PETER PAZMANY CATHOLIC UNIVERSITYConsortiummembersSEMMELWEIS UNIVERSITY, DIALOG CAMPUS PUBLISHER

**Molekuláris bionika és Infobionika Szakok tananyagának komplex fejlesztése konzorciumi keretben

***A projekt az Európai Unió támogatásával, az Európai Szociális Alap társfinanszírozásával valósul meg.

Development of Complex Curricula for Molecular Bionics and Infobionics Programs within a consortial* framework**

Consortium leader

PETER PAZMANY CATHOLIC UNIVERSITY

Consortium members

SEMMELWEIS UNIVERSITY, DIALOG CAMPUS PUBLISHER

The Project has been realised with the support of the European Union and has been co-financed by the European Social Fund ***

PHYSICS FOR NANOBIO-TECHNOLOGY

Principles of Physics for Bionic Engineering

Chapter 12.

Interaction of Matter and Radiation - II

(A nanobio-technológia fizikai alapjai )

(A tér és az anyag kölcsönhatása)

Árpád I. CSURGAY,

Table of Contents

12. Interaction of Matter and Radiation – II 1. A Heuristic Model of a Two-state Atom in

Electromagnetic Field

2. Perturbation of a Stationary State 3. Time-dependent Perturbation

4. Time-evolution Operator – The Propagator

1. A Heuristic Model of a Two-state Atom in Electromagnetic Field

It is often sufficient to focus our attention on just two energy levels of an atom, which are closest to resonance with the

electro-magnetic field, satisfying the energy relation , where is the angular frequency of the

radiating EM field.

In this case the wave function of the atom is

We assume that each energy level corresponds to a single quantum state and

2 1

E  E     2π

 

 

 

 r   r   r

2 2

1( ) 2( ) 1.

c t  c t 

are called ‘level probabilities’, and

are the probability amplitudes, whose dynamics is governed by the Schrödinger equation

First, let us solve the Schrödinger equation for the atom without radiation:

The eigenvalues and the orthonormal eigenfunctions of the Hamiltonian are

2 2

1 and 2

c c

   

1 c t1( ) 1 and 2 c t2( ) 2

   r    r

 

 

j t , .

t t

 

 



r H r

   

01,2  E1,21,2 .

H r r

H0 E E1, 2 and 1

 

 

In electromagnetic field the effect of radiation is significantly weaker than the local field of the nuclei, thus its effect can be considered as perturbation. The Hamiltonian of the atom + radiation system can be approximated by and the Schrödinger equation is

Multiplying the state equation by the complex conjugate of the eigenfunctions, and integrating them on the configuration space, we get

0 perturb

 

H H V

    ^{ }

         

0 perturb

1 1 2 2

j , , ;

, .

t t

t

t c t c t

 

  

  



   

r H V r

r r r

d 1,2( )

j c t ( ) ( ),

E c t V c t

 

where

is called the interaction matrix element between the atom and the radiation field. In most cases it is the potential energy of the atomic dipole in the radiation field

The dipole interaction is distributed over the atomic orbitals, and if the polarization of the field is represented by a unit vector , the field itself by its absolute value E(r,t) , then

, where is the position of the atom.

1,2 1 perturb 2d

V

V 



 

1,2 1 e 2d .

V

V 



ˆ ˆ

( , )t   E( , )t    E( _N, )t

E r e r e r r_N

e ˆ

(Note, that the wavelength of the radiation is much bigger than the size of the atom’s orbital.)

In case of quasi-monochromatic radiation field which is nearly resonant with the atom:

there is a strong synchronous response of the atom to the radiation. Let us introduce new variables

j j

0 0

( _N, ) ( _N, ) e ^t ( _N, ) e ^t, E r t  E r t  ^{ }  E^ r t  ^

21 ; E2 E1,

      

j 1( ) 1( ), 2( ) 2( ) e ^t, a t  c t a t  c t ^

2 2 2 2

1 2 1( ) 2( ) 1.

a  a  c t  c t 

If is close to monochromatic, i.e.

then a so-called RWA (rotating wave approximation) can be use, from which we get the ordinary differential equations for the probability amplitudes

where

If at t = 0 the atom is in its lower energy state, i.e.

the solution is

1 2

2 1

d 1 d 1

j , j ,

d 2 d 2

a a

a a a

t    t     _ 

21 0

2V E / .

  ( _N , )

E r t dE₀ / dt  E₀

1(0) 1, 2(0) 0,

a  a 

j1 2 1

1 1

( ) cos j sin e .

2 2

a t    t  t ^{ }t

where is the detuning-dependent Rabi frequency.

The probabilities of the eigenstates change is time

 

2

j sin 1 e ,

2

a t _{ } _ t_ ^{ } t

 

 

 

2 

   

2

2 2

1

2

2 2

1 1

cos sin ,

2 2

sin 1 . 2

P t t

P t





  

      

     

We see a continuing oscillation between levels one and two

without any steady state (Rabi oscillations). Within RWA the dynamics remain unitary and probability conserving for all values of t.

2 1

1  P  P

 0.5

 

0 2 4 6 8 time

0 0.2 0.4 0.6 0.8 1

Probability (P 2) 0 

 0.5

 

‘Stimulated emission’

We illuminate an atom with a laser, with (the electro-

magnetic wave is resonant with the atoms transition frequency).

The atom is excited from state 1 to state 2.

However, if the atom is in its excited state, another photon of the electromagnetic wave ‘de-excites’ the atom, and it emits a

photon with the same phase and frequency as the EM wave.

 0

 

Rabi oscillation

The probability of finding the atom in the excited state increases over time. This means that at some point in time the probability of finding the atom in its excited state is one.

For as long as this probability is one, you would expect that the atom stays in this state. Nevertheless, the atom is still emitting and absorbing photons through stimulated emission.

This process of emitting and re-absorbing is called Rabi oscillations.

Note: in this approximation the spontaneous emission is neglected.

0 , 0 ¹





2 

π / 2 Pulse

0 , 1 π Pulse

Cl Cl

Cl Proton Carbon

Cl Cl

Cl Proton Carbon

Cl Cl

Cl Proton Carbon

2. Perturbation of a Stationary State

Let us assume that the ‘universe’ is a closed quantum-mechanical system with known stationary eigenvalues and eigenstates, and the external electromagnetic field is weak compared to the

internal forces. The stationary state Hamiltonian is , and from

we get,

the unperturbed energies and stationary eigenstates.

We impose a weak perturbation on the system as

H0

0 0 0

0  n  En  n 

H

0 0 0 0 0 0

1 , 2, , _n ,... and 1 , 2 , , _n ,...,

E E E   

, where 0, if 0.

 

    

H V H H

The perturbed problem is

If the perturbation operator does not depend on time, we call the problem ‘time-independent’ perturbation, if it does, ‘time- dependent’ perturbation.

Time-independent perturbation for

Let us expand the unknowns into a series of









 1

.

0 0 0 0 0

0 _n  E_n _n ; _n  n ; 0 n  E n_n .

H H





0 1 2 ; E E0 E1 E2 ;

           

   

  

2

0 0 1 2

2 2

0 1 2 0 1 2

,

, ,

E E E

     

      

    

      

H V

For  : 0   1

   

   

2

0 0 0 1 0 0 2 1

2

0 0 0 1 1 0 0 2 1 1 2 0

...

...

E E E E E E

      

       

     

      

H H V H V

0

0 0 0 0

1

0 1 0 0 1 1 0

2

0 2 1 0 2 1 1 2 0

,

,

. E

E E

E E E

  

    

     

 

   

    

H

H V

H V

‘Zero-order’ approximation First-order approximation

Multiply it from left by ‘ket’

0

0 _m, 0 .

E  E   m

0 0

1 1 1 .

n m

n n

E n n  V m  E n n   E m

 

k

0 0

1 1 1

k m k km

k  m  E k   k V m  E    E 

1 0 0 ;

m k

k m

k   E E

 V

1 ,

k  m  E  m V m

0 0

0 _{m m mm},

E  E  m V m  E V k₀ m₀ .

m k

  



‘Second-order’ approximation (mutatis mutandis)

2 0

0 ^mn 0 ,

m m m

n m m n

E E V V

E E



  



    

 

0 0

0 0 0 0 0 0 2

2

0 0 2

2

.

k m

k m m k

k n mn m m k m

n m m n m k

k m m k

k m

k m m k

m V k

E E

V V V V

E E E E E E k

V m

E E





 



  



 

 

 

    

 





 







In conclusion, first order approximation of perturbed eigenvalues and eigenstates:

First order approximation: New (perturbed) energy eigenvalues Nonperturb system

(Zero order approximation)

Perturbed system

(First order approximation)

Energy eigenvalues New energy eigenvalues

0 0 n  E nn

H





0

1 1

0

2 2

0

3 3

0

1 1 ,

2 2 ,

3 3 , ,

, E E

E E

E E

E E n n

 

 

 

 

V V V

V

0 1

0 2

0 3

0

, ,

, , ,

E E E E

Nonperturbed system Perturbed system

Orbitals (eigenfunctions) New orbitals (eigenfunctions) 1 , 2 , 3 ,..., n ,...

1 0 0

1 1

2 0 0

2 2

3 0 0

3 3

0 0

1 1 ,

2 2 ,

3 3 ,

,

k k

k k

k k

n

k n n k

k k

E E

k k

E E

k k

E E

k n

n k

E E

















 



 



 



 











V

V

V

V

Start with the eigenvalues and the complete orthonormal set of eigenfunctions generated by the nonperturbed problem. The perturbation operator V is given. For the solution of a

perturbation problem we have to calculate elements of the matrix

Every element is an integral on the configuration space

1 1 1 2 1 3

2 1 2 2 2 3

3 1 3 2 3 3 .

 

 

 

 

 

 

V V V

V V V

V V V

d ...d1 .

n m f

n ^V m 



Example 1: A particle moves in a one-dimensional potential box with a small potential dip.

Treat the potential dip as a perturbation to a regular box. Find the first order approach of the energy of the ground state.

Nonperturbed system Perturbed system

pot

for 0, and 0 for 0

x x

V x

  

   

2

0 2 0

2

2 π

; ( ) sin

n 8 n

h n

E n x x

m 

 

' pot

for 0, and

for 0 / 2

0 for / 2

x x

V b x

x

  



   

  

 0 x

x  x  

b

x  2 0 x

x  x  

ǀ ǀ

The first order approach of the energy of the ground state:

   

 

0 0 0

0 0 0 0

/ 2 / 2

0 0 2

0 0

0 0

/ 2 / 2

2

0 0

1 1 ( ) ( ) ( )d

2 π

( ) ( ) d sin d

2 π 2π

sin d 1 cos d ,

E E x x x x

x b x x x b x

x b x

b x x

 

 









   

 

      

 

   

        



 

 

V V

2 0

0 0 2

1 1 h .

8 2

E E b

  V  m 

Example 2: Hydrogen atom in external electric potential.

Suppose the external electric field is along the z axis, and consider the perturbation Hamiltonian

The first order energy correction for

For the hydrogen atom the ground state is even in z, thus

However, the second order correction term is not zero, it is

negative, and its magnitude is proportional to

' e e .

H  E r z

0 0 0 0

1 ' e 1, 0, 0 e 1, 0, 0 .

E   H    Er   Er

1, 0, 0

n  l  m

 ¹

4 2 2 2

1 e / 8 0 ; _n 1 / ; 0.

E   m  h E  E n E 

 ² 2 2

1 1

,

1, 0, 0 , ,

n n

l m

z n l m

E e

E E







2.

3. Time-dependent perturbation There is a nonperturbed system

and a perturbed system

We know that the solution of the time-varying perturbed system can be expanded as

i.e.

0 0 n  E nn

H





 ^ 

 

H E 

 

0

e j ,

En

t n

n

a t n

 



 

0 0 0

j 0 j j

0 ( ) e j j e e .

n n n

E E E

t t t

n

n n n

a t n ^ n a  E  ^ a ^ 

      

 

 





Let us multiply by ‘bra’ from left

‘Zero-order” approximation First-order approximation

 

d j

e , 1, 2,

dt

knt k

n k n n

a t  



0 0

, .

k n

k n k n

E E

k n

  ^ E 

d 0

d 0.

ak

t 

 ^{1 0}

 

d j

e .

d

k nt k

n k n n

a a t

t

 



k

0 if 1 if

kn

k n k n    ^ k ^ n

The Propagator:

Transition Probability

We have a system in state and we want to know the probability of observing the system at time t, due to perturbation in state

The probability

 

( )t U t t, ( ),t

  

( 0) j

0

( , )

j ( , ) ( , ) e ( , ).

t t t

t t t

t

   ^{ } 

   

 r H

H r r r

( 0) j

( , )0 e .

t t

U t t

 

 ^H

k

Int 0

j1 ( )d

2 , ( ) e .

t

t

k k k

P b b t k

   

 

E

The first-order term allows only direct transitions between and as allowed by the matrix element in E, whereas the second- order term accounts for transitions occurring through all

possible states.

j

Int( ) 0 ( ) 0 e ^k _k ( ),

k E t  k U^E t U  ^{ } E t

1 0

2 1

0 0

j

1 1

2

j j

2 2 1 1

( ) j e ( )d

j e ( )e ( )d ...

k

mk m

t

k k

t

t t

km m

t t

b t k

d

 

   

 

   



 

   

 

  



E

E E

k

For first order perturbation theory, the solution to the differential equation that you get for direct coupling is

No feedback between

If the system is initially prepared in a state and a time-dependent perturbation is turned on and then turned off over the time

interval , then the complex amplitude in the target state is just the Fourier transform of E(t) evaluated at energy gap

j 1

1

( ) j e ^k ( ) (0)

k k

b t b

t

  

   

 E

and k  b_k  b0 .

  ,

 k

 

j

j 2

( ) j e ( ) d ,

1 ˆ

k

k k

t

b t ^{ }



 

 

 







 

  





E

Example 3: A harmonic oscillator is subject to a ‘compression’ that increases its k and later it decreases it back.

The Hamiltonian of the harmonic oscillator is

Let us assume that the compression is Gaussian

   

2

2 2

0

d 1

( ) ,

2 d 2

.

k t x m x

k t k k t

  

  H

 0²

2 0

2

( ) 1e , .

t t k

k t k

m

  

 

  ^{ }

2 0

2

2

2 2 2

0 1

2

d 1 1

2 d 2 2 e

t t

k x k x m x



 

   

H

H0 V(t) k0

k0 k

t

 

k t



If the system is in what is the probability that it is in state

For

0

, 1 , .

n n 2 n

n  E n E   ^ n^ n  

 

H

0 at t  ,

at ?

n t  

 

0 1

0 0

0

j 0

1 1

0, ( ) j e ⁿ d , where .

t n

n n n

t

E E

n  a t    ^{  } V     ^  n

2

2 2 2

2 j 2 2 2 /2

1 1

j j

( ) 0 e e d 2π 0 e .

2 2

n n

a tn k n x k n x



    

  



 



2 2 2 2

2

2 1 2 2 2 1 4

2 2

π π

2 0 2 , ( ) j e , e .

2 2 2

Ω Ω

k k

x a t P Ω

mΩ mΩ k

 

 ^  ^{  }

     

 