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.

Figure 1

(a) Complex energy spectrum (stars) of a one-dimensional (1D) non-Hermitian lattice with chiral symmetry. The effective Hamiltonian $\tilde{H}$ is given by the general expression (1) with the couplings shown in the schematic, where $h=(1+i)g$. Filled and open circles indicate sublattices $A$ and $B$, and the shades on the lattice represent the zero-energy dark state. (b) Trajectories of ${\varepsilon }_n$ (solid lines) as $h$ increases from 0 (squares) to $1.5g$. The lattice and couplings are shown in the schematic.

Figure 2

(a) Complex energy spectrum of a rectangular lattice with non-Hermitian particle-hole symmetry. The system is pumped at the $A$ cavity marked in panel (e) with pump strength ${\gamma }_A=2.0g_x$. The couplings are shown in panel (c) and satisfy $g_y=1.25g_x\in \mathbb{R}$, and the cavity decay rate is given by ${\kappa }_0=0.5g_x$. (b) Same as in panel (a) but with a lower pump strength (${\gamma }_A=1.5g_x$) and no zero modes. (c),(e) Schematics showing the spatial profile and phase distribution of mode 1 in (a). Gaussian packets are used to represent $|{\mathrm{\Psi }}_1{|}^2$ in panel (c). (d), (f) Same as panels (c) and (e) but for mode 4 in panel (a).

Figure 3

Spontaneous restoration of non-Hermitian particle-hole symmetry (b) and the lack of it (a) for modes 1 and 2 in Fig. 2 as the pump strength increases. The laser threshold is reached at ${\gamma }_A={\gamma }_B\approx 1.40g_x$ in panel (a) and ${\gamma }_A\approx 2.02g_x$ in panel (b).

Figure 4

Tunable spatial profile of a zero-mode photonic lattice laser. (a), (c) Spatial profile and phase distribution of mode 5 in Fig. 2 when pumping the $A$ cavity at the upper left corner of the rectangular lattice. (b), (d) The same but when pumping the $A$ cavity marked in panel (d).

Figure 5

Boosted gain of the lasing zero mode (thick black line) after the exceptional point (filled dot). Panels (a) and (b) correspond to the two different pumping configurations in Fig. 4. Thin gray lines show the other 15 supermodes originated from the same single-cavity frequency.

Figure 6

Effect of disorder that break particle-hole symmetry. (a) Histogram of the lasing frequency for 20 000 realizations of ${\omega }_0$ disorder (dark blue bars) and NNN coupling disorder (light blue bars), both with a standard deviation of $g_x/20$ (red solid curve). The pump configuration is the same as in Fig. 5. (b) Same as panel (a) but for the pump configuration in Fig. 5.

Figure 7

(a) Simultaneous restoration of non-Hermitian particle-hole symmetry and breaking of $\mathcal{PT}$ symmetry for modes 5 and 6 in Fig. 2. The pump is applied to the entire sublattice $A$ [filled dots in panel (b)], and the arrows indicate the directions of increasing pump. The spatial profile of mode 5 at the laser threshold (${\gamma }_A\approx 0.55g_x$) is shown in panel (b). (c) $\mathcal{PT}$ symmetry breaking for modes 1, 2, 5, and 6 in Fig. 2 without restoring their particle-hole symmetry. The pump is applied to the eight cavities in the two $2\times 2$ diagonal blocks. The identical spatial profile of modes 1 and 2 at the laser threshold (${\gamma }_A={\gamma }_B\approx 0.59g_x$) is shown in panel (d). (e) Breaking of non-Hermitian particle-hole symmetry for two pairs of modes without restoring their $\mathcal{PT}$ symmetry. The pump profile is the same as in panel (c) and ${\gamma }_A={\gamma }_B\in [6.713.4]g_x$. The identical spatial profile of modes 7 and 8 at the highest pump strength is shown in panel (f).