Robust phase sensing via a nonlinear parity-time symmetric system

Parity-time symmetric systems featuring loss and gain manifest as a form of non-Hermitian Hamiltonian. The inclusion of a nonlinear saturable gain can eliminate the imaginary part of frequency eigenvalues and suppress noise. Consequently, it has been employed to enhance sensing near an exceptional point (EP). A perturbation to the off-diagonal elements of the Hamiltonian leads the frequency eigenvalue to scale with the square-root of the perturbation near an EP. This, however, is associated with a lowered scaling factor, potentially diminishing the sensitivity boosted by the square-root singularity. When a perturbation is introduced to the diagonal elements of the Hamiltonian, the frequency eigenvalue exhibits a cube-root singularity in a two-level system. Nevertheless, the response has a relatively small scaling factor and the cube-root singularity comes at the expense of the sensing dynamic range. An entirely different approach we propose here is to detect the phase difference between the loss and gain resonator instead of focusing on detecting the frequency. We demonstrate theoretically and experimentally that the phase difference reveals a cube-root singularity with an enlarged scaling factor over a wide dynamic range. Experimental results for our wearable capacitive temperature sensors show that the sensitivity of EP phase sensing has enhanced by an order of magnitude across a broad dynamic range relative to that of EP frequency sensing. These findings resolve a longstanding debate on the efficacy of EP-based sensors and pave the way for their practical applications.