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.
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.
A recent paper of Gardiner [Phys. Rev. A 56, 1414 (1997)] introduces a particle-number-conserving Bogoliubov method for the excitation spectrum of a Bose-condensed gas, for use in theories of recently experimentally produced trapped atomic Bose condensates. Gardiner's approach is compared and contrasted with the 1959 Girardeau-Arnowitt theory [Phys. Rev. 113, 755 (1959)], to which it is closely related and which is also fully number conserving. The number-conserving Bogoliubov quasiparticle operators of the Girardeau-Arnowitt theory satisfy Bose commutation relations exactly so long as states with the condensate totally depleted are neglected, whereas those of Gardiner satisfy Bose commutation relations only in an approximation that deteriorates progressively as the condensate is depleted.
The impact of interparticle correlations on the behavior of Bose-Einstein Condensates (BECs) is discussed using two approaches. In the first approach, the wavefunction of a BEC is encoded in the
N-particle sector of an extended "catalytic state". Going to a time-dependent interaction picture, we can organize the effective Hamiltonian by powers of
N−1/2. Requiring the terms of order
N1/2 to vanish, we get the Gross-Pitaevskii Equation. Going to the next order,
N0, we obtain the number-conserving Bogoliubov approximation. Our approach allows one to stay in the Schrödinger picture and to apply many techniques from quantum optics. Moreover, it is easier to track different orders in the Hamiltonian and to generalize to the multi-component case. In the second approach, I consider a state of
N=l×n bosons that is derived by symmetrizing the
n-fold tensor product of an arbitrary
l-boson state. Particularly, we are interested in the pure state case for
l=2, which we call the Pair-Correlated State (PCS). I show that PCS reproduces the number-conserving Bogoliubov approximation; moreover, it also works in the strong interaction regime where the Bogoliubov approximation fails. For the two-site Bose-Hubbard model, I find numerically that the error (measured by the trace distance of the two-particle reduced density matrices) of PCS is less than two percent over the entire parameter space, thus making PCS a bridge between the superfluid and Mott insulating phases. Amazingly, the error of PCS does not increase, in the time-dependent case, as the system evolves for longer times. I derive both time-dependent and -independent equations for the ground state and the time evolution of the PCS ansatz. The time complexity of simulating PCS does not depend on
N and is linear in the number of orbitals in use.
We propose a method to study the time evolution of Bose-Einstein condensed gases perturbed from an initial thermal equilibrium, based on the Wigner representation of the N-body density operator. We show how to generate a collection of random classical fields sampling the initial Wigner distribution in the number conserving Bogoliubov approximation. The fields are then evolved with the time dependent Gross-Pitaevskii equation. We illustrate the method with the damping of a collective excitation of a one-dimensional Bose gas.