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arXiv:quant-ph/0308099 · Physical Review Letters

Semiclassical Time Evolution of the Reduced Density Matrix and Dynamically Assisted Generation of Entanglement for Bipartite Quantum Systems

Philippe Jacquod

Published Apr 13, 2004• 81 citations
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Abstract

Two particles, initially in a product state, become entangled when they come together and start to interact. Using semiclassical methods, we calculate the time evolution of the corresponding reduced density matrix rho(1), obtained by integrating out the degrees of freedom of one of the particles. We find that entanglement generation sensitively depends (i) on the interaction potential, especially on its strength and range, and (ii) on the nature of the underlying classical dynamics. Under general statistical assumptions, and for short-ranged interaction potentials, we find that P(t) decays exponentially fast in a chaotic environment, whereas it decays only algebraically in a regular system. In the chaotic case, the decay rate is given by the golden rule spreading of one-particle states due to the two-particle coupling, but cannot exceed the system's Lyapunov exponent.

DOI
10.1103/physrevlett.92.150403

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Two particles, initially in a product state, become entangled when they come together and start to interact. Using semiclassical methods, we calculate the time evolution of the corresponding reduced density matrix rho(1), obtained by integrating out the degrees of freedom of one of the particles. We find that entanglement generation sensitively depends (i) on the interaction potential, especially on its strength and range, and (ii) on the nature of the underlying classical dynamics. Under general statistical assumptions, and for short-ranged interaction potentials, we find that P(t) decays exponentially fast in a chaotic environment, whereas it decays only algebraically in a regular system. In the chaotic case, the decay rate is given by the golden rule spreading of one-particle states due to the two-particle coupling, but cannot exceed the system's Lyapunov exponent.
We show how a classically vanishing interaction generates entanglement between two initially nonentangled particles, without affecting their classical dynamics. For chaotic dynamics, the rate of entanglement is shown to saturate at the Lyapunov exponent of the classical dynamics as the interaction strength increases. In the saturation regime, the one-particle Wigner function follows classical dynamics better and better as one goes deeper and deeper in the semiclassical limit. This demonstrates that quantum-classical correspondence at the microscopic level does not require coupling to a large number of external degrees of freedom.
The dynamical generation of entanglement in closed bipartite systems is investigated in the semiclassical regime. We consider a model of two particles, initially prepared in a product of coherent states, evolving in time according to a generic Hamiltonian, and derive a formula for the linear entropy of the reduced density matrix using the semiclassical propagator in the coherent-state representation. The formula is explicitly written in terms of quantities that define the stability of classical trajectories of the underlying classical system. The formalism is then applied to the problem of two nonlinearly coupled harmonic oscillators, and the result is shown to be in remarkable agreement with the exact quantum measure of entanglement in the short-time regime. An important by-product of our approach is a unified semiclassical formula, which contemplates both the coherent-state propagator and its complex conjugate.
We analyze the dynamical generation of entanglement in systems of two interacting spins initially prepared in a product of spin coherent states. For arbitrary time-independent Hamiltonians, we derive a semiclassical expression for the purity of the reduced density matrix as a function of time. The final formula, subsidiary to the linear entropy, shows that the short-time dynamics of entanglement depends exclusively on the stability of trajectories governed by the underlying classical Hamiltonian. In addition, this semiclassical measure is shown to reproduce the general properties of its quantum counterpart and give the expected result in the large spin limit. The accuracy of the semiclassical formula is further illustrated in a problem of phase exchange for two particles of spin jjj.
We study dynamical generation of entanglement in bipartite quantum systems, characterized by purity (or linear entropy), and caused by the coupling between the two subsystems. The explicit semiclassical theory of purity decay is derived for integrable classical dynamics of the uncoupled system and for localized (general Gaussian wave packet) initial states. Purity decays as an algebraic function of (time) $\ifmmode\times\else\texttimes\fi{}$ (strength of perturbation), independently of the Planck's constant.
PACS. 03.67.Mn – Entanglement production, characterization, and manipulation. PACS. 05.45.Mt – Quantum chaos; semiclassical methods. PACS. 34.60.+z – Scattering in highly excited states (e.g., Rydberg states). Abstract. – The generation of entanglement produced by a local potential interaction in a bipartite system is investigated. The degree of entanglement is contrasted with the underlying classical dynamics for a Rydberg molecule (a charged particle colliding on a kicked top). En-tanglement is seen to depend on the structure of classical phase-space rather than on the global dynamical regime. As a consequence, regular classical dynamics can in certain circumstances be associated with higher entanglement generation than chaotic dynamics. In addition, quan-tum effects also come into play: for example, partial revivals, which are expected to persist in the semiclassical limit, affect the long-time behaviour of the reduced linear entropy. These results suggest that entanglement may not be a pertinent universal signature of chaos. Entanglement, i.e. the nonseparability intrinsic to composite systems, is one of the most peculiar features of the quantum world. In its most popular form, found for example in the original EPR proposal [1], the entanglement is of geometrical nature. However any usual
The generation of entanglement produced by a local potential interaction in a bipartite system is investigated. The degree of entanglement is contrasted with the underlying classical dynamics for a Rydberg molecule (a charged particle colliding on a kicked top). Entanglement is seen to depend on the structure of classical phase-space rather than on the global dynamical regime. As a consequence, regular classical dynamics can in certain circumstances be associated with higher entanglement generation than chaotic dynamics. In addition, quantum effects also come into play: for example, partial revivals, which are expected to persist in the semiclassical limit, affect the long-time behaviour of the reduced linear entropy. These results suggest that entanglement may not be a pertinent universal signature of chaos.