arXiv:2007.15732v2 [quant-ph] 3 Mar 2021 Noname manuscript No. (will be inserted by the editor) PT-symmetry in compact phase space for a linear Hamiltonian Ivan F. Valtierra · Mario B. Gaeta · Adrian Ortega · Thomas Gorin Received: date / Accepted: date Abstract We study the time evolution of a PT-symmetric, non-Hermitian quantum system for which the associated phase space is compact. We focus on the simplest non-trivial example of such a Hamiltonian, which is linear in the angular momentum operators. In order to describe the evolution of the system, we use a particular disentangling decomposition of the evolution operator, which remains numerically accurate even in the vicinity of the Exceptional Point. We then analyze how the non-Hermitian part of the Hamiltonian affects the time evolution of two archetypical quantum states, coherent and Dicke states. For that purpose we calculate the Husimi distribution or Q function and study its evolution in phase space. For coherent states, the characteristics of the evolution equation of the Husimi function agree with the trajectories of the corresponding angular momentum expectation values. This allows to consider these curves as the trajectories of a classical system. For other types of quantum states, e.g. Dicke states, the equivalence of characteristics and trajectories of expectation values is lost. Keywords PT-Symmetry · Phase Space 1 Introduction and motivation Non-Hermitian quantum systems may still have a real spectrum, if they are PT- symmetric [1]. Since their introduction by Bender and Boettcher in 1998 [2], these systems have found a wide range of applications [3,4,5,6,7]. One of the defining features of such systems is that the corresponding Hamiltonian may have real or complex eigenvalues. In the first case, one says that the PT-symmetric phase is intact, in the second that it is broken. The transition between these phases occurs at so-called “Exceptional Points” (EPs), at which Departamento de Física, Universidad de Guadalajara, Blvd. Gral. Marcelino García Bar- ragán 1421, C.P. 44430, Guadalajara, Jalisco, México.