Entanglement dynamics of two coupled mechanical oscillators in modulated optomechanics
Subhadeep Chakraborty Amarendra K. Sarma
quant-ph cond-mat.mes-hall physics.optics


APS/123-QED





 
 c.subhadeep@iitg.ernet.in

 aksarma@iitg.ernet.in

 Department of Physics, Indian Institute of Technology  Guwahati, Guwahati-781039, Assam, India





















             



We study the entanglement dynamics of two coupled mechanical oscillators, within a modulated optomechanical system. We find that, depending on the strength of the mechanical coupling, one could observe either a stationary or a dynamical behavior of the mechanical entanglement, which is extremely robust against the oscillator temperature. Moreover, we have shown that this entanglement dynamics is strongly related to the stability of the normal modes. Taking mechanical damping effects into account, an analytical expression corresponding to the critical mechanical coupling strength, where the transition from stationary to dynamical entanglement occurs is also reported. The proposed scheme is analysed with experimentally realistic parameters, making it a promising mean to realize macroscopic quantum entanglement within current state-of-the-art experimental setups.




  PACS numbers








Valid PACS appear here
                             

                              
Entanglement dynamics of two coupled mechanical oscillators in modulated optomechanics
    Amarendra K. Sarma
    December 30, 2023
======================================================================================







§ INTRODUCTION


The generation of quantum entanglement between two macroscopic, massive objects has been a task of paramount importance, both in fundamental research, in particular for fundamental tests of quantum mechanics <cit.> and in numerous futuristic potential applications related to quantum computing, quantum information processing, quantum communication and so on <cit.>. Thanks to the fast growing field of cavity optomechanics <cit.>, which provides a versatile platform to prepare such an entangled state in mechanical motions, research in the area has virtually exploded in recent times. Relying on the generic radiation-pressure coupling, much studies have already been reported on the entanglement generation between a cavity field and a mechanical oscillator <cit.>. Besides, based on similar architecture, a lot of emphasis has been currently brought forward to realize entanglement between two macroscopic mechanical oscillators. These studies mostly include: light-to-matter entanglement transfer <cit.>, driving the optical cavity with a two-tone field <cit.>, dissipation induced optomechanical entanglement <cit.> and the reservoir engineered based schemes <cit.>. However, in most of the cases, the possibility of observing such a nonclassical state is seriously hindered by the presence of the environmental noise. Hence a lot of emphasis has currently been put in realizing quantum entanglement at higher bath temperature. 

While the search for robust and hot entanglement <cit.> in optomechanical systems is on, it occurs that modulating an optomechanical system may be a very rewarding proposition in achieving a more robust nonclassicality. For example, in 2009, Mari et al. <cit.> first showed that by gently modulating an optomechanical system, one could not only enhance the degree of squeezing in mechanical quadratures but also improve the stationary entanglement between the cavity field and the nanomechanical resonator. Following Ref. <cit.>, Farace  et al. <cit.> studied the effect of both the amplitude modulation and frequency modulation in an optomechanical system. They showed that there exists an optimal modulation regime where the desired quantum effects can either be enhanced or suppressed. Along this line, Schmidt et al. <cit.> implemented a suitable amplitude modulation scheme in optomechanical circuits for continuous variable (CV) quantum state processing. More recently, Chen  et al. <cit.> has exploited the same amplitude modulation to improve the stationary mechanical entanglement in a double cavity optomechanical system.


In parallel to the developments in optomechanical systems, one modulation scheme of particular interest is the so-called periodic modulation of the coupling strength. It is now well established that by periodically driving the coupling strength with a frequency twice that of the oscillator frequency, one can squeeze the collective quadratures, leading to entanglement generation between the two harmonic oscillators.  In 2010, within a similar framework, two identical harmonic oscillators in contact with two independent thermal baths and coupled via a time periodic driving, Galve  et al. <cit.> first demonstrated the existence of stationary entanglement at a relatively high temperature. Following Ref. <cit.>, Roque et al. <cit.> reported the dynamics of quantum correlations between two coupled harmonic oscillators in contact with a common heat bath. They found that it is not the bath temperature, rather the system parameters to which the entanglement dynamics is more sensitive. However, it should be noted that in their study they could not find any steady-state behavior of the generated entanglement. On the other hand, recently, Chen  et al. <cit.> considered a system of two coupled harmonic oscillators connected via a weak time-dependent coupling. In the absence of any environmental decoherence, they reported that a transition from bounded to unbounded entanglement dynamics occurs when the modulation strength crosses a critical value.

In this work, we theoretically study the entanglement dynamics of two coupled mechanical oscillators, placed within a modulated optomechanical system.  We show that, unlike Ref. <cit.>, both the stationary and dynamical behavior of the mechanical entanglement could be achieved. Moreover, by taking the mechanical damping terms into account, we give an analytical estimation of the critical mechanical coupling strength where the dynamical transition occurs. The rest of the paper is organized as follows. In Sec. <ref> we introduce our physical model of the optomechanical system and derive the equation of motion corresponding to the correlation matrix. In Sec. <ref> we give a detailed discussion of the dynamical behavior of the mechanical entanglement for various mechanical coupling strengths, and, show the connection between the entanglement dynamics and the stability of the normal modes. Finally, we present our concluding remarks in Sec. <ref>.






§ MODEL AND DYNAMICS


The system under consideration, consists of two identical mechanical oscillators placed within an optical cavity. An external laser with a time-dependent amplitude E(t) and frequency ω_l drives the cavity which makes the two oscillators couple indirectly via the radiation-pressure interaction. Besides, there is a direct mechanical coupling between the two oscillators, with a periodically modulated coupling strength λ(t). The schematic of our system is depicted in Fig. 1, and, the Hamiltonian (in a frame rotating with the laser frequency) is written as follows (in the unit of ħ=1): 

    H=Δ_0 a^† a+∑_j=1^2ω_m/2(q_j^2+p_j^2)+g a^† a ∑_j=1^2q_j+λ(t)q_1q_2
    
     +i(E(t)a^†-E^*(t)a).

Here, a (a^†) refers to the annihilation (creation)  operator of the cavity field (with frequency ω_c and decay rate κ), q_j (p_j) is the dimensionless position (momentum)  operator of the j-th mechanical oscillator (with frequency ω_m and damping rate γ_m).  g refers to the strength of the single photon radiation-pressure coupling. In Hamiltonian (<ref>), the first and the second term corresponds to the Hamiltonian of the driven cavity and the mechanical oscillators, respectively, with Δ_0=ω_c-ω_l being the optical detuning. The third term describes the optomechanical interaction between the cavity field and the mechanical oscillators, while the fourth term refers to the bilinear coupling between the two oscillators. Finally, the last term gives the external laser driving.

In addition to this, the system dynamics is unavoidably subjected to the fluctuation-dissipation processes affecting both the cavity field and the mechanical oscillators. Taking all the damping and noise terms into account, the dynamics of the system is fully described by the following set of nonlinear quantum Langevin equations (QLEs):


    q̇_̇j̇   =ω_m p_j,
    ṗ_̇j̇   =-ω_m q_j-g a^† a -λ(t) q_3-j-γ_m p_j+ξ_j(t),
    ȧ   =-{i(Δ_0+g∑_j=1^2q_j)+κ}a+E(t)+√(2κ)a_in,


where a_in is the vacuum input noise operator, with the only nonzero correlation function <cit.>:

    ⟨ a_in(t)a^† _in(t^')⟩=δ(t-t^').

 ξ_j(t) are the stochastic Hermitian Brownian noise operator, with the non-Markovian correlation function given by <cit.>

    ⟨ξ_j(t)ξ_k(t^')⟩=
    δ_jk/2πγ_m/ω_m∫ω e^-iω(t-t^')[(ħω/2K_BT)+1]dω,

(k_B being the Boltzmann constant and T being the temperature of the mechanical oscillators). However, in the limit of large mechanical quality factor Q=ω_m/γ_m≫1, one could well approximate this Brownian noise to a Markovian delta-correlated relation <cit.>:

    ⟨ξ_j(t)ξ_j(t^')+ξ_j(t^')ξ_j(t)⟩ /2≃γ_m(2n_th+1)δ(t-t^'),

with n_th=[ exp(ħω_m/K_BT)-1]^-1 being the number of mean thermal phonons.

Next, when the system is strongly driven to a large classical mean value, we can adopt the standard linearization technique and rewrite each Heisenberg operator as follows: o(t)=⟨ o(t) ⟩+δ o(t) (o=q_j, p_j, a). Here, ⟨ o(t)⟩ refers to the classical c-number mean value and δ o(t) is the zero-mean quantum fluctuation around the classical mean value. The equation of motion corresponding to the classical mean values is given by the following set of nonlinear differential equations:


    ⟨q̇_̇j̇(t) ⟩    =ω_m ⟨ p_j(t)⟩,
    ⟨ṗ_̇j̇(t) ⟩    =-ω_m ⟨ q_j(t) ⟩-g|⟨ a(t) ⟩|^2-λ(t) ⟨ q_3-j(t)⟩
        -γ_m ⟨ p_j(t) ⟩,
    ⟨ȧ(t) ⟩   =-{i(Δ_0+g∑_j=1^2 ⟨ q_j(t) ⟩)+κ}⟨ a(t) ⟩ +E(t).


On the other hand, the dynamics of the quantum fluctuations is governed by the following linearized QLEs, written in a matrix form:

    u̇(̇ṫ)̇=A(t)u(t)+n(t).

Here, u^T(t)=(δ q_1(t),δ p_1(t),δ q_2(t),δ p_2(t),δ X(t),δ Y(t)) is the vector of quadrature fluctuation operators,
n^T(t)=(0,ξ_1(t),0,ξ_2(t),√(2κ)X_in(t),√(2κ)Y_in(t)) is the vector of corresponding noises and

    A(t)=(
     [       0     ω_m       0       0       0       0;    -ω_m    -γ_m   -λ(t)       0  G_x(t)  G_y(t);       0       0       0     ω_m       0       0;   -λ(t)       0    -ω_m    -γ_m  G_x(t)  G_y(t); -G_y(t)       0 -G_y(t)       0      -κ    Δ(t);  G_x(t)       0  G_x(t)       0   -Δ(t)      -κ;         ]),

is the drift matrix. The time-dependent coupling and the detuning terms, respectively, are defined as follows:


    G(t)    =-√(2)g⟨ a(t) ⟩,
    
    G(t)    =G_x(t)+iG_y(t),
    Δ(t)    =Δ_0+g∑_j=1^2⟨ q_j(t)⟩,


It should be noted that in Eq. (<ref>) we have used the quadrature operators for the cavity field with the corresponding Hermitian input noise operators, respectively defined as:
δ X≡(δ a+δ a^†)/√(2), δ Y≡(δ a-δ a^†)/i√(2), and
X_in≡(a_in+a_in^†)/√(2), Y_in≡(a_in-a_in^†)/i√(2).

Due to the above linearized dynamics and the zero-mean Gaussian nature of the quantum noises, the quantum fluctuations in the stable regime evolve to an asymptotic Gaussian state which is completely characterized by its 6×6 correlation matrix, given by:

    V_ij=(⟨ u_i(t)u_j(t)+u_j(t)u_i(t)⟩)/2.

The equation of motion corresponding to the correlation matrix, using Eq. (<ref>) and Eq. (<ref>), can be written as follows:

    V̇(t)=A(t)V(t)+V(t)A^T(t)+D,

where D=Diag[0,γ_m(2n_th+1),0,γ_m(2n_th+1),κ,κ] is the matrix of noise-correlation. Note that, Eq. (<ref>) is an inhomogeneous first-order differential equation with 21 elements which could be numerically solved with the initial condition V(0)=Diag[n_th+1/2,n_th+1/2,n_th+1/2,n_th+1/2,1/2,1/2]. Here, we have assumed that each mechanical oscillators are prepared in their thermal states at temperature T and the cavity field is in it's vacuum state.





§  RESULTS AND DISCUSSIONS


Before proceeding to a direct numerical investigation of the mechanical entanglement, we first specify the exact form of time modulation for both the external driving and the mechanical coupling strength, as follows:

    E(t)=E_0+E_1cos(Ω t),   λ(t)=λ_0cos(Ω t).

Moreover, we choose the following set of parameters for our numerical simulations: Δ_0/ω_m=1.0, κ/ω_m=0.1, γ_m/ω_m=5×10^-4, g/ω_m=1×10^-5, E_0/ω_m=1×10^4, E_1/ω_m=1×10^3, T_0/ω_m=ħ/k_B, Ω/ω_m=2.003 (this particular choice will be justified later) and τ=2π/Ω.

In Fig. 2, we plot the time evolution of the mechanical entanglement E_N (see appendix <ref>) for multiple values of λ_0/ω_m. It can be observed that in absence of the mechanical coupling (λ_0/ω_m=0), the two oscillators exhibits a very small degree of stationary entanglement. However, as soon as the mechanical coupling is introduced, there is a sudden but significant enhancement in E_N at initial time, which finally converges to an asymptotic steady-state value. We note that this enhancement becomes more profound with an increase in coupling strength λ_0/ω_m. Moreover, in the asymptotic regime, we find that the entanglement acquires the same period of modulation (see the right hand panel of Fig. 2). Hence one can identify the degree of the entanglement as the maximum over one period τ=2π/Ω of modulation, defined as follows <cit.>:

    E_N=t∈[𝒯,𝒯+τ]maxE_N(t),

after a long enough time 𝒯≫1/κ,1/γ_m. It is worth mentioning that, with the application of periodically modulated mechanical coupling, λ_0/ω_m=0.005 we obtain a remarkable degree of stationary entanglement, E_N=0.34, as opposed to E_N=0.04 , which is achieved with no direct mechanical coupling.




The dependence of the stationary mechanical entanglement on the oscillator temperature is exhibited in Fig. 3. As expected, it could be seen that the degree of entanglement decreases monotonically with increase in temperature T. However, one should note the improved robustness for the same mechanical entanglement with an increase in the coupling strength λ_0. For example, it is shown that in presence of the mechanical coupling λ_0/ω_m=0.005, the degree of entanglement survives up to a relatively high temperature T/T_0≈0.9, in sharp contrast to the case where entanglement is found to persist upto temperature T/T_0≈0.3 in absence of the mechanical coupling.

Next, in Fig. 4(a), we once again depict the time evolution of E_N, similar to Fig. 2 but with a set of higher values of λ_0/ω_m. Now one could observe that, with increase in λ_0, the entanglement not only grows much faster in time but also decays quickly to zero. Thus, it is evident that depending on the strength of the mechanical coupling, one could achieve a completely different dynamical behaviour of the mechanical entanglement. In order to investigate the role of oscillator temperature on the entanglement dynamics, we have redone the calculations for T=3T_0 and depicted it in Fig. 4(b). It exhibits that with the increase in temperature, E_N decreases and there is a delay in the entanglement formation as well as reduction in survival time for entanglement, compared with its T=0 counterpart. It should be noted that this feature is also reported in Ref. <cit.>. However, if one compares Fig.(3) and Fig.4(b), it is clear that a significant degree of entanglement could be attained at a relatively high temperature, with higher mechanical coupling strength.





Now, to further probe into the entanglement dynamics and the role of the mechanical coupling on it, we introduce the normal modes for the mechanical oscillators as follows:
δ q_±=(δ q_1 ±δ q_2)/√(2), δ p_±=(δ p_1 ±δ p_2)/√(2), and, rewrite the linearized optomechanical Hamiltonian in the following way:

    H^lin=H_++H_-,

where, H_± is given by


    H_+    =Δ/2(δ X^2 + δ Y^2 )+1/2(ω_m δ p_+^2 + (ω_m+λ(t))δ q_+^2)
       -√(2)(G_x δ X + G_y δ G_y)δ q_+,
    
     H_-    =1/2(ω_m δ p_-^2 + (ω_m-λ(t))δ q_-^2).


The above Hamiltonian (<ref>) describes two independent parametric oscillators, one of which (+ mode) is coupled to the cavity field via the usual optomechanical interaction, while the other (- mode) one is completely free. Following a similar procedure, used to obtain Eq. (<ref>), we construct the correlation matrix corresponding to the normal modes and the cavity field.

In order to illustrate the dynamics of the normal modes in the so-called phase-space, in Fig. 5 and 6 we depict the respective Wigner functions (Eq. (<ref>))  at some specific times, for two different mechanical coupling strengths λ_0/ω_m=0.005 and λ_0/ω_m=0.01. We note that the value of the coupling strengths are so chosen to yield two different kinds of entanglement dynamics. The localization of the Wigner function in phase space, for + mode, could be clearly seen in Fig. 5(a)-(c) and 6(a)-(c), and, this localization is independent of the mechanical coupling strength. On the contrary, the - mode exhibits  both the localization and delocalization phenomena depending on the strength of the mechanical coupling. Fig. 5(d)-(f) shows localization of the - mode when we have, λ_0/ω_m=0.005, which is maintained even for a sufficiently long time t/τ=5000. However, as could be seen from Fig. 6(d)-(f), with increase in λ_0/ω_m, the delocalization occurs quickly in time: The Wigner function stretches along the dynamical rotating axis along with a contraction in the perpendicular direction. This feature become quite prominent if one observe the dynamics at a time t/τ=300.45. This clearly signifies a dynamical instability corresponding to the - mode. Therefore, we can infer that the asymptotic nature of the mechanical entanglement is directly related to the instability in the - mode.





Now, to explicitly derive the relationship between the entanglement dynamics and the strength of the mechanical coupling, we focus on the stability of the - mode. Starting form the Hamiltonian (<ref>), we derive the equation of motion corresponding to the - mode, as given below:

    δ̈ ̈q̈_-+(ω_m^2-ω_mλ_0 cosΩ t)δ q_-+γ_mδ̇ ̇q̇_-=0.

The above equation corresponds to a dissipative classical parametric oscillator, with a time modulated mechanical frequency ω_m^2(t)=ω_m^2-ω_mλ_0cos(Ω t). Following a substitution t̃=Ω t/2, we can rewrite the Eq. (<ref>), in the following form:

    δ̈ ̈q̈_-+(ω̃_m-2λ̃_0 cos 2t̃)δ q_-+γ̃_mδ̇ ̇q̇_-=0,

where the dimensionless parameters are defined as follows:

    ω̃_m=4ω_m^2/Ω^2,  λ̃_0=2ω_mλ_0/Ω_2,  γ̃_m=2γ_m/Ω.

Now, defining δ q_-=δ̃ ̃q̃_-e^-γ̃_mt̃/2 and substituting in Eq. (<ref>), we get the standard form of the canonical Mathieu equation

    δ̈̃̈ ̈̃̈q̈̃̈_-+(δ-2ϵcos(2t̃))δ̃ ̃q̃_-=0,

where δ and ϵ are respectively given by δ=4ω_m^2-γ_m^2/Ω^2 and ϵ=2ω_mλ_0/Ω_2. It is clear that, for a modulation frequency Ω^2≈4ω_m^2, we have δ≈1 and ϵ≪1. In this limit, one can neglect all the higher-order terms in the eigenvalues of the Mathieu's equation Eq.  (<ref>) (see appendix <ref> for a better discussion) and obtain:


    α_1(ϵ)≈1+ϵ,
    β_1(ϵ)≈1-ϵ.






The stability of the - mode in the ϵ-δ plane, is depicted in Fig. 7. One can observe that, for the - mode to be stable the following stability criteria must be satisfied:

    β_1(ϵ)=1-ϵ≤δ≤α_1(ϵ)=1+ϵ.

Solving Eq. (<ref>) in terms of the Ω, ω_m and γ_m, we can obtain an analytical expression for the critical mechanical coupling strength λ_0c, given as follows:

    λ_0/ω_m≤λ_0c/ω_m=Ω^2-4ω_m^2+γ_m^2/2ω_m^2.

It should be noted that for the modulation frequency Ω_m=2.003ω_m, one gets the following critical mechanical coupling strength λ_0c/ω_m≈0.0063. This situation is further illustrated in Fig. 7, respectively, for the three distinct mechanical coupling strengths λ_0/ω_m=0.005 (green circle), λ_0/ω_m=0.006 (blue diamond), and λ_0/ω_m=0.007 (red square). We can see that the points corresponding to the aforementioned coupling strengths, respectively, locates in the stable, on the boundary and in the unstable zone of the - mode. This well justifies our previously obtained entanglement dynamics, corresponding to the different sets of the mechanical coupling strengths.




Finally, in Fig. 8 we depict the dependence of the stationary mechanical entanglement on the modulation frequency. It can be seen that the entanglement E_N is quite sensitive to the variation in the modulation frequency, which can be attributed directly to the instability in the - mode. Furthermore, one can observe that the peak of the stationary entanglement is obtained exactly  at Ω/ω_m=2.003, which justifies our initial choice of the modulation frequency.




§ CONCLUSION


In conclusion, we have proposed a scheme to entangle two directly coupled mechanical oscillators, in an optomechanical system. Our scheme exploits the periodic modulation technique, in both the external driving and mechanical coupling strengths. We observe that an abrupt transition from stationary to dynamical mechanical entanglement occurs when the - mode becomes unstable. More importantly, it is shown that in the presence of the mechanical coupling, a significant improvement in the robustness of the generated entanglement could be achieved with respect to the oscillators temperature. Finally, based on the  eigenvalues of the Mathieu's equations, we give an analytical estimations corresponding to the  the critical mechanical coupling strength, where the transition occurs. The feasibility of the chosen parameters, makes our proposed scheme a promising mean to realize macroscopic quantum entanglement within current state-of-the-art experimental setups.



§ ACKNOWLEDGEMENT


S. Chakraborty would like to thank MHRD, Government of India for providing a financial support for his research.




§ APPENDIX



 §.§ Entanglement in Gaussian States




In the context of continuous-variable (CV) quantum information, Gaussian states are of central importance. These are the states with Gaussian Wigner function and are completely characterized by its first and second moment of the field quadrature operators. For any N mode Gaussian state, the vector of the first moments reads R̅=(⟨ R_1 ⟩, ⟨ R_1 ⟩,...,⟨ R_N ⟩, ⟨ R_N ⟩), while the second moments is denoted by the 2N×2N covariance matrix (CM) V of elements

    V_ij=1/2(⟨ R_i R_j+ R_j R_i⟩)-⟨ R_i ⟩⟨ R_j ⟩.




Here, R stands for the 2N dimensional vector of CV operators R^T=(q_1,p_1,q_2,p_2,,...,q_N,p_N). However, by following a local unitary transformation the first moments could be easily adjusted to zero, without affecting any informationally relevant properties. With this consideration, the Wigner function for a N mode Gaussian state could be written as follows <cit.>

    W(R)=1/(2π)^N√(DetV)e^-1/2R^TV^-1R.


Now, to discuss entanglement in CV systems, we consider a very prototypical CV entangled state, i.e. a two-mode Gaussian state. This type of state can be represented by the following covariance matrix

    V_2=(
      [   A   C; C^T   B;     ]),

where A, B and C are 2×2 block matrices, respectively,  describing the local properties mode A, mode B and the intermode correlation between A and B. The degree of entanglement between the two modes is calculated by the so-called logarithmic negativity E_N <cit.>,  defined as:

    E_N=max[0,-ln2ν^-].

Here ν^-≡2^-1/2[Σ(V_2)-√(Σ(V_2)^2-4detV_2)]^1/2 is the smallest symplectic eigenvalue of the partial transpose of V_2 with Σ(V_2)≡det(A)+det(B)-2det(C). A Gaussian state is said to be entangled (E_N>0) if and only if ν^-<1/2 which is equivalent to Simon's necessary and sufficient nonpositive partial transpose criteria <cit.>.



 §.§ Matheu's Equation




The canonical form of Matheu's equation for the parameters δ and ϵ is given by <cit.>

    ÿ+(δ-2ϵcos(2t))y=0.

This equation is a linear second order differential equation, with periodic coefficients. In general, the solution of such equation varies depending on the choice of δ and ϵ. However, it should be noted that to maintain the periodicity of the solution, δ and ϵ must be interrelated. Therefore, one has a set of eigenvalues α_n(ϵ) (n=0,1,2,3,...) and β_n(ϵ) (n=0,1,2,3,...) necessary to yield solution of Eq. (<ref>). For small ϵ, the first three α_n(ϵ) and β_n(ϵ) could be written as follows <cit.>

 
    α_o    =-1/2ϵ^2+7/128ϵ^4+O(ϵ^6), 
    α_1    =1+ϵ-1/8ϵ^2-1/64ϵ^3+O(ϵ^4),
    β_1    =1-ϵ-1/8ϵ^2+1/64ϵ^3+O(ϵ^4).



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