Abstract

We theoretically investigate the properties and temporal evolution of squeezed states generated using degenerate parametric down conversion in lossy cavities. We show that the Lindblad master equation, which governs the evolution of this system, has, as its solution, a squeezed thermal state with an effective temperature and squeezing parameter that depends on time. We derive analytical solutions for the time evolution of quadrature noise, thermal photon number, squeezing parameter, and total photon number under different pumping regimes. We also find the steady state limits of the quadrature noises and discuss the g(2) factor of the generated light inside the cavity in the steady state.

© 2017 Optical Society of America

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  1. P. A. M. Dirac, “The quantum theory of the emission and absorption of radiation,” Proc. R. Soc. A 114, 243–265 (1927).
    [Crossref]
  2. M. Fox, Quantum Optics: An Introduction (Oxford University, 2006).
  3. B. Fain, “Spontaneous emission vs. vacuum fluctuations,” Nuovo Cimento B 68, 73–78 (1982).
    [Crossref]
  4. P. W. Milonni and M.-L. Shih, “Zero‐point energy in early quantum theory,” Am. J. Phys. 59, 684–698 (1992).
    [Crossref]
  5. W. E. Lamb and R. C. Retherford, “Fine structure of the hydrogen atom by a microwave method,” Phys. Rev. 72, 241–243 (1947).
    [Crossref]
  6. H. A. Bethe, “The electromagnetic shift of energy levels,” Phys. Rev. 72, 339–341 (1947).
    [Crossref]
  7. S. Haroche, J. A. Paisner, and A. L. Schawlow, “Hyperfine quantum beats observed in Cs vapor under pulsed dye laser excitation,” Phys. Rev. Lett. 30, 948–951 (1973).
    [Crossref]
  8. H. B. G. Casimir and D. Polder, “The influence of retardation on the London-van der Waals forces,” Phys. Rev. 73, 360–372 (1948).
    [Crossref]
  9. V. Sandoghdar, C. I. Sukenik, E. A. Hinds, and S. Haroche, “Direct measurement of the van der Waals interaction between an atom and its images in a micron-sized cavity,” Phys. Rev. Lett. 68, 3432–3435 (1992).
    [Crossref]
  10. R. X. Adhikari, “Gravitational radiation detection with laser interferometry,” Rev. Mod. Phys. 86, 121–151 (2014).
    [Crossref]
  11. E. Oelker, L. Barsotti, S. Dwyer, D. Sigg, and N. Mavalvala, “Squeezed light for advanced gravitational wave detectors and beyond,” Opt. Express 22, 21106–21121 (2014).
    [Crossref]
  12. S. E. Dwyer, Quantum noise reduction using squeezed states in LIGO, Ph.D. thesis (Massachusetts Institute of Technology, 2013).
  13. R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
    [Crossref]
  14. L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520–2523 (1986).
    [Crossref]
  15. D. F. Walls and P. Zoller, “Reduced quantum fluctuations in resonance fluorescence,” Phys. Rev. Lett. 47, 709–711 (1981).
    [Crossref]
  16. M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, and H. J. Kimble, “Squeezed-state generation by the normal modes of a coupled system,” Phys. Rev. Lett. 59, 198–201 (1987).
    [Crossref]
  17. Z. H. Lu, S. Bali, and J. E. Thomas, “Observation of squeezing in the phase-dependent fluorescence spectra of two-level atoms,” Phys. Rev. Lett. 81, 3635–3638 (1998).
    [Crossref]
  18. P. Grünwald and W. Vogel, “Optimal squeezing in resonance fluorescence via atomic-state purification,” Phys. Rev. Lett. 109, 013601 (2012).
    [Crossref]
  19. M. G. Banaee and J. F. Young, “Squeezed state generation in photonic crystal microcavities,” Opt. Express 16, 20908–20919 (2008).
    [Crossref]
  20. A. Ourjoumtsev, A. Kubanek, M. Koch, C. Sames, P. W. H. Pinkse, G. Rempe, and K. Murr, “Observation of squeezed light from one atom excited with two photons,” Nature 474, 623–626 (2011).
    [Crossref]
  21. M. S. Kim, F. A. M. de Oliveira, and P. L. Knight, “Properties of squeezed number states and squeezed thermal states,” Phys. Rev. A 40, 2494–2503 (1989).
    [Crossref]
  22. Q.-S. Tan, J.-Q. Liao, X. Wang, and F. Nori, “Enhanced interferometry using squeezed thermal states and even or odd states,” Phys. Rev. A 89, 053822 (2014).
    [Crossref]
  23. R. Liu, A. Fang, Y. Zhou, P. Zhang, S. Gao, H. Li, H. Gao, and F. Li, “Enhanced visibility of ghost imaging and interference using squeezed thermal light,” Phys. Rev. A 93, 013822 (2016).
    [Crossref]
  24. H. J. Carmichael, A. S. Lane, and D. F. Walls, “Resonance fluorescence from an atom in a squeezed vacuum,” Phys. Rev. Lett. 58, 2539–2542 (1987).
    [Crossref]
  25. A. S. Parkins and C. W. Gardiner, “Inhibition of atomic phase decays by squeezed light in a microscopic Fabry-Pérot cavity,” Phys. Rev. A 40, 3796–3807 (1989).
    [Crossref]
  26. Z. Ficek and P. D. Drummond, “Two-photon population inversion by squeezed light in a Fabry-Perot microcavity,” Europhys. Lett. 24, 455–460 (1993).
    [Crossref]
  27. Z. Ficek and P. D. Drummond, “Three-level atom in a broadband squeezed vacuum field. I. General theory,” Phys. Rev. A 43, 6247–6257 (1991).
    [Crossref]
  28. Z. Ficek and P. D. Drummond, “Three-level atom in a broadband squeezed vacuum field. II. Applications,” Phys. Rev. A 43, 6258–6271 (1991).
    [Crossref]
  29. A. Joshi and R. R. Puri, “Steady-state behavior of three-level systems in a broadband squeezed bath,” Phys. Rev. A 45, 2025–2030 (1992).
    [Crossref]
  30. G. Milburn, “Interaction of a two-level atom with squeezed light,” Opt. Acta 31, 671–679 (1984).
    [Crossref]
  31. N. P. Georgiades, E. S. Polzik, K. Edamatsu, H. J. Kimble, and A. S. Parkins, “Nonclassical excitation for atoms in a squeezed vacuum,” Phys. Rev. Lett. 75, 3426–3429 (1995).
    [Crossref]
  32. E. S. Polzik, J. Carri, and H. J. Kimble, “Spectroscopy with squeezed light,” Phys. Rev. Lett. 68, 3020–3023 (1992).
    [Crossref]
  33. E. S. Polzik, J. Carri, and H. J. Kimble, “Atomic spectroscopy with squeezed light for sensitivity beyond the vacuum-state limit,” Appl. Phys. B 55, 279–290 (1992).
    [Crossref]
  34. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1997).
  35. J. Johansson, P. Nation, and F. Nori, “QuTiP 2: a Python framework for the dynamics of open quantum systems,” Comput. Phys. Commun. 184, 1234–1240 (2013).
    [Crossref]
  36. We note that Eqs. (14) and (18) are also valid for g→g(t), where g(t) is a real function of time.
  37. H. Fearn and M. Collett, “Representations of squeezed states with thermal noise,” J. Mod. Opt. 35, 553–564 (1988), doi: 10.1080/09500348814550571.
    [Crossref]
  38. M.-A. Lemonde, N. Didier, and A. A. Clerk, “Antibunching and unconventional photon blockade with Gaussian squeezed states,” Phys. Rev. A 90, 063824 (2014).
    [Crossref]
  39. J. Anwar and M. S. Zubairy, “Effect of squeezing on the degenerate parametric oscillator,” Phys. Rev. A 45, 1804–1809 (1992).
    [Crossref]
  40. J. Garrison and R. Chiao, Quantum Optics (Oxford University, 2014).
  41. M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386–1391 (1984).
    [Crossref]

2016 (1)

R. Liu, A. Fang, Y. Zhou, P. Zhang, S. Gao, H. Li, H. Gao, and F. Li, “Enhanced visibility of ghost imaging and interference using squeezed thermal light,” Phys. Rev. A 93, 013822 (2016).
[Crossref]

2014 (4)

Q.-S. Tan, J.-Q. Liao, X. Wang, and F. Nori, “Enhanced interferometry using squeezed thermal states and even or odd states,” Phys. Rev. A 89, 053822 (2014).
[Crossref]

M.-A. Lemonde, N. Didier, and A. A. Clerk, “Antibunching and unconventional photon blockade with Gaussian squeezed states,” Phys. Rev. A 90, 063824 (2014).
[Crossref]

R. X. Adhikari, “Gravitational radiation detection with laser interferometry,” Rev. Mod. Phys. 86, 121–151 (2014).
[Crossref]

E. Oelker, L. Barsotti, S. Dwyer, D. Sigg, and N. Mavalvala, “Squeezed light for advanced gravitational wave detectors and beyond,” Opt. Express 22, 21106–21121 (2014).
[Crossref]

2013 (1)

J. Johansson, P. Nation, and F. Nori, “QuTiP 2: a Python framework for the dynamics of open quantum systems,” Comput. Phys. Commun. 184, 1234–1240 (2013).
[Crossref]

2012 (1)

P. Grünwald and W. Vogel, “Optimal squeezing in resonance fluorescence via atomic-state purification,” Phys. Rev. Lett. 109, 013601 (2012).
[Crossref]

2011 (1)

A. Ourjoumtsev, A. Kubanek, M. Koch, C. Sames, P. W. H. Pinkse, G. Rempe, and K. Murr, “Observation of squeezed light from one atom excited with two photons,” Nature 474, 623–626 (2011).
[Crossref]

2008 (1)

1998 (1)

Z. H. Lu, S. Bali, and J. E. Thomas, “Observation of squeezing in the phase-dependent fluorescence spectra of two-level atoms,” Phys. Rev. Lett. 81, 3635–3638 (1998).
[Crossref]

1995 (1)

N. P. Georgiades, E. S. Polzik, K. Edamatsu, H. J. Kimble, and A. S. Parkins, “Nonclassical excitation for atoms in a squeezed vacuum,” Phys. Rev. Lett. 75, 3426–3429 (1995).
[Crossref]

1993 (1)

Z. Ficek and P. D. Drummond, “Two-photon population inversion by squeezed light in a Fabry-Perot microcavity,” Europhys. Lett. 24, 455–460 (1993).
[Crossref]

1992 (6)

E. S. Polzik, J. Carri, and H. J. Kimble, “Spectroscopy with squeezed light,” Phys. Rev. Lett. 68, 3020–3023 (1992).
[Crossref]

E. S. Polzik, J. Carri, and H. J. Kimble, “Atomic spectroscopy with squeezed light for sensitivity beyond the vacuum-state limit,” Appl. Phys. B 55, 279–290 (1992).
[Crossref]

A. Joshi and R. R. Puri, “Steady-state behavior of three-level systems in a broadband squeezed bath,” Phys. Rev. A 45, 2025–2030 (1992).
[Crossref]

J. Anwar and M. S. Zubairy, “Effect of squeezing on the degenerate parametric oscillator,” Phys. Rev. A 45, 1804–1809 (1992).
[Crossref]

P. W. Milonni and M.-L. Shih, “Zero‐point energy in early quantum theory,” Am. J. Phys. 59, 684–698 (1992).
[Crossref]

V. Sandoghdar, C. I. Sukenik, E. A. Hinds, and S. Haroche, “Direct measurement of the van der Waals interaction between an atom and its images in a micron-sized cavity,” Phys. Rev. Lett. 68, 3432–3435 (1992).
[Crossref]

1991 (2)

Z. Ficek and P. D. Drummond, “Three-level atom in a broadband squeezed vacuum field. I. General theory,” Phys. Rev. A 43, 6247–6257 (1991).
[Crossref]

Z. Ficek and P. D. Drummond, “Three-level atom in a broadband squeezed vacuum field. II. Applications,” Phys. Rev. A 43, 6258–6271 (1991).
[Crossref]

1989 (2)

A. S. Parkins and C. W. Gardiner, “Inhibition of atomic phase decays by squeezed light in a microscopic Fabry-Pérot cavity,” Phys. Rev. A 40, 3796–3807 (1989).
[Crossref]

M. S. Kim, F. A. M. de Oliveira, and P. L. Knight, “Properties of squeezed number states and squeezed thermal states,” Phys. Rev. A 40, 2494–2503 (1989).
[Crossref]

1988 (1)

H. Fearn and M. Collett, “Representations of squeezed states with thermal noise,” J. Mod. Opt. 35, 553–564 (1988), doi: 10.1080/09500348814550571.
[Crossref]

1987 (2)

M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, and H. J. Kimble, “Squeezed-state generation by the normal modes of a coupled system,” Phys. Rev. Lett. 59, 198–201 (1987).
[Crossref]

H. J. Carmichael, A. S. Lane, and D. F. Walls, “Resonance fluorescence from an atom in a squeezed vacuum,” Phys. Rev. Lett. 58, 2539–2542 (1987).
[Crossref]

1986 (1)

L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520–2523 (1986).
[Crossref]

1985 (1)

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[Crossref]

1984 (2)

M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386–1391 (1984).
[Crossref]

G. Milburn, “Interaction of a two-level atom with squeezed light,” Opt. Acta 31, 671–679 (1984).
[Crossref]

1982 (1)

B. Fain, “Spontaneous emission vs. vacuum fluctuations,” Nuovo Cimento B 68, 73–78 (1982).
[Crossref]

1981 (1)

D. F. Walls and P. Zoller, “Reduced quantum fluctuations in resonance fluorescence,” Phys. Rev. Lett. 47, 709–711 (1981).
[Crossref]

1973 (1)

S. Haroche, J. A. Paisner, and A. L. Schawlow, “Hyperfine quantum beats observed in Cs vapor under pulsed dye laser excitation,” Phys. Rev. Lett. 30, 948–951 (1973).
[Crossref]

1948 (1)

H. B. G. Casimir and D. Polder, “The influence of retardation on the London-van der Waals forces,” Phys. Rev. 73, 360–372 (1948).
[Crossref]

1947 (2)

W. E. Lamb and R. C. Retherford, “Fine structure of the hydrogen atom by a microwave method,” Phys. Rev. 72, 241–243 (1947).
[Crossref]

H. A. Bethe, “The electromagnetic shift of energy levels,” Phys. Rev. 72, 339–341 (1947).
[Crossref]

1927 (1)

P. A. M. Dirac, “The quantum theory of the emission and absorption of radiation,” Proc. R. Soc. A 114, 243–265 (1927).
[Crossref]

Adhikari, R. X.

R. X. Adhikari, “Gravitational radiation detection with laser interferometry,” Rev. Mod. Phys. 86, 121–151 (2014).
[Crossref]

Anwar, J.

J. Anwar and M. S. Zubairy, “Effect of squeezing on the degenerate parametric oscillator,” Phys. Rev. A 45, 1804–1809 (1992).
[Crossref]

Bali, S.

Z. H. Lu, S. Bali, and J. E. Thomas, “Observation of squeezing in the phase-dependent fluorescence spectra of two-level atoms,” Phys. Rev. Lett. 81, 3635–3638 (1998).
[Crossref]

Banaee, M. G.

Barsotti, L.

Bethe, H. A.

H. A. Bethe, “The electromagnetic shift of energy levels,” Phys. Rev. 72, 339–341 (1947).
[Crossref]

Boyd, T. L.

M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, and H. J. Kimble, “Squeezed-state generation by the normal modes of a coupled system,” Phys. Rev. Lett. 59, 198–201 (1987).
[Crossref]

Carmichael, H. J.

H. J. Carmichael, A. S. Lane, and D. F. Walls, “Resonance fluorescence from an atom in a squeezed vacuum,” Phys. Rev. Lett. 58, 2539–2542 (1987).
[Crossref]

Carri, J.

E. S. Polzik, J. Carri, and H. J. Kimble, “Spectroscopy with squeezed light,” Phys. Rev. Lett. 68, 3020–3023 (1992).
[Crossref]

E. S. Polzik, J. Carri, and H. J. Kimble, “Atomic spectroscopy with squeezed light for sensitivity beyond the vacuum-state limit,” Appl. Phys. B 55, 279–290 (1992).
[Crossref]

Casimir, H. B. G.

H. B. G. Casimir and D. Polder, “The influence of retardation on the London-van der Waals forces,” Phys. Rev. 73, 360–372 (1948).
[Crossref]

Chiao, R.

J. Garrison and R. Chiao, Quantum Optics (Oxford University, 2014).

Clerk, A. A.

M.-A. Lemonde, N. Didier, and A. A. Clerk, “Antibunching and unconventional photon blockade with Gaussian squeezed states,” Phys. Rev. A 90, 063824 (2014).
[Crossref]

Collett, M.

H. Fearn and M. Collett, “Representations of squeezed states with thermal noise,” J. Mod. Opt. 35, 553–564 (1988), doi: 10.1080/09500348814550571.
[Crossref]

Collett, M. J.

M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386–1391 (1984).
[Crossref]

de Oliveira, F. A. M.

M. S. Kim, F. A. M. de Oliveira, and P. L. Knight, “Properties of squeezed number states and squeezed thermal states,” Phys. Rev. A 40, 2494–2503 (1989).
[Crossref]

Didier, N.

M.-A. Lemonde, N. Didier, and A. A. Clerk, “Antibunching and unconventional photon blockade with Gaussian squeezed states,” Phys. Rev. A 90, 063824 (2014).
[Crossref]

Dirac, P. A. M.

P. A. M. Dirac, “The quantum theory of the emission and absorption of radiation,” Proc. R. Soc. A 114, 243–265 (1927).
[Crossref]

Drummond, P. D.

Z. Ficek and P. D. Drummond, “Two-photon population inversion by squeezed light in a Fabry-Perot microcavity,” Europhys. Lett. 24, 455–460 (1993).
[Crossref]

Z. Ficek and P. D. Drummond, “Three-level atom in a broadband squeezed vacuum field. I. General theory,” Phys. Rev. A 43, 6247–6257 (1991).
[Crossref]

Z. Ficek and P. D. Drummond, “Three-level atom in a broadband squeezed vacuum field. II. Applications,” Phys. Rev. A 43, 6258–6271 (1991).
[Crossref]

Dwyer, S.

Dwyer, S. E.

S. E. Dwyer, Quantum noise reduction using squeezed states in LIGO, Ph.D. thesis (Massachusetts Institute of Technology, 2013).

Edamatsu, K.

N. P. Georgiades, E. S. Polzik, K. Edamatsu, H. J. Kimble, and A. S. Parkins, “Nonclassical excitation for atoms in a squeezed vacuum,” Phys. Rev. Lett. 75, 3426–3429 (1995).
[Crossref]

Fain, B.

B. Fain, “Spontaneous emission vs. vacuum fluctuations,” Nuovo Cimento B 68, 73–78 (1982).
[Crossref]

Fang, A.

R. Liu, A. Fang, Y. Zhou, P. Zhang, S. Gao, H. Li, H. Gao, and F. Li, “Enhanced visibility of ghost imaging and interference using squeezed thermal light,” Phys. Rev. A 93, 013822 (2016).
[Crossref]

Fearn, H.

H. Fearn and M. Collett, “Representations of squeezed states with thermal noise,” J. Mod. Opt. 35, 553–564 (1988), doi: 10.1080/09500348814550571.
[Crossref]

Ficek, Z.

Z. Ficek and P. D. Drummond, “Two-photon population inversion by squeezed light in a Fabry-Perot microcavity,” Europhys. Lett. 24, 455–460 (1993).
[Crossref]

Z. Ficek and P. D. Drummond, “Three-level atom in a broadband squeezed vacuum field. I. General theory,” Phys. Rev. A 43, 6247–6257 (1991).
[Crossref]

Z. Ficek and P. D. Drummond, “Three-level atom in a broadband squeezed vacuum field. II. Applications,” Phys. Rev. A 43, 6258–6271 (1991).
[Crossref]

Fox, M.

M. Fox, Quantum Optics: An Introduction (Oxford University, 2006).

Gao, H.

R. Liu, A. Fang, Y. Zhou, P. Zhang, S. Gao, H. Li, H. Gao, and F. Li, “Enhanced visibility of ghost imaging and interference using squeezed thermal light,” Phys. Rev. A 93, 013822 (2016).
[Crossref]

Gao, S.

R. Liu, A. Fang, Y. Zhou, P. Zhang, S. Gao, H. Li, H. Gao, and F. Li, “Enhanced visibility of ghost imaging and interference using squeezed thermal light,” Phys. Rev. A 93, 013822 (2016).
[Crossref]

Gardiner, C. W.

A. S. Parkins and C. W. Gardiner, “Inhibition of atomic phase decays by squeezed light in a microscopic Fabry-Pérot cavity,” Phys. Rev. A 40, 3796–3807 (1989).
[Crossref]

M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386–1391 (1984).
[Crossref]

Garrison, J.

J. Garrison and R. Chiao, Quantum Optics (Oxford University, 2014).

Georgiades, N. P.

N. P. Georgiades, E. S. Polzik, K. Edamatsu, H. J. Kimble, and A. S. Parkins, “Nonclassical excitation for atoms in a squeezed vacuum,” Phys. Rev. Lett. 75, 3426–3429 (1995).
[Crossref]

Grünwald, P.

P. Grünwald and W. Vogel, “Optimal squeezing in resonance fluorescence via atomic-state purification,” Phys. Rev. Lett. 109, 013601 (2012).
[Crossref]

Hall, J. L.

L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520–2523 (1986).
[Crossref]

Haroche, S.

V. Sandoghdar, C. I. Sukenik, E. A. Hinds, and S. Haroche, “Direct measurement of the van der Waals interaction between an atom and its images in a micron-sized cavity,” Phys. Rev. Lett. 68, 3432–3435 (1992).
[Crossref]

S. Haroche, J. A. Paisner, and A. L. Schawlow, “Hyperfine quantum beats observed in Cs vapor under pulsed dye laser excitation,” Phys. Rev. Lett. 30, 948–951 (1973).
[Crossref]

Hinds, E. A.

V. Sandoghdar, C. I. Sukenik, E. A. Hinds, and S. Haroche, “Direct measurement of the van der Waals interaction between an atom and its images in a micron-sized cavity,” Phys. Rev. Lett. 68, 3432–3435 (1992).
[Crossref]

Hollberg, L. W.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[Crossref]

Johansson, J.

J. Johansson, P. Nation, and F. Nori, “QuTiP 2: a Python framework for the dynamics of open quantum systems,” Comput. Phys. Commun. 184, 1234–1240 (2013).
[Crossref]

Joshi, A.

A. Joshi and R. R. Puri, “Steady-state behavior of three-level systems in a broadband squeezed bath,” Phys. Rev. A 45, 2025–2030 (1992).
[Crossref]

Kim, M. S.

M. S. Kim, F. A. M. de Oliveira, and P. L. Knight, “Properties of squeezed number states and squeezed thermal states,” Phys. Rev. A 40, 2494–2503 (1989).
[Crossref]

Kimble, H. J.

N. P. Georgiades, E. S. Polzik, K. Edamatsu, H. J. Kimble, and A. S. Parkins, “Nonclassical excitation for atoms in a squeezed vacuum,” Phys. Rev. Lett. 75, 3426–3429 (1995).
[Crossref]

E. S. Polzik, J. Carri, and H. J. Kimble, “Spectroscopy with squeezed light,” Phys. Rev. Lett. 68, 3020–3023 (1992).
[Crossref]

E. S. Polzik, J. Carri, and H. J. Kimble, “Atomic spectroscopy with squeezed light for sensitivity beyond the vacuum-state limit,” Appl. Phys. B 55, 279–290 (1992).
[Crossref]

M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, and H. J. Kimble, “Squeezed-state generation by the normal modes of a coupled system,” Phys. Rev. Lett. 59, 198–201 (1987).
[Crossref]

L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520–2523 (1986).
[Crossref]

Knight, P. L.

M. S. Kim, F. A. M. de Oliveira, and P. L. Knight, “Properties of squeezed number states and squeezed thermal states,” Phys. Rev. A 40, 2494–2503 (1989).
[Crossref]

Koch, M.

A. Ourjoumtsev, A. Kubanek, M. Koch, C. Sames, P. W. H. Pinkse, G. Rempe, and K. Murr, “Observation of squeezed light from one atom excited with two photons,” Nature 474, 623–626 (2011).
[Crossref]

Kubanek, A.

A. Ourjoumtsev, A. Kubanek, M. Koch, C. Sames, P. W. H. Pinkse, G. Rempe, and K. Murr, “Observation of squeezed light from one atom excited with two photons,” Nature 474, 623–626 (2011).
[Crossref]

Lamb, W. E.

W. E. Lamb and R. C. Retherford, “Fine structure of the hydrogen atom by a microwave method,” Phys. Rev. 72, 241–243 (1947).
[Crossref]

Lane, A. S.

H. J. Carmichael, A. S. Lane, and D. F. Walls, “Resonance fluorescence from an atom in a squeezed vacuum,” Phys. Rev. Lett. 58, 2539–2542 (1987).
[Crossref]

Lemonde, M.-A.

M.-A. Lemonde, N. Didier, and A. A. Clerk, “Antibunching and unconventional photon blockade with Gaussian squeezed states,” Phys. Rev. A 90, 063824 (2014).
[Crossref]

Li, F.

R. Liu, A. Fang, Y. Zhou, P. Zhang, S. Gao, H. Li, H. Gao, and F. Li, “Enhanced visibility of ghost imaging and interference using squeezed thermal light,” Phys. Rev. A 93, 013822 (2016).
[Crossref]

Li, H.

R. Liu, A. Fang, Y. Zhou, P. Zhang, S. Gao, H. Li, H. Gao, and F. Li, “Enhanced visibility of ghost imaging and interference using squeezed thermal light,” Phys. Rev. A 93, 013822 (2016).
[Crossref]

Liao, J.-Q.

Q.-S. Tan, J.-Q. Liao, X. Wang, and F. Nori, “Enhanced interferometry using squeezed thermal states and even or odd states,” Phys. Rev. A 89, 053822 (2014).
[Crossref]

Liu, R.

R. Liu, A. Fang, Y. Zhou, P. Zhang, S. Gao, H. Li, H. Gao, and F. Li, “Enhanced visibility of ghost imaging and interference using squeezed thermal light,” Phys. Rev. A 93, 013822 (2016).
[Crossref]

Lu, Z. H.

Z. H. Lu, S. Bali, and J. E. Thomas, “Observation of squeezing in the phase-dependent fluorescence spectra of two-level atoms,” Phys. Rev. Lett. 81, 3635–3638 (1998).
[Crossref]

Mavalvala, N.

Mertz, J. C.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[Crossref]

Milburn, G.

G. Milburn, “Interaction of a two-level atom with squeezed light,” Opt. Acta 31, 671–679 (1984).
[Crossref]

Milonni, P. W.

P. W. Milonni and M.-L. Shih, “Zero‐point energy in early quantum theory,” Am. J. Phys. 59, 684–698 (1992).
[Crossref]

Murr, K.

A. Ourjoumtsev, A. Kubanek, M. Koch, C. Sames, P. W. H. Pinkse, G. Rempe, and K. Murr, “Observation of squeezed light from one atom excited with two photons,” Nature 474, 623–626 (2011).
[Crossref]

Nation, P.

J. Johansson, P. Nation, and F. Nori, “QuTiP 2: a Python framework for the dynamics of open quantum systems,” Comput. Phys. Commun. 184, 1234–1240 (2013).
[Crossref]

Nori, F.

Q.-S. Tan, J.-Q. Liao, X. Wang, and F. Nori, “Enhanced interferometry using squeezed thermal states and even or odd states,” Phys. Rev. A 89, 053822 (2014).
[Crossref]

J. Johansson, P. Nation, and F. Nori, “QuTiP 2: a Python framework for the dynamics of open quantum systems,” Comput. Phys. Commun. 184, 1234–1240 (2013).
[Crossref]

Oelker, E.

Orozco, L. A.

M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, and H. J. Kimble, “Squeezed-state generation by the normal modes of a coupled system,” Phys. Rev. Lett. 59, 198–201 (1987).
[Crossref]

Ourjoumtsev, A.

A. Ourjoumtsev, A. Kubanek, M. Koch, C. Sames, P. W. H. Pinkse, G. Rempe, and K. Murr, “Observation of squeezed light from one atom excited with two photons,” Nature 474, 623–626 (2011).
[Crossref]

Paisner, J. A.

S. Haroche, J. A. Paisner, and A. L. Schawlow, “Hyperfine quantum beats observed in Cs vapor under pulsed dye laser excitation,” Phys. Rev. Lett. 30, 948–951 (1973).
[Crossref]

Parkins, A. S.

N. P. Georgiades, E. S. Polzik, K. Edamatsu, H. J. Kimble, and A. S. Parkins, “Nonclassical excitation for atoms in a squeezed vacuum,” Phys. Rev. Lett. 75, 3426–3429 (1995).
[Crossref]

A. S. Parkins and C. W. Gardiner, “Inhibition of atomic phase decays by squeezed light in a microscopic Fabry-Pérot cavity,” Phys. Rev. A 40, 3796–3807 (1989).
[Crossref]

Pinkse, P. W. H.

A. Ourjoumtsev, A. Kubanek, M. Koch, C. Sames, P. W. H. Pinkse, G. Rempe, and K. Murr, “Observation of squeezed light from one atom excited with two photons,” Nature 474, 623–626 (2011).
[Crossref]

Polder, D.

H. B. G. Casimir and D. Polder, “The influence of retardation on the London-van der Waals forces,” Phys. Rev. 73, 360–372 (1948).
[Crossref]

Polzik, E. S.

N. P. Georgiades, E. S. Polzik, K. Edamatsu, H. J. Kimble, and A. S. Parkins, “Nonclassical excitation for atoms in a squeezed vacuum,” Phys. Rev. Lett. 75, 3426–3429 (1995).
[Crossref]

E. S. Polzik, J. Carri, and H. J. Kimble, “Atomic spectroscopy with squeezed light for sensitivity beyond the vacuum-state limit,” Appl. Phys. B 55, 279–290 (1992).
[Crossref]

E. S. Polzik, J. Carri, and H. J. Kimble, “Spectroscopy with squeezed light,” Phys. Rev. Lett. 68, 3020–3023 (1992).
[Crossref]

Puri, R. R.

A. Joshi and R. R. Puri, “Steady-state behavior of three-level systems in a broadband squeezed bath,” Phys. Rev. A 45, 2025–2030 (1992).
[Crossref]

Raizen, M. G.

M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, and H. J. Kimble, “Squeezed-state generation by the normal modes of a coupled system,” Phys. Rev. Lett. 59, 198–201 (1987).
[Crossref]

Rempe, G.

A. Ourjoumtsev, A. Kubanek, M. Koch, C. Sames, P. W. H. Pinkse, G. Rempe, and K. Murr, “Observation of squeezed light from one atom excited with two photons,” Nature 474, 623–626 (2011).
[Crossref]

Retherford, R. C.

W. E. Lamb and R. C. Retherford, “Fine structure of the hydrogen atom by a microwave method,” Phys. Rev. 72, 241–243 (1947).
[Crossref]

Sames, C.

A. Ourjoumtsev, A. Kubanek, M. Koch, C. Sames, P. W. H. Pinkse, G. Rempe, and K. Murr, “Observation of squeezed light from one atom excited with two photons,” Nature 474, 623–626 (2011).
[Crossref]

Sandoghdar, V.

V. Sandoghdar, C. I. Sukenik, E. A. Hinds, and S. Haroche, “Direct measurement of the van der Waals interaction between an atom and its images in a micron-sized cavity,” Phys. Rev. Lett. 68, 3432–3435 (1992).
[Crossref]

Schawlow, A. L.

S. Haroche, J. A. Paisner, and A. L. Schawlow, “Hyperfine quantum beats observed in Cs vapor under pulsed dye laser excitation,” Phys. Rev. Lett. 30, 948–951 (1973).
[Crossref]

Scully, M. O.

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

Shih, M.-L.

P. W. Milonni and M.-L. Shih, “Zero‐point energy in early quantum theory,” Am. J. Phys. 59, 684–698 (1992).
[Crossref]

Sigg, D.

Slusher, R. E.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[Crossref]

Sukenik, C. I.

V. Sandoghdar, C. I. Sukenik, E. A. Hinds, and S. Haroche, “Direct measurement of the van der Waals interaction between an atom and its images in a micron-sized cavity,” Phys. Rev. Lett. 68, 3432–3435 (1992).
[Crossref]

Tan, Q.-S.

Q.-S. Tan, J.-Q. Liao, X. Wang, and F. Nori, “Enhanced interferometry using squeezed thermal states and even or odd states,” Phys. Rev. A 89, 053822 (2014).
[Crossref]

Thomas, J. E.

Z. H. Lu, S. Bali, and J. E. Thomas, “Observation of squeezing in the phase-dependent fluorescence spectra of two-level atoms,” Phys. Rev. Lett. 81, 3635–3638 (1998).
[Crossref]

Valley, J. F.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[Crossref]

Vogel, W.

P. Grünwald and W. Vogel, “Optimal squeezing in resonance fluorescence via atomic-state purification,” Phys. Rev. Lett. 109, 013601 (2012).
[Crossref]

Walls, D. F.

H. J. Carmichael, A. S. Lane, and D. F. Walls, “Resonance fluorescence from an atom in a squeezed vacuum,” Phys. Rev. Lett. 58, 2539–2542 (1987).
[Crossref]

D. F. Walls and P. Zoller, “Reduced quantum fluctuations in resonance fluorescence,” Phys. Rev. Lett. 47, 709–711 (1981).
[Crossref]

Wang, X.

Q.-S. Tan, J.-Q. Liao, X. Wang, and F. Nori, “Enhanced interferometry using squeezed thermal states and even or odd states,” Phys. Rev. A 89, 053822 (2014).
[Crossref]

Wu, H.

L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520–2523 (1986).
[Crossref]

Wu, L.-A.

L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520–2523 (1986).
[Crossref]

Xiao, M.

M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, and H. J. Kimble, “Squeezed-state generation by the normal modes of a coupled system,” Phys. Rev. Lett. 59, 198–201 (1987).
[Crossref]

Young, J. F.

Yurke, B.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[Crossref]

Zhang, P.

R. Liu, A. Fang, Y. Zhou, P. Zhang, S. Gao, H. Li, H. Gao, and F. Li, “Enhanced visibility of ghost imaging and interference using squeezed thermal light,” Phys. Rev. A 93, 013822 (2016).
[Crossref]

Zhou, Y.

R. Liu, A. Fang, Y. Zhou, P. Zhang, S. Gao, H. Li, H. Gao, and F. Li, “Enhanced visibility of ghost imaging and interference using squeezed thermal light,” Phys. Rev. A 93, 013822 (2016).
[Crossref]

Zoller, P.

D. F. Walls and P. Zoller, “Reduced quantum fluctuations in resonance fluorescence,” Phys. Rev. Lett. 47, 709–711 (1981).
[Crossref]

Zubairy, M. S.

J. Anwar and M. S. Zubairy, “Effect of squeezing on the degenerate parametric oscillator,” Phys. Rev. A 45, 1804–1809 (1992).
[Crossref]

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

Am. J. Phys. (1)

P. W. Milonni and M.-L. Shih, “Zero‐point energy in early quantum theory,” Am. J. Phys. 59, 684–698 (1992).
[Crossref]

Appl. Phys. B (1)

E. S. Polzik, J. Carri, and H. J. Kimble, “Atomic spectroscopy with squeezed light for sensitivity beyond the vacuum-state limit,” Appl. Phys. B 55, 279–290 (1992).
[Crossref]

Comput. Phys. Commun. (1)

J. Johansson, P. Nation, and F. Nori, “QuTiP 2: a Python framework for the dynamics of open quantum systems,” Comput. Phys. Commun. 184, 1234–1240 (2013).
[Crossref]

Europhys. Lett. (1)

Z. Ficek and P. D. Drummond, “Two-photon population inversion by squeezed light in a Fabry-Perot microcavity,” Europhys. Lett. 24, 455–460 (1993).
[Crossref]

J. Mod. Opt. (1)

H. Fearn and M. Collett, “Representations of squeezed states with thermal noise,” J. Mod. Opt. 35, 553–564 (1988), doi: 10.1080/09500348814550571.
[Crossref]

Nature (1)

A. Ourjoumtsev, A. Kubanek, M. Koch, C. Sames, P. W. H. Pinkse, G. Rempe, and K. Murr, “Observation of squeezed light from one atom excited with two photons,” Nature 474, 623–626 (2011).
[Crossref]

Nuovo Cimento B (1)

B. Fain, “Spontaneous emission vs. vacuum fluctuations,” Nuovo Cimento B 68, 73–78 (1982).
[Crossref]

Opt. Acta (1)

G. Milburn, “Interaction of a two-level atom with squeezed light,” Opt. Acta 31, 671–679 (1984).
[Crossref]

Opt. Express (2)

Phys. Rev. (3)

W. E. Lamb and R. C. Retherford, “Fine structure of the hydrogen atom by a microwave method,” Phys. Rev. 72, 241–243 (1947).
[Crossref]

H. A. Bethe, “The electromagnetic shift of energy levels,” Phys. Rev. 72, 339–341 (1947).
[Crossref]

H. B. G. Casimir and D. Polder, “The influence of retardation on the London-van der Waals forces,” Phys. Rev. 73, 360–372 (1948).
[Crossref]

Phys. Rev. A (10)

M. S. Kim, F. A. M. de Oliveira, and P. L. Knight, “Properties of squeezed number states and squeezed thermal states,” Phys. Rev. A 40, 2494–2503 (1989).
[Crossref]

Q.-S. Tan, J.-Q. Liao, X. Wang, and F. Nori, “Enhanced interferometry using squeezed thermal states and even or odd states,” Phys. Rev. A 89, 053822 (2014).
[Crossref]

R. Liu, A. Fang, Y. Zhou, P. Zhang, S. Gao, H. Li, H. Gao, and F. Li, “Enhanced visibility of ghost imaging and interference using squeezed thermal light,” Phys. Rev. A 93, 013822 (2016).
[Crossref]

M.-A. Lemonde, N. Didier, and A. A. Clerk, “Antibunching and unconventional photon blockade with Gaussian squeezed states,” Phys. Rev. A 90, 063824 (2014).
[Crossref]

J. Anwar and M. S. Zubairy, “Effect of squeezing on the degenerate parametric oscillator,” Phys. Rev. A 45, 1804–1809 (1992).
[Crossref]

A. S. Parkins and C. W. Gardiner, “Inhibition of atomic phase decays by squeezed light in a microscopic Fabry-Pérot cavity,” Phys. Rev. A 40, 3796–3807 (1989).
[Crossref]

Z. Ficek and P. D. Drummond, “Three-level atom in a broadband squeezed vacuum field. I. General theory,” Phys. Rev. A 43, 6247–6257 (1991).
[Crossref]

Z. Ficek and P. D. Drummond, “Three-level atom in a broadband squeezed vacuum field. II. Applications,” Phys. Rev. A 43, 6258–6271 (1991).
[Crossref]

A. Joshi and R. R. Puri, “Steady-state behavior of three-level systems in a broadband squeezed bath,” Phys. Rev. A 45, 2025–2030 (1992).
[Crossref]

M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386–1391 (1984).
[Crossref]

Phys. Rev. Lett. (11)

N. P. Georgiades, E. S. Polzik, K. Edamatsu, H. J. Kimble, and A. S. Parkins, “Nonclassical excitation for atoms in a squeezed vacuum,” Phys. Rev. Lett. 75, 3426–3429 (1995).
[Crossref]

E. S. Polzik, J. Carri, and H. J. Kimble, “Spectroscopy with squeezed light,” Phys. Rev. Lett. 68, 3020–3023 (1992).
[Crossref]

H. J. Carmichael, A. S. Lane, and D. F. Walls, “Resonance fluorescence from an atom in a squeezed vacuum,” Phys. Rev. Lett. 58, 2539–2542 (1987).
[Crossref]

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[Crossref]

L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520–2523 (1986).
[Crossref]

D. F. Walls and P. Zoller, “Reduced quantum fluctuations in resonance fluorescence,” Phys. Rev. Lett. 47, 709–711 (1981).
[Crossref]

M. G. Raizen, L. A. Orozco, M. Xiao, T. L. Boyd, and H. J. Kimble, “Squeezed-state generation by the normal modes of a coupled system,” Phys. Rev. Lett. 59, 198–201 (1987).
[Crossref]

Z. H. Lu, S. Bali, and J. E. Thomas, “Observation of squeezing in the phase-dependent fluorescence spectra of two-level atoms,” Phys. Rev. Lett. 81, 3635–3638 (1998).
[Crossref]

P. Grünwald and W. Vogel, “Optimal squeezing in resonance fluorescence via atomic-state purification,” Phys. Rev. Lett. 109, 013601 (2012).
[Crossref]

V. Sandoghdar, C. I. Sukenik, E. A. Hinds, and S. Haroche, “Direct measurement of the van der Waals interaction between an atom and its images in a micron-sized cavity,” Phys. Rev. Lett. 68, 3432–3435 (1992).
[Crossref]

S. Haroche, J. A. Paisner, and A. L. Schawlow, “Hyperfine quantum beats observed in Cs vapor under pulsed dye laser excitation,” Phys. Rev. Lett. 30, 948–951 (1973).
[Crossref]

Proc. R. Soc. A (1)

P. A. M. Dirac, “The quantum theory of the emission and absorption of radiation,” Proc. R. Soc. A 114, 243–265 (1927).
[Crossref]

Rev. Mod. Phys. (1)

R. X. Adhikari, “Gravitational radiation detection with laser interferometry,” Rev. Mod. Phys. 86, 121–151 (2014).
[Crossref]

Other (5)

S. E. Dwyer, Quantum noise reduction using squeezed states in LIGO, Ph.D. thesis (Massachusetts Institute of Technology, 2013).

M. Fox, Quantum Optics: An Introduction (Oxford University, 2006).

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

J. Garrison and R. Chiao, Quantum Optics (Oxford University, 2014).

We note that Eqs. (14) and (18) are also valid for g→g(t), where g(t) is a real function of time.

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Figures (7)

Fig. 1.
Fig. 1.

Time dependence of (a) squeezing amplitude and (b) thermal photon number in the case of weak pumping (g=0.8), critical pumping (g=1.0), and strong pumping (g=1.2).

Fig. 2.
Fig. 2.

Quadrature noises and the expectation value of the total photon number as a function of time for (a) weak pumping (g=0.8) and (b) strong pumping (g=1.2).

Fig. 3.
Fig. 3.

Product of the quadrature noises as a function of time for g=1.2 calculated numerically using a number state basis. The number of basis states used in a given calculation is indicated in the legend. The black solid line represents the semi-analytic result, and the dashed line shows the total photon number.

Fig. 4.
Fig. 4.

Steady state second order correlation function (solid line), total photon number (dashed line), and thermal photon number (dotted line) as a function of pump-to-loss ratio, g. The inset shows the second order correlation function for a steady state STS generated with sub-critical pumping (solid) and a SVS (dashed) as a function of the total photon number.

Fig. 5.
Fig. 5.

(a) Quadrature noise in X as a function of time in the strong pumping regime for g=5, 10, 50, and 100. The steady state limits given by Eq. (31) are shown as solid lines, while the solid circles indicate the time ΔX reaches 20% above the corresponding steady state value (i.e., δ=0.2). (b) Quantity (2nth+1) at the threshold times corresponding to δ=0.1 and δ=0.2 as a function of the pump-to-loss ratio, g.

Fig. 6.
Fig. 6.

Plots of the Wigner function for different values of δ and pump-to-loss ratio: (a) g=0.8, δ=0 and (b) g=100, δ=0.2. Note the different scaling.

Fig. 7.
Fig. 7.

Second order correlation function at the threshold times corresponding to δ=0.1 and δ=0.2 as a function of pump-to-loss ratio, g.

Tables (1)

Tables Icon

Table 1. Squeezing Properties of Weak and Strong Pumping Regimes at δ=0 and δ=0.2

Equations (95)

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

H=ωbb+ωPaa+iχeff(2)(b2ab2a),
H=H0+(α(t)γbb+α*(t)γ*bb),
ρ˙=i[H,ρ]+Γ(bρb12bbρ12ρbb),
Uint(t)=eit(γ*α0*bb+γα0bb)/,
S(ξ)|0=|ξ,
ρ˙Γ=i[H,ρ]+(bρb12bbρ12ρbb),
ρ(t)=S(ξ(t))ρT(β(t))S(ξ(t)),
ρT(β(t))=(1eβ(t)ω)eβ(t)H0
nth(t)1eβ(t)ω1,
ρ(t)=S(ξ(t))ρT1/2(nth)O(t)ρT1/2(nth)S(ξ(t)),
O(t)=ρT1/2S(ξ)ρ(t)S(ξ)ρT1/2.
du(t)dt=i(γα(t)eiϕ(t)γ*α*(t)eiϕ(t))Γcs2nth(t)+1,
dϕ(t)dt=2ω+1c2+s2cs(γα(t)eiϕ(t)+γ*α*(t)eiϕ(t)),
dnth(t)dt=Γ(s2nth(t)),
(γα(t)eiϕ(t)+γ*α(t)*eiϕ(t))=0.
dϕ(t)dt=2ω,
Re{|γα0|ei(θϕ0)}=0,
1Γdu(t)dt=g2cs2nth(t)+1,
b=eiωt(X+iY2).
ΔXΔY1.
n=bb=sinh2(u).
ρ=m=0nthm(nth+1)m+1S(ξ)|mm|S(ξ),
(ΔX)2=(2nth+1)e2u,
(ΔY)2=(2nth+1)e2u,
n=bb=nthcosh(2u)+sinh2(u).
g(2)=bbbbbb2.
g(2)=2+(nth+12)2sinh2(2u)(nthcosh(2u)+sinh2(u))2,
g(2)=3+1n,
sinh2(u)=nthss
u=12tanh1(g),
ΔXss=11+g
ΔYss=11g
ΔXssΔYss=11g2,
nthss=11g221g2
nss=g22(1g2).
gss(2)=3+1g2g2,
gss(2)=3+14nthss(nthss+1),
gss(2)=3+12nss.
ΔX=1+δ1+g,
O˙(t)=O˙I(t)+O˙II(t)+O˙III(t)+O˙IV(t).
O˙I=dρT1/2dtρT1/2ρT1/2SρSρT1/2+ρT1/2SρSρT1/2ρT1/2dρT1/2dt,
O˙I={J,O},
J=dρT1/2dtρT1/2=ρT1/2dρT1/2dt.
J=12xdxdt(x1xbb).
J=12xdxdt(nthbb).
O˙II=ρT1/2dSdtSρT1/2(ρT1/2SρSρT1/2)+(ρT1/2SρSρT1/2)ρT1/2SdSdtρT1/2,
O˙II=LO+OL,
L=ρT1/2dSdtSρT1/2,
L=ρT1/2SdSdtρT1/2.
S(ξ)=e12(ξ*b2ξb2),
L=(is2ϕ˙)(bb+12)+12u˙(x1b2eiϕxb2eiϕ)+12icsϕ˙(x1b2eiϕ+xb2eiϕ),
O˙II=(M+iN)O+O(MiN)={M,O}+i[N,O].
O˙III=O˙0+O˙V+O˙L,
O˙0=ρT1/2S(i[ωbb,ρ])SρT1/2
O˙V=ρT1/2S(i[V(t),ρ])SρT1/2,
O˙LΓ=ρT1/2SbρbSρT1/212ρT1/2S(i{bb,ρ})SρT1/2.
O˙0iω=GOOG,
G=ρT1/2SbSρT1/2.
O˙0=iω[P,O]+ω{Q,O}.
O˙V(t)=i[P¯,O]+1{Q¯,O}
O˙L=ΓF12Γ{P,O}i2Γ[Q,O],
F=ρT1/2SbρbSρT1/2=TOT,
T=ρT1/2SbρSρT1/2,
P¯=cs(γαeiϕ+γ*α*eiϕ)2cs(γαeiϕ+γ*α*eiϕ)bb+12(γα(x1+x)c2+γ*α*(x1+x)s2e2iϕ)bb+12(γα(x1+x)s2eiϕ+γ*α*(x1+x)c2)bb,
Q¯=i2(x1x)(γ*α*s2e2iϕ+γαc2)bb+i2(x1x)(γαs2e2iϕ+γ*α*c2)bb,
O˙(t)={J+M+ωQ+1Q¯,O}i[ωP+1P¯N,O]+ΓTOT12Γ{P,O}i2[Q,O].
2J+2M+2ωQ+2Q¯+Γ(TTP)=0.
M=12(L+L)=14u˙(x1x)((b)2eiϕ+b2eiϕ)+14icsϕ˙(x1x)((b)2eiϕb2eiϕ)
N=LL2i=s2ϕ˙(bb+12)14iu˙(x1+x)((b)2eiϕb2eiϕ)+14csϕ˙(x1+x)((b)2eiϕ+b2eiϕ).
G=s2+(c2+s2)bbcs(x1(b)2eiϕ+x(b)2eiϕ),
P=G+G2=s2+(c2+s2)bb12cs(x1+x)((b)2eiϕ+x(b)2eiϕ)
Q=GG2i=12ics(x1x)((b)2eiϕ(b)2eiϕ).
T=x1/2cbx1/2seiϕb,
TT=xc2+(xc2+x1s2)bbcs(b2eiϕ+(b)2eiϕ).
0=1xdxdt(nthbb)+12u˙(x1x)((b)2eiϕ+(b)2eiϕ)+12icsϕ˙((b)2eiϕ(b)2eiϕ)+iωcs(x1x)((b)2eiϕ(b)2eiϕ)+i(x1x)(γαs2e2iϕ+γ*α*c2)bbi(x1x)(γ*α*s2e2iϕ+γαc2)bb+Γ(xc2+(xc2+x1s2)bbcs(b2eiϕ+(b)2eiϕ))Γ(s2+(c2+s2)bb12cs(x1+x)((b)2eiϕ+b2eiϕ)).
χ1=(b)2eiϕ+b2eiϕ,
χ2=i((b)2eiϕb2eiϕ).
χ1iχ2=2(b)2eiϕ,
(b)2=12eiϕ(χ1iχ2),
b2=12eiϕ(χ1+iχ2).
0=1xdxdt(nthbb)+12u˙(x1x)χ1+12csϕ˙(x1x)χ2+ωcs(x1x)χ2+i2(x1x)(γαs2eiϕ+γ*α*c2eiϕ)(χ1+iχ2)i2(x1x)(γ*α*s2eiϕ+γαc2eiϕ)(χ1iχ2)+Γ(xc2+(xc2+x1s2)bbcsχ1)Γ(s2+(c2+s2)bb12cs(x1+x)χ1),
0=F1χ1+F2χ2+F3bb+F4,
F1=12u˙(x1x)+i2(x1x)(γ*α*eiϕγαeiϕ)Γcs+12Γcs(x1+x),
F2=12csϕ˙(x1x)+ωcs(x1x)12(x1x)(c2+s2)(γαeiϕ+γ*α*eiϕ),
F3=1xdxdt+Γ(xc2+x1s2)Γ(c2+s2),
F4=1xdxdtnth+Γ(xc2s2).
F1=F2=F3=F4=0.
u˙=i(γαeiϕγ*α*eiϕ)+2Γcs(x1x)Γcs(x1+x)(x1x),
u˙=i(γαeiϕγ*α*eiϕ)Γcs2nth+1.
12csϕ˙+ωcs12(c2+s2)(γαeiϕ+γ*α*eiϕ)=0,
ϕ˙=2ω+1c2+s2cs(γαeiϕ+γ*α*eiϕ),
1xdxdt=Γ(xc2+x1s2c2s2),
dxdt=Γ(1x)(s2(1x)x).
dnthdt=ddt(x1x)=1(1x)2dxdt,
dnthdt=Γ(s2x1x)=Γ(s2nth),

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