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arXiv:1006.0550 · Physical Review E

Escape behavior of quantum two-particle systems with Coulomb interactions

Tooru Taniguchi, Shin‐ichi Sawada

Published Feb 28, 2011• 30 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreve.83.026208

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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.
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.
Abstract We consider the problem of two interacting particles in one-dimensional harmonic confinement. By considering both particles are subject to the same harmonic curvature, we obtain exact numerical solutions in a straightforward way for any choice of interaction potential. The formulation is then applied in the situations of (i) non-identical particles with different masses m 1 and m 2; (ii) bosons and fermions; (iii) short and long range interactions; and (iv) combining all these ingredients within a time-dependent applied electric field. We analyze the role of the mass imbalance and Pauli's exclusion principle by investigating the formation of bound pairs, the expected value of the separation between the particles and the effects of interaction and exclusion principle on the density distributions.
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.
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.
We study resonances and time decay in a quantum waveguide with two semitransparent barriers. The model is first analyzed as a one-dimensional Schrödinger operator with two delta interactions and then extended to a two-dimensional strip, where transverse Dirichlet modes reduce the problem, channel by channel, to the one-dimensional setting. We derive explicit scattering amplitudes and characterize resonance poles by analytic continuation. In the strong-barrier regime ρ=α−1→0+, the real parts of the resonances approach the eigenvalues of the decoupled cavity, while the imaginary parts are negative and of order α−2, so the lifetimes grow like α2. We also study the time evolution on short, intermediate, and long time scales: the survival probability is quadratic at short times, in accordance with the quantum Zeno effect; the intermediate regime is resonance-dominated and approximately exponential; and the long-time behavior is governed by the low-energy threshold and becomes algebraic. Numerical simulations complement the analysis by confirming these regimes and by showing the breakdown of exponential decay. They also provide an empirical fit for the breakdown time tbr in terms of the barrier strength α and the distance d1 between the barriers.
We study the decay of two repulsively interacting bosons tunnelling through a delta potential barrier by a direct numerical solution of the time-dependent Schrödinger equation. The solutions are analysed according to the regions of particle presence: both particles inside the trap (in–in), one particle in and one particle out (in–out) and both particles outside (out–out). It is shown that the in–in probability is dominated by the exponential decay, and its decay rate is predicted very well from outgoing boundary conditions. Up to a certain range of interaction strength, the decay of in–out probability is dominated by the single-particle decay mode. The decay mechanisms are adequately described by simple models.