Solution of the time-dependent Liouville-von Neumann equation: dissipative evolution (original) (raw)
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Time-dependent solution of the Liouville-von Neumann equation: non-dissipative evolution
Computer Physics Communications, 1991
A mathematical and numerical framework has been worked out to represent the density operator in phase space and to propagate it in time under dissipative conditions. The representation of the density operator is based on the Fourier pseudospectral method which allows a description both in configuration as well as in momentum space. A new propagation scheme which treats the complex eigenvalue structure of the dissipative ate modern computer architecture such as parallelism and vectorization. Comparing the results to closed-form solutions exponentially fast convergence characteristics in phase space as well as in the time propagation is demonstrated. As an example of its usefulness, thenew method has beensuccessfully applied todissipationunderthe constraint ofselection rules. Mare specifically, a harmonic oscillator which relaxes to equilibrium under the constraint of second-order coupling to the bath was studied. The results of the calculation were compared to a mean field approximation developed for this problem. It has been found that this approximation does not capture the essence of the relaxation process. In conclusion, the new method presented is a conceptual tool to model multi-dimensional quantum physical systems which exhibit both relaxation as well as oscillation in an efficient, accurate and convenient manner. ~ ._" n~LI:.L:_. r .A I**' 1ur r " m t >~o~r L~ L L Y
Population equations for quantum systems in contact with dissipation mechanisms
Physical Review A
We discuss the construction of population equations for driven quantum systems in contact with dissipation mechanisms in the limit where the strength of the driving force is suAiciently weak that a suitable Born expansion can be carried out in powers of the coupling constant of the coherent interaction. The Zwanzig projector technique and the application of an appropriate eigenfunction-expansion method due to Weidlich lead to an elegant derivation of population equations. If the decay rates of the irreversible processes allow the application of the Markoff approximation, ordinary first-order differential equations for the level populations can be derived. The transition rates are constructed explicitly in terms of the coherent Liouville operator and the Weidlich eigenfunctions.
Quantum simulation of dissipation for Maxwell equations in dispersive media
arXiv (Cornell University), 2023
The dissipative character of an electromagnetic medium breaks the unitary evolution structure that is present in lossless, dispersive optical media. In dispersive media, dissipation appears in the Schrödinger representation of classical Maxwell equations as a sparse diagonal operator occupying an r-dimensional subspace. A first order Suzuki-Trotter approximation for the evolution operator enables us to isolate the non-unitary operators (associated with dissipation) from the unitary operators (associated with lossless media). The unitary operators can be implemented through qubit lattice algorithm (QLA) on n qubits, based on the discretization and the dimensionality of the pertinent fields. However, the non-unitary-dissipative part poses a challenge both physically and computationally on how it should be implemented on a quantum computer. In this paper, two probabilistic dilation algorithms are considered for handling the dissipative operators. The first algorithm is based on treating the classical dissipation as a linear amplitude damping-type completely positive trace preserving (CPTP) quantum channel where an unspecified environment interacts with the system of interest and produces the non-unitary evolution. Therefore, the combined system-environment is now closed, and must undergo unitary evolution in the dilated space. The unspecified environment can be modeled by just one ancillary qubit, resulting in an implementation scaling of O(2 n−1 n 2) elementary gates for the total system-environment unitary evolution operator. The second algorithm approximates the non-unitary operators by the Linear Combination of Unitaries (LCU). On exploiting the diagonal structure of the dissipation, we obtain an optimized representation of the non-unitary part, which requires O(2 n) elementary gates. Applying the LCU method for a simple dielectric medium with homogeneous dissipation rate, the implementation scaling can be further reduced into O[poly(n)] basic gates. For the particular case of weak dissipation we show that our proposed postselective dilation algorithms can efficiently delve into the transient evolution dynamics of dissipative systems by calculating the respective implementation circuit depth. A connection of our results with the non-linear-in-normalization-only (NINO) quantum channels is also presented.
Dissipation in a quantum-mechanical system
Optical Materials, 2008
We consider a model of a dissipative quantum-mechanical system consisting of weakly coupled quantum and classical subsystems. The classical subsystem is assumed to be infinite, and thus serves as a means to transfer the energy of the quantum subsystem to the infinity (actually, to dissipate the energy). The quantum-classical coupling is treated in the spirit of the mean-field approximation. Solving the equations for the classical subsystem explicitly an effective dissipative Schrö dinger equation for the quantum subsystem is obtained. The proposed method is illustrated by calculating the shape of the nonlinear resonance.
The harmonic oscillator with dissipation within the theory of open quantum systems
1994
Time evolution of the expectation values of various dynamical operators of the harmonic oscillator with dissipation is analitically obtained within the framework of the Lindblad theory for open quantum systems. We deduce the density matrix of the damped harmonic oscillator from the solution of the Fokker-Planck equation for the coherent state representation, obtained from the master equation for the density operator. The Fokker-Planck equation for the Wigner distribution function, subject to either the Gaussian type or the δ-function type of initial conditions, is also solved by using the Wang-Uhlenbeck method. The obtained Wigner functions are two-dimensional Gaussians with different widths.
A minimal coupling method for dissipative quantum systems
2005
Quantum dynamics of a general dissipative system investigated by its coupling to a Klein-Gordon type field as the environment by introducing a minimal coupling method. As an example, the quantum dynamics of a damped three dimensional harmonic oscillator investigated and some transition probabilities indicating the way energy flows between the subsystems obtained. The quantum dynamics of a dissipative two level
International Journal of Quantum Chemistry, 1996
m Ultrafast dissipative dynamics of vibrational degrees of freedom in molecular systems in the condensed phase are studied here. Assuming that the total system is separable into a relevant part and a reservoir, the dynamics of the relevant part can be described by means of a reduced statistical density operator. For a weak or intermediate coupling between the relevant part and the reservoir, it is possible to derive a second-order master equation for this operator. Using a representation of the reduced statistical operator in an appropriate molecular basis set, vibrational dynamics in a variety of potential energy surfaces can be studied. In the numerical calculations, we focus on the dissipative dynamics under the influence of external laser fields. In the first example, vibrational wave-packet dynamics and time-resolved pump-probe spectroscopy of molecular systems with nonadiabatically coupled excited-state potential energy surfaces is presented. In the second part, we show how an intense laser field modifies the wavepacket motion onto two radiatively coupled potential energy surfaces. Finally, the controlled preparation of definite vibrational states in a triatomic molecule with infrared laser pulses is considered taking relaxation and dephasing processes into account. 0 1996 John Wiley & Sons, Inc.
Dynamics of complex quantum systems: dissipation and kinetic equations
Physica E: Low-dimensional Systems and Nanostructures, 2001
We present a microscopic approach to quantum dissipation and sketch the derivation of the kinetic equation describing the evolution of a simple quantum system in interaction with a complex quantum system. A typical quantum complex system is modeled by means of parametric banded random matrices coupled to the subsystem of interest. We do not assume the weak coupling limit and allow for an independent dynamics of the "reservoir". We discuss the reasons for having a new theoretical approach and the new elements introduced by us. The present approach incorporates known limits and previous results, but at the same time includes new cases, previously never derived on a microscopic level. We briefly discuss the kinetic equation and its solution for a particle in the absence of an external field.
The Journal of Chemical Physics, 2011
An approach for treating dissipative, non-adiabatic quantum dynamics in general model systems at finite temperature based on linearizing the density matrix evolution in the forward-backward path difference for the environment degrees of freedom is presented. We demonstrate that the approach can capture both short time coherent quantum dynamics and long time thermal equilibration in an application to excitation energy transfer in a model photosynthetic light harvesting complex. Results are also presented for some nonadiabatic scattering models which indicate that, even though the method is based on a "mean trajectory" like scheme, it can accurately capture electronic population branching through multiple avoided crossing regions and that the approach offers a robust and reliable way to treat quantum dynamical phenomena in a wide range of condensed phase applications.
Quantum Liouville-space trajectories for dissipative systems
2001
In this paper we present a new quantum-trajectory based treatment of quantum dynamics suitable for dissipative systems. Starting from a de Broglie/Bohm-like representation of the quantum density matrix, we derive and define quantum equations-of-motion for Liouville-space trajectories for a generalized system coupled to a dissipative environment. Our theory includes a vector potential which mixes forward and backwards propagating components and non-local quantum potential which continuously produces coherences in the system. These trajectories are then used to propagate an adaptive Lagrangian grid which carries the density matrix, ρ(x, y), and the action, A(x, y), thereby providing a complete hydrodynamic-like description of the dynamics.