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arXiv:quant-ph/0307232 · Journal of Physics A Mathematical and General

Decay in a uniform field: an exactly solvable model

R. M. Cavalcanti, Paola Giacconi A, Roberto Soldati

Published Nov 18, 2003• 25 citations
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Abstract

We investigate the time evolution of the decay (or ionization) probability of a D -dimensional model atom ( D = 1, 2, 3) in the presence of a uniform (i.e., static and homogeneous) background field. The model atom consists in a non-relativistic point particle in the presence of a point-like attractive well. It is shown that the model exhibits infinitely many resonances leading to possible deviations from the naive exponential decay law of the non-decay (or survival) probability of the initial atomic quantum state. Almost stable states exist due to the presence of the attractive interaction, no matter how weak it is. Analytic estimates as well as numerical evaluation of the decay rates are explicitly given and discussed.

DOI
10.1088/0305-4470/36/48/009

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We investigate the time evolution of the decay (or ionization) probability of a D -dimensional model atom ( D = 1, 2, 3) in the presence of a uniform (i.e., static and homogeneous) background field. The model atom consists in a non-relativistic point particle in the presence of a point-like attractive well. It is shown that the model exhibits infinitely many resonances leading to possible deviations from the naive exponential decay law of the non-decay (or survival) probability of the initial atomic quantum state. Almost stable states exist due to the presence of the attractive interaction, no matter how weak it is. Analytic estimates as well as numerical evaluation of the decay rates are explicitly given and discussed.
We investigate the ionization of a 1-D quantum system with a zero-range attractive potential at x = 0 when it is subjected to a time dependent harmonic zero range potential forcing of frequency /spl omega/, strength r, at position x = a. We show that the ionization probability goes to 1 as time t /spl rarr/ /spl infin/ for almost all values of r, /spl omega/, and a. The initially exponential decay of the probability to stay in the bound state is followed at later time by a t/sup -3/ law if r is small and /spl omega/ is not close to resonances. When this is not true the separation between the exponential and power law decay regimes disappears and Fermi's golden rule fails. There exist exceptional sets of parameters where the survival probability never decays to zero and complete ionization is impossible.
Abstract An exact analytical solution to the non-stationary Schrödinger equation is found for a particle confined in a short-range potential in a spatially uniform, time-dependent field. The solution is used to calculate the decay probability of an unstable quantum state initially created in the potential well of a double-barrier system. An exact analytical expression for the decay probability is obtained without assuming weak laser field intensity. Calculations show that, for certain sets of parameters, the entire decay curve oscillates significantly. The physical meaning of the oscillations is revealed, and an estimate of the oscillation period is given. The behavior of the decay probability at short and long times is discussed. If the barrier height tends to zero, then at long times, the decay law is inversely proportional to time. The number of non-decaying particles can be varied over a wide range by changing the length from the emission point to the detector that records the particles.
An analytical solution for the time evolution of decay of two identical noninteracting quantum particles seated initially within a potential of finite range is derived using the formalism of resonant states. It is shown that the wave function, and hence also the survival and nonescape probabilities, for factorized symmetric and entangled symmetric or antisymmetric initial states evolve in a distinctive form along the exponentially decaying and nonexponential regimes. Our findings show the influence of the Pauli exclusion principle on decay. We exemplify our results by solving exactly the sss-wave $\ensuremath{\delta}$ shell potential model.
A detailed analysis of the quantum decay processes shows the survival probability ( t ) can not take that exponential form at any time interval including times smaller than the lifetime τ . We show that for times t ∼ τ and for the times later than τ the form of ( t ) looks as a composition of an oscillating and exponential functions. The amplitude of these oscillations is very small for t ≪ τ and grows with increasing time and depends on the model considered. We also study the survival probability of moving relativistic unstable particles with definite momentum : It turns out that late time deviations of the survival probability of these particles from the exponential-like form of the decay law should occur much earlier than it follows from the classical standard approach resolving itself into replacing time t by t / γ (where γ is the relativistic Lorentz factor) in the formula for the survival probability ( t ).
We study the time evolution of the wave function of a particle bound by an attractive δ-function potential when it is subjected to time-dependent variations of the binding strength (parametric excitation). The simplicity of this model permits certain nonperturbative calculations to be carried out analytically both in one and three dimensions. Thus the survival probability of bound state |θ(t)|2, following a pulse of strength r and duration t, behaves as |θ(t)|2−|θ(∞)|2∼t−α, with both θ(∞) and α depending on r. On the other hand, a sequence of short pulses produces an exponential decay over an intermediate time scale.
While exponential decay is ubiquitous in nature, deviations at both short and long times are dictated by quantum mechanics. Nonexponential decay is known to arise due to the possibility of reconstructing the initial state from the decaying products. We discuss the quantum decay dynamics by tunneling of a many-particle system, characterizing the long-time nonexponential behavior of the nonescape and survival probabilities. The effects of contact interactions and quantum statistics are described. It is found that, whereas for noninteracting bosons the long-time decay follows a power law with an exponent linear in the number of particles NNN, the exponent becomes quadratic in NNN in the fermionic case. The same results apply to strongly interacting many-body systems related by the generalized Bose-Fermi duality. The faster fermionic decay can be traced back to the effective hard-core interactions between particles, which are as well the decaying products, and exhibit spatial antibunching which hinders the reconstruction of the initial unstable state. The results are illustrated with a paradigmatic model of quantum decay from a trap allowing leaks by tunneling, whose dynamics is described exactly by means of an expansion in resonant states.