Engineering atomic quantum reservoirs for photons

We present protocols for creating entangled states of two modes of the electromagnetic field by using a beam of atoms crossing microwave resonators. The atoms are driven by a transverse, classical field and pump correlated photons into (i) two modes of a cavity and (ii) the modes of two distant cavities. The protocols are based on a stochastic dynamics, characterized by random arrival times of the atoms and by random interaction times between atoms and cavity modes. The resulting effective model yields a master equation, whose steady state is an entangled state of the cavity modes. In this respect, the atoms act like a quantum reservoir, pulling the cavity modes into an entangled, Einstein-Podolski-Rosen state, whose degree of entanglement is controlled by the intensity and the frequency of the transverse field. This scheme is robust against stochastic fluctuations in the atomic beam, and it does not require atomic detection nor velocity selection.

Figure 1

Setup of the system for creation of EPR state of (a) two modes of the same cavity and (b) two spatially separated cavities. In both cases atoms from a beam are first prepared in a coherent superposition of two Rydberg states $|g\rangle$ and $|e\rangle$ by a combination of laser and microwave fields. The atoms have random arrival times, and a low pumping rate warrants that at most one atom is inside the resonator at a time [22, 23]. While in the cavities, the dipole transition $|g\rangle \to |e\rangle$ is saturated by a transverse microwave field, thereby pumping on resonance two nondegenerate modes of either one or two cavites, which are led asymptotically to a two-mode squeezed state.

Figure 2

(Left ladder) Energy of photon states $|N\rangle$ (with $N\gg 1$) of the semiclassical field at frequency ${\omega }_L$. (Middle ladder) Corresponding energies of the doublets of states $\{|g,N\rangle ,|e,N-1\rangle \}$, where $\Delta ={\omega }_L-{\omega }_0$ is the corresponding splitting in energy. (Right ladder) Energy of the semiclassical dressed states $|\pm \rangle$, Eq. (23), with energy splitting $d$, Eq. (22). A transition $|+\rangle \to |-\rangle$ is accompanied by absorption (emission) of a photon of frequency ${\omega }_1$ (${\omega }_2$) from (into) the corresponding cavity mode.

Figure 3

Schematic representation of the interaction processes needed in order to prepare the cavity modes in a EPR state. In step 1, the atoms are prepared in the state $|+\rangle$ before crossing the resonator. Inside the resonator they are driven by a classical field, and the atomic frequency is shifted with respect to ${\omega }_L$ according to the energy level scheme displayed at the bottom of the figure. In step 2, the initial state is $|-\rangle$ and the atomic transition frequency is shifted, such that the detuning with the field has opposite sign.

Figure 4

(i) Solid line: total experimetal time $T_{\mathrm{tot}}=2T$ (in seconds) and (ii) dashed line: average photon number per mode at the end of the protocol as a function of $\mu$ [which is controlled by the intensity of the classical field and the detuning according to Eq. (29) and (24)]. The parameters we used are effective coupling $g=$ 125 kHz, interaction time $\mathrm{\tau ̄}=12.5$ $\mu$s, atomic arrival rate 11200 atoms/s.

Figure 5

Logarithmic plot of time $T$ required in each step to reach the EPR state as a function of average interaction time $\mathrm{\tau ̄}$ (in units of ${\Omega }_{\mathrm{b}}^{-1}$). The EPR state here chosen is characterized by $\mu =.95$ (${\mathrm{n̄}}_0=9.3$), and we require that the corresponding fidelity at $T_{\mathrm{tot}}=2T$ is $\mathcal{F}=0.99$. The blue lines represent analytical expressions for fidelities ${\mathcal{F}}_{\mathrm{WC}}$, Eq. (9), and ${\mathcal{F}}_{\mathrm{SC}}$, Eq. (15). The red line has been obtained from numerical solution of Eq. (10), where coefficients $B_n$ were calculated taking $\sigma =0.05{\varphi }_0$ in the Gaussian distribution function of Rabi angles ${\varphi }_0=\mathrm{\tau ̄}{\Omega }_b$. The most efficient regime, in terms of fastest protocols for a given fidelity, lies between weak and strong coupling. The plot shows slower convergence at Rabi angles corresponding to ${\varphi }_0=\pi /\sqrt{j}$ ($j$ positive integer) due to trapping states.

Figure 6

(Top) Sketch of the setup for entangling the modes of two distant resonators. Two atomic beams propagate in both directions and cross the resonators, interacting sequentially with each of them. Here one of the two required steps of the procedure is shown, where the atoms are prepared in the state $|+\rangle$. (Bottom) Energy levels which are coupled inside each resonator. The “ball” represents the atomic occupation. An atom traveling from right to left first emits a photon ${\omega }_2$ into the second resonator and then emits a photon ${\omega }_1$ into the first one, only if it has previously emitted into resonator 2, otherwise it may absorb a photon ${\omega }_1$. An atoms traveling from left to right absorbs or emits photons in the reversed sequence.