We investigate the generation of squeezed state of the mirror motion in a dissipative optomechanical system driven with a strong laser field accompanied with two periodically-modulated lights. Using the density operator approach we calculate the variances of quantum fluctuations around the classical orbits. Both the numerical and analytical results predict that the squeezed state of the mirror motion around its ground state is achievable. Moreover, the obtained squeezed state is robust against the thermal noise because of the strong cooling effect outside the resolved-sideband regime, which arises from the destructive interference of quantum noise.
© 2013 OSA
Quantum effects of a movable mirror coupled to a cavity field via radiation pressure have many potential applications, such as in quantum information processing (QIP) as well as in ultra-sensitive force measurements and in tests of fundamental quantum effects at mesoscopic or macroscopic scales [1–7]. An important resource in optomechanics is the generation of non-classical states of motion around the ground state [8–10]. Especially, achieving squeezed states of a mechanical oscillator, in which the variance of one quadrature of motion is below zero-point value, can enable ultrahigh precision detection at or even below the standard quantum limit [11–14]. Recently, via exploiting the quantum nature of the mechanical interaction between laser light and a membrane mechanical resonator embedded in an optical cavity, the optomechanical squeezing of light has been experimentally demonstrated [15, 16]. These technique developments may offer the possibility to achieve squeezed states of the mechanical oscillator. By now, different schemes for generating squeezed state in optomechanics have been proposed. For example, Clerk et al.  investigated the possibility of generation of a squeezed state via back-action evading measurements, Huang and Agarwal  presented that an optical parametric amplifier inside a cavity can considerably improve the cooling of the micromechanical mirror by radiation pressure and Ian et al.  showed the squeezing of quadrature variance of a mirror assisted by the optical parametric process which is realized by an atomic gas when regarded as a cavity dielectric, and Jähne et al.  investigated the creation of squeezed states of a movable mirror by feeding a squeezed light onto the cavity via the radiation pressure under the assumption of the resolved-sideband limit, etc. In contrast, a different idea without the need of any feedback or control but just by appropriately exploiting time-periodic driving in an optomechanical system, is studied to achieve large degrees of squeezing of a mechanical micromirror [21, 22].
The modulation-assisted driving can give rise to interesting effects and modifications on the quantum dynamics [23, 24], and the employment of the driving has been concerned with the cavity optomechanics in which the motion of a mechanical oscillator modulates the cavity resonance frequency (dispersive coupling). Alternately, the cavity optomechanical system in the sense that the width of the cavity resonance depends on the mechanical displacement (dissipative coupling), was shown to give rise to remarkable quantum noise interference effects which dramatically relax requirements for cooling to the ground state without sideband resolution by Elste et al.  in microwave domain and by Xuereb et al.  in optical domain, respectively. Further, the predicted strong cooling effect outside the resolved-sideband regime will reduce the influence of the thermal noise on the mechanical motion, since the thermal noise will decrease the squeezing of the oscillator. Therefore, it may offer the possibility to obtain the squeezed state of the movable mirror around the ground state in a relatively “hot” environment via implementing the periodic modulations on the dissipative optomechanics.
In this paper, we investigate the generation of squeezed state of the mirror motion in a dissipative optomechanical system, in which the cavity field is driven by an amplitude mildly-modulated light. We obtain the master equation for the cavity-mirror system following the general reservoir theory based on the density operator approach, in which the reservoir variables are adiabatically eliminated by using the reduced density operator for the system in the interaction picture . We first derive the analytical results for the first moments of quadratures in a perturbation method, which are in a good agreement with the asymptotic numerical configurations. Then we carry out the full numerical calculations of the variances of the quantum quadratures around the classical orbits, and find that the squeezed state of the movable mirror is achievable with a much low phonon number occupancy when the modulation frequency is twice the oscillating frequency of the mirror. Finally, we derive the master equation of the mirror by adiabatically eliminating the cavity field due to the weak-coupling strength between them with the experimentally realizable parameters, and the analytical results agree with the numerical results. Moreover, the obtained squeezing can be robust against the thermal noise because of the strong cooling effect in the non-sideband-resolved regime arising from interference of quantum noise.
The paper is arranged as follows: in Sec. 2, the periodically-modulated dissipative optomechanical system is introduced. In Sec. 3, the semiclassical phase orbits are analyzed. In Sec. 4, the full quantum dynamics around the classical orbits is investigated. In Sec. 5, the explicit motion equation for the mirror is obtained via adiabatically eliminating the cavity field and in the last the conclusions are driven.
2. The periodically-modulated dissipative optomechanical system
We consider a dissipative optomechanical system consisted of an effective Fabry-Pérot interferometer [26, 28, 29], sketched in Fig. 1. The movable mirror ℳ oscillates along the x-axis with the frequency ωm and couples to a cavity mode with the resonant frequency ωa via the dispersive and dissipative couplings, which corresponds to the shifts of the cavity’s resonant frequency and damping rate respectively due to its mechanical motion. The full Hamiltonian is a sum of the free cavity Ĥc, free movable mirror Ĥm, free reservoir field ĤR, cavity-reservoir interaction Ĥc-R and cavity-mirror interaction Ĥint Hamiltonians (h̄ = 1):26]. Our main purpose is to stress on the dissipative coupling between the cavity field and the mirror motion and we assume α = 0 that is realizable in the following.
The dispersively and dissipatively coupled optomechanical system has been investigated to cool the mechanical oscillator to its ground state in both the microwave and optical domains in the Heisenberg-Langevin approach [25, 26, 30]. In this paper, we present the dynamics of the cavity-mirror system following the density operator approach, in which the reservoir variables can be adiabatically eliminated by using the reduced density operator for the system. The optical reservoir has two contributions on the cavity field: the c-number part 〈âω〉 and the random fluctuation part δâω. The c-number component is given by periodically-modulated driving field, which is in the form
Following the general reservoir theory , the system-reservoir interaction is given by 𝒱̂(t) = Ĥc-R(t) + Ĥint(t) in the interaction picture with respect to the sum of free Hamiltonians Ĥc, Ĥm, and ĤR. By tracing over the reservoir coordinates under the Born-Markov approximation, the reduced density operator ρ̂s for the cavity-mirror system is governed by32]. Commonly, we may neglect it because it is small and define Δ = ωR − ωa. The Liouvillian operators ℒ̂2 and ℒ̂1 contain the dissipations of the cavity and phonon modes and the interaction between them, which are expressed in the form
3. Analysis of semiclassical phase space orbits
It is very difficult to achieve the full solution for the master equation (4) because of the nonlinear terms and periodic coefficients. We will take the strategy discussed in  in the following, where we first calculate the phase space orbits of the first moments of quadratures and then linearize the quantum dynamics around the asymptotic quasiperiodic orbits.
We introduce the phase and amplitude quadratures for the cavity and mechanical modes,33]
From the master equation (4) with the damping term in Eq. (9), we obtain a group of nonlinear differential equations for the first moments under the factorisation assumption such as , which is expressed as26]: mechanical oscillator effective mass m = 100 ng, frequency ωm = 2π × 103 kHz, and quality factor Q = ωm/(2γm) = 2 × 106, such that the zero-point fluctuation is fm; driving wavelength λc = 1064 nm, effective cavity length L = 7.5 cm, cavity damping rate κc = 2π × 196 kHz, and λ = 0.5 × 10−6. We suppose the initial mean thermal phonon number n̄th = 103 and choose the input power of carrier component equal to PR = 10 mW, where the coherent driving amplitude becomes GHz. The ratio of the modulation sidebands’ amplitude to the carrier’s amplitude ε is equal to 0.3 and the modulation frequency Ω is about twice the natural frequency ωm of the movable mirror. We can numerically solve the Eqs. (10) with the above parameters, and the results are shown by the thin blue solid curves in Fig. 2. One can find that the mean values of each quadrature evolve towards an asymptotic periodic orbit with the same periodicity 2π/Ω of the applied modulation.
When the system is far away from optomechanical instabilities, approximate analytic solutions for these first moments of quadratures can be derived by applying a perturbation method in λ [21, 22]. Thus, we expand the variables Ō(t) into a power seriesEqs. (10), which is presented in appendix A. The differential equations for one set of variables Ōj(t) form a linear inhomogeneous system with τ-periodic coefficients for each j. The asymptotic solution of the variables Ōj(t) is stable and will have the same periodicity of the modulation  due to the existence of the damping. Then, we can expand the variables into Fourier series
In this study, we find that it is sufficient to approximate the asymptotic mean values by truncating the series to the terms with indexes j = 0, 1, 2, 3 and n = −1, 0, 1. The obtained analytical approximations for these first moments are in a good match with the numerical asymptotic quasiperiodic orbits, which are indicated by the green dashed lines in Fig. 2. These analytical approximations are convenient to proceed to linearization on the full quantum dynamics, which can also show a desirable accuracy for the dynamics. The explicit calculations for Ōn,j are presented in detail in appendix A.
4. The full quantum dynamics around the classical orbits
The condition of the large mean values of first moments |Ō(t)| ≫ 1 makes it implementable to treat the system as quantum fluctuations evolving around their classical orbits. Therefore, it is valid to proceed to linearization on the full quantum dynamics by assuming that each operator in the system can be written as a sum of its classical value and a small fluctuation with zero-mean value [21, 22, 35],
In this shift presentation, the quantum behavior of cavity-mirror system is governed by contributions respectively induced by the motion of uncoupled cavity and mirror modes, and the interaction between them, which is determined byEq. (9) just by replacing q̂, p̂ with Q̂, P̂, i.e., [Q̂, [Q̂, ρ̂s]]. The interaction between the cavity and phonon fields is described by
Now we can see that the quantum dynamics of the system will evolve to a zero-mean Gaussian state, for which the second moments dominate. Accordingly, we write the second moments of the cavity-mirror system in a vector form(14), obeys the equation appendix B. The coefficient matrix M̄(t) and term C(t) are both τ-periodic by substituting the approximate values which indeed well approach the asymptotic periodic orbits, and they can also be written as a sum of harmonically oscillating terms Eq. (22) to n = ±2 to calculate the variances of the quantum fluctuations around the classical orbits for this weak modulation. This is because the calculations on the second moments of the variables are adequate up to the first sidebands ±Ω at the condition of Ω ∼ 2ωm, where the dynamics resembles the effect of parametric amplification because the two-phonon average 〈f̂2〉 is nonzero in the rotating wave approximation, leading to the squeezing of the mechanical mode. To be more precise, we present the numerical calculations for X(t) by expanding to the second sidebands ±2Ω. Detailed calculations are shown in appendix B.
In Fig. 3, we plot the time-dependent variances of the mirror position and momentum operators, which are in the form 〈ΔQ̂2〉 = 〈Q̂2〉 − 〈Q̂〉2 and 〈ΔP̂2〉 = 〈P̂2〉 − 〈P̂〉2, respectively. Here 〈Q̂〉 = 0 and 〈P̂〉 = 0 are adopted. The mirror motion is squeezed if either 〈ΔQ̂2〉 or 〈ΔP̂2〉 is less than 1/2 . The initial mean thermal phonon number is 103 and we compare the cases of with and without modulations (ε = 0). For ε = 0, the system becomes the dissipative optomechanics with the monochrome driving [22, 25, 30], where the mirror is allowed to cool down to its ground state at 2Δ = ωm via a destructive interference of quantum noise. The mirror is transferred to a vacuum state and the variances of the mirror position and momentum operators are nearly equal to 0.5 because the thermal noise has little influence on the mirror motion due to the strong cooling effect outside the resolved-sideband regime. The results are shown by the red dashed lines in Fig. 3. For ε = 0.3, the phase and amplitude quadratures get periodically squeezed over time in an opposite phase with the well-chosen Ω around twice the frequency ωm because of the cavity-induced energy shift of the oscillator. Therefore, via appropriately modulating the input light field, quantum squeezing of the quadratures of the mirror motion is achievable in the dissipative optomechanics without the need of any feedback or control. Meanwhile, the motion is closely around its ground state. Such ideas to generate the squeezed state of mechanical resonator have been investigated in the dispersive coupling regime [21, 22, 37, 38]. But a sufficiently strong nonlinearity to overcome the losses and the initial state close to the ground state are required, because the thermal occupation number will decrease the squeezing. The achieved squeezed state here is robust against the thermal noise because the strong cooling effect without requiring the cavity to be in the so-called good cavity limit κc ≪ ωm in this dissipative optomechanics, which arises from the interference of quantum noise, can relax the requirement of much low initial temperature.
We should point out that the effective Hamiltonian Ĥint in Eq. (1) that describes the interaction between the motion of the micromirrror and the electric field is derived in the regime where the displacement x̂0 of the membrane from its average position is small on the scale of a wavelength, i.e., in the Lamb-Dicke regime. The Hamiltonian for the radiation-pressure inter-action in linear order of x̂0. Our calculation contains all the terms in the Eqs. (7), and the terms in the second order of λ is remaining. We may take into account of the quadratic mechanical position coupling between the motion of the mirror and the cavity field that is much smaller compared with linear mechanical position coupling in Lamb-Dicke regime in order to obtain a consistent result. The system-reservoir interaction becomesEq. (7) becomes Eq. (14) is Eqs. (15), (17) and (18) because it is in a higher order of λq̄, which is much smaller 1. We can infer that the quantum effects caused by caused by the quadratic mechanical position coupling are also negligible here.
5. The explicit motion equation for mirror via adiabatically eliminating the cavity variable
In the shift representation, the effective coupling strengths between the cavity fields and the movable mirror denoted by λE and 2λ κc|ā(t)| in Eq. (18) are about 2π × 29 kHz and 2π × 50 kHz with the experimentally realizable parameters, which are much smaller than the cavity damping rate κc. Thus, the cavity variable arrives at the steady state much faster than the mirror variable, and the motion equation of the reduced density operator for the movable mirror can be obtained by adiabatically eliminating the cavity variable in the master equation (14).
In order to see the main contributions on the motion equation of mirror, we first simplify the obtained semiclassical variables in appendix A. For the cavity variable, we find that when we retain the result to the zeroth order in λ, this reduced result can approximately represent the asymptotic orbits. Now we introduce the classical values of the phase and amplitude quadratures for the cavity mode and calculated from the mean value of the annihilation operator ā(t), which isFig. 4, it is shown that the mean values of the quadratures from Eq. (23), which is indicated by the red dotted line, can approximately approach the asymptotic orbit indicated by the blue solid line, which is also equal to the numerical results up to the order of λ3 as discussed in Sec. 3. The mean values of the mirror quadratures q̄(t) and p̄(t) are an order of magnitude lower than the cavity variables x̄(t) and ȳ(t) shown by the numerical values in Fig. 2 because they are nonzero up to the first order in λ. Further, the relation λ|q̄(t)| ≪ 1, which can be also seen from Fig. 2, makes q̄(t) negligible when it appears in Eqs. (15) and (18). In the following, we will display the feasibility of above simplifications on the first moments by comparing the quantum dynamics of the mirror motion governed by the analytical master equation with the full numerical results in Fig. 3.
Now we will neglect the term q̄(t) and take the value of ā(t) in Eq. (23). Then, by adiabatically eliminating the cavity variable and making use of the second-order perturbation method with respect to the effective coupling strengths λE and 2λ κcā(t) in the interaction picture of the Hamiltonian ωmf̂†f̂, the motion equation of the reduced density operator for the movable mirror ρ̂f is given byEq. (15), 𝒟̂f is the superoperator in the same form as Eq. (17), and ℒ̂mρ̂f describes the intrinsic damping of the mirror motion.
In the following, we define δ = Ω − 2ωm and assume δ ≪ (Δ, ωm, Ω) to compensate for the cavity-induced energy shift of the phonon mode. If we neglect the oscillating terms with frequencies ±ωm, ±2ωm,..., the resulting motion equation for the mirror is given by39] is
In general, the heating rate is approximately equal to A(−ωm), in which f (Δ, −ωm) plays a main role due to the small value of ε. When the detuning fulfills the relation Δ = ωm/2, there is f (Δ, −ωm) = 0, which is just the optimal condition for the ground-state cooling of a mechanical oscillator appearing in the dissipative optomechanics with monochromic driving [25, 26]. The heating rate becomes proportional to ε2,21, 22], and the initial mean thermal phonon number should be much small in order to obtain the desired squeezed state because the thermal noise can degrade the squeezing [38,40]. In this periodically-modulated dissipative optomechanical system, the strong cooling effect can guarantee the squeezed state robust against the thermal noise and it may offer the probability to realize the squeezing of the mirror motion in a relatively “hot” environmental temperature.
The full solution of Eq. (31) in the long time limit can be easily calculated, and then the variance of momentum operator can be also obtained, in which 〈f̂〉 = 0 is adopted. The result is shown by the red dashed line in Fig. 5. The analytical result approximately agrees with the numerical result which is indicated by the blue solid line. Thus the result validates the perturbation method we employ.
We now turn to analyze the squeezed state with the numerical values, which are nst = 〈f̂†f̂〉 = 0.065 and 〈f̂2〉 = −0.15 −0.05i with the above experimentally realizable parameters. The final phonon number in the mirror motion is very close to zero, thus the mirror is near its motional ground state. Meanwhile, the mirror motion is squeezed if 〈ΔP̂2〉 < 1/2. When we expand the expression of 〈ΔP̂2〉, we will obtain the relationEqs. (34) is simplified as Eq. (28). These two processes both contribute to the squeezing of the mirror motion, but the directions from these two processes do not coincide with each other. This leads to a destructive interference for squeezing [41, 42].
In the last, we discuss the squeezing of the mirror motion with the increased coupling strength, where the destructive interference of quantum noise becomes not complete. As a result, the cavity will no longer appear to the oscillator as an effective zero-temperature bath, giving rise to nonzero limiting temperature [25, 26]. We follow the procedure of the numerical calculation in Sec. 4 to investigate the variance of the momentum operator 〈ΔP̂2〉 with the increased input power PR while the other parameters are unchanged, and the numerical results are shown in Fig. 6. It shows that in the dissipative optomechanics, the complete destructive interference of quantum noise can help to achieve a better squeezing of the mirror motion. With the increased coupling strength, the photonic excitation of the cavity field will preclude the complete destructive interference and induce an extra thermal phonon excitation, which may decrease the squeezing of the mirror motion.
In summary, we present the generation of squeezed state of the movable mirror motion via exploiting the periodically-modulated driving on the dissipative optomechanical system. The quantum dynamics of the system around the classical orbits are investigated first with a numerical method and then in an analytical approach via adiabatically eliminating the cavity field and employing the second-order perturbation method, and these two results are in a good agreement with each other. Both the results show that by tuning the carrier frequency of the driving fields at the Fano resonance and the modulation frequency twice the oscillating frequency of the mirror, squeezed state of the mirror motion around its motional ground state is achievable. Further, the strong cooling effect outside the resolved-sideband regime arising from the Fano resonance of quantum noise makes the squeezed state robust against the thermal noise, which may offer the possibility to realize the squeezing of the mirror motion in a relatively “hot” environmental temperature.
A. The perturbation calculations of first moments
In this section, we derive the analytical solutions for the first moments of the quadratures by applying the perturbation method in λ. This originates a chained set of equations, and by truncating a finite number of orders j ⩽ jmax, we will achieve desirable results. In the paper, we find that the finally obtained results are in a good match with the asymptotic numerical results when jmax = 3, with the experimentally realizable parameters presented in the main text,.
The variables Ōj(t) form a set of linear differential equations with τ-periodic inhomogeneous terms for each j. These variables can be expanded by the Fourier series in Eq. (12). By substituting the expansions into the nonlinear differential equations (10), at zeroth order in λ we have a group of equations.Eqs. (39).
For the differential equations of the variables in first order of λ, with use of the zeroth-order results we obtain a group of equations.Eqs. (40). At the first order, the cavity quadratures x̄1 and ȳ1 are zero shown by the 3rd and 4th equations.
The differential equations for the variables in the second order of λ can be also achieved in the following with the use of the zeroth- and first- order results, which are given byEqs. (41). Following the same procedure, the equations of the variables in the third order of λ are given by
The perturbation results for the mean values of the quadratures are given by up to λ3. We numerically show that by truncating the oscillating frequencies to n = ±1, the resulting valuesFig. 2. Physically, this is because higher sidebands well fall outside the cavity bandwidth, nΩ > κc, leading to negligible driving effects on the cavity field. Based on these approximate values of first moments, we can investigate the full quantum dynamics of the optomechanical system by assuming small quantum fluctuations around their classical values.
B. The numerical calculation of the second moments X(t)Eqs. (21) and truncating Fourier series of X(t) to n = ±2 in Eq. (22), we obtain a chained set of equations. Eqs. (46), the steady-state solutions for X0,±1 can be calculated by writing the equations in a matrix form Eq. (48), X±2 in Eq. (47) can be also achieved. Finally, the second moments of the variables in the cavity-mirror system are given by
This work is supported by the National Natural Science Foundation of China (Grants No. 11074087 and No. 61275123), the Nature Science Foundation of Wuhan City (Grant No. 201150530149), and the National Basic Research Program of China (Grant No. 2012CB921602).
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