SU(3) phase operators: some solutions and properties

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SU(3) phase operatorssome solutions and
properties

• Hubert de Guise
• Lakehead University

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Collaborators
Luis Sanchez-Soto
Andrei Klimov
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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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The origin the classical harmonic oscillator
Classical harmonic oscillator
Use
Quantize
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Two approaches
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Two approaches

• write operator in polar form

• think of as the exponential of
  a hermitian phase operator

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Two approaches

• write operator in polar form

• think of as the exponential of
  a hermitian phase operator

• Use complementarity condition

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What they have in common

• Look at rather than

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What they have in common

• Look at rather than
• is assumed unitary is
  hermitian

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What they have in common

• Look at rather than
• is assumed unitary is
  hermitian
• Must fix some boundary problems by hand

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SU(2) phase operator
mod(2j1)
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SU(2) phase operator
mod(2j1)
Only one boundary condition
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An example j1
-21mod(3)
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An example j1
-21mod(3)
-21mod(3)
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A short course on su(3)

• There are eight elements in su(3)

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A short course on su(3)

• There are eight elements in su(3)

• There are now two relative phases

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A short course on su(3)

• There are eight elements in su(3)

• There are now two relative phases

• States are of the form

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Commutation relations
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Commutation relations
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Commutation relations
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Commutation relations
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Commutation relations
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Geometry of weight space
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Geometry of weight space
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Geometry of weight space
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Geometry of weight space
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3-dimensional case
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3-dimensional case
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3-dimensional case
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3-dimensional case
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3-dimensional case
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3-dimensional case
NOT an su(3) system
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SU(3) phase operatorspolar decomposition
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Solution 1 commuting solution
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Solution 1 commuting solution
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Solution 1 commuting solution
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Complementaritry
The matrices
form generalized discrete Weyl pairs, in the
sense
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Solution 2 the SU(2) solution
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Higher-dimensional cases

• No commuting solutions

• No complementarity

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Infinite dimensional limit

• The edges are infinitely far

• One can find commuting solutions the phase
  operator commute, and have common eigenstates
  of zero uncertainty

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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