Classical evolution of quantum fluctuations in spin-like systems

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Classical evolution of quantum fluctuations in
spin-like systems
Andrei B. Klimov
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Phase space formulation for spin-like sistems
Invertible map
- Weyl simbol,
is the kernel operator
so that
Density matrix
Wigner function
Stratonovich 1956, Agarwal 1971
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- generators of 2S1 dim irrep of su(2) algebra
Spin operators
Semiclassical limit
Evolution equation
quantum corrections
- Weyl symbol of the Hamiltonian
- Poisson brackets on
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Correspondence rules in
limit
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Semiclassical dynamics of quantum systems
Evolution equation
Solution
Each point of the initial distribution evolves
along classical trajectories
Initial states localized states for which the
norm of quantum corrections is small
Evolution of observables
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limit
Ehrenfest equations in
integration by parts
averaged Heisenberg equations
Second order spin Hamiltonians
semiclassical evolution time
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Quasiclassical description of squeezed states on
the sphere
Generation of squeezed states - non-linear
evolution
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Evolution equation for
Loiuville equation, Classical evolution of
the Wigner function
- classical trajectories
Initial coherent state on the equator
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Wigner evolution and semiclassical
evolution Evolution of the angle of
minimal fluctations and from
semiclassical approximation, S50
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Quasiclassical evolution of entangled satates on
the sphere
Consider two quantum systems H1 y H2
Factorized states Entangled
states Phase space represenatation Maxima
lly entangled states
pure states
and
- plane distribution
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Example
and

Entangled state
Factorized state
(pure states)
Systems of dimension N
- factorized state

- maximilly entangled state
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Generation of entangled states
Factorized state
nonlocal
interactions
Entangled state

Hamiltonian
H1

• classical evolution
• Entanglement
• classical evolution

H2
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Semiclassical orrespondence rules,
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Example Semiclassical initial state where
are coherent states on equator

Evolution equation in semiclassical limit
Solution
where
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