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Controlling the motion of an atom in an optical cavity [Elektronische Ressource] / Thomas Fischer

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Technische Universit¨ at Munc¨ henMax-Planck-Instititut fur¨ QuantenoptikControlling the motion of an atomin an optical cavityThomas FischerVollst¨ andiger Abdruck der von der Fakult¨ at fur¨ Physikder Technischen Universit¨ at Munc¨ henzur Erlangung des akademischen Grades einesDoktors der Naturwissenschaften (Dr. rer. nat.)genehmigten Dissertation.Vorsitzender : Univ.-Prof. Dr. M. KleberPrufer¨ der Dissertation : 1. Hon.-Prof. Dr. G. Rempe2. Univ.-Prof. Dr. Dr. h. c. A. LaubereauDie Dissertation wurde am 27. 6. 2002bei der Technischen Universit¨ at Munc¨ hen eingereichtund durch die Fakult¨ at fur¨ Physik am 18. 12. 2002 angenommen.AbstractAn experiment is described where slow laser-cooled atoms are injected into a high-finesseoptical cavity. Single atoms are trapped in the cavity light field and observed by measuringthe light intensity transmitted through the cavity. Feedback-based methods are realizedto extend the time the atom stays in the cavity. The influence of atoms on the cavitytransmission and the light force on atoms in a cavity are also analyzed theoretically. Thisallows to study the motion of an atom in the cavity and to interpret the experimentaltransmission signals. The current limitations for the storage time are identified and astrategy to overcome them is suggested. Apart from that, a method is proposed whichallows a two-dimensional position measurement of atoms in the cavity.

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Published 01 January 2002
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Technische Universit¨ at Munc¨ hen
Max-Planck-Instititut fur¨ Quantenoptik
Controlling the motion of an atom
in an optical cavity
Thomas Fischer
Vollst¨ andiger Abdruck der von der Fakult¨ at fur¨ Physik
der Technischen Universit¨ at Munc¨ hen
zur Erlangung des akademischen Grades eines
Doktors der Naturwissenschaften (Dr. rer. nat.)
genehmigten Dissertation.
Vorsitzender : Univ.-Prof. Dr. M. Kleber
Prufer¨ der Dissertation : 1. Hon.-Prof. Dr. G. Rempe
2. Univ.-Prof. Dr. Dr. h. c. A. Laubereau
Die Dissertation wurde am 27. 6. 2002
bei der Technischen Universit¨ at Munc¨ hen eingereicht
und durch die Fakult¨ at fur¨ Physik am 18. 12. 2002 angenommen.Abstract
An experiment is described where slow laser-cooled atoms are injected into a high-finesse
optical cavity. Single atoms are trapped in the cavity light field and observed by measuring
the light intensity transmitted through the cavity. Feedback-based methods are realized
to extend the time the atom stays in the cavity. The influence of atoms on the cavity
transmission and the light force on atoms in a cavity are also analyzed theoretically. This
allows to study the motion of an atom in the cavity and to interpret the experimental
transmission signals. The current limitations for the storage time are identified and a
strategy to overcome them is suggested. Apart from that, a method is proposed which
allows a two-dimensional position measurement of atoms in the cavity.
Zusammenfassung
Es wird ub¨ er ein Experiment berichtet, in dem langsame laser-gekuhlte¨ Atome in einen
Resonator hoher Finesse eingebracht werden. In dem Lichtfeld des Resonators werden
einzelne Atome gefangen und beobachtet, indem die durch den Resonator transmittierte
Lichtleistung gemessen wird. Um die Zeit, die ein Atom im Resonator verbleibt, zu
verla¨ngern, werden Verfahren, die auf Ruc¨ kkopplung basieren, realisiert. Der Einfluß von
Atomen auf die Transmission des Resonators und die Lichtkraft, die auf Atome in einem
Resonator wirkt, werden auch theoretisch analysiert. Dies erlaubt es, die Bewegung eines
Atoms im Resonator zu untersuchen und die experimentellen Transmissionssignale zu
interpretieren. Die Faktoren, die die Speicherzeit zur Zeit begrenzen, werden identifiziert
und eine Strategie zu derer Beseitigung wird vorgeschlagen. Außerdem wird eine Methode
vorgeschlagen, die eine zweidimensionale Positionsmessung von Atomen in dem Resonator
erlaubt.
iiiivContents
1 Introduction 1
2 Classical description 7
2.1 Qualitativediscussion.............................. 7
2.2 Clasicalcalculation 10
2.3 Experimentaltransitsignalsofsingleatoms................. 18
3 Quantum description 21
3.1 Model...................................... 2
3.2 Propertiesofthelow-saturationmodel. 25
3.3 Connectiontomeasurablequantities .......... 28
3.4 Steady-stateexpectationvalues................ 30
3.4.1 Atomsinteractingwithasinglemode................. 30
3.4.2 Singleatominteractingwithdegeneratemodes.... 35
3.5 Themomentumdiffusionconstant....................... 37
3.5.1 Singlemode............ 42
3.5.2 Interpretation........ 43
3.6 Thevelocity-dependentforce.................. 45
3.6.1 Singlemode.................... 46
3.7 Far-detunedlimitforasingle-modecavity.......... 48
3.8 Numericalmethodsforlargerpumppower........ 49
4 Measurement of the atomic position 53
v4.1 Single-modecavity............................... 53
4.2 Cavitywithdegeneratemodes.... 56
4.2.1 Numberofrequiredmodes.. 56
4.2.2 Example:Transversalmodesoforder10............... 57
4.2.3 Spatialandtemporalresolution.......... 63
5 Experimental set-up 67
5.1 Theatomicfountain.............................. 67
5.2 Thelasersystem........... 70
5.2.1 Experimentalrequirements.. 70
5.2.2 Lasersetup............................... 71
5.3 Thehigh-finessecavityandthedetectionscheme .... 74
5.3.1 Experimentalrequirements.. 74
5.3.2 Cavityanddetectionsetup....................... 76
6 Control of the atomic motion 83
6.1 Descriptionofthemotionoftheatom..................... 84
6.1.1 Differentcontributionstothelightforce..... 84
6.1.2 Quantitiesdescribingthemotionoftheatom... 85
6.1.3 Choiceofparameters.......................... 89
6.2 Trappinganatom........... 96
6.2.1 Experimentalmethod..... 96
6.2.2 Results....................... 98
6.3 Towardsalongerstoragetime-Fedback.........102
6.3.1 Experimentalmethod.....102
6.3.2 Evaluationandresults.........................105
6.4 Perspectives..............17
7 Conclusion and outlook 121
A Algorithm to determine the exit time of an atom 123
viBibliography 127
Publications 133
Danksagung 135
viiviiiChapter 1
Introduction
In the field of quantum electrodynamics (QED), the quantum properties of light and its
quantized interaction with matter are investigated. This theory is widely used, and has
been thoroughly tested. Some of its early successes are the explanation of spontaneous
emission(Dir27), the calculation of the Lamb shift in atomic hydrogen(Bet47) and of the
magnetic moment of the electron(Rev48), and the prediction of inhibition and enhance-
ment of spontaneous emission of an atom(Pur46). While the first three examples involve
the radiation field of free space, the last effect is due to a change of the mode structure of
the electromagnetic field by imposing new boundary conditions. This can, for example,
be achieved by placing mirrors near to the atom. The interaction of the changed elec-
tromagnetic modes with the atom is, in general, different from the interaction with the
modes in free space. This is, in a wide sense, the subject of a subdomain of QED, the
so-called cavity QED.
Over the years, the manipulation of the interaction between an atom and the elec-
tromagnetic field has been employed in many experiments. As examples may serve the
+micromaser(MWM85; BNB 92), the single-atom laser(ACDF94), and the generation of
+nonclassical states of the light field by its interaction with atoms in a cavity(RTB 91).
All these cavity QED experiments rely on a strong interaction between atoms and modes
of the light field. Strong interaction means that the interaction between an atom and one
mode of the electromagnetic field is stronger than the interaction between the atom or
mode and all the other modes of the radiation field which are responsible for the irre-
versible decay of excitation in the atom or mode. This can be understood as follows: The
interaction between an atom and a mode of the electromagnetic field is strongest when
a transition frequency of the atom is close to the frequency of the mode. In free space,
there is a continuum of modes, and there are many modes near-resonant to the transition
frequency of the atom. The interaction between atom and a mode is proportional to the
amplitude of the mode at the position of the atom. As the modes in free space are not
localized, this amplitude is small. Thus, the atom interacts with many modes, and the
interaction with a single mode is small as compared to the interaction with all the other
12
modes.
In a cavity, the situation is different: The frequencies of the modes supported by the
cavity are discrete. In a suitable setup, only a single cavity mode has a frequency which is
near-resonant to an atomic transition. The cavity modes are localized, leading to a large
amplitude of the mode inside the cavity. In some experiments, especially in the optical
domain, the cavity does not cover the full solid angle, and the atom is still coupled to
the free-space mode. By a very strong localization of the cavity mode, i.e. by using a
very small cavity, the interaction between the atom and the cavity mode can be so strong
that it dominates the interaction of the atom with the other modes of the electromagnetic
field. Because of this dominance, it is very likely that an excited atom emits a photon
into the cavity mode and not into the other modes of the electromagnetic field.
The cavity mode itself is not totally isolated from the environment. In an experiment,
the cavity mirrors have losses. Thus, light cannot be stored in the cavity mode forever,
but will eventually leak out. This can be described as an interaction between the cavity
mode and the modes of the environment (Car93, section 1.3). If the mirror losses are
small,theinteractionbetweenthecavitymodeandtheenvironmentcanbemadesmaller
than the interaction between cavity mode and atom. Then, a photon which is stored in
the mode is absorbed by the atom with a high probability instead of leaking out at the
mirrors.
If the interaction between atom and cavity field is stronger than both the interaction
between atom and environment and the interaction between mode and environment, the
atom-cavity system is said to be strongly coupled. The interaction between mode and
environment can be quantified by the decay rate of the electromagnetic field in the cavity,
κ. The decay rate of the atomic dipole moment, γ, is a measure for the interaction
strength between atom and environment. The atom-mode coupling rate, g,quantifiesthe
interaction between atom and mode, and is given by the interaction energy between a
photon in the mode and an atom divided by h¯. As the amplitude of the cavity mode
changes with position, g depends also on the position of the atom. For strong coupling,
g>κ,γ. In this case, atom and cavity mode cannot be regarded as separate systems any
more, but should be considered one new entity, the atom-cavity system.
In a strongly coupled system, a photon is exchanged many times coherently between
atom and mode before it is lost to the environment. This allows, for example, efficient
transfer of a photon from an atom to a mode and vice versa. This is interesting for quan-
tum information processing (BEZ00), where one would like to have an interface between
a photon to transport information coherently and an atom to store the information. An-
other consequence of strong coupling is that not only a photon, but also quantum noise
can be transferred between the atom and the mode, see section3.5 of this work. This
influences, e.g., the motion of the atom.
Experimentally, a strongly coupled system has been realized in two regimes: the mi-
crowave regime and the optical regime. The first cavity QED experiments have been
performed using highly excited (Rydberg) atoms in an atomic beam interacting with
2