Optomechanical interfaces for hybrid quantum networks

Abstract

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.

cavity optomechanics, radiation pressure, quantum networks, quantum optics, quantum state transfer

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 [3–6]. The unique properties of these quantum systems make them suitable for specific quantum operations [7–9]. 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 [10–12]. 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].

Mechanical degrees of freedom, which have often been overlooked in studies of quantum systems, can play a special role in interfacing disparate quantum systems. The ubiquitous nature of mechanical motion can enable a macroscopic mechanical oscillator to couple to nearly any type of quantum systems, including charge, spin, atomic, and superconducting qubits, as well as to photons at nearly any wavelength [13–28]. As an example, Fig. 1A shows schematically an optomechanical resonator, in which a mechanical oscillator couples simultaneously to an optical cavity and a microwave cavity. In this case, the mechanical oscillator can mediate a quantum state transfer between the optical and microwave fields [29,30]. Figure 1B illustrates an optomechanical transducer, where a spin and a charge qubit couple to respective mechanical oscillators and optomechanical processes including state transfer between the optical and mechanical excitations mediate indirect qubit-light coupling [13].

Figure 1.

Examples of optomechanical interfaces. (a) A mechanical oscillator (red bar in the middle) couples simultaneously to an optical cavity (the Fabre-Perot cavity in blue) and a microwave cavity (the LC circuit in green). (b) A quantum network where spin or charge based qubits and photons are coupled by an optomechanical transducer (adapted with permission from [13]).

Cavity optomechanics was motivated more than thirty years ago in the context of gravitational wave detectors [31–33]. Tremendous progresses have been made since the early experimental demonstration of radiation-pressure induced optical bistability in a Fabry-Perot cavity [34]. The experimental observation of radiation-pressure-induced coherent mechanical oscillations in whispering-gallery optical microresonators about a decade ago [35,36] ushered in intense research efforts on the development of new types of optomechanical resonators such as optomechanical crystals and membrane-in-the-middle or membrane-in-the-evanescent-field cavities [37–39]. Radiation pressure cooling of mechanical oscillators to their motional ground states has been considered theoretically [40–46] and realized experimentally in a number of optomechanical or electromechanical resonators [47–51]. Optomechanical resonators have been used to pursue a variety of coherent optical processes, such as strong coupling between an optical and a mechanical mode [52,53], optomechanically-induced transparency (OMIT) [54–58], optomechanical light storage [59,60], and mechanically-mediated optical wavelength conversion [29,30, 61–65]. Quantum processes such as ponderomotive squeezing and the entanglement between a mechanical oscillator and a microwave field have been demonstrated [66–69]. Optomechanical processes at the single photon level such as photon blockade have also been explored theoretically [70–74].

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 mechanically-mediated 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) = ω₀ + gx, where x is the mechanical displacement and |$g = {{d\omega _c} / {dx}}$| is the optomechanical coupling coefficient. The effective optomechanical coupling rate is then given by |$G = gx_{zpf} \sqrt {n_c }$| where n_c is the intracavity photon number of the optical field and x_zpf 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 so-called 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 = ω₀ ± ω_m.

Figure 2.

Basic processes for cavity optomechanics. (a) Schematic of an optomechanical resonator, in which the circulating optical field in the cavity couples to the mechanical motion of a mirror via radiation pressure. (b) Optomechanical coupling driven by a laser field at the red sideband leads to anti-Stokes scattering, converting a phonon to a photon at the cavity resonance. (c) Optomechanical coupling driven by a laser field at the blue sideband leads to Stokes scattering, generating a photon-phonon pair.

Red sideband coupling

With ω_L = ω₀ − 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 ω₀ (see Fig. 2B). In the limit that κ > G > γ_m, the phonon-to-photon conversion effectively damps the mechanical motion, with a radiation-pressure-induced damping rate given by 4G²/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 |$H_{{\mathop{\rm int}} } = \hbar G(\hat a^ + \hskip-2pt\hat{\hskip2pt b} + \hat a\hskip-2pt\hat{\hskip2pt b}^ + )$|⁠, where |$\hskip-2pt\hat{\hskip2pt b}$| and |$\hat a$| are respectively the annihilation operator for the mechanical displacement and the optical signal field at or near ω₀ [10]. This linearized or mean-field 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 [14,75–77]. 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 = 4C₁C₂/(1 + C₁ + C₂)² (for a detailed derivation, see for example, the supplement in Ref. [61]), where |$C_i = 4G_i^2 /\gamma _m \kappa _i$| (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].

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 Λ-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.

Blue sideband coupling

With ω_L = ω₀ + ω_m, the driving laser can lead to a parametric-down-conversion-like process, generating a photon-phonon pair with a photon at ω₀ 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 [79–81]. The interaction Hamiltonian for the blue sideband coupling in the limit of weak signal field is given by |$H_{{\mathop{\rm int}} } = \hbar G(\hat a^ + \hskip-2pt\hat{\hskip2pt b}^ + + \hat a\hskip-2pt\hat{\hskip2pt b})$|⁠, which has the form of the two-mode squeezing Hamiltonian [10].

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 [82–88]. 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 photon-photon 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, 89–92].

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 mechanically-mediated 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, G₁ and G₂, 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 Λ-type three-level atomic system (see Fig. 3A), we define bright and dark optical “super” modes [17,18,93]. The bright mode, described by

\begin{equation} \hat a_B = (G_1 \hat a_1 + G_2 \hat a_2 )/\tilde G, \end{equation}

(1)

couples to the mechanical mode with an effective rate |$\tilde G = \sqrt {G_1^2 + G_2^2 }$|⁠, where |$\hat a_1$| and |$\hat a_2$| are the annihilation operators for the signal fields in the two respective optical modes. In comparison, the dark mode, described by

\begin{equation} \hat a_D = (G_2 \hat a_1 - G_1 \hat a_2 )/\tilde G, \end{equation}

(2)

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 |$H_{{\mathop{\rm int}} } = \hbar \tilde G(\hat a_B^ + \hskip-2pt\hat{\hskip2pt b} + \hat a_B \hskip-2pt\hat{\hskip2pt b}^ + )$|⁠, 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 “counter-intuitive” pulse sequence (see Fig. 3B), we modulate the coupling rates, G₁ and G₂, 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 order |$\tilde G$|⁠) 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], |$F = [1/(1 + \bar n_h )]\exp [ - \lambda ^2 /(1 + \bar n_h )]$|⁠, which depends on just two positive-definite parameters |$\bar n_h $| and λ [17]. |$\bar n_h $| can be regarded as an effective number of thermal quanta, which quantifies the heating of the state during the transfer protocol. This heating can arise from noise emanating from the cavity and mechanical dissipative baths, as well as from any initial thermal population in the mechanical oscillator. In comparison, λ characterizes the deleterious effects of amplitude decay during the state transfer protocol. Adiabatic passage via the dark-mode can lead to dramatically suppressed |$\bar 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 G₁ = G₂ = 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 |$\pi /(\sqrt 2 G)$| (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 [83–87].

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 [100–102]. 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 [103–105].

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]. Non-reciprocal 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 [111–114].

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.

Figure 4.

Optomechanical light storage [60]. (a) Coherent heterodyne beating between the retrieved and the readout pulse, along with a schematic of the pulse sequence. (b) Spectral response of OMIT (top trace) and light storage (bottom trace). Squares (circles) in the top (bottom) trace are the emission power of the signal (retrieved) pulse as a function of the detuning between the signal field near the WGM resonance and the driving field fixed at the red sideband. The inset illustrates the coupling between a WGM and a radial breathing mode via radiation pressure force. Solid curves are the theoretically calculated responses.

The optomechanical light storage process is closely related to the process of OMIT [54–58]. For OMIT, a single driving pulse serves as both the writing and readout pulses. In this case, the destructive 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, C₂, for mode 2 and with the cooperativity for mode 1 fixed at C₁ = 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].

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 C₁ = C₂ = 0. Solid curves are the theoretically calculated responses. At relatively high C₂, the dip in the spectral response of mode 1 corresponds to optical mode conversion instead of OMIT.

Theoretically, the dark-to-bright population ratio is given by (G₂/G₁)²[1 + C₂ + C₁(κ₁/κ₂)]² [61]. Even with the modest cooperativity used in [61] (C₁ = 1.4 and C₂ = 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-of-principle experiments demonstrate the special role of the dark mode in the optical wavelength conversion process, opening the door to mechanically-mediated 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.

C.H.D. acknowledge financial support from the Strategic Priority Research Program (B) of the Chinese Academy of Sciences (XDB01030200), National Basic Research Program of China (2011CB921200 and 2011CBA00200) and the National Natural Science Foundation of China (61308079), Anhui Provincial Natural Science Foundation (1508085QA08), the Fundamental Research Funds for the Central Universities. Y.D.W. acknowledges support from the Chinese Youth 1000 Talents Program and a grant from NSFC (11434011). HW acknowledges support from US NSF and the DARPA ORCHID program. He also thanks University of Science and Technology of China for the hospitability during his sabbatical leave.