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Physical Review Letters

Unitary Integration of Quantum Liouville-Bloch Equations

A. R. P. Rau

Published Nov 30, 1998• 36 citations
Publisher

Abstract

The Liouville-Bloch equation for a general spin $j$ 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.

DOI
10.1103/physrevlett.81.4785

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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.
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.
Both unitary evolution and the effects of dissipation and decoherence for a general three-level system are of wide interest in quantum optics, molecular physics, and elsewhere. A previous paper presented a technique for solving the time-dependent operator equations involved but under certain restrictive conditions. We now extend our results to a general three-level system with arbitrary time-dependent Hamiltonians and Lindblad operators. Analytical handling of the SU(3) algebra of the eight operators involved leaves behind a set of coupled first-order differential equations for classical functions. Solution of this set gives a complete solution of the quantum problem, without having to invoke rotating-wave or other approximations. Numerical illustrations for multiphoton couplings and quantum control are given.
Four-level systems in quantum optics, and for representing two qubits in quantum computing, are difficult to solve for general time-dependent Hamiltonians. A systematic procedure is presented which combines analytical handling of the algebraic operator aspects with simple solutions of classical, first-order differential equations. In particular, by exploiting $\mathrm{su}(2)\ensuremath{\bigoplus}\mathrm{su}(2)$ and $\mathrm{su}(2)\ensuremath{\bigoplus}\mathrm{su}(2)\ensuremath{\bigoplus}\mathrm{u}(1)$ subalgebras of the full SU(4) dynamical group of the system, the nontrivial part of the final calculation is reduced to a single Riccati (first-order, quadratically nonlinear) equation, itself simply solved. Examples are provided of two-qubit problems from the recent literature, including implementation of two-qubit gates with Josephson junctions.
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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.
This chapter focuses on the time-dependent Schrödinger equation and its solutions for several prototype systems. It provides the basis for discussing and understanding quantum dynamics in condensed phases, however, a full picture can be obtained only by including also dynamical processes that destroy the quantum mechanical phase. Such a full description of quantum dynamics cannot be handled by the Schrödinger equation alone; a more general approach based on the quantum Liouville equation is needed. This important part of the theory of quantum dynamics is discussed in Chapter 10. Given a system characterized by a Hamiltonian Ĥ , the time-dependent Schrödinger equation is For a closed, isolated system Ĥ is time independent; time dependence in the Hamiltonian enters via effect of time-dependent external forces. Here we focus on the earlier case. Equation (1) is a first-order linear differential equation that can be solved as an initial value problem.