Download Chapter 4 - Physics & Astronomy

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
m02 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
m02 2
x
2
• electrons confined in harmonic potential
• restoring force proportional to displacement
• Newtonian dynamics
• dissipation of motion through frictional force
 mx
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
Pt    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
Pt    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  i0t
0
 1 r, t   1
• any state of the two-level atom may hence be written
 r, t   a 1  b 2 exp  i0t
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  i0t
 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  i0t
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  i0t
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  i0t
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  it 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