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Physical Review A

Geometric-phase-propagator approach to time-dependent quantum systems

F. Κ. Diakonos, P. A. Kalozoumis, A. I. Karanikas, Nektarios Manifavas, Peter Schmelcher

Published Jun 18, 2012• 6 citations
Publisher

Abstract

A field-theoretical approach to the scattering off an oscillating quantum system is developed. As a key ingredient it employs the adiabatic eigenstate basis and consists of a perturbative scheme for the calculation of the geometric phases influencing the transmission through the time-dependent potential landscape. The main advantage is the identification of basic diagrams which allow for an immediate interpretation of the underlying elementary physical processes contributing to the scattering and transmission behavior. We apply our method to the simple, but prototypical, problem of transmission through an one-dimensional oscillating $\ensuremath{\delta}$ potential and demonstrate how it enables a deeper understanding of the relevant physical processes.

DOI
10.1103/physreva.85.062110

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A field-theoretical approach to the scattering off an oscillating quantum system is developed. As a key ingredient it employs the adiabatic eigenstate basis and consists of a perturbative scheme for the calculation of the geometric phases influencing the transmission through the time-dependent potential landscape. The main advantage is the identification of basic diagrams which allow for an immediate interpretation of the underlying elementary physical processes contributing to the scattering and transmission behavior. We apply our method to the simple, but prototypical, problem of transmission through an one-dimensional oscillating \ensuremathδ\ensuremath{\delta}\ensuremathδ potential and demonstrate how it enables a deeper understanding of the relevant physical processes.
We develop a field theoretical approach based on the temporary basis description as a tool to investigate the transmission properties of a time-driven quantum device. It employs a perturbative scheme for the calculation of the transmission of a monochromatic beam of particles through the time-dependent set-up. The main advantage of the proposed treatment is that it permits the use of the particle picture for the calculation of the scattering matrix and the transmission coefficient. Therefore the elementary physical processes contributing to the transmission can be identified and interpreted in a transparent way. We apply the method to the simple but prototype problem of transmission through an one-dimensional oscillating delta potential and we demonstrate how it enables a deep understanding of the underlying physical processes.
The study of geometric phase in quantum mechanics has so far been confined to discrete (or continuous) spectra and trace-preserving evolutions. By considering only the transmission channel, a scattering process with internal degrees of freedom is neither a discrete spectrum problem nor a trace-preserving process. We explore the geometric phase in the scattering process by taking only the transmission process into account. We find that the geometric phase can be calculated by the same method as in unitary evolution. The interference visibility depends on the transmission amplitude. The dependence of the geometric phase on the barrier strength and the spin-spin coupling constant is also presented and discussed.
Abstract This review presents time-independent perturbative methods for solving the one-dimensional Schrödinger equation, highlighting representative cases that reveal key aspects of the theory. The focus is on their relevance to quantum computing applications, particularly in systems with finite-dimensional state spaces. The main link between these methods and quantum processing emerges when we consider systems with only a finite number of well-defined energy levels, e.g., two levels corresponding to the ‘zero’ and ‘one’ states of a quantum bit (qubit), for which the Schrödinger equation describes a quantum algorithm corresponding to the unitary evolution of the qubits. Perturbative techniques can then be applied to this evolution to gain deeper insight into the algorithm or to simplify it in a controlled manner, as described in the last sections of this review. We initially discuss representative applications using benchmark cases that exemplify the essential features of perturbation theory while avoiding unnecessary technical complications. The methods presented involve direct approaches to solutions for quantum systems with small potential perturbations, with consideration of indirect and approximation techniques. Key cases examined include the particle in a potential well, the harmonic oscillator, and systems with polynomial or Dirac delta distribution perturbations, each chosen to highlight unique aspects of perturbative analysis. Starting with an overview of the unperturbed Schrödinger equation, we derive canonical solutions and then use perturbative techniques to approximate energy shifts and wavefunction corrections in both non-degenerate and degenerate settings. By combining theoretical discussion with case studies, this review underscores how perturbative methods are applied across a range of elementary quantum mechanical problems especially for applications in the qubit model, which, in many cases of interest, can be reduced to the explicit calculations shown here.
Abstract A generalization of the adiabatic theorem for quantum systems governed by pseudo-Hermitian Hamiltonians and details of its demonstration are given. Introducing a modified time-dependent metric giving a precise description of the quantum unitary evolution where the obtained effective Hamiltonian is observable because it is mean value is real. We show that an eigenstate of a pseudo-Hermitian Hamiltonian slowly transported will acquire a real generalized geometrical phase factor which contains two contributions: the first one corresponds to the conventional Berry’s phase as expected and a new geometrical term that we call the metric geometrical phase. We will apply our results in the cases of the famous time dependent brachistochrone problem and a non-Hermitian displaced harmonic oscillator.
Exactly solvable quantum systems represent a fascinating and critically important class of models in theoretical physics, offering exact solutions to non-trivial problems that often shed light on the behavior of more complex, intractable systems. The remarkable feature of these systems lies not only in their solvability but also in the underlying mathematical structures that facilitate such solutions. This paper explores the concept of a universal algebraic structure that underpins exactly solvable quantum systems. We delve into how diverse models, ranging from one-dimensional spin chains to many-body systems and field theories, exhibit common algebraic symmetries, such as Lie algebras, affine Lie algebras, and quantum groups. These algebraic frameworks provide a unified language for understanding integrability, the existence of an infinite number of conserved quantities, and the ability to construct exact eigenstates. We review the historical development of these ideas, from the Bethe ansatz to the development of the inverse scattering method and quantum groups. Our discussion highlights how these mathematical structures are not merely tools for calculation but fundamental descriptors of the physical reality of these systems, offering profound insights into non-perturbative quantum phenomena. The identification of a universal algebraic architecture suggests a deeper coherence in the landscape of quantum mechanics, paving the way for new discoveries and classifications of quantum models.