Atti della Accademia Peloritana dei Pericolanti. Classe di Scienze Fisiche, Matematiche e Naturali/Atti della Accademia Peloritana dei Pericolanti, Classe di Scienze Fisiche, Matematiche e Naturali
Sampling the time evolution of mixed quantum-classical systems
Quantum mechanics is not logically closed with respect to the classical world.Its formalism unfolds as the quantization of a sub-set of classical Hamiltonians.The interpretation of quantum theory in terms of the measurement process inevitably requires to deal with systems composed by a mixture of both classical and quantum degrees of freedom.Moreover, when energy can flow between the quantum and classical degrees of freedom (i.e., in the case of nonadiabatic dynamics), there are more theoretical difficulties in order to obtain a fully consistent quantum-classical formalism.In order to perform calculations, one can renounce to the usual Lie algebraic structure of well-established physical theories, adopt non-Hamiltonian brackets, and obtain a formalism for the dynamics and statistics of quantum-classical systems that has an affordable computational complexity.Recent progress in the algorithms for the sampling of nonadiabatic dynamics of quantum-classical systems at long time is reviewed here.
Quantum mechanics is not logically closed with respect to the classical world.Its formalism unfolds as the quantization of a sub-set of classical Hamiltonians.The interpretation of quantum theory in terms of the measurement process inevitably requires to deal with systems composed by a mixture of both classical and quantum degrees of freedom.Moreover, when energy can flow between the quantum and classical degrees of freedom (i.e., in the case of nonadiabatic dynamics), there are more theoretical difficulties in order to obtain a fully consistent quantum-classical formalism.In order to perform calculations, one can renounce to the usual Lie algebraic structure of well-established physical theories, adopt non-Hamiltonian brackets, and obtain a formalism for the dynamics and statistics of quantum-classical systems that has an affordable computational complexity.Recent progress in the algorithms for the sampling of nonadiabatic dynamics of quantum-classical systems at long time is reviewed here.
The simulation of the dynamics of quantum systems is very difficult, due to the fact that, in general, it cannot be calculated exactly for interacting many-body systems. Brute force simulations of quantum dynamics are simply not feasible, and approximations need to be made. In many instances a quantum system can be approximated as a quantum-classical system, where only a subsystem of interest is treated quantum mechanically, and the rest is considered as a classical bath. When energy is free to be exchanged between the subsystem and its environment, the dynamics that occur is said to be nonadiabatic. This type of dynamics is challenging to calculate on a computer, as it can lead to large statistical errors at long times. Hence, there is a need for improved algorithms for nonadiabatic dynamics. In this thesis, a recently introduced nonadiabatic sampling scheme [A. Sergi and F. Petruccione, Phys. Rev. E 81, 032101 (2010)] is used to calculate the long-time dynamics of a model system comprising a quantum spin coupled to a bath of harmonic oscillators. Also, various technical aspects of the algorithm are investigated.
Simulation of quantum dynamics for many-body systems is an open area of research. For interacting many-body quantum systems, the computer memory necessary to perform calculations has an astronomical value, so that approximated models are needed to reduce the required computational resources. A useful approximation that can often be made is that of quantum-classical dynamics, where the majority of the degrees are treated classically, while a few of them must be treated quantum mechanically. When energy is exchanged very quickly between the quantum subsystem and classical environment, the dynamics is nonadiabatic. Most theories for nonadiabatic dynamics are unsatisfactory, as they fail to properly describe the quantum backreaction of the subsystem on the environment. However, an approach based on the quantum-classical Liouville equation solves this problem. Even so, nonadiabatic dynamics is difficult to implement on a computer, and longer simulation times are often inaccessible due to statistical error. There is thus a need for improved algorithms for nonadiabatic dynamics. In this thesis, two algorithms that utilise the quantum-classical Liouville equation will be qualitatively and quantitatively compared. In addition, stochastic sampling schemes for nonadiabatic transitions will be studied, and a new sampling scheme is introduced [D. A. Uken et al., Phys. Rev. E. 88, 033301 (2013)] which proves to have a dramatic advantage over existing techniques, allowing far longer simulation times to be calculated reliably.
Quantum-classical mixing is studied by a group-theoretical approach, and a quantum-classical equation of motion is derived. The quantum-classical bracket entering the equation preserves the Lie algebra structure of quantum and classical mechanics, and, therefore, leads to a natural description of interaction between quantum and classical degrees of freedom. The exact formalism is applied to coupled quantum and classical oscillators. Various approximations, such as the mean-field and the multiconfiguration mean-field approaches, which are of great utility in studying realistic multidimensional systems, are derived. Based on the formulation, a natural classification of the previously suggested quantum-classical equations of motion arises, and several problems from earlier works are resolved.
Mixed quantum-classical dynamics is a set of methods often used to understand systems too complex to treat fully quantum mechanically. Many techniques exist for full quantum mechanical evolution on quantum computers, but mixed quantum-classical dynamics are less explored. We present a modular algorithm for general mixed quantum-classical dynamics where the quantum subsystem is coupled with the classical subsystem. We test it on a modified Shin-Metiu model in the first quantization through Ehrenfest propagation. We find that the Time-Dependent Variational Time Propagation algorithm performs well for short-time evolutions and retains qualitative results for longer-time evolutions.
It is shown that application of the time-dependent variational principle to trial wave functions of a particular form results in a description of the quantum system as a classical system consisting of two mutually interacting parts: the ordinary classical analog of the quantum system and a (generally finite) number of additional degrees of freedom. This holds out the prospect of simulating quantum dynamics by coupling the corresponding classical system to additional degrees of freedom, and it is pointed out that such calculations have, in effect, already been performed. Since the trial wave functions that qualify for this treatment may be arbitrarily accurate, there is no limitation to the accuracy of the resulting classical description. In a particular case, the canonical coordinates and momenta for all degrees of freedom of the overall classical system are identified and investigated. In general, certain parameters appearing in the trial wave functions treated in this work may become redundant at certain times, leading to singularities in the Hamiltonian equations of motion. It is shown, however, that in certain such cases, a classical formalism for the quantum dynamics remains valid if the Hamiltonian function is suitably modified. Calculations applying the present Hamiltonian formalism are carried out that demonstrate that it has better convergence properties than an alternative, nonvariational, Hamiltonian technique suggested by previous work. These numerical studies also suggest that the observed spreading of wave packets and development of complicated structure in wave functions may be interpreted classically in terms of energy transfer among the various degrees of freedom in the overall classical system.
Abstract Mixed quantum-classical dynamics is a set of methods often used to understand systems too complex to treat fully quantum mechanically. Many techniques exist for full quantum mechanical evolution on quantum computers, but mixed quantum-classical dynamics are less explored. We present a modular algorithm for general mixed quantum-classical dynamics where the quantum subsystem is coupled with the classical subsystem. We test it on a modified Shin-Metiu model in the first quantization through Ehrenfest propagation. We find that the Time-Dependent Variational Time Propagation algorithm performs well for short-time evolutions and retains qualitative results for longer-time evolutions.