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Vacuum-mediated incoherent processes in coherently prepared media

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Evers,  Jörg
Division Prof. Dr. Christoph H. Keitel, MPI for Nuclear Physics, Max Planck Society;

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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.