# Amplitude-modulation-based atom-mirror entanglement and mechanical squeezing in a hybrid optomechanical system

###### Abstract

We consider a hybrid optomechanical system which is composed of the atomic ensemble and a standard optomechanical cavity driven by the periodically modulated external laser field. We investigate the asymptotic behaviors of Heisenberg operator first moments and clearly show the approaching process between the exact numerical results and analytical solutions. Based on the specific modulation forms of external driving and effective optomechanical coupling, we discuss in detail the atom-mirror entanglement enhancement, respectively. Compared with the constant driving regime, the entanglement can be greatly enhanced with more loose cavity decay rate and is more resistant to the thermal fluctuations of the mechanical bath. The desired form of periodically modulated effective optomechanical coupling can be precisely engineered by the external driving modulation components which can be derived analytically via Laplace transform. Meanwhile, resorting to the quantum interference mechanism caused by atomic ensemble and modulating the external driving appropriately, the mechanical squeezing induced by the periodic modulation can be generated successfully in the unresolved regime.

###### pacs:

42.50.Ct, 42.50.Lc, 42.50.Pq, 07.10.Cm## I Introduction

Recently, cavity optomechanics, as a controllable radiation-pressure interaction interface between cavity field and mechanical motion, has been raised more and more attention and research interests in both experimental and theoretical aspects 2007OE15017172 (); 2009Physics (); 2013AnnPhys525215 (); 2013CPB22114213 (); 2014RMP861391 (); 2018CPB27024204 (). Particularly, with the fast-developing fields of microfabrication and nanotechnology, the significant progress of the experiments about cavity optomechanics has been made. So far, a variety of experimental structures can realize this controllable radiation-pressure interaction: whispering gallery microdisk and microsphere 2009Nature462633 (); 2009NaturPhysics5489 (), dielectric membrance 2008Nature45272 (), nanorod 2009OE1712813 (), silicon photonic waveguide 2008Nature456480 (), and so forth. It is well known that the main potential applications about this subject are to test fundamentals of quantum mechanics for macroscopic systems 2013RMP85471 () and to build quantum sensors for ultrahigh-precision measurements 2012PRA86053806 (); 2014PRA90043825 (); 2015PRA91063827 (); 2015APL106121905 (); 2017PRA95023844 (). The generation of quantum entanglement between macroscopic objects has been a significantly important goal both in fundamental studies 2002JPCM14 (); 2003RMP75715 () and in numerous potential applications related to quantum computation, quantum communication, quantum information processing, etc 2003PRL9013 (); 2005RMP77513 (); 2011PRA84052327 (). To this end, besides the entanglement generation between cavity field and mechanical oscillator 2007PRL98030405 (); 2008PRA78032316 (); 2011PRA84042342 (); 2012PRA86042306 (), many schemes have been put forward to generate the entanglement between two macroscopic oscillators 2009NJP11103044 (); 2006PRL97150403 (); 2011PRL107123601 (); 2013PRA87022318 (); 2014PRA89014302 (); 2015NJP17103037 (); 2016PRA94053807 (); 2017PRA95043819 (); 2017SR72545 (). We also note that the remote quantum entanglement between two mechanical oscillators across two chips that are separated by 20 centimeters has been reported very recently 2018Nature556473 ().

On the other hand, achieving mechanical squeezed state is also a task of paramount importance since it contributes to the realization of ultrahigh precision detection at or even below the standard quantum limit. In recent years, based on the cavity optomechanics, different methods to generate mechanical squeezing have been proposed, including quantum feedback control 2009PRA79052102 (), parametric amplification and weak measurement 2011PRL107213603 (), dissipative optomechanical coupling 2013PRA88013835 (), quadratic optomechanical coupling 2014PRA89023849 (), squeezing transfer from photons to phonons 2016PRA93043844 (), nonlinearity 2015PRA91013834 (); 2016SR624421 (); 2016SR638559 (), etc. In addition, the quantum squeezing of mechanical oscillator has also been observed in experiment 2015Science349952 ().

In parallel with the development in cavity optomechanics, one modulation approach of particular interest is the so-called periodic modulation. The novel phenomena generated by applying periodic modulation to the cavity optomechanics have also been reported 2012NJP14075014 (); 2013OE21020423 (); 2014PRA89023843 (); 2018PRA97042314 (); 2018OE26013783 (); 2018arxiv (); 2009PRL103213603 (); 2012PRA86013820 (); 2011PRA83033820 (); 2018OE26011915 (); 2018PRA97022336 (). By appropriately modulating the time-periodic driving, Mari and Eisert have demonstrated that the entanglement between cavity field and mechanical oscillator can be greatly enhanced and large degrees of mechanical squeezing emerges in the resolved sideband regime 2009PRL103213603 (). By further investigating and analyzing the interplay between the mechanical frequency modulation and input laser intensity modulation, it finds that an interference pattern presents and different choices of the relative phase between two modulations can either enhance or suppress the desired quantum effects in optomechanical system 2012PRA86013820 (). Liao and Law proposed a method of reaching parametric resonance via modulating the driving field amplitude to generate the quadrature squeezing of mirrors 2011PRA83033820 (). The method of simultaneously modulating the radiation-pressure coupling and mechanical spring constant to generate the ponderomotive squeezing and mechanical squeezing in the resolved sideband regime is also proposed recently 2018OE26011915 (). We also note that by combing the periodic modulations for both the driving laser and the mechanical coupling strength simultaneously, entanglement dynamics of two coupled mechanical oscillators is investigated 2018PRA97022336 ().

In this paper, we extend the optomechanical model in Ref. 2008PRA77050307 () to the scenario of periodic modulation. We focus here on enhancing the atom-mirror entanglement for the specific modulation forms of external driving and effective optomechanical coupling, respectively. Compared to the constant driving regime, the periodic modulation greatly enhances the atom-mirror entanglement and its resistance to the thermal fluctuations of the mechanical bath with more loose cavity decay rate. In addition, the desired periodically modulated form of effective optomechanical coupling can be precisely engineered by the external driving modulation components which can be derived analytically via Laplace transform. Meanwhile, besides the entanglement illustrated in Ref. 2008PRA77050307 (), the periodic modulation can also induce mechanical squeezing by appropriately modulating the external driving. Resorting to the quantum interference mechanism caused by atomic ensemble, we generate the mechanical squeezing successfully in the unsolved sideband regime without the need of any feedback or additional squeezed light driving. We depict the Wigner functions at some different specific times and the time evolution of squeezing parameter and find that the mechanical oscillator is always squeezed but, due to the external driving modulation, the direction of squeezing rotates continuously in the phase space with the same period of modulation.

The rest of this paper is organized as follows. In Sec. II, we describe the physical model and obtain the linearized dynamics of the system. In Sec. III, we solve the dynamics of Heisenberg operator first moments analytically with a general periodically modulated amplitude in a perturbative way and derive the equation of motion for the correlation matrix which can completely describe the dynamics of the quantum fluctuations. In Sec. IV, we analyze the asymptotic behaviors of the first moments for the specific modulation form of external driving. We discuss in detail the atom-mirror entanglement enhancement based on the specific modulation forms of external driving and effective optomechanical coupling, respectively, in Sec. V. Via appropriately modulating the external driving, we illustrate the generation of mechanical squeezing in the unresolved sideband regime in Sec. VI. Finally, we present our conclusions in Sec. VII.

## Ii Model and Hamiltonian

The hybrid optomechanical system under consideration is depicted in Fig. 1, in which a cloud of identical two-level atoms (with frequency and decay rake ) is trapped in a standard optomechanical system. An external laser filed with a time-dependent amplitude and frequency drives the optomechanical cavity and the movable mirror modeled as mechanical oscillator with frequency and decay rate is coupled to the optical field via the radiation-pressure interaction. The total Hamiltonian of the system (in the unit of ) is given by the sum of a free evolution term

(1) |

and the interaction term

(2) |

Here, () is the annihilation (creation) operator of the cavity field (with frequency and decay rate ); the collective spin operators of atoms are defined in terms of Pauli Matrices ; and () is the dimensionless position (momentum) operator of the mechanical oscillator, satisfying the standard canonical commutation relation . and refer to the atom-cavity coupling strength and the single-photon radiation-pressure coupling strength, respectively.

Under the conditions of sufficiently large atom number and weak atom-cavity coupling, the dynamics of the atomic polarization can be described in terms of a collective bosonic operator () in the Holstein-Primakoff representation 2008PRA77050307 (); 2015PRA92033841 ()

(3) |

In the rotating frame with respect to laser frequency , the total Hamiltonian is rewritten as follows:

(4) |

where and are, respectively, the cavity and atomic detuning with respect to the laser. is the collective atom-cavity coupling strength. The time-dependent amplitude is imposed the structure of a periodic modulation such that for some of the order of , though the particular form is left unspecified yet.

In addition to the coherent dynamics, the system is also unavoidably affected by the fluctuation-dissipation processes resulting from the environment. Taking all the damping and noise terms into account, the dynamics of the system is completely described by the following set of nonlinear quantum Langevin equations (QLEs) Book1 ():

(6) | |||||

(8) | |||||

(10) | |||||

(11) |

where and are the zero-mean noise operators for cavity and atoms, respectively, with the only nonzero correlation functions

(13) | |||||

(14) |

is the stochastic Hermitian Brownian noise operator describing the dissipative friction forces subjecting to the mechanical oscillator and its non-Markovian correlation function is given by 2001PRA63023812 ()

(15) |

where is the Boltzmann constant and is the temperature of the mechanical bath. However, for a high quality mechanical oscillator with , the colored spectrum of Eq. (15) acquires the Markovian character Note1 ()

(16) |

where is the mean occupation number of the mechanical mode and is the anticommutator.

In general, the set of coupled nonlinear QLEs in Eq. (6) is difficult to solve directly. However, when the system is strongly driven to a large first moment by the external laser, one can rewrite each Heisenberg operator as follows: (), where are quantum fluctuation operators with zero-mean around the classical -number first moments . Thus, the standard linearization techniques can be applied to the nonlinear QLEs in Eq. (6). The equation of motion for Heisenberg operator first moments of system is given by the following set of nonlinear differential equations:

(18) | |||||

(20) | |||||

(22) | |||||

(23) |

The linearized QLEs for the quantum fluctuation operators are

(25) | |||||

(27) | |||||

(29) | |||||

(30) |

and the corresponding linearized system Hamiltonian for the quantum fluctuation operators reads

(33) | |||||

## Iii Dynamics of Heisenberg operator first moments and quantum fluctuations

The classical evolution of the system is governed by the dynamics of the first moments in Eq. (18). One can get the time evolution of the first moments by numerically solving the set of differential equations in Eq. (18) although it is nonlinear. However, when the system is far away from the regions of optomechanical instabilities and multistabilities 2008NJP10095013 (), the radiation pressure coupling can be treated in terms of perturbation. Besides, since , according to Floquet’s theory, the asymptotic solutions of first moments will have the same periodicity of modulation in the long time limit: . Thus one can perform a double expansion of the asymptotic solutions in powers of the radiation pressure coupling strength and in terms of Fourier components:

(34) |

where () are integers and is the fundamental modulation frequency. In addition, the periodic driving amplitude can also be expanded in terms of the similar Fourier series

(35) |

Substituting Eqs. (34) and (35) into Eq. (18), the time-independent coefficients in Eq. (34) are obtained by the following set of recursive relations:

(37) | |||||

(39) | |||||

(40) |

corresponding to the zero-order perturbation with respect to , and for all ,

(42) | |||||

(44) | |||||

(46) | |||||

(47) |

For all the calculations carried out in this paper, we truncated the analytical solutions up to and so that the level of approximation is high enough to agree well with the exact numerical solutions.

On the other hand, as long as the time evolution of the first moments is obtained, the dynamics of the corresponding quantum fluctuations is easy to be solved. To this end, it is convenient to introduce the quadrature operators and corresponding Hermitian input noise operators for the cavity field and atoms, respectively,

(49) | |||||

(51) | |||||

(53) | |||||

(54) |

and the vectors of quadrature fluctuation operators and corresponding noises are

(56) | |||||

(57) |

So the linearized QLEs which govern the dynamics of the quantum fluctuations can be written in a compact form:

(58) |

in which is a time-dependent matrix:

(59) |

where

(60) |

is the effective time-modulated detuning, and and are the real and imaginary parts of the effective optomechanical coupling, respectively,

(61) |

Thanks to the linearized dynamics for the quantum fluctuations and the zero-mean Gaussian nature for the quantum noises, the time evolution of the quantum fluctuations can be completely described by the correlation matrix (CM) whose matrix element is defined by

(62) |

The equation of motion for the CM can be derived from Eqs. (58) and (62) (see Appendix A)

(63) |

where refers to the transpose of , and is a diffusion matrix whose matrix element is related to the noise correlations and defined by

(64) |

from the correlation functions of Eqs. (13) and (16), the diagonal matrix is

(65) |

## Iv Asymptotic behaviors of Heisenberg operator first moments for the specific modulation form of external driving

In the long time asymptotic regime, the time-dependent matrix will have the same periodicity of the modulation, i.e., . Moreover, the stability of the system requires that all of the eigenvalues of have negative real parts for all time . In what follows, the stability condition should be carefully guaranteed anytime.

To proceed, we investigate the asymptotic behaviors of Heisenberg operator first moments for the specific modulation form of external driving. As mentioned above, in the long time limit, the Heisenberg operator first moments () will have the same periodicity of the performed modulation. For simplicity, we restrict the modulation form of external driving only to the lowest-amplitude components: . In Fig. 2, we plot the asymptotic time evolution of the real and imaginary parts of the first moments and . One can clearly see that the first moments and are indeed periodicity in the long time limit. We also check the approximation validity of the analytical solutions in Eqs. (37) and (42) via comparing with the exact numerical results obtained through Eq. (18). We find that, after about 50 modulation periods, the analytical solutions agree with the exact numerical results very well.

Due to the asymptotic periodicity, the trajectories of the first moments in the phase space will converge to a limit cycle in the end. For this reason, to gain more insights about this characteristic and clearly show the asymptotic behaviors, as illustrated in Fig. 3, we plot the phase space trajectories of the first moments for the mechanical oscillator and in the particular time intervals. One can find that, with the periodic modulation proceeding, the trajectories indeed gradually converge to a limit cycle which is well approximated by the analytical prediction. Figure 3 also clearly presents the slow asymptotic process between the exact numerical results and analytical solutions. The system thus obtains the same period of the performed modulation in the long time limit.

## V Atom-mirror entanglement enhancement

### v.1 Atom-mirror entanglement enhancement for the specific modulation form of external driving

In this section, we investigate the atom-mirror entanglement enhancement under the specific modulation form of external driving. Since the asymptotic state of the system is Gaussian, it is very convenient to introduce the logarithmic negativity to measure the entanglement 2002PRA65032314 (); 2004PRA70022318 (), which can be readily computed from the reduced CM for the collective atomic mode and mechanical mode. can be obtained from the full CM by just extracting the first two and last two rows and columns. If the reduced CM is written in the following form:

(66) |

where , , and are subblock matrices of , which describe the local properties of mechanical mode, collective atomic mode, and the intermode correlation between them, respectively, then is defined as

(67) |

with

(69) | |||||

(70) |

The collective atomic mode and mechanical mode are said to be entangled () if and only if which is equivalent to Simon’s necessary and sufficient nonpositive partial transpose criteria 2000PRL842726 ().

For comparison, we first present the steady-state entanglement behavior when there is no periodic modulation (). In Fig. 4(a), we plot versus the time-independent driving amplitude and atom-cavity coupling strength . One can note that the atom-mirror entanglement in the steady state is relative small (one can also see Ref. 2008PRA77050307 ()) and is generated with relatively extreme system parameter (small cavity decay rate ). In Fig. 4(b), we present versus the atom-cavity coupling strength and mean thermal phonon occupation number , which shows that is very susceptible to thermal fluctuations of the mechanical bath and the atom-mirror entanglement can only exist in the case of low mean thermal occupation number.

However, if applying the specific modulation form of external driving () to drive the system, from Fig. 5(a), we note that, after long enough modulation time, not only obtains the same periodicity of the performed periodic modulation, but also is greatly enhanced with a more loose cavity decay rate (). Figure 5(b) shows the asymptotic time evolution of in the presence of thermal phonon number where the atom-mirror entanglement almost vanishes in the case of without performing periodic modulation. As clearly illustrated in Fig. 5(b), we find that, compared to the case of without modulation, the entanglement is also more robust against the thermal fluctuations of the mechanical bath. In addition, in both Figs. 5(a) and 5(b), the entanglement behaviors corresponding to the analytical solutions of first moments in Eq. (18) agree with the exact results obtained from the numerical solutions of first moments in Eqs. (37) and (42) very well.

### v.2 Atom-mirror entanglement enhancement for the specific modulation form of effective optomechanical coupling

In above subsection, we investigate the atom-mirror entanglement enhancement via the specific modulation form of the external driving. In this subsection, we focus on the atom-mirror entanglement enhancement for the specific modulation form of effective optomechanical coupling and further to determine the explicit form of the external driving. To this end, we first assume a simple structure for the asymptotic effective time-dependent optomechanical coupling

(71) |

where and are time-independent positive reals and related to the external driving modulation components . According to Eq. (61), once the effective optomechanical coupling is left specified, we can obtain the first moment of cavity mode . Then the corresponding other first moments of system and the explicit external driving can be analytically derived from Eq. (18) via Laplace transform

(73) | |||||

(75) | |||||

(77) | |||||

(78) |

where

(80) | |||||

(82) | |||||

(84) | |||||

(86) | |||||

(88) | |||||

(89) |

In the long time limit, due to , one has , , and , and the following approximations

(91) | |||||

(93) | |||||

(94) |

Thus, we have

(96) | |||||

(98) | |||||

(100) | |||||

(102) | |||||

(103) |

where the external driving modulation components are

(105) | |||||

(107) | |||||

(108) |

To verify the validity of above approximations, as shown in Fig. 6, we plot the asymptotic time evolution of the real and imaginary parts of effective optomechanical coupling obtained from, respectively, the numerical result by performing the particular form of external driving modulation shown in Eq. (96) and the specified in Eq. (71). The insets of Fig. 6 clearly present the slow approaching process of these two results and after about 30 modulation periods, the desired periodically modulated effective optomechanical coupling in Eq. (71) can be indeed precisely engineered by implementing the external driving modulation components shown in Eq. (105).

Figure 7 shows the atom-mirror entanglement behaviors in the cases of the periodic and constant structures as for the effective optomechanical coupling. From Fig. 7, one can find that, in the constant structure, the atom-mirror entanglement is weak and can only exist in the low mean thermal phonon number limit with the relatively small cavity decay rate yet (). Instead, once applying the periodic structure, the entanglement is not only significantly enhanced with a more loose cavity decay rate (), but also is more resistant to thermal fluctuations of the bath.

## Vi Mechanical squeezing generation in the unresolved sideband regime via modulation

We now turn to investigate the mechanical squeezing generation in the unresolved sideband regime () via periodic modulation. The variances of the quantum fluctuations around the first moments for the mechanical position and momentum operators are in the form and , respectively. The mechanical oscillator is squeezed if either or is less than 1/2. Due to and , so the equation of motion for the CM in Eq. (63) can also completely characterize the time evolution of variances for the mechanical position and momentum operators ( () just is the matrix element of CM ()).

In Fig. 8, we plot the time evolution of variance for the mechanical oscillator position operator from to in the periodic driving modulation regime. As a comparison, we first present the result in the case of without periodic modulation (). For , the system turns into constant driving regime and when the initial mean thermal phonon number , as the blue solid line depicted in Fig. 9, due to the quantum interference mechanism induced by atoms inside the cavity 2015PRA92033841 (); 2009PRA80061803 (); 2015JOSAB322314 (); 2018OE266143 (), the mechanical oscillator can always be retained close to ground state in the total evolution precess even though . Even if the initial mean thermal phonon number , as shown by the red solid line, the mechanical oscillator can still be cooled close to ground state with the time evolution finally. Since the mechanical oscillator is prepared in an approximate vacuum state at the time interval , as the circle lines shown in Fig. 8</