Abstract

A cold gas of polarizable particles moving in the optical potential of a standing wave high finesse optical resonator acts as a dynamic refractive index. For a sufficiently strong cavity pump the optical forces generated by the intra cavity field perturb the particles phase space distribution, which shifts the optical resonance frequency and induces a nonlinear optical response. By help of the corresponding Vlasov equation we predict that beyond the known phenomenon of optical bi-stability one finds regions in parameter space, where no stable stationary solution exists. The atom field dynamics then exhibits oscillatory solutions converging to stable limit cycles of the system. The linearized analytical predictions agree well with corresponding numerical solutions of the full time dependent equations and first experimental observation in both cases.

© 2011 OSA

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  1. D. Kruse, M. Ruder, J. Benhelm, C. von Cube, C. Zimmermann, P. W. Courteille, T. Elsässer, B. Nagorny, and A. Hemmerich, “Cold atoms in a high-Q ring cavity,” Phys. Rev. A 67, 051802 (2003).
    [CrossRef]
  2. M. Khudaverdyan, W. Alt, I. Dotsenko, T. Kampschulte, K. Lenhard, A. Rauschenbeutel, S. Reick, K. Schörner, A. Widera, and D. Meschede, “Controlled insertion and retrieval of atoms coupled to a high-finesse optical resonator,” New J. Phys. 10, 073023 (2008).
    [CrossRef]
  3. P. Domokos and H. Ritsch, “Mechanical effects of light in optical resonators,” J. Opt. Soc. Am. B 20, 1098–1130 (2003).
    [CrossRef]
  4. P. Horak, G. Hechenblaikner, K. M. Gheri, H. Stecher, and H. Ritsch, “Cavity-induced atom cooling in the strong coupling regime,” Phys. Rev. Lett. 79, 4974–4977 (1997).
    [CrossRef]
  5. P. Domokos, P. Horak, and H. Ritsch, “Semiclassical theory of cavity-assisted atom cooling,” J. Phys. B 34, 187–198 (2001).
    [CrossRef]
  6. J. M. Zhang, F. C. Cui, D. L. Zhou, and W. M. Liu, “Nonlinear dynamics of a cigar-shaped bose-einstein condensate in an optical cavity,” Phys. Rev. A 79, 033401 (2009).
    [CrossRef]
  7. S. Ritter, F. Brennecke, K. Baumann, T. Donner, C. Guerlin, and T. Esslinger, “Dynamical coupling between a Bose–Einstein condensate and a cavity optical lattice,” Appl. Phys. B 95, 213–218 (2009).
    [CrossRef]
  8. T. Grießer, H. Ritsch, M. Hemmerling, and G. Robb, “A Vlasov approach to bunching and selfordering of particles in optical resonators,” Eur. Phys. J. D 58, 349–368 (2010).
    [CrossRef]
  9. J. Javaloyes, M. Perrin, G. L. Lippi, and A. Politi, “Self-generated cooperative light emission induced by atomic recoil,” Phys. Rev. A 70, 023405 (2004).
    [CrossRef]
  10. R. Bach, K. Burnett, M. d’Arcy, and S. Gardiner, “Quantum-mechanical cumulant dynamics near stable periodic orbits in phase space: application to the classical-like dynamics of quantum accelerator modes,” Phys. Rev. A 71, 33417 (2005).
    [CrossRef]
  11. S. Gupta, K. L. Moore, K. W. Murch, and D. M. Stamper-Kurn, “Cavity nonlinear optics at low photon numbers from collective atomic motion,” Phys. Rev. Lett. 99, 213601 (2007).
    [CrossRef]
  12. A. Vukics, W. Niedenzu, and H. Ritsch, “Cavity nonlinear optics with few photons and ultracold quantum particles,” Phys. Rev. A 79, 013828 (2009).
    [CrossRef]
  13. F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose–Einstein condensate,” Science 322, 235–238 (2008).
    [CrossRef] [PubMed]
  14. D. Nagy, P. Domokos, A. Vukics, and H. Ritsch, “Nonlinear quantum dynamics of two BEC modes dispersively coupled by an optical cavity,” Eur. Phys. J. D 55, 659–668 (2009).
    [CrossRef]
  15. D. C. Montgomery, Theory of the unmagnetized plasma (Gordon & Breach, 1971).
  16. F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, and T. Esslinger, “Cavity QED with a Bose–Einstein condensate,” Nature 450, 268–271 (2007).
    [CrossRef] [PubMed]
  17. M. Cristiani and J. Eschner (personal communication, 2011).
  18. T. Valenzuela, M. Cristiani, H. Gothe, and J. Eschner, “Cold Ytterbium atoms in high-finesse optical cavities: cavity cooling and collective interactions,” in “Lasers and Electro-Optics 2009 and the European Quantum Electronics Conference. CLEO Europe-EQEC 2009. European Conference on ,” (IEEE, 2009), p. 1.
    [CrossRef]

2010 (1)

T. Grießer, H. Ritsch, M. Hemmerling, and G. Robb, “A Vlasov approach to bunching and selfordering of particles in optical resonators,” Eur. Phys. J. D 58, 349–368 (2010).
[CrossRef]

2009 (4)

J. M. Zhang, F. C. Cui, D. L. Zhou, and W. M. Liu, “Nonlinear dynamics of a cigar-shaped bose-einstein condensate in an optical cavity,” Phys. Rev. A 79, 033401 (2009).
[CrossRef]

S. Ritter, F. Brennecke, K. Baumann, T. Donner, C. Guerlin, and T. Esslinger, “Dynamical coupling between a Bose–Einstein condensate and a cavity optical lattice,” Appl. Phys. B 95, 213–218 (2009).
[CrossRef]

A. Vukics, W. Niedenzu, and H. Ritsch, “Cavity nonlinear optics with few photons and ultracold quantum particles,” Phys. Rev. A 79, 013828 (2009).
[CrossRef]

D. Nagy, P. Domokos, A. Vukics, and H. Ritsch, “Nonlinear quantum dynamics of two BEC modes dispersively coupled by an optical cavity,” Eur. Phys. J. D 55, 659–668 (2009).
[CrossRef]

2008 (2)

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

M. Khudaverdyan, W. Alt, I. Dotsenko, T. Kampschulte, K. Lenhard, A. Rauschenbeutel, S. Reick, K. Schörner, A. Widera, and D. Meschede, “Controlled insertion and retrieval of atoms coupled to a high-finesse optical resonator,” New J. Phys. 10, 073023 (2008).
[CrossRef]

2007 (2)

S. Gupta, K. L. Moore, K. W. Murch, and D. M. Stamper-Kurn, “Cavity nonlinear optics at low photon numbers from collective atomic motion,” Phys. Rev. Lett. 99, 213601 (2007).
[CrossRef]

F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, and T. Esslinger, “Cavity QED with a Bose–Einstein condensate,” Nature 450, 268–271 (2007).
[CrossRef] [PubMed]

2005 (1)

R. Bach, K. Burnett, M. d’Arcy, and S. Gardiner, “Quantum-mechanical cumulant dynamics near stable periodic orbits in phase space: application to the classical-like dynamics of quantum accelerator modes,” Phys. Rev. A 71, 33417 (2005).
[CrossRef]

2004 (1)

J. Javaloyes, M. Perrin, G. L. Lippi, and A. Politi, “Self-generated cooperative light emission induced by atomic recoil,” Phys. Rev. A 70, 023405 (2004).
[CrossRef]

2003 (2)

P. Domokos and H. Ritsch, “Mechanical effects of light in optical resonators,” J. Opt. Soc. Am. B 20, 1098–1130 (2003).
[CrossRef]

D. Kruse, M. Ruder, J. Benhelm, C. von Cube, C. Zimmermann, P. W. Courteille, T. Elsässer, B. Nagorny, and A. Hemmerich, “Cold atoms in a high-Q ring cavity,” Phys. Rev. A 67, 051802 (2003).
[CrossRef]

2001 (1)

P. Domokos, P. Horak, and H. Ritsch, “Semiclassical theory of cavity-assisted atom cooling,” J. Phys. B 34, 187–198 (2001).
[CrossRef]

1997 (1)

P. Horak, G. Hechenblaikner, K. M. Gheri, H. Stecher, and H. Ritsch, “Cavity-induced atom cooling in the strong coupling regime,” Phys. Rev. Lett. 79, 4974–4977 (1997).
[CrossRef]

Alt, W.

M. Khudaverdyan, W. Alt, I. Dotsenko, T. Kampschulte, K. Lenhard, A. Rauschenbeutel, S. Reick, K. Schörner, A. Widera, and D. Meschede, “Controlled insertion and retrieval of atoms coupled to a high-finesse optical resonator,” New J. Phys. 10, 073023 (2008).
[CrossRef]

Bach, R.

R. Bach, K. Burnett, M. d’Arcy, and S. Gardiner, “Quantum-mechanical cumulant dynamics near stable periodic orbits in phase space: application to the classical-like dynamics of quantum accelerator modes,” Phys. Rev. A 71, 33417 (2005).
[CrossRef]

Baumann, K.

S. Ritter, F. Brennecke, K. Baumann, T. Donner, C. Guerlin, and T. Esslinger, “Dynamical coupling between a Bose–Einstein condensate and a cavity optical lattice,” Appl. Phys. B 95, 213–218 (2009).
[CrossRef]

Benhelm, J.

D. Kruse, M. Ruder, J. Benhelm, C. von Cube, C. Zimmermann, P. W. Courteille, T. Elsässer, B. Nagorny, and A. Hemmerich, “Cold atoms in a high-Q ring cavity,” Phys. Rev. A 67, 051802 (2003).
[CrossRef]

Bourdel, T.

F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, and T. Esslinger, “Cavity QED with a Bose–Einstein condensate,” Nature 450, 268–271 (2007).
[CrossRef] [PubMed]

Brennecke, F.

S. Ritter, F. Brennecke, K. Baumann, T. Donner, C. Guerlin, and T. Esslinger, “Dynamical coupling between a Bose–Einstein condensate and a cavity optical lattice,” Appl. Phys. B 95, 213–218 (2009).
[CrossRef]

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

F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, and T. Esslinger, “Cavity QED with a Bose–Einstein condensate,” Nature 450, 268–271 (2007).
[CrossRef] [PubMed]

Burnett, K.

R. Bach, K. Burnett, M. d’Arcy, and S. Gardiner, “Quantum-mechanical cumulant dynamics near stable periodic orbits in phase space: application to the classical-like dynamics of quantum accelerator modes,” Phys. Rev. A 71, 33417 (2005).
[CrossRef]

Courteille, P. W.

D. Kruse, M. Ruder, J. Benhelm, C. von Cube, C. Zimmermann, P. W. Courteille, T. Elsässer, B. Nagorny, and A. Hemmerich, “Cold atoms in a high-Q ring cavity,” Phys. Rev. A 67, 051802 (2003).
[CrossRef]

Cristiani, M.

T. Valenzuela, M. Cristiani, H. Gothe, and J. Eschner, “Cold Ytterbium atoms in high-finesse optical cavities: cavity cooling and collective interactions,” in “Lasers and Electro-Optics 2009 and the European Quantum Electronics Conference. CLEO Europe-EQEC 2009. European Conference on ,” (IEEE, 2009), p. 1.
[CrossRef]

Cui, F. C.

J. M. Zhang, F. C. Cui, D. L. Zhou, and W. M. Liu, “Nonlinear dynamics of a cigar-shaped bose-einstein condensate in an optical cavity,” Phys. Rev. A 79, 033401 (2009).
[CrossRef]

d’Arcy, M.

R. Bach, K. Burnett, M. d’Arcy, and S. Gardiner, “Quantum-mechanical cumulant dynamics near stable periodic orbits in phase space: application to the classical-like dynamics of quantum accelerator modes,” Phys. Rev. A 71, 33417 (2005).
[CrossRef]

Domokos, P.

D. Nagy, P. Domokos, A. Vukics, and H. Ritsch, “Nonlinear quantum dynamics of two BEC modes dispersively coupled by an optical cavity,” Eur. Phys. J. D 55, 659–668 (2009).
[CrossRef]

P. Domokos and H. Ritsch, “Mechanical effects of light in optical resonators,” J. Opt. Soc. Am. B 20, 1098–1130 (2003).
[CrossRef]

P. Domokos, P. Horak, and H. Ritsch, “Semiclassical theory of cavity-assisted atom cooling,” J. Phys. B 34, 187–198 (2001).
[CrossRef]

Donner, T.

S. Ritter, F. Brennecke, K. Baumann, T. Donner, C. Guerlin, and T. Esslinger, “Dynamical coupling between a Bose–Einstein condensate and a cavity optical lattice,” Appl. Phys. B 95, 213–218 (2009).
[CrossRef]

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

F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, and T. Esslinger, “Cavity QED with a Bose–Einstein condensate,” Nature 450, 268–271 (2007).
[CrossRef] [PubMed]

Dotsenko, I.

M. Khudaverdyan, W. Alt, I. Dotsenko, T. Kampschulte, K. Lenhard, A. Rauschenbeutel, S. Reick, K. Schörner, A. Widera, and D. Meschede, “Controlled insertion and retrieval of atoms coupled to a high-finesse optical resonator,” New J. Phys. 10, 073023 (2008).
[CrossRef]

Elsässer, T.

D. Kruse, M. Ruder, J. Benhelm, C. von Cube, C. Zimmermann, P. W. Courteille, T. Elsässer, B. Nagorny, and A. Hemmerich, “Cold atoms in a high-Q ring cavity,” Phys. Rev. A 67, 051802 (2003).
[CrossRef]

Eschner, J.

T. Valenzuela, M. Cristiani, H. Gothe, and J. Eschner, “Cold Ytterbium atoms in high-finesse optical cavities: cavity cooling and collective interactions,” in “Lasers and Electro-Optics 2009 and the European Quantum Electronics Conference. CLEO Europe-EQEC 2009. European Conference on ,” (IEEE, 2009), p. 1.
[CrossRef]

Esslinger, T.

S. Ritter, F. Brennecke, K. Baumann, T. Donner, C. Guerlin, and T. Esslinger, “Dynamical coupling between a Bose–Einstein condensate and a cavity optical lattice,” Appl. Phys. B 95, 213–218 (2009).
[CrossRef]

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

F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, and T. Esslinger, “Cavity QED with a Bose–Einstein condensate,” Nature 450, 268–271 (2007).
[CrossRef] [PubMed]

Gardiner, S.

R. Bach, K. Burnett, M. d’Arcy, and S. Gardiner, “Quantum-mechanical cumulant dynamics near stable periodic orbits in phase space: application to the classical-like dynamics of quantum accelerator modes,” Phys. Rev. A 71, 33417 (2005).
[CrossRef]

Gheri, K. M.

P. Horak, G. Hechenblaikner, K. M. Gheri, H. Stecher, and H. Ritsch, “Cavity-induced atom cooling in the strong coupling regime,” Phys. Rev. Lett. 79, 4974–4977 (1997).
[CrossRef]

Gothe, H.

T. Valenzuela, M. Cristiani, H. Gothe, and J. Eschner, “Cold Ytterbium atoms in high-finesse optical cavities: cavity cooling and collective interactions,” in “Lasers and Electro-Optics 2009 and the European Quantum Electronics Conference. CLEO Europe-EQEC 2009. European Conference on ,” (IEEE, 2009), p. 1.
[CrossRef]

Grießer, T.

T. Grießer, H. Ritsch, M. Hemmerling, and G. Robb, “A Vlasov approach to bunching and selfordering of particles in optical resonators,” Eur. Phys. J. D 58, 349–368 (2010).
[CrossRef]

Guerlin, C.

S. Ritter, F. Brennecke, K. Baumann, T. Donner, C. Guerlin, and T. Esslinger, “Dynamical coupling between a Bose–Einstein condensate and a cavity optical lattice,” Appl. Phys. B 95, 213–218 (2009).
[CrossRef]

Gupta, S.

S. Gupta, K. L. Moore, K. W. Murch, and D. M. Stamper-Kurn, “Cavity nonlinear optics at low photon numbers from collective atomic motion,” Phys. Rev. Lett. 99, 213601 (2007).
[CrossRef]

Hechenblaikner, G.

P. Horak, G. Hechenblaikner, K. M. Gheri, H. Stecher, and H. Ritsch, “Cavity-induced atom cooling in the strong coupling regime,” Phys. Rev. Lett. 79, 4974–4977 (1997).
[CrossRef]

Hemmerich, A.

D. Kruse, M. Ruder, J. Benhelm, C. von Cube, C. Zimmermann, P. W. Courteille, T. Elsässer, B. Nagorny, and A. Hemmerich, “Cold atoms in a high-Q ring cavity,” Phys. Rev. A 67, 051802 (2003).
[CrossRef]

Hemmerling, M.

T. Grießer, H. Ritsch, M. Hemmerling, and G. Robb, “A Vlasov approach to bunching and selfordering of particles in optical resonators,” Eur. Phys. J. D 58, 349–368 (2010).
[CrossRef]

Horak, P.

P. Domokos, P. Horak, and H. Ritsch, “Semiclassical theory of cavity-assisted atom cooling,” J. Phys. B 34, 187–198 (2001).
[CrossRef]

P. Horak, G. Hechenblaikner, K. M. Gheri, H. Stecher, and H. Ritsch, “Cavity-induced atom cooling in the strong coupling regime,” Phys. Rev. Lett. 79, 4974–4977 (1997).
[CrossRef]

Javaloyes, J.

J. Javaloyes, M. Perrin, G. L. Lippi, and A. Politi, “Self-generated cooperative light emission induced by atomic recoil,” Phys. Rev. A 70, 023405 (2004).
[CrossRef]

Kampschulte, T.

M. Khudaverdyan, W. Alt, I. Dotsenko, T. Kampschulte, K. Lenhard, A. Rauschenbeutel, S. Reick, K. Schörner, A. Widera, and D. Meschede, “Controlled insertion and retrieval of atoms coupled to a high-finesse optical resonator,” New J. Phys. 10, 073023 (2008).
[CrossRef]

Khudaverdyan, M.

M. Khudaverdyan, W. Alt, I. Dotsenko, T. Kampschulte, K. Lenhard, A. Rauschenbeutel, S. Reick, K. Schörner, A. Widera, and D. Meschede, “Controlled insertion and retrieval of atoms coupled to a high-finesse optical resonator,” New J. Phys. 10, 073023 (2008).
[CrossRef]

Köhl, M.

F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, and T. Esslinger, “Cavity QED with a Bose–Einstein condensate,” Nature 450, 268–271 (2007).
[CrossRef] [PubMed]

Kruse, D.

D. Kruse, M. Ruder, J. Benhelm, C. von Cube, C. Zimmermann, P. W. Courteille, T. Elsässer, B. Nagorny, and A. Hemmerich, “Cold atoms in a high-Q ring cavity,” Phys. Rev. A 67, 051802 (2003).
[CrossRef]

Lenhard, K.

M. Khudaverdyan, W. Alt, I. Dotsenko, T. Kampschulte, K. Lenhard, A. Rauschenbeutel, S. Reick, K. Schörner, A. Widera, and D. Meschede, “Controlled insertion and retrieval of atoms coupled to a high-finesse optical resonator,” New J. Phys. 10, 073023 (2008).
[CrossRef]

Lippi, G. L.

J. Javaloyes, M. Perrin, G. L. Lippi, and A. Politi, “Self-generated cooperative light emission induced by atomic recoil,” Phys. Rev. A 70, 023405 (2004).
[CrossRef]

Liu, W. M.

J. M. Zhang, F. C. Cui, D. L. Zhou, and W. M. Liu, “Nonlinear dynamics of a cigar-shaped bose-einstein condensate in an optical cavity,” Phys. Rev. A 79, 033401 (2009).
[CrossRef]

Meschede, D.

M. Khudaverdyan, W. Alt, I. Dotsenko, T. Kampschulte, K. Lenhard, A. Rauschenbeutel, S. Reick, K. Schörner, A. Widera, and D. Meschede, “Controlled insertion and retrieval of atoms coupled to a high-finesse optical resonator,” New J. Phys. 10, 073023 (2008).
[CrossRef]

Montgomery, D. C.

D. C. Montgomery, Theory of the unmagnetized plasma (Gordon & Breach, 1971).

Moore, K. L.

S. Gupta, K. L. Moore, K. W. Murch, and D. M. Stamper-Kurn, “Cavity nonlinear optics at low photon numbers from collective atomic motion,” Phys. Rev. Lett. 99, 213601 (2007).
[CrossRef]

Murch, K. W.

S. Gupta, K. L. Moore, K. W. Murch, and D. M. Stamper-Kurn, “Cavity nonlinear optics at low photon numbers from collective atomic motion,” Phys. Rev. Lett. 99, 213601 (2007).
[CrossRef]

Nagorny, B.

D. Kruse, M. Ruder, J. Benhelm, C. von Cube, C. Zimmermann, P. W. Courteille, T. Elsässer, B. Nagorny, and A. Hemmerich, “Cold atoms in a high-Q ring cavity,” Phys. Rev. A 67, 051802 (2003).
[CrossRef]

Nagy, D.

D. Nagy, P. Domokos, A. Vukics, and H. Ritsch, “Nonlinear quantum dynamics of two BEC modes dispersively coupled by an optical cavity,” Eur. Phys. J. D 55, 659–668 (2009).
[CrossRef]

Niedenzu, W.

A. Vukics, W. Niedenzu, and H. Ritsch, “Cavity nonlinear optics with few photons and ultracold quantum particles,” Phys. Rev. A 79, 013828 (2009).
[CrossRef]

Perrin, M.

J. Javaloyes, M. Perrin, G. L. Lippi, and A. Politi, “Self-generated cooperative light emission induced by atomic recoil,” Phys. Rev. A 70, 023405 (2004).
[CrossRef]

Politi, A.

J. Javaloyes, M. Perrin, G. L. Lippi, and A. Politi, “Self-generated cooperative light emission induced by atomic recoil,” Phys. Rev. A 70, 023405 (2004).
[CrossRef]

Rauschenbeutel, A.

M. Khudaverdyan, W. Alt, I. Dotsenko, T. Kampschulte, K. Lenhard, A. Rauschenbeutel, S. Reick, K. Schörner, A. Widera, and D. Meschede, “Controlled insertion and retrieval of atoms coupled to a high-finesse optical resonator,” New J. Phys. 10, 073023 (2008).
[CrossRef]

Reick, S.

M. Khudaverdyan, W. Alt, I. Dotsenko, T. Kampschulte, K. Lenhard, A. Rauschenbeutel, S. Reick, K. Schörner, A. Widera, and D. Meschede, “Controlled insertion and retrieval of atoms coupled to a high-finesse optical resonator,” New J. Phys. 10, 073023 (2008).
[CrossRef]

Ritsch, H.

T. Grießer, H. Ritsch, M. Hemmerling, and G. Robb, “A Vlasov approach to bunching and selfordering of particles in optical resonators,” Eur. Phys. J. D 58, 349–368 (2010).
[CrossRef]

A. Vukics, W. Niedenzu, and H. Ritsch, “Cavity nonlinear optics with few photons and ultracold quantum particles,” Phys. Rev. A 79, 013828 (2009).
[CrossRef]

D. Nagy, P. Domokos, A. Vukics, and H. Ritsch, “Nonlinear quantum dynamics of two BEC modes dispersively coupled by an optical cavity,” Eur. Phys. J. D 55, 659–668 (2009).
[CrossRef]

P. Domokos and H. Ritsch, “Mechanical effects of light in optical resonators,” J. Opt. Soc. Am. B 20, 1098–1130 (2003).
[CrossRef]

P. Domokos, P. Horak, and H. Ritsch, “Semiclassical theory of cavity-assisted atom cooling,” J. Phys. B 34, 187–198 (2001).
[CrossRef]

P. Horak, G. Hechenblaikner, K. M. Gheri, H. Stecher, and H. Ritsch, “Cavity-induced atom cooling in the strong coupling regime,” Phys. Rev. Lett. 79, 4974–4977 (1997).
[CrossRef]

Ritter, S.

S. Ritter, F. Brennecke, K. Baumann, T. Donner, C. Guerlin, and T. Esslinger, “Dynamical coupling between a Bose–Einstein condensate and a cavity optical lattice,” Appl. Phys. B 95, 213–218 (2009).
[CrossRef]

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

F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, and T. Esslinger, “Cavity QED with a Bose–Einstein condensate,” Nature 450, 268–271 (2007).
[CrossRef] [PubMed]

Robb, G.

T. Grießer, H. Ritsch, M. Hemmerling, and G. Robb, “A Vlasov approach to bunching and selfordering of particles in optical resonators,” Eur. Phys. J. D 58, 349–368 (2010).
[CrossRef]

Ruder, M.

D. Kruse, M. Ruder, J. Benhelm, C. von Cube, C. Zimmermann, P. W. Courteille, T. Elsässer, B. Nagorny, and A. Hemmerich, “Cold atoms in a high-Q ring cavity,” Phys. Rev. A 67, 051802 (2003).
[CrossRef]

Schörner, K.

M. Khudaverdyan, W. Alt, I. Dotsenko, T. Kampschulte, K. Lenhard, A. Rauschenbeutel, S. Reick, K. Schörner, A. Widera, and D. Meschede, “Controlled insertion and retrieval of atoms coupled to a high-finesse optical resonator,” New J. Phys. 10, 073023 (2008).
[CrossRef]

Stamper-Kurn, D. M.

S. Gupta, K. L. Moore, K. W. Murch, and D. M. Stamper-Kurn, “Cavity nonlinear optics at low photon numbers from collective atomic motion,” Phys. Rev. Lett. 99, 213601 (2007).
[CrossRef]

Stecher, H.

P. Horak, G. Hechenblaikner, K. M. Gheri, H. Stecher, and H. Ritsch, “Cavity-induced atom cooling in the strong coupling regime,” Phys. Rev. Lett. 79, 4974–4977 (1997).
[CrossRef]

Valenzuela, T.

T. Valenzuela, M. Cristiani, H. Gothe, and J. Eschner, “Cold Ytterbium atoms in high-finesse optical cavities: cavity cooling and collective interactions,” in “Lasers and Electro-Optics 2009 and the European Quantum Electronics Conference. CLEO Europe-EQEC 2009. European Conference on ,” (IEEE, 2009), p. 1.
[CrossRef]

von Cube, C.

D. Kruse, M. Ruder, J. Benhelm, C. von Cube, C. Zimmermann, P. W. Courteille, T. Elsässer, B. Nagorny, and A. Hemmerich, “Cold atoms in a high-Q ring cavity,” Phys. Rev. A 67, 051802 (2003).
[CrossRef]

Vukics, A.

A. Vukics, W. Niedenzu, and H. Ritsch, “Cavity nonlinear optics with few photons and ultracold quantum particles,” Phys. Rev. A 79, 013828 (2009).
[CrossRef]

D. Nagy, P. Domokos, A. Vukics, and H. Ritsch, “Nonlinear quantum dynamics of two BEC modes dispersively coupled by an optical cavity,” Eur. Phys. J. D 55, 659–668 (2009).
[CrossRef]

Widera, A.

M. Khudaverdyan, W. Alt, I. Dotsenko, T. Kampschulte, K. Lenhard, A. Rauschenbeutel, S. Reick, K. Schörner, A. Widera, and D. Meschede, “Controlled insertion and retrieval of atoms coupled to a high-finesse optical resonator,” New J. Phys. 10, 073023 (2008).
[CrossRef]

Zhang, J. M.

J. M. Zhang, F. C. Cui, D. L. Zhou, and W. M. Liu, “Nonlinear dynamics of a cigar-shaped bose-einstein condensate in an optical cavity,” Phys. Rev. A 79, 033401 (2009).
[CrossRef]

Zhou, D. L.

J. M. Zhang, F. C. Cui, D. L. Zhou, and W. M. Liu, “Nonlinear dynamics of a cigar-shaped bose-einstein condensate in an optical cavity,” Phys. Rev. A 79, 033401 (2009).
[CrossRef]

Zimmermann, C.

D. Kruse, M. Ruder, J. Benhelm, C. von Cube, C. Zimmermann, P. W. Courteille, T. Elsässer, B. Nagorny, and A. Hemmerich, “Cold atoms in a high-Q ring cavity,” Phys. Rev. A 67, 051802 (2003).
[CrossRef]

Appl. Phys. B (1)

S. Ritter, F. Brennecke, K. Baumann, T. Donner, C. Guerlin, and T. Esslinger, “Dynamical coupling between a Bose–Einstein condensate and a cavity optical lattice,” Appl. Phys. B 95, 213–218 (2009).
[CrossRef]

Eur. Phys. J. D (2)

T. Grießer, H. Ritsch, M. Hemmerling, and G. Robb, “A Vlasov approach to bunching and selfordering of particles in optical resonators,” Eur. Phys. J. D 58, 349–368 (2010).
[CrossRef]

D. Nagy, P. Domokos, A. Vukics, and H. Ritsch, “Nonlinear quantum dynamics of two BEC modes dispersively coupled by an optical cavity,” Eur. Phys. J. D 55, 659–668 (2009).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Phys. B (1)

P. Domokos, P. Horak, and H. Ritsch, “Semiclassical theory of cavity-assisted atom cooling,” J. Phys. B 34, 187–198 (2001).
[CrossRef]

Nature (1)

F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, and T. Esslinger, “Cavity QED with a Bose–Einstein condensate,” Nature 450, 268–271 (2007).
[CrossRef] [PubMed]

New J. Phys. (1)

M. Khudaverdyan, W. Alt, I. Dotsenko, T. Kampschulte, K. Lenhard, A. Rauschenbeutel, S. Reick, K. Schörner, A. Widera, and D. Meschede, “Controlled insertion and retrieval of atoms coupled to a high-finesse optical resonator,” New J. Phys. 10, 073023 (2008).
[CrossRef]

Phys. Rev. A (5)

D. Kruse, M. Ruder, J. Benhelm, C. von Cube, C. Zimmermann, P. W. Courteille, T. Elsässer, B. Nagorny, and A. Hemmerich, “Cold atoms in a high-Q ring cavity,” Phys. Rev. A 67, 051802 (2003).
[CrossRef]

J. Javaloyes, M. Perrin, G. L. Lippi, and A. Politi, “Self-generated cooperative light emission induced by atomic recoil,” Phys. Rev. A 70, 023405 (2004).
[CrossRef]

R. Bach, K. Burnett, M. d’Arcy, and S. Gardiner, “Quantum-mechanical cumulant dynamics near stable periodic orbits in phase space: application to the classical-like dynamics of quantum accelerator modes,” Phys. Rev. A 71, 33417 (2005).
[CrossRef]

J. M. Zhang, F. C. Cui, D. L. Zhou, and W. M. Liu, “Nonlinear dynamics of a cigar-shaped bose-einstein condensate in an optical cavity,” Phys. Rev. A 79, 033401 (2009).
[CrossRef]

A. Vukics, W. Niedenzu, and H. Ritsch, “Cavity nonlinear optics with few photons and ultracold quantum particles,” Phys. Rev. A 79, 013828 (2009).
[CrossRef]

Phys. Rev. Lett. (2)

S. Gupta, K. L. Moore, K. W. Murch, and D. M. Stamper-Kurn, “Cavity nonlinear optics at low photon numbers from collective atomic motion,” Phys. Rev. Lett. 99, 213601 (2007).
[CrossRef]

P. Horak, G. Hechenblaikner, K. M. Gheri, H. Stecher, and H. Ritsch, “Cavity-induced atom cooling in the strong coupling regime,” Phys. Rev. Lett. 79, 4974–4977 (1997).
[CrossRef]

Science (1)

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

Other (3)

M. Cristiani and J. Eschner (personal communication, 2011).

T. Valenzuela, M. Cristiani, H. Gothe, and J. Eschner, “Cold Ytterbium atoms in high-finesse optical cavities: cavity cooling and collective interactions,” in “Lasers and Electro-Optics 2009 and the European Quantum Electronics Conference. CLEO Europe-EQEC 2009. European Conference on ,” (IEEE, 2009), p. 1.
[CrossRef]

D. C. Montgomery, Theory of the unmagnetized plasma (Gordon & Breach, 1971).

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Figures (7)

Fig. 1
Fig. 1

Schematic drawing of the physical system. The particles are indicated as the shaded cloud in the center. The resonator mode decays at a rate κ and is driven by a laser characterised by the pump parameter η.

Fig. 2
Fig. 2

Normalized steady state photon number versus effective detuning δ/κ for N = 105, U 0 = 0.04κ, η =18κ, ωT = κ ωR = 0.5 · 10−3 κ and a Gaussian velocity distribution.

Fig. 3
Fig. 3

Response curve (red) and regions of instability (shaded) for η = 13κ (a) and η = 18κ (b). Those parts of the response curve that lie inside the unstable region (red dashed), correspond to linearly unstable steady states. The black dashed line separates the parameter space into positive and negative Δ. The intervalls designated A correspond to bistability, the intervall designated B supports no stable steady state at all. All other parameters as in Fig. (2).

Fig. 4
Fig. 4

a: Response curve (stable parts red, unstable parts black dotted) and time averaged normalized photon number (blue circles) versus effective detuning. A deviation from the response curve occurs only inside the unstable intervall [−3.3,0], where the system exhibits limit cycle oscillations. b: Limit cycle frequency versus detuning. Parameters as usual.

Fig. 5
Fig. 5

(a) Photon number as a function of time for δ = −κ (upper row) and δ = −3κ (lower row) and (b) normalized power spectrum. (c) Shape of the limit cycle. All additional parameters as in Fig. (2).

Fig. 6
Fig. 6

Comparison between the reduced (blue) and the Vlasov model Eq. (64) (red). Apart from a phase shift, the results are almost identical. All parameters as in Fig. (2) and Fig. (5).

Fig. 7
Fig. 7

Response curve and numerically determined limit cycle frequency (blue triangles) vs. theoretical frequency from Eq. (53) (green dashed). The blue circles show the time averaged photon number. Parameters: N = 105, kvT = κ, ωR = 10−3 κ and U 0 = 0.05κ.

Equations (66)

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f ( x , v , t ) = m 2 π ħ e i z m v / ħ ρ p , 1 ( x + z 2 , x z 2 , t ) d z ,
f t + v f x + U 0 | α | 2 2 sin ( 2 k x ) ( f ( x , v + v R ) f ( x , v v R ) ) = 0 ,
α ˙ = [ κ + i ( Δ c NU 0 / 2 ) ] α i NU 0 2 α d v 0 L f ( x , v , t ) cos ( 2 k x ) d x + η .
f ( x , v , t ) = f 0 ( v ) + δ f ( x , v , t ) ,
( t + v x ) δ f = U 0 | α | 2 2 sin ( 2 k x ) ( f 0 ( v + v R ) f 0 ( v v R ) ) .
ϕ t + 2 i k v ϕ = i U 0 | α | 2 4 ( F 0 ( v + v R ) F 0 ( v v R ) ) ,
α ˙ = [ κ + i ( Δ c NU 0 / 2 ) ] α i NU 0 2 α Re { ϕ ( v , t ) } d v + η .
ϕ 0 ( v ) = v R U 0 | α 0 | 2 4 1 k v Δ F 0 Δ v ,
Δ F 0 Δ v : = F 0 ( v + v R ) F 0 ( v v R ) 2 v R .
0 = ( κ + i δ ) α 0 i v R NU 0 2 8 k | α 0 | 2 α 0 1 v Δ F 0 Δ v d v + η ,
F ( x ) : = v T F 0 ( x v T ) ,
J ( v R / v T ) : = 1 2 1 x F ( x + v R / v T ) F ( x v R / v T ) v R / v T d x ,
I 0 [ κ 2 + ( δ I 0 ω R NU 0 2 J 4 ω T 2 ) 2 ] = η 2 .
ϕ ( v , t ) = ϕ 0 ( v ) + δ ϕ ( v , t ) ,
α ( t ) = α 0 + δ α ( t ) .
δ ϕ t + 2 i k v δ ϕ = i v R U 0 2 ( α 0 δ α * + α 0 * δ α ) Δ F 0 Δ v ,
δ α ˙ = ( κ + i Δ ) δ α i NU 0 2 α 0 Re { δ ϕ ( v , t ) } d v ,
χ ( s ) : = NU 0 2 v R 8 ( Δ F 0 / Δ v s + 2 i k v Δ F 0 / Δ v s 2 i k v ) d v ,
( s + κ i Δ I 0 χ ( s ) α 0 2 χ ( s ) α 0 * 2 χ ( s ) s + κ + i Δ + I 0 χ ( s ) ) · ( δ α δ α * ) = ( I 1 I 2 ) ,
D ( s ) = ( κ + s ) 2 + Δ 2 2 i Δ I 0 χ ( s ) .
F 0 ( v ) = δ ( v ) ,
ϕ ( v , t ) = ϕ + ( t ) δ ( v v R ) + ϕ ( t ) δ ( v + v R ) + ϕ 0 ( v , t ) ,
ϕ 0 ( v , 0 ) e 2 i k v t d v 0 , as t .
x ¨ + ( 4 ω R ) 2 x = ω R U 0 | α | 2 ,
α ˙ = [ k + i ( Δ c NU 0 / 2 ) ] α i NU 0 2 α x + η ,
χ ( s ) = i ω R NU 0 2 2 1 s 2 + ( 4 ω R ) 2 .
lim ɛ 0 + 1 x y ± i ɛ = P x y i π δ ( x y ) .
χ ( ω ) = NU 0 2 v R 8 k [ 1 i P v Δ F 0 / Δ v ( ω / 2 k 2 ) v 2 d v + π 2 ( Δ F 0 Δ v | ω 2 k Δ F 0 Δ v | ω 2 k ) ] .
F 0 ( v ) = 1 v T π e ( v / v T ) 2 ; J = 2.
Δ F 0 Δ v d F 0 d v
ħ U 0 | α | 2 m v T 2 2 , or 4 U 0 η 2 ω R κ 2 ω T 2 1 ,
D r ( ω ) = κ 2 + Δ 2 ω 2 + NU 0 2 Δ I 0 ω R ω T 2 ( 1 ω ω T Da ( ω 2 ω T ) ) ,
D i ( ω ) = 2 κ ω π NU 0 2 Δ I 0 ω R 2 ω T 2 ω ω T e ( ω 2 ω T ) 2 .
Da ( x ) = e x 2 0 x e t 2 d t .
D r ( ω 0 ) > 0
D i ( ω 0 ) = 0.
I 0 = I 0 crit ( Δ )
δ = Δ NU 0 2 ω R 2 ω T 2 I 0 ,
Δ ( Δ NU 0 2 ω R 2 ω T 2 I 0 crit ( Δ ) , I 0 cirt ( Δ ) )
χ ( t ) : = Re { ϕ ( v , t ) } d v χ 0 + χ 1 cos ( ω t )
| α ( t ) | 2 = : I ( t ) = n I n e i n ω t
I n ( χ 0 , χ 1 ) = η 2 m = J m ( NU 0 χ 1 2 ω ) J m n ( NU 0 χ 1 2 ω ) [ κ + i Δ i m ω ] [ κ i Δ i ( n m ) ω ] ,
( ω + 2 k v ) ϕ n = v R U 0 I n 2 F 0 v .
ϕ n ( v ) = v R U 0 I n 2 ( F 0 ( v ) v P n ω + 2 k v + h n ( v ) δ ( n ω + 2 k v ) ) ,
e i n ω t lim γ 0 + e ( i n ω + γ ) t ,
h n ( v ) = i π F 0 ( v ) v .
χ 0 = v R U 0 I 0 2 k 1 v F 0 ( v ) v d v = U 0 ω R J 2 ω T 2 I 0
χ 1 = U 0 v R 4 k ( I a F 0 ( v ) v d v ω / 2 k + v I b π F 0 v ( ω / 2 k ) ) ,
I b F 0 ( v ) v d v ω / 2 k + v = π I a F 0 v ( ω / 2 k ) .
| NU 0 χ 1 2 ω | 1.
χ 0 = χ 0 ( 0 ) + χ 0 ( 2 ) ( ω ) ( NU 0 χ 1 4 ω ) 2 + ,
X 0 ( 0 ) = U 0 ω R J 2 ω T 2 η 2 κ 2 + Δ 0 2 ,
I 1 = I 1 ( 1 ) ( ω , χ 0 ( 0 ) ) NU 0 χ 1 4 ω + I 1 ( 3 ) ( ω , χ 0 ( 0 ) ) ( NU 0 χ 1 4 ω ) 3 + ,
I 1 ( 1 ) ( ω , χ 0 ( 0 ) ) = 2 η 2 ω Δ 0 ( κ 2 + Δ 0 2 ) ( κ 2 + Δ 0 2 ω 2 + 2 i κ ω ) .
2 κ ω F 0 ( v ) v d v ω / 2 k + v = ( κ 2 + Δ 0 2 ω 2 ) π F 0 v ( ω / 2 k ) ,
ω = κ 2 + Δ 0 2 + 4 κ ω T π .
t ψ ^ = i ħ 2 m 2 x 2 ψ ^ + 1 i ħ V ^ ψ ^ ,
f ^ ( x , p ) = 1 2 π ħ e i z p / ħ ψ ^ ( x z 2 ) ψ ^ ( x + z 2 ) d z ,
f ^ t + p m f ^ x + U ^ ( x , q ) f ^ ( x , p ħ q ) d q = 0 ,
U ^ ( x , q ) : = i 2 π ħ d z [ V ^ ( x + z / 2 ) V ( x z / 2 ) ] e i q z .
f ^ t + p m f ^ x + U 0 2 a ^ a ^ sin ( 2 k x ) [ f ^ ( x , p + ħ k ) f ^ ( x , p ħ k ) ] = 0.
a ^ ˙ = ( κ + i δ ) a ^ i U 0 a ^ 0 L d x f ^ ( x , p ) cos ( 2 k x ) d p + η + a ^ i n .
f t + p m f x + U 0 2 | α | 2 sin ( 2 k x ) [ f ( x , p + ħ k ) f ( x , p ħ k ) ] = 0 ,
α ˙ = ( κ + i δ ) α i U 0 α 2 0 L d x f ( x , p ) cos ( 2 k x ) d p + η .
f ( x , p ± ħ k ) f ( x , p ) ± ħ k f p ( x , p ) + ħ 2 k 2 2 2 f p 2 ( x , p ) .
f t + p m f x + ħ k U 0 | α | 2 sin ( 2 k x ) f p = 0 ,

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