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Optimal Control of Linear Quantum Systems despite Feedback Delay

1/17. International Mini-Workshop : Theoretical Foundations and Applications of Quantum Control, July 11th, 2008. Optimal Control of Linear Quantum Systems despite Feedback Delay. Tokyo Institute of Technology Kazunori Nishio. 2/17. Outline. Physical image of the problem

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Optimal Control of Linear Quantum Systems despite Feedback Delay

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  1. 1/17 International Mini-Workshop : Theoretical Foundations and Applications of Quantum Control, July 11th, 2008 Optimal Control of Linear Quantum Systemsdespite Feedback Delay Tokyo Institute of Technology Kazunori Nishio

  2. 2/17 Outline • Physical image of the problem • Mathematical system model • Optimal feedback control and system analysis • Summary

  3. 3/17 Quantum feedback control • Feedback cooling of atomic motion : photocurrent l.o. atom : Control potential probe Control objective : small

  4. 4/17 Optimal feedback control • Find the control algorithm minimizing the following; : photocurrent l.o. atom : Control potential probe variances of control input

  5. 5/17 Outline • Physical image of the problem • Mathematical system model; linear system • Optimal feedback control and system analysis • Summary

  6. 6/17 Linear quantum systems • System description ( : Boson-Fock space) • Hilbert space • initial state ( : vacuum state (probe)) • time evolution of unitary operator where control law • annihilation and creation processes • Hamiltonian e.g. • system operator coupled to the probe e.g.

  7. 7/17 Linear quantum systems • Time evolution of position and momentum operators e.g. and

  8. 8/17 Linear quantum systems • Output equation • Measured quantity of the probe l.o. ※ annihilation and creation • Output signal (photocurrent) atom probe e.g. and

  9. 9/17 Outline • Physical image of the problem • Mathematical system model • Optimal feedback control and system analysis • Summary

  10. 10/17 Optimal feedback control • Find the control algorithm minimizing the following; : photocurrent l.o. atom : Control potential probe variances of control input

  11. 11/17 Issue on actual implementation • Time delay due to the computational time : photocurrent l.o. atom : Control potential probe Research purpose Optimal feedback control subject to the time delay

  12. 12/17 Optimal feedback control • Optimal control problem Consider the linear quantum system described by , . Then, the problem is to find conditioned on which minimizes the cost functional . covariance control cost Moreover, compute . Here, .

  13. 13/17 Optimal control performance Theorem Consider the optimal control problem for the linear quantum systems. Then, the minimum value of is ‥‥ (1) ① ② ① optimal when no delay, i.e., ② peformance degradation due to the delay are determined by the system parameters and design parameters .

  14. 14/17 Remark ① : Optimal measurement • H. M. Wiseman et al., Phys. Rev. Lett. 94, 070405 (2005) In quantum feedback control, choice of measurement apparatus affects control performance. Existing result for delay-free systems Our result The optimal measurement changes depending on delay length.

  15. 15/17 Remark ② : Performance evaluation • To what extent can measurement improve the performance • Harmonic oscillator (１) If is small, the improvable level does change depending on . linear growth ∵ depends on (2) If is large, the performance cannot be improved so much. improvable level Linear growth of the curve ∵

  16. 16/17 Outline • Physical image of the problem • Mathematical system model • Optimal feedback control and system analysis • Summary

  17. 17/17 Summary • Summary • Derived the optimal feedback controller and optimal performance formula when . • New insight from the optimal performance formula ; • Optimal measurement changes depending on the delay length • Performance evalution for Harmonic oscillator systems Analysis is applicable to other systems.

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