Behavior of three modes of decay channels and their self-energies of elliptic dielectric microcavity

Abstract

The Lamb shift (self-energy) of an elliptic dielectric microcavity is studied. We show that the size of the Lamb shift, which is a small energy shift due to the system-environment coupling in the quantum regime, is dependent on the geometry of the boundary conditions. It shows a global transition depending on the eccentricity of the ellipsis. These transitions can be classified into three types of decay channels known as whispering-gallery modes, stable-bouncing-ball modes, and unstablebouncing-ball modes. These modes are manifested through the Poincaré surface of section with the Husimi distribution function in classical phase space. It is found that the similarity (measured in Bhattacharyya distance) between the Husimi distributions below critical lines of two different modes increases as the difference of their self-energies decreases when the quality factors of the modes are on the same order of magnitude.

Non-Hermiticity and conservation of orthogonal relation in dielectric microcavity

“…It is important to note that the physical meaning of the relation, g g -( ) ( ) R R jj kk , is the relative difference of with interaction of the system-bath at each energy levels, instead of just difference of unperturbed eigenvalues, n n -D D 1 2 . In our previous work [33], we confirmed that the crossings of self-energies of resonances, i.e., g g…”

“…Now we consider the Lamb shift. The Lamb shift, which is a small energy difference between a closed and an open system due to the system-bath interaction, can be also obtained by the effective non-Hermitian Hamiltonian in equation (15) [32,33,45]. That is, it is the difference between the eigenvalues of H S and the real part of the eigenvalues ofĤ eff :…”

Relative Entropy as a Measure of Difference between Hermitian and Non-Hermitian Systems

“…Non-Hermitian Hamiltonians were originated from nuclear physics [ 4 ] in 1958; nowadays, they have been applied to diverse quantum mechanical systems not only in atomic [ 6 ] and solid state physics [ 7 ], but also for optical microcavities [ 8 , 9 ]. Furthermore, non-Hermitian systems exhibit various physical phenomena such as phase rigidity [ 10 ], spontaneous emissions [ 11 , 12 ], parity–time symmetry [ 13 , 14 , 15 ], exceptional points [ 16 , 17 , 18 , 19 ], and Lamb shifts [ 20 , 21 ].…”

“…The Lamb shift describes the small energy difference in a quantum system due to system–bath coupling or vacuum fluctuations [ 22 , 23 ]. The effect was first studied in the case of the hydrogen atom [ 22 ], and recently it has been investigated in metamaterial waveguides [ 24 ], open photonic systems [ 25 ], and optical microcavities [ 20 , 21 ]. In our previous works [ 20 , 21 ], we have employed the Lamb shift as a tool to systemically compare the Hermitian and non-Hermitian systems, by quantifying the difference between the energy eigenvalues of the Hermitian and non-Hermitian systems.…”

“…The effect was first studied in the case of the hydrogen atom [ 22 ], and recently it has been investigated in metamaterial waveguides [ 24 ], open photonic systems [ 25 ], and optical microcavities [ 20 , 21 ]. In our previous works [ 20 , 21 ], we have employed the Lamb shift as a tool to systemically compare the Hermitian and non-Hermitian systems, by quantifying the difference between the energy eigenvalues of the Hermitian and non-Hermitian systems. However, a certain appearance of the Lamb shift and the collective Lamb shift between two systems is very weak when the openness effects mostly appear in the imaginary part of the energy eigenvalues.…”