Coherent Versus Incoherent Dynamics During Bose-Einstein Condensation in Atomic Gases

E. Schrödinger, Ann. Phys. 79, 361 (1926).

M. Born, Z. Phys. 37, 863 (1926); ibid. 38, 803 (1926).

W. Pauli, Z. Phys. 31, 765 (1925).

W. Pauli, Z. Phys. 43, 601 (1927).

S. N. Bose, Z. Phys. 26, 178 (1924).

A. Einstein, Sitz. Kgl. Preuss. Akad. Wiss. (Berlin), 3 (1925).

E. Fermi, Z. Phys. 36, 902 (1926).

L. N. Cooper, Phys. Rev. 104, 1189 (1956).

J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 108, 1175 (1957).

S. R. de Groot, G. J. Hooyman, and C. A. Seldam, Proc. Roy. Soc. London, Ser. A 203, 266 (1950).

R. M. Ziff, G. E. Uhlenbeck, and M. Kac, Phys. Rep. 32, 169 (1977).

F. London, Nature 141, 643 (1938).

L. D. Landau, J. Phys. (U.S.S.R.) 5, 71 (1941).

G. Baym, in Mathematical Methods in Solid State and Superfluid Theory, R. C. Clark and G. H. Derrick (eds.), Oliver and Boyd, Edinburg (1969), p. 121.

We here neglect the fact that the dispersion relation for liquid 4He has a roton minimum, which leads to a more stringent condition on the superfluid velocity than on the basis of the linear, long-wavelength part alone. See A. Griffin, Excitations in a Bose-Condensed Liquid, Cambridge, New York (1993) for more details on the 4He dispersion relation.

J. S. Langer and M. E. Fisher, Phys. Rev. Lett. 19, 560 (1967).

N. N. Bogoliubov, J. Phys. (U.S.S.R.) 11, 23 (1947).

S. T. Beliaev, Zh. Eksp. Teor. Fiz. 34, 417 (1958) [Sov. Phys.-JETP 7, 289 (1958)].

N. M. Hugenholtz and D. Pines, Phys. Rev. 116, 489 (1959).

J. Gavoret and P. Nozières, Ann. Phys. (N.Y.) 28, 349 (1964).

T. D. Lee and C. N. Yang, Phys. Rev. 112, 1419 (1958).

K. Huang and C. N. Yang, Phys. Rev. 105, 767 (1957).

P. C. Hohenberg and P. C. Martin, Ann. Phys. (N.Y.) 34, 291 (1965).

V. N. Popov, Functional Integrals in Quantum Field Theory and Statistical Physics, Reidel, Dordrecht (1983), and references therein.

Yu. A. Nepomnyashchii and N. N. Nepomnyashchii, Zh. Eksp. Teor. Fiz. 75, 976 (1958) [Sov. Phys.-JETP 48, 493 (1958)].

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science 269, 198 (1995).

C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, Phys. Rev. Lett. 75, 1687 (1995); C. C. Bradley, C. A. Sackett, and R. G. Hulet, ibid. 78, 985 (1997).

K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Phys. Rev. Lett. 75, 3969 (1995).

I. F. Silvera and J. T. M. Walraven, Phys. Rev. Lett. 44, 164 (1980).

For an overview of the numerous contributions we refer to T. J. Greytak and D. Kleppner, in New Trends in Atomic Physics, G. Grynberg and R. Stora (eds.), North-Holland, Amsterdam (1984), p. 1125, and I. F. Silvera and J. T. M. Walraven, in Progress in Low Temperature Physics, Vol. 10, D. F. Brewer (ed.), North-Holland, Amsterdam (1986), p. 139.

See Bose-Einstein Condensation, A. Griffin, D. W. Snoke, and S. Stringari (eds.), Cambridge, New York (1995).

H. F. Hess, G. P. Kochanski, J. M. Doyle, N. Masuhara, D. Kleppner, and T. J. Greytak, Phys. Rev. Lett. 59, 672 (1987); N. Masuhara, J. M. Doyle, J. C. Sandberg, D. Kleppner, T. J. Greytak, H. F. Hess, and G. P. Kochanski, ibid. 61, 935 (1988).

R. van Roijen, J. J. Berkhout, S. Jaakkola, and J. T. M. Walraven, Phys. Rev. Lett. 61, 931 (1988).

J. M. Doyle, J. C. Sandberg, I. A. Yu, C. L. Cesar, D. Kleppner, and T. J. Greytak, Phys. Rev. Lett. 67, 603 (1991).

O. J. Luiten, H. G. C. Werij, I. D. Setija, T. W. Hijmans, and J. T. M. Walraven, Phys. Rev. Lett. 70, 544 (1993); I. D. Setija, H. G. C. Werij, O. J. Luiten, M. W. Reynolds, T. W. Hijmans, and J. T. M. Walraven, ibid. 70, 2257 (1993).

After submission of this paper Bose-Einstein condensation was also achieved in spinpolarized atomic hydrogen by the group of D. Kleppner and T. J. Greytak, private communication.

C. W. Gardiner and P. Zoller, Phys. Rev. A 55, 2902 (1997); C. W. Gardiner and P. Zoller (unpublished, cond-mat/9712002) and references therein.

J. Schwinger, J. Math. Phys. 2, 407 (1961).

L. V. Keldysh, Zh. Eksp. Teor. Fiz. 47, 1515 (1964) [Sov. Phys.-JETP 20, 1018 (1965)].

P. Danielewicz, Ann. Phys. (N.Y.) 152, 239 (1984).

K. Chou, Z. Su, B. Hao, and L. Yu, Phys. Rep. 118, 1 (1985).

A. O. Caldeira and A. J. Leggett, Phys. Rev. Lett. 46, 211 (1981); A. O. Caldeira and A. J. Leggett, Ann. Phys. (N.Y.) 149, 374 (1983); ibid. 153, 445 (1984).

See, for example, K. Huang, Statistical Mechanics, Wiley, New York (1987).

A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill, New York (1971). Note that these authors use the more common notation â α(t) and â †α (t) for the bosonic creation and annihilation operators.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge, New York (1995).

J. W. Negele and H. Orland, Quantum Many-Particle Systems, Addison-Wesley, New York (1988).

B. de Wit, private communication.

P. Carruthers and K. S. Dy, Phys. Rev. 147, 214 (1966).

H. Kleinert, Path Integrals in Quantum Mechanics Statistics and Polymer Physics, World Scientific, Singapore (1994).

This follows from the requirement that we want to be able to Fourier transform the Heaviside function.

D. J. Amit, Field Theory, the Renormalization Group, and Critical Phenomena, World Scientific, Singapore (1984).

J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Oxford, New York (1989).

L. P. Kadanoff and G. Baym, Quantum Statistical Mechanics: Green's Function Methods in Equilibrium and Nonequilibrium Problems, Addison-Wesley, New York (1962).

D. C. Langreth and J. W. Wilkins, Phys. Rev. B 6, 3189 (1972).

In agreement with our previous remarks, the notation ε t 0 implies that the time integration can be either from t 0 to ∞ or from t 0 to t.

R. L. Stratonovich, Sov. Phys. Dok. 2, 416 (1958).

J. Hubbard, Phys. Rev. Lett. 3, 77 (1959).

N. G. van Kampen, Stochastic Processes in Physics and Chemistry, North-Holland, Amsterdam (1981).

D. Forster, Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions, Benjamin, Reading (1975).

To make a connection with a large body of knowlede in quantum optics, we note that Eq. (83) explicitly shows that the probability distribution P[ø*, ø t] does not correspond to a P or Q representation of the density matrix, but to a Wigner representation instead. The “diffusion matrix” in the Fokker-Planck equation for P[ø*, ø t] is therefore diagonal and positive, as is explained by C. W. Gardiner, in Quantum Noise, Springer, Berlin (1991), Chapter 6. The same nice feature will also appear in the case of an interacting Bose gas.

M. A. Kastner, Rev. Mod. Phys. 64, 849 (1992) and references therein.

See also H. T. C. Stoof in Bose-Einstein Condensation, A. Griffin, D. W. Snoke, and S. Stringari (eds.), Cambridge, New York (1995), p. 226.

T. W. B. Kibble, J. Phys. A 9, 1387 (1976).

A. H. Guth, Phys. Rev. D 23, 347 (1981).

A. D. Linde, Phys. Lett. B 108, 389 (1982).

W. H. Zurek, Nature 317, 505 (1985).

P. C. Hendry, N. S. Lawson, R. A. M. Lee, P. V. E. McClintock, and C. H. D. Williams, Nature 368, 315 (1994).

E. Levich and Y. Yakhot, Phys. Rev. B 15, 243 (1977); J. Phys. A 11, 2237 (1978).

S. G. Tikhodeev, Zh. Eksp. Teor. Fiz. 97, 681 (1990) [Sov. Phys.-JETP 70, 380 (1990)].

Bose-Einstein condensation has now also been achieved with these gases in the groups of D. J. Heinzen, M. A. Kasevich, L. V. Hau, G. Rempe, W. D. Phillips, T. Hänsch, and C. Salomon, private communication.

H. T. C. Stoof, Phys. Rev. Lett. 66, 3148 (1991); Phys. Rev. A 45, 8398 (1992).

O. J. Luiten, M. W. Reynolds, and J. T. M. Walraven, Phys. Rev. A 53, 381 (1996).

W. Ketterle and N. J. van Druten, Adv. At. Mol. Opt. Phys. 37, 181 (1996).

D. W. Snoke and J. P. Wolfe, Phys. Rev. B 39, 4030 (1989).

B. V. Svistunov, J. Moscow Phys. Soc. 1, 373 (1991).

D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74, 3093 (1995); D. V. Semikoz and I. I. Tkachev (unpublished, hep-ph/9507306).

H. T. C. Stoof, M. Bijlsma, and M. Houbiers, J. Res. Natl. Inst. Stand. Technol. 101, 433 (1996).

H. Shi and A. Griffin, Phys. Rep. 304, 1 (1998).

A. Griffin, Phys. Rev. B 53, 9341 (1996).

M. Bijlsma and H. T. C. Stoof, Phys. Rev. A 55, 498 (1997).

M. Bijlsma and H. T. C. Stoof, Phys. Rev. A 54, 5085 (1996).

B. A. Lippmann and J. Schwinger, Phys. Rev. 79, 469 (1950).

W. Glöckle, The Quantum Mechanical Few-Body Problem, Springer Verlag, Berlin (1983).

K. A. Brückner and K. Sawada, Phys. Rev. 106, 1117 (1957).

I am grateful for a discussion with Paul Julienne about this point.

P. Grüter, D. Ceperley, and F. Laloë, Phys. Rev. Lett. 79, 3549 (1997).

B. D. Josephson, Phys. Lett. 1, 251 (1962). See also P. W. Anderson, Rev. Mod. Phys. 38, 298 (1966).

P. B. Weichmann, Phys. Rev. B 38, 8739 (1988).

T. R. Kirkpatrick and J. R. Dorfman, J. Low Temp. Phys. 58, 301 (1985); ibid. 58, 399 (1985).

U. Eckern, J. Low Temp. Phys. 54, 333 (1984).

Yu. Kagan, B. V. Svistunov, and G. V. Shlyapnikov, Zh. Eksp. Teor. Fiz. 101, 528 (1992) [Sov. Phys.-JETP 75, 387 (1992)].

H.-J. Miesner, D. M. Stamper-Kurn, M. R. Andrews, D. S. Durfee, S. Inouye, and W. Ketterle, Science 279, 1005 (1998).

N. P. Proukakis and K. Burnett, J. Res. Natl. Inst. Stand. Technol. 101, 457 (1996); N. P. Proukakis, K. Burnett, and H. T. C. Stoof, Phys. Rev. A 57, 1230 (1998).

K. Damle, S. N. Majumdar, and S. Sachdev, Phys. Rev. A 54, 5037 (1996).

A. J. Bray, Adv. Phys. 43, 357 (1994).

For a different point of view see Yu. Kagan and B. V. Svistunov, Phys. Rev. Lett. 79, 3331 (1997). Note that the comment in Ref. [6] of this paper is incorrect, because the time derivative term in the action for the condensate density leads, as we have seen explicitly in Sec. III.B, apart from the introduction of the instantaneous chemical potential μ(t) only to a topological term, which affects the boundary conditions but not the Euler-Lagrange equations.

H. Risken, Z. Phys. 186, 85 (1965); ibid. 191, 302 (1966).

H. T. C. Stoof, Phys. Rev. Lett. 78, 768 (1997).

Y. Castin, private communication.

For an equilibrium argument at zero temperature see L. D. Landau and E. M. Lifshitz, Statistical Physics, Pergamon, London (1958) and P. Nozières, in Bose-Einstein Condensation, A. Griffin, D. W. Snoke, and S. Stringari (eds.), Cambridge, New York (1995), p. 15. Nonzero temperatures are discussed in H. T. C. Stoof, Phys. Rev. A 49, 4704 (1995).

W. A. B. Evans and R. I. M. A. Rashid, J. Low Temp. Phys. 11, 93 (1973).

This work is performed in collaboration with A. J. Reijnhart, M. Bijlsma, E. Carlon, and J. O. Indekeu. The mean-field calculation is presented in H. T. C. Stoof, Phys. Rev. A 49, 4704 (1995).

C. A. Sackett and R. G. Hulet, private communication.

P. A. Ruprecht, M. J. Holland, K. Burnett, and M. Edwards, Phys. Rev. A 51, 4704 (1995).

C. W. Gardiner, P. Zoller, R. J. Ballagh, and M. J. Davis, Phys. Rev. Lett. 79, 1793 (1997); C. W. Gardiner, M. D. Lee, R. J. Ballagh, and M. J. Davis, and P. Zoller (unpublished, cond-mat/9801027 and cond-mat/9806295).

H. T. C. Stoof, J. Stat. Phys. 87, 1353 (1997).

V. V. Goldman, I. F. Silvera, and A. J. Leggett, Phys. Rev. B 24, 2870 (1981).

D. A. Huse and E. D. Siggia, J. Low Temp. Phys. 46, 137 (1982).

M. Houbiers and H. T. C. Stoof, Phys. Rev. A 54, 5055 (1996).

T. Bergeman, Phys. Rev. A 55, 3658 (1997).

L. P. Pitaevskii, Sov. Phys. JETP 13, 451 (1961); E. P. Gross, J. Math. Phys. 4, 195 (1963).

Our Fokker-Planck equation in principle also describes the tunneling of the condensate, but since we are dealing with a macroscopic quantum tunneling through a barrier in the infinite dimensional configuration space of the condensate it is more convenient to apply the usual instanton techniques reviewed, for instance, by S. Coleman, in Aspects of Symmetry, Cambridge, New York (1985).

J. A. Freire and D. P. Arovas (unpublished, cond-mat/9803280).

S. Stringari, Phys. Rev. Lett. 77, 2360 (1996).

K. G. Singh and D. S. Rokhsar, Phys. Rev. Lett. 77, 1667 (1996).

M. Edwards, P. A. Ruprecht, K. Burnett, R. J. Dodd, and C. W. Clark, Phys. Rev. Lett. 77, 1671 (1996).

Y. Castin and R. Dum, Phys. Rev. Lett. 77, 5315 (1996).

V. M. Perez-Garcia, H. Michinel, J. I. Cirac, M. Lewenstein, and P. Zoller, Phys. Rev. Lett. 77, 5320 (1996).

R. J. Dodd, M. Edwards, C. J. Williams, C. W. Clarck, M. J. Holland, P. A. Ruprecht, and K. Burnett, Phys. Rev. A 54, 661 (1996).

J. Javanainen, Phys. Rev. A 54, 3722 (1996).

L. You, W. Hoston, and M. Lewenstein, Phys. Rev. A 55, 1581 (1997).

P. Ohberg, E. L. Surkov, I. Tittonen, S. Stenholm, M. Wilkens, and G. V. Shlyapnikov, Phys. Rev. A 56, R3346 (1997).

D. S. Jin, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett. 77, 420 (1996); D. S. Jin, M. R. Matthews, J. R. Ensher, C. E. Wieman, and E. A. Cornell, ibid. 78, 764 (1997).

M.-O. Mewes, M. R. Anderson, N. J. van Druten, D. M. Kurn, D. S. Durfee, C. G. Townsend, and W. Ketterle, Phys. Rev. Lett. 77, 988 (1996).

D. A. W. Hutchinson, E. Zaremba, and A. Griffin, Phys. Rev. Lett. 78, 1842 (1997).

R. J. Dodd, M. Edwards, C. W. Clark, and K. Burnett, Phys. Rev. A 57, 32 (1998).

This work is performed in collaboration with R. L. C. Vink.

For the anisotropic generalization see, for instance, G. Baym and C. J. Pethick, Phys. Rev. Lett. 76, 6 (1996).

A. L. Fetter, Phys. Rev. A 53, 4245 (1996).

For a different calculation of the tunneling rate that however neglects the phase fluctuations of the condensate, see E. V. Shuryak, Phys. Rev. A 54, 3151 (1996).

C. A. Sackett, H. T. C. Stoof, and R. G. Hulet, Phys. Rev. Lett. 80, 2031 (1998).

C. A. Sackett, C. C. Bradley, M. Welling, and R. G. Hulet, Appl. Phys. B 65, 433 (1997).

L. P. Pitaevskii, Phys. Lett. A 221, 14 (1996).

R. G. Hulet, private communication.

The fluctuation-dissipation theorem violating approach of Yu. Kagan, A. E. Muryshev, and G. V. Shlyapnikov (unpublished, cond-mat/9801168) predicts that about 50% of the atoms in the condensate remain after a collapse.

M. Lewenstein and L. You, Phys. Rev. Lett. 77, 3489 (1996).

P. W. Anderson, Phys. Rev. 112, 1900 (1958).

E. Abrahams and T. Tsuneto, Phys. Rev. 152, 416 (1966).

H. Kleinert, Forts. Phys. 26, 565 (1978).

In principle the effective action for a neutral superconductor contains also a topological term, but this does not affect the final result of the following argument because it only leads to a constant shift in the total number of particles. For more details, see M. Stone, Int. J. Mod. Phys. B 9, 1359 (1995) and references therein.

The magnitude of the frequency gap is actually not accurately described by this time-dependent Ginzburg-Landau theory, because the action S[Δ*, Δ] is implicitly only valid for long wavelengths hv F k << |Δ0| and low frequencies hω << |Δ0|.

This may be compared directly with the results of E. M. Wright, D. F. Walls, and J. C. Garrison, Phys. Rev. Lett. 77, 2158 (1996) and K. Mølmer, private communication.

For simplicity we here again neglect a topological term in the action, which essentially plays no role in the following. This is also in agreement with our discussion of the superconductor, where exactly the same topological term was not included in the effective action for the global phase.140

L. H. Thomas, Proc. Camb. Phil. Soc. 23, 542 (1927); E. Fermi, Mat. Natur. 6, 602 (1927).

M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner, Phys. Rep. 106, 121 (1984).

E. Zaremba, A. Griffin, and T. Nikuni, Phys. Rev. A 57, 4695 (1998). See also A. Griffin, W.-C. Wu, and S. Stringari, Phys. Rev. Lett. 78, 1838 (1997); G. M. Kavoulakis, C. J. Pethick, and H. Smith, Phys. Rev. A 57, 2938 (1998); T. Nikuni and A. Griffin, J. Low Temp. Phys. 111, 793 (1998); V. Shenoy and T.-L. Ho, Phys. Rev. Lett. 80, 3895 (1998).

W. Kohn, Phys. Rev. 123, 1242 (1961).

This work is performed in collaboration with M. J. Bijlsma and F. Langeveld.

C. W. Gardiner, Phys. Rev. A 56, 1414 (1997).

Y. Castin and R. Dum, Phys. Rev. A (to be published).

M. J. Bijlsma and H. T. C. Stoof (unpublished, cond-mat/9807051). We would like to point out that Eq. (7) in this preprint contains an error, since the third term on the right-hand side has to be multiplied by an additional factor of 1/π3/2. Unfortunately, this seriously affects the frequencies of the collective modes. The corrected results are therefore shown as the dashed lines in Fig. 5.

E. A. Cornell, private communication.

M. Houbiers and H. T. C. Stoof (unpublished, cond-mat/9804241).

J. R. Anglin and W. H. Zurek (unpublished, cond-mat/9804035).

D. M. Stamper-Kurn, H.-J. Miesner, A. P. Chikkatur, S. Inouye, J. Stenger, and W. Ketterle (unpublished, cond-mat/9805022).

D. A. W. Hutchinson, R. J. Dodd, and K. Burnett (unpublished, cond-mat/9805050).

G. M. Kavoulakis, C. J. Pethick, and H. Smith (unpublished, cond-mat/9804193).