## Abstract

We proposed a scheme for detecting the atom-field coupling constant in the Dicke superradiation regime based on a hybrid cavity optomechanical system assisted by an atomic gas. The critical behavior of the Dicke model was obtained analytically using the spin-coherent-state representation. Without regard to the dynamics of cavity field an analytical formula of one-to-one correspondence between movable mirror’s steady position and atom-field coupling constant for a given number of atoms is obtained. Thus the atom-field coupling constant can be probed by measuring the movable mirror’s steady position, which is another effect of the cavity optomechanics.

© 2012 Optical Society of America

## 1. Introduction

The interplay of light and mechanical motion on the nanoscale has emerged as a new research topic during the past few years. Light interacting with matter can not only be absorbed and emitted by individual atom but also can exert forces on material objects as was predicted by Maxwell. The radiation pressure force of light was first directly observed experimentally in 1901 [1, 2]. Recently there has been a great surge of interest in the application of radiation forces to manipulate the center-of-mass motion of mechanical oscillators which can be used for detecting gravitational waves [3, 4], cooling and reading out micro- and nanomechanical devices towards the quantum regime, even the quantum mechanical ground state for the study of quantum-classical boundary of a mechanical system [5–12]. Furthermore the optomechanical interactions has found more applications in the generation of non-classic state of light and mechanical systems [13–15], realization of quantum entanglement between a micromechanical oscillator and optical cavity field mode [16–18] or a second mechanical oscillator [19–21]. Up to date a number of proposals have been put forward that how atomic systems—such as a trapped ion [22], atom [23], Bose-Einstein condensate (BEC) [24–28], or atomic ensemble [29–32]—could be coupled to a mechanical device. Recently an equivalent optomechanical coupling is realized between one-dimensional interacting bosons and the electromagnetic field in a high-finesse optical cavity or a degenerate Fermi gas in a one-dimensional optical lattice coupled to a cavity [33, 34].

The Dicke model (DM) describing an ensemble of two-level atoms collectively coupled to a single quantized mode of the electromagnetic field exhibits a zero-temperature phase transition at a critical value of the dipole coupling strength. Below such critical coupling strength, the system is in the normal phase in which all the atoms are in their ground state and the field is in vacuum. On the contrary the system is in the superradiant phase which is characterized by a non-zero field and a macroscopic excitation of the matter. The normal-superradiant quantum phase transition (QPT) induced by collective quantum phenomena in atomic physics and quantum optics has been extensively studied since the pioneering works [35–38]. Besides the quantum chaos, ground state entanglement, critical behavior [39–43] correlated with the presence of QPT has also been discussed extensively. During these investigations the most popular method adopted is Holstein-Primakoff (HP) transformation [44].

Our proposal follows along the path in Ref.[45], in which quantum dynamics of a movable mirror coupled to a critical reservoir is investigated. In this article we use the hybrid cavity optomechanical system assisted by an atomic gas to detect the atom-field coupling constant in the Dicke superradiant phase. The paper is organized as follows. In Sec.2, we presented the setup for our scheme. In Set.3 we derived analytically the critical behavior of the Dicke Hamiltonian by means of the spin-coherent-state representation. Then we presented a one-to-one correspondence relation between the atom-field coupling constant and the steady state position of the movable mirror in Set.4 and at the bottom we have a discussion of the experimental feasibility. Finally in Set.5 we have a summary on the full paper.

## 2. Model

We consider an optical Fabry-Perot cavity as shown in Fig. 1, in which the movable mirror is harmonic driven and is much lighter than the other such that the effect of the radiation pressure force can be enhanced. We assume identical coupling constants for all atoms which can be guaranteed by confining the atomic gas in a region of small extent compared with the cavity size *L*, as assumed in the treatment of the Dicke model [35, 39, 40]. The analogous experimental setup has been proposed and analyzed in the Ref. [23], in which only a single atom is considered.

## 3. Dicke hamiltonian

We assume a separation of time scales the cavity field is taken to evolve on a much shorter time scale than the mechanical oscillator so we first consider the Dicke system separately. Consider the collective interaction of *N* identical two-level system with a single mode radiation field inside a lossless cavity. In the long-wavelength limit and dipolar approximation the DM Hamiltonian reads

*ω*is the frequency of radiation field,

*ω*

_{0}is the atomic transition frequency,

*a*

^{†}and

*a*are the creation and annihilation operator of the field mode,

*λ*is the effective atom-field coupling constant,

*J*is the atomic relative population operator,

_{z}*J*

_{±}are the collective atomic raising and lowering operators. They satisfy the SU(2) Lie algebra, where the collective operators are described in terms of standard pauli matrices of each two-level atom

We give a trial wave function being a tensorial product of spin coherent states (SCS) and a boson coherent state |*θ*, *φ*〉 ⊗ |*α*〉, where the boson coherent state is defined by

*j*,−

*j*〉 by the angle

*θ*about the axis

**n**= (sin

*φ*, −cos

*φ*,

**0**) with

*j*=

*N*/2 being the total pseudo-spin value, i.e., where

*R*=

_{θ}_{,φ}*e*

^{−}

^{i}^{θ}

^{J}^{·}

**=**

^{n}*e*

^{−iθ(Jx sinφ−Jy cosφ)}. Thus we have the following eigen equation which allow us to calculate analytically the energy function

*α*=

*μ*+

*iv*with

*μ*,

*v*being real variables. The ground-state energy is the minimum of the energy function

*E*

_{−}(

*α*). Using the variational procedure we have

*λ*<

*λ*the system is in normal phase with mean photon number |

_{c}*α*| = 0, otherwise the system is in superradiation phase with macroscopic mean photon number ${\left|\alpha \right|}^{2}=N{\omega}_{0}^{2}\left({\lambda}^{4}/{\lambda}_{c}^{4}-1\right)/\left(16{\lambda}^{2}\right)$. The expectation value of atomic relative population operator

*J*can be given by

_{z}## 4. Cavity optomechanics assisted by an atomic gas

In the following we are interested in a hybrid cavity optomechanics system assisted by an atomic gas, in which the internal cavity dynamics is not taken into account and the electromagnetic field is nonlinearly coupled to the mechanical vibrational motion of a mirror which is driven harmonically. The Hamiltonian of the optomechanics system can be written as

with $g=\omega /\left(L\sqrt{2M{\omega}_{m}}\right)$ is the nonlinear coupling strength arising from the radiation-pressure of light, where*L*is the distance between the two mirrors,

*ω*is the natural frequency of the a mechanical mode of the movable mirror, and

_{m}*M*is the effective mirror mass. (

*c*

^{†}+

*c*) represents position operator of the movable mirror

*Q*=

*x*(

_{zp}*c*

^{†}+

*c*), where ${x}_{zp}=1/\sqrt{2M{\omega}_{m}}$ is the zero-point fluctuations of the mechanical oscillator. The Hamiltonian

*H*can be rewritten in the coordination-momentum phase space For simplicity we define the dimensionless position

^{OPM}*q*and the momentum variables

*p*for the movable mirror

*q*,

*p*] =

*i*/2. Then the Hamiltonian

*H*reads The quantum stochastic differential equations for this system are given by

^{OPM}*q*(

^{in}*p*) denotes vacuum noise. Note that the form of the stochastic equation for the mirror is that for a zero-temperature, under-damped oscillator and will thus only be valid provided Γ ≪

^{in}*ω*.

_{m}First let us consider the corresponding deterministic semi-classic equations

*q*and

_{s}*p*can be determined by setting Eqs. (16) to zeros

_{s}*q*denotes the new equilibrium position of the movable mirror, which is proportional to the average photon number in the field and in agreement with the results of Ref. [26]. To check whether the steady state itself is stable we linearize the dynamics equation around the steady state. Define the variables

_{s}*iω*, − Γ/2 −

_{m}*iω*) so the steady state is stationary in the absence of the contribution of the radiation pressure force.

_{m}The time-dependent solution to the Eqs. (16) can be obtained according to the theory of non-homogeneous linear differential equations with the initial condition *q*(0) = 1, *p*(0) = 0

*ϕ*= arctan(Γ/2

*ω*) = arctan(1/

_{m}*Q*) and

_{m}*Q*= 2

_{m}*ω*/Γ is the mechanical quality factor of the movable mirror. For the mechanical quality factor

_{m}*Q*≫ 1, the time-dependent solution

_{m}*q*(

*t*) reduces to a simple form

*g*|

*α*|

^{2}/

*ω*. It can be seen from the conclusions in Eqs. (9) that in the normal phase the mean photon number is zero with |

_{m}*α*|

^{2}= 0 and the moving mirror is just a harmonic oscillator with mechanical damping, on the contrary in the superradiation phase with macroscopic mean photon number with ${\left|\alpha \right|}^{2}=N{\omega}_{0}^{2}\left({\lambda}^{4}/{\lambda}_{c}^{4}-1\right)/\left(16{\lambda}^{2}\right)$ the moving mirror is subjected to a superradiation-generated classical driving force in addition to the linear restoring force. The oscillating amplitude of the moving mirror is dependent on the mean photon number and the final steady position is proportional to the mean photon number. The mirror’s position can be measured with high sensitivity within present experimental technology. The main idea of this text is to probe the atom-field coupling constant in the Dicke superradiation regime by measuring the mirror’s steady position. Based on the previous results we come to the following conclusion

The Fig. 3 shows the movable mirror’s steady position *q _{s}* as a function of atom-field coupling constant

*λ*. One can see that the steady position

*q*goes up with the increases of

_{s}*λ*and is proportional to the number of atoms. For a given number of atoms a one-to-one correspondence relation between the movable mirror’s steady position and atom-field coupling constant as shown in Eq. (22) is the main result of this paper. Though in this text the cavity is assumed to be lossless, the dissipation in the cavity field has no considerable effect on our results for the dissipation only causes a shift of the critical point [49]. There has no high demand for mechanical quality factor of the movable mirror in our proposals.

We know that up to date it remains experimentally challenging to move into the superradiant regime for the optical cavity. An effective Dicke model operating in the phase transition regime was proposed based on multilevel atoms and cavity-mediated Raman transitions [49]. And solid qubits coupled to nanomechanical resonator may be of the best prospect for our proposal for, on one hand the solid qubit-oscillator system can reach strong-coupling regime, and on the other hand the direct interaction between qubits such as dipole-dipole interaction or spin-spin interaction may cause a shift of the critical point of QPT and advance the realization of the Dicke superradiation phase transition. We expect our scheme may be realized in future development.

## 5. Conclusion

In conclusion we presented a scheme of detecting the atom-field coupling constant in Dicke superradiation regime by means of hybrid cavity optomechanical system assisted by an atomic gas. The critical behavior of the DM was obtained analytically using the SCS representation. For the movable mirror composing the cavity the DM serves as a structured bath, i.e., in the normal phase of DM the mirror is just a harmonic oscillator with mechanical damping while in the superradiation phase the mirror appears to be driven classically. Analytical formula of one-to-one correspondence between the movable mirror’s steady position and atom-field coupling constant for a given number of atoms is obtained. The results show the steady position of the movable mirror goes up with the increases of atom-field coupling constant and is proportional to the number of atoms. The experimental feasibility is also discussed briefly.

## Acknowledgments

The author Wang thanks Prof. Tiancai Zhang for the valuable discussions about the optical cavity QED system and the anonymous referees for their comments and suggestions that help improve the manuscript. This work was supported by NSFC (Nos. 11075099 and 11047167), Programme of State Key Laboratory of Quantum Optics and Quantum Optics Devices (NO. KF201002) and National Fundamental Fund of Personnel Training (Grant No. J1103210).

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