Non-Hermitian Degeneracies and Unidirectional Reflectionless Atomic Lattices

Light propagation in optical lattices of driven cold atoms exhibits non-Hermitian degeneracies when the first-order modulation amplitudes of real and imaginary parts of the probe susceptibility are manipulated to be balanced. At these degeneracies, one may observe complete unidirectional reflectionless light propagation. This strictly occurs with no gain and can be easily tuned and fully reversed as supported by the transfer-matrix calculations and explained via a coupled-mode analysis.

Figure 1

(a) An ensemble of cold ${}^{87}\mathrm{Rb}$ atoms is loaded into a 1D optical lattice of dipole traps. (b) These atoms interact with a near-resonant probe (${\mathrm{\Omega }}_p$) field, a near-resonant coupling (${\mathrm{\Omega }}_c$) field, and a far-detuned dressing (${\mathrm{\Omega }}_d$) field. (c) Probe and coupling fields travel along the lattice $z$ axis with ${\sigma }^{+}$ and ${\sigma }^{-}$ polarizations, respectively. The dressing field has instead two components nearly counterpropagating at a very small angular offset $\pm {\theta }_d$, both with ${\sigma }^{+}$ polarization, to give a spatial modulation of the light shift ${\delta }_{ds}(z)$ of transition $|2⟩\leftrightarrow |4⟩$, and a third component propagating along the $x$ direction and linearly polarized along $y$ to provide a light shift that is uniform in space (see text).

Figure 2

Plots of ${10}^3\mathrm{Im}({\chi }_{pj})$ and ${10}^3\mathrm{Re}({\chi }_{pj})$ [(a), (b)] vs lattice position $(z-z_j)/a$ and probe detuning ${\mathrm{\Delta }}_p/2\pi$ with ${\gamma }_{12}/2\pi =2.0\mathrm{kHz}$ (increasing ${\gamma }_{12}/2\pi$ up to 20 kHz is immaterial, while for larger values the absorption at each trap center starts to increase somewhat, without qualitatively affecting our results), $\gamma /2\pi =3.0\mathrm{MHz}$, ${\mathrm{\Omega }}_c/2\pi =2.0\mathrm{MHz}$, ${\mathrm{\Omega }}_d/2\pi =16\mathrm{MHz}$, ${\mathrm{\Delta }}_c=0$, ${\mathrm{\Delta }}_d/2\pi =300\mathrm{MHz}$, $a=5d\simeq 400\mathrm{nm}$, $d_{13}=2.0\times {10}^{-29}\mathrm{Cm}$, and $N_0=0.5\times {10}^{12}{\mathrm{cm}}^{-3}$. [(c), (d)] Same as [(a), (b)] plotted at ${\mathrm{\Delta }}_p=0$ for ${\mathrm{\Omega }}_d/2\pi =16\mathrm{MHz}$ (black circles), 22 MHz (red triangles), 28 MHz (blue squares).

Figure 3

Plots of $f_{pa}$ and $f_{pb}$ (a); ${\chi }_{pa}$ and ${\chi }_{pb}$ (b) vs dynamic shift ${\delta }_{d0}/2\pi$ attained from Eq. (4) with the same parameters as in Fig. 2 except ${\mathrm{\Delta }}_p=0$.

Figure 4

Plots of $R_L$ (a) and $R_R$ (b) vs dynamic shift ${\delta }_{d0}/2\pi$ and probe detuning ${\mathrm{\Delta }}_p/2\pi$ for atomic lattices of length $L=0.6\mathrm{mm}$ with the same parameters as in Fig. 2 except $d=a/10$. Plots of $R_L$ (c) and $R_R$ (d) vs dynamic shift ${\delta }_{d0}/2\pi$ with the same parameters as in (a) and (b) except ${\mathrm{\Delta }}_p=0$ and $d=a/10$ for red curves with triangles; $d=a/5$ for blue curves with squares. The two insets show the corresponding reciprocal transmissivity $T=T_L=T_R$.

Figure 5

Real and imaginary parts of ${\lambda }_s^{\pm }$ [(a)–(d)] vs dynamic shift ${\delta }_{d0}/2\pi$ and probe detuning ${\mathrm{\Delta }}_p/2\pi$. Parameters are the same as in Fig. 2 except $d=a/10$ and $L=0.6\mathrm{mm}$. Axes are oppositely oriented to better display the topology of the non-Hermitian degeneracy at ${\mathrm{\Delta }}_p=0$ and ${\delta }_{d0}/2\pi \simeq 1.68\mathrm{MHz}$. $\mathrm{Im}({\lambda }_s^{\pm })$ and $\mathrm{Re}({\lambda }_s^{\pm })$ [(e), (f)] vs dynamic shift ${\delta }_{d0}/2\pi$ at ${\mathrm{\Delta }}_p=0$.