Dynamical approach to quantum optomechanics: Motivation, method, and applications

Abstract

We present an overall summary on a method to deal with quantum dynamics of optomechanical systems. The method is based on the dynamical evolution processes instead of the finally evolved steady states, which are a prerequisite to the standard approach, and well captures the features in optomechanical cooling, entanglement and other scenarios.

Cavity optomechanical systems (OMS) make a feasible platform to realize macroscopic quantumness with various potential applications. A key element for OMS is the radiation pressure on a mechanical resonator, which is nonlinear by modeling such systems as coupling a mechanical mode to a cavity field mode. Previously, most experimental researches in the field focused on cooling the macroscopic mechanical resonator into a quantum state close to its ground state [1]. In this scenario an OMS will finally reach a dynamical stability with its time-independent average cavity field mode $\langle \widehat{a}\rangle =\alpha$ and mechanical mode $\langle \widehat{b}\rangle =\beta$. Then the dynamics of the system can be linearized by expanding the fluctuation of the cavity field (mechanical) mode $\widehat{a}$ ($\widehat{b}$) around $\alpha$ ($\beta$), so that the quantum properties of the OMS can be studied in terms of the linearized dynamical equations. A majority of the previous research works on OMS adopt this approach of linearizing the system dynamics.

If one considers the full dynamics of OMS, which is described by the Hamiltonian ($\hslash \equiv 1$)

$$H=\underset{H_l}{\underbrace{\Delta {\widehat{a}}^{†}\widehat{a}+{\omega }_m{\widehat{b}}^{†}\widehat{b}+iE({\widehat{a}}^{†}-\widehat{a})}}-\underset{H_{om}}{\underbrace{g_m{\widehat{a}}^{†}\widehat{a}(\widehat{b}+{\widehat{b}}^{†})}}$$ (1)

read in a frame rotating at the external drive frequency ${\omega }_L$, where $g_m$ is the optomechanical coupling at the single-photon level, ${\omega }_m$ the mechanical frequency, $\Delta ={\omega }_c-{\omega }_L$ the detuning of the driving field from the resonant cavity frequency ${\omega }_c$, and the amplitude $E$ determines the drive power $\hslash {\omega }_L{E}^2/\kappa$ ($\kappa$ is the cavity damping rate), there will naturally arise one concern—whether the above mentioned time-independent steady states exist under any driving field.

By a straightforward numerical simulation based on the Hamiltonian in Eq. 1, one will see from the examples in Fig. 1 that, even given a continuous-wave (CW) driving field, such steady state exists only when the drive frequency ${\omega }_L$ is lower than the resonant cavity frequency ${\omega }_c$ exactly by the mechanical frequency ${\omega }_m$ (here we only consider the physically meaningful regime where a relatively high $E$ offsets the tiny optomechanical coupling constant $g_m$ in reality). It is more obvious to exclude the steady states for the OMS driven by a pulsed field. The validity of the linearization based on time-independent steady states is, therefore, rather limited.

Fig. 1.

Stabilized mechanical motion${\mathit{X}}_{\mathit{m}}\left(\mathit{t}\right)=\sqrt{\mathit{2}}\mathbf{\text{Re}}\langle \widehat{\mathit{b}}\left(\mathit{t}\right)\rangle$when the OMS is under a CW driving field with varied detuning. The system is chosen with $g_{m}E/{\omega }_m=1.4\kappa$ and ${\omega }_m/{\gamma }_m={10}^6$, where ${\gamma }_m$ is the mechanical damping rate.

A different approach to the quantum dynamics of OMS, which overcomes the non-existence of time-independent steady state, is based on a technique of factorizing dynamical evolution operators [2]. A dynamical process of OMS generally starts from the initial state in equilibrium with the environment, i.e. the product of a cavity vacuum state before pumping and thermal state of a mechanical resonator ($n_{th}$ is the thermal occupation number at a certain temperature):

$$\rho \left(0\right)={|0\rangle }_c\langle 0|...\sum _{n=0}^{\infty }{n}_{\mathit{th}}^n/{(1+n_{\mathit{th}})}^{n+1}{|n\rangle }_m\langle n|$$ (2)

It is extremely difficult to obtain the time evolution from this initial state directly with the following evolution operator in the form of time-ordered exponential (note that $U\left(t\right)$ is not an ordinary unitary operator):

$$U\left(t\right)=\mathcal{T}\exp \{-i{\int }_0^{t}d\tau (H+H_{\mathit{sr}})\left(\tau \right)\}=U_l\left(t\right)\mathcal{T}\exp \left\{i{\int }_0^{t}d\tau U_l^{†}\left(\tau \right)(H_{\mathit{om}}-H_{\mathit{sr}})U_l\left(\tau \right)\right\}$$ (3)

though, by means of the proper Ito’s rules, the stochastic Hamiltonian $H_{sr}$ of system-reservoir couplings leads to the master equation for the associated quantum state and the Heisenberg–Langevin equations for the system operators [3]. The first step to go ahead is the factorization of the evolution operator as in Eq. 3, where $U_l\left(t\right)=\exp \{-i{H}_{l}t\}$ is the action of the linear part $H_l$ of the Hamiltonian in Eq. 1, so that the transformed Hamiltonian $U_l^{†}\left(t\right)H_{om}{U}_l\left(t\right)$ in the second evolution operator takes the form

$$\begin{array}{ccc}& & \underset{H_q\left(t\right)}{\underbrace{g_m[-\mathit{iD}\left(t\right){\widehat{a}}^{†}+i{D}^{*}\left(t\right)\widehat{a}+{\left|D\left(t\right)\right|}^2](e^{-i{\omega }_{m}t}\widehat{b}+e^{i{\omega }_{m}t}{\widehat{b}}^{†})}}\hfill \\ & & +\underset{H_n\left(t\right)}{\underbrace{g_m{\widehat{a}}^{†}\widehat{a}(e^{-i{\omega }_{m}t}\widehat{b}+e^{i{\omega }_{m}t}{\widehat{b}}^{†})}}\hfill \end{array}$$ (4)

where $D\left(t\right)=E(e^{i\Delta t}-1)/\Delta$. It is equivalent to an interaction picture with respect to $H_l$. Meanwhile, the stochastic Hamiltonian $H_{sr}$ is transformed by $U_l\left(t\right)$ to another stochastic Hamiltonian $H_{sr}^{\prime }$ of the same effect after redefining the noise operators [4]. Because the corrections due to the non-commutativity of $H_q$ ($H_{sr}^{\prime }$) and $H_n$ are in the orders of the realistic ratio $g_m/{\omega }_m\ll 1$, the second evolution operator in Eq. 3 can be approximated by the product of the action $U_e\left(t\right)$ due to the sum $H_q+H_{sr}^{\prime }$ with the action $U_n\left(t\right)$ due to the cubic $H_n$. Then we rewrite the expectation value of an evolved system operator $\widehat{O}\left(t\right)$ as (${\rho }_r$ is the reservoir quantum state):

$$\begin{array}{ccc}\hfill \langle \widehat{O}\left(t\right)\rangle & =& {\text{Tr}}_S\left\{\widehat{O}{\text{Tr}}_R\left(U\left(t\right)\rho \left(0\right){\rho }_r{U}^{†}\left(t\right)\right)\right\}\hfill \\ & \approx & {\text{Tr}}_{S,R}\left\{U_e^{†}{U}_l^{†}\left(t\right)\widehat{O}{U}_l{U}_e\left(t\right)U_n\left(t\right)\rho \left(0\right){\rho }_r{U}_n^{†}\left(t\right)\right)\}\hfill \\ & =& {\text{Tr}}_{S,R}\left\{U_e^{†}{U}_l^{†}\left(t\right)\widehat{O}{U}_l{U}_e\left(t\right)\rho \left(0\right){\rho }_r\right\}\hfill \end{array}$$ (5)

in which $U_n\left(t\right)$ leaves the initial state $\rho \left(0\right)$ invariant. The expectation value of the system operator $\widehat{O}\left(t\right)$ effectively evolved by the combined action of $U_e\left(t\right)$ after $U_l\left(t\right)$ can be now obtained with respect to the known initial state $\rho \left(0\right)$. If $\widehat{O}=\widehat{a},\widehat{b}$, we will obtain the linearized dynamical equations (${\widehat{\xi }}_c\left(t\right)$ and ${\widehat{\xi }}_m\left(t\right)$ are the noise terms)

$$\begin{array}{ccc}\hfill \dot{\widehat{a}}& =& -\kappa \widehat{a}+g_{m}D\left(t\right)(e^{-i{\omega }_{m}t}\widehat{b}+e^{i{\omega }_{m}t}{\widehat{b}}^{†})+i\kappa D\left(t\right)\hfill \\ & & +\sqrt{2\kappa }{\widehat{\xi }}_c\left(t\right)\hfill \\ \hfill \dot{\widehat{b}}& =& -{\gamma }_m\widehat{b}+g_m{e}^{i{\omega }_{m}t}[D\left(t\right){\widehat{a}}^{†}-D^{*}\left(t\right)\widehat{a}]\hfill \\ & & +i{g}_m{e}^{i{\omega }_{m}t}{\left|D\left(t\right)\right|}^2+\sqrt{2{\gamma }_m}{\widehat{\xi }}_m\left(t\right)\hfill \end{array}$$ (6)

under the action $U_e\left(t\right)$, which are essential to the study of the concerned quantum dynamical processes. It should be noted that the function $D\left(t\right)$ in Eq. 4 can be found for pulsed and other more complicated drives, so that the approach can deal with more different scenarios of dynamics.

One application of the dynamical approach is the real-time simulation of the optomechanical cooling processes [5], [6]. From Eq. 4 one sees that the term in the form $\lambda \left(t\right){\widehat{a}}^{†}\widehat{b}+h.c.$, the beam-splitter (BS) type coupling that leads to the optomechanical cooling, is enhanced by getting rid of an oscillating phase factor in $\lambda \left(t\right)$ when $\Delta ={\omega }_m$. Then the BS type interaction in Eq. 4 or equivalently in Eq. 6 has the dimensionless magnitude $J=g_{m}E/\left({\omega }_m\kappa \right)$, in contrast to the effective coupling magnitude $g_m\left|\alpha \right|$ in the approach based on the steady cavity field amplitude $\left|\alpha \right|$ [1]. Meanwhile, there exists the squeezing (SQ) type interaction in the form $ϵ\left(t\right){\widehat{a}}^{†}{\widehat{b}}^{†}+h.c.$, which heats the mechanical resonator at the same time. An evolved thermal phonon number $n_m$ observed in a coordinate system co-moving with the mechanical resonator can be found with Eq. 6. Fig. 2 illustrates the general cooling results predicted by the dynamical approach. At a weak coupling, the cooling of a mechanical resonator is continuously improved with the increased the drive power. However, for any mechanical frequency ${\omega }_m$, there exists an optimum $J$ for the best cooling (the terminating points in Fig. 2), beyond which the SQ effect will become more significant. The cooling limit that can only be asymptotically approached under a single CW drive detuned at $\Delta ={\omega }_m$ is the horizontal dashed line to give $n_{m,f}=({\gamma }_m/\kappa )n_{th}$ when ${\omega }_m/\kappa \to \infty$ [5]. In the corresponding strong coupling regime ($J>0.5$ beyond a vertical drop of the dashed curve), the cooling result found in the standard linearization approach is [1]

$$n_{m,f}=\frac{{\gamma }_m}{{\gamma }_m+4\frac{g_m^2{\left|\alpha \right|}^2}{\kappa }}{n}_{th}+\frac{{\gamma }_m}{\kappa }{n}_{th}+\frac{g_m^2{\left|\alpha \right|}^2}{2{\kappa }^2}{\left(\frac{\kappa }{{\omega }_m}\right)}^2$$ (7)

in the resolved sideband regime ${\omega }_m\gg \kappa$, where the cavity thermal occupation is neglected. The predictions by the two different approaches qualitatively meet with a common boundary $({\gamma }_m/\kappa )n_{th}$ that cannot be surpassed by using any driving field. In addition to the capability of simulating the real-time evolution, the dynamical approach can be generalized to other situations without the existence of a steady state, such as cooling under pulsed drive [7], under combined CW and pulsed driving fields [8], or with an extra optical cavity [9]. Some of those improved setups can outperform the cooling limit $({\gamma }_m/\kappa )n_{th}$ under a single drive detuned at $\Delta ={\omega }_m$.

Fig. 2.

Cooling results${\mathit{n}}_{\mathit{m},\mathit{f}}$predicted by the dynamical approach. Here, the dashed curve with a sudden drop in the middle is an exact analytical result obtained in the limit ${\omega }_m/\kappa \to \infty$. We fix the mechanical damping rate with the ratio ${\Gamma }_m={\gamma }_m/\kappa ={10}^{-3}$. The inset shows the cooling result in the weak coupling regime.

Another application is the generation of optomechanical entanglement [10]. For the purpose one needs to enhance the SQ type interaction in Eq. 4 by choosing the driving fields detuned at $\Delta =-{\omega }_m$. Under such blue detuned drive, which excludes time-independent steady state once the amplitude $E$ is beyond the threshold of Hopf bifurcation, the evolution of optomechanical entanglement quantified by logarithmic negativity can be however simulated with Eq. 6 [4]. A further insight provided by the approach is that the criterion $g_{m}E/{\omega }_m\ge {\gamma }_m{n}_{th}$ for generating the entanglement, which is obtained in the limit ${\omega }_m/\kappa \to \infty$ for a CW drive detuned at $\Delta =-{\omega }_m$, clearly reflects the competition between SQ interaction and thermal decoherence. Moreover, the dynamical approach is suitable to deal with the optomechanical entanglement generated beyond the CW drive and a drive on a single optical cavity [11], [12].

The nonlinear and other terms in Eq. 1 can be replaced by those of other physical systems. So far, various processes in the magnomechanical system [13], phonon laser of coupled cavities [14], [15], as well as in atomic ensembles with nonlinearity [16], [17], have been studied in the dynamical approach. Possible applications to more different scenarios are expected.

Some other research works (see, e.g. Liao and Law [18], Liu et al. [19], Shomroni et al. [20]) adopt a linearization based on the expectation values of the system mode operators, especially the real-time ones $\langle \widehat{a}\left(t\right)\rangle$ and $\langle \widehat{b}\left(t\right)\rangle$, to study the dynamical evolutions associated with their fluctuations. Compared to this straightforward approach, there are two necessities to develop the method described here. One is that the ananlytical forms of the evolving $\langle \widehat{a}\left(t\right)\rangle$ cannot be obtained in more general situations, e.g. a pulse drive. The other conceptual reason is that $\langle \widehat{a}\left(t\right)\rangle$ and $\langle \widehat{b}\left(t\right)\rangle$ do not always dominate over the corresponding quantum fluctuations, obviously in a deep quantum regime of a strong optomechanical coupling on the single-photon level or when the cavity frequency shift due to mechanical fluctuations is comparable to the cavity field linewidth $\kappa$. A wider applicability is the main purpose of our dynamical approach.

Instead of reaching equilibrium, an OMS driven by an arbitrary CW field will mostly stabilize in limit cycles (stable oscillations) for its mechanical resonator, together with more complicated oscillation of multiple sidebands for the cavity field. Such non-existence of the time-independent steady state necessitates alternatives to the standard linearization for optomechanical dynamics. The approach summarized here adopts a technique of decomposing the evolution operators for quantum optomechanical processes, so that the system operators follow a linearized effective dynamics reflecting an interplay of the nonlinearity with the external drive, and their expectation values can be evaluated with respect to a known initial quantum state. While many features of quantum optomechanics can be captured, it is still unclear to what extent of nonlinearity the method can be applied. The relevant issues, together with the bigger challenge of genuinely strong nonlinearity, await to be clarified by further study.

Declaration of competing interest

The authors declare that they have no conflicts of interest in this work.

Biographies

Bing He received his Ph.D. degree from the Graduate Center, the City University of New York in 2009. He continued his career as postdoctoral and professional researcher at University Calgary, University of California (Merced) and University of Arkansas. Then, at the end of 2018, he joined Universidad Mayor as an associate professor at Center for Quantum Optics and Quantum Information. His research interests cover quantum nonlinear optics, nonlinear dynamical systems, and quantum information processing.

Qing Lin received his Ph.D. in physics from the CAS Key Laboratory of Quantum Information, University of Science and Technology of China. He is currently a professor at Huaqiao University. He is interested in optomechanical nonlinearity and its applications, quantum optics and computation.