Abstract

We investigate the region of validity of the semiclassical approach to simulating laser cooling. We conclude that for the commonly used πxπy polarization-gradient configuration, the semiclassical approach is valid only for transitions with recoil parameters εr on the order of 104 or less. For the standard laser-cooling transitions only the transitions in Rb and Cs satisfy this condition. For the Doppler and σ+σ polarization-gradient configuration the semiclassical approach is valid for most of the commonly used transitions; however, the expected gain in execution speed compared with quantum Monte Carlo calculations has been realized only in part. A drastic reduction in calculation time is to be expected by implementing an analytical approach to the long-term contribution of the diffusion coefficient.

© 2005 Optical Society of America

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  1. E. te Sligte, B. Smeets, K. M. R. van der Stam, R. W. Herfst, P. van der Straten, H. C. W. Beijerinck, and K. A. H. van Leeuwen, “Atom lithography of Fe,” Appl. Phys. Lett.  85, 4493–4495 (2004).
    [CrossRef]
  2. G. Myszkiewicz, J. Hohlfeld, A. J. Toonen, A. F. Van Etteger, O. I. Shklyarevskii, W. L. Meerts, Th. Rasing, and E. Jurdik, “Laser manipulation of iron for nanofabrication,” Appl. Phys. Lett.  85, 3842–3844 (2004).
    [CrossRef]
  3. E. J. D. Vredenbregt and K. A. H. van Leeuwen, “Laser cooling and trapping visualized,” Am. J. Phys.  71, 760–765 (2003).
    [CrossRef]
  4. G. Nienhuis, P. van der Straten, and S.-Q. Shang, “Operator description of laser cooling below the Doppler limit,” Phys. Rev. A  44, 462–474 (1991).
    [CrossRef] [PubMed]
  5. M. D. Hoogerland, H. F. P. de Bie, H. C. W. Beijerinck, E. J. D. Vredenbregt, K. A. H. van Leeuwen, P. van der Straten, and H. J. Metcalf, “Force, diffusion, and channeling in sub-Doppler laser cooling,” Phys. Rev. A  54, 3206–3218 (1996).
    [CrossRef] [PubMed]
  6. B. R. Mollow, “Pure-state analysis of resonant light scattering: radiative damping, saturation, and multiphoton effects,” Phys. Rev. A  12, 1919–1943 (1975).
    [CrossRef]
  7. R. Dum, P. Zoller, and H. Ritsch, “Monte Carlo simulation of the atomic master equation for spontaneous emission,” Phys. Rev. A  45, 4879–4887 (1992).
    [CrossRef] [PubMed]
  8. R. Blatt, W. Ertmer, P. Zoller, and J. L. Hall, “Atomic-beam cooling: a simulation approach,” Phys. Rev. A  34, 3022–3033 (1986).
    [CrossRef] [PubMed]
  9. M. D. Hoogerland, M. N. H. Wijnands, H. J. Senhorst, H. C. W. Beijerinck, and K. A. H. van Leeuwen, “Photon statistics in resonance fluorescence: Results from an atomic-beam deflection experiment,” Phys. Rev. Lett.  65, 1559–1562 (1990).
    [CrossRef] [PubMed]
  10. J. Dalibard, Y. Castin, and K. Mølmer, “Wave-function approach to dissipative processes in quantum optics,” Phys. Rev. Lett.  68, 580–583 (1992).
    [CrossRef] [PubMed]

2004

E. te Sligte, B. Smeets, K. M. R. van der Stam, R. W. Herfst, P. van der Straten, H. C. W. Beijerinck, and K. A. H. van Leeuwen, “Atom lithography of Fe,” Appl. Phys. Lett.  85, 4493–4495 (2004).
[CrossRef]

G. Myszkiewicz, J. Hohlfeld, A. J. Toonen, A. F. Van Etteger, O. I. Shklyarevskii, W. L. Meerts, Th. Rasing, and E. Jurdik, “Laser manipulation of iron for nanofabrication,” Appl. Phys. Lett.  85, 3842–3844 (2004).
[CrossRef]

2003

E. J. D. Vredenbregt and K. A. H. van Leeuwen, “Laser cooling and trapping visualized,” Am. J. Phys.  71, 760–765 (2003).
[CrossRef]

1996

M. D. Hoogerland, H. F. P. de Bie, H. C. W. Beijerinck, E. J. D. Vredenbregt, K. A. H. van Leeuwen, P. van der Straten, and H. J. Metcalf, “Force, diffusion, and channeling in sub-Doppler laser cooling,” Phys. Rev. A  54, 3206–3218 (1996).
[CrossRef] [PubMed]

1992

R. Dum, P. Zoller, and H. Ritsch, “Monte Carlo simulation of the atomic master equation for spontaneous emission,” Phys. Rev. A  45, 4879–4887 (1992).
[CrossRef] [PubMed]

J. Dalibard, Y. Castin, and K. Mølmer, “Wave-function approach to dissipative processes in quantum optics,” Phys. Rev. Lett.  68, 580–583 (1992).
[CrossRef] [PubMed]

1991

G. Nienhuis, P. van der Straten, and S.-Q. Shang, “Operator description of laser cooling below the Doppler limit,” Phys. Rev. A  44, 462–474 (1991).
[CrossRef] [PubMed]

1990

M. D. Hoogerland, M. N. H. Wijnands, H. J. Senhorst, H. C. W. Beijerinck, and K. A. H. van Leeuwen, “Photon statistics in resonance fluorescence: Results from an atomic-beam deflection experiment,” Phys. Rev. Lett.  65, 1559–1562 (1990).
[CrossRef] [PubMed]

1986

R. Blatt, W. Ertmer, P. Zoller, and J. L. Hall, “Atomic-beam cooling: a simulation approach,” Phys. Rev. A  34, 3022–3033 (1986).
[CrossRef] [PubMed]

1975

B. R. Mollow, “Pure-state analysis of resonant light scattering: radiative damping, saturation, and multiphoton effects,” Phys. Rev. A  12, 1919–1943 (1975).
[CrossRef]

Beijerinck, H. C. W.

E. te Sligte, B. Smeets, K. M. R. van der Stam, R. W. Herfst, P. van der Straten, H. C. W. Beijerinck, and K. A. H. van Leeuwen, “Atom lithography of Fe,” Appl. Phys. Lett.  85, 4493–4495 (2004).
[CrossRef]

M. D. Hoogerland, H. F. P. de Bie, H. C. W. Beijerinck, E. J. D. Vredenbregt, K. A. H. van Leeuwen, P. van der Straten, and H. J. Metcalf, “Force, diffusion, and channeling in sub-Doppler laser cooling,” Phys. Rev. A  54, 3206–3218 (1996).
[CrossRef] [PubMed]

M. D. Hoogerland, M. N. H. Wijnands, H. J. Senhorst, H. C. W. Beijerinck, and K. A. H. van Leeuwen, “Photon statistics in resonance fluorescence: Results from an atomic-beam deflection experiment,” Phys. Rev. Lett.  65, 1559–1562 (1990).
[CrossRef] [PubMed]

Blatt, R.

R. Blatt, W. Ertmer, P. Zoller, and J. L. Hall, “Atomic-beam cooling: a simulation approach,” Phys. Rev. A  34, 3022–3033 (1986).
[CrossRef] [PubMed]

Castin, Y.

J. Dalibard, Y. Castin, and K. Mølmer, “Wave-function approach to dissipative processes in quantum optics,” Phys. Rev. Lett.  68, 580–583 (1992).
[CrossRef] [PubMed]

Dalibard, J.

J. Dalibard, Y. Castin, and K. Mølmer, “Wave-function approach to dissipative processes in quantum optics,” Phys. Rev. Lett.  68, 580–583 (1992).
[CrossRef] [PubMed]

de Bie, H. F. P.

M. D. Hoogerland, H. F. P. de Bie, H. C. W. Beijerinck, E. J. D. Vredenbregt, K. A. H. van Leeuwen, P. van der Straten, and H. J. Metcalf, “Force, diffusion, and channeling in sub-Doppler laser cooling,” Phys. Rev. A  54, 3206–3218 (1996).
[CrossRef] [PubMed]

Dum, R.

R. Dum, P. Zoller, and H. Ritsch, “Monte Carlo simulation of the atomic master equation for spontaneous emission,” Phys. Rev. A  45, 4879–4887 (1992).
[CrossRef] [PubMed]

Ertmer, W.

R. Blatt, W. Ertmer, P. Zoller, and J. L. Hall, “Atomic-beam cooling: a simulation approach,” Phys. Rev. A  34, 3022–3033 (1986).
[CrossRef] [PubMed]

Hall, J. L.

R. Blatt, W. Ertmer, P. Zoller, and J. L. Hall, “Atomic-beam cooling: a simulation approach,” Phys. Rev. A  34, 3022–3033 (1986).
[CrossRef] [PubMed]

Herfst, R. W.

E. te Sligte, B. Smeets, K. M. R. van der Stam, R. W. Herfst, P. van der Straten, H. C. W. Beijerinck, and K. A. H. van Leeuwen, “Atom lithography of Fe,” Appl. Phys. Lett.  85, 4493–4495 (2004).
[CrossRef]

Hohlfeld, J.

G. Myszkiewicz, J. Hohlfeld, A. J. Toonen, A. F. Van Etteger, O. I. Shklyarevskii, W. L. Meerts, Th. Rasing, and E. Jurdik, “Laser manipulation of iron for nanofabrication,” Appl. Phys. Lett.  85, 3842–3844 (2004).
[CrossRef]

Hoogerland, M. D.

M. D. Hoogerland, H. F. P. de Bie, H. C. W. Beijerinck, E. J. D. Vredenbregt, K. A. H. van Leeuwen, P. van der Straten, and H. J. Metcalf, “Force, diffusion, and channeling in sub-Doppler laser cooling,” Phys. Rev. A  54, 3206–3218 (1996).
[CrossRef] [PubMed]

M. D. Hoogerland, M. N. H. Wijnands, H. J. Senhorst, H. C. W. Beijerinck, and K. A. H. van Leeuwen, “Photon statistics in resonance fluorescence: Results from an atomic-beam deflection experiment,” Phys. Rev. Lett.  65, 1559–1562 (1990).
[CrossRef] [PubMed]

Jurdik, E.

G. Myszkiewicz, J. Hohlfeld, A. J. Toonen, A. F. Van Etteger, O. I. Shklyarevskii, W. L. Meerts, Th. Rasing, and E. Jurdik, “Laser manipulation of iron for nanofabrication,” Appl. Phys. Lett.  85, 3842–3844 (2004).
[CrossRef]

Meerts, W. L.

G. Myszkiewicz, J. Hohlfeld, A. J. Toonen, A. F. Van Etteger, O. I. Shklyarevskii, W. L. Meerts, Th. Rasing, and E. Jurdik, “Laser manipulation of iron for nanofabrication,” Appl. Phys. Lett.  85, 3842–3844 (2004).
[CrossRef]

Metcalf, H. J.

M. D. Hoogerland, H. F. P. de Bie, H. C. W. Beijerinck, E. J. D. Vredenbregt, K. A. H. van Leeuwen, P. van der Straten, and H. J. Metcalf, “Force, diffusion, and channeling in sub-Doppler laser cooling,” Phys. Rev. A  54, 3206–3218 (1996).
[CrossRef] [PubMed]

Mollow, B. R.

B. R. Mollow, “Pure-state analysis of resonant light scattering: radiative damping, saturation, and multiphoton effects,” Phys. Rev. A  12, 1919–1943 (1975).
[CrossRef]

Mølmer, K.

J. Dalibard, Y. Castin, and K. Mølmer, “Wave-function approach to dissipative processes in quantum optics,” Phys. Rev. Lett.  68, 580–583 (1992).
[CrossRef] [PubMed]

Myszkiewicz, G.

G. Myszkiewicz, J. Hohlfeld, A. J. Toonen, A. F. Van Etteger, O. I. Shklyarevskii, W. L. Meerts, Th. Rasing, and E. Jurdik, “Laser manipulation of iron for nanofabrication,” Appl. Phys. Lett.  85, 3842–3844 (2004).
[CrossRef]

Nienhuis, G.

G. Nienhuis, P. van der Straten, and S.-Q. Shang, “Operator description of laser cooling below the Doppler limit,” Phys. Rev. A  44, 462–474 (1991).
[CrossRef] [PubMed]

Rasing, Th.

G. Myszkiewicz, J. Hohlfeld, A. J. Toonen, A. F. Van Etteger, O. I. Shklyarevskii, W. L. Meerts, Th. Rasing, and E. Jurdik, “Laser manipulation of iron for nanofabrication,” Appl. Phys. Lett.  85, 3842–3844 (2004).
[CrossRef]

Ritsch, H.

R. Dum, P. Zoller, and H. Ritsch, “Monte Carlo simulation of the atomic master equation for spontaneous emission,” Phys. Rev. A  45, 4879–4887 (1992).
[CrossRef] [PubMed]

Senhorst, H. J.

M. D. Hoogerland, M. N. H. Wijnands, H. J. Senhorst, H. C. W. Beijerinck, and K. A. H. van Leeuwen, “Photon statistics in resonance fluorescence: Results from an atomic-beam deflection experiment,” Phys. Rev. Lett.  65, 1559–1562 (1990).
[CrossRef] [PubMed]

Shang, S.-Q.

G. Nienhuis, P. van der Straten, and S.-Q. Shang, “Operator description of laser cooling below the Doppler limit,” Phys. Rev. A  44, 462–474 (1991).
[CrossRef] [PubMed]

Shklyarevskii, O. I.

G. Myszkiewicz, J. Hohlfeld, A. J. Toonen, A. F. Van Etteger, O. I. Shklyarevskii, W. L. Meerts, Th. Rasing, and E. Jurdik, “Laser manipulation of iron for nanofabrication,” Appl. Phys. Lett.  85, 3842–3844 (2004).
[CrossRef]

Smeets, B.

E. te Sligte, B. Smeets, K. M. R. van der Stam, R. W. Herfst, P. van der Straten, H. C. W. Beijerinck, and K. A. H. van Leeuwen, “Atom lithography of Fe,” Appl. Phys. Lett.  85, 4493–4495 (2004).
[CrossRef]

te Sligte, E.

E. te Sligte, B. Smeets, K. M. R. van der Stam, R. W. Herfst, P. van der Straten, H. C. W. Beijerinck, and K. A. H. van Leeuwen, “Atom lithography of Fe,” Appl. Phys. Lett.  85, 4493–4495 (2004).
[CrossRef]

Toonen, A. J.

G. Myszkiewicz, J. Hohlfeld, A. J. Toonen, A. F. Van Etteger, O. I. Shklyarevskii, W. L. Meerts, Th. Rasing, and E. Jurdik, “Laser manipulation of iron for nanofabrication,” Appl. Phys. Lett.  85, 3842–3844 (2004).
[CrossRef]

van der Stam, K. M. R.

E. te Sligte, B. Smeets, K. M. R. van der Stam, R. W. Herfst, P. van der Straten, H. C. W. Beijerinck, and K. A. H. van Leeuwen, “Atom lithography of Fe,” Appl. Phys. Lett.  85, 4493–4495 (2004).
[CrossRef]

van der Straten, P.

E. te Sligte, B. Smeets, K. M. R. van der Stam, R. W. Herfst, P. van der Straten, H. C. W. Beijerinck, and K. A. H. van Leeuwen, “Atom lithography of Fe,” Appl. Phys. Lett.  85, 4493–4495 (2004).
[CrossRef]

M. D. Hoogerland, H. F. P. de Bie, H. C. W. Beijerinck, E. J. D. Vredenbregt, K. A. H. van Leeuwen, P. van der Straten, and H. J. Metcalf, “Force, diffusion, and channeling in sub-Doppler laser cooling,” Phys. Rev. A  54, 3206–3218 (1996).
[CrossRef] [PubMed]

G. Nienhuis, P. van der Straten, and S.-Q. Shang, “Operator description of laser cooling below the Doppler limit,” Phys. Rev. A  44, 462–474 (1991).
[CrossRef] [PubMed]

Van Etteger, A. F.

G. Myszkiewicz, J. Hohlfeld, A. J. Toonen, A. F. Van Etteger, O. I. Shklyarevskii, W. L. Meerts, Th. Rasing, and E. Jurdik, “Laser manipulation of iron for nanofabrication,” Appl. Phys. Lett.  85, 3842–3844 (2004).
[CrossRef]

van Leeuwen, K. A. H.

E. te Sligte, B. Smeets, K. M. R. van der Stam, R. W. Herfst, P. van der Straten, H. C. W. Beijerinck, and K. A. H. van Leeuwen, “Atom lithography of Fe,” Appl. Phys. Lett.  85, 4493–4495 (2004).
[CrossRef]

E. J. D. Vredenbregt and K. A. H. van Leeuwen, “Laser cooling and trapping visualized,” Am. J. Phys.  71, 760–765 (2003).
[CrossRef]

M. D. Hoogerland, H. F. P. de Bie, H. C. W. Beijerinck, E. J. D. Vredenbregt, K. A. H. van Leeuwen, P. van der Straten, and H. J. Metcalf, “Force, diffusion, and channeling in sub-Doppler laser cooling,” Phys. Rev. A  54, 3206–3218 (1996).
[CrossRef] [PubMed]

M. D. Hoogerland, M. N. H. Wijnands, H. J. Senhorst, H. C. W. Beijerinck, and K. A. H. van Leeuwen, “Photon statistics in resonance fluorescence: Results from an atomic-beam deflection experiment,” Phys. Rev. Lett.  65, 1559–1562 (1990).
[CrossRef] [PubMed]

Vredenbregt, E. J. D.

E. J. D. Vredenbregt and K. A. H. van Leeuwen, “Laser cooling and trapping visualized,” Am. J. Phys.  71, 760–765 (2003).
[CrossRef]

M. D. Hoogerland, H. F. P. de Bie, H. C. W. Beijerinck, E. J. D. Vredenbregt, K. A. H. van Leeuwen, P. van der Straten, and H. J. Metcalf, “Force, diffusion, and channeling in sub-Doppler laser cooling,” Phys. Rev. A  54, 3206–3218 (1996).
[CrossRef] [PubMed]

Wijnands, M. N. H.

M. D. Hoogerland, M. N. H. Wijnands, H. J. Senhorst, H. C. W. Beijerinck, and K. A. H. van Leeuwen, “Photon statistics in resonance fluorescence: Results from an atomic-beam deflection experiment,” Phys. Rev. Lett.  65, 1559–1562 (1990).
[CrossRef] [PubMed]

Zoller, P.

R. Dum, P. Zoller, and H. Ritsch, “Monte Carlo simulation of the atomic master equation for spontaneous emission,” Phys. Rev. A  45, 4879–4887 (1992).
[CrossRef] [PubMed]

R. Blatt, W. Ertmer, P. Zoller, and J. L. Hall, “Atomic-beam cooling: a simulation approach,” Phys. Rev. A  34, 3022–3033 (1986).
[CrossRef] [PubMed]

Am. J. Phys.

E. J. D. Vredenbregt and K. A. H. van Leeuwen, “Laser cooling and trapping visualized,” Am. J. Phys.  71, 760–765 (2003).
[CrossRef]

Appl. Phys. Lett.

E. te Sligte, B. Smeets, K. M. R. van der Stam, R. W. Herfst, P. van der Straten, H. C. W. Beijerinck, and K. A. H. van Leeuwen, “Atom lithography of Fe,” Appl. Phys. Lett.  85, 4493–4495 (2004).
[CrossRef]

G. Myszkiewicz, J. Hohlfeld, A. J. Toonen, A. F. Van Etteger, O. I. Shklyarevskii, W. L. Meerts, Th. Rasing, and E. Jurdik, “Laser manipulation of iron for nanofabrication,” Appl. Phys. Lett.  85, 3842–3844 (2004).
[CrossRef]

Phys. Rev. A

G. Nienhuis, P. van der Straten, and S.-Q. Shang, “Operator description of laser cooling below the Doppler limit,” Phys. Rev. A  44, 462–474 (1991).
[CrossRef] [PubMed]

M. D. Hoogerland, H. F. P. de Bie, H. C. W. Beijerinck, E. J. D. Vredenbregt, K. A. H. van Leeuwen, P. van der Straten, and H. J. Metcalf, “Force, diffusion, and channeling in sub-Doppler laser cooling,” Phys. Rev. A  54, 3206–3218 (1996).
[CrossRef] [PubMed]

B. R. Mollow, “Pure-state analysis of resonant light scattering: radiative damping, saturation, and multiphoton effects,” Phys. Rev. A  12, 1919–1943 (1975).
[CrossRef]

R. Dum, P. Zoller, and H. Ritsch, “Monte Carlo simulation of the atomic master equation for spontaneous emission,” Phys. Rev. A  45, 4879–4887 (1992).
[CrossRef] [PubMed]

R. Blatt, W. Ertmer, P. Zoller, and J. L. Hall, “Atomic-beam cooling: a simulation approach,” Phys. Rev. A  34, 3022–3033 (1986).
[CrossRef] [PubMed]

Phys. Rev. Lett.

M. D. Hoogerland, M. N. H. Wijnands, H. J. Senhorst, H. C. W. Beijerinck, and K. A. H. van Leeuwen, “Photon statistics in resonance fluorescence: Results from an atomic-beam deflection experiment,” Phys. Rev. Lett.  65, 1559–1562 (1990).
[CrossRef] [PubMed]

J. Dalibard, Y. Castin, and K. Mølmer, “Wave-function approach to dissipative processes in quantum optics,” Phys. Rev. Lett.  68, 580–583 (1992).
[CrossRef] [PubMed]

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Figures (6)

Fig. 1
Fig. 1

Sub-Doppler force for different J values in the crossed linear-polarization configuration, calculated with the SC method. The saturation parameter is s = 2 , and detuning is δ = 2 Γ .

Fig. 2
Fig. 2

Integrand of the stimulated part of the diffusion coefficient for J = 1 J = 2 . The part due to a spontaneous decay of the excited state decays on a timescale of Γ 1 . The part due to a pumping over the ground-state sublevels decays on a timescale of 100 Γ 1 . The decay rate of the latter part will decrease with [ J ( J + 1 ) ] 1 . The transverse velocity in this particular case is ν = Γ k .

Fig. 3
Fig. 3

Stimulated (solid curve) and spontaneous (dotted curve) part of the diffusion coefficient for crossed linear polarizations for a saturation parameter s = 2 and a detuning δ = 2 Γ .

Fig. 4
Fig. 4

Velocity distributions for different recoil parameters ϵ r = k 2 2 m Γ = 0.01 down to 0.0005 as indicated in the figure, with saturation parameter s = 2 and detuning δ = 2 Γ for a J = 1 J = 2 transition. The solid curves are the SC data; the dotted curves are the QMC data.

Fig. 5
Fig. 5

Velocity distribution for the σ + σ configuration. Saturation parameters is s = 2 , detuning is δ = 2 Γ , and recoil parameter is ϵ = 0.005 .

Fig. 6
Fig. 6

Force and diffusion profile for π x π y and σ + σ configurations. The recoil velocity for this particular recoil parameter is indicated.

Tables (1)

Tables Icon

Table 1 Recoil Parameters of the Transitions of Some of the Commonly Used Atomic Species in Experiments with Laser-Cooling Techniques

Equations (24)

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

t W ( v ) = 1 M v F ( v ) W ( v ) + 1 M 2 2 v v : D ( v ) W ( v ) ,
F ( v ) = f ( v )
2 D ( v ) = 0 d τ [ f ( t ) f ( t + τ ) f ( t ) f ( t + τ ) + f ( t + τ ) f ( t ) f ( t + τ ) f ( t ) ] .
E ( r , t ) = E + ( r ) exp ( i ω t ) + E ( r ) exp ( i ω t ) .
R = μ eg E + ,
d σ g g d t = Γ β Q β σ e e Q β + i R σ e g i σ g e R ,
d σ e e d t = Γ σ e e + i R σ g e i σ e g R ,
d σ g e d t = ( Γ 2 + i δ ) σ g e + i R σ e e i σ g g R ,
d σ e g d t = ( Γ 2 i δ ) σ e g + i R σ g g i σ e e R ,
e J e m J e Q β g J g m J g = C i .
D sp = 1 2 d n ̂ g ( n ̂ ) ( k ) 2 n ̂ n ̂ ,
g ( n ̂ ) = 3 Γ 8 π u ̂ n ̂ Tr u ̂ * Q σ ¯ e e Q u ̂
Q x = 1 2 ( Q 1 Q 1 ) , Q y = 1 i 2 ( Q 1 + Q 1 ) , Q z = Q 0 .
σ ( t + τ ) = U ( t + τ , t ) σ ( t ) .
f st ( t ) f st ( t + τ ) = Tr U ( t + τ ) [ σ ¯ ( t ) f st ( t ) ] f st ( t + τ ) ,
f st ( t + τ ) f st ( t ) = Tr f st ( t + τ ) U ( t + τ ) [ f st ( t ) σ ¯ ( t ) ] .
H ̂ = p ̂ 2 2 M + H ̂ 0 A + H ̂ 0 F + H ̂ I ( t ) ,
H ̂ I ( t ) = μ eg * E ̂ a ̂ + H.C.
Ψ ( r , t ) = Ψ 0 ( r , t ) + n = 1 Ψ n ( r , t ) .
1 Ψ 0 ( t ) 2 = 0 t W ( t ) d t = Γ 0 t C e 0 ( t ) 2 d t .
1 Ψ 0 ( t ) 2 = Υ .
i d d t C g m g j ( t ) = [ 2 2 M ( j k + k 0 ) 2 ] C g m g j ( t ) + q = ± 1 Ω e g * 2 j g m g 1 q j e ( m g q ) [ ϵ q + * C e ( m g q ) j + 1 ( t ) + ϵ q * C e ( m g q ) j 1 ( t ) ] ,
i d d t C e m e j ( t ) = [ 2 2 M ( j k + k 0 ) 2 ( Δ + i Γ 2 ) ] C e m e j ( t ) + q = ± 1 Ω e g 2 j g ( m e + q ) 1 q j e ( m e ) [ ϵ q + C g ( m e + q ) j 1 ( t ) + ϵ q C g ( m e + q ) j + 1 ( t ) ] .
t W ( v , t ) = 2 ε r v F ( v ) W ( v , t ) + 4 ε r 2 2 v 2 D ( v ) W ( v , t ) .

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