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 coher- ent 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 spa- tially 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 optomechan- ical system, which consists of an optical polarizer and a Fabry-Pérot γm cavity with one movable mirror. Exploiting the polarization of pho- tons 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., j~eli and j~e$i, which correspond to the vertical (TE) and horizontal (TM) modes of the Fabry-Pérot cavity [55]. Therefore, Fig. 1. (a) Schematic diagram of a polarized-light-driven optome- an arbitrary linearly polarized light can be thought as a superposition chanical system, which consists of an optical polarizer and a Fabry- of these orthogonal patterns, i.e., whose unit vector is represented Pérot cavity with a movable mirror. (b) Frequency spectrogram of a by j~ei = cosq j~eli + sinq j~e$i, with q being the angle between the vector optomechanical system in panel (a), with q 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 q, one cal mode. The orthogonal cavity modes with degenerate resonance can coherently manipulate the spatial distribution of a linearly po- frequency wc and decay rate k; the frequency of the mirror is wm; larized optical field. In addition, because light could exert radiation the driving frequency is wL, and Dc = wc − wL 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¯wL(aˆlaˆl + aˆ$aˆ$), the Hamiltonian of the polarized-light-driven optomechanical system where j =l,$, gm = wm/Qm is the mechanical damping rate with in is given by Qm the quality factor of the movable mirror; aˆ j are the zero-mean h¯w input vacuum noise operators for the orthogonal cavity modes, and ˆ m 2 2 † † ˆ H = (pˆ + qˆ ) + h¯ ∑ Dcaˆ j aˆ j − g0aˆ j aˆ jqˆ + Hdr, they satisfy the following correlation functions [56] 2 j=l,$ p in,† in 0 ˆ † ∗ haˆ (t)aˆ j (t )i = 0, Hdr = ih¯ 2k ∑ aˆ j S j − aˆ jS j , (1) j j=l,$ in in,† 0 0 haˆ j (t)aˆ j (t )i = d (t −t ). (3) † where aˆ j (aˆ j ) is the annihilation (creation) operator of the orthogonal cavity modes with degenerate resonance frequency wc and decay rate Moreover, xˆ denotes the Brownian noise operator for the mechanical k; 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 wm; Dc = wc −wL 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 w is obtained Z L ˆ ˆ 0 gm dw 0 h¯w Uˆ iHˆ t h hx (t)x (t )i = exp[−iw(t −t )] coth + 1 , by applying the unitary transformationp = exp[ 0 /¯] (see, e.g., wm 2p 2kBT 2 2 Ref. [36]). S = jSlj + jS$j = P/h¯wL 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 q $ q where kB is the Boltzmann constant and T is the environment temper- and horizontal modes, respectively. ature of the mechanical mode. The noise operator xˆ(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, xˆ(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 p in aˆ˙ j = (−iDc + ig0qˆ − k)aˆ j + 2kP/wLh~e jj~ei + 2kaˆ , j 0 0 hxˆ(t)xˆ(t )i ' gm(2nm + 1)d (t −t ), (5) qˆ˙ = wm pˆ, ˙ † ˆ −1 pˆ = −wmqˆ − gm pˆ + g0 ∑ aˆ j aˆ j + x, (2) where nm = [exp[(h¯wm/kBT )] − 1] is the mean thermal phonon j=l,$ number.