Cavity QED quantum phase gates for a single longitudinal mode of the intracavity field

A single three-level atom driven by a longitudinal mode of a high-$Q$ cavity is used to implement two-qubit quantum phase gates for the intracavity field. The two qubits are associated with the zero- and one-photon Fock states of each of the two opposite circular polarization states of the field. The three-level atom mediates the conditional phase gate provided the two polarization states and the atom interact in a V-type configuration and the two-photon resonance condition is satisfied. Microwave and optical implementations are discussed with gate fidelities being evaluated against several decoherence mechanisms such as atomic velocity fluctuations or the presence of a weak magnetic field. The use of coherent states for both polarization states is investigated to assess the entanglement capability of the proposed quantum gates.

Figure 1

(a) V-type three-level configuration under investigation. Here $g_{+}\left(g_{-}\right)$ is the vaccuum Rabi frequency of the coupling between the atom and the right- (left-)hand circularly polarized field, and ${\Delta }_{+}\left({\Delta }_{-}\right)$ is the corresponding cavity-transition detuning. (b) Dressed-state picture of (a) for $g_{+}=g_{-}(\equiv g),{\omega }_{ac}={\omega }_{bc}(\equiv {\omega }_0)$, and ${\Delta }_{+}={\Delta }_{-}(\equiv \Delta )$. $\mid B_{01}⟩\left(\mid D_{01}⟩\right)$ is the bright (dark) state combination of $\mid a,0,1⟩$ and $\mid b,1,0⟩$ (see text).

Figure 2

Time evolution of the fidelity $F$ for both quantum phase gates (solid curve for ${\widehat{U}}_1$ and dotted curve for ${\widehat{U}}_2$). (a) $\Delta =0$, and (b) $\Delta =2.35g$.

Figure 3

Fidelity of the first ${\widehat{U}}_1$ gate in Table in the parameter plane atomic velocity versus the intensity of a stray uniform magnetic field along the cavity axis. The parameter settings are $\Delta =0,g=2\pi \times 50\mathrm{kHz}$, and an effective cavity length of $L=4\mathrm{cm}$.

Figure 4

Concurrence between the two polarization states of the intracavity field (restricted to the computational basis) as a function of the mean photon number of both polarizations. Parameters correspond to the first ${\widehat{U}}_1$ gate sorted in Table .