Bloch Oscillations in Complex Crystals with $\mathcal{P}\mathcal{T}$ Symmetry

Bloch oscillations in complex lattices with $\mathcal{P}\mathcal{T}$ symmetry are theoretically investigated with specific reference to optical Bloch oscillations in photonic lattices with gain or loss regions. Novel dynamical phenomena with no counterpart in ordinary lattices, such as nonreciprocal Bloch oscillations related to violation of the Friedel’s law of Bragg scattering in complex potentials, are highlighted.

Figure 1

BO in the complex lattice of Eq. (2) for $\alpha =0.3$ and for an index gradient (a) $F=-3.52{\mathrm{m}}^{-1}$ and (b) $F=3.52{\mathrm{m}}^{-1}$. Other parameter values are $a=6\mu \mathrm{m}$, $\lambda =633\mathrm{nm}$, $V_0=0.001$, and $n_s=1.42$. The figures show the evolution of field amplitude $|\psi (x,z)|$ and beam power $P=\int dx|\psi (x,z){|}^2$ (normalized to the input power) versus propagation distance $z$. The BO period is $z_B=3\mathrm{cm}$.

Figure 2

Same as Fig. 1, but for $\alpha ={\alpha }_c=1$ and (a) $F=-7.03{\mathrm{m}}^{-1}$, (b) $F=7.03{\mathrm{m}}^{-1}$. Other parameter values are as in Fig. 1, except for $V_0=2\times {10}^{-4}$. Note that BO disappear for a negative index gradient $F$.

Figure 3

BO in a complex binary lattice ($a=10\mu \mathrm{m}$). (a) Profiles of $V_R$ and $V_I$, and (b) corresponding band diagram for $\alpha =0.01$ (real and imaginary parts of energy $E$). (c) Resonance curves in the ($\Delta /f$, $g/f$) plane corresponding to a real-valued WS ladder spectrum. (d) Breathing BO mode (single waveguide input excitation) for $F=1.618{\mathrm{m}}^{-1}$, corresponding to a Bloch period $z_B\simeq 39.1\mathrm{mm}$.