Optimal Dynamical Decoherence Control Goren Gordon Gershon

Optimal Dynamical Decoherence Control Goren Gordon , Gershon Kurizki Weizmann Institute of Science, Israel Daniel Lidar University of Southern California, USA QEC 07 USC Los Angeles, USA Dec. 17 -21, 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 Keller et al. Nature 431, 1075 (2004) Ion trap Cold atom in (imperfect) optical lattice 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)

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, S 75 (2007) [review] system+ modulation bath coupling Fidelity of an initial excited state: Average modified decoherence rate Reservoir response (memory) function Phase modulation

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, S 75 (2007) [review] Time-domain Ft( ) Frequency-domain G( ) System-bath coupling spectrum Spectral modulation intensity No modulation (Golden Rule)

Universal dynamical decoherence control formalism Single-qubit decoherence control (Gordon et al. J. Phys. B, 40, S 75 (2007)) Decay due to finite-temperature bath coupling Proper dephasing 1 2 Multi-qudit entanglement preservation Imposing DFS by dynamical modulation Entanglement death and resuscitation G( ) (Gordon, unpublished) Dephasing control during quantum computation (Gordon & Kurizki, PRA 76, 042310 (2007)) A (Gordon & Kurizki, PRL 97, 110503 (2006)) B

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): (Gordon et al. J. Phys. B, 40, S 75 (2007)) AC-Stark shift Resonant field amplitude

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 58 2733 (1998) Shiokawa & Lidar PRA 69 030302(R) (2004) Vitali & Tombesi PRA 65 012305 (2001) Agarwal, Scully, Walther PRA 63, 044101 (2001)

Numerical results Compare optimal modulation to Bang-Bang (BB) control: 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)

Numerical results Compare optimal modulation to Bang-Bang (BB) control: 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) DD condition

Numerical results Optimal pulse shape X F. T.

Numerical results Optimal pulse shape

Conclusions • Dynamical decoupling and Bang-Bang modulations environment-insensitive, i. e. ignore coupling spectrum are • 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 finitetemperatures • Extensions to multi-partite deocherence and entanglement optimal control underway… “Know thy enemy” Thank you !!!