Impossibility of faithfully storing single photons with the three-pulse photon echo

The three-pulse photon echo is a well-known technique to store intense light pulses in an inhomogeneously broadened atomic ensemble. This protocol is attractive because it is relatively simple and it is well suited for the storage of multiple temporal modes. Furthermore, it offers very long storage times, greater than the phase relaxation time. Here, we consider the three-pulse photon echo in both two- and three-level systems as a potential technique for the storage of light at the single-photon level. By explicit calculations, we show that the ratio between the echo signal corresponding to a single-photon input and the noise is smaller than one. This severely limits the achievable fidelity of the quantum state storage, making the three-pulse photon echo unsuitable for single-photon quantum memory.

Figure 1

Basic scheme of inhomogeneously broadened two-level systems with a ground state $|1\rangle$ and an excited state $|2\rangle .$ The spectral atomic distribution is supposed to have a Gaussian shape with the characteristic width ${\Delta }_0$. The interaction of the atomic ensemble with a laser pulse is parametrized by the Rabi frequency $\Omega$ and by the detuning $\Delta .$

Figure 2

Pulse sequence associated with the three-pulse photon echo in two-level atoms. The first pulse corresponding to ${\Omega }_1$ interacts with the atoms at time $t_1$ in order to be stored. The two next pulses, ${\Omega }_2$ and ${\Omega }_3$, applied at times $t_2$ and $t_3$, respectively, correspond to rephasing pulses. The echo signal is expected at time $t_4=t_3+t_2-t_1.$

Figure 3

Basic scheme of three-level systems with an excited state $|2\rangle$ and two ground states $|1\rangle$ and $|3\rangle$ that exhibits an inhomogeneous broadening on the spin transition. As in the two-level scheme, the atomic spectral distribution is supposed to be Gaussian with characteristic width ${\Delta }_0$. The interaction of the atomic ensemble with a Raman laser pulse is parametrized by the effective Rabi frequency ${\Omega }_i^r=\sqrt{{\Omega }^2+{\Omega }^{\prime 2}}$ and by the two-photon detuning ${\Delta }_r.$

Figure 4

Pulse sequence for the three-pulse photon echo in three-level systems. The first pulse, resonant with the transition $|1\rangle$–$|2\rangle$, is the one to store. It is immediately followed by a $\pi$ pulse at time $t_1$ on the transition $|2\rangle$–$|3\rangle$ to transfer the optical coherence into a spin coherence. The next two pulses, corresponding to ${\Omega }_2^r$ and ${\Omega }_3^r$, which are Raman pulses, force the ensemble of dipoles to rephase. The resulting coherences are read out at time $t_4=t_3+t_2-t_1$ by a $\pi$ pulse on the transition $|2\rangle$–$|3\rangle .$ The echo is emitted immediately after this final $\pi$ pulse.