Decays of Unstable Quantum Systems

Abstract

This paper is a pedagogical yet critical introduction to the quantum description of unstable systems, mostly at the level of a graduate quantum mechanics course. Quantum decays appear in many different fields of physics, and their description beyond the exponential approximation is the source of technical and conceptual challenges. In this article, we present both general methods that can be adapted to a large class of problems, and specific elementary models to describe phenomena like photo-emission, beta emission and tunneling-induced decays. We pay particular attention to the emergence of exponential decay; we analyze the approximations that justify it, and we present criteria for its breakdown. We also present a detailed model for non-exponential decays due to resonance, and an elementary model describing decays in terms of particle-detection probabilities. We argue that the traditional methods for treating decays face significant problems outside the regime of exponential decay, and that the exploration of novel regimes of current interest requires new tools. * anastop@physics.upatras.gr arXiv:1808.03798v2 [quant-ph] 8 Dec 2018Probability currents. An alternative elementary description of decays is available, whenever we can associate a probability-current operatorĴ J J(x x x, t) to one of the decay products. For example, for non-relativistic particles of mass m satisfying Schrödinger's equation with HamiltonianĤ, the current operator iŝ J J J(x x x, t) = 1 2m e

Tunneling-Induced Decay of Wave Packets Localized Beyond an Asymmetric Potential Barrier

“…While the exponential decay law occurs everywhere in nature, microscopic systems should be described by quantum mechanics and the exponential decay cannot be valid for all times. In particular, it undoubtedly fails when the time of evolution is either long [3,[8][9][10] or short [9][10][11][12]. The exponential decay law is commonly associated with the survival probability that a system, initially prepared in an unstable state, will be still in the same state after some time [8][9][10].…”

“…In particular, it undoubtedly fails when the time of evolution is either long [3,[8][9][10] or short [9][10][11][12]. The exponential decay law is commonly associated with the survival probability that a system, initially prepared in an unstable state, will be still in the same state after some time [8][9][10]. The survival probability yields a quadratic behavior at short times, exponential law at intermediate times, and power law at longer times [1][2][3][4][5][6][7][8][9][10][11][12].…”

“…The exponential decay law is commonly associated with the survival probability that a system, initially prepared in an unstable state, will be still in the same state after some time [8][9][10]. The survival probability yields a quadratic behavior at short times, exponential law at intermediate times, and power law at longer times [1][2][3][4][5][6][7][8][9][10][11][12]. The above-mentioned features of time evolution are the consequences of rather general considerations assuring that the quantum decay law can be described by this three-step function.…”

Open quantum systems and applications to the quantum information theory

“…Eq. (4.32) is similar for the equation for the persistence amplitude of an unstable quantum state in the random phase approximation [7]. In fact, the two kernelsγ 0 andγ r are similar to the ones that appear in the evolution of a pair of atomic qubits interacting with the EM field [56].…”

“…In particular, the van Hove limit may not be physical relevant in cases where the open system dynamics are characterized by long time scales other than the dissipation time. This occurs for example, if the environment is characterized by resonance frequencies or thresholds [7]. In this paper, we present another case of failure of the Markovian approximate, due to the time-scale of traversal time in a bipartite system with components at large separation.…”