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arXiv:1511.02728 · The Journal of Chemical Physics

Multiconfigurational quantum propagation with trajectory-guided generalized coherent states

Adriano Grigolo, Thiago F. Viscondi, Marcus A. M. de Aguiar

Published Mar 4, 2016• 12 citations
arXivHTMLPDF

Abstract

A generalized version of the coupled coherent states method for coherent states of arbitrary Lie groups is developed. In contrast to the original formulation, which is restricted to frozen-Gaussian basis sets, the extended method is suitable for propagating quantum states of systems featuring diversified physical properties, such as spin degrees of freedom or particle indistinguishability. The approach is illustrated with simple models for interacting bosons trapped in double- and triple-well potentials, most adequately described in terms of SU(2) and SU(3) bosonic coherent states, respectively.

DOI
10.1063/1.4942926

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A generalized version of the coupled coherent states method for coherent states of arbitrary Lie groups is developed. In contrast to the original formulation, which is restricted to frozen-Gaussian basis sets, the extended method is suitable for propagating quantum states of systems featuring diversified physical properties, such as spin degrees of freedom or particle indistinguishability. The approach is illustrated with simple models for interacting bosons trapped in double- and triple-well potentials, most adequately described in terms of SU(2) and SU(3) bosonic coherent states, respectively.
A previous initial value coherent state representation is applied to Gaussian wave packet propagation on one-dimensional and two-dimensional double well potentials. The method uses a basis of trajectory guided frozen Gaussian coherent states, sampled from a Monte Carlo ensemble, for numerical solutions of the quantum time dependent integro-differential Schrödinger equation. Accurate tunneling splittings are obtained. Comparisons are also made with corresponding results obtained by the Herman–Kluk semiclassical initial value representation wave packet method.
Previously, we introduced two versions of the Multiconfigurational Ehrenfest (MCE) approach to high dimensional quantum dynamics. It has been shown that the first version, MCEv1, converges well to the existing benchmarks for high dimensional model systems. At the same time, it was found that the second version, MCEv2, had more difficulty converging in some regimes. As MCEv2 is particularly suited for direct dynamics, it is important to facilitate its convergence. This paper investigates an efficient method of basis set sampling, called Quantum Superposition Sampling (QSS), which dramatically improves the performance of the MCEv2 approach. QSS is tested on the spin-boson model, often used for modeling of open quantum systems. It is also shown that the quantum subsystem in the spin-boson model can be conveniently treated with the help of two level system coherent states. Generalized coherent states, which combine two level system coherent states for the description of the system and Gaussian coherent states for description of the bath, are introduced. Various forms of quantum equations of motion in the basis of generalized coherent states can be developed by analogy with known quantum dynamics equations in the basis of Gaussian coherent states; in particular, the multiconfigurational Ehrenfest method becomes a version of coupled generalized coherent states, and QSS can then be viewed as a generalization of a sampling method known for the existing coupled coherent states method which uses Gaussian coherent states.
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.
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.
We find that a technique analogous to hybrid mechanics previously developed for Gaussian coherent states and molecular vibrations, can be extended to SU(2) coherent states describing coupled qubits. This formulation reveals a sparse structure of the propagator, leading to a computationally efficient scheme while maintaining numerical accuracy. As a result, the method enables simulations of larger qubit systems on classical computers. In particular, we demonstrate accurate real-time propagation for systems of up to 21 qubits, which would be computationally demanding using straightforward quantum-dynamics approaches.
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.