Abstract

An ideal superradiant laser on an optical clock transition of noninteracting cold atoms is predicted to exhibit an extreme frequency stability and accuracy far below mHz-linewidth. In any concrete setup sufficiently many atoms have to be confined and pumped within a finite cavity mode volume. Using a magic wavelength lattice minimizes light shifts and allows for almost uniform coupling to the cavity mode. Nevertheless, the atoms are subject to dipole-dipole interaction and collective spontaneous decay which compromises the ultimate frequency stability. In the high density limit the Dicke superradiant linewidth enhancement will broaden the laser line and nearest neighbor couplings will induce shifts and fluctuations of the laser frequency. We estimate the magnitude and scaling of these effects by direct numerical simulations of few atom systems for different geometries and densities. For Strontium in a regularly filled magic wavelength configuration atomic interactions induce small laser frequency shifts only and collective spontaneous emission weakly broadens the laser. These interactions generally enhance the laser sensitivity to cavity length fluctuations but for optimally chosen operating conditions can lead to an improved synchronization of the atomic dipoles.

© 2014 Optical Society of America

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  1. T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. Martin, L. Chen, J. Ye, “A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6, 687–692 (2012).
    [CrossRef]
  2. D. Meiser, J. Ye, D. Carlson, M. Holland, “Prospects for a millihertz-linewidth laser,” Phys. Rev. Lett. 102, 163601 (2009).
    [CrossRef] [PubMed]
  3. J. Bohnet, Z. Chen, J. Weiner, D. Meiser, M. Holland, J. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484, 78–81 (2012).
    [CrossRef] [PubMed]
  4. J. G. Bohnet, Z. Chen, J. M. Weiner, K. C. Cox, J. K. Thompson, “Active and passive sensing of collective atomic coherence in a superradiant laser,” Phys. Rev. A 88, 013826 (2013).
    [CrossRef]
  5. V. Vuletic, “Atomic physics: An almost lightless laser,” Nature 484, 43–44 (2012).
    [CrossRef] [PubMed]
  6. M. Xu, D. Tieri, M. Holland, “Simulating Open Quantum Systems using the Simple Lie Group SU (4),” arXiv preprint arXiv:1302.6284 (2013).
  7. K. Henschel, J. Majer, J. Schmiedmayer, H. Ritsch, “Cavity QED with an ultracold ensemble on a chip: Prospects for strong magnetic coupling at finite temperatures,” Phys. Rev. A 82, 033810 (2010).
    [CrossRef]
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    [CrossRef] [PubMed]
  9. P. Horak, K. Gheri, H. Ritsch, “Quantum dynamics of a single-atom cascade laser,” Phys. Rev. A 51, 3257–3266 (1995).
    [CrossRef] [PubMed]
  10. M. D. Swallows, M. Bishof, Y. Lin, S. Blatt, M. J. Martin, A. M. Rey, J. Ye, “Suppression of collisional shifts in a strongly interacting lattice clock,” Science 331, 1043–1046 (2011).
    [CrossRef] [PubMed]
  11. M. Gross, S. Haroche, “Superradiance: An essay on the theory of collective spontaneous emission,” Phys. Rep. 93, 301–396 (1982).
    [CrossRef]
  12. R. Bonifacio, P. Schwendimann, F. Haake, “Quantum statistical theory of superradiance. I,” Phys. Rev. A 4, 302 (1971).
    [CrossRef]
  13. N. E. Rehler, J. H. Eberly, “Superradiance,” Phys. Rev. A 3, 1735–1751 (1971).
    [CrossRef]
  14. H. Zoubi, “Collective light emission of a finite-size atomic chain,” Europhys. Lett. 100, 24002 (2012).
    [CrossRef]
  15. J. MacGillivray, M. Feld, “Theory of superradiance in an extended, optically thick medium,” Phys. Rev. A 14, 1169–1189 (1976).
    [CrossRef]
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    [CrossRef]
  18. S. Inouye, A. Chikkatur, D. Stamper-Kurn, J. Stenger, D. Pritchard, W. Ketterle, “Superradiant Rayleigh scattering from a Bose-Einstein condensate,” Science 285, 571–574 (1999).
    [CrossRef] [PubMed]
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    [CrossRef]
  20. M. Takamoto, F. Hong, R. Higashi, H. Katori, “An optical lattice clock,” Nature 435, 321–324 (2005).
    [CrossRef] [PubMed]
  21. L. Casperson, “Spectral narrowing in double-pass superradiant lasers,” Opt. Commun. 8, 85–87 (1973).
    [CrossRef]
  22. R. Lehmberg, “Radiation from an N-atom system. I. General formalism,” Phys. Rev. A 2, 883–888 (1970).
    [CrossRef]
  23. Z. Ficek, R. Tanaś, S. Kielich, “Quantum beats and superradiant effects in the spontaneous emission from two nonidentical atoms,” Physica A: Stat. Mech. Appl. 146, 452–482 (1987).
    [CrossRef]
  24. Z. Ficek, R. Tanaś, “Entangled states and collective nonclassical effects in two-atom systems,” Phys. Rep. 372, 369–443 (2002).
    [CrossRef]
  25. P. Meystre, M. Sargent, Elements of Quantum Optics (Springer-Verlag, 1990).
  26. R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93, 99–110 (1954).
    [CrossRef]
  27. G. Campbell, A. Ludlow, S. Blatt, J. Thomsen, M. Martin, M. de Miranda, T. Zelevinsky, M. Boyd, J. Ye, S. Diddams, T. Heavner, T. Parker, S. Jefferts, “The absolute frequency of the 87Sr optical clock transition,” Metrologia 45, 539–548 (2008).
    [CrossRef]
  28. B. Bloom, T. Nicholson, J. Williams, S. Campbell, M. Bishof, X. Zhang, W. Zhang, S. Bromley, J. Ye, “An optical lattice clock with accuracy and stability at the 10−18 level,” Nature 506, 71–75 (2014).
    [CrossRef] [PubMed]

2014 (1)

B. Bloom, T. Nicholson, J. Williams, S. Campbell, M. Bishof, X. Zhang, W. Zhang, S. Bromley, J. Ye, “An optical lattice clock with accuracy and stability at the 10−18 level,” Nature 506, 71–75 (2014).
[CrossRef] [PubMed]

2013 (1)

J. G. Bohnet, Z. Chen, J. M. Weiner, K. C. Cox, J. K. Thompson, “Active and passive sensing of collective atomic coherence in a superradiant laser,” Phys. Rev. A 88, 013826 (2013).
[CrossRef]

2012 (5)

V. Vuletic, “Atomic physics: An almost lightless laser,” Nature 484, 43–44 (2012).
[CrossRef] [PubMed]

T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. Martin, L. Chen, J. Ye, “A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6, 687–692 (2012).
[CrossRef]

J. Bohnet, Z. Chen, J. Weiner, D. Meiser, M. Holland, J. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484, 78–81 (2012).
[CrossRef] [PubMed]

H. Zoubi, “Collective light emission of a finite-size atomic chain,” Europhys. Lett. 100, 24002 (2012).
[CrossRef]

L. Ostermann, H. Zoubi, H. Ritsch, “Cascaded collective decay in regular arrays of cold trapped atoms,” Opt. Express 20, 29634–29645 (2012).
[CrossRef]

2011 (1)

M. D. Swallows, M. Bishof, Y. Lin, S. Blatt, M. J. Martin, A. M. Rey, J. Ye, “Suppression of collisional shifts in a strongly interacting lattice clock,” Science 331, 1043–1046 (2011).
[CrossRef] [PubMed]

2010 (1)

K. Henschel, J. Majer, J. Schmiedmayer, H. Ritsch, “Cavity QED with an ultracold ensemble on a chip: Prospects for strong magnetic coupling at finite temperatures,” Phys. Rev. A 82, 033810 (2010).
[CrossRef]

2009 (1)

D. Meiser, J. Ye, D. Carlson, M. Holland, “Prospects for a millihertz-linewidth laser,” Phys. Rev. Lett. 102, 163601 (2009).
[CrossRef] [PubMed]

2008 (1)

G. Campbell, A. Ludlow, S. Blatt, J. Thomsen, M. Martin, M. de Miranda, T. Zelevinsky, M. Boyd, J. Ye, S. Diddams, T. Heavner, T. Parker, S. Jefferts, “The absolute frequency of the 87Sr optical clock transition,” Metrologia 45, 539–548 (2008).
[CrossRef]

2005 (1)

M. Takamoto, F. Hong, R. Higashi, H. Katori, “An optical lattice clock,” Nature 435, 321–324 (2005).
[CrossRef] [PubMed]

2002 (1)

Z. Ficek, R. Tanaś, “Entangled states and collective nonclassical effects in two-atom systems,” Phys. Rep. 372, 369–443 (2002).
[CrossRef]

1999 (2)

S. Inouye, A. Chikkatur, D. Stamper-Kurn, J. Stenger, D. Pritchard, W. Ketterle, “Superradiant Rayleigh scattering from a Bose-Einstein condensate,” Science 285, 571–574 (1999).
[CrossRef] [PubMed]

M. Moore, P. Meystre, “Theory of superradiant scattering of laser light from Bose-Einstein condensates,” Phys. Rev. Lett. 83, 5202–5205 (1999).
[CrossRef]

1995 (1)

P. Horak, K. Gheri, H. Ritsch, “Quantum dynamics of a single-atom cascade laser,” Phys. Rev. A 51, 3257–3266 (1995).
[CrossRef] [PubMed]

1993 (1)

F. Haake, M. I. Kolobov, C. Fabre, E. Giacobino, S. Reynaud, “Superradiant laser,” Phys. Rev. Lett. 71, 995–998 (1993).
[CrossRef] [PubMed]

1987 (1)

Z. Ficek, R. Tanaś, S. Kielich, “Quantum beats and superradiant effects in the spontaneous emission from two nonidentical atoms,” Physica A: Stat. Mech. Appl. 146, 452–482 (1987).
[CrossRef]

1982 (1)

M. Gross, S. Haroche, “Superradiance: An essay on the theory of collective spontaneous emission,” Phys. Rep. 93, 301–396 (1982).
[CrossRef]

1976 (1)

J. MacGillivray, M. Feld, “Theory of superradiance in an extended, optically thick medium,” Phys. Rev. A 14, 1169–1189 (1976).
[CrossRef]

1973 (2)

N. Skribanowitz, I. P. Herman, J. C. MacGillivray, M. S. Feld, “Observation of Dicke superradiance in optically pumped HF gas,” Phys. Rev. Lett. 30, 309–312 (1973).
[CrossRef]

L. Casperson, “Spectral narrowing in double-pass superradiant lasers,” Opt. Commun. 8, 85–87 (1973).
[CrossRef]

1971 (2)

R. Bonifacio, P. Schwendimann, F. Haake, “Quantum statistical theory of superradiance. I,” Phys. Rev. A 4, 302 (1971).
[CrossRef]

N. E. Rehler, J. H. Eberly, “Superradiance,” Phys. Rev. A 3, 1735–1751 (1971).
[CrossRef]

1970 (1)

R. Lehmberg, “Radiation from an N-atom system. I. General formalism,” Phys. Rev. A 2, 883–888 (1970).
[CrossRef]

1954 (1)

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

Bishof, M.

B. Bloom, T. Nicholson, J. Williams, S. Campbell, M. Bishof, X. Zhang, W. Zhang, S. Bromley, J. Ye, “An optical lattice clock with accuracy and stability at the 10−18 level,” Nature 506, 71–75 (2014).
[CrossRef] [PubMed]

M. D. Swallows, M. Bishof, Y. Lin, S. Blatt, M. J. Martin, A. M. Rey, J. Ye, “Suppression of collisional shifts in a strongly interacting lattice clock,” Science 331, 1043–1046 (2011).
[CrossRef] [PubMed]

Blatt, S.

M. D. Swallows, M. Bishof, Y. Lin, S. Blatt, M. J. Martin, A. M. Rey, J. Ye, “Suppression of collisional shifts in a strongly interacting lattice clock,” Science 331, 1043–1046 (2011).
[CrossRef] [PubMed]

G. Campbell, A. Ludlow, S. Blatt, J. Thomsen, M. Martin, M. de Miranda, T. Zelevinsky, M. Boyd, J. Ye, S. Diddams, T. Heavner, T. Parker, S. Jefferts, “The absolute frequency of the 87Sr optical clock transition,” Metrologia 45, 539–548 (2008).
[CrossRef]

Bloom, B.

B. Bloom, T. Nicholson, J. Williams, S. Campbell, M. Bishof, X. Zhang, W. Zhang, S. Bromley, J. Ye, “An optical lattice clock with accuracy and stability at the 10−18 level,” Nature 506, 71–75 (2014).
[CrossRef] [PubMed]

Bohnet, J.

J. Bohnet, Z. Chen, J. Weiner, D. Meiser, M. Holland, J. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484, 78–81 (2012).
[CrossRef] [PubMed]

Bohnet, J. G.

J. G. Bohnet, Z. Chen, J. M. Weiner, K. C. Cox, J. K. Thompson, “Active and passive sensing of collective atomic coherence in a superradiant laser,” Phys. Rev. A 88, 013826 (2013).
[CrossRef]

Bonifacio, R.

R. Bonifacio, P. Schwendimann, F. Haake, “Quantum statistical theory of superradiance. I,” Phys. Rev. A 4, 302 (1971).
[CrossRef]

Boyd, M.

G. Campbell, A. Ludlow, S. Blatt, J. Thomsen, M. Martin, M. de Miranda, T. Zelevinsky, M. Boyd, J. Ye, S. Diddams, T. Heavner, T. Parker, S. Jefferts, “The absolute frequency of the 87Sr optical clock transition,” Metrologia 45, 539–548 (2008).
[CrossRef]

Bromley, S.

B. Bloom, T. Nicholson, J. Williams, S. Campbell, M. Bishof, X. Zhang, W. Zhang, S. Bromley, J. Ye, “An optical lattice clock with accuracy and stability at the 10−18 level,” Nature 506, 71–75 (2014).
[CrossRef] [PubMed]

Campbell, G.

G. Campbell, A. Ludlow, S. Blatt, J. Thomsen, M. Martin, M. de Miranda, T. Zelevinsky, M. Boyd, J. Ye, S. Diddams, T. Heavner, T. Parker, S. Jefferts, “The absolute frequency of the 87Sr optical clock transition,” Metrologia 45, 539–548 (2008).
[CrossRef]

Campbell, S.

B. Bloom, T. Nicholson, J. Williams, S. Campbell, M. Bishof, X. Zhang, W. Zhang, S. Bromley, J. Ye, “An optical lattice clock with accuracy and stability at the 10−18 level,” Nature 506, 71–75 (2014).
[CrossRef] [PubMed]

Carlson, D.

D. Meiser, J. Ye, D. Carlson, M. Holland, “Prospects for a millihertz-linewidth laser,” Phys. Rev. Lett. 102, 163601 (2009).
[CrossRef] [PubMed]

Casperson, L.

L. Casperson, “Spectral narrowing in double-pass superradiant lasers,” Opt. Commun. 8, 85–87 (1973).
[CrossRef]

Chen, L.

T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. Martin, L. Chen, J. Ye, “A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6, 687–692 (2012).
[CrossRef]

Chen, Z.

J. G. Bohnet, Z. Chen, J. M. Weiner, K. C. Cox, J. K. Thompson, “Active and passive sensing of collective atomic coherence in a superradiant laser,” Phys. Rev. A 88, 013826 (2013).
[CrossRef]

J. Bohnet, Z. Chen, J. Weiner, D. Meiser, M. Holland, J. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484, 78–81 (2012).
[CrossRef] [PubMed]

Chikkatur, A.

S. Inouye, A. Chikkatur, D. Stamper-Kurn, J. Stenger, D. Pritchard, W. Ketterle, “Superradiant Rayleigh scattering from a Bose-Einstein condensate,” Science 285, 571–574 (1999).
[CrossRef] [PubMed]

Cox, K. C.

J. G. Bohnet, Z. Chen, J. M. Weiner, K. C. Cox, J. K. Thompson, “Active and passive sensing of collective atomic coherence in a superradiant laser,” Phys. Rev. A 88, 013826 (2013).
[CrossRef]

de Miranda, M.

G. Campbell, A. Ludlow, S. Blatt, J. Thomsen, M. Martin, M. de Miranda, T. Zelevinsky, M. Boyd, J. Ye, S. Diddams, T. Heavner, T. Parker, S. Jefferts, “The absolute frequency of the 87Sr optical clock transition,” Metrologia 45, 539–548 (2008).
[CrossRef]

Dicke, R. H.

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

Diddams, S.

G. Campbell, A. Ludlow, S. Blatt, J. Thomsen, M. Martin, M. de Miranda, T. Zelevinsky, M. Boyd, J. Ye, S. Diddams, T. Heavner, T. Parker, S. Jefferts, “The absolute frequency of the 87Sr optical clock transition,” Metrologia 45, 539–548 (2008).
[CrossRef]

Eberly, J. H.

N. E. Rehler, J. H. Eberly, “Superradiance,” Phys. Rev. A 3, 1735–1751 (1971).
[CrossRef]

Fabre, C.

F. Haake, M. I. Kolobov, C. Fabre, E. Giacobino, S. Reynaud, “Superradiant laser,” Phys. Rev. Lett. 71, 995–998 (1993).
[CrossRef] [PubMed]

Feld, M.

J. MacGillivray, M. Feld, “Theory of superradiance in an extended, optically thick medium,” Phys. Rev. A 14, 1169–1189 (1976).
[CrossRef]

Feld, M. S.

N. Skribanowitz, I. P. Herman, J. C. MacGillivray, M. S. Feld, “Observation of Dicke superradiance in optically pumped HF gas,” Phys. Rev. Lett. 30, 309–312 (1973).
[CrossRef]

Ficek, Z.

Z. Ficek, R. Tanaś, “Entangled states and collective nonclassical effects in two-atom systems,” Phys. Rep. 372, 369–443 (2002).
[CrossRef]

Z. Ficek, R. Tanaś, S. Kielich, “Quantum beats and superradiant effects in the spontaneous emission from two nonidentical atoms,” Physica A: Stat. Mech. Appl. 146, 452–482 (1987).
[CrossRef]

Gheri, K.

P. Horak, K. Gheri, H. Ritsch, “Quantum dynamics of a single-atom cascade laser,” Phys. Rev. A 51, 3257–3266 (1995).
[CrossRef] [PubMed]

Giacobino, E.

F. Haake, M. I. Kolobov, C. Fabre, E. Giacobino, S. Reynaud, “Superradiant laser,” Phys. Rev. Lett. 71, 995–998 (1993).
[CrossRef] [PubMed]

Grebing, C.

T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. Martin, L. Chen, J. Ye, “A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6, 687–692 (2012).
[CrossRef]

Gross, M.

M. Gross, S. Haroche, “Superradiance: An essay on the theory of collective spontaneous emission,” Phys. Rep. 93, 301–396 (1982).
[CrossRef]

Haake, F.

F. Haake, M. I. Kolobov, C. Fabre, E. Giacobino, S. Reynaud, “Superradiant laser,” Phys. Rev. Lett. 71, 995–998 (1993).
[CrossRef] [PubMed]

R. Bonifacio, P. Schwendimann, F. Haake, “Quantum statistical theory of superradiance. I,” Phys. Rev. A 4, 302 (1971).
[CrossRef]

Hagemann, C.

T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. Martin, L. Chen, J. Ye, “A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6, 687–692 (2012).
[CrossRef]

Haroche, S.

M. Gross, S. Haroche, “Superradiance: An essay on the theory of collective spontaneous emission,” Phys. Rep. 93, 301–396 (1982).
[CrossRef]

Heavner, T.

G. Campbell, A. Ludlow, S. Blatt, J. Thomsen, M. Martin, M. de Miranda, T. Zelevinsky, M. Boyd, J. Ye, S. Diddams, T. Heavner, T. Parker, S. Jefferts, “The absolute frequency of the 87Sr optical clock transition,” Metrologia 45, 539–548 (2008).
[CrossRef]

Henschel, K.

K. Henschel, J. Majer, J. Schmiedmayer, H. Ritsch, “Cavity QED with an ultracold ensemble on a chip: Prospects for strong magnetic coupling at finite temperatures,” Phys. Rev. A 82, 033810 (2010).
[CrossRef]

Herman, I. P.

N. Skribanowitz, I. P. Herman, J. C. MacGillivray, M. S. Feld, “Observation of Dicke superradiance in optically pumped HF gas,” Phys. Rev. Lett. 30, 309–312 (1973).
[CrossRef]

Higashi, R.

M. Takamoto, F. Hong, R. Higashi, H. Katori, “An optical lattice clock,” Nature 435, 321–324 (2005).
[CrossRef] [PubMed]

Holland, M.

J. Bohnet, Z. Chen, J. Weiner, D. Meiser, M. Holland, J. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484, 78–81 (2012).
[CrossRef] [PubMed]

D. Meiser, J. Ye, D. Carlson, M. Holland, “Prospects for a millihertz-linewidth laser,” Phys. Rev. Lett. 102, 163601 (2009).
[CrossRef] [PubMed]

M. Xu, D. Tieri, M. Holland, “Simulating Open Quantum Systems using the Simple Lie Group SU (4),” arXiv preprint arXiv:1302.6284 (2013).

Hong, F.

M. Takamoto, F. Hong, R. Higashi, H. Katori, “An optical lattice clock,” Nature 435, 321–324 (2005).
[CrossRef] [PubMed]

Horak, P.

P. Horak, K. Gheri, H. Ritsch, “Quantum dynamics of a single-atom cascade laser,” Phys. Rev. A 51, 3257–3266 (1995).
[CrossRef] [PubMed]

Inouye, S.

S. Inouye, A. Chikkatur, D. Stamper-Kurn, J. Stenger, D. Pritchard, W. Ketterle, “Superradiant Rayleigh scattering from a Bose-Einstein condensate,” Science 285, 571–574 (1999).
[CrossRef] [PubMed]

Jefferts, S.

G. Campbell, A. Ludlow, S. Blatt, J. Thomsen, M. Martin, M. de Miranda, T. Zelevinsky, M. Boyd, J. Ye, S. Diddams, T. Heavner, T. Parker, S. Jefferts, “The absolute frequency of the 87Sr optical clock transition,” Metrologia 45, 539–548 (2008).
[CrossRef]

Katori, H.

M. Takamoto, F. Hong, R. Higashi, H. Katori, “An optical lattice clock,” Nature 435, 321–324 (2005).
[CrossRef] [PubMed]

Kessler, T.

T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. Martin, L. Chen, J. Ye, “A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6, 687–692 (2012).
[CrossRef]

Ketterle, W.

S. Inouye, A. Chikkatur, D. Stamper-Kurn, J. Stenger, D. Pritchard, W. Ketterle, “Superradiant Rayleigh scattering from a Bose-Einstein condensate,” Science 285, 571–574 (1999).
[CrossRef] [PubMed]

Kielich, S.

Z. Ficek, R. Tanaś, S. Kielich, “Quantum beats and superradiant effects in the spontaneous emission from two nonidentical atoms,” Physica A: Stat. Mech. Appl. 146, 452–482 (1987).
[CrossRef]

Kolobov, M. I.

F. Haake, M. I. Kolobov, C. Fabre, E. Giacobino, S. Reynaud, “Superradiant laser,” Phys. Rev. Lett. 71, 995–998 (1993).
[CrossRef] [PubMed]

Legero, T.

T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. Martin, L. Chen, J. Ye, “A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6, 687–692 (2012).
[CrossRef]

Lehmberg, R.

R. Lehmberg, “Radiation from an N-atom system. I. General formalism,” Phys. Rev. A 2, 883–888 (1970).
[CrossRef]

Lin, Y.

M. D. Swallows, M. Bishof, Y. Lin, S. Blatt, M. J. Martin, A. M. Rey, J. Ye, “Suppression of collisional shifts in a strongly interacting lattice clock,” Science 331, 1043–1046 (2011).
[CrossRef] [PubMed]

Ludlow, A.

G. Campbell, A. Ludlow, S. Blatt, J. Thomsen, M. Martin, M. de Miranda, T. Zelevinsky, M. Boyd, J. Ye, S. Diddams, T. Heavner, T. Parker, S. Jefferts, “The absolute frequency of the 87Sr optical clock transition,” Metrologia 45, 539–548 (2008).
[CrossRef]

MacGillivray, J.

J. MacGillivray, M. Feld, “Theory of superradiance in an extended, optically thick medium,” Phys. Rev. A 14, 1169–1189 (1976).
[CrossRef]

MacGillivray, J. C.

N. Skribanowitz, I. P. Herman, J. C. MacGillivray, M. S. Feld, “Observation of Dicke superradiance in optically pumped HF gas,” Phys. Rev. Lett. 30, 309–312 (1973).
[CrossRef]

Majer, J.

K. Henschel, J. Majer, J. Schmiedmayer, H. Ritsch, “Cavity QED with an ultracold ensemble on a chip: Prospects for strong magnetic coupling at finite temperatures,” Phys. Rev. A 82, 033810 (2010).
[CrossRef]

Martin, M.

T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. Martin, L. Chen, J. Ye, “A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6, 687–692 (2012).
[CrossRef]

G. Campbell, A. Ludlow, S. Blatt, J. Thomsen, M. Martin, M. de Miranda, T. Zelevinsky, M. Boyd, J. Ye, S. Diddams, T. Heavner, T. Parker, S. Jefferts, “The absolute frequency of the 87Sr optical clock transition,” Metrologia 45, 539–548 (2008).
[CrossRef]

Martin, M. J.

M. D. Swallows, M. Bishof, Y. Lin, S. Blatt, M. J. Martin, A. M. Rey, J. Ye, “Suppression of collisional shifts in a strongly interacting lattice clock,” Science 331, 1043–1046 (2011).
[CrossRef] [PubMed]

Meiser, D.

J. Bohnet, Z. Chen, J. Weiner, D. Meiser, M. Holland, J. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484, 78–81 (2012).
[CrossRef] [PubMed]

D. Meiser, J. Ye, D. Carlson, M. Holland, “Prospects for a millihertz-linewidth laser,” Phys. Rev. Lett. 102, 163601 (2009).
[CrossRef] [PubMed]

Meystre, P.

M. Moore, P. Meystre, “Theory of superradiant scattering of laser light from Bose-Einstein condensates,” Phys. Rev. Lett. 83, 5202–5205 (1999).
[CrossRef]

P. Meystre, M. Sargent, Elements of Quantum Optics (Springer-Verlag, 1990).

Moore, M.

M. Moore, P. Meystre, “Theory of superradiant scattering of laser light from Bose-Einstein condensates,” Phys. Rev. Lett. 83, 5202–5205 (1999).
[CrossRef]

Nicholson, T.

B. Bloom, T. Nicholson, J. Williams, S. Campbell, M. Bishof, X. Zhang, W. Zhang, S. Bromley, J. Ye, “An optical lattice clock with accuracy and stability at the 10−18 level,” Nature 506, 71–75 (2014).
[CrossRef] [PubMed]

Ostermann, L.

Parker, T.

G. Campbell, A. Ludlow, S. Blatt, J. Thomsen, M. Martin, M. de Miranda, T. Zelevinsky, M. Boyd, J. Ye, S. Diddams, T. Heavner, T. Parker, S. Jefferts, “The absolute frequency of the 87Sr optical clock transition,” Metrologia 45, 539–548 (2008).
[CrossRef]

Pritchard, D.

S. Inouye, A. Chikkatur, D. Stamper-Kurn, J. Stenger, D. Pritchard, W. Ketterle, “Superradiant Rayleigh scattering from a Bose-Einstein condensate,” Science 285, 571–574 (1999).
[CrossRef] [PubMed]

Rehler, N. E.

N. E. Rehler, J. H. Eberly, “Superradiance,” Phys. Rev. A 3, 1735–1751 (1971).
[CrossRef]

Rey, A. M.

M. D. Swallows, M. Bishof, Y. Lin, S. Blatt, M. J. Martin, A. M. Rey, J. Ye, “Suppression of collisional shifts in a strongly interacting lattice clock,” Science 331, 1043–1046 (2011).
[CrossRef] [PubMed]

Reynaud, S.

F. Haake, M. I. Kolobov, C. Fabre, E. Giacobino, S. Reynaud, “Superradiant laser,” Phys. Rev. Lett. 71, 995–998 (1993).
[CrossRef] [PubMed]

Riehle, F.

T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. Martin, L. Chen, J. Ye, “A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6, 687–692 (2012).
[CrossRef]

Ritsch, H.

L. Ostermann, H. Zoubi, H. Ritsch, “Cascaded collective decay in regular arrays of cold trapped atoms,” Opt. Express 20, 29634–29645 (2012).
[CrossRef]

K. Henschel, J. Majer, J. Schmiedmayer, H. Ritsch, “Cavity QED with an ultracold ensemble on a chip: Prospects for strong magnetic coupling at finite temperatures,” Phys. Rev. A 82, 033810 (2010).
[CrossRef]

P. Horak, K. Gheri, H. Ritsch, “Quantum dynamics of a single-atom cascade laser,” Phys. Rev. A 51, 3257–3266 (1995).
[CrossRef] [PubMed]

Sargent, M.

P. Meystre, M. Sargent, Elements of Quantum Optics (Springer-Verlag, 1990).

Schmiedmayer, J.

K. Henschel, J. Majer, J. Schmiedmayer, H. Ritsch, “Cavity QED with an ultracold ensemble on a chip: Prospects for strong magnetic coupling at finite temperatures,” Phys. Rev. A 82, 033810 (2010).
[CrossRef]

Schwendimann, P.

R. Bonifacio, P. Schwendimann, F. Haake, “Quantum statistical theory of superradiance. I,” Phys. Rev. A 4, 302 (1971).
[CrossRef]

Skribanowitz, N.

N. Skribanowitz, I. P. Herman, J. C. MacGillivray, M. S. Feld, “Observation of Dicke superradiance in optically pumped HF gas,” Phys. Rev. Lett. 30, 309–312 (1973).
[CrossRef]

Stamper-Kurn, D.

S. Inouye, A. Chikkatur, D. Stamper-Kurn, J. Stenger, D. Pritchard, W. Ketterle, “Superradiant Rayleigh scattering from a Bose-Einstein condensate,” Science 285, 571–574 (1999).
[CrossRef] [PubMed]

Stenger, J.

S. Inouye, A. Chikkatur, D. Stamper-Kurn, J. Stenger, D. Pritchard, W. Ketterle, “Superradiant Rayleigh scattering from a Bose-Einstein condensate,” Science 285, 571–574 (1999).
[CrossRef] [PubMed]

Sterr, U.

T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. Martin, L. Chen, J. Ye, “A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6, 687–692 (2012).
[CrossRef]

Swallows, M. D.

M. D. Swallows, M. Bishof, Y. Lin, S. Blatt, M. J. Martin, A. M. Rey, J. Ye, “Suppression of collisional shifts in a strongly interacting lattice clock,” Science 331, 1043–1046 (2011).
[CrossRef] [PubMed]

Takamoto, M.

M. Takamoto, F. Hong, R. Higashi, H. Katori, “An optical lattice clock,” Nature 435, 321–324 (2005).
[CrossRef] [PubMed]

Tanas, R.

Z. Ficek, R. Tanaś, “Entangled states and collective nonclassical effects in two-atom systems,” Phys. Rep. 372, 369–443 (2002).
[CrossRef]

Z. Ficek, R. Tanaś, S. Kielich, “Quantum beats and superradiant effects in the spontaneous emission from two nonidentical atoms,” Physica A: Stat. Mech. Appl. 146, 452–482 (1987).
[CrossRef]

Thompson, J.

J. Bohnet, Z. Chen, J. Weiner, D. Meiser, M. Holland, J. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484, 78–81 (2012).
[CrossRef] [PubMed]

Thompson, J. K.

J. G. Bohnet, Z. Chen, J. M. Weiner, K. C. Cox, J. K. Thompson, “Active and passive sensing of collective atomic coherence in a superradiant laser,” Phys. Rev. A 88, 013826 (2013).
[CrossRef]

Thomsen, J.

G. Campbell, A. Ludlow, S. Blatt, J. Thomsen, M. Martin, M. de Miranda, T. Zelevinsky, M. Boyd, J. Ye, S. Diddams, T. Heavner, T. Parker, S. Jefferts, “The absolute frequency of the 87Sr optical clock transition,” Metrologia 45, 539–548 (2008).
[CrossRef]

Tieri, D.

M. Xu, D. Tieri, M. Holland, “Simulating Open Quantum Systems using the Simple Lie Group SU (4),” arXiv preprint arXiv:1302.6284 (2013).

Vuletic, V.

V. Vuletic, “Atomic physics: An almost lightless laser,” Nature 484, 43–44 (2012).
[CrossRef] [PubMed]

Weiner, J.

J. Bohnet, Z. Chen, J. Weiner, D. Meiser, M. Holland, J. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484, 78–81 (2012).
[CrossRef] [PubMed]

Weiner, J. M.

J. G. Bohnet, Z. Chen, J. M. Weiner, K. C. Cox, J. K. Thompson, “Active and passive sensing of collective atomic coherence in a superradiant laser,” Phys. Rev. A 88, 013826 (2013).
[CrossRef]

Williams, J.

B. Bloom, T. Nicholson, J. Williams, S. Campbell, M. Bishof, X. Zhang, W. Zhang, S. Bromley, J. Ye, “An optical lattice clock with accuracy and stability at the 10−18 level,” Nature 506, 71–75 (2014).
[CrossRef] [PubMed]

Xu, M.

M. Xu, D. Tieri, M. Holland, “Simulating Open Quantum Systems using the Simple Lie Group SU (4),” arXiv preprint arXiv:1302.6284 (2013).

Ye, J.

B. Bloom, T. Nicholson, J. Williams, S. Campbell, M. Bishof, X. Zhang, W. Zhang, S. Bromley, J. Ye, “An optical lattice clock with accuracy and stability at the 10−18 level,” Nature 506, 71–75 (2014).
[CrossRef] [PubMed]

T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. Martin, L. Chen, J. Ye, “A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6, 687–692 (2012).
[CrossRef]

M. D. Swallows, M. Bishof, Y. Lin, S. Blatt, M. J. Martin, A. M. Rey, J. Ye, “Suppression of collisional shifts in a strongly interacting lattice clock,” Science 331, 1043–1046 (2011).
[CrossRef] [PubMed]

D. Meiser, J. Ye, D. Carlson, M. Holland, “Prospects for a millihertz-linewidth laser,” Phys. Rev. Lett. 102, 163601 (2009).
[CrossRef] [PubMed]

G. Campbell, A. Ludlow, S. Blatt, J. Thomsen, M. Martin, M. de Miranda, T. Zelevinsky, M. Boyd, J. Ye, S. Diddams, T. Heavner, T. Parker, S. Jefferts, “The absolute frequency of the 87Sr optical clock transition,” Metrologia 45, 539–548 (2008).
[CrossRef]

Zelevinsky, T.

G. Campbell, A. Ludlow, S. Blatt, J. Thomsen, M. Martin, M. de Miranda, T. Zelevinsky, M. Boyd, J. Ye, S. Diddams, T. Heavner, T. Parker, S. Jefferts, “The absolute frequency of the 87Sr optical clock transition,” Metrologia 45, 539–548 (2008).
[CrossRef]

Zhang, W.

B. Bloom, T. Nicholson, J. Williams, S. Campbell, M. Bishof, X. Zhang, W. Zhang, S. Bromley, J. Ye, “An optical lattice clock with accuracy and stability at the 10−18 level,” Nature 506, 71–75 (2014).
[CrossRef] [PubMed]

Zhang, X.

B. Bloom, T. Nicholson, J. Williams, S. Campbell, M. Bishof, X. Zhang, W. Zhang, S. Bromley, J. Ye, “An optical lattice clock with accuracy and stability at the 10−18 level,” Nature 506, 71–75 (2014).
[CrossRef] [PubMed]

Zoubi, H.

Europhys. Lett. (1)

H. Zoubi, “Collective light emission of a finite-size atomic chain,” Europhys. Lett. 100, 24002 (2012).
[CrossRef]

Metrologia (1)

G. Campbell, A. Ludlow, S. Blatt, J. Thomsen, M. Martin, M. de Miranda, T. Zelevinsky, M. Boyd, J. Ye, S. Diddams, T. Heavner, T. Parker, S. Jefferts, “The absolute frequency of the 87Sr optical clock transition,” Metrologia 45, 539–548 (2008).
[CrossRef]

Nat. Photonics (1)

T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. Martin, L. Chen, J. Ye, “A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6, 687–692 (2012).
[CrossRef]

Nature (4)

J. Bohnet, Z. Chen, J. Weiner, D. Meiser, M. Holland, J. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484, 78–81 (2012).
[CrossRef] [PubMed]

V. Vuletic, “Atomic physics: An almost lightless laser,” Nature 484, 43–44 (2012).
[CrossRef] [PubMed]

B. Bloom, T. Nicholson, J. Williams, S. Campbell, M. Bishof, X. Zhang, W. Zhang, S. Bromley, J. Ye, “An optical lattice clock with accuracy and stability at the 10−18 level,” Nature 506, 71–75 (2014).
[CrossRef] [PubMed]

M. Takamoto, F. Hong, R. Higashi, H. Katori, “An optical lattice clock,” Nature 435, 321–324 (2005).
[CrossRef] [PubMed]

Opt. Commun. (1)

L. Casperson, “Spectral narrowing in double-pass superradiant lasers,” Opt. Commun. 8, 85–87 (1973).
[CrossRef]

Opt. Express (1)

Phys. Rep. (2)

Z. Ficek, R. Tanaś, “Entangled states and collective nonclassical effects in two-atom systems,” Phys. Rep. 372, 369–443 (2002).
[CrossRef]

M. Gross, S. Haroche, “Superradiance: An essay on the theory of collective spontaneous emission,” Phys. Rep. 93, 301–396 (1982).
[CrossRef]

Phys. Rev. (1)

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

Phys. Rev. A (7)

R. Lehmberg, “Radiation from an N-atom system. I. General formalism,” Phys. Rev. A 2, 883–888 (1970).
[CrossRef]

R. Bonifacio, P. Schwendimann, F. Haake, “Quantum statistical theory of superradiance. I,” Phys. Rev. A 4, 302 (1971).
[CrossRef]

N. E. Rehler, J. H. Eberly, “Superradiance,” Phys. Rev. A 3, 1735–1751 (1971).
[CrossRef]

P. Horak, K. Gheri, H. Ritsch, “Quantum dynamics of a single-atom cascade laser,” Phys. Rev. A 51, 3257–3266 (1995).
[CrossRef] [PubMed]

J. MacGillivray, M. Feld, “Theory of superradiance in an extended, optically thick medium,” Phys. Rev. A 14, 1169–1189 (1976).
[CrossRef]

K. Henschel, J. Majer, J. Schmiedmayer, H. Ritsch, “Cavity QED with an ultracold ensemble on a chip: Prospects for strong magnetic coupling at finite temperatures,” Phys. Rev. A 82, 033810 (2010).
[CrossRef]

J. G. Bohnet, Z. Chen, J. M. Weiner, K. C. Cox, J. K. Thompson, “Active and passive sensing of collective atomic coherence in a superradiant laser,” Phys. Rev. A 88, 013826 (2013).
[CrossRef]

Phys. Rev. Lett. (4)

D. Meiser, J. Ye, D. Carlson, M. Holland, “Prospects for a millihertz-linewidth laser,” Phys. Rev. Lett. 102, 163601 (2009).
[CrossRef] [PubMed]

F. Haake, M. I. Kolobov, C. Fabre, E. Giacobino, S. Reynaud, “Superradiant laser,” Phys. Rev. Lett. 71, 995–998 (1993).
[CrossRef] [PubMed]

N. Skribanowitz, I. P. Herman, J. C. MacGillivray, M. S. Feld, “Observation of Dicke superradiance in optically pumped HF gas,” Phys. Rev. Lett. 30, 309–312 (1973).
[CrossRef]

M. Moore, P. Meystre, “Theory of superradiant scattering of laser light from Bose-Einstein condensates,” Phys. Rev. Lett. 83, 5202–5205 (1999).
[CrossRef]

Physica A: Stat. Mech. Appl. (1)

Z. Ficek, R. Tanaś, S. Kielich, “Quantum beats and superradiant effects in the spontaneous emission from two nonidentical atoms,” Physica A: Stat. Mech. Appl. 146, 452–482 (1987).
[CrossRef]

Science (2)

S. Inouye, A. Chikkatur, D. Stamper-Kurn, J. Stenger, D. Pritchard, W. Ketterle, “Superradiant Rayleigh scattering from a Bose-Einstein condensate,” Science 285, 571–574 (1999).
[CrossRef] [PubMed]

M. D. Swallows, M. Bishof, Y. Lin, S. Blatt, M. J. Martin, A. M. Rey, J. Ye, “Suppression of collisional shifts in a strongly interacting lattice clock,” Science 331, 1043–1046 (2011).
[CrossRef] [PubMed]

Other (2)

M. Xu, D. Tieri, M. Holland, “Simulating Open Quantum Systems using the Simple Lie Group SU (4),” arXiv preprint arXiv:1302.6284 (2013).

P. Meystre, M. Sargent, Elements of Quantum Optics (Springer-Verlag, 1990).

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Figures (10)

Fig. 1
Fig. 1

Schematics of a lattice laser setup. A transversely pumped (pumping rate R) finite atomic ensemble with dipole-dipole couplings Ωij and collective spontaneous emission Γij inside an optical resonator with a loss rate of κ

Fig. 2
Fig. 2

Stationary photon number as a function of the pump strength R and the spontaneous decay rate Γ for collectively pumped and collectively decaying atoms (a), individually pumped but collectively decaying atoms (b) and individually pumped and individually decaying atoms (c)

Fig. 3
Fig. 3

Output spectrum of a fully collective laser with different atom numbers N for Γ = κ/20 and R = κ/5 compared to the empty cavity linewidth (N = 0), absolute (a) and normalized (b)

Fig. 4
Fig. 4

Stationary operation of a four atom laser on a square lattice. (a) photon number, (b) atomic inversion, where the black line indicates equal population of the excited and the ground state, (c) g2(0) function, with the white line at g(2)(0) = 1 representing a coherent state

Fig. 5
Fig. 5

Photon number (a), atomic inversion (b) and g2 function (c) of the laser as a function of the pump strength R for different atomic arrangements and a fixed spontaneous decay rate Γ = 0.2κ

Fig. 6
Fig. 6

Photon number (a), atomic inversion (b) and g2 function (c) of the laser as a function of the pump strength R for a square of different lattice constants d and a fixed spontaneous decay rate Γ = 0.2κ

Fig. 7
Fig. 7

Laser linewidth (a) and frequency shift for a square atom arrangement at different distances as a function of the pump strength for a fixed atomic decay rate of Γ = 0.2κ

Fig. 8
Fig. 8

Laser linewidth (a) and frequency shift (b) for different geometric configurations but same lattice constant (d = λ0/10) as a function of the pump strength for a fixed atomic decay rate of Γ = 0.2κ

Fig. 9
Fig. 9

Average photon number for atoms on a square with d = λ0/10 (a) and d = λmagic/2 (b) for variable cavity detuning and an atomic decay rate Γ = 0.2κ

Fig. 10
Fig. 10

Frequency shift for a square atom configuration with d = λ0/10 (a) and d = λmagic/2 (b) for variable detuning and a fixed atomic decay rate of Γ = 0.2κ. The dashed line represents δa/Γ = −1 and the solid line corresponds to δa/Γ = 1

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

ρ t = i [ ρ , H ] + cd [ ρ ] + pump [ ρ ] + cav [ ρ ] = [ ρ ] ,
H = ω 0 2 i σ i z + i j Ω i j σ i + σ j + ω c a a + H int ,
H int = g i ( a σ i + + a σ i )
cd [ ρ ] = 1 2 i , j Γ i j ( 2 σ i ρ σ j + σ i + σ j ρ ρ σ i + σ j )
pump [ ρ ] = R 2 i ( 2 σ i + ρ σ i σ i σ i + ρ ρ σ i σ i + )
cav [ ρ ] = κ ( 2 a ρ a a a ρ ρ a a ) .
Γ i j = 3 Γ 2 F ( k 0 r i j ) Ω i j = 3 Γ 4 G ( k 0 r i j )
F ( ξ ) = ( 1 cos 2 θ ) sin ξ ξ + ( 1 3 cos 2 θ ) ( cos ξ ξ 2 sin ξ ξ 3 ) , G ( ξ ) = ( 1 cos 2 θ ) cos ξ ξ + ( 1 3 cos 2 θ ) ( sin ξ ξ 2 + cos ξ ξ 3 ) ,
S ( ω , t ) = e i ω τ a ( t + τ ) a ( t ) d τ .

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