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

### Submission summary

 As Contributors: Jeong Ryeol Choi Preprint link: scipost_202010_00030v1 Date submitted: 2020-10-31 09:32 Submitted by: Choi, Jeong Ryeol Submitted to: SciPost Physics Core Academic field: Physics Specialties: Quantum Physics 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 quantum-information protocols in quantum computation, quantum networks, and other domains in quantum science.

###### Current status:
Has been resubmitted

### Submission & Refereeing History

Resubmission scipost_202010_00030v2 on 4 April 2021

Submission scipost_202010_00030v1 on 31 October 2020

## Reports on this Submission

### Report

The author calculates analytically the linear, von Neumann and Rényi entanglement entropies for a system composed by two coupled harmonic oscillators with time dependent parameters in the adiabatic regime. The harmonic oscillators describe a mechanical resonator and a optical cavity as in Ref.[2]. The entanglement entropies increase as the coupling strength increases, as expected.
The calculations are sound and elegant, I recommend publication in SciPost Physics after the requested changes are addressed.

### Requested changes

1) validity of the adiabatic approximation: the author should quantify what "the variation of $\varphi(t)$ over time is sufficiently slow" (as written after eq. (19)) means. In particular the author should justify analytically or numerically in which parameters regime the adiabatic approximation is valid, and check that the parameters used in Sec. 5 and Sec. 6 fulfill it.

2) the derivative of $\beta(t)$ does not appear in eq.(17), is this an exact result or it depends on some approximation? if it depends on some approximation it should be pointed out in the text.

3) a more self-contained description of the system in Sec.2 may be useful to the reader. At the moment the main reference for the description of the system is Ref.[2].

4) the entanglement entropies are calculated for the ground states. It is not clear if both systems can actually be in the ground state since the optical cavity is driven by a laser. The author should justify that.

5) Can the author give an idea of what would change for excited states?

• validity: good
• significance: good
• originality: good
• clarity: good
• formatting: excellent
• grammar: good

Author Jeong Ryeol Choi on 2020-12-27
(in reply to Report 2 on 2020-12-08)

<Response to the comment of the reviewer 2>
In the revised version, I will improve the manuscript regarding the report of the reviewer 2.

### Weaknesses

1) The analysis is very similar to what has been presented in previous works.

2) No real physical motivation for the studied problem of a modulated optical cavity frequency.

3) Not applicable to realistic optomechanical systems with dissipation.

### Report

In this paper different entanglement measures for a linearized optomechanical system are evaluated. Analytic expressions for the linear entropy, the von Neumann entropy and the Renyi entropy are derived from the exact evolution of the wavefunction of the combined system. The resulting entropies are then investigated for a system with a modulated frequency and for different coupling strengths.

The generation of entangled states in optomechanical systems is a well-studied problem, which in the considered linearized regime reduces to the study of the entanglement of a two-mode Gaussian state. Although in the quantum optics literature one usually doesn't evaluate the von Neumann entropy or the Renyi entropy explicitly, also the general expressions for these entropies in a two-mode system are known. Indeed, the current analysis very similar to what is already presented, for example, in Ref. [30]. Since only the coherent evolution is considered the whole problem discussed in this paper reduces the evaluation of the wavefunction of two coupled oscillators. Although the analytic results for the different entropies etc. might still be involved, their derivation is conceptually straightforward.

Despite the claim of providing more insights into entanglement properties of optomechanical systems, the actual findings have little relevance for such systems. The important influence of dissipation or thermal noise, but also the preparation of the initial state or the readout of the entanglement are not discussed at all. The study of a modulated cavity frequency is not motivated and no connection to any quantum information processing applications is made.

In summary, I fail to see a significant new theoretical result or an important physical insight for the field of optomechanics. Therefore, I cannot recommend a publication of this work in SciPost.

• validity: high
• significance: poor
• originality: low
• clarity: ok
• formatting: good
• grammar: excellent

Author Jeong Ryeol Choi on 2020-12-27
(in reply to Report 1 on 2020-11-25)

<1. Response to the comment "Indeed, the current analysis very similar to what is already presented, for example, in Ref. [30].">
The mathematical evaluations given in Ref. [30] is based on the simple rotation method. I have pointed out the weak point of the simple rotation method in my manuscript (see from line 8 on page 4 to the last line on the same page). Usually, the simple rotation method is inapplicable unless there are some restriction(s) in the coupled oscillatory systems.
The system adopted in Ref. [30] is a simple case where the masses of the two oscillatory subsystems are identical to each other: m_1 = m_2 = 1. In this restricted case, the simple rotation method is applicable. However, for the case that m_1 and m_2 are not equal to each other, such a simple rotation method is no longer applicable (see the appended file for detailed verification of this). In addition to this, the simple rotation method is also not applicable to the system in the present work, as mentioned in my manuscript with a verification.
In order to overcome this difficulty, I have adopted much more powerful method which is the unitary transformation method in my manuscript instead of the simple rotation method.

<2. Response to the comment "Not applicable to realistic optomechanical systems with dissipation.">
In the revised version, I will improve the manuscript in such a way that it can also be applicable to dissipative optomechanical systems.

<3. Response to the comment "No real physical motivation for the studied problem of a modulated optical cavity frequency.">
In the revised version, I will provide physical motivation for the modulation of the optical cavity frequency.