Steady State Entanglement beyond Thermal Limits

Classical engines turn thermal resources into work, which is maximized for reversible operations. The quantum realm has expanded the range of useful operations beyond energy conversion, and incoherent resources beyond thermal reservoirs. This is the case of entanglement generation in a driven-dissipative protocol, which we hereby analyze as a continuous quantum machine. We show that for such machines the more irreversible the process, the larger the concurrence. Maximal concurrence and entropy production are reached for the hot reservoir being at negative effective temperature, beating the limits set by classic thermal operations on an equivalent system.

Figure 1

(a) Elementary model of a driven-dissipative quantum thermal machine with diamondlike internal level structure and degenerate symmetric ($S$) and antisymmetric ($A$) states; the collective eigenstates are assumed to be connected to two independent reservoirs, defined through their effective temperatures ${\mathcal{T}}_S={\omega }_0/{\beta }_S$ and ${\mathcal{T}}_A={\omega }_0/{\beta }_A$, respectively. (b) Steady state concurrence, Eq. (4), plotted against ${\beta }_A$ and ${\beta }_S$, respectively. White dashed lines mark the thermal region (${\beta }_{A,S}\ge 0$); the red dashed curves show the contour line for the limiting value $C=1/3$. (c) Rate of entropy production in steady state, Eq. (6), plotted in the same $({\beta }_A,{\beta }_S)$ plane as (b) and normalized to ${\omega }_0$ and ${\mathrm{\Gamma }}^{+}$.

Figure 2

(a) Quantum optical scheme of the full cavity QED model representing a pair of two-level emitters coupled to the same lossy cavity mode and incoherently pumped by an external drive, and the corresponding level scheme of the effective model after adiabatic elimination of the cavity degree of freedom. (b) Comparison between the steady state concurrence calculated numerically for the full quantum optical model, Eq. (8), and analytically for the effective model, Eq. (2) with parameters as in Eq. (9), as a function of the cavity dissipation rate for the ideal case of negligible qubits relaxation rate ($\gamma =0$, at pumping strength $p=0.0002g$). (c) Same comparison for finite $\gamma$ and varying pumping strength: $p=0.0002g$, $p=0.001g$, $p=0.005g$. Here, the qubit-cavity coupling rate is chosen to be $g/{\omega }_0=0.001$.