A “square-root” method for the density matrix and its applications to Lindblad operators

Abstract

The evolution of open systems, subject to both Hamiltonian and dissipative forces, is studied by writing the nm element of the time (t) dependent density matrix in the formThe so called "square root factors", the γ(t)'s, are non-square matrices and are averaged over A systems (α) of the ensemble. This square-root description is exact. Evolution equations are then postulated for the γ(t) factors, such as to reduce to the Lindblad-type evolution equations for the diagonal terms in the density matrix. For the off-diagonal terms they differ from the Lindblad-equations. The "square root factors" γ(t) are not unique and the equations for the γ(t)'s depend on the specific representation chosen. Two criteria can be suggested for fixing the choice of γ(t)'s one is simplicity of the resulting equations and the other has to do with the reduction of the difference between the γ(t) formalism and the Lindblad-equations. When the method is tested on cases which have been previously treated by other methods, our results agree with them. Examples chosen are (i) molecular systems, such that are either periodically driven 1 near level degeneracies, for which we calculate the decoherence occurring in multiple Landau-Zener transition, or else when undergoing descent around conical intersections in the potential surfaces, (ii) formal dissipative systems with Lindblad-type operators representing either a non-Markovian process or a two-state system coupled to bosons.Attractive features of the present factorization method are complete positivity, the no higher than linear increase of the implementation effort with the number of states involved and the introduction of randomness only at the start of the process.

Non-stochastic matrix Schrödinger equation for open systems

“…It is clear, that TDVP equations based on NOSSE definitions [Eqs. (20) and ( 21)] conserve the total energy of an isolated system, whereas the DM counterparts [Eqs. (18) and ( 19)] do not.…”

“…18) and ( 19)] and NOSSE [Eqs. (20) and ( 21)] formalisms. A basis set of 41 Gaussians is used in both cases.…”

“…Another approach is propagation of a square root of the density matrix. 20 This approach was developed only for the Lindblad type of QME, 5 and even within the Lindblad scope the resulting equations of motion are not fully equivalent to the initial QME. Yet another alternative has been suggested through constructing an approximate, so-called surrogate Hamiltonian that contains system and selected environmental DOF.…”

“…However, the propagation is stochastic, and simulations must be converged in the number of stochastic trajectories. Another approach is propagation of a square-root of the density matrix 20 . This approach was developed only for the Lindblad type of QME 5 , and even within the Lindblad scope the resulting equations of motion are not fully equivalent to the initial QME.…”

The wave operator representation of quantum and classical dynamics

“…For example, the recently developed Ensemble Rank Truncation method (ERT) has at its heart a method for representing a Lindbladian evolution of a density in terms of a weighted sum of wavefunctions [40]. The wave operator has also been used for foundational research [41][42][43], but here we extend this to demonstrate that when combined with purification techniques from quantum information, it provides a natural bridge between the Hilbert space representation of quantum dynamics, the phase space Wigner representation, as well as KvN dynamics [44]. Through this, we are able to derive not only a consistent interpretation of mixed states in the Wigner representation, but also establish a connection between the commonly utilised phase space methods of quantum chemistry, and quantum information.…”