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New Journal of Physics

Quantum entanglement in coupled harmonic oscillator systems: from micro to macro

Jhih-Yuan Kao, Chung-Hsien Chou

Published Jul 1, 2016• 11 citations
Publisher

Abstract

We investigate the entanglement dynamics of several models of coupled harmonic oscillators, whereby a number of properties concerning entanglement have been scrutinized, such as how the environment affects entanglement of a system, and death and revival of entanglement. Among them, there are two models for which we are able to vary their particle numbers easily by assuming identicalness, thereby examining how the particle number affects entanglement. We have found that the upper bound of entanglement between identical oscillators is approximately inversely proportional to the particle number.

DOI
10.1088/1367-2630/18/7/073001

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We investigate the entanglement dynamics of several models of coupled harmonic oscillators, whereby a number of properties concerning entanglement have been scrutinized, such as how the environment affects entanglement of a system, and death and revival of entanglement. Among them, there are two models for which we are able to vary their particle numbers easily by assuming identicalness, thereby examining how the particle number affects entanglement. We have found that the upper bound of entanglement between identical oscillators is approximately inversely proportional to the particle number.
The coupled quantum harmonic oscillator is one of the most researched and important model systems in quantum optics and quantum informatics. This system is often investigated for quantum entanglement in the environment. As a result, such studies are complex and can only be carried out using numerical methods that do not reveal the general pattern of such systems. In this work, the external environment is considered to be two independent particles interacting with coupled harmonic oscillators. It is shown that such a system has an exact analytical solution to the dynamic Schrödinger equation. The analysis of this solution is carried out, and the main parameters of this system are revealed. The solutions obtained can be used to study more complex systems and their quantum entanglement.
We study the entanglement dynamics of a system consisting of a large number of coupled harmonic oscillators in various configurations and for different types of nearest-neighbour interactions. For a one-dimensional chain, we provide compact analytical solutions and approximations to the dynamical evolution of the entanglement between spatially separated oscillators. Key properties such as the speed of entanglement propagation, the maximum amount of transferred entanglement and the efficiency for the entanglement transfer are computed. For harmonic oscillators coupled by springs, corresponding to a phonon model, we observe a non-monotonic transfer efficiency in the initially prepared amount of entanglement, i.e. an intermediate amount of initial entanglement is transferred with the highest efficiency. In contrast, within the framework of the rotating-wave approximation (as appropriate, e.g. in quantum optical settings) one finds a monotonic behaviour. We also study geometrical configurations that are analogous to quantum optical devices (such as beamsplitters and interferometers) and observe characteristic differences when initially thermal or squeezed states are entering these devices. We show that these devices may be switched on and off by changing the properties of an individual oscillator. They may therefore be used as building blocks of large fixed and pre-fabricated but programmable structures in which quantum information is manipulated through propagation. We discuss briefly possible experimental realizations of systems of interacting harmonic oscillators in which these effects may be confirmed experimentally
We explore in this paper ways to qualify and quantify the quantum entanglement E(M)E(M)E(M) between two macroscopic objects by way of model studies. Knowing that a macroscopic object is a composite, how does one determine E(M)E(M)E(M) in terms of the entanglements between its micro-constituents E(μ)E(μ)E(μ)? We assert that the notion of `levels of structure', the coupling strength between constituents in different levels, and the use of collective variables in each level are all pertinent factors. We consider two types of coupling, each constituent particle is coupled to only one other particle (1-to-1) versus it coupled to all particles (1-to-all). In the 1-1 case with pairwise interactions of equal strength, the entanglement is independent of the number of constituent particles NNN in the macroscopic object. In the 1-to-all case the relative coordinates are decoupled and the center of mass (CoM) coupling scales with NNN. We provide a proof of the conditions whereby the CoM variable decouples, a cause for the special role the CoM variable plays in the entanglement between the two such macroscopic objects. This qualitative behavior is largely not affected by fluctuations in the interaction strength. We also analyzed the entanglement pattern of 4 coupled oscillators in two pairs, representing the two objects A and B (or two adjacent levels of structure), each with two constituents. By assigning different coupling strengths we can investigate the interplay of inter-level entanglement with intra- level interactions. From the entanglement dynamics of the 4-oscillator system with varying coupling strength we see the entanglement between constituents meeting sudden death while the CoM variables may sustain over longer times. This offers another way to determine under what conditions quantum entanglement between macroscopic objects can persist.
We study the entanglement dynamics of a system consisting of a large number of coupled harmonic oscillators in various configurations and for different types of nearest-neighbour interactions. For a one-dimensional chain, we provide compact analytical solutions and approximations to the dynamical evolution of the entanglement between spatially separated oscillators. Key properties such as the speed of entanglement propagation, the maximum amount of transferred entanglement and the efficiency for the entanglement transfer are computed. For harmonic oscillators coupled by springs, corresponding to a phonon model, we observe a non-monotonic transfer efficiency in the initially prepared amount of entanglement, i.e. an intermediate amount of initial entanglement is transferred with the highest efficiency. In contrast, within the framework of the rotating-wave approximation (as appropriate, e.g. in quantum optical settings) one finds a monotonic behaviour. We also study geometrical configurations that are analogous to quantum optical devices (such as beamsplitters and interferometers) and observe characteristic differences when initially thermal or squeezed states are entering these devices. We show that these devices may be switched on and off by changing the properties of an individual oscillator. They may therefore be used as building blocks of large fixed and pre-fabricated but programmable structures in which quantum information is manipulated through propagation. We discuss briefly possible experimental realizations of systems of interacting harmonic oscillators in which these effects may be confirmed experimentally.
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.
We study the dynamics of the entanglement between two oscillators that are initially prepared in a general two-mode Gaussian state and evolve while coupled to the same environment. In a previous paper, we showed that there are three qualitatively different dynamical phases for the entanglement in the long-time limit: sudden death, sudden death and revival, and no sudden death [Paz and Roncaglia, Phys. Rev. Lett. 100, 220401 (2008)]. Here we generalize and extend those results along several directions: We analyze the fate of entanglement for an environment with a general spectral density providing a complete characterization of the evolution for Ohmic, sub-Ohmic, and super-Ohmic environments. We consider two different models for the interaction between the system and the environment (one where the coupling is through position and another where the coupling is symmetric in position and momentum). Finally, we show that for nonresonant oscillators, the final entanglement is independent of the initial state and may be nonzero at very low temperatures.