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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 Outline Universal dynamical decoherence control formalism Brief overview of

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Optimal dynamical decoherence control l.jpg

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


Outline l.jpg
Outline

  • Universal dynamical decoherence

    control formalism

  • Brief overview of

    Calculus of Variations

  • Analytical derivation of equation

    for optimal modulation

  • Numerical results

  • Conclusions


Decoherence scenarios l.jpg
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)


Universal dynamical decoherence control formalism l.jpg
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


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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)


Universal dynamical decoherence control formalism6 l.jpg

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))


Brief overview of calculus of variations l.jpg
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 l.jpg
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))


Analytical derivation of optimal modulation9 l.jpg
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


Analytical derivation of optimal modulation10 l.jpg
Analytical derivation of optimal modulation

Euler-Lagrange equation for optimal modulation

Using the energy constraint,

one can obtain:

Equation for Optimal Modulation


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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:


Numerical results12 l.jpg
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:


Numerical results13 l.jpg
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


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Numerical results

Optimal pulse shape

X

F. T.


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Numerical results

Optimal pulse shape


Conclusions l.jpg
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”


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