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arXiv:cond-mat/0512359 · Laser Physics

Images of a Bose-Einstein condensate in position and momentum space

J. Dziarmaga, Krzysztof Sacha

Published Dec 1, 2006• 1 citation
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Abstract

In the Bogoliubov theory, a condensate initially prepared in its ground state described by a stationary Bogoliubov vacuum and later perturbed by a time-dependent potential or interaction strength evolves into a time-dependent excited state which is a dynamical Bogoliubov vacuum. The dynamical vacuum has a simple diagonal form in a time-dependent orthonormal basis of single-particle modes. This diagonal representation leads to a Gaussian probability distribution for possible density-measurement outcomes in position and momentum space.

DOI
10.1134/s1054660x06120188

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In the Bogoliubov theory, a condensate initially prepared in its ground state described by a stationary Bogoliubov vacuum and later perturbed by a time-dependent potential or interaction strength evolves into a time-dependent excited state which is a dynamical Bogoliubov vacuum. The dynamical vacuum has a simple diagonal form in a time-dependent orthonormal basis of single-particle modes. This diagonal representation leads to a Gaussian probability distribution for possible density-measurement outcomes in position and momentum space.
Evolution of a Bose–Einstein condensate subject to a time-dependent external perturbation can be described by a time-dependent number-conserving Bogoliubov theory: a condensate initially in its ground state Bogoliubov vacuum evolves into a time-dependent excited state which can be formally written as a time-dependent Bogoliubov vacuum annihilated by time-dependent quasiparticle annihilation operators. We prove that any Bogoliubov vacuum can be brought to a diagonal form in a time-dependent orthonormal basis. This diagonal form is tailored for simulations of quantum measurements on an excited condensate. As an example we work out phase imprinting of a dark soliton followed by a density measurement.
We consider the Bogoliubov vacuum state in the number-conserving Bogoliubov theory proposed by Castin and Dum [Phys. Rev. A 57 , 3008 (1998)]. We show that, in the particle representation, the vacuum can be written in a simple diagonal form. The vacuum state can describe the stationary N -particle ground state of a condensate in a trap, but it can also represent a dynamical state when, for example, a Bose-Einstein condensate initially prepared in the stationary ground state is subject to a time-dependent perturbation. In both cases the diagonal form of the Bogoliubov vacuum can be obtained by basically diagonalizing the reduced single-particle density matrix of the vacuum. We compare N -body states obtained within the Bogoliubov theory with the exact ground states in a 3-site Bose-Hubbard model. In this example, the Bogoliubov theory fails to accurately describe the stationary ground state in the limit when N → ∞ but a small fraction of depleted particles is kept constant.
As is well-known, in Bogoliubov's theory of an interacting Bose gas the ground state of the Hamiltonian H^=∑k≠0H^k\hat{H}=\sum_{\bf k\neq 0}\hat{H}_{\bf k}H^=∑k=0​H^k​ is found by diagonalizing each of the Hamiltonians H^k\hat{H}_{\bf k}H^k​ corresponding to a given momentum mode k{\bf k}k independently of the Hamiltonians H^k′(≠k)\hat{H}_{\bf k'(\neq k)}H^k′(=k)​ of the remaining modes. We argue that this way of diagonalizing H^\hat{H}H^ may not be adequate, since the Hilbert spaces where the single-mode Hamiltonians H^k\hat{H}_{\bf k}H^k​ are diagonalized are not disjoint, but have the k=0{\bf k}=0k=0 in common. A number-conserving generalization of Bogoliubov's method is presented where the total Hamiltonian H^\hat{H}H^ is diagonalized directly. When this is done, the spectrum of excitations changes from a gapless one, as predicted by Bogoliubov's method, to one which has a finite gap in the k→0k\to 0k→0 limit.
Quantum fluctuations in time-dependent, harmonically trapped Bose-Einstein condensates are studied within Bogoliubov theory. An eigenmode expansion of the linear field operators permits the diagonalization of the Bogoliubov--de Gennes equation for a stationary condensate. When trap frequency or interaction strength is varied, the inhomogeneity of the background gives rise to off-diagonal coupling terms between different modes. This coupling is negligible for low energies, i.e., in the hydrodynamic regime, and an effective space-time metric can be introduced. The influence of the intermode coupling will be demonstrated in an example, where I calculate the quasiparticle number for a quasi-one-dimensional Bose-Einstein condensate subject to an exponential sweep of interaction strength and trap frequency.