Double two-photon coalescence in a controlled propagation

The propagation, storage, and release of two quantized light fields in an optically dressed atomic medium in the tripod configuration are studied. Using the formalism of spinor dark-state polaritons, the probabilities are obtained of finding outgoing photons of two possible localizations and two possible types, i.e., having different frequencies or polarizations. An additional interaction at the storage stage is applied which modifies the atomic coherences due to the stored photons; this results in a modification of the states of the outgoing photons. The conditions are examined for the photons' spatial and type coalescence typical of the Hong-Ou-Mandel effect, i.e., a zero probability of finding the outgoing photons having not only different localizations but also types. These results can possibly be observed in rubidium vapors.

Figure 1

Level and coupling scheme. The ground states $b_1$ and $b_2$ are populated and the corresponding coherence is nonzero.

Figure 2

The probabilities of finding two photons of any type at the position $x_1$, solid red line; two photons at the position $x_2$, the same line (the two probabilities coincide); one photon of any type at $x_1$ and one photon of any type at $x_2$, dashed green line, as a function of the population ${\sigma }_{b_1{b}_1}$ of the state $b_1$. Note the zero minimum of the noncoalescence probability for ${\sigma }_{b_1{b}_1}=\frac{1}{2}$. The coherence ${\sigma }_{b_1{b}_2}=0.3$; the values of other data given in the text.

Figure 3

The probabilities of finding two photons of field 1 at any position, solid red line; two photons of field 2 at any position, the same line (the two probabilities coincide); one photon of field 1 and one photon of field 2 at any position, dashed green line, as a function of the population ${\sigma }_{b_1{b}_1}$ of the state $b_1$. The data as in Fig. 2.

Figure 4

The probabilities of finding two photons of field 1 at any position, solid red line; two photons of field 2 at any position, the same line (the two probabilities coincide); one photon 1 and one photon 2 at any position, dashed green line, as a function of the imaginary part of $\gamma$, with its real part equal to zero, for equal populations of the states $b_1$ and $b_2$. The data as in Fig. 2.

Figure 5

The probabilities of finding two photons of field 1, solid red line; one photon of field 1 and one photon of field 2, dashed green line; two photons of field 2, dotted blue line, at the position $y_1+z_1$ as a function of the imaginary part of $\gamma$, with its real part equal to zero, for equal populations of the states $b_1$ and $b_2$. The probabilities given by the three curves sum up to the value $\frac{1}{2}$. The data as in Fig. 2.

Figure 6

The probability of finding one photon 1 and one photon 2 at either location as a function of the real and imaginary parts of $\gamma$, for equal populations of the states $b_1$ and $b_2$. The data as in Fig. 2.