Dissipatively driven entanglement of two macroscopic atomic ensembles

Up to now, the lifetime of experimentally demonstrated entangled states has been limited due to their fragility under decoherence and dissipation. Therefore, they are created under strict isolation conditions. In contrast, new approaches harness the coupling of the system to the environment, which drives the system into the desired state. Following these ideas, we present a robust method for generating steady-state entanglement between two distant atomic ensembles. The proposed scheme relies on the interaction of the two atomic systems with the common vacuum modes of an electromagnetic field which act as an engineered environment. We develop the theoretical framework for two-level systems, including dipole-dipole interactions, and complement it by considering its implementation in multilevel ground states. Based on these results, the realization of entanglement generation by engineered dissipation has been experimentally demonstrated [H. Krauter, C. A. Muschik, K. Jensen, W. Wasilewski, J. M. Petersen, J. I. Cirac, and E. S. Polzik, e-print arXiv:1006.4344].

Figure 1

Setup for creating steady-state entanglement between two atomic ensembles separated by a distance $\mathbf{R}$. Both ensembles are placed in magnetic fields $\mathbf{B}$, which are oriented along $\stackrel{\wedge }{\mathbf{x}}$. A strong $\stackrel{\wedge }{\mathbf{y}}$-polarization laser beam (shown in green color) propagates along $\stackrel{\wedge }{\mathbf{z}}$ and couples the atomic system to the environment that consists of vacuum modes of the copropagating electromagnetic field in $\stackrel{\wedge }{\mathbf{x}}$ polarization (depicted in red). The interaction between atoms and light modes is illustrated in Fig. 2.

Figure 2

Atomic level schemes. (a) A magnetic field which is oriented along $\stackrel{\wedge }{\mathbf{x}}$ (see Fig. 1) causes a Zeeman splitting $\Omega$ of atomic ground-state levels $|\uparrow \rangle$ and $|\downarrow \rangle$ and defines the quantization axis. A strong $\stackrel{\wedge }{\mathbf{y}}$ polarized coherent field with detuning $\Delta$ drives transitions $|\uparrow \rangle \to |e_{\downarrow }\rangle$ and $|\downarrow \rangle \to |e_{\uparrow }\rangle$. Coupling to the vacuum modes of the electromagnetic field gives rise to transitions $|e_{\downarrow }\rangle \to |\downarrow \rangle$ and $|e_{\uparrow }\rangle \to |\uparrow \rangle$. (b) A static electric field is applied to the second ensemble such that the energy difference between the ground and excited states is enhanced by $2\Delta$.

Figure 3

Steady-state entanglement ${\xi }_{\infty }$ in a two-level system vs $Z=(|\mu |-|\nu |){}^{-1}$ for optical depth $d=30$ per ensemble. The horizontal black line indicates the separable limit. (For separable states $\xi ⩾1$, the smaller $\xi$, the higher the amount of entanglement.) The lowest line (violet) depicts ${\xi }_{\infty }$ for purely radiative dephasing ${\Gamma }_{\mathrm{d}}^{\mathrm{add}}=0$. The next curves show in ascending order ${\Gamma }_d^{\mathrm{add}}=2\Gamma$ (blue), ${\Gamma }_{\mathrm{d}}^{\mathrm{add}}=5\Gamma$ (green), ${\Gamma }_{\mathrm{d}}^{\mathrm{add}}=10\Gamma$ (red), and ${\Gamma }_{\mathrm{d}}^{\mathrm{add}}=20\Gamma$ (orange), where $\Gamma$ is the single-particle decay rate.

Figure 4

Steady-state entanglement ${\xi }_{\infty }^{\mathrm{opt}}$ for optimal squeezing parameter $Z$ vs pump parameter $x$. (Two-level model, $d=30$.) The curves correspond in ascending order to ${\Gamma }_{\mathrm{d}}^{\mathrm{add}}=15$ (gray), ${\Gamma }_{\mathrm{d}}^{\mathrm{add}}=20\Gamma$ (orange), ${\Gamma }_{\mathrm{d}}^{\mathrm{add}}=25\Gamma$ (brown), ${\Gamma }_{\mathrm{d}}^{\mathrm{add}}=30\Gamma$ (pink), and ${\Gamma }_{\mathrm{d}}^{\mathrm{add}}=35\Gamma$ (light green). The inset shows the steady-state polarization $P_{2,\infty }$ vs $Z$ in the absence of pump fields $x=0$ (dashed line) and for $x=5$ (solid line).

Figure 5

Implementation of the proposed scheme in ${}^{133}\mathrm{Cs}$ ensembles. The atomic samples are placed in $\stackrel{\wedge }{\mathbf{x}}$-polarized magnetic fields and interact with a $\stackrel{\wedge }{\mathbf{y}}$-polarized probe field propagating along $\stackrel{\wedge }{\mathbf{z}}$, as shown in Fig. 1. The relevant two-level subsystem is encoded in the $6S_{1/2}$ ground state, $|\uparrow \rangle \equiv |4,\pm 4\rangle$ and $|\downarrow \rangle \equiv |4,\pm 3\rangle$, in the first and second ensemble and are coupled to the excited states in $6P_{3/2}$. Only desired transitions are shown. Level splittings are not represented true to scale. (In a magnetic field of $1$ G, the Zeeman shift of magnetic sublevels is $~$${10}^5$ Hz while the hyperfine and fine splitting is on the order of ${10}^8$ Hz and ${10}^{14}$, respectively.)

Figure 6

General scheme for the realization of long-lived entanglement between alkali-metal ensembles. The polarization of the probe field is assumed to be parallel to the applied magnetic field. The ground state $S_{1/2}$ consists of two manifolds with total spin $F=I+1/2$ and $F^{\prime }=I-1/2$, where $I$ is the nuclear spin. The states $|\uparrow \rangle$ and $|\downarrow \rangle$ are encoded in the outermost levels of the $F=I+1/2$ ground-state manifold. We identify $|\uparrow \rangle \equiv |F,F\rangle$ and $|\downarrow \rangle \equiv |F,F-1\rangle$ for the first atomic ensemble and $|\uparrow \rangle \equiv |F,-F\rangle$ and $|\downarrow \rangle \equiv |F,-F+1\rangle$ for the second one. Only desired transitions are shown. In the presence of strong pump fields, the amount of entanglement generated can be estimated by a simplified three-level model including the state $|h\rangle \equiv |F^{\prime },\pm F^{\prime }\rangle$, for the first and second ensemble.

Figure 7

Steady-state entanglement ${\xi }_{\mathrm{expt},\infty }$ between two ${}^{133}\mathrm{Cs}$ ensembles vs strength of repump fields $x_{\mathrm{repump}}$ (see main text, Sec. 4b and Appendix ). The plots depict data for ensembles with optical depth $d=30$, blue detuned probe light with $\Delta =700$ MHz, and different values for the nonradiative dephasing rate ${\Gamma }_{\mathrm{d}}^{\mathrm{add}}$. The curves correspond in ascending order to ${\Gamma }_{\mathrm{d}}^{\mathrm{add}}=0$ (violet), ${\Gamma }_{\mathrm{d}}^{\mathrm{add}}=2$ (blue), ${\Gamma }_d^{\mathrm{add}}=5$ (green), ${\Gamma }_{\mathrm{d}}^{\mathrm{add}}=10$ (red), and ${\Gamma }_{\mathrm{d}}^{\mathrm{add}}=20$ (orange).

Figure 8

Quasi-steady-state entanglement of ${}^{133}\mathrm{Cs}$ ensembles vs time in units $1/\Gamma$ in the absence of repump fields ($x_{\mathrm{repump}}=0$; all other parameters take values as in Fig. 7). The fast entanging dynamics results in a drop in ${\xi }_{\mathrm{expt}}(t)$ (indicating the creation of an inseparable state), but since particle losses are not counteracted by repump fields, this additional slow dynamics eventually causes ${\xi }_{\mathrm{expt}}(t)$ to rise. Hence, a quasi-steady state is produced. The behavior on long time scales is determined by the stationary state given by Eq. (15) superposed by slow multilevel dynamics.