Measurement of photons number in a quantum cavity is very difficult and the photons number is changed after each measurement. Recently, many efforts have been done for the nondemolition measurement methods. Haroche et al. succeed in recognizing existence or nonexistence of one photon in a quantum cavity. In this paper, we employ their experimental setup for a quantum nondemolition measurement and pump a coherent state in their quantum cavity. In this case, we could detect more photons in the quantum cavity by a measurement of a displaced Wigner function. It is also shown that the measurement of more than one photon is possible by the Haroche method by measuring just one point of displaced Wigner function. Furthermore, if the cavity field is filled by a superposition of two number states, the average number of photons within the cavity would be measurable. We show that their setup is also suitable to apply for the measurement of the squeezing parameter for the squeezed state of photons number in the quantum cavity successfully. 1. Introduction The formulation of quantum mechanics in phase space was proposed by Wigner [1]. This formulation is very useful in various fields of physics including quantum mechanics [2, 3], quantum optics [4–6], and condensate matter [7, 8]. The physical concepts are extractable from Wigner function. Wigner function may take negative value for a quantum state. The existence of negative or interference of Wigner function is a nonclassicality indicator for quantum systems [9–11]. On the other hand, Wigner function is a measurable quantity. Many authors introduced methods to measure Wigner function for trapped ions [12], photonic number states in quantum cavity [13–15], Schrodinger cat state, and coherent state [16]. Bertet et al. measure a complete Wigner function for the vacuum and a single photon state [17]. Lutterbach and Davidovich presented a method to measure the Wigner distribution function of photonic state in a quantum cavity field [18, 19]. They used an experimental ingenious setup which was made by one high Q-factor and two low Q-factor cavities. Nogues et al. (members of Haroche group) measured the Wigner distribution functions of electromagnetic fields in a cavity with the number states and at origin of phase space [20]. The Wigner distribution function at the origin of phase space is positive for and negative for . Therefore, the sign of measured Wigner distribution function, itself, gives us the number of photons in the cavity and its value is not important [20]. So, if there are more than one photon, it would not
References
[1]
E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Physical Review, vol. 40, no. 5, pp. 749–759, 1932.
[2]
A. G. Baker Jr., “Formulation of quantum mechanics based on the quasi-probability distribution induced on phase space,” Physical Review, vol. 109, no. 6, pp. 2198–2206, 1958.
[3]
K. C. Zachos, B. D. Fairlie, and L. T. Curtright, Quantum Mechanics in Phase Space, World Scientific, Singapore, 2005.
[4]
R. J. Glauber, “The quantum theory of optical coherence,” Physical Review, vol. 130, no. 6, pp. 2529–2539, 1963.
[5]
R. J. Glauber, “Coherent and incoherent states of the radiation field,” Physical Review, vol. 131, no. 6, pp. 2766–2788, 1963.
[6]
M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge University Press, Cambridge, UK, 1997.
[7]
P. Deuar and P. D. Drummond, “First-principles quantum dynamics in interacting Bose gases. I. The positive P representation,” Journal of Physics A, vol. 39, no. 5, article 1163, 2006.
[8]
P. D. Drummond, P. Deuar, and K. V. Kheruntsyan, “Canonical bose gas simulations with stochastic gauges,” Physical Review Letters, vol. 92, no. 4, Article ID 040405, 4 pages, 2004.
[9]
G. M. Benedict and A. Czirják, “Wigner functions, squeezing properties, and slow decoherence of a mesoscopic superposition of two-level atoms,” Physical Review A, vol. 60, no. 5, pp. 4034–4044, 1999.
[10]
P. Faldi, A. Czirják, B. Molnár, and G. M. Benedict, “Formation of schr？dinger-cat states in the morse potential: wigner function picture,” Optics Express, vol. 10, no. 8, pp. 376–381, 2002.
[11]
P. Sadeghi, S. Khademi, and S. Nasiri, “Nonclassicality indicator for the real phase-space distribution functions,” Physical Review A, vol. 82, no. 1, Article ID 012102, 8 pages, 2010.
[12]
D. Leibfried, D. M. Meekhof, B. E. King, C. Monroe, W. M. Itano, and D. J. Wineland, “Experimental determination of the motional quantum state of a trapped atom,” Physical Review Letters, vol. 77, no. 21, pp. 4281–4285, 1996.
[13]
K. Banaszek and K. Wódkiewicz, “Direct probing of quantum phase space by photon counting,” Physical Review Letters, vol. 76, no. 23, pp. 4344–4347, 1996.
[14]
S. Wallentowitz and W. Vogel, “Unbalanced homodyning for quantum state measurements,” Physical Review A, vol. 53, no. 6, pp. 4528–4533, 1996.
[15]
C. Guerlin, J. Bernu, S. Deléglise et al., “Progressive field-state collapse and quantum non-demolition photon counting,” Nature, vol. 448, pp. 889–893, 2007.
[16]
S. Deléglise, I. Dotsenko, C. Sayrin et al., “Reconstruction of non-classical cavity field states with snapshots of their decoherence,” Nature, vol. 455, no. 7212, pp. 510–514, 2008.
[17]
P. Bertet, A. Auffeves, P. Maioli et al., “Direct measurement of the wigner function of a one-photon fock state in a cavity,” Physical Review Letters, vol. 89, no. 20, Article ID 200402, 4 pages, 2002.
[18]
L. G. Lutterbach and L. Davidovich, “Method for direct measurement of the wigner function in cavity QED and ion traps,” Physical Review Letters, vol. 78, no. 13, pp. 2547–2550, 1997.
[19]
L. G. Lutterbach and L. Davidovich, “Non-classical states of the electromagnetic field in cavity QED,” Optics Express, vol. 3, no. 4, pp. 147–153, 1998.
[20]
G. Nogues, A. Rauschenbeutel, S. Osnaghi et al., “Measurement of a negative value for the wigner function of radiation,” Physical Review A, vol. 62, no. 5, Article ID 054101, 4 pages, 2000.
[21]
K. E. Cahill and R. J. Glauber, “Density operators and quasiprobability distributions,” Physical Review, vol. 177, no. 5, pp. 1882–1902, 1969.
[22]
M. G. Benedict and A. Czirjak, “Generalized parity and quasi-probability density functions,” Journal of Physics A, vol. 28, no. 16, article 4599, 1995.
[23]
C. Gerry and P. Knight, Introductory Quantum Optics, Cambridge University Press, Cambridge, UK, 2004.
[24]
G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, Elseveir Academic Press, Burlington, Mass, USA, 2005.
[25]
J. I. Cirac and P. Zoller, “Preparation of macroscopic superpositions in many-atom systems,” Physical Review A, vol. 50, no. 4, pp. R2799–R2802, 1994.
[26]
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,” Physical Review Letters, vol. 55, no. 22, pp. 2409–2412, 1985.
[27]
L. Ulf, Measuring the Quantum State of Light, Cambridge University Press, Cambridge, UK, 1997.
[28]
N. G. de Almeida, R. M. Serra, C. J. Villas-B？as, and M. H. Y. Moussa, “Engineering squeezed states in high-Q cavities,” Physical Review A, vol. 69, no. 3, Article ID 035802, 4 pages, 2004.
