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Optimal Control of Linear Quantum Systems despite Feedback Delay

Explore the theoretical foundations and applications of quantum control in linear systems, focusing on feedback delay challenges. Delve into optimal feedback control algorithms and system analysis. Discover insights on measurement apparatus performance in quantum feedback control. Uncover the impact of time delay on control potential and probe variances.

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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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