Transcript NonLinear_Suseptibility

Quantum Optics
Quantum-Mechanical Approach to the
Nonlinear Optical Susceptibility,
Density Matrix
by Tsukanov Roman
1
Outline
1. Introduction and definitions
2. Perturbation solution for 1st 2nd and 3rd
order susceptibilities
3. Density matrix formalism
4. Density matrix calculation for 1st and 2nd
order susceptibility
5. Expansion terms representation by
Feynman diagrams
2
Introduction
•
•
Nonlinear optics – field which studies the phenomena which occurs as
a consequence of the modification of the optical properties of a material
by the presence of light
Only laser light is sufficiently intense to modify the opt. properties of
material system
We look at the dependence of dipole moment per unit volume P of a material
system upon the strength E of the applied field
in case of linear optics:
~
(1) ~
P (t )   E (t )
Where (1) is the linear susceptibility.
In non linear optics we express P as a power series of E:
~
~
~
~
~
~
~
P (t )   (1) E (t )   ( 2) E 2 (t )   (3) E 3 (t )  ...  P (1) (t )  P ( 2) (t )  P (3) (t )  ...
Where (2) and (3) are the 2nd and 3rd order nonlinear optical susceptibilities,
we will see that they depend on the frequencies of applied field
3
Formal Definition of the Nonlinear
Susceptibility
We want to consider the general case of a material with dispersion and loss.
Then the susceptibility becomes a complex quantity relating the complex
amplitudes of the electric field and polarization
~
~
E (r , t )   ' E n (r , t )
n
prime – summation over positive frequenves only
We represent E as the sum of its positive and neg. frequency parts
~
~ () ~ ()
En  En  E n
Where
~ ()
E n  E n e  i n t
~ (  ) ~ (  )*
En  En
 E n* e imt
by requiring complex conjugate we get the physical field E to be real
4
Formal Definition of the Nonlinear
Susceptibility
definition: An-slowly varying field amplitude
E n  An e ikn r
Then the total field
~
E (r , t )   ' An e i ( kn r nt )
n
new notation:
En  E ( n )
An  A( n )
En ( n )  E * ( n )
An ( n )  A* ( n )
using new notations we can write the field in the more compact form
~
~
E (r , t )   E ( n )e int   A( n )e i ( kn r nt )
n
n
where the summation runs over all frequencies
5
Formal Definition of the Nonlinear
Susceptibility
According to the new definitions, the field is given by
~
E (r , t )   cos( k  r  t )
where
1 ikr
e ,
2
1
A( )   ,
2
E ( ) 
1 ikr
e
2
1
A( )  
2
E ( ) 
In similar way we define the nonlinear polarization
~
Pn (r , t )   P( n )e int
where the summation runs over the
positive and negative frequencies
n
And finally we define the components of the second order susceptibility as const.
of proportionality relating the amplitude to the product of field amplitude
( 2)
Pi ( n   m )    ijk
( n   m ,  n ,  m ) E j ( n )E k ( m )
jk nm
6
Motivation
Our goal is to calculate explicit expressions for the
Nonlinear optical susceptibility.
1.
2.
3.
These expressions display the functional form of the
nonlinear susceptibility and hence show how the
susceptibility depends on material parameters such as
dipole transition moment and atomic energy levels
These expressions show the internal symmetries of
the susceptibility
These expressions can be used to obtain numerical
values of the non-linear susceptibilities
7
Schematic representation of the interaction
processes
We may consider the interaction in terms of the exchange of photons between
The various frequency components of the field
In the single quantum process three photons of frequency  are destroyed
And a single photon of frequency 3 is simultaneously created
If one of the real atomic levels is close to one of the virtual levels, the coupling
Between the radiation and the atom is particularly strong and the nonlinear
Susceptibility becomes large.
3rd harmonic generation in terms
of virtual levels (a) and with real
atomic levels indicated (b)
Solid lines represents the eigenlevels of the atom and dashed lines –virtual
levels. They represent the combined energy of one of the energy eigenstates
of the atom and of one or more photons of the radiation field.
8
Schematic representation of the interaction
processes
Example – enhancing the efficiency of the 3rd harmonic
generation
(a) – one photon transition Is nearly resonant, (b) – two photon transition is
Nearly resonant, (c) – the three photon transition is nearly resonant
The method shown in part (b) is the preferred way, because:
In (a) the incident field experiences linear absorption and can be rapidly attenuated
As it propagates through the medium
In (c) the generated field experiences linear absorption.
In (b) the two photon absorption occurs with much lower efficiency than one-photon
9
process
Schrödinger Equation Calculation of the Nonlinear
Optical Susceptibility
We assume that all of the properties of the atomic system can be described in terms of
the atomic wave function (r,t), which is the solution to the time-dependent
Schrödinger equation

i
 Hˆ 
t
Hˆ  Hˆ 0  Vˆ (t )
Hˆ 0 is a free atom Hamiltonia n,Vˆ (t )  the int eraction
of the atom with the electromagnetic field
~
ˆ
ˆ
V (t )     E (t )
  erˆ(t )
~
iw t
E (t )   E ( p )e p
p
The sum runs over the positive and negative frequency components
10
Energy Eigenstates
The solution:
 n ( r , t )  u n ( r )e
 i n t
After substituting to the Shrodinger equation we see that un(r) must
satisfy the eigenvalue equation (time independent S. equation)
Hˆ 0 un (r )  En un (r ), En  n
The solution is chosen in a such manner that they form a complete,
orthonormal set:
*
3
u
u
d
 m n r   mn
11
Perturbation solution to Shrodinger Equation
For the general case in which the atom is exposed to an electromagnetic
field, Schrödinger's equation

i
 Hˆ 
t
cannot be solved exactly.
So we will use the perturbation theory in order to solve it
Hˆ  Hˆ 0  Vˆ (t )
where  is a continuously varying parameter ranging from zero to unity
that characterizes the strength of the interaction;
The value =1 corresponds to the actual physical situation.
Then the desired solution may be written in the form
 (r , t )   (0) (r , t )   (1) (r , t )  2 ( 2) (r , t )  ...
12
Perturbation solution to Shrodinger Equation
After substitution we require that the Schrödinger eq. will be fulfilled for
Each order separately. We obtain the set of equations:
 ( 0)
i
 Hˆ 0 ( 0)
t( N )

i
 Hˆ 0 ( N )  Vˆ ( N 1)
t
N  1, 2, 3, ......
under assumption that initially the atom is in the ground state, we get

( 0)
(r , t )  u g (r )e
 iEg t / 
The Nth order contribution to the wave function may be represented as
(remember the Schrödinger picture)
 N (r , t )   al( N ) (t )ul (r )e i t
l
l
13
Perturbation solution to Shrodinger Equation
Then
i  a l( N ) u l (r )e  il t 
l
 il t
( N 1) ˆ
a
V
u
(
r
)
e
 l
l
l
We multiply each side of the equation by u* and integrate over all space,
The result is
i t
a m( N )  (i) 1  al( N 1)Vml e
ml
,
l
ml  m  l
Vml  um Vˆ ul 
As we see – once al(N-1) are determined, by integration we can get the next order
t
a m( N ) (t )  (i) 1   dt 'Vml (t ' ) al( N 1) e imlt '
l

14
Perturbation solution to Schrödinger Equation
We are interested to determine the linear, second order and third order optical
Susceptibilities. For this we need to determine the probability amplitudes
a
a
( 2)
m
1
a (t )  3

( 3)
(1)
m
 mg  E ( p ) i (mg  p ) t
1
(t )  
e
 p  mg   p
1
(t )  2


pqr mn

pq
m
[  nm  E ( q )][  mg  E ( p )]
( ng   p   q )( mg   p )
e
i ( n g  p  q ) t
[ n  E ( r )][  nm  E ( q )][  mg  E ( p )]
(g   p   q   r )( ng   p   q )( mg   p )
e
i (g  p  q r )
15
Linear Susceptibility
Now we want to use the results to determine linear optical properties of a
material system.
The expectation value of the electric dipole moment is given by
~
p   ˆ  
Where  is given by perturbation expansion with =1.The linear contribution to
( 0)
(1)
(1)
( 0)
<p> is given by
~
 p  
1
~
p (1)  
 p

  


 gm [  mg  E ( p )] i t [  mg  E ( p )] mg i t
(
e

e )

 mg   p
 mg   p
m
p
p
We may formally replace p by -p in the second term, then
1
(1)
~
 p  
 p
 gm [  mg  E ( p )] [  mg  E ( p )] mg i t
(

)e

 mg   p
 mg   p
m
p
16
Linear Susceptibility
Linear polarization
~
P (1)  N  ~
p (1) 
~
P (1)   P (1) ( p ) exp( i p t )
p
Linear susceptibility defined through the relation
Pi (1) ( p )    ij(1) E j ( p )
j
then
N
 ij ( p ) 

i
j
j
i
 gm
 mg
 gm
 mg
(
 *
)

 mg   p
m  mg   p
The first and the second terms are the resonant and anti resonant
Contributions to the susceptibility
17
Second Order Susceptibility
~
p 2   0 ˆ  2     1 ˆ  1     2 ˆ  0 
Then
Now we may replace q by -q in the second term, q by -q and p by -p in the
third term
18
Second Order Susceptibility
We perform similar steps like we did while deriving 1st order:
~
P ( 2)  N  ~
p ( 2) 
~
P ( 2)   P ( 2) ( r ) exp( i r t )
r
After introducing into the standard definition of the 2nd order susceptibility
( 2)
Pi ( 2)    ijk
( p   q ,  q ,  p ) E j ( q )E k ( p )
jk
pq
We obtain the following result
PI-intrinsic permutation symmetry (averages the expression over
Both permutations of the frequencies  and q of the applied fields)
19
Second Order Susceptibility
We may look at the energy level diagram in order to show where the levels
m and n have to be located in order for each term to become resonant
For the case of highly non resonant excitation ng and mg can be taken
To be real, and the expression can by simplified further
Where we used the full permutation operator, defined as the expression summed
Over all permutations of the frequencies p , q and -
The statement that can be made is: the non linear susceptibility of a lossless
Medium possesses full permutation symmetry
20
Third Order Susceptibility
Now we want to treat the 3rd Order.
The dipole moment per atom, correct to third order in perturbation theory is
~
p 3   0 ˆ  3     1 ˆ  2     2 ˆ  1     3 ˆ  0 
Then
21
Third Order Susceptibility
We can replace the values of p, q and r by their negatives in those
Expressions where the complex conjugate of a field amplitude appears.
Then
22
Third Order Susceptibility
~
P ( 3)  N  ~
p ( 3) 
~
P (3)   P (3) ( s ) exp( i s t )
Like before we let
s
3rd
The definition of the
order susceptibility
( 3)
Pk ( p   q   r )    kji
( ,  r ,  q ,  p ) E j ( r ) Ei ( q ) E h ( p )
hij pqr
And the result for the 3rd order is
23
Third Order Susceptibility
The illustration of the locations of the resonances we may see in the figure
Case of highly non resonant excitation – permutation symmetry
24
Density Matrix Formalism
•
A density matrix, or density operator, is used in quantum theory to
describe the statistical state of a quantum system.
•
The formalism was introduced by John von Neumann in 1927.
•
The need for a statistical description via density matrices arises
because it is not possible to describe a quantum mechanical system
that undergoes general quantum operations such as measurement,
using exclusively states represented by ket vectors.
•
In general a system is said to be in a mixed state, except in the case
the state is not reducible to a combination of other statistical states. In
that case it is said to be in a pure state.
25
Density Matrix Formalism
Using the D.M. Formalism we may treat effects, such as collisional
broadening of the atomic resonances, that cannot be treated by the
simpler theoretical formalism based on the atomic wave function
26
Density Matrix Formalism - Introduction
if the system is known to be in a particular state s then s describes all the
Physical properties of the system. Also this wave function obeys the S. E.:
i
 s (r , t ) ˆ
 H s (r , t )
t
We assume that H can be represented as
Hˆ  Hˆ 0  Vˆ (t )
Where H0 - - the Hamiltonian for a free atom and V(t) – an interaction
field-atom. We make explicit use of the fact that the energy eigenstates
of the free atom Hamiltonian H0 form a complete set of basis functions.
 s (r , t )   C ns (t )u n (r ),
n
Hˆ 0 u n (r )  E n u n (r ),
*
3
u
u
d
 m n r   mn
27
Density Matrix Formalism - Introduction
The coefficient C(t) gives the probability evolution that the atom,
Which is known to be in state s, is in energy eigenstate n at time t.
To determine the time evolution we introduce the expansion into
The Schrödinger equation to obtain
dCns (t )
i
u n (r )   C ns (t ) Hˆ u n (r )
dt
n
n
We multiply by u*m and integrate over all space
i
d s
C m (t )   H mn C ns (t ), where
dt
n
H mn   u m* Hˆ u n d 3 r
The expectation value in terms of the wave function of the system
(the 3rd postulate of quantum mechanics):
 A   *s Aˆ  d 3 r   s Aˆ  s  s A s 
s
28
Density Matrix Formalism - Introduction
In terms of probability amplitudes Cns(t) we obtain
 A   C ms*C ns Amn ,
mn
Amn  u m Aˆ u n   u m * Aˆu n d 3 r
As long as the initial state of the system and the Hamiltonian operator H
For the system are known, the formalism described above is capable of
Providing a complete description of time evolution of the system and all of
Its observable properties.
BUT there are circumstances under which the state of system is not known
In a precise manner.
We define the elements of the density matrix of the system by
 nm   p(s)Cms*Cns  Cm* Cn
s
29
Density Matrix Formalism
Interpretation of matrix elements
Diagonal elements = probabilities
Off-diagonal elements = "coherences"
(provide info. about relative phase)
30
Density Matrix Formalism
The Density matrix formalism is useful because it can be used to
Calculate the expectation value of any observable quantity
 A    p( s ) C ms*C ns Amn
s
nm
Integrating the notation we have used
 A     nm Amn
nm
The last expression may be simplified as follows

nm
nm
Amn   (  nm Amn )   ( ˆAˆ ) nn  tr ( ˆAˆ )
n
m
n
then
 A   tr ( ˆAˆ )
31
Density Matrix Formalism
In order to determine how any expectation value evolves in time, it is necessary
Only to determine how the density matrix evolves in time. So we need to
Differentiate the equation
s* s
 nm   p( s)Cm Cn
s
And we will get
 nm
s
s*
dC
dC
dp( s) s* s
n

C m C n   p( s)(C ms*
 m C ns )
dt
dt
dt
s
s
Now we assume that p(s) does not vary in time, so the first term vanishes
and use the Schrödinger equation for the probability amplitudes for the
second term evaluation
dC ns  i s*
C

C m  H n Cs ,
dt


s*
m
dC ms*  i s
C

C n  Hm Cs* .
dt


s
n
32
Density Matrix Formalism
After substitution
 nm   p( s)
s
i
s
s*
s* s
p
(
s
)(
C
C
H

C

n 
m
m C H n )
 
Using the density matrix notation we may write
 nm 
 nm
i
(  n Hm  H n m )

 

i
i ˆ
ˆ
ˆ
 ( ˆH  Hˆ ) nm 
H , ˆ



nm
The last equation describes how the density matrix evolves in time as the
Result of interactions that are included in H
33
Density Matrix Formalism
•Till now we found how the DM evolves in time as a result of interactions
that are included in H.
• But in addition there are interactions that change the state of the system
and cannot conveniently be included in H.
• One of the ways to include such an effects in the formalism is to add
phenomenological damping terms to the equation of motion. Then
 nm 

i ˆ
H , ˆ


eq


(



nm
nm
nm
nm )
And the meaning Is that nm relaxes to its equilibrium value nmeq
with decay rate 
The additional physical assumption is that
eq
 nm
 0,
for n  m
This means that thermal excitation is incoherent process and cannot produce
any coherent superpositions of atomic states
34
Density Matrix - Example
Two-Level Atom:
There are only two atomic states a and b interacting appreciably with the
incident optical field.
The wave function describing s state is given by
35
Density Matrix - Example
DM for the atom is given by
The dipole moment operator
 0
ˆ  
 ba
 ab 
,  ij  i  ezˆ j
0 
The expectation value of the dipole moment is given by

   tr( ˆˆ )

   tr( ˆˆ )   ab  ba   ba  ab
As seen the expectation of the dipole moment depend upon the off-diagonal
36
Elements of the density matrix
Perturbation Solution of the Density Matrix
Equation of Motion
The density matrix equation of motion with phenomenological inclusion of
Damping is
 nm 

i ˆ
H , ˆ


nm
eq
  nm (  nm   nm
)
This equation cannot be solved exactly for physical systems of interest and
We should use the perturbavite technique for solving it:
( 0)
(1)
( 2)
 nm   nm
  nm
 2  nm
 ...
Hˆ  Hˆ 0  Vˆ (t ),
~
ˆ
V   ˆ  E (t ),
ˆ  erˆ
We suppose that V is
given by the electric
dipole approximation
37
Perturbation Solution of the Density Matrix
Equation of Motion
We require that the expansion of  will be the solution of the original equation
For any value of the parameter , so the coefficients of each power of  must
Satisfy the equation separately (for derivation see Appendix 1). Then
( 0)
( 0)
( 0)
eq
 nm
 inm  nm
  nm (  nm
  nm
)




i ˆ (0)
  (inm   nm )   V ,  nm

i ˆ (1)
( 2)
( 2)

 nm  (inm   nm )  nm  V ,  nm

(1)
nm
(1)
nm
38
Perturbation Solution of the Density
Matrix Equation of Motion
We use the same assumtion we used before  incoherenc e of the thermal exitation precesses :
(0)
eq
 nm
  nm
eq
 nm
0
for n  m
(1)
(1)
 nm
(t )  S nm
(t )e (i
nm   nm ) t
(1)
(1)  ( i
 nm
(t )  (i nm   nm ) S nm
e
then after substitution
 i ˆ (0)
(1)
S nm

V , ˆ nm e (i  )t

Or

S


i ˆ
(0)
ˆ
V
(
t
'
),



t
(1)
nm

nm

nm
nm   nm ) t
(1)  ( i nm  nm ) t
 S nm
e
nm
e (inm  nm )t ' dt '
39
Perturbation Solution of the Density Matrix
Equation of Motion
Then after substitution S into 

i ˆ
 (t )  
V (t ' ), ˆ ( 0)


t
(1)
nm

( inm  nm )( t '  t )
e
dt '
nm
All the higher-order corrections to the density matrix can be obtained
By appropriate index change
(1)
(q)
 nm
(t )   nm
(t ) on the left  hand side
ˆ (0)  ˆ ( q 1) on the right  hand side
40
Density Matrix of the Linear Susceptibility
 (t )  e
(1)
nm
 ( i n m   n m ) t

i ˆ
(0)
ˆ
V
(
t
'
),

 
t

( i n m  n m ) t '
e
dt '
nm
Easy to show
 i p t
(1)
( 0)
 nm
(t )   1 (  mm
 nm E ( p )e
(0)
  nn )
p ( nm   p )  i nm
41
Density Matrix of the Linear Susceptibility
(1)
 ˆ (t )  tr( ˆ (1) ˆ )    nm
 mn
nm
 i p t
 mn [  nm  E ( p )]e
1
(0)
(0)
   (  mm   nn )
( nm   p )  i nm
nm
p
We may decompose the dipole moment int o the frequency components :
 ˆ (t )     ( p ) e
 i p t
p
definition of the linear susceptibility is
P( p )  N  ˆ ( p )   (1) ( p )  E ( p )
Then
 mn  nm
N
(0)
(0)
 ( p )   (  mm   nn )
 nm
( nm   p )  i nm
(1)
42
Density Matrix of the Linear Susceptibility
By few operations and simplifying assumption that all of the population
Is in one level (typically the ground state - a)
(0)
(0)
 aa
 1,  mm
 0 for m  a
N
 ( p ) 

(1)
ij
i
i


 an
 naj
 na
 anj
n  (   )  i  (   )  i 
 na
p
na
na
p
na 

As we may see the first term is resonant for positive frequencies p .
The second one is antiresonant and can be dropped when p is
Close to one of the resonance frequencies of the atom. Then to good
Approximation the linear susceptibility is given by
i
j


N
N i j ( na   p )  i na
(1)
an na
 ij ( p ) 
  an  na
2
 ( na   p )  i na 
( na   p ) 2   na
43
Density Matrix Calculation of the
Second Order Susceptibility

( 2)
nm
(t )  e
 ( i nm  nm ) t

i ˆ
(1)
ˆ
V
(
t
'
),



t

nm
e (inm  nm )t ' dt '
After the calculatio ns


( 2)
nm


pq



[  m  E ( p )][ m  E ( p )]
2
[( nm   p   q )  i nm ][(m   p )  i m ]
[  n  E ( p )][ m  E ( p )]
(0)
( 0)   nn
2
(0)
 mm
 ( 0)
[( nm   p   q )  i nm ][( n   p )  i n ]
 K nm e
  e  i (   ) t
p
q


 i (  p  q ) t
pq
The expectation value of the atomic dipole moment is
 ˆ    nm  mn
nm
 ˆ     ( r )  e ir t
r
44
Density Matrix Calculation of the
Second Order Susceptibility
We will with to look at atomic dipole moment oscillating at frequency p+ q
 ˆ ( p   q )    K nm  mn
nm
pq
And nonlinear polarizati on is given by
P 2 ( p   q )  N  ( p   q )  N   K nm  mn
nm
pq
The nonlinear susceptibility is defined
( 2)
Pi ( 2)    ijk
( p   q ,  q ,  p ) E j ( q )E k ( p )
jk
pq
45
Density Matrix Calculation of the Second
Order Susceptibility
By comparison of the equalities we obtain
m, n and  are dummy indices then can be replaced, so the susceptibility
May be recast in the form
46
Density Matrix Calculation of the Second
Order Susceptibility
47
Density Matrix Calculation of the Second
Order Susceptibility
ˆ   
 nm  n ˆ m
48
References
1. R. W. Boyd, Nonlinear Optics (1992)
2. A. Maitland and M.H. Dunn, Laser
Physics (1969)
49
APPENDIX 1
Perturbation Solution of the Density Matrix
Equation of Motion
 nm 

i ˆ
H , ˆ


eq


(



nm
nm
nm
nm )
Hˆ  Hˆ 0  Vˆ (t )
H 0,nm  E n nm
Hˆ , ˆ 
0
nm
 ( Hˆ 0 ˆ  ˆHˆ 0 ) nm   Hˆ 0,n ˆm  ˆ n Hˆ 0,m

  ( E n n m   n m E m )  E n  nm  E m  nm

 ( E n  E n )  nm
 nm
En  Em


 nm  inm  nm 
i
eq
(
V



V
)


(



 n m n m nm nm nm )
 
( 0)
(1)
( 2)
 nm   nm
  nm
 2  nm
 ...
50
APPENDIX 1
Perturbation Solution of the Density Matrix
Equation of Motion

( 0)
nm

eq
nm
eq
 nm
 0 for n  m
 nm  
( 0)
nm
 
(1)
nm
 
2
( 2)
nm
 ...
( 0)
( 0)
( 0)
eq
 nm
 inm  nm
  nm (  nm
  nm
)




i ˆ (0)
  (inm   nm )   V ,  nm

i
( 2)
( 2)
 nm
 (inm   nm )  nm
 Vˆ ,  (1) nm

(1)
nm
(1)
nm
51
Appendix 2
Centrosymmetric medium
• Centrosymmetric medium is the medium which
displays the inversion symmetry
–
In such a medium the nonlinear optical interactions cannot
occur (because the matrix elements of µ in the expression of
optical susceptibility 2 are equal to zero)
– Different gases, liquids, amorphous solids and even many
crystals do display inversion symmetry, so 2 vanishes for
them
– On the other hand, third order nonlinear optical interactions
(those described by a 3 susceptibility) can occur both for
centrosymmetric and non centrosymmetric media
52