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Physica Scripta

Fast driven quantum systems: interaction picture and boundary conditions

N. P. Tretyakov

Published Sep 7, 2021• 0 citations
Publisher

Abstract

Abstract The work continues systematic applications of the method developed by Di Martino and Facchi (2015) for obtaining, for systems with variable boundary conditions, an equivalent Schrödinger equation with constant boundary conditions. From the resulting equation with rapidly varying terms, an effective equation is then derived for the averaged (slowly varying) wave functions. In this work, we obtain this effective equation for rapidly driven systems in the interaction picture (in first order in reverse frequency of external driving). This is the most suitable picture for systems with external driving. As an example, an extended two-state fast driven system is considered. As a second example, we consider the one-dimensional motion of a particle in a potential box with high-frequency oscillations of its width. Oscillations of width lead to additional terms in the effective equation proportional not only to the second derivative of the potential, but to the first derivative as well. In the case of width oscillations, compared to the oscillations of the box as a whole, the changes are even more significant. In our examples, phenomena such as slow averaged oscillations, resonance, the appearance of volcano potentials are observed, as is often the case in driven systems.

DOI
10.1088/1402-4896/ac2184

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Abstract The work continues systematic applications of the method developed by Di Martino and Facchi (2015) for obtaining, for systems with variable boundary conditions, an equivalent Schrödinger equation with constant boundary conditions. From the resulting equation with rapidly varying terms, an effective equation is then derived for the averaged (slowly varying) wave functions. In this work, we obtain this effective equation for rapidly driven systems in the interaction picture (in first order in reverse frequency of external driving). This is the most suitable picture for systems with external driving. As an example, an extended two-state fast driven system is considered. As a second example, we consider the one-dimensional motion of a particle in a potential box with high-frequency oscillations of its width. Oscillations of width lead to additional terms in the effective equation proportional not only to the second derivative of the potential, but to the first derivative as well. In the case of width oscillations, compared to the oscillations of the box as a whole, the changes are even more significant. In our examples, phenomena such as slow averaged oscillations, resonance, the appearance of volcano potentials are observed, as is often the case in driven systems.
The method of the effective Schrodinger equation applied to the one-dimensional problem of a moving particle in a potential box with a potential function and high-frequency time-dependent boundary conditions is investigated. The method of converting the initial equation written for a system with time-dependent boundary conditions to an equation describing a system in a constant region is briefly presented. The resulting system can be considered as a particle in a potential box, located in the field of some effective potential, different from the initial one. Next we consider a particular case of wall motion, with the subsequent construction of the effective Schrodinger equation. Rapid oscillations of the potential box as a whole (i.e., without changing its width) lead to an additional term in the effective equation proportional to the second derivative of the potential. This, in particular, can lead to the formation of potential wells at potential peaks in which bound states of a particle are possible. In the considered case, it was assumed that the oscillation amplitudes are small in comparison with the characteristic dimensions of the system.
Effective Schrödinger equations, governing the mean movement of a system influenced by fast oscillating external perturbation, are derived. The method represents a direct and straightforward quantum extension of the Kapitza’s approach provided in the Schrödinger picture. First-order terms (over inverse frequency of external driving) in effective Hamiltonians that cannot be eliminated by a unitary transformation, have been derived. Additional terms, in comparison with a standard equation, are similar to those found in the classical case and they may be responsible for a variety of new interesting effects.
A quantum system driven by a high-frequency periodic perturbation is studied. By using the asymptotic expansions in terms of inverse powers of the driving frequency, a class of unitary time-dependent canonical transformations is defined which renders the transformed Hamiltonians time independent. One representative of those Hamiltonians is the quasienergy operator, which is explicitly derived up to the fourth order. The classical limit of the theory and the possibility of separating the mean-motion Hamiltonian is discussed. It is shown that this separation can consistently be carried out to higher orders only in the case of a uniform external force. The application of the method to unperturbed systems with regular confining and singular potentials is discussed by considering the examples of harmonic oscillator, particle in the double-well potential, and particle in the Coulomb potential.
The work presented in this dissertation is centered around the phenomena of correlations between members of a system, and how these correlations lead to unique effects which the system is in a non-equilibrium state. The systems described in this work are divided into two sections: dynamical systems and networks. The work on dynamical systems is focused on how correlated systems can gradually evolve when strongly driven. The work on networks is focused on how percolation is affected by correlations.Dynamical systems focuses on the motion of particles in the presence of an external driving force, specifically focusing on nonlinear effects observable in three different examples of correlated systems which are strongly driven by an external field which is rapidly oscillating in time. The first system is a classical system of coupled oscillators, the second is a quantum system with avoided level crossings and the third is the conduction level of a semiconductor. All three systems are strongly driven and specifically driven far from an initial equilibrium. It is posited that these strongly driven systems will reach quasi-stationary states, where the effects of the external driving can be classified on two time scales, slow evolution and rapid oscillation. The classical system is studied as a classical field theory, the effects of the strong driving force is observed in the qualitative properties of the system's effective potential. In the presence of a strong driving force this effective potential will exhibit points whose stability will change depending on the parameters of the driving force. In the quantum systems the effects of the external driving are seen in the Floquet quasienergy spectrum. In the third system, the strong driving force results in a renormalization of the bandwidth. If the driving force is strong enough the band can be completely inverted. Again working in the Floquet representation, the inverted band will exhibit multiplication of the driving frequency, specifically the third harmonic will become the dominant frequency and the fundamental frequency will be suppressed. Correlations in networks have many different aspects, correlations can affect how a network is formed (via attachment models) or how a random process is able to percolate across a random graph. The effects of correlated percolation is discussed in the second part of this dissertation, where a process can affect single sites on a graph or a cluster of sites simultaneously. It is shown that a lower bound on the percolation threshold can be established, and this bound is dependent on the size of clusters present.
The problem of controlling the quantum state of a system is investigated using a time varying potential. As a concrete example we study the problem of a particle in a box with a periodically oscillating infinite square-well potential, from which we obtain results that can be applied to systems with periodically oscillating boundary conditions. We derive an analytic expression for the frequencies of resonance between states, and against standard intuition, we show how to use this behavior to control the quantum state of the system at will. In particular, we offer as an example the transition from the ground state to the first excited state of the square well potential. At first sight, it may be counter intuitive that we can control the state of such a quantum Hamiltonian, as the Schrödinger equation conserves the norm of the wave function. In this manuscript, we show how that can be achieved.