Optomechanical interfaces for hybrid quantum networks

Recent advances on optical control of mechanical motion in an optomechanical resonator have stimulated strong interests in exploring quantum behaviors of otherwise classical, macroscopic mechanical systems and especially in exploiting mechanical degrees of freedom for applications in quantum information processing. In an optomechanical resonator, an optically- active mechanical mode can couple to any of the optical resonances supported by the resonator via radiation pressure. This unique property leads to a remarkable phenomenon: mechanically-mediated conversion of optical fields between vastly different wavelengths. The resulting optomechanical interfaces can play a special role in a hybrid quantum network, enabling quantum communication between disparate quantum systems. In this review, we introduce the basic concepts of optomechanical interactions and discuss recent theoretical and experimental progresses in this field. A particular emphasis is on taking advantage of mechanical degrees of freedom, while avoiding detrimental effects of thermal mechanical motion.


INTRODUCTION
In a quantum network, local quantum nodes consisting of matter qubits generate, process, and store quantum information. An optical network couples together distant quantum nodes, transferring or distributing quantum information among these nodes via photons [1,2]. Such a network can play a major role in quantum communication and quantum computing and can also serve as a platform for exploring and understanding quantum many body interactions. A variety of quantum systems including both atomic and solid state systems, such as trapped ions, superconducting circuits, and spins in diamond or silicon, have emerged as promising candidates for matter qubits [3][4][5][6]. The unique properties of these quantum systems make them suitable for specific quantum operations [7][8][9]. For example, spins in diamond can serve as long-lived quantum memories, while superconducting circuits can enable rapid information processing. A formidable challenge for developing a hybrid quantum network that can incorporate and take advantage of disparate quantum systems is to enable quantum communication between these systems.
An effective approach for mediating interactions between different types of quantum systems emerged recently from a seemingly unrelated field, cavity optomechanics, which explores the interactions between the circulating optical fields and the mechanical motion in an optomechanical resonator [10][11][12]. The optomechanical interactions take place via either the radiation pressure force induced by the optical fields or processes such as Brillouin scattering. These interactions can lead to quantum state transfer between relevant optical and mechanical systems and can also generate quantum entanglement between these systems [10,11].
In this review, we focus on recent theoretical and experimental studies on the state transfer between optical and mechanical modes and on the mechanically-mediated state transfer between optical fields. These advances lay the ground work for using optomechanical interfaces to enable quantum communication in a hybrid quantum network. We introduce in Section 2 the basic optomechanical processes and outline in Section 3 two theoretical proposals for overcoming thermal mechanical noise that can severely limit the fidelity of mechanicallymediated quantum processes. Section 4 describes recent experimental progresses using a silica microsphere as a model optomechanical resonator. Future directions and opportunities are briefly discussed in Section 5.

BASIC OPTOMECHANICAL PROCESSES
We use a Fabry-Perot optical cavity with a moving mirror as a generic optomechanical resonator to introduce basic optomechanical processes. As illustrated in Fig. 2a, radiation pressure of the circulating optical field in the cavity exerts a force on the moving mirror. The displacement of the mirror in turn induces a frequency shift of the optical cavity mode, with ω c (x) = ω 0 + g x, where x is the mechanical displacement and g = d ω c /d x is the optomechanical coupling coefficient. The effective optomechanical coupling rate is then given by G = g x zp f √ n c 512 Natl Sci Rev, 2015, Vol. 2, No. 4 Figure 3. Mechanically-mediated state transfer between two optical modes. (a) Schematic of two optical modes coupling to a mechanical oscillator via respective red sideband coupling. In analogy to a -type three-level system, a mechanically-dark super optical mode can form in this three mode system. (b) Three schemes for the mechanically-mediated optical state transfer: double swap, adiabatic passage of dark mode, and hybrid schemes [94]. Top panel: the time-dependence of the two optomechanical coupling rates in each scheme. Lower panel: schematics illustrating how the transfer works in each scheme. Here the blue, red, and green dots represent (respectively) cavity 1, the mechanical oscillator, and cavity 2; the wave packet in cavity 1 represents the state to be transferred. Arrows indicate the optomechanical couplings, with the magnitude indicating their magnitude.

REVIEW
where n c is the intracavity photon number of the optical field and x zp f is the zero-point fluctuation of the mechanical mode. This type of optomechanical coupling can induce an inter-conversion or swap between optical and mechanical excitations and can also generate entanglement between the optical and mechanical modes, thus enabling a variety of coherent or quantum optical processes. For a more detailed understanding of these dynamical processes, we assume that the mechanical frequency, ω m , is large compared with both the cavity decay rate κ and the mechanical damping rate γ m (this is the socalled resolved sideband regime) and that the optomechanical coupling is driven by a classical laser field at the red or blue sideband of the cavity resonance, i.e., with the laser at ω L = ω 0 ± ω m .

Red sideband coupling
With ω L = ω 0 − ω m , the driving laser can convert the mechanical vibration to an optical signal field at the cavity resonance via anti-Stokes scattering, i.e., converting a phonon at ω m to a photon at ω 0 (see Fig. 2b). In the limit that κ > G > γ m , the phonon-to-photon conversion effectively damps the mechanical motion, with a radiationpressure-induced damping rate given by 4G 2 /κ. This radiation-pressure-induced damping has been used with remarkable success for laser cooling of the mechanical motion. The interaction Hamiltonian for the red sideband coupling in the limit of a weak signal field is given by H int = G (â +b +âb + ), wherê b andâ are respectively the annihilation operator for the mechanical displacement and the optical signal field at or near ω 0 [10]. This linearized or meanfield Hamiltonian has the form of the well-known beam-splitter Hamiltonian in quantum optics. Under suitable conditions, the red sideband coupling can map or swap quantum states between the optical mode and the mechanical oscillator [14,[75][76][77]. Mechanically-mediated state transfer between two optical modes can also take place, if the two optical modes couple to the same mechanical oscillator through respective red sideband coupling (see Fig. 3a). In the steady state and with no internal optical loss, the photon conversion efficiency is given by χ = 4C 1 C 2 /(1 + C 1 + C 2 ) 2 (for a detailed derivation, see for example, the supplement in Ref. [61]), where C i = 4G 2 i /γ m κ i (with i = 1, 2) is the optomechanical cooperativity, a dimensionless parameter that characterizes the strength of the optomechanical coupling [15,17,18,61]. In this limit, near unity optical mode conversion can be achieved with large and equal cooperativity for the two optomechanical coupling processes. This condition has also been termed optomechanical impedance matching [15].

Blue sideband coupling
With ω L = ω 0 + ω m , the driving laser can lead to a parametric-down-conversion-like process, generating a photon-phonon pair with a photon at ω 0 and a phonon at ω m via Stokes scattering (see Fig. 2c). This process leads to amplification of the mechanical vibration and serves as a mechanism for generating self-sustained mechanical oscillations [35,36,78]. The pair generation process can also induce entanglement between the optical and mechanical modes [79][80][81]. The interaction Hamiltonian for the blue sideband coupling in the limit of weak signal field is given by H int = G (â +b + +âb), which has the form of the two-mode squeezing Hamiltonian [10]. Dong et al. 513 Entanglement between two optical modes can be generated, if optical mode 1 couples to the mechanical oscillator via blue sideband coupling, while optical mode 2 couples to the mechanical oscillator via red sideband coupling [82][83][84][85][86][87][88]. On a heuristic level, the entanglement of the two optical modes can be viewed as a two-step process: the two-mode squeezing process generates a correlated phonon-photon pair and the beam-splitter process swaps the phonon state to mode 2, generating a correlated photonphoton pair of the two optical modes. In addition, optically-mediated entanglement of mechanical systems and the control of an optomechanical system with squeezed optical fields have also been proposed [81,[89][90][91][92].

OVERCOMING THERMAL MECHANICAL MOTION
A technical challenge for using mechanical degrees of freedom in applications such as quantum information processing is to overcome the detrimental effects of thermal motion inherent in a mechanical system. For the mechanically-mediated quantum state transfer between optical fields discussed in Section 2, a conceptually straightforward approach is a double swap, converting the optical field in mode 1 to a mechanical excitation, followed by the conversion of the mechanical excitation to an optical field in mode 2 (see Fig. 3b) [14]. This double swap process is strongly affected by thermal phonons. A brute force approach to avoid this complication is to cool the mechanical system to its motional ground state. Two different, but closely related, approaches proposed recently can, however, enable mechanicallymediated quantum state transfer even in the presence of thermal phonons. The first approach is based on the adiabatic passage of a dark mode and is analogous to the Stimulated Raman adiabatic passage (STIRAP) of dark states in an atomic system [17,18,93]. The second approach returns the mechanical system to its initial state after the completion of the relevant quantum operation [94], which resembles the Sorensen-Molmer mechanism for entanglement of trapped ions in a thermal environment [95,96]. Both approaches aim to take advantage of mechanical degrees of freedom, while avoiding the detrimental effects of thermal mechanical motion.

Optomechanical dark mode
For the three-mode optomechanical system in Fig. 3a, two optical modes couple to the same mechanical mode, where the optomechanical coupling is driven by two strong laser fields with effective coupling rates, G 1 and G 2 , and with frequencies that are each ω m below the respective optical cavity resonance. In analogy to bright and dark atomic superposition states in a -type three-level atomic system (see Fig. 3a), we define bright and dark optical "super" modes [17,18,93]. The bright mode, described byâ couples to the mechanical mode with an effective rateG = G 2 1 + G 2 2 , whereâ 1 andâ 2 are the annihilation operators for the signal fields in the two respective optical modes. In comparison, the dark mode, described bŷ is decoupled from the mechanical mode due to a cancellation in the relevant optomechanical coupling. The effective interaction Hamiltonian for the three-mode system can now be written as H int = G (â + Bb +â Bb + ), leading to the formation of coupled optical-mechanical modes or normal modes. Optical state transfer from mode 1 to mode 2 can be performed adiabatically via the dark mode, without exciting the mechanical mode. In a "counterintuitive" pulse sequence (see Fig. 3b), we modulate the coupling rates, G 1 and G 2 , in time, such that mode 1 is the dark mode at t = 0 and mode 2 is the dark mode at the end of the protocol. For the adiabatic state transfer, which means no excitation of the mechanical oscillator, the transfer rate needs to be small compared with the energetic gap (of orderG ) separating the dark and bright modes. Note that the dark mode is only completely immune from thermal mechanical noise when the optical cavities have the same decay rate (symmetric dissipation) since otherwise, the optical damping process can mix the dark and bright super modes. Nevertheless, the dark mode approach can remain highly effective even when there are deviations from the symmetry condition.
The transfer of a Gaussian state (a pure state, whose Wigner function is Gaussian) between two optical modes can be characterized with a simple analytic expression for the Ulhmann fidelity [94], F = [1/(1 +n h )] exp[−λ 2 /(1 +n h )], which depends on just two positive-definite parametersn h and λ [17].n h can be regarded as an effective number of thermal quanta, which quantifies the heating of the state during the transfer protocol. This heating can arise from noise emanating from the cavity and mechanical dissipative baths, as well as from any initial thermal population in the REVIEW mechanical oscillator. In comparison, λ characterizes the deleterious effects of amplitude decay during the state transfer protocol. Adiabatic passage via the dark-mode can lead to dramatically suppressed n h . It should be noted that for realistic experimental parameters, perfect adiabatic limit can never be attained [94]. As a result, the fidelity of the optical state transfer can be highly sensitive to the initial thermal population in the mechanical oscillator. In this regard, although high-fidelity optical state transfer can take place in a thermal environment via adiabatic passage of the dark mode, precooling of the mechanical oscillator to its ground state is still needed.

Sorensen-Molmer mechanism
Another approach to circumvent effects of thermal mechanical motion is to return the mechanical oscillator to its initial state upon the completion of the state transfer protocol, thus disentangling the mechanical oscillator from the rest of the system. In limit of G 1 = G 2 = G and with equal cavity decay rates, optical state transfer can be achieved with the mechanical system returned to its initial state in a duration of π/( √ 2G ) (see Fig. 3b) [94]. This approach, which was initially termed the "hybrid" scheme, resembles a mechanism that Sorensen and Molmer proposed in their earlier work to entangle trapped ions in a thermal environment [95,97]. With comparable peak coupling rates, the hybrid scheme is faster than the adiabatic passage scheme and can thus perform better in terms of avoiding degradation due to cavity damping. Furthermore, the Sorensen-Molmer mechanism can be completely immune to initial thermal phonon population in the mechanical oscillator, in contrast to the adiabatic passage of the dark mode discussed above [94].
In order to overcome the thermal mechanical noise, the two transient approaches discussed above require that the peak cooperativity should exceed the number of thermal phonons contributed by the bath. Under steady state operations, the large cooperativity can lead to radiation pressure cooling of the mechanical oscillator close to its motional ground state. However, transient approaches, such as the Sorensen-Molmer mechanism, do not involve or require ground state cooling. The use of transient operations also avoids or minimizes optically-induced heating inherent in most optomechanical systems, which is crucial for the experimental realization of the desired quantum operations. Both the Sorensen-Molmer mechanism and the dark mode process have also been successfully exploited for the generation of mechanically-mediated optical entanglement [83][84][85][86][87].

EXPERIMENTAL PROGRESSES
In this section, we highlight the experimental realization of mechanically-mediated wavelength conversion as well as coherent inter-conversion between optical and mechanical excitations at room temperature, using a silica microsphere with a diameter near 30 μm as a simple model system for a multimode optomechanical resonator [98]. Similar studies have also been carried out in optomechanical resonators such as silica toroids, SiN microdisks, and optomechanical crystals [53,62,63]. Coherent conversion between optical and microwave fields, which is important for coupling superconducting circuits to optical systems, has also been demonstrated recently [29,30]. State transfer between mechanical and microwave excitations has been realized in a quantum regime, with the mechanical system near its motional ground state [99]. Optical wavelength conversion at the single-photon level via traditional nonlinear optical processes has also been demonstrated [100][101][102]. In related studies, coupling of two mechanical modes to a common optical mode has led to the demonstration of optically-mediated mechanical normal modes and Bogoliubov mechanical modes [103][104][105].
Surface acoustic waves (SAWs) or propagating phonon modes have also been used for optomechanical studies. SAWs have been coupled to superconducting qubits [106]. Brillouin cooling of SAWs has been demonstrated [107]. Cavity optomechanics of SAWs in fluidic devices with potential for biological applications has been reported [108]. Nonreciprocal optomechanical storage using SAWs as well as Brillouin scattering induced transparency of SAWs has also been realized in silica microspheres [109,110]. In addition, parity-time symmetric processes have also been exploited for optomechanical processes as well as for non-reciprocal propagation of light [111][112][113][114].

Optomechanical light storage
In a silica microsphere, high-finesse optical whispering gallery modes (WGMs) form via total internal reflection along the curved surface. The WGMs can couple to the radial-breathing mechanical vibration of the sphere through radiation pressure force along the radial direction, as illustrated in an inset in Fig. 4 [98]. The coherent inter-conversion between an excitation in an optical WGM and that in a mechanical breathing mode can manifest as the storage of an optical pulse in the mechanical oscillator. For the storage process, we used a writing pulse, at the red sideband of the WGM resonance (with λ∼ 800 nm), to convert a signal pulse at the WGM resonance to a mechanical oscillation (with ω m = 161 MHz). For the retrieval process, we used a readout pulse, at the same frequency as the writing pulse, to convert the mechanical excitation back to an optical pulse, as demonstrated by the coherent heterodyne beating between the readout and the retrieved pulse shown in Fig. 4a [59,60]. The storage lifetime is determined by the mechanical damping time, which is of order several μs for a breathing mode in the silica microsphere.
The optomechanical light storage process is closely related to the process of OMIT [54][55][56][57][58]. For OMIT, a single driving pulse serves as both the writing and readout pulses. In this case, the destruc-tive interference between the input signal field and the retrieved optical field can prevent the excitation of the optical cavity. The OMIT resembles the well-known process of electromagnetically-induced transparency (EIT) [115], studied extensively in a variety of atomic and solid state systems. Optical analogue of EIT and especially its relation to the Autler-Townes splitting have also been carefully examined in a recent study [116]. Figure 4b compares the spectral responses of OMIT and optomechanical light storage obtained under similar experimental conditions, demonstrating the close correspondence between the transparency dip of the OMIT and the anti-Stokes resonance of the optomechanical light storage [60].

Optical wavelength conversion via a dark mode
The coherent inter-conversion between the optical and mechanical excitations discussed above can enable the mechanically-mediated state transfer between two optical modes. By coupling two optical WGMs in a silica microsphere to one of its mechanical breathing modes as shown in Fig. 3a, earlier studies have demonstrated optical wavelength conversion via the mechanically-dark super optical mode in a classical regime. These studies highlight two key signatures of the dark mode excitation [61]. First of all, these is no OMIT for the dark mode since this mode is decoupled from the mechanical oscillator. Secondly, as a superposition of the two individual optical modes, the excitation of the dark mode naturally leads to the coherent conversion of excitations between the two optical modes. As shown by the spectral responses plotted in Fig. 5, with increasing cooperativity, C 2 , for mode 2 and with the cooperativity for mode 1 fixed at C 1 = 1.4, a diminishing OMIT contribution is accompanied by an increasing conversion of the input signal field from mode 1 to mode 2. The wavelength or mode conversion process is both coherent and bidirectional [65]. The maximum photon conversion efficiency achieved is near 10%, limited by the relatively small cooperativity used and also by the internal optical loss in the experiments [61,65].
Theoretically, the dark-to-bright population ratio is given by (G 2 /G 1 ) 2 [1 + C 2 + C 1 (κ 1 /κ 2 )] 2 [61]. Even with the modest cooperativity used in [61] (C 1 = 1.4 and C 2 = 3.5), the dark mode fraction in the experiment already reaches 99%. Additional experiments further confirm that as expected, an increasing excitation of the dark mode leads to the suppression of the mechanical excitation. Overall, these proof-ofprinciple experiments demonstrate the special role 516 Natl Sci Rev, 2015, Vol. 2, No. 4 REVIEW Figure 5. Optical wavelength conversion via a dark mode [61]. The left and right panels show respectively steady-state optical emissions from mode 1 (λ ∼ 637 nm) and mode 2 (λ ∼ 800 nm), obtained as a function of the detuning between an input signal field near the mode 1 resonance and a driving field fixed at the red sideband of mode 1. The emission power is normalized to that obtained at the mode 1 resonance with C 1 = C 2 = 0. Solid curves are the theoretically calculated responses. At relatively high C 2 , the dip in the spectral response of mode 1 corresponds to optical mode conversion instead of OMIT.
of the dark mode in the optical wavelength conversion process, opening the door to mechanicallymediated optical state transfer in a quantum regime in a thermal environment.

CONCLUSION
Mechanical or mechanics-related coupling has emerged as a promising platform for interfacing disparate quantum systems. Considerable experimental progresses have already been made in optomechanical systems. Mechanically-mediated optical wavelength conversion, including that between optical and microwave fields, have been successfully demonstrated in a classical regime. With further advance in the design and engineering of optomechanical resonators, near unity conversion efficiency is achievable. Furthermore, transient schemes such as those that incorporate the Sorensen-Molmer mechanism can provide an avenue to realize quantum state transfer at an elevated temperature without precooling the mechanical system to its motional ground state, which is highly desirable for practical optomechanical interfaces. The mechanical degrees of freedom, which have often been overlooked in modern physics, are attracting ample attentions in the pursuit of quantum information processing.