Enhanced Generative Adversarial Networks
for Unseen Word Generation from EEG Signals
[This work was supported by Institute for Information & Communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No.2021-0-02068, Artificial Intelligence Innovation Hub; No. 2019-0-00079, Artificial Intelligence Graduate School Program(Korea University)).

Young-Eun Lee
Dept. of Brain and Cognitive Engineering

Korea University

Seoul, Republic of Korea

Seo-Hyun Lee
Dept. of Brain and Cognitive Engineering

Korea University

Seoul, Republic of Korea

Soowon Kim
Dept. of Artificial Intelligence

Korea University

Seoul, Republic of Korea

Jung-Sun Lee
Dept. of Artificial Intelligence

Korea University

Seoul, Republic of Korea

Deok-Seon Kim
Dept. of Artificial Intelligence

Korea University

Seoul, Republic of Korea

Seong-Whan Lee
Dept. of Artificial Intelligence

Korea University

Seoul, Republic of Korea

Recent advances in brain-computer interface (BCI) technology, particularly based on generative adversarial networks (GAN), have shown great promise for improving decoding performance for BCI. Within the realm of Brain-Computer Interfaces (BCI), GANs find application in addressing many areas. They serve as a valuable tool for data augmentation, which can solve the challenge of limited data availability, and synthesis, effectively expanding the dataset and creating novel data formats, thus enhancing the robustness and adaptability of BCI systems.
Research in speech-related paradigms has significantly expanded, with a critical impact on the advancement of assistive technologies and communication support for individuals with speech impairments.
In this study, GANs were investigated, particularly for the BCI field, and applied to generate text from EEG signals.
The GANs could generalize all subjects and decode unseen words, indicating its ability to capture underlying speech patterns consistent across different individuals. The method has practical applications in neural signal-based speech recognition systems and communication aids for individuals with speech difficulties.

Keywords–generative adversarial network, brain-computer interface, electroencephalogram, imagined speech;

§ INTRODUCTION

In recent years, brain-computer interfaces (BCIs) have emerged as promising new areas of research, providing a means for humans to interact with external devices or control environments via brain signals [1]. Electroencephalography (EEG), which captures electrical activity by non-invasively attaching electrodes on the scalp, stands out among the several approaches for BCI research [2]. EEG offers an electrical measurement of cerebral activity without any surgical process of implanting electrodes, making it a significant source of information for applications involving brain signals [3]. BCIs based on EEG have been investigated for a variety of uses, including the control of motor activities, communication, and cognitive assessment [4, 5]. Despite the low signal-to-noise ratio of non-invasive EEG signals, several applications employing EEG have been explored owing to their ease of use and practical advantages [6].

Generative adversarial networks (GANs) have emerged as a seminal framework in the field of machine learning, computer vision, and BCI. Originally proposed by Goodfellow et al. [7], GANs have revolutionized the generation of realistic data by pitting a generator network against a discriminator network in a minimax game, resulting in the generation of data that closely mimics the underlying distribution. GANs have found wide-ranging applications, including image synthesis, speech synthesis, style transfer, and data augmentation, making them a fundamental tool in contemporary artificial intelligence research [8].

Furthermore, GANs have found notable applications within the domain of EEG, where they are employed for both data augmentation and signal reconstruction [9]. In the context of EEG data, data augmentation techniques using GANs play a pivotal role in expanding the size and diversity of available datasets [10, 11]. By generating synthetic EEG recordings that closely resemble real-world data, GANs enable improved training of machine learning models, thereby enhancing the models' ability to generalize and perform robustly on unseen data.

Additionally, GANs are leveraged in EEG signal reconstruction tasks, where the goal is to restore corrupted or missing segments of EEG recordings. This application is particularly valuable in scenarios where EEG data may be affected by noise or artifacts [12, 13], such as in clinical settings or during real-time monitoring [14]. GANs, through their generative capabilities, can effectively reconstruct EEG signals, contributing to improved signal quality and the extraction of meaningful information, which is crucial for various neuroscientific and clinical applications [15, 16]. The dual role of GANs in EEG data augmentation and signal reconstruction underscores their versatility and growing significance in the field of BCI.

Brain-to-text system translates the brain signals into text. The character embedding is generated from the brain signals, given the target text generated from the voice. The overall framework of the GAN-based text generation model consists of a generator and discriminator with pretrained ASR. The generator converts the embedding vector $z$ of the EEG from multiple speakers into character embedding vector $x$. The discriminator distinguishes the character embedding vector as real or fake based on adversarial learning.

BCI-based communication systems have been developed in various ways, including exogenous paradigms such as event-related potential and steady-state visually evoked potential and intuitive paradigms such as spoken speech and imagined speech.
Recent studies have investigated the use of brain signals including electrocorticography (ECoG) [17], and stereoEEG (sEEG) to reconstruct text or speech using machine learning and deep learning models. Willett et al. [18] developed a novel handwriting communication system based on intracortical BCI by reconstructing pen trajectories of written alphabets using a recurrent neural network. ECoG studies have explored speech synthesis from brain signals, using linear and nonlinear regression approaches and recurrent neural networks during spoken/mimed speech [19]. Studies on sEEG have shown potential for reconstructing speech using a multiple input convolutional neural network or a unit selection approach [20]. However, invasive techniques like as ECoG and sEEG entail risks and challenges to be applied for non-patient users. Therefore, non-invasive signals such as functional magnetic resonance imaging and EEG is gaining attention to be investigated, despite their inferior performance [21, 15, 22, 23]. Recently, Lee et al. [15] demonstrated the potential of speech synthesis at the word level from EEG during spoken and imagined speech. This study has yielded promising findings; however, it is notable that the training time required for the proposed model was considerable, and a substantial amount of calibration data for each subject was required for optimal performance.

In this study, our primary objective was to explore the practical use of GANs within the field of BCI. Specifically, we focused on the intricate task of reconstructing text-based output directly from brain signals, aiming to advance the capabilities of BCI systems in generating textual information from neural data, thereby contributing to the broader domain of human-computer interaction and assistive technologies.

§ MATERIALS AND METHODS

§.§ Generative Adversarial Networks

In the context of our GAN-based approach, a generator network, denoted as $G$, learns to map input vectors $z$ from a latent space to synthetic data samples that resemble the target mel-spectrogram from voice $x$. The discriminator network, denoted as $D$, is simultaneously trained to distinguish between real brain signals and those generated by $G$. Through a min-max game, $G$ aims to generate increasingly realistic signals to deceive $D$, while $D$ strives to become more proficient at discrimination. This adversarial training process continues until an equilibrium is reached, where $G$ generates brain signals that are indistinguishable from real ones.

§.§ Adversarial Loss

The central component of the GAN framework is the adversarial loss, which comprises a generator loss ($L_G$) and a discriminator loss ($L_D$). The generator loss encourages $G$ to create synthetic signals that are convincingly realistic, while the discriminator loss guides $D$ in effectively distinguishing between real and synthetic data. The generator loss is minimized when $G$ produces signals that successfully fool the discriminator, and the discriminator loss is minimized when it accurately discriminates between real and generated signals. This competitive interplay between $L_G$ and $L_D$ drives the refinement of both networks, ultimately leading to the generation of brain signals that closely mimic the underlying distribution of real data.
Here is the equation of loss:

\begin{equation}
L_D = \mathbb{E}[\log(D(x))] + \mathbb{E}[\log(1 - D(G(z))],
\end{equation}

\begin{equation}
L_G = \mathbb{E}[\log(D(G(z))].
\end{equation}

§.§ Dataset

This study involved EEG signals and voice recordings of twenty-one subjects while they performed speech. The recordings were captured using scalp EEG consisting 64-channel electrode array placed on the scalp, in conjunction with a microphone to record voices, which was synchronized with the EEG signals. The subjects were instructed to speak in accordance with the instructions displayed on the screen. They spoke thirteen distinct words including “ambulance," “light," “TV," “water," “pain," “hello," “toilet," “clock," “yes" “stop," “help me," “thank you," and rest, in the same manner as the study [24]. Four trials were rhythmically given following each random auditory cue indicating the word or phrase the subject should speak. To minimize potential visual or auditory artifacts, the experiment was designed to capture EEG signals without any stimuli present. The study received approval from the Institutional Review Board and was conducted in accordance with the principles outlined in the Declaration of Helsinki. Prior to commencing the study, all subjects provided informed consent in adherence to ethical standards.

§.§ Text Generation from Brain Signals

Fig. <ref> provides an overview of our proposed framework, which receives EEG signals from multiple speakers as input and generates corresponding text as output. It includes embedding processes, a generator, and a discriminator.

For each subject, the dataset was split into five folds with a fixed random seed used for training, validation, and test sets. In order to make the unseen word for evaluating the scalability of the model, the word, `stop', was removed from the training dataset because all the phonemes composing the word were already included in the 11 words/phrases used for training. Thus, the model was trained using 11 words/phrases and a silent phase as the training set, while 12 words/phrases including the unseen word and a silent phase were used for validation and testing.

§.§ Preprocessing

Preprocessing methods were applied to the recorded raw EEG data, including downsampling to 250 Hz, band-pass filtering between 0.5 and 125 Hz, notch filtering at 60 and 120 Hz, and re-referencing using a common average reference method. To eliminate potential noise from movement and sound, automatic electrooculography and electromyography removal methods were employed to remove ocular and muscular artifacts [25, 26]. Moreover, the EEG signals in the high-gamma frequency band were selected for training the model and data analysis. The dataset was epoched into 2-second segments and baseline correction was executed 500 ms before speaking.
Preprocessing procedures, as outlined in lee et al. [27], were implemented using MATLAB-based toolboxes including OpenBMI Toolbox [28] and EEGLAB [29].

The proposed framework employs embedding processes to generate representative feature vectors from EEG signals. This process involves using the common spatial pattern (CSP) technique and log-variance functions to extract spatio-temporal features from the EEG signals. CSP is a widely used mathematical approach in the area of BCI that separates a multivariate signal into subcomponents with the greatest variance fluctuations across two windows [30]. Feature embedding was performed through time-wise computing of CSP patterns, with each EEG segment divided into 16 time points. The multi-CSP algorithm was used to calculate eight patterns per class, with the pattern size for each time point set to 104. The signals from each channel were processed to mean normalization to illustrate the temporal fluctuations of the embedding characteristics.

Spatial patterns of synchronization and de-synchronization of EEG signals for six different words. The color indicates the power of each channel across all trials.

§ RESULTS AND DISCUSSION

§.§ Results

We evaluated the performance of our generated text from brain signals using the traditional metric: character error rate (CER).
The performance of seen words and unseen words for generating text from brain signals was shown in CER. The CER of seen words was $61.8 \pm 8.5$ and the CER of unseen words was $83.3 \pm 3.9$.
We observed that our model was able to generalize to totally unseen subjects who were not included in the training set, demonstrating its ability to capture underlying patterns of speech that are consistent across different individuals.
The results show that our method has the capacity to generalize to new subjects and speech patterns when given a small amount of calibration data, where data collection may be challenging due to limitations in sample size or data variability.
Our findings demonstrate that the proposed model was able to learn global speech characteristics using data acquired from multiple speakers. Furthermore, our approach may have important practical applications, such as in the development of speech recognition systems or communication aids for individuals with speech impairments.

§.§ Spatial Analysis

To investigate differences in brain activity during speech, we conducted an analysis of spatial and spectral features, as shown in Fig. <ref>. In the spatial characteristics, our analysis revealed that prominent synchronization in the central lobe and de-synchronization in the temporal lobe were observed [31]. In terms of the spectral features, we discovered that the high-gamma frequency range between 30 Hz to 120 Hz was dominantly synchronized, which is selected to analyze. In addition, we observed that the spatial characteristics of each text revealed significant differences, with each text exhibiting its distinct pattern of brain activation. These findings suggest that different aspects of speech may engage distinct neural networks, and the observed patterns of brain activation may reflect the cognitive and motor processes underlying speech production.

§.§ Limitation and Future Works

In our study, we employed a limited dataset consisting of only thirteen words, which offers a challenge in generating an extensive range of texts from brain signals. While we attempted to train the model to generate the unseen word by incorporating all the phonemes present in the unseen word, the outcomes were not satisfactory enough.
We plan to extend our research in the future by employing a sentence-based dataset that contains a large number of phonemes that should be capable of generating various words. In addition, we will utilize our proposed approach for generating text from the brain signals during the imagined speech, which may serve as a communication method for individuals who experience speech impairments [32].

§ CONCLUSION

In this paper, we investigated GAN for BCI, and presented the generative adversarial networks for BCI, in particular, for speech-related brain signals.
Our approach employs embedding processes, a generator, and a discriminator to learn the underlying patterns of speech and generate text at the word level with high fidelity. The model was able to learn global speech characteristics, demonstrating that our approach can generalize to unseen subjects and words, with a low error rate.
Additionally, our spatial analysis revealed significant differences in brain activity during speaking, indicating distinct neural networks are engaged in different aspects of speech production. Despite the limited amount of words in the training dataset, our method has the potential to be applied in real-world applications such as speech recognition systems and communication aids for those with speech impairments.
Future research will focus on expanding our dataset to include more words and sentences, further improving the framework's generalization ability.

[1]
U. Chaudhary, N. Birbaumer, and A. Ramos-Murguialday, “Brain–computer interfaces for communication and rehabilitation,” Nature Reviews Neurology, vol. 12, no. 9, pp. 513–525, 2016.

[2]
K.-H. Thung et al., “Conversion and time–to–conversion predictions of mild cognitive impairment using low–rank affinity pursuit denoising and matrix completion,” Med. Image Anal., vol. 45, pp. 68–82, 2018.

[3]
M. Angrick et al., “Towards closed-loop speech synthesis from stereotactic EEG: A unit delection approach,” in IEEE Int. Conf. Acoustics, Speech, and Signal Processing (ICASSP), pp. 1296–1300.

[4]
R. Mane et al., “FBCNet: A multi-view convolutional neural network for brain-computer interface,” arXiv preprint arXiv:2104.01233, 2021.

[5]
M. Lee, C.-B. Song, G.-H. Shin, and S.-W. Lee, “Possible effect of binaural beat combined with autonomous sensory meridian response for inducing sleep,” Front. Hum. Neurosci., vol. 13, pp. 425–440, 2019.

[6]
Y.-E. Lee, N.-S. Kwak, and S.-W. Lee, “A real-time movement artifact removal method for ambulatory brain-computer interfaces,” IEEE Transactions on Neural Systems and Rehabilitation Engineering, vol. 28, no. 12, pp. 2660–2670, 2020.

[7]
I. Goodfellow et al., “Generative adversarial nets,” vol. 27.

[8]
J.-S. Bang, M.-H. Lee, S. Fazil, C. Guan, and S.-W. Lee, “Spatio–spectral feature representation for motor imagery classification using convolutional neural networks,” IEEE Trans. Neural Netw. Learn. Syst., vol. 33, no. 7, pp. 3038–3049, 2021.

[9]
K. G. Hartmann, R. T. Schirrmeister, and T. Ball, “EEG-GAN: Generative adversarial networks for electroencephalograhic (EEG) brain signals,” arXiv preprint arXiv:1806.01875, 2018.

[10]
S. Panwar, P. Rad, T.-P. Jung, and Y. Huang, “Modeling EEG data distribution with a Wasserstein generative adversarial network to predict RSVP events,” IEEE Transactions on Neural Systems and Rehabilitation Engineering, vol. 28, no. 8, pp. 1720–1730, 2020.

[11]
S. M. Abdelfattah, G. M. Abdelrahman, and M. Wang, “Augmenting the size of EEG datasets using generative adversarial networks,” in Int. Joint Conf. Neural Networks (IJCNN), pp. 1–6.

[12]
Y.-E. Lee, M. Lee, and S.-W. Lee, “Reconstructing ERP signals using generative adversarial networks for mobile brain-machine interface,” arXiv preprint arXiv:2005.08430, 2020.

[13]
P. Sawangjai et al., “EEGANet: Removal of ocular artifacts from the EEG signal using generative adversarial networks,” IEEE Journal of Biomedical and Health Informatics, vol. 26, no. 10, pp. 4913–4924, 2021.

[14]
H.-I. Suk, S. Fazli, J. Mehnert, K.-R. Müller, and S.-W. Lee, “Predicting BCI subject performance using probabilistic spatio-temporal filters,” PLoS One, vol. 9, no. 2, p. e87056, 2014.

[15]
Y.-E. Lee, S.-H. Lee, S.-H. Kim, and S.-W. Lee, “Towards voice reconstruction from EEG during imagined speech,” in AAAI Conference on Artificial Intelligence (AAAI).

[16]
J. Tang, A. LeBel, S. Jain, and A. G. Huth, “Semantic reconstruction of continuous language from non-invasive brain recordings,” Nature Neuroscience, pp. 1–9, 2023.

[17]
M. Angrick et al., “Speech synthesis from ECoG using densely connected 3D convolutional neural networks,” Journal of Neural Engineering, vol. 16, no. 3, p. 036019, 2019.

[18]
F. R. Willett, D. T. Avansino, L. R. Hochberg, J. M. Henderson, and K. V. Shenoy, “High-performance brain-to-text communication via handwriting,” Nature, vol. 593, no. 7858, pp. 249–254, 2021.

[19]
C. Herff et al., “Generating natural, intelligible speech from brain activity in motor, premotor, and inferior frontal cortices,” Frontiers in Neuroscience, vol. 13, p. 1267, 2019.

[20]
G. K. Anumanchipalli, J. Chartier, and E. F. Chang, “Speech synthesis from neural decoding of spoken sentences,” Nature, vol. 568, no. 7753, pp. 493–498, 2019.

[21]
L. A. Ross, S. Molholm, J. S. Butler, V. A. Del Bene, and J. J. Foxe, “Neural correlates of multisensory enhancement in audiovisual narrative speech perception: a fMRI investigation,” Neuroimage, vol. 263, p. 119598, 2022.

[22]
S.-H. Lee, M. Lee, J.-H. Jeong, and S.-W. Lee, “Towards an EEG-based intuitive BCI communication system using imagined speech and visual imagery,” in Conf. Proc. IEEE Int. Conf. Syst. Man Cybern. (SMC), pp. 4409–4414.

[23]
S. Kim, Y.-E. Lee, S.-H. Lee, and S.-W. Lee, “Diff-E: Diffusion-based learning for decoding imagined speech EEG,” in Interspeech.

[24]
S.-H. Lee, M. Lee, and S.-W. Lee, “Neural decoding of imagined speech and visual imagery as intuitive paradigms for BCI communication,” IEEE Trans. Neural Syst. Rehabil. Eng., vol. 28, no. 12, pp. 2647–2659, 2020.

[25]
J. Kim et al., “Abstract representations of associated emotions in the human brain,” J. Neurosci., vol. 35, no. 14, pp. 5655–5663, 2015.

[26]
H. Gómez-Gauchía and F. Peinado, “Automatic customization of non-player characters using players temperament,” in International Conference on Technologies for Interactive Digital Storytelling and Entertainment, pp. 241–252.

[27]
Y.-E. Lee, G.-H. Shin, M. Lee, and S.-W. Lee, “Mobile BCI dataset of scalp-and ear-EEGs with ERP and SSVEP paradigms while standing, walking, and running,” Scientific Data, vol. 8, no. 1, pp. 1–12, 2021.

[28]
M.-H. Lee et al., “EEG dataset and OpenBMI toolbox for three BCI paradigms: an investigation into BCI illiteracy,” GigaScience, vol. 8, no. 5, p. giz002, 2019.

[29]
A. Delorme and S. Makeig, “EEGLAB: An open source toolbox for analysis of single-trial EEG dynamics including independent component analysis,” Journal of Neuroscience Methods, vol. 134, no. 1, pp. 9–21, 2004.

[30]
D. Devlaminck, B. Wyns, M. Grosse-Wentrup, G. Otte, and P. Santens, “Multisubject learning for common spatial patterns in motor-imagery BCI,” Computational Intelligence and Neuroscience, vol. 2011, 2011.

[31]
S.-B. Lee et al., “Comparative analysis of features extracted from EEG spatial, spectral and temporal domains for binary and multiclass motor imagery classification,” Inf. Sci., vol. 502, pp. 190–200, 2019.

[32]
K.-T. Kim, C. Guan, and S.-W. Lee, “A subject-transfer framework based on single-trial EMG analysis using convolutional neural networks,” IEEE Trans. Neural Syst. Rehabil. Eng., vol. 28, no. 1, pp. 94–103, 2019.


plus1sp

[1]Le-Man Kuang [1]Hui Jing

# Vector optomechanical entanglement

Ying Li  Key Laboratory of Low-Dimensional Quantum Structures and Quantum
Control of Ministry of Education, Department of Physics and Synergetic
Innovation Center for Quantum Effects and Applications, Hunan Normal
University, Changsha 410081, China. These authors contributed equally to this
work.  Ya-Feng Jiao  Key Laboratory of Low-Dimensional Quantum Structures and
Quantum Control of Ministry of Education, Department of Physics and Synergetic
Innovation Center for Quantum Effects and Applications, Hunan Normal
University, Changsha 410081, China. These authors contributed equally to this
work.  Jing-Xue Liu  Key Laboratory of Low-Dimensional Quantum Structures and
Quantum Control of Ministry of Education, Department of Physics and Synergetic
Innovation Center for Quantum Effects and Applications, Hunan Normal
University, Changsha 410081, China.  Adam Miranowicz  Institution, Department,
City, Country of second author and third author, Institute of Spintronics and
Quantum Information, Faculty of Physics, Adam Mickiewicz University, 61-614
Poznań, Poland  Yun-Lan Zuo  Key Laboratory of Low-Dimensional Quantum
Structures and Quantum Control of Ministry of Education, Department of Physics
and Synergetic Innovation Center for Quantum Effects and Applications, Hunan
Normal University, Changsha 410081, China.

###### Abstract

The polarizations of optical fields, besides field intensities, provide more
degrees of freedom to manipulate coherent light-matter interactions. Here we
propose how to achieve a coherent switch of optomechanical entanglement in a
polarized-light-driven cavity system. We show that by tuning the polarizations
of the driving field, the effective optomechanical coupling can be well
controlled and, as a result, quantum entanglement between the mechanical
oscillator and the optical transverse electric (TE) mode can be coherently and
reversibly switched to that between the same phonon mode and the optical
transverse magnetic (TM) mode. This ability of switching optomechanical
entanglement with such a vectorial device can be important for building a
quantum network being capable of efficient quantum information interchanges
between processing nodes and flying photons.

##  1 Introduction

Vector beams, characterized by the ability to tailor light by polarization
control, are important for both fundamental researches and practical
applications in optics and photonics [1, 2, 3]. Manipulating the polarization
of vector beams, for examples, provides efficient ways to realize optical
trapping or imaging [4, 5, 6], material processing [7, 8], optical data
storage [9], sensing [10], and nonlinearity enhancement [11]. Compared with
conventional scalar light sources, vector beams provide more degrees of
freedom to regulate coherent light-matter interactions, that is, manipulating
the coupling intensity by tuning spatial polarization distributions of an
optical field [12, 13, 14, 15]. Thus, vector beams have been used in a variety
of powerful devices, such as optoelectrical [16, 17] or optomechanical systems
[18, 19], metamaterial structures [20, 21], and atomic gases [22]. In a recent
experiment, through vectorial polarization control, coherent information
transfer from photons to electrons was demonstrated [16]. In the quantum
domain, mature techniques have been developed for creating entangled photons
with vector beams [23, 24], and by using the destructive interference of the
two excitation pathways of a polarized quantum-dot cavity, unconventional
photon blockade effect was observed very recently [25]. We also note that the
single- and multi-photon resources with well-defined polarization properties,
which are at the core of chiral quantum optics, have also been addressed via
metasurfaces [26, 27]. In addition, a recent experiment shows that the hybrid
light–mechanical states with a vectorial nature could also be achieved by
using levitated optomechanical system [28]. However, as far as we know, the
possibility of generating and switching macroscopic entanglement between light
and motion via polarization control has not yet been explored.

A peculiar property of quantum entanglement, that measuring one part of the
entangled elements allows to determine the state of the other, makes it a key
resource for quantum technologies, ranging from quantum information processing
[29, 30] to quantum sensing [31, 32]. Recently, entanglement-based secure
quantum cryptography has also been achieved at a distance over 1,120
kilometres [33]. So far, quantum entanglement has been observed in diverse
systems involving photons, ions, atoms, and superconducting qubits [34, 35].
Quantum effects such as entanglement has also been studied at macroscopic
scales and even in biological systems [36]. In parallel, the rapidly emerging
field of cavity optomechanics (COM), featuring coherent coupling of motion and
light [37, 38], has provided a vital platform for engineering macroscopic
quantum objects [39, 40, 41]. Very recently, quantum correlations at room
temperature were even demonstrated between light and a $40\,\textrm{kg}$
mirror, circumventing the standard quantum limit of measurement [42]. Quantum
entanglement between propagating optical modes, between optical and mechanical
modes, or between massive mechanical oscillators have all been realized in COM
systems [43, 45, 46, 44, 47, 48, 49, 50, 51, 52, 53]. In view of these rapid
advances, COM devices have becomes one of the promising candidates to operate
as versatile quantum nodes processing or interchanging information with flying
photons in a hybrid quantum network.

Here, based on a COM system, we propose how to achieve a coherent switch of
quantum entanglement of photons and phonons through polarization control. We
show that the intracavity field intensity and the associated COM entanglement
can be coherently manipulated by adjusting the polarization of a driving
laser. This provides an efficient way to manipulate the light-motion coupling,
which is at the core of COM-based quantum technologies. Besides the specific
example of COM entanglement switch, our work can also serve as the first step
towards making vectorial COM devices with various structured lights, such as
Bessel-Gauss beams, cylindrical beams or the Poincaré beams [55, 54, 1], where
the optical polarization distribution is spatially inhomogeneous. Our work can
also be extended to various COM systems realized with e.g., cold atoms,
magnomechanical devices, and optoelectrical circuits [56, 57].

##  2 Vectorial quantum dynamics

As shown in Figure 1, we consider a polarized-light-driven optomechanical
system, which consists of an optical polarizer and a Fabry-Pérot cavity with
one movable mirror. Exploiting the polarization of photons rather than solely
their intensity, has additional advantages for controlling light-matter
interactions. Here, to describe the polarization of an optical field, it is
convenient to introduce a set of orthogonal basis vectors, i.e.,
$|\vec{\textrm{e}}_{\updownarrow}\rangle$ and
$|\vec{\textrm{e}}_{\leftrightarrow}\rangle$, which correspond to the vertical
(TE) and horizontal (TM) modes of the Fabry-Pérot cavity [58]. Therefore, an
arbitrary linearly polarized light can be thought as a superposition of these
orthogonal patterns, i.e., whose unit vector is represented by
$|\vec{\textrm{e}}\rangle=\cos\theta\,|\vec{\textrm{e}}_{\updownarrow}\rangle+\sin\theta\,|\vec{\textrm{e}}_{\leftrightarrow}\rangle$,
with $\theta$ being the angle between the polarization of the linearly
polarized light and the vertical mode [see Figure 1(b)]. In this situation, by
adjusting the polarization angle $\theta$, one can coherently manipulate the
spatial distribution of a linearly polarized optical field. In addition,
because light could exert radiation pressure on the movable mirror, both
spatial components of the linearly polarized light would experience an
optomechanical interaction. Then, in a rotating frame with respect to
$\hat{H}_{0}=\hbar\omega_{L}(\hat{a}_{\updownarrow}^{\dagger}\hat{a}_{\updownarrow}+\hat{a}_{\leftrightarrow}^{\dagger}\hat{a}_{\leftrightarrow})$,
the Hamiltonian of the polarized-light-driven optomechanical system is given
by

$\displaystyle\hat{H}=\dfrac{\hbar\omega_{m}}{2}(\hat{p}^{2}\\!+\\!\hat{q}^{2})+\hbar\\!\sum_{j=\updownarrow,\leftrightarrow}\left(\Delta_{c}\hat{a}_{j}^{\dagger}\hat{a}_{j}\\!-\\!g_{0}\hat{a}_{j}^{\dagger}\hat{a}_{j}\hat{q}\right)+\hat{H}_{\mathrm{dr}},$
$\displaystyle\hat{H}_{\mathrm{dr}}=i\hbar\sqrt{2\kappa}\sum_{j=\updownarrow,\leftrightarrow}\left(\hat{a}_{j}^{\dagger}{S}_{j}-\hat{a}_{j}{S}_{j}^{\ast}\right),$
(1)

where $\hat{a}_{j}$ $(\hat{a}_{j}^{\dagger})$ is the annihilation (creation)
operator of the orthogonal cavity modes with degenerate resonance frequency
$\omega_{c}$ and decay rate $\kappa$; $\hat{p}$ and $\hat{q}$ are,
respectively, the dimensionless momentum and position operators of the mirror
with mass $m$ and frequency $\omega_{m}$; $\Delta_{c}=\omega_{c}-\omega_{L}$
is optical detuning between the cavity mode and the driving field; $g_{0}$ is
the single-photon optomechanical coupling coefficient for both orthogonal
cavity modes. The frame rotating with driving frequency $\omega_{L}$ is
obtained by applying the unitary transformation
$\hat{U}=\exp[i\hat{H}_{0}t/\hbar]$ (see, e.g., Ref. [37]).
$S=|S_{\updownarrow}|^{2}+|S_{\leftrightarrow}|^{2}=\sqrt{P/\hbar\omega_{L}}$
denotes the amplitude of a linearly polarized driving field with an input
laser power $P$, where $S_{\updownarrow}=S\cos\theta$ and
$S_{\leftrightarrow}=S\sin\theta$ are the projections of $S$ onto the vertical
and horizontal modes, respectively.

Fig. 1: (a) Schematic diagram of a polarized-light-driven optomechanical
system, which consists of an optical polarizer and a Fabry-Pérot cavity with a
movable mirror. (b) Frequency spectrogram of a vector optomechanical system in
panel (a), with $\theta$ being the angle between the polarization of the
linearly polarized light and the vertical mode. The orthogonal cavity modes
with degenerate resonance frequency $\omega_{c}$ and decay rate $\kappa$; the
frequency of the mirror is $\omega_{m}$; the driving frequency is
$\omega_{L}$, and $\Delta_{c}=\omega_{c}-\omega_{L}$ is optical detuning
between the cavity mode and the driving field.

By considering the damping and the corresponding noise term of both optical
and mechanical modes, the quantum Langevin equations (QLEs) of motion
describing the dynamics of this system are obtained as

$\displaystyle\dot{\hat{a}}_{j}$
$\displaystyle=(-i\Delta_{c}+ig_{0}\hat{q}-\kappa)\hat{a}_{j}+\sqrt{2\kappa}S_{j}+\sqrt{2\kappa}\hat{a}_{j}^{\textrm{in}},$
$\displaystyle\dot{\hat{q}}$ $\displaystyle=\omega_{m}\hat{p},$
$\displaystyle\dot{\hat{p}}$
$\displaystyle=-\omega_{m}\hat{q}-\gamma_{m}\hat{p}+g_{0}\sum_{j=\updownarrow,\leftrightarrow}\hat{a}_{j}^{\dagger}\hat{a}_{j}+\hat{\xi},$
(2)

where $j=\updownarrow,\leftrightarrow$, $\gamma_{m}=\omega_{m}/Q_{m}$ is the
mechanical damping rate with $Q_{m}$ the quality factor of the movable mirror;
$\hat{a}_{j}^{\textrm{in}}$ are the zero-mean input vacuum noise operators for
the orthogonal cavity modes, and they satisfy the following correlation
functions [59]

$\displaystyle\langle\hat{a}_{j}^{\textrm{in},\dagger}(t)\hat{a}_{j}^{\textrm{in}}(t^{\prime})\rangle=0,$
$\displaystyle\langle\hat{a}_{j}^{\textrm{in}}(t)\hat{a}_{j}^{\textrm{in},\dagger}(t^{\prime})\rangle=\delta(t-t^{\prime}).$
(3)

Moreover, $\hat{\xi}$ denotes the Brownian noise operator for the mechanical
mode, resulting from the coupling of the mechanical mode with the
corresponding thermal environment. It satisfies the following correlation
function [60]

$\displaystyle\langle\hat{\xi}(t)\hat{\xi}(t^{\prime})\rangle=\dfrac{\gamma_{m}}{\omega_{m}}\int\dfrac{d\omega}{2\pi}e^{-i\omega(t-t^{\prime})}\left[\coth\left(\dfrac{\hbar\omega}{2k_{B}T}\right)+1\right],$
(4)

where $k_{B}$ is the Boltzmann constant and $T$ is the environment temperature
of the mechanical mode. The noise operator $\hat{\xi}(t)$ models the
mechanical Brownian motion as, in general, a non-Markovian process. However,
in the limit of a high mechanical quality factor $Q_{m}\gg 1$, $\hat{\xi}(t)$
can be faithfully considered as Markovian, and, then, its correlation function
is reduced to

$\displaystyle\langle\hat{\xi}(t)\hat{\xi}(t^{\prime})\rangle\simeq\gamma_{m}(2n_{m}+1)\delta(t-t^{\prime}),$
(5)

where $n_{m}=[\textrm{exp}[(\hbar\omega_{m}/k_{B}T)]-1]^{-1}$ is the mean
thermal phonon number.

Setting all the derivatives in QLEs (2) as zero leads to the steady-state mean
values of the optical and the mechanical modes

$\displaystyle\alpha_{j}=$
$\displaystyle\dfrac{\sqrt{2\kappa}}{i\Delta+\kappa}S_{j}\leavevmode\nobreak\
\leavevmode\nobreak\ (j=\updownarrow,\leftrightarrow),$ $\displaystyle q_{s}=$
$\displaystyle\dfrac{g_{0}}{\omega_{m}}\left(|\alpha_{\updownarrow}|^{2}+|\alpha_{\leftrightarrow}|^{2}\right),\leavevmode\nobreak\
\leavevmode\nobreak\ p_{s}=0,$ (6)

where $\Delta=\Delta_{c}-g_{0}q_{s}$ is the effective optical detuning. Under
the condition of intense optical driving, one can expand each operator as a
sum of its steady-state mean value and a small quantum fluctuation around it,
i.e.,
$\hat{a}_{j}=\alpha_{j}+\delta\hat{a}_{j},\hat{q}=q_{s}+\delta\hat{q},\hat{p}=p_{s}+\delta\hat{p}$.
Then, by defining the following vectors of quadrature fluctuations and
corresponding input noises :
$u(t)=(\delta\hat{X}_{\updownarrow},\delta\hat{Y}_{\updownarrow},\delta\hat{X}_{\leftrightarrow},\delta\hat{Y}_{\leftrightarrow},\delta\hat{q},\delta\hat{p})^{\textit{T}}$,
$v(t)=(\sqrt{2\kappa}\hat{X}_{\updownarrow}^{\textrm{in}},\sqrt{2\kappa}\hat{Y}_{\updownarrow}^{\textrm{in}},\sqrt{2\kappa}\hat{X}_{\leftrightarrow}^{\textrm{in}},\sqrt{2\kappa}\hat{Y}_{\leftrightarrow}^{\textrm{in}},0,\hat{\xi})^{\textit{T}}$,
with the components:

$\displaystyle\delta\hat{X}_{j}$
$\displaystyle=\dfrac{1}{\sqrt{2}}(\delta\hat{a}_{j}^{\dagger}+\delta\hat{a}_{j}),$
$\displaystyle\delta\hat{Y}_{j}$
$\displaystyle=\dfrac{i}{\sqrt{2}}(\delta\hat{a}_{j}^{\dagger}-\delta\hat{a}_{j}),$
$\displaystyle\hat{X}_{j}^{\textrm{in}}$
$\displaystyle=\dfrac{1}{\sqrt{2}}(\hat{a}_{j}^{\textrm{in}\dagger}+\hat{a}_{j}^{\textrm{in}}),$
$\displaystyle\hat{Y}_{j}^{\textrm{in}}$
$\displaystyle=\dfrac{i}{\sqrt{2}}(\hat{a}_{j}^{\textrm{in}\dagger}-\hat{a}_{j}^{\textrm{in}}),$
(7)

one can obtain a set of linearized QLEs, which can be written in a compact
form as

$\displaystyle\dot{u}(t)=Au(t)+v(t),$ (8)

where

$A=\begin{pmatrix}-\kappa&\Delta&0&0&-G_{\updownarrow}^{y}&0\\\
-\Delta&-\kappa&0&0&G_{\updownarrow}^{x}&0\\\
0&0&-\kappa&\Delta&-G_{\leftrightarrow}^{y}&0\\\
0&0&-\Delta&-\kappa&G_{\leftrightarrow}^{x}&0\\\ 0&0&0&0&0&\omega_{m}\\\
G_{\updownarrow}^{x}&G_{\updownarrow}^{y}&G_{\leftrightarrow}^{x}&G_{\leftrightarrow}^{y}&-\omega_{m}&-\gamma_{m}\end{pmatrix}.$
(9)

The linearized QLEs indicate that the effective COM coupling rate
$G_{j}\equiv\sqrt{2}g_{0}\alpha_{j}=G_{j}^{x}+iG_{j}^{y}$, can be
significantly enhanced by increasing the intracavity photons. Moreover, the
solution of the linearized QLEs (8) is given by

$\displaystyle u(t)=M(t)u(0)+\int_{0}^{t}\textrm{d}\tau\,M(\tau)v(t-\tau),$
(10)

where $M(t)=\exp(At)$. When all of the eigenvalues of the matrix $A$ have
negative real parts, the system is stable and reaches its steady state,
leading to $M(\infty)=0$ and

$\displaystyle
u_{i}(\infty)=\int_{0}^{\infty}\textrm{d}\tau\,\sum_{k}M_{ik}(\tau)v_{k}(t-\tau).$
(11)

The stability conditions can usually be derived by applying the Routh-Hurwitz
criterion [61]. In our following numerical simulations, we have confirmed that
the chosen parameters in this paper can keep the COM system in a stable
regime.

Because of the linearized dynamics of the QLEs and the Gaussian nature of the
input noises, the steady state of the quantum fluctuations, independently of
any initial conditions, can finally evolve into a tripartite continuous
variable (CV) Gaussian state, which is fully characterized by a $6\times 6$
stationary correlation matrix (CM) $V$ with the components

Fig. 2: Polarization-controlled optomechanical entanglement switch within
intracavity mode. (a) The logarithmic negativity $E_{N,j}$ of the TE and TM
modes versus the scaled optical detuning $\Delta_{c}$ for different values of
the polarization angle $\theta$. (b) The logarithmic negativity $E_{N,j}$ of
the TE and TM modes versus the polarization angle $\theta$ in polar
coordinates, with the optical detuning $\Delta_{c}/\omega_{m}=1$.
$E_{N\\!,\,\updownarrow}$ and $E_{N\\!,\,\leftrightarrow}$ demonstrate a
complementary distribution with the variation of the polarization angle
$\theta$. (c) Density plot of $E_{N,j}$ as a function of the scaled optical
detuning $\Delta_{c}$ and the polarization angle $\theta$. By tuning the
polarization angle, an adjustable coherent COM entanglement switch between the
TE and TM modes could be achieved. (d) The effective COM coupling
$|G_{\updownarrow}|$ of the TE mode versus the scaled optical detuning
$\Delta_{c}$ and the polarization angle $\theta$.

$\displaystyle
V_{ij}=\big{\langle}u_{i}(\infty)u_{j}(\infty)\\!+\\!u_{j}(\infty)u_{i}(\infty)\big{\rangle}/2.$
(12)

Using the solution of the steady-state QLEs, we can obtain the correlation
matrix

$\displaystyle V=\int_{0}^{\infty}\textrm{d}\tau\,M(\tau)DM^{T}(\tau),$ (13)

where

$\displaystyle
D\\!=\\!\textrm{Diag}\,[\kappa,\kappa,\kappa,\kappa,0,\gamma_{m}(2n_{m}\\!+\\!1)],$
(14)

is the diffusion matrix, which is defined through $\langle
n_{i}(\tau)n_{j}(\tau^{\prime})\\!+\\!n_{j}(\tau^{\prime})n_{i}(\tau)\rangle/2\\!=\\!D_{ij}\delta(\tau-\tau^{\prime})$.
When the stability condition is fulfilled, the dynamics of the steady-state
correlation matrix is determined by the Lyapunov equation [45]:

$\displaystyle AV+VA^{\textit{T}}=-D.$ (15)

As seen from Eq. (15), the Lyapunov equation is linear and can
straightforwardly be solved, thus allowing us to derive the correlation matrix
$V$ for any values of the relevant parameters. However, the explicit form of
$V$ is complicated and is not be reported here.

##  3 Switching optomechanical entanglement by tuning the polarization of
vector light

To explore the polarization-controlled coherent switch of the steady-state COM
entanglement, we adopt the logarithmic negativity, $E_{N}$, for quantifying
the bipartite distillable entanglement between different degrees of freedom of
our three-mode Gaussian state [62]. In the CV case, $E_{N}$ can be defined as

$\displaystyle E_{N}=\max\,\left[0,-\ln\left(2\tilde{\nu}_{-}\right)\right],$
(16)

where

$\displaystyle\tilde{\nu}_{-}\\!=\\!2^{-1/2}\big{\\{}\Sigma(V_{bp})-\big{[}\Sigma(V_{bp})^{2}-4\det\,V_{bp}\big{]}^{1/2}\big{\\}}^{1/2},$
(17)

with

$\displaystyle\Sigma(V_{bp})=\det\,A+\det\,B-2\det\,C.$ (18)

Here $\tilde{\nu}_{-}$ is the minimum symplectic eigenvalue of the partial
transpose of the reduced $4\times 4$ CM $V_{bp}$. By tracing out the rows and
columns of the uninteresting mode in $V$, the reduced CM $V_{bp}$ can be given
in a $2\times 2$ block form

$\displaystyle V_{bp}=\left(\begin{matrix}A&C\\\
C^{\textit{T}}&B\end{matrix}\right).$ (19)

Equation (16) indicates that the COM entanglement emerges only when
$\tilde{\nu}_{-}<1/2$, which is equivalent to the Simon's necessary and
sufficient entanglement nonpositive partial transpose criterion (or the
related Peres-Horodecki criterion) for certifying bipartite CV distillable
entanglement in Gaussian states [63]. Therefore, $E_{N}$, quantifying the
amount by which the Peres-Horodecki criterion is violated, is an efficient
entanglement measure that is widely used when studying bipartite entanglement
in a multi-mode system.

In Figure 2, the logarithmic negativity $E_{N\\!,\,j}$ and the associated
effective COM coupling rate $G_{j}$ with the intracavity field are shown as a
function of the optical detuning $\Delta_{c}$ with respect to different
polarization angle $\theta$. Here, $E_{N\\!,\,\updownarrow}$ and
$E_{N\\!,\,\leftrightarrow}$ are used to denote the case of the COM
entanglement with respect to the TE and TM modes, respectively. For ensuring
the stability and experimental feasibility, the parameters are moderately
chosen as follows: $m\\!=\\!50\,\textrm{ng}$,
$\lambda\\!=\\!810\,\textrm{nm}$, $\omega_{m}/2\pi\\!=\\!10\,\textrm{MHz}$,
$g_{0}/2\pi\\!=\\!68.5\,\textrm{Hz}$,
$Q_{c}\\!=\\!\omega_{c}/\kappa\\!=\\!4.94\times 10^{7}$,
$Q_{m}\\!=\\!\omega_{m}/\gamma_{m}\\!=\\!10^{5}$,
$\mathrm{T}\\!=\\!400\,\textrm{mK}$, and $P\\!=\\!30\,\textrm{mW}$. Note that
$Q_{c}$ is typically $10^{5}-10^{10}$ [37, 64], and $Q_{m}$ is typically
$10^{5}-10^{6}$ [65] in Fabry-Pérot cavities. Using a whispering-gallery-mode
COM system, much higher value of $Q_{c}$ is achievable, reaching even up to
$10^{12}$ [66]. The polarization angle $\theta$, which can be tuned by
rotating the orientations of the optical polarizer, is usually used to control
the spatial amplitude and phase of a linearly polarized driving field, with
values in a period ranging from $0$ to $2\pi$ [67]. In particular, $\theta=0$
($\theta=\pi/2$) corresponds to the case with a vertically (horizontally)
polarized pump field applied to drive the TE (TM) optical mode.

Fig. 3: Polarization-controlled optomechanical entanglement within the cavity
output mode. (a) The logarithmic negativity of the TE mode,
$E_{N\\!,\,\updownarrow}$ , as a function of the polarization angle $\theta$
in polar coordinates for different values of the inverse bandwidth
$\varepsilon$, with the central frequency $\Omega/\omega_{m}=-1$. (b)
$E_{N\\!,\,\updownarrow}$ versus the central frequency $\Omega$ and the
polarization angle $\theta$ for different values of $\varepsilon$. There is an
optimal value of $\theta$ for achieving the maximum COM entanglement within
the cavity output mode.

Specifically, as shown in Figure 2(a)–(c), the COM entanglement is present
only within a finite interval of the values of $\Delta_{c}$ around
$\Delta_{c}\simeq\omega_{m}$, and the spectral offset of the logarithmic
negativity peak is due to the radiation pressure induced redshift of the
cavity mode (see, e.g., Ref. [45, 68]). In fact, the COM entanglement survives
only with a tiny thermal noise occupancy. Therefore, at the light-motion
resonance $\Delta/\omega_{m}\simeq 1$, the COM interaction could significantly
cool the mechanical mode and, simultaneously, leads to a considerable COM
entanglement. In addition, the logarithmic negativities
$E_{N\\!,\,\leftrightarrow}$ and $E_{N\\!,\,\updownarrow}$ always demonstrate
a complementary distribution with the variation of the polarization angle
$\theta$, indicating that an adjustable COM entanglement conversion between
the TE and the TM modes would be implemented by a coherent polarization
control. The underlying physics of this phenomenon can be understood as
follows. In the polarization-controlled COM system, the field intensity of a
the TE and TM modes is dependent on the spatial distribution of the linearly
polarized driving field [15]. Correspondingly, the strength of the effective
COM coupling rate $|G_{j}|$ now also relies on polarization angle $\theta$. As
shown in Figure 2(c) and 2(d), it can be clearly seen that
$E_{N\\!,\,\updownarrow}$ achieves its maximum value for the optimal value of
$|G_{\updownarrow}|$. Therefore, the distribution of the COM entanglement with
respect to the TE and TM modes could be manipulated by tuning the polarization
angle. In practical aspect, the ability to generate an adjustable entanglement
conversion between subsystems of a compound COM system would provide another
degree of freedom for quantum optomechanical information processing.

For practical applications, the COM entanglement with the intracavity field is
hard to be directly accessed and utilized. In order to verify the generated
COM entanglement, an essential step is to perform homodyne or heterodyne
detections to the cavity output field, which allow to measure the
corresponding CM $V^{\textrm{out}}$. Specifically, the quantum correlations in
$V^{\textrm{out}}$, that involve optical quadratures, can be directly read out
by homodyning the cavity output. However, accessing the mechanical quadratures
typically requires to map the mechanical motion to a weak probe field first,
which then can be read out by applying a similar homodyne procedure of the
probe field.

Now, by applying the standard input-output relations, the output field of this
compound COM system is given by

$\displaystyle a_{j}^{\mathrm{out}}(t)=\sqrt{2}\delta
a_{j}(t)-a_{j}^{\mathrm{in}}(t),$ (20)

where the optical output field has the same non-zero correlation function as
the input field $\delta a_{j}(t)$, i.e., $[a_{j}^{\mathrm{out}}(t)$,
$a_{j}^{\mathrm{out}}(t^{\prime})^{\dagger}]=\delta(t-t^{\prime})$. As
discussed in detail in Ref. [69], by selecting different time or frequency
intervals from the continuous output field $a_{j}^{\mathrm{out}}(t)$, one can
extract a set of independent optical modes by means of spectral filters. Here,
for convenience, we consider the case where only a single output mode of the
TE and TM cavity field is detected. Therefore, in terms of a causal filter
function $g(s)$, the output field can be rewritten as

$\displaystyle
a_{1j}^{\mathrm{out}}(t)=\int_{-\infty}^{t}ds\,g_{j}(t-s)a_{j}^{\mathrm{out}}(s).$
(21)

In the frequency domain, $a_{1j}^{\mathrm{out}}$ takes the following form

$\displaystyle\tilde{a}_{1j}^{\mathrm{out}}(\omega)=\int_{-\infty}^{\infty}\frac{dt}{\sqrt{2\pi}}a_{1j}^{\mathrm{out}}(t)\textrm{exp}(i\omega
t)=\sqrt{2\pi}\tilde{g}_{j}(\omega)a_{1j}^{\mathrm{out}}(\omega),$ (22)

where $\tilde{g}_{j}(\omega)$ is the Fourier transform of the filter function.
An explicit example of an orthonormal set of the causal filter functions is
given by [70]

$\displaystyle
g_{j}(t)=\frac{\Lambda(t)-\Lambda(t-\tau)}{\sqrt{\tau}}\textrm{exp}(-i\Omega
t),$ (23)

and $g_{j}=g_{j}^{x}+ig_{j}^{y}$, where $g_{j}^{x}$ $(g_{j}^{y})$ is the real
(imaginary) part of $g_{j}$, $\Lambda(t)$ denotes the Heaviside step function,
with $\Omega$ and $\tau^{-1}$ being the central frequency and the bandwidth of
the causal filter, respectively. Here, $\tilde{g}(\omega)$ is given by

$\displaystyle\tilde{g}_{j}(\omega)=\sqrt{\frac{\tau}{2\pi}}\textrm{exp}[i(\omega-\Omega)\tau/2]\frac{\sin[(\omega-\Omega)\tau/2]}{(\omega-\Omega)\tau/2}.$
(24)

The COM entanglement for the cavity output mode is verified via the CM defined
as follows, i.e.,

$\displaystyle\mathrm{V}_{kl}^{\mathrm{out}}(t)=\frac{1}{2}\big{\langle}u_{k}^{\mathrm{out}}(t)u_{l}^{\mathrm{out}}(t)+u_{l}^{\mathrm{out}}(t)u_{k}^{\mathrm{out}}(t)\big{\rangle},$
(25)

where $u^{\mathrm{out}}(t)=(\delta X_{\updownarrow}^{\mathrm{out}},\delta
Y_{\updownarrow}^{\mathrm{out}},\delta
X_{\leftrightarrow}^{\mathrm{out}},\delta
Y_{\leftrightarrow}^{\mathrm{out}},\delta q,\delta p)^{\textit{T}}$ is the
vector form by the mechanical position and momentum fluctuations and by the
(canonical) position quadrature,

$\displaystyle
X_{j}^{\mathrm{out}}(t)=\big{\\{}a_{1j}^{\mathrm{out}}(t)+\big{[}a_{1j}^{\mathrm{out}}(t)\big{]}^{\dagger}\big{\\}}/\sqrt{2},$
(26)

and the momentum quadrature,

$\displaystyle
Y_{j}^{\mathrm{out}}(t)=\big{\\{}a_{1j}^{\mathrm{out}}(t)-\big{[}a_{1j}^{\mathrm{out}}(t)\big{]}^{\dagger}\big{\\}}/i\sqrt{2},$
(27)

of the optical output modes. From the input-output relation in Equation (20),
one can obtain

$\displaystyle
u_{i}^{\mathrm{out}}(t)=\int_{-\infty}^{t}ds\,\mathcal{T}_{i}(t-s)[u(s)-v^{\prime}(s)],$
(28)

where

$\displaystyle v^{\prime}(t)$
$\displaystyle=\frac{1}{\sqrt{2\kappa}}(X_{\updownarrow}(t),Y_{\updownarrow}(t),X_{\leftrightarrow}(t),Y_{\leftrightarrow}(t),0,0)^{T},$
(29)

is analogous to the noise vector $v(t)$ in Equation (8) but without noise
acting on the mechanical mode. We have also introduced the matrix,

$\displaystyle\mathcal{T}(t)=\left(\begin{smallmatrix}f_{\updownarrow}^{x}&-f_{\updownarrow}^{y}&0&0&0&0\\\
f_{\updownarrow}^{y}&f_{\updownarrow}^{x}&0&0&0&0\\\
0&0&f_{\leftrightarrow}^{x}&-f_{\leftrightarrow}^{y}&0&0\\\
0&0&f_{\leftrightarrow}^{y}&f_{\leftrightarrow}^{x}&0&0\\\
0&0&0&0&\delta(t)&0&\\\ 0&0&0&0&0&\delta(t)\end{smallmatrix}\right),$ (30)

where $f_{j}^{x}=\sqrt{2\kappa}g_{j}^{x},f_{j}^{y}=\sqrt{2\kappa}g_{j}^{y}$.
Using the Fourier transform and the correlation function of the noises, one
can derive the following general expression for the stationary output
correlation matrix, which is the counterpart of the intracavity relation of
Equation (13):

$\displaystyle V^{\mathrm{out}}=$ $\displaystyle\int
d\omega\,\tilde{T}(\omega)\left[\tilde{M}(\omega)+\frac{P_{\mathrm{out}}}{2\kappa}\right]$
$\displaystyle\times
D(\omega)\left[\tilde{M}^{\dagger}(\omega)+\frac{P_{\mathrm{out}}}{2\kappa}\right]\tilde{T}^{\dagger}(\omega),$
(31)

where $P_{\mathrm{out}}=\mathrm{Diag}[1,1,1,1,0,0]$ is the projector onto the
two-dimensional space associated with the output quadratures, and

$\displaystyle\tilde{M}^{\mathrm{ext}}(\omega)=(i\omega+A)^{-1},$ (32)

and

$D(\omega)=\begin{pmatrix}\kappa&0&0&0&0&0\\\ 0&\kappa&0&0&0&0\\\
0&0&\kappa&0&0&0\\\ 0&0&0&\kappa&0&0\\\ 0&0&0&0&0&0\\\
0&0&0&0&0&N_{m}\end{pmatrix},$ (33)

where $N_{m}=(\gamma_{m}\omega/\omega_{m})\coth(\hbar\omega/2k_{B}T)$. A
deeper understanding of the general expression for $V^{\mathrm{out}}$ in
Equation (31) is obtained by multiplying the terms in the integral, one
obtains

$\displaystyle V^{\mathrm{out}}$ $\displaystyle=\int
d\omega\,\tilde{T}(\omega)\tilde{M}^{\dagger}(\omega)D(\omega)\tilde{M}(\omega)\tilde{T}^{\dagger}(\omega)+\frac{P_{\mathrm{out}}}{2}$
$\displaystyle\quad+\int
d\omega\,\tilde{T}(\omega)\big{[}\tilde{M}(\omega)R_{\mathrm{out}}+R_{\mathrm{out}}\tilde{M}^{\dagger}(\omega)\big{]}\tilde{T}^{\dagger}(\omega),$
(34)

where
$R_{\mathrm{out}}=P_{\mathrm{out}}D(\omega)/\kappa=D(\omega)P_{\mathrm{out}}/\kappa$,
for simplicity, we have defined [68]

$\displaystyle P_{\mathrm{out}}\equiv
2\int\frac{d\omega}{4\kappa^{2}}\tilde{T}(\omega)P_{\mathrm{out}}D(\omega)P_{\mathrm{out}}\tilde{T}^{\dagger}(\omega).$
(35)

Equation (34) clearly shows the quantum correlations of the quadrature
operators in the output field, where the first integral term stems from the
intracavity COM interaction, the second term gives the contribution the
correlations of the noise operators, and the last term describes the
interactions between the intracavity mode and the optical input field.

Fig. 4:  The role of thermal effects and quality factors on the generation and
manipulation of COM entanglement. (a) The logarithmic negativity of the TE
mode, $E_{N\\!,\,\updownarrow}$ , as a function of the environment temperature
$T$ and the polarization angle $\theta$, with the central frequency
$\Omega/\omega_{m}=-1$ and $\varepsilon=10$. (b) $E_{N\\!,\,\updownarrow}$
versus the central frequency $\Omega$ and the environment temperature $T$,
with the polarization angle $\theta=0$ and $\varepsilon=10$. (c) Density plot
of $E_{N\\!,\,\updownarrow}$ versus the polarization angle $\theta$ and the
quality factor $Q$, with the optical detuning $\Delta_{c}/\omega_{m}=0.6$. (d)
$E_{N\\!,\,\updownarrow}$ as a function of the scaled optical detuning
$\Delta_{c}$ and the quality factor $Q$, with $\theta=0$

.

Then, by numerically calculating the CM $V^{\mathrm{out}}$ and the associated
logarithmic negativity $E_{N\\!,\,j}$, one can detect and verify the generated
intracavity COM entanglement at the cavity output. Here, as a specific example
shown in Figure 3, we studied the COM entanglement between the TE output mode
and mechanical mirror. Also, as discussed in detail in Ref. [68], the
generated intracavity COM entanglement is mostly carried by the lower-
frequency Stokes sideband of the output field. Therefore, it would be better
to choose an input field with the center frequency at $\Omega=-\omega_{m}$,
which can usually be implemented by using the filter function of Equation
(23). Specifically, for $\varepsilon=1$, as shown in Figure 3(a), the
variation of COM entanglement with the polarization angle at the cavity output
field is similar to that of its intracavity counterpart. However, by
optimizing the value of $\varepsilon$ or, equivalently, the detection
bandwidth $\tau$, $E_{N\\!,\,\updownarrow}$ would achieve much higher value
than that of the intracavity field, implying a significant enhancement of COM
entanglement at the output field. In addition, for particular detection
bandwidth, there is also an optimal value of the polarization angle for which
the COM entanglement of the output field achieves its maximum value [see
Figure 3(b) for more details]. The physical origin of the enhancement of COM
entanglement at cavity output field results from the formation of quantum
correlations between the intracavity mode and the optical input field may
cancel the destructive effects of the input noises. The process of detecting
entanglement at the output field is related to entanglement distillation,
whose basic idea is originating from extracting, from an ensemble of pairs of
non-maximally entangled qubits, a smaller number of pairs with a higher degree
of entanglement [71]. In this situation, it is possible that one can further
manipulate and optimize the polarization-controlled COM entanglement at the
output field, which can be useful for quantum communications [72, 73], quantum
teleportation [74], and loophole-free Bell test experiment [75].

Besides, as discussed in details in Ref. [68], the thermal phonon excitations
could greatly affect the quantum correlations between photons and phonons,
and, thus, tend to destroy the generated COM entanglement. The mean thermal
phonon number, which is monotonically increasing with the environment
temperature, is assumed to be $n_{m}\simeq 833$ in our former discussions.
Here, by computing $E_{N\\!,\,\updownarrow}$ as a function of temperature $T$,
as shown in Figure 4, we confirmed that the conversion of COM entanglement
through polarization control could still exist when considering larger mean
thermal phonon numbers. To clearly see this phenomenon, we plot the
logarithmic $E_{N\\!,\,\updownarrow}$ of the CV bipartite system formed by the
mechanical mode and the cavity output mode centered around the Stokes sideband
$\Omega=-\omega_{m}$ versus polarization angle $\theta$ and temperature $T$ in
Figure 4(a) and set $\theta=0$. We also consider $E_{N\\!,\,\updownarrow}$ as
a function of temperature $T$ and the cavity frequency $\Omega$ in Figure
4(b). It is seen that for $\varepsilon=10$, the polarization-controlled COM
entanglement can still be observed at the cavity output field below a critical
temperature $T_{c}\approx 2\,\textrm{K}$, corresponding to $n_{m}\simeq 4160$.
Furthermore, the quality factor $Q$ of the cavity mode could also affect the
implementation of the COM entanglement switch. As shown in Figure 4(c)-4(d),
for a certain environment temperature, e.g., $T=400\,\textrm{mK}$, the degree
of COM entanglement is suppressed by decreasing the value of $Q$, and it is
seen that the minimum $Q$ factor required for the COM entanglement switch
process is approximately $Q\simeq 3\times 10^{7}$.

Finally, we remark that based on experimentally feasible parameters, not only
all bipartite entanglements but also the genuine tripartite entanglement can
be created and controlled in our system. In fact, we have already confirmed
that the tripartite entanglement, measured by the minimum residual contangle
[76, 77], can reach its maximum at the optimal polarization angle
$\theta=\pi/4$. However, the bipartite purely optical entanglement of TE and
TM modes is typically very weak due to the indirect coupling of these two
optical modes, and thus the resulting tripartite entanglement is also weak in
this system. Nevertheless, we note that the techniques of achieving stationary
entanglement between two optical fields and even strong tripartite
entanglement are already well available in current COM experiments [44, 48,
49, 50, 51], and our work here provides a complementary way to achieve
coherent switch of the COM entanglement via polarization control.

##  4 Conclusion

In summary, we have proposed how to manipulate the light-motion interaction in
a COM resonator via polarization control, which enables the ability of
coherently switching COM entanglement in such a device. We note that the
ability to achieve coherent switch of COM entanglement is useful for a wide
range of entanglement-based quantum technologies, such as quantum information
processing [30], quantum routing [78], or quantum networking [79]. Also, our
work reveals the potential of engineering various quantum effects by tuning
the optical polarizations, such as mechanical squeezing [80], quantum state
transfer [81], and asymmetric Einstein-Podolsky-Rosen steering [82]. Although
we have considered here a specific case of linearly polarized field whose
spatial distribution is homogeneous, we can envision that future developments
with inhomogeneous vector beams can further facilitate more appealing quantum
COM techniques, such as opto-rotational entanglement [83] or polarization-
tuned topological energy transfer [84]. In a broader view, our findings shed
new lights on the marriage of vectorial control and quantum engineering,
opening up the way to control quantum COM states by utilizing synthetic
optical materials [37, 85].

Author contribution: All the authors have accepted responsibility for the
entire content of this submitted manuscript and approved submission.

Research funding: This work is supported by the National Natural Science
Foundation of China (Grants No. 11935006 and No. 11774086) and the Science and
Technology Innovation Program of Hunan Province (Grant No. 2020RC4047). L.-M.
K. is supported by the National Natural Science Foundation of China (NSFC,
Grants No. 11775075 and No. 11434011). A. M. was supported by the Polish
National Science Centre (NCN) under the Maestro Grant No.
DEC-2019/34/A/ST2/00081.

Conflict of interest statement: The authors declare no conflicts of interest
regarding this article.

## Bibliography

  * [1] J. Chen, C. Wan, and Q. Zhan, “Vectorial Optical Fields: Recent Advances and Future Prospects,” Sci. Bull., vol. 63, pp. 54–74, 2018.
  * [2] C. Rosales-Guzmán, B. Ndagano, and A. Forbes, “A Review of Complex Vector Light Fields and Their Applications,” J. Opt., vol. 20, p. 123001, 2018.
  * [3] A. Z. Goldberg et al., “Quantum concepts in optical polarization,” Adv. Opt. Photon., vol. 13, p. 1–73, 2021.
  * [4] F. Walter, G. Li, C. Meier, S. Zhang, and T. Zentgraf, “Ultrathin Nonlinear Metasurface for Optical Image Encoding,” Nano Letters, vol. 17, pp. 3171–3175, 2017
  * [5] Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express, vol. 18, pp. 10828–10833, 2010.
  * [6] R. Chen, K. Agarwal, C. J. R. Sheppard, and X. Chen, “Imaging using cylindrical vector beams in a high-numerical-aperture microscopy system,” Opt. Lett., vol. 38, pp. 3111–3114, 2013.
  * [7] J. Ahn et al., “Optically Levitated Nanodumbbell Torsion Balance and GHz Nanomechanical Rotor,” Phys. Rev. Lett., vol. 121, p. 033603, 2018.
  * [8] W. Liu, D. Dong, H. Yang, Q. Gong, and K. Shi, “Robust and High-Speed Rotation Control in Optical Tweezers by Using Polarization Synthesis Based on Heterodyne Interference,” Opto-Electron Adv, vol. 3, p. 200022, 2020.
  * [9] M. Xian, Y. Xu, X. Ouyang, Y. Cao, S. Lan, and X. Li, “Segmented Cylindrical Vector Beams for Massively-Encoded Optical Data Storage,” Sci. Bull, vol. 65, pp. 2072–2079, 2020.
  * [10] M. Neugebauer, P. Woźniak, A. Bag, G. Leuchs, and P. Banzer, “Polarization-controlled directional scattering for nanoscopic position sensing,” Nat. Commun., vol. 7, p. 11286, 2016.
  * [11] X. Zhang et al., “Symmetry-breaking-induced nonlinear optics at a microcavity surface,” Nat. Photonics, vol. 13, pp. 21–24, 2019.
  * [12] T. P. Purdy, R. W. Peterson, and C. A. Regal, “Observation of Radiation Pressure Shot Noise on a Macroscopic Object,” Science, vol. 339, pp. 801–804, 2013.
  * [13] L. Li et al., “Polarimetric Parity-Time Symmetry in a Photonic System,” Light: Sci. Appl., vol. 9, p. 169, 2020.
  * [14] Q. Zhan, “Cylindrical Vector Beams: From Mathematical Concepts to Applications,” Adv. Opt. Photon., vol. 1, pp. 1–57, 2009.
  * [15] J. Wang, F. Castellucci, and S. Franke-Arnold, “Vectorial Light– Matter Interaction: Exploring Spatially Structured Complex Light Fields,” AVS Quantum Sci., vol. 2, p. 031702, 2020.
  * [16] S. Sederberg et al., “Vectorized Optoelectronic Control and Metrology in a Semiconductor,” Nat. Photonics, vol. 14, pp. 680–685, 2020.
  * [17] N. Malossi et al., “Sympathetic Cooling of a Radio-Frequency LC Circuit to Its Ground State in an Optoelectromechanical System,” Phys. Rev. A, vol. 103, p. 033516, 2021.
  * [18] L. He, H. Li, and M. Li, “Optomechanical Measurement of Photon Spin Angular Momentum and Optical Torque in Integrated Photonic Devices,” Sci. Adv., vol. 2, p. e1600485, 2016.
  * [19] G. Xu et al., “Spontaneous symmetry breaking of dissipative optical solitons in a two-component Kerr resonator,” Nat. Commun., vol. 12, p.4023, 2021.
  * [20] Y. Kan, S. K. H. Andersen, F. Ding, S. Kumar, C. Zhao, and S. I. Bozhevolnyi, “Metasurface-Enabled Generation of Circularly Polarized Single Photons,” Adv. Mater., vol. 32, p. 1907832, 2020.
  * [21] S. Buddhiraju, A. Song, G. T. Papadakis, and S. Fan, “Nonreciprocal Metamaterial Obeying Time-Reversal Symmetry,” Phys. Rev. Lett., vol. 124, p. 257403, 2020.
  * [22] L. Zhu et al., “A Dielectric Metasurface Optical Chip for the Generation of Cold Atoms,” Sci. Adv., vol. 6, p. eabb6667, 2020.
  * [23] V. D’Ambrosio et al., “Entangled Vector Vortex Beams,” Phys. Rev. A, vol. 94, p. 030304, 2016.
  * [24] A. Villar et al., “Entanglement Demonstration on Board a Nano-Satellite,” Optica, vol. 7, pp. 734–737, 2020.
  * [25] H. J. Snijders et al., “Observation of the Unconventional Photon Blockade,” Phys. Rev. Lett., vol. 121, p. 043601, 2018.
  * [26] T. Stav et al., “Quantum Entanglement of the Spin and Orbital Angular Momentum of Photons Using Metamaterials,” Science, vol. 361, pp. 1101–1104, 2018.
  * [27] K. Wang et al., “Quantum metasurface for multiphoton interference and state reconstruction,” Science, vol. 361, pp. 1104–1108, 2018.
  * [28] Ranfagni, A., Vezio, P., Calamai, M. et al. Vectorial polaritons in the quantum motion of a levitated nanosphere. Nat. Phys. (2021). https://doi.org/10.1038/s41567-021-01307-y
  * [29] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys., vol. 81, pp. 865–942, 2009.
  * [30] K. Stannigel et al., “Optomechanical Quantum Information Processing with Photons and Phonons,” Phys. Rev. Lett., vol. 109, p. 013603, 2012.
  * [31] R. Fleury, D. Sounas, and A. Alù, “An invisible acoustic sensor based on parity-time symmetry,” Nat. Commun., vol. 6, p. 5905, 2015.
  * [32] X. Guo et al., “Distributed quantum sensing in a continuous-variable entangled network,” Nat. Phys., vol. 16, pp. 281–284, 2020.
  * [33] J. Yin et al., “Entanglement-based secure quantum cryptography over 1,120 kilometres,” Nature, vol. 582, pp. 501–505, 2020.
  * [34] J. Volz et al., “Observation of Entanglement of a Single Photon with a Trapped Atom,” Phys. Rev. Lett., vol. 96, p. 030404, 2006.
  * [35] A. N. Cleland and M. R. Geller, “Superconducting Qubit Storage and Entanglement with Nanomechanical Resonators,” Phys. Rev. Lett., vol. 93, p. 070501, 2004.
  * [36] B. Adams and F. Petruccione, “Quantum Effects in the Brain: A Review,” AVS Quantum Sci., vol. 2, p. 022901, 2020.
  * [37] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity Optomechanics,” Rev. Mod. Phys., vol. 86, pp. 1391–1452, 2014.
  * [38] E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-Coherent Coupling of a Mechanical Oscillator to an Optical Cavity Mode,” Nature, vol. 482, pp. 63–67, 2012.
  * [39] D. C. Moore and A. A. Geraci, “Searching for New Physics Using Optically Levitated Sensors,” Quantum Sci. Technol., vol. 6, p. 014008, 2021.
  * [40] Y. Qin et al., “Nonlinearity-induced nanoparticle circumgyration at sub-diffraction scale,” Nat. Commun., vol. 12, p. 3722, 2021.
  * [41] F. Lecocq and J., “Quantum Nondemolition Measurement of a Nonclassical State of a Massive Object,” Phys. Rev. X, vol. 5, p. 041037, 2015.
  * [42] H. Yu, L. McCuller, M. Tse, N. Kijbunchoo, L. Barsotti, and N. Mavalvala, “Quantum Correlations between Light and the Kilogram-Mass Mirrors of LIGO,” Nature, vol. 583, pp. 43–47, 2020.
  * [43] R. Riedinger et al., “Non-classical correlations between single photons and phonons from a mechanical oscillator,” Nature, vol. 530, pp. 313–316, 2016.
  * [44] J. Chen, M. Rossi, D. Mason, and A. Schliesser, “Entanglement of propagating optical modes via a mechanical interface,” Nat. Commun., vol. 11, p. 943, 2020.
  * [45] D. Vitali et al., “Optomechanical Entanglement between a Movable Mirror and a Cavity Field,” Phys. Rev. Lett., vol. 98, p. 030405, 2007.
  * [46] Y.-F. Jiao, S.-D. Zhang, Y.-L. Zhang, A. Miranowicz, L.-M. Kuang, and H. Jing, “Nonreciprocal Optomechanical Entanglement against Backscattering Losses,” Phys. Rev. Lett., vol. 125, p. 143605, 2020.
  * [47] T. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Entangling Mechanical Motion with Microwave Fields,” Science, vol. 342, pp. 710–713, 2013.
  * [48] S. Barzanjeh et al., “Stationary Entangled Radiation from Micromechanical Motion,” Nature, vol. 570, pp. 480–483, 2019.
  * [49] L. M. de Lépinay, C. F. Ockeloen-Korppi, M. J. Woolley, and M. A. Sillanpää, “Quantum mechanics–free subsystem with mechanical oscillators,” Science, vol. 372, pp. 625–629, 2021.
  * [50] C. F. Ockeloen-Korppi et al., “Stabilized entanglement of massive mechanical oscillators,” Nature, vol. 556, pp. 478–482, 2018\.
  * [51] R. Riedinger et al., “Remote quantum entanglement between two micromechanical oscillators,” Nature, vol. 556, pp. 473–477, 2018.
  * [52] J.-Q. Liao, Q.-Q. Wu, and F. Nori, “Entangling two macroscopic mechanical mirrors in a two-cavity optomechanical system,” Phys. Rev. A, vol. 89, p. 014302, Jan. 2014.
  * [53] J.-Q. Liao and L. Tian, “Macroscopic Quantum Superposition in Cavity Optomechanics,” Phys. Rev. Lett., vol. 116, p. 163602, 2016.
  * [54] S. Chen, X. Zhou, Y. Liu, X. Ling, H. Luo, and S. Wen, “Generation of Arbitrary Cylindrical Vector Beams on the Higher Order Poincaré Sphere,” Opt. Lett., vol. 39, pp. 5274–5276, 2014.
  * [55] A. V. Poshakinskiy and A. N. Poddubny, “Optomechanical circulator with a polaritonic microcavity,” Phys. Rev. B, vol. 104, p. 085303, 2021.
  * [56] Y. Nambu et al., “Observation of Magnon Polarization,” Phys. Rev. Lett., vol. 125, p. 027201, 2020.
  * [57] X.-X. Hu et al., “Noiseless Photonic Non-Reciprocity via Optically-Induced Magnetization,” Nat. Commun., vol. 12, p. 2389, 2021.
  * [58] H. Xiong, Y.-M. Huang, L.-L. Wan, and Y. Wu, “Vector Cavity Optomechanics in the Parameter Configuration of Optomechanically Induced Transparency,” Phys. Rev. A, vol. 94, p. 013816, 2016.
  * [59] C. W. Gardiner and P. Zoller, Quantum Noise. Berlin: Springer, 2000.
  * [60] V. Giovannetti and D. Vitali, “Phase-Noise Measurement in a Cavity with a Movable Mirror Undergoing Quantum Brownian Motion,” Phys. Rev. A, vol. 63, p. 023812, 2001.
  * [61] E. X. DeJesus and C. Kaufman, “Routh-Hurwitz Criterion in the Examination of Eigenvalues of a System of Nonlinear Ordinary Differential Equations,” Phys. Rev. A, vol. 35, pp. 5288–5290, 1987.
  * [62] G. Adesso, A. Serafini, and F. Illuminati, “Extremal Entanglement and Mixedness in Continuous Variable Systems,” Phys. Rev. A, vol. 70, p. 022318, 2004.
  * [63] R. Simon, “Peres-Horodecki Separability Criterion for Continuous Variable Systems,” Phys. Rev. Lett., vol. 84, pp. 2726–2729, 2000.
  * [64] P. Verlot et al., “Scheme to Probe Optomechanical Correlations between Two Optical Beams Down to the Quantum Level,” Phys. Rev. Lett., vol. 102, p. 103601, 2009.
  * [65] I. Galinskiy, Y. Tsaturyan, M. Parniak, and E. S. Polzik, “Phonon Counting Thermometry of an Ultracoherent Membrane Resonator near Its Motional Ground State,” Optica, vol. 7, pp. 718–725, 2020.
  * [66] V. Huet et al., “Millisecond Photon Lifetime in a Slow-Light Microcavity,” Phys. Rev. Lett., vol. 116, p. 133902, 2016.
  * [67] M. N. Mohd Nasir, S. Bakhtiari Gorajoobi, G. Senthil Murugan, and M. N. Zervas, “Polarization Effects in Optical Microresonators,” J. Opt. Soc. Am. B, vol. 36, pp. 705–716, 2019.
  * [68] C. Genes, A. Mari, P. Tombesi, and D. Vitali, “Robust Entanglement of a Micromechanical Resonator with Output Optical Fields,” Phys. Rev. A, vol. 78, p. 032316, 2008.
  * [69] D. Vitali, P. Cañizares, J. Eschner, and G. Morigi, “Time-Separated Entangled Light Pulses from a Single-Atom Emitter,” New J. Phys., vol. 10, p. 033025, 2008.
  * [70] X.-B. Yan, Z.-J. Deng, X.-D. Tian, and J.-H. Wu, “Entanglement Optimization of Filtered Output Fields in Cavity Optomechanics,” Opt. Express, vol. 27, pp. 24393–24402, 2019.
  * [71] P. G. Kwiat, S. Barraza-Lopez, A. Stefanov, and N. Gisin, “Experimental Entanglement Distillation and ‘Hidden’ Non-Locality,” Nature, vol. 409, pp. 1014–1017, 2001.
  * [72] H.-J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, “Quantum Repeaters: The Role of Imperfect Local Operations in Quantum Communication,” Phys. Rev. Lett., vol. 81, pp. 5932–5935, 1998.
  * [73] F. Xu, X. Ma, Q. Zhang, H.-K. Lo, and J.-W. Pan, “Secure quantum key distribution with realistic devices,” Rev. Mod. Phys., vol. 92, p. 025002, 2020.
  * [74] D. Llewellyn et al., “Chip-to-chip quantum teleportation and multi-photon entanglement in silicon,” Nat. Phys., vol. 16, pp. 148–153, 2020.
  * [75] R. García-Patrón, J. Fiurášek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, “Proposal for a Loophole-Free Bell Test Using Homodyne Detection,” Phys. Rev. Lett., vol. 93, p. 130409, 2004.
  * [76] G. Adesso and F. Illuminati, “Entanglement in continuous-variable systems: recent advances and current perspectives,” J. Phys. A: Math. Theor., vol. 40, pp. 7821–7880, 2007.
  * [77] J. Li, S.-Y. Zhu, and G., “Magnon-Photon-Phonon Entanglement in Cavity Magnomechanics,” Phys. Rev. Lett., vol. 121, p. 203601, 2018.
  * [78] Y. Yuan et al., “A Fully Phase-Modulated Metasurface as An Energy-Controllable Circular Polarization Router,” Adv. Sci., vol. 7, p. 2001437, 2020.
  * [79] H. J. Kimble, “The quantum internet,” Nature, vol. 453, pp. 1023–1030, 2008.
  * [80] E. E. Wollman et al., “Quantum squeezing of motion in a mechanical resonator,” Science, vol. 349, pp. 952–955, 2015.
  * [81] P. Kurpiers et al., “Deterministic quantum state transfer and remote entanglement using microwave photons,” Nature, vol. 558, pp. 264–267, 2018.
  * [82] S. Armstrong et al., “Multipartite Einstein–Podolsky–Rosen steering and genuine tripartite entanglement with optical networks,” Nat. Phys, vol. 11, pp. 167–172, 2015.
  * [83] H. Jing, D. S. Goldbaum, L. Buchmann, and P. Meystre, “Quantum Optomechanics of a Bose-Einstein Antiferromagnet,” Phys. Rev. Lett., vol. 106, p. 223601, 2011.
  * [84] H. Xu, L. Jiang, A. A. Clerk, and J. G. E. Harris, “Nonreciprocal Control and Cooling of Phonon Modes in an Optomechanical System,” Nature, vol. 568, pp. 65–69, 2019.
  * [85] Y. Chen et al., “Synthetic Gauge Fields in a Single Optomechanical Resonator,” Phys. Rev. Lett., vol. 126, p. 123603, 2021.