Gapless mean-field theory of Bose-Einstein condensates

Hutchinson, D. A. W.; Burnett, K.; Dodd, R. J.; Morgan, S. A.; Rusch, M.; Zaremba, E.; Proukakis, N. P.; Edwards, Mark; Clark, Charles W.

doi:10.1088/0953-4075/33/19/302

Cited by 88 publications

(196 citation statements)

References 30 publications

(64 reference statements)

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“…On the other hand, the anomalous density profiles seem to have no structure at the centre of the trap for weak interactions. This is in contradiction with what was found in the literature [36,43] where the HFB-BdG approximation and the classical-field trajectories of the Projected GrossPitaevskii equation were used, and where these densities are found to have a "dip".…”

Section: Introductioncontrasting

confidence: 99%

“…(3.4), the dots denote time derivatives and we have introduced the quantities It is well known that in the HFB theory the issues of the ultraviolet divergence of the anomalous density arise from the zero-point occupation of quasiparticle modes [36].…”

Section: Application Of the Tdhfb Formalism To Trapped Bose Gasesmentioning

confidence: 99%

“…Among the theoretical investigations of the anomalous density, we can cite in particular those of Hutchinson et al [36] and Giorgini [37] based on the mean field HFBBdG approximation. To go beyond the mean field, Fedichev et al [38] and Proukakis et al [39,40] developed a finite temperature perturbation theory using an HFB basis.…”

Section: Introductionmentioning

confidence: 99%

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Anomalous density for Bose gases at finite temperature

Boudjemâa¹,

Benarous²

2011

Phys. Rev. A

102

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We analyze the behavior of the anomalous density as function of the radial distance at different temperatures in a variational framework. We show that the temperature dependence of the anomalous density agrees with the Hartree-Fock-Bogoliubov (HFB) calculations. Comparisons between the normal and the anomalous fractions at low temperature show that the latter remains higher and consequently the neglect of the anomalous density may destabilize the condensate. These results are compatible with those of Yukalov. Surprisingly, the study of the anomalous density in terms of the interaction parameter shows that the dip in the central density is destroyed for sufficiently weak interactions. We explain this effect.

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Section: Introductioncontrasting

confidence: 99%

Section: Application Of the Tdhfb Formalism To Trapped Bose Gasesmentioning

confidence: 99%

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Anomalous density for Bose gases at finite temperature

Boudjemâa¹,

Benarous²

2011

Phys. Rev. A

102

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“…where H GP = −(1/2)∇ 2 + V ext (r, t) + g|ψ| 2 accounts for the "classical" GP terms in non-dimensional units, n ′ (r, t) denotes the non-condensate density andm 0 (r, t) the anomalous mean-field average [505][506][507][508]. Furthermore, taking into account that atomic collisions happen within the gas (and not in vacuum), one has to modify the inter-atomic interactions by a contact potential with a position-dependent amplitude: g → g[1 +m 0 (r)/ψ 2 (r)] [509][510][511].…”

Section: Beyond Mean-field Descriptionmentioning

confidence: 99%

Nonlinear waves in Bose–Einstein condensates: physical relevance and mathematical techniques

2008

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Abstract. The aim of the present review is to introduce the reader to some of the physical notions and of the mathematical methods that are relevant to the study of nonlinear waves in Bose-Einstein Condensates (BECs). Upon introducing the general framework, we discuss the prototypical models that are relevant to this setting for different dimensions and different potentials confining the atoms. We analyze some of the model properties and explore their typical wave solutions (plane wave solutions, bright, dark, gap solitons, as well as vortices). We then offer a collection of mathematical methods that can be used to understand the existence, stability and dynamics of nonlinear waves in such BECs, either directly or starting from different types of limits (e.g., the linear or the nonlinear limit, or the discrete limit of the corresponding equation). Finally, we consider some special topics involving more recent developments, and experimental setups in which there is still considerable need for developing mathematical as well as computational tools.PACS numbers: 03.75. Kk, 03.75.Lm, 05.45 • EP: Ermakov-Pinney (Equation)• GP: Gross-Pitaevskii (Equation)• KdV: Korteweg-de Vries (Equation)• LS: Lyapunov-Schmidt (Technique)• MT: Magnetic Trap• NLS: Nonlinear Schrödinger (Equation)• NPSE: Non-polynomial Schrödinger Equation IntroductionThe phenomenon of Bose-Einstein condensation is a quantum phase transition originally predicted by Bose [1] and Einstein [2,3] in 1924. In particular, it was shown that below a critical transition temperature T c , a finite fraction of particles of a boson gas (i.e., whose particles obey the Bose statistics) condenses into the same quantum state, known as the Bose-Einstein condensate (BEC). Although BoseEinstein condensation is known to be a fundamental phenomenon, connected, e.g., to superfluidity in liquid helium and superconductivity in metals (see, e.g., Ref.[4]), BECs were experimentally realized 70 years after their theoretical prediction: this major achievement took place in 1995, when different species of dilute alkali vapors confined in a magnetic trap (MT) were cooled down to extremely low temperatures [5][6][7], and has already been recognized through the 2001 Nobel prize in Physics [8,9]. This first unambiguous manifestation of a macroscopic quantum state in a manybody system sparked an explosion of activity, as reflected by the publication of several thousand papers related to BECs since then. Nowadays there exist more than fifty experimental BEC groups around the world, while an enormous amount of theoretical work has followed and driven the experimental efforts, with an impressive impact on many branches of Physics.From a theoretical standpoint, and for experimentally relevant conditions, the static and dynamical properties of a BEC can be described by means of an effective mean-field equation known as the Gross-Pitaevskii (GP) equation [10,11]. This is a variant of the famous nonlinear Schrödinger (NLS) equation [12] (incorporating an external potential used to confine th...

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“…In the case that there is still a condensate, the total density in the region where n c (r) = 0 is approximately constant just below the critical temperature, so that the meanfield energy only constitutes a near constant shift to the trapping potential and, hence, only slightly alters the eigenfrequencies of the trap. We would briefly like to draw comparison with the three-dimensional case where the frequency spectrum looks very similar [19,34]. The striking difference is the breathing mode which is temperature dependent in three dimensions, whereas it is a feature of the two-dimensional system to have breathing oscillations with a universal energy of 2hω ⊥ .…”

Section: B Condensate Excitationsmentioning

confidence: 99%

Coherence properties of the two-dimensional Bose-Einstein condensate

Gies

Hutchinson

2004

Phys. Rev. A

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We present a detailed finite-temperature Hartree-Fock-Bogoliubov (HFB) treatment of the twodimensional trapped Bose gas. We highlight the numerical methods required to obtain solutions to the HFB equations within the Popov approximation, the derivation of which we outline. This method has previously been applied successfully to the three-dimensional case and we focus on the unique features of the system which are due to its reduced dimensionality. These can be found in the spectrum of low-lying excitations and in the coherence properties. We calculate the Bragg response and the coherence length within the condensate in analogy with experiments performed in the quasione-dimensional regime [Richard et al., Phys. Rev. Lett. 91, 010405 (2003)] and compare to results calculated for the one-dimensional case. We then make predictions for the experimental observation of the quasicondensate phase via Bragg spectroscopy in the quasi-two-dimensional regime.

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Gapless mean-field theory of Bose-Einstein condensates

Cited by 88 publications

References 30 publications

Anomalous density for Bose gases at finite temperature

Anomalous density for Bose gases at finite temperature

Nonlinear waves in Bose–Einstein condensates: physical relevance and mathematical techniques

Coherence properties of the two-dimensional Bose-Einstein condensate

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