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Physical Review A

Entanglement dynamics of many-body systems: Analytical results

A. L. de Paula, J. G. G. de Oliveira, J. G. Peixoto de Faria, Dagoberto S. Freitas, Maria Carolina Nemes

Published Feb 5, 2014• 13 citations
Publisher

Abstract

We consider a many-body system consisting of a harmonic oscillator linearly coupled to $N$ others and solve the corresponding dynamical problem analytically. Initially, the main oscillator is prepared in a superposition of two coherent states and the $N$ others in the ground states. We divide this system in arbitrary three partitions and the entanglement dynamics between any of these partitions is quantified. We show that the residual entanglement of the tripartite system is always present. Besides, the concurrence of any two partitions depends on the overlap between the two coherent states forming the initial superposition of the main oscillator and the distribution of excitations in the partitions. Finally, we obtain a hierarchy of entanglement between the central oscillator and partitions formed by the other oscillators. For long times the excitations of the main oscillator are completely transferred to the $N$ others and these are found entangled.

DOI
10.1103/physreva.89.022303

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We consider a many-body system consisting of a harmonic oscillator linearly coupled to NNN others and solve the corresponding dynamical problem analytically. Initially, the main oscillator is prepared in a superposition of two coherent states and the NNN others in the ground states. We divide this system in arbitrary three partitions and the entanglement dynamics between any of these partitions is quantified. We show that the residual entanglement of the tripartite system is always present. Besides, the concurrence of any two partitions depends on the overlap between the two coherent states forming the initial superposition of the main oscillator and the distribution of excitations in the partitions. Finally, we obtain a hierarchy of entanglement between the central oscillator and partitions formed by the other oscillators. For long times the excitations of the main oscillator are completely transferred to the NNN others and these are found entangled.
In this work, we extend the discussion that began in Ref. 16 [A. L. de Paula, Jr., J. G. G. de Oliveira, Jr., J. G. P. de Faria, D. S. Freitas and M. C. Nemes, Phys. Rev. A 89 (2014) 022303] to deal with the dynamics of the concurrence of a many-body system. In that previous paper, the discussion was focused on the residual entanglement between the partitions of the system. The purpose of the present contribution is to shed some light on the dynamical properties of entanglement among the environment oscillators. We consider a system consisting of a harmonic oscillator linearly coupled to N others and solve the corresponding dynamical problem analytically. We divide the environment into two arbitrary partitions and the entanglement dynamics between any of these partitions is quantified and it shows that in the case when excitations in each partition are equal, the concurrence reaches the value 1 and the two partitions of the environment are maximally entangled. For long times, the excitations of the main oscillator are completely transferred to environment and the environment oscillators are found entangled.
We studied the entanglement dynamics of a many body system. The model that we used consists of a harmonic oscillator (main oscillator) linearly coupled to N others. Initially, the main oscillator is prepared in a superposition of two coherent states and the N others in the ground states. From this initial state we obtain the temporal evolution of the system analytically. We divide this system in three arbitrary partitions and we quantify the entanglement, by concurrence, between any two partitions. We show that the concurrence of any two partitions depends on the overlap between the two coherent states forming the initial superposition of the main oscillator and the distribution of excitations in the partitions. We also show that the residual entanglement is always present. Finally, we obtain a hierarchy of entanglement between the main oscillator and partitions formed by the N others. For long times, the excitations of the main oscillator are completely transferred to the N others and these are found entangled.
We investigate the quantum entanglement dynamics of two coupled harmonic oscillators with a time-dependent interaction. Using the Lewis-Riesenfeld invariant method, we derive the exact analytical wave functions without any perturbative or adiabatic approximations and combine this with a phase-space analysis using the Wigner function to provide a complete description of the system's quantum state evolution. We obtain general expression form for the purity and the linear entropy SL=1−PS_L=1-\mathcal{P}SL​=1−P for arbitrary excitation numbers (n,m)(n,m)(n,m), allows a systematic study of entanglement for a large class of quantum states. We show that the entanglement dynamics is very sensitive to the interplay between the detuning parameters θθθ and ϑ2\vartheta_2ϑ2​, the frequency parameter β0β_0β0​ and the coupling strength εεε: the increase of detuning takes the system from slow irregular oscillations to fast and regular periodic behavior, and the stronger coupling systematically enhances both the amplitude and the average value of the linear entropy. Most importantly, for the resonance case ω1=ω2=1ω_1=ω_2=1ω1​=ω2​=1 and strong couplings ε≈0.99ε\approx 0.99ε≈0.99, the system shows robust undamped synchronized periodic oscillations of the linear entropy for all quantum states considered, indicating preserved quantum coherence without saturation. Our findings demonstrate that linear entropy is a sensitive and practical entanglement witness, and we establish explicit analytical relations between the coupling parameters of the system and its entanglement properties, which are directly relevant to quantum information processing and the control of quantum correlations in continuous-variable systems.
In many applications of quantum optics, nonlinear physics, molecular chemistry and biophysics, one can encounter models in which the coupled quantum harmonic oscillator provides an explanation for many physical phenomena and effects. In general, these are harmonic oscillators coupled via coordinates and momenta, which can be represented as H^=∑i=12p^i22mi+miωi22xi2+H^int, where the interaction of two oscillators H^int=ik1x1p^2+ik2x2p^1+k3x1x2−k4p^1p^2. Despite the importance of this system, there is currently no general solution to the Schrödinger equation that takes into account arbitrary initial states of the oscillators. Here, this problem is solved in analytical form, and it is shown that the probability of finding the system in any states and quantum entanglement depends only on one coefficient R∈(0,1) for the initial factorizable Fock states of the oscillator and depends on two parameters R∈(0,1) and ϕ for arbitrary initial states. These two parameters R∈(0,1) and ϕ include the entire set of variables of the system under consideration.
Quantum entanglement serves as a key phenomenon in understanding correlations in many-body systems, but analytical results remain scarce for coupled three-body oscillators. In this work, we address this gap by introducing a geometrical diagonalization approach that constrains Euler angles, thereby reducing the degrees of freedom in the entanglement analysis. It consists of deriving analytical expressions for linear entropy and purity under the bipartitions (x∣yz)(x|yz)(x∣yz), (y∣xz)(y|xz)(y∣xz), and (xy∣z)(xy|z)(xy∣z) using the Wigner function framework. Our results indicate that excitations in any oscillator basically enhance the redistribution of correlations across the system. The mixing angle θθθ governs entanglement intensity, ranging from separability to maximal correlation. Moreover, we reveal the symmetry relations, notably SLy[(n,m,l),θ]=SLz[(n,m,l),−θ]S_{Ly}[(n,m,l),θ]=S_{Lz}[(n,m,l),-θ]SLy​[(n,m,l),θ]=SLz​[(n,m,l),−θ] and an intrinsic symmetry within (x∣yz)(x|yz)(x∣yz). Hence, we clarify how excitation levels and mixing angles create and enhance entanglement in the three coupled harmonic oscillators.