Microcavity BluesFor cavities (point defects)frequency-domain has its drawbacks:• Best methods compute lowest-! bands, but Nd supercells have Nd modes below the cavity mode — expensive• Best methods are for Hermitian operators, but losses requires non-HermitianTime-Domain Eigensolvers(finite-difference time-domain = FDTD)Simulate Maxwell’s equations on a discrete grid,+ absorbing boundaries (leakage loss)• Excite with broad-spectrum dipole ( ) source"!Response is manysharp peaks,one peak per modecomplex !n[ Mandelshtam,J. Chem. Phys. 107, 6756 (1997) ]signal processingdecay rate in time gives lossEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE0501001502002503003504004500 0.5 1 1.5 2 2.5 3 3.5 4Signal Processing is Trickycomplex !n?signal processingEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE-1-0.8-0.6-0.4-0.200.20.40.60.80 1 2 3 4 5 6 7 8 9 10Decaying signal (t)Lorentzian peak (!)FFTa common approach: least-squares fit of spectrumfit to:! A("#"0)2+ $2E E E EEEEE E E E05000100001500020000250003000035000400000.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5Fits and UncertaintyEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE-1-0.8-0.6-0.4-0.200.20.40.60.810 1 2 3 4 5 6 7 8 9 10Portion of decaying signal (t)Unresolved Lorentzian peak (!)actualsignalportionproblem: have to run long enough to completely decayThere is a better way, which gets complex ! to > 10 digitsUnreliability of Fitting ProcessE E E E E EE E EE E EEEEEEEEEEEEEEEEEEEEEEEEEEE EEE EE E E EE E E E E0200400600800100012000.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5! = 1+0.033i! = 1.03+0.025isum of two peaksResolving two overlapping peaks isnear-impossible 6-parameter nonlinear fit(too many local minima to converge reliably)Sum of two Lorentzian peaks (!)There is a betterway, which getscomplex !for both peaksto > 10 digitsQuantum-inspired signal processing (NMR spectroscopy):Filter-Diagonalization Method (FDM)[ Mandelshtam, J. Chem. Phys. 107, 6756 (1997) ]Given time series yn, write:! yn= y(n"t) = ake#i$kn"tk%…find complex amplitudes ak & frequencies !kby a simple linear-algebra problem!Idea: pretend y(t) is autocorrelation of a quantum system: ! ˆ H "= ih##t"say:! yn="(0)"(n#t) ="(0)ˆ U n"(0)time-!t evolution-operator: ! ˆ U = e"iˆ H #t / hFilter-Diagonalization Method (FDM)[ Mandelshtam, J. Chem. Phys. 107, 6756 (1997) ]! yn="(0)"(n#t) ="(0)ˆ U n"(0) ! ˆ U = e"iˆ H #t / hWe want to diagonalize U: eigenvalues of U are ei!!t…expand U in basis of |#(n!t)>:! Um,n="(m#t)ˆ U "(n#t) ="(0)ˆ U mˆ U ˆ U n"(0) = ym + n +1Umn given by yn’s — just diagonalize known matrix!Filter-Diagonalization Summary[ Mandelshtam, J. Chem. Phys. 107, 6756 (1997) ]Umn given by yn’s — just diagonalize known matrix!A few omitted steps: —Generalized eigenvalue problem (basis not orthogonal) —Filter yn’s (Fourier transform):small bandwidth = smaller matrix (less singular) • resolves many peaks at once • # peaks not known a priori • resolve overlapping peaks • resolution >> Fourier