Quantum dynamics in ultracold atomic physics

Abstract

We review recent developments in the theory of quantum dynamics in ultracold atomic physics, including exact techniques and methods based on phase-space mappings that are applicable when the complexity becomes exponentially large. Phase-space representations include the truncated Wigner, positive-P and general Gaussian operator representations which can treat both bosons and fermions. These phase-space methods include both traditional approaches using a phase-space of classical dimension, and more recent methods that use a non-classical phase-space of increased dimensionality. Examples used include quantum Einstein–Podolsky–Rosen (EPR) entanglement of a four-mode BEC, time-reversal tests of dephasing in single-mode traps, BEC quantum collisions with up to 10⁶ modes and 10⁵ interacting particles, quantum interferometry in a multi-mode trap with nonlinear absorption, and the theory of quantum entropy in phase-space. We also treat the approach of variational optimization of the sampling error, giving an elementary example of a nonlinear oscillator.

Keywords

quantum / dynamics / atomic / ultracold / collisions / entanglement / phase-space

References

[1] D. J. Heinzen, R. Wynar, P. D. Drummond, and K. V. Kheruntsyan, Phys. Rev. Lett., 2000, 84(22): 5029

[2] E. A. Donley, N. R. Claussen, S. T. Thompson, and C. E. Wieman, Nature, 2002, 417(6888): 529

[3] A. E. Leanhardt, T. A. Pasquini, M. Saba, A. Schirotzek, Y. Shin, D. Kielpinski, D. E. Pritchard, and W. Ketterle, Science, 2003, 301(5639): 1513

[4] S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge: Cambridge University Press, 1995

[5] H. J. Metcalf and P. van der Straten, Laser Cooling and Trapping, New York: Springer-Verlag, 1999: 339

[6] M. Schellekens, R. Hoppeler, A. Perrin, J. V. Gomes, D. Boiron, A. Aspect, and C. I. Westbrook, Science, 2005, 310(5748): 648

[7] A. Robert, O. Sirjean, A. Browaeys, J. Poupard, S. Nowak, D. Boiron, C. I. Westbrook, and A. Aspect, Science, 2001, 292(5516): 461

[8] J. M. Vogels, K. Xu, and W. Ketterle, Phys. Rev. Lett., 2002, 89: 020401, arXiv: cond-mat /0203286, 2002

[9] F. A. Berezin, The Method of Second Quantization, New York: Academic Press, 1966

[10] P. D. Drummond and S. J. Carter, J. Opt. Soc. Am. B, 1987, 4(10): 1565

[11] D. I. Choi and Q. Niu, Phys. Rev. Lett., 1999, 82(10): 2022

[12] J. H. Denschlag, J. E. Simsarian, H. Haeffner, C. McKenzie, A. Browaeys, D. Cho, K. Helmerson, S. L. Rolston, and W. D. Phillips, J. Phys. B, 2002, 35: 3095, arXiv:cond-mat /0206063, 2002

[13] J. Hubbard, Proc. Roy. Soc. A, 1963, 276(1365): 238

[14] R. P. Feynman, Int. J. Theor. Phys., 1982, 21(6-7): 467

[15] Q. Y. He, M. D. Reid, T. G. Vaughan, C. Gross, M. Oberthaler, and P. D. Drummond, Phys. Rev. Lett., 2011, 106(12): 120405

[16] T. J. Haigh, A. J. Ferris, and M. K. Olsen, Opt. Commun., 2010, 283(18): 3540

[17] P. D. Drummond, R. M. Shelby, S. R. Friberg, and Y. Yamamoto, Nature, 1993, 365(6444): 307

[18] V. Giovannetti, S. Mancini, D. Vitali, and P. Tombesi, Phys. Rev. A, 2003, 67(2): 022320

[19] R. Dong, J. Heersink, J. I. Yoshikawa, O. Glockl, U. L. Andersen, and G. Leuchs, New J. Phys., 2007, 9(11): 410

[20] W. P. Bowen, N. Treps, R. Schnabel, and P. K. Lam, Phys. Rev. Lett., 2002, 89(25): 253601

[21] C. W. Gardiner and P. Zoller, Quantum Noise, 2nd Ed., Berlin: Springer-Verlag, 2000

[22] E. P. Wigner, Phys. Rev., 1932, 40(5): 749

[23] R. J. Glauber, Phys. Rev., 1963, 131(6): 2766

[24] E. C. G. Sudarshan, Phys. Rev. Lett., 1963, 10(7): 277

[25] K. Husimi, Proc. Phys.-Math. Soc. Jpn., 1940, 22: 264

[26] S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby, Phys. Rev. Lett., 1987, 58(18): 1841

[27] S. J. Carter and P. D. Drummond, Phys. Rev. Lett., 1991, 67(27): 3757

[28] P. D. Drummond and A. D. Hardman, Europhys. Lett., 1993, 21(3): 279

[29] P. D. Drummond, R. M. Shelby, S. R. Friberg, and Y. Yamamoto, Nature, 1993, 365(6444): 307

[30] P. D. Drummond and J. F. Corney, J. Opt. Soc. Am. B, 2001, 18(2): 139

[31] M. Rosenbluh and R. M. Shelby, Phys. Rev. Lett., 1991, 66(2): 153

[32] J. Heersink, V. Josse, G. Leuchs, and U. L. Andersen, Opt. Lett., 2005, 30(10): 1192

[33] J. F. Corney, P. D. Drummond, J. Heersink, V. Josse, G. Leuchs, and U. L. Andersen, Phys. Rev. Lett., 2006, 97(2): 023606

[34] M. J. Steel, M. K. Olsen, L. I. Plimak, P. D. Drummond, S. M. Tan, M. J. Collet, D. F. Walls, and R. Graham, Phys. Rev. A, 1998, 58(6): 4824

[35] P. B. Blakie, A. S. Bradley, M. J. Davis, R. J. Ballagh, and C. W. Gardiner, Adv. Phys., 2008, 57: 363

[36] M. Egorov, R. P. Anderson, V. Ivannikov, B. Opanchuk, P. Drummond, B. V. Hall, and A. I. Sidorov, Phys. Rev. A, 2011, 84(2): 021605(R)

[37] R. Glauber, Phys. Rev., 1963, 131(6): 2766

[38] K. Cahill and R. Glauber, Phys. Rev. A, 1999, 59(2): 1538

[39] J. F. Corney and P. D. Drummond, Phys. Rev. A, 2003, 68: 23, arXiv:quant-ph/0308064, 2003

[40] K. E. Cahill and R. J. Glauber, Phys. Rev., 1969, 177(5): 1857

[41] J. E. Moyal, Math. Proc. Camb. Philos. Soc., 1949, 45(01): 99

[42] P. D. Drummond and A. D. Hardman, Europhys. Lett., 1993, 21(3): 279

[43] M. Steel, M. K. Olsen, L. I. Plimak, P. D. Drummond, S. Tan, M. J. Collett, D. Walls, and R. Graham, Phys. Rev. A, 1998, 58(6): 4824

[44] A. Sinatra, C. Lobo, and Y. Castin, J. Phys. B, 2002, 35(17): 3599

[45] M. W. Jack, Phys. Rev. Lett., 2002, 89(14): 140402

[46] A. Sørensen, L. M. Duan, J. I. Cirac, and P. Zoller, Nature, 2001, 409(6816): 63

[47] D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen, Phys. Rev. A, 1994, 50(1): 67

[48] A. Kaufman, R. P. Anderson, T. Hanna, E. Tiesinga, P. Julienne, and D. Hall, Phys. Rev. A, 2009, 80(5): 050701

[49] P. D. Drummond and C. W. Gardiner, J. Phys. A, 1980, 13(7): 2353

[50] P. D. Drummond and P. Deuar, J. Opt. B, 2003, 5: S281, arXiv:cond-mat/0309514, 2003

[51] M. R. Dowling, P. D. Drummond, M. J. Davis, and P. Deuar, Phys. Rev. Lett., 2005, 94(13): 130401

[52] P. Deuar and P. D. Drummond, Phys. Rev. Lett., 2007, 98(12): 120402

[53] J. M. Vogels, K. Xu, and W. Ketterle, Phys. Rev. Lett., 2002, 89(2): 020401

[54] V. Krachmalnicoff, J. C. Jaskula, M. Bonneau, V. Leung, G. B. Partridge, D. Boiron, C. I. Westbrook, P. Deuar, P. Ziń, M. Trippenbach, and K. V. Kheruntsyan, Phys. Rev. Lett., 2010, 104(15): 150402

[55] P. Deuar and P. Drummond, Phys. Rev. Lett., 2007, 98: 120402, arXiv:cond-mat/0607831, 2006

[56] J. F. Corney and P. D. Drummond, Phys. Rev. A, 2003, 68(6): 63822

[57] J. F. Corney and P. D. Drummond, Phys. Rev. B, 2006, 73(12): 125112

[58] T. Aimi and M. Imada, J. Phys. Soc. Jpn., 2007, 76(11): 113708

[59] A. Rényi, in: Proc. Fourth Berkeley Symp. Math. Stat. and Probability., Vol. I, Berkeley, CA: University of California Press, 1961: 547-561

[60] L. E. C. Rosales-Zárate and P. D. Drummond, Phys. Rev. A, 2011, 84(4): 042114

[61] K. E. Cahill and R. J. Glauber, Phys. Rev. A, 1999, 59(2): 1538

[62] J. Frenkel, Wave Mechanics: Advanced General Theory, 1st Ed., Oxford: Clarendon Press, 1934

[63] H. D. Meyer, F. Gatti, and G. A. Worth (Eds.), Multidimensional Quantum Dynamics: MCTDH Theory and Applications, Chichester: Wiley-VCH, 2009: 442

[64] O. Alon, A. Streltsov, and L. Cederbaum, Phys. Rev. A, 2008, 77(3): 33613

[65] H. Everett, Rev. Mod. Phys., 1957, 29(3): 454

[66] T. Fabcic, J. Main, and G. Wunner, J. Chem. Phys., 2008, 128(4): 044116

[67] A. N. Tikhonov, A. S. Leonov, and A. G. Yagola, Nonlinear Ill-posed Problems, London: Chapman & Hall, 1990

[68] G. J. Milburn, Phys. Rev. A, 1986, 33(1): 674

[69] M. Greiner, O. Mandel, T. W. Hänsch, and I. Bloch, Nature, 2002, 419(6902): 51

[70] P. Deuar and P. D. Drummond, Phys. Rev. A, 2002, 66(3): 33812

[71] P. Deuar and P. D. Drummond, J. Phys. A, 2006, 39(11): 2723

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