Transcript Dynamical properties of BEC gas

Hartree-Fock approximation for BEC
revisited
Jürgen Bosse
Freie Universität Berlin
Panjab University, Chandigarh
3rd March, 2014
Overview
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Introduction
Thermodynamic Variational Principle
Review: HFA for T >Tc
T ≤ Tc : Exc modified by <dN02>
HFA including ground-state fluctuations
T ≤ Tc : chemical potential
Variational Principle
A+B = - b ( H - m Nop )
A = - b ( H r - m Nop )
grand-canonical potential
e.g., interacting bosons
in a trap
1. Calculate GCP-upper bound using reference hamiltonian of single-particle type
2. Find effective hamiltonian (HF) from extremum conditions for upper-bound
Variational Principle
A+B = - b ( H - m Nop )
A = - b ( H r - m Nop )
grand-canonical potential
e.g., interacting bosons
in a trap
1. Calculate GCP-upper bound using reference hamiltonian of single-particle type
2. Find effective hamiltonian (HF) from extremum conditions for upper-bound
Calculation of GCP-upper bound
average occupation number
of state | fk >
inadequate
for
bosons
in
condensed
phase
(T ≤ Tc)
Calculation of GCP-upper bound
average occupation number
of state | fk >
inadequate
for
bosons
in
condensed
phase
(T ≤ Tc)
HF hamiltonian from extremum conditions
normal system
HFA
HF hamiltonian from extremum conditions
normal system
HFA
Calculation of GCP-upper bound
ground-state fluctuations
modify Exc
<Nkk>
average occupation number
of state | fk >
<Nkk> <Nll>
[{ej},{fj}, N0, f0]
(1-clk)
Huse & Siggia (1982)
Uniform Gas of s=0 Bosons Interacting via Repulsive Contact
2ng
ng
ideal
interacting
Fluctuation
effect on
chemical
potential
appears to be
small
N=200 bosons in isotropic harmonic trap
Summary and Outlook
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HF-hamiltonian for BEC phase derived by
accounting not only for exchange but also for
(ground-state) correlation in Exc
N0 and dN0 identified as unknowns to be determined
from additional sources