Light transport in $\mathcal{PT}$-invariant photonic structures with hidden symmetries

We introduce a recursive bosonic quantization technique for generating classical parity-time ($\mathcal{PT}$) photonic structures that possess hidden symmetries and higher-order exceptional points. We study light transport in these geometries and we demonstrate that perfect state transfer is possible only for certain initial conditions. Moreover, we show that for the same propagation direction, left and right coherent transports are not symmetric with field amplitudes following two different trajectories. A general scheme for identifying the conservation laws in such $\mathcal{PT}$-symmetric photonic networks is also presented.

Figure 1

(a, b) A schematic of waveguide (cavity) arrays that can be used to emulate Eq. (3) when $N=1$ and $N=3/2$, respectively. The waveguides (cavities) are represented by the nodes, and the network connectivity [diagonal (off-diagonal) values of $\Omega$] is indicated in the figure. Distances between elements are not shown to scale.

Figure 2

A schematic of the coupled photonic structures that correspond to two- and three-boson representations of ${\widehat{H}}_3^{\left(2\right)}$ are shown in (a) and (b), respectively. Similar to Fig. 1, the diagonal and off-diagonal elements are also indicated in the figure. Distances between elements are not shown to scale.

Figure 3

(a, b) The intensity distribution (or probability amplitudes) $|c_n{|}^2$ associated with the degenerate eigenmodes ${|1,0,2\rangle }_D$ and ${|0,2,1\rangle }_D$ when $\kappa =1$ and $\gamma =1$, while (c) and (d) illustrate the same modes when $\kappa =1$ and $\gamma =3$. Note that the onset of $\mathcal{PT}$ phase transition is $\gamma =2\kappa$. The transparent cylinders represent the waveguide structure.

Figure 4

Depicts boson occupation numbers of some degenerate eigenstates (each degenerate pair is schematically shown in one row) associated with the Hamiltonian ${\widehat{H}}_{D,3}^{\left(2\right)}=-\varepsilon \left({\widehat{q}}_1^{†}{\widehat{q}}_1-{\widehat{q}}_3^{†}{\widehat{q}}_3\right)$, where $\varepsilon =\hslash \sqrt{4{\kappa }^2-{\gamma }^2}$. The eigenvalue levels (analogous to energy levels in atoms) are shown by thick yellow lines, while individual bosons are represented using red spheres. For instance, it is clear that bosonic occupation configurations shown in the first row have a null eigenvalue. The action of hidden symmetry operators $A_{1,2}$ is also indicated in the figure. Evidently, even higher-order degeneracies will arise for Hamiltonians associated with more than three sites.

Figure 5

(a) Light transport in the array shown in Fig. 2 under the initial excitations: $c_n\left(0\right)={\delta }_{n,1}$. Note that perfect state transfer from $c_1$ to $c_6$ occurs after a propagation distance $l_1$. (b) Depicts the revival dynamics after one full cycle where coherent transport in the opposite direction (from $c_6$ to $c_1$) occurs after $l_1\ne l_2$. Note that the optical power levels depend on the transport direction. (c) Demonstrates the absence of PST when the initial excitation is $c_n\left(0\right)={\delta }_{n,2}$. The transparent cylinders in (a) represent the individual waveguides. The array numbering scheme is also indicated in (a).

Figure 6

Schematics of possible implementation of $\mathcal{PT}$ networks using (a) Er/Yr planar waveguide technology, (b) two-dimensional Er/Yr laser-written waveguide channels, and (c) semiconductor microring resonators. Details about material systems and principle of operations are presented in the text.