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Physical review. B, Condensed matter

One-dimensional quantum transport in the presence of traps

P. E. Parris

Published Sep 1, 1989• 25 citations
Publisher

Abstract

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}${t}^{1/4}$. 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.

DOI
10.1103/physrevb.40.4928

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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 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.
This work shows that dynamical features typical of full random matrices can be observed also in the simple finite one-dimensional (1D) noninteracting Anderson model with nearest-neighbor couplings. In the thermodynamic limit, all eigenstates of this model are exponentially localized in configuration space for any infinitesimal on-site disorder strength W. But this is not the case when the model is finite and the localization length is larger than the system size L, which is a picture that can be experimentally investigated. We analyze the degree of energy-level repulsion, the structure of the eigenstates, and the time evolution of the finite 1D Anderson model as a function of the parameter ξ∝(W^{2}L)^{-1}. As ξ increases, all energy-level statistics typical of random matrix theory are observed. The statistics are reflected in the corresponding eigenstates and also in the dynamics. We show that the probability in time to find a particle initially placed on the first site of an open chain decays as fast as in full random matrices and much faster than when the particle is initially placed far from the edges. We also see that at long times, the presence of energy-level repulsion manifests in the form of the correlation hole. In addition, our results demonstrate that the hole is not exclusive to random matrix statistics, but emerges also for W=0, when it is in fact deeper.
The time behavior of a wave packet, transmitted through a potential barrier, is investigated for all times assuming it to be initially located outside the asymmetric barrier. The survival probability of the initial state, which characterizes the decay of the unstable quantum system in three time domains, is discussed. The presence of initial, exponential, and long-time regimes is a general feature of the decay process. We analytically found the three crucial points of time that separate these regimes. These characteristic times are set by simple formulae, including momentum distribution. It is then shown that the radius of convergence of the cumulant expansion for the survival probability is always small. A long-time behavior of survival probability is described by the asymptotic series whose coefficients decrease with increasing particle energy but at different rates. To be specific, we have examined the exactly solvable model of tunneling through a finite range triangular barrier.
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.
One dimensional chains coupled to reservoirs prosent decoherence phenomena in electron transport. Impurities, which are randomly distributed over the chain, lead to the Anderson localization phenomenon, regardless of the degree of disorder. However, this effect c an be modified if the chain includes non-linear effects of impurities, leading to a self-trapping of the cha rge density. In this paper, we estimate the temporal dynamics of site-occupation probability as a function of the non-linear parameter that characterizes the impurity and the amplitude of hopping between adjacent sites. The excistence of a self-trapping phenomenon, which produce solitons, is demonstrated. The electron production depends on a critical exponent, which is a function of the dimensionality of the considered chain.
We consider coherent exciton transport modeled by continuous-time quantum walks on long-range interacting cycles (LRICs), which are constructed by connecting all the two nodes of distance m in the cycle graph. LRIC has a symmetric structure and can be regarded as the extensions of the cycle graph (nearest-neighboring lattice). For small values of m , the classical and quantum return probabilities show power law behavior p(t) approximately t;{-0.5} and pi(t) approximately t;{-1} , respectively. However, for large values of m , the classical and quantum efficiency scales as p(t) approximately t;{-1} and pi(t) approximately t;{-2} . We give a theoretical explanation of this transition using the method of stationary phase approximation. In the long time limit, depending on the network size N and parameter m , the limiting probability distributions of quantum transport show various patterns. When the network size N is an even number, we find an asymmetric transition probability of quantum transport between the initial node and its opposite node. This asymmetry depends on the precise values of N and m . Finally, we study the transport processes in the presence of traps and find that the survival probability decays faster on networks of large m .