Superfluidity and phase transitions in a resonant Bose gas

Annals of Physics

Volumn 323, Issue 10, 2008, Pages 2376-2451

  Radzihovsky, Leo ^a   Weichman, Peter B ^b   Park, Jae I ^a,c  

^a UNIVERSITY OF COLORADO   (United States)

^b BAE SYSTEMS   (United States)

^c NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY   (United States)

Author keywords

BCS; BEC; Bose Einstein; Condensate; Feshbach; Quantum phase transition; Squeezed; Superfluidity; Vortices

Indexed keywords

EID: 50149103766     PISSN: 00034916     EISSN: 1096035X     Source Type: Journal    
DOI: 10.1016/j.aop.2008.05.008     Document Type: Article

References (111)

1. Feshbach H. Ann. Phys. 19 (1962) 287
2. Timmermans E., Tommasini P., Hussein M., and Kerman A. Phys. Rep. 315 (1999) 199
3. Wynar R., Freeland R.S., Han D.J., Ryu C., and Heinzen D.J. Science 287 (2000) 1016
4. Donley E.A., Claussen N.R., Thompson S.T., and Wieman C.E. Nature 417 (2002) 529
5. Mark M., Kraemer T., Herbig J., Chin C., Naegerl H.-C., and Grimm R. Europhys. Lett. 69 (2005) 706
6. Mark M., Ferlaino F., Knoop S., Danzl J.G., Kraemer T., Chin C., Naegerl H.-C., and Grimm R. Phys. Rev. A 76 (2007) 042514
7. Jochim S., Bartenstein M., Altmeyer A., Hendl G., Riedl S., Chin C., Hecker Denschlag J., and Grimm R. Science 302 (2003) 2101
8. Greiner M., Regal C.A., and Jin D.S. Nature 426 (2003) 537
9. Zwierlein M.W., Stan C.A., Schunck C.H., Raupach S.M.F., Gupta S., Hadzibabic Z., and Ketterle W. Phys. Rev. Lett. 91 (2003) 250401
10. Bardeen J., Cooper L.N., and Schrieffer J.R. Phys. Rev. 108 (1957) 1175
11. Sa de Melo C.A.R., Randeria M., and Engelbrecht J.R. Phys. Rev. Lett. 71 (1993) 3202
12. Timmermans E., Furuya K., Milonni P.W., and Kerman A.K. Phys. Lett. A 285 (2001) 228
13. Ohashi Y., and Griffin A. Phys. Rev. Lett. 89 (2002) 130402
14. Milstein J.N., Kokkelmans S.J.J.M.F., and Holland M.J. Phys. Rev. A 66 (2002) 043604
15. Andreev A.V., Gurarie V., and Radzihovsky L. Phys. Rev. Lett. 93 (2004) 130402
16. Chen Q., Stajic J., and Levin K. Phys. Rep. 412 (2005) 1
17. Gurarie V., and Radzihovsky L. Ann. Phys. 322 (2007) 2-119
18. Eagles D.M. Phys. Rev. 186 (1969) 456
19. Leggett A.J. J. Phys. (Paris) Colloq. 41 (1980) C7-C19
20. Nozieres P., and Schmitt-Rink S. J. Low Temp. Phys. 59 (1985) 195
21. Greiner, Regal C.A., and Jin D.S. Phys. Rev. Lett. 94 (2005) 070403
22. Ho T.-L. Phys. Rev. Lett. 92 (2004) 090402
23. Carlson J., and Reddy S. Phys. Rev. Lett. 95 (2005) 060401
24. A. Bulgac, J.E. Drut, P. Magierski, cond-mat/0701786.
25. Burovski E., Prokof'ev N., Svistunov B., and Troyer M. Phys. Rev. Lett. 96 (2006) 160402
26. Nishida Y., and Son D.T. Phys. Rev. Lett. 97 (2006) 050403
27. Nikolic P., and Sachdev S. Phys. Rev. A 75 (2007) 033608
28. Veillette M.Y., Sheehy D.E., and Radzihovsky L. Phys. Rev. A 75 (2007) 043614
29. A summary of our results has appeared in:. Radzihovsky L., Park J., and Weichman P.B. Phys. Rev. Lett. 92 (2004) 160402
30. Romans M.W.J., Duine R.A., Sachdev S., and Stoof H.T.C. Phys. Rev. Lett. 93 (2004) 020405
31. The problem of particle and particle pair condensation in a system of attractive bosons was first considered, within a zero temperature variational wavefunction approach, by. Nozieres P., and Saint James D. J. Phys. (Paris) 43 (1982) 1133 References to previous literature on boson pairing can also be found there
32. Penrose O. Phil. Mag. 42 (1951) 1373
33. In the experiments the bound state actually exists in a high vibrational state. Observation of a quasi-equilibrium molecular condensate requires that the lifetime of this state be much larger than collisional equilibration time of the vapor. Only then can the lower vibrational states to which this state must eventually decay be neglected.
34. Petrov D.S., Salomon C., and Shlyapnikov G.V. Phys. Rev. Lett. 93 (2004) 090404
35. Donley E.A., Claussen N.R., Cornish S.L., Roberts J.L., Cornell E.A., and Wieman C.E. Nature 412 (2001) 295
36. AM in Fig. 17).
37. Goral K., Koehler T., Gardiner S.A., Tiesinga E., and Julienne P.S. J. Phys. B 37 (2004) 3457
38. Search C.P., and Meystre P. Phys. Rev. Lett. 93 (2004) 140405
39. Wynar R., Freeland R.S., Han D.J., Ryu C., and Heinzen D.J. Science 287 (2000) 1016
40. Kokkelmans S.J.J.M.F., and Holland M.J. Phys. Rev. Lett. 89 (2002) 180401
41. Roberts J.L., Claussen N.R., Burke Jr. J.P., Greene C.H., Cornell E.A., and Wieman C.E. Phys. Rev. Lett. 81 (1998) 5109
42. Efimov V.N. Sov. J. Nucl. Phys. 12 (1971) 589
43. The authors thank V. Gurarie for discussion on this point.
44. In the noninteracting BEC limit (α = 0) the ASF will not be accompanied by a molecular condensate as it is for a finite Feshbach resonance coupling. As discussed in Section 4.2, for a finite α a molecular condensate will necessarily appear in the ASF state, but will be small in the weakly interacting limit.
45. Folling S., Widera A., Miller T., Gerbier F., and Bloch I. Phys. Rev. Lett. 97 (2006) 060403
46. Campbell G.K., Mun J., Boyd M., Medley P., Leanhardt A.E., Marcassa L., Pritchard D.E., and Ketterle W. Science 313 (2006) 649
47. R.A. Barankov, C. Lannert, S. Vishveshwara, cond-mat/0611126 and references therein.
48. 2 symmetry in going from molecular to atomic superfluid phases. Both the steady state and non-equilibrium properties of this system have been extensively studied: see, e.g., H.J. Kimble, Quantum fluctuations in quantum optics-squeezing and related phenomena, Les Houches, Session LIII, 1990 (Elsevier, 1992) and references therein.
49. Orzel C., Tuchman A.K., Fenselau M.L., Yasuda M., and Kasevich M.A. Science 291 (2001) 2386
50. Gradshteyn I.S., and Rhyzhik I.M. Table of Integrals, Series, Products (1980), Academic Press
51. It will be seen in Section 4 that this transition may also be driven first-order in certain regions of the phase diagram (see Figs. 2, 10, and 11). This possibility has been explored previously within a renormalization group framework [52,51], where runaway flows generated by the coupling between the incipient atomic order and fluctuations in the molecular order parameter, are interpreted in this fashion. Near the point (critical endpoint) where all three phases meet, this is also found unambiguously even at the mean field level.
52. Lee Y.W., and Lee Y.L. Phys. Rev. B 70 (2004) 224506
53. Frey E., and Balents L. Phys. Rev. B 55 (1997) 1050
54. Stenger J., Inouye S., Chikkatur A.P., Stamper-Kurn D.M., Pritchard D.E., and Ketterle W. Phys. Rev. Lett. 82 (1999) 4569
55. Stamper-Kurn D.M., Chikkatur A.P., Gorlitz A., Inouye S., Gupta S., Pritchard D.E., and Ketterle W. Phys. Rev. Lett. 83 (1999) 2876
56. Chin C., Bartenstein M., Altmeyer A., Riedle S., Jochim S., Hecker Denschlag J., and Grimm R. Science 305 (2004) 1128
57. Schunck C.H., Shin Y., Schirotzek A., Zwierlein M.W., and Ketterle W. Science 316 (2007) 867
58. Shin Y., Schunck C.H., Schirotzek A., and Ketterle W. Phys. Rev. Lett. 99 (2007) 090403
59. Bruun G.M., and Baym G. Phys. Rev. B 74 (2006) 033623
60. In the presence of a hyperfine interaction, open and closed channels are coupled, and the inter-channel transition is accompanied by (an angular momentum compensating) nuclear spin flip. The coupling of the latter to the external magnetic field is much smaller than that of the electrons (by the ratio of the nuclear to the Bohr magneton) and therefore plays a negligible role in the Feshbach resonance parameters.
61. Landau L.D., and Lifshitz E.M. Quantum Mechanics: Non-relativistic Theory (1977), Permagon, New York
62. At low energies the same scattering amplitude structure arises for two atoms interacting through a potential with a large potential barrier separating a short-range attractive minimum from the rapidly decaying repulsive part at infinity. For a sufficiently deep minimum, a (negative energy) bound state (molecule) exists. For a sufficiently shallow minimum, a (positive energy) quasi-bound state exists with a finite lifetime (resonance). However, there is also an intermediate range in which there is no bound state, but the real part of the energy pole is negative, and the metastable molecule interpretation fails.
63. Sheehy D.E., and Radzihovsky L. Ann. Phys. 322 (2007) 1790
64. † pairs that diverges in size as resonance is approached. Thus, such a model captures all further renormalization, including critical fluctuations. In particular, the multi-body parameters in over(H, ^), Eq. 2.1 should all be treated as constants throughout the neighborhood of the Feshbach resonance.
65. †, the standard derivation of the coherent state formulation goes through essentially without change. See, e.g.,. Negele J.W., and Orland H. Quantum Many-Particle Systems (1988), Addison-Wesley
66. Such an order-parameter classification of the phases by the types of broken symmetries is incomplete. There are systems that exhibit (so-called) "topological" phase transitions that do not break any global symmetry and therefore are not associated with any local order parameter. Probably the best known example of this is the Kosterlitz-Thouless transition in a two-dimensional XY-model [. Kosterlitz M., and Thouless D.J. J. Phys. C6 (1973) 1181 ], where the ordered phase, while exhibiting a finite supefluid density and power-law correlations (that distinguish it from the fully disordered paramagnetic state) does not break any global symmetries, and is not characterized by a local order parameter
67. Zinn-Justin J. Quantum Field Theory and Critical Phenomena (2002), Clarendon Press
68. de Gennes P.G., and Prost J. The Physics of Liquid Crystals. 2nd ed. (1995), Oxford
69. In 1d, for attractive interactions, the energy functional also admits bright soliton solutions that (for a range of parameters) minimize the energy: V. Gurarie, unpublished. Focusing here on sufficiently large nonresonant repulsive interactions, these inhomogeneous solutions we will be neglected.
70. Chaikin P.M., and Lubensky T.C. Principles of Condensed Matter Physics (1995), Cambridge University Press
71. Radzihovsky L., and Toner J. Phys. Rev. Lett. 75 (1995) 4752
72. Radzihovsky L., and Toner J. Phys. Rev. E 57 (1998) 1832
73. (d - 2)2 / 2 in the critical temperature. For a detailed analysis of the crossover between ideal and interacting superfluid criticality, see:. Weichman P.B., Rasolt M., Fisher M.E., and Stephen M.J. Phys. Rev. B 33 (1986) 4632 as well as Ref. [78] below
74. See, e.g.,. Ziff R.M., Uhlenbeck G.E., and Kac M. Phys. Rep. 32 (1977) 169
75. See, e.g.,. Erdélyi A. Higher Transcendental Functions (1953), McGraw-Hill, New York Vol. 1, Secs. 1.10 and 1.11
76. c 0 still setting a valid reference temperature.
77. The nonlinear relation (5.16) between the canonical and grand canonical variables is an example of the very general feature of Fisher renormalization: see:. Fisher M.E. Phys. Rev. 176 (1968) 257 This relation will also be altered by interaction effects very close to the critical line: see footnote [69] above
78. Anderson M.H., Ensher J.R., Matthews M.R., Wieman C.E., and Cornell E.A. Science 269 (1995) 198
79. Davis K.B., Mewes M.-O., Andrews M.R., van Druten N.J., Durfee D.S., Kurn D.M., and Ketterle W. Phys. Rev. Lett. 75 (1995) 3969
80. †, associated with a particle and a hole and the fact that it is governed by two (first-order) Heisenberg equations.
81. 0 σ. Such corrections can also be derived from the leading (quadratic) corrections to the free energy-see Section 6.4.1.
82. More detailed discussion of how the extremum condition for the order parameter mixes different orders in perturbation theory about the weakly interacting limit, and how to ensure consistency order by order for any given quantity, may be found in:. Weichman P.B. Phys. Rev. B 38 (1988) 8739
83. In the presence of a trapping potential, the spectrum should be observably discrete for excitation wavelengths comparable to the system size, and will depend on the trap shape.
84. k σ at higher momenta.
85. For d ≥ 4 a similar analysis may be performed, but one now needs to impose an ultraviolet (large k) cutoff, which physically arises from the vanishing of the scattering coefficients at large k.
86. Hohenberg P.C. Phys. Rev. 158 (1967) 383
87. Mermin N.D., and Wagner H. Phys. Rev. Lett. 17 (1966) 1133
88. Fisher M.E., Barber M.N., and Jasnow D. Phys. Rev. A 8 (1973) 1111
89. 2 Ising symmetry of the MSF. Further discussion of this, and related, issues can be found in Appendix A.
90. See, e.g.,. Popov V.N. Functional Integrals in Quantum Field Theory and Statistical Physics (1983), D. Riedel, Dordrecht especially Chapter 6
91. 2, obtaining a local current-current interaction, that (because of the two extra powers of the gradient) at long wavelengths is subdominant to the local quartic coupling.
92. Bergman D.J., and Halperin B.I. Phys. Rev. B 13 (1976) 2145
93. The connection to a compressible Ising model can be made even closer by a duality transformation. See for example Ref. [52].
94. Halperin B.I., Lubensky T.C., and Ma S.K. Phys. Rev. Lett. 32 (1974) 292
95. Fisher M.E. Rev. Mod. Phys. 46 (1974) 597
96. Girardeau M., and Arnowitt R. Phys. Rev. 113 (1959) 755
97. Feynman R.P. Statistical Mechanics. Second ed. (1998), Perseus Books Group
98. Despite this particle gap, it can be shown that the required Goldstone mode appears in the spectrum of the two particle bound states:. Coniglio A., and Marinaro M. Il Nuovo Cimento XLVIII B (1967) 262
99. The single-particle energy spectrum should indeed be gapped in the MSF phase. The fact that it is gapped even in the ASF phase is an artifact of the variational calculation [30], and can be cured by more sophisticated treatments [78,86]. In particular, the usual Bogoliubov approximation for g > 0 yields gapless excitations, but the variational approach, which includes additional higher order terms in the Hamiltonian coefficients (8.20) to account for the possibility of molecular order, reopens the gap. Reclosing it requires careful inclusion of further higher order terms.
100. Fisher M.E. Phys. Rev. 176 (1968) 257
101. Dasgupta C., and Halperin B.I. Phys. Rev. Lett. 47 (1981) 1556
102. For 2d systems that do spontaneously break a continuous symmetry see, for example, L. Radzihovsky, Anisotropic and Heterogeneous Polymerized Membranes, in "Statistical Mechanics of Membranes and Surfaces," (World Scientific, edited by D.R. Nelson, T. Piran and S. Weinberg), as well as Ref. [68], and references therein.
103. Fisher M.P.A., and Lee D.-H. Phys. Rev. B 39 (1989) 2756
104. In a different context, this was independently discussed in. Klinkhamer F.R., and Volovik G.E. JETP Lett. 80 (2004) 343
105. The existence of highly anisotropic dumbbell 2 π atomic vortices allows for the possibility of liquid crystal [65,67] normal phases. These can, in principle, appear when the anisotropic 2 π vortices proliferate but exhibit some form of (e.g., smectic or nematic) spatial order, that distinguishes the resulting liquid-crystal normal state from a fully disordered normal state, where vortices form a homogeneous, isotropic liquid.
106. 1 is not), and a fully disordered, isotropic phase. These are isomorphic to the ASF, MSF and normal phases, respectively, and (as shown by Grinstein and Lee) the polar and nematic phases are separated by an Ising deconfinement transition characterized by the roughenning and loss of line tension of the domain wall across which θ jumps by π; L. Radzihovsky and J. Park, unpublished.
107. Balents L., Fisher M.P.A., and Nayak C. Int. J. Mod. Phys. B 12 (1998) 1033
108. Balents L., Fisher M.P.A., and Nayak C. Phys. Rev. B 60 (1999) 1654
109. ma mb.
110. 1 / 3.
111. Feynman R.P., and Hibbs A.R. Quantum Mechanics and Path Integrals (1965), McGraw-Hill, New York