Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.
Hendra I. Nurdin (ANU) TexPoint fonts used in EMF.  Read the TexPoint manual before you delete this box.:  A A
Outline of talk <ul><li>Quick reminder: Linear quantum stochastic systems </li></ul><ul><li>Synthesis via quantum feedback...
Linear quantum stochastic systems <ul><li>An (Fabry-Perot) optical cavity </li></ul>Non-commuting Wiener processes Quantum...
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...
Synthesis of linear quantum systems <ul><li>“ Divide and conquer” – Construct the system as a suitable interconnection of ...
An earlier synthesis theorem <ul><li>The  G j ’s are  one degree  (single mode) of freedom oscillators with appropriate pa...
Realization of direct coupling Hamiltonians <ul><li>Many-to-many quadratic interaction Hamiltonian  can be realized, in pr...
Quantum feedback networks
Quantum feedback networks <ul><li>Quantum feedback networks are not Markov due to the time delays for propagation of field...
Approximate direct interaction via field-mediated interactions <ul><li>Idea: Use field-mediated feedback connections  to a...
Model matrix
Concatenated model matrix  In channel 1 In channel 2 Out channel 1 Out channel 2
Connecting input and output, and reduced Markov model (Out channel 1 connected to In channel 2) (Series product) Gough & J...
Synthesis via quantum feedback networks <ul><li>Suppose we wish to realize  G sys =( I ,  L ,  H ) with    and  </li></ul>...
Synthesis via quantum feedback networks <ul><li>Define  G j , j= 1,…,  n,  and  G  as in the diagram below: </li></ul>
Synthesis via quantum feedback networks <ul><li>Summary of results: </li></ul><ul><ul><li>One can always find  L jk , S jk...
Synthesis example
Synthesis example Quantum optical circuit based on Nurdin, James & Doherty, SIAM J. Control. Optim., 48(4), pp. 2686–2718,...
Concluding remarks <ul><li>Linear quantum stochastic systems  can be approximately synthesized by a suitable quantum feedb...
That’s all folks THANK YOU FOR LISTENING!
From linear quantum stochastic systems to cavity QED systems Linear quantum stochastic system Cavity QED system
Upcoming SlideShare
Loading in …5
×

Synthesis of linear quantum stochastic systems via quantum feedback networks

1,040 views

Published on

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

Published in: Technology
  • Be the first to comment

  • Be the first to like this

Synthesis of linear quantum stochastic systems via quantum feedback networks

  1. 1. Hendra I. Nurdin (ANU) TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A
  2. 2. Outline of talk <ul><li>Quick reminder: Linear quantum stochastic systems </li></ul><ul><li>Synthesis via quantum feedback networks </li></ul><ul><li>Synthesis example </li></ul><ul><li>Concluding remarks </li></ul>
  3. 3. Linear quantum stochastic systems <ul><li>An (Fabry-Perot) optical cavity </li></ul>Non-commuting Wiener processes Quantum Brownian motion
  4. 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. 5. Synthesis of linear quantum systems <ul><li>“ Divide and conquer” – Construct the system as a suitable interconnection of simpler quantum building blocks, i.e., a quantum network, as illustrated below: </li></ul>Wish to realize this system ( S , L , H ) ? ? ? ? ? ? Network synthesis Quantum network Input fields Output fields Input fields Output fields
  6. 6. An earlier synthesis theorem <ul><li>The G j ’s are one degree (single mode) of freedom oscillators with appropriate parameters determined using S , L and H. </li></ul><ul><li>The H jk ’s are certain bilinear interaction Hamiltonian between G j and G k determined using S , L, and H. </li></ul>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. 7. Realization of direct coupling Hamiltonians <ul><li>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. </li></ul>Complicated in general, are there alternatives?
  8. 8. Quantum feedback networks
  9. 9. Quantum feedback networks <ul><li>Quantum feedback networks are not Markov due to the time delays for propagation of fields. </li></ul><ul><li>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). </li></ul>
  10. 10. Approximate direct interaction via field-mediated interactions <ul><li>Idea: Use field-mediated feedback connections to approximate a direct interaction for small time delays. </li></ul>Feedback interconnections to approximate direct interactions
  11. 11. Model matrix
  12. 12. Concatenated model matrix In channel 1 In channel 2 Out channel 1 Out channel 2
  13. 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. 14. Synthesis via quantum feedback networks <ul><li>Suppose we wish to realize G sys =( I , L , H ) with and </li></ul><ul><li>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. </li></ul>
  15. 15. Synthesis via quantum feedback networks <ul><li>Define G j , j= 1,…, n, and G as in the diagram below: </li></ul>
  16. 16. Synthesis via quantum feedback networks <ul><li>Summary of results: </li></ul><ul><ul><li>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 . </li></ul></ul><ul><ul><li>L jk , S jk , and R j can be computed explicitly. </li></ul></ul><ul><ul><li>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 . </li></ul></ul>
  17. 17. Synthesis example
  18. 18. Synthesis example Quantum optical circuit based on Nurdin, James & Doherty, SIAM J. Control. Optim., 48(4), pp. 2686–2718, 2009.
  19. 19. Concluding remarks <ul><li>Linear quantum stochastic systems can be approximately synthesized by a suitable quantum feedback network for small time delays between interconnections. </li></ul><ul><li>Main idea is to approximate direct interaction Hamiltonians by field-mediated interconnections. </li></ul><ul><li>Direct interactions and field-mediated interactions can be combined to form a hybrid synthesis method. </li></ul><ul><li>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. </li></ul>
  20. 20. That’s all folks THANK YOU FOR LISTENING!
  21. 21. From linear quantum stochastic systems to cavity QED systems Linear quantum stochastic system Cavity QED system

×