PhD thesis in English

Appendix B Time-dependent variational analysisFor completeness, here we present details of the time-dependent variational analysisused in Chapter 5, which was originally introduced in Refs. [104, 105].We start from the Lagrangian (4.16), assuming a time-dependent interactiong(t). For the Gaussian variational Ansatz (5.1), we calculateψ G∂ψG∗∂t∂ψ G∂σ− ψ G∗∂ψG∂t= −2 i N(t) 2 ∑σ=x,y,z( )∂ψ G∗ (σ − σ2∂σ = 0 )N(t)2 + (ϕu 4 σ + 2σφ σ ) 2 expσ() (σ ˙ϕ σ + σ 2 ˙φσ exp − ∑(− ∑σ=x,y,zσ=x,y,z)(σ − σ 0 ) 2,u 2 σ)(σ − σ 0 ) 2,where we have introduced the notation N(t) 2 = (π 3/2 u x (t)u y (t)u z (t)) −1 , and σ ∈{x, y, z}. Using the following Gaussian integralsN(t) 2 ∫N(t) 2 ∫N(t) 4 ∫d⃗r σ expd⃗r σ 2 expd⃗r exp(((we calculate the GP LagrangianL GP= ∑σ=x,y,z+ 22M+ ∑σ=x,y,z− ∑σ=x,y,z− ∑σ=x,y,z− ∑σ=x,y,z)(σ − σ 0 ) 2u 2 σ)(σ − σ 0 ) 2u 2 σ)2(σ − σ 0 ) 2u 2 σ(σ 0 ˙ϕ σ + 1 )( )u22 σ + 2σ02 ˙φσ∑σ=x,y,zMω 2 σ2( 12u 2 σ= u σ ,= 1 2=u 2 σ( )u2σ + 2σ02 ,1(2π) 3/2 u x u y u z,+ ϕ 2 σ + 4ϕ σφ σ σ 0 + 2φ 2 σ u2 σ + 4φ2 σ σ2 0( ) u2σ2 + σ2 0+ g(t)21.(2π) 3/2 u x u y u z)142