Network Synthesis of Linear Dynamical Quantum Stochastic Systems Hendra Nurdin (ANU) Matthew James...
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Transcript of Network Synthesis of Linear Dynamical Quantum Stochastic Systems Hendra Nurdin (ANU) Matthew James...
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!