Optomechanical circuits for nanomechanical continuous variable quantum state processing

The field of cavity optomechanics studies the interaction between light and mechanical motion, with promising prospects in fundamental tests of quantum physics, ultrasensitive detection and applications in quantum information processing (see [1–3] for reviews). One particularly promising platform consists of 'optomechanical crystals', in which strongly localized optical and vibrational modes are implemented in a photonic crystal structure [4]. So far, several interesting possibilities have been pointed out that would make use of multi-mode setups designed on this basis. For example, suitably engineered setups may coherently convert phonons to photons [5], and collective nonlinear dynamics might be observed in optomechanical arrays [6]. Moreover, optomechanical systems, in general, have been demonstrated to furnish the basic ingredients for writing quantum information from the light field into the long-lived mechanical modes [7–9]. The recent success in ground state laser cooling [10, 11] has opened the door to coherent quantum dynamics in optomechanical systems.

In this paper, we propose a general scheme for continuous-variable quantum state processing [12] utilizing the vibrational modes of such structures. We show how entanglement and state-transfer operations can be applied selectively to pairs of modes, by suitable intensity modulation of a single incoming laser beam. We discuss limitations for entanglement generation and transfer fidelity, and show how to pick suitable designs to address these challenges.

We will first restrict our attention to a single optical mode coupled to many mechanical modes, such that the following standard optomechanical Hamiltonian describes the photon field $\skew4\hat {a}$, the phonons $\skew4\hat {b}_{l}$ of different localized vibrational modes (l = 1,2,...,N), their mutual coupling and the laser drive:

$$\begin{equation} \skew4\hat{H}=-\hbar\Delta\skew4\hat{a}^{\dagger}\skew4\hat{a}+\sum_{l}\hbar\Omega_{l}\hat{b}^{\dagger}_{l}\hat{b}_{l}-\hbar\skew4\hat{a}^{\dagger}\skew4\hat{a}\sum_{l}g_{0}^{(l)}(\hat{b}^{\dagger}_{l}+\hat{b}_{l})+\hbar\alpha_{L}(\skew4\hat{a}+\skew4\hat{a}^{\dagger})+\cdots\,. \end{equation} \noindent \tag{ 1 }$$

We are working in a frame rotating at the laser frequency ω_L with detuning Δ = ω_L − ω_cav. Here, α_L denotes the coupling to the drive, which is proportional to the laser amplitude. We omitted to explicitly write down the coupling to the photon and phonon baths, with damping rates κ and Γ_l, respectively, although these will be included in our treatment. The bare (single-photon) coupling constants g^(l)₀ depend on the overlap between the optical and mechanical mode functions and are of the order of ω_cavx_ZPF/L, where x_ZPF = (ℏ/2m_lΩ_l)^1/2 is the mechanical zero-point amplitude. In photonic crystals the effective cavity length L reaches down to wavelength dimensions (see figure 1 for the illustration of a setup).

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After the application of the standard procedure of splitting off the coherent optical amplitude induced by the laser, $\skew4\hat {a}=\alpha +\delta \skew4\hat {a}$, and omitting terms quadratic in $\delta \skew4\hat {a}$ (valid for strong drive) to the Hamiltonian, we recover the linearized optomechanical coupling,

$$\begin{equation} \skew4\hat{H}_{\mathrm{int}}=-\hbar(\delta\skew4\hat{a}+\delta\skew4\hat{a}^{\dagger})\sum_{l}g_{l}(\hat{b}^{\dagger}_{l}+\hat{b}_{l})\,=-\hbar(\delta\skew4\hat{a}+\delta\skew4\hat{a}^{\dagger})\skew4\hat{Y}. \end{equation} \tag{ 2 }$$

Here the dressed couplings g_l = g^(l)₀α can be tuned via the laser intensity, i.e. the circulating photon number: $|\alpha |=\sqrt {\bar {n}_{\mathrm {phot}}}$ ($\alpha \in \mathbb {R}$ without loss of generality). We can now eliminate the driven cavity field (noting that $\delta \skew4\hat {a}$ is in the ground state) by second-order perturbation theory: at large detuning $\left |\Delta \right |\gg \Omega _{l}$ the energy scales for phonons and photons separate, and we retain a fully coherent, light-induced mechanical coupling between all the vibrational modes,

$$\begin{equation} \skew4\hat{H}_{\mathrm{int}}^{\mathrm{eff}}=\hbar\frac{\skew4\hat{Y}^{2}}{\Delta}=2\hbar\sum_{l,k}J_{lk}(t)\skew4\hat{X}_{l}\skew4\hat{X}_{k}, \end{equation} \tag{ 3 }$$

where $\skew4\hat {X}_{l}\equiv \skew4\hat {b}_{l}+\skew4\hat {b}^{\dagger }_{l}$ is the mechanical displacement in units of x_ZPF. For this to be valid, we have to fulfill κ ≪ |Δ|, which prevents unwanted transitions. Equation (3) may be viewed as a 'collective optical spring' effect, coupling all the mechanical displacements. The couplings J_lk(t) = g_lg_k/2Δ can be changed in situ and in a time-dependent manner via the laser intensity or the detuning. In the numerical simulations, we take $g_{0}^{(l)}=g_{0}x_{\mathrm {zpf}}^{(l)}/x_{\mathrm {zpf}}^{(1)}=g_{0}\sqrt {\Omega _{1}/\Omega _{l}}$ and assume all modes to have equal masses. This feature will be crucial for our approach described below. Note that if multiple optical modes are driven, the corresponding coupling constants will add.

In general, the couplings in equation (3) will induce quantum state transfer between mutually resonant mechanical modes and entanglement at low temperatures (usually with the help of optomechanical laser cooling). The generation of entanglement between two mechanical modes based on the bilinear interaction, equation (3), has been analyzed in [13–19] or in [19–22] for entanglement between the light field and the phononic mode. However, these schemes are not easily scalable to many mechanical modes, mainly because it is not possible to address the modes that are to be entangled. Moreover, these schemes produce an entangled steady state that is sensitive to thermal fluctuations. We will describe a different scheme that allows mode-selective entanglement and is particularly suited for multi-mode setups. Its robustness against thermal influence is enhanced, since it employs parametric instabilities for entanglement generation.

In contrast to the schemes mentioned above, we have in mind a multi-mode situation for continuous variable quantum information processing and are interested in an efficient approach to selectively couple arbitrary pairs of modes, both for entanglement and for state transfer. There are several desiderata to address for a suitable optomechanical architecture of that style: one should be able to (i) switch couplings in time, (ii) easily select pairs for operations, (iii) get by with only one laser (or a limited number), (iv) achieve large enough operation speeds to beat decoherence and (v) scale to a reasonably large number of modes.

3.1. Frequency-selective operations

Static couplings as in equation (3) could be used for selective pair-wise operations if one were able to shift locally the mechanical frequency to bring the two respective modes into resonance. In principle, this is achievable via the optical spring effect, but would require local addressing with independent laser beams. This could prove challenging in a micron-scale photonic crystal, severely hampering scalability.

Instead, we propose to employ frequency-selective operations, by modulating the laser intensity (and thus J) in time. Entanglement generation by parametric driving has recently been analyzed in various contexts, including superconducting circuits [23], trapped ions [24], general studies of entanglement in harmonic oscillators [25–27], optomechanical state transfer and entanglement between the motion of a trapped atom and a mechanical oscillator [28] and entanglement between mechanical and radiation modes [29]. Parametric driving can also lead to mechanical squeezing in optomechanical systems [30].

Let us now consider two modes with coupling $2\hbar J(t)(\skew4\hat {X}_{1}+\skew4\hat {X}_{2})^{2}$. Note that for the purposes of our discussion we set J_lk = J for the sake of simplicity. The results mentioned below, however, remain valid for the general case of unequal couplings, which is also used for the numerical simulations. The time-dependent coupling is achieved by modulating the laser intensity at a frequency ω of the order of the mechanical frequencies. For ω ≪ |Δ| the circulating photon number follows adiabatically |α(t)|² = |α_max|² cos²(ωt), and we have J(t) = Jcos²(ωt) = J[1 + cos(2ωt)]/2. The resulting time-dependent light-induced mechanical coupling can be broken down into several contributions, whose relative importance will be determined by the drive frequency ω. The static terms, $\hbar J(\skew4\hat {X}_{1}+\skew4\hat {X}_{2})^{2}$, shift the oscillator frequencies by δΩ_j = 2J, and give rise to an off-resonant coupling that is ineffective for |Ω₁ − Ω₂| ≫ J, but gains influence for |Ω₁ − Ω₂| ≲ J. In a realistic setup this $\skew4\hat {X}_{1}\skew4\hat {X}_{2}$ interaction might be enhanced due to intrinsically present phonon tunneling between distinct vibrational modes, which could easily be included in the analysis, since it is of the same structure. Moreover, the static terms contain single mode squeezing terms $\skew4\hat {b}_{l}^{\dagger }\skew4\hat {b}_{l}^{\dagger }+\skew4\hat {b}_{l}\skew4\hat {b}_{l}$ that are always off-resonant and of negligible influence. For the oscillating terms

$$\begin{equation} \hbar J(\mathrm{e}^{2\mathrm{i}\omega t}+\mathrm{e}^{-2\mathrm{i}\omega t})(\hat{b}_{1}+\hat{b}_{1}^{\dagger})(\hat{b}_{2}+\hat{b}_{2}^{\dagger}), \end{equation} \noindent \tag{ 4 }$$

there are three important cases. A mechanical beam-splitter (state-transfer) interaction is selected for a laser drive modulation frequency ω = (Ω₁ − Ω₂)/2. In the interaction picture with respect to Ω₁ and Ω₂, the resonant part of the full Hamiltonian reads

$$\begin{eqnarray*} &&\skew4\hat{H}_{\mathrm{b.s.}}=\hbar J(\hat{b}_{2}^{\dagger}\hat{b}_{1}+\hat{b}_{1}^{\dagger}\hat{b}_{2}). \end{eqnarray*} \noindent$$

In contrast, for ω = (Ω₁ + Ω₂)/2, we obtain from equation (4) a two-mode squeezing (non-degenerate parametric amplifier) Hamiltonian,

$$\begin{eqnarray*} &&\skew4\hat{H}_{\mathrm{ent}}=\hbar J(\hat{b}_{1}\hat{b}_{2}+\hat{b}_{1}^{\dagger}\hat{b}_{2}^{\dagger}), \end{eqnarray*} \noindent$$

which can lead to efficient entanglement between the modes. Finally, ω = Ω_j selects the squeezing interaction for a given mode,

$$\begin{eqnarray*} &&\skew4\hat{H}_{\mathrm{sq}}=\hbar(J/2)(\hat{b}_{j}^{2}+\hat{b}_{j}^{\dagger2}), \end{eqnarray*} \noindent$$

out of the full Hamiltonian. These laser-tunable, frequency-selective mechanical interactions are the basic ingredients for the architecture we will develop and analyze here. Furthermore, combinations of these Hamiltonians can be constructed when the laser intensity is modulated with multiple frequencies. Note that one has to keep in mind that the modulation also generates a time-dependent radiation pressure force, ℏg₀|α_max|²cos²(ωt)/x_ZPF, which leads to a coherent driving of each of the mechanical modes for ω ≈ Ω_l/2. For the parameters we choose here, these processes are off-resonant, however.

3.2. Limitations

We now address the constraining factors for the operation fidelity, both for the two-mode and the multi-mode case. At higher drive powers (as needed for fast operations), the frequency–time uncertainty implies that the different processes need not be resonant exactly, with an allowable spread $\left |\delta \omega \right |\lesssim J$. The parametric instabilities occur for |ω − (Ω_i + Ω_j)/2| < J. Once these intervals start to overlap, process selectivity is lost and the fidelity suffers. At low operation speeds, quantum dissipation and thermal fluctuations will limit the fidelity. This is the essential problem faced by a multi-mode setup, and we will discuss possible remedies further below. The schematic situation for the example of three modes is illustrated in figure 2.

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In order to analyze decoherence and dissipation, we employ a Lindblad master equation to evolve the joint state of the mechanical modes. The evolution of any expectation value can be derived from the master equation and is governed by

$$\begin{eqnarray*} &&\frac{\mathrm{d}}{\mathrm{d}t}\langle \skew4\hat{A}\rangle = \frac{1}{\mathrm{i}\hbar}\langle [\skew4\hat{A},\skew4\hat{H}]\rangle +\sum_{j}(\bar{n}_{j}+1)\Gamma_{j}\langle \mathcal{R}[\hat{b}_{j}^{\dagger}]\skew4\hat{A}\rangle +\sum_{j}\bar{n}_{j}\Gamma_{j}\langle \mathcal{R}[\hat{b}_{j}]\skew4\hat{A}\rangle. \end{eqnarray*}$$

Here $\skew4\hat {H}$ describes the vibrational modes and already contains the effective interaction (3), with modulated time-dependent couplings. Γ_j are the damping rates of the vibrational modes, and $\bar {n}_{j}$ their equilibrium occupations at the bulk temperature. Note that we effectively added the light-induced decoherence rate Γ^φ_opt ≈ g²₀α²κ/Δ² = 2J(κ/Δ) to the intrinsic rate $\Gamma \bar {n}$ [31]. Γ^φ_opt is suppressed by a factor κ/|Δ| and can be arbitrarily small for larger detuning (at the expense of higher photon number α² to keep the same J). In the numerical simulation, we use J and Δ as independent parameters. Note, however, that the sign of the detuning affects the sign of J. The first dissipative term describes damping (spontaneous and induced emission), and the second dissipative term refers to absorption of thermal fluctuations. The relaxation superoperators are defined by $\mathcal {R}[\skew4\hat {b}^{\dagger }]\skew4\hat {A}=\skew4\hat {b}^{\dagger }\skew4\hat {A}\skew4\hat {b}-\skew4\hat {b}^{\dagger }\skew4\hat {b}\skew4\hat {A}/2-\skew4\hat {A}\skew4\hat {b}^{\dagger }\skew4\hat {b}/2$ (in contrast to the equation for $\skew4\hat {\rho }$). For the quadratic Hamiltonian studied here, the equations for the correlators remain closed and are sufficient to describe the Gaussian states produced in the evolution.

We evaluate the logarithmic negativity

$$\begin{equation} E_{\mathcal{N}}(\skew4\hat{\rho})=\mathrm{log}_{2}\left\Vert \skew4\hat{\rho}_{AB}^{T_{A}}\right\Vert _{1} \end{equation}\noindent \tag{ 5 }$$

as a measure of entanglement for any two given modes (A and B), where $\skew4\hat {\rho }_{AB}$ is the state of these two modes, and the partial transpose T_A acts on A only. For Gaussian states, $E_{\mathcal {N}}$ can be calculated via the symplectic eigenvalues of the position–momentum covariance matrix [32].

In figure 3, we show the simulation results for entangling two out of three vibrational modes. The entanglement saturates at later times, while the phonon number grows exponentially. We plot the results at the fixed time t = 5.6/J, since there the logarithmic negativity has already saturated. One clearly sees the features predicted above (figure 2), i.e. the unwanted overlap between entanglement processes at higher driving strengths (figures 3(a) and (b)). Increasing the vibrational frequency spacing suppresses these unwanted effects (figure 3(b)). In figure 3(c), the threshold $J=\Gamma \bar {n}$ for entanglement generation at finite temperature is evident, as is the loss of entanglement at large J. Finally, figures 3(d) and (e) show the dependence on temperature and mechanical quality factor, indicating that this scheme should be feasible for realistic experimental parameters (see below).

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3.3. Larger arrays

Having evenly spaced mechanical frequencies is impossible, because the state transfers between adjacent modes would all be addressed at the same modulation frequency. Any simple layout that offers selectivity and avoids resonance overlap seems to require a frequency interval that grows exponentially with the number N of modes. Hence, another approach is needed for large N.

The scheme (figure 4) that solves this challenge involves an auxiliary mode at Ω_aux, removed in frequency from the array of evenly spaced 'memory' modes in [Ω_min, Ω_max]. All operations will take place between a selected memory mode and the auxiliary mode. Then, the state-transfer resonances are in the band [(Ω_aux − Ω_max)/2,(Ω_aux − Ω_min)/2], and entanglement is addressed within [(Ω_min + Ω_aux)/2,(Ω_max + Ω_aux)/2]. To make this work, one needs to fulfill the mild constraint 2Ω_max − Ω_min < Ω_aux < 2Ω_min, where the upper limit prevents unintended driving of the modes caused by the modulated radiation pressure force (see above). Starting with an arbitrary multi-mode state, state transfer between two memory modes is performed in three steps (swapping 1–aux, aux–2 and aux–1), as is entanglement (swap 1–aux, entangle aux–2 and swap aux–1). Note that this overhead does not grow with the number of memory modes. Figure 5 shows the transfer of a squeezed state from the auxiliary mode to a memory mode.

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Several such arrays could be connected in a two-dimensional (2D) scheme by linking their auxiliaries with an optical mode (figure 4(c)). The spectrum of the auxiliaries can be chosen as in the introductory example (figure 2), which is allowed by the above constraint and enables selective operations between the auxiliaries. State transfer between memory modes of distinct arrays is possible in five steps (swap mem₁–aux₁, swap mem₂–aux₂, swap aux–aux, swap aux₁–mem₁ and swap aux₂–mem₂) as well as entanglement (swap mem₁–aux₁, swap mem₂–aux₂, entangle aux–aux, swap aux₁–mem₁ and swap aux₂–mem₂). Note that the transfer from the memory modes to the auxiliaries can be done in parallel; hence the scheme can be effectively completed in three steps and there is no time delay compared to the single array with auxiliary. Moreover, one might employ more sophisticated transfer schemes [33–35] to improve the transfer fidelities. Two blocks can be connected by introducing a higher order auxiliary that couples to both auxiliary arrays. State transfer can then be performed between auxiliary modes from different blocks. In principle, many blocks can be connected when their auxiliary modes are stringed together in a chain via higher order auxiliaries.

Regarding the experimental implementation, in principle any optomechanical system with long-lived mechanical modes can be used. One promising platform is 'optomechanical crystals' as introduced by Painter and co-workers [4] that feature vibrational defect cavities in the GHz regime with experimentally accessible Q ∼ 10⁶ [36]. These would be very well suited to the scheme presented here, owing to their design flexibility, particularly of 2D structures, and the all-integrated approach, as well as the very large optomechanical coupling strength. Given the currently achieved coupling strength [11, 36] g₀/2π ∼ 1 MHz, a detuning of Δ/Ω = 10 and around 2000 cavity photons (reached in recent experiments), we can estimate the induced coupling to approach the damping rate, J ∼ Γ. This corresponds to the threshold for coherent operations, provided one were to cool down the bath to k_BT_bath < ℏΩ. This is, in principle, possible (at 20 mK), but will likely run into practical difficulties due to the re-heating of the structure via spurious photon absorption or other effects. At finite bath temperatures corresponding to a thermal occupation $\bar {n}\sim k_{\mathrm {B}}T_{\mathrm {bath}}/\hbar \Omega$, the light intensity must be increased by a factor of $\bar {n}$ towards $J\gg \Gamma \bar {n}$. Initially, the vibrational ground state would be prepared via laser cooling, as demonstrated in [11].

In these devices, localized vibrational and optical modes can be produced at engineered defects in a periodic array of holes cut into a free-standing substrate. Adjacent optical and vibrational modes are coupled via tunneling. The typical photon tunnel coupling for modes spaced apart by several lattice constants is [6] in the range of several THz. Thus, hybridized optical modes will form, one of which can be selected via the laser driving frequency while the others remain idle. The vibrational modes' frequencies can be different or equal, in which case delocalized hybridized mechanical modes are produced. A 'snowflake' crystal made of connected triangles (honeycomb lattice) supports wave guides (line defects) and localized defect modes with optomechanical interaction [5]. Placing point defects (heavier triangles/thicker bridges) in the structure, a tight binding analysis indicates that the mechanical frequency spectrum (figure 4(b)) can be generated.

Utilizing the phononic modes of an optomechanical system for processing continuous variable quantum information has a number of desirable aspects in comparison to other systems such as optical modes or cold atomic vapors. First, the phononic modes can be integrated on a chip and the devices are thus naturally scalable. The Gaussian operations discussed above can be applied easily, and the decoherence times are already reasonable, although admittedly worse than for cold atomic vapors.

We mention another option for improving the fidelity: optimal control techniques [25] could be employed to numerically optimize the pulse shape J(t).

Finally, an essential ingredient will be the readout. We have pointed out [37] how to produce a quantum-non-demolition readout of the mechanical quadratures in an optomechanical setup. A laser beam (detuning Δ = 0) is amplitude-modulated at the mechanical frequency Ω_j of one of the modes. The reflected light carries information about only one quadrature $\mathrm {e}^{\mathrm {i}\varphi }\skew4\hat {b}_{j}+\mathrm {e}^{-\mathrm {i}\varphi }\skew4\hat {b}_{j}^{\dagger }$. Its phase φ is selected by the phase of the amplitude modulation, while the measurement back-action perturbs solely the other quadrature. Different modes can be read out simultaneously, and the covariance matrix may be thus obtained in repeated experimental runs. Taking measurement statistics for continuously varied quadrature phases would also allow us to perform full quantum-state tomography and thereby ultimately process tomography. Alternatively, short pulses may be used for readout (and manipulation) [38].

The scheme described here would enable coherent scalable nanomechanical state processing in optomechanical arrays. It can form the basis for generating arbitrary entangled mechanical Gaussian multi-mode states such as continuous variable cluster states [39]. An interesting application would be to investigate the decoherence of such states due to the correlated quantum noise acting on the nanomechanical modes. Moreover, recent experiments have shown, in principle, how arbitrary states can be written from the light field into the mechanics [7–9]. These could then be manipulated by the interactions described here. Alternatively, for very strong coupling g₀ > κ, non-Gaussian mechanical states [40–42] could be produced, and the induced nonlinear interactions (see, e.g., [43, 44]) could potentially open the door to universal quantum computation with continuous variables [12] in these systems.

We acknowledge the ITN Cavity Quantum Optomechanics, an ERC Starting Grant, the DFG Emmy-Noether program and the DARPA ORCHID for funding.