Quantum control with Lyapunov function and bang–bang solution in the optomechanics system

Abstract

We propose a quantum control scheme with the help of Lyapunov control function in the optomechanics system. The principle of the idea is to design suitable control fields to steer the Lyapunov control function to zero as t→∞ while the quantum system is driven to the target state. Such an evolution makes no limit on the initial state and one needs not manipulate the laser pulses during the evolution. To prove the effectiveness of the scheme, we show two useful applications in the optomechanics system: one is the cooling of nanomechanical resonator and the other is the quantum fluctuation transfer between membranes. Numerical simulation demonstrates that the perfect and fast cooling of nanomechanical resonator and quantum fluctuation transfer between membranes can be rapidly achieved. Besides, some optimizations are made on the traditional Lyapunov control waveform and the optimized bang–bang control fields makes Lyapunov function V decrease faster. The optimized quantum control scheme can achieve the same goal with greater efficiency. Hence, we hope that this work may open a new avenue of the experimental realization of cooling mechanical oscillator, quantum fluctuations transfer between membranes and other quantum optomechanics tasks and become an alternative candidate for quantum manipulation of macroscopic mechanical devices in the near future.

Graphical abstract

Keywords

bang–bang solution / quantum control / Lyapunov control / optomechanics system

1 Introduction

As an important platform of quantum mechanics, the quantum optomechanics system has been widely investigated because of its possible applications in quantum communication and quantum information processing [1, 2]. Due to the splendid characteristics, several nontrivial quantum phenomena have been realized in optomechanical systems, including near-ground-state cooling [3, 4], strong-coupling effects [5], squeezing of a mechanical oscillator [6–9], and so on. Utilizing these phenomena, lots of schemes have been proposed to realize quantum information tasks under the optomechanical physical model in the past decades [10–59].

In the quantum information tasks [10–59], cooling a mechanical resonator to its ground state is the foundation and precondition for the rest of work in quantum optomechanics system. How to cool a mechanical resonator has attracted much attention and various schemes have been proposed [11–16]. For example, in 2008, Genes et al. provided a general framework to describe cooling of a micromechanical oscillator to its quantum ground state by means of radiation-pressure coupling with a driven optical cavity [11]. Lately, Nunnenkamp et al. derived an effective master equation describing two-phonon cooling of the mechanical oscillator and demonstrated how to achieve mechanical squeezing by driving the cavity with two beams [12]. Recently, Peterson et al. explored the radiation pressure of light and successfully sideband cooled a micromechanical membrane resonator to the quantum back action limit [14]. Among different kinds of cooling schemes [11–16], the sideband cooling is a common cooling method where the decay rate of the cavity mode should be much less than the mechanical frequency. This is a strict restriction since a high quality factor of the cavity is needed, or the Stokes heating cannot be well suppressed and the cooling comes to nothing. Therefore, the strict restriction is hard to implement in most physical systems since a perfect cavity is difficult to achieve experimentally. So it is hard to find a suitable quantum control method to realize the sideband cooling in most physical systems.

On the other hand, some significant schemes have been proposed that it is possible to utilize optomechanical coupling in various applications without cooling the mechanical oscillator to its ground state [44–46]. For example, Dong et al. experimentally demonstrated an adiabatic transfer of optical fields between two optical modes of a silica resonator without cooling the mechanical oscillator to its ground state [44]. Garg et al. presented a scheme for the adiabatic transfer of average fluctuations in the phonon number between two membranes in an optical cavity [45]. Soon after, by invariant-based inverse engineering [46–51], Chen et al. designed classical driving fields to transfer quantum fluctuations between two suspended membranes in an optomechanical cavity system [46]. However, these previously reported works [44–46] require a certain initial state or a strict adiabatic condition is satisfied. These constraints may make it difficult to find a suitable quantum control method to realize quantum information processes and obtain a good experimental result.

Thus, whether it is needed to cool the mechanical oscillator to its ground state before we utilize optomechanical coupling in various applications, some constraints always hinder the efficiency of quantum information tasks. So it is important to find a suitable quantum control method in optomechanics system with a wide range of applications and few constraints.

In order to solve the problem, various schemes such as quantum optimal control, quantum Lyapunov control and Lie group decompositions have been used to design control laws to drive quantum systems to target states or to realize some specific operations [36–43, 46, 60–63]. Among these quantum control schemes [36–43, 46, 60–63], quantum Lyapunov control [40–43, 46, 60–63] plays an important role in coherently manipulating quantum systems because it offers a simple and effective way to design control fields. A significant advantage of the Lyapunov method is that the designed control laws would not make the closed-loop system divergent. In addition, the calculation of control fields for Lyapunov control is much easier since it does not need iteration. Another merit of Lyapunov control is that the shape of control fields is flexible. So the Lyapunov control method is a flexible and widely applicable control method.

In quantum Lyapunov control, the function V , called Lyapunov function of quantum states, is specified to design time-varying control fields [60–63]. Then, with the help of control fields, the system converges to the target state given by \dot{V}=0 while V decreases to its minimum value. Lyapunov control scheme does not need to qualify a particular initial state. Besides, different definitions of Lyapunov control function may offer different evolution paths. Thus, manipulating the quantum information process with Lyapunov control is quite flexible.

In this paper, we present a quantum control scheme based on Lyapunov control function V = T r(P ρ) [60] to manipulate the evolution in the optomechanics system. (We denote the state of the target resonator by ρ and P is an Hermitian operator and assumed to be positive semidefinite in order to satisfy the standard requirement for a Lyapunov control function, V≥0.) For the sake of convenience, we call this scheme as Lyapunov-based control scheme in the rest of paper. In the scheme, different kinds of quantum information processes can be realized depending on the different definitions of Lyapunov control functions V = T r(P ρ).

Whereafter, we give two applications to verify the validity of Lyapunov-based control scheme. One is to realize the cooling of mechanical oscillator in linear coupling (LC) circuit. Different from the previously reported sideband cooling schemes [16] where the decay rate of the cavity mode should be much less than the mechanical frequency, the Lyapunov-based control scheme could realize the cooling of mechanical oscillator in the LC circuit with relatively loose parameter selection. The other application is to transfer the quantum fluctuation between two membranes in the cavity optomechanics system. The quantum fluctuations can be coherently and deterministically transferred from one mechanical oscillator to the other with the help of Lyapunov-based control scheme. The evolution process can start with a random initial state and do not have to meet the adiabatic condition. This clearly opens up an avenue of quantum communication between two truly mesoscopic systems. However, cooling of the resonators and the fluctuation transfer between membranes are just two applications of our control method. There are many other aspects to which our scheme can be applied, just like reaching the actual motion of the mechanical membranes [64]. We will give more applications of the control method in the following work.

Furthermore, in the Lyapunov-based control scheme, the amplitude of the control fields f_n(t) is proportional to T_n(t) and will still change during the whole evolution process. For better practical implementation of the scheme, we need to manipulate the control fields at each instance of time and compensate the positive or negative effect caused by time delay in experiments. Therefore, this kind of amplitude may be difficult to control in realistic situations and errors would be introduced in the complexity regulation because the control fields are a continuous function of time. Thus, this is a challenging task for practical applications. To handle this problem, we use the bang–bang solutions [63] of control fields’ waveform to optimize the Lyapunov-based control scheme. For convenience, we call this optimized control scheme as bang–bang-based control scheme in the rest of paper. Then, we make a numerical simulation of the fluctuation transfer between two membranes. The simulation shows that it is possible to realize a perfect fluctuation transfer between membranes and the bang–bang-based control scheme makes the evolution time significantly reduced, compared with the Lyapunov-based control scheme. Besides, the optimized design has the bang–bang type control form which may be easier to realize experimentally [65]. Hence, we hope that this scheme could be a good candidate for the experimental realization of cooling mechanical oscillator, quantum fluctuations transfer between membranes and other quantum optomechanics tasks in the near future.

The paper is organized as follows: In Section 2, we first review the principle of quantum Lyapunov control theory and choose the suitable Lyapunov control function V = T r(P ρ) to design the Lyapunov-based control fields. To prove the effectiveness of the Lyapunov-based control scheme, two applications are displayed in Section 3 and Section 4, respectively. That is, in Section 3, we consider a LC circuit to cool a nanomechanical resonator to its ground state with the help of Lyapunov-based control scheme. Then, in Section 4, we utilize the Lyapunov-based control scheme to realize the transfer of quantum fluctuations between two membranes in a membrane-in-the-middle (MIM) cavities system. In Section 5, we optimize the Lyapunov-based control scheme with the help of bang–bang solution for better efficiency and experimental feasibility. Finally, we summarize the work in Section 6.

2 The quantum Lyapunov control design

In this section, we propose the quantum control scheme (Lyapunov-based control scheme) based on Lyapunov control function V [42, 60, 63] in the optomechannics system. Firstly, we start with a brief introduction of Lyapunov control function. The definition of quantum Lyapunov control was proposed in the early 2000s as a good candidate for realizing state transfer [40]. Soon after, this strategy has been widely studied both in theory and applications because it offers a simple and effective way to design control fields. Generally, in quantum Lyapunov control theory, the total Hamiltonian can be written in two parts, the free Hamiltonian H₀ and the external control Hamiltonian H_c(t)={\displaystyle {\sum }_{n=1}^k}{f}_n(t)H_n [42, 62]. The Lyapunov control function V of quantum states is specified to design time-varying control fields. These corresponding additional control fields should be designed to ensure \dot{V}\le 0.

However, there are many ways to design control fields to achieve this goal and designing Lyapunov functions from some special geometric or physical meanings is a good approach. Here we consider the following form of Lyapunov control function V = T r(P ρ), where P is an Hermitian operator and assumed to be positive semidefinite in order to satisfy the standard requirement for a Lyapunov control function, V≥0 [63].

A controlled quantum system can be modeled in different ways, either as a closed system evolving unitarily governed by a Hamiltonian or as an open system coupling to its environment. We first consider a closed system and its dynamics as Markovian. Therefore the dynamics obeys the Markovian master equation,

\begin{array}{}\text{(1)}& \begin{array}{l}\dot{\rho }=-\text{i}[H\mathrm{,}\rho ]\mathrm{,}\\ H=H_0+H_c\mathrm{.}\end{array}\end{array}

Before we design the control fields, the time derivative of V needs to be calculated as [63]

\begin{array}{}\text{(2)}& \begin{array}{ll}\dot{V}\hfill & =Tr(-\text{i}P[H_0+{\displaystyle \sum _{n=1}^k}{f}_n(t)H_n\mathrm{,}\rho ])\hfill \\ \hfill & =Tr(-\text{i}\rho [P\mathrm{,}{H}_0])+{\displaystyle \sum _{n=1}^k}{f}_n(t)Tr(-\text{i}\rho [P\mathrm{,}{H}_n])\hfill \\ \hfill & ={\displaystyle \sum _{n=1}^m}{f}_n(t)T_n(t)\mathrm{,}\hfill \end{array}\end{array}

where T_n(t) = T r(−iρ[P, H_n]) is a real function of ρ, H_n, and P . For simplification, we have used the assumption that [P, H₀] = 0, which can be achieved by constructing P using the eigenvectors of H₀. A simple and conventional way to design the control fields is to let f_n(t) =−KT_n with K > 0. In this case,\dot{V}(\rho )=-{\displaystyle {\sum }_{n=1}^k}K{T}_n{(t)}^2\le 0, so the system converges to the target state ρ_t given by \dot{V}=0 while V decreases to its minimum value.

After introducing the quantum Lyapunov control theory, we will display two applications of Lyapunov-based control scheme in the next two sections.

3 The cooling of resonators

The cooling of resonators has become an active research topic in recent years due to its potential applications in detecting extremely small displacement and observing quantum phenomena of macroscopic mechanical objects [63, 66, 67]. Inspired from the works mentioned in Refs. [62, 63], we will introduce how to utilize the Lyapunov-based control scheme to cool a nanomechanical resonator to its ground state in an open system.

3.1 Physical system and the effective Hamiltonian

When studying the cooling of mechanical resonator, it is not suitable to use the closed system as a cooling environment. This is not a reasonable consideration because the losses and dephasing mechanisms and, most importantly, the flux of energy from the environment to the system have a great influence on the whole cooling process. For more reliable discussion, we restrict our discussion to a N -dimensional open quantum system and consider its dynamics as Markovian.

Consider the circuit in Fig. 1, where a nanomechanical resonator capacitively couples with two charge islands of linear coupling (LC) oscillators, respectively [67]. The nanomechanical resonator interacts with the circuit with movable membranes. In order to find out more intuitively where the mechanical resonator is, we use a dotted line around the mechanical resonator to emphasize it in Fig. 1. Each LC oscillator includes a capacitor C₁₍₂₎ and an inductor L₁₍₂₎. We denote the vibrational displacement of the nanomechanical mode as x₁₍₂₎ and the annihilation operators for the LC oscillators and the nanomechanical resonators by b₁, b₂ and a, respectively, with x_{1(2)}=\delta x_0^{1(2)}(a+a^{†}) and x_0^{1(2)} being the quantum displacement of each side. The coupling capacitance can be expressed as C_{x1(x2)}=C_{x0}^{1(2)}+C_{x1(x2)}^{\prime }{x}_{1(2)} where C_{x1(x2)}^{\prime }=\frac{\partial C_{x1(x2)}}{\partial x_{1(2)}}. Applying a Lagrangian approach for quantum circuits, the total Hamiltonian of the resonator is given by ( \hslash =1) [67]

\begin{array}{}\text{(3)}& \begin{array}{ll}\hfill & H=H_0+H_{c1}+H_{c2}\mathrm{,}\hfill \\ \hfill & H_0=\omega a^{†}a\mathrm{,}\hfill \\ \hfill & H_{c1}=\frac{{(P_{{\text{Φ}}_1}+v_{c1}{C}_{x1}^{\prime }{x}_1)}^2}{2(C_1+C_{g1}+C_{x0}^1+C_{x1}^{\prime }{x}_1)}+\frac{{\varphi }_1^2}{2L_1}\mathrm{,}\hfill \\ \hfill & H_{c2}=\frac{{(P_{{\text{Φ}}_2}+v_{c2}{C}_{x2}^{\prime }{x}_2)}^2}{2(C_2+C_{g2}+C_{x0}^2+C_{x2}^{\prime }{x}_2)}+\frac{{\varphi }_2^2}{2L_2}\mathrm{,}\hfill \end{array}\end{array}

where ω is the frequency of nanomechanical resonator, P_Φ is the conjugate momentum of the charge island which describes the total charge on the island. This kind of expression is not easy to calculate the effective Hamiltonian of the system, so we use Ω₁₍₂₎ to replace {[L_{1(2)}(C_{1(2)}+C_{g1(g2)}+C_{x0}^{1(2)})]}^{-1/2} to define the frequency of the LC oscillator. In this case, the Hamiltonian of the LC oscillators including the coupling can be rewritten as [67]

\begin{array}{}\text{(4)}& \begin{array}{ll}\hfill & H_{c1}={\text{Ω}}_1{b}_1^{†}{b}_1-g_{r1}(a+a^{†})b_1{b}_1^{†}-\text{i}{g}_{l1}(a+a^{†})(b_1+b_1^{†})\mathrm{,}\hfill \\ \hfill & H_{c2}={\text{Ω}}_2{b}_2^{†}{b}_2-g_{r2}(a+a^{†})b_2{b}_2^{†}-\text{i}{g}_{l2}(a+a^{†})(b_2+b_2^{†})\mathrm{,}\hfill \\ \hfill & g_{r1}=\frac{{\text{Ω}}_1{C}_{x1}^{\prime }\delta x_0^1}{2(C_1+C_{g1}+C_{x0}^1)}\mathrm{,}\hfill \\ \hfill & g_{r2}=\frac{{\text{Ω}}_2{C}_{x2}^{\prime }\delta x_0^2}{2(C_2+C_{g2}+C_{x0}^2)}\mathrm{,}\hfill \\ \hfill & g_{l1}=v_{c1}\sqrt{{\text{Ω}}_1/2(C_1+C_{g1}+C_{x0}^1)}(C_{x1}^{\prime }\delta x_0^1)\mathrm{,}\hfill \\ \hfill & g_{l2}=v_{c2}\sqrt{{\text{Ω}}_2/2(C_2+C_{g2}+C_{x0}^2)}(C_{x2}^{\prime }\delta x_0^2)\mathrm{,}\hfill \end{array}\end{array}

which includes radiation-pressure-like coupling with the coupling constant g_r1(r2) and Bogoliubov linear coupling with the coupling constant g_l1(l2). When we choose a suitable voltage of v_c1 = v_c2 = 300 mV, the rate of g_l/g_r≈10⁴ and the radiation-pressure-like coupling can be ignored. Thus, we can deliver the effective Hamiltonian of the system as

\begin{array}{}\text{(5)}& \begin{array}{l}H=H_0+H_c(t)\mathrm{,}\\ H_0={\text{Ω}}_1{b}_1^{†}{b}_1+{\text{Ω}}_2{b}_2^{†}{b}_2+\omega a^{†}a\mathrm{,}\\ H_c(t)=g_{l1}(t)(b_1+b_1^{†})(a+a^{†})\\ +g_{l2}(t)(b_2+b_2^{†})(a+a^{†})\mathrm{,}\end{array}\end{array}

where (b_1+b_1^{†})\mathrm{,}(b_2+b_2^{†}) and (a + a^†) are the position operators of the LC oscillators and the nanomechanical resonator, respectively. One can modify the coupling rate g_l1(l2)(t) by changing the voltage of the LC circuit [66, 67].

Therefore the dynamics obeys the Markovian master equation

\begin{array}{}\text{(6)}& \begin{array}{cc}\dot{\rho }=\hfill & -\text{i}[H\mathrm{,}\rho ]+\mathcal{P}\mathrm{,}\hfill \\ \mathcal{P}=\hfill & {\kappa }_{b_1}\overline{n}\mathcal{L}(b_1^{†})+{\kappa }_{b_1}(\overline{n}+1)\mathcal{L}(b_1)\hfill \\ \hfill & +{\kappa }_{b_2}\overline{n}\mathcal{L}(b_2^{†})+{\kappa }_{b_2}(\overline{n}+1)\mathcal{L}(b_2)\hfill \\ \hfill & +{\kappa }_a\overline{n}\mathcal{L}(a^{†})+{\kappa }_a(\overline{n}+1)\mathcal{L}(a)\mathrm{,}\hfill \\ H=\hfill & H_0+H_c(t)\mathrm{,}\hfill \\ \mathcal{L}(o)=\hfill & 2o\rho o^{†}-o^{†}o\rho -\rho o^{†}o\mathrm{,}\hfill \end{array}\end{array}

where {\kappa }_{b_1} , {\kappa }_{b_2} and κ_a are the decay rate of the LC oscillators and the nanomechanical resonator, respectively. Here, \overline{n} represents the temperature of the thermal bath coupled to the LC oscillators and the nanomechanical resonator.

3.2 Corresponding Lyapunov control function and control fields

For the sake of convenience, we denote the state of the target nanomechanical resonator by ρ and use this setting in the rest of paper. Based on the Lyapunov function V = T r(P ρ) and the Hamiltonian in Eq. (5) where the free(internal) Hamiltonian is H_0={\text{Ω}}_1{b}_1^{†}{b}_1+{\text{Ω}}_2{b}_2^{†}{b}_2+\omega a^{†}a. Similar to Ref. [63], here we set the Hermitian operator P as P = a^†a, the corresponding Lyapunov control function is [63]

\begin{array}{}\text{(7)}& V(\rho )=Tr(a^{†}a\rho )=\langle n_a\rangle \mathrm{.}\end{array}

In this case, we choose the mean phonon number of the target nanomechanical resonator as the Lyapunov control function V , which is non-negative and becomes zero when the target system is cooled to its ground state. Similar to the procedure in Section 2, we should calculate the time derivative of V according to Eqs. (5)–(7),

\begin{array}{}\text{(8)}& \begin{array}{l}\dot{V}=Tr\{a^{†}a(-\text{i}[H_0+H_c(t)\mathrm{,}\rho ]+\mathcal{P})\}\\ =f_1(t)T_1(t)+f_2(t)T_2(t)+Tr(a^{†}a\mathcal{P})\mathrm{,}\\ \begin{array}{cc}{T}_1(t)& =Tr\{-\text{i}\rho [a^{†}a\mathrm{,}(b_1+b_1^{†})(a+a^{†})]\}\\ & =Tr\{-\text{i}\rho (a^{†}-a)(b_1+b_1^{†})\}\mathrm{,}\hfill \end{array}\\ \begin{array}{ll}{T}_2(t)\hfill & =Tr\{-\text{i}\rho [a^{†}a\mathrm{,}(b_2+b_2^{†})(a+a^{†})]\}\hfill \\ \hfill & =Tr\{-\text{i}\rho (a^{†}-a)(b_2+b_2^{†})\}\mathrm{,}\hfill \end{array}\end{array}\end{array}

where

\begin{array}{}\text{(9)}& \begin{array}{ll}Tr(a^{†}a\mathcal{P})\hfill & =Tr[a^{†}a({\kappa }_a\overline{n}\mathcal{L}(a^{†})+{\kappa }_a(\overline{n}+1)\mathcal{L}(a)]\hfill \\ \hfill & =Tr[2{\kappa }_a\overline{n}\rho a{a}^{†}-2{\kappa }_a(\overline{n}+1)\rho a^{†}a]\hfill \\ \hfill & =Tr[2{\kappa }_a\rho (\overline{n}-a^{†}a)]\mathrm{.}\hfill \end{array}\end{array}

It is noticed that when we realize the cooling of resonator, the temperature of the thermal bath \overline{n} is generally considered to be higher than the temperature of the target resonator 〈a^†a〉. So the term Tr[2{\kappa }_a\rho (\overline{n}-a^{†}a)] in Eq. (9) should be a negative number. Then, we set the time-dependent coupling strength as f₁₍₂₎(t) =−KT₁₍₂₎ (t) with K being a positive constant [63]. In this case, the time derivative of Lyapunov control function \dot{V} = T r{a^†a(−i[H₀ + H_c(t), ρ] + \mathcal{P})}≤0. Because the time derivative of Lyapunov control function V is less than or equal to zero and we choose the mean phonon number of the target resonator as the Lyapunov control function V , the phonon number of target resonator will decrease monotonically. Meanwhile, the target nanomechanical resonator will be gradually cooled to the ground state. Next we will use the the Markovian master equation in Eq. (6) to perform numerical simulation.

3.3 Analysis and numerical simulations

In this subsection, numerical simulation was used to prove the effectiveness of the cooling of resonators with Lyapunov-based control scheme. The simulations are performed in the Fock space. For the sake of simplicity, here we truncate the Fock space of each oscillator up to 15-Fock-states. In order to choose the suitable phonon number truncation point, we compared the simulations with 20-Fock-state truncation and the simulations with 15-Fock-state truncation, the improvement of 20-Fock-state truncation is not significant, so the simulations with 15-Fock-state truncation are reasonable.

Firstly, we assume the target nanomechanical resonator is initially in a thermal state with average phonon number \langle n_{b_1}\rangle = 3 and the auxiliary system is prepared in its ground state. With suitable parameters are chosen as ω = 50 MHz, Ω_b1 = 10ω, Ω_b2 = 15ω, g₁ = g₂ = 0.05ω, K = 0.01, \stackrel{-}{n} = 1, κ_b1 = κ_b2 = κ_a = 10⁻⁵ω, the nanomechanical resonator can be rapidly cooled to its ground state.

The corresponding Lyapunov control fields versus time are shown in Fig. 2. It is worth noting that to avoid T_n(t) = 0 at the initial time, we set the control fields g_l1(t) and g_l2(t) start with a nonzero small number g_l1(0) = g_l2(0) = 0.1ω = 5 MHz. The red and blue solid lines in Fig. 2 show that the control fields will work during the evolution and gradually vanish in the end. So the system will become stable again.

As shown in Fig. 3, the dash blue line represents the cooling process in a open system. The phonon number 〈n_a〉 of the nanomechanical resonator begins to decrease at t = 20 and quickly reduce to 〈n_s〉 = 0.05 at t = 50. Comparing the red solid line which represents the cooling process in a closed system and the blue dotted line in Fig. 3, we find that the decay of the mechanical resonator and the thermal noise from environment will affect the cooling process to some extent. However, the cooling of the mechanical resonator with the Lyapunov-based control method can still be achieved when we take the open environment into consideration. The results show that Lyapunov-based control scheme could be a good candidate to cool a nanomechanical resonator to its ground state in current experimental environment.

4 Fluctuation transfer between membranes

In 2010, Tian et al. proposed that high-fidelity conversion can be realized for states with small photon numbers in systems with experimentally achievable parameters. The pulsed conversion process makes it possible to maintain high conversion fidelity at elevated bath temperatures for states with small photon numbers and in systems with experimentally achievable parameters. An optomechanical interface that converts quantum states between optical fields with different wavelengths opens up a new and promising avenue for interfacing hybrid quantum systems and networks [10]. Then, Garg et al. presented a scheme for the adiabatic transfer of average fluctuations in the phonon number between two membranes in an optical cavity [45]. Soon after, by invariant-based inverse engineering, Chen et al. designed classical driving fields to transfer quantum fluctuations between two suspended membranes in an optomechanical cavity system [46]. Thus, the exchange of the energy fluctuations between two mechanical systems may pose as a possible quantum communication protocol. This clearly opens up an avenue of quantum communication between two truly mesoscopic systems.

Inspired from the previous works [10, 44–46], we propose a scheme based on Lyapunov control function V to transfer the energy fluctuations between two membranes. Different from the reported works [44–46] where a certain initial state or strict adiabatic condition is needed, Lyapunov-based control scheme could realize the energy fluctuations transfer with a random initial state. Besides, there’s no requirement for the strict adiabatic condition. So the scheme is much more universal and easier to realize under the current experimental conditions.

4.1 Physical system and the derivation of effective linearized Hamiltonian

As shown in Fig. 4, the cavity optomechanical system contains a single-mode cavity of frequency ω_c and two suspended membranes. For convenience, we mark the two suspended membranes as the first and the second membrane, respectively. The decay rate and the frequency of the two membranes are γ_m,k and ω_m,k (k = 1, 2). Then, a monochromatic wave Ω is added to drive the modes a. In this case, the Hamiltonian for the system reads (\hslash =1)

\begin{array}{}\text{(10)}& \begin{array}{ll}\hfill & H=H_0+H_I+H_D\mathrm{,}\hfill \\ \hfill & H_0={\omega }_c{a}^{†}a+{\displaystyle \sum _{k=\mathrm{1,2}}}{\omega }_{m\mathrm{,}k}{b}_k^{†}{b}_k\mathrm{,}\hfill \\ \hfill & H_I=-g_1{a}^{†}a(b_1+b_1^{†})+g_2{a}^{†}a(b_2+b_2^{†})\mathrm{,}\hfill \\ \hfill & H_D=\text{Ω}a{\text{e}}^{\text{i}{\omega }_{l}t}+\text{H}\text{.c}\mathrm{..}\hfill \end{array}\end{array}

In Eq. (10), a is the annihilation operator for the cavity mode with corresponding frequency ω_c, while b_k (k = 1, 2) is the annihilation operator of the mechanical mode with respective frequency ω_m,k , and g₁(g₂) determines the optomechanical coupling for the first (second) membrane with cavity mode. In the rotating frame of laser frequency (∆= ω_c−ω_l), the Hamiltonian H takes the following form

\begin{array}{}\text{(11)}& \begin{array}{ll}H=\hfill & \text{Δ}{a}^{†}a+\text{Ω}(a+a^{†})+{\displaystyle \sum _{k=\mathrm{1,2}}}{\omega }_{m\mathrm{,}k}{b}_k^{†}{b}_k\hfill \\ \hfill & -g_1(a^{†}a)(b_1+b_1^{†})+g_2(a^{†}a)(b_2+b_2^{†})\mathrm{.}\hfill \end{array}\end{array}

In order to study the dynamics of the cavity mode and the membranes, we use the standard linearization procedure [2], in which one expands all the bosonic operators as a sum of the average values and the zero-mean fluctuation as follows: a→α + δa and b_k→β_k + δb_k , where α and β_k are generally complex and denote the steady-state values of the annihilation operators. Then, an effective Hamiltonian for the operator fluctuations (derivation is given in Appendix C) is given as

\begin{array}{}\text{(12)}& H_{\text{eff}}=(-g_1{\alpha }^{*}\delta b_1^{†}\delta a+g_2{\alpha }^{*}\delta b_2^{†}\delta a)+\text{H}\text{.c}\mathrm{.,}\end{array}

where

\begin{array}{}\text{(13)}& \alpha =\frac{\text{Ω}}{-{\text{Δ}}^{\prime }+\text{i}\gamma /2}\mathrm{.}\end{array}

4.2 Corresponding Lyapunov control function and control fields

Based on the linearized Hamiltonian of the system in Eq. (C6) where the free (internal) Hamiltonian is \begin{array}{l}{H}_{\mathrm{0,}line}={\text{Δ}}^{\prime }\delta a^{†}\delta a+{\displaystyle {\sum }_{k=\mathrm{1,2}}{\omega }_{m\mathrm{,}k}\delta b_k^{†}\delta b_k}\hfill \end{array} , we utilize the Lyapunov control function V = T r(P ρ) to achieve the fluctuation transfer between membranes. The Hermitian operator P is chosen as P=\delta b_2^{†}\delta b_2-\delta b_1^{†}\delta b_1 and the corresponding Lyapunov control function is

\begin{array}{}\text{(14)}& V(\rho )=Tr(\delta b_2^{†}\delta b_2\rho -\delta b_1^{†}\delta b_1\rho )\mathrm{.}\end{array}

Namely, we choose the mean fluctuation excitations from mode b₂ to minus the mean fluctuation excitations from mode b₁ as the Lyapunov control function. In order to obtain the concrete expression form of effective control Hamiltonian H_c(t) and design the corresponding control fields, we calculate the time derivative of V as

\begin{array}{}\text{(15)}& \begin{array}{ll}\hfill & \dot{V}={\displaystyle \sum _{n=1}^k}{f}_n(t)T_n(t)\mathrm{,}\hfill \\ \hfill & T_n(t)=Tr\{-\text{i}\rho [\delta b_2^{†}\delta b_2-\delta b_1^{†}\delta b_1\mathrm{,}{H}_n]\}\mathrm{,}\hfill \\ \hfill & H_1=-g_1{\alpha }^{*}\delta b_1^{†}\delta a+\text{H}\text{.c}\text{.,}\hfill \\ \hfill & H_2=g_2{\alpha }^{*}\delta b_2^{†}\delta a+\text{H}\text{.c}\text{.,}\hfill \\ \hfill & T_1=Tr\{-\text{i}\rho (g_1{\alpha }^{*}\delta b_1\delta a^{†}-g_1\alpha \delta b_1^{†}\delta a)\}\mathrm{,}\hfill \\ \hfill & T_2=Tr\{-\text{i}\rho (-g_2{\alpha }^{*}\delta b_2\delta a^{†}+g_2\alpha \delta b_2^{†}\delta a)\}\mathrm{.}\hfill \end{array}\end{array}

When f_n(t) is chosen to keep \dot{V}\le 0 , the mean fluctuation excitations transfer from mode b₁ to mode b₂. After setting the time-dependent coupling strength as f_n(t) =−KT_n(t) with K a positive constant, we obtain the control Hamiltonian H_c(t) as

\begin{array}{}\text{(16)}& \begin{array}{ll}{H}_c(t)\hfill & ={\displaystyle \sum _{n=1}^k}{f}_n(t)H_n\hfill \\ \hfill & =-K{T}_1{H}_1-K{T}_2{H}_2\mathrm{.}\hfill \end{array}\end{array}

With control Hamiltonian H_c(t) in Eq. (16), the mean fluctuation excitations transfer from mode b₁ to mode b₂ while V decreases to its minimum value.

4.3 Analysis and numerical simulations

In this subsection, we show in details how to transfer fluctuation excitations from mode b₁ to mode b₂ with the help of Lyapunov-based control scheme. After linearizing the Hamiltonian of the system, we only consider the average quantum fluctuation instead of the actual motion of the mechanical membranes. Assuming that the decay rates of the cavity and membranes are much smaller than their fundamental frequencies such that there is no significant decay of the photons during the transfer process. (We verify that the moderate values of decay rates of the subcavity modes and the membranes do not affect the transfer process much in Appendix A.)

The simulations are performed in the Fock space. For the sake of simplicity, here we also truncate the Fock space of each oscillator up to 15-Fock-states and the dissipation of each resonator is ignored. The reason is similar to that mentioned in Section 3.

Firstly, we assume the first resonator is initially in a thermal state with average phonon number \langle n_{b_1}\rangle = 3.35 and the auxiliary system is prepared in its ground state. For the sake of convenience, we define the final average excitation fluctuation of the second membrane \langle \delta b_2^{†}\delta b_2\rangle = F as the fidelity of the transfer process. For each case, the control field g(t) starts with a nonzero small number to avoid f_n(t) at the initial time.

With suitable parameters, the Lyapunov-based control fields versus time are shown in Fig. 5. From Fig. 5 we can see that with the evolution going on, the control function f₁(t)(f₂(t)) gradually decreases and finally disappears at the end of the evolution. This condition guarantees the stability of the system because the control fields will vanish after a spell of time. Finally, the system is only affected by the original Hamiltonian H_eff.

As shown in Fig. 6, we display the evolution of the average excitation fluctuations of the two membranes by solving numerically the Langevin equations. The fluctuation in the first membrane is gradually transferred to the second one after we apply the Lyapunov-based control fields in Eq. (16) to the system. Such a transfer of fluctuations between two membranes is not perfect because we should take the effect caused by time delay into consideration.

4.4 Time delay effect

In this subsection, we should take the non-negligible (positive or negative) time delay into consideration for better practical implementation of the scheme. Here we set these time delays may stem from actuators and electronic devices in the control loop [68].

Consider a cavity optomechanics system with a free evolution Hamiltonian H_0,delay and a control Hamiltonian H_c,delay . For the sake of generality, the free Hamiltonian H_0,delay can be written as

\begin{array}{}\text{(17)}& H_{\mathrm{0,}delay}={\displaystyle \sum _j^{\infty }\delta a^{†}\delta a+}{\displaystyle \sum _j^{\infty }{\omega }_{m\mathrm{,}j}\delta b_j^{†}\delta b_j\mathrm{.}}\end{array}

Then we choose the target state as the ground state |ϕ〉 of H_0,delay . The control Hamiltonian H_c,delay is fully connected, which means for any m and n, we have 〈ϕ_n|H_c,delay |ϕ_m〉≠ 0.

It is obvious that the controlled dynamic is affected by the delay through the control field f (t−τ ) significantly. In this case, the control problem would be much more complex and difficult to solve than that without time delay. Thus, we shall use the control fidelity defined by F=\langle \delta b_2^{†}{b}_2\rangle to quantify the effect caused by time delay. After taking the time delay into consideration, the control fidelity F becomes a function of the delay time τ . When the delay time τ is small enough to be viewed as perturbation, the fidelity can be expanded up to the second order in τ as \begin{array}{l}F(t\mathrm{,}\tau )=F(t\mathrm{,}\tau =0)+\frac{\partial F(t\mathrm{,}\tau )}{\partial \tau }{|}_{\tau =0}\cdot \tau +\hfill \end{array} \frac{1}{2}\frac{{\partial }^{2}F(t\mathrm{,}\tau )}{\partial {\tau }^2}{|}_{\tau =0}\cdot {\tau }^2+\cdots· [69]. Here we should notice that the delay time τ only connects with the control field f_n(t), so we further simplify the function of the control fidelity F and have \frac{\partial F}{\partial \tau }=\frac{\partial F}{\partial f}\frac{\partial f}{\partial \tau }\text{and}\frac{{\partial }^{2}F}{\partial {\tau }^2}=\frac{{\partial }^{2}F}{\partial f^2}\frac{\partial f^2}{\partial {\tau }^2}. Considering that \frac{\partial F}{\partial f} remains unchanged for the control system regardless of the time delay, so we find \frac{\partial f}{\partial \tau } plays a crucial role in the time delay effects. With a small time delay τ , the control field f (t−τ ) can be expanded up to second order in {\tau }^2\text{ as }f(t-\tau )\approx f(t)-\frac{\partial f(t)}{\partial t}\tau +\frac{1}{2}\frac{{\partial }^{2}f(t)}{\partial t^2}{\tau }^2. It is obvious that in the presence of time delay, the final control fidelity depends not only on the delay time, but also on the initial states. The control fidelity is sensitive to the time delays so it is quite a challenge to realize the perfect fluctuation transfer between membranes with Lyapunov-based control scheme when we take the time delay into consideration [69].

5 Bang–bang solutions of control fields’ waveform

5.1 The bang–bang design of waveform

In the Lyapunov-based control scheme, the amplitude of the control fields f_n(t) is proportional to T_n(t) and will still change during the whole evolution process. For better practical implementation of the scheme, we need to manipulate the control fields at each instance of time and compensate the positive or negative effect caused by time delay in experiments. Therefore, this kind of amplitude may be difficult to control in realistic situations and errors would be introduced in the complexity regulation because the control fields are a continuous function of time. Thus, this is a challenging task for practical applications. One solution to this problem is to replace the control field by a square pulse train called bang–bang control fields.

Bang–bang control is a kind of feedback control method that switches suddenly between two states. Because of simplicity or convenience, bang–bang solutions are ubiquitous with applications including atom cooling [70, 71], transport of trapped-ion qubits [72–74], ground-state preparation [75], and so on. Besides, this method can be further extended to solve other problems such as the simplification of control fields’ waveform. In this case, the bang–bang control is restricted to switch suddenly between a lower bound and an upper bound. Therefore, we try to use the bang–bang solutions to simplify the Lyapunov control fields and call the optimized fields as optimized bang–bang control fields. Similar to Ref. [63], we consider the following factors about the amplitude of the control fields f_n(t): (i) Weak amplitude would lead to long evolution time. (ii) Strong external control fields may destroy the approximate condition. (iii) Strong fields may disturb neighboring quantum systems that we do not want to be involved in. Based on these factors, we set a constraint on the strength of each field as Q, and the simplified control fields become [63]

After using the corresponding bang–bang optimization process in Eq. (18) to modify the Lyapunov control field’s waveform, the simplified control Hamiltonian H_c(t) becomes

This design has the bang–bang type waveform and may be easy to realize experimentally [69]. The unwilling effect caused by time delay is successfully eliminated because there is no need to change the amplitude of the control fields at each instance of time. Besides, the bang–bang-based control scheme could maximize the use of energy because the strength of each field is always at the max constraint value |Q| during the whole evolution. Therefore, we cannot help but wonder if the optimized bang–bang control fields could effectively shorten the time of evolution. In order to find the answer, we use bang–bang-based control scheme to realize the fluctuation transfer between membranes and make a comparison between the optimized bang–bang control fields and the Lyapunov control fields in the following numerical simulation.

5.2 Analysis and numerical simulations

In this subsection, we first show in details how to transfer fluctuation excitations from mode b₁ to mode b₂ with the help of bang–bang-based control scheme. Then, in order to find out whether the bang–bang-based control scheme has an advantage in terms of evolution rate, we make a comparison between the transfer effects caused by the bang–bang-based control scheme and by the Lyapunov-based control scheme.

Similar to that in Section 4, the simulations are performed in the Fock space. Firstly, we assume the first resonator b₁ is initially in a thermal state with average phonon number \langle n_{b_1}\rangle = 3.35 and the auxiliary system is prepared in its ground state. For the sake of convenience, we define the final average excitation fluctuation of the second membrane F=\langle \delta b_2^{†}{b}_2\rangle as the fidelity of the transfer process. For each case, the control field g(t) starts with a nonzero small number to avoid f_n(t) at the initial time.

As shown in Fig. 7, the optimized bang–bang control fields have the bang–bang waveform with maximum strength Q = 0.06. (According to Fig. 5, the strength of each control field is no more than 0.06 so we set the maximum strength of each field as |Q| = 0.06.) The strength of each field is always at the max constraint value |Q| = 0.06 during the whole evolution, so the optimized bang–bang control fields make full use of control fields’ energy and are easy to realize experimentally. When the final average excitation fluctuation of the second membrane F=\langle \delta b_2^{†}{b}_2\rangle is larger than 0.99, we turn off the control fields and the system becomes stable.

After optimizing the control fields, in order to learn about the fluctuation transfer effect of the bang–bang-based control scheme, we simulate the evolution of the average excitation fluctuations of the two membranes by solving numerically the Langevin equations. As shown in Fig. 8, complete fluctuation transfer between membranes can be achieved. The fluctuation in the first membrane is gradually transferred to the other one after we apply the bang–bang-based control fields in Eq. (19) to the system.

Noted that when the evolution arrives t = 0.8, the modes b₁ and b₂ almost reach 0 and 1, respectively, which means the process of transfer has basically completed. There is no need of the further change of f₂(t). Meanwhile, when the system is near the target state, the control fields almost makes no sense to the evolution. So we can add a judgment statement when we write the code, if the the mode b₂ > 0.99, we turn off the control fields f₁(t) and f₂(t) and the fluctuation transfer is obtained.

In the following, in order to demonstrate the advantages of the bang–bang-based control scheme over the traditional one, we display the average excitation fluctuations of the second membrane for these two kinds of control scheme in Fig. 9.

Figure 9 shows that while the blue dotted line reaches 1, the red solid line only reaches 0.7. It is obvious that once we use the bang–bang-based control scheme, the efficiency of fluctuation transfer has been significantly improved, compared with the Lyapunov-based control scheme. Because the bang–bang-based control scheme leads to a faster decrease of Lyapunov control function V and the system will evolve to the target state with a faster evolution speed. Besides, the optimized bang–bang control fields are easy to realize in current experiments without causing operational errors. So it is seen that the simplified control fields can be a good candidate for realistic experimental environment.

Interestingly, when we use the optimized bang–bang control fields to drive the system, these control fields may oscillate with very high frequency at the end of evolution. These oscillations are closely associated with the initial state, the strength of control fields, and the distance between the actual and target states. In bang–bang-based control scheme, f_n(t) is designed to take the value Q or−Q, which makes the state oscillate at almost every step of simulation. While in Lyapunov-based control scheme, the control field f_n(t) decreases to zero with T_n(t)→0. Nevertheless, this problem can be solved by averaging the control fields over a proper time period and using the reshaped fields instead of the oscillating one [63]. By doing so, the same control fields and results would be obtained.

6 Conclusion

In this paper, we have proposed a quantum control scheme based on Lyapunov control function V . To demonstrate the effectiveness of Lyapunov-based control scheme, we realize the cooling of mechanical oscillator in LC circuit and successfully transfer the fluctuation between membranes without meeting strict adiabatic conditions or specific initial state in cavity optomechanics system. The simulations show that the implementation of Lyapunov-based control scheme will drive the system gradually to evolve to the target state with the decrease of Lyapunov control function V . However, in the Lyapunov-based control scheme, the amplitude of the control fields f_n(t) is proportional to T_n(t) and will change during the evolution. To realize such fields in experiments, we need to manipulate the control fields at each instance of time and compensate the positive or negative effects caused by time delay. Therefore, we try to use the bang–bang solutions to simplify the control fields. Some optimizations are made on the Lyapunov-based control waveform and the bang–bang-based control scheme makes Lyapunov control function V decrease faster. So the evolution time is effectively reduced. In addition to this, the optimized design of control fields makes this scheme easier to implement under current experimental conditions because the unwilling effect caused by time delay is successfully eliminated. There is no need to change the amplitude of the control fields at each instance of time so the errors introduced in the complexity regulation could be reduced.

It is worth emphasizing that the control theory is not limited to cooling of the resonators and the fluctuation transfer between membranes, and can also be utilized to study the actual motion of the mechanical membranes and other quantum information tasks. Hence, we hope that the work may open a new avenue of the experimental realization of cooling mechanical oscillator, quantum fluctuations transfer between membranes and other quantum information tasks in the near future.

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