1. The Theory of Effective Hamiltonians for Detuned Systems
Universität Ulm, 18 November 2005
The Theory of Effective Hamiltonians for Detuned Systems
Daniel F. V. JAMES
Department of Physics, University of Toronto,
60, St. George St., Toronto,
Ontario M5S 1A7, CANADA

2. Detuned Systems • Example: Two level system, detuned field
• Interaction Picture Hamiltonian:
• BUT: we know what really happens is the A.C. Stark shift, i.e.:
• Is there a systematic way to get Heff from HI (preferably without all that tedious mucking about with adiabatic elimination)?

3. Time Averaged Dynamics: Definitions
Unitary time evolution operator
(1)
Time-Averaged evolution operator
(2)
Filter Function (real valued)
Define the effective Hamiltonian by:
(3)

4. General Expression I
(4)
Use a perturbative series for U and Heff:
(5)

5. General Expression II
(5)
(6a)
(6b)
etc...

6. What’s wrong with this result?
Hamiltonians have to be Hermitian!
where
This is easy to fix:
(7)
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.

7. General Expression III
Definition of a real averaging process implies:
and so, (AT BLOODY LAST):
(8)
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 Schrödinger’s equation with this Hamiltonian gives a result that involves all orders of the perturbation parameter

9. Harmonic Hamiltonians + Low Pass Filter
important special case:
Harmonic Hamiltonians + Low Pass Filter
Suppose we have a Hamiltonian made up of a sum of harmonic terms:
(9a)
(9b)
And the time averaging has the effect of removing all frequencies ≥ min{m}, so that

10. Eq.(8):
(10)
where:
Ref: D. F. V. James, Fortschritte der Physik 48, (2000); Related results: Average Hamiltonians (NMR); C. Cohen-Tannoudji J Dupont-Roc and G. Grynberg, Atom-Photon Interactions (Wiley, 1992), pp

11. Example 1: AC Stark Shifts
i.e.:

12. Example 2: Raman Processes
A.C. Stark shifts (again!)
Raman Transitions

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14. Example 3: Quantum A.C. Stark Shift
one trapped ion
laser
“job security factor”: D.F.V. James, Appl. Phys. B 66, 181 (1998).
C. d’Helon 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).

15. • Lamb-Dicke approximation:
• “carrier” term
• red sideband:
• blue sideband:
• low pass filter excludes oscillations at p, hence:

16. What about two ions?
big-ass laser

17. • nearly resonant with the CM (p=1) mode
• Define a collective spin operator
• “carrier”, red and blue sideband terms:
new term: wasn’t there for single ion

18. Hence the effective Hamilton is
Quantum A.C. Stark shift again: BORING!
Couples the two ions: VERY INTERESTING!!!
• Add another laser (with negative detuning): Quantum A.C. Stark shifts cancel, but coupling term is doubled: Mølmer-Sørensen gåtë
• Take a closer butchers at the coupling term and it looks like spin-spin coupling: Quantum Simulations

19. Conclusions
where:
• The time-averaged dynamics of a system with a harmonic Hamiltonian of the form:
Is described by an effective Hamiltonian given by:
• Quantum Simulations are a lot easier than Porras and Cirac said.