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arXiv:quant-ph/0601041 · Physical Review A

Loss of purity by wave-packet scattering at low energies

Jia Wang, C. K. Law, M.-C. Chu

Published Mar 9, 2006• 26 citations
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Abstract

We study the quantum entanglement produced by a head-on collision between two Gaussian wave packets in three-dimensional space. By deriving the two-particle wave function modified by $s$-wave scattering amplitudes, we obtain an approximate analytic expression of the purity of an individual particle. The loss of purity provides an indicator of the degree of entanglement. In the case the wave packets are narrow in momentum space, we show that the loss of purity is solely controlled by the ratio of the scattering cross section to the transverse area of the wave packets.

DOI
10.1103/physreva.73.034302

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We study the quantum entanglement produced by a head-on collision between two Gaussian wave packets in three-dimensional space. By deriving the two-particle wave function modified by sss-wave scattering amplitudes, we obtain an approximate analytic expression of the purity of an individual particle. The loss of purity provides an indicator of the degree of entanglement. In the case the wave packets are narrow in momentum space, we show that the loss of purity is solely controlled by the ratio of the scattering cross section to the transverse area of the wave packets.
Einstein, Podolsky, and Rosen discussed their paradox in terms of measuring the positions or momenta of two particles. These degrees of freedom can become entangled upon scattering, but how much entanglement can be created in this process? Here we address this question using fully coherent calculations of bipartite scattering in three-dimensional space, quantifying entanglement by the inverse of the single particle purity. We show that the standard plane-wave description of scattering fails to capture the entanglement properties, due to the essential role of quantum uncertainty in the initial state. For a more realistic description of a scattering setup, we find that the entanglement scales linearly with the scattering cross section, including strong enhancement near shape resonances, for sufficiently narrow initial momentum dispersion. We highlight the differences between scattering in one and higher spatial dimensions and discuss how the generation of motional entanglement can be detected in experiments. Our results open the way to probing, controlling, and eventually using entanglement in quantum collisions.
The scattering cross section is the effective area of collision when two particles collide. Quantum mechanically, it is a measure of the probability for a specific process to take place. Employing wave packets to describe the scattering process, we compute the entanglement entropy in 2-to-2 scattering of particles in a general setting using the S -matrix formalism. Applying the optical theorem, we show that the linear entropy E 2 is given by the elastic cross section σ el in unit of the transverse size L 2 of the wave packet, E 2 ∼ σ el / L 2 , when the initial states are not entangled. The result allows for dual interpretations of the entanglement entropy as an area and as a probability. Since σ el is generally believed, and observed experimentally, to grow with the collision energy s in the high energy regime, the result suggests a “second law” of entanglement entropy for high energy collisions. Furthermore, the Froissart bound places an upper limit on the entropy growth.
We study the entanglement creation in the low-energy scattering of two particles in three dimensions, for a general class of interaction potentials that are not required to be spherically symmetric. The incoming asymptotic state, before the collision, is a product of two normalized Gaussian states. After the scattering, the particles are entangled. We take as a measure of the entanglement the purity of one of them. We provide a rigorous explicit computation, with error bound, of the leading order of the purity at low energy. The entanglement depends strongly on the difference of the masses. It takes its minimum when the masses are equal, and it increases rapidly with the difference of the masses. It is quite remarkable that the anisotropy of the potential gives no contribution to the leading order of the purity, in spite of the fact that entanglement is a second-order effect.
We study the generation of quantum entanglement in two-particle scattering processes. When two wave packets collide, we discover that entanglement grows appreciably because of the interference between the incident and scattered waves. After the collision, quantum entanglement is mainly governed by the scattering phase shift and the widths of the incident wave packets. For interaction potentials that exhibit scattering resonance, we show that quantum entanglement can be enhanced near resonance energies.
We present universal relations between entanglement entropy, which quantifies the quantum correlation between subsystems, and the cross section, which is the primary observable for high-energy particle scattering, by employing a careful formulation of wave packets for the incoming particles. For 2-to-2 elastic scattering with no initial entanglement and subdividing the system along particle labels, we show that both the Rényi and Tsallis entropies in the final states are directly proportional to the elastic cross section in units of the transverse size for the initial wave packets, which is then interpreted as the elastic scattering probability. The relations do not depend on the underlying dynamics of the quantum field theory and are valid to all orders in coupling strengths. Furthermore, computing quantum correlations between momentum and nonkinematic data leads to entanglement entropies expressed as various semi-inclusive elastic cross sections. Our result gives rise to a novel “area law” for entanglement entropy in a two-body system.
We consider multiple collisions of quantum wave packets in one dimension. The system under investigation consists of an impenetrable wall and two hard-core particles with very different masses. The lighter particle bounces between the heavier one and the wall. Both particles are initially represented by narrow Gaussian wave packets. A complete analytical solution of this problem is presented. The idea of the method used is to decompose the two-particle wave function into a continuous superposition of terms (channels), such that the multiple collisions within each channel do not lead to subsequent entanglement between the two particles. For each channel, the time evolution of the two-particle wave function is completely determined by the motion of the corresponding classical pointlike particles; therefore the whole quantum problem is reduced to a classical calculation. The calculation based on the above method reveals the following unexpected result: The entanglement between the two particles first increases with time due to the collisions, but then it begins to decrease, disappearing completely when the light particle becomes too slow to catch up with the heavy one.