Variational Wave Function for Inhomogeneous Bose–Einstein Condensate with 3/2-Body Correlations

Abstract

We construct a variational wave function for inhomogeneous weakly interacting Bose-Einstein condensates beyond the mean-field approximation by incorporating 3/2-body correlations. From our numerical results calculated for a system trapped by a one-dimensional harmonic oscillator, the 3/2-body correlations give a contribution comparable to the meanfield energy toward lowering the ground-state energy.

Variational Approach to Many-Body Problems Incorporating Many-Body Effects at Finite Temperature

“…That is, it is considered that our formalism is a natural extension of the zero-temperature formalism. [17][18][19] On the other hand, we note that Eq. (A·14) cannot be used to solve the self-consistent equations because A (1) k (ε) expressed by Eq.…”

“…15,16) In recent years, however, we constructed a variational wave function for the ground state of weakly interacting bosons that self-consistently incorporates the dynamical 3/2-body processes. [17][18][19] Using it, we found that these processes contribute to the ground state energy with the same order as the mean-field energies. Thus, the 3/2-body correlations should be incorporated self-consistently, even in the collisionless regime.…”

“…We then apply it to the weak-coupling Bose-Einstein condensed phase and include the 3/2-body correlation self-consistently. Unlike the zero-temperature formalism, [17][18][19] it is difficult to evaluate the expectation values of Bogoliubov's quasiparticle operators. 20,21) We evaluate them in terms of Green's function of these operators expressed by the variational parameters.…”

“…(A•14) in the limit of T → 0 is identical to Φ| γ † k γk |Φ in the zero-temperature formalism. If we introduce the Lehmann representation for Green's function in coordinate space, we can check the connection to the zero-temperature formalism in an inhomogeneous system 19) in the same manner. That is, it is considered that our formalism is a natural extension of the zero-temperature formalism.…”