Anti-$\mathcal{PT}$-symmetric Kerr gyroscope

Non-Hermitian systems can exhibit unconventional spectral singularities called exceptional points (EPs). Various EP sensors have been fabricated in recent years, showing strong spectral responses to external signals. Here we propose how to achieve a nonlinear anti-parity-time ($\mathcal{APT}$) gyroscope by spinning an optical resonator. We show that, in the absence of any nonlinearity, the sensitivity or optical mode splitting of the linear device can be magnified up to 3 orders than that of the conventional device without EPs. Remarkably, the $\mathcal{APT}$ symmetry can be broken when including the Kerr nonlinearity of the materials and, as the result, the detection threshold can be significantly lowered, i.e., much weaker rotations which are well beyond the ability of a linear gyroscope can now be detected with the nonlinear device. Our work shows the powerful ability of $\mathcal{APT}$ gyroscopes in practice to achieve ultrasensitive rotation measurement.

Spectral Properties of a Non-Self-Adjoint Differential Operator with Block-Triangular Operator Coefficients

In this chapter, the Sturm-Liouville equation with block-triangular, increasing at infinity operator potential is considered. A fundamental system of solutions is constructed, one of which decreases at infinity, and the second increases. The asymptotic behavior at infinity was found out. The Green’s function and the resolvent for a non-self-adjoint differential operator are constructed. This allows to obtain sufficient conditions under which the spectrum of this non-self-adjoint differential operator is real and discrete. For a non-self-adjoint Sturm-Liouville operator with a triangular matrix potential growing at infinity, an example of operator having spectral singularities is constructed.