Goren Gordon, Gershon Kurizki Weizmann Institute of Science, Israel Daniel Lidar University of Southern California, USA QEC07 USC Los Angeles, USA Dec.

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Presentation transcript:

Goren Gordon, Gershon Kurizki Weizmann Institute of Science, Israel Daniel Lidar University of Southern California, USA QEC07 USC Los Angeles, USA Dec , 2007

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

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

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

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

Universal dynamical decoherence control formalism Single-qubit decoherence 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 & Kurizki, PRL 97, (2006)) (Gordon & Kurizki, PRA 76, (2007)) ( Gordon et al. J. Phys. B, 40, S75 (2007)) G(  ) A B (Gordon, unpublished)

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:

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

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

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

Numerical results Compare optimal modulation to Bang-Bang (BB) control: Viola & Lloyd PRA (1998) Shiokawa & Lidar PRA (R) (2004) Vitali & Tombesi PRA (2001) Agarwal, Scully, Walther PRA 63, (2001)

Numerical results Compare optimal modulation to Bang-Bang (BB) control: Viola & Lloyd PRA (1998) Shiokawa & Lidar PRA (R) (2004) Vitali & Tombesi PRA (2001) Agarwal, Scully, Walther PRA 63, (2001)

Numerical results Compare optimal modulation to Bang-Bang (BB) control: Viola & Lloyd PRA (1998) Shiokawa & Lidar PRA (R) (2004) Vitali & Tombesi PRA (2001) Agarwal, Scully, Walther PRA 63, (2001) DD condition

Numerical results Optimal pulse shape F. T. X

Numerical results Optimal pulse shape

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… “Know thy enemy” Thank you !!!