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Light and Matter Quantum electrodynamics Tim Freegarde School of Physics & Astronomy University of Southampton How light interacts with matter • atoms and molecules are polarized by applied fields P 0 E D 0E P m02 2 x 2 0 1 E 0 r E • induced polarization modifies field propagation: refractive index; absorption 2 Lorentz theory of atomic polarization • bound or massive nuclei m02 2 x 2 • electrons confined in harmonic potential • restoring force proportional to displacement • Newtonian dynamics • dissipation of motion through frictional force mx Ne2 m P 2 E 2 0 i 3 Lorentz theory of atomic polarization • complex dielectric constant Ne 0 m 1 2 0 2 i Re 2 • real part: refractive index =0.050 1 0 Im 0 freq • imaginary part: (absorption) 4 Lorentz theory of atomic polarization • complex dielectric constant Ne 0 m 1 2 0 2 i Re 2 • real part: refractive index =0.500 =0.050 =0.075 =0.100 =0.125 =0.150 =0.175 =0.200 =0.225 =0.250 =0.275 =0.300 =0.325 =0.350 =0.375 =0.400 =0.425 =0.450 =0.475 1 0 Im 0 freq • imaginary part: absorption 5 Lorentz theory of atomic polarization • complex dielectric constant Ne 0 m 1 2 0 2 i Re 2 • real part: refractive index =0.200 1 0 Im 0 L freq • imaginary part: absorption • ‘stop band’ from 0 to L: strong attenuation even for small 6 Lorentz theory of atomic polarization • complex dielectric constant Ne 0 m 1 2 0 2 i Re 2 • real part: refractive index =0.500 =0.050 =0.075 =0.100 =0.125 =0.150 =0.175 =0.200 =0.225 =0.250 =0.275 =0.300 =0.325 =0.350 =0.375 =0.400 =0.425 =0.450 =0.475 1 0 Im 0 freq • imaginary part: absorption • ‘stop band’ from 0 to L: strong attenuation even for small 7 Causality and the dispersion relations • causality: effect follows cause E t Pt 0 t T E T dT time • causality: must obey t 0 0 time E time 8 the Kramers-Krönig dispersion relations t • causality: effect follows cause Pt 0 t T E T dT • Kramers-Krönig relations relate the real and imaginary parts of () • if 1 i 2, then 2 2 1 1 2 d 2 0 2 1 1 2 d 2 2 0 9 Implication for all dielectrics • evaluate 1 as →0 1 0 1 2 0 2 d • if 1 ≠ 1, there must be frequencies at which 1 ≠ 0 (absorption) • dielectrics cannot be transparent at all wavelengths 10 Application to a single sharp absorption • suppose a single absorption at = 0 2 C 0 • Kramers-Krönig then gives 2 C 0 1 1 d 2 2 0 2C 0 1 02 2 Re 1 0 Im 0 freq 11 Quantum description of atomic polarization • spatial part of eigenfunctions given by 1 and 2 energy • full time-dependent eigenfunctions therefore 2 r, t 2 exp i0t 0 1 r, t 1 • any state of the two-level atom may hence be written r, t a 1 b 2 exp i0t 0 2 1 12 Quantum description of atomic polarization • spatial part of eigenfunctions given by 1 and 2 • full time-dependent eigenfunctions therefore 2 r, t 2 exp i0t 1 r, t 1 write time-dependent Schrödinger equation for two-level atom insert energy of interaction with oscillating electric field • any state of the two-level atom may hence be written r, t a 1 b 2 exp i0t reduce to coupled equations for a(t) and b(t) 13 Quantum description of atomic polarization eigenfunctions given • spatial part of by 1 and 2 2 i eigenfunctions therefore V • full time-dependent t 2m 2 2 r, t 2 exp i0t V r, t e x E0 cos t r, t 1 1 write time-dependent Schrödinger equation for two-level atom insert energy of interaction with oscillating electric field • any state of the two-level atom may hence be written r, t a 1 b 2 exp i0t reduce to coupled equations for a(t) and b(t) 14 Quantum description of atomic polarization 1 x/a0 2 x/a0 r, t 1 exp it 2 0 • electron density depends upon relative phase of superposition components 15 Atomic polarization • response of massive electrons to applied electric field • resonant frequency due to confining potential of electrons in atom • electron displacement leads to atomic polarization =0.500 =0.050 =0.075 =0.100 =0.125 =0.150 =0.175 =0.200 =0.225 =0.250 =0.275 =0.300 =0.325 =0.350 =0.375 =0.400 =0.425 =0.450 =0.475 Re 1 0 Im 0 freq • frequency-dependent amplitude and phase lag of response related by causality • Newtonian and quantum mechanical models give same result 16