Thesis (MSc)--Stellenbosch University, 2017ENGLISH ABSTRACT : Dynamic correlations hψ| O1(t1)O2(t2)|ψi of quantum observables are useful quantities
for the study of quantum dynamics. Attempts at measuring these correlations
are however complicated, due to the measurement backaction (wave function collapse)
incurred during measurements at the early time t1. We propose a noninvasive
measurement protocol, based on a weak ancilla target coupling, which reduces this
backaction at t1. We show that both real and imaginary parts of the desired correlation
can be extracted through appropriate choices of the initial ancilla state and
of the ancilla target coupling Hamiltonian. The protocol is applicable to arbitrary
(pseudo)spin systems with arbitrary (non)equilibrium initial states. Errors arising
in experimental implementations are analysed, and we show that deviations from
the desired correlation can be minimised through an optimal choice of the ancilla
target coupling time. Implementation in linear ion trap experiments is discussed.
We derive the positive-operator-valued measure which describes the noninvasive
measurement at t1. For dynamic correlations of single-site spin-1/2 observables, this
operator formalism shows that measurement backaction is of no concern. Real parts
can be obtained with projective measurements of the target at t1 and t2. Imaginary
parts are obtained by performing a local rotation of the target at t1, followed
by a projective measurement at t2. These ancilla-free protocols are theoretically
simpler than the noninvasive measurement protocol, but remain experimentally
challenging. Rotations and projections performed at t1 may be subject to noise,
which propagates into the measured correlation. We use Lieb-Robinson theory to
bound the size of the resulting error terms. An analysis of the spatio-temporal
behaviour of these errors provides guidance for experimental implementation of the
ancilla-free measurement protocols.AFRIKAANSE OPSOMMING : Dinamiese korrelasies hψ| O1(t1)O2(t2)|ψi van kwantum waarneembares is nuttig in
die studie van kwantum dinamika. Die meting van hierdie hoeveelhede word egter
gekompliseer deur die meetingsterugreaksie (gol unksie ineenstorting), wat tydens
die meting by die vroeër tyd t1 plaasvind. Ons stel 'n nie-ingrypende metingsprotokol
voor, gebaseer op 'n swak ancilla teikenstelsel koppeling, wat die meetingsterugreaksie
by t1 verminder. Ons toon dat die reële en imaginêre dele van
die korrelasiefunksie deur geskikte keuses van die ancilla begintoestand en van die
ancilla teikenstelsel koppeling bepaal kan word. Die protokol is van toepassing op
(pseudo)spin stelsels met arbitrêre nie-ewewig begintoestande. Foute wat in eksperimentele
implementering ontstaan word geanaliseer. Ons toon dat afwykings van
die gewenste korrelasie geminimeer kan word deur 'n optimale ancilla teikenstelsel
koppelingstyd. Implementering in liniêre ioonputte word bespreek.
Ons herlei die positiewe operator-waardige maat wat die nie-ingrypende meting
by t1 beskryf. Hierdie formalisme toon dat die meetingsterugreaksie van geen belang
is vir dinamiese korrelasies van enkelpunt spin-1/2 waarneembares nie. Die
reële deel kan deur projektiewe meetings by t1 en t2 bepaal word. Die imaginêre
deel word verkry deur 'n lokale rotasie by t1, gevolg deur 'n projektiewe meting
by t2. Hierdie ancilla-vrye protokolle is teoreties eenvoudiger as die nie-ingrypende
metingsprotokol, maar eksperimentele implementering bly uitdagend. Rotasies en
projeksies wat by t1 uitgevoer word kan onder steurings ly, wat dan ook die gemete
korrelasies a ekteer. Ons gebruik Lieb-Robinson teorie om die grootte van die resulterende
foute te begrens. 'n Analise van die foute se tyd en ruimtelike gedrag
bied leiding vir die eksperimentele implementering van die ancilla-vrye metingsprotokoll