The quantum dynamics and grand canonical thermodynamics of many-mode (one-, two-, and three-dimensional) interacting Bose gases are simulated from first principles. The model uses a lattice Hamiltonian based on a continuum second-quantized model with two-particle interactions, external potential, and interactions with an environment, with no further approximations. The interparticle potential can be either an (effective) delta function as in Bose-Hubbard models, or extended with a shape resolved by the lattice. Simulations are of a set of stochastic equations that in the limit of many realizations correspond exactly to the full quantum evolution of the many-body systems. These equations describe the evolution of samples of the gauge P distribution of the quantum state, details of which are developed. Conditions under which general quantum phase-space representations can be used to derive stochastic simulation methods are investigated in detail, given the criteria: 1) The simulation corresponds exactly to quantum mechanics in the limit of many trajectories. 2) The number of equations scales linearly with system size, to allow the possibility of efficient first-principles quantum mesoscopic simulations. 3) All observables can be calculated from one simulation. 4) Each stochastic realization is independent to allow straightforward use of parallel algorithms. Special emphasis is placed on allowing for simulation of open systems. In contrast to typical Monte Carlo techniques based on path integrals, the phase-space representation approach can also be used for dynamical calculations. Two major (and related) known technical stumbling blocks with such stochastic simulations are instabilities in the stochastic equations, and pathological trajectory distributions as the boundaries of phase space are approached. These can (and often do) lead to systematic biases in the calculated observables. The nature of these problems are investigated in detail. Many phase-space distributions have, however, more phase-space freedoms than the minimum required for exact correspondence to quantum mechanics, and these freedoms can in many cases be exploited to overcome the instability and boundary term problems, recovering an unbiased simulation. The stochastic gauge technique, which achieves this in a systematic way, is derived and heuristic guidelines for its use are developed. The gauge P representation is an extension of the positive P distribution, which uses coherent basis states, but allows a variety of useful stochastic gauges that are used to overcome the stability problems. Its properties are investigated, and the resulting equations to be simulated for the open interacting Bose gas system are derived. The dynamics of the following many-mode systems are simulated as examples: 1) Uniform one-dimensional and two-dimensional Bose gases after the rapid appearance of significant two-body collisions (e.g. after entering a Feshbach resonance). 2) Trapped bosons, where the size of the trap is of the same order as the range of the interparticle potential. 3) Stimulated Bose enhancement of scattered atom modes during the collision of two Bose-Einstein condensates. The grand canonical thermodynamics of uniform one-dimensional Bose gases is also calculated for a variety of temperatures and collision strengths. Observables calculated include first to third order spatial correlation functions (including at finite interparticle separation) and momentum distributions. The predicted phenomena are discussed. Improvements over the positive P distribution and other methods are discussed, and simulation times are analyzed for Bose-Hubbard lattice models from a general perspective. To understand the behavior of the equations, and subsequently optimize the gauges for the interacting Bose gas, single- and coupled two-mode dynamical and thermodynamical models of interacting Bose gases are investigated in detail. Directions in which future progress can be expected are considered. Lastly, safeguards are necessary to avoid biased averages when exponentials of Gaussian-like trajectory distributions are used (as here), and these are investigated.

The quantum dynamics and grand canonical thermodynamics of many-mode (one-, two-, and three-dimensional) interacting Bose gases are simulated from first principles. The model uses a lattice Hamiltonian based on a continuum second-quantized model with two-particle interactions, external potential, and interactions with an environment, with no further approximations. The interparticle potential can be either an (effective) delta function as in Bose-Hubbard models, or extended with a shape resolved by the lattice. Simulations are of a set of stochastic equations that in the limit of many realizations correspond exactly to the full quantum evolution of the many-body systems. These equations describe the evolution of samples of the gauge P distribution of the quantum state, details of which are developed. Conditions under which general quantum phase-space representations can be used to derive stochastic simulation methods are investigated in detail, given the criteria: 1) The simulation corresponds exactly to quantum mechanics in the limit of many trajectories. 2) The number of equations scales linearly with system size, to allow the possibility of efficient first-principles quantum mesoscopic simulations. 3) All observables can be calculated from one simulation. 4) Each stochastic realization is independent to allow straightforward use of parallel algorithms. Special emphasis is placed on allowing for simulation of open systems. In contrast to typical Monte Carlo techniques based on path integrals, the phase-space representation approach can also be used for dynamical calculations. Two major (and related) known technical stumbling blocks with such stochastic simulations are instabilities in the stochastic equations, and pathological trajectory distributions as the boundaries of phase space are approached. These can (and often do) lead to systematic biases in the calculated observables. The nature of these problems are investigated in detail. Many phase-space distributions have, however, more phase-space freedoms than the minimum required for exact correspondence to quantum mechanics, and these freedoms can in many cases be exploited to overcome the instability and boundary term problems, recovering an unbiased simulation. The stochastic gauge technique, which achieves this in a systematic way, is derived and heuristic guidelines for its use are developed. The gauge P representation is an extension of the positive P distribution, which uses coherent basis states, but allows a variety of useful stochastic gauges that are used to overcome the stability problems. Its properties are investigated, and the resulting equations to be simulated for the open interacting Bose gas system are derived. The dynamics of the following many-mode systems are simulated as examples: 1) Uniform one-dimensional and two-dimensional Bose gases after the rapid appearance of significant two-body collisions (e.g. after entering a Feshbach resonance). 2) Trapped bosons, where the size of the trap is of the same order as the range of the interparticle potential. 3) Stimulated Bose enhancement of scattered atom modes during the collision of two Bose-Einstein condensates. The grand canonical thermodynamics of uniform one-dimensional Bose gases is also calculated for a variety of temperatures and collision strengths. Observables calculated include first to third order spatial correlation functions (including at finite interparticle separation) and momentum distributions. The predicted phenomena are discussed. Improvements over the positive P distribution and other methods are discussed, and simulation times are analyzed for Bose-Hubbard lattice models from a general perspective. To understand the behavior of the equations, and subsequently optimize the gauges for the interacting Bose gas, single- and coupled two-mode dynamical and thermodynamical models of interacting Bose gases are investigated in detail. Directions in which future progress can be expected are considered. Lastly, safeguards are necessary to avoid biased averages when exponentials of Gaussian-like trajectory distributions are used (as here), and these are investigated.