Non-Hermitian Hamiltonian and Lamb shift in circular dielectric microcavity

Abstract

We study the normal modes and quasi-normal modes (QNMs) in circular dielectric microcavities through non-Hermitian Hamiltonian, which come from the modifications due to system-environment coupling. Differences between the two types of modes are studied in detail, including the existence of resonances tails. Numerical calculations of the eigenvalues reveal the Lamb shift in the microcavity due to its interaction with the environment. We also investigate relations between the Lamb shift and quantized angular momentum of the whispering gallery mode as well as the refractive index of the microcavity. For the latter, we make use of the similarity between the Helmholtz equation and the Schrödinger equation, in which the refractive index can be treated as a control parameter of effective potential. Our result can be generalized to other open quantum systems with a potential term.

Non-Hermiticity and conservation of orthogonal relation in dielectric microcavity

“…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 :…”

“…where y ñ | j S is an eigenvector of non-Hermitian HamiltonianĤ eff in open quantum system and c ñ | j B is a resonance tail only localized in bath omitting the homogeneous solution (or the plane-wave term) bñ | B (see details in appendix A). In the case of dielectric microcavity, y ñ | j S correspond to eigenmode of the inner part of the cavity and c ñ | j B to the emission pattern of the outer-part of it [32], respectively. Then the inner product between two eigenvectors of the total Hamiltonian will be given by means that when the inner product of eigenvectors ofĤ eff , y y á ñ | S j k S , increases, that of resonance tail, c c á ñ | B j k B , must vary in a way to cancel out y y á ñ | S j k S for preserving orthogonal relation in total Hilbert space.…”

“…This non-Hermitian Hamiltonian has generally complex eigenvalues under the Feshbach projectionoperator (FPO) formalism: (see [32] and the references therein. )…”

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.…”