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

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

Linear stochastic systems

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

Oscillator mode

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

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 3 Y 2 = y 3 + i y 4 2m’-1 Y m’ = y 2m’-1 + i y 2m’ S Quadratic HamiltonianLinear coupling operator Scattering matrix S B1B1 B2B2 BmBm

Linear quantum stochastic dynamics

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

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

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?

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

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

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: (S,L,H)(S,L,H) ? ? ? ? ? ? Network synthesis Quantum network Input fields Output fields Input fields Output fields Wish to realize this system

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

Concatenation product G1G1 G2G2

Series product G1G1 G2G2

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

Direct interaction Hamiltonians GjGj GkGk H jk G G1G1 G2G2 H 12 G GnGn H 2n H 1n... d dd d

A network synthesis theorem G1G1 G2G2 G3G3 GnGn H 12 H 23 H 13 H 2n H 3n H 1n G = (S,L,H) A(t)y(t) 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

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,1/2x T Rx) can be realized, or both one degree of freedom oscillators of the form G’ = (I,Kx,1/2x T Rx) and any static passive network S can be realized –The direct interaction Hamiltonians {H jk } can be realized

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

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)

Realization of linear couplings 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 Two mode squeezer Beam splitter

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

Realization of direct coupling Hamiltonians A direct interaction Hamiltonian between two cavity modes a 1 and a 2 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

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!

Synthesis example

H TMS2 = 5ia 1 * a 2 * + h.c. H DPA = ia 1 * a 2 * + h.c. H TMS1 = 2ia 1 * a 2 * + h.c. Coefficient = 4 Coefficient = 100 H BS1 = -10ia 1 * b + h.c. a 1 = (q 1 + p 1 )/2 a 2 = (q 2 + p 2 )/2 b is an auxiliary cavity mode H BS2 = -ia 1 * a 2 + h.c.

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

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)

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

That’s all folks THANK YOU FOR LISTENING!