Numerical studies on the boundary entanglement in an optomechanical phonon laser system
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Qing-Xia Meng1,2,3, Zhi-Jiao Deng,1,3,∗, Shi-Wei Cui1,31 Department of Physics, College of Liberal Arts and Sciences, National University of Defense Technology, Changsha 410073, China 2 Northwest Institute of Nuclear Technology, Xi’an 710024, Shaanxi, China 3 Interdisciplinary Center for Quantum Information, National University of Defense Technology, Changsha 410073, China
First author contact:∗ Author to whom any correspondence should be addressed. Received:2020-08-3Revised:2020-08-31Accepted:2020-09-11Online:2020-10-27
Abstract In our previous work (Meng et al 2020 Phys. Rev. A 101 023838), we discover the phenomenon that the quantum entanglement on the driving threshold line remains a constant in a three-mode optomechanical phonon laser system. In this paper, to find the conditions under which the constant boundary entanglement shows up, we explicitly study how this boundary entanglement depends on various parameters through numerical integrations. The results show that the necessary and sufficient condition is a resonant frequency match between the optical frequency difference and the mechanical vibrational frequency, and this constant value is proportional to the multiplication of the square of the optomechanical coupling strength and the resonant driving threshold power. Keywords：boundary entanglement;optomechanics;phonon laser
PDF (524KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Qing-Xia Meng, Zhi-Jiao Deng, Shi-Wei Cui. Numerical studies on the boundary entanglement in an optomechanical phonon laser system. Communications in Theoretical Physics[J], 2020, 72(11): 115101- doi:10.1088/1572-9494/abb7db
Optomechanical systems [1], which deal with nonlinear interaction between a radiation field and mechanical motions, have attracted huge research interest. They can be used to study the mutual control and influences between photon and phonon degrees of freedom [2, 3], mutual conversion between optical and acoustic signals and all possible applications [4]. Usually, the single-photon optomechanical coupling is quite small; a cavity with a driving laser is needed to amplify this interaction. When the driving is strong enough, i.e. above a certain threshold, the system will settle into the so-called self-sustained mechanical oscillations [5], which is essentially a limit cycle [6] from the perspective of nonlinear dynamics, also referred to as a phonon laser [7, 8]. The phonon laser, in the context of optomechanical systems, has been observed in several experiments [9 –11].
When the temperature goes down, the influence of quantum fluctuations becomes prominent and various quantum properties will also appear. An interesting question is how the quantum properties would change in the phonon laser system from below to above the threshold? We have studied this problem based on an experimental three-mode optomechanical phonon laser model [12]. The most striking finding is that the quantum entanglement on the threshold line keeps as a constant and it is quite robust against thermal phonon noise, as a strong indication of transitions from stable fixed points to limit cycles. However, we utilize a resonant condition between the optical energy difference and the mechanical vibrational frequency in our calculation. Whether it is a necessary condition to retain a constant boundary entanglement remains a question in our mind.
In this paper, we will numerically study the boundary entanglement by changing various parameters, aiming to find out under what circumstances can we obtain a constant boundary entanglement. Calculations show that as long as we keep the resonant condition, the entanglement on the threshold line remains a constant, and other parameters might only change the value of the constant. In other words, the resonant condition is the necessary and sufficient condition for maintaining a constant boundary entanglement. Due to the calculation complexity, we can only resort to numerical integrations. Our paper is organized as follows. Firstly, we review the physical model and the key procedures used to obtain the boundary entanglement in [12]. Then, we concretely study how the boundary entanglement changes with different parameters, trying to search the conditions for constant boundary entanglement. Finally, we summarize our results.
The optomechanical phonon laser system discussed in [12] is based on the experiment in [9]. The dynamics of the whole system are described by a set of nonlinear quantum Langevin equations (written in the rotating frame with respect to the laser frequency ωL ) [12, 13],$ \begin{eqnarray}\begin{array}{rcl}{\mathop{\hat{c}}\limits^{\cdot }}_{1} & = & \left({\rm{i}}({\rm{\Delta }}-J)-\displaystyle \frac{\kappa }{2}\right){\hat{c}}_{1}+\displaystyle \frac{{\rm{i}}g}{2}({\hat{c}}_{1}-{\hat{c}}_{2})\hat{q}+\displaystyle \frac{{\rm{\Lambda }}}{\sqrt{2}}+\sqrt{\kappa }{\hat{c}}_{{in},1}\\ {\mathop{\hat{c}}\limits^{\cdot }}_{2} & = & \left({\rm{i}}({\rm{\Delta }}+J)-\displaystyle \frac{\kappa }{2}\right){\hat{c}}_{2}-\displaystyle \frac{{\rm{i}}g}{2}({\hat{c}}_{1}-{\hat{c}}_{2})\hat{q}+\displaystyle \frac{{\rm{\Lambda }}}{\sqrt{2}}+\sqrt{\kappa }{\hat{c}}_{{in},2}\\ \mathop{\hat{q}}\limits^{\cdot } & = & {\omega }_{m}\hat{p}\\ \mathop{\hat{p}}\limits^{\cdot } & = & \displaystyle \frac{g}{2}\left({\hat{c}}_{1}^{\dagger }{\hat{c}}_{1}+{\hat{c}}_{2}^{\dagger }{\hat{c}}_{2}-{\hat{c}}_{1}^{\dagger }{\hat{c}}_{2}-{\hat{c}}_{2}^{\dagger }{\hat{c}}_{1}\right)-{\omega }_{m}\hat{q}-{\gamma }_{m}\hat{p}+\hat{\xi },\end{array}\end{eqnarray}$ where, ${\hat{c}}_{1}=\tfrac{1}{\sqrt{2}}({\hat{a}}_{1}+{\hat{a}}_{2})$, ${\hat{c}}_{2}=\tfrac{1}{\sqrt{2}}({\hat{a}}_{1}-{\hat{a}}_{2})$ are two optical supermodes, where two optical local modes ${\hat{a}}_{1}$, ${\hat{a}}_{2}$ with the mutual tunneling rate J, have the same frequency ωa . One of them, i.e. ${\hat{a}}_{1}$ mode, is driven by a laser with frequency ωL and amplitude Λ, while the other one, i.e. ${\hat{a}}_{2}$ mode, is coupled by a strength g with mechanical mode through radiation pressure force. The operators $\hat{q}=\tfrac{1}{\sqrt{2}}({\hat{b}}^{\dagger }+\hat{b})$, $\hat{p}=\tfrac{1}{\sqrt{2}{\rm{i}}}(\hat{b}-{\hat{b}}^{\dagger })$ represent the dimensionless position and momentum of the mechanical mode with frequency ωm, respectively. The parameters ${\rm{\Delta }}={\omega }_{L}-{\omega }_{a}$ denote the laser detuning from the cavity resonance, κ is the optical intensity decay rate, and γm is the mechanical damping rate. The operators ${\hat{c}}_{{in},1}$, ${\hat{c}}_{{in},2}$ are the vacuum radiation input noise with zero means. The Brownian noise operator $\hat{\xi }$ with a zero mean value, satisfies $\tfrac{1}{2}\left\langle \hat{\xi }(t)\hat{\xi }({t}^{{}^{{\prime} }})+\hat{\xi }({t}^{{}^{{\prime} }})\hat{\xi }(t)\right\rangle ={\gamma }_{m}(2\overline{n}+1)\delta (t-{t}^{{}^{{\prime} }})$ in the limit of the high mechanical quality factor [14], where $\overline{n}\,={\left(\exp (\tfrac{{\hslash }{\omega }_{m}}{{k}_{B}T})-1\right)}^{-1}$ is the mean thermal phonon number.
The generation of the phonon laser is through a typical parametric down conversion process. If the optical frequency difference 2J between these two supermodes is near resonant with the mechanical vibrational frequency ωm, i.e. 2J ≃ωm, an efficient driving of c1 mode with ${\rm{\Delta }}\simeq J$ could lead to amplification of the mechanical mode via the interaction term ${\hat{c}}_{1}^{\dagger }{\hat{c}}_{2}\hat{b}+{\hat{c}}_{1}{\hat{c}}_{2}^{\dagger }{\hat{b}}^{\dagger }$, which means that one photon disappears in c1 mode accompanying the simultaneous birth of a photon in c2 mode and a phonon. Coherent oscillation (i.e. mechanical lasing) would occur in the mechanical mode once above a certain driving threshold. Moreover, this two-mode squeezing interaction term will inevitably cause the quantum entanglement between the optical c2 mode and the mechanical mode [15 –18].
The equation (1 ) is generally difficult to solve; however, in the regime of weak coupling $g\ll \kappa $ and moderate driving Λ, it can be solved by the mean-field approximation [1]. The idea is to separate the quantum operators into two parts $\hat{O}=\langle \hat{O}\rangle +\delta \hat{O}$, where $\langle \hat{O}\rangle \equiv O$ is the mean field describing the classical behavior of the system, and $\delta \hat{O}$ is the quantum fluctuation around the classical orbit. The threshold for lasing can be analyzed by the classical mean-field equations [12],$ \begin{eqnarray}\begin{array}{rcl}{\mathop{c}\limits^{\cdot }}_{1} & = & \left[{\rm{i}}({\rm{\Delta }}-J)-\displaystyle \frac{\kappa }{2}\right]{c}_{1}+\displaystyle \frac{{\rm{i}}g}{2}({c}_{1}-{c}_{2})q+\displaystyle \frac{{\rm{\Lambda }}}{\sqrt{2}},\\ {\mathop{c}\limits^{\cdot }}_{2} & = & \left[{\rm{i}}({\rm{\Delta }}+J)-\displaystyle \frac{\kappa }{2}\right]{c}_{2}-\displaystyle \frac{{\rm{i}}g}{2}({c}_{1}-{c}_{2})q+\displaystyle \frac{{\rm{\Lambda }}}{\sqrt{2}},\\ \mathop{q}\limits^{\cdot } & = & {\omega }_{m}p,\\ \mathop{p}\limits^{\cdot } & = & -{\omega }_{m}q-{\gamma }_{m}p+\displaystyle \frac{1}{2}g({c}_{1}^{\ast }{c}_{1}+{c}_{2}^{\ast }{c}_{2}-{c}_{1}^{\ast }{c}_{2}-{c}_{2}^{\ast }{c}_{1}).\end{array}\end{eqnarray}$ The basic picture is that when the driving is not very strong, the mechanical mode is only being pushed and its equilibrium position gets shifted, corresponding to a stable fixed point, and while increasing the driving power above a certain threshold, the mechanical mode oscillates at the mechanical frequency ωm besides reaching a new equilibrium position, corresponding to an unstable fixed point evolving into a limit cycle. To find the laser threshold is to find the boundary between stable and unstable fixed points. The fixed points in equation (2 ) are the solutions after allowing all the first-order derivatives $\mathop{O}\limits^{\cdot }$ to be 0. Their stability can be judged by the linearized Langevin equations for the quantum fluctuation operators [12],$ \begin{eqnarray}\mathop{u}\limits^{\cdot }(t)=S(t)u(t)+n(t),\end{eqnarray}$ where we have defined ${u}^{T}(t)=(\delta {\hat{X}}_{1}(t),\delta {\hat{Y}}_{1}(t),\delta {\hat{X}}_{2}(t),\delta {\hat{Y}}_{2}(t),\delta \hat{q}(t),\delta \hat{p}(t))$ with quadrature operators $\delta {\hat{X}}_{j}=\tfrac{1}{\sqrt{2}}(\delta {\hat{c}}_{j}+\delta {\hat{c}}_{j}^{\dagger })$, $\delta {\hat{Y}}_{j}=\tfrac{1}{\sqrt{2}{\rm{i}}}(\delta {\hat{c}}_{j}-\delta {\hat{c}}_{j}^{\dagger })$, and n (t) represents the input noise operators. Equation (3 ) is derived from equation (1 ) by the standard linearization procedure. The dynamics of the Jacobian matrix S depend on the time evolution of equation (2 ) [12, 19]. The two equations (2 ) and (3 ) are combined to predict the quantum dynamics of the whole system in the mean-field approximation. To judge the stability of the fixed point, we evaluate matrix S at the fixed point. It is stable only if all the eigenvalues of matrix S have negative real parts.
To examine the quantum properties in the system, we adopt logarithmic negativity [20] to measure the quantum entanglement between optical c2 mode and the mechanical mode. Since equation (3 ) is linear, the properties of quantum fluctuations for Gaussian input noises are fully characterized by the covariance matrix V [21], with its elements defined by ${V}_{{ij}}=\tfrac{1}{2}(\langle {u}_{i}(t){u}_{j}(t)+{u}_{j}(t){u}_{i}(t)\rangle )$ . The entanglement is related to the covariance matrix between c2 mode and the mechanical mode, which is a submatrix of V . The logarithmic negativity EN can be obtained by a formula concerning this submatrix [12, 22]. Here, we only focus on the boundary entanglement, i.e. the entanglement between c2 mode and the mechanical mode when the system works at the phonon laser threshold. Due to the numerical discreteness, the points can never be exactly on the boundary. The points we choose are quite close to the boundary from the side of the stable region. In the numerical integrations, we start with random initial values for V, c1, c2, q, and p until EN reaches a steady final value.
Our previous work [12] discovers that the entanglement on the boundary line keeps as a constant, as shown in figure 1 (b), which is based on a two-dimensional phase diagram with a specific set of system parameters given in figure 1 (a) and, in particular, the resonant condition 2J =ωm is employed. The boundary line takes the shape of a single peak pointing to the left. Although the positions of the fixed points in the parameter space are generally quite different along the boundary, the entanglement on this boundary line remains a constant. In the following, we discuss how the boundary entanglement changes with all the related parameters. For simplicity, we set the temperature T =0, i.e. $\overline{n}=0$ . The two-dimensional phase diagram is plotted with respect to the driving strength Λ and driving detuning Δ, as in figure 1 (a). The J /κ =10 can be fixed for comparison, so the remaining parameters are ωm /κ, g /κ, γm /κ .
Figure 1.
New window|Download| PPT slide Figure 1.(a) A phase diagram of the system with the parameters J /κ =10,ωm /κ =20, g /κ =0.02, γm /κ =0.01, and its corresponding boundary entanglement at zero temperature is depicted in panel (b).
In figure 2, we change the value of ωm from resonant to nonresonant with regard to the optical energy difference 2J . The boundary line in the phase diagram develops from a single peak to double peaks pointing to the left. For the resonant case, the peak is at Δ/κ =10, while for the nonresonant case, the peaks are at Δ/κ =10 and ${\rm{\Delta }}/\kappa \,=10-(2J-{\omega }_{m})/\kappa $, respectively. It is easy to understand that Δ=J is most efficient for 2J =ωm, so the instability of this driving detuning occurs at the lowest driving power, as clearly seen in figure 1 (a). However, for $2J\ne {\omega }_{m}$, besides the resonant driving of c1 mode at Δ=J, driving c1 at the frequency that matches the resonant down conversion ${\hat{c}}_{1}{\hat{c}}_{2}^{\dagger }{\hat{b}}^{\dagger }$ is also favorable. One of the peaks in figure 2 (b) at Δ/κ =6 is not shown due to the same vertical range as in figure 1 (a). We compare the boundary entanglement with different values of ωm /κ in figure 2 (c). The entanglement of the nonresonant case starts to drop down when ${\rm{\Delta }}/\kappa \gt 10.5$ ; the larger the detuning $(2J-{\omega }_{m})/\kappa $, the faster it decreases, which shows that the constant boundary entanglement is only valid for resonant conditions.
Figure 2.
New window|Download| PPT slide Figure 2.Phase diagrams of the system with the same parameters as in figure 1 except the mechanical vibration frequency ${\omega }_{m}/\kappa =18$ in panel (a) and ωm /κ =16 in panel (b). A comparison of the boundary entanglement with different ωm /κ is shown in panel (c).
The influence from γm /κ is demonstrated in figure 3 . With the increase in the mechanical damping rate, it needs more driving power to become unstable. Therefore, the threshold value Λth moves to the right, and the peak gets much sharper. From the calculation in [12], we could estimate the threshold value Λth,0 at resonant driving of c1 mode, i.e. Δ=J . It is determined by demanding the effective mechanical damping rate ${\gamma }_{\mathrm{eff}}\,={\gamma }_{m}+{\gamma }_{\mathrm{opt}}=0$ [23], where γopt is the optomechanical damping rate induced by the radiation pressure force. The expression for γopt is quite cumbersome. Comparing the threshold ${\gamma }_{\mathrm{opt},0}$ at Δ=J with that of figure 1 (a), ${\gamma }_{\mathrm{opt},0}\propto {g}^{2}{{\rm{\Lambda }}}_{\mathrm{th},0}^{2}$ . In figure 1 (a), we could read out $g{{\rm{\Lambda }}}_{\mathrm{th},0}/{\kappa }^{2}=0.02\ast 5$, so to increase γm /κ would require us to increase ${g}^{2}{{\rm{\Lambda }}}_{\mathrm{th},0}^{2}/{\kappa }^{4}$ by the same factor. With γm /κ =0.02 (or 0.05), g /κ unchanged, ${{\rm{\Lambda }}}_{\mathrm{th},0}/\kappa =5\ast \sqrt{2}\,\simeq 7$ (or $5\ast \sqrt{5}\simeq 11.2$ ). The boundary entanglement is perfectly kept as a constant, as seen in figure 3 (c), and its values for γm /κ =0.01, 0.02, and 0.05 are approximately 0.0149, 0.0292, and 0.0731, respectively, which implies that the boundary EN is proportional to γm /κ or, more essentially, is proportional to ${g}^{2}{{\rm{\Lambda }}}_{\mathrm{th},0}^{2}/{\kappa }^{4}$ . The increase in the boundary entanglement with respect to γm /κ is beneficial from the increase in the threshold driving power.
Figure 3.
New window|Download| PPT slide Figure 3.Phase diagrams of the system with the same parameters as in figure 1 except the mechanical damping rate γm /κ =0.02 in panel (a) and γm /κ =0.05 in panel (b). A comparison of the boundary entanglement with different γm /κ is shown in panel (c).
Finally, we analyze the impact from g /κ . The bigger the coupling strength, the less driving power is required for the instability. As seen in figures 4 (a) and (b), with the increase in g /κ, the peak moves to the left and gets wider. By the same argument as above, since γm /κ is kept unchanged, we could deduce that the value of ${g}^{2}{{\rm{\Lambda }}}_{\mathrm{th},0}^{2}/{\kappa }^{4}$ at Δ=J is the same as that in figure 1 (a). So, with g /κ =0.01 (or 0.05), ${{\rm{\Lambda }}}_{\mathrm{th},0}/\kappa =0.1/0.01=10$ (or 0.1/0.05=2). The entanglement horizontal lines in figure 4 (c) with different g /κ coincide with each other, which is consistent with the conclusion that the boundary entanglement is proportional to ${g}^{2}{{\rm{\Lambda }}}_{\mathrm{th},0}^{2}/{\kappa }^{4}$ . The two factors g and ${{\rm{\Lambda }}}_{\mathrm{th},0}$ compensate each other to avoid the effects on the boundary entanglement.
Figure 4.
New window|Download| PPT slide Figure 4.Phase diagrams of the system with the same parameters as in figure 1 except the optomechanical coupling strength g /κ =0.01 in panel (a) and g /κ =0.05 in panel (b). A comparison of the boundary entanglement with different g /κ is shown in panel (c).
In summary, we have numerically studied how various parameters influence the boundary entanglement in an optomechanical phonon laser system. The calculations show that as long as the resonant condition is met, i.e. the optical frequency difference equals the mechanical vibrational frequency, the entanglement on the threshold line remains constant. It is proportional to γm /κ, but does not depend on g /κ . Essentially, this entanglement constant is proportional to the multiplication of the square of the optomechanical coupling strength and the resonant driving threshold power. These results should be useful to experimentalists who are interested in the threshold entanglement in lasers. The underlying physics as to why it can be a constant is very hard to obtain just by numerical methods, but definitely deserves further study. We hope other researchers will look into this question. We are also trying to look for the answers from simpler models.
Acknowledgments
This work is supported by the National Natural Science Foundation of China under Grant Nos. 11574398 and 61632021, the National Basic Research Program of China under Grant No. 2016YFA0301903, and by the Natural Science Foundation of Hunan Province of China under Grant Nos. 2018JJ2467 and 2018JJ1031.
,1,3,∗, Shi-Wei Cui1,31 Department of Physics, College of Liberal Arts and Sciences, 
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