Quantum theory for generation of nonclassical photon pairs by a medium with collective atomic memory

We present the quantum theory for creation of collective atomic memory and generation of nonclassically correlated photon pairs from an ensemble via the protocol of Duan et al. [Nature (London) 414, 413 (2001)]. The temporal evolution of photon numbers, photon statistics, and cross-correlation between the Stokes and anti-Stokes fields is found by solving the equation of motion for atomic spin-wave excitations. We consider a low-finesse cavity model with collectively enhanced signal-to-noise ratio, which remains still considerably large in the free-space limit. Our results describe analytically the dependence of quantum correlations on spin decoherence time and time delay between the write and read lasers and reproduce the observed data very well including the generated pulse shapes, strong violation of Cauchy-Schwarz inequality and conditional generation of anti-Stokes single-photon pulse. The approach we developed may be used also for quantum description of storage and retrieval of quantum information, especially when the statistical properties of nonclassical pulses are studied.

Figure 1

(Color online) (a) Atomic level structure for emission of Stokes and anti-Stokes photons in off-resonant Raman configuration with one photon detunings ${\Delta }_1$ and ${\Delta }_2$, respectively. (b) Interaction diagram in the case of one Stokes-photon emmission, where in initial state ${\Psi }_0$ all atoms are prepared in the ground state ∣1⟩. In intermediate stage, the atomic ensemble is in a collective state ${\Psi }_1$ with symmetric distribution of one atom excitation. (c) Time sequence for the write and read pulses having durations $T_W$ and $T_R$. ${\tau }_d$ is time delay between the pulses.

Figure 2

(Color online) (a) Stokes photon flux $d{n}_1^{\left(\mathrm{out}\right)}∕dt$ as a function of time for different values of write pulse intensity. The curves are labeled with the corresponding values of $\alpha$. The total number of Stokes photons emitted from the cavity are determined by the areas of the corresponding peaks. The duration of write pulse $T_W=1.6\mu \mathrm{s}$. (b) Anti-Stokes photon flux $d{n}_2^{\left(\mathrm{out}\right)}∕dt$ for fixed number of output Stokes photons $n_1^{\left(\mathrm{out}\right)}\left(T_W\right)=3$ and different values of read laser intensity. The read pulse duration $T_R=1\mu \mathrm{s}$. The time delay ${\tau }_d=1.4\mu \mathrm{s}$ and ${\gamma }_c=0.03{\left(\mu \mathrm{s}\right)}^{-1}$.

Figure 3

(Color online) (a) Average number of spin-wave excitations $N_{\mathrm{sp}}=⟨S^{+}\left(t\right)S\left(t\right)⟩$ as a function of time for fixed number of detected Stokes photons $n_1^{\left(\mathrm{out}\right)}\left(T_W\right)=3$ and the values of read laser intensity, as in Fig. 2b. ${\gamma }_c=0.03{\left(\mu \mathrm{s}\right)}^{-1}$. (b) The same, but ${\gamma }_c=0.3{\left(\mu \mathrm{s}\right)}^{-1}$.