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SU(3) phase operators: some solutions and properties. Hubert de Guise Lakehead University. Collaborators:. Luis Sanchez-Soto Andrei Klimov. Summary. Polar decomposition : can be easily generalized but many “free parameters” - PowerPoint PPT Presentation
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SU(3) phase operators:some solutions
and properties
Hubert de Guise
Lakehead University
Collaborators:
Luis Sanchez-Soto Andrei Klimov
Summary
• Polar decomposition:– can be easily generalized but many “free
parameters”– Normally yields non-commuting phase
operators
• Complementarity:
– cannot be easily generalized but no “free parameters”
– Normally yields commuting phase operators
The origin: the classical harmonic oscillator
Classical harmonic oscillator: 2 2 21 1
2 2H p m x
m
Use:2 2
*, 2 2
m i m ia x p a x p
m m
( ) i t i i ta t Ae A e e
Quantize: †*1 1, a a a a
Two approaches
Two approaches
†expa i a a• write operator in polar form:
• think of as the exponential of a hermitian phase operator
exp i
Two approaches
†expa i a a• write operator in polar form:
• think of as the exponential of a hermitian phase operator
exp i
• Use complementarity condition:
, , exp exp
,
N a a N i i
N
What they have in common
• Look at rather than exp i
What they have in common
• Look at rather than
• is assumed unitary: is hermitian
exp i
exp i
What they have in common
• Look at rather than
• is assumed unitary: is hermitian
• Must fix some “boundary” problems by hand
exp i
exp i
SU(2) phase operator
ˆexp , 1 ,m
i j m j m
ˆ ˆ ˆ,exp expzL i i
mod(2j+1)
SU(2) phase operator
0 1
0 1 1ˆˆexp ,
0 1 1
0 0 0*
z
j
j
i L
j
j
ˆexp , 1 ,m
i j m j m
ˆ ˆ ˆ,exp expzL i i
mod(2j+1)
Only one “boundary” condition
An example: j=1
1 0 0 0 1 0
ˆ ˆ0 0 0 , exp 0 0 1 ,
0 0 1 01 0zL i
ˆexp 1,1 1,0 1,0 1, 1 1, 1 1, 2i -2=1mod(3)
An example: j=1
1 0 0 0 1 0
ˆ ˆ0 0 0 , exp 0 0 1 ,
0 0 1 01 0zL i
ˆexp 1,1 1,0 1,0 1, 1 1, 1 1, 2i -2=1mod(3)
0 1 0
ˆ ˆ,exp 0 0 1
0 02zL i
-2=1mod(3)
A short course on su(3)
• There are eight elements in su(3)
†
† †1 1 1 2 2
† †2 2 2 3 3
ˆ ˆ: 1, 2,3
ˆ ˆ ˆ ˆ:
ˆ ˆ ˆ ˆ:
ij i jC a a i j
h a a a a
h a a a a
A short course on su(3)
• There are now two relative phases
• There are eight elements in su(3)
†
† †1 1 1 2 2
† †2 2 2 3 3
ˆ ˆ: 1, 2,3
ˆ ˆ ˆ ˆ:
ˆ ˆ ˆ ˆ:
ij i jC a a i j
h a a a a
h a a a a
A short course on su(3)
• There are now two relative phases
• There are eight elements in su(3)
†
† †1 1 1 2 2
† †2 2 2 3 3
ˆ ˆ: 1, 2,3
ˆ ˆ ˆ ˆ:
ˆ ˆ ˆ ˆ:
ij i jC a a i j
h a a a a
h a a a a
• States are of the form 1 2 3n n n
Commutation relations
Commutation relations
12 , C
Commutation relations
12 23,CC
Commutation relations
12 23 13,C CC
Commutation relations
13 23C 0,C
Geometry of weight space
Geometry of weight space
3,0,0
2,0,1
1,1,1
Geometry of weight space
12C
3,0,0
2,0,1
1,1,1
Geometry of weight space
12C
3,0,0
2,0,1
1,1,1
23C
3-dimensional case
1,0,0
0,1,00,0,1
3-dimensional case
1,0,0
0,1,00,0,1
3-dimensional case
1,0,0
0,1,00,0,1
3-dimensional case
1,0,0
0,1,00,0,1
L
3-dimensional case
1,0,0
0,1,00,0,1
L
3-dimensional case
1,0,0
0,1,00,0,1
NOT
an su(3) system
SU(3) phase operators:polar decomposition
12 23
0 1 0 0 0 0
C 0 0 0 C 0 0 1
0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0
0 0 1 0
* *
= 0 0 1 0 0 0
0 0 0 0 0 0 0 1
* *
* * * *
Solution 1: commuting solution
Solution 1: commuting solution
12
23
2
2
0 1 0
0
0
0
0
0
0
0
0
0 1
0
i
i
e
e
Solution 1: commuting solution
12
23
2
2
0 1 0
0
0
0
0
0
0
0
0
0 1
0
i
i
e
e
23 13 2312 1212 23 12 23 13, 0i i ii ie e e e e
ComplementaritryThe matrices
12 1
13 2
2 212 1
13 12 23 12 2
0 1 0 1
0 0
0 0
0 0 1 1
0 0
0 0
i ih
i ih
E e H e
E E E e H e
form generalized discrete Weyl pairs, in the sense
kij k k ijE H H E
Solution 2: the SU(2) solution
12
23
1 0
0
0 1 0
0
0
0
0 0 1
1
1 0
0 1 0
i
i
e
e
23 13 2312 1212 23 12 23 13, 0i i ii ie e e e e
Higher-dimensional cases
• No commuting solutions
• No complementarity
Infinite dimensional limit
• The edges are infinitely far
•One can find commuting solutions: the phase operator commute, and have common eigenstates of zero uncertainty
Summary
• Polar decomposition:– can be easily generalized but many “free
parameters”– Normally yields non-commuting phase
operators
• Complementarity:
– cannot be easily generalized but no “free parameters”
– Normally yields commuting phase operators
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