$\mathcal{PT}$ -symmetric quantum oscillator in an optical cavity

Abstract

The quantum harmonic oscillator with parity-time (PT ) symmetry, obtained from the ordinary (Hermitian) quantum harmonic oscillator by an imaginary displacement of the spatial coordinate, provides an important and exactly-solvable model to investigate non-Hermitian extension of the Ehrenfest theorem. Here it is shown that transverse light dynamics in an optical resonator with off-axis longitudinal pumping can emulate a PT -symmetric quantum harmonic oscillator, providing an experimentally accessible system to investigate non-Hermitian coherent state propagation.

Efficient Quantum Simulation of an Anti-P-Pseudo-Hermitian Two-Level System

“…A class of Hamiltonians with PT-symmetry, which commute with the joint operation parity P and time-reversal operator T , was found to be able to keep the eigenvalues of H real in the exact PT phase [ 23 ]. Since then, PT quantum mechanics has been extensively investigated theoretically [ 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 ] and experimentally [ 12 , 13 , 14 , 45 , 46 , 47 , 48 ]. In fact, P -pseudo-Hermiticity was pointed out to be a sufficient and necessary condition to keep the spectrum of a non-Hermitian Hamiltonian purely real [ 49 , 50 , 51 ], and both the theory and applications are developed further [ 52 , 53 , 54 , 55 , 56 , 57 ].…”

Anyonic $\boldsymbol{\mathcal{PT}}$ symmetry, drifting potentials and non-Hermitian delocalization

“…and which is a self-orthogonal mode, corresponding to an exceptional point of H 0 at the edge of the continuum [41].…”

“…This happens even for a stationary (i.e. not drifting) potential and does not require anyonic PT symmetry (see, for example, [41]). In the latter case a large value of G ∞ arises because of the non-Hermitian delocalization of the bound state as v → v c , i.e.…”

“…which is a self-orthogonal mode, corresponding to an exceptional point of H 0 at the edge of the continuum [41]. Figure 4 shows the behavior of the power growth G ∞ for the potential well with ϕ = π/3 anyonic phase versus the drift velocity v for a few increasing values of the complex space displacement iδ.…”

Non-Hermitian multimode interference

“…An example of selfimaging and breakdown of mirror-imaging in the PTsymmetric Pöschl-Teller waveguide is shown in Fig.2. The non-orthogonality of modes is responsible for an enhanced sensitivity of the system to perturbations, and is measured by the Petermann factor [32,34,40]…”