We show that the time-averaged density matrix which involves the solution of any Liouville equation with a specified initial condition is also the steady-state solution of an appropriately constructed Bloch equation taken in the weak-damping limit. This leads to new explicit expressions for the time-averaged density matrix which, unlike expressions in general use, give a completely analytic relation between the time average and the initial condition and contain no numerically determined eigenvector components. We apply this approach to driven multilevel systems to find new expressions for the time-averaged level populations in a three-level system. A strong initial-condition dependence is exhibited even for an equally spaced resonantly driven system. Analytic forms specifically showing a Stark-shifted multiphoton resonance and the associated line shape are also derived.
We show that the time-averaged density matrix which involves the solution of any Liouville equation with a specified initial condition is also the steady-state solution of an appropriately constructed Bloch equation taken in the weak-damping limit. This leads to new explicit expressions for the time-averaged density matrix which, unlike expressions in general use, give a completely analytic relation between the time average and the initial condition and contain no numerically determined eigenvector components. We apply this approach to driven multilevel systems to find new expressions for the time-averaged level populations in a three-level system. A strong initial-condition dependence is exhibited even for an equally spaced resonantly driven system. Analytic forms specifically showing a Stark-shifted multiphoton resonance and the associated line shape are also derived.
We consider the density matrix for weakly damped quantum oscillators. It is proved that the locally time-averaged density matrix derived from the full density matrix satisfies a closed set of equations, which reduce under specified conditions to population rate equations. Rate coefficients in these equations include contributions from collision-induced transitions and internal collisionless transitions explicitly arranged as sequential processes. We compare solutions of the new rate equations with the full density matrix for a four-level driven system. It is shown that despite some notable success in matching locally time-averaged quantities by extending the Wilcox-Lamb formalism to the weak-damping limit, the new equations are more naturally suited for this limit. In contrast with the Wilcox-Lamb formalism, we show that the new equations give a representation of the local time average whose accuracy is limited only by the degree of time-scale separation between collisions and internal dynamics.
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.
We derive a closed-form analytical expression for the expectation values of a damped Kerr nonlinear oscillator initialized in a coherent state. Starting from the exact Liouville-space solution of the Lindblad master equation, we specialize to coherent-state initial conditions, obtaining explicit time-dependent expressions for arbitrary normally ordered expectation values. The resulting expressions depend only on the system parameters and the initial coherent-state amplitude, requiring neither numerical integration nor evolution in a truncated Fock space. We verify the analytical expressions against master-equation simulations over a range of nonlinear interaction strengths. The closed-form solution provides an efficient alternative to numerical simulation while remaining free of Fock-space truncation error. Finally, we note an extension of the underlying Lie-algebraic structure that may provide a useful starting point for obtaining exact solutions to a broader class of open quantum optical systems.
A Floquet-Liouville supermatrix (FLSM) approach is presented for nonperturbative treatment of the time development of the density-matrix operator of atoms and molecules exposed to intense polychromatic fields. By extending the many-mode Floquet theory (MMFT) recently developed, the time-dependent Liouville equation for the density matrix of quantum systems undergoing relaxations (due to radiative decays and collisional dampings, etc.) can be transformed into an equivalent time-independent non-Hermitian FLSM eigenvalue problem. This yields a numerically stable and computationally efficient approach for the unified treatment of nonresonant and resonant, one- and multiple-photon, steady-state and transient phenomena in nonlinear optical processes, much beyond the conventional rotating-wave-approximation (RWA) method. Connections of the FLSM approach to the MMFT in the limit of zero relaxations are also made, providing the understanding of the physical significance of FLSM supereigenvalues and eigenvectors. In addition to the exact FLSM formalism, we have also presented higher-order perturbative results, based on the extension of the generalized Van Vleck (GVV) nearly degenerate perturbation theory, appropriate for somewhat weaker fields and near-resonant processes, but beyond the RWA limit. The implementation of the GVV method in the time-independent Floquet-Liouvillian allows the reduction of the infinite-dimensional FLSM into a finite-dimensional GVV-Liouville matrix, from which essential analytical results are readily obtained. As an illustration of the usefulness of the new formalism, we extend both the FLSM and the GVV methods to a formal study of the multiphoton-induced resonance fluorescence spectra of two-level systems subject to purely radiation relaxations. Both the time-averaged power spectrum and the time-dependent physical spectrum are exploited in details, and novel new features in intense fields are pointed out.
How does a driven system with many energy levels approach its steady state? Insights are gained by studying a system with three energy levels when the ground state is excited by a laser. The time-dependent occupation probabilities of the three energy levels show students how the system develops in time. The occupation probabilities come from the numerical solution of the Liouville-von Neumann equations for the density operator matrix elements when relaxation is included. A combination of the Interaction Picture, the Rotating Wave Approximation, and the assumption of resonance permit the eigenvalues of the Liouville-von Neumann equations to be found numerically and in closed-form in certain limits. The two methods are complementary and help students understand time-dependent systems. In addition, the eigenvalues allow the short-time and the long-time occupation probabilities to be connected to the relaxation parameters and the magnitude of the laser's electric field. Thus, this model three-level system illuminates how a driven system behaves over time and provides guidance for students studying time-dependent systems.
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.