and all-optical partial PBS in the vector optomechanical system which
contains two orthogonal polarized cavity modes with degenerate frequency. The
results show that one can manipulate the polarization states of different
output fields by tuning the polarization angle of the pumping field and the
system function as partial PBS when the pump laser polarizes vertically or
horizontally. As a significant application of the tunable PBS, we propose a
scheme of implementing quantum walks in resonator arrays without the aid of
other auxiliary systems. Furthermore, we investigate the optomechanically
induced Faraday effect in the vector optomechanical system which enables
arbitrary tailoring of the input lights and the behaviors of polarization
angles of the output fields in the under couple, critical couple, and over
couple regimes. Our findings prove the optomechanical system is a potential
platform to manipulate the polarization states in multimode resonators and
boost the process of applications related to polarization modulation.

##  I INTRODUCTION

The manipulation of arbitrary polarization states is of significant
fundamental and applied relevance to a variety of research fields such as
quantum communication networks [1] and quantum optics [2]. Polarization beam
splitter (PBS) [3, 4, 5] plays a significant role in polarization selection.
Varies of PBS schemes have been proposed based on Mach-Zehnder interferometer
[6], photonic crystal fiber structure [7], binary blazed grating coupler [8],
multimode interference [9], and asymmetrical directional coupler [10].
However, previous works barely report a continuously tunable PBS and a general
platform that provides tunability of polarization states is needed.

Quantum walks (QW), the quantum correspondence of classical random walks, is
proved to be a versatile platform to implement quantum algorithms and
simulations[11, 12, 13, 14, 15, 16, 17, 18, 19]. QW has been developed in
various physical system such as nuclear magnetic resonance [20, 21], coupled
waveguides [22, 23, 24], trapped ions [25, 26], and photonic systems [27, 28,
29, 30]. Recently, the QW exhibits various topological phases [31, 32, 33, 34]
and demonstrates fascinating topological phenomena [35, 36, 37, 38]. However,
there is short of schemes that using the internal degrees of freedom as coin
states in resonator arrays without the aid of other auxiliary systems.

High-quality whispering gallery mode (WGM) microcavities [39] have potential
value in investigating fundamental physics and practical technologies such as
cavity optomechanics [40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], low-
threshold lasing [52, 53, 54, 55, 56, 57], quantum sensing [58, 59, 60, 61,
62, 63, 64, 65, 66], and nonlinear optics [67, 42, 68, 69, 70] due to their
ability to enhance light-matter interactions. Characterized by exploring the
radiation pressure interaction between optical modes and mechanical modes,
optomechanics exhibits rich physical phenomena such as optomechanically
induced transparency (OMIT) [41, 48, 42, 71], absorption (OMIA) [48, 72, 73],
and optomechanically induced Faraday effect (OMIFE) [74]. These effects enable
a new degree of light control and achieve arbitrary tailoring of the input
lights in optomechanical systems. Further, the additional degree of light
control allows varies of applications including state transfer [75, 76, 77,
78, 79], optical routing [80, 81, 82], and entanglement generation [83, 84,
85]. Besides progressing in many applications such as frequency comb
generation [86, 87] and light storage [47, 88], optoemchanical systems
provides a promising platform to study polarization behaviors.

In this paper, we theoretically propose a concise, continuously tunable and
all-optical partial PBS in the vector optomechanical system which contains two
optical modes coupling with the same mechanical mode. Since the effective
refractive indexes are polarization depend in the resonators, the two optical
modes with orthogonal polarizations and degenerate frequency can be achieved.
In this content, we consider the pump and probe fields are both linearly
polarized. We study the transmission spectra of different ports with different
polarizations. Specifically, when the included angle between the polarization
vector of the pump field and the horizontal mode equals to 0, the output field
of port 2 polarizes vertically only while the polarization of the output field
of port 4 is parallel to the horizontal mode in the case of resonance. Thus,
the vector optomechanical system function as PBS and it turns over the result
when tuning the polarization of pump field from horizontal to vertical. As a
significant application of the tunable PBS, we propose a scheme of
implementing QW in resonator arrays without the aid of other auxiliary
systems. Furthermore, OMIFE enables arbitrary tailoring of the input fields in
the system and we investigate the polarization behaviors of the output fields
in the under couple, critical couple, and over couple regimes. We believe that
our findings evidence the optomechanical system is a potential platform to
manipulate the polarization states in multimode resonators and boost the
process of applications related to polarization modulation.

This article is organized as follows: In Sec.II, we demonstrate the basic
model and the dynamical equations. We study the transmission spectra in
Sec.III. We show the OMIFE in Sec.IV. Conclusion is given in Sec.V.

##  II MODEL AND DYNAMICAL EQUATIONS

The vector optomechanical model we proposed is illustrated in Fig. 1 which
contains two degenerate optical modes, with degenerate frequency $\omega_{c}$
and decay rate $\kappa$, coupling with the same mechanical mode characterized
by frequency $\omega_{m}$ and the damping constant $\Gamma_{m}$. The
Hamiltonian of our system pumped by the linearly optical field can be
described by ($\hbar=1$)

Figure 1: (a) Schematic of the vector optomechanical system. (b) Frequency
spectrogram of the vector optomechanical system, which is composed of two
degenerate cavity modes with orthogonal polarizations. The pump and probe
fields are both linearly polarized and the included angle between the
polarization vector of the pump (probe) field and the horizontal mode is
$\theta$ ($\alpha$).

$\displaystyle H$ $\displaystyle=H_{free}+H_{int}+H_{drive}+H_{probe},$ (1)

where

$\displaystyle H_{free}=$
$\displaystyle\omega_{c}(a_{\updownarrow}^{\dagger}a_{\updownarrow}+a_{\leftrightarrow}^{\dagger}a_{\leftrightarrow})+\omega_{m}b^{\dagger}b,$
$\displaystyle H_{int}=$ $\displaystyle
ga_{\updownarrow}^{\dagger}a_{\updownarrow}(b^{\dagger}+b)+ga_{\leftrightarrow}^{\dagger}a_{\leftrightarrow}(b^{\dagger}+b),$
$\displaystyle H_{drive}=$ $\displaystyle
i\epsilon_{p\updownarrow}\sqrt{\kappa_{ex1}}e^{-i\omega_{p}t}a_{\updownarrow}^{\dagger}+i\epsilon_{p\leftrightarrow}\sqrt{\kappa_{ex1}}e^{-i\omega_{p}t}a_{\leftrightarrow}^{\dagger}+H.c.,$
$\displaystyle H_{probe}=$ $\displaystyle
i\epsilon_{r\updownarrow}\sqrt{\kappa_{ex1}}e^{-i\omega_{r}t}a_{\updownarrow}^{\dagger}+i\epsilon_{r\leftrightarrow}\sqrt{\kappa_{ex1}}e^{-i\omega_{r}t}a_{\leftrightarrow}^{\dagger}+H.c.,$
(2)

$H_{free}$ describes the free Hamiltonian of the optomechanical system,
$a_{j}$ and $a_{j}^{\dagger}$ (for $j=\updownarrow,\leftrightarrow$) are the
annihilation and creation operators of the optical mode, $\updownarrow$ and
$\leftrightarrow$ label the vertical and horizontal polarization axes
respectively. The mechanical annihilation and creation operators are denoted
by $b$ and $b^{\dagger}$. $H_{int}$ characterizes the interaction Hamiltonian
of the system with the single-photon optomechanical coupling strength $g$.
$H_{drive}$ implies the two degenerate optical modes are driven by external
fields with strength $\epsilon_{pj}$ and frequency $\omega_{p}$. As
illustrated in Fig. 1(b), $\theta$ describes the included angle between the
polarization vector of the driving field and the horizontal axes. Hence
$\epsilon_{p\updownarrow}=\epsilon_{p}\sin(\theta)$,
$\epsilon_{p\leftrightarrow}=\epsilon_{p}\cos(\theta)$ where
$\epsilon_{p}=\sqrt{P_{in}/(\hbar\omega_{p})}$ and $P_{in}$ is the input power
of the driving field. $\kappa_{ex1}$ denotes the external loss rate between
the optical mode $a_{j}$ and the fiber. $H_{probe}$ describes the probe laser
characterized by strength $\epsilon_{rj}$ and frequency $\omega_{r}$.
$\epsilon_{r\updownarrow}$ and $\epsilon_{r\leftrightarrow}$ satisfy
$\epsilon_{r\updownarrow}/\epsilon_{r\leftrightarrow}=\tan(\alpha)$ where
$\alpha$ denotes the angle between the polarization vector of the probe laser
and the horizontal axes. $\epsilon_{r}=\sqrt{P_{r}/(\hbar\omega_{r})}$ and
$P_{r}$ denotes the input power of the probe field. In the rotating frame with
the driving fields and after following the standard linearization procedure,
the linearized equations of the fluctuation parts are expressed as

$\displaystyle\frac{\mathrm{d}a_{\updownarrow}}{\mathrm{d}t}$
$\displaystyle=-(i\Delta+\frac{\kappa}{2})a_{\updownarrow}-iG_{\updownarrow}b+\sqrt{\kappa_{ex1}}\epsilon_{r\updownarrow}e^{-i\delta
t},$ (3) $\displaystyle\frac{\mathrm{d}a_{\leftrightarrow}}{\mathrm{d}t}$
$\displaystyle=-(i\Delta+\frac{\kappa}{2})a_{\leftrightarrow}-iG_{\leftrightarrow}b+\sqrt{\kappa_{ex1}}\epsilon_{r\leftrightarrow}e^{-i\delta
t},$ (4) $\displaystyle\frac{\mathrm{d}b}{\mathrm{d}t}$
$\displaystyle=-(i\omega_{m}+\frac{\Gamma_{m}}{2})b-iG_{\updownarrow}a_{\updownarrow}-iG_{\leftrightarrow}a_{\leftrightarrow}.$
(5)

Here, $\Delta=\omega_{c}-\omega_{p}$ represents the detuning between the
optical mode and the driving field. $\delta=\omega_{r}-\omega_{p}$ is the
detuning between the probe laser and the control field. $G_{\updownarrow}$
($G_{\leftrightarrow}$) is the effective optomechanical coupling strength
between the vertical(horizontal) optical mode and the mechanical mode. The
solutions of the Eq. 3-Eq. 5 are given by

$\displaystyle a_{\updownarrow}$
$\displaystyle=\frac{\sqrt{\kappa_{ex1}}\epsilon_{r\updownarrow}-iG_{\updownarrow}b}{\beta_{1}},$
(6) $\displaystyle a_{\leftrightarrow}$
$\displaystyle=\frac{\sqrt{\kappa_{ex1}}\epsilon_{r\leftrightarrow}-iG_{\leftrightarrow}b}{\beta_{1}},$
(7) $\displaystyle b$
$\displaystyle=-\frac{i\sqrt{\kappa_{ex1}}(G_{\updownarrow}\epsilon_{r\updownarrow}+G_{\leftrightarrow}\epsilon_{r\leftrightarrow})}{\beta_{m}\beta_{1}+G_{\updownarrow}^{2}+G_{\leftrightarrow}^{2}},$
(8)

$\beta_{1}=i\Delta+\kappa/2$ and $\beta_{m}=i\omega_{m}+\Gamma_{m}/2$. The
output fields of the optomechanical system can be obtained by adopting the
input-output relation $\epsilon_{out}=\epsilon_{in}-\sqrt{\kappa_{ex}}a$.
Specifically, the output field of port 2 and port 4 are expressed as

$\displaystyle\vec{\epsilon}_{2out}$
$\displaystyle=-\sqrt{\kappa_{ex2}}a_{\updownarrow}\vec{e}_{\updownarrow}-\sqrt{\kappa_{ex2}}a_{\leftrightarrow}\vec{e}_{\leftrightarrow},$
(9) $\displaystyle\vec{\epsilon}_{4out}$
$\displaystyle=(\epsilon_{r\updownarrow}-\sqrt{\kappa_{ex1}}a_{\updownarrow})\vec{e}_{\updownarrow}+(\epsilon_{r\leftrightarrow}-\sqrt{\kappa_{ex1}}a_{\leftrightarrow})\vec{e}_{\leftrightarrow}.$
(10)

Here, $\vec{e}_{\updownarrow}$ and $\vec{e}_{\leftrightarrow}$ are the unit
vectors of the vertical mode and the horizontal mode, respectively.

##  III TRANSMISSION SPECTRA ANALYSIS

Figure 2: Transmissions of different ports with different polarizations as a
function of $\theta/\pi$ and $\delta/\omega_{m}$: (a) port 2 with vertical
polarization, (b) port 2 with horizontal polarization, (c) port 4 with
vertical polarization, and (d) port 4 with horizontal polarization. (e) and
(f) illustrate the transmissions of port 2 and port 4 when $\theta=0$. (g) and
(h) illustrate the transmissions of port 2 and port 4 when $\theta=\pi/2$. The
parameters used in this system are
$\kappa_{0\updownarrow}/2\pi=\kappa_{0\leftrightarrow}/2\pi=\kappa_{0}/2\pi=1$
MHz, $\kappa_{ex1}/2\pi=9$ MHz, $\kappa_{ex2}/2\pi=8$ MHz,
$\omega_{m}/2\pi=90.47$ MHz, $\Gamma_{m}/2\pi=22$ kHz,
$G_{\updownarrow}/2\pi=5.5\times sin(\theta)$ MHz,
$G_{\leftrightarrow}/2\pi=5.5\times cos(\theta)$ MHz, $c=3\times 10^{8}$ m/s,
$\lambda=1550$ nm, $P_{r}=20\mu$W, and $\alpha=\pi/4$.

In Eq. 9 and Eq. 10, higher-order sidebands are not considered, one can obtain
the normalized transmission coefficients of different polarizations out of
port 2 and port 4, i.e., $tran_{2\updownarrow}$, $tran_{2\leftrightarrow}$,
$tran_{4\updownarrow}$, and $tran_{4\leftrightarrow}$. The normalized
transmission coefficients link input to output modes,

$\displaystyle\left[\begin{array}[]{c}\epsilon_{2out\updownarrow}\\\
\epsilon_{2out\leftrightarrow}\\\ \epsilon_{4out\updownarrow}\\\
\epsilon_{4out\leftrightarrow}\end{array}\right]=\left[\begin{array}[]{ccccc}tran_{2\updownarrow}&0&0&0\\\
0&tran_{2\leftrightarrow}&0&0\\\ 0&0&tran_{4\updownarrow}&0\\\
0&0&0&tran_{4\leftrightarrow}\end{array}\right]\times\left[\begin{array}[]{c}\epsilon_{r\updownarrow}\\\
\epsilon_{r\leftrightarrow}\\\ \epsilon_{r\updownarrow}\\\
\epsilon_{r\leftrightarrow}\end{array}\right]$ (23)
$\displaystyle=\left[\begin{array}[]{ccccc}-\frac{\sqrt{\kappa_{ex2}}a_{\updownarrow}}{\epsilon_{r\updownarrow}}&0&0&0\\\
0&-\frac{\sqrt{\kappa_{ex2}}a_{\leftrightarrow}}{\epsilon_{r\leftrightarrow}}&0&0\\\
0&0&1-\frac{\sqrt{\kappa_{ex1}}a_{\updownarrow}}{\epsilon_{r\updownarrow}}&0\\\
0&0&0&1-\frac{\sqrt{\kappa_{ex1}}a_{\leftrightarrow}}{\epsilon_{r\leftrightarrow}}\end{array}\right]\times\left[\begin{array}[]{c}\epsilon_{r\updownarrow}\\\
\epsilon_{r\leftrightarrow}\\\ \epsilon_{r\updownarrow}\\\
\epsilon_{r\leftrightarrow}\end{array}\right],$ (32)

where $\epsilon_{2out\updownarrow}$ ($\epsilon_{4out\updownarrow}$) and
$\epsilon_{2out\leftrightarrow}$ ($\epsilon_{4out\leftrightarrow}$) are the
projections of $\vec{\epsilon}_{2out}$ ($\vec{\epsilon}_{4out}$) onto the
vertical and horizontal modes, respectively. Furthermore, the transmission
rate is the square of the corresponding normalized transmission coefficient.
For instance, the transmission rate of the vertical field out of port 2 is
$T_{2\updownarrow}=|tran_{2\updownarrow}|^{2}$.

As analysis above, the transmission of different ports with different
polarizations can be tuned by changing the related parameter values. The
results show that the vector optomechanical system function as tunable PBS for
some specific parameters. Considering the experimental feasibility [82], the
parameters used in this system are the intrinsic decay rate of the two optical
modes
$\kappa_{0\updownarrow}/2\pi=\kappa_{0\leftrightarrow}/2\pi=\kappa_{0}/2\pi=1$
MHz, $\kappa_{ex1}/2\pi=9$ MHz, $\kappa_{ex2}/2\pi=8$ MHz,
$\omega_{m}/2\pi=90.47$ MHz, $\Gamma_{m}/2\pi=22$ kHz,
$G_{\updownarrow}/2\pi=5.5\times sin(\theta)$ MHz,
$G_{\leftrightarrow}/2\pi=5.5\times cos(\theta)$ MHz, $c=3\times 10^{8}$ m/s,
$\lambda=1550$ nm, the power of the probe field $P_{r}=20\mu$W, and
$\alpha=\pi/4$.

The transmission spectra of port 2 and port 4 with different polarizations are
demonstrated in Fig. 2. Fig. 2 (a) - Fig. 2 (d) illustrate
$T_{2\updownarrow}$, $T_{2\leftrightarrow}$, $T_{4\updownarrow}$, and
$T_{4\leftrightarrow}$ as a function of the included polarization angle of the
pump laser $\theta$ in the unit of $\pi$ and the detuning $\delta$ in the unit
of $\omega_{m}$, respectively. It is clear that the transmission rate changes
periodically with $\theta$ and the period is $\pi$ no matter which port the
output belongs to or which polarization the output field is. Note that the
transmission rate in Fig. 2 exceeds 1 in some regions which never happens in
regular transmission spectra. The physical interpretation is there is Faraday
effect induced by optomechanics in the vector system. The polarization of the
incident probe laser experiences rotation related to the polarization angle of
the pump field. The details about optomechanical induced Faraday effect(OMIFE)
can be found in Sec. IV.

It is interesting that the optomechanical vector system can function as
tunable PBS when $\theta$ equals to some specific values. Fig. 2 (e) and Fig.
2 (f) demonstrate the transmission spectra of port 2 and port 4 when
$\theta=0$. If the detuning between the pump laser and the probe field
$\delta$ equals to the mechanical frequency $\omega_{m}$, the output field of
port 2 polarizes vertically only and has no projection onto the horizontal
mode. On the contrary, the polarization of the output field of port 4 is
parallel to the horizontal mode. The physics behind the phenomenon is that
when $\theta=0$ there is driving field only for the horizontal mode
$a_{\leftrightarrow}$. Due to the interference effect between two pathways,
optomechanical induced transparence(OMIT) emerges for the horizontal mode. The
first pathway is the probe photons excite optical mode $a_{\leftrightarrow}$
and couple to the output port 4 and the other one is the photons generated by
the sideband transition through the optomechanical interaction are coupled out
of the cavity. For the vertical mode $a_{\updownarrow}$, there is no driving
field and the OMIT cannot be observed. As expected, the transmission rate
$T_{4\updownarrow}$ exhibits a Lorenz curve. In parallel, Fig. 2 (g) and Fig.
2 (h) show the PBS can turn over the result when tuning the value of $\theta$
to $\pi/2$.

Figure 3: The transmission of different ports with different polarizations as
a function of $\kappa_{ex2}/\kappa_{ex20}$ and $\delta/\omega_{m}$. (a)-(e)
are in the case of $\theta=0$ and (f)-(j) are in the case of $\theta=\pi/2$.
(e) and (j) indicate the transmission rate of different ports with vertical or
horizontal polarization when tuning the value of $\kappa_{ex2}$ in the case of
resonance $\delta=\omega_{m}$. $\kappa_{ex20}/2\pi=8$ MHz and
$\kappa_{ex1}=\kappa_{ex2}+\kappa_{0}$. The other parameters are the same as
that in Fig. 2.

Note that the transmission of port 2 with vertical polarization in the case of
$\delta=\omega_{m}$ and $\theta=0$ is not 1 due to the presence of the loss in
the system. For practical applications, the loss of different polarization
states is also should be manipulated to meet experimental requirements. We
investigate the impact of the coupling rate $\kappa_{ex2}$ on the transmission
rates as shown in Fig. 3. To make sure the polarization of port 2 and port 4
is either vertical or horizontal only, the critical couple condition
$\kappa_{ex1}=\kappa_{ex2}+\kappa_{0}$ should be maintained. Take $\theta=0$
for example, the results show the value of $\kappa_{ex2}$ has a big impact on
the linewidth of the output fields of port 2 and port 4, which has been
demonstrated by Fig. 3(a) and Fig. 3(c). Under the condition of not changing
the polarization state of the output of port 2 and port 4, the transmission
rate of the resonance of $\delta=\omega_{m}$ varies from 0 to 0.9 by tuning
the value of $\kappa_{ex2}$ which can be realized by changing the distance
between the below fiber and the resonator are shown in Fig. 1(a). Fig. 3(e)
plots the transmission rate of port 2 and port 4 with different polarization
in the resonance of $\delta=\omega_{m}$. It is evident that the transmission
of the vertical polarization field of port 2 can be adjusted and so does the
loss while the loss of the other polarization state of port 2 and port 4 is
maintained. Correspondingly, the loss of the horizontal polarization field of
port 4 can also be tuned in the case of $\theta=\pi/2$ and
$\delta=\omega_{m}$.

As mentioned above, the transmission rate in Fig. 2 may exceed 1 in some
regions while the total transmission rate of port 2 or port 4 will not. Unlike
Eq. 32, the transmission rates of port 2 and port 4 are given by

$\displaystyle T_{2}$
$\displaystyle=\frac{|-\sqrt{\kappa_{ex2}}a_{\updownarrow}|^{2}+|-\sqrt{\kappa_{ex2}}a_{\leftrightarrow}|^{2}}{|\epsilon_{r}|^{2}},$
(33) $\displaystyle T_{4}$
$\displaystyle=\frac{|\epsilon_{r\updownarrow}-\sqrt{\kappa_{ex1}}a_{\updownarrow}|^{2}+|\epsilon_{r\leftrightarrow}-\sqrt{\kappa_{ex1}}a_{\leftrightarrow}|^{2}}{|\epsilon_{r}|^{2}}.$
(34)

Figure 4: The total transmission rate as a function of $\theta/\pi$ and
$\delta/\omega_{m}$: (a) port 2 $T_{2}$ and (b) port 4 $T_{4}$. (c) The total
transmission rate $T_{2}$ and $T_{4}$ as a function of $\delta/\omega_{m}$
with different value of $\theta$. To make it clear, note that the transmission
rates when $\theta=0.75\pi$ and $\theta=1.25\pi$ have -$\omega_{m}$ and
$\omega_{m}$ shift in the $\delta$ axis, respectively. The other parameters
are the same as that in Fig. 2.

Fig. 4 shows the total transmission rate of port 2 and port 4 as a function of
$\theta/\pi$ and $\delta/\omega_{m}$. Similar to Fig. 2 (a) - Fig. 2 (d), the
total transmission rates feature a period with $\pi$ as $\theta$ increases. In
the domain of Fig. 4 (a) and Fig. 4 (b), the total transmission rates $T_{2}$
and $T_{4}$ are always between 0 and 1. Fig. 4 (c) presents the transmission
spectra with different value of $\theta$ and are marked by different markers.
Note that the markers are corresponding to the markers in Fig. 4 (a) and Fig.
4 (b) according to the value of $\theta$. Further, we have shift the
transmission spectra when $\theta=0.75\pi$ and $\theta=1.25\pi$ with amount of
-$\omega_{m}$ and $\omega_{m}$ in the axis of $\delta$ to make it clear. For
port 4, the transmission spectrum varies from a typical Lorenz curve to OMIT
as $\theta$ changes from 0.75 $\pi$ to 1.25 $\pi$. It can be inferred that the
angle of polarization of pump field has a big impact on the optomechanical
interference effect and influences the transmission spectrum further.

With the tunable polarization beam splitter as the vector optomechanical
system functions in hand, one can design the QW scheme in whispering-gallery-
mode resonator arrays with reasonable arrangement. One of the straightforward
ways to construct QW in resonator arrays is using the polarization states of
the photon as coin states to determine which side of the cavity it will go
into for the next step. Fig. 5(a) shows an alternating scheme to implement QW
which is governed by the operator $U=SC(\theta(x))$, with
$S=\sum_{x}(|x\rangle\langle x+1|\bigotimes|0\rangle\langle
0|+|x\rangle\langle x-1|\bigotimes|1\rangle\langle 1|)$ the conditional
transition operator and the position-dependent coin operator can be expressed
as

$\displaystyle C(\theta(x))$ $\displaystyle=I_{x}\bigotimes P(\theta(x)),$
$\displaystyle I_{x}$ $\displaystyle=\sum_{x}|x\rangle\langle x|,$ (35)
$\displaystyle P(\theta(x))$
$\displaystyle=\left(\begin{array}[]{cc}cos(\theta(x))&sin(\theta(x))\\\
sin(\theta(x))&-cos(\theta(x))\end{array}\right).$ (38)

$x$ is the position of the walker and $\\{|0\rangle,|1\rangle\\}$ are the two
orthogonal coin states corresponding to the vertical and horizontal
polarization of the photons, respectively. $I_{x}$ is the identity operator.
$P(\theta(x))$ indicates there is a rotation for the coin states after every
step and can be realized by half wavelength plate in our scheme. $\theta(x)$
depicts the rotation angle of the half wavelength plate dependent on the
position of the walker for the generality.

For the parameters in Fig. 2, the transmission of $T_{2\updownarrow}$ is
$90\%$ and the transmission of $T_{4\leftrightarrow}$ is $100\%$ when
$\theta=0$. Considering the loss of the system, there are position and
polarization dependent loss operator $L$ after the condition operator $S$ and
the coin operator $C$ in each step which can be shown as

$\displaystyle L=I_{x}\bigotimes\left(\begin{array}[]{cc}l_{1x}&0\\\
0&l_{2x}\end{array}\right),0\leq l_{1x},l_{2x}\leq 1.$ (41)

Fig. 5(b) and Fig. 5(c) show the probability distribution of the first 6 steps
and the standard deviation of the first 15 steps of QW in the passive
resonator arrays. Unlike classical random walks, the probability of the edge
position is much higher than the probability of $x=0$ for the QW case. It is
inferred that the behavior of the walks in the resonator arrays matches with
the quantum case. The most important difference between the QW and the
classical random walks is that the standard deviation of the QW is
proportional to the number of the steps $s$ while the classical random walks
is proportional to $\sqrt{s}$. To make it clear, Fig. 5(c) demonstrates the
standard deviation of the quantum case (the blue solid line), the classical
case (the red solid line) and the case in resonator arrays (the triangle
markers). It can be found that the behavior of this case is similar to the
quantum case, which solids the walks in our case is indeed the QW. Notice that
the standard deviation of the first few steps close to the classical case
because of the loss of the vector system.

The results in Fig. 3 show that the loss of the photons with different
polarization is adjustable in the vector system, which provides an alternating
way to implement PT-symmetric QW [36] in the resonator arrays. Implementation
of the PT-symmetric discrete-time QW allows us to observe different
topological phases and have potential value in designing topological device
taking advantages of the robustness of these phases to a variety of
perturbations including impurities, decoherence, interactions, and explicit
breaking of symmetries.

Figure 5: (a) Schematic of implementation of QW in optomechanical system. HWP:
half wavelength plate. (b) The probability distribution of the QW in resonator
arrays with the walkers starting from $x=0$ and the coin state chosen to be
$(|H\rangle+|V\rangle)/\sqrt{2}$ for the first 6 steps. (c) The standard
deviation of the QW (blue solid line), the classical random walks (the red
solid line) and the QW in the resonator arrays (triangle markers) for the
first 15 steps. The parameters are the same as that in Fig. 2.

##  IV OPTOMECHANICALLY INDUCED FARADAY EFFECT

As mentioned above, there is OMIFE in the vector system. Fig. 6 (b) depicts
the schematic of the input and output field polarization angles in the model.
In Fig. 6, we fix the polarization angle of the probe field at $\pi/4$. Note
the polarization angle of port 2 and port 4 are $\beta_{2}$ and $\beta_{4}$,
which satisfy

$\displaystyle tan(\beta_{2})$
$\displaystyle=\frac{a_{\updownarrow}}{a_{\leftrightarrow}},$ (42)
$\displaystyle tan(\beta_{4})$
$\displaystyle=\frac{\epsilon_{r\updownarrow}-\sqrt{\kappa_{ex1}}a_{\updownarrow}}{\epsilon_{r\leftrightarrow}-\sqrt{\kappa_{ex1}}a_{\leftrightarrow}}.$
(43)

The parameter values in Fig. 2 indicate the coupling between the cavity and
the fiber is in the critical couple regime for both the optical modes as
$\kappa_{ex1}=\kappa_{ex2}+\kappa_{0}$. It is necessary to explore the
behaviors of the polarization angles in the under couple regime and in the
over couple regime. Fig. 6 (a) and Fig. 6 (c) demonstrate the polarization
angles of output field of port 2 and port 4 as a function of $\theta/\pi$ and
$\Delta\kappa_{ex2}/\kappa_{ex20}$. In the $\Delta\kappa_{ex2}/\kappa_{ex20}$
axis we pick three values, i.e. 0.25, 0, and -0.25, which are in the under
couple, critical couple, and over couple regime, respectively. It can be found
that the behavior of the polarization angle of the output field of port 2 is
maintained when changing the value of $\Delta\kappa_{ex2}/\kappa_{ex20}$ while
the behavior of $\beta_{4}$ varies in the three coupling regimes. Furthermore,
the corresponding polarization angle behaviors are shown in Fig. 6 (d) - Fig.
6 (f). The polarization of the output field of port 2 is always perpendicular
to the polarization of the pump field. The reason is one can always construct
a pair of new modes ($a_{//}$ and $a_{\bot}$) whose polarization are parallel
to and perpendicular to the polarization of the pump field with
$a_{\updownarrow}$ and $a_{\leftrightarrow}$. Due to OMIT effect the
polarization of the output field of port 4 is parallel to the polarization of
the pump field while the polarization of the output field of port 2 is
parallel to the polarization of the pump field in the critical couple regime.
In the under couple and over couple regimes the transmission of $a_{//}$ of
port 4 is always 1 while the transmission of $a_{\bot}$ is not 0 anymore in
the case of resonance. So the behaviors in the two regimes of $\beta_{4}$ is
different from the critical couple regime. For port 2 the amplitude of
transmission of $a_{\bot}$ will not change the polarization of the output
field of port 2 as the transmission of $a_{//}$ is always zero in the three
regimes under the condition of $\delta=\omega_{m}$. Due to Faraday effect in
the vector optomechanical system, the polarization angles of port 2 and port 4
can be adjusted rapidly by tuning the polarization angle of the pump field and
the coupling between the cavity and the fiber.

Figure 6: The angle between the polarization of the output field and the
horizontal mode $\beta/\pi$ as a function of $\theta/\pi$ and
$\Delta\kappa_{ex2}/\kappa_{ex20}$: (a) port 2 and (c) port 4. (b) Schematic
of the input and output field polarization angles in the vector optomechanical
system. The angle $\beta/\pi$ of different ports as a function of $\theta$
with different value of $\Delta\kappa_{ex2}/\kappa_{ex20}$: (d) 0.25, (e) 0,
and (f) -0.25. $\kappa_{ex20}/2\pi=8$ MHz. The other parameters are the same
as that in Fig. 2.

##  V CONCLUSION

We have demonstrated a concise vector optomechanical system, consisting of two
degenerate optical modes coupling with the same mechanical mode, which is a
promising platform for continuously and all-optical tuning PBS. By changing
the polarization angle of the pump field, one can control the polarization
angles of different output ports. Furthermore, we study the OMIFE in this
system and the different behaviors of polarization angles of the output field
in the under couple, critical couple, and over couple regimes. Considering the
feasibility of experiments we choose the values of the parameters from the
previous experimental literature [82]. In this paper, we only focus on the
case both the pump field and the probe field are linearly polarized. The input
fields with circular polarization may lead to other interesting phenomenons
which give more opportunities to complex polarization manipulations. As a
significant application of the tunable PBS, an optional scheme of implementing
QW in resonator arrays without the aid of other auxiliary systems is proposed
in this paper. Furthermore, taking advantage of the tunable loss of the PBS,
one can design novel QW platforms to detect and observe topological phases
with a reasonable arrangement of the passive resonators. Our results prove the
optomechanical system is a potential platform to manipulate the polarization
states of the output fields and boost the process of applications of the
optomechanical system.

###### Acknowledgements.

We are grateful to Yuhao Qin for useful discussion and generous help about the
code. This work is supported by the National Natural Science Foundation of
China (61727801, 12025401 and U1930402); National Key Research and Development
Program of China (2017YFA0303700); The Key Research and Development Program of
Guangdong province (2018B030325002); Beijing Advanced Innovation Center for
Future Chip (ICFC); Tsinghua University Initiative Scientific Research
Program.

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