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Optimal Dynamical Decoherence Control

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  1. Optimal Dynamical Decoherence Control Goren Gordon , Gershon Kurizki Weizmann Institute of Science, Israel Daniel Lidar University of Southern California, USA QEC07 USC Los Angeles, USA Dec. 17-21, 2007

  2. Outline • Universal dynamical decoherence control formalism • Brief overview of Calculus of Variations • Analytical derivation of equation for optimal modulation • Numerical results • Conclusions

  3. Decoherence Scenarios Ion trap Cold atom in (imperfect) optical lattice Keller et al. Nature 431, 1075 (2004) Häffner et al. Nature 438 643 (2005) Jaksch et al. PRL 82, 1975 (1999) Mandel et al. Nature 425, 937 (2003) Ion in cavity Kreuter et al. PRL 92 203002 (2004)

  4. Universal dynamical decoherence control formalism Kofman & Kurizki, Nature 405, 546(2000); PRL 87, 270405 (2001); PRL 93, 130406(2004) Gordon, Erez and Kurizki, J. Phys. B, 40, S75 (2007) [review] system+ modulation bath coupling Fidelity of an initial excited state: Average modified decoherence rate Reservoir response (memory) function Phase modulation

  5. Universal dynamical decoherence control formalism Kofman & Kurizki, Nature 405, 546(2000); PRL 87, 270405 (2001); PRL 93, 130406(2004) Gordon, Erez and Kurizki, J. Phys. B, 40, S75 (2007) [review] Time-domain Frequency-domain Ft() System-bath coupling spectrum G() Spectral modulation intensity No modulation (Golden Rule)

  6. 2 1 1 2 G() B A Universal dynamical decoherence control formalism • Single-qubitdecoherence control • Decay due to finite-temperature bath coupling • Proper dephasing • Multi-qudit entanglement preservation • Imposing DFS by dynamical modulation • Entanglement death and resuscitation • Dephasing control during quantum computation (Gordon et al. J. Phys. B, 40, S75 (2007)) (Gordon & Kurizki, PRL 97, 110503 (2006)) (Gordon, unpublished) (Gordon & Kurizki, PRA 76, 042310 (2007))

  7. Brief overview of Calculus of Variations Want to minimize the functional: With the constraint: The procedure: 1. Solve Euler-Lagrange equation Get solution: 2. Insert the solution to the constraint: Get 3. Get solution as a function of the constraint:

  8. Analytical derivation of optimal modulation Resonant field amplitude AC-Stark shift Want to minimize the average modified decoherence rate: With the energy constraint (a given modulation energy): (Gordon et al. J. Phys. B, 40, S75 (2007))

  9. Analytical derivation of optimal modulation Want to minimize the average modified decoherence rate: With the energy constraint (a given modulation energy): Use notation: Euler-Lagrange equation for optimal modulation

  10. Analytical derivation of optimal modulation Euler-Lagrange equation for optimal modulation Using the energy constraint, one can obtain: Equation for Optimal Modulation

  11. Numerical results Viola & Lloyd PRA 58 2733 (1998) Shiokawa & Lidar PRA 69 030302(R) (2004) Vitali & Tombesi PRA 65 012305 (2001) Agarwal, Scully, Walther PRA 63, 044101 (2001) Compare optimal modulation to Bang-Bang (BB) control:

  12. Numerical results Viola & Lloyd PRA 58 2733 (1998) Shiokawa & Lidar PRA 69 030302(R) (2004) Vitali & Tombesi PRA 65 012305 (2001) Agarwal, Scully, Walther PRA 63, 044101 (2001) Compare optimal modulation to Bang-Bang (BB) control:

  13. Numerical results Viola & Lloyd PRA 58 2733 (1998) Shiokawa & Lidar PRA 69 030302(R) (2004) Vitali & Tombesi PRA 65 012305 (2001) Agarwal, Scully, Walther PRA 63, 044101 (2001) Compare optimal modulation to Bang-Bang (BB) control: DD condition

  14. Numerical results Optimal pulse shape X F. T.

  15. Numerical results Optimal pulse shape

  16. Conclusions Dynamical decoupling and Bang-Bang modulations are environment-insensitive, i.e. ignore coupling spectrum Optimal modulation “reshapes” (chirps) the pulse to minimize spectral overlap of the system-bath coupling and modulation spectra Current results using universal dynamical decoherence control are also applicable to decay and proper-dephasing, at finite- temperatures Extensions to multi-partite deocherence and entanglement optimal control underway… Thank you !!! “Know thy enemy”