The Theory of Effective Hamiltonians for Detuned Systems

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The Theory of Effective Hamiltonians for Detuned
Systems
Universität Ulm, 18 November 2005
Daniel F. V. JAMES
Department of Physics, University of
Toronto, 60, St. George St., Toronto, Ontario
M5S 1A7, CANADA Email dfvj_at_physics.utoronto.ca
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Detuned Systems
Example Two level system, detuned field
Is there a systematic way to get Heff from HI
(preferably without all that tedious mucking
about with adiabatic elimination)?
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Time Averaged Dynamics Definitions

• Unitary time evolution operator

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General Expression I
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General Expression II
(5)
etc...
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• Hamiltonians have to be Hermitian!

• This can be justified by deriving a master
  equation
• excluded part of the frequency domain takes role
  of reservoir
• Lindblat equation with unitary part given by (7)
• Neglect dephasing effects.

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General Expression III

• Result is independent of lower limit in integral
  for V1(t).

• Also applies statistical averages over a
  stationary ensemble.

• This is NOT a perturabtive theory.

-YES, we have used perturbation theory with
reckless abandon, BUT
-Solving Schrödingers equation with this
Hamiltonian gives a result that involves all
orders of the perturbation parameter
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Harmonic Hamiltonians Low Pass Filter
important special case

• Suppose we have a Hamiltonian made up of a sum of
  harmonic terms

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Ref D. F. V. James, Fortschritte der Physik 48,
823-837 (2000) Related
results Average Hamiltonians (NMR) C.
Cohen-Tannoudji J Dupont-Roc and G. Grynberg,
Atom-Photon Interactions (Wiley, 1992), pp. 38-48.
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Example 1 AC Stark Shifts
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Example 2 Raman Processes
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Example 3 Quantum A.C. Stark Shift
C. dHelon and G. Milburn, Phys. Rev. A 54, 5141
(1996) S. Schneider et al., J. Mod Opt. 47, 499
(2000) F. Schmidt-Kaler et al, Europhys. Lett.
65, 587 (2004).
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What about two ions?
big-ass laser
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Hence the effective Hamilton is
Add another laser (with negative detuning)
Quantum A.C. Stark shifts cancel, but coupling
term is doubled Mølmer-Sørensen gåtë
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Conclusions
Quantum Simulations are a lot easier than
Porras and Cirac said.