Cooling atom-cavity systems into entangled states

Generating entanglement by simply cooling a system into a stationary state which is highly entangled has many advantages. Schemes based on this idea are robust against parameter fluctuations, tolerate relatively large spontaneous decay rates, and achieve high fidelities independent of their initial state. A possible implementation of this idea in atom-cavity systems has recently been proposed by Kastoryano et al., [Kastoryano et al., Phys. Rev. Lett. 106, 090502 (2011).]. Here we propose an improved entanglement cooling scheme for two atoms inside an optical cavity which achieves higher fidelities for comparable single-atom cooperativity parameters $C$. For example, we predict fidelities above $90\%$ even for $C$ as low as 20 without having to detect photons.

Figure 1

(a) Experimental setup to cool two atoms inside an optical cavity into a maximally entangled state. Here, $\Gamma$ and $\kappa$ denote the spontaneous atom and cavity decay rates while ${\Omega }_0$ and ${\Omega }_1$ are the relevant laser Rabi frequencies. (b) Level scheme of a single-atom. The 1–2 transition couples resonantly with coupling constant $g$ to the cavity field. The spontaneously decay rates for the 0–2 and the 1–2 transitions are ${\Gamma }_0$ and ${\Gamma }_1$ with $\Gamma ={\Gamma }_0+{\Gamma }_1$.

Figure 2

The toy-model level scheme. The 0–2 and the 1–3 transitions are driven by a laser field with Rabi frequency $\Omega$ and a detuning $\Delta$ with respect to the 0–2 transition. The excited atomic states both decay spontaneously into $|0\rangle$ and $|1\rangle$ with a decay rate $\Gamma /2$.

Figure 3

Logarithmic plot of the time dependence of the distance $1-F$ from the target state $|0\rangle$ for different values of $\Omega /\Delta$ and $\Gamma /\Delta$. The system is initially in $|1\rangle$. In the upper plot we have $\Gamma =0.2\Delta$. In the lower plot we have $\Omega =0.05\Delta$. The thick lines are the numerical solutions of the rate equations in Eq. (10). The thin lines illustrate the analytical solution in Eq. (21).

Figure 4

Contour plot which shows the stationary state fidelity $F$ in Eq. (12) as a function of $\Omega /\Delta$ and $\Gamma /\Delta$.

Figure 5

Contour plot which shows the cooling rate ${\gamma }_c$ in Eq. (19) as a function of $\Omega /\Delta$ and $\Gamma /\Delta$.

Figure 6

(a) Logarithmic plot of the time dependence of the distance $1-F$ from the target state $|0\rangle$ for the case where the system is initially in $|1\rangle$ and where $\Gamma =0.2\Delta$ and $\Omega =0.05\Delta$. (b) Same as (a) but for a time-dependent laser pulse with a Rabi frequency $\Omega \left(t\right)$ as in Eq. (22) with ${\Omega }_0=0.05\Delta$. (c) Theoretical minimum for $1-F$ obtained from Eq. (23) for $\Gamma =0.2\Delta$.

Figure 7

Level configuration showing the laser driving, Rabi frequencies, and detunings experienced by the target state $|+,0\rangle$ in the dressed state. For simplicity, we show only the least detuned couplings.

Figure 8

Level configuration showing all the resonantly driven transitions in the dressed state picture, their Rabi frequencies, and their detunings in the subspace with zero or one excitation in $|2\rangle$ or the cavity mode. (a) Symmetric subspace with ground states $|00,0\rangle$, $|+,0\rangle$, and $|11,0\rangle$. (b) Antisymmetric subspace with ground state $|-,0\rangle$.

Figure 9

Stationary state fidelity $F$ of the proposed entangling scheme for different spontaneous decay rates $\kappa$ and $\Gamma$ and $\Omega =0.03g$ obtained from a quantum jump simulation. Here $\kappa$ and $\Gamma$ are varied such that the cooperativity parameter $C$ remains constant at $C=25$.

Figure 10

Curve a: Logarithmic plot of the time dependence of the distance $1-F$ from the target state $|0\rangle$ for the case where the system is initially in $|00,0\rangle$ and where $\kappa =\Gamma =0.2g$ and $\Omega =0.03g$. Curve b: Same as curve a but for a time-dependent laser pulse with a Rabi frequency $\Omega \left(t\right)$ as in Eq. (39) with ${\Omega }_0=0.015g$.

Figure 11

Stationary state fidelity $F$ of the proposed entangling scheme as a function of the cooperativity parameter $C$ for $\Omega =0.03g$ and $\kappa =2\Gamma$. (a) Numerical solution of the time evolution of the system using the quantum jump approach. (b) Analytical result in Eq. (37).