Gaussian entanglement induced by an extended thermal environment

We study stationary entanglement between three harmonic oscillators which are dipole coupled to a one-dimensional or a three-dimensional bosonic environment. The analysis of the open-system dynamics is performed with generalized quantum Langevin equations which we solve exactly in a Fourier representation. The focus lies on Gaussian bipartite and tripartite entanglement induced by the highly non-Markovian interaction mediated by the environment. This environment-induced interaction represents an effective many-party interaction with a spatial long-range feature: A main finding is that the presence of a passive oscillator is detrimental for stationary two-mode entanglement. Furthermore, our results indicate that the environment-induced entanglement mechanism corresponds to uncontrolled feedback which is predominantly coherent at low temperatures and for moderate oscillator-environment coupling as compared to the oscillator frequency.

Figure 1

Oscillator-environment configuration considered in this paper. Three oscillators are confined to the direction indicated by the arrow; in the 1D arrangement (a) the oscillators move only in the $x$ direction, while in the 3D configuration (b), the oscillators move only along the $z$ direction. The interaction between the oscillators is mediated by a bosonic field, which also causes decoherence and quantum dissipation.

Figure 2

Stationary two-mode entanglement measured by the logarithmic negativity (27) as a function of oscillator distance $R$ and temperature $T$ for a (a) 1D and a (b) 3D environment. The oscillator frequencies are ${\omega }_{\mathcal{A}}=7.2\Omega$, and ${\omega }_{\mathcal{B}}=13.2\Omega$, while the dissipation is $\gamma =5\Omega$.

Figure 3

Stationary two-mode entanglement in the linear arrangement quantified by the logarithmic negativities $E_N\left({\rho }_{\mathcal{A}\mathcal{C}}\right)$ (black solid line), $E_N\left({\rho }_{\mathcal{A}\mathcal{B}}\right)$ (red dashed), and $E_N\left({\rho }_{\mathcal{B}\mathcal{C}}\right)$ (blue dashed-dotted) for $R{\omega }_c/c=0.933$. Temperature and damping are $k_{B}T/\hslash {\omega }_c=0.026$, $\gamma =5\Omega$, respectively, while the frequencies are ${\omega }_{\mathcal{A}}=7.2\Omega$, ${\omega }_{\mathcal{B}}=10.1\Omega$, and ${\omega }_{\mathcal{C}}=13.2\Omega$, where $\Omega =1$ GHz. The asymmetry between $E_N\left({\rho }_{\mathcal{A}\mathcal{B}}\right)$ and $E_N\left({\rho }_{\mathcal{B}\mathcal{C}}\right)$ is a consequence of choosing different oscillator frequencies. The entanglement between $\mathcal{A}$ and $\mathcal{B}$ is less sensitive to a moderate increase of temperature (not shown), because it involves the oscillators with the highest frequencies. The inset is a zoom that demonstrates the small quadratic increase of $E_N\left({\rho }_{\mathcal{A}\mathcal{C}}\right)$.

Figure 4

(a) Stationary two-mode entanglement measured by the logarithmic negativities $E_N\left({\rho }_{\mathcal{A}\mathcal{C}}\right)$ (black solid line), $E_N\left({\rho }_{\mathcal{A}\mathcal{B}}\right)$ (red dashed line), and $E_N\left({\rho }_{\mathcal{B}\mathcal{C}}\right)$ (blue dashed-dotted line) for the triangular geometry with $R{\omega }_c/c=0.167$ as a function of the displacement $r$. All other parameters are as in Fig. 3. The inset provides an extended picture of $E_N\left({\rho }_{\mathcal{A}\mathcal{C}}\right)$, where the dotted line marks the value in the absence of oscillator $\mathcal{B}$. (b) Phase diagram for fixed $R$ and various values of $r$ as a function of coupling strength $\gamma$ and temperature $T$. In the shaded areas, the oscillators $\mathcal{A}$ and $\mathcal{B}$ exhibit stationary entanglement. The outer blue line marks the limit $r\to \infty$, which is equivalent to the absence of oscillator $\mathcal{B}$. As oscillator $\mathcal{B}$ comes closer, the area with entanglement shrinks.

Figure 5

Phase diagram of the separability classes for the linear configuration as a function of temperature and position $r$ of oscillator $\mathcal{B}$. All other parameters are as in Fig. 3.

Figure 6

(a) Separability phase diagram for the equilateral triangular for the oscillator frequencies and coupling strengths used in Fig. 4. (b) Quantum fidelity between the stationary state and the thermal canonical state as a function of the temperature for the distances $R{\omega }_c/c=0.066$ (blue dashed line) and $R{\omega }_c/c=2.367$ (pink dashed-dotted line).