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Physical Review Letters

Unitary Integration: A Numerical Technique Preserving the Structure of the Quantum Liouville Equation

Brad A. Shadwick, Walter F. Buell

Published Dec 29, 1997• 28 citations
Publisher

Abstract

The quantum Liouville equation for an $n$-level atomic system driven by external fields has a nontrivial kinematic structure; the quantities $\mathrm{tr}{\ensuremath{\rho}}^{j}$, $j\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1,2,\dots{},n$ remain constant in time, independent of the Hamiltonian. These invariants are physically significant; the qualitative character of the solution depends on their existence. A generic numerical method will not, in general, preserve these invariants. We present a numerical technique which evolves the density matrix via unitary transformations thus exactly preserving these invariants to all orders in the time step.

DOI
10.1103/physrevlett.79.5189

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The quantum Liouville equation for an nnn-level atomic system driven by external fields has a nontrivial kinematic structure; the quantities tr\ensuremathρj\mathrm{tr}{\ensuremath{\rho}}^{j}tr\ensuremathρj, j=1,2,…,nj\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1,2,\dots{},nj=1,2,…,n remain constant in time, independent of the Hamiltonian. These invariants are physically significant; the qualitative character of the solution depends on their existence. A generic numerical method will not, in general, preserve these invariants. We present a numerical technique which evolves the density matrix via unitary transformations thus exactly preserving these invariants to all orders in the time step.
Unitary integration is a numerical method that preserves the structure of the quantum Liouville equation by evolving the density via unitary transformations. Unitary integrators preserve the kinematic invariants C(sub j) = trp(sup j), j = 1,..., n to all orders in the time step. Here we extend unitary integration to weakly dissipative systems. We apply the technique of operator splitting, using a unitary integrator for the Hamiltonian evolution and a conventional integrator for the dissipative piece. In this way, we guarantee that all dissipation and decoherence (variation of the C(sub j)) is due to the new non-Hamiltonian terms and not to any numerical artifacts. We illustrate the method with examples.
The Liouville-Bloch equation for a general spin jjj in time-dependent external fields is solved as a unitary product of exponential operators. Operator noncommutativity is explicitly handled so as to leave behind a set of first-order differential equations for classical functions of time. Of these, only one is nontrivial and, once solved, the remaining yield to simple quadrature. Exact solutions or perturbative expansions of the classical equations can be used for analytical or numerical integration of the quantal equations.
Dissipation and decoherence, and the evolution from pure to mixed states in quantum physics, are handled through master equations for the density matrix. By embedding elements of this matrix in a higher-dimensional Liouville-Bloch equation, the methods of unitary integration are adapted to solve for the density matrix as a function of time, including the nonunitary effects of dissipation and decoherence. The input requires only solutions of classical, initial value time-dependent equations. Results are illustrated for a damped, driven two-level system.
In this paper, the Dyson series corresponding to the time-varying Hamiltonian of a finite dimensional quantum mechanical system is expanded in terms of products of exponentials of a complete basis of commutator superoperators in the corresponding Liouville space. The Cayley-Hamilton theorem and the Wei-Norman formula allow to express explicitly the functional relation between the Dyson series and the product of exponentials via a set of first order differential equations. Since the method is structure preserving, it can be used for the exact unitary integration of the driven Liouville-von Neumann equation.
Dissipation and decoherence, and the evolution from pure to mixed states in quantum physics, are handled through master equations for the density matrix. Master equations such as the Lindblad equation preserve the trace of this matrix. Viewing them as first-order time-dependent operator equations for the elements of the density matrix, a unitary integration procedure can be adapted to solve for these elements. A simple model for decoherence preserves the Hermiticity of the density matrix. A single, classical Riccati equation is the only one requiring numerical handling to obtain a full solution of the quantum evolution. The procedure is general, valid for any number of levels, but is illustrated here for a three-level system with two driving fields. For various choices of the initial state, we study the evolution of the system as a function of the amplitudes, relative frequencies, and phases of the driven fields and of the strength of the decoherence. The monotonic growth of the entropy is followed as the system evolves from a pure to a mixed state. An example is provided by the n=3n=3n=3 states of the hydrogen atom in a time-dependent electric field, such degenerate manifolds affording an analytical solution.