Abstract

A model of microcavity semiconductor lasers in which both the cavity field and the gain medium are quantized is presented. The equation of motion for the elements of the reduced density matrix for the field in the photon number representation is developed and numerically solved to find the steady-state photon number distribution and the laser linewidth for a variety of operating conditions. For typical semiconductor microcavity operating conditions, the intensity noise is smaller and the laser linewidth is larger for lasers with a larger fraction of spontaneous emission into the cavity mode. However, for very-low-loss microcavity lasers, if the rate of spontaneous emission into the cavity mode exceeds the loss rate, the laser can appear to turn on at pump rates for which the gain medium is not inverted. In this anamolous regime, the laser intensity noise increases with an increased fraction of spontaneous emission into the cavity mode.

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  1. M. O. Scully and W. E. Lamb, Jr., "Quantum theory of an optical maser. I. General theory," Phys. Rev. 159, 208-226 (1967).
    [CrossRef]
  2. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge U. Press, 1997).
  3. P. Meystre and M. Sargent III, Elements of Quantum Optics (Springer-Verlag, 1991).
  4. W. W. Chow, S. W. Koch, and M. Sargent III, Semiconductor Laser Physics (Springer, 1993).
  5. G. P. Agrawal and N. K. Dutta, Long-Wavelength Semiconductors (Van Nostrand Reinhold, 1986).
  6. H. Haug and H. Haken, "Theory of noise in semiconductor laser emission," Z. Phys. 204, 262-275 (1967).
    [CrossRef]
  7. H. Haug, "Quantum-mechanical rate equations for semiconductor lasers," Phys. Rev. 184, 338-248 (1969).
    [CrossRef]
  8. H. Haug and S. W. Koch, "Semiconductor laser theory with many-body effects," Phys. Rev. A 39, 1887-1898 (1989).
    [CrossRef] [PubMed]
  9. P. R. Rice and H. J. Carmichael, "Photon statistics of a cavity-QED laser: a comment on the laser-phase-transition analogy," Phys. Rev. A 50, 4318-4329 (1994).
    [CrossRef] [PubMed]
  10. C. W. Gardiner and A. Eschmann, "Master-equation theory of semiconductor lasers," Phys. Rev. A 51, 4982-4995 (1995).
    [CrossRef] [PubMed]
  11. Y. Yamamoto, "AM and FM quantum noise in semiconductor lasers--Part I: Theoretical analysis," IEEE J. Quantum Electron. QE-19, 34-36 (1983).
    [CrossRef]
  12. A. Imamoglu and Y. Yamamoto, "Turnstile device for heralded single photons: Coulomb blockade of electron and hole tunneling in quantum confined p-i-n heterojunctions," Phys. Rev. Lett. 72, 210-213 (1994).
    [CrossRef] [PubMed]
  13. R. Jin, D. Boggavarapu, M. Sargent III, P. Meystre, H. A. Gibbs, and G. Khitrova, "Photon-number correlations near the threshold of microcavity lasers in the weak-coupling regime," Phys. Rev. A 49, 4038-4042 (1994).
    [CrossRef] [PubMed]
  14. L. M. Pedrotti, M. Sokol, and P. R. Rice, "Linewidth of four-level microcavity lasers," Phys. Rev. A 59, 2295-2301 (1999).
    [CrossRef]
  15. U. Mohideen, R. E. Slusher, F. Jahnke, and S. W. Koch, "Semiconductor microcavity linewidths," Phys. Rev. Lett. 73, 1785-1788 (1994).
    [CrossRef] [PubMed]
  16. R. E. Slusher, A. F. J. Levi, U. Mohideen, S. L. McCall, S. J. Pearton, and R. A. Logan, "Threshold characteristics of semiconductor microdisk lasers," Appl. Phys. Lett. 63, 1310-1312 (1993).
    [CrossRef]
  17. R. E. Slusher (Lucent Technologies, Bell Laboratories, Murray Hill, N.J. 07974, USA) and U. Mohideen (Department of Physics, University of California Riverside, Riverside, Calif. 92521, USA) (personal communication, 1995).

1999 (1)

L. M. Pedrotti, M. Sokol, and P. R. Rice, "Linewidth of four-level microcavity lasers," Phys. Rev. A 59, 2295-2301 (1999).
[CrossRef]

1995 (1)

C. W. Gardiner and A. Eschmann, "Master-equation theory of semiconductor lasers," Phys. Rev. A 51, 4982-4995 (1995).
[CrossRef] [PubMed]

1994 (4)

A. Imamoglu and Y. Yamamoto, "Turnstile device for heralded single photons: Coulomb blockade of electron and hole tunneling in quantum confined p-i-n heterojunctions," Phys. Rev. Lett. 72, 210-213 (1994).
[CrossRef] [PubMed]

R. Jin, D. Boggavarapu, M. Sargent III, P. Meystre, H. A. Gibbs, and G. Khitrova, "Photon-number correlations near the threshold of microcavity lasers in the weak-coupling regime," Phys. Rev. A 49, 4038-4042 (1994).
[CrossRef] [PubMed]

U. Mohideen, R. E. Slusher, F. Jahnke, and S. W. Koch, "Semiconductor microcavity linewidths," Phys. Rev. Lett. 73, 1785-1788 (1994).
[CrossRef] [PubMed]

P. R. Rice and H. J. Carmichael, "Photon statistics of a cavity-QED laser: a comment on the laser-phase-transition analogy," Phys. Rev. A 50, 4318-4329 (1994).
[CrossRef] [PubMed]

1993 (1)

R. E. Slusher, A. F. J. Levi, U. Mohideen, S. L. McCall, S. J. Pearton, and R. A. Logan, "Threshold characteristics of semiconductor microdisk lasers," Appl. Phys. Lett. 63, 1310-1312 (1993).
[CrossRef]

1989 (1)

H. Haug and S. W. Koch, "Semiconductor laser theory with many-body effects," Phys. Rev. A 39, 1887-1898 (1989).
[CrossRef] [PubMed]

1983 (1)

Y. Yamamoto, "AM and FM quantum noise in semiconductor lasers--Part I: Theoretical analysis," IEEE J. Quantum Electron. QE-19, 34-36 (1983).
[CrossRef]

1969 (1)

H. Haug, "Quantum-mechanical rate equations for semiconductor lasers," Phys. Rev. 184, 338-248 (1969).
[CrossRef]

1967 (2)

M. O. Scully and W. E. Lamb, Jr., "Quantum theory of an optical maser. I. General theory," Phys. Rev. 159, 208-226 (1967).
[CrossRef]

H. Haug and H. Haken, "Theory of noise in semiconductor laser emission," Z. Phys. 204, 262-275 (1967).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal and N. K. Dutta, Long-Wavelength Semiconductors (Van Nostrand Reinhold, 1986).

Boggavarapu, D.

R. Jin, D. Boggavarapu, M. Sargent III, P. Meystre, H. A. Gibbs, and G. Khitrova, "Photon-number correlations near the threshold of microcavity lasers in the weak-coupling regime," Phys. Rev. A 49, 4038-4042 (1994).
[CrossRef] [PubMed]

Carmichael, H. J.

P. R. Rice and H. J. Carmichael, "Photon statistics of a cavity-QED laser: a comment on the laser-phase-transition analogy," Phys. Rev. A 50, 4318-4329 (1994).
[CrossRef] [PubMed]

Chow, W. W.

W. W. Chow, S. W. Koch, and M. Sargent III, Semiconductor Laser Physics (Springer, 1993).

Dutta, N. K.

G. P. Agrawal and N. K. Dutta, Long-Wavelength Semiconductors (Van Nostrand Reinhold, 1986).

Eschmann, A.

C. W. Gardiner and A. Eschmann, "Master-equation theory of semiconductor lasers," Phys. Rev. A 51, 4982-4995 (1995).
[CrossRef] [PubMed]

Gardiner, C. W.

C. W. Gardiner and A. Eschmann, "Master-equation theory of semiconductor lasers," Phys. Rev. A 51, 4982-4995 (1995).
[CrossRef] [PubMed]

Gibbs, H. A.

R. Jin, D. Boggavarapu, M. Sargent III, P. Meystre, H. A. Gibbs, and G. Khitrova, "Photon-number correlations near the threshold of microcavity lasers in the weak-coupling regime," Phys. Rev. A 49, 4038-4042 (1994).
[CrossRef] [PubMed]

Haken, H.

H. Haug and H. Haken, "Theory of noise in semiconductor laser emission," Z. Phys. 204, 262-275 (1967).
[CrossRef]

Haug, H.

H. Haug and S. W. Koch, "Semiconductor laser theory with many-body effects," Phys. Rev. A 39, 1887-1898 (1989).
[CrossRef] [PubMed]

H. Haug, "Quantum-mechanical rate equations for semiconductor lasers," Phys. Rev. 184, 338-248 (1969).
[CrossRef]

H. Haug and H. Haken, "Theory of noise in semiconductor laser emission," Z. Phys. 204, 262-275 (1967).
[CrossRef]

Imamoglu, A.

A. Imamoglu and Y. Yamamoto, "Turnstile device for heralded single photons: Coulomb blockade of electron and hole tunneling in quantum confined p-i-n heterojunctions," Phys. Rev. Lett. 72, 210-213 (1994).
[CrossRef] [PubMed]

Jahnke, F.

U. Mohideen, R. E. Slusher, F. Jahnke, and S. W. Koch, "Semiconductor microcavity linewidths," Phys. Rev. Lett. 73, 1785-1788 (1994).
[CrossRef] [PubMed]

Jin, R.

R. Jin, D. Boggavarapu, M. Sargent III, P. Meystre, H. A. Gibbs, and G. Khitrova, "Photon-number correlations near the threshold of microcavity lasers in the weak-coupling regime," Phys. Rev. A 49, 4038-4042 (1994).
[CrossRef] [PubMed]

Khitrova, G.

R. Jin, D. Boggavarapu, M. Sargent III, P. Meystre, H. A. Gibbs, and G. Khitrova, "Photon-number correlations near the threshold of microcavity lasers in the weak-coupling regime," Phys. Rev. A 49, 4038-4042 (1994).
[CrossRef] [PubMed]

Koch, S. W.

U. Mohideen, R. E. Slusher, F. Jahnke, and S. W. Koch, "Semiconductor microcavity linewidths," Phys. Rev. Lett. 73, 1785-1788 (1994).
[CrossRef] [PubMed]

H. Haug and S. W. Koch, "Semiconductor laser theory with many-body effects," Phys. Rev. A 39, 1887-1898 (1989).
[CrossRef] [PubMed]

W. W. Chow, S. W. Koch, and M. Sargent III, Semiconductor Laser Physics (Springer, 1993).

Lamb, W. E.

M. O. Scully and W. E. Lamb, Jr., "Quantum theory of an optical maser. I. General theory," Phys. Rev. 159, 208-226 (1967).
[CrossRef]

Levi, A. F. J.

R. E. Slusher, A. F. J. Levi, U. Mohideen, S. L. McCall, S. J. Pearton, and R. A. Logan, "Threshold characteristics of semiconductor microdisk lasers," Appl. Phys. Lett. 63, 1310-1312 (1993).
[CrossRef]

Logan, R. A.

R. E. Slusher, A. F. J. Levi, U. Mohideen, S. L. McCall, S. J. Pearton, and R. A. Logan, "Threshold characteristics of semiconductor microdisk lasers," Appl. Phys. Lett. 63, 1310-1312 (1993).
[CrossRef]

McCall, S. L.

R. E. Slusher, A. F. J. Levi, U. Mohideen, S. L. McCall, S. J. Pearton, and R. A. Logan, "Threshold characteristics of semiconductor microdisk lasers," Appl. Phys. Lett. 63, 1310-1312 (1993).
[CrossRef]

Meystre, P.

R. Jin, D. Boggavarapu, M. Sargent III, P. Meystre, H. A. Gibbs, and G. Khitrova, "Photon-number correlations near the threshold of microcavity lasers in the weak-coupling regime," Phys. Rev. A 49, 4038-4042 (1994).
[CrossRef] [PubMed]

P. Meystre and M. Sargent III, Elements of Quantum Optics (Springer-Verlag, 1991).

Mohideen, U.

U. Mohideen, R. E. Slusher, F. Jahnke, and S. W. Koch, "Semiconductor microcavity linewidths," Phys. Rev. Lett. 73, 1785-1788 (1994).
[CrossRef] [PubMed]

R. E. Slusher, A. F. J. Levi, U. Mohideen, S. L. McCall, S. J. Pearton, and R. A. Logan, "Threshold characteristics of semiconductor microdisk lasers," Appl. Phys. Lett. 63, 1310-1312 (1993).
[CrossRef]

R. E. Slusher (Lucent Technologies, Bell Laboratories, Murray Hill, N.J. 07974, USA) and U. Mohideen (Department of Physics, University of California Riverside, Riverside, Calif. 92521, USA) (personal communication, 1995).

Pearton, S. J.

R. E. Slusher, A. F. J. Levi, U. Mohideen, S. L. McCall, S. J. Pearton, and R. A. Logan, "Threshold characteristics of semiconductor microdisk lasers," Appl. Phys. Lett. 63, 1310-1312 (1993).
[CrossRef]

Pedrotti, L. M.

L. M. Pedrotti, M. Sokol, and P. R. Rice, "Linewidth of four-level microcavity lasers," Phys. Rev. A 59, 2295-2301 (1999).
[CrossRef]

Rice, P. R.

L. M. Pedrotti, M. Sokol, and P. R. Rice, "Linewidth of four-level microcavity lasers," Phys. Rev. A 59, 2295-2301 (1999).
[CrossRef]

P. R. Rice and H. J. Carmichael, "Photon statistics of a cavity-QED laser: a comment on the laser-phase-transition analogy," Phys. Rev. A 50, 4318-4329 (1994).
[CrossRef] [PubMed]

Sargent, M.

R. Jin, D. Boggavarapu, M. Sargent III, P. Meystre, H. A. Gibbs, and G. Khitrova, "Photon-number correlations near the threshold of microcavity lasers in the weak-coupling regime," Phys. Rev. A 49, 4038-4042 (1994).
[CrossRef] [PubMed]

P. Meystre and M. Sargent III, Elements of Quantum Optics (Springer-Verlag, 1991).

W. W. Chow, S. W. Koch, and M. Sargent III, Semiconductor Laser Physics (Springer, 1993).

Scully, M. O.

M. O. Scully and W. E. Lamb, Jr., "Quantum theory of an optical maser. I. General theory," Phys. Rev. 159, 208-226 (1967).
[CrossRef]

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge U. Press, 1997).

Slusher, R. E.

U. Mohideen, R. E. Slusher, F. Jahnke, and S. W. Koch, "Semiconductor microcavity linewidths," Phys. Rev. Lett. 73, 1785-1788 (1994).
[CrossRef] [PubMed]

R. E. Slusher, A. F. J. Levi, U. Mohideen, S. L. McCall, S. J. Pearton, and R. A. Logan, "Threshold characteristics of semiconductor microdisk lasers," Appl. Phys. Lett. 63, 1310-1312 (1993).
[CrossRef]

R. E. Slusher (Lucent Technologies, Bell Laboratories, Murray Hill, N.J. 07974, USA) and U. Mohideen (Department of Physics, University of California Riverside, Riverside, Calif. 92521, USA) (personal communication, 1995).

Sokol, M.

L. M. Pedrotti, M. Sokol, and P. R. Rice, "Linewidth of four-level microcavity lasers," Phys. Rev. A 59, 2295-2301 (1999).
[CrossRef]

Yamamoto, Y.

A. Imamoglu and Y. Yamamoto, "Turnstile device for heralded single photons: Coulomb blockade of electron and hole tunneling in quantum confined p-i-n heterojunctions," Phys. Rev. Lett. 72, 210-213 (1994).
[CrossRef] [PubMed]

Y. Yamamoto, "AM and FM quantum noise in semiconductor lasers--Part I: Theoretical analysis," IEEE J. Quantum Electron. QE-19, 34-36 (1983).
[CrossRef]

Zubairy, M. S.

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge U. Press, 1997).

Appl. Phys. Lett. (1)

R. E. Slusher, A. F. J. Levi, U. Mohideen, S. L. McCall, S. J. Pearton, and R. A. Logan, "Threshold characteristics of semiconductor microdisk lasers," Appl. Phys. Lett. 63, 1310-1312 (1993).
[CrossRef]

IEEE J. Quantum Electron. (1)

Y. Yamamoto, "AM and FM quantum noise in semiconductor lasers--Part I: Theoretical analysis," IEEE J. Quantum Electron. QE-19, 34-36 (1983).
[CrossRef]

Phys. Rev. (2)

M. O. Scully and W. E. Lamb, Jr., "Quantum theory of an optical maser. I. General theory," Phys. Rev. 159, 208-226 (1967).
[CrossRef]

H. Haug, "Quantum-mechanical rate equations for semiconductor lasers," Phys. Rev. 184, 338-248 (1969).
[CrossRef]

Phys. Rev. A (5)

H. Haug and S. W. Koch, "Semiconductor laser theory with many-body effects," Phys. Rev. A 39, 1887-1898 (1989).
[CrossRef] [PubMed]

P. R. Rice and H. J. Carmichael, "Photon statistics of a cavity-QED laser: a comment on the laser-phase-transition analogy," Phys. Rev. A 50, 4318-4329 (1994).
[CrossRef] [PubMed]

C. W. Gardiner and A. Eschmann, "Master-equation theory of semiconductor lasers," Phys. Rev. A 51, 4982-4995 (1995).
[CrossRef] [PubMed]

R. Jin, D. Boggavarapu, M. Sargent III, P. Meystre, H. A. Gibbs, and G. Khitrova, "Photon-number correlations near the threshold of microcavity lasers in the weak-coupling regime," Phys. Rev. A 49, 4038-4042 (1994).
[CrossRef] [PubMed]

L. M. Pedrotti, M. Sokol, and P. R. Rice, "Linewidth of four-level microcavity lasers," Phys. Rev. A 59, 2295-2301 (1999).
[CrossRef]

Phys. Rev. Lett. (2)

U. Mohideen, R. E. Slusher, F. Jahnke, and S. W. Koch, "Semiconductor microcavity linewidths," Phys. Rev. Lett. 73, 1785-1788 (1994).
[CrossRef] [PubMed]

A. Imamoglu and Y. Yamamoto, "Turnstile device for heralded single photons: Coulomb blockade of electron and hole tunneling in quantum confined p-i-n heterojunctions," Phys. Rev. Lett. 72, 210-213 (1994).
[CrossRef] [PubMed]

Z. Phys. (1)

H. Haug and H. Haken, "Theory of noise in semiconductor laser emission," Z. Phys. 204, 262-275 (1967).
[CrossRef]

Other (5)

R. E. Slusher (Lucent Technologies, Bell Laboratories, Murray Hill, N.J. 07974, USA) and U. Mohideen (Department of Physics, University of California Riverside, Riverside, Calif. 92521, USA) (personal communication, 1995).

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge U. Press, 1997).

P. Meystre and M. Sargent III, Elements of Quantum Optics (Springer-Verlag, 1991).

W. W. Chow, S. W. Koch, and M. Sargent III, Semiconductor Laser Physics (Springer, 1993).

G. P. Agrawal and N. K. Dutta, Long-Wavelength Semiconductors (Van Nostrand Reinhold, 1986).

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Figures (12)

Fig. 1
Fig. 1

Circles represent different k states. Filled circles represent an electron in a particular k state while open circles represent a hole in that k state. The state labels refer to the pair of conduction band–valence band occupation states at a given value of k.

Fig. 2
Fig. 2

Comparison of our model and the standard semiclassical model for the parameters of Table 1. The solid curve corresponds to the predictions of our quantum model, and the dotted curve results from a standard semiclassical model. The points labeled on the solid curve are referenced in Fig. 3 below. The plot uses the parameters of Table 1 with 2 g 2 γ = 0.00375 Γ .

Fig. 3
Fig. 3

Photon number probability distribution ρ n , n below and above threshold for the parameters of Fig. 2. The curve labels refer to specific pumps and photon numbers labeled in Fig. 2.

Fig. 4
Fig. 4

Photon number (solid curves) and variance over the mean (dotted curves) as a function of pump rate for different values of 2 g 2 γ . For each case the peak of the pump threshold occurs approximately at the peak of the variance-over-mean-photon-number curve. The curve with the smallest pump threshold has 2 g 2 γ = 0.01 Γ , the curve with the intermediate pump threshold has 2 g 2 γ = 0.00375 Γ , and the curve with the largest pump threshold has 2 g 2 γ = 0.0025 Γ .

Fig. 5
Fig. 5

Mean photon number (solid curve) and g ( 2 ) ( 0 ) 1 (dotted curve) as functions of pump rate. The plot uses the parameters of Fig. 2.

Fig. 6
Fig. 6

Second-order correlation function versus mean photon number. The solid curve is for 2 g 2 γ = 0.0025 Γ , the large-dotted curve is for 2 g 2 γ = 0.00375 Γ , and the small-dotted curve is for 2 g 2 γ = 0.01 Γ . Note that the intensity noise decreases with increased fraction of spontaneous emission into the cavity mode.

Fig. 7
Fig. 7

Laser linewidth as a function of mean photon number for semiconductor lasers with different values of 2 g 2 γ . The solid curve is the ‘standard’ linewidth C 2 n ¯ , the curve with large open dots is for 2 g 2 γ = 0.0075 Γ , the curve with filled squares is for 2 g 2 γ = 0.005 Γ , the curve with small filled dots is for 2 g 2 γ = 0.00375 Γ , and the dotted curve is for 2 g 2 γ = 0.0025 Γ .

Fig. 8
Fig. 8

Fit of the semiconductor model to the experimental data of Jin et al. [13]. Here K = Δ n 2 n ¯ 2 = n ¯ ( g ( 2 ) ( 0 ) 1 ) . The triangles represent the experimental output power, and the circles are the experimental values of K. The solid curves represent the predictions of the semiconductor model for these quantities. The theoretical curves use the parameters of Table 1 and 2 g 2 γ = 0.0015 Γ .

Fig. 9
Fig. 9

Comparison of the noise properties of three-level atomic and semiconductor microlasers. In both figures, the dotted curve corresponds to a three-level gain medium with β = 0.0065 and the gain volume and cavity loss rate given in Table 1. The solid curve corresponds to the semiconductor gain medium with the parameters listed in Table 1 and with 2 g 2 γ = 0.0015 Γ ( β = 0.0015 ) .

Fig. 10
Fig. 10

Second-order correlation function and mean photon number as functions of the pump rate. The curves correspond to a cavity loss rate of 7.4 × 10 8 s and 2 g 2 γ = Γ . The transparency pump rate is 4.4 × 10 13 s and is marked by the jump discontinuity in the curve. Note that the apparent laser turn-on occurs at a pump rate of about 3.7 × 10 13 s , which is significantly less than the transparency pump rate.

Fig. 11
Fig. 11

Second-order correlation function and mean photon number as functions of the pump rate. The curves correspond to a cavity loss rate of 7.4 × 10 9 s and 2 g 2 γ = Γ . The transparency pump rate is 4.4 × 10 13 s , whereas the apparent laser turn-on appears at about 3.8 × 10 13 s .

Fig. 12
Fig. 12

Second-order correlation function versus mean photon number in the anomalous regime. In this plot the cavity loss rate is 7 . 410 8 s . The solid curve is for 2 g 2 γ = 1.5 Γ , the dashed curve is for 2 g 2 γ = Γ , and the dotted curve is for 2 g 2 γ = 0.5 Γ . Note that, in this regime, the intensity noise increases with increased fraction of spontaneous emission into the cavity mode.

Tables (1)

Tables Icon

Table 1 VCSEL Parameters Used in the Plots of Section 3

Equations (60)

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

α , β , n , m , k p α n , β m k α n k β m k .
ρ ̇ n n = k ρ ̇ n n k = k ( ρ ̇ 3 n , 3 n k + ρ ̇ 2 n , 2 n k + ρ ̇ 1 n , 1 n k + ρ ̇ 0 n , 0 n k ) .
ρ ̇ n n = k ( ρ ̇ 3 n , 3 n k + ρ ̇ 0 n , 0 n k ) .
H k = k H C k + H F + k V k .
H C k = E 3 k 3 k 3 k + E 0 k 0 k 0 k ,
H F = ν ( a a + 1 2 ) ,
V k = g ( a 3 k 0 k + a 0 k 3 k ) .
ρ ̇ = i [ H , ρ ] + dissipation + pump.
ρ ̇ 3 n , 3 n k = P k ρ n , n Γ ρ 3 n , 3 n k i 3 n [ H , ρ ] 3 n .
ρ ̇ 3 n , 3 n k = P k ρ n , n Γ ρ 3 n , 3 n k i g n + 1 ( ρ 0 n + 1 , 3 n k ρ 3 n , 0 n + 1 k ) .
ρ ̇ 0 n + 1 , 0 n + 1 k = P k ρ n + 1 n + 1 k + Γ ρ 3 n + 1 , 3 n + 1 k + i g n + 1 ( ρ 0 n + 1 , 3 n k ρ 3 n , 0 n + 1 k ) .
ρ ̇ 0 n + 1 , 3 n k = ( γ i ( ω k ν ) ) ρ 0 n + 1 , 3 n k + i Ω n ( ρ 3 n , 3 n k ρ 0 n + 1 , 0 n + 1 k ) ,
ρ ̇ 3 n , 0 n + 1 = ( γ + i ( ω k ν ) ) ρ 0 n + 1 , 3 n k i Ω n ( ρ 3 n , 3 n k ρ 0 n + 1 , 0 n + 1 k ) .
ρ ̇ 3 n , 3 n k = P k ρ n , n Γ ρ 3 n , 3 n k 2 Ω n 2 γ L ( ω k ) ( ρ 3 n , 3 n k ρ 0 n + 1 , 0 n + 1 k ) ,
ρ ̇ 0 n + 1 , 0 n + 1 k = P k ρ n , n + Γ ρ 3 n , 3 n k + 2 Ω n 2 γ L ( ω k ) ( ρ 3 n , 3 n k ρ 0 n + 1 , 0 n + 1 k ) .
L ( ω k ) = γ 2 γ 2 + ( ω k ν ) 2
ρ ̇ n n = k ( ρ ̇ 3 n , 3 n k + ρ ̇ 0 n , 0 n k ) C n ρ n , n + C ( n + 1 ) ρ n + 1 , n + 1 .
ρ ̇ n n = k [ a n 1 k ( ρ 3 n 1 , 3 n 1 k ρ 0 n , 0 n k ) a n k ( ρ 3 n , 3 n k ρ 0 n + 1 , 0 n + 1 k ) ] C n ρ n , n + C ( n + 1 ) ρ n + 1 , n + 1 .
a n k = 2 Ω n 2 γ L ( ω k ) .
ρ 3 n , 3 n k = f e n ( k ) f h n ( k ) ρ n , n ,
ρ 0 n , 0 n k = ( 1 f e n ( k ) ) ( 1 f h n ( k ) ) ρ n , n .
f e , h n ( k ) = 1 e ( ϵ e , h k μ e , h n ) ( k B T ) + 1 ,
ρ ̇ n , n = A n + A n 1 + C ( n + 1 ) ρ n + 1 , n + 1 C n ρ n , n ,
A n = k a n k [ f e n ( k ) f h n ( k ) ρ n , n ( 1 f e n + 1 ( k ) ) ( 1 f h n + 1 ( k ) ) ρ n + 1 , n + 1 ] .
A n = C ( n + 1 ) ρ n + 1 , n + 1 .
ρ n + 1 , n + 1 = ( 2 g 2 γ ) k L ( ω k ) f e n ( k ) f h n ( k ) C + ( 2 g 2 γ ) k L ( ω k ) ( 1 f e n + 1 ( k ) ) ( 1 f h n + 1 ( k ) ) ρ n , n .
k f e n ( k ) = k f h n ( k ) .
R = n , k ρ ̇ 3 n , 3 n k = n , k [ P k ρ n , n Γ ρ 3 n , 3 n k 2 g 2 ( n + 1 ) γ L ( ω k ) ( ρ 3 n , 3 n k ρ 0 n + 1 , 0 n + 1 k ) ] .
R = P k ( ( Γ + ( 2 g 2 γ ) L ( ω k ) ) f e n 0 ( k ) f h n 0 ( k ) + n 0 ( 2 g 2 γ ) L ( ω k ) ( f e n 0 ( k ) + f h n 0 ( k ) 1 ) ) .
k G ( k ) ( V 2 π 2 ) ( 2 m r 2 ) 3 2 0 ϵ G ( ϵ ) d ϵ .
ϵ = 2 k 2 2 m r = ϵ e , h k ( m e , h m r ) .
f e , h n ( ϵ ) = 1 e ( ( ϵ m r m e , h μ e n ) k B T ) + 1 .
0 ϵ f e n ( ϵ ) d ϵ = 0 ϵ f h n ( ϵ ) d ϵ .
I 1 n = ( V 2 π 2 ) ( 2 m r 2 ) 3 2 0 f e n ( ϵ ) f h n ( ϵ ) ϵ d ϵ ,
I 2 n = ( V 2 π 2 ) ( 2 m r 2 ) 3 2 0 L ( ϵ ) ( f e n ( ϵ ) + f h n ( ϵ ) 1 ) ϵ d ϵ ,
I 3 n = ( V 2 π 2 ) ( 2 m r 2 ) 3 2 0 L ( ϵ ) f e n ( ϵ ) f h n ( ϵ ) ϵ d ϵ .
ρ n + 1 , n + 1 = ( 2 g 2 γ ) I 3 n C ( 2 g 2 γ ) ( I 3 n + 1 + I 2 n + 1 ) ρ n , n ,
n 0 = P Γ I 1 n 0 ( 2 g 2 γ ) I 3 n 0 ( 2 g 2 γ ) I 2 n 0 .
β = ( 2 g 2 γ ) I 3 n Γ I 1 n + ( 2 g 2 γ ) I 3 n .
β = 2 g 2 γ Γ + 2 g 2 γ .
n ¯ = n n ρ n , n .
g ( 2 ) ( 0 ) 1 = 1 n ¯ ( Δ n 2 n ¯ 2 1 ) .
Δ n 2 = n ( n n ¯ ) 2 ρ n , n .
E ( z , t ) = E 0 ( a + a ) sin ( K z ) .
E = E 0 sin ( K z ) n = 1 ( n + 1 ρ n , n + 1 + c.c. ) .
ρ n , n + 1 = ρ n , n ρ n + 1 , n + 1 e D t 2 .
E = E 0 sin ( K z ) e D t 2 n = 1 ( ρ n , n ρ n + 1 , n + 1 n + 1 + c.c. ) .
ρ ̇ n , n + 1 = k ( ρ ̇ 3 n , 3 n + 1 k + ρ ̇ 0 n , 0 n + 1 ) + C ( n + 1 ) ( n + 2 ) ρ n + 1 , n + 1 C ( n + 1 2 ) ρ n , n + 1 ,
ρ ̇ 3 n , 3 n + 1 k = i 3 n [ H , ρ k ] 3 n + 1 = i g ( n + 1 ρ 0 n + 1 , 3 n + 1 k n + 2 ρ 3 n , 0 n + 2 k ) .
ρ ̇ 0 n + 1 , 3 n + 1 k = ( γ i ( ω k ν ) ) ρ 0 n + 1 , 3 n + 1 k i g ( n + 1 ρ 3 n , 3 n + 1 k n + 2 ρ 0 n + 1 , 0 n + 2 k ) .
ρ 0 n + 1 , 3 n + 1 k = i g γ i ( ω k ν ) ( n + 1 ρ 3 n , 3 n + 1 k n + 2 ρ 0 n + 1 , 0 n + 2 k ) .
ρ 3 n , 0 n + 2 k = i g γ + i ( ω k ν ) ( n + 1 ρ 0 n + 1 , 0 n + 2 k n + 2 ρ 3 n , 3 n + 1 k ) .
ρ ̇ 3 n , 3 n + 1 k = 2 g 2 γ ( n + 1 ) L ( ω k ) ρ 3 n , 3 n + 1 k g 2 γ + i ( ω k ν ) ρ 3 n , 3 n + 1 k + 2 g 2 γ ( n + 1 ) ( n + 2 ) L ( ω k ) ρ 0 n + 1 , 0 n + 2 k .
ρ ̇ 0 n , 0 n + 1 k = 2 g 2 γ n ( n + 1 ) L ( ω k ) ρ 3 n 1 , 3 n k 2 g 2 γ n L ( ω k ) ρ 0 n , 0 n + 1 k g 2 γ i ( ω k ν ) ρ 0 n , 0 n + 1 k .
ρ ̇ n , n + 1 = k { 2 g 2 γ ( n + 1 ) L ( ω ν ) ρ 3 n , 3 n + 1 k g 2 γ + i ( ω ν ) ρ 3 n , 3 n + 1 k + 2 g 2 γ ( n + 1 ) ( n + 2 ) L ( ω k ) ρ 0 n + 1 , 0 n + 2 k + 2 g 2 γ n ( n + 1 ) L ( ω k ) ρ 3 n 1 , 3 n k 2 g 2 γ n L ( ω k ) ρ 0 n , 0 n + 1 k g 2 γ i ( ω ν ) ρ 0 n , 0 n + 1 k } + C ( n + 1 ) ( n + 2 ) ρ n + 1 , n + 2 C ( n + 1 2 ) ρ n , n + 1 .
ρ 3 n , 3 n + 1 k = ρ 3 n , 3 n k ρ 3 n + 1 , 3 n + 1 k e D t 2 ,
ρ 0 n , 0 n + 1 k = ρ 0 n , 0 n k ρ 0 n + 1 , 0 n + 1 k e D t 2 .
D = 4 g 2 γ [ ( n ¯ + 3 2 ) F 1 n ¯ + ( n ¯ + 1 2 ) F 2 n ¯ n ¯ ( n ¯ + 1 ) ρ n ¯ 1 , n ¯ 1 ρ n ¯ + 1 , n ¯ + 1 F 1 n ¯ 1 ( n ¯ + 1 ) ( n ¯ + 2 ) a ρ n ¯ + 2 , n ¯ + 2 ρ n ¯ , n ¯ F 2 n ¯ + 1 ] + 2 C [ n ¯ + 1 2 ( n ¯ + 1 ) ( n ¯ + 2 ) ρ n + 2 , n + 2 ρ n , n ] .
F 1 n = ( V 2 π 2 ) ( 2 m r 2 ) 3 2 0 f e n ( ϵ ) f h n ( ϵ ) f e n + 1 ( ϵ ) f h n + 1 ( ϵ ) ϵ L ( ϵ ) d ϵ ,
F 2 n = ( V 2 π 2 ) ( 2 m r 2 ) 3 2 0 ( 1 f e n ( ϵ ) ) ( 1 f h n ( ϵ ) ) ( 1 f e n + 1 ( ϵ ) ) ( 1 f h n + 1 ( ϵ ) ) L ( ϵ ) ϵ d ϵ .

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