Entanglement Generated by Dissipation and Steady State Entanglement of Two Macroscopic Objects

Entanglement is a striking feature of quantum mechanics and an essential ingredient in most applications in quantum information. Typically, coupling of a system to an environment inhibits entanglement, particularly in macroscopic systems. Here we report on an experiment where dissipation continuously generates entanglement between two macroscopic objects. This is achieved by engineering the dissipation using laser and magnetic fields, and leads to robust event-ready entanglement maintained for 0.04 s at room temperature. Our system consists of two ensembles containing about ${10}^{12}$ atoms and separated by 0.5 m coupled to the environment composed of the vacuum modes of the electromagnetic field. By combining the dissipative mechanism with a continuous measurement, steady state entanglement is continuously generated and observed for up to 1 h.

Figure 1

(a) Collective dissipation modes and atomic levels: two spatially separated atomic ensembles interact with the environment composed of the vacuum modes of the electromagnetic field. The coupling is driven by the $y$-polarized laser beam. The engineered collective dissipation is due to photons scattered in the forward $z$ direction. Internal level scheme of the atoms: the effective two-level systems $|\uparrow ⟩$ and $|\downarrow ⟩$ are two magnetic sublevels Zeeman shifted by a magnetic field applied in the $x$ direction, which defines the quantization axis. Atoms in the two ensembles are initialized in opposite spin states. The laser beam off-resonantly couples these levels to the excited states and to the electromagnetic vacuum modes. Because of the Zeeman shift of the ground state levels, photons are emitted into the upper and lower sidebands (shown in blue and red) leading to collective spin flips $J_{\pm }$. (b) Geometry of the experiment. The $S_2$ detector signal processed by the lock-in amplifier (LI) is used to determine the atomic quantum spin components $J_{y,z}$ as described in the text. The optical pumping scheme is also shown.

Figure 2

Entanglement generated by dissipation (a)–(c) and steady state entanglement (d). (a) Time evolution of ${\Sigma }_J(t)/(2|⟨J_x(0)⟩|)$ (blue line) and $⟨J_x(t)⟩/⟨J_x(0)⟩$ (gray line). The theoretical fits (full and dashed black lines) are based on the parameters $d=55$ (optical density), ${\Gamma }_{\mathrm{col}}=0.002{\mathrm{ms}}^{-1}$ (collisional rate), and $\tilde{\Gamma }=0.193{\mathrm{ms}}^{-1}$ (dephasing rate [22]). The rates for driving field induced transitions $|4,\pm 3⟩\to |4,\pm 4⟩$ and $|4,\pm 4⟩\to |4,\pm 3⟩$ are given by ${\mu }^2\Gamma$ and ${\nu }^2\Gamma$, respectively, where $\Gamma =0.002{\mathrm{ms}}^{-1}$. (b) Entanglement $\xi (t)$ versus time in ms. Blue data points correspond to the results shown in (a). Data points in orange are obtained for a lower optical depth ($d=35$). The other parameters used in the fits take the same values as in (a). $\xi (t)<1$ certifies the creation of an inseparable state. The relevant pulse sequence is shown below. The data taken in the absence of the driving field (black points) show no entanglement. (c) Dissipative entanglement generation in the presence of the pump field which incoherently transfers atomic population from undesired levels within $F=4$ back to the two-level subsystem. The pump rate is ${\Gamma }_{\mathrm{pump}}=0.168{\mathrm{ms}}^{-1}$. $d=37$, the fitting parameters ${\Gamma }_{\mathrm{col}}$ and $\Gamma$ take the same values as in (a) and $\tilde{\Gamma }=0.233{\mathrm{ms}}^{-1}$. The inset shows the evolution of $\xi (t)$ after the driving field is switched off after entanglement is generated by dissipation. (d) Entanglement $\xi (t)$ for different initial conditions. The upper curves show a purely dissipative evolution. The lower curves, the entanglement generated by dissipation combined with the measurement. Points on the right represent an average over measurements of 1 h where atoms were kept in a steady state. The used exponential time mode functions are depicted in the pulse sequence and are described in the text. (e) Schematic illustration of entanglement generation and verification. The signal from the detector $D$ for times $t>T$ is used for verification of entanglement in (a)–(c). In (d) the signal taken at $t<T$ is given to the verifier as additional information.