Abstract

A cold atomic gas with an inverted population on a transition coupled to a field mode of an optical resonator constitutes a generic model of a laser. For quasi-continuous operation, external pumping, trapping and cooling of the atoms is required to confine them in order to achieve enough gain inside the resonator. As inverted atoms are high-field seekers in blue detuned light fields, tuning the cavity mode to the blue side of the atomic gain transition allows for combining lasing with stimulated cavity cooling and dipole trapping of the atoms at the antinodes of the laser field. We study such a configuration using a semiclassical description of particle motion along the cavity axis. In extension of earlier work we include free space atomic and cavity decay as well as atomic dipole-dipole interactions and their corresponding forces. We show that for a proper choice of parameters even in the bad cavity limit the atoms can create a sufficiently strong field inside the resonator such that they are trapped and cooled via the superradiant lasing action with less than one photon on average inside the cavity.

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  11. A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112(6), 1940–1949 (1958).
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  13. T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. J. Martin, L. Chen, and J. Ye, “A sub-40-mhz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6(10), 687–692 (2012).
    [Crossref]
  14. D. Meiser, J. Ye, D. R. Carlson, and M. J. Holland, “Prospects for a millihertz-linewidth laser,” Phys. Rev. Lett. 102(16), 163601 (2009).
    [Crossref]
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  18. M. Xu, D. A. Tieri, and M. J. Holland, “Simulating open quantum systems by applying su(4) to quantum master equations,” Phys. Rev. A 87(6), 062101 (2013).
    [Crossref]
  19. K. Henschel, J. Majer, J. Schmiedmayer, and H. Ritsch, “Cavity qed with an ultracold ensemble on a chip: Prospects for strong magnetic coupling at finite temperatures,” Phys. Rev. A 82(3), 033810 (2010).
    [Crossref]
  20. R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93(1), 99–110 (1954).
    [Crossref]
  21. T. Salzburger and H. Ritsch, “Atomic self-trapping induced by single-atom lasing,” Phys. Rev. Lett. 93(6), 063002 (2004).
    [Crossref]
  22. T. Salzburger and H. Ritsch, “Lasing and cooling in a finite-temperature cavity,” Phys. Rev. A 74(3), 033806 (2006).
    [Crossref]
  23. M. Xu, S. B. Jäger, S. Schütz, J. Cooper, G. Morigi, and M. J. Holland, “Supercooling of atoms in an optical resonator,” Phys. Rev. Lett. 116(15), 153002 (2016).
    [Crossref]
  24. R. Lehmberg, “Radiation from an N-atom system. I. General formalism,” Phys. Rev. A 2(3), 883–888 (1970).
    [Crossref]
  25. Z. Ficek and R. Tanaś, “Entangled states and collective nonclassical effects in two-atom systems,” Phys. Rep. 372(5), 369–443 (2002).
    [Crossref]
  26. H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger, “Cold atoms in cavity-generated dynamical optical potentials,” Rev. Mod. Phys. 85(2), 553–601 (2013).
    [Crossref]
  27. R. R. Puri, Mathematical Methods of Quantum Optics (Springer Berlin Heidelberg, 2001).
  28. S. Krämer, D. Plankensteiner, L. Ostermann, and H. Ritsch, “Quantumoptics. jl: A julia framework for simulating open quantum systems,” Comput. Phys. Commun. 227, 109–116 (2018).
    [Crossref]

2018 (2)

M. A. Norcia, R. J. Lewis-Swan, J. R. Cline, B. Zhu, A. M. Rey, and J. K. Thompson, “Cavity-mediated collective spin-exchange interactions in a strontium superradiant laser,” Science 361(6399), 259–262 (2018).
[Crossref]

S. Krämer, D. Plankensteiner, L. Ostermann, and H. Ritsch, “Quantumoptics. jl: A julia framework for simulating open quantum systems,” Comput. Phys. Commun. 227, 109–116 (2018).
[Crossref]

2017 (2)

S. L. Campbell, R. Hutson, G. Marti, A. Goban, N. D. Oppong, R. McNally, L. Sonderhouse, J. Robinson, W. Zhang, B. Bloom, and J. Ye, “A fermi-degenerate three-dimensional optical lattice clock,” Science 358(6359), 90–94 (2017).
[Crossref]

J. M. Weiner, K. C. Cox, J. G. Bohnet, and J. K. Thompson, “Phase synchronization inside a superradiant laser,” Phys. Rev. A 95(3), 033808 (2017).
[Crossref]

2016 (2)

M. A. Norcia, M. N. Winchester, J. R. Cline, and J. K. Thompson, “Superradiance on the millihertz linewidth strontium clock transition,” Sci. Adv. 2(10), e1601231 (2016).
[Crossref]

M. Xu, S. B. Jäger, S. Schütz, J. Cooper, G. Morigi, and M. J. Holland, “Supercooling of atoms in an optical resonator,” Phys. Rev. Lett. 116(15), 153002 (2016).
[Crossref]

2014 (1)

2013 (3)

M. Xu, D. A. Tieri, and M. J. Holland, “Simulating open quantum systems by applying su(4) to quantum master equations,” Phys. Rev. A 87(6), 062101 (2013).
[Crossref]

J. G. Bohnet, Z. Chen, J. M. Weiner, K. C. Cox, and J. K. Thompson, “Active and passive sensing of collective atomic coherence in a superradiant laser,” Phys. Rev. A 88(1), 013826 (2013).
[Crossref]

H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger, “Cold atoms in cavity-generated dynamical optical potentials,” Rev. Mod. Phys. 85(2), 553–601 (2013).
[Crossref]

2012 (3)

V. Vuletic, “An almost lightless laser,” Nature 484(7392), 43–44 (2012).
[Crossref]

T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. J. Martin, L. Chen, and J. Ye, “A sub-40-mhz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6(10), 687–692 (2012).
[Crossref]

J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, and J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484(7392), 78–81 (2012).
[Crossref]

2010 (2)

D. Meiser and M. Holland, “Steady-state superradiance with alkaline-earth-metal atoms,” Phys. Rev. A 81(3), 033847 (2010).
[Crossref]

K. Henschel, J. Majer, J. Schmiedmayer, and H. Ritsch, “Cavity qed with an ultracold ensemble on a chip: Prospects for strong magnetic coupling at finite temperatures,” Phys. Rev. A 82(3), 033810 (2010).
[Crossref]

2009 (1)

D. Meiser, J. Ye, D. R. Carlson, and M. J. Holland, “Prospects for a millihertz-linewidth laser,” Phys. Rev. Lett. 102(16), 163601 (2009).
[Crossref]

2007 (1)

2006 (1)

T. Salzburger and H. Ritsch, “Lasing and cooling in a finite-temperature cavity,” Phys. Rev. A 74(3), 033806 (2006).
[Crossref]

2004 (1)

T. Salzburger and H. Ritsch, “Atomic self-trapping induced by single-atom lasing,” Phys. Rev. Lett. 93(6), 063002 (2004).
[Crossref]

2002 (1)

Z. Ficek and R. Tanaś, “Entangled states and collective nonclassical effects in two-atom systems,” Phys. Rep. 372(5), 369–443 (2002).
[Crossref]

1994 (1)

S. Kuppens, M. Van Exter, and J. Woerdman, “Quantum-limited linewidth of a bad-cavity laser,” Phys. Rev. Lett. 72(24), 3815–3818 (1994).
[Crossref]

1993 (1)

F. Haake, M. I. Kolobov, C. Fabre, E. Giacobino, and S. Reynaud, “Superradiant laser,” Phys. Rev. Lett. 71(7), 995–998 (1993).
[Crossref]

1970 (1)

R. Lehmberg, “Radiation from an N-atom system. I. General formalism,” Phys. Rev. A 2(3), 883–888 (1970).
[Crossref]

1958 (1)

A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112(6), 1940–1949 (1958).
[Crossref]

1954 (1)

R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93(1), 99–110 (1954).
[Crossref]

Audoin, C.

J. Vanier and C. Audoin, The Quantum Physics of Atomic Frequency Standards (CRC, 1989).

Blatt, S.

Bloom, B.

S. L. Campbell, R. Hutson, G. Marti, A. Goban, N. D. Oppong, R. McNally, L. Sonderhouse, J. Robinson, W. Zhang, B. Bloom, and J. Ye, “A fermi-degenerate three-dimensional optical lattice clock,” Science 358(6359), 90–94 (2017).
[Crossref]

Bohnet, J. G.

J. M. Weiner, K. C. Cox, J. G. Bohnet, and J. K. Thompson, “Phase synchronization inside a superradiant laser,” Phys. Rev. A 95(3), 033808 (2017).
[Crossref]

J. G. Bohnet, Z. Chen, J. M. Weiner, K. C. Cox, and J. K. Thompson, “Active and passive sensing of collective atomic coherence in a superradiant laser,” Phys. Rev. A 88(1), 013826 (2013).
[Crossref]

J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, and J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484(7392), 78–81 (2012).
[Crossref]

Boyd, M. M.

Brennecke, F.

H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger, “Cold atoms in cavity-generated dynamical optical potentials,” Rev. Mod. Phys. 85(2), 553–601 (2013).
[Crossref]

Campbell, S. L.

S. L. Campbell, R. Hutson, G. Marti, A. Goban, N. D. Oppong, R. McNally, L. Sonderhouse, J. Robinson, W. Zhang, B. Bloom, and J. Ye, “A fermi-degenerate three-dimensional optical lattice clock,” Science 358(6359), 90–94 (2017).
[Crossref]

Carlson, D. R.

D. Meiser, J. Ye, D. R. Carlson, and M. J. Holland, “Prospects for a millihertz-linewidth laser,” Phys. Rev. Lett. 102(16), 163601 (2009).
[Crossref]

Chen, L.

T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. J. Martin, L. Chen, and J. Ye, “A sub-40-mhz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6(10), 687–692 (2012).
[Crossref]

Chen, Z.

J. G. Bohnet, Z. Chen, J. M. Weiner, K. C. Cox, and J. K. Thompson, “Active and passive sensing of collective atomic coherence in a superradiant laser,” Phys. Rev. A 88(1), 013826 (2013).
[Crossref]

J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, and J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484(7392), 78–81 (2012).
[Crossref]

Cline, J. R.

M. A. Norcia, R. J. Lewis-Swan, J. R. Cline, B. Zhu, A. M. Rey, and J. K. Thompson, “Cavity-mediated collective spin-exchange interactions in a strontium superradiant laser,” Science 361(6399), 259–262 (2018).
[Crossref]

M. A. Norcia, M. N. Winchester, J. R. Cline, and J. K. Thompson, “Superradiance on the millihertz linewidth strontium clock transition,” Sci. Adv. 2(10), e1601231 (2016).
[Crossref]

Cooper, J.

M. Xu, S. B. Jäger, S. Schütz, J. Cooper, G. Morigi, and M. J. Holland, “Supercooling of atoms in an optical resonator,” Phys. Rev. Lett. 116(15), 153002 (2016).
[Crossref]

Cox, K. C.

J. M. Weiner, K. C. Cox, J. G. Bohnet, and J. K. Thompson, “Phase synchronization inside a superradiant laser,” Phys. Rev. A 95(3), 033808 (2017).
[Crossref]

J. G. Bohnet, Z. Chen, J. M. Weiner, K. C. Cox, and J. K. Thompson, “Active and passive sensing of collective atomic coherence in a superradiant laser,” Phys. Rev. A 88(1), 013826 (2013).
[Crossref]

Davila-Rodriguez, J.

H. Leopardi, J. Davila-Rodriguez, J. Sherman, F. Quinlan, S. Diddams, and T. Fortier, “Absolute frequency comb comparisons and the measurement of optical atomic clock transitions,” in CLEO: Science and Innovations, (Optical Society of America, 2018), pp. SM1L–4.

Dicke, R. H.

R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93(1), 99–110 (1954).
[Crossref]

Diddams, S.

H. Leopardi, J. Davila-Rodriguez, J. Sherman, F. Quinlan, S. Diddams, and T. Fortier, “Absolute frequency comb comparisons and the measurement of optical atomic clock transitions,” in CLEO: Science and Innovations, (Optical Society of America, 2018), pp. SM1L–4.

Domokos, P.

H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger, “Cold atoms in cavity-generated dynamical optical potentials,” Rev. Mod. Phys. 85(2), 553–601 (2013).
[Crossref]

Esslinger, T.

H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger, “Cold atoms in cavity-generated dynamical optical potentials,” Rev. Mod. Phys. 85(2), 553–601 (2013).
[Crossref]

Fabre, C.

F. Haake, M. I. Kolobov, C. Fabre, E. Giacobino, and S. Reynaud, “Superradiant laser,” Phys. Rev. Lett. 71(7), 995–998 (1993).
[Crossref]

Ficek, Z.

Z. Ficek and R. Tanaś, “Entangled states and collective nonclassical effects in two-atom systems,” Phys. Rep. 372(5), 369–443 (2002).
[Crossref]

Foreman, S. M.

Fortier, T.

H. Leopardi, J. Davila-Rodriguez, J. Sherman, F. Quinlan, S. Diddams, and T. Fortier, “Absolute frequency comb comparisons and the measurement of optical atomic clock transitions,” in CLEO: Science and Innovations, (Optical Society of America, 2018), pp. SM1L–4.

Giacobino, E.

F. Haake, M. I. Kolobov, C. Fabre, E. Giacobino, and S. Reynaud, “Superradiant laser,” Phys. Rev. Lett. 71(7), 995–998 (1993).
[Crossref]

Goban, A.

S. L. Campbell, R. Hutson, G. Marti, A. Goban, N. D. Oppong, R. McNally, L. Sonderhouse, J. Robinson, W. Zhang, B. Bloom, and J. Ye, “A fermi-degenerate three-dimensional optical lattice clock,” Science 358(6359), 90–94 (2017).
[Crossref]

Grebing, C.

T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. J. Martin, L. Chen, and J. Ye, “A sub-40-mhz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6(10), 687–692 (2012).
[Crossref]

Haake, F.

F. Haake, M. I. Kolobov, C. Fabre, E. Giacobino, and S. Reynaud, “Superradiant laser,” Phys. Rev. Lett. 71(7), 995–998 (1993).
[Crossref]

Hagemann, C.

T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. J. Martin, L. Chen, and J. Ye, “A sub-40-mhz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6(10), 687–692 (2012).
[Crossref]

Henschel, K.

K. Henschel, J. Majer, J. Schmiedmayer, and H. Ritsch, “Cavity qed with an ultracold ensemble on a chip: Prospects for strong magnetic coupling at finite temperatures,” Phys. Rev. A 82(3), 033810 (2010).
[Crossref]

Holland, M.

D. Meiser and M. Holland, “Steady-state superradiance with alkaline-earth-metal atoms,” Phys. Rev. A 81(3), 033847 (2010).
[Crossref]

Holland, M. J.

M. Xu, S. B. Jäger, S. Schütz, J. Cooper, G. Morigi, and M. J. Holland, “Supercooling of atoms in an optical resonator,” Phys. Rev. Lett. 116(15), 153002 (2016).
[Crossref]

M. Xu, D. A. Tieri, and M. J. Holland, “Simulating open quantum systems by applying su(4) to quantum master equations,” Phys. Rev. A 87(6), 062101 (2013).
[Crossref]

J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, and J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484(7392), 78–81 (2012).
[Crossref]

D. Meiser, J. Ye, D. R. Carlson, and M. J. Holland, “Prospects for a millihertz-linewidth laser,” Phys. Rev. Lett. 102(16), 163601 (2009).
[Crossref]

Huang, X.

Hutson, R.

S. L. Campbell, R. Hutson, G. Marti, A. Goban, N. D. Oppong, R. McNally, L. Sonderhouse, J. Robinson, W. Zhang, B. Bloom, and J. Ye, “A fermi-degenerate three-dimensional optical lattice clock,” Science 358(6359), 90–94 (2017).
[Crossref]

Jäger, S. B.

M. Xu, S. B. Jäger, S. Schütz, J. Cooper, G. Morigi, and M. J. Holland, “Supercooling of atoms in an optical resonator,” Phys. Rev. Lett. 116(15), 153002 (2016).
[Crossref]

Kessler, T.

T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. J. Martin, L. Chen, and J. Ye, “A sub-40-mhz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6(10), 687–692 (2012).
[Crossref]

Kolobov, M. I.

F. Haake, M. I. Kolobov, C. Fabre, E. Giacobino, and S. Reynaud, “Superradiant laser,” Phys. Rev. Lett. 71(7), 995–998 (1993).
[Crossref]

Kraemer, S.

Krämer, S.

S. Krämer, D. Plankensteiner, L. Ostermann, and H. Ritsch, “Quantumoptics. jl: A julia framework for simulating open quantum systems,” Comput. Phys. Commun. 227, 109–116 (2018).
[Crossref]

Kuppens, S.

S. Kuppens, M. Van Exter, and J. Woerdman, “Quantum-limited linewidth of a bad-cavity laser,” Phys. Rev. Lett. 72(24), 3815–3818 (1994).
[Crossref]

Legero, T.

T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. J. Martin, L. Chen, and J. Ye, “A sub-40-mhz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6(10), 687–692 (2012).
[Crossref]

Lehmberg, R.

R. Lehmberg, “Radiation from an N-atom system. I. General formalism,” Phys. Rev. A 2(3), 883–888 (1970).
[Crossref]

Leopardi, H.

H. Leopardi, J. Davila-Rodriguez, J. Sherman, F. Quinlan, S. Diddams, and T. Fortier, “Absolute frequency comb comparisons and the measurement of optical atomic clock transitions,” in CLEO: Science and Innovations, (Optical Society of America, 2018), pp. SM1L–4.

Lewis-Swan, R. J.

M. A. Norcia, R. J. Lewis-Swan, J. R. Cline, B. Zhu, A. M. Rey, and J. K. Thompson, “Cavity-mediated collective spin-exchange interactions in a strontium superradiant laser,” Science 361(6399), 259–262 (2018).
[Crossref]

Ludlow, A. D.

Maier, T.

Majer, J.

K. Henschel, J. Majer, J. Schmiedmayer, and H. Ritsch, “Cavity qed with an ultracold ensemble on a chip: Prospects for strong magnetic coupling at finite temperatures,” Phys. Rev. A 82(3), 033810 (2010).
[Crossref]

Marti, G.

S. L. Campbell, R. Hutson, G. Marti, A. Goban, N. D. Oppong, R. McNally, L. Sonderhouse, J. Robinson, W. Zhang, B. Bloom, and J. Ye, “A fermi-degenerate three-dimensional optical lattice clock,” Science 358(6359), 90–94 (2017).
[Crossref]

Martin, M. J.

T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. J. Martin, L. Chen, and J. Ye, “A sub-40-mhz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6(10), 687–692 (2012).
[Crossref]

McNally, R.

S. L. Campbell, R. Hutson, G. Marti, A. Goban, N. D. Oppong, R. McNally, L. Sonderhouse, J. Robinson, W. Zhang, B. Bloom, and J. Ye, “A fermi-degenerate three-dimensional optical lattice clock,” Science 358(6359), 90–94 (2017).
[Crossref]

Meiser, D.

J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, and J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484(7392), 78–81 (2012).
[Crossref]

D. Meiser and M. Holland, “Steady-state superradiance with alkaline-earth-metal atoms,” Phys. Rev. A 81(3), 033847 (2010).
[Crossref]

D. Meiser, J. Ye, D. R. Carlson, and M. J. Holland, “Prospects for a millihertz-linewidth laser,” Phys. Rev. Lett. 102(16), 163601 (2009).
[Crossref]

Morigi, G.

M. Xu, S. B. Jäger, S. Schütz, J. Cooper, G. Morigi, and M. J. Holland, “Supercooling of atoms in an optical resonator,” Phys. Rev. Lett. 116(15), 153002 (2016).
[Crossref]

Norcia, M. A.

M. A. Norcia, R. J. Lewis-Swan, J. R. Cline, B. Zhu, A. M. Rey, and J. K. Thompson, “Cavity-mediated collective spin-exchange interactions in a strontium superradiant laser,” Science 361(6399), 259–262 (2018).
[Crossref]

M. A. Norcia, M. N. Winchester, J. R. Cline, and J. K. Thompson, “Superradiance on the millihertz linewidth strontium clock transition,” Sci. Adv. 2(10), e1601231 (2016).
[Crossref]

Notcutt, M.

Oppong, N. D.

S. L. Campbell, R. Hutson, G. Marti, A. Goban, N. D. Oppong, R. McNally, L. Sonderhouse, J. Robinson, W. Zhang, B. Bloom, and J. Ye, “A fermi-degenerate three-dimensional optical lattice clock,” Science 358(6359), 90–94 (2017).
[Crossref]

Ostermann, L.

S. Krämer, D. Plankensteiner, L. Ostermann, and H. Ritsch, “Quantumoptics. jl: A julia framework for simulating open quantum systems,” Comput. Phys. Commun. 227, 109–116 (2018).
[Crossref]

T. Maier, S. Kraemer, L. Ostermann, and H. Ritsch, “A superradiant clock laser on a magic wavelength optical lattice,” Opt. Express 22(11), 13269–13279 (2014).
[Crossref]

Plankensteiner, D.

S. Krämer, D. Plankensteiner, L. Ostermann, and H. Ritsch, “Quantumoptics. jl: A julia framework for simulating open quantum systems,” Comput. Phys. Commun. 227, 109–116 (2018).
[Crossref]

Puri, R. R.

R. R. Puri, Mathematical Methods of Quantum Optics (Springer Berlin Heidelberg, 2001).

Quinlan, F.

H. Leopardi, J. Davila-Rodriguez, J. Sherman, F. Quinlan, S. Diddams, and T. Fortier, “Absolute frequency comb comparisons and the measurement of optical atomic clock transitions,” in CLEO: Science and Innovations, (Optical Society of America, 2018), pp. SM1L–4.

Rey, A. M.

M. A. Norcia, R. J. Lewis-Swan, J. R. Cline, B. Zhu, A. M. Rey, and J. K. Thompson, “Cavity-mediated collective spin-exchange interactions in a strontium superradiant laser,” Science 361(6399), 259–262 (2018).
[Crossref]

Reynaud, S.

F. Haake, M. I. Kolobov, C. Fabre, E. Giacobino, and S. Reynaud, “Superradiant laser,” Phys. Rev. Lett. 71(7), 995–998 (1993).
[Crossref]

Riehle, F.

T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. J. Martin, L. Chen, and J. Ye, “A sub-40-mhz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6(10), 687–692 (2012).
[Crossref]

Ritsch, H.

S. Krämer, D. Plankensteiner, L. Ostermann, and H. Ritsch, “Quantumoptics. jl: A julia framework for simulating open quantum systems,” Comput. Phys. Commun. 227, 109–116 (2018).
[Crossref]

T. Maier, S. Kraemer, L. Ostermann, and H. Ritsch, “A superradiant clock laser on a magic wavelength optical lattice,” Opt. Express 22(11), 13269–13279 (2014).
[Crossref]

H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger, “Cold atoms in cavity-generated dynamical optical potentials,” Rev. Mod. Phys. 85(2), 553–601 (2013).
[Crossref]

K. Henschel, J. Majer, J. Schmiedmayer, and H. Ritsch, “Cavity qed with an ultracold ensemble on a chip: Prospects for strong magnetic coupling at finite temperatures,” Phys. Rev. A 82(3), 033810 (2010).
[Crossref]

T. Salzburger and H. Ritsch, “Lasing and cooling in a finite-temperature cavity,” Phys. Rev. A 74(3), 033806 (2006).
[Crossref]

T. Salzburger and H. Ritsch, “Atomic self-trapping induced by single-atom lasing,” Phys. Rev. Lett. 93(6), 063002 (2004).
[Crossref]

Robinson, J.

S. L. Campbell, R. Hutson, G. Marti, A. Goban, N. D. Oppong, R. McNally, L. Sonderhouse, J. Robinson, W. Zhang, B. Bloom, and J. Ye, “A fermi-degenerate three-dimensional optical lattice clock,” Science 358(6359), 90–94 (2017).
[Crossref]

Salzburger, T.

T. Salzburger and H. Ritsch, “Lasing and cooling in a finite-temperature cavity,” Phys. Rev. A 74(3), 033806 (2006).
[Crossref]

T. Salzburger and H. Ritsch, “Atomic self-trapping induced by single-atom lasing,” Phys. Rev. Lett. 93(6), 063002 (2004).
[Crossref]

Schawlow, A. L.

A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112(6), 1940–1949 (1958).
[Crossref]

Schmiedmayer, J.

K. Henschel, J. Majer, J. Schmiedmayer, and H. Ritsch, “Cavity qed with an ultracold ensemble on a chip: Prospects for strong magnetic coupling at finite temperatures,” Phys. Rev. A 82(3), 033810 (2010).
[Crossref]

Schütz, S.

M. Xu, S. B. Jäger, S. Schütz, J. Cooper, G. Morigi, and M. J. Holland, “Supercooling of atoms in an optical resonator,” Phys. Rev. Lett. 116(15), 153002 (2016).
[Crossref]

Sherman, J.

H. Leopardi, J. Davila-Rodriguez, J. Sherman, F. Quinlan, S. Diddams, and T. Fortier, “Absolute frequency comb comparisons and the measurement of optical atomic clock transitions,” in CLEO: Science and Innovations, (Optical Society of America, 2018), pp. SM1L–4.

Sonderhouse, L.

S. L. Campbell, R. Hutson, G. Marti, A. Goban, N. D. Oppong, R. McNally, L. Sonderhouse, J. Robinson, W. Zhang, B. Bloom, and J. Ye, “A fermi-degenerate three-dimensional optical lattice clock,” Science 358(6359), 90–94 (2017).
[Crossref]

Sterr, U.

T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. J. Martin, L. Chen, and J. Ye, “A sub-40-mhz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6(10), 687–692 (2012).
[Crossref]

Tanas, R.

Z. Ficek and R. Tanaś, “Entangled states and collective nonclassical effects in two-atom systems,” Phys. Rep. 372(5), 369–443 (2002).
[Crossref]

Thompson, J. K.

M. A. Norcia, R. J. Lewis-Swan, J. R. Cline, B. Zhu, A. M. Rey, and J. K. Thompson, “Cavity-mediated collective spin-exchange interactions in a strontium superradiant laser,” Science 361(6399), 259–262 (2018).
[Crossref]

J. M. Weiner, K. C. Cox, J. G. Bohnet, and J. K. Thompson, “Phase synchronization inside a superradiant laser,” Phys. Rev. A 95(3), 033808 (2017).
[Crossref]

M. A. Norcia, M. N. Winchester, J. R. Cline, and J. K. Thompson, “Superradiance on the millihertz linewidth strontium clock transition,” Sci. Adv. 2(10), e1601231 (2016).
[Crossref]

J. G. Bohnet, Z. Chen, J. M. Weiner, K. C. Cox, and J. K. Thompson, “Active and passive sensing of collective atomic coherence in a superradiant laser,” Phys. Rev. A 88(1), 013826 (2013).
[Crossref]

J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, and J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484(7392), 78–81 (2012).
[Crossref]

Tieri, D. A.

M. Xu, D. A. Tieri, and M. J. Holland, “Simulating open quantum systems by applying su(4) to quantum master equations,” Phys. Rev. A 87(6), 062101 (2013).
[Crossref]

Townes, C. H.

A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112(6), 1940–1949 (1958).
[Crossref]

Van Exter, M.

S. Kuppens, M. Van Exter, and J. Woerdman, “Quantum-limited linewidth of a bad-cavity laser,” Phys. Rev. Lett. 72(24), 3815–3818 (1994).
[Crossref]

Vanier, J.

J. Vanier and C. Audoin, The Quantum Physics of Atomic Frequency Standards (CRC, 1989).

Vuletic, V.

V. Vuletic, “An almost lightless laser,” Nature 484(7392), 43–44 (2012).
[Crossref]

Weiner, J. M.

J. M. Weiner, K. C. Cox, J. G. Bohnet, and J. K. Thompson, “Phase synchronization inside a superradiant laser,” Phys. Rev. A 95(3), 033808 (2017).
[Crossref]

J. G. Bohnet, Z. Chen, J. M. Weiner, K. C. Cox, and J. K. Thompson, “Active and passive sensing of collective atomic coherence in a superradiant laser,” Phys. Rev. A 88(1), 013826 (2013).
[Crossref]

J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, and J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484(7392), 78–81 (2012).
[Crossref]

Winchester, M. N.

M. A. Norcia, M. N. Winchester, J. R. Cline, and J. K. Thompson, “Superradiance on the millihertz linewidth strontium clock transition,” Sci. Adv. 2(10), e1601231 (2016).
[Crossref]

Woerdman, J.

S. Kuppens, M. Van Exter, and J. Woerdman, “Quantum-limited linewidth of a bad-cavity laser,” Phys. Rev. Lett. 72(24), 3815–3818 (1994).
[Crossref]

Xu, M.

M. Xu, S. B. Jäger, S. Schütz, J. Cooper, G. Morigi, and M. J. Holland, “Supercooling of atoms in an optical resonator,” Phys. Rev. Lett. 116(15), 153002 (2016).
[Crossref]

M. Xu, D. A. Tieri, and M. J. Holland, “Simulating open quantum systems by applying su(4) to quantum master equations,” Phys. Rev. A 87(6), 062101 (2013).
[Crossref]

Ye, J.

S. L. Campbell, R. Hutson, G. Marti, A. Goban, N. D. Oppong, R. McNally, L. Sonderhouse, J. Robinson, W. Zhang, B. Bloom, and J. Ye, “A fermi-degenerate three-dimensional optical lattice clock,” Science 358(6359), 90–94 (2017).
[Crossref]

T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. J. Martin, L. Chen, and J. Ye, “A sub-40-mhz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6(10), 687–692 (2012).
[Crossref]

D. Meiser, J. Ye, D. R. Carlson, and M. J. Holland, “Prospects for a millihertz-linewidth laser,” Phys. Rev. Lett. 102(16), 163601 (2009).
[Crossref]

A. D. Ludlow, X. Huang, M. Notcutt, T. Zanon-Willette, S. M. Foreman, M. M. Boyd, S. Blatt, and J. Ye, “Compact, thermal-noise-limited optical cavity for diode laser stabilization at $1{\times }10^{-15}$1×10−15,” Opt. Lett. 32(6), 641–643 (2007).
[Crossref]

Zanon-Willette, T.

Zhang, W.

S. L. Campbell, R. Hutson, G. Marti, A. Goban, N. D. Oppong, R. McNally, L. Sonderhouse, J. Robinson, W. Zhang, B. Bloom, and J. Ye, “A fermi-degenerate three-dimensional optical lattice clock,” Science 358(6359), 90–94 (2017).
[Crossref]

Zhu, B.

M. A. Norcia, R. J. Lewis-Swan, J. R. Cline, B. Zhu, A. M. Rey, and J. K. Thompson, “Cavity-mediated collective spin-exchange interactions in a strontium superradiant laser,” Science 361(6399), 259–262 (2018).
[Crossref]

Comput. Phys. Commun. (1)

S. Krämer, D. Plankensteiner, L. Ostermann, and H. Ritsch, “Quantumoptics. jl: A julia framework for simulating open quantum systems,” Comput. Phys. Commun. 227, 109–116 (2018).
[Crossref]

Nat. Photonics (1)

T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. J. Martin, L. Chen, and J. Ye, “A sub-40-mhz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6(10), 687–692 (2012).
[Crossref]

Nature (2)

V. Vuletic, “An almost lightless laser,” Nature 484(7392), 43–44 (2012).
[Crossref]

J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, and J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484(7392), 78–81 (2012).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rep. (1)

Z. Ficek and R. Tanaś, “Entangled states and collective nonclassical effects in two-atom systems,” Phys. Rep. 372(5), 369–443 (2002).
[Crossref]

Phys. Rev. (2)

A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112(6), 1940–1949 (1958).
[Crossref]

R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93(1), 99–110 (1954).
[Crossref]

Phys. Rev. A (7)

T. Salzburger and H. Ritsch, “Lasing and cooling in a finite-temperature cavity,” Phys. Rev. A 74(3), 033806 (2006).
[Crossref]

J. G. Bohnet, Z. Chen, J. M. Weiner, K. C. Cox, and J. K. Thompson, “Active and passive sensing of collective atomic coherence in a superradiant laser,” Phys. Rev. A 88(1), 013826 (2013).
[Crossref]

R. Lehmberg, “Radiation from an N-atom system. I. General formalism,” Phys. Rev. A 2(3), 883–888 (1970).
[Crossref]

D. Meiser and M. Holland, “Steady-state superradiance with alkaline-earth-metal atoms,” Phys. Rev. A 81(3), 033847 (2010).
[Crossref]

J. M. Weiner, K. C. Cox, J. G. Bohnet, and J. K. Thompson, “Phase synchronization inside a superradiant laser,” Phys. Rev. A 95(3), 033808 (2017).
[Crossref]

M. Xu, D. A. Tieri, and M. J. Holland, “Simulating open quantum systems by applying su(4) to quantum master equations,” Phys. Rev. A 87(6), 062101 (2013).
[Crossref]

K. Henschel, J. Majer, J. Schmiedmayer, and H. Ritsch, “Cavity qed with an ultracold ensemble on a chip: Prospects for strong magnetic coupling at finite temperatures,” Phys. Rev. A 82(3), 033810 (2010).
[Crossref]

Phys. Rev. Lett. (5)

S. Kuppens, M. Van Exter, and J. Woerdman, “Quantum-limited linewidth of a bad-cavity laser,” Phys. Rev. Lett. 72(24), 3815–3818 (1994).
[Crossref]

M. Xu, S. B. Jäger, S. Schütz, J. Cooper, G. Morigi, and M. J. Holland, “Supercooling of atoms in an optical resonator,” Phys. Rev. Lett. 116(15), 153002 (2016).
[Crossref]

T. Salzburger and H. Ritsch, “Atomic self-trapping induced by single-atom lasing,” Phys. Rev. Lett. 93(6), 063002 (2004).
[Crossref]

D. Meiser, J. Ye, D. R. Carlson, and M. J. Holland, “Prospects for a millihertz-linewidth laser,” Phys. Rev. Lett. 102(16), 163601 (2009).
[Crossref]

F. Haake, M. I. Kolobov, C. Fabre, E. Giacobino, and S. Reynaud, “Superradiant laser,” Phys. Rev. Lett. 71(7), 995–998 (1993).
[Crossref]

Rev. Mod. Phys. (1)

H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger, “Cold atoms in cavity-generated dynamical optical potentials,” Rev. Mod. Phys. 85(2), 553–601 (2013).
[Crossref]

Sci. Adv. (1)

M. A. Norcia, M. N. Winchester, J. R. Cline, and J. K. Thompson, “Superradiance on the millihertz linewidth strontium clock transition,” Sci. Adv. 2(10), e1601231 (2016).
[Crossref]

Science (2)

M. A. Norcia, R. J. Lewis-Swan, J. R. Cline, B. Zhu, A. M. Rey, and J. K. Thompson, “Cavity-mediated collective spin-exchange interactions in a strontium superradiant laser,” Science 361(6399), 259–262 (2018).
[Crossref]

S. L. Campbell, R. Hutson, G. Marti, A. Goban, N. D. Oppong, R. McNally, L. Sonderhouse, J. Robinson, W. Zhang, B. Bloom, and J. Ye, “A fermi-degenerate three-dimensional optical lattice clock,” Science 358(6359), 90–94 (2017).
[Crossref]

Other (3)

H. Leopardi, J. Davila-Rodriguez, J. Sherman, F. Quinlan, S. Diddams, and T. Fortier, “Absolute frequency comb comparisons and the measurement of optical atomic clock transitions,” in CLEO: Science and Innovations, (Optical Society of America, 2018), pp. SM1L–4.

R. R. Puri, Mathematical Methods of Quantum Optics (Springer Berlin Heidelberg, 2001).

J. Vanier and C. Audoin, The Quantum Physics of Atomic Frequency Standards (CRC, 1989).

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Figures (8)

Fig. 1.
Fig. 1. Schematic of our model. We consider a confined ensemble of two-level atoms moving along the axis of a cavity resonantly coupled to a single mode with amplitude $g(r_i)$. The atoms directly interact via resonant dipole-dipole coupling inducing pairwise energy exchange $\Omega _{ij}$ and collective decay with decay rates $\Gamma _{ij}$. A uniform transverse pump mechanism individually excites atoms at rate $R$, while the cavity loses photons at rate $\kappa$.
Fig. 2.
Fig. 2. Exemplary trajectories of three particles and their time-averaged kinetic energy loss. In figure (a) an initially untrapped particle (green and orange line) is slowed down by cavity cooling until it is trapped, whereas all other particles in the examples remain close to their initial trapping position near a field antinode. The set of initial momenta is $p_0 \simeq [-1.78, 3.92, 2.83]\hbar k_{\textrm {a}}$ in (a) and $p_0 \simeq [1.00, -0.73, 1.18]\hbar k_{\textrm {a}}$ in (b). In (c), we see that the time-averaged relative kinetic energy does not vary that much in time as opposed to the momentary kinetic energy exhibiting trapped oscillatory motion. For better visibility we normalize the kinetic energy to its initial value. The parameters are $N=3$, $\omega _{\textrm {r}} = 0.1 \Gamma$, $\Delta = 10\Gamma$, $g = 5\Gamma$, $\kappa = 10\Gamma$ and $R = 8\Gamma$ for all three figures.
Fig. 3.
Fig. 3. Cycle-averaged relative kinetic energy change after an evolution time of 500 atomic lifetimes as function of various operating parameters. We show scans over $\Delta$ and $R$ in (a),(b) and (c) and vary $g$ and $R$, respectively, in (d),(e) and (f) for different $\omega _{\textrm {r}}$. The parameters when kept constant are $N=3$, $\Delta = 5\Gamma$, $g = 5\Gamma$, and $\kappa = 10\Gamma$ and $t=500/\Gamma$.
Fig. 4.
Fig. 4. Comparison of motional cooling in time with and without direct dipole interaction. We compare the motional energy loss for collectively interacting (solid lines) and independent atoms (dashed lines) showing $\bar {E}_{\textrm {kin}}^{\mathrm {rel}}$ for both cases. The independent case describes atoms far apart from each other. The parameters are $N = 3$, $\Delta = 5\Gamma$, $g = 5\Gamma$, $\kappa = 10\Gamma$ and $R = 8\Gamma$ for both, $\omega _{\textrm {r}} = 0.1 \Gamma$ and $\omega _{\textrm {r}} = 1 \Gamma$.
Fig. 5.
Fig. 5. Properties of the emitted laser light as a function of different system parameters. We depict scans for the peak frequency shift $\delta _0$ from the atomic resonance in (a) and (b), the laser linewidth $\gamma$ in (c) and (d), the average photon number $n$ in (e) and (f), and the bunching parameter $g^{(2)}(0)$ in (g) and (h). We focus on values of $\Delta \geq 5\Gamma$, since below this threshold the particle motion shows few stable trajectories only (see Fig. 8). The remaining parameters are the same as in Fig. 3.
Fig. 6.
Fig. 6. Appearance of a second maximum in the spectrum for large atom cavity detuning. For detunings larger than $\Delta = 25 \Gamma$ a second peak emerges at the cavity resonance to the right-hand side of the atomic peak. For $\Delta = 50 \Gamma$ this peak is almost completely separated. This shifts the average of the emitted intensity towards the cavity resonance. We call the offset from the atomic transition frequency $\delta _{\textrm {a}}$ and the one from the cavity resonance $\delta _{\textrm {c}}$. The parameters are $N=3$, $\omega _{\textrm {r}} = 0.1 \Gamma$, $g = 5\Gamma$, $\kappa = 10\Gamma$ and $R = 10\Gamma$ for both, $\Delta = 25\Gamma$ and $\Delta = 50\Gamma$.
Fig. 7.
Fig. 7. Lorentzian fit of the normalized spectrum. The calculated spectrum is fitted with a three-parameter Lorentzian function. We call the FWHM $\gamma$ and the offset to the atomic resonance frequency $\delta _0$. The parameters are the same as in Fig. 2(b).
Fig. 8.
Fig. 8. Percentage of completely stable trajectories. For $\Delta \leq 0$ there are no stable trajectories, because the atoms are heated and leave their initial trap. The fixed parameters are the same as in Fig. 3.

Equations (30)

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H = Δ a a + i = 1 N g ( r i ) [ a σ i + + a σ i ] + i , j : i j Ω i j σ i + σ j ,
ρ ˙ = i [ H , ρ ] + L [ ρ ] .
L [ ρ ] = L p u m p [ ρ ] + L c a v [ ρ ] + L c d [ ρ ] ,
L p u m p [ ρ ] = R 2 i ( 2 σ i + ρ σ i σ i σ i + ρ ρ σ i σ i + ) ,
L c a v [ ρ ] = κ ( 2 a ρ a a a ρ ρ a a ) ,
L c d [ ρ ] = 1 2 i j Γ i j ( 2 σ i ρ σ j + σ i + σ j ρ ρ σ i + σ j ) .
Ω i j = 3 Γ 4 [ ( 1 cos 2 Θ ) cos ( k a r i j ) k a r i j ( 1 3 cos 2 Θ ) ( sin ( k a r i j ) ( k a r i j ) 2 + cos ( k a r i j ) ( k a r i j ) 3 ) ] ,
Γ i j = 3 Γ 2 [ ( 1 cos 2 Θ ) sin ( k a r i j ) k a r i j + ( 1 3 cos 2 Θ ) ( cos ( k a r i j ) ( k a r i j ) 2 sin ( k a r i j ) ( k a r i j ) 3 ) ] .
r ˙ i = p i m = 2 ω r p i k a 2 ,
p ˙ i = r i [ g ( r i ) a σ i + + a σ i + j : j i 2 Ω i j Re { σ i + σ j } ] .
E kin ( t ) = i p i ( t ) 2 2 m ,
E ¯ kin r e l ( t ) = E ¯ kin ( t ) E ¯ kin ( 0 ) ,
S ( ω ) = 2 Re { 0 d τ e i ω τ g ( 1 ) ( τ ) } .
n = a a ,
g ( 2 ) ( 0 ) = a a a a a a 2 .
p e = i = 1 N σ i + σ i
H tot = H 0 + i g ( r ^ i c ) ( a σ i + σ i + a ) + k , λ ω k b k , λ b k , λ + i k , λ g k , λ ( b k , λ σ i e i k r ^ i + H.c. ) + i p ^ i 2 2 m ,
H 0 := ω c a a + ω a i σ i + σ i
ρ ˙ tot = i [ H tot , ρ tot ] .
ρ tot ( t ) ρ acf ( t ) ρ m ( t ) .
ρ ˙ acf = i tr m ( [ H tot , ρ tot ] ) = i [ H acf , ρ acf ] .
H acf := H 0 + i g ( r i c ( t ) ) ( a σ i + σ i + a ) + k , λ ω k b k , λ b k , λ + i k , λ g k , λ ( b k , λ σ i e i k r i ( t ) + H.c. ) ,
r i ( t ) = r ^ i ( t ) = tr ( r ^ i ρ m ( t ) ) .
ρ ˙ = tr f ( ρ ˙ acf ) = i [ H , ρ ] + L cd [ ρ ] ,
p ˙ i = tr ( p ^ i ρ ˙ tot ) = i tr ( p ^ i [ H tot , ρ tot ] )
p ˙ i = r i j : j i ( 2 Ω i j Re { σ i + σ j } + Γ i j Im { σ i + σ j } ) .
g ( τ ) = a ( t + τ ) a ( t ) .
g ( τ ) = tr ( U ( t ) U ( τ ) a U ( τ ) a U ( t ) ρ tot ( 0 ) ) = tr ( a U ( τ ) a ρ tot U ( τ ) ) ,
g ( τ ) = tr ( a ρ ¯ tot ( τ ) ) .
ρ ¯ tot ( τ ) = U acf ( τ ) a ρ acf ( t ) U acf ( τ ) U m ( τ ) ρ m ( t ) U m ( τ ) = ρ ¯ acf ( τ ) ρ m ( t + τ ) ,

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