Network Synthesis of Linear Dynamical Quantum Stochastic

Systems

Hendra Nurdin (ANU)

Matthew James (ANU)

Andrew Doherty (U. Queensland)

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)

Spe

ctra

l den

sity

0

Linear quantum stochastic systems

x = (q1,p1,q2,p2,…,qn,pn)T

A1 = w1+iw2

A2 = w3+iw4

Am=w2m-1+iw2m

Y1 = y1 + i y2

Y2 = y33 + i y4

Ym’ = y2m’-12m’-1 + i y2m’

S

Quadratic Hamiltonian Linear coupling operator Scattering matrix S

B1

B2

Bm

Linear quantum stochastic dynamics

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 experimentJames, 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)

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Networksynthesis

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

G1

G2

Series product

G1 G2

Two useful decompositions

(S,0,0) (I,L,H)

(S,L,H)

(I,S*L,H) (S,0,0)

(S,L,H)

(S,0,0)

Static passive network

Direct interaction Hamiltonians

Gj Gk

HjkG

G1G2

H12G

Gn

H2n

H1n

. . .

d

d d

d

A network synthesis theorem

G1 G2 G3 Gn

H12

H23

H13

H2n

H3n

H1nG = (S,L,H)

A(t) y(t)

• 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

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

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 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

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!

Synthesis example

Synthesis example

HTMS2 = 5ia1* a2

* + h.c.

HDPA = ia1* a2

* + h.c.

HTMS1 = 2ia1* a2

* + h.c.

Coefficient = 4

Coefficient =100

HBS1 = -10ia1* b + h.c.

a1 = (q1 + p1)/2a2 = (q2 + p2)/2

b is an auxiliary cavity mode

HBS2 = -ia1* a2 + 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:0806.4448, 2008

That’s all folks

THANK YOU FOR LISTENING!