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On the exact unitary integration of time-varying quantum Liouville equations.

Claudio Altafini

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

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.

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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.
For almost 75 years, the general solution for the Schrödinger equation was assumed to be generated by an exponential or a time-ordered exponential known as the Dyson series. We study the unitarity of a solution in the case of a singular Hamiltonian and provide a new methodology that is not based on the assumption that the underlying space is L2(R). Then, an alternative operator for generating the time evolution that is manifestly unitary is suggested, regardless of the choice of Hamiltonian. The new construction involves an additional positive operator that normalizes the wave function locally and allows us to preserve unitarity, even when dealing with infinite dimensional or non-normed spaces. Our considerations show that Schrödinger and Liouville equations are, in fact, two sides of the same coin and together they provide a unified description for unbounded quantum systems.
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.
Starting from our idea of combining the Feynman path integral spirit and the Dyson series kernel, we find an explicit and general form of time evolution operator that is a ccc-number function and a power series of perturbation including all order approximations in the unperturbed Hamiltonian representation. Based on it, we obtain an exact solution of the Schrödinger equation in general quantum systems independent of time. Comparison of our exact solution with the existed perturbation theory makes some features and significance of our exact solution clear. The conclusions expressly indicate that our exact solution is obviously consistent with the usual time-independent perturbation theory at any order approximation, it explicitly calculates out the expanding coefficients of the unperturbed state in the non-perturbation method, and it fully solves the recurrence equation of the expansion coefficients of final state in the unperturbed Hamiltonian representation from a view of time-dependent perturbation theory. At the same time, the exact solution of the von Neumann equation is also given. Our results can be thought of as theoretical developments of quantum dynamics, and are helpful for understanding the dynamical behavior and related subjects of general quantum systems in both theory and application. Our exact solution, together with its sequence studies on perturbation theory [An Min Wang, quant-ph/0611217] and open system dynamics [An Min Wang, quant-ph/0601051] can be used to establish the foundation of theoretical formulism of quantum mechanics in general quantum systems. Further applications of our exact solution to quantum theory can be expected.
The Wei-Norman technique allows to express the solution of a system of linear non-autonomous differential equations in terms of product of exponentials. In particular it enables to find a time-ordered product of exponentials by solving a set of nonlinear differential equations. The method has numerous theoretical and computational advantages, in particular in optimal control theory. We show that in the unitary case, i.e.\ when the solution of the linear system is given by a unitary evolution operator, the nonlinear system can be by an appropriate choice of ordering reduced to a hierarchy of matrix Riccati equations. Our findings have a particular significance in quantum control theory since pure quantum evolution is unitary.
The quantum Liouville equation for an nnn-level atomic system driven by external fields has a nontrivial kinematic structure; the quantities $\mathrm{tr}{\ensuremath{\rho}}^{j}$, 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.
The unitary operators U(t), describing the quantum time evolution of systems with a time-dependent Hamiltonian, can be constructed in an explicit manner using the method of time-dependent invariants. We clarify the role of Lie-algebraic techniques in this context and elaborate the theory for SU(2) and SU(1,1). We show that the constructions known as Magnus expansion and Wei-Norman expansion correspond with different representations of the rotation group. A simpler construction is obtained when representing rotations in terms of Euler angles. The many applications are postponed to Part II of the paper.