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Quantum revivals and carpets in some exactly solvable systems

Loinaz, W, T. J. Newman

Published Jan 1, 1999• 15 citations
OpenAlex

Abstract

We consider the revival properties of quantum systems with an eigenspectrum E_{n} ~ n^{2}, and compare them with the simplest member of this class - the infinite square well. In addition to having perfect revivals at integer multiples of the revival time t_{R}, these systems all enjoy perfect fractional revivals at quarterly intervals of t_{R}. A closer examination of the quantum evolution is performed for the Scarf potentials, and comparison is made with the infinite square well using quantum carpets.

Related papers

Quantum revivals and carpets in some exactly solvable systems

Will Loinaz A, T. J. Newman

We consider the revival properties of quantum systems with an eigenspectrum E n n 2 , and compare them with the simplest member of this class - the infinite square well. In addition to having perfect revivals at integer multiples of the revival time t R , these systems all enjoy perfect fractional revivals at quarterly intervals of t R . A closer examination of the quantum evolution is performed for the Pöschel-Teller and Rosen-Morse potentials, and comparison is made with the infinite square well using quantum carpets.

arXiv:quant-ph/9902039•Journal of Physics A Mathematical and General•Dec 2, 1999•23 citations

Decay of wave packet revivals in the asymmetric infinite square well

Todd Keene Timberlake, Seth Camp

In the infinite square well, any wave function will return to its initial state at integer multiples of the revival time. Most quantum systems do not exhibit perfect revivals, but some exhibit partial revivals in which the wave function returns close to its initial state. Subsequent partial revivals usually deteriorate in quality. We discuss the reasons for the perfect revivals in the infinite square well and how a small change in the potential disrupts the revivals. As an example, we examine partial revivals of a Gaussian wave packet in an infinite square well with a step. First-order and second-order perturbation theory show that the rate at which revivals decay depends on the location of the step.

American Journal of Physics•May 13, 2011•3 citations

Quantum carpets: A probe to identify wave packet fractional revivals

Iqra Yausaf, Shahid Iqbal

Quantum carpets- in position and momentum space- woven by the self-interference of de Broglie wave of an atom or an electron, trapped in an infinitely deep potential well, are explained. The recurrence of self-similar structures in designs of these carpets mimics the phenomena of quantum revivals and fractional revivals. We identify fractional revivals of various order by means of these space-time and momentum-time interference patterns.

arXiv:1607.07496•Jul 25, 2016•0 citations

Analytical investigation of revival phenomena in the finite square-well potential

David L. Aronstein, Carlos R. Stroud

We present an analytical investigation of revival phenomena in the finite square-well potential. The classical motion, revival, and super-revival time scales are derived exactly for wave packets excited in the finite well. These time scales exhibit a richer dependence on wave-packet energy and on potential-well depth than has been found in other quantum systems: They explain, for example, the difficulties in exciting wave packets with strong classical features at the bottom of a finite well, or with clearly resolved super-revivals in a shallow well. In the proper regions of validity, the time scales predict the instances of wave-packet reformation extremely accurately. Revivals at the bottom of the well are explored as a ``universal'' limit of the general theory, which offers the clearest connection with the series of fractional and full revivals seen in the dynamics of the infinite square-well potential.

Physical Review A•Jul 6, 2000•31 citations

A Foray into Quantum Dynamics

Ross C. O’Connell

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 $a_m$ that govern fractional revivals, a demonstration that $Ψ_{cl}$, 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.

arXiv:quant-ph/0212092•Dec 16, 2002•1 citation

Fractional revivals, multiple-Schrödinger-cat states, and quantum carpets in the interaction of a qubit with N qubits

Shane Dooley, Timothy P. Spiller

We study the dynamics of a system comprised of a single qubit interacting equally with $N$ qubits (a ``spin star'' system). Although this model can be solved exactly, the exact solution does not give much intuition for the dynamics of the model. Here, we find an approximation that gives some insight into the dynamics for a particular class of initial spin-coherent states of the $N$ qubits. We find an effective Hamiltonian for the system that is a finite Kerr (one-axis twisting) Hamiltonian for the $N+1$ qubits. The initial spin-coherent state evolves to spin-squeezed states on short time scales, and to ``multiple-Schr\"odinger-cat'' states (superpositions of many spin-coherent states) on longer time scales, a manifestation of the phenomenon of fractional revivals of the initial state. The evolution of the system is visualized with phase-space plots ($Q$ functions) that, when plotted against time, reveal a ``quantum carpet'' pattern. Of particular interest is the fact that our approximation captures the qualitative features of the model even for small values of $N$. This suggests the possibility of observing the phenomenon of fractional revival in this model for systems of few qubits.

arXiv:1404.4296•Physical Review A•Jul 14, 2014•13 citations
We consider the revival properties of quantum systems with an eigenspectrum E_{n} ~ n^{2}, and compare them with the simplest member of this class - the infinite square well. In addition to having perfect revivals at integer multiples of the revival time t_{R}, these systems all enjoy perfect fractional revivals at quarterly intervals of t_{R}. A closer examination of the quantum evolution is performed for the Scarf potentials, and comparison is made with the infinite square well using quantum carpets.
We consider the revival properties of quantum systems with an eigenspectrum E n n 2 , and compare them with the simplest member of this class - the infinite square well. In addition to having perfect revivals at integer multiples of the revival time t R , these systems all enjoy perfect fractional revivals at quarterly intervals of t R . A closer examination of the quantum evolution is performed for the Pöschel-Teller and Rosen-Morse potentials, and comparison is made with the infinite square well using quantum carpets.
In the infinite square well, any wave function will return to its initial state at integer multiples of the revival time. Most quantum systems do not exhibit perfect revivals, but some exhibit partial revivals in which the wave function returns close to its initial state. Subsequent partial revivals usually deteriorate in quality. We discuss the reasons for the perfect revivals in the infinite square well and how a small change in the potential disrupts the revivals. As an example, we examine partial revivals of a Gaussian wave packet in an infinite square well with a step. First-order and second-order perturbation theory show that the rate at which revivals decay depends on the location of the step.
We present an analytical investigation of revival phenomena in the finite square-well potential. The classical motion, revival, and super-revival time scales are derived exactly for wave packets excited in the finite well. These time scales exhibit a richer dependence on wave-packet energy and on potential-well depth than has been found in other quantum systems: They explain, for example, the difficulties in exciting wave packets with strong classical features at the bottom of a finite well, or with clearly resolved super-revivals in a shallow well. In the proper regions of validity, the time scales predict the instances of wave-packet reformation extremely accurately. Revivals at the bottom of the well are explored as a ``universal'' limit of the general theory, which offers the clearest connection with the series of fractional and full revivals seen in the dynamics of the infinite square-well potential.
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.
We study the dynamics of a system comprised of a single qubit interacting equally with NNN qubits (a ``spin star'' system). Although this model can be solved exactly, the exact solution does not give much intuition for the dynamics of the model. Here, we find an approximation that gives some insight into the dynamics for a particular class of initial spin-coherent states of the NNN qubits. We find an effective Hamiltonian for the system that is a finite Kerr (one-axis twisting) Hamiltonian for the N+1N+1N+1 qubits. The initial spin-coherent state evolves to spin-squeezed states on short time scales, and to ``multiple-Schr\"odinger-cat'' states (superpositions of many spin-coherent states) on longer time scales, a manifestation of the phenomenon of fractional revivals of the initial state. The evolution of the system is visualized with phase-space plots (QQQ functions) that, when plotted against time, reveal a ``quantum carpet'' pattern. Of particular interest is the fact that our approximation captures the qualitative features of the model even for small values of NNN. This suggests the possibility of observing the phenomenon of fractional revival in this model for systems of few qubits.