We present the efficient and universal numerical method for simulation of interacting quantum gas kinetics on a finite momentum lattice, based on the Boltzmann equation for occupation numbers. Usually, the study of models with two-particle interaction generates the excessive amount of terms in the equations essentially limiting the possible system size. Here we employ the original analytical transformation to decrease the scaling index of the amount of calculations. As a result, lattice sizes as large as 48x48x48 can be simulated, allowing to study realistic problems with complex interaction models. The method was applied to the simulation of weakly interacting Fermi and Bose gases where we calculated the relaxation times depending on the momentum and temperature.
We present the efficient and universal numerical method for simulation of interacting quantum gas kinetics on a finite momentum lattice, based on the Boltzmann equation for occupation numbers. Usually, the study of models with two-particle interaction generates the excessive amount of terms in the equations essentially limiting the possible system size. Here we employ the original analytical transformation to decrease the scaling index of the amount of calculations. As a result, lattice sizes as large as 48x48x48 can be simulated, allowing to study realistic problems with complex interaction models. The method was applied to the simulation of weakly interacting Fermi and Bose gases where we calculated the relaxation times depending on the momentum and temperature.
We present results of simulations of a quantum Boltzmann master equation (QBME) describing the kinetics of a dilute Bose gas confined in a trapping potential in the regime of Bose condensation. The QBME is the simplest version of a quantum kinetic master equation derived in previous work. We consider two cases of trapping potentials: a three-dimensional square-well potential with periodic boundary conditions and an isotropic harmonic oscillator. We discuss the stationary solutions and relaxation to equilibrium. In particular, we calculate particle distribution functions, fluctuations in the occupation numbers, the time between collisions, and the mean occupation numbers of the one-particle states in the regime of onset of Bose condensation.
Ultracold bosonic gases in optical lattices are strongly correlated quantum systems similar to solids. The strong correlation between the electrons in a solid on the one hand, and the bosonic atoms in optical lattice on the other, exhibit various quantum phenomena like insulation, conductivity, localization of electrons and atoms, respectively. Controlled by the intensity of the lattice laser, the ultracold bosonic gas can be transferred from a regime with superfluid character for shallow lattices into a regime of strong correlations, the Mott insulator. As an additional external parameter besides the lattice depth, one can generate spatial inhomogeneities by superimposing an additional standing wave (so-called two-color superlattices), which gives rise to localization effects or the formation of a Bose-glass phase. In the present work, numerical simulations are employed in order to investigate characteristic signatures of the quantum phases in the low-energy excitation spectrum of one-dimensional systems. We simulate temporal small amplitude modulations of the optical lattice in analogy to experiments, and evaluate the response of the system from the time-evolved initial state. The lattice systems are described in the framework of the Bose-Hubbard model. For the evaluation of the time-evolved state, we employ several numerical methods. We analyze systems of small size (6 particles on 6 sites) using an exact time-evolution by integration of the time-dependent Schrödinger equation. The formulation of an importance truncation scheme enables us to retain only the relevant components of the model space in the strongly correlated regime and, thus, allows for the investigation of systems with 10 particles on 10 sites using exact time-evolution. Based on this method, we present results of the Mott-insulating regime as well as for the Bose-glass phase. Furthermore, we employ particle-hole methods, which allow for the treatment of systems with experimental lattice sizes and particle numbers. Starting from the equation of motion method we adapt the Tamm-Dancoff approximation as well as the random-phase approximation for the occupation number representation of the Bose-Hubbard model. We present results of simulations of up to 50 particles on 50 sites and discuss the impact of the lattice depth on the low-energy excitations (U-resonance). Moreover, the impact of a two-color superlattice and the variation of its amplitude is 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.
Quantum-computing ideas are applied to the practical and ubiquitous problem of fluid dynamics simulation. Hence, this paper addresses two separate areas of physics: quantum mechanics and fluid dynamics (or specifically, the computational simulation of fluid dynamics). The quantum algorithm is called a quantum lattice gas. An analytical treatment of the microscopic quantum lattice-gas system is carried out to predict its behavior at the mesoscopic scale. At the mesoscopic scale, a lattice Boltzmann equation with a nonlocal collision term that depends on the entire system wave function, governs the dynamical system. Numerical results obtained from an exact simulation of a one-dimensional quantum lattice model are included to illustrate the formalism. A symbolic mathematical method is used to implement the quantum mechanical model on a conventional work- station The numerical simulation indicates that classical viscous damping is not present in the one-dimensional quantum lattice-gas system.
Understanding the effect of correlations in interacting many-body systems is one of the main challenges in quantum mechanics. While the general problem can only be addressed by approximate methods and numerical simulations, in some limiting cases, it is amenable to exact solutions. I will present a family of exact solutions for strongly interacting 1D quantum gases in arbitrary external potential, valid for bosons, multicomponent fermions and mixtures. It allows us to obtain experimental observables such as the density profiles and momentum distribution at all momentum scales, beyond the Luttinger liquid approach. It also predicts the exact quantum dynamics at all the times, including the small oscillation regime yielding the collective modes of the system and the large quench regime where the system parameters are changed considerably. The solution can be extended to describe finite-temperature conditions, spin, and magnetization effects. In particular I will present large-size calculations of the frequency- and momentum-resolved spectral function of strongly repulsive lattice bosons and a symmetry-breaking exact solution for a Bose-Bose mixture. Our approach directly describes the experiments with trapped ultracold atoms where the strongly correlated regime in one dimension has been achieved.
This document aims to describe the methodology of quantum lattice gas algorithm based quantum physics simulations with details on implementation using graphics processing units. A variety of examples illustrating the general application of these methods are included. These examples are used to highlight topics such as scaling, potentials, momentums, multiple particle and dimensions, and various Hamiltonians in quantum simulations. Although intended to fullfill the doctor of philosophy degree requirement, this document is well suited as an introduction to this approach to modeling physics. In recent times parallelization has significantly improved the available computational power. Graphics processing units (GPUs) are a shining example of this, with thousands of compute cores and continually improved general computational capabilities, these processors can outshine their central processing unit (CPU) counterparts by orders of magnitude. Presented here are a number of examples of efficient computational methods for simulating physics using GPUs. Our quantum simulation method involves the quantum lattice gas model of quantum computation [59, 68, 64], however much of the parallel technology implementation is applicable to other numerical approaches. The quantum lattice gas model of computation is based on a unitary stream-collide quantum algorithm applied repeatedly over time upon an ordered array of qubits [53, 56, 68, 61, 63, 67]. The stream-collide quantum algorithm represents local quantum evolution of the system of quantum particles whose motion is constrained to reside on a spacetime lattice. Each particle in the system undergoes a quantum walk, interacting locally with other particles in the system that may reside on the same point. We will first mathematically describe the basic idea of quantum lattice gas algorithms. Next we will discuss GPUs and the basic features needed for implementation. We will then describe several examples of quantum lattice gas algorithm approach to illustrate the various applications and implementation challenges. Of particular interest is a look at modeling time-dependent many-body quantum mechanics using quantum lattice gas methods. For practical purposes we focus on the case of two fermions governed by the nonrelativistic Schrödinger equation. Numerical comparisons to analytic prediction show the high accuracy of these methods. Quantum lattice gas methods are chosen as a unitary algorithm for time dependent quantum simulation as they are efficient, accurate, and avoid the Fermi sign problem. Our findings from two-particle simulations under various conditions include the physics of the one and two particle systems diverging in their respective time-dependent behavior with increasing entanglement. We also confirm the multiple particle extension of the Ehrenfest theorem. Using the examples discussed throughout this document, we will then describe our specific computational implementation utilizing the computational power of GPUs through Nvidia’s compute unified device architecture (CUDA) language. Here we will discuss the open source software our group is developing based on these methods, along with some implementation verification and debugging concepts. Education is another application peripherally connected to this work and our software. The applications of this type of modeling as a learning tool is not fully explored yet, but the opportunities are promising.