Continuous Measurement and Stochastic Methods in Quantum Optical Systems

2.4 MB·English

by  Robert L. Cook

Continuous Measurement and Stochastic
Methods in Quantum Optical Systems
3
1
0
2
by

n
a
J
Robert Lawrence Cook
5
2

]
h
p B.S., University of California Santa Cruz, 2003
-
t
n
a
u
q
[

1
v
3
DISSERTATION
9
1
6
.
1
Submitted in Partial Fulfillment of the
0
3
Requirements for the Degree of
1
:
v
i Doctor of Philosophy
X
r Physics
a
The University of New Mexico
Albuquerque, New Mexico
May 2013
iii
(cid:13)c 2013, Robert Lawrence Cook
iv
Dedication
Kylie, I promise we’ll take a walk when this is all over.
v
Acknowledgments
First of all I’d like to thank my most recent and final advisor Ivan Deutch. When
I started at UNM 10 years ago I had no idea what I wanted to study, only that
a masters program seemed better than a job at the latest flying-Starbucks. It was
your undergraduate quantum mechanics lectures that showed me how strange and
rich the quantum world can be and they ultimately set me on the path to where I
am today. I will be forever grateful for your help and guidance though the bumpier
parts of my graduate career. I also have to thank Brad Chase. Without you this
dissertation would have taken a very different form. Prior to reading the epic works
of van Handel et al. I would never have guessed that I’d become an advocate for
mathematical formalism. To Ben Baragiola I thank you for your friendship, enthusi-
asm and willingness to talk though a problem. And to Heather Partner I will always
be grateful for your support and camaraderie on the roller coaster ride that started
at Los Alamos, ran through UNM and ended in Sandia.
In my latest academic home of Room 30, I need to thank Carlos Riofr´ıo for your
friendship, warmth and immediate inclusion into Deutsch group, Josh Combes for
your shared enthusiasm for QSDEs, Leigh Norris for your kind hearted adoption of
the luckiest goldfish on the planet, and Vaibhav Madhok for just being Vaibhav.
To the rest of Deutsch group - Bob Keating, Charlie Baldwin, and Krittika Goya -
thanks for listening to me prattle on in group meeting about stochastic calculus and
statistical estimation. I hope I didn’t bore you too much. In the greater quantum
information group I need to thank Professors Carl Caves and Andrew Landahl, cur-
rent and former CQuIC students Jonas Anderson, Chris Cesare, Seth Merkel, Iris
Reichenbach, Alexandre Tacla, Zhang Jiang, Matthias Lang, and Shashank Pandey.
I must also thank Vicky Bird for feeding us so well during arxiv review. From my
short tenure at Sandia national labs I need to thank Cort Johnson, Dan Stick, Todd
Barrick, DaveMoehring, FranciscoBenito, PeterSchwindt, Yuan-YuJau, MikeMan-
gan, Tom Hamilton, and Grant Biedermann for the help and support as I learned
that cryogenic experiments are not for me. I will never forget the time spent working
with Roy Keyes, Tom Jones, Thomas Loyd, and Paul Martin. While we may not
have gotten a lot done we had a whole lot of fun doing it. To Laura Zschaechner
thanks for being a good friend and a shoulder to cry on. And finally I’d like thank
my parents and family for their love and support.
vi
Continuous Measurement and Stochastic
Methods in Quantum Optical Systems
by
Robert Lawrence Cook
B.S., University of California Santa Cruz, 2003
Ph.D., Physics, University of New Mexico, 2013
Abstract
This dissertation studies the statistics and modeling of a quantum system probed by
a coherent laser field. We focus on an ensemble of qubits dispersively coupled to a
travelingwavelightfield. Thefirstresearchtopicexploresthequantummeasurement
statistics of a quasi-monochromatic laser probe. We identify the shortest timescale
that successive measurements approximately commute. Our model predicts that for
a probe in the near infrared, noncommuting measurement effects are apparent for
subpicosecond times.
The second dissertation topic attempts to find an approximation to a conditional
master equation, which maps identical product states to identical product states.
Through a technique known as projection filtering, we find such a equation for an
ensemble of qubits experiencing a diffusive measurement of a collective angular mo-
mentum projection, in addition to global rotations. We then test the quality of the
approximation through numerical simulations. This measurement model is known
to be entangling and without the rotations we find poor agreement between the ex-
act and approximate predictions. However, in the presence of strong randomized
vii
rotations, the approximation reproduces the exact expectation values to within 95%
accuracy.
The final topic applies the projection filter to the problem of state reconstruc-
tion. We find an initial state estimate based on a single continuous measurement
of an identically prepared atomic ensemble. Given the ability to make a continuous
collective measurement and simultaneously applying time varying controls, it is pos-
sible to find an accurate estimate given based upon a single measurement realization.
Previous experiments implementing this method found high fidelity estimates, but
were ultimately limited by decoherence. Here we explore the fundamental limits of
this protocol by studying an idealized model for pure qubits, which is limited only
by measurement backaction. This ultimately makes the measurement statistics a
nonlinear function of the initial state. Via the projection filter, we find an efficiently
computed approximation to the log-likelihood function. Using the exact dynamics to
produce simulated measurements, we then numerically search for a maximum like-
lihood estimate based on the approximate expression. We ultimately find that our
estimation technique nearly achieves an average fidelity bound set by an optimum
POVM.
viii
Contents
1 Introduction 1
1.0.1 A note on quantum foundations . . . . . . . . . . . . . . . . . 10
1.1 An executive summary . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1.1 Quantum optics and quantum stochastic differential equations 12
1.1.2 Classical and quantum probability theory . . . . . . . . . . . . 13
1.1.3 Projection filtering for qubit ensembles . . . . . . . . . . . . . 16
1.1.4 Qubit state reconstruction . . . . . . . . . . . . . . . . . . . . 23
2 Quantum Optics and Quantum Stochastic Differential Equations 29
2.1 Quantum Stochastic Process in Optical Fields . . . . . . . . . . . . . 30
2.1.1 Free space quantization . . . . . . . . . . . . . . . . . . . . . 32
2.2 Wave Packets, Fock Space and Stochastic
Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2.1 Wave packets . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2.2 Weyl operators . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Contents ix
2.2.3 Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.2.4 Abasisindependentexpressionforthewavepacketinnerproduct 43
2.2.5 Fock space and stochastic srocesses . . . . . . . . . . . . . . . 45
2.2.6 Localized wave packets and stochastic processes . . . . . . . . 47
2.3 Paraxial Envelopes and Measurable Pulses . . . . . . . . . . . . . . . 49
2.3.1 Paraxial wave packets in the time domain . . . . . . . . . . . 52
2.3.2 The measurable subspace . . . . . . . . . . . . . . . . . . . . . 54
2.4 The one-dimensional limit . . . . . . . . . . . . . . . . . . . . . . . . 57
2.5 Quantum Wiener processes and the
continuous-time decomposition . . . . . . . . . . . . . . . . . . . . . 60
2.5.1 The continuous-time tensor decomposition . . . . . . . . . . . 62
2.5.2 The quantum Wiener process . . . . . . . . . . . . . . . . . . 65
2.5.3 The units of quantum noise . . . . . . . . . . . . . . . . . . . 67
2.6 Systems Interacting with Quantum Noise . . . . . . . . . . . . . . . 68
2.6.1 Quantum white noise in paraxial wave packets . . . . . . . . . 73
2.6.2 The scattering process . . . . . . . . . . . . . . . . . . . . . . 76
2.6.3 The limiting stochastic propagator . . . . . . . . . . . . . . . 78
2.6.4 A simple 1D example . . . . . . . . . . . . . . . . . . . . . . . 80
2.7 The Faraday Interaction . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.7.1 The quadratic Faraday interaction . . . . . . . . . . . . . . . . 86
Contents x
3 Classical and Quantum Probability Theory 89
3.1 Classical Probability Theory . . . . . . . . . . . . . . . . . . . . . . 90
3.1.1 Stochastic processes and random variables . . . . . . . . . . . 94
3.1.2 Expectation values, the conditional expectation, and measur-
ability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.1.3 Special processes - time-adaption and martingales . . . . . . . 103
3.1.4 The Wiener process . . . . . . . . . . . . . . . . . . . . . . . 105
3.2 Quantum Probability Theory . . . . . . . . . . . . . . . . . . . . . . 107
3.2.1 Embedding the quantum into the classical . . . . . . . . . . . 107
3.2.2 Quantum probability . . . . . . . . . . . . . . . . . . . . . . 111
3.2.3 The quantum conditional expectation . . . . . . . . . . . . . . 114
3.2.4 The conditional expectation and generalized measurements . . 116
3.3 Quantum Filtering Theory . . . . . . . . . . . . . . . . . . . . . . . 118
3.4 The Conditional Master Equation . . . . . . . . . . . . . . . . . . . 122
3.4.1 The innovation process . . . . . . . . . . . . . . . . . . . . . 124
3.4.2 The Ito¯ correction in the conditional master equation . . . . . 126
3.4.3 The conditional Schro¨dinger equation . . . . . . . . . . . . . 129
4 Projection Filtering for Qubit Ensembles 132
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.1.1 An introduction to differential projections . . . . . . . . . . . 134
Contents xi
4.1.2 The conditional master equation . . . . . . . . . . . . . . . . . 135
4.2 Differential Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 137
4.2.1 Tangent spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 138
4.2.2 Riemannian Metrics and orthogonal projections . . . . . . . . 139
4.2.3 Differentials on abstract manifolds . . . . . . . . . . . . . . . 141
4.2.4 Stochastic calculus on differential manifolds . . . . . . . . . . 143
4.3 The Bloch Sphere as a Riemannian Manifold . . . . . . . . . . . . . . 145
4.3.1 Projecting the unconditional master equation . . . . . . . . . 146
4.4 Projections in the tensor product submanifold . . . . . . . . . . . . . 148
4.4.1 The metric in spherical coordinates . . . . . . . . . . . . . . . 149
4.4.2 Calculating collective operator inner products . . . . . . . . . 151
4.4.3 The spherical projection of the CME . . . . . . . . . . . . . . 157
4.5 The Projection Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
4.5.1 Special cases for the projection filter . . . . . . . . . . . . . . 163
4.6 Simulations and Performance . . . . . . . . . . . . . . . . . . . . . . 165
4.6.1 Simulation parameters . . . . . . . . . . . . . . . . . . . . . . 167
4.6.2 Spin squeezing comparisons . . . . . . . . . . . . . . . . . . . 167
4.6.3 Squeezing simulations . . . . . . . . . . . . . . . . . . . . . . 169
4.6.4 Projection filter simulations . . . . . . . . . . . . . . . . . . . 172
5 Qubit State Reconstruction 178