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 [512]. Furthermore the optomechanical interactions has found more applications in the generation of non-classic state of light and mechanical systems [1315], realization of quantum entanglement between a micromechanical oscillator and optical cavity field mode [1618] or a second mechanical oscillator [1921]. 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) [2428], or atomic ensemble [2932]—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 [3538]. Besides the quantum chaos, ground state entanglement, critical behavior [3943] 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.

 

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.

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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)
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±
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))

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
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
where Rθ = eiθJ·n = e(Jx sinφJy cosφ). Thus we have the following eigen equation
Jn|θ,φ=j|θ,φ
which allow us to calculate analytically the energy function
E±(α)=ω(μ2+v2)±N2ω02+(4λμN)2
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.
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
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
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, 47]. 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], 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.

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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)
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
For simplicity we define the dimensionless position q and the momentum variables p for the movable mirror
q=Mωm2Qp=12MωmP
which satisfies the commutation relation [q, p] = i/2. Then the Hamiltonian HOPM reads
HOPM=ωm2(p2+q2)2gaaq
The quantum stochastic differential equations for this system are given by
q˙=ωmpΓ2q+Γqinp˙=ωmq+2gaaΓ2p+Γpin
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
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
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]. 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
Then we have
ddt(δqδp)=(Γ2ωmωmΓ2)(δqδp)
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 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

q(t)=eΓ2t[cosωmt2g|α|2Γ24+ωm2cos(ωmtϕ)]+2g|α|2/ωm1+Γ24ωm2=eΓ2t[cosωmt2g|α|2/ωm1Qm2+1cos(ωmttan(1Qm))]+2g|α|2/ωm1Qm2+1
where ϕ = arctan(Γ/2ωm) = arctan(1/Qm) and Qm = 2ωm/Γ is the mechanical quality factor of the movable mirror. For the mechanical quality factor Qm ≫ 1, the time-dependent solution q(t) reduces to a simple form
q(t)=(12g|α|2ωm)eΓ2tcosωmt+2g|α|2ωm
which is typical of under-damped oscillator. For sufficient long relaxing time the moving mirror comes to its steady position 2g|α|2 /ωm. It can be seen from the conclusions in Eqs. (9) that in the normal phase the mean photon number is zero with |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 |α|2=Nω02(λ4/λc41)/(16λ2) 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
qs=2g|α|2ωm=Nωω02xzp8ωmλ2L(λ4λc41)

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]. 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.

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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).

References and links

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References

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  1. P. Lebedew, “Experimental examination of light pressure,” Ann. Phys. (Leipzig) 6, 433–458 (1901).
  2. E. F. Nichols and G. F. Hull, “A preliminary communication on the pressure of heat and light radiation,” Phys. Rev. 13, 307 (1901).
  3. T. Corbitt and N. Mavalvala, “Quantum noise in gravitational-wave interferometers,” J. Opt. B: Quantum Semiclass. Opt 6, S675–S683 (2004).
    [Crossref]
  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]
  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]
  6. P. F. Cohadon, A. Heidmann, and M. Pinard, “Cooling of a mirror by radiation pressure,” Phys. Rev. Lett. 833174 (1999).
    [Crossref]
  7. F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
    [Crossref] [PubMed]
  8. I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
    [Crossref] [PubMed]
  9. T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321, 1172–1176 (2008).
    [Crossref] [PubMed]
  10. J. Chan, T. P. Mayer Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
    [Crossref] [PubMed]
  11. D. Teufe, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
    [Crossref]
  12. K. C. Schwab and M. L. Roukes, “Putting mechanics into quantum mechanics,” Physics Today 58, 36–42 (2005).
    [Crossref]
  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]
  14. W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett. 91, 130401 (2003).
    [Crossref] [PubMed]
  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]
  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]
  17. C. Genes, D. Vitali, and P. Tombesi, “Emergence of atom-light-mirror entanglement inside an optical cavity,” Phys. Rev. A 77, 050307 (2008).
    [Crossref]
  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]
  19. M. Ludwig, K. Hammerer, and F. Marquardt, “Entanglement of mechanical oscillators coupled to a nonequilibrium environment,” Phys. Rev. A 82, 012333 (2010).
    [Crossref]
  20. K. Børkje, A. Nunnenkamp, and S. M. Girvin, “Proposal for entangling remote micromechanical oscillators via optical measurements,” Phys. Rev. Lett. 107, 123601 (2011).
    [Crossref] [PubMed]
  21. S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401 (2002).
    [Crossref] [PubMed]
  22. L. Tian and P. Zoller, “Coupled ion-nanomechanical systems,” Phys. Rev. Lett. 93, 266403 (2004).
    [Crossref]
  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]
  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]
  25. F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein condensate,” Science 322, 235–238 (2008).
    [Crossref] [PubMed]
  26. A. B. Bhattacherjee, “Cavity quantum optomechanics of ultracold atoms in an optical lattice: normal-mode splitting,” Phys. Rev. A 80, 043607 (2009).
    [Crossref]
  27. D. Hunger, S. Camerer, T. W. Hänsch, D. König, J. P. Kotthaus, J. Reichel, and P. Treutlein, “Resonant coupling of a Bose-Einstein condensate to a micromechanical oscillator,” Phys. Rev. Lett. 104, 143002 (2010).
    [Crossref] [PubMed]
  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]
  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]
  30. Y. Chang and C. P. Sun, “Analog of the electromagnetically-induced-transparency effect for two nanomechanical or micromechanical resonators coupled to a spin ensemble,” Phys. Rev. A 83, 053834 (2011).
    [Crossref]
  31. G. Chen, Y. Zhang, L. Xiao, J.-Q. Liang, and S. Jia, “Strong nonlinear coupling between an ultracold atomic ensemble and a nanomechanical oscillator,” Opt. Express 18, 23016–23023 (2010).
    [Crossref] [PubMed]
  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]
  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. 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]
  35. R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93, 99–110 (1954).
    [Crossref]
  36. K. Hepp and E. H. Lieb, “On the superradiant phase transition for molecules in a quantized radiation field: the Dicke maser model,” Annals Phys.(N.Y.) 76, 360–404 (1973).
    [Crossref]
  37. Y. K. Wang and F. T. Hioes, “Phase transition in the Dicke model of superradiance,” Phys. Rev. A 7, 831–836 (1973).
    [Crossref]
  38. F. T. Hioes, “Phase transitions in some generalized Dicke models of superradiance,” Phys. Rev. A 8, 1440–1445 (1973).
    [Crossref]
  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. 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]

2011 (9)

J. Chan, T. P. Mayer Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
[Crossref] [PubMed]

D. Teufe, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[Crossref]

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]

K. Børkje, A. Nunnenkamp, and S. M. Girvin, “Proposal for entangling remote micromechanical oscillators via optical measurements,” Phys. Rev. Lett. 107, 123601 (2011).
[Crossref] [PubMed]

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]

Y. Chang and C. P. Sun, “Analog of the electromagnetically-induced-transparency effect for two nanomechanical or micromechanical resonators coupled to a spin ensemble,” Phys. Rev. A 83, 053834 (2011).
[Crossref]

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]

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]

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

2010 (6)

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]

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]

G. Chen, Y. Zhang, L. Xiao, J.-Q. Liang, and S. Jia, “Strong nonlinear coupling between an ultracold atomic ensemble and a nanomechanical oscillator,” Opt. Express 18, 23016–23023 (2010).
[Crossref] [PubMed]

D. Hunger, S. Camerer, T. W. Hänsch, D. König, J. P. Kotthaus, J. Reichel, and P. Treutlein, “Resonant coupling of a Bose-Einstein condensate to a micromechanical oscillator,” Phys. Rev. Lett. 104, 143002 (2010).
[Crossref] [PubMed]

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

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]

2009 (3)

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]

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

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]

2008 (5)

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]

F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein condensate,” Science 322, 235–238 (2008).
[Crossref] [PubMed]

C. Genes, D. Vitali, and P. Tombesi, “Emergence of atom-light-mirror entanglement inside an optical cavity,” Phys. Rev. A 77, 050307 (2008).
[Crossref]

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321, 1172–1176 (2008).
[Crossref] [PubMed]

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]

2007 (7)

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]

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

F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[Crossref] [PubMed]

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
[Crossref] [PubMed]

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]

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]

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]

2006 (2)

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]

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]

2005 (2)

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

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

2004 (2)

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

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

2003 (3)

W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett. 91, 130401 (2003).
[Crossref] [PubMed]

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]

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

2002 (1)

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

1999 (1)

P. F. Cohadon, A. Heidmann, and M. Pinard, “Cooling of a mirror by radiation pressure,” Phys. Rev. Lett. 833174 (1999).
[Crossref]

1997 (1)

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]

1973 (3)

K. Hepp and E. H. Lieb, “On the superradiant phase transition for molecules in a quantized radiation field: the Dicke maser model,” Annals Phys.(N.Y.) 76, 360–404 (1973).
[Crossref]

Y. K. Wang and F. T. Hioes, “Phase transition in the Dicke model of superradiance,” Phys. Rev. A 7, 831–836 (1973).
[Crossref]

F. T. Hioes, “Phase transitions in some generalized Dicke models of superradiance,” Phys. Rev. A 8, 1440–1445 (1973).
[Crossref]

1954 (1)

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

1940 (1)

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

1901 (2)

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

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

Allman, M. S.

D. Teufe, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[Crossref]

Ansmann, M.

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]

Ashhab, S.

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]

Aspelmeyer, M.

J. Chan, T. P. Mayer Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
[Crossref] [PubMed]

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]

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]

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]

Bäuerle, D.

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]

Bhattacherjee, A. B.

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

Bialczak, R. C.

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]

Blaser, F.

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]

Bohm, H. R.

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]

Böhm, H. R.

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]

Børkje, K.

K. Børkje, A. Nunnenkamp, and S. M. Girvin, “Proposal for entangling remote micromechanical oscillators via optical measurements,” Phys. Rev. Lett. 107, 123601 (2011).
[Crossref] [PubMed]

Bose, S.

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]

Bouwmeester, D.

W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett. 91, 130401 (2003).
[Crossref] [PubMed]

Brandes, T.

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

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]

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

Brennecke, F.

F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein condensate,” Science 322, 235–238 (2008).
[Crossref] [PubMed]

Camerer, S.

D. Hunger, S. Camerer, T. W. Hänsch, D. König, J. P. Kotthaus, J. Reichel, and P. Treutlein, “Resonant coupling of a Bose-Einstein condensate to a micromechanical oscillator,” Phys. Rev. Lett. 104, 143002 (2010).
[Crossref] [PubMed]

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]

Carmichael, H. J.

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]

Chan, J.

J. Chan, T. P. Mayer Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
[Crossref] [PubMed]

Chang, Y.

Y. Chang and C. P. Sun, “Analog of the electromagnetically-induced-transparency effect for two nanomechanical or micromechanical resonators coupled to a spin ensemble,” Phys. Rev. A 83, 053834 (2011).
[Crossref]

Chen, G.

Chen, J. P.

F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[Crossref] [PubMed]

Chen, Y.

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]

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]

Chiara, G. D.

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

Cicak, K.

D. Teufe, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[Crossref]

Cleland, A. N.

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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).
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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).
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J. Chan, T. P. Mayer Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
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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).
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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).
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K. Børkje, A. Nunnenkamp, and S. M. Girvin, “Proposal for entangling remote micromechanical oscillators via optical measurements,” Phys. Rev. Lett. 107, 123601 (2011).
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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).
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P. F. Cohadon, A. Heidmann, and M. Pinard, “Cooling of a mirror by radiation pressure,” Phys. Rev. Lett. 833174 (1999).
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Polzik, E. S.

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]

Primakoff, H.

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

Rehbein, H.

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]

Reichel, J.

D. Hunger, S. Camerer, T. W. Hänsch, D. König, J. P. Kotthaus, J. Reichel, and P. Treutlein, “Resonant coupling of a Bose-Einstein condensate to a micromechanical oscillator,” Phys. Rev. Lett. 104, 143002 (2010).
[Crossref] [PubMed]

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]

Ritter, S.

F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein condensate,” Science 322, 235–238 (2008).
[Crossref] [PubMed]

Roukes, M. L.

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

Safavi-Naeini, A. H.

J. Chan, T. P. Mayer Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
[Crossref] [PubMed]

Santos, J. P.

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]

Schwab, K. C.

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]

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]

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

Semião, F. L.

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]

Sigg, D.

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]

Simmonds, R. W.

D. Teufe, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[Crossref]

Simon, C.

W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett. 91, 130401 (2003).
[Crossref] [PubMed]

Singh, S.

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]

Sirois, A. J.

D. Teufe, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[Crossref]

Steinke, S. K.

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]

Sun, C. P.

Y. Chang and C. P. Sun, “Analog of the electromagnetically-induced-transparency effect for two nanomechanical or micromechanical resonators coupled to a spin ensemble,” Phys. Rev. A 83, 053834 (2011).
[Crossref]

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]

Sun, Q.

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]

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]

Tasgin, M. E.

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]

Teufe, D.

D. Teufe, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[Crossref]

Tian, L.

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

Tombesi, P.

C. Genes, D. Vitali, and P. Tombesi, “Emergence of atom-light-mirror entanglement inside an optical cavity,” Phys. Rev. A 77, 050307 (2008).
[Crossref]

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]

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

Treutlein, P.

D. Hunger, S. Camerer, T. W. Hänsch, D. König, J. P. Kotthaus, J. Reichel, and P. Treutlein, “Resonant coupling of a Bose-Einstein condensate to a micromechanical oscillator,” Phys. Rev. Lett. 104, 143002 (2010).
[Crossref] [PubMed]

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]

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]

Vahala, K. J.

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321, 1172–1176 (2008).
[Crossref] [PubMed]

Vedral, V.

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]

Vengalattore, M.

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]

Vitali, D.

C. Genes, D. Vitali, and P. Tombesi, “Emergence of atom-light-mirror entanglement inside an optical cavity,” Phys. Rev. A 77, 050307 (2008).
[Crossref]

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]

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

Wallquist, M.

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]

Wang, H.

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]

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Y. K. Wang and F. T. Hioes, “Phase transition in the Dicke model of superradiance,” Phys. Rev. A 7, 831–836 (1973).
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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]

Whitcomb, S.

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]

Whittaker, J. D.

D. Teufe, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[Crossref]

Wilson-Rae, I.

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
[Crossref] [PubMed]

Wipf, C.

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]

Xiao, L.

Xie, X. C.

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]

Yang, H.

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]

Ye, J.

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]

Zeilinger, A.

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]

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]

Zhang, W.

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]

Zhang, Y.

Zhou, L.

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]

Zoller, P.

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]

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]

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

Zwerger, W.

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
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Ann. Phys. (Leipzig) (1)

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

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J. Opt. B: Quantum Semiclass. Opt (1)

T. Corbitt and N. Mavalvala, “Quantum noise in gravitational-wave interferometers,” J. Opt. B: Quantum Semiclass. Opt 6, S675–S683 (2004).
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Nature (4)

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]

J. Chan, T. P. Mayer Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
[Crossref] [PubMed]

D. Teufe, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[Crossref]

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]

Opt. Express (1)

Phys. Rev. (3)

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[Crossref]

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

Phys. Rev. A (18)

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]

N. Lambert, C. Emary, and T. Brandes, “Entanglement and entropy in a spin-boson quantum phase transition,” Phys. Rev. A 71, 053804 (2005).
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G. D. Chiara, M. Paternostro, and G. M. Palma, “Entanglement detection in hybrid optomechanical systems,” Phys. Rev. A 83, 052324–052329 (2011).
[Crossref]

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]

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]

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]

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).
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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]

C. Genes, D. Vitali, and P. Tombesi, “Emergence of atom-light-mirror entanglement inside an optical cavity,” Phys. Rev. A 77, 050307 (2008).
[Crossref]

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]

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

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]

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]

Y. Chang and C. P. Sun, “Analog of the electromagnetically-induced-transparency effect for two nanomechanical or micromechanical resonators coupled to a spin ensemble,” Phys. Rev. A 83, 053834 (2011).
[Crossref]

Y. K. Wang and F. T. Hioes, “Phase transition in the Dicke model of superradiance,” Phys. Rev. A 7, 831–836 (1973).
[Crossref]

F. T. Hioes, “Phase transitions in some generalized Dicke models of superradiance,” Phys. Rev. A 8, 1440–1445 (1973).
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A. B. Bhattacherjee, “Cavity quantum optomechanics of ultracold atoms in an optical lattice: normal-mode splitting,” Phys. Rev. A 80, 043607 (2009).
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C. Emary and T. Brandes, “Chaos and the quantum phase transition in the Dicke model,” Phys. Rev. E 67, 066203 (2003).
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Phys. Rev. Lett. (16)

E. K. Irish, “Generalized rotating-wave approximation for arbitrarily large coupling,” Phys. Rev. Lett. 99, 173601 (2007).
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W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett. 91, 130401 (2003).
[Crossref] [PubMed]

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]

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]

D. Hunger, S. Camerer, T. W. Hänsch, D. König, J. P. Kotthaus, J. Reichel, and P. Treutlein, “Resonant coupling of a Bose-Einstein condensate to a micromechanical oscillator,” Phys. Rev. Lett. 104, 143002 (2010).
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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).
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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).
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S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401 (2002).
[Crossref] [PubMed]

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

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]

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]

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).
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[Crossref] [PubMed]

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
[Crossref] [PubMed]

Physics Today (1)

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

Science (2)

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321, 1172–1176 (2008).
[Crossref] [PubMed]

F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein condensate,” Science 322, 235–238 (2008).
[Crossref] [PubMed]

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Figures (3)

Fig. 1
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.
Fig. 2
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.
Fig. 3
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.

Equations (22)

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H D M = ω a a + ω 0 J z + λ N ( J + + J ) ( a + a )
[ J + , J ] = 2 J z , [ J z , J ± ] = ± J ±
J u = 1 2 i = 1 N σ u ( i ) ( u = x , y , z ) J ± = 1 2 i = 1 N ( σ x ( i ) ± i σ y ( i ) )
a | α = α | α
| θ , φ = R θ , φ | j , j
J n | θ , φ = j | θ , φ
E ± ( α ) = ω ( μ 2 + v 2 ) ± N 2 ω 0 2 + ( 4 λ μ N ) 2
E ( α ) μ = 0 , E ( α ) v = 0.
α = { 0 , λ < λ c N ω 0 2 ( λ 4 / λ c 4 1 ) 16 λ 2 , λ > λ c
J z = θ , φ | R J z R | θ , φ = { N 2 , λ < λ c N λ 2 2 λ c 2 , λ > λ c
H O P M = ω m c c g a a ( c + c )
H O P M = P 2 2 M + M ω m 2 2 Q 2 ω L a a Q
q = M ω m 2 Q p = 1 2 M ω m P
H O P M = ω m 2 ( p 2 + q 2 ) 2 g a a q
q ˙ = ω m p Γ 2 q + Γ q in p ˙ = ω m q + 2 g a a Γ 2 p + Γ p in
q ˙ = ω m p Γ 2 q p ˙ = ω m q + 2 g | α | 2 Γ 2 p
q s = 2 g | α | 2 / ω m 1 + Γ 2 4 ω m 2 p s = Γ q s 2 ω m = g | α | 2 Γ / ω m 2 1 + Γ 2 4 ω m 2
δ q ( t ) = q ( t ) q s δ p ( t ) = p ( t ) p s
d d t ( δ q δ p ) = ( Γ 2 ω m ω m Γ 2 ) ( δ q δ p )
q ( t ) = e Γ 2 t [ cos ω m t 2 g | α | 2 Γ 2 4 + ω m 2 cos ( ω m t ϕ ) ] + 2 g | α | 2 / ω m 1 + Γ 2 4 ω m 2 = e Γ 2 t [ cos ω m t 2 g | α | 2 / ω m 1 Q m 2 + 1 cos ( ω m t tan ( 1 Q m ) ) ] + 2 g | α | 2 / ω m 1 Q m 2 + 1
q ( t ) = ( 1 2 g | α | 2 ω m ) e Γ 2 t cos ω m t + 2 g | α | 2 ω m
q s = 2 g | α | 2 ω m = N ω ω 0 2 x z p 8 ω m λ 2 L ( λ 4 λ c 4 1 )

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