Statistics of resonance states in a weakly open chaotic microwave cavity

Complexness of eigenfunctions was studied using the effective Hamiltonian formalism & RMT Proportionality between the average complexness parameter and the variance of the resonance width Exact probability distribution of the complexness parameter derived in the GOE case Spatially continuously distributed losses in a chaotic cavity described by a Random Matrix model with a ﬁnite number M of coupling chann els, w h ich co nst itute s a var iable parameter in the cavity.


Statistics of the spectral widths
Statistics of the spectral widths are well-known from the regime of isolated resonances to the overlapping regime.

Γ ∆
The spectral widths are distributed χ 2 Using Gaussian coupling amplitudes: Regime of isolated resonances Γ ∆ CLOSED CHAOTIC SYSTEM with TRS

Properties of the eigenfunctions
The Þeld may be viewed as a random Gaussian variable (M.V. Berry J. Phys A: Math.Gen. 10, 2083 (1977) ) The system is described by a real Hermitian Hamiltonian eigenfunctions are real and orthogonal: Direct observation of BerryÕ s hypothesis in a chaotic optical Þber: 056223 (2002).

OPEN CHAOTIC SYSTEM
Properties of the eigenfunctions may be viewed as independent random Gaussian variables (R. Pnini and B. Shapiro, Phys.Rev. E, 54 (1996) R1032.) The system is described by a non Hermitian Hamiltonian The eigenfunctions are complex:

Real part of the Þeld
Imaginary part of the Þeld and bi-orthogonal: Properties of the complex Þeld The Petermann Factor The enhancement factor called the Petermann factor characterizes the non orthogonality of the resonance modes

Schalow-Townes line width
True fundamental limit of line width Γ P out output power spectral width of the passive cavity Properties of the complex Þeld The phase rigidity ρ 2 Y. -H Kim . et al., P.R.L. 94, 036804, (2005) Measurement of Long-Range Wave-Function Correlations in an Open Microwave Billiard

Properties of the complex Þeld
The complexness parameter : O. Lobkis andR. Weaver, JASA 108, 1480 (2000) Complex modal statistics in a reverberant dissipative body Barthélemy et. al., Euro. Phys. Lett. 70, 162 (2005) Inhomogeneous resonance broadening and statistics of complex functions in a chaotic microwave cavity Through a statistical ray model:

Relations between Phase rigidity, Petermann factor and Complexness parameter
For a given resonance state, the quantities are closely related The eigenfunction statistics are investigated via the complexness parameter Using the complexness parameter: In the weak coupling regime: Complexness of elastic modes of a chaotic silicon wafer Legrand, F. Mortessagne, and P. Sebbah, PRE 80 (2009) The N-level model Effective Hamiltonian in the eigenbasis of H: Eigenvalues: statistics correspond to the closed case are distributed

Non-overlapping regime
with Sum of correlated random variables Eigenvectors: Expression of the complexness parameter: is the eigenbasis of the closed part where The N-level model ( 2) The Weak coupling regime: then Average Complexness parameter and widths ßuctuations (1) and also: Average Complexness parameter and width ßuctuations (2) where: Z p = γ p cos 2 θ np Integrating over γ θ and in the deÞnition yields: The average of the complexness factor has a logarithmic divergence Summation of the perturbation series is required 2-point correlation function at small distance: considering the continuous limit: Distribution function of X for GOE But... for all practical purposes, regularized by: Probability distribution of the complexness parameter for GOE +,-./,01.,23!24!5 5 678!/9-2373:9-!*""!.2!%""# ;<* = 70-<!>)' f ∈ [4; 5] MHz f ∈ [14.7; 15.7] MHz {ψ n } Statistics of open chaotic wave systems Description of the statistical models Statistics of the complexness parameter for the N-level Distribution function for GOE The N-level model The 2-level model Statistics of spectral widths (a brief reminder) Properties of the complex Þeld Petermann factor, Phase rigidity and Complexness parameter Average Complexness parameter and widths ßuctuations Statistics of a 2D chaotic open microwave cavity FEM numerical results with Ohmic losses at the boundary Distribution of ) and Imaginary (right) components of the 500th resonance state solved by FEM with Comsol TM Complex eigenvalues ωn = ω n − iζ n /2 Usual correspondance Properties of the complex Þeld open chaotic wave systems Description of the statistical models Statistics of the complexness parameter for the N-level Distribution function for GOE The N-level model The 2-level model Statistics of spectral widths (a brief reminder)