Fidelity of photon propagation in electromagnetically induced transparency in the presence of four-wave mixing

We study the effects of four-wave mixing (4WM) in a quantum memory scheme based on electromagnetically induced transparency (EIT). We treat the problem of field propagation on the quantum mechanical level, which allows us to calculate the fidelity of propagation for a quantum light pulse such as a single photon. While 4WM can be beneficial for classical, all-optical information storage, the quantum noise associated with the signal amplification and idler generation is, in general, detrimental for a quantum memory. We identify a range of parameters where 4WM makes a single-photon quantum memory impossible.

Figure 1

Level scheme for EIT memory with four-wave mixing (4WM). A double-$\Lambda$ scheme, with one $\Lambda$ being the signal field ${\widehat{a}}_S$ and strong control field $\Omega$ giving standard EIT, and the second $\Lambda$ far-detuned from resonance made up of the same control field acting on the $|g\rangle -|e^{\prime }\rangle$ transition and the idler field ${\widehat{a}}_I$ generated by 4WM.

Figure 2

Plot of the transmission coefficient as a function of frequency for EIT with 4WM [solid (blue) line] and without 4WM [dashed (red) line]. Parameters were chosen to emphasize the spectral behavior.

Figure 3

Logarithmic plot of the signal [solid (blue) line] and idler [dashed (red) line] amplitudes for parameters $\Omega =0.1{\gamma }_{ge}$, $\Delta /{\gamma }_{ge}=33$, and $\eta =1$ as a function of the effective 4WM optical depth, assuming no initial idler field and normalized to the amplitude of the initial signal field. Note that both become equal and grow exponentially for high optical depths. Inset: Linear plot showing the the low-optical-depth behavior of the amplitudes.

Figure 4

Plot of the signal delay time, normalized to the standard EIT delay time $D{\gamma }_{ge}/{\left|\Omega \right|}^2$ [solid (blue) line], and the signal transmission frequency width, normalized to the standard EIT transmission window ${\left|\Omega \right|}^2/\left({\gamma }_{ge}\sqrt{D}\right)$ [dashed (red) line], as a function of the effective 4WM optical depth, with $\Omega =0.1{\gamma }_{ge}$, $\Delta /{\gamma }_{ge}=33$, and $\eta =1$.

Figure 5

Plot of the signal field at the beginning of the medium [dot-dashed (black) line], after an effective optical depth of $x=0.75$ [solid (blue) line], and after $x=3$ [dashed (red) line], normalized to the initial pulse amplitude. Parameters are $\Omega =0.1{\gamma }_{ge}$, $\Delta /{\gamma }_{ge}=33$, and $\eta =1$.

Figure 6

Plot of the number of noise photons produced as a function of the effective 4WM optical depth for $\Omega =0.1{\gamma }_{ge}$, $\Delta /{\gamma }_{ge}=33$, and $\eta =1$. Note that it becomes larger than 1 slightly before when $x>1/2$.

Figure 7

Number of photons affected by spontaneous emission [solid (red) line] showing at what effective 4WM optical depth it becomes larger than 1, assuming a single-photon input. For comparison the number of vacuum noise photons [dashed (blue) line] is also plotted. In both cases the parameters are $\Omega =0.1{\gamma }_{ge}$, $\Delta /{\gamma }_{ge}=33$, and $\eta =1$.

Figure 8

Plot of fidelity for the single-photon case as a function of the effective 4WM optical depth for $\Omega =0.1{\gamma }_{ge}$, $\Delta /{\gamma }_{ge}=30$, and $\eta =1$. In the limit $\Delta {\omega }_0/\Delta {\omega }_S$, $\Delta {\omega }_0/\Delta {\omega }_I\to 0$.

Figure 9

Plot of the fidelity of four-wave mixing [FWM; solid (blue) line] and standard EIT [dashed (red) line] fidelities as a function of the effective optical depth in the presence of linear losses, under the assumption that the signal field is always well within the transmission window. The losses are taken to match the 4WM gain present at the effective optical depth of $x=2$. The remaining parameters are $\Delta /{\gamma }_{ge}=30$, $\Omega =0.1{\gamma }_{ge}$, and $\eta =1$.