Quantum correlations and energy currents across three dissipative oscillators

We present a study that addresses both the stationary properties of the energy current and quantum correlations in a three-mode chain subjected to Ohmic and super-Ohmic dissipations. An extensive numerical analysis shows that the mean value and the fluctuations of the energy current remain insensitive to the emergence of a rich variety of quantum correlations, such as two-mode discord and entanglement and bipartite three-mode and genuine tripartite entanglement. The discussion of the numerical results is based on the derived expressions for the stationary properties in terms of the two-time correlation functions of the oscillator operators, which carry the quantum correlations. Interestingly, we show that quantum discord can be enhanced by considering both initially squeezed thermal bath states and imposing temperature gradients.

Figure 1

Schematic representation of the chain composed of the three oscillators coupled to independent heat baths, with temperatures $T_{\mathcal{L}}, T_{\mathcal{C}}$, and $T_{\mathcal{R}}$. ${\widehat{j}}_{iB}(i=\mathcal{L},\mathcal{C},\mathcal{R})$ indicates the energy current from the heat bath to the $i\mathrm{th}$ oscillator and ${\widehat{j}}_{ij}$ the energy current from the $j\mathrm{th}$ to the $i\mathrm{th}$ oscillator, in the case of $T_{\mathcal{L}}>T_{\mathcal{C}}>T_{\mathcal{R}}$. $k$ is the coupling constant between first-neighbor oscillators, and $g_{i\mu }$ the coupling constant between the $i$-th chain oscillator and the $\mu \mathrm{th}(\mu =1,\cdots ,N)$ oscillator of the bath.

Figure 2

(a) The two-mode $\mathcal{L}|\mathcal{R}$ entanglement (labels on the left), and the $\mathcal{C}|\mathcal{R}$ and $\mathcal{L}|\mathcal{C}$ entanglements (labels on the right) as a function of the temperature gradient $\mathrm{\Delta }T$. (b) The average of the total energy current across the chain as function of the temperature gradient. On both panels the orange solid line corresponds to Ohmic dissipation and the blue dashed line to super-Ohmic dissipation. The temperature gradient must satisfy $\mathrm{\Delta }T/T>-1$ to prevent the temperature of the central oscillator from becoming negative. We have fixed $\delta \omega /\mathrm{\Omega }=0.5, k/m{\mathrm{\Omega }}^2=1.8$, and $k_{B}T/\hslash \mathrm{\Omega }\simeq 0.27$.

Figure 3

The two-mode entanglement $E_N\left({\widehat{\rho }}_{\mathcal{R}\mathcal{C}}\right)$ (a) and the stationary energy current $〈{\widehat{j}}_{\mathcal{R}\mathcal{C}}〉$ (units of $\hslash {\mathrm{\Omega }}^2$) (b) in terms of the temperature gradient $\mathrm{\Delta }T$ and the coupling strength $k$ under Ohmic dissipation. The system parameters are the same as in Fig. 2.

Figure 4

(a) The criteria ${\mathcal{T}}_{23}$ and ${\mathcal{T}}_{33}$ as a function of the temperature gradient for Ohmic dissipation. (b) The time evolution of the fluctuations of the total current under Ohmic dissipation, in a system that is genuine entangled $(\mathrm{\Delta }T/T=-0.95)$ (black solid line), bipartite three-mode entangled $(\mathrm{\Delta }T/T=1.9)$ (blue dashed line), and likely separable in the three possible bipartitions $(\mathrm{\Delta }T/T=4.3)$ (red dot-dashed line). The remaining parameters are $k/m{\mathrm{\Omega }}^2=2, \delta \omega /\mathrm{\Omega }=0.5$, and $k_{B}T/\hslash \mathrm{\Omega }\simeq 0.27$.

Figure 5

Density plot of the energy current correlations $K_{JJ}(t,t)$ (units of ${\hslash }^2{\mathrm{\Omega }}^4$) as a function of the temperature gradient $\mathrm{\Delta }T$ and the coupling strength $k$ for Ohmic dissipation. A similar result is obtained for super-Ohmic dissipation. The black dashed line delimits the states in which the hierarchy ${\mathcal{T}}_{3,3}$ changes from positive to negative. The parameters are the same as in Fig. 4.

Figure 6

(a) The right discord as a function of the detuning $\delta \omega$ for the temperature gradients $\mathrm{\Delta }T/T=3.8$ and $0.95$ (black dot-dashed line). (b) The average interaction energy $\langle {\widehat{H}}_I\rangle$, see Eqs. (2) and (39), in terms of $\delta \omega$. (c) The right discord at resonance $(\delta \omega =0)$, as a function of the temperature gradient. The black dot-dashed line gives both the Ohmic and super-Ohmic dissipative discord for an initially squeezed central reservoir, with $r_{\mathcal{C}}=1(r_{\mathcal{R}}=r_{\mathcal{L}}=0)$. In the three panels the orange solid and blue dashed lines correspond to Ohmic and super-Ohmic dissipations, respectively, and the parameters are $k/m{\mathrm{\Omega }}^2=1.8$ and $k_{B}T/\hslash \mathrm{\Omega }\simeq 0.53$.

Figure 7

The discord $D^{\leftarrow }\left({\widehat{\rho }}_{\mathcal{R}\mathcal{C}}\right)$ (a) and the energy current $〈{\widehat{j}}_{\mathcal{R}\mathcal{C}}〉$ (units of $\hslash {\mathrm{\Omega }}^2$) (b) between the central and right oscillators in the resonant system under Ohmic dissipation. Similar results are obtained for the discord and the energy current between the $\mathcal{C}$ and $\mathcal{L}$ oscillators, under both Ohmic and super-Ohmic dissipations. The parameters are the same as in Fig. 6.