Abstract

In this paper, we consider the interaction of the nonlinear coherent states (CSs) on a sphere with a three-level atom. Since these generalized CSs depend on the curvature of the sphere, this model enables us to investigate the curvature effects of the physical space. By using the time-dependent state of the atom-field system, we first study the curvature effects on the occupation probabilities of the atomic levels. We especially study the relation between the revival time of the atomic occupation probabilities and the curvature. Then, to study the curvature effects on the dynamical properties of the cavity field, we consider photon distributions, correlation functions, and Mandel parameters of the field. The cavity field in this atom-field system exhibits nonclassical features which depend on the curvature of the physical space.

© 2013 Optical Society of America

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    [CrossRef]
  44. N. Ashby, “Relativity in the global positioning system,” Living Rev. Relativity 6, 1–45 (2003).
    [CrossRef]
  45. H. Müller, A. Peters, and S. Chu, “A precision measurement of the gravitational redshift by the interference of matter waves,” Nature 463, 926–929 (2010).
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2013

A. Mahdifar, “Coherent states for nonlinear two-boson realization of the isotropic oscillator algebra on a sphere,” Int. J. Geom. Methods Mod. Phys. 10, 1350028 (2013).
[CrossRef]

2012

A. Mahdifar, R. Roknizadeh, and M. H. Naderi, “Detection of the spatial curvature effects through physical phenomena: the nonlinear coherent states approach,” Int. J. Geom. Methods Mod. Phys. 9, 1250009 (2012).
[CrossRef]

2010

Sh. Barzanjeh, M. H. Naderi, and M. Soltanolkotabi, “Dynamical behaviours of the nonlinear atom-field interaction in the presence of classical gravity: f-deformation approach,” J. Phys. A 43, 375304 (2010).
[CrossRef]

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, “Optics in curved space,” Phys. Rev. Lett. 105, 143901 (2010).
[CrossRef]

H. Müller, A. Peters, and S. Chu, “A precision measurement of the gravitational redshift by the interference of matter waves,” Nature 463, 926–929 (2010).
[CrossRef]

C. W. Chou, “Optical clocks and relativity,” Science 329, 1630–1633 (2010).
[CrossRef]

2008

S. Batz and U. Peschel, “Linear and nonlinear optics in curved space,” Phys. Rev. A 78, 043821 (2008).
[CrossRef]

T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science 319, 1367–1370 (2008).
[CrossRef]

2007

M. Mohammadi, M. H. Naderi, and M. Soltanolkotabi, “Quantum statistical properties of the Jaynes–Cummings model in the presence of a classical homogeneous gravitational field,” J. Phys. A 40, 1377–1393 (2007).
[CrossRef]

2006

A. Mahdifar, R. Roknizadeh, and M. H. Naderi, “Geometric approach to nonlinear coherent states using the Higgs model for harmonic oscillator,” J. Phys. A 39, 7003–7014 (2006).
[CrossRef]

2005

R. Schutzhold and W. G. Unruh, “Hawking radiation in an electromagnetic waveguide?” Phys. Rev. Lett. 95, 031301 (2005).
[CrossRef]

J. Gravesen, M. Willatzen, and L. L. Y. Voon, “Quantum-mechanical particle confined to surfaces of revolution—truncated cone and elliptic torus case studies,” Phys. Scr. 72, 105–111 (2005).
[CrossRef]

J. Gravesen, M. Willatzen, and L. L. Y. Voon, “Schrödinger problems for surfaces of revolution—the finite cylinder as a test example,” J. Math. Phys. 46, 012107 (2005).
[CrossRef]

2003

C. Barcelo, S. Liberati, and M. Visser, “Probing semiclassical analog gravity in Bose–Einstein condensates with widely tunable interactions,” Phys. Rev. A 68, 053613 (2003).
[CrossRef]

P. O. Fedichev and U. R. Fischer, “Gibbons–Hawking effect in the sonic de sitter space-time of an expanding Bose–Einstein-condensed gas,” Phys. Rev. Lett. 91, 240407 (2003).
[CrossRef]

N. Ashby, “Relativity in the global positioning system,” Living Rev. Relativity 6, 1–45 (2003).
[CrossRef]

2002

R. Schutzhold and W. G. Unruh, “Gravity wave analogues of black holes,” Phys. Rev. D 66, 044019 (2002).
[CrossRef]

U. Leonhardt, “A laboratory analogue of the event horizon using slow light in an atomic medium,” Nature 415, 406–409 (2002).
[CrossRef]

V. V. Nesvizhevsky, H. G. Börner, A. K. Petukhov, H. Abele, S. Baeler, F. J. Rue, Th. Stöferle, A. Westphal, A. M. Gagarski, G. A. Petrov, and A. V. Strelkov, “Quantum states of neutrons in the Earth’s gravitational field,” Nature 415, 297–299 (2002).
[CrossRef]

2000

I. Bloch, T. W. Hänsch, and T. Esslinger, “Measurement of the spatial coherence of a trapped Bose gas at the phase transition,” Nature 403, 166–170 (2000).
[CrossRef]

P. Aniello, V. I. Man’ko, G. Marmo, S. Solimeno, and F. Zaccaria, “On the coherent states, displacement operators and quasidistributions associated with deformed quantum oscillators,” J. Opt. B 2, 718–725 (2000).
[CrossRef]

1998

M. Encinosa and B. Etemadi, “Energy shifts resulting from surface curvature of quantum nanostructures,” Phys. Rev. A 58, 77–81 (1998).
[CrossRef]

1997

V. I. Man’ko, G. Marmo, E. C. G. Sudarshan, and F. Zaccaria, “f-oscillators and nonlinear coherent states,” Phys. Scr. 55, 528–541 (1997).
[CrossRef]

1996

R. L. de Matos Filho and W. Vogel, “Nonlinear coherent states,” Phys. Rev. A 54, 4560–4563 (1996).
[CrossRef]

1995

B. Deb, G. Gangopadhyay, and D. Sh. Ray, “Generation of a class of arbitrary two-mode field states in a cavity,” Phys. Rev. A 51, 2651–2653 (1995).
[CrossRef]

W. Vogel and R. L. de Matos Filho, “Nonlinear Jaynes–Cummings dynamics of a trapped ion,” Phys. Rev. A 52, 4214–4217 (1995).
[CrossRef]

1994

A. I. Solomon, “A characteristic functional for deformed photon phenomenology,” Phys. Lett. A 196, 29–34 (1994).
[CrossRef]

J. Katriel and A. I. Solomon, “Nonideal lasers, nonclassical light, and deformed photon states,” Phys. Rev. A 49, 5149–5151 (1994).
[CrossRef]

P. Shanta, S. Chaturvedi, V. Srinivasan, and R. Jagannathan, “Unified approach to the analogues of single-photon and multiphoton coherent states for generalized bosonic oscillators,” J. Phys. A 27, 6433–6442 (1994).
[CrossRef]

1993

K. Vogel, V. M. Akulin, and W. P. Schleich, “Quantum state engineering of the radiation field,” Phys. Rev. Lett. 71, 1816–1819 (1993).
[CrossRef]

1991

W. K. Lai, V. Buzek, and P. L. Knight, “Interaction of a three-level atom with an SU(2) coherent state,” Phys. Rev. A 44, 2003–2012 (1991).
[CrossRef]

1985

H.-I. Yoo and J. H. Eberly, “Dynamical theory of an atom with two or three levels interacting with quantized cavity fields,” Phys. Rep. 118, 239–337 (1985).
[CrossRef]

1981

W. G. Unruh, “Experimental black-hole evaporation?” Phys. Rev. Lett. 46, 1351–1353 (1981).
[CrossRef]

R. C. T. da Costa, “Quantum mechanics of a constrained particle,” Phys. Rev. A 23, 1982–1987 (1981).
[CrossRef]

1979

D. Walsh, R. F. Carswell, and R. J. Weymann, “0957 + 561 A, B: twin quasistellar objects or gravitational lens?” Nature 279, 381–384 (1979).
[CrossRef]

1963

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
[CrossRef]

R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
[CrossRef]

R. J. Glauber, “Photon correlations,” Phys. Rev. Lett. 10, 84–86 (1963).
[CrossRef]

1960

R. V. Pound and G. A. Rebka, “Apparent weight of photons,” Phys. Rev. Lett. 4, 337–341 (1960).
[CrossRef]

1920

F. W. Dyson, A. S. Eddington, and C. Davidson, “A determination of the deflection of light by the Sun’s gravitational field, from observations made at the total eclipse of 29 May 1919,” Philos. Trans. R. Soc. Lond. A 220, 291–333 (1920).
[CrossRef]

Abele, H.

V. V. Nesvizhevsky, H. G. Börner, A. K. Petukhov, H. Abele, S. Baeler, F. J. Rue, Th. Stöferle, A. Westphal, A. M. Gagarski, G. A. Petrov, and A. V. Strelkov, “Quantum states of neutrons in the Earth’s gravitational field,” Nature 415, 297–299 (2002).
[CrossRef]

Akulin, V. M.

K. Vogel, V. M. Akulin, and W. P. Schleich, “Quantum state engineering of the radiation field,” Phys. Rev. Lett. 71, 1816–1819 (1993).
[CrossRef]

Aniello, P.

P. Aniello, V. I. Man’ko, G. Marmo, S. Solimeno, and F. Zaccaria, “On the coherent states, displacement operators and quasidistributions associated with deformed quantum oscillators,” J. Opt. B 2, 718–725 (2000).
[CrossRef]

Antoine, J.-P.

S. Twareqe Ali, J.-P. Antoine, and J.-P. Gazeau, Coherent States, Wavelets and Their Generalizations (Springer-Verlag, 2000).

Ashby, N.

N. Ashby, “Relativity in the global positioning system,” Living Rev. Relativity 6, 1–45 (2003).
[CrossRef]

Baeler, S.

V. V. Nesvizhevsky, H. G. Börner, A. K. Petukhov, H. Abele, S. Baeler, F. J. Rue, Th. Stöferle, A. Westphal, A. M. Gagarski, G. A. Petrov, and A. V. Strelkov, “Quantum states of neutrons in the Earth’s gravitational field,” Nature 415, 297–299 (2002).
[CrossRef]

Barcelo, C.

C. Barcelo, S. Liberati, and M. Visser, “Probing semiclassical analog gravity in Bose–Einstein condensates with widely tunable interactions,” Phys. Rev. A 68, 053613 (2003).
[CrossRef]

Barzanjeh, Sh.

Sh. Barzanjeh, M. H. Naderi, and M. Soltanolkotabi, “Dynamical behaviours of the nonlinear atom-field interaction in the presence of classical gravity: f-deformation approach,” J. Phys. A 43, 375304 (2010).
[CrossRef]

Batz, S.

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, “Optics in curved space,” Phys. Rev. Lett. 105, 143901 (2010).
[CrossRef]

S. Batz and U. Peschel, “Linear and nonlinear optics in curved space,” Phys. Rev. A 78, 043821 (2008).
[CrossRef]

Bloch, I.

I. Bloch, T. W. Hänsch, and T. Esslinger, “Measurement of the spatial coherence of a trapped Bose gas at the phase transition,” Nature 403, 166–170 (2000).
[CrossRef]

Börner, H. G.

V. V. Nesvizhevsky, H. G. Börner, A. K. Petukhov, H. Abele, S. Baeler, F. J. Rue, Th. Stöferle, A. Westphal, A. M. Gagarski, G. A. Petrov, and A. V. Strelkov, “Quantum states of neutrons in the Earth’s gravitational field,” Nature 415, 297–299 (2002).
[CrossRef]

Buzek, V.

W. K. Lai, V. Buzek, and P. L. Knight, “Interaction of a three-level atom with an SU(2) coherent state,” Phys. Rev. A 44, 2003–2012 (1991).
[CrossRef]

Carswell, R. F.

D. Walsh, R. F. Carswell, and R. J. Weymann, “0957 + 561 A, B: twin quasistellar objects or gravitational lens?” Nature 279, 381–384 (1979).
[CrossRef]

Chaturvedi, S.

P. Shanta, S. Chaturvedi, V. Srinivasan, and R. Jagannathan, “Unified approach to the analogues of single-photon and multiphoton coherent states for generalized bosonic oscillators,” J. Phys. A 27, 6433–6442 (1994).
[CrossRef]

Chou, C. W.

C. W. Chou, “Optical clocks and relativity,” Science 329, 1630–1633 (2010).
[CrossRef]

Chu, S.

H. Müller, A. Peters, and S. Chu, “A precision measurement of the gravitational redshift by the interference of matter waves,” Nature 463, 926–929 (2010).
[CrossRef]

da Costa, R. C. T.

R. C. T. da Costa, “Quantum mechanics of a constrained particle,” Phys. Rev. A 23, 1982–1987 (1981).
[CrossRef]

Davidson, C.

F. W. Dyson, A. S. Eddington, and C. Davidson, “A determination of the deflection of light by the Sun’s gravitational field, from observations made at the total eclipse of 29 May 1919,” Philos. Trans. R. Soc. Lond. A 220, 291–333 (1920).
[CrossRef]

de Matos Filho, R. L.

R. L. de Matos Filho and W. Vogel, “Nonlinear coherent states,” Phys. Rev. A 54, 4560–4563 (1996).
[CrossRef]

W. Vogel and R. L. de Matos Filho, “Nonlinear Jaynes–Cummings dynamics of a trapped ion,” Phys. Rev. A 52, 4214–4217 (1995).
[CrossRef]

Deb, B.

B. Deb, G. Gangopadhyay, and D. Sh. Ray, “Generation of a class of arbitrary two-mode field states in a cavity,” Phys. Rev. A 51, 2651–2653 (1995).
[CrossRef]

Dreisow, F.

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, “Optics in curved space,” Phys. Rev. Lett. 105, 143901 (2010).
[CrossRef]

Dyson, F. W.

F. W. Dyson, A. S. Eddington, and C. Davidson, “A determination of the deflection of light by the Sun’s gravitational field, from observations made at the total eclipse of 29 May 1919,” Philos. Trans. R. Soc. Lond. A 220, 291–333 (1920).
[CrossRef]

Eberly, J. H.

H.-I. Yoo and J. H. Eberly, “Dynamical theory of an atom with two or three levels interacting with quantized cavity fields,” Phys. Rep. 118, 239–337 (1985).
[CrossRef]

Eddington, A. S.

F. W. Dyson, A. S. Eddington, and C. Davidson, “A determination of the deflection of light by the Sun’s gravitational field, from observations made at the total eclipse of 29 May 1919,” Philos. Trans. R. Soc. Lond. A 220, 291–333 (1920).
[CrossRef]

Encinosa, M.

M. Encinosa and B. Etemadi, “Energy shifts resulting from surface curvature of quantum nanostructures,” Phys. Rev. A 58, 77–81 (1998).
[CrossRef]

Esslinger, T.

I. Bloch, T. W. Hänsch, and T. Esslinger, “Measurement of the spatial coherence of a trapped Bose gas at the phase transition,” Nature 403, 166–170 (2000).
[CrossRef]

Etemadi, B.

M. Encinosa and B. Etemadi, “Energy shifts resulting from surface curvature of quantum nanostructures,” Phys. Rev. A 58, 77–81 (1998).
[CrossRef]

Fedichev, P. O.

P. O. Fedichev and U. R. Fischer, “Gibbons–Hawking effect in the sonic de sitter space-time of an expanding Bose–Einstein-condensed gas,” Phys. Rev. Lett. 91, 240407 (2003).
[CrossRef]

Fischer, U. R.

P. O. Fedichev and U. R. Fischer, “Gibbons–Hawking effect in the sonic de sitter space-time of an expanding Bose–Einstein-condensed gas,” Phys. Rev. Lett. 91, 240407 (2003).
[CrossRef]

Gagarski, A. M.

V. V. Nesvizhevsky, H. G. Börner, A. K. Petukhov, H. Abele, S. Baeler, F. J. Rue, Th. Stöferle, A. Westphal, A. M. Gagarski, G. A. Petrov, and A. V. Strelkov, “Quantum states of neutrons in the Earth’s gravitational field,” Nature 415, 297–299 (2002).
[CrossRef]

Gangopadhyay, G.

B. Deb, G. Gangopadhyay, and D. Sh. Ray, “Generation of a class of arbitrary two-mode field states in a cavity,” Phys. Rev. A 51, 2651–2653 (1995).
[CrossRef]

Gazeau, J.-P.

S. Twareqe Ali, J.-P. Antoine, and J.-P. Gazeau, Coherent States, Wavelets and Their Generalizations (Springer-Verlag, 2000).

Glauber, R. J.

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
[CrossRef]

R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
[CrossRef]

R. J. Glauber, “Photon correlations,” Phys. Rev. Lett. 10, 84–86 (1963).
[CrossRef]

Gravesen, J.

J. Gravesen, M. Willatzen, and L. L. Y. Voon, “Quantum-mechanical particle confined to surfaces of revolution—truncated cone and elliptic torus case studies,” Phys. Scr. 72, 105–111 (2005).
[CrossRef]

J. Gravesen, M. Willatzen, and L. L. Y. Voon, “Schrödinger problems for surfaces of revolution—the finite cylinder as a test example,” J. Math. Phys. 46, 012107 (2005).
[CrossRef]

Hänsch, T. W.

I. Bloch, T. W. Hänsch, and T. Esslinger, “Measurement of the spatial coherence of a trapped Bose gas at the phase transition,” Nature 403, 166–170 (2000).
[CrossRef]

Haroche, S.

S. Haroche and J.-M. Raimond, Exploring the Quantum, Atoms, Cavities, and Photons (Oxford University, 2006).

Hill, S.

T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science 319, 1367–1370 (2008).
[CrossRef]

Jagannathan, R.

P. Shanta, S. Chaturvedi, V. Srinivasan, and R. Jagannathan, “Unified approach to the analogues of single-photon and multiphoton coherent states for generalized bosonic oscillators,” J. Phys. A 27, 6433–6442 (1994).
[CrossRef]

Katriel, J.

J. Katriel and A. I. Solomon, “Nonideal lasers, nonclassical light, and deformed photon states,” Phys. Rev. A 49, 5149–5151 (1994).
[CrossRef]

Klauder, J. R.

J. R. Klauder and B. S. Skagerstam, Coherent States, Applications in Physics and Mathematical Physics (World Scientific, 1985).

Knight, P. L.

W. K. Lai, V. Buzek, and P. L. Knight, “Interaction of a three-level atom with an SU(2) coherent state,” Phys. Rev. A 44, 2003–2012 (1991).
[CrossRef]

Kuklewicz, C.

T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science 319, 1367–1370 (2008).
[CrossRef]

Lai, W. K.

W. K. Lai, V. Buzek, and P. L. Knight, “Interaction of a three-level atom with an SU(2) coherent state,” Phys. Rev. A 44, 2003–2012 (1991).
[CrossRef]

Leonhardt, U.

T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science 319, 1367–1370 (2008).
[CrossRef]

U. Leonhardt, “A laboratory analogue of the event horizon using slow light in an atomic medium,” Nature 415, 406–409 (2002).
[CrossRef]

Liberati, S.

C. Barcelo, S. Liberati, and M. Visser, “Probing semiclassical analog gravity in Bose–Einstein condensates with widely tunable interactions,” Phys. Rev. A 68, 053613 (2003).
[CrossRef]

Longhi, S.

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, “Optics in curved space,” Phys. Rev. Lett. 105, 143901 (2010).
[CrossRef]

Mahdifar, A.

A. Mahdifar, “Coherent states for nonlinear two-boson realization of the isotropic oscillator algebra on a sphere,” Int. J. Geom. Methods Mod. Phys. 10, 1350028 (2013).
[CrossRef]

A. Mahdifar, R. Roknizadeh, and M. H. Naderi, “Detection of the spatial curvature effects through physical phenomena: the nonlinear coherent states approach,” Int. J. Geom. Methods Mod. Phys. 9, 1250009 (2012).
[CrossRef]

A. Mahdifar, R. Roknizadeh, and M. H. Naderi, “Geometric approach to nonlinear coherent states using the Higgs model for harmonic oscillator,” J. Phys. A 39, 7003–7014 (2006).
[CrossRef]

Man’ko, V. I.

P. Aniello, V. I. Man’ko, G. Marmo, S. Solimeno, and F. Zaccaria, “On the coherent states, displacement operators and quasidistributions associated with deformed quantum oscillators,” J. Opt. B 2, 718–725 (2000).
[CrossRef]

V. I. Man’ko, G. Marmo, E. C. G. Sudarshan, and F. Zaccaria, “f-oscillators and nonlinear coherent states,” Phys. Scr. 55, 528–541 (1997).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Marmo, G.

P. Aniello, V. I. Man’ko, G. Marmo, S. Solimeno, and F. Zaccaria, “On the coherent states, displacement operators and quasidistributions associated with deformed quantum oscillators,” J. Opt. B 2, 718–725 (2000).
[CrossRef]

V. I. Man’ko, G. Marmo, E. C. G. Sudarshan, and F. Zaccaria, “f-oscillators and nonlinear coherent states,” Phys. Scr. 55, 528–541 (1997).
[CrossRef]

Mohammadi, M.

M. Mohammadi, M. H. Naderi, and M. Soltanolkotabi, “Quantum statistical properties of the Jaynes–Cummings model in the presence of a classical homogeneous gravitational field,” J. Phys. A 40, 1377–1393 (2007).
[CrossRef]

Müller, H.

H. Müller, A. Peters, and S. Chu, “A precision measurement of the gravitational redshift by the interference of matter waves,” Nature 463, 926–929 (2010).
[CrossRef]

Naderi, M. H.

A. Mahdifar, R. Roknizadeh, and M. H. Naderi, “Detection of the spatial curvature effects through physical phenomena: the nonlinear coherent states approach,” Int. J. Geom. Methods Mod. Phys. 9, 1250009 (2012).
[CrossRef]

Sh. Barzanjeh, M. H. Naderi, and M. Soltanolkotabi, “Dynamical behaviours of the nonlinear atom-field interaction in the presence of classical gravity: f-deformation approach,” J. Phys. A 43, 375304 (2010).
[CrossRef]

M. Mohammadi, M. H. Naderi, and M. Soltanolkotabi, “Quantum statistical properties of the Jaynes–Cummings model in the presence of a classical homogeneous gravitational field,” J. Phys. A 40, 1377–1393 (2007).
[CrossRef]

A. Mahdifar, R. Roknizadeh, and M. H. Naderi, “Geometric approach to nonlinear coherent states using the Higgs model for harmonic oscillator,” J. Phys. A 39, 7003–7014 (2006).
[CrossRef]

Nesvizhevsky, V. V.

V. V. Nesvizhevsky, H. G. Börner, A. K. Petukhov, H. Abele, S. Baeler, F. J. Rue, Th. Stöferle, A. Westphal, A. M. Gagarski, G. A. Petrov, and A. V. Strelkov, “Quantum states of neutrons in the Earth’s gravitational field,” Nature 415, 297–299 (2002).
[CrossRef]

Nolte, S.

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, “Optics in curved space,” Phys. Rev. Lett. 105, 143901 (2010).
[CrossRef]

Perelomov, A. P.

A. P. Perelomov, Generalized Coherent States and their Applications (Springer, 1986).

Peschel, U.

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, “Optics in curved space,” Phys. Rev. Lett. 105, 143901 (2010).
[CrossRef]

S. Batz and U. Peschel, “Linear and nonlinear optics in curved space,” Phys. Rev. A 78, 043821 (2008).
[CrossRef]

Peters, A.

H. Müller, A. Peters, and S. Chu, “A precision measurement of the gravitational redshift by the interference of matter waves,” Nature 463, 926–929 (2010).
[CrossRef]

Petrov, G. A.

V. V. Nesvizhevsky, H. G. Börner, A. K. Petukhov, H. Abele, S. Baeler, F. J. Rue, Th. Stöferle, A. Westphal, A. M. Gagarski, G. A. Petrov, and A. V. Strelkov, “Quantum states of neutrons in the Earth’s gravitational field,” Nature 415, 297–299 (2002).
[CrossRef]

Petukhov, A. K.

V. V. Nesvizhevsky, H. G. Börner, A. K. Petukhov, H. Abele, S. Baeler, F. J. Rue, Th. Stöferle, A. Westphal, A. M. Gagarski, G. A. Petrov, and A. V. Strelkov, “Quantum states of neutrons in the Earth’s gravitational field,” Nature 415, 297–299 (2002).
[CrossRef]

Philbin, T. G.

T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science 319, 1367–1370 (2008).
[CrossRef]

Pound, R. V.

R. V. Pound and G. A. Rebka, “Apparent weight of photons,” Phys. Rev. Lett. 4, 337–341 (1960).
[CrossRef]

Raimond, J.-M.

S. Haroche and J.-M. Raimond, Exploring the Quantum, Atoms, Cavities, and Photons (Oxford University, 2006).

Ray, D. Sh.

B. Deb, G. Gangopadhyay, and D. Sh. Ray, “Generation of a class of arbitrary two-mode field states in a cavity,” Phys. Rev. A 51, 2651–2653 (1995).
[CrossRef]

Rebka, G. A.

R. V. Pound and G. A. Rebka, “Apparent weight of photons,” Phys. Rev. Lett. 4, 337–341 (1960).
[CrossRef]

Robertson, S.

T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science 319, 1367–1370 (2008).
[CrossRef]

Roknizadeh, R.

A. Mahdifar, R. Roknizadeh, and M. H. Naderi, “Detection of the spatial curvature effects through physical phenomena: the nonlinear coherent states approach,” Int. J. Geom. Methods Mod. Phys. 9, 1250009 (2012).
[CrossRef]

A. Mahdifar, R. Roknizadeh, and M. H. Naderi, “Geometric approach to nonlinear coherent states using the Higgs model for harmonic oscillator,” J. Phys. A 39, 7003–7014 (2006).
[CrossRef]

Rue, F. J.

V. V. Nesvizhevsky, H. G. Börner, A. K. Petukhov, H. Abele, S. Baeler, F. J. Rue, Th. Stöferle, A. Westphal, A. M. Gagarski, G. A. Petrov, and A. V. Strelkov, “Quantum states of neutrons in the Earth’s gravitational field,” Nature 415, 297–299 (2002).
[CrossRef]

Schleich, W. P.

K. Vogel, V. M. Akulin, and W. P. Schleich, “Quantum state engineering of the radiation field,” Phys. Rev. Lett. 71, 1816–1819 (1993).
[CrossRef]

Schultheiss, V. H.

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, “Optics in curved space,” Phys. Rev. Lett. 105, 143901 (2010).
[CrossRef]

Schutzhold, R.

R. Schutzhold and W. G. Unruh, “Hawking radiation in an electromagnetic waveguide?” Phys. Rev. Lett. 95, 031301 (2005).
[CrossRef]

R. Schutzhold and W. G. Unruh, “Gravity wave analogues of black holes,” Phys. Rev. D 66, 044019 (2002).
[CrossRef]

Schwinger, J.

J. Schwinger, Quantum Theory of Angular Momentum (Academic, 1965).

Shanta, P.

P. Shanta, S. Chaturvedi, V. Srinivasan, and R. Jagannathan, “Unified approach to the analogues of single-photon and multiphoton coherent states for generalized bosonic oscillators,” J. Phys. A 27, 6433–6442 (1994).
[CrossRef]

Skagerstam, B. S.

J. R. Klauder and B. S. Skagerstam, Coherent States, Applications in Physics and Mathematical Physics (World Scientific, 1985).

Solimeno, S.

P. Aniello, V. I. Man’ko, G. Marmo, S. Solimeno, and F. Zaccaria, “On the coherent states, displacement operators and quasidistributions associated with deformed quantum oscillators,” J. Opt. B 2, 718–725 (2000).
[CrossRef]

Solomon, A. I.

J. Katriel and A. I. Solomon, “Nonideal lasers, nonclassical light, and deformed photon states,” Phys. Rev. A 49, 5149–5151 (1994).
[CrossRef]

A. I. Solomon, “A characteristic functional for deformed photon phenomenology,” Phys. Lett. A 196, 29–34 (1994).
[CrossRef]

Soltanolkotabi, M.

Sh. Barzanjeh, M. H. Naderi, and M. Soltanolkotabi, “Dynamical behaviours of the nonlinear atom-field interaction in the presence of classical gravity: f-deformation approach,” J. Phys. A 43, 375304 (2010).
[CrossRef]

M. Mohammadi, M. H. Naderi, and M. Soltanolkotabi, “Quantum statistical properties of the Jaynes–Cummings model in the presence of a classical homogeneous gravitational field,” J. Phys. A 40, 1377–1393 (2007).
[CrossRef]

Srinivasan, V.

P. Shanta, S. Chaturvedi, V. Srinivasan, and R. Jagannathan, “Unified approach to the analogues of single-photon and multiphoton coherent states for generalized bosonic oscillators,” J. Phys. A 27, 6433–6442 (1994).
[CrossRef]

Stephani, H.

H. Stephani, General Relativity (Cambridge, 1996).

Stöferle, Th.

V. V. Nesvizhevsky, H. G. Börner, A. K. Petukhov, H. Abele, S. Baeler, F. J. Rue, Th. Stöferle, A. Westphal, A. M. Gagarski, G. A. Petrov, and A. V. Strelkov, “Quantum states of neutrons in the Earth’s gravitational field,” Nature 415, 297–299 (2002).
[CrossRef]

Strelkov, A. V.

V. V. Nesvizhevsky, H. G. Börner, A. K. Petukhov, H. Abele, S. Baeler, F. J. Rue, Th. Stöferle, A. Westphal, A. M. Gagarski, G. A. Petrov, and A. V. Strelkov, “Quantum states of neutrons in the Earth’s gravitational field,” Nature 415, 297–299 (2002).
[CrossRef]

Sudarshan, E. C. G.

V. I. Man’ko, G. Marmo, E. C. G. Sudarshan, and F. Zaccaria, “f-oscillators and nonlinear coherent states,” Phys. Scr. 55, 528–541 (1997).
[CrossRef]

Szameit, A.

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, “Optics in curved space,” Phys. Rev. Lett. 105, 143901 (2010).
[CrossRef]

Tünnermann, A.

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, “Optics in curved space,” Phys. Rev. Lett. 105, 143901 (2010).
[CrossRef]

Twareqe Ali, S.

S. Twareqe Ali, J.-P. Antoine, and J.-P. Gazeau, Coherent States, Wavelets and Their Generalizations (Springer-Verlag, 2000).

Unruh, W. G.

R. Schutzhold and W. G. Unruh, “Hawking radiation in an electromagnetic waveguide?” Phys. Rev. Lett. 95, 031301 (2005).
[CrossRef]

R. Schutzhold and W. G. Unruh, “Gravity wave analogues of black holes,” Phys. Rev. D 66, 044019 (2002).
[CrossRef]

W. G. Unruh, “Experimental black-hole evaporation?” Phys. Rev. Lett. 46, 1351–1353 (1981).
[CrossRef]

Visser, M.

C. Barcelo, S. Liberati, and M. Visser, “Probing semiclassical analog gravity in Bose–Einstein condensates with widely tunable interactions,” Phys. Rev. A 68, 053613 (2003).
[CrossRef]

Vogel, K.

K. Vogel, V. M. Akulin, and W. P. Schleich, “Quantum state engineering of the radiation field,” Phys. Rev. Lett. 71, 1816–1819 (1993).
[CrossRef]

Vogel, W.

R. L. de Matos Filho and W. Vogel, “Nonlinear coherent states,” Phys. Rev. A 54, 4560–4563 (1996).
[CrossRef]

W. Vogel and R. L. de Matos Filho, “Nonlinear Jaynes–Cummings dynamics of a trapped ion,” Phys. Rev. A 52, 4214–4217 (1995).
[CrossRef]

Voon, L. L. Y.

J. Gravesen, M. Willatzen, and L. L. Y. Voon, “Schrödinger problems for surfaces of revolution—the finite cylinder as a test example,” J. Math. Phys. 46, 012107 (2005).
[CrossRef]

J. Gravesen, M. Willatzen, and L. L. Y. Voon, “Quantum-mechanical particle confined to surfaces of revolution—truncated cone and elliptic torus case studies,” Phys. Scr. 72, 105–111 (2005).
[CrossRef]

Walsh, D.

D. Walsh, R. F. Carswell, and R. J. Weymann, “0957 + 561 A, B: twin quasistellar objects or gravitational lens?” Nature 279, 381–384 (1979).
[CrossRef]

Westphal, A.

V. V. Nesvizhevsky, H. G. Börner, A. K. Petukhov, H. Abele, S. Baeler, F. J. Rue, Th. Stöferle, A. Westphal, A. M. Gagarski, G. A. Petrov, and A. V. Strelkov, “Quantum states of neutrons in the Earth’s gravitational field,” Nature 415, 297–299 (2002).
[CrossRef]

Weymann, R. J.

D. Walsh, R. F. Carswell, and R. J. Weymann, “0957 + 561 A, B: twin quasistellar objects or gravitational lens?” Nature 279, 381–384 (1979).
[CrossRef]

Willatzen, M.

J. Gravesen, M. Willatzen, and L. L. Y. Voon, “Schrödinger problems for surfaces of revolution—the finite cylinder as a test example,” J. Math. Phys. 46, 012107 (2005).
[CrossRef]

J. Gravesen, M. Willatzen, and L. L. Y. Voon, “Quantum-mechanical particle confined to surfaces of revolution—truncated cone and elliptic torus case studies,” Phys. Scr. 72, 105–111 (2005).
[CrossRef]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Yoo, H.-I.

H.-I. Yoo and J. H. Eberly, “Dynamical theory of an atom with two or three levels interacting with quantized cavity fields,” Phys. Rep. 118, 239–337 (1985).
[CrossRef]

Zaccaria, F.

P. Aniello, V. I. Man’ko, G. Marmo, S. Solimeno, and F. Zaccaria, “On the coherent states, displacement operators and quasidistributions associated with deformed quantum oscillators,” J. Opt. B 2, 718–725 (2000).
[CrossRef]

V. I. Man’ko, G. Marmo, E. C. G. Sudarshan, and F. Zaccaria, “f-oscillators and nonlinear coherent states,” Phys. Scr. 55, 528–541 (1997).
[CrossRef]

Int. J. Geom. Methods Mod. Phys.

A. Mahdifar, R. Roknizadeh, and M. H. Naderi, “Detection of the spatial curvature effects through physical phenomena: the nonlinear coherent states approach,” Int. J. Geom. Methods Mod. Phys. 9, 1250009 (2012).
[CrossRef]

A. Mahdifar, “Coherent states for nonlinear two-boson realization of the isotropic oscillator algebra on a sphere,” Int. J. Geom. Methods Mod. Phys. 10, 1350028 (2013).
[CrossRef]

J. Math. Phys.

J. Gravesen, M. Willatzen, and L. L. Y. Voon, “Schrödinger problems for surfaces of revolution—the finite cylinder as a test example,” J. Math. Phys. 46, 012107 (2005).
[CrossRef]

J. Opt. B

P. Aniello, V. I. Man’ko, G. Marmo, S. Solimeno, and F. Zaccaria, “On the coherent states, displacement operators and quasidistributions associated with deformed quantum oscillators,” J. Opt. B 2, 718–725 (2000).
[CrossRef]

J. Phys. A

A. Mahdifar, R. Roknizadeh, and M. H. Naderi, “Geometric approach to nonlinear coherent states using the Higgs model for harmonic oscillator,” J. Phys. A 39, 7003–7014 (2006).
[CrossRef]

P. Shanta, S. Chaturvedi, V. Srinivasan, and R. Jagannathan, “Unified approach to the analogues of single-photon and multiphoton coherent states for generalized bosonic oscillators,” J. Phys. A 27, 6433–6442 (1994).
[CrossRef]

M. Mohammadi, M. H. Naderi, and M. Soltanolkotabi, “Quantum statistical properties of the Jaynes–Cummings model in the presence of a classical homogeneous gravitational field,” J. Phys. A 40, 1377–1393 (2007).
[CrossRef]

Sh. Barzanjeh, M. H. Naderi, and M. Soltanolkotabi, “Dynamical behaviours of the nonlinear atom-field interaction in the presence of classical gravity: f-deformation approach,” J. Phys. A 43, 375304 (2010).
[CrossRef]

Living Rev. Relativity

N. Ashby, “Relativity in the global positioning system,” Living Rev. Relativity 6, 1–45 (2003).
[CrossRef]

Nature

H. Müller, A. Peters, and S. Chu, “A precision measurement of the gravitational redshift by the interference of matter waves,” Nature 463, 926–929 (2010).
[CrossRef]

I. Bloch, T. W. Hänsch, and T. Esslinger, “Measurement of the spatial coherence of a trapped Bose gas at the phase transition,” Nature 403, 166–170 (2000).
[CrossRef]

V. V. Nesvizhevsky, H. G. Börner, A. K. Petukhov, H. Abele, S. Baeler, F. J. Rue, Th. Stöferle, A. Westphal, A. M. Gagarski, G. A. Petrov, and A. V. Strelkov, “Quantum states of neutrons in the Earth’s gravitational field,” Nature 415, 297–299 (2002).
[CrossRef]

D. Walsh, R. F. Carswell, and R. J. Weymann, “0957 + 561 A, B: twin quasistellar objects or gravitational lens?” Nature 279, 381–384 (1979).
[CrossRef]

U. Leonhardt, “A laboratory analogue of the event horizon using slow light in an atomic medium,” Nature 415, 406–409 (2002).
[CrossRef]

Philos. Trans. R. Soc. Lond. A

F. W. Dyson, A. S. Eddington, and C. Davidson, “A determination of the deflection of light by the Sun’s gravitational field, from observations made at the total eclipse of 29 May 1919,” Philos. Trans. R. Soc. Lond. A 220, 291–333 (1920).
[CrossRef]

Phys. Lett. A

A. I. Solomon, “A characteristic functional for deformed photon phenomenology,” Phys. Lett. A 196, 29–34 (1994).
[CrossRef]

Phys. Rep.

H.-I. Yoo and J. H. Eberly, “Dynamical theory of an atom with two or three levels interacting with quantized cavity fields,” Phys. Rep. 118, 239–337 (1985).
[CrossRef]

Phys. Rev.

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
[CrossRef]

R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
[CrossRef]

Phys. Rev. A

J. Katriel and A. I. Solomon, “Nonideal lasers, nonclassical light, and deformed photon states,” Phys. Rev. A 49, 5149–5151 (1994).
[CrossRef]

R. C. T. da Costa, “Quantum mechanics of a constrained particle,” Phys. Rev. A 23, 1982–1987 (1981).
[CrossRef]

M. Encinosa and B. Etemadi, “Energy shifts resulting from surface curvature of quantum nanostructures,” Phys. Rev. A 58, 77–81 (1998).
[CrossRef]

W. K. Lai, V. Buzek, and P. L. Knight, “Interaction of a three-level atom with an SU(2) coherent state,” Phys. Rev. A 44, 2003–2012 (1991).
[CrossRef]

W. Vogel and R. L. de Matos Filho, “Nonlinear Jaynes–Cummings dynamics of a trapped ion,” Phys. Rev. A 52, 4214–4217 (1995).
[CrossRef]

R. L. de Matos Filho and W. Vogel, “Nonlinear coherent states,” Phys. Rev. A 54, 4560–4563 (1996).
[CrossRef]

C. Barcelo, S. Liberati, and M. Visser, “Probing semiclassical analog gravity in Bose–Einstein condensates with widely tunable interactions,” Phys. Rev. A 68, 053613 (2003).
[CrossRef]

S. Batz and U. Peschel, “Linear and nonlinear optics in curved space,” Phys. Rev. A 78, 043821 (2008).
[CrossRef]

B. Deb, G. Gangopadhyay, and D. Sh. Ray, “Generation of a class of arbitrary two-mode field states in a cavity,” Phys. Rev. A 51, 2651–2653 (1995).
[CrossRef]

Phys. Rev. D

R. Schutzhold and W. G. Unruh, “Gravity wave analogues of black holes,” Phys. Rev. D 66, 044019 (2002).
[CrossRef]

Phys. Rev. Lett.

R. Schutzhold and W. G. Unruh, “Hawking radiation in an electromagnetic waveguide?” Phys. Rev. Lett. 95, 031301 (2005).
[CrossRef]

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, “Optics in curved space,” Phys. Rev. Lett. 105, 143901 (2010).
[CrossRef]

P. O. Fedichev and U. R. Fischer, “Gibbons–Hawking effect in the sonic de sitter space-time of an expanding Bose–Einstein-condensed gas,” Phys. Rev. Lett. 91, 240407 (2003).
[CrossRef]

R. V. Pound and G. A. Rebka, “Apparent weight of photons,” Phys. Rev. Lett. 4, 337–341 (1960).
[CrossRef]

W. G. Unruh, “Experimental black-hole evaporation?” Phys. Rev. Lett. 46, 1351–1353 (1981).
[CrossRef]

R. J. Glauber, “Photon correlations,” Phys. Rev. Lett. 10, 84–86 (1963).
[CrossRef]

K. Vogel, V. M. Akulin, and W. P. Schleich, “Quantum state engineering of the radiation field,” Phys. Rev. Lett. 71, 1816–1819 (1993).
[CrossRef]

Phys. Scr.

J. Gravesen, M. Willatzen, and L. L. Y. Voon, “Quantum-mechanical particle confined to surfaces of revolution—truncated cone and elliptic torus case studies,” Phys. Scr. 72, 105–111 (2005).
[CrossRef]

V. I. Man’ko, G. Marmo, E. C. G. Sudarshan, and F. Zaccaria, “f-oscillators and nonlinear coherent states,” Phys. Scr. 55, 528–541 (1997).
[CrossRef]

Science

T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science 319, 1367–1370 (2008).
[CrossRef]

C. W. Chou, “Optical clocks and relativity,” Science 329, 1630–1633 (2010).
[CrossRef]

Other

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

J. Schwinger, Quantum Theory of Angular Momentum (Academic, 1965).

S. Haroche and J.-M. Raimond, Exploring the Quantum, Atoms, Cavities, and Photons (Oxford University, 2006).

H. Stephani, General Relativity (Cambridge, 1996).

A. P. Perelomov, Generalized Coherent States and their Applications (Springer, 1986).

J. R. Klauder and B. S. Skagerstam, Coherent States, Applications in Physics and Mathematical Physics (World Scientific, 1985).

S. Twareqe Ali, J.-P. Antoine, and J.-P. Gazeau, Coherent States, Wavelets and Their Generalizations (Springer-Verlag, 2000).

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Figures (11)

Fig. 1.
Fig. 1.

Mean number of photons in the first mode (left) and in the second mode (right) of the state |μs versus λ for N=20 (solid line), N=30 (dotted line) and N=40 (dashed line), with μ=1.0.

Fig. 2.
Fig. 2.

Energy diagram of a three-level atom in the Λ configuration interacting with two quantized cavity modes.

Fig. 3.
Fig. 3.

Time evolution of the atomic occupation probabilities when the initial field is in the two-mode sphere CS with a maximum total N of 40 photons with gb/ga=1 and μ=1.0; (a) λ=0 and (b) λ=0.1.

Fig. 4.
Fig. 4.

As in Fig. 2 with gb/ga=2; (a) λ=0 and (b) λ=0.1.

Fig. 5.
Fig. 5.

Dependence of tr on the curvature, for N=10 (solid line), N=20 (dotted line), and N=40 (dashed line), with μ=1.0 and gb/ga=2.

Fig. 6.
Fig. 6.

Cross-correlation function versus gat for μ=1, N=40, and gb/ga=1. The dashed curve corresponds to λ=0.1, the dotted curve to λ=0.02, and the solid curve to λ=0.0.

Fig. 7.
Fig. 7.

Cross-correlation function versus gat for μ=1, N=40, and gb/ga=2. The upper and bottom curves, respectively, show g2(t,λ) for λ=0.1 and for λ=0.0.

Fig. 8.
Fig. 8.

Mandel parameter of the first mode of the field versus gat for μ=1, N=40, and gb/ga=1; (a) λ=0 and (b) λ=0.1.

Fig. 9.
Fig. 9.

Mandel parameter of the second mode of the field versus gat for μ=1, N=40, and gb/ga=1; (a) λ=0 and (b) λ=0.1.

Fig. 10.
Fig. 10.

As in Fig. 8 with gb/ga=2; (a) λ=0 and (b) λ=0.1.

Fig. 11.
Fig. 11.

As in Fig. 9 with gb/ga=2; (a) λ=0 and (b) λ=0.1.

Equations (34)

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A^=a^f(n^),A^=f(n^)a^,n^=a^a^,
[J^0,J^±]=±J^±,[J^+,J^]=2J^0h(λ,N,J^0),
h(λ,N,J^0)=1+λ(1+λ4)1/2(N+1)λ2[2J^02N(N2+1)14].
J^+=c1(λ)+c2(λ)[n^12+n^2(n^2+2)]a^1a^2,J^=a^1a^2c1(λ)+c2(λ)[n^12+n^2(n^2+2)],J^0=12(n^1n^2),
c1(λ)=1+λ(1+λ4)1/2(N+1)+λ2[N(N2+1)+14],
c2(λ)=12λ2.
|μs=M1n=0N(Nn)[g(λ,n)]!μn|n,Nn,
M2=n=0N(Nn){[g(λ,n)]!}2(|μ|2)n.
[g(λ,0)]!=1,[g(λ,n)]!=g(λ,n)[g(λ,n1)]!,
g(λ,n)=(λ(N+1n)+1+λ2/4)×(λn+1+λ2/4).
n^a|n,Nn=n|n,Nn,n^b|n,Nn=(Nn)|n,Nn,n,Nn|m,Nm=δnm,n=0N|n,Nnn,Nn|=1.
Pf(na,nb;λ)=|na,nb|μs|2=M2(Nna)[g(λ,na)!]2|μ|2naδnb,Nna.
Pfa(n;λ)=na|ρ^a|na=M2(Nn)[g(λ,n)!]2|μ|2n,
Pfb(n;λ)=Pfa(Nn;λ).
n^a=M2n=0N(Nn)[g(λ,n)!]2|μ|2nn,
n^b=M2n=0N(Nn)[g(λ,n)!]2|μ|2n(Nn).
H^I=ga(σ^10a^+a^σ^01)+gb(σ^12b^+b^σ^21),
|ψ(0)f=na,nbNAna,nb(λ)|na,nb,
|ψ(t)=na,nbAna,nb(λ){C0(na,nb;t)|0;na,nb+C1(na,nb;t)|1;na1,nb+C2(na,nb;t)|2;na1,nb+1}.
iC˙0=Ωna1C1,iC˙1=Ωna1C0+ΩnbC2,C˙2=ΩnbC1,
Ωna=ga[na+1]12,Ωnb=gb[nb+1]12.
P0(na,nb;t)=Ωnb4Ωna1,nb4+2Ωna12Ωnb2Ωna1,nb4cos(Ωna1,nbt)+Ωna14Ωna1,nb4cos2(Ωna1,nbt),P1(na,nb;t)=Ωna12Ωna1,nb2sin2(Ωna1,nbt),P2(na,nb;t)=Ωna12Ωnb2Ωna1,nb4[12cos(Ωna1,nbt)+cos2(Ωna1,nbt)],
Ωna,nb=[Ωna2+Ωnb2]12=[na+1+gbga(nb+1)]12ga.
P˜i(t,λ)=na,nbPf(na,nb;λ)Pi(na,nb;t),i=0,1,2,
(Ωna1,NnaΩna,Nna1)tr=2π.
tr4πΩna,Nna1gb2ga2.
Pf(na,nb;t,λ)=Pf(na,nb;λ)P0(na,nb;t)+Pf(na+1,nb;λ)P1(na+1,nb;t)+Pf(na+1,nb1;λ)P2(na+1,nb1;t).
Pfa(na;t,λ)=nbPf(na,nb;t,λ),Pfb(nb;t,λ)=naPf(na,nb;t,λ).
Pf(na,nb;t,λ)=Pf(na,Nna;λ)×P0(na,Nna,t)δnb,Nna+Pf(na+1,Nna1;λ)×P1(na+1,Nna1;t)δnb,Nna1+Pf(na+1,Nna1;λ)×P2(na+1,Nna1;t)δnb,Nna,
Pfa(n;t,λ)=Pfa(n;λ)P0(n,Nn;t)+Pfa(n+1;λ)[1P0(n+1,Nn1;t)],Pfb(n;t,λ)=Pfb(n1;λ)P2(Nn+1,n1;t)+Pfb(n;λ)[1P2(Nn,n;t)],
g2(t;λ)=nanbtnatnbt.
n^an^bt=na,nbnanbPf(na,nb;t,λ),n^at=nanaPfa(na;t,λ),n^bt=nbnbPfb(nb;t,λ).
Qi(t)=(Δni(t))2n^i(t)n^i(t),
Q¯=limT1T0TQ(t)dt

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