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Universe

Effects of Impurities and Deformations on Electronic Effective Mass in Quantum Revival Time within the Infinite Square Well

Cleverson Filgueiras, L. H. C. Borges, Moisés Rojas

Published Jun 20, 2024• 1 citation
Publisher

Abstract

Quantum revival phenomena, wherein the wave function of a quantum system periodically returns to its initial state after evolving in time, are investigated in this study. Focusing on electrons confined within a quantum box with an impurity, both weak- and strong-coupling regimes are explored, revealing intricate relationships between impurity parameters and temporal dynamics. This investigation considers the influence of impurity position, impurity strength, and external factors such as aluminum concentration, temperature and hydrostatic pressure on classical periods and revival times. Through analytical derivations and graphical analyses, this study elucidates the sensitivity of quantum revivals to these parameters, providing valuable insights into the fundamental aspects of quantum mechanics. While no specific physical applications are discussed, the findings offer implications for quantum heat engines and other quantum-based technologies, emphasizing the importance of understanding quantum revivals in confined quantum systems.

DOI
10.3390/universe10060269

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Quantum revival phenomena, wherein the wave function of a quantum system periodically returns to its initial state after evolving in time, are investigated in this study. Focusing on electrons confined within a quantum box with an impurity, both weak- and strong-coupling regimes are explored, revealing intricate relationships between impurity parameters and temporal dynamics. This investigation considers the influence of impurity position, impurity strength, and external factors such as aluminum concentration, temperature and hydrostatic pressure on classical periods and revival times. Through analytical derivations and graphical analyses, this study elucidates the sensitivity of quantum revivals to these parameters, providing valuable insights into the fundamental aspects of quantum mechanics. While no specific physical applications are discussed, the findings offer implications for quantum heat engines and other quantum-based technologies, emphasizing the importance of understanding quantum revivals in confined quantum systems.
A particle of mass M moves in an infinite square well of width L (the “particle in a box”). Classically, the motion has period L2M/E, which depends on the initial condition through the energy E. Quantum mechanically, any wave function repeats exactly with period 4ML2/πℏ, independent of the initial condition. Given this qualitative difference, how can the classical motion possibly be the limit of the quantal time development? The resolution of this paradox involves the difference between the exact revival (recurrence) of the wave function and the approximate periodicity of expectation values such as 〈x(t)〉. (The latter may recur an odd integral number of times before the full wave function recurs.) The period of the expectation values does depend on the initial condition and can possess the expected classical limit. [An Appendix demonstrates that, under suitably quasiclassical conditions, the quantal time evolution of 〈x(t)〉 passes over to the classical result not only in period, but also in its exact functional form. Another Appendix proves four theorems concerning state-dependent exact revival times.]
We analyze the effect of impurity on the work output and efficiency of quantum Otto and quantum Carnot heat cycles, modeled as a single quantum particle in an infinite square well potential, which is the working substance. We solve this quantum mechanical system perturbatively up to first and second order in strength of the impurity for strong- and weak-coupling regimes, respectively. We derive the analytical expressions of work and efficiency for the strong-coupling regime to the first order in the strength parameter. The threshold value of the strength parameter in weak coupling is obtained up to which the numerical result agrees with the perturbative result for a repulsive and attractive impurity. To our surprise, an embedded impurity unlocks new operational phases in the system, such as a quantum heat engine, quantum refrigerator, and quantum cold pump. In addition, the efficiency of the quantum Otto heat engine is seen to reach Carnot efficiency for some parameter regimes. The cooling power and coefficient of performance of the quantum refrigerator and quantum cold pump are nontrivially affected by the impurity.
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.
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.
Quantum mechanical objects or nano-objects have been proposed as bits for information storage. While time-averaged properties of magnetic, quantum-mechanical particles have been extensively studied experimentally and theoretically, experimental investigations of the real time evolution of magnetization in the quantum regime were not possible until recent developments in pump–probe techniques. Here we investigate the quantum dynamics of effective spin systems by means of analytical and numerical treatments. Particular attention is paid to the quantum revival time and its relation to the magnetization tunneling. The quantum revival time has been initially defined as the recurrence time of a total wave-function. Here we show that the quantum revivals of wave-functions and expectation values in spin systems may be quite different which gives rise to a more sophisticated definition of the quantum revival within the realm of experimental research. Particularly, the revival times for integer spins coincide which is not the case for half-integer spins. Furthermore, the quantum revival is found to be shortest for integer ratios between the on-site anisotropy and an external magnetic field paving the way to novel methods of anisotropy measurements. We show that the quantum tunneling of magnetization at avoided level crossing is coherent to the quantum revival time of expectation values, leading to a connection between these two fundamental properties of quantum mechanical spins.