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.
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.
Thermostated time evolutions are on a firm ground and widely used in classical molecular dynamics (MD) simulations. 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. However, similar methods for quantum MD schemes are still lacking. Given the rich dynamical behavior of ultracold trapped quantum gases depending on the value of the s-wave scattering length, it is timely to investigate how classical thermostating methods can be combined with powerful approximate quantum dynamics schemes to deal with interacting quantum systems at finite temperature. In this work, the popular method of Nose and Hoover to create canonically distributed positions and momenta in classical MD simulations is generalized to a genuine quantum system of infinite dimensionality. We show that for the quantum harmonic oscillator, the equations of motion in terms of coherent states may be modified in a Nose-Hoover manner to mimic the coupling of the system to a thermal bath and create a quantum canonical ensemble. The method is developed initially for a single particle and then generalized to the case of an arbitrary number of identical quantum particles, involving entangled distribution functions. The resulting isothermal equations of motion for bosons and fermions contain additional terms leading to Bose-attraction and Pauli-blocking, respectively. Questions of ergodicity are discussed for different coupling schemes. In the many-particle case, the superiority of the Nose-Hoover technique to a Langevin approach is demonstrated. In addition, the work contains an investigation of the Grilli-Tosatti thermostating method applied to the harmonic oscillator, and calculations for quantum wavefunctions moving with a time-invariant shape in a harmonic potential.
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.
Quantum rate processes in condensed phase systems are often computed by combining quantum and classical descriptions of the dynamics. An algorithm for simulating the quantum-classical Liouville equation, which describes the dynamics of a quantum subsystem coupled to a classical bath, is presented in this paper. The algorithm is based on a Trotter decomposition of the quantum-classical propagator, in conjunction with Monte Carlo sampling of quantum transitions, to yield a surface-hopping representation of the dynamics. An expression for the nonadiabatic propagator that is responsible for quantum transitions and associated bath momentum changes is derived in a form that is convenient for Monte Carlo sampling and exactly conserves the total energy of the system in individual trajectories. The expectation values of operators or quantum correlation functions can be evaluated by initial sampling of quantum states and use of quantum-classical Liouville dynamics for the time evolution. The algorithm is tested by calculations on the spin-boson model, for which exact quantum results are available, and is shown to reproduce the exact results for stronger nonadiabatic coupling and much longer times using fewer trajectories than other schemes for simulating quantum-classical Liouville dynamics.
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 modelling of quantum heat transfer processes at the nanoscale is crucial for the development of energy harvesting and molecular electronic devices. Herein, we adopt a mixed quantum-classical description of a device, in which the open subsystem of interest is treated quantum mechanically and the surrounding heat baths are treated in a classical-like fashion. By introducing such a mixed quantum-classical description of the composite system, one is able to study the heat transfer between the subsystem and bath from a closed system point of view, thereby avoiding simplifying assumptions related to the bath time scale and subsystem-bath coupling strength. In particular, we adopt the full counting statistics approach to derive a general expression for the moment generating function of heat in systems whose dynamics are described by the quantum-classical Liouville equation (QCLE). From this expression, one can deduce expressions for the dynamics of the average heat and heat current, which may be evaluated using numerical simulations. Due to the approximate nature of the QCLE, we also find that the steady state fluctuation symmetry holds up to order ℏ for systems whose subsystem-bath couplings and baths go beyond bilinear and harmonic, respectively. To demonstrate the approach, we consider the nonequilibrium spin boson model and simulate its time-dependent average heat and heat current under various conditions.
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).