## A scheme for detecting the atom-field coupling constant in the Dicke superradiation regime using hybrid cavity optomechanical system |

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

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]

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]

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]

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]

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]

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]

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]

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]

## 2. Model

*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. 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]

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]

## 3. Dicke hamiltonian

*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 where

*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

*θ*,

*φ*〉 ⊗ |

*α*〉, where the boson coherent state is defined by 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**= (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 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 The critical point of phase transition for the ground state can be obtained exactly and given by where

*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

*J*can be given by 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

_{z}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]

## 4. Cavity optomechanics assisted by an atomic gas

*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

*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 which satisfies the commutation relation [

*q*,

*p*] =

*i*/2. Then the Hamiltonian

*H*reads The quantum stochastic differential equations for this system are given by where Γ/2 is the damping rate for the movable mirror and

^{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}*q*and

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

_{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

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

*iω*, − Γ/2 −

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

_{m}*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 which is typical of under-damped oscillator. For sufficient long relaxing time the moving mirror comes to its steady position 2

*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

*q*as a function of atom-field coupling constant

_{s}*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

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]

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]

## 5. Conclusion

## Acknowledgments

## References and links

1. | P. Lebedew, “Experimental examination of light pressure,” Ann. Phys. (Leipzig) |

2. | E. F. Nichols and G. F. Hull, “A preliminary communication on the pressure of heat and light radiation,” Phys. Rev. |

3. | T. Corbitt and N. Mavalvala, “Quantum noise in gravitational-wave interferometers,” J. Opt. B: Quantum Semiclass. Opt |

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. |

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 |

6. | P. F. Cohadon, A. Heidmann, and M. Pinard, “Cooling of a mirror by radiation pressure,” Phys. Rev. Lett. |

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. |

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. |

9. | T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science |

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 |

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 |

12. | K. C. Schwab and M. L. Roukes, “Putting mechanics into quantum mechanics,” Physics Today |

13. | S. Bose, K. Jacobs, and P. L. Knight, “Preparation of nonclassical states in cavities with a moving mirror,” Phys. Rev. A |

14. | W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett. |

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. |

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. |

17. | C. Genes, D. Vitali, and P. Tombesi, “Emergence of atom-light-mirror entanglement inside an optical cavity,” Phys. Rev. A |

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 |

19. | M. Ludwig, K. Hammerer, and F. Marquardt, “Entanglement of mechanical oscillators coupled to a nonequilibrium environment,” Phys. Rev. A |

20. | K. Børkje, A. Nunnenkamp, and S. M. Girvin, “Proposal for entangling remote micromechanical oscillators via optical measurements,” Phys. Rev. Lett. |

21. | S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. |

22. | L. Tian and P. Zoller, “Coupled ion-nanomechanical systems,” Phys. Rev. Lett. |

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. |

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. |

25. | F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein condensate,” Science |

26. | A. B. Bhattacherjee, “Cavity quantum optomechanics of ultracold atoms in an optical lattice: normal-mode splitting,” Phys. Rev. A |

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. |

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 |

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 |

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 |

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 |

32. | K. Hammerer, M. Aspelmeyer, E. S. Polzik, and P. Zoller, “Establishing Einstein-Poldosky-Rosen channels between nanomechanics and atomic ensembles,” Phys. Rev. Lett. |

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 |

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 |

35. | R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. |

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.) |

37. | Y. K. Wang and F. T. Hioes, “Phase transition in the Dicke model of superradiance,” Phys. Rev. A |

38. | F. T. Hioes, “Phase transitions in some generalized Dicke models of superradiance,” Phys. Rev. A |

39. | C. Emary and T. Brandes, “Quantum chaos triggered by precursors of a quantum phase transition: the Dicke model,” Phys. Rev. Lett. |

40. | C. Emary and T. Brandes, “Chaos and the quantum phase transition in the Dicke model,” Phys. Rev. E |

41. | N. Lambert, C. Emary, and T. Brandes, “Entanglement and entropy in a spin-boson quantum phase transition,” Phys. Rev. A |

42. | G. D. Chiara, M. Paternostro, and G. M. Palma, “Entanglement detection in hybrid optomechanical systems,” Phys. Rev. A |

43. | G. Chen, J. Li, and J.-Q. Liang, “Critical property of the geometric phase in the Dicke model,” Phys. Rev. A |

44. | T. Holstein and H. Primakoff, “Field dependence of the intrinsic domain magnetization of a ferromagnet,” Phys. Rev. |

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 |

46. | E. K. Irish, “Generalized rotating-wave approximation for arbitrarily large coupling,” Phys. Rev. Lett. |

47. | S. Ashhab and F. Nori, “Qubit-oscillator systems in the ultrastrong-coupling regime and their potential for preparing nonclassical states,” Phys. Rev. A |

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 |

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 |

**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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### References

- 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).
- T. Corbitt and N. Mavalvala, “Quantum noise in gravitational-wave interferometers,” J. Opt. B: Quantum Semiclass. Opt6, S675–S683 (2004). [CrossRef]
- 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]
- 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,” Nature444, 67–70 (2006). [CrossRef] [PubMed]
- P. F. Cohadon, A. Heidmann, and M. Pinard, “Cooling of a mirror by radiation pressure,” Phys. Rev. Lett.833174 (1999). [CrossRef]
- 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. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science321, 1172–1176 (2008). [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,” Nature478, 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,” Nature475, 359–363 (2011). [CrossRef]
- K. C. Schwab and M. L. Roukes, “Putting mechanics into quantum mechanics,” Physics Today58, 36–42 (2005). [CrossRef]
- S. Bose, K. Jacobs, and P. L. Knight, “Preparation of nonclassical states in cavities with a moving mirror,” Phys. Rev. A56, 4175 (1997). [CrossRef]
- 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]
- C. Genes, D. Vitali, and P. Tombesi, “Emergence of atom-light-mirror entanglement inside an optical cavity,” Phys. Rev. A77, 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. A83, 052117 (2011). [CrossRef]
- M. Ludwig, K. Hammerer, and F. Marquardt, “Entanglement of mechanical oscillators coupled to a nonequilibrium environment,” Phys. Rev. A82, 012333 (2010). [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. 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]
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