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Physical Review A

Relation of the entanglement entropy and uncertainty product in ground states of coupled anharmonic oscillators

Ning Ning Chung, Chen Er, Yong Siah Teo, Lock Yue Chew

Published Jul 16, 2010• 12 citations
Publisher

Abstract

We explore quantitatively the fundamental connection between the entropy and local uncertainty product as an entanglement measure within the lowest energy eigenstate of a class of two-coupled anharmonic oscillator systems. We find that these two quantum aspects approach a unique relation as the entanglement between the oscillators becomes large if the potential of the first collective mode displays a generic single-well potential structure. This implies that the uncertainty product can be a good choice for the characterization of entanglement when the entropy is sufficiently large. Interestingly, although this relation is robust against the anharmonic perturbation, we find that it can be altered effectively by a quantum catastrophe.

DOI
10.1103/physreva.82.014101

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We explore quantitatively the fundamental connection between the entropy and local uncertainty product as an entanglement measure within the lowest energy eigenstate of a class of two-coupled anharmonic oscillator systems. We find that these two quantum aspects approach a unique relation as the entanglement between the oscillators becomes large if the potential of the first collective mode displays a generic single-well potential structure. This implies that the uncertainty product can be a good choice for the characterization of entanglement when the entropy is sufficiently large. Interestingly, although this relation is robust against the anharmonic perturbation, we find that it can be altered effectively by a quantum catastrophe.
We have extended the applicability of the two-step approach to higher-dimensional problem, and have further developed it to evaluate the quantum dynamics and stationary states of a general class of coupled anharmonic oscillator system. This new approach has enabled us to investigate the topic of continuous variable entanglement within these coupled oscillator systems with good numerical efficiency. In particular, we have applied it to study the relation between entanglement, squeezing, and quantum uncertainties, of the systems’ ground states. Interestingly, while the relation between quadrature squeezing and entanglement depends sensitively on the nonlinear potential, the influence of nonlinear perturbation on the entropy-uncertainty relation is minimal, especially when the entanglement between the two oscillators is large. We have also explored into the dynamics of entanglement where we have found that entanglement dynamics can depend completely on the global classical dynamical regime, without being influenced by the local classical behavior.
We have generalized the two-step approach to the solution of systems of NNN coupled quantum anharmonic oscillators. By using the squeezed vacuum state of each individual oscillator, we construct the tensor product state, and obtain the optimal squeezed vacuum product state through energy minimization. We then employ this optimal state and its associated bosonic operators to define a basis set to construct the Heisenberg matrix. The diagonalization of the matrix enables us to obtain the energy eigenvalues of the coupled oscillators. In particular, we have applied our formalism to determine the eigenenergies of systems of two coupled quantum anharmonic oscillators perturbed by a general polynomial potential, as well as three and four coupled systems. Furthermore, by performing a first-order perturbation analysis about the optimal squeezed vacuum product state, we have also examined into the squeezing properties of two coupled oscillator systems.
We study entanglement in a system comprising two coupled quartic oscillators. It is shown that the entanglement, as measured by the von Neumann entropy, increases with the classical chaos parameter for generic chaotic eigenstates. We consider certain isolated periodic orbits whose bifurcation sequence affects a class of strongly scarred quantum eigenstates, called the channel localized states. For these states, the entanglement is a local minima in the vicinity of a pitchfork bifurcation but is a local maxima near an antipitchfork bifurcation. We place these results in the context of the close connections that may exist between entanglement measures and conventional measures of localization. We also point to an interesting near degeneracy that arises in the spectrum of reduced density matrices of certain states as an interplay between localization and symmetry.
We investigate the quantum entanglement dynamics of undriven anharmonic (nonlinear) oscillators with quartic potentials. We first consider the indirect interaction between two such nonlinear oscillators mediated by a third, linear oscillator and show that this leads to a time-varying entanglement of the oscillators, the entanglement being strongly influenced by the nonlinear oscillator dynamics. In the presence of dissipation, the role of nonlinearity is strongly manifested in the steady-state dynamics of the indirectly coupled anharmonic oscillators. We further illustrate the effect of nonlinearities by studying the coupling between an electromagnetic field in a cavity and one movable mirror which is modelled as a nonlinear oscillator. For this case, we present a full analytical treatment, which is valid in a regime where both the nonlinearity and the coupling due to radiation pressure are weak. We show that, without the need of any conditional measurements on the cavity field, the state of the movable mirror is non-classical as a result of the combined effect of the intrinsic nonlinearity and the radiation-pressure coupling. This interaction is also shown to be responsible for squeezing the movable mirror's position quadrature beyond the minimum uncertainty state even when the mirror is initially prepared in its ground state.
The time evolution of a quantum particle’s product of uncertainties in position and momentum is calculated when it is coupled with an external source. We have used a simple toy model where the particle is subject to a harmonic potential and coupled with an equivalent harmonic oscillator via a linear term. It is found that the long-time-averaged product is an increasing function of the coupling strength. It diverges when one of the eigenmodes of the coupled system goes soft, with the singular term twice of that for the stationary state. Generally, there is a jump of finite size for this quantity when a small coupling is turned on, compared to the uncoupled case. Similar behaviors have also been found for the von Neumann entanglement entropy, which is calculated exactly using a covariance matrix formalism. We find that the mode-interference plays an important role in the main features of this work.
In this paper, we investigate into the numerical and analytical relationship between the dynamically generated quadrature squeezing and entanglement within a coupled harmonic oscillator system. The dynamical relation between these two quantum features is observed to vary monotically, such that an enhancement in entanglement is attained at a fixed squeezing for a larger coupling constant. Surprisingly, the maximum attainable values of these two quantum entities are found to consistently equal to the squeezing and entanglement of the system ground state. In addition, we demonstrate that the inclusion of a small anharmonic perturbation has the effect of modifying the squeezing versus entanglement relation into a nonunique form and also extending the maximum squeezing to a value beyond the system ground state.