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Physical Review Letters

One-dimensional trapping kinetics at zero temperature

P. E. Parris

Published Mar 20, 1989• 21 citations
Publisher

Abstract

The asymptotic decay of the survival probability is calculated for a quantum particle moving at zero temperature on a one-dimensional tight-binding chain possessing randomly placed irreversible traps of strength \ensuremath{\gamma}. The survival probability exhibits a decay, P(t)\ensuremath{\sim}exp(-${\mathrm{At}}^{1/4}$), which is slower than that associated with a diffusing particle.

DOI
10.1103/physrevlett.62.1392

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The asymptotic decay of the survival probability is calculated for a quantum particle moving at zero temperature on a one-dimensional tight-binding chain possessing randomly placed irreversible traps of strength \ensuremath{\gamma}. The survival probability exhibits a decay, P(t)\ensuremath{\sim}exp(-At1/4{\mathrm{At}}^{1/4}At1/4), which is slower than that associated with a diffusing particle.
A study is made of the survival probability P(t) for a quantum particle moving coherently on an ordered one-dimensional chain containing randomly placed irreversible traps in concentration q. We consider two separate models of the trapping process, focusing on the long-time limit of each. In the first model, intended to describe substitutional traps, the trapping impurities act as disruptive absorbing barriers which prevent further motion along the chain. For this model it is shown that -ln[P(t)]\ensuremath{\sim}t1/4{t}^{1/4}t1/4. The second model is intended to describe interstitial trapping impurities which do not disrupt the transport. It is argued for this case that Anderson localization of the quasiparticle wave functions will occur in the otherwise ordered chain as a result of the disorder introduced by the trapping impurities. In addition, it is suggested that the asymptotic decay of P(t) will be dominated by slowly decaying long-wavelength modes associated with asymptotically large segments of the chain which are free of traps. Our analysis predicts that the asymptotic decay will be of the same form as that which obtains in the substitutional model. Numerical results that we have performed support this prediction.
The long time behaviour of the survival probability of the initial state and its dependence on the initial states are considered, for the one-dimensional free quantum particle. We derive the asymptotic expansion of the time evolution operator at long times, in terms of the integral operators. This enables us to obtain the asymptotic formula for the survival probability of the initial state ψ( x ), which is assumed to decrease sufficiently rapidly at large | x |. We then show that the behaviour of the survival probability at long times is determined by that of the initial state ψ at zero momentum k = 0. Indeed, it is proved that the survival probability can exhibit the various power decays such as t −2 m −1 for an arbitrary non-negative integer m as t → ∞, corresponding to the initial states with the condition ( k ) = O ( k m ) as k → 0.
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 ).
The dynamics of the survival probability of quantum walkers on a one-dimensional lattice with random distribution of absorbing immobile traps is investigated. The survival probability of quantum walkers is compared with that of classical walkers. It is shown that the time dependence of the survival probability of quantum walkers has a piecewise stretched exponential character depending on the density of traps in numerical and analytical observations. The crossover between the quantum analogues of the Rosenstock and Donsker–Varadhan behavior is identified.
Quantum escapes of two particles with Coulomb interactions from a confined one-dimensional region to a semi-infinite lead are discussed by using the probability of finding all particles within the confined region, that is, the survival probability, in comparison with free particles. By taking into account the quantum effects of two identical particles, such as the Pauli exclusion principle, it is shown analytically that for two identical free fermions (bosons), the survival probability decays asymptotically in power ~t(-10) (~t(-6)) as a function of time t, although it decays in power ~t(-3) for one free particle. On the other hand, for two particles with attractive Coulomb interactions it is shown numerically that the survival probability decays in power ~t(-3) after a long time. Moreover, for two particles with repulsive Coulomb interactions it decays exponentially in time ~exp (-αt) with a constant α, which is almost independent of the initial energy of particles.
Quantum walks are interesting models with potential applications to quantum algorithms and physical processes such as photosynthesis. In this paper, we study two models of one-dimensional quantum walks, namely, quantum walks with a moving absorbing wall and quantum walks with one stationary and one moving absorbing wall. For the former, we calculate numerically the survival probability, the rate of change of average position, and the rate of change of standard deviation of the particle's position in the long time limit for different wall velocities. Moreover, we also study the asymptotic behavior and the dependence of the survival probability on the initial particle's state. While for the latter, we compute the absorption probability of the right stationary wall for different velocities and initial positions of the left wall boundary. The results for these two models are compared with those obtained for the classical model. The difference between the results obtained for the quantum and classical models can be attributed to the difference in the probability distributions.