The Bloch Vector - Stony Brook University
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Transcript The Bloch Vector - Stony Brook University
The Bloch Vector
Dan Li
SUNY-Stony Brook
12/11/2002
Outline
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 0n (r ) Enn (r )
(r , t ) Ck (t )k (r )eik 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
ia t
Heg' (t ) cos(kz l t )
eE0
Rabi Frequency
Laser Detuning
l a
er g
i
dCe (t )
Cg (t ) H eg' (t )eiat
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
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
Its Components:
e
r3 Ce Cg
2
g
r2 i(Cg C C Ce )
r1 Cg Ce CgCe
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
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