Skip to main content Accessibility help
×
Hostname: page-component-7c8c6479df-94d59 Total loading time: 0 Render date: 2024-03-28T20:55:37.160Z Has data issue: false hasContentIssue false

6 - Effects of Interactions on Bose-Einstein Condensation

from Part II - General Topics

Published online by Cambridge University Press:  18 May 2017

R. P. Smith
Affiliation:
Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge, CB3 0HE, UK
Nick P. Proukakis
Affiliation:
Newcastle University
David W. Snoke
Affiliation:
University of Pittsburgh
Peter B. Littlewood
Affiliation:
University of Chicago
Get access

Summary

Bose-Einstein condensation is a unique phase transition in that it is not driven by interparticle interactions, but can theoretically occur in an ideal gas, purely as a consequence of quantum statistics. This chapter addresses the question, ‘How is this ideal Bose gas condensation modified in the presence of interactions between the particles?’ This seemingly simple question turns out to be surprisingly difficult to answer. Here we outline the theoretical background to this question and discuss some recent measurements on ultracold atomic Bose gases that have sought to provide some answers.

Introduction

Unlike the vast majority of phase transitions, Bose-Einstein condensation (BEC) is not driven by interparticle interactions but can theoretically occur in an ideal (noninteracting) gas, purely as a consequence of quantum statistics. However, in reality, interactions are needed for a Bose gas to remain close to thermal equilibrium. It is thus interesting to discuss if something close to ideal gas BEC can be observed in a real system and what happens in the vicinity of the BEC transition in the presence of interparticle interactions. These simple questions have not been easy to answer, either theoretically or experimentally.

The theoretical foundations for studying the effect of interactions on Bose condensed systems were laid over half a century ago by Bogoliubov [1], Penrose and Onsager [2], and Belieav [3], among others. These works initially focused on zerotemperature properties and were extended to nonzero temperature in the pioneering papers of Lee, Huang, and Yang [4, 5, 6, 7]. At that time, the main experimental system was liquid He-4 in which the inter-particle interactions are strong, making connections with theory difficult. The realisation in 1995 of BEC in weakly interacting ultracold atomic gases [8, 9] thus opened up the possibility to experimentally revisit some of these long-discussed questions. This was further aided by, among other advances, the use of Feshbach resonances to tune the interaction strength in atomic gases [10, 11]. Thus, in the last twenty years the study of ultracold Bose gases has been very successful (see Chapter 3 for an insightful account of selected topics and surprising developments over this exciting period). However, the fact that, until recently, ultracold atoms were confined using harmonic potentials has hindered the study of the BEC transition itself.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2017

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Bogoliubov, N. N. 1947. On the theory of superfluidity. J. Phys. (USSR), 11, 23.Google Scholar
[2] Penrose, O., and Onsager, L. 1956. Bose–Einstein condensation and liquid helium. Phys. Rev., 104, 576.Google Scholar
[3] Beliaev, S. T. 1958. Sov. Phys. JETP, 34, 323.
[4] Lee, T. D., Huang, K., and Yang, C. N. 1957. Eigenvalues and eigenfunctions of a Bose system of hard spheres and its low-temperature properties. Phys. Rev., 106, 1135–1145.Google Scholar
[5] Lee, T. D., and Yang, C. N. 1957. Many-body problem in quantum mechanics and quantum statistical mechanics. Phys. Rev., 105, 1119–1120.Google Scholar
[6] Lee, T. D., and Yang, C. N. 1958. Low-temperature behavior of a dilute Bose system of hard spheres. I. Equilibrium properties. Phys. Rev., 112, 1419–1429.Google Scholar
[7] Huang, K., Yang, C. N., and Luttinger, J. M. 1957. Imperfect Bose gas with hardsphere interaction. Phys. Rev., 105, 776–784.Google Scholar
[8] Anderson, M. H., Ensher, J. R., Matthews, M. R., Wieman, C. E., and Cornell, E. A. 1995. Observation of Bose-Einstein condensation in a dilute atomic vapor. Science, 269, 198.Google Scholar
[9] Davis, K. B., Mewes, M. O., Andrews, M. R., van Druten, N. J., Durfee, D. S., Kurn, D. M., and Ketterle, W. 1995. Bose-Einstein condensation in a gas of sodium atoms. Phys. Rev. Lett., 75, 3969–3973.Google Scholar
[10] Inouye, S., Andrews, M., Stenger, J., Miesner, H. J., Stamper-Kurn, D. M., and Ketterle, W. 1998. Observation of Feshbach resonances in a Bose-Einstein condensate. Nature, 392, 151.Google Scholar
[11] Courteille, Ph., Freeland, R. S., Heinzen, D. J., van Abeelen, F. A., and Verhaar, B. J. 1998. Observation of a Feshbach resonance in cold atom scattering. Phys. Rev. Lett., 81, 69–72.Google Scholar
[12] Smith, R. P., and Hadzibabic, Z. 2013. Physics of Quantum Fluids. New York: Springer. Chap. Effects of interactions on Bose-Einstein condensation of an atomic gas, pages 341–359.
[13] Dalfovo, F. S., Pitaevkii, L. P., Stringari, S., and Giorgini, S. 1999. Theory of Bose- Einstein condensation in trapped gases. Rev. Mod. Phys., 71, 463.Google Scholar
[14] Pethick, C. J., and Smith, H. 2002. Bose-Einstein Condensation in Dilute Gases. Cambridge: Cambridge University Press.
[15] Pitaevskii, L., and Stringari, S. 2003. Bose-Einstein Condensation. Oxford: Oxford University Press.
[16] Andersen, J. O. 2004. Theory of the weakly interacting Bose gas. Rev. Mod. Phys., 76(2), 599–639.Google Scholar
[17] Huang, K. 1987. Statistical Mechanics. New York: Wiley.
[18] Popov, V. N. 1987. Functional Integrals and Collective Modes. Cambridge: Cambridge University Press.
[19] Prokof'ev, N., Ruebenacker, O., and Svistunov, B. 2004. Weakly interacting Bose gas in the vicinity of the normal-fluid–superfluid transition. Phys. Rev. A, 69, 053625.Google Scholar
[20] Arnold, P., and Moore, G. 2001. BEC transition temperature of a dilute homogeneous imperfect Bose gas. Phys. Rev. Lett., 87, 120401.Google Scholar
[21] Baym, G., Blaizot, J.-P., Holzmann, M., Lalöe, F., and Vautherin, D. 2001. Bose- Einstein transition in a dilute interacting gas. Eur. Phys. J. B, 24, 107–124.Google Scholar
[22] Holzmann, M., Fuchs, J. N., Baym, G., Blaizot, J. P., and Lalöe, F. 2004. Bose- Einstein transition temperature in a dilute repulsive gas. Comptes Rendus Physique, 5, 21.Google Scholar
[23] Kashurnikov, V. A., Prokof'ev, N. V., and Svistunov, B. V. 2001. Critical temperature shift in weakly interacting Bose gas. Phys. Rev. Lett., 87, 120402.Google Scholar
[24] Campostrini, M., Hasenbusch, M., Pelissetto, A., and Vicari, E. 2006. Theoretical estimates of the critical exponents of the superfluid transition in 4He by lattice methods. Phys. Rev. B, 74, 144506.Google Scholar
[25] Burovski, E., Machta, J., Prokof'ev, N., and Svistunov, B. 2006. High-precision measurement of the thermal exponent for the three-dimensiona. XY universality class. Phys. Rev. B, 74, 132502.Google Scholar
[26] Tammuz, N., Smith, R. P., Campbell, R. L. D., Beattie, S., Moulder, S., Dalibard, J., and Hadzibabic, Z. 2011. Can a Bose gas be saturated. Phys. Rev. Lett., 106, 230401.Google Scholar
[27] Gaunt, A. L., Schmidutz, T. F., Gotlibovych, I., Smith, R. P., and Hadzibabic, Z. 2013. Bose-Einstein condensation of atoms in a uniform potential. Phys. Rev. Lett., 110, 200406.Google Scholar
[28] Schmidutz, T. F., Gotlibovych, I., Gaunt, A. L., Smith, R. P., Navon, N., and Hadzibabic, Z. 2014. Quantum Joule-Thomson effect in a saturated homogeneous Bose gas. Phys. Rev. Lett., 112, 040403.Google Scholar
[29] Giorgini, S., Pitaevskii, L. P., and Stringari, S. 1996. Condensate fraction and critical temperature of a trapped interacting Bose gas. Phys. Rev. A, 54, R4633.Google Scholar
[30] Gaunt, A., and Smith, R. P. Private communication.
[31] Houbiers, M., Stoof, H. T. C., and Cornell, E. A. 1997. Critical temperature of a trapped Bose gas: mean-field theory and fluctuations. Phys. Rev. A, 56, 2041.Google Scholar
[32] Holzmann, M., Krauth, W., and Naraschewski, M. 1999. Precision Monte Carlo test of the Hartree-Fock approximation for a trapped Bose gas. Phys. Rev. A, 59, 2956–2961.Google Scholar
[33] Arnold, P., and Tomášik, B. 2001. Tc for trapped dilute Bose gases: a second-order result. Phys. Rev. A, 64, 053609.Google Scholar
[34] Davis, M. J., and Blakie, P. B. 2006. Critical temperature of a trapped Bose gas: comparison of theory and experiment. Phys. Rev. Lett., 96, 060404.Google Scholar
[35] Zobay, O. 2009. Phase transition of trapped interacting Bose gases. Laser Physics, 19, 700–724.Google Scholar
[36] Smith, R. P., Campbell, R. L. D., Tammuz, N., and Hadzibabic, Z. 2011. Effects of interactions on the critical temperature of a trapped Bose gas. Phys. Rev. Lett., 106, 250403.Google Scholar
[37] Ensher, J. R., Jin, D. S., Matthews, M. R., Wieman, C. E., and Cornell, E. A. 1996. Bose-Einstein condensation in a dilute gas: measurement of energy and ground-state occupation. Phys. Rev. Lett., 77, 4984.Google Scholar
[38] Gerbier, F., Thywissen, J. H., Richard, S., Hugbart, M., Bouyer, P., and Aspect, A. 2004. Critical temperature of a trapped, weakly interacting Bose gas. Phys. Rev. Lett., 92, 030405.Google Scholar
[39] Meppelink, R., Rozendaal, R. A., Koller, S. B., Vogels, J. M., and van der Straten, P. 2010. Thermodynamics of Bose-Einstein–condensed clouds using phase-contrast imaging. Phys. Rev. A, 81, 053632.Google Scholar
[40] Donner, T., Ritter, S., Bourdel, T., Ottl, A., Köhl, M., and Esslinger, T. 2007. Critical behavior of a trapped interacting Bose gas. Science, 315(5818), 1556–1558.Google Scholar
[41] Navon, N., Gaunt, A. L., Smith, R. P., and Hadzibabic, Z. 2015. Critical dynamics of spontaneous symmetry breaking in a homogeneous Bose gas. Science, 347(6218), 167–170.Google Scholar
[42] Hohenberg, P. C., and Halperin, B. I. 1977. Theory of dynamic critical phenomena. Rev. Mod. Phys., 49(Jul), 435–479.Google Scholar
[43] Rem, B. S., Grier, A. T., Ferrier-Barbut, I., Eismann, U., Langen, T., Navon, N., Khaykovich, L., Werner, F., Petrov, D. S., Chevy, F., and Salomon, C. 2013. Lifetime of the Bose gas with resonant interactions. Phys. Rev. Lett., 110, 163202.Google Scholar
[44] Fletcher, R. J., Gaunt, A. L., Navon, N., Smith, R. P., and Hadzibabic, Z. 2013. Stability of a unitary Bose gas. Phys. Rev. Lett., 111, 125303.Google Scholar
[45] Makotyn, P., Klauss, C. E., Goldberger, D. L., Cornell, E. A., and Jin, D. S. 2014. Universal dynamics of a degenerate unitary Bose gas. Nature Physics, 10, 116–119.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×