QUANTUM FEEDBACK NETWORKS

John Gough

Aberystwyth

IMA, Minneapolis, 11th April 2016

It is only in relatively recent times that we though it was possible to control quantum systems!

“One never realizes experiments with a single electron or an atom or a small molecule. In thought experiments, one assumes that sometimes this is possible; invariably, this leads to ridiculous consequences... One may say that one does not realize experiments with single particles, more than one raises ichthyosaurs in the zoo.”

E. Schrödinger, “Are there quantum jumps?,” Br. J. Philosophy Sci., vol. 3, 109–123, 1952.

Forward to present times: Serge Haroche and David Wineland win the Nobel Prize in Physics for Experiments utilizing Quantum Control

I. Dotsenko, M. Mirrahimi, M. Brune, S. Haroche, J.-M. Raimond, and P. RouchonQuantum feedback by discrete quantum nondemolition measurements: Towardson-demand generation of photon-number statesPhys. Rev. A 80, 013805 (2009)

• Evidently a great deal of engineering has gone into getting these experiments to work.

• But our interest is in the design of the controllers!

Where is Quantum Technology Going?Proposed applications:

• Quantum sensors;• Quantum secure communication;• Quantum simulation;• Quantum computing

… these range from low-hanging fruit to the search for the holy grail.

Ray Beausoleil : We need the quantum analogue of the hearing aid!

Perhaps we need to be looking more closely at where nanotechnology is going – and at where the computer industry wants to go!

Networks and Feedback Control

• Measurement BasedFeedback Control

• Coherent FeedbackControl

Quantum Inputs and Outputs

Lamb Model / Caldeira-Leggett / Ford-Kac-Mazur / Thirring-Schwabl / Yurke-Denker

Non-Markov Models and Markov Limits

Gardiner-Collett

Input-output relations

Spectral Density

Quantum Markovian Models

The “wires” are quantum fields and may carry a multiplicity.

SLH Formalism

• Quantum white noise

• Hamiltonian H

• Coupling/Collapse Operators L

• Scattering Operator S

Quantum Stochastic Models• Single input – no scattering (S=1 ,L , H)

Wick-ordered form:

Heisenberg Picture

Lindblad Generator

Input-Output Relations

Quantum Stochastic Models• Two inputs – pure scattering (S ,L=O , H=O)

Wick-ordered form:

Heisenberg Picture

Input-Output Relations

Quantum Stochastic Models• General (S ,L , H) case

Wick-ordered form:

Heisenberg Picture

Lindblad Generator

Input-Output Relations

Quantum Networks

• How to connect models?

• Cascaded models

• Algebraic loops

• Feedback Control

The Series Product

Network Rule # 1 Open loop systems in parallel

Network Rule # 2 Feedback Reduction Formula

Properties of the Feedback Reduction Formula• Mathematically a Schur complement of the matrix of coefficient operators:

• Equivalently formulated as a fractional linear transformation.

• Independent of the order of edge-elimination.

The Network Rules are implemented ina workflow capture package QHDL

QHDL (MabuchiLab)N. Tezak, et al., (2012) Phil. Trans. Roy. Soc. A, 370, 5270.

Coherent Quantum Feedback Control

Adiabatic Elimination• An important model simplification split the systems into slow and fast subspaces

• Mathematical this is also a Schur complement of the model matrix G

• It commutes with feedback reduction!

Autonomous Quantum Error Correction

Set up in QHDL

Network rules yield the overall “SLH”From which we deduce the master equation

Thank You For Your Attention,

& Keep Spreading the Tapes!