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#### Vacuum-mediated incoherent processes in coherently prepared media

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Evers, J. (2004). Vacuum-mediated incoherent processes in coherently prepared media. PhD Thesis, Albert-Ludwigs-Universität, Freiburg i. Br.

Cite as: http://hdl.handle.net/11858/00-001M-0000-0011-8AD8-6

##### Abstract

The interaction of matter with light is of seminal importance to many fundamental
and more applied physical processes. The current understanding of the underlying
physics, however, does not only allow to describe or predict the optical properties of
various media. In addition, in the past few decades, dramatic success has been achieved
in preparing, modifying, or controlling the matter-light interaction, in some cases almost
at will. One approach to control this interaction is to appropriately change the
medium itself, such as in waveguides, photonic crystals, or media with a negative index
of refraction. Especially in atomic and molecular physics, however, often a di erent
approach is used. There, the properties of a given medium are altered by external
influences such as electromagnetic fields. In particular the invention of the laser has
led to many fascinating and often counter intuitive observations [1]. Few examples for
this have been termed as lasing without inversion, slowing or stopping of light, electromagnetically
induced transparency or absorption, the suppression or enhancement of
spontaneous emission, quantum beats, the adiabatic passage of atomic population, superfluorescence,
the enhancement of (non-)linear susceptibilities, or the control of the
index of refraction. While the multitude of the predicted and observed features seems
very diverse, it turns out that many of these e ects share common physical mechanisms,
some of which may be summarized as coherence or interference phenomena
(according to Feynman, interference “has in it the heart of quantum mechanics”) [2].
Thus it is not surprising that the laser as a source of coherent light is important for
many of the above schemes.
One of the above e ects, electromagnetically induced transparency (EIT), is a good
example for the coherence or interference mechanisms that will be of particular interest
throughout this work. The simplest setup for EIT consists of an atom with three
relevant energy levels in configuration, i.e. with two lower states which may be
excited by external laser fields to a common upper state [3]. One of the two lower states
is coupled by a near-resonant “driving” laser field to the upper state (with detuning d
between field frequency and transition frequency), which induces coherences between
the atomic states and thus prepares the atom. The transition from the other lower state
to the upper state is taken as a “probe” transition driven by a near-resonant probe laser
field with detuning p. One then studies the dependence of the absorption of the probe
laser field on the probe laser detuning p. This spectrum of course also depends on
the parameters of the preparation laser field. The most striking result is that the atom
becomes completely transparent at a certain frequency of the probe laser field, which
is given by the so-called two-photon resonance condition p = d. This vanishing
absorption is also observed for probe laser frequencies where the atom would absorb
strongly without the additional driving laser field - hence the name electromagnetically
induced transparency. The absorption canceling is accompanied by a steep slope of
the index of refraction for the probe field, which gives rise to many applications. The
usual interpretation of EIT is that the laser fields create a coherence between the two
lower states which then allows for a canceling of the excitation amplitudes to the upper
state under two-photon resonance condition.
1
INTRODUCTION
Somewhat related to the success of coherent interactions, in the recent past, considerable
attention has also been devoted to the study of incoherent processes, such as the
spontaneous emission of light by an atom mediated by the surrounding vacuum field.
One of the reasons for this interest is that the decoherence due to incoherent processes
is one of the major limitations to many schemes of current theoretical and experimental
interest, especially to those relying on coherence e ects. A prominent example for this
is the area of quantum information, computation, or communication, where generally
an almost perfect control of the dynamics of the considered system is required [4].
This is immediately clear if one considers that a qubit, the basic information storage
unit typically made out of two quantum states, requires the two quantum states to be
perfectly stable in order to store the information with certainty. However, there are
many other examples, one of which is high-frequency lasing, which usually requires a
population inversion on an atomic transition with a high transition frequency. This
becomes exceedingly di cult with increasing transition frequency due to spontaneous
emission to lower states. Of course, there are also incoherent processes which are not
mediated by the surrounding vacuum, such as collisions or phonon interactions. The
work in this subject area can be summarized as an e ort to inhibit or circumvent the
disturbing incoherent processes, which traditionally have been considered inevitable.
Ultimately, the goal is thus either to have a convenient control parameter to stop and
start the influence of the incoherent processes at will, or to devise schemes which do
not su er from the incoherent processes.
Incoherent processes, however, are not undesirable a priori. For example, the detection
of atoms often relies on the fluorescence emitted during a spontaneous emission.
Laser cooling of atoms or ions typically requires the momentum kick acting on the
particle during a spontaneous emission. There are also schemes which make explicit
use of the incoherent processes, such as dissipation-assisted quantum gates or schemes
involving a cavity in the bad-cavity limit. A lossy cavity, for example, allows to observe
spontaneous-emission interference between two atomic transitions with orthogonal
dipole moments by pre-selecting the same dominant spontaneous decay mode for
the two transitions [5]. In addition, it has been shown that some of the coherence
or interference e ects have corresponding counterparts involving incoherent processes.
For example, multiple pathways required for quantum interference can also be induced
by incoherent driving fields.
These di erent sides of the incoherent processes are also reflected in the current
work and naturally divide the thesis into three parts (I-III). The three parts are selfcontained
in the sense that the respective historical context and the required common
prerequisites are given in each of the parts separately. These general introductions are
then augmented by more specific discussions at the beginning of each of the chapters
(1-6). The first part deals with spontaneous emission with the aim of controlling or
suppressing the irreversible incoherent evolution due to the emission. In the second
part, we discuss mechanical e ects of the matter-light interaction with the emphasis
on ground state laser cooling of trapped ions. Here, on the one hand, the momentum
transfer caused by the spontaneous decay is crucial in order to cool the trapped
particle, but on the other hand, the system needs to be prepared suitably in order to
avoid the usual cooling limit due to the recoil of uncontrolled spontaneous emission
events. In the last part, we make explicit use of incoherent relaxation, as we discuss
the resonance fluorescence spectrum of laser-driven few-level atoms. Even though it
may seem paradoxical on first sight, we propose the incoherent part of the fluorescence
spectrum as an interesting candidate for high-precision spectroscopy. After this brief
overview, in the following, we give a more detailed summary of the topics discussed in
this thesis:
In the first chapter I.1, the idea is to use quantum interference e ects to inhibit
the spontaneous emission from the upper to the lower state in an atomic two-level
system. For this, multiple transition pathways from the upper to the lower state are
2
INTRODUCTION
required, whose amplitudes cancel each other. In order to induce the various transition
pathways, a strong coherent low-frequency field is applied to the atom, where by “low
frequency” we mean that the field frequency ¯! is smaller than the transition decay
width. As the applied field is strong, the atom may decay both on the direct onephoton
transition, as well as on multi-photon transitions from the upper to the lower
state. The photons emitted spontaneously on the relevant transition pathways have
similar frequencies up to di erences of order ¯!, which is within the transition width.
Thus the pathways cannot be distinguished, and quantum interference is possible.
It is important to note that the scheme does not require near-degenerate transitions
with non-orthogonal dipole moments as many other schemes involving quantum interference.
First, we derive the e ective Hamiltonian for the system including up to
three-photon processes without using the rotating-wave approximation for the lowfrequency
field. This Hamiltonian may also serve as starting point for other studies
involving multiphoton processes in atomic few-level systems. In the second part, this
Hamiltonian is applied to study the decay dynamics of a two-level system subject to
an intense low-frequency field. Finally, we discuss results of this simulation based
on rubidium atoms and show that the spontaneous emission in this system may be
suppressed substantially.
The scheme described in the second chapter I.2 is similar to the one in the first
chapter in that it also discusses interference e ects. The realization of the interference
e ects, however, is di erent: While the interference is induced by a coherent driving
field in the first chapter, here the interfering pathways are induced by incoherent
pump fields [6]. We study the inelastic spectral intensity emitted in the spontaneous
decay of two near-degenerate atomic states to a common ground state, where the
external incoherent pump fields couple the two levels to an upper state. The analysis
focuses on the interplay of the interference induced by the incoherent pump fields with
the interference between the two spontaneous decay channels, and we find that the
incoherent relaxation processes are altered by the external incoherent pump fields.
While in this setup the control possibilities with incoherent fields are not as good as
with comparable setups using coherent fields, the interference induced by incoherent
processes can still have a considerable e ect on the emission spectrum.
In the third chapter I.3, we combine the collective e ects occurring in a sample of
nearby atoms with coherence and interference phenomena usually studied in singleatom
systems. Typical advantages of collections of atoms over single-atom setups
are e.g. a rapid system evolution and almost complete population transfer between
various system states [7]. For single-atom systems, the spontaneous decay usually
prohibits the complete transfer of population to stable states except for the ground
state. Coherent control schemes, on the other hand, feature the availability of sensitive
and convenient control parameters. Thus we analyze a sample of three-level atoms in
V - or - configuration in the Dicke setup, where both transitions are driven by strong
coherent fields. Additionally, the atoms are subject to external control parameters such
as the relative phase of the driving fields, the strength of additional incoherent pump
rates, the frequency spacing between the two nearby atomic states or the temperature
of the surrounding vacuum. We discuss the steady-state population distribution of
the sample and show that the collective quantum dynamics can be controlled with
the help of the above-mentioned external parameters to a great extend. We also show
that the presented scheme features a rapid system evolution as required for many
applications. As possible implementation, we discuss the absorption and fluorescence
emission properties of the atomic sample which may be used to construct e.g. fast
optical switching devices.
The fourth chapter II.4 discusses mechanical e ects of the matter-light interaction,
with the emphasis on laser cooling. Especially the cooling of trapped ions to the ground
state of the trapping potential is a crucial step in the preparation of the medium for
many current experiments [8]. The technique discussed in this chapter relies on the
3
INTRODUCTION
fact that the trapped particle acquires a change in momentum during an interaction
with a photon. This of course also holds for the spontaneous emission, which for cold
particles induces a motion similar to a random walk due to the statistical distribution
of the emission directions, and thus gives rise to a finite cooling limit, the Doppler limit.
In order to circumvent this, we propose a scheme where—in addition to the cooling
laser field—other coupling laser fields are used to design the absorption spectrum
of the atom such that unwanted transitions leading to a heating of the system are
suppressed. The scheme makes use of double electromagnetically induced transparency
(EIT) in order to allow for a complete suppression of the cooling laser field absorption
at certain frequencies. Then, on average and in leading order of the so-called Lamb-
Dicke expansion, the trapped particle is only excited together with a decrease in the
motional quantum number, such that no unwanted spontaneous emissions can occur.
Therefore, the scheme allows to e ciently cool the system to the motional ground
state with almost complete ground state occupation. We provide analytical results
based on a rate equation description of the system, and augment this analysis by
numerical studies of the full cooling dynamics of trapped ions using both quantum
Monte-Carlo simulations and a numerical integration of the system master equation.
As examples, we discuss the cooling of 199Hg+, 171Yb+, and 40Ca+ ions. Finally, we
discuss the extension to multiple-EIT which allows to cool at di erent trap frequencies
simultaneously. This is of interest e.g. for setups with di erent axial and radial trap
frequencies or for the cooling of ion strings.
The final part on high-precision spectroscopy has two chapters. The fifth chapter
III.5 discusses relativistic and radiative corrections to the resonance fluorescence spectrum
of strongly driven few-level atoms, thus combining ideas from quantum optics and
quantum electrodynamics. At lowest order, the Mollow spectrum, i.e. the resonance
fluorescence spectrum of a strongly laser-driven two-level atom, is a classic textbook
example in theoretical quantum optics. The spectrum can easily be understood in
terms of the so-called dressed states, which are the eigenstates of the interaction picture
Hamiltonian of the atom-field system [9]. The usual quantum optical analysis,
however, involves several approximations in order to reveal the relevant physical processes
qualitatively, and does not reach the accuracy obtained in current high-precision
experiments. Quantum electrodynamics (QED), on the other hand, does allow to obtain
the most accurate theoretical predictions obtained so far [10], but is not well-suited
to treat dynamical processes such as the Mollow spectrum. The reason is that QED
relies on a perturbative expansion in the light-matter expansion. In the famous example
of the self-energy shift of a bound electron due to virtual interactions with the
surrounding vacuum field [11, 12], the matter-light coupling is weak, such that already
the leading-order perturbation yields quite accurate results. For the Mollow-spectrum,
however, the driving laser field is strong, such that a summation over many orders of
the perturbation series would be required to obtain satisfying results. This is hopeless,
as the perturbations are complicated already at the lowest order. In order to resolve
this problem, we first use an ansatz from quantum optics by transferring the system
Hamiltonian to the dressed state picture. In doing so, the interaction of the atom with
the driving laser field is accounted for to all orders. Then we apply the usual QED
analysis to the dressed states of the system rather than to the bare states. Among
other, we evaluate corrections to the approximations made in transforming the system
to the dressed state picture, as well as relativistic and radiative corrections up to
relative order (Z )2 and (Z )2, respectively. Here, Z is the nuclear charge number,
is the fine structure constant, and the powers of Z and indicate the order of the
perturbative expansion considered in the analysis. In a numerical analysis, we provide
complete results for the hydrogen 1S−2Pj and 1S−3Pj (j = 1/2, 3/2) transitions, where
the latter case requires a generalization of the analysis to a three level system because
of an additional decay channel via the 2S state. As an application, the outcome of
such experiments would allow for a sensitive test of the validity of the dressed-state
basis as the natural description of the combined atom-laser system.
4
INTRODUCTION
The sixth chapter III.6 deals with more fundamental questions related to the interpretation
of the complex energy shift acquired by a bound electron due to the virtual
interactions with the surrounding vacuum. In leading order of the self-energy (oneloop
corrections), the self-energy has a real part, which contributes to the Lamb shift,
and an imaginary part, which is interpreted as the inverse lifetime of the given atomic
state. This interpretation, however, becomes problematic if one evaluates the nexthigher
order corrections (two-loop order). Some of the two-loop corrections contain
products of two one-loop contributions [13]. In these contributions, the product of
the imaginary parts of the two one-loop corrections becomes real and contributes to
the energy shift rather than to the inverse lifetime. This problem is related to the
fact that in the usual treatment of the self-energy the finite lifetime of the involved
states is neglected initially. We evaluate the problematic contributions of the squared
decay rates in order to gauge the magnitude of the e ect and interpret some of these
contributions as o -resonant corrections to the photon scattering cross section.