Qubit-assisted squeezing of the mirror motion in a dissipative optomechanical cavity system
We investigate a hybrid system consisting of an atomic ensemble trapped inside a dissipative optomechanical cavity assisted with the perturbative oscillator-qubit coupling. It is shown that such a hybrid system is very suitable for generating stationary squeezing of the mirror motion in the long-time limit under the unresolved sideband regime. Based on the approaches of master equation and covariance matrix, we discuss the respective squeezing effects in detail and find that in both approaches, simplifying the system dynamics with adiabatic elimination of the highly dissipative cavity mode is very effective. In the approach of master equation, we find that the squeezing is a resulting effect of cooling process and is robust against the thermal fluctuations of the mechanical mode. While in the covariance matrix approach, we can obtain the analytical result of the steady-state mechanical position variance from the reduced dynamical equation approximately. Finally, we compare the two different approaches and find that they are completely equivalent for the stationary dynamics. The scheme may be meaningful for the possible ultraprecise quantum measurement involved mechanical squeezing.
pacs:42.50.Dv, 42.50.Ct, 42.50.Pq, 07.10.Cm
Many significant progresses have been achieved with the recent advance of cavity optomechanics over the last few years 2014RMP861391 (). Examples include ground-state cooling of the mechanical mode 2007PRL99093901 (); 2007PRL99093902 (); 2014PRA90053841 (); 2015SC58516 (); 2018PRA98023816 (), macroscopic entanglement between two spatially separated movable mirrors 2014PRA89023843 (); 2015SC58518 (); 2018FP13130319 (), optical multistability behavior 2016PRA93023844 (); 2017SC60010311 (), and so on.
Thereinto, generation of non-classical states of motion around the ground state based on the cavity optomechanical system is one of the most effective methods to study the quantum effects at mesoscopic or macroscopic scales. Specifically, the quantum squeezing associated with the mechanical motion, reduction of the quantum fluctuation in its position or momentum below the quantum noise limit, is not only of significant importance for testing the quantum fundamental theory 2012PT6529 (), such as exploring the quantum-classical boundary 1991PT4436 (), but also has widely potential applications, such as the detection of gravitational waves 1980RMP52341 (); 1992Science256325 (); 1999PT5244 (). Thus, achieving squeezed state in mechanical oscillator (mirror) is a greatly desired goal.
To this end, several well-known methods and techniques to generate squeezing of the mechanical mode were proposed 1991PRL673665 (); 2009PRL103213603 (); 2011PRA83033820 (); 2013OE21020423 (); 2018OE26013783 (); 2013PRA88063833 (); 2009PRA79063819 (); 2010PRA82033811 (); 2016PRA93043844 (); 2015PRA91013834 (); 2010PRA82021806R (); 2014PRA89023849 (); 2018PRA97043619 (); 2018PRA98023807 (); 2008APL92133102 (); 2015PRL115243601 (). One of the early most outstanding schemes was to modulate the frequency of the oscillator 1991PRL673665 (). Nevertheless, although this is simplest, it is not easy to utilize for many different types of mechanical systems. Subsequently, the alternative methods based on the cavity optomechanical system to overcome this drawback have been proposed. Examples include modulation of the external driving laser 2009PRL103213603 (); 2011PRA83033820 (); 2013OE21020423 (); 2018OE26013783 (); adoption of one red detuned and the other blue detuned two-zone driving sources 2013PRA88063833 (); direct squeezing transfer from the squeezed external driving field or squeezed cavity field generated by the parametric amplifier inside the cavity to the oscillator 2009PRA79063819 (); 2010PRA82033811 (); 2016PRA93043844 (); use of the Duffing nonlinearity 2015PRA91013834 (), etc. While concentrating on the linear radiation pressure interaction, the squeezing of the mechanical mode in quadratically coupled optomechanical system has also been investigated. In this case, one can drive the cavity with two beams 2010PRA82021806R () and use the bang-bang control technique to kick the mechanical mode 2014PRA89023849 (). Meanwhile, we have noted very recently that the effects of the cooperations between the squeezed field driving and quadratic optomechanical coupling 2018PRA97043619 () and between the periodically modulated driving and parametric driving 2018PRA98023807 () on the generation of mechanical squeezing are investigated. The stronger mechanical squeezing can be viewed as the joint effect in the cooperation regime.
In fact, the basic mechanism for creating mechanical squeezing is to introduce a parametric coupling for the motional degree of freedom of the oscillator. The Hamiltonian takes the form (where and are the annihilation and creation operators of the oscillator) and the corresponding evolution operator is a squeezed operator so that the squeezing can be achieved effectively QuantumOptics (). Therefore, a significantly interesting question is how the parametric coupling can be reached in cavity optomechanical system. Very fortunately, we have noted that this type of parametric coupling has been used to enhance the quantum correlations in optomechanical system and it can be introduced by perturbatively coupling a single qubit to the mechanical oscillator 2018AOP39239 (). In addition, the photon blockade and two-color optomechanically induced transparency in this kind of model have been discussed in detail 2015PRA92033806 (); 2014PRA90023817 (). Meanwhile, the oscillator-qubit coupling can also be realized in experiments successfully based on the superconducting quantum circuit system 2009Nature459960 (); 2018PRA98023821 ().
On the other hand, as we all know, the master equation is a powerful tool to study the evolution of a practical quantum system dynamics in quantum theory QuantumOptics (). However, since the dynamics of fluctuations is linearized and the noises are Gaussian in general optomechanical system, it is greatly convenient to introduce the covariance matrix to study the system dynamics 2009PRL103213603 (); 2007PRL98030405 (); 2014PRA89023843 (). But to our knowledge, the dynamical results obtained from the two different approaches have not been compared until now.
In this paper, we study the squeezing effect of mechanical oscillator induced by the oscillator-qubit coupling in a hybrid system consisting of an atomic ensemble trapped inside a dissipative optomechanical cavity. We discuss the mechanical squeezing in detail based on the approaches of master equation and covariance matrix, respectively. In the approach of master equation, we eliminate the highly dissipative cavity mode adiabatically and obtain the effective Hamiltonian. By solving the master equation numerically, we find that the steady-state squeezing of mechanical oscillator can be generated successfully in the long-time limit. We also demonstrate that the squeezing is the resulting effect of cooling process. By numerically and dynamically deriving the optimal effective detuning simultaneously, we check the cooling effects when the mechanical oscillator is prepared in a thermal state with certain mean thermal phonon number initially.
As to the approach of covariance matrix, by eliminating the highly dissipative cavity mode adiabatically, the dynamical equation of covariance matrix can be reduced as the one of covariance matrix, which significantly simplifies the system dynamics. In the appropriate parameter regime, the analytical solution of the steady-state variance for the oscillator position can be obtained approximately. Finally, we make a clear comparison for these two different approaches. We find that the steady-state dynamics in the long-time limit obtained from the two different approaches are completely equivalent.
This paper is organized as follows. In Sec. II, we introduce the hybrid system model under consideration and derive the Hamiltonian of the system. In Sec. III, we discuss the squeezing effect of mechanical oscillator in detail based on the approaches of master equation and covariance matrix, respectively. In Sec. IV, we give a brief discussion about the implementation of present scheme with the circuit-QED system. Finally, a conclusion is given in Sec. V.
Ii System and model
The system under consideration is schematically shown in Fig. 1, where a cloud of identical two-level atoms (with frequency and decay rate ) is trapped in a dissipative optomechanical cavity (with frequency and decay rate ). An external laser field with time-independent amplitude and frequency drives the optomechanical cavity and the movable mirror coupled with a qubit is modeled as the mechanical oscillator with frequency and damping rate . The mechanical oscillator is coupled to the cavity field via the radiation-pressure interaction. The Hamilton of the system is given by (in the unit of )
in which () is the annihilation (creation) operator of the cavity field, are the collective spin Pauli operators of atoms, and () is the dimensionless position (momentum) operator of the mechanical oscillator, satisfying the standard canonical commutation relation . and represent, respectively, the atom-cavity coupling strength and the single-photon radiation-pressure coupling strength. In Hamiltonian Eq. (3), the first three terms in first line correspond to the free Hamiltonian of the driven cavity, atoms, and mechanical oscillator, respectively. The fourth term refers to the Hamiltonian for the qubit-oscillator interaction, where is the coupling strength. As to the generation of this term, we will make a discussion finally. The first two terms in second line describe the coupling between atoms and cavity field and the optomechanical interaction between the cavity field and mechanical oscillator, respectively. The last term gives the driving of the cavity by an external laser field.
The spin operators of the atoms can be described in terms of a collective bosonic operator, . For the sufficiently large atom number and weak atom-cavity couping, 2015PRA92033841 (). In the rotating frame with respect to laser frequency , the Hamiltonian can be rewritten as
where and are, respectively, the cavity and atomic detuning with respect to the external driving laser. is the collective atom-cavity coupling strength.
In the following, we will discuss the squeezing effect of the movable mirror in detail based on the approaches of master equation and covariance matrix, respectively.
Iii Discussion of the squeezing for the movable mirror
iii.1 The approach of master equation
To discuss the squeezing of the movable mirror based on the approach of master equation, it is better to introduce the annihilation and creation operators of the mechanical oscillator
In terms of and , the Hamiltonian in Eq. (4) can be rewritten as
where and .
In general, besides the coherent dynamics, the quantum systems will also be inevitably coupled to their environments. Taking all the damping and noise effects into account, the evolution of the system can be completely described by the following nonlinear quantum Langevin equations (QLEs)
where , , and are the noise operators for the cavity field, mechanical oscillator, and atoms, respectively, which have zero mean value and satisfy the following correlation functions
in which is the mean thermal phonon number. Here is the environment temperature of mechanical reservoir and is the Boltzmann constant.
The strong driving on the cavity leads to the large amplitudes for the cavity field, mechanical mode, and atoms. Thus, the standard linearization procedure can be applied to simplify the dynamical equations. To this end, we express the operators in Eq. (8) as the sum of their mean values and quantum fluctuations, i.e., . Hence, the dynamical equation corresponding to the mean values is given by the following set of nonlinear differential equations:
On the other hand, the dynamics of the quantum fluctuations is governed by the following linearized QLEs:
Via solving Eq. (18) numerically, we plot the time evolution of the real and imaginary parts of the cavity mode mean value and the mechanical mode mean value in Fig. 2. From Fig. 2, we can find that the real and imaginary parts of the mean values reach their steady states quickly and the real part is much larger than the imaginary part ( and ). As a consequence, we can make the following approximations safely:
where and represent, respectively, the steady state mean values of cavity mode and mechanical mode.
So the Hamiltonian of the system for the quantum fluctuations corresponding to Eq. (23) can be written as
in which is the effective cavity detuning and is the effective optomechanical coupling strength.
Under the parameter regimes and , the cavity mode can be eliminated adiabatically and the solution of the fluctuation operator at the time scale can be obtain (see Appendix A)
where is the modified noise operator.
where and represent the modified noise terms. The effective parameters for mechanical frequency , optomechanical coupling , bilinear strength , detuning , and decay rate are defined as, respectively,
Therefore, the effective Hamiltonian corresponding to QLEs about mechanical mode and atom mode in Eq. (31) is
iii.1.2 Generation of the mechanical squeezing
We now introduce the quadrature operators for the mechanical mode and , so the variance of the quadrature operator is determined by
where is the system density operator, the dynamics of which is completely governed by the following master equation
in which is the standard Lindblad superoperators.
According to the Heisenberg uncertainty principle, the product of the variances and satisfies the following inequality,
where . Thus if either or is below , the state of the movable mirror exhibits the behavior of quadrature squeezing.
In Fig. 3, we present the time evolution of the variance for the quadrature operator with the original linearized Hamiltonian in Eq. (28). We find that the variance finally converges to a steady-state value below after the transitory oscillation. Moreover, to check the validity for the adiabatic elimination of cavity mode , it is very necessary to solve the effective master equation
where is the effective decay rate of atoms, as given in Eq. (34). We also give the time evolution of the variance in this case in Fig. 3 and find that the numerical results obtained from the original linearized Hamiltonian and effective Hamiltonian agree well. Thus, in the present parameter regime, simplifying the system dynamics with adiabatic elimination of cavity mode is valid.
Next, we present steady-state variance for the quadrature operator versus the cavity decay rate and atom-cavity coupling strength in Fig. 4. One notes that the squeezing of movable mirror can be generated successfully even in the high dissipative optomechanical cavity , as long as the coupling strength is appropriate. This originates from the strong enough atom-cavity coupling effectively suppresses the undesired effect of cavity dissipation on the mechanical squeezing.
iii.1.3 The optimal effective detuning
In Eq. (36), the last term describes a parametric-amplification process and plays the paramount role in the generation of squeezing. While the third term describes an effective optomechanical coupling process that leads to cooling and heating of the mechanical mode simultaneously. As is well known, to reveal the quantum effects including mechanical squeezing at the macroscopic level, it is a prerequisite to suppress the heating process as soon as possible.
In Fig. 5, we present the dependence of steady-state variance for the quadrature operator on the effective detuning in the case of different mean thermal numbers. We find that, as is expected, the more mean thermal phonon number exists, the larger steady-state variance will become, but there is an optimal effective detuning point . This is because at this point, the heating process of mechanical mode is strongly suppressed. Thus, the destructive effect of thermal noises on the squeezing of movable mirror is almost non-existent. Next, to give more insight of the physical mechanism, we analyze the optimal effective detuning from the system dynamics.
We first apply the squeezing transformation with the squeezing parameter (see Appendix B)
to the effective Hamiltonian in Eq. (36). In this transformation,
The transformed effective Hamiltonian is thus given by
In the interaction picture with respect to the free parts , in Eq. (43) is transformed to
In Fig. 6, we show the energy-level diagram of above Hamiltonian in the resonant condition clearly. We find that the cooling of mechanical mode corresponding to the anti-Stokes process can be significantly enhanced due to the resonant interaction. While the heating corresponding to the Stokes process is strongly suppressed since the detuning is much larger than the coupling strength (in the present parameter regime, ). The resonant condition just is the optimal effective detuning in Fig. 5.
To further check the cooling effect of mechanical mode in the optimal effective detuning condition, there is necessary to present the evolution of the mean phonon number. Since the mechanical oscillator is initially at thermal equilibrium with its environment, it is physically reasonable to assume the mechanical oscillator is prepared in the initial thermal state with certain mean thermal phonon number . Here is the Fock basis. In Fig. 7, we plot the time evolution of the mean phonon number corresponding to the optimal effective detuning when the initial state of the mechanical oscillator is a thermal state. One can see that the final mean phonon number can be less than 1, therefore, which provides a prerequisite for the reveal of squeezing effect.
iii.2 The approach of covariance matrix
iii.2.1 Dynamical equation for covariance matrix
According to Eq. (4), the set of nonlinear QLEs with system operators , , , and is
where is the stochastic Hermitian Brownian noise operator which describes the dissipative friction forces subjecting to the mechanical oscillator. Its non-Markovian correlation function is given by 2001PRA63023812 ()
However, as to the case of (a high quality mechanical oscillator), only the resonant noise components at frequency dominantly affect the dynamics of the mechanical oscillator. Thus the colored spectrum of Eq. (53) can be simplified as the Markovian process and the correlation function becomes
Exploiting above similar linearization procedure, the equation of motion corresponding to the classical mean values about , , , and is given by
and the set of linearized QLEs for the quantum fluctuation operators , , , and is
By introducing the quadrature operators for the cavity field, atoms, and their input noises:
and the vectors of all quadratures and noises:
the linearized QLEs for the quantum fluctuation operators in Eq. (63) can be rewritten as
where is a 66 time-dependent matrix
Here, is the effective time-modulated detuning and and are, respectively, the real and imaginary parts of the effective optomechanical coupling .
Due to the above linearized dynamics and the zero-mean Gaussian nature for the quantum noises, the quantum fluctuations in the stable regime will evolve to an asymptotic Gaussian state which can be characterized by the covariance matrix completely
where denotes the transpose of and is the matrix of noise correlation. Equation (82) is an inhomogeneous first-order differential equation with 21 elements which can be numerically solved with the initial condition of covariance matrix .
iii.2.2 Time evolution of variance for the mirror position
In Fig. 8, we show the time evolution of the real and imaginary parts of the mirror position mean value and the cavity mode mean value . We find that the real and imaginary parts of the mean values reach the steady states quickly and the real part is much larger than the imaginary part ( and ). Thus, we can make the approximations as above subsection
where and denote the steady state mean values of the mirror position and cavity mode, respectively.
By numerically solving the dynamical equation about covariance matrix in Eq. (82) under above approximations, we plot the time evolution of variance for the mirror position in Fig. 9. From Fig. 9, one notes that the variance also finally reaches its steady-state value below .
In fact, exploiting the similar means of eliminating the cavity mode adiabatically, we can obtain the dynamical equation about the reduced covariance matrix :
iii.2.3 Variance for the mirror position in the steady-state regime
When the system reaches the steady state, the reduced covariance matrix is dominated by the following Lyapunov equation
Equation (86) can be analytically solved in the parameter regime with the negligible mechanical damping and the variance for the mirror position in the steady state is given by