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Quantum-field-theoretical approach to phase-space techniques: Symmetric Wick theorem and multitime Wigner representation. (English) Zbl 1360.81228

Summary: In this work we present the formal background used to develop the methods used in earlier works to extend the truncated Wigner representation of quantum and atom optics in order to address multi-time problems. Analogs of Wick’s theorem for the Weyl ordering are verified. Using the Bose-Hubbard chain as an example, we show how these may be applied to constructing a mapping of the system in question to phase space. Regularisation issues and the reordering problem for the Heisenberg operators are addressed.

MSC:

81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
81T99 Quantum field theory; related classical field theories
81V80 Quantum optics
81V45 Atomic physics
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References:

[1] Berg, B.; Plimak, L. I.; Polkovnikov, A.; Olsen, M. K.; Fleischhauer, M.; Schleich, W. P., Phys. Rev. A, 80, 033624 (2009)
[2] Werner, M. J.; Drummond, P. D., J. Comput. Phys., 132, 312 (1997)
[3] Wolf, E.; Mandel, L., Optical Coherence and Quantum Optics (1995), Cambridge University Press
[4] Vasil’ev, A. N., Functional Methods in Quantum Field Theory and Statistical Physics (1998), Gordon and Breach
[6] Plimak, L. I.; Stenholm, S., Ann. Phys. (N.Y.), 327, 2691 (2012)
[7] Polkovnikov, A., Ann. Phys. (N.Y.), 325, 1790 (2010)
[8] Steel, M. J.; Olsen, M. K.; Plimak, L. I.; Drummond, P. D.; Tan, S. M.; Collett, M. J.; Walls, D. F.; Graham, R., Phys. Rev. A, 58, 4824 (1998)
[9] Olsen, M. K.; Plimak, L. I., Phys. Rev. A, 68, 031603 (2003)
[10] Olsen, M. K., Phys. Rev. A, 69, 013601 (2004)
[11] Olsen, M. K.; Bradley, A. S.; Cavalcanti, S. B., Phys. Rev. A, 70, 033611 (2004)
[12] Olsen, M. K.; Bradley, A. S., Opt. Comm., 282, 3924 (2009)
[13] Johnsson, M. T.; Hope, J. J., Phys. Rev. A, 75, 043619 (2007)
[14] Jain, P.; Bradley, A. S.; Gardiner, C. W., Phys. Rev. A, 76, 023617 (2007)
[15] Ferris, A. J., Phys. Rev. A, 77, 012712 (2008)
[16] Hoffmann, S. E.; Corney, J. F.; Drummond, P. D., Phys. Rev. A, 78, 013622 (2008)
[17] Corney, J. F., Phys. Rev. A, 78, 023831 (2008)
[18] Olsen, M. K.; Davis, M. J., Phys. Rev. A, 73, 063618 (2006)
[19] Ferris, A. J.; Olsen, M. K.; Davis, M. J., Phys. Rev. A, 79, 043634 (2009)
[20] Shrestha, U.; Javanainen, J.; Ruostekoski, J., Phys. Rev. A, 79, 043617 (2009)
[21] Midgley, S. L.W., Phys. Rev. A, 79, 053632 (2009)
[22] Opanchuk, B., Europhys. Lett., 97, 5003 (2012)
[23] Sau, J. D., Phys. Rev. A, 80, 023622 (2009)
[24] Mathey, L.; Polkovnikov, A., Phys. Rev. A, 80, 041601 (2009)
[25] Mathey, L.; Polkovnikov, A., Phys. Rev. A, 81, 033605 (2010)
[26] Martin, A. D.; Ruostekoski, J., Phys. Rev. Lett., 104, 194102 (2010)
[27] Chianca, C. V.; Olsen, M. K., Phys. Rev. A, 83, 043607 (2011)
[28] Chianca, C. V.; Olsen, M. K., Phys. Rev. A, 84, 043636 (2011)
[29] Opanchuk, B., Phys. Rev. A, 86, 023625 (2012)
[30] Kubo, R., J. Phys. Soc. Jap., 12, 570 (1957)
[31] Kubo, R.; Toda, M.; Hashitsume, N., Statistical Physics II: Nonequilibrium Statistical Mechanics (1985), Springer
[32] Plimak, L. I.; Fleischhauer, M.; Olsen, M. K.; Collett, M. J., Phys. Rev. A, 67, 013812 (2003)
[33] Plimak, L. I.; Stenholm, S., Ann. Phys. (N.Y.), 323, 1963 (2008)
[34] Plimak, L. I.; Stenholm, S., Ann. Phys. (N.Y.), 323, 1989 (2008)
[35] Plimak, L. I.; Stenholm, S., Ann. Phys. (N.Y.), 324, 600 (2009)
[36] Konstantinov, O. V.; Perel, V. I., Zh. Eksp. Theor. Phys.. Zh. Eksp. Theor. Phys., Sov. Phys. JETP, 12, 142 (1961)
[37] Keldysh, L. V., Zh. Eksp. Theor. Phys.. Zh. Eksp. Theor. Phys., Sov. Phys. JETP, 20, 1018 (1965)
[38] Wyld, H. W., Ann. Phys. (N.Y.), 14, 143 (1961)
[39] Zinn-Justin, J., Quantum Field Theory and Critical Phenomena (1989), Oxford University Press
[40] Vasil’ev, A. N., The Field Theoretic Renormalization Group in Critical Behavior Theory and Stochastic Dynamics (2004), CRC Press · Zbl 1140.82019
[41] Pawula, R. F., Phys. Rev. A, 162, 186 (1967)
[42] Plimak, L. I.; Olsen, M. K.; Fleischhauer, M.; Collett, M. J., Europhys. Lett., 56, 372 (2001)
[43] Plimak, L. I., Phys. Rev. A, 50, 2120 (1994)
[44] Schwinger, J. S., J. Math. Phys., 2, 407 (1961)
[45] Kamenev, A.; Levchenko, A., Adv. Phys., 58, 197 (2009), Also available as e-print arXiv:0901.3586v3
[46] Jaksch, D.; Bruder, C.; Cirac, J. I.; Gardiner, C. W.; Zoller, P., Phys. Rev. Lett., 81, 3108 (1998)
[47] Plimak, L. I.; Fleischhauer, M.; Walls, D. F., Europhys. Lett., 43, 641 (1998)
[48] Plimak, L. I.; Collett, M. J.; Walls, D. F.; Fleischhauer, M., (Casalbuoni, R.; etal., Proceedings of the Sixth International Conference on Path Integrals from peV to TeV (1999), World Scientific: World Scientific London), 241
[49] Schweber, S., An Introduction to Relativistic Quantum Field Theory (2005), Dover · Zbl 0111.43102
[50] Bogoliubov, N. N.; Shirkov, D. V., Introduction to the Theory of Quantized Fields (1980), Wiley: Wiley NY · Zbl 0925.81002
[51] Gardiner, C. W., Quantum Noise (1991), Springer-Verlag: Springer-Verlag Berlin · Zbl 0773.60095
[52] Olsen, M. K.; Plimak, L. I.; Collett, M. J.; Walls, D. F., Phys. Rev. A, 62, 023802 (2000)
[53] Hori, T., Progr. Theoret. Phys., 7, 378 (1952)
[54] Plimak, L. I.; Walls, D. F., Phys. Rev. A, 50, 2627 (1994)
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