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Open AccessArticle

Self-Consistent Derivation of the Modified Gross–Pitaevskii Equation with Lee–Huang–Yang Correction

1
Dipartimento di Fisica e Astronomia “Galileo Galilei” and CNISM, Università di Padova, Via Marzolo 8, 35131 Padova, Italy
2
Istituto Nazionale di Ottica (INO) del Consiglio Nazionale delle Ricerche (CNR), Via Nello Carrara 1, 50019 Sesto Fiorentino, Italy
Appl. Sci. 2018, 8(10), 1998; https://doi.org/10.3390/app8101998
Received: 23 September 2018 / Revised: 14 October 2018 / Accepted: 17 October 2018 / Published: 21 October 2018
(This article belongs to the Special Issue Optical Properties of Confined Quantum Systems)
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. View Full-Text
Keywords: Bose–Einstein condensation; quantum field theory; Gross–Pitaevskii equation Bose–Einstein condensation; quantum field theory; Gross–Pitaevskii equation
MDPI and ACS Style

Salasnich, L. Self-Consistent Derivation of the Modified Gross–Pitaevskii Equation with Lee–Huang–Yang Correction. Appl. Sci. 2018, 8, 1998.

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