Low-loss negative refraction by laser-induced magnetoelectric cross coupling

We discuss the feasibility of negative refraction with reduced absorption in coherently driven atomic media. Coherent coupling of an electric and a magnetic dipole transition by laser fields induces magnetoelectric cross coupling and negative refraction at dipole densities which are considerably smaller than necessary to achieve a negative permeability. At the same time the absorption gets minimized due to destructive quantum interference and the ratio of negative refraction index to absorption becomes orders of magnitude larger than in systems without coherent cross coupling. The proposed scheme allows for a fine tuning of the refractive index. We derive a generalized expression for the impedance of a medium with magnetoelectric cross coupling and show that impedance matching to vacuum can easily be achieved. Finally we discuss the tensorial properties of the medium response and derive expressions for the dependence of the refractive index on the propagation direction.

Figure 1

A simplistic three-level system which employs electromagnetically induced cross coupling. $E$ and $B$ are the electric and magnetic components of the probe field and ${\gamma }_2$ and ${\gamma }_3$ are the decay rates out of levels $|2⟩$ and $|3⟩$. ${\Omega }_c$ denotes the Rabi frequency of an applied field which couples levels $|2⟩$ and $|3⟩$.

Figure 2

(Color online) Modification of the level scheme of Fig. 1. The ground state $|1⟩$ is substituted by the dark state $|D⟩=\left({\Omega }_2|1⟩-{\Omega }_1|4⟩\right)/\sqrt{{\Omega }_1^2+{\Omega }_2^2}$ of the three-level $\Lambda$-type subsystem formed by levels $\left\{|1⟩,|4⟩,|5⟩\right\}$.

Figure 3

Five-level scheme for the implementation of negative refraction via electromagnetically induced cross coupling. ${\Omega }_1$, ${\Omega }_2$, and ${\Omega }_c$ denote Rabi frequencies of the external laser fields. The magnetic dipole transition $|2⟩\text{−}|1⟩$ and the electric dipole transition $|3⟩\text{−}|4⟩$ are coupled by ${\Omega }_c$ to induce chirality. The ground state of the system is formed by the dark state $|D⟩=\left({\Omega }_2|1⟩-{\Omega }_1|4⟩\right)/\sqrt{{\Omega }_1^2+{\Omega }_2^2}$ of the subsystem $\left\{|1⟩,|4⟩,|5⟩\right\}$.

Figure 4

(Color online) Real (solid red lines) and imaginary (dashed blue lines) parts of the electric $\left({\alpha }^{EE}\right)$ and magnetic $\left({\alpha }^{BB}\right)$ polarizabilities as well as the cross-coupling parameters $\left({\alpha }^{EB},{\alpha }^{BE}\right)$ for arbitrary but the same units of ${\text{cm}}^3$ for ${\Omega }_c={10}^4{\gamma }_2{e}^{i\pi /2}$.

Figure 5

(Color online) Deviation (16) of real and imaginary parts of the exact polarizabilities ${\alpha }^{IJ}\left(E,B\right)$ compared to the linear response results for two different probe-field Rabi frequencies: ${\Omega }_E=137{\Omega }_B={\gamma }_2$ (real: solid red lines; imaginary: dashed blue lines) and ${\Omega }_E=137{\Omega }_B=10\cdot {\gamma }_2$ (real: dashed-dotted green lines; imaginary: dotted orange lines).

Figure 6

(Color online) Real (solid red lines) and imaginary (dashed blue lines) parts of the electric $\left({\alpha }^{EE}\right)$ and magnetic $\left({\alpha }^{BB}\right)$ polarizabilities as well as the cross-coupling parameters $\left({\alpha }^{EB},{\alpha }^{BE}\right)$ for arbitrary but the same units of ${\text{cm}}^3$ for ${\Omega }_c={10}^4{\gamma }_2{e}^{i\pi /2}$. In contrast to Fig. 4 additional homogeneous broadenings according to Eq. (19) with ${\gamma }_p={10}^3{\gamma }_2$ apply.

Figure 7

(Color online) Real (solid red lines) and imaginary (dashed blue lines) parts of the refractive index, including local field effects for two different densities.

Figure 8

(Color online) (a) Real (solid red line) and imaginary (dashed blue line) parts of the refractive index for $\varrho =5\times {10}^{16}{\text{cm}}^{-3}$. Compared to Fig. 7b here ${\tilde{\rho }}_{41}^{\left(0\right)}=0$ applies. (b) Real (solid red line) and imaginary (dashed blue line) parts of the refractive index as a function of the phase $\varphi$ of the coupling Rabi frequency ${\Omega }_c$.

Figure 9

(Color online) Real part of (a) the permittivity $\epsilon$ (solid red line) as well as the imaginary parts of ${\xi }_{EH}$ (dashed blue line) and ${\xi }_{HE}$ (dotted green line) and (b) real (solid red line) and imaginary (dashed blue line, $\times 50$) parts of the refractive index, as well as the real part of the permeability (dotted green line) as a function of the logarithm of the density ${\text{log}}_{10}\varrho$.

Figure 10

(Color online) Real (solid red line) and imaginary (dashed blue line) parts of the refractive index, including local field effects for a density of $\varrho =5\times {10}^{17}{\text{cm}}^{-3}$.

Figure 11

(Color online) The figure of merit $\mathcal{F}$ for densities $\varrho =5\times {10}^{17}{\text{cm}}^{-3}$ (solid red line) and $\varrho =5\times {10}^{16}{\text{cm}}^{-3}$ (dashed blue line) as a function of the detuning $\Delta$.

Figure 12

(Color online) Real (solid red line) and imaginary (dashed blue line) parts of the refractive index as a function of the coupling field Rabi frequency $\left|{\Omega }_c\right|$ relative to the radiative decay rate ${\gamma }_3$ for $\varrho =1.56\times {10}^{17}{\text{cm}}^{-3}$.

Figure 13

(Color online) Real (solid red line) and imaginary (dashed blue line) parts of the refractive index as well as the real (dashed-dotted green line) and imaginary (dotted orange line) parts of the impedance $Z_2^{-1}$ for $\varrho =1.56\times {10}^{17}{\text{cm}}^{-3}$.

Figure 14

(Color online) For a direction of incidence other than the direction of the coupling field vector ${\mathbf{E}}_c$ numerous additional angle-dependent couplings occur (dashed green). The coupling Rabi frequencies are given by ${\Omega }_c^{ab}=⟨3,a|\mathbf{d}\cdot {\mathbf{E}}_c|2,b⟩/\hslash$ with $a,b∊\left\{+,0,-\right\}$.

Figure 15

(Color online) Real (solid red line) and imaginary (dashed blue line) parts of refractive index (45) as a function of $\theta$.