Alessandro Sergi

Quantum-classical dynamics of nonadiabatic chemical reactions

A reactive flux correlation function formalism for the calculation of rate constants for mixed quantum-classical systems undergoing nonadiabatic dynamics is presented. The linear response formalism accounts for the stationarity of the equilibrium density under quantum-classical dynamics and expresses the rate constant in terms of an ensemble of surface-hopping trajectories. Calculations are carried out on a model two-level system coupled to a nonlinear oscillator which is in turn coupled to a harmonic heat bath. Relevant microscopic species variables for this system include two stable states, corresponding to the ground state adiabatic surface, as well as another species corresponding to the excited state surface. The time-dependent rate constants for the model are evaluated in the adiabatic limit, where the dynamics is confined to the ground Born–Oppenheimer surface, and these results are compared with calculations that account for nonadiabatic transitions among the system states.

Non-Hermitian quantum dynamics of a two-level system and models of dissipative environments

We consider a non-Hermitian Hamiltonian in order to effectively describe a two-level system (TLS) coupled to a generic dissipative environment. The total Hamiltonian of the model is obtained by adding a general anti-Hermitian part, depending on four parameters, to the Hermitian Hamiltonian of a tunneling TLS. The time evolution is formulated and derived in terms of the normalized density operator of the model, different types of decays are revealed and analyzed. In particular, the population difference and coherence are defined and calculated analytically. We have been able to mimic various physical situations with different properties, such as dephasing, vanishing population difference and purification.

Quantum-classical limit of quantum correlation functions.

A quantum-classical limit of the canonical equilibrium time correlation function for a quantum system is derived. The quantum-classical limit for the dynamics is obtained for quantum systems comprising a subsystem of light particles in a bath of heavy quantum particles. In this limit the time evolution of operators is determined by a quantum-classical Liouville operator, but the full equilibrium canonical statistical description of the initial condition is retained. The quantum-classical correlation function expressions derived here provide a way to simulate the transport properties of quantum systems using quantum-classical surface-hopping dynamics combined with sampling schemes for the quantum equilibrium structure of both the subsystem of interest and its environment

Non-Hamiltonian commutators in quantum mechanics.

The symplectic structure of quantum commutators is first unveiled and then exploited to describe generalized non-Hamiltonian brackets in quantum mechanics. It is easily recognized that quantum-classical systems are described by a particular realization of such a bracket. In light of previous work, this paper explains a unified approach to classical and quantum-classical non-Hamiltonian dynamics. In order to illustrate the use of non-Hamiltonian commutators, it is shown how to define thermodynamic constraints in quantum-classical systems. In particular, quantum-classical Nosé-Hoover equations of motion and the associated stationary density matrix are derived. The non-Hamiltonian commutators for both Nosé-Hoover chains and Nosé-Andersen (constant-pressure, constant-temperature) dynamics are also given. Perspectives of the formalism are discussed.

Numerical and analytical approach to the quantum dynamics of two coupled spins in bosonic baths

The quantum dynamics of a spin chain interacting with multiple bosonic baths is described in a mixed Wigner-Heisenberg representation. The formalism is illustrated by simulating the time evolution of the reduced density matrix of two coupled spins, where each spin is also coupled to its own bath of harmonic oscillators. In order to prove the validity of the approach, an analytical solution in the Born-Markov approximation is found. The agreement between the two methods is shown.

Nonadiabatic reaction rates for dissipative quantum-classical systems

The dynamics of a quantum system which is directly coupled to classical degrees of freedom is investigated. The classical degrees of freedom are in turn coupled to a classical bath whose detailed dynamics is not of interest. The resulting quantum-classical evolution equations are dissipative as a result of coupling to the classical heat bath. The dissipative quantum-classical dynamics is used to study nonadiabatic chemical reactions and compute their rates. The reactive flux correlation formalism for the calculation of nonadiabatic rate constants is generalized to dissipative quantum-classical dynamics and implemented in terms of averages over surface-hopping Langevin trajectory segments. The results are illustrated for a simple quantum-classical two-state model. The techniques developed in this paper can be applied to complex classical environments encountered, for example, in proton and electron transfer processes in the condensed phase where local environmental degrees of freedom must be treated explicitly but the remainder of the environment can be treated simply as a heat bath.

Comparison and unification of non-Hermitian and Lindblad approaches with applications to open quantum optical systems

We compare two approaches to open quantum systems, namely, the non-Hermitian dynamics and the Lindblad master equation. In order to deal with more general dissipative phenomena, we propose the unified master equation that combines the characteristics of both of these approaches. This allows us to assess the differences between them as well as to clarify which observed features come from the Lindblad or the non-Hermitian part, when it comes to experiment. Using a generic two-mode single-atom laser system as a practical example, we analytically solve the dynamics of the normalized density matrix operator. We study the two-level model in a number of cases (depending on parameters and types of dynamics), compute different observables and study their physical properties. It turns out that one can not only able to describe the different types of damping in dissipative quantum optical systems but also mimic the undamped anharmonic oscillatory phenomena which happen in quantum systems with more than two levels (while staying within the framework of the analytically simple two-mode approximation).

On the geometry and entropy of non-Hamiltonian phase space

We analyse the equilibrium statistical mechanics of canonical, non-canonical and non-Hamiltonian equations of motion, throwing light on the peculiar geometric structure of phase space. Some fundamental issues regarding time translation and phase space measure are clarified. In particular, we emphasize that a phase space measure should be defined by means of the Jacobian of the transformation between different kinds of coordinates since such a determinant is different from zero in the non-canonical case even if the phase space compressibility is null. Instead, the Jacobian determinant associated with phase space flows is unity whenever non-canonical coordinates lead to a vanishing compressibility, so its use for defining a measure may not always be correct. To better illustrate this point, we derive a mathematical condition for defining non-Hamiltonian phase space flows with zero compressibility. The Jacobian determinant associated with the time evolution in phase space is very useful for analysing time translation invariance. The proper definition of a phase space measure is particularly important when defining the entropy functional in the canonical, non-canonical, and non-Hamiltonian cases. We show how the use of relative entropies can circumvent some subtle problems that are encountered when dealing with continuous probability distributions and phase space measures. Finally, a maximum (relative) entropy principle is formulated for non-canonical and non-Hamiltonian phase space flows.

Deterministic constant-temperature dynamics for dissipative quantum systems

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 Nose–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 Nose–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 Nose–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.

Reversible integrators for basic extended system molecular dynamics

Starting from a family of equations of motion for the dynamics of extended systems whose trajectories sample constant pressure and temperature ensemble distributions (Ferrario, M., 1993, in Computer Simulation in Chemical Physics, edited by M. P. Allen and D. J. Tildesley (Dordrecht: Kluwer)), explicit time reversible integration schemes are derived through a straightforward Trotter factorization of the dynamic Liouville propagator, along the lines first described by Tuckerman, M., Martyna, G. J., and Berne, B. J., 1992, J. chem. Phys., 97, 1990. The original Andersens constant-pressure dynamics are recovered in the limit of zero coupling with the Nose thermostat. Reversible integration schemes are derived as a generalization of the velocity Verlet algorithm, with direct handling of the velocity dependent forces in such a way that both predictions and relative iterative corrections are not required. For the sake of clarity both the equations of motion and the Trotter factorization are kept to the basic l...