network synthesis of linear dynamical quantum stochastic systems hendra nurdin (anu) matthew james...
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Network Synthesis of Linear Dynamical Quantum Stochastic
Systems
Hendra Nurdin (ANU)
Matthew James (ANU)
Andrew Doherty (U. Queensland)
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Outline of talk
• Linear quantum stochastic systems
• Synthesis theorem for linear quantum stochastic systems
• Construction of arbitrary linear quantum stochastic systems
• Concluding remarks
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Linear stochastic systems
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Linear quantum stochastic systems
An (Fabry-Perot) optical cavity
Non-commuting Wiener processes
Quantum Brownian motion
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Oscillator mode
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Lasers and quantum Brownian motion
f
O(GHz)+
O(MHz)
Spe
ctra
l den
sity
0
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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
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Linear quantum stochastic dynamics
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Linear quantum stochastic dynamics
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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
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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
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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?
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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
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Linear electrical network synthesis• Consider the following state-space representation:
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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
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Challenge
• The synthesis must be such that structural constraints of linear quantum stochastic systems are satisfied
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Concatenation product
G1
G2
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Series product
G1 G2
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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
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Direct interaction Hamiltonians
Gj Gk
HjkG
G1G2
H12G
Gn
H2n
H1n
. . .
d
d d
d
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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
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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
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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
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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)
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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
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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
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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
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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!
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Synthesis example
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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.
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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
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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)
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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
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That’s all folks
THANK YOU FOR LISTENING!