Spontaneous $\mathcal{PT}$ symmetry breaking in Dirac-Kronig-Penney crystals

We introduce a non-Hermitian $\mathcal{PT}$ invariant extension of the Dirac-Kronig-Penney model, describing the motion of a Dirac quasiparticle in a locally periodic sequence of imaginary $\delta$-Dirac barriers and wells, and propose its optical realization using superstructure fiber Bragg gratings with alternating regions of optical gain and absorption. For the infinite crystal, we determine the band structure and show that the $\mathcal{PT}$ phase is always broken. For a finite crystal, we derive analytical expressions for reflection and transmission probabilities, and show that the $\mathcal{PT}$ phase is unbroken below a finite threshold of the $\delta$-barrier area. In the proposed optical realization, the onset of $\mathcal{PT}$ symmetry breaking in the finite crystal corresponds to the lasing condition for the grating superstructures.

Figure 1

(a) Schematic behavior of the complex vector potential $V\left(x\right)$ in the unit cell for the $\mathcal{PT}$ invariant DKP crystal and (b) its optical realization based on a superstructure FBG with alternating regions of gain and absorption. The $2\times 2$ transfer matrix ${\mathcal{M}}^{\left(\text{cell}\right)}$ of the unit cell, describing the propagation of the wave function $\psi$ from $x=0$ to $2d$, is obtained as the ordered product of the four matrices ${\mathcal{M}}_{-}$, ${\mathcal{M}}_0$, ${\mathcal{M}}_{+}$, and ${\mathcal{M}}_0$ shown in (a) and describing propagation in the four sections of the unit cell with $V\left(x\right)$ constant.

Figure 2

$\mathcal{PT}$ symmetry breaking point $Q_c$ of the finite DKP crystal for $md=1$ and for increasing number $N$ of unit cells (dots). The solid curve is the approximate value of $Q_c$ as given by Eq. (37).

Figure 3

Behavior of (a) spectral transmittance and spectral reflectance for (b) left- and (c) right-side particle incidence in a $\mathcal{PT}$ symmetric DKP crystal made of $N=5$ unit cells for increasing values of $Q$ and for $m=d=1$. Curve 1: $Q=0$; curve 2: $Q=0.05$; curve 3: $Q=0.1$; curve 4: $Q=0.15$. The $\mathcal{PT}$ symmetry breaking is attained at $Q_c\simeq 0.1556$.

Figure 4

Behavior of spectral transmittance (solid curve), numerically computed using Eqs. ((3, 4, 5)), in a crystal composed by $N=5$ cells for parameter values $m=1$, $d=1$, $a=0.2$, and $V_1=0.5$. The dashed curve depicts the behavior of the spectral transmittance as computed for the $\delta$-Dirac limit of the potential wells and barriers with an area $Q=V_{1}a=0.1$.