Discrete squeezed states for finite-dimensional spaces

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 76, Issue 3, 2007, Pages

  Marchiolli, Marcelo A ^a   Ruzzi, Maurizio ^a   Galetti, Diógenes ^a  

^a SÃO PAULO STATE UNIVERSITY UNESP   (Brazil)

Author keywords

[No Author keywords available]

Indexed keywords

QUANTUM THEORY; TOMOGRAPHY;

DISCRETE VARIABLES; INTERFERENCE EFFECTS; QUANTUM INFORMATION THEORY; SQUEEZING EFFECTS;

ATOMIC PHYSICS;

EID: 34848866848     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.76.032102     Document Type: Article

References (74)

1. E. Schrödinger, Naturwiss. NATWAY 0028-1042 10.1007/BF01507634 14, 664 (1926).
2. E. H. Kennard, Z. Phys. ZEPYAA 0044-3328 10.1007/BF01391200 44, 326 (1927).
3. C. G. Darwin, Proc. R. Soc. London, Ser. A PRLAAZ 0950-1207 117, 258 (1927).
4. R. Loudon and P. L. Knight, J. Mod. Opt. JMOPEW 0950-0340 34, 709 (1987);
5. V. V. Dodonov, J. Opt. B: Quantum Semiclassical Opt. JOBOFD 1464-4266 10.1088/1464-4266/4/1/201 4, R1 (2002).
6. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, UK, 2000).
7. S. L. Braunstein and P. van Loock, Rev. Mod. Phys. RMPHAT 0034-6861 10.1103/RevModPhys.77.513 77, 513 (2005).
8. W. K. Wootters, Ann. Phys. (N.Y.) APNYA6 0003-4916 10.1016/0003-4916(87) 90176-X 176, 1 (1987);
9. K. S. Gibbons, M. J. Hoffman, and W. K. Wootters, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.70.062101 70, 062101 (2004).
10. D. Galetti and A. F. R. de Toledo Piza, Physica A PHYADX 0378-4371 10.1016/0378-4371(88)90219-1 149, 267 (1988).
11. O. Cohendet, P. Combe, M. Sirugue, and M. Sirugue-Collin, J. Phys. A JPHAC5 0305-4470 10.1088/0305-4470/21/13/012 21, 2875 (1988).
12. P. Leboeuf and A. Voros, J. Phys. A JPHAC5 0305-4470 10.1088/0305-4470/ 23/10/017 23, 1765 (1990).
13. D. Galetti and A. F. R. de Toledo Piza, Physica A PHYADX 0378-4371 10.1016/0378-4371(92)90213-A 186, 513 (1992);
14. D. Galetti and M. A. Marchiolli, Ann. Phys. (N.Y.) APNYA6 0003-4916 10.1006/aphy.1996.0079 249, 454 (1996).
15. T. Opatrný, D.-G. Welsch, and V. Bužek, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.53.3822 53, 3822 (1996).
16. A. Luis and J. Peřina, J. Phys. A JPHAC5 0305-4470 10.1088/0305-4470/31/5/012 31, 1423 (1998).
17. T. Hakioglu, J. Phys. A JPHAC5 0305-4470 10.1088/0305-4470/31/33/008 31, 6975 (1998).
18. A. Takami, T. Hashimoto, M. Horibe, and A. Hayashi, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.64.032114 64, 032114 (2001).
19. N. Mukunda, S. Chatuverdi, and R. Simon, Phys. Lett. A PYLAAG 0375-9601 10.1016/j.physleta.2003.11.053 321, 160 (2004).
20. A. Vourdas, Rep. Prog. Phys. RPPHAG 0034-4885 10.1088/0034-4885/67/3/R03 67, 267 (2004);
21. A. B. Klimov, L. L. Sánchez-Soto, and H. de Guise, J. Phys. A JPHAC5 0305-4470 10.1088/0305-4470/38/12/015 38, 2747 (2005);
22. A. B. Klimov, L. L. Sánchez-Soto, and H. de Guise, J. Opt. B: Quantum Semiclassical Opt. JOBOFD 1464-4266 10.1088/1464-4266/7/9/008 7, 283 (2005);
23. A. B. Klimov and C. Muñoz, J. Opt. B: Quantum Semiclassical Opt. JOBOFD 1464-4266 10.1088/1464-4266/7/12/022 7, S588 (2005);
24. A. B. Klimov, C. Muñoz, and J. L. Romero, J. Phys. A JPHAC5 0305-4470 10.1088/0305-4470/39/46/016 39, 14471 (2006).
25. M. Ruzzi, M. A. Marchiolli, and D. Galetti, J. Phys. A JPHAC5 0305-4470 10.1088/0305-4470/38/27/010 38, 6239 (2005);
26. M. A. Marchiolli, M. Ruzzi, and D. Galetti, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.72.042308 72, 042308 (2005).
27. A. Vourdas, J. Phys. A JPHAC5 0305-4470 10.1088/0305-4470/39/7/R01 39, R65 (2006).
28. J. Schwinger, Advanced Book Classics: Quantum Kinematics and Dynamics (Addison-Wesley Publishing Company, New York, 1991);
29. Quantum Mechanics: Symbolism of Atomic Measurements (Springer-Verlag, Berlin, 2001).
30. K. E. Cahill and R. J. Glauber, Phys. Rev. PHRVAO 0031-899X 10.1103/PhysRev.177.1857 177, 1857 (1969);
31. K. E. Cahill and R. J. Glauber, Phys. Rev. PHRVAO 0031-899X 10.1103/PhysRev.177.1882 177, 1882 (1969).
32. S. S. Mizrahi and D. Galetti, Physica A PHYADX 0378-4371 153, 567 (1988) and references therein.
33. C. Miquel, J. P. Paz, M. Saraceno, E. Knill, R. Laflamme, and C. Negrevergne, Nature (London) NATUAS 0028-0836 10.1038/nature00801 418, 59 (2002);
34. J. P. Paz and A. J. Roncaglia, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.68.052316 68, 052316 (2003);
35. J. P. Paz, A. J. Roncaglia, and M. Saraceno, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.69.032312 69, 032312 (2004).
36. R. Jozsa, D. Robb, and W. K. Wootters, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.49.668 49, 668 (1994);
37. N. J. Cerf and C. Adami, Physica D PDNPDT 0167-2789 10.1016/S0167- 2789(98)00045-1 120, 62 (1998);
38. V. Vedral, Rev. Mod. Phys. RMPHAT 0034-6861 10.1103/RevModPhys.74.197 74, 197 (2002);
39. A. Galindo and M. A. Martin-Delgado, Rev. Mod. Phys. RMPHAT 0034-6861 10.1103/RevModPhys.74.347 74, 347 (2002);
40. M. Keyl, Phys. Rep. PRPLCM 0370-1573 10.1016/S0370-1573(02)00266-1 369, 431 (2002);
41. H. Fan, J. Phys. A JPHAC5 0305-4470 10.1088/0305-4470/36/48/010 36, 12081 (2003);
42. D. W. Berry and B. C. Sanders, J. Phys. A JPHAC5 0305-4470 10.1088/0305-4470/36/49/008 36, 12255 (2003);
43. A. Peres and D. R. Terno, Rev. Mod. Phys. RMPHAT 0034-6861 10.1103/RevModPhys.76.93 76, 93 (2004).
44. In order to establish a pattern notation for the Jacobi theta functions in the next sections, we will adopt the Vilenkin's notation throughout this paper. Note that a rich material on this specific class of special functions and their respective properties can be found in R. Bellman, A Brief Introduction to Theta Functions (Holt Rinehart and Winston, New York, 1961);
45. W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer-Verlag, New York, 1966);
46. N. J. Vilenkin and A. U. Klimyk, Representation of Lie Groups and Special Functions: Simplest Lie Groups, Special Functions and Integral Transforms (Kluwer Academic, Dordrecht, 1992);
47. E. T. Whittaker and G. N. Watson, Cambridge Mathematical Library: A Course of Modern Analysis (Cambridge University Press, New York, 2000).
48. M. L. Mehta, J. Math. Phys. JMAPAQ 0022-2488 10.1063/1.527619 28, 781 (1987).
49. M. Ruzzi, J. Math. Phys. JMAPAQ 0022-2488 10.1063/1.2209770 47, 063507 (2006).
50. M. Ruzzi, J. Phys. A JPHAC5 0305-4470 10.1088/0305-4470/35/7/320 35, 1763 (2002) and references therein.
51. J. R. Klauder, J. Math. Phys. JMAPAQ 0022-2488 10.1063/1.1704035 4, 1058 (1963);
52. J. R. Klauder and B. S. Skagerstam, Coherent States, Applications in Physics and Mathematical Physics (World Scientific, Singapore, 1985).
53. From the historical point of view, to our knowledge, the first definition of squeezed-vacuum state in a finite-dimensional Hilbert space was established in the literature by A. Miranowicz, W. Leońsky, S. Dyrting, and R. Tanaś, Acta Phys. Slov. APSVCO 0323-0465 46, 451 (1996). In particular, the authors proposed a generation scheme for this quantum state which is based on cavity-QED experiments involving a cavity with nonlinear Kerr medium pumped by an external second-order parametric process. It must be stressed, however, that such theoretical approach presents inherent mathematical restrictions on the right choice of reference state and associated problems with the correct definition of the unitary squeeze operator into the finite-dimensional context.
54. M. A. Marchiolli, S. S. Mizrahi, and V. V. Dodonov, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.56.4278 56, 4278 (1997);
55. M. A. Marchiolli, S. S. Mizrahi, and V. V. Dodonov, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.57.3885 57, 3885 (1998).
56. F. Dell'Anno, S. De Siena, and F. Illuminati, Phys. Rep. PRPLCM 0370-1573 10.1016/j.physrep.2006.01.004 428, 53 (2006).
57. S. S. Mizrahi and J. Daboul, Physica A PHYADX 0378-4371 189, 635 (1992);
58. S. S. Mizrahi and M. A. Marchiolli, Physica A PHYADX 0378-4371 10.1016/0378-4371(93)90100-I 199, 96 (1993).
59. H. Araki and E. H. Lieb, Commun. Math. Phys. CMPHAY 0010-3616 10.1007/BF01646092 18, 160 (1970);
60. E. H. Lieb and M. B. Ruskai, J. Math. Phys. JMAPAQ 0022-2488 10.1063/1.1666274 14, 1938 (1973).
61. Note that the mathematical procedure established in Ref. for the continuum limit permits us to show that E [H;s] coincides exactly with E [H;λ]. Following Wehrl's conjecture [A. Wehrl, Rev. Mod. Phys. RMPHAT 0034-6861 10.1103/RevModPhys.50.221 50, 221 (1978);
62. A. Wehrl, Rep. Math. Phys. RMHPBE 0034-4877 10.1016/0034-4877(79)90070-3 16, 353 (1979)]
63. which was subsequently proved by Lieb [E. H. Lieb, Commun. Math. Phys. CMPHAY 0010-3616 10.1007/BF01940328 62, 35 (1978)], for any wave packet in the continuous coherent-state representation the minimum value that E [H;λ] can reach is 1, namely, λ (0,) we obtain E [H;λ] ≥1. However, if one considers both the discrete squeezed-state representations here studied, it is possible to verify indirectly that, in principle, min (E [H;s]) is dependent on the Hilbert-space dimensionality.
64. Complexity, Entropy and the Physics of Information, edited by, W. H. Zurek, Santa Fe Institute Studies in the Sciences of Complexity Vol. VIII, Proceedings of the Workshop on "Complexity, Entropy, and the Physics of Information," 1989, in Santa Fe, New Mexico (Addison-Wesley, Redwood City, CA, 1990).
65. C. M. Caves, Phys. Rev. D PRVDAQ 0556-2821 10.1103/PhysRevD.23.1693 23, 1693 (1981).
66. H. P. Yuen, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.13.2226 13, 2226 (1976).
67. W. Schleich and J. A. Wheeler, J. Opt. Soc. Am. B JOBPDE 0740-3224 4, 1715 (1987);
68. W. Schleich and J. A. Wheeler, Nature (London) NATUAS 0028-0836 10.1038/326574a0 326, 574 (1987);
69. W. Schleich, D. F. Walls, and J. A. Wheeler, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.38.1177 38, 1177 (1988);
70. A. Vourdas and R. M. Weiner, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.36.5866 36, 5866 (1987);
71. V. V. Dodonov, A. B. Klimov, and V. I. Man'ko, Phys. Lett. A PYLAAG 0375-9601 10.1016/0375-9601(89)90398-8 134, 211 (1989).
72. W. Miller, Encyclopedia of Mathematics and its Applications: Symmetry and Separation of Variables (Addison-Wesley Publishing Company, Reading, MA, 1977).
73. M. S. Kim, F. A. M. de Oliveira, and P. L. Knight, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.40.2494 40, 2494 (1989).
74. M. Ruzzi, M. A. Marchiolli, and D. Galetti (unpublished).