Light propagation through closed-loop atomic media beyond the multiphoton resonance condition

The light propagation of a probe field pulse in a four-level double-lambda type system driven by laser fields that form a closed interaction loop is studied. The finite frequency width of the probe pulse requires a time-dependent analysis beyond the multiphoton resonance assumption. We apply a Floquet decomposition to the equations of motion to solve this time-dependent problem and to identify the different scattering processes contributing to the medium response. We find that the response oscillating in phase with the probe field is phase-independent. The phase dependence arises from a scattering of the coupling fields into the probe field mode at a frequency which in general differs from the probe field frequency. In particular for short pulses with a large frequency width, inducing a closed loop interaction contour may lead to a distortion of the pulse shape via this phase-sensitive scattering. Finally, we demonstrate that both the closed loop and the nonclosed loop configuration allow for sub- and superluminal light propagation with small absorption or even gain, where one of the coupling field Rabi frequencies acts as a control parameter that enables one to switch between sub- and superluminal light propagation.

Figure 1

(Color online) The four-level double-$\Lambda$ type schemes considered in the analysis. (a) is driven by four laser fields indicated by the red arrows with Rabi frequencies $g_{ij}$ ($i∊\{3,4\}$, $j∊\{1,2\}$) that form a closed interaction loop. This loop gives rise to a dependence on the relative phase of the different laser fields, and in general makes a time-independent steady state of the system dynamics impossible. (b) is the corresponding level scheme where one of the driving fields has been removed $(g_{32}=0)$. Here, the laser fields do not form a closed loop. The spontaneous decays with rates ${\gamma }_{ij}$ ($i∊\{1,2\}$, $j∊\{3,4\}$) are denoted by the wiggly green lines.

Figure 2

(Color online) Interpretation of the different contributions to the probe field susceptibility in terms of transition pathways. (a) represents the interaction loop leading to a scattering of the driving fields into the probe field mode. (b) is the direct scattering of the probe field off of the probe transition. (c) shows a counter-rotating term. The solid red arrows indicate coupling field transitions, the dashed blue line is a probe field interaction.

Figure 3

(Color online) Real (blue solid line) and imaginary (red dashed line) parts of the atomic density matrix element ${\widehat{\rho }}_{41}$ in a reference frame oscillating in phase with the probe field as a function of the Raman detuning ${\delta }_{12}={\Delta }_{31}-{\Delta }_{32}={\Delta }_{41}-{\Delta }_{42}$. The real part corresponds to the probe field dispersion, whereas the imaginary part describes the absorptive properties. The relative initial phase difference between the four applied laser fields is ${\varphi }_0=0$ in (a), ${\varphi }_0=\pi$ in (b), and ${\varphi }_0=\pi ∕2$ in (c). The common parameters are $2{\gamma }_{13}=2{\gamma }_{14}=2{\gamma }_{23}=2{\gamma }_{24}={\gamma }_0$, and thus ${\Gamma }_{13}={\Gamma }_{14}={\Gamma }_{23}={\Gamma }_{24}={\gamma }_0$, ${\Gamma }_{12}=0.0$, and ${\Gamma }_{34}=2{\gamma }_0$. The detunings are ${\Delta }_{32}={\Delta }_{42}=0$, and the Rabi frequencies are chosen as $g_{31}=g_{32}=g_{42}=0.6{\gamma }_0$ and $g_{41}=0.01{\gamma }_0$.

Figure 4

(Color online) Real (blue solid line) and imaginary (red dashed line) parts of the contribution to the Floquet decomposition representing the direct scattering of the probe field off of the probe transition, $g_{41}{\left[{\tilde{R}}_1\right]}_{13}$ [see Eq. (24)]. This figure depicts one of the processes contributing to the results in Fig. 3, and the parameters are the same as in this figure. Note that this particular contribution is independent of the phase ${\varphi }_0$, such that only one subfigure is shown, consistent with the interpretation in Sec. 2D.

Figure 5

(Color online) Real (blue solid line) and imaginary (red dashed line) parts of the contribution to the Floquet decomposition representing a counter-rotating contribution, $g_{41}^{*}{\left[{\tilde{R}}_{-1}\right]}_{13}\exp \left(2i\Phi \right)$ [see Eq. (24)]. This figure depicts one of the processes contributing to the results in Fig. 3, and the parameters are the same as in this figure. This particular contribution depends on the phase $2{\varphi }_0$, such (a) shows the result for ${\varphi }_0=0,\pi$ and (b) for $\varphi =\pi ∕2$, consistent with the interpretation in Sec. 2D.

Figure 6

(Color online) Real (blue solid line) and imaginary (red dashed line) parts of the contribution to the Floquet decomposition representing the direct scattering of the probe field off of the probe transition, $g_{41}{\left[{\tilde{R}}_1\right]}_{13}$ [see Eq. (24)]. The susceptibility is plotted against the probe field detuning ${\Delta }_{41}$. This corresponds to a calculation relevant for the evaluation of the group velocity, but violates the multiphoton resonance condition and thus requires a Floquet analysis. The parameters in (a) are ${\Delta }_{31}={\Delta }_{32}={\Delta }_{42}=0$, $2{\gamma }_{13}=2{\gamma }_{14}=2{\gamma }_{23}=2{\gamma }_{24}={\gamma }_0$, $g_{31}=1.8{\gamma }_0$, $g_{32}=0.2{\gamma }_0$, $g_{42}=0.5{\gamma }_0$, and $g_{41}=0.01{\gamma }_0$. (b) shows the case ${\Delta }_{31}=10{\gamma }_0$, ${\Delta }_{32}={\Delta }_{42}=0$, $2{\gamma }_{13}=2{\gamma }_{14}=2{\gamma }_{23}=2{\gamma }_{24}={\gamma }_0$, $g_{31}=g_{32}=0.1{\gamma }_0$, $g_{42}=0.5{\gamma }_0$, and $g_{41}=0.01{\gamma }_0$.

Figure 7

(Color online) Real (a) and imaginary (b) parts of the scaled contribution to the Floquet decomposition representing the direct scattering of the probe field off of the probe transition, ${\left[{\tilde{R}}_1\right]}_{13}$ [see Eq. (24)]. The susceptibility is plotted against the probe field detuning ${\Delta }_{41}$ with other detunings chosen as ${\Delta }_{31}={\Delta }_{42}=0$. One of the Rabi frequencies is chosen as zero, $g_{32}=0$, such that the setup corresponds to the one shown in Fig. 1b. The solid blue line is for $g_{31}=0.7{\gamma }_0$, the red dashed line for $g_{31}=0.85{\gamma }_0$, and the green dash-dotted line for $g_{31}=1.5{\gamma }_0$. The other parameters are $2{\gamma }_{13}=2{\gamma }_{14}=2{\gamma }_{23}=2{\gamma }_{24}={\gamma }_0$, $g_{42}=0.2{\gamma }_0$, and $g_{41}=0.01{\gamma }_0$.

Figure 8

(Color online) Same as in Fig. 7, but with parameters $g_{31}=0.7{\gamma }_0$ for the solid blue line, $g_{31}=0.85{\gamma }_0$ for the red dashed line, and $g_{31}=0.85{\gamma }_0$ for the green dash-dotted line. The other parameters are ${\Delta }_{42}=-5{\gamma }_0$, ${\Delta }_{31}=0$, $g_{32}=0$, $2{\gamma }_{13}=2{\gamma }_{14}=2{\gamma }_{23}=2{\gamma }_{24}={\gamma }_0$, $g_{42}=0.6{\gamma }_0$, and $g_{41}=0.01{\gamma }_0$. The solid vertical line indicates ${\Delta }_{41}=4{\gamma }_0$.