Light Stops at Exceptional Points

Tamar Goldzak, Alexei A. Mailybaev, and Nimrod Moiseyev

Phys. Rev. Lett. 120, 013901 – Published 3 January 2018

Abstract

Almost twenty years ago, light was slowed down to less than $10^{- 7}$ of its vacuum speed in a cloud of ultracold atoms of sodium. Upon a sudden turn-off of the coupling laser, a slow light pulse can be imprinted on cold atoms such that it can be read out and converted into a photon again. In this process, the light is stopped by absorbing it and storing its shape within the atomic ensemble. Alternatively, the light can be stopped at the band edge in photonic-crystal waveguides, where the group speed vanishes. Here, we extend the phenomenon of stopped light to the new field of parity-time ( $P T$ ) symmetric systems. We show that zero group speed in $P T$ symmetric optical waveguides can be achieved if the system is prepared at an exceptional point, where two optical modes coalesce. This effect can be tuned for optical pulses in a wide range of frequencies and bandwidths, as we demonstrate in a system of coupled waveguides with gain and loss.

Received 29 September 2017

DOI:https://doi.org/10.1103/PhysRevLett.120.013901

Physics Subject Headings (PhySH)

Light propagation, transmission & absorption Waveguide arrays

Atomic, Molecular & Optical

Authors & Affiliations

Tamar Goldzak¹, Alexei A. Mailybaev^2,*, and Nimrod Moiseyev^1,†

¹Schulich Faculty of Chemistry and Faculty of Physics, Technion—Israel Institute of Technology, Haifa 32000, Israel
²Instituto Nacional de Matemática Pura e Aplicada—IMPA, 22460-320 Rio de Janeiro, Brazil

^*alexei@impa.br
^†nimrod@tx.technion.ac.il

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Vol. 120, Iss. 1 — 5 January 2018

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Images

Figure 1
Coupling between the gain guided mode and the loss guided mode provides a $P T$ -symmetric system with the refractive index profile such that $n (x) = n^{*} (- x)$ . A control parameter $α$ defines the gain and loss strength in the WGs as $Im n = \pm α / k$ with $k = ω / c$ . In simulations of Figs. 3 and 4, we use the WGs of width $W = 1.25 μ m$ and the distance between them $D = 1.25 μ m$ . Analogous setup [15] with different parameters has been used in the first experiment [8], which demonstrated the $P T$ symmetry in optics.
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Figure 2
(a) Adiabatic change of the gain-loss parameter from $α = 0$ at $t = 0$ to $α_{EP} = 1$ at final time $t = 2000 / c$ ( $c$ is the light speed in vacuum) in a system with representative parameters $k = 0.5$ , $n_{w} = 1.6$ , and $δ = 0.5$ (arb. uints). (b) Temporal evolution of the center $z_{c}$ of the Gaussian pulse, stopping when $α$ reaches the value $α_{EP} = 1$ at EP. The pulse is prepared initially in antisymmetric mode of the system with no gain or loss with the mean propagation number $β = 0.8$ and standard deviation $σ = 0.01$ . (c) Pulse envelope $| ψ_{1} |$ in the first WG at times $c t = 0, 250, \dots, 2000$ , which correspond to squares in the upper figures. At the three latest times, the group speed vanishes and the corresponding graphs collapse to a single curve demonstrating the full-stop of a pulse. A backward change of the gain-loss parameter brings the optical signal to its original mobile form.
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Figure 3
(a) The propagation constant $β$ as a function of $k = ω / c$ at two different values of the gain-loss parameter: Hermitian system with $α = 0$ (dashed black lines: upper symmetric and lower antisymmetric modes) and non-Hermitian $P T$ -symmetric system with $α = 0.15 μ m^{- 1}$ (solid blue line). Two modes of the $P T$ -symmetric system coalesce at the EP marked with a red circle. Inset shows enlarged vicinity of the EP at $k_{EP} = 0.6414 μ m^{- 1}$ and $β_{EP} = 0.851 μ m^{- 1}$ . The infinite derivative $d β / d k$ at the EP yields the vanishing group velocity $v_{g} = c (d β / d k)^{- 1}$ . (b) Real and imaginary parts of the complex eigenfunction for the $P T$ -symmetric system at the EP. This mode satisfies the self-orthogonality condition $\int ψ^{2} d x = 0$ . Grey vertical regions in the background show positions of the two coupled WGs.
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Figure 4
Contour plot of the light power $| ϕ (x, z, t) |^{2}$ for the Gaussian wave packets at the initial time $t = 0$ and the final time $t = 10 ps$ . In both plots, the pulse has the mean propagating constant $β = 0.851 μ m^{- 1}$ with standard deviation $σ = 0.002 μ m^{- 1}$ . Grey regions in the background show positions of the two coupled WGs. (a) Fully stopped pulse centered exactly at the EP in the $P T$ -symmetric system for $α = 0.15 μ m^{- 1}$ . (b) The antisymmetric mode in the Hermitian case with no gain and loss ( $α = 0$ ). One can see that the pulse dislocates for about 1.4 mm in the Hermitian case, while it does not move at all when prepared at the EP in the $P T$ -symmetric system.
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