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 matrixS 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 Inputfields Output fields Inputfields Output fields 6. An earlier synthesis theorem

TheG j s areone degree(single mode) of freedom oscillators with appropriate parameters determined usingS ,LandH.

TheH jk s are certain bilinear interaction Hamiltonian betweenG jandG kdetermined usingS ,L,andH.

Nurdin, James & Doherty, SIAM J. Control. Optim., 48(4), pp. 26862718, 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 Hamiltoniancan 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. 11091132, 2009).

10. Approximate direct interaction via field-mediated interactions

Idea: Use field-mediated feedback connectionsto approximate a direct interaction for small time delays.

Feedback interconnections to approximate directinteractions 11. Model matrix 12. Concatenated model matrixIn 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. 11091132, 2009; IEEE-TAC , 54(11), pp. 25302544, 2009 14. Synthesis via quantum feedback networks

Suppose we wish to realizeG sys =( I ,L ,H ) withand

LetG jk = ( S jk , L jk ,0) forjk,andG jj = ( I, K j x j ,x j T R j x j ), forj ,k= 1 , ,n , withL jkandR jto be determined,L jk =K jk x jhaving multiplicity 1, andS jka complex number with | S jk |=1.

15. Synthesis via quantum feedback networks

DefineG j , j= 1,,n,andGas in the diagram below:

16. Synthesis via quantum feedback networks

Summary of results:

• One can always findL jk , S jk( j k ) andR jsuch that when the output field associated withL jkis connected to the input field associated withL kj , for allj ,k =1, ,nandj k , and for small time delays in these connections, thenGas constructed approximatesG sys .

• L jk , S jk , andR jcan be computed explicitly.

• In the limit of zero time delays the interconnections viaL kjandL jk( j k ) realizes the direct interaction HamiltonianH 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. 26862718, 2009. 19. Concluding remarks

Linear quantum stochastic systemscan 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. Thats all folks THANK YOU FOR LISTENING! 21. From linear quantum stochastic systems to cavity QED systems Linear quantum stochastic system Cavity QED system