The static fluctuation approximation (SFA) is applied to compute the thermodynamic properties of a trapped two-dimensional (2D) interacting hard-sphere (HS) Bose gas in the weakly and strongly interacting regime. A mean-field approach involving a variational wave function is used to compute the mean-field energy as a function of temperature for each harmonic oscillator (HO) state plugged into the SFA technique. In the variational approach, a parameter α is introduced into the harmonic oscillator wave function in order to take into account the changes in the width when the repulsive interactions between the bosons are increased. In the weakly interacting regime, below the critical temperature, the total energy of all HO states (evaluated by our model) matches the noninteracting result very well. However, beyond the critical temperature, we "fit" our energies to the classical limit for 2D bosons in a trap by using a suitably proposed weighting function. We compare our results to earlier results of mean-field theory. Further, we evaluate the density matrix arising from correlations between the HO orbitals.
Instantaneous collapse of the wave function upon measurement of a single particle is one of the postulates of the Copenhagen interpretation of quantum mechanics. However, what happens when a many-body system in a macroscopic coherent state is measured one particle at a time? Here, we consider successive measurements of individual spins from a spin Bose condensate that starts in a Schrödinger cat state. When the spin measurements are done one particle at a time, the collapse of the spin condensate is not instantaneous but leads to probabilities for spin measurement that strongly depend on the previous measurements. What is surprising is that an almost complete collapse occurs in very few measurements. Even in a large system, a single cat component is emphasized quite quickly in the sequence of measurements. We examine the process by analysis of a simple two-Fock-state cat, as well as a cat state that has many components. Justification is given for our theoretical measurement process.
The dynamic depletion of a trapped one-dimensional Bose-Einstein condensate (BEC) that is driven by laser stirring is numerically explored using beyond mean-field methods. For this purpose, the multi-configurational time-dependent Hartree method for bosons (MCTDHB) [Alon \ea, Phys. Rev. A {\bf 77}, 033613 (2008)] is applied. In order to induce the depletion, the BEC is excited by a negative Gaussian potential (dimple) whose depth is modulated with time. The BEC is examined in various trapping geometries, with different interactions, and the condensate depletion is recorded as a function of time. A general power--law trap is considered that can be experimentally generated and shaped by the holographic methods of Bruce \ea\ [Bruce \ea, Phys. Rev. A {\bf 84}, 053410 (2011)]. The chief goal is to explore the interplay between trapping geometry and interactions in defining the depletion dynamics. It is chiefly found, that the details of these depletion dynamics are unpredictable and determined by a combination of the principle dimple depth, trap, and interactions. One significant feature of this work is that quite a number of plateaus is reached in the aforementioned dynamics.
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