Berry phase in controlled light propagation and storage

It is shown that during light storage in an atomic medium in the $\mathrm{\Lambda }$ configuration, with not only the amplitude of the control field but also its phase changing adiabatically, a photon gains a Berry (geometric) phase. In the case of the tripod configuration with two probe fields the Berry phase is replaced by a $2\times 2$ matrix. The probe fields are shown to be superpositions of two modes, each of them being characterized not only by its own velocity but also by its own Berry phase. If after light storage photons are released backwards, the contributions of the two modes interfere and the distribution of the outgoing photons can be steered by changing the difference between the Berry phases of the modes, due to the choice of the control field at the storage and release stages. In particular, one can turn a single photon of one of the probe fields into a photon of the other field or essentially modify coherent states of the incoming pulses.

Figure 1

Level and coupling scheme for the tripod configuration. The scheme corresponding to the $\mathrm{\Lambda }$ configuration is obtained by skipping level $b_2$ and the coupling ${\mathrm{\Omega }}_2$ and by replacing the index $b_1$ by $b$.

Figure 2

The trajectory in the parameter space of the control field: the normalized amplitude $|{\mathrm{\Omega }}_3|/{\mathrm{\Omega }}_{30}$ as a function of the phase $\chi$.

Figure 3

The Berry phases ${\beta }_1$, solid red line, and ${\beta }_2$, dashed green line, as functions of the initial amplitude ${\mathrm{\Omega }}_{30}$ of the control field. The value $2\chi \left(t_1\right)$ has been subtracted [cf. Eq. (43)].

Figure 4

The probability of obtaining (i) a single photon of type 2 if a single photon of type 1 has been stored, solid red line; (ii) two photons of the same type if one photon of type 1 and one photon of type 2 have been stored, green dashed line, as a function of the amplitude ${\mathrm{\Omega }}_{30}$ of the control field.

Figure 5

The mean value $n_{1av}={\left|\varsigma \right|}^2$ of the number of outgoing photons of type 1 as a function of the amplitude ${\mathrm{\Omega }}_{30}$ of the control field for $\alpha =1;{\alpha }^{\prime }=10$, solid red line, ${\alpha }^{\prime }=10i$, dashed green line, ${\alpha }^{\prime }=-10i$, short-dashed blue line, ${\alpha }^{\prime }=5$, dotted violet line.