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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 9 — Apr. 23, 2012
  • pp: 10106–10114
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A scheme for detecting the atom-field coupling constant in the Dicke superradiation regime using hybrid cavity optomechanical system

Yueming Wang, Bin Liu, Jinling Lian, and Jiuqing Liang  »View Author Affiliations


Optics Express, Vol. 20, Issue 9, pp. 10106-10114 (2012)
http://dx.doi.org/10.1364/OE.20.010106


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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 OSA

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

1. P. Lebedew, “Experimental examination of light pressure,” Ann. Phys. (Leipzig) 6, 433–458 (1901).

, 2

2. E. F. Nichols and G. F. Hull, “A preliminary communication on the pressure of heat and light radiation,” Phys. Rev. 13, 307 (1901).

]. 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

3. T. Corbitt and N. Mavalvala, “Quantum noise in gravitational-wave interferometers,” J. Opt. B: Quantum Semiclass. Opt 6, S675–S683 (2004). [CrossRef]

, 4

4. T. Corbitt, Y. Chen, E. Innerhofer, H. Müller-Ebhardt, D. Ottaway, H. Rehbein, D. Sigg, S. Whitcomb, C. Wipf, and N. Mavalvala, “An all-optical trap for a gram-scale mirror,” Phys. Rev. Lett. 98, 150802 (2007). [CrossRef] [PubMed]

], 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

5. S. Gigan, H. R. Bohm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006). [CrossRef] [PubMed]

12

12. K. C. Schwab and M. L. Roukes, “Putting mechanics into quantum mechanics,” Physics Today 58, 36–42 (2005). [CrossRef]

]. Furthermore the optomechanical interactions has found more applications in the generation of non-classic state of light and mechanical systems [13

13. S. Bose, K. Jacobs, and P. L. Knight, “Preparation of nonclassical states in cavities with a moving mirror,” Phys. Rev. A 56, 4175 (1997). [CrossRef]

15

15. F. Khalili, S. Danilishin, H. Miao, H. Müller-Ebhardt, H. Yang, and Y. Chen, “Preparing a mechanical oscillator in non-Gaussian quantum states,” Phys. Rev. Lett. 105, 070403 (2010). [CrossRef] [PubMed]

], realization of quantum entanglement between a micromechanical oscillator and optical cavity field mode [16

16. D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98, 030405 (2007). [CrossRef] [PubMed]

18

18. L. Zhou, Y. Han, J. Jing, and W. Zhang, “Entanglement of nanomechanical oscillators and two-mode fields induced by atomic coherence,” Phys. Rev. A 83, 052117 (2011). [CrossRef]

] or a second mechanical oscillator [19

19. M. Ludwig, K. Hammerer, and F. Marquardt, “Entanglement of mechanical oscillators coupled to a nonequilibrium environment,” Phys. Rev. A 82, 012333 (2010). [CrossRef]

21

21. S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401 (2002). [CrossRef] [PubMed]

]. Up to date a number of proposals have been put forward that how atomic systems—such as a trapped ion [22

22. L. Tian and P. Zoller, “Coupled ion-nanomechanical systems,” Phys. Rev. Lett. 93, 266403 (2004). [CrossRef]

], atom [23

23. K. Hammerer, M. Wallquist, C. Genes, M. Ludwig, F. Marquardt, P. Treutlein, P. Zoller, J. Ye, and H. J. Kimble, “Strong coupling of a mechanical oscillator and a single atom,” Phys. Rev. Lett. 103, 063005 (2009). [CrossRef] [PubMed]

], Bose-Einstein condensate (BEC) [24

24. P. Treutlein, D. Hunger, S. Camerer, T. W. Hänsch, and J. Reichel, “Bose-Einstein condensate coupled to a nanomechanical resonator on an atom chip,” Phys. Rev. Lett. 99, 140403 (2007). [CrossRef] [PubMed]

28

28. S. K. Steinke, S. Singh, M. E. Tasgin, P. Meystre, K. C. Schwab, and M. Vengalattore, “Quantum-measurement backaction from a Bose-Einstein condensate coupled to a mechanical oscillator,” Phys. Rev. A 84, 023841 (2011). [CrossRef]

], or atomic ensemble [29

29. H. Ian, Z. R. Gong, Y. X. Liu, C. P. Sun, and F. Nori, “Cavity optomechanical coupling assisted by an atomic gas,” Phys. Rev. A 78, 013824 (2008). [CrossRef]

32

32. K. Hammerer, M. Aspelmeyer, E. S. Polzik, and P. Zoller, “Establishing Einstein-Poldosky-Rosen channels between nanomechanics and atomic ensembles,” Phys. Rev. Lett. 102, 020501 (2009). [CrossRef] [PubMed]

]—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

33. Q. Sun, X.-H. Hu, W. M. Liu, X. C. Xie, and A.-C. Ji, “Effect on cavity optomechanics of the interaction between a cavity field and a one-dimensional interacting bosonic gas,” Phys. Rev. A 84, 023822 (2011). [CrossRef]

, 34

34. Q. Sun, X.-H. Hu, A.-C. Ji, and W. M. Liu, “Dynamics of a degenerate Fermi gas in a one-dimensional optical lattice coupled to a cavity,” Phys. Rev. A 83, 043606 (2011). [CrossRef]

].

Our proposal follows along the path in Ref.[45

45. J. P. Santos, F. L. Semião, and K. Furuya, “Probing the quantum phase transition in the Dicke model through mechanical vibrations,” Phys. Rev. A 82, 063801 (2010). [CrossRef]

], 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

35. R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93, 99–110 (1954). [CrossRef]

, 39

39. C. Emary and T. Brandes, “Quantum chaos triggered by precursors of a quantum phase transition: the Dicke model,” Phys. Rev. Lett. 90, 044101 (2003). [CrossRef] [PubMed]

, 40

40. C. Emary and T. Brandes, “Chaos and the quantum phase transition in the Dicke model,” Phys. Rev. E 67, 066203 (2003). [CrossRef]

]. The analogous experimental setup has been proposed and analyzed in the Ref. [23

23. K. Hammerer, M. Wallquist, C. Genes, M. Ludwig, F. Marquardt, P. Treutlein, P. Zoller, J. Ye, and H. J. Kimble, “Strong coupling of a mechanical oscillator and a single atom,” Phys. Rev. Lett. 103, 063005 (2009). [CrossRef] [PubMed]

], in which only a single atom is considered.

Fig. 1 A collective of N two-level atoms interact with a single-mode quantized cavity field. The movable cavity mirror is in a harmonic motion due to a linear restoring force from the spring.

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
HDM=ωaa+ω0Jz+λN(J++J)(a+a)
(1)
where ω is the frequency of radiation field, ω0 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, Jz is the atomic relative population operator, J± are the collective atomic raising and lowering operators. They satisfy the SU(2) Lie algebra,
[J+,J]=2Jz,[Jz,J±]=±J±
(2)
where the collective operators are described in terms of standard pauli matrices of each two-level atom
Ju=12i=1Nσu(i)(u=x,y,z)J±=12i=1N(σx(i)±iσy(i))
(3)

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
a|α=α|α
(4)
and the SCS, or an arbitrary Dicke states, can be created by rotating the ground state |j,− j〉 by the angle θ about the axis n = (sinφ, −cosφ, 0) with j = N/2 being the total pseudo-spin value, i.e.,
|θ,φ=Rθ,φ|j,j
(5)
where Rθ = eiθJ·n = e(Jx sinφJy cosφ). Thus we have the following eigen equation
Jn|θ,φ=j|θ,φ
(6)
which allow us to calculate analytically the energy function
E±(α)=ω(μ2+v2)±N2ω02+(4λμN)2
(7)
Here we defined the boson coherent state 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
E(α)μ=0,E(α)v=0.
(8)
The critical point of phase transition for the ground state can be obtained exactly and given by
α={0,λ<λcNω02(λ4/λc41)16λ2,λ>λc
(9)
where λc=ωω0/2 which is a well known result in the zero-temperature QPT for DM. For the atom-field coupling constant λ < λc the system is in normal phase with mean photon number |α| = 0, otherwise the system is in superradiation phase with macroscopic mean photon number |α|2=Nω02(λ4/λc41)/(16λ2). The expectation value of atomic relative population operator Jz can be given by
Jz=θ,φ|RJzR|θ,φ={N2,λ<λcNλ22λc2,λ>λc
(10)
The results of Eqs. (9) and (10) are shown graphically in Fig. 2, in which the expectation values of the number of photons and the number of atoms in excited states exhibit a sudden increase from zero. The critical behavior is in agreement with that reached by means of Holstein-Primakoff series expansion of the Dicke Hamiltonian truncated to second order in terms of the ratio between the number of excited atoms to the total number of atoms, which is assumed to be a very small quantity. But however the SCS representation is valid for arbitrary atomic number. Furthermore it is worthwhile to point out that the rotating wave approximation (RWA) is not made here as usually for RWA is only valid in the case of weak coupling and near-resonance but within present experiment technology the strong coupling even deep-strong coupling between qubit and resonator can be realized in solid-state system [46

46. E. K. Irish, “Generalized rotating-wave approximation for arbitrarily large coupling,” Phys. Rev. Lett. 99, 173601 (2007). [CrossRef] [PubMed]

, 47

47. S. Ashhab and F. Nori, “Qubit-oscillator systems in the ultrastrong-coupling regime and their potential for preparing nonclassical states,” Phys. Rev. A 81, 042311 (2010). [CrossRef]

]. As an example superconducting Josephson junction-based qubits and superconducting resonant cavities have emerged as the ideal realization of quantum two-level systems interacting with a single mode of the electromagnetic spectrum, which can reach the deep-strong coupling regime [48

48. M. Hofheinz, E. M. Weig, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. D. OConnell, H. Wang, J. M. Martinis, and A. N. Cleland, “Generation of Fock states in a superconducting quantum circuit,” Nature 454, 310–314 (2008). [CrossRef] [PubMed]

], where the RWA breaks down.

Fig. 2 The expectation values of the number of photons and the number of atoms in excited states per atom as a function of atom-field coupling constant λ with ω = ω0 = 2λc.

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
HOPM=ωmccgaa(c+c)
(11)
with g=ω/(L2Mωm) is the nonlinear coupling strength arising from the radiation-pressure of light, where L is the distance between the two mirrors, ωm is the natural frequency of the a mechanical mode of the movable mirror, and M is the effective mirror mass. (c + c) represents position operator of the movable mirror Q = xzp(c + c), where xzp=1/2Mωm is the zero-point fluctuations of the mechanical oscillator. The Hamiltonian HOPM can be rewritten in the coordination-momentum phase space
HOPM=P22M+Mωm22Q2ωLaaQ
(12)
For simplicity we define the dimensionless position q and the momentum variables p for the movable mirror
q=Mωm2Qp=12MωmP
(13)
which satisfies the commutation relation [q, p] = i/2. Then the Hamiltonian HOPM reads
HOPM=ωm2(p2+q2)2gaaq
(14)
The quantum stochastic differential equations for this system are given by
q˙=ωmpΓ2q+Γqinp˙=ωmq+2gaaΓ2p+Γpin
(15)
where Γ/2 is the damping rate for the movable mirror and qin(pin) 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 ≪ ωm.

First let us consider the corresponding deterministic semi-classic equations
q˙=ωmpΓ2qp˙=ωmq+2g|α|2Γ2p
(16)
The steady state values qs and ps can be determined by setting Eqs. (16) to zeros
qs=2g|α|2/ωm1+Γ24ωm2ps=Γqs2ωm=g|α|2Γ/ωm21+Γ24ωm2
(17)
where qs 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

26. A. B. Bhattacherjee, “Cavity quantum optomechanics of ultracold atoms in an optical lattice: normal-mode splitting,” Phys. Rev. A 80, 043607 (2009). [CrossRef]

]. To check whether the steady state itself is stable we linearize the dynamics equation around the steady state. Define the variables
δq(t)=q(t)qsδp(t)=p(t)ps
(18)
Then we have
ddt(δqδp)=(Γ2ωmωmΓ2)(δqδp)
(19)
The eigenvalues of the linear dynamics are then found to be (−Γ/2 + m, − Γ/2 − m) so the steady state is stationary in the absence of the contribution of the radiation pressure force.

The Fig. 3 shows the movable mirror’s steady position qs as a function of atom-field coupling constant λ. One can see that the steady position qs goes up with the increases of 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

49. F. Dimer, B. Estienne, A. S. Parkins, and H. J. Carmichael, “Proposed realization of the Dicke-model quantum phase transition in an optical cavity QED system,” Phys. Rev. A 75, 013804 (2007). [CrossRef]

]. There has no high demand for mechanical quality factor of the movable mirror in our proposals.

Fig. 3 The movable mirror’s steady position qs as a function of atom-field coupling constant λ for a given number of atoms. qs is in unit of xzp/L. The solid line (red), dashed line (blue) and dotted line (black) represents N =5, 10 and 20 respectively. The system parameters are ω = ω0 = 10ωm and λc = 1.

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

49. F. Dimer, B. Estienne, A. S. Parkins, and H. J. Carmichael, “Proposed realization of the Dicke-model quantum phase transition in an optical cavity QED system,” Phys. Rev. A 75, 013804 (2007). [CrossRef]

]. 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).

References and links

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2.

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24.

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29.

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34.

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C. Emary and T. Brandes, “Quantum chaos triggered by precursors of a quantum phase transition: the Dicke model,” Phys. Rev. Lett. 90, 044101 (2003). [CrossRef] [PubMed]

40.

C. Emary and T. Brandes, “Chaos and the quantum phase transition in the Dicke model,” Phys. Rev. E 67, 066203 (2003). [CrossRef]

41.

N. Lambert, C. Emary, and T. Brandes, “Entanglement and entropy in a spin-boson quantum phase transition,” Phys. Rev. A 71, 053804 (2005). [CrossRef]

42.

G. D. Chiara, M. Paternostro, and G. M. Palma, “Entanglement detection in hybrid optomechanical systems,” Phys. Rev. A 83, 052324–052329 (2011). [CrossRef]

43.

G. Chen, J. Li, and J.-Q. Liang, “Critical property of the geometric phase in the Dicke model,” Phys. Rev. A 74, 054101 (2006). [CrossRef]

44.

T. Holstein and H. Primakoff, “Field dependence of the intrinsic domain magnetization of a ferromagnet,” Phys. Rev. 58, 1098–1113 (1940). [CrossRef]

45.

J. P. Santos, F. L. Semião, and K. Furuya, “Probing the quantum phase transition in the Dicke model through mechanical vibrations,” Phys. Rev. A 82, 063801 (2010). [CrossRef]

46.

E. K. Irish, “Generalized rotating-wave approximation for arbitrarily large coupling,” Phys. Rev. Lett. 99, 173601 (2007). [CrossRef] [PubMed]

47.

S. Ashhab and F. Nori, “Qubit-oscillator systems in the ultrastrong-coupling regime and their potential for preparing nonclassical states,” Phys. Rev. A 81, 042311 (2010). [CrossRef]

48.

M. Hofheinz, E. M. Weig, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. D. OConnell, H. Wang, J. M. Martinis, and A. N. Cleland, “Generation of Fock states in a superconducting quantum circuit,” Nature 454, 310–314 (2008). [CrossRef] [PubMed]

49.

F. Dimer, B. Estienne, A. S. Parkins, and H. J. Carmichael, “Proposed realization of the Dicke-model quantum phase transition in an optical cavity QED system,” Phys. Rev. A 75, 013804 (2007). [CrossRef]

OCIS Codes
(220.4880) Optical design and fabrication : Optomechanics
(270.0270) Quantum optics : Quantum optics
(270.6630) Quantum optics : Superradiance, superfluorescence

ToC Category:
Quantum Optics

History
Original Manuscript: February 10, 2012
Revised Manuscript: April 10, 2012
Manuscript Accepted: April 11, 2012
Published: April 18, 2012

Citation
Yueming Wang, Bin Liu, Jinling Lian, and Jiuqing Liang, "A scheme for detecting the atom-field coupling constant in the Dicke superradiation regime using hybrid cavity optomechanical system," Opt. Express 20, 10106-10114 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-9-10106


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