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The Journal of Chemical Physics

Quantum dynamics using a discretized coherent state representation: An adaptive phase space method

L. Mauritz Andersson

Published Jul 15, 2001• 33 citations
Publisher

Abstract

We introduce a discretized coherent state representation (DCSR) for quantum dynamics. Expansion of a wave function in the nonorthogonal slightly overcomplete set is made with an identity operator computed using an iterative refinement method. Calculating the inverse of the overlap matrix is not necessary. The result is an accurate and efficient representation, where you only put basis functions in the region of phase space where the wave function is nonvanishing. Compared to traditional spatial grid methods, fewer grid points are needed. The DCSR can be viewed as an application of the Weyl–Heisenberg frame and extends it into a useful computational method. A scheme for fully quantum mechanical propagation is constructed and applied to the realistic problem of highly excited vibration in the heavy diatomic molecule Rb2. Compared to split-operator propagation in a conventional spatial grid, an order of magnitude longer time steps can be taken and fewer grid points are needed. The computational effort scales linearly with the number of basis functions. Nonreflecting boundary conditions are a natural property of the representation and is illustrated in a model of predissociation.

DOI
10.1063/1.1380204

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We introduce a discretized coherent state representation (DCSR) for quantum dynamics. Expansion of a wave function in the nonorthogonal slightly overcomplete set is made with an identity operator computed using an iterative refinement method. Calculating the inverse of the overlap matrix is not necessary. The result is an accurate and efficient representation, where you only put basis functions in the region of phase space where the wave function is nonvanishing. Compared to traditional spatial grid methods, fewer grid points are needed. The DCSR can be viewed as an application of the Weyl–Heisenberg frame and extends it into a useful computational method. A scheme for fully quantum mechanical propagation is constructed and applied to the realistic problem of highly excited vibration in the heavy diatomic molecule Rb2. Compared to split-operator propagation in a conventional spatial grid, an order of magnitude longer time steps can be taken and fewer grid points are needed. The computational effort scales linearly with the number of basis functions. Nonreflecting boundary conditions are a natural property of the representation and is illustrated in a model of predissociation.
Properties of a discretized coherent state representation (DCSR) and its connection to Gabor frame analysis are discussed. The DCSR approach was recently shown (Andersson L M 2001 J. Chem. Phys. 115 1158) to yield a practical computational scheme for quantum dynamics, and an iterative scheme for finding the identity operator was proposed. In the present work, we suggest a proof of fast convergence of the iterative scheme for computing the canonical dual to any given countable frame in a Hilbert space. The method of frames is concerned with the use of a non-orthogonal, over-complete set of functions for expansion of an arbitrary function. We also introduce the concept of 'representations of the identity operator' and show how to expand arbitrary vectors using the frame elements, without explicit diagonalization to an orthonormal basis. Numerical examples that illustrate the method are shown.
Numerical solutions of the quantum time-dependent integro-differential Schrödinger equation in a coherent state Husimi representation are investigated. Discretization leads to propagation on a grid of nonorthogonal coherent states without the need to invert an overlap matrix, with the further advantage of a sparse Hamiltonian matrix. Applications are made to the evolution of a Gaussian wave packet in a Morse potential. Propagation on a static rectangular grid is fast and accurate. Results are also presented for a moving rectangular grid, guided at its center by a mean classical path, and for a classically guided moving grid of individual coherent states taken from a Monte Carlo ensemble.
The finite-element discrete variable representation is proposed to express the nonequilibrium Green's function for strongly inhomogeneous quantum systems. This method is highly favorable against a general basis approach with regard to numerical complexity, memory resources, and computation time. Its flexibility also allows for an accurate representation of spatially extended Hamiltonians and thus opens the way toward a direct solution of the two-time Schwinger-Keldysh-Kadanoff-Baym equations on spatial grids, including, for example, the description of highly excited states in atoms. As benchmarks, we compute and characterize, in Hartree-Fock and second Born approximations, the ground states of the He atom, the H2{\mathrm{H}}_{2}H2​ molecule, and the LiH molecule in one spatial dimension. Thereby, the ground-state and binding energies, densities, and bond lengths are compared with the direct solution of the time-dependent Schr\"odinger equation.
Quantum dynamics calculations scale steeply with system size, rendering calculations on a classical computer intractable for complex systems. While quantum computing offers a natural solution, its application to nuclear quantum dynamics has been scarce. Here, we present a quantum algorithm to study photodissociation dynamics on quantum computers, benchmarked on the NOCl molecule. The wave function is propagated via a split-operator method, utilizing the quantum Fourier transform and unitary transformation matrix to switch representations. To impose outgoing boundary conditions on a truncated grid, we use a nonunitary absorbing potential propagator, implemented through a dilation scheme. The photodissociation cross section is calculated from the autocorrelation function, which is extracted using the Hadamard test. Our quantum computing results agree well with benchmarks under ideal conditions, and we further demonstrate that the algorithm is robust to noise and statistical sampling errors, indicating its promise for application of noisy devices for quantum dynamics studies.
In this Chapter we study the Weyl–Heisenberg group in the Schrödinger representation in arbitrary dimension n. One shows that it operates in the Hilbert space of quantum states (and on quantum operators) as a phase-space translation. Then applying it to a Schwartz class state of arbitrary profile we get a set of generalized coherent states. When we apply the Weyl–Heisenberg translation operator to the ground state of the n-dimensional Harmonic Oscillator, one gets the standard coherent states introduced by Schrödinger [233] (Naturwissenshaften 14:664–666, 1926) in the early days of quantum mechanics (1926). Later the coherent states have been extensively studied by Glauber [118, 119] (Phys. Rev. 131:2766–2788, 1963; Phys. Rev. 130:2529–2539, 1963) for the purpose of quantum optics and it seems that their name comes back to this work. The standard coherent states have been generalized by Perelomov [208] (Generalised Coherent States and Their Applications, 1986) to more general Lie groups than the Weyl–Heisenberg group. We also introduce the usual creation and annihilation operators in dimension n which are very convenient for the study of coherent states. We show that coherent states constitute a non-orthogonal over-complete system which yields a resolution of the identity operator in the Hilbert space and which allows a computation of the Hilbert–Schmidt norm and of the trace of respectively Hilbert–Schmidt class and trace-class operators. We study their time-evolution for the quantum Harmonic Oscillator hamiltonian and show that a time evolved coherent state located around phase-space point z is up to a phase a coherent state located around the phase-space point ztz_tzt​, where ztz_tzt​ is the phase-space point of the classical flow governed by the Harmonic Oscillator. This property was described by Schrödinger as the non-spreading of the time evolution of coherent states under the quantum Harmonic Oscillator dynamics. We also show how to go from the Schrödinger to the Fock–Bargmann representation using the standard coherent states.
System-specific coherent states are constructed based on the formulation of supersymmetric quantum mechanics for arbitrary quantum systems. By regarding the superpotential as a generalized displacement variable, we identity the ground state of a quantum system as the minimizer of the supersymmetric Heisenberg uncertainty product. A special case is the ground state of the standard harmonic oscillator. One constructs standard coherent states by applying a shift operator to a 'fiducial function', taken as the ground state Gaussian. By analogy, we use the ground state for any other system as a new fiducial function, generating from its shifts new dynamically-adapted, overcomplete coherent states. The discretized system-specific coherent states can serve as a dynamically-adapted basis for bound state calculations. Accurate computational results for the Morse potential, the double well potential and the two-dimensional anharmonic oscillator systems demonstrate that the system-specific coherent states can provide rapidly-converging approximations for excited state energies and wave functions.