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Entropic analysis of optomechanical entanglement for a nanomechanical resonator coupled to an optical cavity field
by Jeong Ryeol Choi
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Authors (as registered SciPost users):  Jeong Ryeol Choi 
Submission information  

Preprint Link:  scipost_202010_00030v2 (pdf) 
Date submitted:  20210404 06:33 
Submitted by:  Choi, Jeong Ryeol 
Submitted to:  SciPost Physics Core 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We investigate entanglement dynamics for a nanomechanical resonator coupled to an optical cavity field through the analysis of the associated entanglement entropies. The effects of time variation of several parameters, such as the optical frequency and the coupling strength, on the evolution of entanglement entropies are analyzed. We consider three kinds of entanglement entropies as the measures of the entanglement of subsystems, which are the linear entropy, the von Neumann entropy, and the Renyi entropy. The analytic formulae of these entropies are derived in a rigorous way using wave functions of the system. In particular, we focus on time behaviors of entanglement entropies in the case where the optical frequency is modulated by a small oscillating factor. We show that the entanglement entropies emerge and increase as the coupling strength grows from zero. The entanglement entropies fluctuate depending on the adiabatic variation of the parameters and such fluctuations are significant especially in the strong coupling regime. Our research may deepen the understanding of the optomechanical entanglement, which is crucial in realizing hybrid quantuminformation protocols in quantum computation, quantum networks, and other domains in quantum science.
Author comments upon resubmission
I am resubmitting a paper entitled "Entropic analysis of optomechanical entanglement for a nanomechanical resonator coupled to an optical cavity field" in SciPost Physics Core.
I agree with this submission.
This work is original research and has not been published or submitted for publication elsewhere.
I declare no conflict of interests.
Author: Jeong Ryeol Choi
Author’s contact information:
Affiliation: Department of Nanoengineering, Kyonggi University, Yeongtonggu, Suwon, Kyeonggido, 16227, Republic of Korea
Email 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 changes
<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 [3237]. Quantum effects of optomechanical systems can be practically enhanced by periodic modulations of the frequencies [3436]. 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 optomechanicalcooling 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 [3840] 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 [4951].
END
Current status:
Reports on this Submission
Report #1 by Anonymous (Referee 3) on 2021514 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202010_00030v2, delivered 20210514, doi: 10.21468/SciPost.Report.2914
Report
While the Author answered reasonably to my points of the first review, I find that his description of dissipative effects is quite lacking.
If I follow correctly what the Author does: the dumping therms are "phenomenologically" added to the Hamiltonian, and then approximated away obtaining the same result of the previous version. So, nothing changes except some new "phenomenological terms" are added and removed without the possibility to even understand if they were correct. If this is the case, I think that this must be improved before pubblication.
Requested changes
1 Improve or remove the description of the description of the dissipative effects.
Author: Jeong Ryeol Choi on 20210521 [id 1444]
(in reply to Report 1 on 20210514)I will revise the manuscript according to the reviewer's comment.