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Stoof, H. T. C.

doi:10.1023/a:1021897703053

Cited by 209 publications

(331 citation statements)

References 74 publications

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“…Other approaches to incorporate the thermal cloud include Stoof's nonequilibrium theory based on quantum kinetic theory [514][515][516][517] and Gardiner-Zoller's quantum kinetic master equation using techniques borrowed from quantum optics [518][519][520][521][522][523][524]. Also, efforts to include stochastic effects have been considered in Ref.…”

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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Section: Beyond Mean-field Descriptionmentioning

confidence: 99%

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

2008

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“…Thus, the true value of the interaction matrix element ought to decrease much faster with energy than the value obtained from the simple hard core delta-potential approximation. It is well known that indiscriminate use of the energy independent approximation leads to a non physical ultra-violet divergence [28,35,44].…”

Section: A Irreducible Tensor Fieldsmentioning

confidence: 99%

“…To account for these differences that distinguish the present experimental situation from the homogeneous Bose-Einstein gas [20][21][22][23], a growing number of equilibrium and nonequilibrium kinetic theories have been recently presented [24][25][26][27][28][29]. However, the effort to go beyond the mean-field description of the Gross-Pitaevskii equation [30] is considerable.…”

Section: Introductionmentioning

confidence: 99%

Reversible and irreversible evolution of a condensed bosonic gas

2000

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We have formulated a kinetic theory for a condensed atomic gas in a trap, i. e., a generalized Gross-Pitaevskii equation, as well as a quantum-Boltzmann equation for the normal and anomalous fluctuations [R. Walser et al., Phys. Rev. A, 59, 3878 (1999)]. In this article, the theory is applied to the case of an isotropic configuration and we present numerical and analytical results for the reversible real-time propagation, as well as irreversible evolution towards equilibrium.

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“…A more systematic treatment of interactions requires a T-matrix formulation [36] but in order to simplify the discussion we will only consider the local s-wave interaction at this stage. It will become clear that the method can be straightforwardly generalized to include the T-matrix resummation of the interaction.…”

Section: The Modelmentioning

confidence: 99%

“…The functional integral formulation is particularly convenient as it can be generalized to study nonequilibrium phenomena by implementing the Schwinger-Keldysh formulation [30][31][32][33][34][35] at the level of the path integrals. Recently, Stoof [36] has advanced the Schwinger-Keldysh formulation of nonequilibrium field theory to study the dynamics of condensation in Bose-condensed gases.…”

Section: Introductionmentioning

confidence: 99%

Nonequilibrium Relaxation of Bose–Einstein Condensates: Real-Time Equations of Motion and Ward Identities

et al. 2002

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We present a field-theoretical method to obtain consistently the equations of motion for small amplitude condensate perturbations in a homogeneous Bose-condensed gas directly in real time. It is based on linear response, and combines the SchwingerKeldysh formulation of nonequilibrium quantum field theory with the NambuGor'kov formalism of quasiparticle excitations in the condensed phase and the tadpole method in quantum field theory. This method leads to causal equations of motion that allow to study the nonequilibrium evolution as an initial value problem. It also allows to extract directly the Ward identities, which are a consequence of the underlying gauge symmetry and which in equilibrium lead to the HugenholtzPines theorem. An explicit one-loop calculation of the equations of motion beyond the Hartree-Fock-Bogoliubov approximation reveals that the nonlocal, absorptive contributions to the self-energies corresponding to the Beliaev and Landau damping processes are necessary to fulfill the Ward identities in or out of equilibrium. It is argued that a consistent implementation at low temperatures must be based on the loop expansion, which is shown to fulfill the Ward identities order by order in perturbation theory.

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Untitled

Cited by 209 publications

References 74 publications

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

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

Reversible and irreversible evolution of a condensed bosonic gas

Nonequilibrium Relaxation of Bose–Einstein Condensates: Real-Time Equations of Motion and Ward Identities

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