A novel method is introduced in order to treat the dissipative dynamics of quantum systems interacting with a bath of classical degrees of freedom. The method is based upon an extension of the Nosè–Hoover chain (constant temperature) dynamics to quantum-classical systems. Both adiabatic and nonadiabatic numerical calculations on the relaxation dynamics of the spin-boson model show that the quantum-classical Nosè–Hoover chain dynamics represents the thermal noise of the bath in an accurate and simple way. Numerical comparisons, both with the constant-energy calculation and with the quantum-classical Brownian motion treatment of the bath, show that the quantum-classical Nosè–Hoover chain dynamics can be used to introduce dissipation in the evolution of a quantum subsystem even with just one degree of freedom for the bath. The algorithm can be computationally advantageous in modelling, within computer simulation, the dynamics of a quantum subsystem interacting with complex molecular environments.
A novel method is introduced in order to treat the dissipative dynamics of quantum systems interacting with a bath of classical degrees of freedom. The method is based upon an extension of the Nosè–Hoover chain (constant temperature) dynamics to quantum-classical systems. Both adiabatic and nonadiabatic numerical calculations on the relaxation dynamics of the spin-boson model show that the quantum-classical Nosè–Hoover chain dynamics represents the thermal noise of the bath in an accurate and simple way. Numerical comparisons, both with the constant-energy calculation and with the quantum-classical Brownian motion treatment of the bath, show that the quantum-classical Nosè–Hoover chain dynamics can be used to introduce dissipation in the evolution of a quantum subsystem even with just one degree of freedom for the bath. The algorithm can be computationally advantageous in modelling, within computer simulation, the dynamics of a quantum subsystem interacting with complex molecular environments.
When modelling real physical systems, one should always consider the influence of the environment.The study of open quantum systems aims precisely at the assessment of the role of the environment in the evolution of these systems.The computational study of these systems is in general a formidable task.However, in many instances one can approximate the environment by means of classical dynamics.For such cases, when the environment is in a thermal state, one can generalize non-Hamiltonian equations of motion to quantum-classical dynamics in order to control the thermodynamic temperature.Such a generalization is achieved within the partial Wigner representation of quantum mechanics, and leads to a quantum-classical evolution of the system.We have studied how to simulate a thermal bath, by means of the least possible number of additional extended variables [1].This has been achieved upon reformulating the Nosé-Hoover Power (NHP) thermostat in quantum-classical theory.When applied to the dissipative evolution of a quantum spin the NHP thermostat, obtained numerical results in agreement with those obtained using Nosè-Hoover chains.However, since a fewer number of variables are needed to achieve the correct sampling of the canonical distribution at equilibrium, the NHP thermostat promises to be better suited for the simulation of low dimensional open quantum system on discrete grids.Following this the quantum dynamics of a XXZ two-spin chain interacting with multiple bosonic baths have been studied, within the mixed Wigner-Heisenberg representation of quantum mechanics.In particular, we have simulated the dynamics of the reduced density matrix ρ s (t), along with the time evolution of the quantum entanglement of the two spins via the concurrence C (ρ).
This manuscript aims to illustrate a quantum-classical dissipative theory (suited to be converted to effective algorithms for numerical simulations) within the long-term project of studying molecular processes in the brain. Other approaches, briefly sketched in the text, have advocated the need to deal with both quantum and classical dynamic variables when studying the brain. At variance with these other frameworks, the manuscript’s formalism allows us to explicitly treat the classical dynamical variables. The theory must be dissipative not because of formal requirements but because brain processes appear to be dissipative at the molecular, physiological, and high functional levels. We discuss theoretically that using Brownian dynamics or the Nosè-Hoover-Chain thermostat to perform computer simulations provides an effective way to introduce an arrow of time for open quantum systems in a classical environment. In the future, We plan to study classical models of neurons and astrocytes, as well as their networks, coupled to quantum dynamical variables describing, e.g., nuclear and electron spins, HOMO and LUMO orbitals of phenyl and indole rings, ion channels, and tunneling protons.
We study a new model for treating quantum dissipative systems, in which the bath is modeled as a collection of spins coupled to the system of interest. We develop a quasiclassical method to study this model, approximating the quantum Heisenberg equations by the classical ones, supplemented with stochastic initial conditions carefully chosen so that the results obtained from the classical equations are as close as possible to the quantum results. Using this method we compare the dynamics of such a spin–spin system with that of a spin–boson system, in which the bath is modeled as a collection of harmonic oscillators. We verify numerically that when the system-bath coupling is spread over many bath spins (the Brownian motion limit), the spin–spin model can be mapped on the spin–boson model (although with a temperature dependent spectral density). We also demonstrate that the two dissipative models are qualitatively very different in a non-Brownian motion regime.
Many open quantum systems encountered in both natural and synthetic situations are embedded in classical-like baths. Often, the bath degrees of freedom may be represented in terms of canonically conjugate coordinates, but in some cases they may require a non-canonical or non-Hamiltonian representation. Herein, we review an approach to the dynamics and statistical mechanics of quantum subsystems embedded in either non-canonical or non-Hamiltonian classical-like baths which is based on operator-valued quasi-probability functions. These functions typically evolve through the action of quasi-Lie brackets and their associated Quantum-Classical Liouville Equations, or through quasi-Lie brackets augmented by dissipative terms. Quasi-Lie brackets possess the unique feature that, while conserving the energy (which the Noether theorem links to time-translation symmetry), they violate the time-translation symmetry of their algebra. This fact can be heuristically understood in terms of the dynamics of the open quantum subsystem. We then describe an example in which a quantum subsystem is embedded in a bath of classical spins, which are described by non-canonical coordinates. In this case, it has been shown that an off-diagonal open-bath geometric phase enters into the propagation of the quantum-classical dynamics. Next, we discuss how non-Hamiltonian dynamics may be employed to generate the constant-temperature evolution of phase space degrees of freedom coupled to the quantum subsystem. Constant-temperature dynamics may be generated by either a classical Langevin stochastic process or a Nosé–Hoover deterministic thermostat. These two approaches are not equivalent but have different advantages and drawbacks. In all cases, the calculation of the operator-valued quasi-probability function allows one to compute time-dependent statistical averages of observables. This may be accomplished in practice using a hybrid Molecular Dynamics/Monte Carlo algorithms, which we outline herein.
The method of Nosé and Hoover1,2 to create canonically distributed positions and momenta in classical molecular dynamics simulations is frequently used. Hamilton's equations of motion are supplemented by time-dependent pseudofriction terms that convert the microcanonical isoenergetic time evolution into a canonical isothermal time evolution, thus permitting the calculation of canonical ensemble averages by time averaging. We show that for one quantum particle in an external harmonic oscillator, the equations of motion in terms of coherent states can easily be modified in an analogous manner to mimic the coupling of the system to a thermal bath and create a quantum canonical ensemble.3 The method is generalised to a system of two identical quantum particles. In the resulting equations of motion, one obtains an additional attractive term for bosons and a repulsive term for fermions in the dynamics of the pseudofriction coefficients, leading to a correctly sampled thermal weight.
In this dissertation the formulation of various integration algorithms is studied, with a view to simulate quantum-classical systems in contact with a thermal bath. In particular focus is given to the constant temperature dynamics of the Nos e-Hoover, Nos e-Hoover Chain and Nos e-Hoover Power thermostat schemes. Through the use of the time symmetric Trotter factorisation of the Liouville operator, algorithms are derived that are both time-reversible and measure-preserving. The efficiency of these algorithms is tested via the constant temperature simulation of a low-dimensional harmonic system. In addition The Nos e-Hoover Power thermostat was then extended to the quantum-classical case. The damping of a tunnelling spin coupled to a thermalised harmonic mode was simulated and the results are presented.