Abstract
We employ the relative entropy as a measure to quantify the difference of eigenmodes between Hermitian and non-Hermitian systems in elliptic optical microcavities. We have found that the average value of the relative entropy in the range of the collective Lamb shift is large, while that in the range of self-energy is small. Furthermore, the weak and strong interactions in the non-Hermitian system exhibit rather different behaviors in terms of the relative entropy, and thus it displays an obvious exchange of eigenmodes in the elliptic microcavity.
Highlights
The Hermitian property of observables is one of the fundamental principles in quantum physics.All the eigenvalues of a given Hermitian system are real and its eigenmodes corresponding to separable eigenvalues are orthogonal to each other
A convenient and effective method to deal with an open system is the application of a non-Hermitian Hamiltonian model [2,3] in terms of the Feshbach projection-operator (FPO) formalism [4], in which the FPO divides the total Hermitian system into two subspaces of a non-Hermitian subsystem and a complementary subsystem known as bath
We studied the relative entropy to quantify the difference of eigenmodes between the Hermitian and non-Hermitian systems in an elliptic optical microcavity
Summary
The Hermitian property of observables is one of the fundamental principles in quantum physics. The Hamiltonian operator of a non-Hermitian system typically has complex eigenvalues, and its eigenmodes corresponding to different eigenvalues are bi-orthogonal [5] contrary to those in the Hermitian cases. Hamiltonian in a closed system S, known as a Hermitian system As mentioned above, it always has real eigenvalues and its eigenmodes with different eigenvalues are orthogonal. Without the complete isolation of the system, it is natural to assume that the system interacts with its bath B; the resulting system is open In this case, the open system can be effectively described by a non-Hermitian Hamiltonian [2,3]. To simplify the consideration of strong and weak interactions, we assume that the effective (non-Hermitian) Hamiltonian HNH is symmetric and the off-diagonal (i.e., coupling) terms are real. The eigenvalues of HNH show a repulsion in the imaginary part while a crossing occurs in the real part in the case of 2δ0 < |Im(ω1 ) − Im(ω2 )|, i.e., the weak interaction [46]
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