Bringing Entanglement to the High Temperature Limit

Decoherence due to contact with a hot environment typically restricts quantum phenomena to the low temperature limit, $k_{B}T/\hslash \omega \ll 1$ ($\hslash \omega$ is the typical energy of the system). Here we report the existence of a nonequilibrium state for two coupled, parametrically driven, dissipative harmonic oscillators which, contrary to generalized intuition, has stationary entanglement at high temperatures. This clarifies the role of temperature and could lighten the burden on quantum experiments requiring delicate precooling setups.

Figure 1

(a) The system is formed by two linearly coupled oscillators, initially thermalized due to each of them being dissipatively coupled to an environment at temperature $T$. Driving the coupling (or frequency) sinusoidally leads to the production of entanglement even at very high temperatures. (b) Entanglement phase diagram for the case without driving. The state thermalizes to a state with no entanglement (white) unless the temperature is below the quantum limit $k_{B}T<\hslash \omega$. We used $\gamma =0.0005\omega$. (c) Wigner phase space representation of the normal modes. They are sufficiently squeezed along orthogonal directions, so the oscillators are entangled ($E_N\simeq 0.7$). The parameters are $k_{B}T/\hslash \omega =5$, $\gamma =0.005\omega$, $c_1=0.5m{\omega }^2$, while the snapshot has been taken at time $\omega t=12.5$. The projections have been drawn to guide the eye.

Figure 2

Time evolution of entanglement towards its stationary value at (a) different environmental temperatures $k_{B}T/\hslash \omega =250$ [grey (red)], 300 [light grey (green)], and 350 [dark grey (blue)], with a damping of $\gamma =0.005\omega$, driving amplitude of $c_1=0.5m{\omega }^2$, and driving frequency ${\omega }_d=2\times 0.998\omega$. The stationary regime is reached in a reasonable time with a significative amount of entanglement. Notice that for an oscillator with frequency $\omega =21\mathrm{GHz}$ the values of temperature are directly given in Kelvin, and this would imply observation of entanglement at room temperature. (b) Now the temperature is kept fix, $k_{B}T/\hslash \omega =5$, with the same parameters, while the damping parameter is varied: $\gamma =0.005\omega$ [grey (red)], $0.01\omega$ [light grey (green)], $0.02\omega$ [dark grey (blue)].

Figure 3

Phase diagram of entanglement in the presence of parametric driving. We compare the condition (6) [lines] with the exact time evolution [dots] for different bath couplings $\gamma =0.005\omega$ [dark grey (blue) triangles], $0.001\omega$ [light grey (green) circles] and $0.0005\omega$ [grey (red) squares]. Inset: time evolution for different initial conditions, namely, a two-mode squeezed vacuum state (dotted curves) with squeezing parameter $r=0$ [grey (red)], 0.5 [dark grey (blue)], and 1 [light grey (green)], as compared to that of an initial thermal state (black). They all converge after some tens of periods. The parameters here are $\gamma =0.001\omega$, $c_1=0.2{\omega }^2$, ${\omega }_d=2\times 0.998\omega$, and $k_{B}T/\hslash \omega =10$.