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Physical Review A

Fractional revival “matrix mechanics” in quantum billiards

Bernd Rohwedder

Published Jul 21, 2008• 1 citation
Publisher

Abstract

Certain classes of integrable quantum billiards provide a versatile arena for time-domain atom optics experiments. In order to learn how to fully exploit the fractional revival behavior that governs these kinds of systems, we here develop a convenient theoretical framework for their description. As an illustrative example, we present a billiard-internal realization of Grover's quantum search algorithm. The considered devices are scalable and could be used for the construction of quantum computers using trapped neutral atoms.

DOI
10.1103/physreva.78.012340

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Certain classes of integrable quantum billiards provide a versatile arena for time-domain atom optics experiments. In order to learn how to fully exploit the fractional revival behavior that governs these kinds of systems, we here develop a convenient theoretical framework for their description. As an illustrative example, we present a billiard-internal realization of Grover's quantum search algorithm. The considered devices are scalable and could be used for the construction of quantum computers using trapped neutral atoms.
The phenomenon of fractional self-imaging can be used to implement atom-optical components in the interior of a rectangular quantum billiard. Their temporal concatenation may lead to complex "atom chips" operating in the time-domain. As an elementary example, we discuss a billiard internal Mach-Zehnder interferometer that can be transformed into an all-optical switch or controllable beam splitter for atoms. The inherent coherence and reversibility of its dynamics also provides a simple means to realize a Welcher Weg/Quantum erasure experiment. The whole concept is shown to be robust with respect to realistic wall-softness effects and feasible within the present state of the art.
Summary form only given. The motion of a particle inside a closed area surrounded by reflecting walls can be integrable or chaotic, depending on the shape of the boundary. These billiards serve as a good and simple model to study chaotic dynamics both classically and quantum mechanically. Recently, we realized an "atom-optics billiard", where ultra-cold atoms are trapped inside a dark optical trap, whose shape can be varied to create different atom dynamics. In this work we describe how scattering and wall softness influence the dynamics of the atoms, and destroy or create stability.
The use of wave packet to analyze the dynamics of quantum mechanical systems is an increasingly important method to the study of the classical-quantum correspondence.Using the quantum Gaussian wave packet analysis method,we calculate the autocorrelation function of the rectangular billiard,the peak positions of the autocorrelation function match well with the periods of the classical periodic orbits,which show that the period of the classical orbits can be produced by the time-dependent quantum wave packet method.We also discuss wave packet revivals and fractional revivals in the rectangular billiard,the results show that there are exact revival for all wave packet at each revival time.We find additional cases of exact revivals with short revival times for zero-momentum wave packets initially located at special symmetry point inside the billiard.
Abstract Position-dependent-mass systems are of great importance in many physical situations, such as the transport of charge carriers in semiconductors with non-uniform composition and in the theory of many-body interactions in condensed matter. Here we investigate, numerically and analytically, the phenomenon of fractional revivals in such systems, which is a generic characteristic manifested by the wave-packet evolution in bounded Hamiltonian systems. Identifying the fractional revivals using specific probes is an important task in the theory of quantum measurement and sensing. We numerically simulate the temporal evolution of probability density and information entropy density, which manifest self-similarly recurring interference patterns, namely, quantum carpets. Our numerical results show that the quantum carpets not only serve as an effective probe for recognizing the fractional revivals of various order but they efficiently describe the effect of spatially-varying mass on the structure of fractional revivals, which is manifested as a symmetry breaking in their designs.
Arrays of identical neutral atoms trapped in optical tweezers are a promising candidate for use in quantum computing. These platforms are highly scalable to large numbers of qubits and neutral atoms boost several attractive features as long coherence times and the possibility to be entangled via strong dipole-dipole interactions by driving them to highly excited Rydberg states. \nThe Thesis work is developed inside the framework of the QRydDemo project, whose aim in the next few years is to realize a neutral atom quantum processor with several hundred qubits. The smallest building blocks for the quantum computer are one and two-qubit gates: to entangle two atoms in the quantum register, a controlled-phase (CZ) gate will be implemented by shining fine-tuned laser pulses onto them. In this work, after giving a theoretical description of the Hamiltonian of two neutral atoms in the quantum register, a numerical simulation of this system is exploited to reproduce the behavior of the two-qubit CZ gate. Realistic effects are taken into account as finite temperature, imperfect Rydberg blockade, or decay out of the Rydberg state. A protocol with constant pulses is analyzed and its optimal parameters are found through classical optimizers. Then, time-dependent pulses are introduced and the optimal pulses are found through the optimal control algorithm dCRAB in an open-loop optimization. For the experimental realization of the gate, this analysis is of pivotal importance to know in which aspect more effort has to be put to maximize the experimental precision of the operation and thus improving the performance of the whole device \n
The dynamics of a quantum mechanical particle in a time-independent potential are found to contain many interesting phenomena. These are direct consequences of the (typical) existence of more than one time scale governing the problem. This gives rise to full revivals of initial wavepackets, fractional revivals (multiple wavepackets appearing at fractions of the revival time) and the striking quantum carpets. A variety of analytic techniques are used to consider the interference that gives rise to these phenomena while skirting calculations involving cross-terms. Novel results include a new theorem on the weighting coefficients ama_mam​ that govern fractional revivals, a demonstration that ΨclΨ_{cl}Ψcl​, the function that governs the distribution and features of these fractional revivals, really does behave classically, a treatment of the wavepacket dephasing in the infinite square well by means of the Poisson summation formula, and a correct analysis of the spatial distribution of intermode traces. Also, this work presents a coherent treatment of these phenomena, which before now did not exist.