Polarization phase gate with a tripod atomic system

We analyze the nonlinear optical response of a four-level atomic system driven into a tripod configuration. The large cross-Kerr nonlinearities that occur in such a system are shown to produce nonlinear phase shifts of order $\pi$. Such a substantial shift may be observed in a cold atomic gas in a magneto-optical trap where it could be feasibly exploited towards the realization of a polarization quantum phase gate. The experimental feasibility of such a gate is here examined in detail.

Figure 1

Energy level scheme for a tripod. Probe and trigger fields have Rabi frequencies ${\Omega }_P$ and ${\Omega }_T$ and polarizations ${\sigma }_{+}$ and ${\sigma }_{-}$. The pump Rabi frequency is $\Omega$ while ${\delta }_j={\omega }_0-{\omega }_j-{\omega }_j^{\left(L\right)}$ denote the laser (frequency ${\omega }_j^{\left(L\right)}$) detunings from the respective transitions $\mid j⟩\leftrightarrow \mid 0⟩$.

Figure 2

Probe absorption and dispersion $\chi \left({\delta }_1\right)=\hslash {ϵ}_0{\chi }_P∕\left(\mathcal{N}{\mid {\mathbf{\mu }}_P\mid }^2\right)={\left({\rho }_{10}∕{\Omega }_P\right)}_{ss}\left[{\gamma }^{-1}\right]$ (upper frame) vs the probe detuning ${\delta }_1∕\gamma$ when ${\delta }_3=0.1\gamma$ and ${\delta }_2=0.1\gamma$. Trigger absorption and dispersion $\chi \left({\delta }_3\right)=\hslash {ϵ}_0{\chi }_T∕\left(\mathcal{N}{\mid {\mathbf{\mu }}_T\mid }^2\right)={\left({\rho }_{30}∕{\Omega }_T\right)}_{ss}\left[{\gamma }^{-1}\right]$ (lower frame) vs the trigger detuning ${\delta }_3∕\gamma$ when ${\delta }_1=0.1\gamma$ and ${\delta }_2=0.1\gamma$. In both cases we take the Rabi frequencies as ${\Omega }_P={\Omega }_T=0.1\gamma$, $\Omega =\gamma$.

Figure 3

Probe absorption and dispersion $\chi \left({\delta }_1\right)=\hslash {ϵ}_0{\chi }_P∕\left(\mathcal{N}{\mid {\mathbf{\mu }}_P\mid }^2\right)={\left({\rho }_{10}∕{\Omega }_P\right)}_{ss}\left[{\gamma }^{-1}\right]$ (upper frame) vs the probe detuning ${\delta }_1∕\gamma$ when ${\delta }_3=10.02\gamma$ and ${\delta }_2=10\gamma$. Trigger absorption and dispersion $\chi \left({\delta }_3\right)=\hslash {ϵ}_0{\chi }_T∕\left(\mathcal{N}{\mid {\mathbf{\mu }}_T\mid }^2\right){\left({\rho }_{30}∕{\Omega }_T\right)}_{ss}\left[{\gamma }^{-1}\right]$ (lower frame) vs the trigger detuning ${\delta }_3∕\gamma$ when ${\delta }_1=10.01\gamma$ and ${\delta }_2=10\gamma$. In both cases we take the Rabi frequencies as ${\Omega }_P={\Omega }_T=\gamma$, $\Omega =4.5\gamma$.

Figure 4

Probe absorption (scaled) at the center of probe transparency window, plotted against the dephasing rate, for ${\Omega }_P={\Omega }_T=\gamma$, $\Omega =4.5\gamma$, ${\delta }_j=0$.