Multipartite entanglement in harmonic oscillators subjected to dissipative quantum dynamics

Abstract

A meaningful description of the low-energy dynamics of multipartite entanglement is provided for harmonic oscillator systems in general dissipative scenarios. Without doing any central approximation, this mainly relies on a set of reasonable physical statements on the environment and system-environment interaction consistent with a linear analysis of the open-system dynamics. We rst investigate the inseparability properties of arbitrary entangled states, which generally entails an optimization procedure of certain functional de ned in the in nitedimensional Hilbert space of continuous-variable systems. Using a Gaussian-like assumption analogous to that used in the derivation of Gaussian entanglement of formation, we derive a computational-e cient form of such functional which considerable simpli es the optimization task. This consists on a hierarchy of separability criteria that permit in a uni ed way to characterize k-partite entanglement of broad classes of Gaussian and non- Gaussian states. The strength of the criteria is proven by showing that they satisfactorily reproduce previous results from the theory of entanglement, like PPT criterion applied to arbitrary two-mode and pure three-mode Gaussian states. The separability criteria allow to monitor the transient evolution of multipartite entanglement under the environmental in uence, this permitted to show that non-Gaussian entanglement may be as robust against harmful dissipative e ects and thermal noise as Gaussian one. We devote special attention to the stationary properties of tripartite entanglement when the system oscillators are in contact with either a common environment or independent environments at initial di erent temperatures. For the former dissipative scheme, it is shown that the environment mediates an e ective many-party interaction with a spatial long-range feature between system oscillators, which is able to generate tripartite entanglement whereas two-mode entanglement is degraded. Regarding the second scheme, it is illustrated that thermal non-equilibrium conditions result in a rich variety of quantum correlations, however a temperature gradient eventually destroys the entanglement shared by the system oscillators as a consequence of the growth of thermal noise. Finally, the stationary entanglement and energy current across a nite harmonic chain are studied in detail by doing an extensive numerical analysis of both phenomena for a broad range of the parameters (i.e., temperatures, oscillator frequencies, and oscillator coupling strengths). Our numerical ndings are discussed in terms of the derived energy current expressions, these show an explicit dependence on the two-time correlation functions (between oscillator operators) which carry quantum correlations.

Una detallada descripción de la dinámica de bajas energías del entrelazamiento multipartito es proporcionada para sistemas armónicos en una gran variedad de escenarios disipativos. Sin hacer ninguna aproximación central, esta descripción yace principalmente sobre un conjunto razonable de hipótesis acerca del entorno e interacción entorno-sistema, ambas consistente con un análisis lineal de la dinámica disipativa.

En la primera parte se deriva un criterio de inseparabilidad capaz de detectar el entrelazamiento k-partito de una extensa clase de estados gausianos y no-gausianos en sistemas de variable continua. Este criterio se emplea para monitorizar la dinámica transitiva del entrelazamiento, mostrando que los estados no-gausianos pueden ser tan robustos frente a los efectos disipativos como los gausianos. Especial atención se dedicada a la dinámica estacionaria del entrelazamiento entre tres osciladores interaccionando con el mismo entorno o diferentes entornos a distintas temperaturas. Este estudio contribuye a dilucidar el papel de las correlaciones cuánticas en el comportamiento de la corrientes energéticas.