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QUANTUM COHERENT FEEDBACK

John Gough

Quantum Structures, Information & Control, AberystwythQUANTUM CYBERNETICS & CONTROL

Nottingham University, January 2015

John Gough QUANTUM COHERENT FEEDBACK

What is Cybernetics? A transdisciplinary approach for exploring regulatory systems, their

structures, constraints, and possibilities. Cybernetics is relevant to the study of systems, such as mechanical,

physical, biological, cognitive, and social systems. Cybernetics is applicable when a system being analyzed is

involved in a closed signaling loop; that is, where action by the system generates some change in its environment

and that change is reflected in that system in some manner (feedback) that triggers a system change, originally

referred to as a ”circular causal” relationship. (Wikipedia)

Figure : Norbert Wiener (1948): cybernetics is ”the scientific study ofcontrol and communication in the animal and the machine”.

Root: Greek κυβερν meaning Helmsman, see also governor!


What is Feedback?

If you have to ask, you’ll never know!


What is Feedback? If you have to ask, you’ll never know!


Quantum Measurement-based Feedback Control

Partial information about the state of high Q cavity modes isobtained by measuring Rydberg atoms that are passed one-by-onethrough the cavity. The measurement results can be used to applya feedback action on the cavity mode.

I. Dotsenko, M. Mirrahimi, M. Brune, S. Haroche, J.-M. Raimond, and P. Rouchon, Quantum feedback by discrete

quantum nondemolition measurements: Towards on-demand generation of photon-number states Phys. Rev. A 80,

013805 (2009)


(Autonomous) Quantum Coherent Feedback Control


Why networks?

Processing of (classical) information on chip using photonics, YuriVlasov, CLEO 2012.

On chip quantum information processing: Old and New quantumteleportation devices (A. Furusawa)


Connections through direct coupling

Given an system with Hamiltonian HS on Hilbert space hS , couplethe system directly to a second system (the governor) with Hilbertspace hG .

System Governor

The total evolution on hS ⊗ hG is of the form

H = HS ⊗ 1G + 1S ⊗ HG + V .

Design problems of this type first promoted by Seth Lloyd.


Connections mediated by quantum fields

We begin with quantum Markov models. Each component isrepresented a system (S , L,H) as a single component with inputand output field:

System Hamiltonian H.

Coupling operator L between the system and the field.

Scattering operator S , unitary.

Figure : Gardiner’s input formalism Figure : (S,L,H)


We consider formal “white noise” processes

[b (t) , b† (s)] = δ (t − s)

with

B (t) =

∫ t

0b (s) ds, B† (t) =

∫ t

0b† (s) ds.

It is possible to build a non-commutative version of the Ito calculus(Hudson and Parthasarathy, 1984; Gardiner and Collett,1985) on the Fock space over L2[0,∞) with respect todifferentials dB (t) and dB† (t), and we have

dB (t) dB† (t) = dt.


A unitary system + noise dynamics:

dU =

{L⊗ dB† − L† ⊗ dB −

(1

2L†L + iH

)⊗ dt

}U

future pointing differentials!

The flow of system observables jt (X ) = U† (t) [X ⊗ 1] U (t):

Heisenberg-Langevin equations of motion

djt (X ) = jt (LX ) dt + jt([X , L])dB† + jt([L†,X

])dB,

where LX = 12 [L†,X ]L + 1

2 L† [X , L]− i [X ,H].

Implies the master equation for %t : 〈jt(X )〉 = tr{%tX}.John Gough QUANTUM COHERENT FEEDBACK

For example, we may have an optical cavity, H = ~ωa†a, withcoupling

L =√γa.

Figure : Absorption of field quanta Figure : emission of field quanta

Leads to

d%tdt

= γ

(a%ta

† − 1

2a†a%t −

1

2%ta†a

)+ iω[%t , a

†a].


Linear Quantum Systems (Yanagisawa and Kimura, 2003)


Handling quantum signals coherently!

Beamsplitters: The S in the ”SLH” ...[Bout

1

Bout2

]=

[S11 S12

S21 S22

] [B1

B2

].


Cascades: Systems in Series

Output fields Bout (t) = U†t (1⊗ B (t))Ut

dBout (t) = jt(S)dB (t) + jt(L)dt.

We generalize the notion of cascade introduced by H.J. Carmichael†.

dB(2)out = S2dB

(2)in + L2dt

= S2(S1dB(1)in + L1dt) + L2dt

= S2S1B(1)in + (S2L1 + L2)dt

† H.J. Carmichael, Phys. Rev. Lett., 70(15):2273 2276, 1993.


The Series Product

The Series Product† gives the rule for nonlinear cascaded quantumMarkov systems in the instantaneous feedforward limit isequivalent to the single component:

(S2, L2,H2) C (S1, L1,H1) =(S2S1, L2 + S2L1,H1 + H2 + Im

{L†2S2L1

}).


Bilinear Control Hamiltonian∗

Based on H. M. Wiseman and G. J.Milburn. All-optical versuselectro-optical quantum-limitedfeedback.Phys. Rev. A, 49(5):41104125, 1994.

(I , u (t) , 0) C (−I , 0, 0) C (I , L, 0) C (−I , 0, 0) C

(I ,−u (t) , 0) C (I , L, 0) = (I , 0,H (t))

where

H (t) = Im{L†u(t)} =1

2iL†u (t)− 1

2iLu (t)∗ .

∗ J. G., Construction of bilinear control Hamiltonians using the seriesproduct and quantum feedback Phys. Rev. A 78, 052311 (2008)


Components in-loop

Back to the model considered by Yanagisawa & Kimura:Example

beamsplitter S =

[S11 S12

S21 S22

],

and in-loop component (S0, L0, 0):

dB2 = S0dBout2 + L0dt = S0(S21dB1 + S22dB2) + L0dt

⇒ dBout1 = S11dB1 + S12dB2 ≡ S0dB1 + L0dt

where

S0 = S11 + S12(I − S0S22)−1S0S21, L0 = S12(I − S22)−1S0L0.

Equivalent component (S0, L0, H0):


Networks Rules

More generally how do we buildarbitrary networks from multiplecomponents.

Rule # 1: Open loop system in parallel

�nj=1 (Sj , Lj ,Hj) =

S1 0 0

0. . . 0

0 0 Sn

, L1

...Ln

,H1 + · · ·+ Hn

.


Feedback Reduction Formula:

Rule # 2: Open loop system in parallel Feedback Reduction

B =

[Be

Bi

]L =

[Le

Li

]

S =

[See Sei

Sie Sii

]X

external external

internal

S fb = See + Sei

(X−1 − Sii

)−1Sie ,

Lfb = Le + Sei

(X−1 − Sii

)−1Li ,

H fb = H + ImL†eSei

(X−1 − Sii

)−1Li

+ ImL†i Sii

(X−1 − Sii

)−1Li .


QHDL: a hardware description language for QFNs

Computer-aided schematic capture workflow for modelling andsimulating multi-component photonic circuit

Figure : Pseudo-NAND circuit schematic (a) as created with scheme andits device symbol embedded as a component in an SR-NAND-latchcircuit (b). John Gough QUANTUM COHERENT FEEDBACK

Quantum filtering

Non-demolition SchemeConditional expectation ontothe measurement algebra Jtgenerated by J(s), 0 ≤ s ≤ t.

πt(X ) = E[U∗t (X ⊗ I )Ut | Jt ]

(Quadrature Measurement)

dπt(X ) = πt(LX )dt + {πt(XL + L∗X )− πt(L + L∗)πt(X )}×[dJ(t)− πt(L + L∗)dt].

(Photon Counting Measurement)

dπt(X ) = πt(LX )dt +

{πt(L∗XL)

πt(L∗L)− πt(X )

}×[dJ(t)− πt(L∗L)dt].


The Separation Principle

To minimize an expectation cost based on a fixed measurementprocedure, one has a Hamiltonian-Jacobi-Bellman principle1.There is a quantum analog of the separation principle for controllerdesign2.

1 J. G., V.P. Belavkin, O.G. Smolyanov, Hamilton-Jacobi-Bellman equations for Quantum Filtering and Control, J.Opt. B: Quantum Semiclass. Opt. 7 S237-S244 (2005)

2 L. Bouten, R van Handel, Quantum Stochastics and Information: Statistics, Filtering and Control (V. P. Belavkin

and M. I. Guta, eds.), World Scientific, (2008)


Risk Sensitive Measurement-Based Feedback Control

The design of the optimal controller can be split into

the estimation stage (calculating the least squares estimatefor the state, i.e. the conditional state %t),

the actuation stage (determining the control policy based on%t).

However a LEQG (linear exponential quadratic gaussian) versionalso exists in the quantum case3.

The optimal estimate %t depends on the past measurement recordand the control objective!

3 M.R. James, Risk-Sensitive Optimal Control of Quantum Systems, Physical Review A, 69, 032108 (2004)

M.R. James, A Quantum Langevin Formulation of Risk-Sensitive Optimal Control, J. Opt. B: Quantum Semiclass.

Opt. 7 S198–S207 (2005)


Non-Vacuum input

J.G., Guofeng Zhang, Generating Nonclassical Quantum InputField States with Modulating Filters, arXiv:1404.3866


Quantum Memories:

H.I. Nurdin, J.G., Modular Quantum Memories Using PassiveLinear Optics and Coherent Feedback, arXiv:1409.7473, to appearQuantum Info. Processing.


Quantum Coherent Control in Quantum Transport


Quantum Feedback Networks in Quantum Transport


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