Brooding over bosons, wave packets and Bose - Einstein correlations, we present a generic quantum mechanical system that contains arbitrary number of bosons characterized by wave-packets and that can undergo a Bose-Einstein condensation either by cooling, or increasing the number density of bosons, or by increasing the overlap of the multi-boson wave-packet states, achieved by changing the size of the single-particle wave-packets. We show that the n-particle correlations may mimic coherent or chaotic behaviour for certain limiting wave-packet sizes. Effects of complete n-particle symmetrization are included. The resulting weights which fluctuate between 1 and n! are summed up with the help of a formal analogy between the considered wave-packet system and an already explored multi-boson plane-wave system. We solve the model analytically in the highly condensed and in the rare gas limiting cases, numerically in the intermediate cases. The relevance of the model to multi-pion production in high energy heavy ion physics as well as to the Bose-Einstein condensation of atomic vapours is discussed. As a by-product, a new class of probability distribution functions is obtained, event-by-event fluctuations of single-particle momentum distributions are predicted and the critical density for the onset of pion-lasing in high energy heavy ion collisions is derived.
Brooding over bosons, wave packets and Bose - Einstein correlations, we present a generic quantum mechanical system that contains arbitrary number of bosons characterized by wave-packets and that can undergo a Bose-Einstein condensation either by cooling, or increasing the number density of bosons, or by increasing the overlap of the multi-boson wave-packet states, achieved by changing the size of the single-particle wave-packets. We show that the n-particle correlations may mimic coherent or chaotic behaviour for certain limiting wave-packet sizes. Effects of complete n-particle symmetrization are included. The resulting weights which fluctuate between 1 and n! are summed up with the help of a formal analogy between the considered wave-packet system and an already explored multi-boson plane-wave system. We solve the model analytically in the highly condensed and in the rare gas limiting cases, numerically in the intermediate cases. The relevance of the model to multi-pion production in high energy heavy ion physics as well as to the Bose-Einstein condensation of atomic vapours is discussed. As a by-product, a new class of probability distribution functions is obtained, event-by-event fluctuations of single-particle momentum distributions are predicted and the critical density for the onset of pion-lasing in high energy heavy ion collisions is derived.
A recent experiment [Deng et al., Nature (London) 398, 218 (1999)] demonstrated four-wave mixing of matter wave packets created from a Bose-Einstein condensate. The experiment utilized light pulses to create two high-momentum wave packets via Bragg diffraction from a stationary Bose-Einstein condensate. The high-momentum components and the initial low-momentum condensate interact to form a new momentum component due to the nonlinear self-interaction of the bosonic atoms. We develop a three-dimensional quantum- mechanical description, based on the slowly-varying-envelope approximation, for four-wave mixing in Bose-Einstein condensates using the time-dependent Gross-Pitaevskii equation. We apply this formulation to describe the experimental observations and to make predictions. We examine the role of phase-modulation, momentum, and energy conservation (i.e., phase matching), and particle number conservation in four-wave mixing of matter waves, and develop simple models for understanding our numerical results.
Many useful properties of dilute Bose gases at ultralow temperature are predicted precisely by the (mean-field) product-state Ansatz, in which all particles are in the same quantum state. Yet, in situations where particle-particle correlations become important, the product Ansatz fails. To include correlations nonperturbatively, we consider a new set of states: the particle-correlated state of $N=l\ifmmode\times\else\texttimes\fi{}n$ bosons is derived by symmetrizing the n-fold product of an l-particle quantum state. Quantum correlations of the l-particle state ``spread out'' to any subset of the N bosons by symmetrization. The particle-correlated states can be simulated efficiently for large N, because their parameter spaces, which depend on l, do not grow with n. Here we formulate and develop in great detail the pure-state case for l=2, where the many-body state is constructed from a two-particle pure state. These paired wave functions, which we call pair-correlated states (PCS), were introduced by A. J. Leggett [Rev. Mod. Phys. 73, 307 (2001)] as a particle-number-conserving version of the Bogoliubov approximation. We present an iterative algorithm that solves for the reduced (marginal) density matrices (RDMs), i.e., the correlation functions, associated with PCS in time O(N). The RDMs can also be derived from the normalization factor of PCS, which is derived analytically in the large-N limit. To test the efficacy of PCS, we analyze the ground state of the two-site Bose-Hubbard model by minimizing the energy of the PCS state, both in its exact form and in its large-N approximate form, and comparing with the exact ground state. For N=1000, the relative errors of the ground-state energy for both cases are within ${10}^{\ensuremath{-}5}$ over the entire parameter region from a single condensate to a Mott insulator. We present numerical results that suggest that PCS might be useful for describing the dynamics in the strongly interacting regime.
Many useful properties of dilute Bose gases at ultra-low temperature are predicted precisely by the product-state ansatz, in which all particles are in the same quantum state. As particle-particle correlations become important, however, the product ansatz begins to fail. We consider a new set of states, which constitute a natural generalization of the product ansatz; the particle-correlated state of N=l×n identical particles is derived by symmetrizing the n-fold product of an l-particle quantum state. The particle-correlated states can be simulated efficiently for large N, because their parameter spaces, which depend on l, do not grow with n. Here we pay special attention to the pure-state case for l=2, where the many-body state is constructed from a two-particle pure state. These paired wave functions for bosons, which we call pair-correlated states (PCS), were introduced by Leggett [Rev. Mod. Phys. 73, 307 (2001)] as a particle-number-conserving version of the Bogoliubov approximation. We present an iterative algorithm that solves the reduced (marginal) density matrices (RDMs)---these are the correlation functions---associated with PCS in time O(N). The RDMs can also be derived from the normalization factor of PCS, which is derived analytically in the large N limit. To test the efficacy of our theory, we analyze the ground state of the two-site Bose-Hubbard model by minimizing the energy of the PCS state, both in its exact form and in its large-N approximate form. The relative errors of the ground state energy for both cases are within 10−5 for N=1000 particles over the entire parameter region from a single condensate to a Mott insulator. Moreover, we present numerical results that suggest that PCS might be useful for describing the dynamics in the strongly interacting regime.
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 calculate the energy fluctuation of a truly finite number of interacting bosons and study the role of interaction. Although the ideal Bose gas in thermodynamic limit is an exactly solvable problem and analytic expression of various fluctuation measures exists, the experimental Bose-Einstein condensation (BEC) is a nontrivial many-body problem. We employ a two-body correlated basis function and utilize the realistic van der Waals interaction. We calculate the energy fluctuation (△E2) of the interacting trapped bosons and plot kB2T2△E2 as a function of TcT. In the classical limit △E2 is related to the specific heat per particle cv through the relation △E2=kBT2cv. We have obtained a distinct hump in kB2T2△E2 around the condensation point for three-dimesional harmonically trapped Bose gas when the particle number $N\ensuremath{\simeq}5000$ and above which corresponds to the second-order phase transition. However for finite-size interacting bosons ($N\ensuremath{\simeq}$ a few hundred) the hump is not sharp, and the maximum in kB2T2△E2 can be interpreted as a smooth increase in the scaled fluctuation below Tc and then a decrease above Tc. To illustrate the justification we also calculate cv, which exhibits the same feature, which leads to the conjecture that for finite-sized interacting bosons phase transition is ruled out.
The role of repulsive interactions in statistical systems of Bose particles is investigated. Three different phenomenological frameworks are considered: a mean-field model, an excluded volume model, and a model with a medium-dependent effective mass. All three models are tuned to yield similar equations of state, with only minor deviations from the ideal Bose gas at small chemical potentials. Our analysis indicates, however, that these models lead to qualitatively different results for the Bose-Einstein condensation phenomenon. We discuss the different aspects of this phenomenon, namely, an onset of the Bose-Einstein condensation, particle number fluctuations, and a behavior of the Bose condensate. The obtained results can be helpful for interpreting the lattice QCD data at small temperature and large isospin chemical potential and the data on multiple pion production in high-energy nuclear collisions.