Symmetry-protected zero-mode laser with a tunable spatial profile

We propose to utilize symmetry-protected zero modes of a photonic lattice to realize a single-mode, fixed-frequency, and spatially tunable laser. These properties are the consequence of the underlying non-Hermitian particle-hole symmetry, with which the energy spectrum satisfies $\varepsilon_m=-\varepsilon_n^*$. Unlike in the Hermitian case, the symmetric phase of particle-hole symmetry is no longer restricted to $\varepsilon=0$ but extends along the imaginary-$\varepsilon$ axis, which is set by the single-cavity frequency and symmetry-protected against position and coupling disorder of the photonic lattice. By selectively pumping different cavities in the photonic lattice, we control the spontaneous symmetry restoration process, which provides a convenient method to tune the spatial profile of the laser without changing its frequency.