Dynamic entanglement transfer in a double-cavity optomechanical system
We give a theoretical study of a double-cavity system in which a mechanical resonator beam is coupled to two cavity modes on both sides through radiation pressures. The indirect coupling between the cavities via the resonator sets up a correlation in the optomechanical entanglements between the two cavities with the common resonator. This correlation initiates an entanglement transfer from the intracavity photon-phonon entanglements to an intercavity photon-photon entanglement. Using numerical solutions, we show two distinct regimes of the optomechanical system, in which the indirect entanglement either builds up and eventually saturates or undergoes a death-and-revival cycle, after a time lapse for initiating the cooperative motion of the left and right cavity modes.
pacs:42.50.Wk, 03.65.Ud, 42.50.Lc
Cavity optomechanical systems kippenberg08 () arise from the classical Fabry-Perot interferometer vaughan89 () by replacing one of the fixed sidewalls with a cantilever or double-clamped beam cleland96 (); meyer88 (); kleckner06 (). The one-dimensional degree of freedom introduced by the movable mechanical element adds a free resonator mode to the cavity system and allows this mode to interact with the cavity field through radiation pressure on the reflectively coated mechanical resonator. Regarded as a micromirror, this resonator can be feedback-controlled through the cavity field, on which numerous cooling protocols have been conceived and experimentally demonstrated in the last decade metzger04 (); naik06 (); arcizet06 (); ian08-1 (); liu13 ().
The degree of control in this hybrid cavity-micromirror system can be further enhanced when the micromirror is replaced by a double-face reflective membrane thompson08 (); jayich08 (). If a second optical cavity is coupled to it on the opposite side of the existing cavity, a two-mode or double-cavity optomechanical system with enhanced nonlinearity is formed pinard05 (); miao09 (); naeini11 (); ludwig12 (). Entanglement-wise, though it was observed that the enhanced squeezing resulted from the nonlinear coupling helps generate static entangled state of distant mirrorspinard05 (), the dynamic property of entanglement between the two cavities is less well-understood.
Recent studies reveal that the dynamics of phonon-photon entanglement plays an important role in defining the system characteristics, such as the transitions between oscillation modes wang14 (); ying14 (), robustness against noisy environment tian13 (), sudden death and revival of states ian08 (); chang09 (); lin14 (), and optimal entanglement ydwang13 (). In this article, we study the dynamics of the entanglements in a double-cavity optomechanical system where each photon mode in the two opposite cavities is, structure-wise, symmetrically coupled to a common mechanical resonator mode via radiation pressures, albeit asymmetric coupling strengths and driving powers are generally assumed. Our main concern is to determine how the cavity-resonator entanglements vitali07 () can be transferred to the indirectly coupled cavities over time.
We show here such an entanglement transfer is possible in a double-cavity optomechanical system through measuring the entanglements in logarithmic negativities among the component pairs. In particular, the negativity is computed through determining the symplectic eigenvalues of a covariance matrix that relates the fluctuations of all six quadratures of the system’s main components. This method is standard in the literature of dynamic entanglement but we have generalized it to apply on a covariance matrix. We observe that the successful generation of entanglement transfer only requires a single-sided driving laser and that the transfer patterns can be distinctively categorized into two groups for the different operating regimes assumed by optomechanical system.
Moreover, all the logarithmic negativities computed exhibit a time delay before the first appearance of a non-zero value. This time point signifies the initiation of cooperative motions among the three components in the optomechanical system, showing the transient response of the system to the external driving lasers as a whole. Nonetheless, the indirect entanglement between the left and the right cavity is apparent in all cases, thereby facilitating a mechanism for entanglement relay through cascaded cavities although the cavities are physically not directly coupled. Such a mechanism would be useful to quantum information processing, especially in terms of non-adiabatic quantum state transfer palomaki13 (); zhang03 (), and would provide a physical means to realize cavity arrays or resonator waveguides for transmitting information encoded in a quantum state zhou08 (); gong08 (); xuereb12 ().
In Sec. II, we give a detailed description of the double-cavity model. The equations of motions are derived under the Heisenberg picture in Sec. III and the steady-state solutions are calculated to give proof of the sufficiency of single-sided driving. After the covariance matrix of the fluctuations is introduced, the entanglements among all component pairs are computed numerically and analyzed in Sec. IV. The conclusions are given finally in Sec. V.
Ii Double optomechanical cavity
The proposed double-cavity optomechanical system is illustrated in Fig. 1, in which a mechanical resonator with reflective coatings on both sides receives the radiation pressures from both the cavity on the left side (L) and the cavity on the right side (R). The total Hamiltonian thus consists of three parts, which reads (), respectively,
The part accounts for the free Hamiltonians of the resonator and the cavities, the latter being regarded as bosonic modes of frequencies and . We associate a pair of annihilation and creation operators and for each bosonic mode, where indexes the cavity side, either left or right . We assume the frequencies and to be different in general according to the asymmetric cavity lengths and finesses assumed. The leakage rates are defined correspondingly from these parameters: . The mechanical resonator is described by the conjugate pair and , along with its oscillation mode frequency of and its mechanical damping rate of .
The part accounts for the phonon-photon interactions derived from the radiation pressures. The radiation pressure from each cavity side results from the deformation of the cavity volume due to the displacement of the middle resonator, which shifts the resonance frequencies of each cavity modes. Expending this frequency shift to first order, i.e. , the radiation pressure term sets the optomechanical coupling on the left side with strength , where is the effective mass of the resonator mode. The same derivation applies to the right cavity, giving the coupling strength but the leading sign would be opposite to as the common resonator has its displacement follow opposite directions for the two radiation pressures. Between the two cavities, there is no direct coupling.
The part accounts for the two external driving lasers with frequency and frequency . The driving strength of each laser is related to the input laser power and the leakage by . Note that we assume an asymmetric setting for the double-cavity system: the radiation pressures from the two sides are not identical and the cavities are unequally driven.
To study the indirect entanglement across the two cavity modes, we begin with the dynamics of the three components in the double-cavity optomechanical system through deriving a set of nonlinear Langevin equations. We carry out this step by finding the Heisenberg equations of the operators from the Hamiltonian in E. (1)-(3) and introducing phenomenologically the relaxation terms and their associative Brownian noise terms. The Langevin equations under the rotating frames of reference read
where () is the static detuning of the left (right) cavity field from the left (right) driving laser. The zero-mean fluctuation terms obey the correlation relation
The mechanical mode is under the influence of stochastic Brownian noise that satisfies in general the non-Markovian auto-correlation relation with a colored spectrum:
where is the Boltzmann constant and is the temperature of the mechanical bath. However, for a high quality mechanical resonator with , this non-Markovian process can be approximated as a Markovian one, where its fluctuation-dissipation relation can be asymptotically simplified to Giovannetti01 (); Fabre94 ():
where is the mean occupation number of the mechanical mode. This simplified Markovian relation will be assumed in the calculation of the entanglements.
Iii Dynamics and entanglement
iii.1 Steady states
In a single-cavity optomechanical system, the radiation pressure contributes the nonlinear photon number term in the Langevin equation of the mirror momentum in Eq. (4), leading to a multistability of the coordinate with three nonzero steady states. For a double-cavity case here, the second radiation pressure by the other cavity contributes a similar term in the equation. Under the asymmetric setting, the two pressure terms are not commensurate and the number of steady states of increases to five. The steady states are given by
where the plus (minus) signs in the second equation refers to the left (right) cavity.
For entanglement generation, it is necessary for the equation set (7)-(8) to have non-zero steady states. Therefore, a single-cavity optomechanical system usually requires an external driving laser (i.e., non-zero value of ) to drive the mechanical resonator out of its zero steady states at equilibrium position. However, for optomechanical systems with double-sided cavities, one external driving laser at either end of the cavities is sufficient to drive the mechanical resonator out of its zero position, in which case Eq. (7) would fall back to the single-cavity case of three roots.
In addition, we can observe that even when the double cavities have exactly symmetrical setup, i.e. identical laser driving amplitudes (), radiation pressures , and cavity lengths, the differing signs before to be taken by and in Eq. (8) allows the cavities to admit non-zero steady states. This is because the two cavity modes are constructively interfering with each other at the interface of the mechanical resonator through their indirect interactions of radiation pressures. In other words, even though the radiation pressures are exerted along opposite directions, the dynamic -phase difference between the cavities fields, reflected in the Hamiltonian Eq. (2) as the generator of the cavity motion, render the radiation pressures out of phase to favor the generation of entanglement. Given the symmetric setting where and in addition to the identities in driving amplitudes and radiation pressures, the condition for the steady-state equations to admit real roots is the inequality among the system parameters
Its derivation is given in Appendix A. Finding as the Young’s modulus of the resonator () and that the cavities have sufficient finesses (), the above criterion is met in most scenarios and the validity of entanglement generation is almost guaranteed.
For the symmetric setting, we expect the patterns of entanglement generations between either end of the cavity modes and the mechanical resonator to be qualitatively similar and differ only quantitatively in their variations over time. Deviating from this setting, the increase in asymmetry among the system parameters would increase the qualitative difference between the patterns of entanglements. We demonstrate these effect later in Sec. IV.
iii.2 Entanglement measure
Theoretically, the entanglements in terms of logarithmic negativity are computed through the fluctuations of the cavity quadratures about the steady states obtained from Eqs. (7)-(8). That is, we define the dimensionless quadratures of the two cavity fields as
and the corresponding input noise operators accordingly. Then taking as the vector operator for all the quadratures in the optomechanical system, we expand it to first-order using a c-number steady-state value and a zero-mean fluctuation operator . In addition, the nonlinear terms are linearized assuming in the expansion: and , while the higher-order products of the fluctuations are ignored.
The Langevin equations in Eq. (4) with the first-order expansion gives a coupled system of differential equations about the noise operators, enabling the coupling between the fluctuations of the two cavity fields and the mechanical resonator and thus the generation of entanglement between the two optical modes. Note that even though we have linearized the equations for these operators, eliminating the mechanical quadratures and in Eq. (4) will lead to equations of and nonlinearly related to and . This implies that the indirect entanglement between the quadratures of the left and the right cavities follow a nonlinear form in time.
In the following, instead of solving the coupled equations analytically, we follow the standard numerical approach adopted by the current researches on dynamic entanglement wang14 (); ying14 (); vitali07 (). The difference here is that we have a 6-component vector over the six quadratures of the tripartite optomechanical system instead of the usual 4-component vector. Similarly extending the input-noise vector to the 6-component ), we write the time-dependent inhomogeneous equations of motion as , where
In the matrix, and are the real and the imaginary parts of the scaled coupling constants . Along with the oscillation of the mechanical resonator, the dynamic detunings of the two cavities are defined as
where the plus (minus) sign corresponds to the left (right) cavity.
When the tripartite system is stable, it reaches a unique steady state, independently from the initial condition. Then given any arbitrary steady state, the fluctuations about it is fully characterized by its covariance matrix of the pairwise correlations among the quadratures, which obeys the equation . The diagonal elements of the are, in order, auto-correlations of the quadratures of the resonator, the left, and the right cavity mode. Hence, is the diagonal matrix for the corresponding damping and leakage rates responsible for the fluctuations. The multiple fluctuation-dissipation relations defined in Sec. II are therefore encapsulated in the relation . From its evolution equation, the covariance matrix can be written as a block-matrix
where each block represents matrix. The blocks on the diagonal indicate the variance within each subsystem (the resonator , the left cavity mode , and the right cavity mode ), while the off-diagonal blocks indicate covariance across different subsystems, i.e. the correlations between two components that describe their entanglement property.
To compute the pairwise entanglements, we reduce the covariance matrix to a submatrix . There are three such cases of the submatrix : (i) if the indices and for the element are confined to the set , the submatrix is formed by the first four rows and columns of and corresponds to the covariance between the resonator mode and the left cavity mode. Similarly, (ii) if the indices run over , is the covariance matrix of the resonator and the right cavity mode. (iii) If the indices run over , designates the covariance between the two opposite cavity modes. Summarizing, the submatrix can be written as
where , , and index the subsystems in the optomechanical cavity. The entanglement measured by logarithmic negativity is computed through a process known as symplectic diagonalization of each submatrix , where the entanglement properties are contained in the symplectic eigenvalues of the diagonalized matrix. If we write the diagonalized matrix as , then the eigenvalues along the diagonal read plenio07 ()
Denoting the state of a bipartite subsystem in the tripartite optomechanical cavity as , the negativity is defined as
where indicates the trace norm of the partial transposition of Vidal02 (). Taking as the minimum symplectic eigenvalue of the covariance matrix, is equivalent to after the diagonalization. Hence, the negativity is a decreasing function of and we usually write and take its logarithmic value as a measure of the entanglement Adesso04 (). This logarithmic negativity has the expression
In other words, the symplectic eigenvalue completely quantifies the quantum entanglement between each pair of components in the system. The necessary condition for showing a bipartite subsystem is entangled is that the symplectic eigenvalue retains a value less than one, which is equivalent to the inequality Simon00 ().
Iv Entanglement transfer
iv.1 Delayed build-up
To measure the entanglements, the noise terms , , , , and that appear in the variance matrix of Eq. (14) are taken as random variables of zero-mean Gaussian processes. The entanglements measured in logarithmic negativities are plotted against time for each of submatrices given in Eq. (15) to discern the entanglement transfer. We found similar transfer patterns over a range of parameters close to the experimentsZhang10 (). One typical case is shown here in Fig. 2, where from top to bottom we plot, respectively, between the left cavity and the resonator, between the right cavity and the resonator, and finally between the left and the right cavities.
For comparison, two cases are plotted for each entanglement pair: the blue ones denote the symmetric case and the red ones denote the asymmetric case. For the symmetric case, we adopt for the mechanical resonator a quality factor , resonance frequency MHZ, and effective mass ng ; for the cavities, we take cavity length at mm with finesse and cavity mode wavelength of nm. We set the power of the driving lasers at 70, which is detuned from the cavity mode at . For the asymmetric case plotted in red, we have adjusted the right cavity to a length of mm, which consequently affects the cavity leakage and the coupling amplitude between the driving and the cavity, while the length of the left cavity and other parameters remain unchanged.
We observe from Fig. 2 that there are two phases in the entanglement evolution. The initial phase is a period of zero , showing a delay in the formation of entanglement. The latter phase is a gradual build-up until certain saturation is reached. While the entanglement generations between either cavity and the mechanical resonator are smooth, that between the two cavities are oscillating or quasi-oscillating because of the nonlinear nature of the radiation pressure coupling wang14 (). Averaging out the oscillation, we see the patterns in the build-up of entanglement are identical to those between the cavity and the resonator. In addition, the delay periods among all three pairs coincide, demonstrating the transfer of cavity-resonator entanglement to intercavity entanglement and showing that distant entanglement is possible if the distant objects are indirectly coupled.
The delay in the entanglement build-up, during which assumes zero value, corresponds to the negativity in Eq. (17) taking a nonphysical negative value. We can interpret this delay period as the time duration when the three components in the tripartite system spend to establish their cooperation, which like the effect of superradiance depends strongly on the resonance linewidths. Comparing the delays for the symmetric and the asymmetric cases from Fig. 2(a) and (b), we see the similar inverse proportionality in the entanglement delay on the cavity leakage rates , i.e., . When the cavities are setup symmetrically, we measure the delays in both Fig. 2(a) and (b) at about ; when they are setup asymmetrically with , we observe for the left cavity being greater than its counterpart at the right side, at a difference of in time for a difference about in cavity linewidths.
iv.2 Death and revival
The influences of asymmetric parameter setup for the cavities are not only reflected in the delays of entanglement generation, but also in the entanglement pattern itself. In Fig. 3, we show a typical example with entanglements generated in a pattern distinctly differently from those in Fig. 2. The entanglements measured in logarithmic negativity are again plotted from top to bottom, respectively, for the three component pairs discussed above, but with driving laser powers increased to and cavity finesses decreased to . The left and the right cavity lengths remain in an asymmetric setup of mm and mm, respectively, and the rest of parameters are kept identical to those in Sec. IVA.
While the cavity-resonator entanglements for the two cavities follow the pattern of build-up to saturation after a time delay, which is similar to those of Fig. 2, the intercavity entanglement does not but otherwise oscillate over a death-revival cycle. Because of the inverse proportionality of the time delay to the cavity linewidths, the plots show a shortened delay and a reduced discrepancy between the delays in the left and the right cavity-resonator entanglements due to the decrease in cavity finesses.
On closer inspection, we can see the build-up in (a) and (b) are sharper and less gradual than their counterparts in Sec. IVA and the absolute negativity they can obtain are much smaller, especially for the left cavity. Even for the right cavity, its entanglement with the resonator declines shortly after a peak value, making all three plots assume essentially different characteristics than those of Fig. 2. This distinction can be attributed to the strong dependence of the operating regimes of optomechanical systems on external driving power and cavity finesse. In a single optomechanical cavity, it is reflected as periodic and quasiperiodic motions of the resonator wang14 (); in the double optomechanical cavity here, it is reflected as the resonator being driven monotonically in-phase (Fig. 2(c)) and driven periodically in-phase and out-of-phase (Fig. 3(c)) with the left and right cavities.
To summarize, we have studied the dynamic transfer of quantum entanglement from those within two cavity-resonator pairs to that between these two cavities inside a double-cavity optomechanical system. We numerically solved a coupled set of Heisenberg-Langevin equations to show the generation of quantum entanglements between each pair of the components under an experimentally accessible set of parameters. We find that the entanglement of the indirectly coupled cavities is built up over time in a pattern similar to those of the directly entangled cavity-resonator pairs, verifying the entanglement transfer. The similarities are accentuated by the almost identical characteristic delays and rising patterns but the entanglement transfer would be suppressed by the asymmetries in the two cavities. The asymmetries also differentiates the initiation times of the cavity-resonator entanglements, which leads to our speculation that the tripartite system is undergoing a cooperation process similar to that of superradiance before the emergence of the entanglement. To understand such a transient effect in a multipartite system demands a detailed analysis of the Heisenberg-Langevin equation set, which we shall leave to future studies, but we have seen here that dynamic entanglement is not only a measure of quantum information, but also a useful tool to dissect the cooperative motions of microscopic systems.
Acknowledgements.R. Z. is supported by the National Natural Science Foundation of China under Grant No. 61463016 and 61340029, Program for New Century Excellent Talents in University under Grant No. NCET-13-0795, Landing project of science and technique of colleges and universities of Jiangxi Province under Grant No. KJLD14037, Project of International Cooperation and Exchanges of Jiangxi Province under Grant No. 20141BDH80007. H. I. is supported by the FDCT of Macau under grant 013/2013/A1, University of Macau under grants MRG022/IH/2013/FST and MYRG2014-00052-FST, and National Natural Science Foundation of China under Grant No. 11404415.
Appendix A Steady states of symmetrical double cavity optomechanical system
Substituting Eq. (8) into Eq. (7) and cancelling the factor on both sides of the equation, which implies the trivial solution being one of the steady state in the symmetrical cavity setup, we arrive at the quartic equation
Lacking the odd-order terms in , the roots of the equation can be solved directly through quadratic formula. Since , the real roots exist only when:
i) is real, i.e. the discriminant being non-negative, which gives
and ii) the quadratic root to Eq. (18) is non-negative.
To satisfy the latter, we have to consider two cases:
ii-1) when , the square root of the determinant could take either the positive or the negative value. For the negative case, it is required that
or ii-2) for the postive case or when , it is required that
When two cases of condition (ii) are combined with condition (i), we see case (ii-1) impose a very stringent constraint on the admissible values of : between zero and . Case (ii-2) is more inclusive, which is what we are interested in here. Since it always holds that , when the inequality of Eq. (21) holds, the first condition in Eq. (19) is automatically satisfied.
To simplify the study, we confine our investigation in the positive domain of the detuning , for which Eq. (21) can be further reduced to
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