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arXiv:1309.1130 · Journal of Modern Optics

Evolution of anN-level system via automated vectorization of the Liouville equations and application to optically controlled polarization rotation

M. S. Shahriar, Ye Wang, Subramanian Krishnamurthy, Yongming Tu, G. S. Pati, S. Tseng

Published Jan 27, 2014• 41 citations
arXivHTMLPDF

Abstract

The Liouville equation governing the evolution of the density matrix for an atomic/molecular system is expressed in terms of a commutator between the density matrix and the Hamiltonian, along with terms that account for decay and redistribution. To find solutions of this equation, it is convenient first to reformulate the Liouville equation by defining a vector corresponding to the elements of the density operator, and determining the corresponding time-evolution matrix. For a system of N energy levels, the size of the evolution matrix is N2 × N2. When N is very large, evaluating the elements of these matrices becomes very cumbersome. We describe a novel algorithm that can produce the evolution matrix in an automated fashion for an arbitrary value of N. As a non-trivial example, we apply this algorithm to a 15-level atomic system used for producing optically controlled polarization rotation. We also point out how such a code can be extended for use in an atomic system with arbitrary number of energy levels.

DOI
10.1080/09500340.2013.865806

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The Liouville equation governing the evolution of the density matrix for an atomic/molecular system is expressed in terms of a commutator between the density matrix and the Hamiltonian, along with terms that account for decay and redistribution. To find solutions of this equation, it is convenient first to reformulate the Liouville equation by defining a vector corresponding to the elements of the density operator, and determining the corresponding time-evolution matrix. For a system of N energy levels, the size of the evolution matrix is N2 × N2. When N is very large, evaluating the elements of these matrices becomes very cumbersome. We describe a novel algorithm that can produce the evolution matrix in an automated fashion for an arbitrary value of N. As a non-trivial example, we apply this algorithm to a 15-level atomic system used for producing optically controlled polarization rotation. We also point out how such a code can be extended for use in an atomic system with arbitrary number of energy levels.
Abstract Based oh the properties of Lie algebras, in this work we develop a general framework to linearize the von Neumann equation rendering it in a suitable form for quantum simulations. Departing from the conventional method of expanding the density matrix in the Liouville space formed by matrices unit we express the von Neumann equation in terms of Pauli strings. This provides several advantages related to the quantum tomography of the density matrix and the formulation of the unitary gates that generate the time evolution. The use of Pauli strings facilitates the quantum tomography of the density matrix whose elements are purely real. As for any other basis of Hermitian matrices, this eliminates the need to calculate the phase of the complex entries of the density matrix. This approach also enables to express the evolution operator as a sequence of commuting Hamiltonian gates of Pauli strings that can readily be synthetized using Clifford gates. Additionally, the fact that these gates commute with each other along with the unique properties of the algebra formed by Pauli strings allows to avoid the use of Trotterization hence considerably reducing the circuit depth. The algorithm is demonstrated for three Hamiltonians using the IBM noisy quantum circuit simulator.
The arguments of Chapter 2, as well as of those of subsequent chapters, have as their foundation the formalism based on the state vector of a system of which the evolution is described by the Schrödinger equation. In fact, such an approach is badly suited to the case in which the coupling between an atom and its environment (for example through collisions with other atoms or spontaneous emission into formerly empty modes of the electromagnetic field) cannot be neglected. If the correlations induced by these interactions between the atom and its environment do not concern us and we are only interested in the evolution of the atom, the formalism of the density matrix must be employed. This provides a description at all times of the state of the atom, although a state vector for the atom alone cannot be defined. In this formalism the effect of the environment on the atom is accounted for by the introduction of suitable relaxation terms (Section 2C.1) in the equation of evolution of the density matrix. An important application of the density matrix is to the case of a two-level atomic system for which the relaxation terms lead to its deexcitation to a level of lower energy. We shall show that in this case the density matrix can be represented by a vector, known as the Bloch vector , which will allow us to give simple geometrical pictures of the evolution of the system.
An adiabatic approximation for the Liouville density-matrix equation which includes decay terms is developed. The adiabatic approximation employs the eigenvectors of the non-normal Liouville operator. The approximation is valid when there exists a complete set of eigenvectors of the non-normal Liouville operator (i.e., the eigenvectors span the density-matrix space), the time rate of change of the Liouville operator is small, and an auxiliary matrix is nonsingular. Numerical examples are presented involving efficient population transfer in a molecule by stimulated Raman scattering, with the intermediate level of the molecule decaying on a time scale that is fast compared with the pulse durations of the pump and Stokes fields. The adiabatic density-matrix approximation can be simply used to determine the density matrix for atomic or molecular systems interacting with cw electromagnetic fields when spontaneous emission or other decay mechanisms prevail.
The starting point of the classical description of motion is the Newton equations that yield a phase space trajectory (rN (t), pN (t)) for a given initial condition (rN (0), pN (0)). Alternatively one may describe classical motion in the framework of the Liouville equation (Section (1.2.2)) that describes the time evolution of the phase space probability density f (rN , pN ; t). For a closed system fully described in terms of a well specified initial condition, the two descriptions are completely equivalent. Probabilistic treatment becomes essential in reduced descriptions that focus on parts of an overall system, as was demonstrated in Sections 5.1–5.3 for equilibrium systems, and in Chapters 7 and 8 that focus on the time evolution of classical systems that interact with their thermal environments. This chapter deals with the analogous quantum mechanical problem. Within the limitations imposed by its nature as expressed, for example, by Heisenbergtype uncertainty principles, the Schrödinger equation is deterministic. Obviously it describes a deterministic evolution of the quantum mechanical wavefunction. The analog of the phase space probability density f (rN , pN ; t) is now the quantum mechanical density operator (often referred to as the “density matrix”), whose time evolution is determined by the quantum Liouville equation. Again, when the system is fully described in terms of a well specified initial wavefunction, the two descriptions are equivalent. The density operator formalism can, however, be carried over to situations where the initial state of the system is not well characterized and/or a reduced description of part of the overall system is desired. Such situations are considered later in this chapter.
In many cases, measurements are performed on “mixed” ensembles of realizations of a system, which cannot be associated with a single vector in its Hilbert space. In these cases the state of the system is represented by a proper “density operator.” It is instructive to associate density operators with state vectors in a vector space, termed Liouville’s space, where the Schrödinger equation is reformulated as the Liouville–Von Neumann equation. An important consequence of this equation is that the density operator of a system at equilibrium must commute with its Hamiltonian. Namely, the matrix representation of the density operator in the basis of Hamiltonian eigenstates is diagonal, where the diagonal elements are the relative populations of the system’s Hamiltonian eigenstates. The equilibrium populations are revealed by maximizing the (Von Neumann) entropy, subject to given constraints. We derive the equilibrium density operator explicitly for the cases of canonical and grand canonical ensembles.
Abstract The n‐level quantum system density‐operators evolution is not usually given by the Liouville equation, but by a more general one. This equation must keep density‐operators trace, hermiticity, and positiveness. These three conditions restrict the available kinds of evolution equations. In this paper we investigate linear equations for systems without memory effects. By using the first two conditions and the formalism we introduced in earlier papers, the evolution equation takes the form of a first order differential equation concerning the n2 – 1 dimension “coherence‐vector.” The third one is the essential object of this paper. Moreover, we obtain a cononical splitting of this equation into four parts that may be separately studied.