Synthesis of linear quantum stochastic systems via quantum feedback networks
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Synthesis of linear quantum stochastic systems via quantum feedback networks

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Presented at the 48th IEEE Conference on Decision and Control (CDC), Shanghai, China, Dec. 16-18, 2009.

Presented at the 48th IEEE Conference on Decision and Control (CDC), Shanghai, China, Dec. 16-18, 2009.

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  • I am now going to introduce a class of quantum systems that are called linear quantum stochastic systems, these types of systems appear in quantum optics. Starting with a very simple example of such a system: An optical cavity (explain optical cavity).
  • More general linear quantum stochastic systems are as shown in the following. Explain figure, especially quadratures of A i . Some outputs may be ignored, thus number of outputs need not be equal to the number of outputs
  • Port strategy from classical network synthesis
  • Spare slide, whip out for any necessary additional explanations

Synthesis of linear quantum stochastic systems via quantum feedback networks Presentation Transcript

  • 1. Hendra I. Nurdin (ANU) TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A
  • 2. Outline of talk
    • Quick reminder: Linear quantum stochastic systems
    • Synthesis via quantum feedback networks
    • Synthesis example
    • Concluding remarks
  • 3. Linear quantum stochastic systems
    • An (Fabry-Perot) optical cavity
    Non-commuting Wiener processes Quantum Brownian motion
  • 4. Linear quantum stochastic systems x = ( q 1 ,p 1 ,q 2 ,p 2 ,…, q n ,p n ) T A 1 = w 1 +iw 2 A 2 = w 3 +iw 4 A m =w 2m-1 +iw 2m Y 1 = y 1 + i y 2 Y 2 = y 3 + i y 4 Y m’ = y 2m’-1 + i y 2m’ S Quadratic Hamiltonian Linear coupling operator Scattering matrix S B 1 B 2 B m
  • 5. Synthesis of linear quantum systems
    • “ Divide and conquer” – Construct the system as a suitable interconnection of simpler quantum building blocks, i.e., a quantum network, as illustrated below:
    Wish to realize this system ( S , L , H ) ? ? ? ? ? ? Network synthesis Quantum network Input fields Output fields Input fields Output fields
  • 6. An earlier synthesis theorem
    • The G j ’s are one degree (single mode) of freedom oscillators with appropriate parameters determined using S , L and H.
    • The H jk ’s are certain bilinear interaction Hamiltonian between G j and G k determined using S , L, and H.
    Nurdin, James & Doherty, SIAM J. Control. Optim., 48(4), pp. 2686–2718, 2009. G 1 G 2 G 3 G n H 12 H 23 H 13 H 2n H 3n H 1n G = ( S , L , H ) A(t) y(t)
  • 7. Realization of direct coupling Hamiltonians
    • Many-to-many quadratic interaction Hamiltonian can be realized, in principle, by simultaneously implementing the pairwise quadratic interaction Hamiltonians { H jk }, for instance as in the configuration shown on the right.
    Complicated in general, are there alternatives?
  • 8. Quantum feedback networks
  • 9. Quantum feedback networks
    • Quantum feedback networks are not Markov due to the time delays for propagation of fields.
    • In the limit as all time delays go to zero one can recover an effective reduced Markov model (Gough & James, Comm. Math. Phys., 287, pp. 1109–1132, 2009).
  • 10. Approximate direct interaction via field-mediated interactions
    • Idea: Use field-mediated feedback connections to approximate a direct interaction for small time delays.
    Feedback interconnections to approximate direct interactions
  • 11. Model matrix
  • 12. Concatenated model matrix In channel 1 In channel 2 Out channel 1 Out channel 2
  • 13. Connecting input and output, and reduced Markov model (Out channel 1 connected to In channel 2) (Series product) Gough & James, Comm. Math. Phys. , 287, pp. 1109–1132, 2009; IEEE-TAC , 54(11), pp. 2530–2544, 2009
  • 14. Synthesis via quantum feedback networks
    • Suppose we wish to realize G sys =( I , L , H ) with and
    • Let G jk = ( S jk , L jk , 0) for j ≠ k, and G jj = ( I, K j x j , ½ x j T R j x j ), for j , k = 1 , …, n , with L jk and R j to be determined, L jk =K jk x j having multiplicity 1, and S jk a complex number with | S jk |=1.
  • 15. Synthesis via quantum feedback networks
    • Define G j , j= 1,…, n, and G as in the diagram below:
  • 16. Synthesis via quantum feedback networks
    • Summary of results:
      • One can always find L jk , S jk ( j ≠ k ) and R j such that when the output field associated with L jk is connected to the input field associated with L kj , for all j , k =1, …, n and j ≠ k , and for small time delays in these connections, then G as constructed approximates G sys .
      • L jk , S jk , and R j can be computed explicitly.
      • In the limit of zero time delays the interconnections via L kj and L jk ( j ≠ k ) realizes the direct interaction Hamiltonian H jk =x j T R jk x k .
  • 17. Synthesis example
  • 18. Synthesis example Quantum optical circuit based on Nurdin, James & Doherty, SIAM J. Control. Optim., 48(4), pp. 2686–2718, 2009.
  • 19. Concluding remarks
    • Linear quantum stochastic systems can be approximately synthesized by a suitable quantum feedback network for small time delays between interconnections.
    • Main idea is to approximate direct interaction Hamiltonians by field-mediated interconnections.
    • Direct interactions and field-mediated interactions can be combined to form a hybrid synthesis method.
    • Additional results available in: H. I. Nurdin, “Synthesis of linear quantum stochastic systems via quantum feedback networks,” accepted for IEEE-TAC, preprint: arXiv:0905.0802, 2009.
  • 20. That’s all folks THANK YOU FOR LISTENING!
  • 21. From linear quantum stochastic systems to cavity QED systems Linear quantum stochastic system Cavity QED system