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Quantum Optics
Ottica Quantistica
Fabio De Matteis
[email protected]
Sogene room D007 - phone 06 7259 4521
Didatticaweb
didattica.uniroma2.it/files/index/insegnamento/164488-Ottica-Quantistica
Quantum optics
• Quantum optics deals with phenomena
that can only be explained by treating light
as a stream of photons
• Light-matter interaction is the only way to
«experience» light properties
Model
Matter
Light
Classical
Hertzian dipoles
Waves
Semi-classical
Quantized
Waves
Quantum
Quantized
Photons
F. De Matteis
Quantum Optics
1/34
On the Light side
• Tight binding between Light and Matter
• We are matter, light is a mean to
investigate matter properties
• Effect of light on matter (photoelectric
effect, state transition, diffraction grating,
ecc.)
• Let’s start adopting light’s point of view
F. De Matteis
Quantum Optics
2/34
Light as electromagnetic radiation
Electric displacement field
Electric field E
Electric dipole moment
per unit volume

 

D   0 E  P   0 r E


P   0 E  r  1  
Electric susceptibility
Magnetic (induction) field B
F. De Matteis
Quantum Optics
B = µ0 H
3/34
Light as electromagnetic radiation
No free charges r=0 nor currents j=0
Maxwell Equation

 1E
2
E 2 2
c t
2
Wave Equation
F. De Matteis
Quantum Optics
4/34
Light as electromagnetic radiation
Perfectly conductive walls
z
L
Tangential component of
E field is vanishing
y
x
Independently from
volume, shape and
nature.
Matter plays a role, however. Confinement
F. De Matteis
Quantum Optics
5/34
Field Modes
z
L
y
x

Ex(r,t)Ex(t)cos(
kxx)sin(
kyy)sin(
kzz)

Ey(r,t)Ey(t)sin(
kxx)cos(
kyy)sin(
kzz)

Ez(r,t)Ez(t)sin(
kxx)sin(
kyy)cos(
kzz)
kx nx/L ky ny/L kz nz/L
nx,ny,nz 0,1
,2,3
,
Stationary Waves
F. De Matteis
Quantum Optics
6/34
Field Modes x=0 (x=L)
z
L
y
x

Ex(r,t)Ex(t)cos(
kxx)sin(
kyy)sin(
kzz)

Ey(r,t)Ey(t)sin(
kxx)cos(
kyy)sin(
kzz)

Ez(r,t)Ez(t)sin(
kxx)sin(
kyy)cos(
kzz)
Only normal component different from 0
kx nx/L ky ny/L kz nz/L
E x  0 E  0
nx,ny,nz 0
,1
,2,3
,
E 0

 E//  0
Ez  0 
y

E0
Stationary Waves
F. De Matteis
Quantum Optics
7/34
Field Modes (ny=nz=0)
z
L
y
x

Ex(r,t)Ex(t)cos(
kxx)sin(
kyy)sin(
kzz)

Ey(r,t)Ey(t)sin(
kxx)cos(
kyy)sin(
kzz)

Ez(r,t)Ez(t)sin(
kxx)sin(
kyy)cos(
kzz)
kx nx/L ky ny/L kz nz/L
nx,ny,nz 0,1
,2,3
,Not more than one can be
null at once 
(Otherwise E  0 )
Stationary Waves
F. De Matteis
Quantum Optics
8/34
Field Modes
 
k E 0
E
2 polarization for each
k value
kz
k
Divergence eq.
/L
_
How many field modes for each
frequency interval  : +d
ky
kx
F. De Matteis
Quantum Optics
9/34
N modes?
Spectral density of modes
_
Number of lattice point in the first
octant of a spherical shell defined by
radius k : k+dk
_ Each point occupies a
volume (/L)3
kz
/L
_ 2 polarizzation for each k


2
1
1
k
 4π  k 2 dk
 2 = 2 dk
3
8
π
π / L
ky
kx
F. De Matteis
Quantum Optics
10/34
N modes?
Spectral density of modes
_
Number of lattice point in the first
octant of a spherical shell defined by
radius k : k+dk
k=/c
_ Each point occupies a
volume (/L)3
kz
/L
_ 2 polarizzation for each k


2
1
1
k
 4π  k 2 dk
 2 = 2 dk
3
8
π
π / L
2

r
d
 2 3d


c
F. De Matteis
Quantum Optics
ky
kx
11/34
Energy of harmonic oscillator field
Time dependency of
e.m. field


2 2
2
2
 E = c k x  k y  k z E
2


E
(t)
2

E
(t)
2

t
2
 
 
 E  
B
t
  i 2

k Ee
 B
F. De Matteis




E
(
t)
E

i
t)
0exp(
Harmonic Oscillator
E0 =
Quantum Optics
c k = ω
B0
k
B0
 0 0
 cB0
12/34
Energy of harmonic oscillator field
Time dependency of
e.m. field


2 2
2
2
 E = c k x  k y  k z E
2


E
(t)
2

E
(t)
2

t
2
E0 =
B0
 0 0
 cB0


c k = ω


E
(
t)
E

i
t)
0exp(
Harmonic Oscillator


 2
 2 1 2 
1
1
 ε0 E + B   dV =
ε0 E(r t) dV 


2 cavity 
μ0
2 cavity

Cycle-average theorem
1

  n +  ω
2

1
Re A  Re B = Re AB*
2
F. De Matteis
Quantum Optics
“First quantization”
13/34
Planck’s Law
At thermal equilibrium* Temperature T
Excitation probability of nth-state
n
n
n
U
n



P


1

U

U
n
n
U

Let’s set U = exp(- ħ/kBT)
n =  nPn = 1  U U  nU


exp
E
T
nk
B
P

n


exp
E
T

nk
B
n 1

n
n
1/(1-U)
U
1
1

= 1 =
1  U U  1 exp( ω / k BT)  1
*Once more we need to resort to some matter. Thermalization
F. De Matteis
Quantum Optics
14/34
Mean Energy Density WT()
W
(

)
d


n


r
d


T

1




 23d

exp(


/k
T
)

1 
c
B
2
Mean number of excited photons (for  mode)
Photon energy (for  mode)
Mode density in the interval  ÷ d
F. De Matteis
Quantum Optics
15/34
Mean Energy Density WT()
Classic Limit
(Rayleigh 1900)
k BT  ω
ω2
WT (ω)  2 3 k BT
π c
Wien’s displacement law
ω
1
WT (ω) = ω  2 3
π c exp( ω / k BT)  1
2


0
max  2.8 k BT
Stefan-Boltzmann’s
Law
π k
4
WT (ω)d 
T
15c 3 3
2
4
B
ω3
WT ( )  2 3 exp( ω / k BT) ω  k BT
π c
At low temperature
Wien’s Formula
F. De Matteis
Quantum Optics
16/34
Fluctuations in photon number
We stated the probability
Absorption and emission will
distribution of the mode
cause the fluctuation of the
occupation for the cavity
photon number in each mode of
the radiation field in the cavity
field
with characteristic times


exp

E
T
nk
B
P
n


exp
E
T

nk
B
n
1
n=
exp ( ω / k BT)  1
Neglecting for now the nature of the
characteristic times, we can infer some
general properties making use of the
ergodic theorem
F. De Matteis
Quantum Optics
17/34
Fluctuations in photon number
Un
n


Pn =
=
1

U

U
 U = exp(- ħ/kBT)
n
U
n
n =  nPn = 1  U U  nU
n
n 1
n
U

1U
nn
Pn =
1  n 1n
n(n  1) =  n(n  1) Pn = 1  U U 2  n(n  1)U n  2 
n
n
2
2 
 1  U U
U 2
3
 1 
n
2
n U  1  U U 2 1 1  U  
2
2
 U 
 2 1
  2!n
1U 
F. De Matteis
Quantum Optics
18/34
Fluctuations in photon number
The r-th factorial moment is
defined
n(n  1)( n  2)  (n  r  1)  r!(n ) r
The root mean square
deviation Dn of the
distribution is
Dn2
 n 2  n 
2
Therefore the second moment is
n(n  1)  n  n 
Dn 2  (n 2  2nn  n 2 )  n 2  n 2
 n  n  2(n )
Dn 
2
2
n=
1
exp ( ω / k BT)  1
F. De Matteis
2
n2  n
The fluctuation is always larger than the mean value
Dn  n  1 2
Quantum Optics
(n  1)
19/34
Emission and Absorption
ħ=E2-E1
An electron in an atom can make
transition between two energy state
absorbing or emitting a photon of
frequency  = DE/ħ with DE = E2 –
N2 +1 EE22
Energy
E1 energy difference between the
two levels.
The processes that can occur are:
ħ
N1-1 E11
Absorption
Spontaneous Emission
Stimulated Emission
F. De Matteis
An electron occupying the lower energy state E1 in
presence of a photon of energy ħ= E2-E1 can be
excited to a level E2 absorbing the energy of the photon.
Quantum Optics
20/34
Emission and Absorption
ħ=E2-E1
An electron in an atom can make
transition between two energy state
absorbing or emitting a photon of
frequency  = DE/ħ with DE = E2 –
Energy
N2-1 E2
E1 energy difference between the
two levels.
ħ
The processes that can occur are:
N1+1 E1
Absorption
Spontaneous Emission
Stimulated Emission
F. De Matteis
An electron occupying the higher energy state E2 can
decay to the lower energy state (E1) releasing the
energy difference as a photon of energy ħ= E2-E1 and
a random direction (k)
Quantum Optics
21/34
Emission and Absorption
ħ=E2-E1
An electron in an atom can make
transition between two energy state
absorbing or emitting a photon of
frequency  = DE/ħ with DE = E2 –
E1 energy difference between the
two levels.
N2-1 E2
Energy
ħ
ħ
ħ
The processes that can occur are:
N11+1 E
E11
An electron occupying the higher energy state E2 can decay
Absorption
to the lower energy state (E1) releasing the energy difference
as a photon of energy ħ= E2-E1 Differently from the
Spontaneous Emission
previous case the process is stimulated by the presence of a
photon. The process is coherent, the emitted photon is
Stimulated Emission
coherent in phase and direction (k) with the stimulating one
F. De Matteis
Quantum Optics
22/34
Einstein’s coefficients
At thermal equilibrium the transition rate
from state E1 to E2 has to be equal to
that from state E2 to E1.
N1 number of atoms per unit of volume with
energy E1,
Absorption rate proportional to N1 and to the energy density at frequency 
W to promote the transition.
N1 WT B12
N1 W B21 with B21 constant called coefficient of stimulated emission.
N2 A21 with A21 constant called spontaneous emission.
The coefficients B12, B21 and A21 are called Einstein’s coefficients.
F. De Matteis
Quantum Optics
23/34
Thermal Equilibrium
At equilibrium the
processes must
equilibrate:


N
W
(
)
B

N
W
(
)
B

N
A
1
T
12
2
T
21
2
21

N
A
A
B
2
21
21
21
W
(
)


T




N
B

N
B
B
B
N
N

1
1
12
2
21
12
21
1
2
The population of a generic energy


g

E
T
jexp
j k
B
level j of a system at thermal
N
N
j
0
equilibrium is expressed by Boltzmann


g

E
T

iexp
ik
B
i
statistic:
N
g1
1
E2E1 kBT
 exp
N
g2
2
g1
kBT
 exp
g2
F. De Matteis
Quantum Optics
Nj population density of jlevel of energy Ej
N0 total population density
gj j-level degeneracy.
24/34
Is Thermal Radiation Choerent?
WT (ω) =

A21 / B21
8πω3 
1
equilibrium WT (
 )
Thermal
=
3
3 
g1B12 / g 2 B21 exp ω / k BT   1 c h equal
ωblack
/ k BT body
1 
 exp to
8πω h
A21 =
c3
B21
3
g1 B12 = g 2 B21
3
with h refraction index
of the medium.
Ratio between rate of
A21
spontaneous emission
R=
=e
WT (ω) B21
and stimulated emission
at thermal equilibrium
R~1 if kT~ħ
F. De Matteis
Quantum Optics
ω
k BT
1
 1
n
T=12000 K
25/34
Other no-thermal radiation
A21
R=
=e
WT (ω) B21
Visible
l~600nm
~3x1015 s-1
r~3,4x104d m-3
ω
k BT
1
 1  1
n
Which energy density does
it need in order to get a
ratio R~1 ?
A21
 3
W d =
d = 2 3 d  10 14 d Jm 3
B21
 c
I d  c h W d  3 106 d Wm 2
For a typical linewidth ~10-2 nm
or d~21010 s-1
Mercury lamp
CW laser
Pulsed laser
F. De Matteis
I (W/m2)
104
105
1013
I  2 105 Wm 2  20 Wcm 2
E (V/m)
103
104
108
Quantum Optics
n/V (m-3)
1014
1015
1023
Photons/mode
10-2
1010
1018
26/34
Optical excitation of atoms
Atomic level population achieved by light irradiation (N2(t=0)=0)
Thin cavity crossed by a light beam (negligible light intensity losses)
d
N1   d N 2  N1BW  N 2 BW  N 2 A21 
dt
dt
N  N1  N 2

B  B12
 g1  B  B
 g 
21
2

N2  N

BW
1  e 2 BW  At
A  2BW

Atoms are lifted into the excited state
 energy is stored in the atomic
system
For BW>>A system reaches saturation
N2=N/2
Powerful lasers
F. De Matteis
Quantum Optics
27/34
Optical excitation of atoms
When the light is switched off (t=0), the atomic system relaxes to its
ground state (thermal equilibrium)
The energy stored in the matter is re-emitted as photons.
d
N 2   N 2 A21
dt
N 2  N 20e  At  N 20et  R
Reciprocal of A is the radiative
lifetime of the transition.
F. De Matteis
Quantum Optics
28/34
Absorption
Let’s consider a collimated monochromatic beam of unitary area flowing
through an absorbing medium with a single transition between level E1 and
E2.
The intensity variation of the beam as a function of the distance will be:
D
I
(
x
)
I
(
x

D
x
)

I
(
x
)
For a homogeneous medium DI(x) is proportional to the intensity I(x) and to
the travelled distance D(x). Hence DI(x) = -I(x)Dx with  absorption
coefficient.
Writing the differential equation:
I(x)
I(x+Dx)
dI
(x)


I(x)
dx
and by integration, we obtain:
x
I(x) I0e
Dx
F. De Matteis
where I0 is the input radiative
intensity.
Quantum Optics
29/34
Macroscopic theory of absorption
When the e.m. wave propagate in a dielectric medium it generates a polarizzation field
P. For a not too intense field (linear response regime)


P =  0 E
 0  8.854 10 12 F m 1
where  is the linear elettric susceptibility
The electric displacement vector D is connected to the electric field E by
 


~
D  P   0 E   0 E
With the generalization of the dispersion relation
The susceptibility is a complex quantity:
kc  2 = 1    ~
     i 
We define the square root of the dielectric coefficient as complex refractive index
~
  1    h  ik
where h is the refractive index and k is the extinction coefficient.
F. De Matteis
Quantum Optics
30/34
Macroscopic Theory of Absorption
Let’s skip to a travelling plane wave solution rather than a stazionary one
 h
 k
exp ikz  t   exp i  z  t  
 c
 c
W 
1 ~
 0 E
2
2

1
 0h 2 E
2

z

B
E h  ik

E
v
c
2
The intensity I of the electromagnetic wave, defined as the energy crossing the
unit area in unit of time, is represented
by the value, averaged over a cycle, of

 
the flux of the Poynting vector
S = 1 0 E  B


1
I  S   0 ch E
2
2

c
h
W
The dependence on the space-time variables of all fields is that of a plane wave
propagating along the z-axis, i.e. the intensity is
I ( z )  I 0 exp   z    2 k  c
Where I0 is the intensity at z=0 and  is the absorption coefficient
F. De Matteis
Quantum Optics
31/34
Microscopic Theory of Absorption
Relation between absorption coefficient (macro) and Einstein’s coefficients
(atomicmicro)
Einstein’s Coefficients deal with em radiation incident on a 2 level system in
vacuum
W represents em energy density in the dielectric. Therefore we must
substitute WW/h2
dnk
net loss of photons for
= N 1 B12 Wk h 2  N 2 B21 Wk h 2 =
kmode per unit of
dt
volume
 g2

  N 1  N 2  B21 Wk h 2
 
h
1
h
B = E ; W (ω) = ε h E (r t) = I
g
 1

c
2
c
N 2 A21  
2
2
0
Wk 
  I(x) 
 =
= ( ωnk ) = 
t
t
t  c / h 
  I(x)  x   I(x)  c 

 = 
 = I(x)
x  c / h  t x  c / h  h x
F. De Matteis
Quantum Optics
0
k
0
k
I(x)
Wk(t)
I(x+dx)
dx
32/34
Absorption Coefficient
 g2

 N1  N 2  B21ω
Absorption
g


I = 1
I(x) = αI(x) Coefficient
x
ch
at thermal equilibrium (g2/g1)N1>N2
hence the coefficient is positive.
Population Inversion (g2/g1)N1<N2 → negative absorption
coefficient
Increment of the intensity passing through the medium
I = I 0exp(Kx)
F. De Matteis
Quantum Optics

g 2  ω21

K =  N 2  N1  B21
g1 
ch

33/34
Absorption Coefficient

d
N1   d N 2   N1B12 I hc  N 2 B21 I hc  N 2 A21 


dt
dt
N  N1  N 2
In stationary condition, the two level
populations does not vary.
B21  B
B12 
g2
B
g1
 g2

NA
 N1  N 2   g 2 g1
A  1 g 2 g1 B I ch
 g1

1  g1 
g1  BI  
NBω


 1  
I=


I  g 2  g 2  chA  x
ch
For all ordinary light beams the second term in brackets is
negligible with respect to the first one
g
ω
K = 1 NB
 2k / c
g2
ch
F. De Matteis
Quantum Optics
34/34