Transcript The Bloch Vector - Stony Brook University

The Bloch Vector
Dan Li
SUNY-Stony Brook
12/11/2002
Outline
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Time-Dependent Perturbation Theory
Two-Level Problem
The Solution for the Problem
Bloch Vector
The Animations
Acknowledgments
References
Time-Dependent Perturbation Theory
H  (r , t )  i
.,
n
H0
 (r , t )
t
n (r )
H 0n (r )  Enn (r )
 (r , t )   Ck (t )k (r )eik t
k
 j (r )
i
dC j (t )
dt
  Ck (t ) H 'jk (t )e
i jk t
k
H 'jk (t )   j H ' (t ) k
 jk  ( j  k )
Two-Level Problem

To truncate the summation to just two terms
single ground and excited state
i
dCg (t )
dt
 Ce (t ) H (t )e
'
ge
 ia t
Heg' (t )   cos(kz  l t )
eE0
Rabi Frequency

Laser Detuning
  l  a
er g
i
dCe (t )
 Cg (t ) H eg' (t )eiat
dt
a  eg
The Solution for the Problem

Equations:
d 2Cg (t )
dt
2
 i
dCg (t )
dt
2

Cg (t )  0
4
d 2Ce (t )
dCe (t ) 2
 i

Ce (t )  0
dt 2
dt
4
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Solutions:
'

'
t  i sin t )ei t / 2
2
'
2

'
Ce (t )  i sin te  i t / 2
'
2
Cg (t )  (cos
 '  2   2
The Solution for the Problem (2)
The probability for the atom to be in the ground and excited state
for Ω=γ and δ=γ. Time is in unit of 1/ γ.
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
2
4
6
8
10
2
4
6
8
10
R
Bloch Vector
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Its Components:

e
r3  Ce  Cg
2

g
r2  i(Cg C  C Ce )
r1  Cg Ce  CgCe
2
1
2
0.6
0.8
0.6
0.4
-0.2
0.2
-0.4
0.4
2
4
6
8
10 -0.6
-0.2
0.2
2
4
6
8
-0.4
-0.8
-0.6
10
-1
4
6
8
10
Bloch Vector (2)

The Bloch vector obeys
dR
  R
dt
The vector  has the three components:
Re( H ge' )
Im( H ge' )

Animations
Acknowledgments
I am very grateful to Professor Metcalf. I got
much inspiration in the discussion with him. He is a
strict but nice man. He gives much help as possible.
I also want to thank to those guys in Prof. Metcalf’s
group.
References
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1, Review of Quantum Mechanics
2, Geometrical Representation of the Schrodinger Equation
for Solving Maser Problem,
R. P. Feynman et.al. Journal of Applied Physics P49
Vol.28, No.1 Jan. 1957
3, Mathematica: A system for doing mathematics by
computer. 2nd-Edition. AddisonWesley Publishing Company, Inc. 1991
My Webpage:
http://grad.physics.sunysb.edu/~dli/Courses/OpticsRot
ation/FinalReportBlochVector.htm