Optimal Control of the

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Optimal Control of the Non-Linear Schrodinger
Equation. Application to Loading a BEC in an
Optical Lattice. Shlomo E. Sklarz and David
J. Tannor
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Outline

• Optimal Control of BECs
• Formulation of the Optimal Control problem
• Krotov Non-Linear iterative method.
• Optimization Results
• Theoretical treatment Flat Phase Loading of a
  BEC onto an Optical Lattice.
• Collaboration I. Friedler, Y.B. Band, C.J.
  Williams

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Optimal Control of Bose Einstein Condensates
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Formulation of OCT for the NLSE

• Formulation of OCT problem
• Minimize Objective
• Equation of motion
• Control K(t)Magnetic Trap Force Constant

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Minimization under constraint

• Unconstrained objective
• cdj/dy is the well known lagrange multiplier
• Familiar example - momentum p in classical
  mechanics
• Additional perspective - Krotov minimum principal

Adjoined constraint
Lagrange multiplier
Original Objective
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Krotov minimum principal
Minimizing unconstrained J yields lower bound
on minimum of constrained J.
Constrained y
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Improvement problem

• Given an initial process (y0,u0),
• find an improved process (y,u)
• such that
• J(y,u) lt J(y0,u0)

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Krotov Improvement Method Central point Use
freedom in choice of f(t,y) to secure current
yield.

• Choose f(t,y) such thatJy,u0 f(t,y) be
  maximum at y0 gt Current y0 is the worst of
  possible histories
• Freely choose new u to minimize Jy,u
  f. gtAny change in y caused by u can only
  improve minimization

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Choosing f(y,t) A.I Konnov V.A. Krotov, Auto.
Remote Cont. 60, 1427 (1999) Linear f
ltcygt c set to enforce 1st derivative dJ/dy00
gt dc/dt-?H/?y
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Iterative step
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Results of Optimization S.E. Sklarz D.J.
Tannor, PRA 66, 053620 (2002) Trap
Force-Constant is Dynamically controlled to
minimize Variance of Phase at final time.
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Flat Phase Loading of a BEC onto an Optical
Lattice.S.E. Sklarz, I. Friedler, D.J. Tannor,
Y.B. Band and C.J. Williams, PRA 66, 053620
(2002).

• Dynamics of a BEC wave function in an Harmonic
  potential
• BEC Franck-Condon picture
• Thomas-Fermi Ansatz Free parameters w, b, c.
  pwb
• Hamiltonian Set of Eq.
• Effective Potential

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The effect of switching on the Optical Lattice

• Complicated local structure
• Population builds up in wells.
• Tunneling between wells
• Simple global structure
• Stretched Thomas-Fermi shape
• New effective U?UnU.

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Conclusions
1) OCT formalism derived for NLSE. 2) Concrete
OCT algorithm presented. 3) Methodology
successfully applied to physical problem Flat
Phase Loading of a BEC onto an Optical
Lattice. 4) Analytical theory developed for
obtaining a stationary flat phase
(BEC-Franck-Condon). 5) Many Applications to
systems governed by NLSE, such as soliton fiber
optics.