Topics in quantum measurement and quantum noise

118 Pages·1998·0.873 MB·English

by  Kurt Jacobs

8
TOPICS IN
9
9 QUANTUM MEASUREMENT AND QUANTUM NOISE
1

t
c
O

6

2
v

5
1
0
By
0
1
8
9
h/ Kurt Jacobs
p
-
t
n
a
u
q
:
v
i
X A thesis submitted to the University of London
for the degree of Doctor of Philosophy
r
a The Blackett Laboratory
Imperial College
June 1998
AnIndianandawhitemanfelltotalking. Thewhite
man drew a circle in the sand and a larger circle to
surround it.
“See”, he said, pointing to the small circle “that is
what the Indian knows, and that”, pointing to the
larger circle, “is what the white man knows.”
The Indian was silent for many minutes, and then
slowlyhepointedwithhisarmtotheeastandturned
and waved to the west.
“And that, white man”, he said, “is what neither of
us knows”.
ii
Acknowledgments
I wouldlike to thank Prof. P. L. Knightfor his supervision,patience and encouragementduring
the entire length of my PhD research, and for providing helpful comments and suggestions on my
thesis drafts. Working in the quantum optics group at Imperial College has been both challenging
andenjoyable. I wouldliketo thankalsothe manycolleagueswithwhomIhaveworkedthroughout
my time here, in particular Vlatko Vedral, Mike Rippin, Sougato Bose and MartinPlenio. Working
with them has been a pleasure and a privilege. I would also like to thank Howard Wiseman, Gerd
Breitenbach, Ilkka Tittonen, Sze Tan and Andrew Doherty. Their suggestions and advice, albeit
mainly at a distance, was nevertheless valuable and greatly appreciated.
Therearemanyfriendswhohavecontributedtomytimehereinwaysincalculable. Inparticular
I would like to thank Peter,Nat, Nick, Brody and Dave for being there in spirit, and Mike, Brooke,
Elaine,RysandJaimieforbeingthereinperson. Veryspecialthanksgoto myparents,Sandra,Ian
and Lorraine, for always being there.
Iwouldalsolike tothank the AssociationofCommonwealthUniversitiesforthe Commonwealth
Scholarship which allowed me to study in England, and the New Zealand Vice-Chancellors’ Com-
mitteefortheEdward&IsabelKidsonScholarshipandtheL.B.WoodTravellingScholarship,both
of which provided very useful additional support for travel.
iii
List of Publications
For this thesis I have selected from among my research those publications primarily concerned
with quantum measurement theory. In the following chronologicallist, those on which this thesis is
based are asterisked.
1. K.Jacobs,P.Tombesi,M.J.CollettandD.F.Walls,‘AQNDMeasurementofPhotonNumber
using Radiation Pressure’, Phys. Rev. A 49, 1961, (1994).
2. G. J. Milburn, K. Jacobs and D. F. Walls,‘Quantum Limited Measurements with the Atomic
Force Microscope’, Phys. Rev. A 50, 5256 (1994).
3. K. Jacobs, M. J. Collet, H. M. Wiseman, S. M. Tan and D. F. Walls,‘Force Measurement via
Dark State Cooling’, Phys. Rev. A 54, 2260 (1996).
4. *K.JacobsandP.L.Knight,‘ConditionalProbabilitiesforaSinglePhotonataBeamSplitter’,
Phys Rev. A 54, R3738 (1996).
5. K.Jacobs,‘A Model for the Production of Regular FluorescentLight fromCoherently Driven
Atoms’, J. of Mod. Optics 44, 1475 (1997).
6. *K.Jacobs,P.L. KnightandV.Vedral, ‘Determining the State ofa Single CavityMode from
Photon Statistics’, J. of Mod. Optics, 44, 2427 (1997).
7. S. Bose, K. Jacobs and P. L. Knight, ‘Preparation of non-classical states in a cavity with a
moving mirror’, Phys Rev. A 56, 4175 (1997).
8. V. Vedral, M. B. Plenio,K. Jacobsand P. L. Knight,‘Statistical Inference, Destinguishability
of Quantum States, and Quantum Entanglement’, Phys Rev. A 56, 4452 (1997).
9. *K. Jacobs and P. L. Knight, ‘Linear Quantum Trajectories: Applications to Continuous
Projection Measurements’, Phys. Rev. A 57, 2301 (1998).
10. S. Bose, K. Jacobs and P. L. Knight, ‘A Scheme to Probe the Decoherence of a Macroscopic
Object’, (in submission).
11. *I. Tittonen, K. Jacobs and H. M. Wiseman, ‘Quantum noise in the position measurement of
a cavity mirror undergoing Brownian motion’, (in submission).
iv
Abstract
In this thesis we consider primarily the dynamics of quantum systems subjected to continuous
observation. Inthe Schr¨odingerpicture the evolutionof a continuouslymonitoredquantumsystem,
referredtoasa‘quantumtrajectory’,maybedescribedbyastochasticequationforthestatevector.
We present a method of deriving explicit evolution operators for linear quantum trajectories, and
apply this to a number of physical examples of varying mathematical complexity.
In the Heisenberg picture evolution resulting from continuous observation may be described by
quantumLangevinequations. We usethis methodto calculatethe noisespectrumthatresultsfrom
acontinuousobservationofthepositionofamovingmirror,andexaminethepossibilityofdetecting
the noise resulting from the quantum back-action of the measurement.
In addition to the work on continuous measurement theory, we also consider the problem of
reconstructingthe state of a quantum systemfrom a set of measurements. We present a scheme for
determining the state of a single cavity mode from the photon statistics measured both before and
after an interaction with one or two two-level atoms.
v
vi
Contents
Acknowledgments iii
List of Publications iv
Abstract v
List of Figures ix
1 Introduction 1
1.1 Overview of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Quantum Mechanics and Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Breaking the Rules of Probability Theory . . . . . . . . . . . . . . . . . . . . 2
1.2.2 Entanglement and Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.3 Entanglement and Generalised Measurements . . . . . . . . . . . . . . . . . . 7
1.2.4 Entanglement and Non-Locality . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Open Quantum Systems: Master Equations . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Open Quantum Systems: Quantum Trajectories. . . . . . . . . . . . . . . . . . . . . 12
1.4.1 Direct Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4.2 Homodyne Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4.3 Other Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4.4 A Continuous Measurement of Position . . . . . . . . . . . . . . . . . . . . . 17
2 Quantum Trajectories and Quantum noise 21
2.1 Stochastic Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.1 The Wiener Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.2 The PoissonProcess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Quantum Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.1 Quantum Trajectories and Generalised Measurements . . . . . . . . . . . . . 26
2.2.2 Linear and Non-linear Formulations . . . . . . . . . . . . . . . . . . . . . . . 28
2.3 Quantum Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3 Evolution Operators for Linear Quantum Trajectories 35
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 The Wiener Process: General Method . . . . . . . . . . . . . . . . . . . . . . . . . . 36
vii
3.2.1 A QND Measurement of Photon-Number . . . . . . . . . . . . . . . . . . . . 37
3.2.2 A Measurement of Momentum in a Linear Potential . . . . . . . . . . . . . . 39
3.2.3 A Quadrature Measurement with a General Quadratic Hamiltonian . . . . . 42
3.2.4 An Open Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3 Extension to the PoissonProcess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.3.1 A Damped Cavity with Coherent Driving . . . . . . . . . . . . . . . . . . . . 47
3.3.2 A Damped Cavity with a Kerr Non-linearity . . . . . . . . . . . . . . . . . . 50
3.4 Possibilities for Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4 Quantum Noise in a Cavity with a Moving Mirror 53
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 The System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3 Phase Modulation Detection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.4 Solving the System Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.5 The Power Spectral Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.5.1 Comparison with Standard Brownian Motion . . . . . . . . . . . . . . . . . . 64
4.5.2 Comparison with Homodyne Detection. . . . . . . . . . . . . . . . . . . . . . 64
4.5.3 The Error in a Measurement of Position . . . . . . . . . . . . . . . . . . . . . 64
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5 Determining the State of a Single Cavity Mode 69
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.2 Preliminary Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.3 Measuring quantum states with a one-atom interaction. . . . . . . . . . . . . . . . . 74
5.4 Measuring quantum states with a two-atom interaction. . . . . . . . . . . . . . . . . 76
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6 Conclusion 79
A An Introduction to the Input-Output Relations 81
A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
A.2 The Lagrangianand Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
A.2.1 The Standard Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
A.2.2 The Input-Output Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . 84
A.2.3 The Hamiltonian and Quantisation . . . . . . . . . . . . . . . . . . . . . . . . 87
A.3 Input and Output Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
A.4 Input-Output Relations for a Damped Cavity . . . . . . . . . . . . . . . . . . . . . . 92
B Exponentials of P and Q 95
C Atomic Detection Probabilities 99
Bibliography 102
viii
List of Figures
3.1 Conditional uncertainty for a QND measurement of photon-number. . . . . . . . . . 39
4.1 A driven optical cavity with a moving mirror . . . . . . . . . . . . . . . . . . . . . . 56
4.2 Error in a position measurement of a moving cavity mirror . . . . . . . . . . . . . . 66
A.1 A small system interacting with an external field which extends from x = 0 out to
infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
ix
x