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The Theory of Effective Hamiltonians for Detuned Systems

Universit ät 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 Email: dfvj@physics.utoronto.ca. • Interaction Picture Hamiltonian:.

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The Theory of Effective Hamiltonians for Detuned Systems

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  1. 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 Email: dfvj@physics.utoronto.ca

  2. • Interaction Picture Hamiltonian: • BUT: we know what really happens is the A.C. Stark shift, i.e.: 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)?

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

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

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

  6. What’s wrong with this result? • This is easy to fix: (7) where • 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.

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

  8. (9a) (9b) • And the time averaging has the effect of removing all frequencies ≥ min{m}, so that important special case: Harmonic Hamiltonians + Low Pass Filter • Suppose we have a Hamiltonian made up of a sum of harmonic terms:

  9. Eq.(8): 0 0 0 0 0 (10) where: Ref: D. F. V. James, Fortschritte der Physik48, 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.

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

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

  12. Wales’s Grand Slam, 2005 5th February 2005 Wales 11 - 9 England 12th February 2005 Italy 8 - 38 Wales 26th February 2005 France 18 - 24 Wales 13th March 2005 Scotland 22 - 46 Wales 19th March 2005 Wales 32 - 20 Ireland

  13. one trapped ion laser “job security factor”: D.F.V. James, Appl. Phys. B 66, 181 (1998). Example 3: Quantum A.C. Stark Shift C. d’Helon and G. Milburn, Phys. Rev. A54, 5141 (1996); S. Schneider et al., J. Mod Opt.47, 499 (2000); F. Schmidt-Kaler et al, Europhys. Lett.65, 587 (2004).

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

  15. What about two ions? big-ass laser

  16. • 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

  17. Quantum A.C. Stark shift again: BORING! Couples the two ions: VERY INTERESTING!!! • Take a closer butchers at the coupling term and it looks like spin-spin coupling: Quantum Simulations 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

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

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