Coherent Vectorial Switch of Optomechanical Entanglement

Research Article Photonics Research 1

Coherent vectorial switch of optomechanical entanglement

YING LI1,*,Y A-FENG JIAO1,*,J ING-XUE LIU1,A DAM MIRANOWICZ2,Y UN-LAN ZUO1,L E-MAN KUANG1,†, AND HUI JING1,‡

1Key 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 2Institute of Spintronics and Quantum Information, Faculty of Physics, Adam Mickiewicz University, 61-614 Poznan,´ Poland *Co-ﬁrst authors with equal contribution. †[email protected] ‡Corresponding author:[email protected]

Compiled July 20, 2021

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. © 2021 Optical Society of America http://dx.doi.org/10.1364/ao.XX.XXXXXX

1. INTRODUCTION macroscopic entanglement between light and motion via polarization control has not yet been explored. Vector beams, characterized by the ability to tailor light by polarization A peculiar property of quantum entanglement, that measuring one control, are important for both fundamental researches and practical part of the entangled elements allows to determine the state of the applications in optics and photonics [1–3]. Manipulating the polariza- other, makes it a key resource for quantum technologies, ranging from tion of vector beams, for examples, provides efficient ways to realize quantum information processing [28, 29] to quantum sensing [30, 31]. optical trapping or imaging [4–6], material processing [7, 8], opti- Recently, entanglement-based secure quantum cryptography has also cal data storage [9], sensing [10], and nonlinearity enhancement [11]. been achieved at a distance over 1,120 kilometres [32]. So far, quantum Compared with conventional scalar light sources, vector beams provide entanglement has been observed in diverse systems involving photons, more degrees of freedom to regulate coherent light-matter interac- ions, atoms, and superconducting qubits [33, 34]. Quantum effects arXiv:2107.08384v1 [quant-ph] 18 Jul 2021 tions, that is, manipulating the coupling intensity by tuning spatial such as entanglement has also been studied at macroscopic scales and polarization distributions of an optical field [12–15]. Thus, vector even in biological systems [35]. In parallel, the rapidly emerging field beams have been used in a variety of powerful devices, such as opto- of cavity optomechanics (COM), featuring coherent coupling of mo- electrical [16, 17] or optomechanical systems [18, 19], metamaterial tion and light [36, 37], has provided a vital platform for engineering structures [20, 21], and atomic gases [22]. In a recent experiment, macroscopic quantum objects [38–40]. Very recently, quantum cor- through vectorial polarization control, coherent information transfer relations at room temperature were even demonstrated between light from photons to electrons was demonstrated [16]. In the quantum and a 40kg mirror, circumventing the standard quantum limit of mea- domain, mature techniques have been developed for creating entangled surement [41]. Quantum entanglement between propagating optical photons with vector beams [23, 24], and by using the destructive in- modes, between optical and mechanical modes, or between massive terference of the two excitation pathways of a polarized quantum-dot mechanical oscillators have all been realized in COM systems [42–50]. cavity, unconventional photon blockade effect was observed very re- In view of these rapid advances, COM devices have becomes one of the cently [25]. We also note that the single- and multi-photon resources promising candidates to operate as versatile quantum nodes processing with well-defined polarization properties, which are at the core of chiral or interchanging information with flying photons in a hybrid quantum quantum optics, have also been addressed via metasurfaces [26, 27]. network. However, as far as we know, the possibility of generating and switching Here, based on a COM system, we propose how to achieve a coher- Research Article Photonics Research 2 ent switch of quantum entanglement of photons and phonons through (a) polarization control. We show that the intracavity field intensity and κ γm the associated COM entanglement can be coherently manipulated by adjusting the polarization of a driving laser. This provides an efficient ωL way to manipulate the light-motion coupling, which is at the core ωc ωm of COM-based quantum technologies. Besides the specific example S of COM entanglement switch, our work can also serve as the first step towards making vectorial COM devices with various structured Polarizer θ lights, such as Bessel-Gauss beams, cylindrical beams or the Poincaré beams [1, 51, 52], where the optical polarization distribution is spatially inhomogeneous. Our work can also be extended to various COM (b) Cavity mode systems realized with e.g., cold atoms, magnomechanical devices, and TE Polarization optoelectrical circuits [53, 54]. θ vector Mechanical TM 2. VECTORIAL QUANTUM DYNAMICS mode κ As shown in Fig.1, we consider a polarized-light-driven optomechanical system, which consists of an optical polarizer and a Fabry-Pérot γm cavity with one movable mirror. Exploiting the polarization of photons rather than solely their intensity, has additional advantages for Δc controlling light-matter interactions. Here, to describe the polarization Pump field of an optical field, it is convenient to introduce a set of orthogonal ωm ωL ωc basis vectors, i.e., |~eli and |~e↔i, which correspond to the vertical (TE) and horizontal (TM) modes of the Fabry-Pérot cavity [55]. Therefore, Fig. 1. (a) Schematic diagram of a polarized-light-driven optome- an arbitrary linearly polarized light can be thought as a superposition chanical system, which consists of an optical polarizer and a Fabry- of these orthogonal patterns, i.e., whose unit vector is represented Pérot cavity with a movable mirror. (b) Frequency spectrogram of a by |~ei = cosθ |~eli + sinθ |~e↔i, with θ being the angle between the vector optomechanical system in panel (a), with θ being the angle polarization of the linearly polarized light and the vertical mode [see between the polarization of the linearly polarized light and the verti- Fig.1(b)]. In this situation, by adjusting the polarization angle θ, one cal mode. The orthogonal cavity modes with degenerate resonance can coherently manipulate the spatial distribution of a linearly po- frequency ωc and decay rate κ; the frequency of the mirror is ωm; larized optical field. In addition, because light could exert radiation the driving frequency is ωL, and ∆c = ωc − ωL is optical detuning pressure on the movable mirror, both spatial components of the lin- between the cavity mode and the driving field early polarized light would experience an optomechanical interaction. ˆ † † Then, in a rotating frame with respect to H0 = h¯ωL(aˆlaˆl + aˆ↔aˆ↔), the Hamiltonian of the polarized-light-driven optomechanical system where j =l,↔, γm = ωm/Qm is the mechanical damping rate with in is given by Qm the quality factor of the movable mirror; aˆ j are the zero-mean h¯ω input vacuum noise operators for the orthogonal cavity modes, and ˆ m 2 2 † † ˆ H = (pˆ + qˆ ) + h¯ ∑ ∆caˆ j aˆ j − g0aˆ j aˆ jqˆ + Hdr, they satisfy the following correlation functions [56] 2 j=l,↔ √ in,† in 0 ˆ † ∗ haˆ (t)aˆ j (t )i = 0, Hdr = ih¯ 2κ ∑ aˆ j S j − aˆ jS j , (1) j j=l,↔ in in,† 0 0 haˆ j (t)aˆ j (t )i = δ (t −t ). (3) † where aˆ j (aˆ j ) is the annihilation (creation) operator of the orthogonal cavity modes with degenerate resonance frequency ωc and decay rate Moreover, ξˆ denotes the Brownian noise operator for the mechanical κ; pˆ and qˆ are, respectively, the dimensionless momentum and position mode, resulting from the coupling of the mechanical mode with the cor- operators of the mirror with mass m and frequency ωm; ∆c = ωc −ωL is responding thermal environment. It satisfies the following correlation optical detuning between the cavity mode and the driving field; g0 is the function [57] single-photon optomechanical coupling coefficient for both orthogonal cavity modes. The frame rotating with driving frequency ω is obtained Z L ˆ ˆ 0 γm dω 0 h¯ω Uˆ iHˆ t h hξ (t)ξ (t )i = exp[−iω(t −t )] coth + 1 , by applying the unitary transformation√ = exp[ 0 /¯] (see, e.g., ωm 2π 2kBT 2 2 Ref. [36]). S = |Sl| + |S↔| = P/h¯ωL denotes the amplitude of (4) a linearly polarized driving field with an input laser power P, where S = Scos and S = Ssin are the projections of S onto the vertical l θ ↔ θ where kB is the Boltzmann constant and T is the environment temper- and horizontal modes, respectively. ature of the mechanical mode. The noise operator ξˆ(t) models the By considering the damping and the corresponding noise term of mechanical Brownian motion as, in general, a non-Markovian process. both optical and mechanical modes, the quantum Langevin equations However, in the limit of a high mechanical quality factor Qm 1, ξˆ(t) (QLEs) of motion describing the dynamics of this system are obtained can be faithfully considered as Markovian, and, then, its correlation as function is reduced to √ p in aˆ˙ j = (−i∆c + ig0qˆ − κ)aˆ j + 2κP/ωLh~e j|~ei + 2κaˆ , j 0 0 hξˆ(t)ξˆ(t )i ' γm(2nm + 1)δ (t −t ), (5) qˆ˙ = ωm pˆ, ˙ † ˆ −1 pˆ = −ωmqˆ − γm pˆ + g0 ∑ aˆ j aˆ j + ξ, (2) where nm = [exp[(h¯ωm/kBT )] − 1] is the mean thermal phonon j=l,↔ number. Research Article Photonics Research 3

Setting all the derivatives in QLEs (2) as zero leads to the steady- Using the solution of the steady-state QLEs, we can obtain the correla- state mean values of the optical and the mechanical modes tion matrix √ Z ∞ 2κ V = dτ M(τ)DMT (τ), (13) α j = S j ( j =l,↔), i∆ + κ 0 g0 2 2 where qs = |αl| + |α↔| , ps = 0, (6) ωm D=Diag[κ,κ,κ,κ,0,γm(2nm+1)], (14) where ∆ = ∆c − g0qs is the effective optical detuning. Under the 0 condition of intense optical driving, one can expand each opera- is the diffusion matrix, which is defined through hni(τ)n j(τ ) + 0 0 tor as a sum of its steady-state mean value and a small quantum n j(τ )ni(τ)i/2 = Di jδ (τ − τ ). When the stability condition is ful- fluctuation around it, i.e., aˆ j = α j + δaˆ j,qˆ = qs + δqˆ, pˆ = ps + δ pˆ. filled, the dynamics of the steady-state correlation matrix is determined Then, by defining the following vectors of quadrature fluctuations and by the Lyapunov equation [44]: T corresponding input noises : u(t) = (δXˆl,δYˆl,δXˆ↔,δYˆ↔,δqˆ,δ pˆ) , √ √ √ √ AV VAT −D ˆ in ˆ in ˆ in ˆ in ˆ T + = . (15) v(t) = ( 2κXl , 2κYl , 2κX↔, 2κY↔,0,ξ ) , with the components: As seen from Eq. (15), the Lyapunov equation is linear and can straight- 1 † i † forwardly be solved, thus allowing us to derive the correlation matrix δXˆ j = √ (δaˆ + δaˆ j), δYˆj = √ (δaˆ − δaˆ j), 2 j 2 j V for any values of the relevant parameters. However, the explicit form of V is complicated and is not be reported here. in 1 in† in in i in† in Xˆ j = √ (aˆ + aˆ j ), Yˆj = √ (aˆ − aˆ j ), (7) 2 j 2 j 3. SWITCHING OPTOMECHANICAL ENTANGLEMENT one can obtain a set of linearized QLEs, which can be written in a BY TUNING THE POLARIZATION OF VECTOR LIGHT compact form as To explore the polarization-controlled coherent switch of the steady- u˙(t) = Au(t) + v(t), (8) state COM entanglement, we adopt the logarithmic negativity, EN , for quantifying the bipartite distillable entanglement between different where degrees of freedom of our three-mode Gaussian state [59]. In the CV  y  case, EN can be defined as −κ ∆ 0 0 −Gl 0    x  EN = max [0,−ln(2ν˜−)], (16) −∆ −κ 0 0 Gl 0     y  where  0 0 −κ ∆ −G↔ 0  A =  . (9)  x  −1/2 2 1/2 1/2  0 0 −∆ −κ G↔ 0  ν˜− =2 Σ(Vbp) − Σ(Vbp) − 4det Vbp , (17)      0 0 0 0 0 ωm  with   x y x y Gl G G↔ G↔ −ωm −γm l Σ(Vbp) = det A + det B − 2det C. (18) The linearized√ QLEs indicate that the effective COM coupling rate x y Here ν˜− is the minimum symplectic eigenvalue of the partial transpose G j ≡ 2g0α j = G + iG , can be significantly enhanced by increas- j j of the reduced 4 × 4 CM Vbp. By tracing out the rows and columns of ing the intracavity photons. Moreover, the solution of the linearized the uninteresting mode in V, the reduced CM Vbp can be given in a QLEs (8) is given by 2 × 2 block form Z t   u(t) = M(t)u(0) + dτ M(τ)v(t − τ), (10) AC 0 Vbp =  . (19) CT B 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 Equation (16) indicates that the COM entanglement emerges only state, leading to M(∞) = 0 and when ν˜− < 1/2, which is equivalent to the Simon’s necessary and Z ∞ sufficient entanglement nonpositive partial transpose criterion (or the ui(∞) = dτ ∑Mik(τ)vk(t − τ). (11) related Peres-Horodecki criterion) for certifying bipartite CV distillable 0 k entanglement in Gaussian states [60]. Therefore, EN , quantifying The stability conditions can usually be derived by applying the Routh- the amount by which the Peres-Horodecki criterion is violated, is an Hurwitz criterion [58]. In our following numerical simulations, we efficient entanglement measure that is widely used when studying have confirmed that the chosen parameters in this paper can keep the bipartite entanglement in a multi-mode system. COM system in a stable regime. In Fig.2, the logarithmic negativity EN, j and the associated effec- Because of the linearized dynamics of the QLEs and the Gaussian tive COM coupling rate G j with the intracavity field are shown as nature of the input noises, the steady state of the quantum fluctua- a function of the optical detuning ∆c with respect to different polar- tions, independently of any initial conditions, can finally evolve into ization angle θ. Here, EN,l and EN,↔ are used to denote the case of a tripartite continuous variable (CV) Gaussian state, which is fully the COM entanglement with respect to the TE and TM modes, re- characterized by a 6 × 6 stationary correlation matrix (CM) V with the spectively. For ensuring the stability and experimental feasibility, the components parameters are moderately chosen as follows: m=50ng, λ =810nm, 7 ωm/2π =10MHz, g0/2π =68.5Hz, Qc =ωc/κ =4.94 × 10 , Qm = 5 ωm/γm =10 , T=400mK, and P=30mW. Note that Qc is typically V = u (∞)u (∞)+u (∞)u (∞) /2. (12) 5 10 5 6 i j i j j i 10 − 10 [36, 61], and Qm is typically 10 − 10 [62] in Fabry-Pérot Research Article Photonics Research 4

(a) 0.12 θ = 0 θ = π/8 θ = π/4 θ = 3π/8 θ = π/2 EN,↕ EN,↔ 0.08 EN,↕ EN,↔ EN EN,↕ 0.04 EN,↔ EN,↔ EN,↕ EN,↔ EN,↕ 0 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Δc / ωm Δc / ωm Δc / ωm Δc / ωm Δc / ωm

(b) (c) (d) | G↕ | m 0˚ Δc / ωm = 1 / ω 1.2 EN,↔ EN 0.1 0 0.7 θ m ω

/ 0.08 0.8 c

Δ 0 0.4

270˚ 90˚ EN,↕ 0.04 1.2 0.1

0.06 m ω

/ 0.8 c

Δ 1.2 2 0 0.4 Δ 0.8 c 1 EN,↕ / ω 0 1 2 (π) m 0.4 0 θ 180˚ EN,↔ θ (π)

Fig. 2. Polarization-controlled optomechanical entanglement switch using intracavity mode. (a) The logarithmic negativity EN versus the scaled optical detuning ∆c/ωm for θ = 0, θ = π/8, θ = π/4, θ = 3π/8, θ = π/2 for the TE and TM modes, COM entanglement always is generated around ∆c/ωm ' 1. (b) In polar coordinates, set ∆c/ωm = 1, the logarithmic negativity EN as a function of the phase for the TE and TM modes, COM entanglement for the TE and TM modes demonstrates a complementary distribution with the variation of the polarization angle θ. (c) EN as a function of the optical detuning for the TE and TM modes in a period of 0 to 2π, indicating that an adjustable COM entanglement conversion between the TE and TM modes would be implemented by coherent polarization control. (d) The effective COM coupling |G j| versus the optical detuning ∆c and the phase θ to the TE mode, EN,l achieves its maximum value for the optimal value of effective COM coupling rate |Gl|. See text for details of the other parameters. cavities. Using a whispering-gallery-mode COM system, much higher angle. In practical aspect, the ability to generate an adjustable entan- 12 value of Qc is achievable, reaching even up to 10 [63]. The polar- glement conversion between subsystems of a compound COM system ization angle θ, which can be tuned by rotating the orientations of would provide another degree of freedom for quantum optomechanical the optical polarizer, is usually used to control the spatial amplitude information processing. and phase of a linearly polarized driving field, with values in a period For practical applications, the COM entanglement with the intra- ranging from 0 to 2π [64]. In particular, θ = 0 (θ = π/2) corresponds cavity field is hard to be directly accessed and utilized. In order to to the case with a vertically (horizontally) polarized pump field applied verify the generated COM entanglement, an essential step is to perform to drive the TE (TM) optical mode. homodyne or heterodyne detections to the cavity output field, which al- low to measure the corresponding CM V out. Specifically, the quantum Specifically, as shown in Figs.2(a)–2(c), the COM entanglement is correlations in V out, that involve optical quadratures, can be directly ' present only within a finite interval of the values of ∆c around ∆c ωm, read out by homodyning the cavity output. However, accessing the me- resulting from the fact that the COM interaction in the red-detuned chanical quadratures typically requires to map the mechanical motion regime could significantly cool the mechanical mode around the light- to a weak probe field first, which then can be read out by applying a motion resonance and simultaneously entanglement survives only with similar homodyne procedure of the probe field. a tiny thermal noise occupancy. In addition, the logarithmic negativi- Now, by applying the standard input-output relations, the output ties EN,↔ and EN,l always demonstrate a complementary distribution with the variation of the polarization angle θ, indicating that an ad- field of this compound COM system is given by justable COM entanglement conversion between the TE and the TM √ modes would be implemented by a coherent polarization control. The out in a j (t) = 2δa j(t) − a j (t), (20) 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 where the optical output field has the same non-zero correlation func- out out 0 † 0 linearly polarized driving field [15]. Correspondingly, the strength of tion as the input field δa j(t), i.e., [a j (t), a j (t ) ] = δ (t −t ). As the effective COM coupling rate |G j| now also relies on polarization discussed in detail in Ref. [65], by selecting different time or frequency out angle θ. As shown in Figs.2(c) and2(d), it can be clearly seen that intervals from the continuous output field a j (t), one can extract a set EN,l achieves its maximum value for the optimal value of |Gl|. There- of independent optical modes by means of spectral filters. Here, for fore, the distribution of the COM entanglement with respect to the convenience, we consider the case where only a single output mode of TE and TM modes could be manipulated by tuning the polarization the TE and TM cavity field is detected. Therefore, in terms of a causal Research Article Photonics Research 5

(a) (b) Here, g˜(ω) is given by Ω / ωm = -1 EN,↕ 0 0.4 r ε = 10 τ sin[(ω − Ω)τ/2] EN,↕ 0˚ 2 g˜ j(ω) = exp[i(ω − Ω)τ/2] . (24) θ 2π (ω − Ω)τ/2 0.4 The COM entanglement for the cavity output mode is veriﬁed via the CM deﬁned as follows, i.e.,

0.2 π ) ( 270˚ 90˚ 1 0.2 1 θ Vout(t) = uout(t)uout(t) + uout(t)uout(t) , (25) ε = 1 kl 2 k l l k ε = 10 out out out out out T where u (t) = (δXl ,δYl ,δX↔ ,δY↔ ,δq,δ p) is the vector 0 form by the mechanical position and momentum ﬂuctuations and by 180˚ the (canonical) position quadrature, ε = 20 2 √ 0˚ out out out † Xj (t) = a1 j (t) + a1 j (t) / 2, (26) EN,↕ 0.4 θ and the momentum quadrature, 0.2 † √

π ) out out out

( Y (t) = a (t) − a (t) i 2, (27) 270˚ 1 0.2 j 1 j 1 j / 90˚ θ of the optical output modes. From the input-output relation in Eq. (20), ε = 20 one can obtain Z t 0 out 0 ui (t) = dsTi(t − s)[u(s) − v (s)], (28) 180˚ ε −∞ 2 = 30 where 0˚ EN,↕ θ 0 1 T v (t) = √ (Xl(t),Yl(t),X↔(t),Y↔(t),0,0) , (29) 2κ π ) ( 0.4 1 0.2 is analogous to the noise vector v(t) in Eq. (8) but without noise acting

0.2 θ 270˚ 90˚ on the mechanical mode. We have also introduced the matrix, x y  fl − fl 0 0 0 0  y x fl fl 0 0 0 0 0  x y  ε = 30 T (t) =  0 0 f↔ − f↔ 0 0 , (30) -1.3 -1 -0.7  0 0 f y f x 0 0   ↔ ↔  180˚ 0 0 0 0 δ (t) 0 Ω / ωm 0 0 0 0 0 δ (t) √ √ x x y y Fig. 3. Polarization-controlled optomechanical entanglement with where f j = 2κg j, f j = 2κg j. Using the Fourier transform and the the cavity output mode. (a) In polar coordinates, setting Ω/ωm = −1, correlation function of the noises, one can derive the following general EN as a function of the phase for the TE mode at different values expression for the stationary output correlation matrix, which is the of its inverse bandwidth ε. (b) The Logarithmic negativity EN as a counterpart of the intracavity relation of Eq. (13): function of the optical detuning and the phase at ε = 10,20,30 for the Z Pout Pout TE mode, demonstrating an optimal value of the polarization angle V out = dω T˜ (ω)M˜ (ω) + × D(ω)M˜ †(ω) + T˜ †(ω), 2κ 2κ for which the COM entanglement of the output field could achieve (31) its maximum value. See text for details of the parameters. where Pout = Diag[1,1,1,1,0,0] is the projector onto the two- dimensional space associated with the output quadratures, and filter function g(s), the output field can be rewritten as M˜ ext(ω) = (iω + A)−1, (32) Z t out out and a1 j (t) = dsg j(t − s)a j (s). (21)   −∞ κ 0 0 0 0 0   out   In the frequency domain, a1 j takes the following form 0 κ 0 0 0 0      Z ∞ √ 0 0 κ 0 0 0  out dt out out D(ω) =  , (33) a˜1 j (ω) = √ a1 j (t)exp(iωt) = 2πg˜ j(ω)a1 j (ω), (22)   −∞ 2π 0 0 0 κ 0 0      0 0 0 0 0 0  where g˜ j(ω) is the Fourier transform of the filter function. An explicit   example of an orthonormal set of the causal filter functions is given 0 0 0 0 0 Nm by [66] where Nm = (γmω/ωm)coth(h¯ω/2kBT ). A deeper understanding of out θ(t) − θ(t − τ) the general expression for V in Eq. (31) is obtained by multiplying g j(t) = √ exp(−iΩt), (23) the terms in the integral, one obtains τ Z out † † Pout x y x y V = dω T˜ (ω)M˜ (ω)D(ω)M˜ (ω)T˜ (ω) + and g j = g j + ig j, where g j (g j) is the real (imaginary) part of g j, 2 −1 Z θ(t) denotes the Heaviside step function, with Ω and τ being the † † + dω T˜ (ω) M˜ (ω)Rout + RoutM˜ (ω) T˜ (ω), (34) central frequency and the bandwidth of the causal filter, respectively. Research Article Photonics Research 6

(a) detection bandwidth, there is also an optimal value of the polarization m = Ω / ω -1 angle for which the COM entanglement of the output field achieves EN,↕ its maximum value [see Fig.3(b) for more details]. The physical 0.4 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 0.2 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- 2 0 maximally entangled qubits, a smaller number of pairs with a higher 2 degree of entanglement [68]. In this situation, it is possible that one 1 T 1 can further manipulate and optimize the polarization-controlled COM (K) entanglement at the output field, which can be useful for quantum (π) 0 0 θ communications [69, 70], quantum teleportation [71], and loophole- (b) free Bell test experiment [72]. θ = 0 Besides, as discussed in details in Ref. [67], the thermal phonon ex- EN,↕ citations could greatly affect the quantum correlations between photons 0.4 and phonons, and, thus, tend to destroy the generated COM entanglement. The mean thermal phonon number, which is monotonically in- creasing with the environment temperature, is assumed to be nm ' 833 0.2 in our former discussions. Here, by computing EN,l as a function of temperature T, as shown in Fig.4, we confirmed that the conversion of COM entanglement through polarization control could still exist 2 0 when considering larger mean thermal phonon numbers. To clearly 2 see this phenomenon, we plot the logarithmic EN,l of the CV bipartite T 1 system formed by the mechanical mode and the cavity output mode (K) 1 m centered around the Stokes sideband Ω = −ωm versus polarization / ω 0 0 Ω angle θ and temperature T in Fig.4(a) and set θ = 0. We also consider EN,l as a function of temperature T and the cavity frequency Ω in Fig. 4. The effect of thermal phonon excitations on the COM en- Fig.4(b). It is seen that for ε = 10, the polarization-controlled COM tanglement on output field. (a) Logarithmic negativity EN of the entanglement can still be observed at the cavity output field below a mechanical and TE output modes centered around Ω/ωm = −1 ver- critical temperature Tc ≈ 2K, corresponding to nm ' 4160. sus temperature and phase for ε = 10. (b) Logarithmic negativity EN Finally, we remark that based on experimentally feasible parameters, as a function of the center frequency Ω and temperature T for ε = 10 not only all bipartite entanglements but also the genuine tripartite at θ = 0. See text for details of the parameters. 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 [73, 74], can reach its maximum at where Rout = PoutD(ω)/κ = D(ω)Pout/κ, for simplicity, we have the optimal polarization angle θ = π/4. However, the bipartite purely defined [67] optical entanglement of TE and TM modes is typically very weak due to Z the indirect coupling of these two optical modes, and thus the resulting dω † Pout ≡ 2 T˜ (ω)PoutD(ω)PoutT˜ (ω). (35) tripartite entanglement is also weak in this system. Nevertheless, we 4κ2 note that the techniques of achieving stationary entanglement between Equation (34) clearly shows the quantum correlations of the quadrature two optical fields and even strong tripartite entanglement are already operators in the output field, where the first integral term stems from well available in current COM experiments [43, 47–50], and our work the intracavity COM interaction, the second term gives the contribution here provides a complementary way to achieve coherent switch of the the correlations of the noise operators, and the last term describes the COM entanglement via polarization control. interactions between the intracavity mode and the optical input field. Then, by numerically calculating the CM V out and the associated 4. CONCLUSION logarithmic negativity EN, j, one can detect and verify the generated intracavity COM entanglement at the cavity output. Here, as a specific In summary, we have proposed how to manipulate the light-motion example shown in Fig.3, we studied the COM entanglement between interaction in a COM resonator via polarization control, which enables the TE output mode and mechanical mirror. Also, as discussed in detail the ability of coherently switching COM entanglement in such a device. in Ref. [67], the generated intracavity COM entanglement is mostly We note that the ability to achieve coherent switch of COM entan- carried by the lower-frequency Stokes sideband of the output field. glement is useful for a wide range of entanglement-based quantum Therefore, it would be better to choose an input field with the center technologies, such as quantum information processing [29], quantum frequency at Ω = −ωm, which can usually be implemented by using routing [75], or quantum networking [76]. Also, our work reveals the the filter function of Eq. (23). Specifically, for ε = 1, as shown in potential of engineering various quantum effects by tuning the opti- Fig.3(a), the variation of COM entanglement with the polarization cal polarizations, such as mechanical squeezing [77], quantum state angle at the cavity output field is similar to that of its intracavity transfer [78], and asymmetric Einstein-Podolsky-Rosen steering [79]. counterpart. However, by optimizing the value of ε or, equivalently, Although we have considered here a specific case of linearly polarized the detection bandwidth τ, EN,l would achieve much higher value field whose spatial distribution is homogeneous, we can envision that than that of the intracavity field, implying a significant enhancement future developments with inhomogeneous vector beams can further of COM entanglement at the output field. In addition, for particular facilitate more appealing quantum COM techniques, such as opto- Research Article Photonics Research 7 rotational entanglement [80] or polarization-tuned topological energy angular momentum and optical torque in integrated photonic devices,” transfer [81]. In a broader view, our findings shed new lights on the Sci. Adv. 2, e1600485 (2016). marriage of vectorial control and quantum engineering, opening up 19. G. Xu, A. Nielsen, B. Garbin, L. Hill, G.-L. Oppo, J. Fatome, S. G. the way to control quantum COM states by utilizing synthetic optical Murdoch, S. Coen, and M. Erkintalo, “Spontaneous symmetry breaking materials [36, 82]. of dissipative optical solitons in a two-component Kerr resonator,” Nat. Commun 12, 4023 (2021). Funding. L.-M. K. is supported by the National Natural Sci- 20. Y. Kan, S. K. H. Andersen, F. Ding, S. Kumar, C. Zhao, and S. I. ence Foundation of China (NSFC, Grants No. 11775075 and No. Bozhevolnyi, “Metasurface-Enabled Generation of Circularly Polarized 11434011). H. J. is supported by the NSFC (Grants No. 11935006 and Single Photons,” Adv. Mater. 32, 1907832 (2020). No. 11774086) and the Science and Technology Innovation Program 21. S. Buddhiraju, A. Song, G. T. Papadakis, and S. Fan, “Nonreciprocal of Hunan Province (Grant No. 2020RC4047). A. M. was supported Metamaterial Obeying Time-Reversal Symmetry,” Phys. Rev. Lett. 124, by the Polish National Science Centre (NCN) under the Maestro Grant 257403 (2020). No. DEC-2019/34/A/ST2/00081. 22. L. Zhu, X. Liu, B. Sain, M. Wang, C. Schlickriede, Y. Tang, J. Deng, Disclosures. The authors declare no conflicts of interest. K. Li, J. Yang, M. Holynski, S. Zhang, T. Zentgraf, K. Bongs, Y.-H. Lien, and G. Li, “A dielectric metasurface optical chip for the generation of cold atoms,” Sci. Adv. 6, eabb6667 (2020). REFERENCES 23. V. D’Ambrosio, G. Carvacho, F. 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