First order perturbation analysis of atom lifetime undergoing dipole transition due to presence of resonating ato

Power, E. A.

English

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, 
THE EFFECT ON THE LIFETIME OF AN ATOM 
UNDERGOING A DIPOLE TRANSITION DUE TO TBE 
* 
PRESENCE OF A RESONATING ATOM. 
Edwin A. Pwer 
University College, Department of Mathematics 
London, England 
and 
University of Wisconsin, Theoretical Chemistry Institute 
Madison, Wisconsin 
Rie modification of the transition rate from the 
Einstein A coefficient for an atomic dipole transition 
due to the presence of a similar atom is obtained without 
the use of damping theory. 
4urPcrz -  
* 
This research was supported by National Aeronautics and 
Space Administration Grant NsG-275-62. 
https://ntrs.nasa.gov/search.jsp?R=19660030104 2020-03-16T17:56:47+00:00Z
2 
The purpose of t h i s  note i s  t o  show how t h e  e f f e c t  of 
t h e  presence of a s i m i l a r  atom i n  i t s  ground state on t h e  l i f e -  
t i m e  of an atom i n  i t s  f i r s t  excited state i s  e a s i l y  obtained by 
f i r s t  order per turba t ion  theory. This i s  i n  exact analogy with 
t h e  c a l c u l a t i o n  of t h e  Eins te in  A - c o e f f i c i e n t  and does not 
need damping theory. Dicke") pointed out t h e  changes i n  l i f e -  
t i m e  t o  be expected on symmetry arguments when the  two atoms are 
separated by a d is tance  R < 3: ( i s  t h e  reduced d ipole  
wave length) but s t i l l  g rea t e r  than t h e  atomic s ize .  Then the  
two atoms see e s e n t i a l l y  the  same amplitude of t he  electric f i e l d  
and d ipole  s e l e c t i o n  r u l e s  apply t o  t h e  decay of t h e  combined 
system. 
has a l i f e t i m e  one ha l f  of t he  i s o l a t e d  atom. 
Hutchinson and J&~neka(~)  have computed t h e  apparent l i f e t i m e  as 
a func t ion  of R by using va r i an t s  of Hei t ler ' s  damping theory . 
. .  
The antisymmetric s t a t e  fs' i tAble t h e  symmetric state 
Stephen") and 
(4) 
H e r e  we c a l c u l a t e  t h e  l i f e t ime  by f i r s t  order per turba t ion  theory 
i n  a manner completely analogous t o  obtaining the  E ins t e in  A 
c o e f f i c i e n t  f o r  a r a d i a t i n g  i so l a t ed  atom. 
The i n t e r a c t i o n  Hamiltonian f o r  d ipole  t r a n s i t i o n s  i s  
a re  t h e  dipole moment operator 
, 
where (1) and 
of t h e  two systems. I 2 (5) i s  t h e  t ransverse  e l e c t r i c  f i e l d  
vec tor  which is  expanded i n t o  t h e  Fourier Se r i e s  i n  t h e  volume V 
- 
A=$L 
w = kc is the circular frequency, k the wave vector. 
e(A) (k) is a unit vector along the direction of polarization 
for the gave-vector k and polarization = 1 or 2. Due to 
the transverse nature of the photon 
- 
yu- c 
.-. 
which gives the two possible values of 1 . Finally a(A) (k) 
1 c 
and a(lZt(k) are the annihilation and creation operators for 
L. 
photons of mode k, and satisfy the commutation relations 
r, 
Since the interaction (1) is linear in the electric field 
first order perturbation theory is sufficient to give the lowest 
order non-vanishing matrix element for the transition from 
to If) . Now from the golden rule 
2% k' x 
From t h e  dipole approximation inherent i n  t h e  i n t e r a c t i o n  (1) 
n e i t h e r  t h e  v a r i a t i o n  of 
of e over molecule (2). 
ei kaL1 over molecule (1) nor t h a t  
i L. r.2 
Thus n 
4 
The atomic f ina l  state i s  j u s t  the  two atoms i n  t h e i r  ground 
state 
while t h e  i n i t i a l  state i s  e i t h e r  of t he  states 
so 
Now from the transverse nature of the photon, equation 3, 
+ 
The angular integral is elementary 
and 
6 
Finally using the necessary hypothesis that the two atoms are 
identical 
When R r t  $ we are left with the Dicke result; using 
Tij(o) = $ $9 from equation (13) 
ShkR 
for R >> 4- 
= & (Einstein) - ( 2 0 )  
On the other hand for large R r z j  (kR)-d;j - (19) 
J 
'L - 
and so '.]if + = w;$ = fLc'(p1 
ACKNOWLEDGEMENTS 
It is a pleasure to thank Dr. William J. Meath for 
helpful discuss ions. 
REFERENCES 
1. R.H. Dicke, Phys. Rev. 93, 99 (1954) 
2. M.J. Stephen, J. Chem. Phys. , 40, 669 (1964) 
3. 
4. W. Heitler, The Quantum Theory of Radiation, (Oxford 
-
D.A. Hutchinson and H.F. Hameka, J. Chem. Phys., 41, 2006 (1964) 
University Press, London 1954) pp. 163-174. 


The Effect on the Lifetime of an Atom Undergoing a Dipole Transition Due to the Presence of a Resonating Atom

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