Abstract
We consider a dilute and ultracold bosonic gas of weakly-interacting atoms. Within the framework of quantum field theory, we derive a zero-temperature modified Gross–Pitaevskii equation with beyond-mean-field corrections due to quantum depletion and anomalous density. This result is obtained from the stationary equation of the Bose–Einstein order parameter coupled to the Bogoliubov–de Gennes equations of the out-of-condensate field operator. We show that, in the presence of a generic external trapping potential, the key steps to get the modified Gross–Pitaevskii equation are the semiclassical approximation for the Bogoliubov–de Gennes equations, a slowly-varying order parameter and a small quantum depletion. In the uniform case, from the modified Gross–Pitaevskii equation, we get the familiar equation of state with Lee–Huang–Yang correction.
Highlights
In 1924, Bose and Einstein introduced the concept of Bose–Einstein statistics and Bose–Einstein condensation, i.e., the macroscopic occupation of the lowest single-particle state of a system of bosons [1,2]
Gross–Pitaevskii equation are the semiclassical approximation for the Bogoliubov–de Gennes equations, a slowly-varying order parameter and a small quantum depletion
Experiments with atomic gases reported evidence of beyond-mean-field effects on the equation of state of repulsive bosons [10,11]. These experimental results are quite well reproduced [12] by a modified Gross–Pitaevskii equation, which includes a beyond-mean-field correction that is the local version of the Lee–Huang–Yang term
Summary
In 1924, Bose and Einstein introduced the concept of Bose–Einstein statistics and . Experiments with atomic gases reported evidence of beyond-mean-field effects on the equation of state of repulsive bosons [10,11] These experimental results are quite well reproduced [12] by a modified Gross–Pitaevskii equation, which includes a beyond-mean-field correction that is the local version of the Lee–Huang–Yang term. The presence of a generic external trapping potential U (r) is circumvented by adopting a semiclassical approximation for the Bogoliubov–de Gennes equations of the fluctuating quantum field [23,24] In this way, at zero temperature, we obtain the local density ñ(r) of the out-of-condensate bosons as a function of the classical field ψ0 (r) and the corresponding equation for ψ0 (r), that is the stationary modified Gross–Pitaevskii equation with beyond-mean-field terms.
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