Adiabatic entanglement in two-atom cavity QED

We analyze the problem of a single mode field interacting with a pair of two level atoms. The atoms enter and exit the cavity at different times. Instead of using constant coupling, we use time-dependent couplings which represent the spatial dependence of the mode. Although the system evolution is adiabatic for most of the time, a previously unstudied energy crossing plays a key role in the system dynamics when the atoms have a time delay. We show that conditional atom-cavity entanglement can be generated, while for large photon numbers the entangled system has a behavior which can be mapped onto the single atom Jaynes-Cummings model. Exploring the main features of this system we propose simple and fairly robust methods for entangling atoms independently of the cavity, for quantum state mapping, and for implementing SWAP and controlled-NOT (CNOT) gates with atomic qubits.

Figure 1

Top: The coupling functions ${\eta }_1\left(\tau \right)$ (solid) and ${\eta }_2\left(\tau \right)$ (dashed) with respect to $\tau$. Bottom: The adiabatic energies $E_1\left(\tau \right)$ (solid), $E_2\left(\tau \right)$ (long dashed), $E_3\left(\tau \right)$ (dashed), and $E_4\left(\tau \right)$ (chain) with respect to $\tau$. The parameters are $\delta =1.0$ and $n=0$.

Figure 2

Top: The projection ${\left|⟨{\Psi }_1^{\prime }\left(\tau \right)|\Psi \left(\tau \right)⟩\right|}^2$ with respect to time. The parameters are $n=0$, $\delta =1.0$, and $g\sigma =30$. The initial state was $|{\Psi }_1^{\prime }\left(-{\tau }_0\right)⟩$, and $|\Psi \left(\tau \right)⟩$ is the solution for the Schrödinger equation. Note the narrow range of the vertical axis. Bottom: The imaginary part of $⟨{\Psi }_1^{\prime }\left({\tau }_0\right)|\Psi \left({\tau }_0\right)⟩\phantom{⟩}$ with respect to $g\sigma$. The parameters are $n=0$ and $\delta =1.0$.

Figure 3

The matrix element $Q_n^{13}\left(\tau ,\delta \right)$, Eq. (37), with respect to $\tau$ and $\delta =0.5$, 1.0, and 1.25. The photon number was $n=0$ and the step $d\tau =0.01$.

Figure 4

The function $\delta {ϵ}_3\left(\tau \right)$ with respect to $\tau$ and different $g\sigma$: $g\sigma =5.0$ (solid), $g\sigma =10.0$ (long dashed), and $g\sigma =20$ (dashed). The other parameters are $n=0$ and $\delta =1.0$.

Figure 5

The function $\delta {ϵ}_3\left(\tau \right)$ with respect to $\tau$ and different $n$: $n=0$ (solid), $n=5$ (long dashed), and $n=10$ (dashed). The parameters are $g\sigma =30$ and $\delta =1.0$.

Figure 6

The function $\delta {ϵ}_3\left(\tau \right)$ for $\tau \to \infty$ with respect to $n$. Top: $g\sigma =10$ (solid), $g\sigma =20$ (long dashed), and $g\sigma =30$ (dashed). Bottom: $g\sigma =50$. For both plots we have $\delta =1.2$.

Figure 7

The fidelity for a maximally entangled state with respect to variations in the interaction time $\Delta \sigma$ and different variations in the delay time: 0 (solid), $0.05\Delta t$ (dashed), and $0.1\Delta t$ (dot). The parameters $\sigma$ and $\Delta t$ correspond to $\delta =1.0$, ${\varphi }_{-1}=2m\pi$, and ${\theta }_1=0$.