Phase space theory of Bose-Einstein condensates and time-dependent modes

Journal Article · Mon Oct 15 00:00:00 EDT 2012 · Annals of Physics (New York)

DOI:https://doi.org/10.1016/J.AOP.2012.06.005· OSTI ID:22157102

Dalton, B.J., E-mail: [e-mail]

A phase space theory approach for treating dynamical behaviour of Bose-Einstein condensates applicable to situations such as interferometry with BEC in time-dependent double well potentials is presented. Time-dependent mode functions are used, chosen so that one, two, Horizontal-Ellipsis highly occupied modes describe well the physics of interacting condensate bosons in time dependent potentials at well below the transition temperature. Time dependent mode annihilation, creation operators are represented by time dependent phase variables, but time independent total field annihilation, creation operators are represented by time independent field functions. Two situations are treated, one (mode theory) is where specific mode annihilation, creation operators and their related phase variables and distribution functions are dealt with, the other (field theory) is where only field creation, annihilation operators and their related field functions and distribution functionals are involved. The field theory treatment is more suitable when large boson numbers are involved. The paper focuses on the hybrid approach, where the modes are divided up between condensate (highly occupied) modes and non-condensate (sparsely occupied) modes. It is found that there are extra terms in the Ito stochastic equations both for the stochastic phases and stochastic fields, involving coupling coefficients defined via overlap integrals between mode functions and their time derivatives. For the hybrid approach both the Fokker-Planck and functional Fokker-Planck equations differ from those derived via the correspondence rules, the drift vectors are unchanged but the diffusion matrices contain additional terms involving the coupling coefficients. Results are also presented for the combined approach where all the modes are treated as one set. Here both the Fokker-Planck and functional Fokker-Planck equations are exactly the same as those derived via the correspondence rules. However, although the Ito stochastic field equations are also unchanged, the Ito equations for the stochastic phases contain an extra classical term involving the coupling coefficients. - Highlights: Black-Right-Pointing-Pointer Wigner, P+ representations used for condensate and non-condensate bosons, respectively. Black-Right-Pointing-Pointer Time dependence of modes leads to novel Fokker-Planck and Ito equations. Black-Right-Pointing-Pointer Field theory and separate mode treatments.

• OSTI ID: 22157102
• Journal Information: Annals of Physics (New York), Vol. 327, Issue 10; Other Information: Copyright (c) 2012 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); ISSN 0003-4916
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ANNIHILATION OPERATORS
BOSE-EINSTEIN CONDENSATION
BOSONS
COUPLING
CREATION OPERATORS
DIFFUSION
DISTRIBUTION FUNCTIONS
FIELD EQUATIONS
FIELD THEORIES
FOKKER-PLANCK EQUATION
INTERFEROMETRY
PHASE SPACE
POTENTIALS
STOCHASTIC PROCESSES
TIME DEPENDENCE
TRANSITION TEMPERATURE