Abstract

We construct deformed photon-added nonlinear coherent states (DPANCSs) by application of the deformed creation operator upon the nonlinear coherent states obtained as eigenstates of the deformed annihilation operator and by application of a deformed displacement operator upon the vacuum state. We evaluate some statistical properties like the Mandel parameter, Husimi, and Wigner functions for these states and analyze their differences; we give closed analytical expressions for them. We found a profound difference in the statistical properties of the DPANCSs obtained from the two abovementioned generalizations.

© 2013 Optical Society of America

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    [CrossRef]
  31. O. de los Santos-Sánchez and J. Récamier, “The f-deformed Jaynes–Cummings model and its nonlinear coherent states,” J. Phys. B 45, 015502 (2012).
    [CrossRef]
  32. R. Roknizadeh and M. K. Tavassoly, “The construction of some important classes of generalized coherent states: the nonlinear coherent states method,” J. Phys. A 37, 8111–8127 (2004).
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  35. B. C. Sanders and G. J. Milburn, “Complementarity in a quantum nondemolition measurement,” Phys. Rev. A 39, 694–702 (1989).
    [CrossRef]
  36. B. Yurke and D. Stoler, “Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion,” Phys. Rev. Lett. 57, 13–16 (1986).
    [CrossRef]
  37. R. Román-Ancheyta, O. de los Santos-Sánchez, and J. Récamier, “Ladder operators and coherent states for nonlinear potentials,” J. Phys. A 44, 435304 (2011).
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  40. H. M. Cessa and P. Knight, “Series representation of quantum-field quasiprobabilities,” Phys. Rev. A 48, 2479–2481 (1993).
    [CrossRef]
  41. H. M. Cessa, “Decoherence in atom-field interactions: a treatment using superoperator techniques,” Phys. Rep. 432, 1–41 (2006).
    [CrossRef]
  42. A. Royer, “Wigner function as the expectation value of a parity operator,” Phys. Rev. A 15, 449–450 (1977).
    [CrossRef]
  43. F. A. M. de Oliveira, M. S. Kim, P. L. Knight, and V. Buzek, “Properties of displaced number states,” Phys. Rev. A 41, 2645 (1990).
    [CrossRef]
  44. M. Boiteux and A. Levelut, “Semicoherent states,” J. Phys. A 6, 589–596 (1973).
    [CrossRef]
  45. R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical analogue of displaced Fock states and quantum correlations in Glauber–Fock photonic lattices,” Phys. Rev. Lett. 107, 103601 (2011).
    [CrossRef]
  46. A. I. Lvovsky and S. A. Babichev, “Synthesis and tomographic characterization of the displaced Fock state of light,” Phys. Rev. A 66, 011801 (2002).
    [CrossRef]
  47. F. Ziesel, T. Ruster, A. Walther, H. Kaufmann, S. Dawkins, K. Singer, F. Schmidt-Kaler, and U. G. Poschinger, “Experimental creation and analysis of displaced number states,” J. Phys. B 46, 104008 (2013).
    [CrossRef]

2013 (1)

F. Ziesel, T. Ruster, A. Walther, H. Kaufmann, S. Dawkins, K. Singer, F. Schmidt-Kaler, and U. G. Poschinger, “Experimental creation and analysis of displaced number states,” J. Phys. B 46, 104008 (2013).
[CrossRef]

2012 (4)

2011 (5)

R. Román-Ancheyta, O. de los Santos-Sánchez, and J. Récamier, “Ladder operators and coherent states for nonlinear potentials,” J. Phys. A 44, 435304 (2011).
[CrossRef]

O. de los Santos-Sánchez and J. Récamier, “Nonlinear coherent states for nonlinear systems,” J. Phys. A 44, 145307 (2011).
[CrossRef]

O. Safaeian and M. K. Tavassoly, “Deformed photon-added nonlinear coherent states and their non-classical properties,” J. Phys. A 44, 225301 (2011).
[CrossRef]

T. Kiesel, W. Vogel, M. Bellini, and A. Zavatta, “Nonclassicality quasiprobability of single-photon-added thermal states,” Phys. Rev. A 83, 032116 (2011).
[CrossRef]

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical analogue of displaced Fock states and quantum correlations in Glauber–Fock photonic lattices,” Phys. Rev. Lett. 107, 103601 (2011).
[CrossRef]

2010 (1)

M. Barbieri, N. Spagnolo, M. G. Genoni, F. Ferreyrol, R. Blandino, M. G. A. Paris, P. Grangier, and R. Tualle-Brouri, “Non-Gaussianity of quantum states: an experimental test on single-photon-added coherent states,” Phys. Rev. A 82, 063833 (2010).
[CrossRef]

2009 (2)

J. Lee, J. Kim, and H. Nha, “Demonstrating higher-order nonclassical effects by photon-added classical states: realistic schemes,” J. Opt. Soc. Am. B 26, 1363–1369 (2009).
[CrossRef]

A. Zavatta, V. Parigi, M. S. Kim, H. Jeong, and M. Bellini, “Experimental demonstration of the bosonic commutation relation via superpositions of quantum operations on thermal light fields,” Phys. Rev. Lett. 103, 140406 (2009).
[CrossRef]

2008 (2)

M. S. Kim, H. Jeong, A. Zavatta, V. Parigi, and M. Bellini, “Scheme for proving the bosonic commutation relation using single-photon interference,” Phys. Rev. Lett. 101, 260401 (2008).
[CrossRef]

J. Récamier, M. Gorayeb, W. L. Mochán, and J. L. Paz, “Nonlinear coherent states and some of their properties,” Int. J. Theor. Phys. 47, 673–683 (2008).
[CrossRef]

2006 (3)

J. Récamier, W. L. Mochán, M. Gorayeb, J. L. Paz, and R. Jáuregui, “Uncertainty relations for a deformed oscillator,” Int. J. Mod. Phys. B 20, 1851–1859 (2006).
[CrossRef]

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83–86 (2006).
[CrossRef]

H. M. Cessa, “Decoherence in atom-field interactions: a treatment using superoperator techniques,” Phys. Rep. 432, 1–41 (2006).
[CrossRef]

2005 (2)

A. Zavatta, S. Viciani, and M. Bellini, “Single-photon excitation of a coherent state: catching the elementary step of stimulated light emission,” Phys. Rev. A 72, 023820 (2005).
[CrossRef]

J. M. Raimond, T. Meunier, P. Bertet, S. Gleyzes, P. Maioli, A. Auffeves, G. Nogues, M. Brune, and S. Haroche, “Probing a quantum field in a photon box,” J. Phys. B 38, S535–S550 (2005).
[CrossRef]

2004 (3)

R. Roknizadeh and M. K. Tavassoly, “The construction of some important classes of generalized coherent states: the nonlinear coherent states method,” J. Phys. A 37, 8111–8127 (2004).
[CrossRef]

J. Wenger, R. Tualle-Brouri, and P. Grangier, “Non-Gaussian statistics from individual pulses of squeezed light,” Phys. Rev. Lett. 92, 153601 (2004).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660–662 (2004).
[CrossRef]

2002 (2)

V. V. Dodonov, “Non classical states in quantum optics: a squeezed review of the first 75 years,” J. Opt. B 4, R1–R33 (2002).
[CrossRef]

A. I. Lvovsky and S. A. Babichev, “Synthesis and tomographic characterization of the displaced Fock state of light,” Phys. Rev. A 66, 011801 (2002).
[CrossRef]

2000 (1)

B. Roy and P. Roy, “New nonlinear coherent states and some of their nonclassical properties,” J. Opt. B 2, 65–68 (2000).
[CrossRef]

1999 (1)

S. Sivakumar, “Photon-added coherent states as nonlinear coherent states,” J. Phys. A 32, 3441–3447 (1999).
[CrossRef]

1998 (1)

1997 (1)

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 (2)

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

R. R. Puri and G. S. Agarwal, “SU(1,1) coherent states defined via a minimum-uncertainty product and an equality of quadrature variances,” Phys. Rev. A 53, 1786–1790 (1996).
[CrossRef]

1995 (1)

H. Moya-Cessa, “Generation and properties of superpositions of displaced Fock states,” J. Mod. Opt. 42, 1741–1754 (1995).
[CrossRef]

1993 (1)

H. M. Cessa and P. Knight, “Series representation of quantum-field quasiprobabilities,” Phys. Rev. A 48, 2479–2481 (1993).
[CrossRef]

1991 (1)

G. S. Agarwal and K. Tara, “Nonclassical properties of states generated by the excitations on a coherent state,” Phys. Rev. A 43, 492–497 (1991).
[CrossRef]

1990 (2)

W.-M. Zhang, D. H. Feng, and R. Gilmore, “Coherent states: theory and some applications,” Rev. Mod. Phys. 62, 867–927 (1990).
[CrossRef]

F. A. M. de Oliveira, M. S. Kim, P. L. Knight, and V. Buzek, “Properties of displaced number states,” Phys. Rev. A 41, 2645 (1990).
[CrossRef]

1989 (1)

B. C. Sanders and G. J. Milburn, “Complementarity in a quantum nondemolition measurement,” Phys. Rev. A 39, 694–702 (1989).
[CrossRef]

1986 (1)

B. Yurke and D. Stoler, “Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion,” Phys. Rev. Lett. 57, 13–16 (1986).
[CrossRef]

1980 (1)

P. D. Drummond and D. F. Walls, “Quantum theory of optical bistability. I. Nonlinear polarisability model,” J. Phys. A 13, 725–741 (1980).
[CrossRef]

1979 (1)

1977 (1)

A. Royer, “Wigner function as the expectation value of a parity operator,” Phys. Rev. A 15, 449–450 (1977).
[CrossRef]

1973 (1)

M. Boiteux and A. Levelut, “Semicoherent states,” J. Phys. A 6, 589–596 (1973).
[CrossRef]

1963 (3)

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

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

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

Agarwal, G. S.

R. R. Puri and G. S. Agarwal, “SU(1,1) coherent states defined via a minimum-uncertainty product and an equality of quadrature variances,” Phys. Rev. A 53, 1786–1790 (1996).
[CrossRef]

G. S. Agarwal and K. Tara, “Nonclassical properties of states generated by the excitations on a coherent state,” Phys. Rev. A 43, 492–497 (1991).
[CrossRef]

Auffeves, A.

J. M. Raimond, T. Meunier, P. Bertet, S. Gleyzes, P. Maioli, A. Auffeves, G. Nogues, M. Brune, and S. Haroche, “Probing a quantum field in a photon box,” J. Phys. B 38, S535–S550 (2005).
[CrossRef]

Babichev, S. A.

A. I. Lvovsky and S. A. Babichev, “Synthesis and tomographic characterization of the displaced Fock state of light,” Phys. Rev. A 66, 011801 (2002).
[CrossRef]

Barbieri, M.

M. Barbieri, N. Spagnolo, M. G. Genoni, F. Ferreyrol, R. Blandino, M. G. A. Paris, P. Grangier, and R. Tualle-Brouri, “Non-Gaussianity of quantum states: an experimental test on single-photon-added coherent states,” Phys. Rev. A 82, 063833 (2010).
[CrossRef]

Bellini, M.

T. Kiesel, W. Vogel, M. Bellini, and A. Zavatta, “Nonclassicality quasiprobability of single-photon-added thermal states,” Phys. Rev. A 83, 032116 (2011).
[CrossRef]

A. Zavatta, V. Parigi, M. S. Kim, H. Jeong, and M. Bellini, “Experimental demonstration of the bosonic commutation relation via superpositions of quantum operations on thermal light fields,” Phys. Rev. Lett. 103, 140406 (2009).
[CrossRef]

M. S. Kim, H. Jeong, A. Zavatta, V. Parigi, and M. Bellini, “Scheme for proving the bosonic commutation relation using single-photon interference,” Phys. Rev. Lett. 101, 260401 (2008).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, “Single-photon excitation of a coherent state: catching the elementary step of stimulated light emission,” Phys. Rev. A 72, 023820 (2005).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660–662 (2004).
[CrossRef]

Bertet, P.

J. M. Raimond, T. Meunier, P. Bertet, S. Gleyzes, P. Maioli, A. Auffeves, G. Nogues, M. Brune, and S. Haroche, “Probing a quantum field in a photon box,” J. Phys. B 38, S535–S550 (2005).
[CrossRef]

Blandino, R.

M. Barbieri, N. Spagnolo, M. G. Genoni, F. Ferreyrol, R. Blandino, M. G. A. Paris, P. Grangier, and R. Tualle-Brouri, “Non-Gaussianity of quantum states: an experimental test on single-photon-added coherent states,” Phys. Rev. A 82, 063833 (2010).
[CrossRef]

Boiteux, M.

M. Boiteux and A. Levelut, “Semicoherent states,” J. Phys. A 6, 589–596 (1973).
[CrossRef]

Brune, M.

J. M. Raimond, T. Meunier, P. Bertet, S. Gleyzes, P. Maioli, A. Auffeves, G. Nogues, M. Brune, and S. Haroche, “Probing a quantum field in a photon box,” J. Phys. B 38, S535–S550 (2005).
[CrossRef]

Buzek, V.

F. A. M. de Oliveira, M. S. Kim, P. L. Knight, and V. Buzek, “Properties of displaced number states,” Phys. Rev. A 41, 2645 (1990).
[CrossRef]

Cessa, H. M.

H. M. Cessa, “Decoherence in atom-field interactions: a treatment using superoperator techniques,” Phys. Rep. 432, 1–41 (2006).
[CrossRef]

H. M. Cessa and P. Knight, “Series representation of quantum-field quasiprobabilities,” Phys. Rev. A 48, 2479–2481 (1993).
[CrossRef]

Christodoulides, D. N.

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical analogue of displaced Fock states and quantum correlations in Glauber–Fock photonic lattices,” Phys. Rev. Lett. 107, 103601 (2011).
[CrossRef]

Dawkins, S.

F. Ziesel, T. Ruster, A. Walther, H. Kaufmann, S. Dawkins, K. Singer, F. Schmidt-Kaler, and U. G. Poschinger, “Experimental creation and analysis of displaced number states,” J. Phys. B 46, 104008 (2013).
[CrossRef]

de los Santos-Sánchez, O.

O. de los Santos-Sánchez and J. Récamier, “The f-deformed Jaynes–Cummings model and its nonlinear coherent states,” J. Phys. B 45, 015502 (2012).
[CrossRef]

O. de los Santos-Sánchez and J. Récamier, “Nonlinear coherent states for nonlinear systems,” J. Phys. A 44, 145307 (2011).
[CrossRef]

R. Román-Ancheyta, O. de los Santos-Sánchez, and J. Récamier, “Ladder operators and coherent states for nonlinear potentials,” J. Phys. A 44, 435304 (2011).
[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]

de Oliveira, F. A. M.

F. A. M. de Oliveira, M. S. Kim, P. L. Knight, and V. Buzek, “Properties of displaced number states,” Phys. Rev. A 41, 2645 (1990).
[CrossRef]

Dodonov, V. V.

V. V. Dodonov, “Non classical states in quantum optics: a squeezed review of the first 75 years,” J. Opt. B 4, R1–R33 (2002).
[CrossRef]

Dreisow, F.

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical analogue of displaced Fock states and quantum correlations in Glauber–Fock photonic lattices,” Phys. Rev. Lett. 107, 103601 (2011).
[CrossRef]

Drummond, P. D.

P. D. Drummond and D. F. Walls, “Quantum theory of optical bistability. I. Nonlinear polarisability model,” J. Phys. A 13, 725–741 (1980).
[CrossRef]

Du, J.-M.

Feng, D. H.

W.-M. Zhang, D. H. Feng, and R. Gilmore, “Coherent states: theory and some applications,” Rev. Mod. Phys. 62, 867–927 (1990).
[CrossRef]

Ferreyrol, F.

M. Barbieri, N. Spagnolo, M. G. Genoni, F. Ferreyrol, R. Blandino, M. G. A. Paris, P. Grangier, and R. Tualle-Brouri, “Non-Gaussianity of quantum states: an experimental test on single-photon-added coherent states,” Phys. Rev. A 82, 063833 (2010).
[CrossRef]

Genoni, M. G.

M. Barbieri, N. Spagnolo, M. G. Genoni, F. Ferreyrol, R. Blandino, M. G. A. Paris, P. Grangier, and R. Tualle-Brouri, “Non-Gaussianity of quantum states: an experimental test on single-photon-added coherent states,” Phys. Rev. A 82, 063833 (2010).
[CrossRef]

Gilmore, R.

W.-M. Zhang, D. H. Feng, and R. Gilmore, “Coherent states: theory and some applications,” Rev. Mod. Phys. 62, 867–927 (1990).
[CrossRef]

Glauber, R. J.

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

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

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

Gleyzes, S.

J. M. Raimond, T. Meunier, P. Bertet, S. Gleyzes, P. Maioli, A. Auffeves, G. Nogues, M. Brune, and S. Haroche, “Probing a quantum field in a photon box,” J. Phys. B 38, S535–S550 (2005).
[CrossRef]

Gorayeb, M.

J. Récamier, M. Gorayeb, W. L. Mochán, and J. L. Paz, “Nonlinear coherent states and some of their properties,” Int. J. Theor. Phys. 47, 673–683 (2008).
[CrossRef]

J. Récamier, W. L. Mochán, M. Gorayeb, J. L. Paz, and R. Jáuregui, “Uncertainty relations for a deformed oscillator,” Int. J. Mod. Phys. B 20, 1851–1859 (2006).
[CrossRef]

Grangier, P.

M. Barbieri, N. Spagnolo, M. G. Genoni, F. Ferreyrol, R. Blandino, M. G. A. Paris, P. Grangier, and R. Tualle-Brouri, “Non-Gaussianity of quantum states: an experimental test on single-photon-added coherent states,” Phys. Rev. A 82, 063833 (2010).
[CrossRef]

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83–86 (2006).
[CrossRef]

J. Wenger, R. Tualle-Brouri, and P. Grangier, “Non-Gaussian statistics from individual pulses of squeezed light,” Phys. Rev. Lett. 92, 153601 (2004).
[CrossRef]

Haroche, S.

J. M. Raimond, T. Meunier, P. Bertet, S. Gleyzes, P. Maioli, A. Auffeves, G. Nogues, M. Brune, and S. Haroche, “Probing a quantum field in a photon box,” J. Phys. B 38, S535–S550 (2005).
[CrossRef]

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

Heinrich, M.

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical analogue of displaced Fock states and quantum correlations in Glauber–Fock photonic lattices,” Phys. Rev. Lett. 107, 103601 (2011).
[CrossRef]

Hu, L.-Y.

Jáuregui, R.

J. Récamier, W. L. Mochán, M. Gorayeb, J. L. Paz, and R. Jáuregui, “Uncertainty relations for a deformed oscillator,” Int. J. Mod. Phys. B 20, 1851–1859 (2006).
[CrossRef]

Jeong, H.

A. Zavatta, V. Parigi, M. S. Kim, H. Jeong, and M. Bellini, “Experimental demonstration of the bosonic commutation relation via superpositions of quantum operations on thermal light fields,” Phys. Rev. Lett. 103, 140406 (2009).
[CrossRef]

M. S. Kim, H. Jeong, A. Zavatta, V. Parigi, and M. Bellini, “Scheme for proving the bosonic commutation relation using single-photon interference,” Phys. Rev. Lett. 101, 260401 (2008).
[CrossRef]

Jia, F.

Kaufmann, H.

F. Ziesel, T. Ruster, A. Walther, H. Kaufmann, S. Dawkins, K. Singer, F. Schmidt-Kaler, and U. G. Poschinger, “Experimental creation and analysis of displaced number states,” J. Phys. B 46, 104008 (2013).
[CrossRef]

Keil, R.

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical analogue of displaced Fock states and quantum correlations in Glauber–Fock photonic lattices,” Phys. Rev. Lett. 107, 103601 (2011).
[CrossRef]

Kiesel, T.

T. Kiesel, W. Vogel, M. Bellini, and A. Zavatta, “Nonclassicality quasiprobability of single-photon-added thermal states,” Phys. Rev. A 83, 032116 (2011).
[CrossRef]

Kim, J.

Kim, M. S.

A. Zavatta, V. Parigi, M. S. Kim, H. Jeong, and M. Bellini, “Experimental demonstration of the bosonic commutation relation via superpositions of quantum operations on thermal light fields,” Phys. Rev. Lett. 103, 140406 (2009).
[CrossRef]

M. S. Kim, H. Jeong, A. Zavatta, V. Parigi, and M. Bellini, “Scheme for proving the bosonic commutation relation using single-photon interference,” Phys. Rev. Lett. 101, 260401 (2008).
[CrossRef]

F. A. M. de Oliveira, M. S. Kim, P. L. Knight, and V. Buzek, “Properties of displaced number states,” Phys. Rev. A 41, 2645 (1990).
[CrossRef]

Knight, P.

H. M. Cessa and P. Knight, “Series representation of quantum-field quasiprobabilities,” Phys. Rev. A 48, 2479–2481 (1993).
[CrossRef]

Knight, P. L.

F. A. M. de Oliveira, M. S. Kim, P. L. Knight, and V. Buzek, “Properties of displaced number states,” Phys. Rev. A 41, 2645 (1990).
[CrossRef]

Laurat, J.

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83–86 (2006).
[CrossRef]

Lee, C. T.

Lee, J.

Levelut, A.

M. Boiteux and A. Levelut, “Semicoherent states,” J. Phys. A 6, 589–596 (1973).
[CrossRef]

Lvovsky, A. I.

A. I. Lvovsky and S. A. Babichev, “Synthesis and tomographic characterization of the displaced Fock state of light,” Phys. Rev. A 66, 011801 (2002).
[CrossRef]

Maioli, P.

J. M. Raimond, T. Meunier, P. Bertet, S. Gleyzes, P. Maioli, A. Auffeves, G. Nogues, M. Brune, and S. Haroche, “Probing a quantum field in a photon box,” J. Phys. B 38, S535–S550 (2005).
[CrossRef]

Man’ko, V. I.

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, “Sub-Poissonian photon statistics in resonance fluorescence,” Opt. Lett. 4, 205–207 (1979).
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995), p. 1102.

Marmo, 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]

Meunier, T.

J. M. Raimond, T. Meunier, P. Bertet, S. Gleyzes, P. Maioli, A. Auffeves, G. Nogues, M. Brune, and S. Haroche, “Probing a quantum field in a photon box,” J. Phys. B 38, S535–S550 (2005).
[CrossRef]

Milburn, G. J.

B. C. Sanders and G. J. Milburn, “Complementarity in a quantum nondemolition measurement,” Phys. Rev. A 39, 694–702 (1989).
[CrossRef]

Mochán, W. L.

J. Récamier, M. Gorayeb, W. L. Mochán, and J. L. Paz, “Nonlinear coherent states and some of their properties,” Int. J. Theor. Phys. 47, 673–683 (2008).
[CrossRef]

J. Récamier, W. L. Mochán, M. Gorayeb, J. L. Paz, and R. Jáuregui, “Uncertainty relations for a deformed oscillator,” Int. J. Mod. Phys. B 20, 1851–1859 (2006).
[CrossRef]

Moya-Cessa, H.

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical analogue of displaced Fock states and quantum correlations in Glauber–Fock photonic lattices,” Phys. Rev. Lett. 107, 103601 (2011).
[CrossRef]

H. Moya-Cessa, “Generation and properties of superpositions of displaced Fock states,” J. Mod. Opt. 42, 1741–1754 (1995).
[CrossRef]

Nha, H.

Nogues, G.

J. M. Raimond, T. Meunier, P. Bertet, S. Gleyzes, P. Maioli, A. Auffeves, G. Nogues, M. Brune, and S. Haroche, “Probing a quantum field in a photon box,” J. Phys. B 38, S535–S550 (2005).
[CrossRef]

Nolte, S.

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical analogue of displaced Fock states and quantum correlations in Glauber–Fock photonic lattices,” Phys. Rev. Lett. 107, 103601 (2011).
[CrossRef]

Ourjoumtsev, A.

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83–86 (2006).
[CrossRef]

Parigi, V.

A. Zavatta, V. Parigi, M. S. Kim, H. Jeong, and M. Bellini, “Experimental demonstration of the bosonic commutation relation via superpositions of quantum operations on thermal light fields,” Phys. Rev. Lett. 103, 140406 (2009).
[CrossRef]

M. S. Kim, H. Jeong, A. Zavatta, V. Parigi, and M. Bellini, “Scheme for proving the bosonic commutation relation using single-photon interference,” Phys. Rev. Lett. 101, 260401 (2008).
[CrossRef]

Paris, M. G. A.

M. Barbieri, N. Spagnolo, M. G. Genoni, F. Ferreyrol, R. Blandino, M. G. A. Paris, P. Grangier, and R. Tualle-Brouri, “Non-Gaussianity of quantum states: an experimental test on single-photon-added coherent states,” Phys. Rev. A 82, 063833 (2010).
[CrossRef]

Paz, J. L.

J. Récamier, M. Gorayeb, W. L. Mochán, and J. L. Paz, “Nonlinear coherent states and some of their properties,” Int. J. Theor. Phys. 47, 673–683 (2008).
[CrossRef]

J. Récamier, W. L. Mochán, M. Gorayeb, J. L. Paz, and R. Jáuregui, “Uncertainty relations for a deformed oscillator,” Int. J. Mod. Phys. B 20, 1851–1859 (2006).
[CrossRef]

Perez-Leija, A.

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical analogue of displaced Fock states and quantum correlations in Glauber–Fock photonic lattices,” Phys. Rev. Lett. 107, 103601 (2011).
[CrossRef]

Poschinger, U. G.

F. Ziesel, T. Ruster, A. Walther, H. Kaufmann, S. Dawkins, K. Singer, F. Schmidt-Kaler, and U. G. Poschinger, “Experimental creation and analysis of displaced number states,” J. Phys. B 46, 104008 (2013).
[CrossRef]

Puri, R. R.

R. R. Puri and G. S. Agarwal, “SU(1,1) coherent states defined via a minimum-uncertainty product and an equality of quadrature variances,” Phys. Rev. A 53, 1786–1790 (1996).
[CrossRef]

Raimond, J. M.

J. M. Raimond, T. Meunier, P. Bertet, S. Gleyzes, P. Maioli, A. Auffeves, G. Nogues, M. Brune, and S. Haroche, “Probing a quantum field in a photon box,” J. Phys. B 38, S535–S550 (2005).
[CrossRef]

Raimond, J.-M.

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

Récamier, J.

O. de los Santos-Sánchez and J. Récamier, “The f-deformed Jaynes–Cummings model and its nonlinear coherent states,” J. Phys. B 45, 015502 (2012).
[CrossRef]

O. de los Santos-Sánchez and J. Récamier, “Nonlinear coherent states for nonlinear systems,” J. Phys. A 44, 145307 (2011).
[CrossRef]

R. Román-Ancheyta, O. de los Santos-Sánchez, and J. Récamier, “Ladder operators and coherent states for nonlinear potentials,” J. Phys. A 44, 435304 (2011).
[CrossRef]

J. Récamier, M. Gorayeb, W. L. Mochán, and J. L. Paz, “Nonlinear coherent states and some of their properties,” Int. J. Theor. Phys. 47, 673–683 (2008).
[CrossRef]

J. Récamier, W. L. Mochán, M. Gorayeb, J. L. Paz, and R. Jáuregui, “Uncertainty relations for a deformed oscillator,” Int. J. Mod. Phys. B 20, 1851–1859 (2006).
[CrossRef]

Ren, G.

Roknizadeh, R.

R. Roknizadeh and M. K. Tavassoly, “The construction of some important classes of generalized coherent states: the nonlinear coherent states method,” J. Phys. A 37, 8111–8127 (2004).
[CrossRef]

Román-Ancheyta, R.

R. Román-Ancheyta, O. de los Santos-Sánchez, and J. Récamier, “Ladder operators and coherent states for nonlinear potentials,” J. Phys. A 44, 435304 (2011).
[CrossRef]

Roy, B.

B. Roy and P. Roy, “New nonlinear coherent states and some of their nonclassical properties,” J. Opt. B 2, 65–68 (2000).
[CrossRef]

Roy, P.

B. Roy and P. Roy, “New nonlinear coherent states and some of their nonclassical properties,” J. Opt. B 2, 65–68 (2000).
[CrossRef]

Royer, A.

A. Royer, “Wigner function as the expectation value of a parity operator,” Phys. Rev. A 15, 449–450 (1977).
[CrossRef]

Ruster, T.

F. Ziesel, T. Ruster, A. Walther, H. Kaufmann, S. Dawkins, K. Singer, F. Schmidt-Kaler, and U. G. Poschinger, “Experimental creation and analysis of displaced number states,” J. Phys. B 46, 104008 (2013).
[CrossRef]

Safaeian, O.

O. Safaeian and M. K. Tavassoly, “Deformed photon-added nonlinear coherent states and their non-classical properties,” J. Phys. A 44, 225301 (2011).
[CrossRef]

Sanders, B. C.

B. C. Sanders and G. J. Milburn, “Complementarity in a quantum nondemolition measurement,” Phys. Rev. A 39, 694–702 (1989).
[CrossRef]

Schmidt-Kaler, F.

F. Ziesel, T. Ruster, A. Walther, H. Kaufmann, S. Dawkins, K. Singer, F. Schmidt-Kaler, and U. G. Poschinger, “Experimental creation and analysis of displaced number states,” J. Phys. B 46, 104008 (2013).
[CrossRef]

Singer, K.

F. Ziesel, T. Ruster, A. Walther, H. Kaufmann, S. Dawkins, K. Singer, F. Schmidt-Kaler, and U. G. Poschinger, “Experimental creation and analysis of displaced number states,” J. Phys. B 46, 104008 (2013).
[CrossRef]

Sivakumar, S.

S. Sivakumar, “Photon-added coherent states as nonlinear coherent states,” J. Phys. A 32, 3441–3447 (1999).
[CrossRef]

Spagnolo, N.

M. Barbieri, N. Spagnolo, M. G. Genoni, F. Ferreyrol, R. Blandino, M. G. A. Paris, P. Grangier, and R. Tualle-Brouri, “Non-Gaussianity of quantum states: an experimental test on single-photon-added coherent states,” Phys. Rev. A 82, 063833 (2010).
[CrossRef]

Stoler, D.

B. Yurke and D. Stoler, “Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion,” Phys. Rev. Lett. 57, 13–16 (1986).
[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.

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical analogue of displaced Fock states and quantum correlations in Glauber–Fock photonic lattices,” Phys. Rev. Lett. 107, 103601 (2011).
[CrossRef]

Tara, K.

G. S. Agarwal and K. Tara, “Nonclassical properties of states generated by the excitations on a coherent state,” Phys. Rev. A 43, 492–497 (1991).
[CrossRef]

Tavassoly, M. K.

O. Safaeian and M. K. Tavassoly, “Deformed photon-added nonlinear coherent states and their non-classical properties,” J. Phys. A 44, 225301 (2011).
[CrossRef]

R. Roknizadeh and M. K. Tavassoly, “The construction of some important classes of generalized coherent states: the nonlinear coherent states method,” J. Phys. A 37, 8111–8127 (2004).
[CrossRef]

Tualle-Brouri, R.

M. Barbieri, N. Spagnolo, M. G. Genoni, F. Ferreyrol, R. Blandino, M. G. A. Paris, P. Grangier, and R. Tualle-Brouri, “Non-Gaussianity of quantum states: an experimental test on single-photon-added coherent states,” Phys. Rev. A 82, 063833 (2010).
[CrossRef]

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83–86 (2006).
[CrossRef]

J. Wenger, R. Tualle-Brouri, and P. Grangier, “Non-Gaussian statistics from individual pulses of squeezed light,” Phys. Rev. Lett. 92, 153601 (2004).
[CrossRef]

Viciani, S.

A. Zavatta, S. Viciani, and M. Bellini, “Single-photon excitation of a coherent state: catching the elementary step of stimulated light emission,” Phys. Rev. A 72, 023820 (2005).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660–662 (2004).
[CrossRef]

Vogel, W.

T. Kiesel, W. Vogel, M. Bellini, and A. Zavatta, “Nonclassicality quasiprobability of single-photon-added thermal states,” Phys. Rev. A 83, 032116 (2011).
[CrossRef]

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

Walls, D. F.

P. D. Drummond and D. F. Walls, “Quantum theory of optical bistability. I. Nonlinear polarisability model,” J. Phys. A 13, 725–741 (1980).
[CrossRef]

Walther, A.

F. Ziesel, T. Ruster, A. Walther, H. Kaufmann, S. Dawkins, K. Singer, F. Schmidt-Kaler, and U. G. Poschinger, “Experimental creation and analysis of displaced number states,” J. Phys. B 46, 104008 (2013).
[CrossRef]

Wenger, J.

J. Wenger, R. Tualle-Brouri, and P. Grangier, “Non-Gaussian statistics from individual pulses of squeezed light,” Phys. Rev. Lett. 92, 153601 (2004).
[CrossRef]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995), p. 1102.

Xu, Y.-J.

Yu, H.-J.

Yurke, B.

B. Yurke and D. Stoler, “Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion,” Phys. Rev. Lett. 57, 13–16 (1986).
[CrossRef]

Zaccaria, F.

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]

Zavatta, A.

T. Kiesel, W. Vogel, M. Bellini, and A. Zavatta, “Nonclassicality quasiprobability of single-photon-added thermal states,” Phys. Rev. A 83, 032116 (2011).
[CrossRef]

A. Zavatta, V. Parigi, M. S. Kim, H. Jeong, and M. Bellini, “Experimental demonstration of the bosonic commutation relation via superpositions of quantum operations on thermal light fields,” Phys. Rev. Lett. 103, 140406 (2009).
[CrossRef]

M. S. Kim, H. Jeong, A. Zavatta, V. Parigi, and M. Bellini, “Scheme for proving the bosonic commutation relation using single-photon interference,” Phys. Rev. Lett. 101, 260401 (2008).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, “Single-photon excitation of a coherent state: catching the elementary step of stimulated light emission,” Phys. Rev. A 72, 023820 (2005).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660–662 (2004).
[CrossRef]

Zhang, W.-M.

W.-M. Zhang, D. H. Feng, and R. Gilmore, “Coherent states: theory and some applications,” Rev. Mod. Phys. 62, 867–927 (1990).
[CrossRef]

Zhang, Z.-M.

Ziesel, F.

F. Ziesel, T. Ruster, A. Walther, H. Kaufmann, S. Dawkins, K. Singer, F. Schmidt-Kaler, and U. G. Poschinger, “Experimental creation and analysis of displaced number states,” J. Phys. B 46, 104008 (2013).
[CrossRef]

Int. J. Mod. Phys. B (1)

J. Récamier, W. L. Mochán, M. Gorayeb, J. L. Paz, and R. Jáuregui, “Uncertainty relations for a deformed oscillator,” Int. J. Mod. Phys. B 20, 1851–1859 (2006).
[CrossRef]

Int. J. Theor. Phys. (1)

J. Récamier, M. Gorayeb, W. L. Mochán, and J. L. Paz, “Nonlinear coherent states and some of their properties,” Int. J. Theor. Phys. 47, 673–683 (2008).
[CrossRef]

J. Mod. Opt. (1)

H. Moya-Cessa, “Generation and properties of superpositions of displaced Fock states,” J. Mod. Opt. 42, 1741–1754 (1995).
[CrossRef]

J. Opt. B (2)

V. V. Dodonov, “Non classical states in quantum optics: a squeezed review of the first 75 years,” J. Opt. B 4, R1–R33 (2002).
[CrossRef]

B. Roy and P. Roy, “New nonlinear coherent states and some of their nonclassical properties,” J. Opt. B 2, 65–68 (2000).
[CrossRef]

J. Opt. Soc. Am. B (5)

J. Phys. A (7)

R. Román-Ancheyta, O. de los Santos-Sánchez, and J. Récamier, “Ladder operators and coherent states for nonlinear potentials,” J. Phys. A 44, 435304 (2011).
[CrossRef]

R. Roknizadeh and M. K. Tavassoly, “The construction of some important classes of generalized coherent states: the nonlinear coherent states method,” J. Phys. A 37, 8111–8127 (2004).
[CrossRef]

P. D. Drummond and D. F. Walls, “Quantum theory of optical bistability. I. Nonlinear polarisability model,” J. Phys. A 13, 725–741 (1980).
[CrossRef]

O. de los Santos-Sánchez and J. Récamier, “Nonlinear coherent states for nonlinear systems,” J. Phys. A 44, 145307 (2011).
[CrossRef]

S. Sivakumar, “Photon-added coherent states as nonlinear coherent states,” J. Phys. A 32, 3441–3447 (1999).
[CrossRef]

O. Safaeian and M. K. Tavassoly, “Deformed photon-added nonlinear coherent states and their non-classical properties,” J. Phys. A 44, 225301 (2011).
[CrossRef]

M. Boiteux and A. Levelut, “Semicoherent states,” J. Phys. A 6, 589–596 (1973).
[CrossRef]

J. Phys. B (3)

F. Ziesel, T. Ruster, A. Walther, H. Kaufmann, S. Dawkins, K. Singer, F. Schmidt-Kaler, and U. G. Poschinger, “Experimental creation and analysis of displaced number states,” J. Phys. B 46, 104008 (2013).
[CrossRef]

O. de los Santos-Sánchez and J. Récamier, “The f-deformed Jaynes–Cummings model and its nonlinear coherent states,” J. Phys. B 45, 015502 (2012).
[CrossRef]

J. M. Raimond, T. Meunier, P. Bertet, S. Gleyzes, P. Maioli, A. Auffeves, G. Nogues, M. Brune, and S. Haroche, “Probing a quantum field in a photon box,” J. Phys. B 38, S535–S550 (2005).
[CrossRef]

Opt. Lett. (1)

Phys. Rep. (1)

H. M. Cessa, “Decoherence in atom-field interactions: a treatment using superoperator techniques,” Phys. Rep. 432, 1–41 (2006).
[CrossRef]

Phys. Rev. (2)

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
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R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
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Phys. Rev. A (11)

R. R. Puri and G. S. Agarwal, “SU(1,1) coherent states defined via a minimum-uncertainty product and an equality of quadrature variances,” Phys. Rev. A 53, 1786–1790 (1996).
[CrossRef]

G. S. Agarwal and K. Tara, “Nonclassical properties of states generated by the excitations on a coherent state,” Phys. Rev. A 43, 492–497 (1991).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, “Single-photon excitation of a coherent state: catching the elementary step of stimulated light emission,” Phys. Rev. A 72, 023820 (2005).
[CrossRef]

T. Kiesel, W. Vogel, M. Bellini, and A. Zavatta, “Nonclassicality quasiprobability of single-photon-added thermal states,” Phys. Rev. A 83, 032116 (2011).
[CrossRef]

M. Barbieri, N. Spagnolo, M. G. Genoni, F. Ferreyrol, R. Blandino, M. G. A. Paris, P. Grangier, and R. Tualle-Brouri, “Non-Gaussianity of quantum states: an experimental test on single-photon-added coherent states,” Phys. Rev. A 82, 063833 (2010).
[CrossRef]

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

H. M. Cessa and P. Knight, “Series representation of quantum-field quasiprobabilities,” Phys. Rev. A 48, 2479–2481 (1993).
[CrossRef]

B. C. Sanders and G. J. Milburn, “Complementarity in a quantum nondemolition measurement,” Phys. Rev. A 39, 694–702 (1989).
[CrossRef]

A. Royer, “Wigner function as the expectation value of a parity operator,” Phys. Rev. A 15, 449–450 (1977).
[CrossRef]

F. A. M. de Oliveira, M. S. Kim, P. L. Knight, and V. Buzek, “Properties of displaced number states,” Phys. Rev. A 41, 2645 (1990).
[CrossRef]

A. I. Lvovsky and S. A. Babichev, “Synthesis and tomographic characterization of the displaced Fock state of light,” Phys. Rev. A 66, 011801 (2002).
[CrossRef]

Phys. Rev. Lett. (6)

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical analogue of displaced Fock states and quantum correlations in Glauber–Fock photonic lattices,” Phys. Rev. Lett. 107, 103601 (2011).
[CrossRef]

B. Yurke and D. Stoler, “Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion,” Phys. Rev. Lett. 57, 13–16 (1986).
[CrossRef]

M. S. Kim, H. Jeong, A. Zavatta, V. Parigi, and M. Bellini, “Scheme for proving the bosonic commutation relation using single-photon interference,” Phys. Rev. Lett. 101, 260401 (2008).
[CrossRef]

A. Zavatta, V. Parigi, M. S. Kim, H. Jeong, and M. Bellini, “Experimental demonstration of the bosonic commutation relation via superpositions of quantum operations on thermal light fields,” Phys. Rev. Lett. 103, 140406 (2009).
[CrossRef]

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

J. Wenger, R. Tualle-Brouri, and P. Grangier, “Non-Gaussian statistics from individual pulses of squeezed light,” Phys. Rev. Lett. 92, 153601 (2004).
[CrossRef]

Phys. Scr. (1)

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]

Rev. Mod. Phys. (1)

W.-M. Zhang, D. H. Feng, and R. Gilmore, “Coherent states: theory and some applications,” Rev. Mod. Phys. 62, 867–927 (1990).
[CrossRef]

Science (2)

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83–86 (2006).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660–662 (2004).
[CrossRef]

Other (2)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995), p. 1102.

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

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Figures (7)

Fig. 1.
Fig. 1.

Probability distributions Pk,m with m=1 and α=3 for a coherent state (yellow); for a NLCS obtained as eigenstate of the deformed annihilation operator (blue); and for a NLCS obtained by the deformed displacement operator acting upon the vacuum state (green).

Fig. 2.
Fig. 2.

Mandel parameter Q with m=0, 1 as a function of α=(x,0) for a m-PACS |α(m) (red, purple), and deformed m-photon-added NLCSs |α(m),f (dark blue, blue) and |ζ(α),m (dark green, green) with the parameter χ/ω0=0.15.

Fig. 3.
Fig. 3.

Husimi Q-function Q[|α(1),fα(1),f|](z) with α=1.1 for a nonlinear PACS |α(1),f, and a coherent state |z with z=x+iy and χ/ω0=0.15.

Fig. 4.
Fig. 4.

Husimi Q-function Q[|ζ(α),1ζ(α),1|](z) with α=1.1, for a nonlinear PACS |ζ(α),1, and a coherent state |z with z=x+iy and χ/ω0=0.15.

Fig. 5.
Fig. 5.

Wigner function Wf,Am=1(α) with a density matrix corresponding to a NLCS |β(1),f obtained as eigenstate of the annihilation operator with β=1.1, χ/ω0=0.15.

Fig. 6.
Fig. 6.

Wigner function Wf,Dm=1(α) with a density matrix corresponding to a NLCS |ζ(β),1 obtained by displacement of the vacuum state with β=1.1, χ/ω0=0.15.

Fig. 7.
Fig. 7.

Wigner function Wf,Am(α) with a density matrix corresponding to a photon-added NLCS |β(m),f obtained as eigenstate of the deformed annihilation operator with β=0.5, χ/ω0=0.15 and m=4.

Equations (36)

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|α(m)=Nα(m)a^m|α=Nα(m)k=0m(mk)k!α*mk|α,k,
Nα(m)=1/(α|a^ma^m|α)1/2=1/(Lm(|α|2)m!)1/2,
|α(m)=exp(|α|2/2)[Lm(|α|2)m!]1/2n=0αn(n+m)!n!|n+m.
|α(m),f=A^m|α,f(α,f|A^mA^m|α,f)1/2,
|α,f=Nfn=0αnn!f(n)!|n.
A^m|α,f=Nfn=0αn(n+m)!n![f(n)!]2f(n+m)!|n+m,
|α(m),f=Nfα,f|A^mA^m|α,f1/2n=0αn(n+m)!n!×[f(n+m)]![f(n)!]2|n+m.
HD=ΩA^A^
HD=Ωn^f2(n^).
f2(n^)=1+χΩn^,
HD=Ωn^+χn^2ω0[(1χ/ω0)n^+(χ/ω0)n^2],
H=ωn^+χn^k
|α,f=Nf(α)n=0(ω0χχ)n/2αnn!(ω0/χ)n|n,
|α(m),f=(Nfm(α))1/2n=0αn(ω0χχ)n/2×(m+1)n(ω0/χ+m)nn!(ω0/χ)n|n+m
Pk,m(α)=|Nfm(α)|1(ω0χχ)kmk!(ω0/χ)k×|α|2(km)m!(ω0/χ)m[(km)!]2[(ω0/χ)km]2.
[A^,A^]=1+χω0χ+2χω0χn^,[A^,n^]=A^,[A^,n^]=A^
|ζ(α)=(1|ζ(α)|2)ω0/2χn=0(ω0/χ)nn!ζ(α)n|n,
|ζ(α),m=(Nζ(α)m)1/2n=0ζ(α)n×(m+1)n(ω0/χ+m)nn!|n+m
Pk,m(ζ(α))=|k|ζ(α),m|2=1Nζ(α)mk!(ω0/χ)k|ζ(α)|2(km)m!(ω0/χ)m[(km)!]2.
Q=n^2n^2n^.
Q[ρ](α)=1π2d2λe(αλ*α*λ)Can[ρ](λ),
Q[ρ](α)=1πTr[ρ|αα|]=1πα|ρ|α,
Q[ρ](α)=1π0|D(α)ρD(α)|0=1πTr[|00|D(α)ρD(α)].
Q[|αα|](z)=1πz|αα|z=1πexp(|zα|2),
Q[|α(m)α(m)|](z)=1πz|α(m)α(m)|z=1π|z|2mm!Lm(|α|2)exp(|zα|2).
Q[|α(m),fα(m),f|](z)=1πz|α(m),fα(m),f|z,
Q[|α(m),fα(m),f|](z)=1πe|z|2Nαm,f|z|2mm!×|n=0(z*α)nn!(ω0χχ)n/2(ω0/χ+m)n(ω0/χ)n|2
Q[|ζ(α),mζ(α),m|](z)=1πe|z|2Nζ(α)m|z|2mm!×|n=0(z*ζ(α))nn!(ω0/χ+m)n|2
W(α)=2πk=0(1)kα,k|ρ^|α,k
|α,k=D^(α)|k=exp(|α|2/2)n=0(k!n!)1/2αknLnkn(|α|2)|n,
W(α)=2πk=0(1)kα,k|ββ|α,k=2πexp(2|βα|2).
Wf(α)=2πk=0(1)kα,k|β,fβ,f|α,k,
Wf,Am(α)=2πk=0(1)kα,k|β(m),fβ(m),f|α,k,
Wf,Am(α)=2e|α|2πNfm(β)k=0(1)k|α|2(mk)|l=0k(|α|2)ll!×n=0(α*β)nn!k!m!(ω0χχ)n/2(n+mkl)×(ω0/χ+m)n(ω0/χ)n|2.
Wf,Dm(α)=2πk=0(1)kα,k|ζ(β),mζ(β),m|α,k
Wf,Dm(α)=2e|α|2πNζ(β)mk=0(1)k|α|2(mk)|l=0k(|α|2)ll!×n=0[α*ζ(β)]nn!k!m!(n+mkl)(ω0/χ+m)n|2.

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