Abstract
The phase transition to a Bose-Einstein condensate is unusual in that it is not necessarily driven by inter-particle interactions but can occur in an ideal gas as a result of a purely statistical saturation of excited states. However, interactions are necessary for any system to reach thermal equilibrium and so are required for condensation to occur in finite time. In this chapter we review the role of interactions in Bose-Einstein condensation, covering both theory and experiment. We focus on measurements performed on harmonically trapped ultracold atomic gases, but also discuss how these results relate to the uniform-system case, which is more theoretically studied and also more relevant for other experimental systems.
We first consider interaction strengths for which the system can be considered sufficiently close to equilibrium to measure thermodynamic behaviour. In particular we discuss the effects of interactions both on the mechanism of condensation (namely the saturation of the excited states) and on the critical temperature at which condensation occurs. We then discuss in more detail the conditions for the equilibrium thermodynamic measurements to be possible, and the non-equilibrium phenomena that occur when these conditions are controllably violated by tuning the strength of interactions in the gas.
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Notes
- 1.
In the recently observed Bose-Einstein condensation of a photon gas [5], there is no direct interaction between the light particles. However the interaction with the material environment, which ensures thermalisation, leads to a second-order interaction between the photons.
- 2.
- 3.
Finite-size corrections slightly reduce the ideal-gas critical temperature, by \(k_{\rm B}\varDelta T_{c}^{0} =-\zeta(2)/(2 \zeta(3)) \hbar\omega_{m}\approx- 0.684 \hbar\omega_{m}\), where ω m is the algebraic mean of the trapping frequencies [9].
- 4.
This approach does not take into account the modification of the excitation spectrum due to the presence of the condensate, which is included in more elaborate MF theories such as those of Bogoliubov [11] and Popov [12] (see also [9]). However, it is often sufficient to give the correct leading order MF results.
- 5.
Note that gn 0(r)=max{μ 0−V(r),0}.
- 6.
The shift of the critical point can be equivalently expressed as ΔT c (N) or ΔN c (T), with \(\varDelta N_{c}(T)/N_{\rm c}^{0}\approx-3 \varDelta T_{c}/T_{c}^{0}\).
- 7.
The size of the central critical region is r c ∼(a/λ 0)R T , where \(R_{T} = \sqrt{k_{\rm B}T / m\omega^{2}}\) is the thermal radius of the cloud [30].
- 8.
Note that B 2 is not just a constant but contains logarithmic corrections in a/λ 0 [23]. We neglect these in our discussion since they are not discernible at the current level of experimental precision.
- 9.
The non-interacting equation of state (16.3) cannot be expanded about D c in βμ, but rather in \(\sqrt{-\beta\mu}\); up to first order this expansion gives \(D=D_{c}-2\sqrt{\pi}\sqrt{-\beta\mu}\). This scaling goes some way in explaining the qualitative difference between (16.19) and (16.22) although it cannot be used quantitatively.
- 10.
Note that this relationship is closely related to (16.13).
- 11.
This scaling holds for any distance from the critical point given by \((\mu- \mu_{c})(\lambda _{0}/a)^{2} = {\rm const}\). By applying it to the MF critical point we neglect the logarithmic corrections to \(\mu _{c}^{\rm MF}- \mu_{c}\), which are so far not experimentally observable.
References
A. Einstein, Sitz.ber./Phys. Kl. Preuss. Akad. Wiss. 1, 3 (1925)
K. Huang, Statistical Mechanics (Wiley, New York, 1987)
C. Pethick, H. Smith, Bose–Einstein Condensation in Dilute Gases (Cambridge University Press, Cambridge, 2002)
L. Pitaevskii, S. Stringari, Bose–Einstein Condensation (Oxford University Press, Oxford, 2003)
J. Klaers, J. Schmitt, F. Vewinger, M. Weitz, Nature 468, 545 (2010)
N. Tammuz, R.P. Smith, R.L.D. Campbell, S. Beattie, S. Moulder, J. Dalibard, Z. Hadzibabic, Phys. Rev. Lett. 106, 230401 (2011)
R.P. Smith, R.L.D. Campbell, N. Tammuz, Z. Hadzibabic, Phys. Rev. Lett. 106, 250403 (2011)
R.P. Smith, N. Tammuz, R.L.D. Campbell, M. Holzmann, Z. Hadzibabic, Phys. Rev. Lett. 107, 190403 (2011)
F. Dalfovo, S. Giorgini, L.P. Pitaevskii, S. Stringari, Rev. Mod. Phys. 71, 463 (1999)
J.O. Andersen, Rev. Mod. Phys. 76, 599 (2004)
N.N. Bogoliubov, J. Phys. (USSR) 11, 23 (1947)
V.N. Popov, Functional Integrals and Collective Modes (Cambridge University Press, Cambridge, 1987)
R.L.D. Campbell, R.P. Smith, N. Tammuz, S. Beattie, S. Moulder, Z. Hadzibabic, Phys. Rev. A 82, 063611 (2010)
M. Zaccanti, B. Deissler, C. D’Errico, M. Fattori, M. Jona-Lasinio, S. Mueller, G. Roati, M. Inguscio, G. Modugno, Nat. Phys. 5, 586 (2009)
W. Ketterle, D.S. Durfee, D.M. Stamper-Kurn, Bose–Einstein condensation in atomic gases, in Proceedings of the International School of Physics Enrico Fermi, vol. CXL (IOS Press, Amsterdam, 1999)
F. Gerbier, J.H. Thywissen, S. Richard, M. Hugbart, P. Bouyer, A. Aspect, Phys. Rev. A 70, 013607 (2004)
T.D. Lee, C.N. Yang, Phys. Rev. 105, 1119 (1957)
T.D. Lee, C.N. Yang, Phys. Rev. 112, 1419 (1958)
M. Bijlsma, H.T.C. Stoof, Phys. Rev. A 54, 5085 (1996)
G. Baym, J.P. Blaizot, M. Holzmann, F. Laloë, D. Vautherin, Phys. Rev. Lett. 83, 1703 (1999)
M. Holzmann, W. Krauth, Phys. Rev. Lett. 83, 2687 (1999)
J.D. Reppy, B.C. Crooker, B. Hebral, A.D. Corwin, J. He, G.M. Zassenhaus, Phys. Rev. Lett. 84, 2060 (2000)
P. Arnold, G. Moore, Phys. Rev. Lett. 87, 120401 (2001)
V.A. Kashurnikov, N.V. Prokof’ev, B.V. Svistunov, Phys. Rev. Lett. 87, 120402 (2001)
M. Holzmann, G. Baym, J.P. Blaizot, F. Laloë, Phys. Rev. Lett. 87, 120403 (2001)
G. Baym, J.P. Blaizot, M. Holzmann, F. Laloë, D. Vautherin, Eur. Phys. J. B 24, 107 (2001)
H. Kleinert, Mod. Phys. Lett. B 17, 1011 (2003)
M. Holzmann, J.N. Fuchs, G.A. Baym, J.P. Blaizot, F. Laloë, C. R. Phys. 5, 21 (2004)
S. Giorgini, L.P. Pitaevskii, S. Stringari, Phys. Rev. A 54, R4633 (1996)
P. Arnold, B. Tomášik, Phys. Rev. A 64, 053609 (2001)
M. Houbiers, H.T.C. Stoof, E.A. Cornell, Phys. Rev. A 56, 2041 (1997)
M. Holzmann, W. Krauth, M. Naraschewski, Phys. Rev. A 59, 2956 (1999)
M.J. Davis, P.B. Blakie, Phys. Rev. Lett. 96, 060404 (2006)
O. Zobay, Laser Phys. 19, 700 (2009)
J.R. Ensher, D.S. Jin, M.R. Matthews, C.E. Wieman, E.A. Cornell, Phys. Rev. Lett. 77, 4984 (1996)
F. Gerbier, J.H. Thywissen, S. Richard, M. Hugbart, P. Bouyer, A. Aspect, Phys. Rev. Lett. 92, 030405 (2004)
R. Meppelink, R.A. Rozendaal, S.B. Koller, J.M. Vogels, P. van der Straten, Phys. Rev. A 81, 053632 (2010)
M. Holzmann, G. Baym, Phys. Rev. Lett. 90, 040402 (2003)
N. Prokof’ev, O. Ruebenacker, B. Svistunov, Phys. Rev. A 69, 053625 (2004)
C.R. Monroe, E.A. Cornell, C.A. Sackett, C.J. Myatt, C.E. Wieman, Phys. Rev. Lett. 70, 414 (1993)
M. Arndt, M.B. Dahan, D. Guéry-Odelin, M.W. Reynolds, J. Dalibard, Rev. Mod. Phys. 79, 625 (1997)
N.R. Newbury, C.J. Myatt, C.E. Wieman, Phys. Rev. A 51, R2680 (1995)
G.M. Kavoulakis, C.J. Pethick, H. Smith, Phys. Rev. A 61, 053603 (2000)
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Smith, R.P., Hadzibabic, Z. (2013). Effects of Interactions on Bose-Einstein Condensation of an Atomic Gas. In: Bramati, A., Modugno, M. (eds) Physics of Quantum Fluids. Springer Series in Solid-State Sciences, vol 177. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37569-9_16
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