Network Synthesis of Linear Dynamical Quantum Stochastic Systems

1. Network Synthesis of Linear Dynamical Quantum Stochastic Systems Hendra Nurdin (ANU) Matthew James (ANU) Andrew Doherty (U. Queensland) TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA

2. Outline of talk • Linear quantum stochastic systems • Synthesis theorem for linear quantum stochastic systems • Construction of arbitrary linear quantum stochastic systems • Concluding remarks

3. Linear stochastic systems

4. Linear quantum stochastic systems An (Fabry-Perot) optical cavity Quantum Brownian motion Non-commuting Wiener processes

5. Oscillator mode

6. Lasers and quantum Brownian motion O(MHz) Spectral density f 0 O(GHz)+

7. Linear quantum stochastic systems A1 = w1+iw2 B1 Y1 = y1 + i y2 S x = (q1,p1,q2,p2,…, qn,pn)T A2 = w3+iw4 B2 Y2 = y3 + i y4 Am=w2m-1+iw2m Bm Ym’ = y2m’-1 + i y2m’ Quadratic Hamiltonian Linear coupling operator Scattering matrix S

8. Linear quantum stochastic dynamics

9. Linear quantum stochastic dynamics

10. Physical realizability and structural constraints A, B, C, D cannot be arbitrary. Assume S = I. Then the system is physically realizable if and only if

11. Motivation: Coherent control • Control using quantum signals and controllers that are also quantum systems • Strategies include: Direct coherent control not mediated by a field (Lloyd) and field mediated coherent control (Yanagisawa & Kimura, James, Nurdin & Petersen, Gough and James, Mabuchi) Mabuchi coherent control experiment James, Nurdin & Petersen, IEEE-TAC

12. Coherent controller synthesis • We are interested in coherent linear controllers: • They are simply parameterized by matrices • They are relatively more tractable to design • General coherent controller design methods may produce an arbitrary linear quantum controller • Question: How do we build general linear coherent controllers?

13. Linear electrical network synthesis • We take cues from the well established classical linear electrical networks synthesis theory (e.g., text of Anderson and Vongpanitlerd) • Linear electrical network synthesis theory studies how an arbitrary linear electrical network can be synthesized by interconnecting basic electrical components such as capacitors, resistors, inductors, op-amps etc

14. Linear electrical network synthesis • Consider the following state-space representation:

15. Network synthesis Input fields Output fields Input fields ? Output fields (S,L,H) ? ? ? ? ? Quantum network 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

16. Challenge • The synthesis must be such that structural constraints of linear quantum stochastic systems are satisfied

17. Concatenation product G1 G2

18. Series product G1 G2

19. Two useful decompositions (S,L,H) (S,0,0) (I,L,H) (S,0,0) (S,L,H) Static passive network (I,S*L,H) (S,0,0)

20. Direct interaction Hamiltonians d Hjk G Gj Gk G d d H12 H2n . . . G2 Gn G1 d H1n

21. A network synthesis theorem G = (S,L,H) H1n H13 H23 A(t) y(t) G1 G2 G3 Gn H3n H12 H2n • The Gj’s are one degree (single mode) of freedom oscillators with appropriate parameters determined using S, L and H • The Hjk’s are certain bilinear interaction Hamiltonian between Gj and Gk determined using S, L and H

22. A network synthesis theorem • According to the theorem, an arbitrary linear quantum system can be realized if • One degree of freedom open quantum harmonic oscillators G = (S,Kx,1/2xTRx) can be realized, or both one degree of freedom oscillators of the form G’ = (I,Kx,1/2xTRx) and any static passive network S can be realized • The direct interaction Hamiltonians {Hjk} can be realized

23. A network synthesis theorem • The synthesis theorem is valid for any linear open Markov quantum system in any physical domain • For concreteness here we explore the realization of linear quantum systems in the quantum optical domain. Here S can always be realized so it is sufficient to consider oscillators with identity scattering matrix

24. Realization of the R matrix • The R matrix of a one degree of freedom open oscillator can be realized with a degenerate parametric amplifier (DPA) in a ring cavity structure (in a rotating frame at half-pump frequency)

25. Realization of linear couplings Two mode squeezer • Linear coupling of a cavity mode a to a field can be (approximately) implemented by using an auxiliary cavity b that has much faster dynamics and can adiabatically eliminated • Partly inspired by a Wiseman-Milburn scheme for field quadrature measurement • Resulting equations can be derived using the Bouten-van Handel-Silberfarb adiabatic elimination theory Beam splitter

26. Realization of linear couplings • An alternative realization of a linear coupling L = αa + βa*for the case α> 0 and α> |β| is by pre- and post-processing with two squeezers Squeezers

27. Realization of direct coupling Hamiltonians • A direct interaction Hamiltonian between two cavity modes a1 and a2 of the form:can be implemented by arranging the two ring cavities to intersect at two points where a beam splitter and a two mode squeezer with suitable parameters are placed

28. Realization of direct coupling Hamiltonians • Many-to-many quadratic interaction Hamiltonian can be realized, in principle, by simultaneously implementing the pairwise quadratic interaction Hamiltonians {Hjk}, for instance as in the configuration shown on the right Complicated in general!

29. Synthesis example

30. Synthesis example a1 = (q1 + p1)/2 a2 = (q2 + p2)/2 b is an auxiliary cavity mode Coefficient =100 HBS1 = -10ia1* b + h.c. HBS2 = -ia1* a2 + h.c. HTMS1 = 2ia1* a2* + h.c. HTMS2 = 5ia1* a2* + h.c. Coefficient = 4 HDPA= ia1* a2* + h.c.

31. Conclusions • A network synthesis theory has been developed for linear dynamical quantum stochastic systems • The theory allows systematic construction of arbitrary linear quantum systems by cascading one degree of freedom open quantum harmonic oscillators • We show in principle how linear quantum systems can be systematically realized in linear quantum optics

32. Recent and future work • Alternative architectures for synthesis (recently submitted) • Realization of quantum linear systems in other physical domains besides quantum optics (monolithic photonic circuits?) • New (small scale) experiments for coherent quantum control • Applications (e.g., entanglement distribution)

33. To find out more… • Preprint: H. I. Nurdin, M. R. James and A. C. Doherty, “Network synthesis of linear dynamical quantum stochastic systems,” arXiv:0806.4448, 2008

34. That’s all folks THANK YOU FOR LISTENING!