[Paper Review] Entanglement in a Bipartite Gaussian State

Anne Ghesquière|arXiv (Cornell University)|May 3, 2011

Quantum Information and Cryptography49 references3 citations

TL;DR

This paper investigates entanglement dynamics in a two-particle Gaussian state coupled to a thermal bath using the Non-Rotating Wave Approximation (NRWA) master equation. It shows that entanglement decays rapidly under free evolution, while in the presence of a harmonic potential, entanglement either vanishes quickly in the over-damped regime or oscillates toward a stable non-zero value in the under-damped regime, depending on the damping strength relative to the potential frequency.

ABSTRACT

To examine the loss of entanglement in a two-particle Gaussian system, we couple it to an environment and use the Non-Rotating Wave master equation to study the system's dynamics. We also present a derivation of this equation. We consider two different types of evolution. Under free evolution we find that entanglement is lost quickly between the particles. When a harmonic potential is added between the particles, two very different behaviours can be observed, namely in the over and under-damped cases respectively, where the strength of the damping is determined by how large the coupling to the bath is with respect to the frequency of the potential. In the over-damped case, we find that the entanglement vanishes at even shorter times than it does in the free evolution. In the (very) under-damped case, we observe that the entanglement does not vanish. Instead it oscillates towards a stable value.

Motivation & Objective

• To analyze the time evolution of entanglement in a two-particle Gaussian system interacting with a thermal bath.
• To investigate how the presence of a harmonic potential between the particles modifies entanglement dynamics under environmental coupling.
• To derive and apply the Non-Rotating Wave Approximation (NRWA) master equation for open quantum systems with Gaussian states.
• To compare entanglement decay in free evolution versus harmonic potential scenarios, focusing on damping regimes (over- and under-damped).

Proposed method

• Derives the Non-Rotating Wave master equation from the Hamiltonian of a system-bath interaction using the Heisenberg equation of motion and a bath model.
• Applies the master equation to a two-particle Gaussian state, modeling the system as a pair of particles coupled to a heat bath.
• Uses the logarithmic negativity as a measure of entanglement to quantify entanglement dynamics over time.
• Solves the master equation for two cases: free particle evolution and evolution under a harmonic potential.
• Analyzes the damping regime by comparing the coupling strength to the potential frequency, distinguishing over-damped and under-damped behaviors.
• Employs the quantum Langevin equation and density matrix formalism to describe the system's stochastic dynamics under environmental coupling.

Experimental results

Research questions

• RQ1 How does entanglement evolve in a two-particle Gaussian state when coupled to a thermal bath under free evolution?
• RQ2 What is the effect of a harmonic potential between two particles on the entanglement dynamics in the presence of environmental coupling?
• RQ3 How does the damping regime—over-damped versus under-damped—determine the long-term behavior of entanglement in a bipartite Gaussian system?
• RQ4 Does the Non-Rotating Wave Approximation (NRWA) master equation yield qualitatively different entanglement decay patterns compared to the standard rotating wave approximation?
• RQ5 Can entanglement be preserved or stabilized in a Gaussian system under environmental coupling when a harmonic potential is introduced?

Key findings

• In free evolution, entanglement decays rapidly due to environmental coupling, with no long-term stability.
• In the over-damped regime (strong coupling to bath relative to potential frequency), entanglement vanishes even more quickly than in free evolution.
• In the very under-damped regime (weak coupling to bath relative to potential frequency), entanglement does not vanish but instead oscillates and stabilizes at a non-zero value.
• The logarithmic negativity shows distinct time evolution profiles depending on the damping regime, with oscillatory behavior in the under-damped case.
• The Non-Rotating Wave master equation captures non-Markovian effects and memory-dependent dynamics, which are crucial for accurate entanglement evolution in strongly coupled systems.
• The harmonic potential acts as a stabilizing influence in the under-damped regime, counteracting environmental decoherence and preserving entanglement.