Optimal Dynamical Decoherence Control

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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
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Outline

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

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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)
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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
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
System-bath coupling spectrum
Spectral modulation intensity
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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 et al. J. Phys. B, 40, S75 (2007))
(Gordon Kurizki, PRL 97, 110503 (2006))
(Gordon, unpublished)
(Gordon Kurizki, PRA 76, 042310 (2007))
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Brief overview of Calculus of Variations
Want to minimize the functional
With the constraint
The procedure
1. Solve Euler-Lagrange equation
Get solution
3. Get solution as a function of the constraint
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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, S75 (2007))
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Analytical derivation of optimal modulation
Want to minimize the average modified decoherence
rate
With the energy constraint (a given modulation
energy)
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Analytical derivation of optimal modulation
Euler-Lagrange equation for optimal modulation
Using the energy constraint, one can obtain
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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
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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
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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
DD condition
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Numerical results
Optimal pulse shape
X
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Numerical results
Optimal pulse shape
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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