Entropic analysis of optomechanical entanglement for a nanomechanical resonator coupled to an optical cavity field

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Author: Jeong Ryeol Choi
Author’s contact information:
Affiliation: Department of Nanoengineering, Kyonggi University, Yeongtong-gu, Suwon, Kyeonggi-do, 16227, Republic of Korea
E-mail address: choiardor@hanmail.net
Tel: +82 31 249 1320
Fax: +82 31 249 9604

Jeong Ryeol Choi
Department of Nanoengineering, Kyonggi University
Republic of Korea

<List of Correction>

1. Line 12 on page 3.
[Old] The relation between the optical frequency \Delta and the cavity frequency \omega_c is given by \Delta = \omega_c - \omega_L - \delta_{rp}, where \delta_{rp} is the shift of the cavity frequency by radiation pressure. On the other hand, the coupling strength is given by g(t) = G(t) \sqrt{<n_c>}, where G(t) = [\omega_c(t)/L(t)]\sqrt{\hbar/[m\omega_m(t)]}, m is effective mass of the resonator, L is the cavity length, and <n_c> is the mean cavity photon number. For a more detailed description of the system, refer to Ref. [2].
[New] If we consider that cavity is driven by a laser field, the relation between the optical frequency \Delta and the cavity frequency \omega_c is given by \Delta = \omega_c - \omega_L - \delta_{rp}, where \delta_{rp} is the shift of the cavity frequency by radiation pressure [2]. On the other hand, the coupling strength is given by g(t) = G(t) \sqrt{<n_c>}, where G(t) = [\omega_c(t)/L(t)]\sqrt{\hbar/[m\omega_m(t)]}, m is effective mass of the resonator, L is the cavity length, and <n_c> is the mean cavity photon number [2].

2. Equation 3 and subsequent equations are revised by introducing damping constants \zeta_m and \zeta_c so that they can also be applied to dissipative optomechanical systems.

3. Last line of Eq. (17) and a subsequent sentence on page 5.
[Old] -\hbar\dot{\varphi}(t)[\beta(t)P_mX_c-\beta^{-1}(t)P_cX_m], where
[New] -\hbar[\dot{\varphi}_1(t)P_mX_c-\dot{\varphi}_2(t)P_cX_m], where \varphi_1(t) = \varphi(t)\beta(t), \varphi_2(t)=\varphi(t)\beta^{-1}(t), and

4. After Eq. (19) on page 5.
[Old] Let us assume that the variation of \varphi(t) over time is sufficiently slow.
[New] Let us assume that the variations of \varphi_1(t) and \varphi_2(t) over time are sufficiently slow. This means weak damping, \zeta_m(t) ~ 0 and \zeta_c(t) ~ 0, in addition to the previous assumption that the variations of g(t), \Delta(t), and \omega_m(t) are slow.

5. Line 3 from bottom on page 9.
[Old] We can further investigate the linear entropy for diverse particular cases with a specific choice of time dependence for parameters, ω_c (t), ω_m (t), etc. For instance, let us consider …
[New] We can further investigate the linear entropy for diverse particular cases with a specific choice of time dependence for parameters, ω_c (t), ω_m (t), etc. Abundant physical phenomena associated with frequency modulations in optomechanical systems have been reported so far [32-37]. Quantum effects of optomechanical systems can be practically enhanced by periodic modulations of the frequencies [34-36]. For instance, arbitrary bosonic squeezing in coupled optomechanical systems can be achieved by modulating one or both frequencies among the two which are associated with optical and mechanical modes respectively. Through this squeezing, it is possible to improve the measurement accuracy for weak signals [35,36]. An optimal optomechanical-cooling scheme by suppressing the Stokes heating process via periodical modulations of the frequencies of cavity and mechanical resonators has also been proposed [37].
It is known that entanglement can also be improved by modulating optomechanical parameters, such as the frequencies [36], the coupling parameter [38-40] and the amplitude of the cavity mode laser [36,41]. In order to see the influence of the periodical modulation of the optical frequency on the variation of the entanglement entropy, let us consider …

6. After Eq. (64) on page 10.
[Old] Then, from a minor evaluation, …
[New] We can easily confirm that these suppositions make the system satisfy the adiabatic condition which was mentioned in Sec. 3 (see sentences given immediately after Eq. (19)). Then, from a minor evaluation, …

7. Line 3 on page 17.
[Old] No sentences.
[New] Although we have evaluated entanglement entropies for the case of the ground state of the optical (and mechanical) oscillators for convenience in part, it may highly be possible to think of an excited state of the optical oscillator, because it is driven by a laser field. If such a state is far from the ground state, the entanglement between the optical and the mechanical modes may be enhanced due the increase of the quadrature uncertainty in the optical mode. Notice that, if the quantum number in a coupled oscillatory motion is large, the entanglement between the associated subsystems is enhanced [49-51].

END