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Analytic model for elastic scattering losses from colliding condensates
M. Trippenbach
P. Zin, J. Chwedenczuk, A. Perez
Physics Department, Warsaw University,
Yehuda Band
Chemistry Department, Ben Gurion University
Outline

Spontaneous & Stimulated – processes

Examples from Quantum Optics

Examples from BEC (elastic scattering)

Stochastic Approach

Benchmark – our simple quantum model to study
properties of the scattered atoms
Parametric down conversion
*


H   aa a a
Superradiance
Bragg Scattering
Elastic collisions
Model - 1
2
ˆ
p
ˆ  (r , t )
ˆ (r , t )  g d 3r
ˆ  (r , t )
ˆ  (r , t )
ˆ (r , t )
ˆ (r , t )  H .c.
Hˆ   d 3r

e
e

2m
ˆ (r , t )   (r , t )   (r , t )  
ˆ (r , t )

Q
Q
e
2
ˆ
p
ˆ (r , t )
ˆ (r , t )  g d 3r
ˆ  (r , t )
ˆ  (r , t ) (r , t ) (r , t )  H .c.
Hˆ   d r

e
e
Q
Q
 e
2m
3

e
Model - 2
2
ˆ
p
ˆ (r , t )
ˆ (r , t )  g d 3r
ˆ  (r , t )
ˆ  (r , t ) (r , t ) (r , t )  H .c.
Hˆ   d r

e
e
Q
Q
 e
2m
3

e
2

i

Q
t
1

 Q (r , t ) 
exp  iQx 
 2
3/ 2 3

2 R
2m 2 R

N
 iQ 2t 1
Q (r , t )Q (r , t )  3 / 2 3 exp  
 2

2 R
m
R

N
2


Qt 
2
2 
 y z 
 x 
m 

 
2
 2

Qt

  
2
2
 
x  y  z  
 m   

Model - 3
d ˆ
2 ˆ

ˆ
i e (r , t )  
e (r , t )  2 gQ (r , t ) Q (r , t )e (r , t )
dt
2m
 iQ 2t 1
Q (r , t )Q (r , t )  3 / 2 3 exp  
 2

2 R
m
R

N
 e (r , t ) 
R
n ,l
n ,l , m
2
 2

Qt

  
2
2
 
x  y  z  
 m   


(r )Yl ,m ( ,  ) a (t ) n ,l ,m
Summary

We presented FULL quantum model (multimodel)
to study how spontaneously initiated emission can
get into stimulated mode
 Scattered atoms form a squeezed state that
can be viewed as a multi-component
condensate.
 Not only are we able to calculate the
dynamics of mode occupation, but also the
full statistics of scattered atoms.
 Test ground for other (stochastic) methods