Parametric and nonparametric magnetic response enhancement via electrically induced magnetic moments

The realization of negative refraction in atomic gases requires a strong magnetic response of the atoms. Current proposals for such systems achieve a parametric enhancement of the magnetic response by one inverse power of the fine-structure constant $\alpha$, but still rely on high gas densities. To enable further enhancement, here we study in detail the magnetic and electric response to a probe field interacting with three-level atoms in ladder configuration. In our first model, the three transitions are driven by a control field and the electric and magnetic components of the probe field, giving rise to a closed interaction loop. In a reference model, the coherent driving is replaced by an incoherent pump field. A time-dependent analysis of the closed-loop system enables us to identify the different contributions to the medium response and to determine the enhancement mechanism. Based on these results, we show that in addition to the parametric enhancement of ${\alpha }^{-1}\sim 137$, our system allows for a further nonparametric enhancement of the magnetic response by almost two orders of magnitude, relaxing the requirement for a high vapor density.

Figure 1

(Color online) (a) Three-level system driven by coherent fields in loop configuration. The probe field components ${\Omega }_{21}$ and ${\Omega }_{32}$ are denoted by red solid double arrows, the coupling field ${\Omega }_{31}$ by purple solid double arrows. Spontaneous emission is indicated by the wiggly green arrows. The transition $\mid 1⟩\leftrightarrow \mid 2⟩$ couples to the magnetic component, while transition $\mid 2⟩\leftrightarrow \mid 3⟩$ couples to the electric component of the same probe field. (b) Reference system obtained by replacing the coherent control field by an incoherent, bidirectional pumping, indicated by the green dashed double arrow.

Figure 2

(Color online) Real (red solid curve) and imaginary (blue dashed curve) parts of the direct terms in the magnetic and electric susceptibilities, respectively. The top row shows ${\widehat{\varrho }}_{21}^{(1,0)}$, the bottom row ${\widehat{\varrho }}_{32}^{(0,1)}$. (a) Closed-loop system and (b) incoherently pumped system as shown in Fig. 1.

Figure 3

(Color online) Dependence of the expansion coefficients ${\widehat{\varrho }}_{32}^{(0,1)}$ in the closed loop (black solid line) and in the incoherently pumped system (green long-dashed line) as well as ${\widehat{\varrho }}_{21}^{(1,0)}$ for the closed-loop (blue dash-dotted) and the incoherently pumped (red dashed) systems on the strength of the control field and the incoherent pump rate, respectively. Shown are only the imaginary parts of the coefficients evaluated at their absolute maxima. For the incoherent configuration, the maximum of $\mathrm{Im}\left({\widehat{\varrho }}_{21}^{(1,0)}\right)$ is always at ${\Delta }_3=0$, and the maximum of $\mathrm{Im}\left({\widehat{\varrho }}_{32}^{(0,1)}\right)$ is at ${\Delta }_2=0$. The coefficients are scaled by $\gamma$ to obtain a unitless quantity and ${\widehat{\varrho }}_{21}^{(1,0)}$ is scaled by a factor of ${10}^{-4}$. In this figure, negative values indicate amplification, positive values absorption.

Figure 4

(Color online) Real (red solid line) and imaginary (blue dashed curve) parts of the expansion coefficients (a) ${\widehat{\varrho }}_{21}^{(1,0)}$ and (b) ${\widehat{\varrho }}_{32}^{(0,1)}$ in Eqs. (15). The curves are drawn for ${\Delta }_1=2\gamma$ in the closed-loop system. The coefficients are scaled by $\gamma$. The parameters are as in Fig. 2a, except for the nonvanishing detuning of the control field.

Figure 5

(Color online) Interpretation of the different scattering processes that contribute to ${\tilde{\varrho }}_{21}$ in the case of $\Delta =0$. (a) ${\tilde{R}}_0$ contributes to the cross term in ${\tilde{\varrho }}_{21}$, and thus to the chirality ${\tilde{\xi }}_{HE}$. (b) ${\tilde{R}}_1$ contributes to the direct term, i.e., to the magnetic susceptibility. (c) ${\tilde{R}}_{-1}$ is of higher order in either one of the probe field Rabi frequencies and therefore negligible in our calculation.

Figure 6

(Color online) Interpretation of the different scattering processes that contribute to ${\tilde{\varrho }}_{32}$ in the case of $\Delta =0$. (a) ${\tilde{R}}_0$ contributes to the direct term in ${\tilde{\varrho }}_{32}$, and thus to the electric susceptibility. (b) ${\tilde{R}}_1$ is of higher order in either one of the probe field Rabi frequencies and can therefore be neglected. (c) ${\tilde{R}}_{-1}$ contributes to one of the cross terms, i.e., to one of the chirality coefficients.

Figure 7

(Color online) Real (red solid line) and imaginary (blue dashed) parts of the expansion coefficient ${\widehat{\varrho }}_{21}^{(0,1)}$ in the closed-loop system [see Eqs. (15)]. ${\widehat{\varrho }}_{32}^{(1,0)}$ is not shown, since it is virtually identical to $-{\widehat{\varrho }}_{21}^{(0,1)}$. The coefficients correspond to cross terms and determine ${\tilde{\xi }}_{HE}$ and ${\tilde{\xi }}_{EH}$, respectively. The plotted coefficients are phase dependent; in this figure all phases are set to zero. The variable $\sigma =({\Delta }_3-{\Delta }_2)∕2={\omega }_2-({\omega }_3+{\omega }_1)∕2$ denotes the shift of the eigenfrequency of ∣2⟩ with respect to the average frequency of ∣1⟩ and ∣3⟩. Then, for all values of $\sigma$, the multiphoton resonance condition $\Delta =0$ is satisfied for a fixed probe field frequency ${\omega }_p$.

Figure 8

(Color online) Nonparametric enhancement of the magnetic response. The parameters are as in Fig. 7, but with ${\Omega }_{31}=\alpha \gamma$. Note the different scale of the frequency axis.

Figure 9

(Color online) Example level scheme for magnetic response enhancement independent of the probe field frequency. ${\Omega }_1$ and ${\Omega }_2$ are coherent coupling fields with frequencies ${\omega }_1$ and ${\omega }_2$, respectively. $E$ and $B$ are the electric and magnetic components of the probe field with frequency ${\omega }_p$. Here, the closed-loop path contains an absorption and an emission of a probe field photon such that the multiphoton detuning $\Delta ={\omega }_1-{\omega }_2$ can be satisfied for arbitrary probe field frequencies.