Quantum kinetic theory. V. Quantum kinetic master equation for mutual interaction of condensate and noncondensate

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 61, Issue 3, 2000, Pages 336011-3360126

  Gardiner, C W ^a   Zoller, P ^b  

^a VICTORIA UNIVERSITY OF WELLINGTON   (New Zealand)

^b UNIVERSITY OF INNSBRUCK   (Austria)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0011767290     PISSN: 10502947     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (52)

1. M. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E.A. Cornell, Science 269, 198 (1995).
2. C.C. Bradley, C.A. Sackett, J.J. Tollet, and R. Hulet, Phys. Rev. Lett. 75, 1687 (1995).
3. K.B. Davis, M.-O. Mewes, MR. Andrews, N.J. van Druten, D.S. Durfee, D.M. Kurn, and W. Ketterle, Phys. Rev. Lett. 75, 3969 (1995).
4. For a current using of condensates, see the BEC Home Page, URL http://amo.phy.gasou.edu/bec.html/
5. N.N. Bogoliubov, J. Phys. (Moscow) 11, 23 (1947); reprinted in D. Pines, The Many Body Problem (Benjamin, New York, 1962).
6. N.N. Bogoliubov, J. Phys. (Moscow) 11, 23 (1947); reprinted in D. Pines, The Many Body Problem (Benjamin, New York, 1962).
7. A.L. Fetter, Ann. Phys. (N.Y.) 70, 67 (1972); Phys. Rev. A 53, 4245 (1996); M. Lewenstein and L. You, ibid. 53, 909 (1996); L. You, W. Hoston, and M. Lewenstein, ibid. 55, R1581 (1997); M. Lewenstein and L. You, Phys. Rev. Lett. 77, 3489 (1996).
8. A.L. Fetter, Ann. Phys. (N.Y.) 70, 67 (1972); Phys. Rev. A 53, 4245 (1996); M. Lewenstein and L. You, ibid. 53, 909 (1996); L. You, W. Hoston, and M. Lewenstein, ibid. 55, R1581 (1997); M. Lewenstein and L. You, Phys. Rev. Lett. 77, 3489 (1996).
9. A.L. Fetter, Ann. Phys. (N.Y.) 70, 67 (1972); Phys. Rev. A 53, 4245 (1996); M. Lewenstein and L. You, ibid. 53, 909 (1996); L. You, W. Hoston, and M. Lewenstein, ibid. 55, R1581 (1997); M. Lewenstein and L. You, Phys. Rev. Lett. 77, 3489 (1996).
10. A.L. Fetter, Ann. Phys. (N.Y.) 70, 67 (1972); Phys. Rev. A 53, 4245 (1996); M. Lewenstein and L. You, ibid. 53, 909 (1996); L. You, W. Hoston, and M. Lewenstein, ibid. 55, R1581 (1997); M. Lewenstein and L. You, Phys. Rev. Lett. 77, 3489 (1996).
11. A.L. Fetter, Ann. Phys. (N.Y.) 70, 67 (1972); Phys. Rev. A 53, 4245 (1996); M. Lewenstein and L. You, ibid. 53, 909 (1996); L. You, W. Hoston, and M. Lewenstein, ibid. 55, R1581 (1997); M. Lewenstein and L. You, Phys. Rev. Lett. 77, 3489 (1996).
12. C.W. Gardiner, Phys. Rev. A 56, 1414 (1997); R. Dum and Y. Castin, ibid. 57, 3008 (1998).
13. C.W. Gardiner, Phys. Rev. A 56, 1414 (1997); R. Dum and Y. Castin, ibid. 57, 3008 (1998).
14. V.L. Ginzburg and L.P. Pitaevskii, Zh. Éksp. Teor. Fiz. 34, 1240 (1958) [Sov. Phys. JETP 7, 858 (1958)]; E.P. Gross, J. Math. Phys. 4, 195 (1963).
15. V.L. Ginzburg and L.P. Pitaevskii, Zh. Éksp. Teor. Fiz. 34, 1240 (1958) [Sov. Phys. JETP 7, 858 (1958)]; E.P. Gross, J. Math. Phys. 4, 195 (1963).
16. V.L. Ginzburg and L.P. Pitaevskii, Zh. Éksp. Teor. Fiz. 34, 1240 (1958) [Sov. Phys. JETP 7, 858 (1958)]; E.P. Gross, J. Math. Phys. 4, 195 (1963).
17. A.A. Abrikosov, L.P. Gorkov, and I.E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics (Dover New York, 1963); E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, Vol. 9 of Landau and Lifshitz Course of Theoretical Physics (Pergamon Press, Oxford, 1980), Pt. 2.
18. A.A. Abrikosov, L.P. Gorkov, and I.E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics (Dover New York, 1963); E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, Vol. 9 of Landau and Lifshitz Course of Theoretical Physics (Pergamon Press, Oxford, 1980), Pt. 2.
19. A. Griffin, Excitations in a Bose-Condensed Liquid (Cambridge University Press, Cambridge, 1993).
20. In this paper will use the notation QKI for [12], QKII for [13], QKIII for [14], and QKIV for [15].
21. C.W. Gardiner and P. Zoller, Phys. Rev. A 55, 2902 (1997).
22. D. Jaksch, C.W. Gardiner, and P. Zoller, Phys. Rev. A 56, 575 (1997).
23. C.W. Gardiner and P. Zoller, Phys. Rev. A 58, 536 (1998).
24. D. Jaksch, C.W. Gardiner, and P. Zoller, Phys. Rev. A 58, 1450 (1997).
25. C.W. Gardiner, P. Zoller, R.J. Ballagh, and M.J. Davis, Phys. Rev. Lett. 79, 1793 (1997).
26. C.W. Gardiner, M.D. Lee, R.J. Ballagh, M.J. Davis, and P. Zoller, Phys. Rev. Lett. 81, 5266 (1998).
27. L.W. Nordheim, Proc. R. Soc. London, Ser. A 119, 689 (1928); F. Bloch, Z. Phys. 52, 555 (1928); S. Kikuchi and L. Nordheim, ibid. 60, 652 (1930); E.A. Uehling and G.E. Uhlenbeck, Phys. Rev. 43, 552 (1933).
28. L.W. Nordheim, Proc. R. Soc. London, Ser. A 119, 689 (1928); F. Bloch, Z. Phys. 52, 555 (1928); S. Kikuchi and L. Nordheim, ibid. 60, 652 (1930); E.A. Uehling and G.E. Uhlenbeck, Phys. Rev. 43, 552 (1933).
29. L.W. Nordheim, Proc. R. Soc. London, Ser. A 119, 689 (1928); F. Bloch, Z. Phys. 52, 555 (1928); S. Kikuchi and L. Nordheim, ibid. 60, 652 (1930); E.A. Uehling and G.E. Uhlenbeck, Phys. Rev. 43, 552 (1933).
30. L.W. Nordheim, Proc. R. Soc. London, Ser. A 119, 689 (1928); F. Bloch, Z. Phys. 52, 555 (1928); S. Kikuchi and L. Nordheim, ibid. 60, 652 (1930); E.A. Uehling and G.E. Uhlenbeck, Phys. Rev. 43, 552 (1933).
31. E. Levich and V. Yakhot, Phys. Rev. B 15, 243 (1977); J. Phys. A 11, 2237 (1978); D.W. Snoke and J.P. Wolfe, Phys. Rev. B 39, 4030 (1989); B. Svistunov, J. Mosc. Phys. Soc. 1, 373 (1991); H.T.C. Stoof, Phys. Rev. Lett. 66, 3148 (1991); Yu.M. Kagan, B.V. Svistunov, and G.V. Shlyapnikov, Sov. Phys. JETP 75, 387 (1992); D.V. Semikoz and I.I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995); H.T.C. Stoof, Phys. Rev. A 49, 3824 (1994); Bose-Einstein Condensation, edited by A. Griffin, D.W. Snoke, and S. Stringari (Cambridge University Press, Cambridge, 1995).
32. E. Levich and V. Yakhot, Phys. Rev. B 15, 243 (1977); J. Phys. A 11, 2237 (1978); D.W. Snoke and J.P. Wolfe, Phys. Rev. B 39, 4030 (1989); B. Svistunov, J. Mosc. Phys. Soc. 1, 373 (1991); H.T.C. Stoof, Phys. Rev. Lett. 66, 3148 (1991); Yu.M. Kagan, B.V. Svistunov, and G.V. Shlyapnikov, Sov. Phys. JETP 75, 387 (1992); D.V. Semikoz and I.I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995); H.T.C. Stoof, Phys. Rev. A 49, 3824 (1994); Bose-Einstein Condensation, edited by A. Griffin, D.W. Snoke, and S. Stringari (Cambridge University Press, Cambridge, 1995).
33. E. Levich and V. Yakhot, Phys. Rev. B 15, 243 (1977); J. Phys. A 11, 2237 (1978); D.W. Snoke and J.P. Wolfe, Phys. Rev. B 39, 4030 (1989); B. Svistunov, J. Mosc. Phys. Soc. 1, 373 (1991); H.T.C. Stoof, Phys. Rev. Lett. 66, 3148 (1991); Yu.M. Kagan, B.V. Svistunov, and G.V. Shlyapnikov, Sov. Phys. JETP 75, 387 (1992); D.V. Semikoz and I.I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995); H.T.C. Stoof, Phys. Rev. A 49, 3824 (1994); Bose-Einstein Condensation, edited by A. Griffin, D.W. Snoke, and S. Stringari (Cambridge University Press, Cambridge, 1995).
34. E. Levich and V. Yakhot, Phys. Rev. B 15, 243 (1977); J. Phys. A 11, 2237 (1978); D.W. Snoke and J.P. Wolfe, Phys. Rev. B 39, 4030 (1989); B. Svistunov, J. Mosc. Phys. Soc. 1, 373 (1991); H.T.C. Stoof, Phys. Rev. Lett. 66, 3148 (1991); Yu.M. Kagan, B.V. Svistunov, and G.V. Shlyapnikov, Sov. Phys. JETP 75, 387 (1992); D.V. Semikoz and I.I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995); H.T.C. Stoof, Phys. Rev. A 49, 3824 (1994); Bose-Einstein Condensation, edited by A. Griffin, D.W. Snoke, and S. Stringari (Cambridge University Press, Cambridge, 1995).
35. E. Levich and V. Yakhot, Phys. Rev. B 15, 243 (1977); J. Phys. A 11, 2237 (1978); D.W. Snoke and J.P. Wolfe, Phys. Rev. B 39, 4030 (1989); B. Svistunov, J. Mosc. Phys. Soc. 1, 373 (1991); H.T.C. Stoof, Phys. Rev. Lett. 66, 3148 (1991); Yu.M. Kagan, B.V. Svistunov, and G.V. Shlyapnikov, Sov. Phys. JETP 75, 387 (1992); D.V. Semikoz and I.I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995); H.T.C. Stoof, Phys. Rev. A 49, 3824 (1994); Bose-Einstein Condensation, edited by A. Griffin, D.W. Snoke, and S. Stringari (Cambridge University Press, Cambridge, 1995).
36. E. Levich and V. Yakhot, Phys. Rev. B 15, 243 (1977); J. Phys. A 11, 2237 (1978); D.W. Snoke and J.P. Wolfe, Phys. Rev. B 39, 4030 (1989); B. Svistunov, J. Mosc. Phys. Soc. 1, 373 (1991); H.T.C. Stoof, Phys. Rev. Lett. 66, 3148 (1991); Yu.M. Kagan, B.V. Svistunov, and G.V. Shlyapnikov, Sov. Phys. JETP 75, 387 (1992); D.V. Semikoz and I.I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995); H.T.C. Stoof, Phys. Rev. A 49, 3824 (1994); Bose-Einstein Condensation, edited by A. Griffin, D.W. Snoke, and S. Stringari (Cambridge University Press, Cambridge, 1995).
37. E. Levich and V. Yakhot, Phys. Rev. B 15, 243 (1977); J. Phys. A 11, 2237 (1978); D.W. Snoke and J.P. Wolfe, Phys. Rev. B 39, 4030 (1989); B. Svistunov, J. Mosc. Phys. Soc. 1, 373 (1991); H.T.C. Stoof, Phys. Rev. Lett. 66, 3148 (1991); Yu.M. Kagan, B.V. Svistunov, and G.V. Shlyapnikov, Sov. Phys. JETP 75, 387 (1992); D.V. Semikoz and I.I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995); H.T.C. Stoof, Phys. Rev. A 49, 3824 (1994); Bose-Einstein Condensation, edited by A. Griffin, D.W. Snoke, and S. Stringari (Cambridge University Press, Cambridge, 1995).
38. E. Levich and V. Yakhot, Phys. Rev. B 15, 243 (1977); J. Phys. A 11, 2237 (1978); D.W. Snoke and J.P. Wolfe, Phys. Rev. B 39, 4030 (1989); B. Svistunov, J. Mosc. Phys. Soc. 1, 373 (1991); H.T.C. Stoof, Phys. Rev. Lett. 66, 3148 (1991); Yu.M. Kagan, B.V. Svistunov, and G.V. Shlyapnikov, Sov. Phys. JETP 75, 387 (1992); D.V. Semikoz and I.I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995); H.T.C. Stoof, Phys. Rev. A 49, 3824 (1994); Bose-Einstein Condensation, edited by A. Griffin, D.W. Snoke, and S. Stringari (Cambridge University Press, Cambridge, 1995).
39. E. Levich and V. Yakhot, Phys. Rev. B 15, 243 (1977); J. Phys. A 11, 2237 (1978); D.W. Snoke and J.P. Wolfe, Phys. Rev. B 39, 4030 (1989); B. Svistunov, J. Mosc. Phys. Soc. 1, 373 (1991); H.T.C. Stoof, Phys. Rev. Lett. 66, 3148 (1991); Yu.M. Kagan, B.V. Svistunov, and G.V. Shlyapnikov, Sov. Phys. JETP 75, 387 (1992); D.V. Semikoz and I.I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995); H.T.C. Stoof, Phys. Rev. A 49, 3824 (1994); Bose-Einstein Condensation, edited by A. Griffin, D.W. Snoke, and S. Stringari (Cambridge University Press, Cambridge, 1995).
40. A. Griffin, Phys. Rev. B 53, 9341 (1996). This paper summarizes the various "standard" ways of using the Hartree-Fock-Bogoliubov method, and discusses the problems that can occur. See also Refs. [6,25,26] and S.A. Morgan, S. Choi, and K. Burnett, Phys. Rev. A 57, 3818 (1998).
41. A. Griffin, Phys. Rev. B 53, 9341 (1996). This paper summarizes the various "standard" ways of using the Hartree-Fock- Bogoliubov method, and discusses the problems that can occur. See also Refs. [6,25,26] and S.A. Morgan, S. Choi, and K. Burnett, Phys. Rev. A 57, 3818 (1998).
42. H.T.C. Stoof, J. Low Temp. Phys. 114, 11 (1999).
43. R. Walser, J. Williams, J. Cooper, and M. Holland, Phys. Rev. A 59, 3878 (1999).
44. E. Zaremba, A. Griffin, and T. Nikuni, Phys. Rev. A 57, 4695 (1998).
45. A. Griffin, in Bose-Einstein Condensation in Atomic Gases, edited by M. Inguscio (IOS Press, Amsterdam, 1999).
46. N.P. Proukakis, K. Burnett, and H.T.C. Stoof, Phys. Rev. A 57, 1230 (1998) gives an excellent exposition of the issues which must be addressed in this approximation.
47. F. Dalfovo, S. Giorgini, L.P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 (1999).
48. L. You, W. Hoston, and M. Lewenstein, Phys. Rev. A 55, R1581 (1997) present and solve these equations in the form most appropriate to current condensates.
49. R. Graham, Phys. Rev. Lett. 81, 5262 (1998).
50. P.D. Drummond and J.F. Corney, Phys. Rev. A 60, R2661 (1999).
51. M. Girardeau and R. Arnowitt, Phys. Rev. 113, 755 (1959); M. Girardeau, Phys. Rev. A 58, 775 (1998).
52. M. Girardeau and R. Arnowitt, Phys. Rev. 113, 755 (1959); M. Girardeau, Phys. Rev. A 58, 775 (1998).