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Filtering and Control in Quantum Optics
een wetenschappelijke proeve op het gebied van de
Natuurwetenschappen, Wiskunde en Informatica
Proefschrift
ter verkrijging van de graad van doctor
aan de Radboud Universiteit Nijmegen
op gezag van de Rector Magnificus prof. dr. C.W.P.M. Blom,
volgens het besluit van het College van Decanen
in het openbaar te verdedigen op maandag 29 november 2004
des morgens om 10.30 precies
door
Lucas Martinus Bouten
geboren op 11 oktober 1976
te Helden
Promotor: Prof. Dr. G.J. Heckman
Copromotor: Dr. J.D.M. Maassen
Manuscriptcommissie:
Prof. Dr. V.P. Belavkin (University of Nottingham)
Prof. Dr. R.D. Gill (University of Utrecht)
Prof. Dr. B. Ku¨mmerer (University of Darmstadt)
ISBN 90-9018790-1
Acknowledgments
The theme for this thesis, applying the tools of quantum probability to problems in quan-
tum optics, was provided by Hans Maassen, who has been my supervisor for the past
four years. Apart from sharing his deep insight in quantum mechanics, I am grateful to
Hans for his patience and encouragement in finding my own way in research. I would like
to extend these words of thanks to M˘ada˘lin Gu¸t˘a, my colleague in Nijmegen in the first
two years, and a good friend in close scientific contact afterwards. I have benefited a lot
from the seminar on operator algebras organised by our little group in Nijmegen for the
first two years. It grew to become the ”QRandom seminar” in collaboration with Richard
Gill’s groups in Eindhoven and Utrecht afterwards.
The last four years I have been in very pleasant and friendly contact with Gert Heckman,
who has advised and encouraged me on numerous occasions. I am grateful to Gert for
the many enriching discussions on mathematics we have had over the years. Gert has also
encouraged me to attend the seminar on noncommutative geometry organised by Klaas
Landsman and Eric Opdam in Amsterdam, which was a very stimulating experience.
In the second year Hans sent me to Oregon (USA) on a three month visit to Howard
Carmichael. This visit proved to be a turning point for me. Howard introduced me to
the stochastic Schro¨dinger equations and he also provided me with literature pointing me
in the direction of quantum filtering theory. Apart from showing me a new direction in
research, I am grateful to him for the many things he taught me about quantum optics
and for his kind hospitality.
In the last year I met Slava Belavkin who invited me to Nottingham. I am thankful to
him for the many discussions we had on quantum filtering theory. I am also grateful to
him for introducing me to optimal control theory in a quantum setting, a line of research
in which I would like to continue in the future. I thank Slava and Martin Lindsay for
arranging the nice time spent in Nottingham.
I am indebted to Andreas Buchleitner for organising the many workshops I attended
at the Max Planck Institut in Dresden. I also thank him for his advice in rewriting
the introduction to the paper that makes up Chapter 3 of this thesis. I am thankful
to Burkhard Ku¨mmerer for his inspiring lectures at the ”Coherent Evolution in Noisy
Environments” summer school in Dresden, and for his interest and encouragement in the
work I have done.
I thank Joost and Stefan, with whom I shared an office, and many friends for the pleasant
atmosphere of the past four years. I thank my parents for their support.
Contents
1 Introduction 1
1.1 Some spectral theory and quantum probability . . . . . . . . . . . . . . . . 1
1.2 Stochastic calculus on Fock spaces . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Quantum optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Filtering and control, outline of results . . . . . . . . . . . . . . . . . . . . . 15
2 The Davies process of resonance fluorescence 21
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2 The dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 Guichardet space and integral-sum kernels . . . . . . . . . . . . . . . . . . . 25
2.4 The Davies process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5 Quantum trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.6 A renewal process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3 Stochastic Schro¨dinger equations 37
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
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3.2 The Davies process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3 Homodyne detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4 Conditional expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.5 The dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.6 Quantum stochastic calculus . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.7 Belavkin’s stochastic Schro¨dinger equations . . . . . . . . . . . . . . . . . . 56
3.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4 Squeezing enhanced control 65
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2 The dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3 Quantum stochastic calculus . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.4 The Belavkin equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.5 Control: the essentially commutative case . . . . . . . . . . . . . . . . . . . 78
4.6 Control without squeezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.7 Squeezed states and their calculus . . . . . . . . . . . . . . . . . . . . . . . 82
4.8 Control with squeezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Summary 93
Samenvatting 95
Curriculum Vitae 97
Chapter 1
Introduction
Sixty years after its invention by the Japanese mathematician Kiyosi Itˆo, stochastic ana-
lysis has found a wide range of applications varying from pure mathematics to physics,
enginering, biology and economics. Stochastic differential equations are an important tool
for solving problems in pricing options at the stock market, finding solutions to boundary
value problems, filtering signals in electrical enginering and for solving many other prob-
lems. In 1984 Hudson and Parthasarathy published a paper [54] extending the definition
of Itˆo’s stochastic integral and its subsequent calculus to the non-commutative world of
quantum mechanics. Thegoal of this thesis is to apply this generalized stochastic calculus
to problems in quantum optics.
1.1 Some spectral theory and quantum probability
The question of the mathematical foundations of quantum mechanics was settled by von
Neumann in a series of papers written between 1927 and 1932, culminating in his book
”Mathematische Grundlagen der Quantenmechanik” [87]. In this series of articles von
Neumanndeveloped spectraltheory fornormaloperatorsonaHilbertspace, i.e. operators
that commute with their adjoint. Avoiding tedious considerations regarding domains of
operators we will now first focus on spectral theory for bounded normal operators on a
separable Hilbert space H.
Denote by (H) the algebra of all bounded operators on H and let be a subset of (H).
B S B
We call the set := R (H); SR = RS S the commutant of in (H). A
′
S { ∈ B ∀ ∈ S} S B
von Neumann algebra on H is a -subalgebra of (H) such that equals its double
∗
A B A
commutant, i.e. = . It follows from von Neumann’s double commutant theorem, cf.
′′
A A
1
2 CHAPTER 1. INTRODUCTION
[55], that a von Neumann algebra is a C -subalgebra, i.e. a -subalgebra closed in the
∗ ∗
operator norm topology, of (H) that is closed even in the weak operator topology.
B
It immediately follows from = that the identity 1 (H) is an element of the von
′′
A A ∈ B
Neumann algebra . A state on is a linear map ρ : C such that ρ is positive in
A A A →
the sense that ρ(A A) 0 for all A and such that ρ is normalised ρ(1) = 1. A state
∗
≥ ∈ A
is called normal if it is weak operator continuous on the unit ball of . The following
A
theorem, see [56] sections 9.3, 9.4 and 9.5 for a proof, is at the heart of spectral theory.
Theorem 1.1.1: Let be a commutative von Neumann algebra and ρ a normal state on
C
. Then there is a probability space (Ω,Σ,P) such that is -isomorphic to L (Ω,Σ,P),
∗ ∞
C C
the space of all bounded measurable functions on Ω. Furthermore, if we denote the -
∗
isomorphism between and L (Ω,Σ,P) by C C we have
∞
C 7→ •
ρ(C)= C P(dω), C ,
ω
∈ C
ZΩ
where C denotes the function C evaluated at ω, i.e. C = C (ω).
ω ω
• •
Since a normal operator, i.e. an operator that commutes with its adjoint, generates a
commutative von Neumann algebra the above theorem can be applied. The simplest
example of a normal operator is a Hermitian operator A on a finite dimensional Hilbert
space. The above theorem then states that A is equivalent with a function A and that
•
the algebra generated by A is isomorphic to the algebra generated by this function A .
•
Indeed, after diagonalisation, A is equivalent with the function that maps i, standing for
diagonal entry number i, to its corresponding eigenvalue. Note that the above theorem
also covers simultaneous diagonalisation of several commuting Hermitian operators.
Given a probability space (Ω,Σ,P), we can study the commutative von Neumann algebra
:= L (Ω,Σ,P), acting on the Hilbert space L2(Ω,Σ,P) by pointwise multiplication,
∞
B
equipped with the normal state ρ given by expectation with respect to the measure P.
Thepair( ,ρ)faithfullyencodestheprobabilityspace(Ω,Σ,P)[70]. Indeed,theσ-algebra
B
Σ can be reconstructed (up to equivalence of sets with symmetric difference 0, a point on
which we will not dwell here) as the set of projections in , i.e. the set of characteristic
B
functions of sets in Σ, and the probability measure is given by acting with the state ρ on
this set of projections. We conclude that studying commutative von Neumann algebras
equipped with normal states is the same as studying probability spaces. This motivates
the definition of a non-commutative or quantum probability space as a von Neumann
algebra equipped with a normal state.
From thepointofviewof physicsitisviolation ofBell’s inequalities [21]inforinstancethe
Aspect experiment [6], that motivates the study of non-commutative probability. See for
instance [64]fora clear exposition of thepointthatthis violation can notbeaccounted for
