Coupled Coherent States for Indistinguishable Bosons in the Second Quantisation Representation

James A. Green, Dmitrii V. Shalashilin

Published Nov 15, 2018• 0 citations

Abstract

Computer simulations of many-body quantum dynamics of indistinguishable particles is a challenging task for computational physics. In this paper we demonstrate that the method of coupled coherent states (CCS) developed previously for multidimensional quantum dynamics can be used to study indistinguishable bosons in the second quantisation formalism. To prove its validity, the technique termed here coupled coherent states for indistinguishable bosons (CCSB) is tested on two model problems. The first is a system-bath problem consisting of a tunnelling mode coupled to a harmonic bath, previously studied by CCS and other methods in distinguishable representation in 20 dimensions. The harmonic bath is comprised of identical oscillators, and may be second quantised for use with CCSB. The cross-correlation function for the dynamics of the system and Fourier transform spectrum compare extremely well with a benchmark calculation, and are in much better agreement than prior methods of studying the problem. The second model problem involves 100 bosons in a shifted harmonic trap. Oscillations in the 1-body density are calculated and shown to compare favourably to a multiconfigurational time-dependent Hartree for bosons calculation, demonstrating the applicability of the method as a new formally exact way to study the quantum dynamics of Bose-Einstein condensates.

Related papers

Simulation of the quantum dynamics of indistinguishable bosons with the method of coupled coherent states

James A. Green, Dmitrii V. Shalashilin

Computer simulations of many-body quantum dynamics of indistinguishable particles is a challenging task for computational physics. In this paper we demonstrate that the method of coupled coherent states (CCS) developed previously for multidimensional quantum dynamics of distinguishable particles can be used to study indistinguishable bosons in the second-quantization formalism. To prove its validity, the technique termed here coupled coherent states for indistinguishable bosons (CCSB) is tested on two model problems. The first is a system-bath problem consisting of a tunneling mode coupled to a harmonic bath, previously studied by CCS and other methods in distinguishable representation in 20 dimensions. The harmonic bath is comprised of identical oscillators, and may be second quantized for use with CCSB, so that this problem may be thought of as a bosonic bath with an impurity. The cross-correlation function for the dynamics of the system and Fourier transform spectrum compare extremely well with a benchmark calculation, which none of the prior methods of studying the problem achieved. The second model problem involves 100 bosons in a shifted harmonic trap. Breathing oscillations in the one-body density are calculated and shown to compare favorably to a multiconfigurational time-dependent Hartree for bosons calculation, demonstrating the applicability of the method as a new formally exact way to study the quantum dynamics of Bose-Einstein condensates.

arXiv:1811.06293Physical Review AJul 10, 20198 citations

Development and Application of New Numerical Extensions to the Coupled Coherent States Family of Multidimensional Quantum Dynamics Methods

James Alexander Green

The coupled coherent states method has demonstrated itself as an accurate and efficient method of studying the quantum dynamics of various systems. In recent years, its applicability has been extended by incorporating a number of new numerical expansions and modifications to generate a closely related family of methods. In this thesis, two new augmentations are developed to further broaden the scope of problems that are able to be treated. The first of these is a 2-layer extension of coupled coherent states, capable of providing an increased mathematical description of a degree or degrees of freedom within a quantum mechanical system, as well as beneficial numerical and scalability properties. The newly developed method is tested on a model system-bath Hamiltonian consisting of a tunnelling mode governed by an asymmetric double well potential coupled to a harmonic bath. It is found to compare well to previous methods of studying the Hamiltonian, as well as a benchmark calculation on the system conducted in this thesis, and demonstrate the beneficial numerical and scalability properties expected. The second development is to extend coupled coherent states to treat systems of indistinguishable bosons in the second quantisation representation. The method is tested on the same Hamiltonian as the 2-layer coupled coherent states scheme, where the harmonic bath is second quantised as it is comprised of oscillators of the same frequency, so they may be thought of as indistinguishable. Exploiting this symmetry property is found to be extremely advantageous, with remarkable agreement to the benchmark calculation. The method is then tested on a model Hamiltonian consisting of 100 bosons in a shifted harmonic trap, with oscillations in the 1-body density calculated. The results are found to compare favourably with a multiconfigurational time-dependent Hartree for bosons calculation that is equivalent to the Gross-Pitaevskii equation, providing impetus for future studies on systems of Bose-Einstein condensates. The existing ab initio multiple cloning extension of coupled coherent states for nonadiabatic dynamics is also used to study the ultrafast photodissociation of 2-ethylpyrrole. The results are compared to experimental data, and a novel insight into the dissociation mechanism is obtained, with it shown to be composed of a two step process. Firstly, molecules that are able to dissociate immediately over the barrier along the N-H coordinate do so in < 50 fs, and this is followed by a second slower dissociation process from molecules that must sample the potential energy surface before finding a way around the barrier. This is not observed experimentally due to the temporal widths of the laser pulses obscuring the dynamics in the < 50 fs window.

Mar 1, 20180 citations

Numerically exact quantum dynamics for indistinguishable particles: The multilayer multiconfiguration time-dependent Hartree theory in second quantization representation

Haobin Wang, Michael Thoss

A new theory is proposed to accurately simulate quantum dynamics in systems of identical particles. It is based on the second quantization formalism of many-body quantum theory, in which the Fock space is represented by occupation-number states. Within this representation the overall Fock space can be formally decomposed into smaller subspaces, and the wave function can be expressed as a multilayer multiconfiguration Hartree expansion involving subvectors in these subspaces. The theory unifies the multilayer multiconfiguration time-dependent Hartree theory for both distinguishable and indistinguishable particles. Specific formulations are given for systems of identical fermions, bosons, and combinations thereof. Practical implementations are discussed, especially for the case of fermions, to include the operator algebra that enforces the symmetry of identical particles. The theory is illustrated by a numerical example on vibrationally coupled electron transport.

The Journal of Chemical PhysicsJul 13, 2009182 citations

Numerically exact quantum dynamics of bosons with time-dependent interactions of harmonic type

Axel U. J. Lode, Kaspar Sakmann, Ofir E. Alon +2 more

The exactly solvable quantum many-particle model with harmonic one- and two-particle interaction terms is extended to include time dependency. We show that when the external trap potential and interparticle interaction have a time dependency, the numerically exact solutions of the corresponding time-dependent many-boson Schr\"odinger equation are still available. We use these exact solutions to benchmark the recently developed multiconfigurational time-dependent Hartree method for bosons (MCTDHB) [Phys. Rev. Lett. 99, 030402 (2007); Phys. Rev. A 77, 033613 (2008)]. In particular, we benchmark the MCTDHB method for (i) the ground state; (ii) the breathing many-body dynamics activated by a quench scenario where the interparticle interaction strength is suddenly turned on to a finite value; (iii) the nonequilibrium dynamic for driven scenarios where both the trap- and interparticle-interaction potentials are time-dependent. Excellent convergence of the ground state and dynamics is demonstrated. The great relevance of the self-consistency and time adaptivity, which are the intrinsic features of the MCTDHB method, is demonstrated by contrasting the MCTDHB predictions and those obtained within the standard full configuration interaction method spanning a Fock space of the same size, but utilizing as one-particle basis set the fixed-shape eigenstates of the one-particle potential. Connections of the model's results to ultracold Bose-Einstein condensed systems are addressed.

arXiv:1207.5128Physical Review ADec 6, 2012117 citations

Basis set approach to the quantum dissipative dynamics: Application of the multiconfiguration time-dependent Hartree method to the spin-boson problem

Haobin Wang

The feasibility of using a basis set approach to the study of quantum dissipative dynamics is investigated for the spin-boson model, a system of two discrete states linearly coupled to a harmonic bath. The infinite Hamiltonian is discretized to a finite number of degrees of freedom. Traditional basis set approach, in a multiconfiguration time-dependent Hartree context, is used to solve the time-dependent Schrödinger equations by explicitly including all the degrees of freedom (“system”+“bath”). Quantities such as the reduced density matrix are then evaluated via a quadrature summation/Monte Carlo procedure over a certain number of time-dependent wave functions. Numerically exact results are obtained by systematically increasing the number of bath modes used to represent the condensed phase environment, as well as other variational parameters (number of basis functions, configurations, etc.). The potential of the current method is briefly discussed.

The Journal of Chemical PhysicsDec 8, 2000108 citations

Benchmark calculation for tunnelling through a multidimensional asymmetric double well potential

James A. Green, Dmitrii V. Shalashilin

A benchmark calculation is presented for the quantum dynamics of tunnelling through a multidimensional asymmetric double well potential. A model Hamiltonian is used with a 1-dimensional tunnelling mode coupled to an ( M − 1)-dimensional harmonic bath, a system-bath problem. The benchmark calculation uses a basis set expansion of the wavefunction, with separate basis functions for the system and bath. Indistinguishability of configurations is exploited to greatly reduce the expense of the calculation, and a fully converged result is achieved. Comparison is offered to existing quantum dynamical methods that have tested this model problem, and further benchmark results not previously studied are presented.

Chemical Physics LettersNov 1, 20157 citations