Time-Domain Eigensolvers
Microcavity Blues
(finite-difference time-domain = FDTD)
Simulate Maxwell’s equations on a discrete grid,
+ absorbing boundaries (leakage loss)
For cavities (point defects)
frequency-domain has its drawbacks:
• Excite with broad-spectrum dipole ( ) source
• 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-Hermitian
"!
Response is many
sharp peaks,
one peak per mode
signal processing
complex !n
[ Mandelshtam,
J. Chem. Phys. 107, 6756 (1997) ]
decay rate in time gives loss
Signal Processing is Tricky
Fits and Uncertainty
signal processing
problem: have to run long enough to completely decay
complex !n
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a common approach: least-squares fit of spectrum
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There is a better way, which gets complex ! to > 10 digits
!
!
Quantum-inspired signal processing (NMR spectroscopy):
Unreliability of Fitting Process
Filter-Diagonalization Method (FDM)
Resolving two overlapping peaks is
near-impossible 6-parameter nonlinear fit
[ Mandelshtam, J. Chem. Phys. 107, 6756 (1997) ]
(too many local minima to converge reliably)
Given time series yn, write:
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#
Hˆ " = ih "
#t
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Idea: pretend y(t) is autocorrelation of a quantum system:
! = 1.03+0.025i
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…find complex amplitudes ak & frequencies !k
by a simple linear-algebra problem!
There is a better
way, which gets
complex !
for both peaks
to > 10 digits
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y n = y(n"t) = % ak e#i$ k n"t
say:
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Sum of two Lorentzian peaks (!)
time-!t evolution-operator:
ˆ
Uˆ = e"iH#t / h
y n = " (0) " (n#t) = " (0) Uˆ n " (0)
!
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Filter-Diagonalization
Method (FDM)
[ Mandelshtam, J. Chem. Phys. 107, 6756 (1997) ]
y n = " (0) " (n#t) = " (0) Uˆ n " (0)
ˆ
Uˆ = e"iH#t / h
We want to diagonalize U: eigenvalues of U are ei!!t
…expand U in basis of!|#(n!t)>:
U m,n
= " (m#t) Uˆ " (n#t) = " (0) Uˆ mUˆ Uˆ n " (0) = y m +n +1
Umn 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 uncertainty