Extension of Noether’s theorem in ${\cal P}{\cal T}$-symmetry systems and its experimental demonstration in an optical setup

Abstract

Noether’s theorem is one of the fundamental laws in physics, relating the symmetry of a physical system to its constant of motion and conservation law. On the other hand, there exist a variety of non-Hermitian parity-time (${\cal P}{\cal T}$)-symmetric systems, which exhibit novel quantum properties and have attracted increasing interest. In this work, we extend Noether’s theorem to a class of significant ${\cal P}{\cal T}$-symmetry systems for which the eigenvalues of the ${\cal P}{\cal T}$-symmetry Hamiltonian ${{\hat H}_{{\cal P}{\cal T}}}$ change from purely real numbers to purely imaginary numbers, and introduce a generalized expectation value of an operator based on biorthogonal quantum mechanics. We find that the generalized expectation value of a time-independent operator is a constant of motion when the operator presents a standard symmetry in the ${\cal P}{\cal T}$-symmetry unbroken regime, or a chiral symmetry in the ${\cal P}{\cal T}$-symmetry broken regime. In addition, we experimentally investigate the extended Noether’s theorem in ${\cal P}{\cal T}$-symmetry single-qubit and two-qubit systems using an optical setup. Our experiment demonstrates the existence of the constant of motion and reveals how this constant of motion can be used to judge whether the ${\cal P}{\cal T}$-symmetry of a system is broken. Furthermore, a novel phenomenon of masking quantum information is first observed in a ${\cal P}{\cal T}$-symmetry two-qubit system. This study not only contributes to full understanding of the relation between symmetry and conservation law in ${\cal P}{\cal T}$-symmetry physics, but also has potential applications in quantum information theory and quantum communication protocols.