Quantum state transfer of superposed multi-photon states via phonon-induced dynamic resonance in an optomechanical system

Quantum state transfer (QST), the faithful transmission of quantum states from one place to another, plays a pivotal role in quantum information processing and communication [1, 2, 3, 4, 5]. Such a process has been pursued through different methods. Quantum teleportation [6, 7, 8, 9, 10] exploits entanglement between distant subsystems and classical communication of measurement outcomes. Alternatively, double-swap protocol [11] facilitates state transfer through the exchange of flying qubits. Another notable approach involves adiabatic passage via photonic dark states [12], akin to stimulated Raman adiabatic passage [13, 14, 15], providing yet another avenue for efficient quantum state transfer. Inspired by these proposals, a variety of theoretical studies and practical realizations have been explored, spanning diverse quantum systems, such as atoms [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], spins [33, 34, 35, 36, 37, 38, 39], ions [40, 41, 42, 43, 44, 45], superconducting circuits [46, 47, 48, 49, 50, 51, 52, 53, 54, 55], solid qubits [56, 57, 58, 59], and optics [60, 61, 62].

Optomechanical system (OMS), a composite system of light and mechanical modes interacting by radiation pressure force [63], has emerged as a potential platform for quantum information tasks. Near-ground state cooling for mechanical resonators and strong optomechanical coupling were investigated [64, 65, 66, 67, 68, 69]. These facilitated the successful realization [70, 71, 72] of an optical-to-mechanical QST [73, 74, 75, 76, 77, 78, 79, 80]. A notable feature of OMS is that the coupling is independent of the resonant properties of the intra-cavity component, enabling it to function as a quantum interface for QST across systems with significant energy gaps. Both approaches of double-swap and adiabatic passage were employed to transfer a quantum state between optical modes of different frequencies in a three-mode OMS [81, 82, 83, 84, 85], with experimental demonstrations validating their efficacy [86, 87, 88, 89]. Efforts to improve these schemes were also explored, including the utilization of hybrid solutions [90, 91, 92] and shortcut-to-adiabaticity techniques [93, 94, 95]. Additionally, QST between two mechanical resonators coupled to a common cavity mode [96, 97, 98, 99], across distant optomechanical sites [100, 101], and in hybrid atom-optomechanical systems [103, 104, 105, 106, 107] were investigated. Moreover, progress of controlling entanglement in OMS [108] opens a promising route towards realizing quantum teleportation in OMS [109, 110, 111, 112, 113, 114]. However, these works on QST in OMS rely on the linearized cavity-mechanical interaction [63], limiting the amplitude of quantum states to transfer [77, 78].

In this work, we propose a model to transfer quantum states between two coupled optical cavities via a mechanical oscillator, operating in a nonlinear regime. In particular, we focus on a three-mode OMS consisting of two cavities coupled to a movable in-between mirror via optomechanical interactions [115], while also interacting each other. Unlike prior approaches reliant on beam-splitter interactions for the transfer mechanism, we exploit the dynamic resonance transfer [116]. Here, the motion of the mechanical oscillator induces dynamic resonance between two cavities, by which a quantum state is transferred between the cavity modes of (initially) different wavelengths without necessitating linearization. To the end we require high amplitude limit of oscillator, weak coupling between optical cavities, and adiabatic approximation. The high amplitude limit, achievable by preparing an initial state in a large number of photons, allows to approximately neglect quantum fluctuations and thermal noise inherent in the mechanical oscillator. As a result, there is an advantage experimental that it does not need to initialize the mechanical mode in its ground state nor to maintain extremely low bath temperatures, as typically required in a double-swap approach [81]. The conditions of weak optical coupling and adiabaticity, readily attained through appropriate parameterization of the system, enable the passive transfer assisted by the dynamic resonance, in other words, no external control fields are demanded except the preparation of initial state. Our approach is expected to be practical compared to the adiabatic passage method, as in Refs. [83, 84]. Under the conditions, we demonstrate the near-perfect transfer of high-amplitude quantum states between the cavities. We employ Fock states, displaced squeezed states, and Schrödinger cat states, which are at the heart of quantum optics and applications for quantum information processing [117, 118, 119, 120, 121].

The paper is organized as follows. In Section II, we introduce our system and derive its dynamics. Section III discusses the conditions requisite for achieving phonon-induced dynamic resonance. Subsequently, in Section IV, we assess the efficiency of our model by analyzing the transfer fidelity of quantum states. Summaries of our results are represented in Section V. In addition, Appendix A offers a detailed exposition of adiabatic approximation and Appendix B provides explicit expressions for the transfer fidelity of displaced squeezed states.

We present an optomechanical model tailored for QST between two optical cavities. This model consists of two optical Fabry-Perot cavities that share a two-sided mirror, as depicted in Fig. 1. The in-between mirror of mass $m$ is allowed to vibrate with frequency $\omega_{m}$ around its equilibrium position, located by $q_{0}\neq 0$ from the middle of the end mirrors. If the equilibrium position were located exactly at the middle, the two cavities would have the same fundamental frequency $\omega_{0}=2\pi c/\lambda_{0}$ with wavelength $\lambda_{0}$. Here, the position is located by $q_{0}>0$ to the left with $q_{0}\ll\lambda_{0}$. The fundamental frequencies in the left and right cavities are therefore different by $\Delta\omega=\omega_{1}-\omega_{2}$, where $\omega_{1(2)}=2\pi c/\lambda_{1(2)}$ with $\lambda_{1}=\lambda_{0}-2q_{0}$ and $\lambda_{2}=\lambda_{0}+2q_{0}$. The vibrating mirror, called a mechanical oscillator (or simply an oscillator), is assumed to be slightly transparent so as to couple the two cavities with a coupling constant $g>0$. The effective Hamiltonian in the rotating frame at the frequency $\omega_{0}$ is given by (in the unit of $\hbar=1$) [115]

$\displaystyle\hat{H}_{\text{eff}}=g(\hat{a}_{1}^{\dagger}\hat{a}_{2}+\hat{a}_{1}\hat{a}_{2}^{\dagger})+2\kappa_{0}b_{0}(1-\frac{\hat{b}+\hat{b}^{\dagger}}{2b_{0}})\Delta\hat{n}+\omega_{m}\hat{b}^{\dagger}\hat{b}.$

(1)

Here, $\hat{a}_{1(2)}$ and $\hat{b}$ are the annihilation operators of the left (right) cavity and the oscillator, respectively, and $\Delta\hat{n}=\hat{n}_{1}-\hat{n}_{2}$ with the number operators $\hat{n}_{j}=\hat{a}^{\dagger}_{j}\hat{a}_{j}$ for $j=1,2$. The optomechanical coupling constant $\kappa_{0}=2x_{m}\omega_{0}/\lambda_{0}$ and the dimensionless shift $b_{0}=q_{0}/2x_{m}$ with $x_{m}=(\hbar/2m\omega_{m})^{1/2}$ being the mechanical zero-point fluctuation amplitude [63]. The first term in Eq. (1) describes the optical coupling between cavities, the second term is the optomechanical coupling between the oscillator and the cavities, and the third is the free energy of the oscillator. We note that $\hat{H}_{\text{eff}}$ does not explicitly depend on time $t$.

In addition, we consider the coherent control of the cavity amplitudes by external fields $\alpha_{j}(t)$ with the interaction Hamiltonian $\hat{H}_{\text{coh}}(t)=\sum_{j}\alpha_{j}(t)\hat{a}^{\dagger}_{j}+\text{h.c.}$. The cavities and oscillator are also affected by damping and noise processes due to their coupling with the environment. Taking into account control and dissipation terms, the Langevin equations for the system operators are then written by [122]

$$i\dot{{\bf v}}(t)={\bf M}(t){\bf v}(t)+{\bf f}_{\text{coh}}(t)+i{\bf f}_{\text{in}}(t).$$

(2)

Here, ${\bf v}(t)=\big{(}\hat{a}_{1}(t),\hat{a}_{2}(t),\hat{b}(t)\big{)}^{T}$ is the annihilation operator vector and $\dot{{\bf v}}=d{\bf v}/dt$. Vectors ${\bf f}_{\text{coh}}(t)=\big{(}\alpha_{1}(t),\alpha_{2}(t),-\kappa_{0}\Delta\hat{n}(t)\big{)}^{T}$ and ${\bf f}_{\text{in}}(t)=\left(\sqrt{2\gamma_{1}}\hat{s}_{1}^{\text{in}}(t),\sqrt{2\gamma_{2}}\hat{s}_{2}^{\text{in}}(t),\sqrt{2\gamma_{m}}\hat{s}_{m}^{\text{in}}(t)\right)^{T}$ are coherent controlling and noise fields, respectively. $\gamma_{1,2,m}$ are the decay rates of cavities 1, 2, and oscillator $m$. We assume the reservoirs are independent of one another with their correlations $\langle\hat{s}^{\text{in}}_{j}(t)\hat{s}^{\text{in}{\dagger}}_{j}(t^{\prime})\rangle=\left(n^{\text{th}}_{j}+1\right)\delta(t-t^{\prime})$, where $n^{\text{th}}_{j}$ (for $j=1,2,m$) are the thermal average numbers. The dynamic matrix ${\bf M}(t)$ is given by

$${\bf M}(t)=\begin{pmatrix}\hat{\omega}(t)-i\gamma_{1}&g&0\\ g&-\hat{\omega}(t)-i\gamma_{2}&0\\ 0&0&\omega_{m}-i\gamma_{m}\end{pmatrix},$$

(3)

where the frequency operator $\hat{\omega}(t):=2\kappa_{0}b_{0}\left[1-\big{(}\hat{b}(t)+\hat{b}^{\dagger}(t)\big{)}/2b_{0}\right]$ is a function of the oscillator’s operators $\hat{b}$ and $\hat{b}^{\dagger}$.

It is worth noting that the dynamic equation in Eq. (2) is nonlinear. Nevertheless, if the oscillator is treated classically and oscillates slowly enough ($\omega_{m}\ll g$), the dynamics simplify to a beam-splitter-type interaction between two cavities with time-dependent frequencies. This interaction becomes resonant when the frequencies of the two become mutually equal depending on the oscillator’s amplitude. This phenomenon, known as phonon-induced dynamic resonance [116], is discussed in more detail in the next section.

In this section, we show how our model enables the manifestation of phonon-induced dynamic resonance between two cavities. We begin by providing an intuitive description from a classical perspective, followed by the comprehensive solution of quantum dynamics. Meanwhile, we also explore the specific conditions necessary to establish phonon-induced dynamic resonance, providing a detailed discussion of their significance.

We assume a weak coupling between optical cavities, characterized by $g\ll\delta\omega=(\omega_{1}-\omega_{2})/2\approx 2\kappa_{0}b_{0}$, ensuring that the two cavities are initially in far-off resonance. In contrast, under strong optical coupling, direct hopping between the cavities dominates, leading to a conventional quantum state exchange [123]. We also assume the oscillator frequency $\omega_{m}\ll g$. Let us suppose that the left cavity is coherently excited by turning on and off the external driving fields in a short time $T_{p}\simeq 2\pi/\delta\omega\ll 2\pi/g\ll 2\pi/\omega_{m}$. The driving fields are switched off at $t=0$ onward. In this way, we prepare the initial quantum state $\hat{\rho}(0)$ of the optomechanical system, while leaving the the oscillator’s state unchanged, $\langle\hat{b}(-T_{p})\rangle\approx\langle\hat{b}(0)\rangle=0$. In other words, during the preparation, the oscillator is nearly immobilized at its initial equilibrium position. Consequently, the two cavities remain far from resonance and do not exchange energy (photons). Starting from $t=0$, as the oscillator is allowed to vibrate, the radiation pressure by the left cavity pushes the oscillator toward a new equilibrium position at the right. The oscillator then approaches the middle of the two cavities, depending on the excited energy in the left cavity, so that the cavities become resonant and exchange photons. The resonance condition is temporally satisfied with the oscillator’s position, so-called a phonon-induced dynamic resonance [116].

The oscillator’s equilibrium position of average $b_{\text{eq}}=\kappa_{0}\langle\Delta\hat{n}\rangle/\omega_{m}$ is determined by the radiation pressures or the photon numbers of the cavity fields. The initial photon-number average in the left cavity, denoted as $\bar{n}$, needs to be so large that the dynamic resonance condition can be fulfilled with $2b_{\text{eq}}>b_{0}$, or equivalently, $\bar{n}>n_{\text{thr}}$, where the threshold number of photons $n_{\text{thr}}=\omega_{m}b_{0}/2\kappa_{0}$. In this regime of dynamic resonance and the time duration much shorter than $2\pi/\gamma_{m}$ (i.e., $\gamma_{m}\ll\omega_{m}$), one may neglect the quantum fluctuation $\delta\hat{b}$ of the oscillator if the following condition is satisfied [124]

$\displaystyle\sqrt{\langle\delta\hat{b}^{\dagger}\delta\hat{b}\rangle}\ll\left|b-b_{\text{eq}}\right|\simeq b_{\text{eq}}.$

(4)

Here we expand $\hat{b}=b+\delta\hat{b}$, where the average amplitude $b=\langle\hat{b}\rangle$ is the solution to $i\dot{b}=\omega_{m}(b-b_{\text{eq}})$. The high amplitude limit in Eq. (4), as assumed throughout this paper, is that the average amplitude from the equilibrium, $|b-b_{\text{eq}}|$, is much larger than the quantum fluctuation [124]. In terms of the number fluctuation of cavities, the condition in Eq. (4) is reformulated as

$$\delta\Delta n\equiv\sqrt{\langle(\Delta\hat{n}-\langle\Delta\hat{n}\rangle)^{2}\rangle}\ll\langle\Delta\hat{n}\rangle.$$

(5)

In this limit, the frequency operator $\hat{\omega}(t)$ in Eq. (3), a function of $\hat{b}$, is reduced to the scalar function $\omega(t)$ of the average $b$, i.e., $\hat{\omega}(t)$ is replaced by $\omega(t)=\delta\omega(1-b_{r}(t)/b_{0})$, where $b_{r}$ is the real part of $b$. In the oscillator’s equation, we replace the operators by their averages, i.e., $\hat{b}(t)$ by $b(t)$ and $\Delta\hat{n}(t)$ by $\langle\Delta\hat{n}(t)\rangle$. We also remove the noise operator $\sqrt{2\gamma_{m}}\hat{s}^{\text{in}}_{m}(t)$ by assuming the thermal phonon number $n^{\text{th}}_{m}\ll|b-b_{\text{eq}}|^{2}$. The dynamic equation (2) becomes, divided into two types,

	$\displaystyle i\dot{\textbf{a}}(t)$	$\displaystyle=$	$\displaystyle\textbf{M}_{a}(t)\textbf{a}(t)+i{\bf f}^{\text{in}}_{a}(t),$		(6)
	$\displaystyle i\dot{b}(t)$	$\displaystyle=$	$\displaystyle(\omega_{m}-i\gamma_{m})b(t)-\kappa_{0}\langle\Delta\hat{n}(t)\rangle,$		(7)

where ${\bf a}(t)=\big{(}\hat{a}_{1}(t),\hat{a}_{2}(t)\big{)}^{T}$ and ${\bf f}^{\text{in}}_{a}(t)=\left(2\sqrt{\gamma_{1}}\hat{s}^{\text{in}}_{1}(t),\sqrt{2\gamma_{2}}\hat{s}^{\text{in}}_{2}(t)\right)^{T}$ are vectors for the cavities, and

$${\bf M}_{a}(t)=\begin{pmatrix}\omega(t)-i\gamma_{1}&g\\ g&-\omega(t)-i\gamma_{2}\end{pmatrix}.$$

(8)

The absence of the coherent terms $\alpha_{1,2}(t)$ for the cavities in Eq. (6) is due to the assumption that the diabatic processes of initializing the cavities are performed in a very short time $T_{p}$.

To apply adiabatic approximation to the cavity dynamics with the phonon-induced dynamic resonance, we assume $\omega_{m}\delta\omega/g^{2}\ll 1$ (see Appendix A). This adiabatic condition, when combined with the weak optical coupling, results in

$$\frac{\omega_{m}}{g}\ll\frac{g}{\delta\omega}\ll 1.$$

(9)

Accordingly, Eq. (6) is transformed, as in Appendix A, to the dynamic equation for eigen modes of annihilation operators ${\bf a}_{\pm}(t)=\big{(}\hat{a}_{+}(t),\hat{a}_{-}(t)\big{)}^{T}$,

$$i\dot{{\bf a}}_{\pm}(t)={\bf\Lambda}(t){\bf a}_{\pm}(t)+i{\bf f}^{\text{in}}_{\pm}(t),$$

(10)

where ${\bf a}_{\pm}(t)={\bf W}^{T}(t){\bf a}(t)$, ${\bf f}^{\text{in}}_{\pm}(t)={\bf W}^{T}(t){\bf f}^{\text{in}}_{a}(t)$, and ${\bf\Lambda}(t)={\bf W}^{-1}(t){\bf M}_{a}(t){\bf W}(t)$ is a diagonal matrix with diagonal elements $\lambda_{\pm}(t)$. For $\gamma_{1}=\gamma_{2}$, the eigenvalues are given by $\lambda_{\pm}(t)=\pm\epsilon(t)-i(\gamma_{1}+\gamma_{2})/2$, where $\epsilon(t)=\sqrt{\omega^{2}(t)+g^{2}}$. The transformation ${\bf W}(t)$ is expressed as

$${\bf W}(t)=\begin{pmatrix}c(t)&-s(t)\\ s(t)&c(t)\end{pmatrix}$$

(11)

with $c(t)=\sqrt{\left(1+\omega(t)/\epsilon(t)\right)/2}$ and $s(t)=\sqrt{\left(1-\omega(t)/\epsilon(t)\right)/2}$. The solution to Eq. (10) is given by

$${\bf a}_{\pm}(t)=e^{-i{\bf\Xi}(t,0)}{\bf a}_{\pm}(0)+\int^{t}_{0}d\tau e^{-i{\bf\Xi}(t,\tau)}{\bf f}^{\text{in}}_{\pm}(\tau),$$

(12)

where ${\bf\Xi}(t,\tau)=\int^{t}_{\tau}dt^{\prime}{\bf\Lambda}(t^{\prime})$. It is remarkable that the two coupled cavities are described by two independent eigen modes. On the other hand, each eigen mode remains coupled with the oscillator in a way that its frequency depends temporally on the amplitude of oscillator, which is the solution to Eq. (7),

	$\displaystyle b(t)$	$\displaystyle=$	$\displaystyle e^{-i\xi_{m}(t,0)}b(0)+i\int^{t}_{0}d\tau e^{-i\xi_{m}(t,\tau)}\kappa_{0}\langle\Delta\hat{n}(\tau)\rangle$		(13)
		$\displaystyle\approx$	$\displaystyle e^{-i\xi_{m}(t,0)}b(0)+i\int^{t}_{0}d\tau e^{-i\xi_{m}(t,\tau)}\kappa_{0}(\omega(\tau)/\epsilon(\tau))\langle\Delta\hat{n}_{\pm}(\tau)\rangle,$

where $\xi_{m}(t,\tau)=(\omega_{m}-i\gamma_{m})(t-\tau)$, $\Delta\hat{n}_{\pm}(\tau)=\hat{n}_{+}(\tau)-\hat{n}_{-}(\tau)$ with $\hat{n}_{\pm}=\hat{a}_{\pm}^{\dagger}\hat{a}_{\pm}$. Here we neglected the terms including $\hat{a}_{\pm}^{\dagger}(\tau)\hat{a}_{\mp}(\tau)$, by the rotating wave approximation, as they are fast rotating with $|\epsilon(\tau)|\gg|\omega_{m}-i\gamma_{m}|\simeq\omega_{m}$.

Figure 2 presents optical frequencies in terms of $b_{r}/b_{0}$, with real amplitude $b_{r}=\text{Re}[b]$, which shows a typical feature of the adiabatic process. For zero optical coupling of $g=0$, the width of the left (right) cavity is linearly increased (decreased) when the oscillator moves to the right. As a result, the frequency of cavity mode $\hat{a}_{1}$ ($\hat{a}_{2}$) is linearly decreased (increased), leading to a frequency crossing at $b_{r}/b_{0}=1$. When $g>0$, a gap emerges between the two eigen-mode frequencies, characterized by the order of $g/\delta\omega$. It’s essential for this order to be small enough, say $10^{-2}$, that the two frequencies nearly touch each other.

Given $g/\delta\omega=10^{-2}$ and $\omega_{m}/g=10^{-3}$, we illustrate the phonon-induced dynamic resonant transfer by presenting the semiclassical solution of $b(t)/b_{0}$ on phase space of $(\text{Re}[b],\text{Im}[b])$ in Fig. 3 and photon populations of cavities in Fig. 4. The phase-space trajectory of the oscillator (in Fig. 3) starts at the origin and moves clock-wise to the resonant position of $b_{r}/b_{0}=1$ if $\bar{n}>n_{\text{thr}}$. Around here, all the photons are transferred to the right cavity mode (as seen in Fig. 4) and the oscillator moves along the other larger circle. Turning back to the resonant position of $b_{r}/b_{0}=1$, all the photons are transferred back to the left mode (as seen in Fig. 4) and the oscillator moves along the original circle.

The phonon-induced dynamic resonance in a three-mode optomechanical system is at the heart of our approach to QST. The nearly perfect population transfer between two cavities is achievable when the key assumptions are satisfied. Specifically, the high amplitude limit is feasible with the large number of photons (as indicated in the subsequent section). The conditions of adiabaticity and weak optical coupling are met by appropriately configuring the system’s parameters, for instance, setting $\delta\omega\simeq 10^{2}g$, $g\simeq 10^{3}\omega_{m}$, and $\bar{n}/n_{\text{thr}}<10$, which are experimentally available [125, 126].

In this section, we propose two schemes to transfer quantum states between two cavities. Initially, the cavities are prepared in a composite state $|\Psi_{0}\rangle=|\psi\rangle\otimes|0\rangle$, where the left cavity is in the state $|\psi\rangle$ and the right cavity is in the vacuum (ground) state $|0\rangle$. The local state $|\psi\rangle$ is to be transferred from the left cavity to the right through the phonon-induced dynamic resonance. In the first scheme, we aim that the cavities are in a fixed target state $|\Psi_{\text{fix}}\rangle=|0\rangle\otimes|\psi\rangle$, similar to the conventional state-swapping. In the second scheme, we introduce a moving target state $|\Psi_{\text{mov}}\rangle=|0\rangle\otimes|\psi_{\text{m}}(t)\rangle$, where the transferred state $|\psi_{\text{m}}(t)\rangle$ depend on time in terms of the phase. Under the assumptions required to establish phonon-induced dynamic resonance, we derive expressions for the transfer fidelity (section IV.1) and evaluate the efficiency of our schemes by employing Fock states, displaced squeezed states, and Schrödinger cat states (section IV.2). These states, fundamental to quantum optics, are pivotal for applications in quantum information processing.

We propose a novel model to transfer quantum states between two optical cavities mediated by a mechanical oscillator in a three-mode optomechanical system. The system operates in the nonlinear regime of optomechanical interaction and with parameters set to satisfy three key assumptions: high amplitude limit of mechanical oscillator, weak coupling between optical cavities, and adiabatic approximation. These assumptions are essential to establish phonon-induced dynamic resonance, the heart of our approach to quantum state transfer.

We examine two schemes: the first involves the fixed target state, akin to conventional state-swapping, while the second assumes the moving target state whose phase evolves over time. Our results demonstrate that both schemes can achieve nearly perfect fidelity once population transfer via phonon-induced dynamic resonance is reliable. In the first scheme, the transfer fidelity rapidly oscillates with time. However, by introducing a phase shifter in the second scheme, the oscillation is mitigated, enabling nearly perfect fidelity to be maintained over extended periods, comparable to the timescale of the mechanical oscillator.

A defining strength of our scheme is its capacity to transfer high-amplitude quantum states, a capability unattainable in approaches relying on linearized optomechanical interactions. Quantum states with large numbers of photons tend to be resilient to specific types of noise and errors including thermal noise, shot noise and photon losses, enhancing the overall fidelity of quantum state transfer [133]. Efficiently transferring such high-amplitude quantum states is essential for large-scale quantum computing based on continuous-variable systems [134, 135].

In this work, the efficiency of quantum state transfer is analyzed under the assumption that quantum fluctuations and thermal noises of the mechanical oscillator are negligible. Furthermore, damping and noise effects are excluded to enable the derivation of analytical expressions for fidelity. An investigation that includes all of these effects will be addressed in future studies.

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (2023M3K5A1094813).