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arXiv:cond-mat/0410287

The Gross-Pitaevskii equations and beyond for inhomogeneous condensed bosons

G. G. N. Angilella, Saverio Bartalini, F. S. Cataliotti, Herrera, I., Norman Henry March, R. Pucci

Published Oct 12, 2004• 0 citations

Abstract

A simple derivation of the static Gross-Pitaevskii (GP) equation is given from an energy variational principle. The result is then generalized heuristically to the time-dependent GP form. With this as background, a number of different experimental areas explored very recently are reviewed, in each case contact being established between the measurements and the predictions of the GP equations. The various limitations of these equations as used on dilute inhomogeneous condensed Boson atomic gases are then summarized, reference also being made to the fact that there is no many-body wave function underlying the GP formulation. This then leads into a discussion of a recently proposed integral equation, derived by taking the Bogoliubov-de Gennes equation as starting point. Some limitations of the static GP differential equation are thereby removed, though it is a matter of further study to determine whether a correlated wave function exists as underpinning for the integral equation formulation.

DOI: 10.48550/arxiv.cond-mat/0410287

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After a brief introduction to the static Gross–Pitaevskii (GP) differential equation for the ground state of a Bose–Einstein condensed gas and an example of its use in describing a vortex array in a trapped condensate, a recent generalization is reviewed. By starting from the Bogoliubov–de Gennes equations for superfluid fermions, an integral equation results. This is shown to contain the static GP equation as a limiting case, by means of a low-order gradient expansion of the order parameter. The time-dependent GP approximation is then presented and illustrated by a numerical example concerning drop emission under gravity from a gaseous condensate in an optical lattice. Further selected topics of present interest relating more generally to the dynamics of trapped superfluid gases concern the effects of temperature on collective excitations in a Bose gas and the fingerprints that these excitations in a gas of paired fermions contain of the crossover from a Bardeen–Cooper–Schrieffer correlated assembly to a Bose–Einstein condensate of composite bosons. Finally, some directions are pointed in which interaction of first-principles theory with experiment should prove fruitful.

We derive the time-independent Gross-Pitaevskii equation at zero temperature for condensed bosons, which form as bound-fermion pairs when the mutual fermionic attractive interaction is sufficiently strong, from the strong-coupling limit of the Bogoliubov-de Gennes equations that describe superfluid fermions in the presence of an external potential. Three-body corrections to the Gross-Pitaevskii equation are also obtained by our approach. Our results are relevant to the recent advances with ultracold fermionic atoms in a trap.

The Hamiltonian of a moving atom in electromagnetic fields includes velocity- dependent terms. We show that the leading velocity dependence emerges systematically in the non-relativistic limit from a scheme firmly based on the relativistic invariance of the energy-momentum stress tensor of the coupled matter-fields system. We then extend the Hamiltonian to the many-body situation suitable for describing a Bose-Einstein condensate (BEC). From first principles, we use the equation of motion for the condensate wavefunction to obtain an extended version of the Gross-Pitaevskii (GP) equation and an equation for the internal states of the atoms. It is shown that laser fields modify the GP equation by inclusion of convective terms involving a Rontgen interaction plus a term coupling the centre of mass momentum to the Poynting vector. We also obtain the modified Maxwell equations for the electromagnetic fields coupled to the BEC involving the average velocity of the atoms.

We introduce a non-commutative generalization of the Gross-Pitaevskii equation for one-dimensional quantum gasses and quantum liquids. This generalization is obtained by applying the time-dependent variational principle to the variational manifold of continuous matrix product states. This allows for a full quantum description of many body system —including entanglement and correlations— and thus extends significantly beyond the usual mean-field description of the Gross-Pitaevskii equation, which is known to fail for (quasi) one-dimensional systems. By linearizing around a stationary solution, we furthermore derive an associated generalization of the Bogoliubov – de Gennes equations. This framework is applied to compute the steady state response amplitude to a periodic perturbation of the potential.