Recent development of spectral analysis of the Liouville-von Neumann equation has revealed the fact that irreversibility is a rigorous dynamical property of Poincaré non-integrable systems with an infinite degrees of freedom interacting among each other through resonance coupling. In the present work we discuss this role of resonance in some examples of matter-field coupling systems for both classical and quantum mechanics: the one is a classical motion of a charged particle in electromagnetic waveguide, and the other is the decoherence problem of quantum matter-field interacting systems. In the first part of this dissertation, we study an accelerated motion of a charged classical dipole molecule with frequency ω1 inside the rectangular waveguide. If the particle is in free space, it is well known that its accelerated motion will eventually stop by radiating the field through the resonance interaction. This result is the so-called radiation damping. For the case in the waveguide, there are two possible situations, due to the existence of the cut-off frequency ωc of the waveguide. Under the cut-off frequency electromagnetic wave cannot propagate inside the waveguide. The stability of the dipole depends on the relation between ω1 and ωc. For ω1 < ωc, the dipole cannot resonate with the field. This corresponds to the Poincaré integrable system. For this case the dipole keeps its accelerated motion without emitting the radiating field. Therefore the radiation damping of the dipole molecules is suppressed inside the waveguide under the absence of resonance interaction. The motion of this steady state somewhat resembles a quantum ground state. We show that this steady state is dressed by electromagnetic field. The overlap of the dressing field leads to a force analogous to van der Waals force in quantum mechanics. The critical frequency determined by ω1 = ωc gives a critical size of the waveguide. For heavy molecules, such as HCl, this is of order 10−5m. We show that the size of the dressing field is the same order of the size of the waveguide. Hence we have a macroscopic size of the dressing in the waveguide. For ω1 > ωc, the dipole can resonate with the field, and the system becomes non-integrable in the sense of Poincaré. As a result, the accelerated motion eventually stop by emitting the resonance field. This corresponds to the problem of classical radiation damping. We show that there is non-negligible deviation of exponential decay in a short time scale of the order t ∼ 1/ω1. This corresponds to the quantum Zeno effect, well known in quantum unstable systems. After this period, the dipole decays exponentially in time by emitting the resonance field. We found by choosing ωc very close to ω1, we can increase the decay rate 105 times faster than the case where the dipole is in the free space, at the same time the emitted field travel 10−4 times slower than the speed of light. This is again a consequence of the existence of the cut-off frequency in the waveguide. Indeed, the cut-off frequency leads to a non-linear dispersion relation for the electromagnetic field. To some extent, the electromagnetic field is sticky inside the waveguide. Due to the large decay rate and slower speed of light, the size of the wavepacket emitted by the dipole is significantly small (about 10cm for HCl). This is even smaller than the quantum case in free space, where the wavepacket of the field emitted by the decay of electron in hydrogen atom is about 1m. In the second part of this dissertation, we study a quantum matter-field coupled system. We focus our attention on the problem of quantum decoherence in a system of a particle coupled with a field, the Hamiltonian of which has a similar structure to the problem of classical radiation damping mentioned above. We apply the complex spectral representation of the Liouville-von Neumann operator that gives a rigorous approach to the irreversible processes. We focus our attention on the time evolution of the field, which is commonly neglected in phenomenological approaches to the decoherence problem. We found a signature of decoherence in the field which has a characteristic time dependence proportional to t that comes from the secular effect between the particle and the field through the resonance interaction that breaks time-symmetry.
Recent development of spectral analysis of the Liouville-von Neumann equation has revealed the fact that irreversibility is a rigorous dynamical property of Poincaré non-integrable systems with an infinite degrees of freedom interacting among each other through resonance coupling. In the present work we discuss this role of resonance in some examples of matter-field coupling systems for both classical and quantum mechanics: the one is a classical motion of a charged particle in electromagnetic waveguide, and the other is the decoherence problem of quantum matter-field interacting systems. In the first part of this dissertation, we study an accelerated motion of a charged classical dipole molecule with frequency ω1 inside the rectangular waveguide. If the particle is in free space, it is well known that its accelerated motion will eventually stop by radiating the field through the resonance interaction. This result is the so-called radiation damping. For the case in the waveguide, there are two possible situations, due to the existence of the cut-off frequency ωc of the waveguide. Under the cut-off frequency electromagnetic wave cannot propagate inside the waveguide. The stability of the dipole depends on the relation between ω1 and ωc. For ω1 < ωc, the dipole cannot resonate with the field. This corresponds to the Poincaré integrable system. For this case the dipole keeps its accelerated motion without emitting the radiating field. Therefore the radiation damping of the dipole molecules is suppressed inside the waveguide under the absence of resonance interaction. The motion of this steady state somewhat resembles a quantum ground state. We show that this steady state is dressed by electromagnetic field. The overlap of the dressing field leads to a force analogous to van der Waals force in quantum mechanics. The critical frequency determined by ω1 = ωc gives a critical size of the waveguide. For heavy molecules, such as HCl, this is of order 10−5m. We show that the size of the dressing field is the same order of the size of the waveguide. Hence we have a macroscopic size of the dressing in the waveguide. For ω1 > ωc, the dipole can resonate with the field, and the system becomes non-integrable in the sense of Poincaré. As a result, the accelerated motion eventually stop by emitting the resonance field. This corresponds to the problem of classical radiation damping. We show that there is non-negligible deviation of exponential decay in a short time scale of the order t ∼ 1/ω1. This corresponds to the quantum Zeno effect, well known in quantum unstable systems. After this period, the dipole decays exponentially in time by emitting the resonance field. We found by choosing ωc very close to ω1, we can increase the decay rate 105 times faster than the case where the dipole is in the free space, at the same time the emitted field travel 10−4 times slower than the speed of light. This is again a consequence of the existence of the cut-off frequency in the waveguide. Indeed, the cut-off frequency leads to a non-linear dispersion relation for the electromagnetic field. To some extent, the electromagnetic field is sticky inside the waveguide. Due to the large decay rate and slower speed of light, the size of the wavepacket emitted by the dipole is significantly small (about 10cm for HCl). This is even smaller than the quantum case in free space, where the wavepacket of the field emitted by the decay of electron in hydrogen atom is about 1m. In the second part of this dissertation, we study a quantum matter-field coupled system. We focus our attention on the problem of quantum decoherence in a system of a particle coupled with a field, the Hamiltonian of which has a similar structure to the problem of classical radiation damping mentioned above. We apply the complex spectral representation of the Liouville-von Neumann operator that gives a rigorous approach to the irreversible processes. We focus our attention on the time evolution of the field, which is commonly neglected in phenomenological approaches to the decoherence problem. We found a signature of decoherence in the field which has a characteristic time dependence proportional to t that comes from the secular effect between the particle and the field through the resonance interaction that breaks time-symmetry.
The interaction between matter and fields is a classical problem. Still there are difficulties. Time symmetry is broken. So the radiation damping belongs to a class of phenomena, which includes transport properties. We propose a radically different approach based on the extension of the dynamics of integrable systems. We consider a simple model, a harmonic oscillator interacting with a field. For integrable systems, it is well known that there exists a unitary transformation U. However, in the radiation damping we have resonances between the action of the particle and the actions of the field. This makes the system an example of Poincar\'e nonintegrable systems. We extend the unitary operator to a new star-unitary operator $\ensuremath{\Lambda}.$ This changes the dynamical description of radiation damping. Once we know the Hamiltonian, we can of course write the Hamilton equations. But we have the possibility to go to new descriptions. The invertible $\ensuremath{\Lambda}$ transformation gives many new aspects, which are hidden in the initial description. For example, we show that there are fluctuations, and that there exists an irreducible probability description. The transformation to $\ensuremath{\Lambda}$ representation corresponds to a transformation to Markovian probability equations. We can always come back to the initial representation by the inverse transformation. We have verified this remarkable prediction by detailed numerical calculations. We need the $\ensuremath{\Lambda}$ transformation to obtain the definition of a dressed unstable mode, which has a well-defined lifetime. In the initial representation there are various time scales and there is no strictly exponential lifetime. The situation is the same as the one we have studied in the quantum case in recent papers.
Electromagnetic interactions of a single charged particle in the presence of damping are investigated theoretically in this work. Classically, by considering the explicit example of a particle in a constant magnetic field, it is shown that the minimal coupling prescription is inadequate for the infinitely large number of Hamiltonians from which the equations of motion in the presence of damping are derivable by Hamilton's equations. The use of minimal coupling procedure to switch on electromagnetic interactions in phenomenological quantum equations of damped motion is shown to lead to incorrect equations in the classical limit. It is necessary to consider the microscopic models of damping to allow for electromagnetic interactions in these systems. Two microscopic models for damping involving a linear coupling with a heat bath of oscillators are considered in detail and it is shown that correct Heisenberg equations of motion follow upon elimination of bath degrees of motion so that damping and Lorentz force do not affect each other. However, the field couples to the mechanical momentum as mv-(e/c)A rather than to the canonical momentum. The linearly coupled quantal case is analogous to the classical situation. The Schr\"odinger-Langevin equation contains an extra term Fr A\ensuremath{\cdot}dr'; however, the gauge invariance character and the Aharonov-Bohm effect are unaffected. Two nonlinear models of damping are studied by considering the exchange of bosons in the different states of the central particle. In this situation the Lorentz force is not recovered in the classical limit.
Over the past ten years, our group has investigated the effects of confinement on atoms inside metallic micron–sized cavities in order to elucidate some basic phenomena in the field of cavity quantum electrodynamics (QED). The first of these was the inhibition of spontaneous emission from an atom inside a cavity. This was followed by a laser spectroscopic measurement of the van der Waals interaction between a single Rydberg atom and a gold cavity, which showed that a simple electrostatic model of the atom–cavity interaction is correct when the cavity is small enough. More recently, the retarded Casimir–Polder force was measured between a ground state sodium atom and a large cavity, demonstrating that the van der Waals potential fails at long enough range and that the vacuum fluctuations of the field then have an important role in the interaction of the atom with the cavity. Our group is now pushing forward these investigations to study cavities whose walls have losses and dispersion, where the theory of cavity QED is significantly more complicated. With real surfaces, we have to deal with the complex dielectric response ε(ω) of the material, which exhibits frequency–dependent absorption and dispersion. One particularly interesting case is when a downward transition in the atom is resonant with an excitation of the cavity walls. This opens a new branch for the atomic decay: as an alternative to creating a photon within the space surrounded by the cavity walls the atomic decay can now create an electromagnetic excitation of the walls themselves. Another novel feature of our experiments is that the Bohr frequencies of the atom are close to the kT/h, where T is room temperature. We therefore expect to be able to measure effects associated with QED at finite temperature; in other words, to study how the blackbody radiation affects our experiments. By conducting experiments with real surfaces, we hope to elucidate and perhaps simplify the theoretical models used to describe these systems.
The quantum description of the electromagnetic (EM) field, even in the absence of sources, reveals that the vacuum exhibits fluctuations in the energy and of the components of the field, known as quantum vacuum fluctuations. The coupling of these quantum fluctuations of the vacuum to electric charges leads to experimentally observable phenomena, such as the spontaneous emission of an excited atom, the Lamb effect, the Casimir effect, and the van der Waals interactions, which have been historically fundamental for the confirmation of the existence of these fluctuations. The optical response of simple atoms and the EM interaction between two atoms constitute the simplest realistic phenomena for studying these effects. This thesis focuses on the fundamental study of the properties of interatomic interactions, addressing aspects such as their dependence on the interatomic distance, their time dependence, and the emergence of nonreciprocal interatomic forces due to parity breaking. In particular, the detailed analysis of the electric dipole forces in a system of two atoms, one of which is initially excited, is carried out, along with the study of the implication of the resonant interactions in the appearance of collective effects in absorption and emission processes of many-atom systems. Due to the inherently time-dependent nature of the resonant forces between excited atoms, a time-dependent Hamiltonian approach is employed. The atomic interaction processes are represented with time-ordered diagrams. It is obtained that the atoms are subjected to resonant conservative forces, directed along the interatomic axis, which exert a net force on the system. This result implies an apparent violation of the action-reaction principle due to the lack of conservation of the kinetic momentum of the atoms. However, this net force on the atomic system is compensated by the time derivative of the transversal momentum of the EM field, associated with the resonant photons that mediate the interaction, ensuring the conservation of the total momentum of the system. In addition to these conservative forces, there are nonconservative forces which are a consequence of the time derivative of the so-called longitudinal momentum of the electromagnetic field and contain components perpendicular to the interatomic axis. We analyze the particular case of the interaction between identical atoms. In this scenario, the net force on the system becomes time-dependent. Furthermore, states with degenerate energy arise and evolve coherently through the periodic transfer of resonant photons between the atoms. This implies that in these degenerate states, there are contributions from interatomic interactions that cannot be expressed as an energy shift in the atomic wave functions. Consequently, these interactions give rise to collective phenomena observable in photon absorption and emission processes. At the same time, the strength of these forces is markedly superior to that calculated for different atoms, which makes this scenario a more favorable set-up for the experimental observation of the nonreciprocal forces and the associated coherent phenomena. Concerning collective effects and their application in the design of multi-atomic qubits, these phenomena have significant implications for the scalability and connectivity of such systems. To describe these collective properties, we have formulated a Hamiltonian that incorporates both the interaction processes, i.e., those that give rise to dynamic coherence and incoherence effects, and the stationary interaction processes that renormalize the energy of the atomic states. It is found that the photon absorption probability is determined by the ratio between the interatomic interaction and the intensity of the laser that excites the system. On the other hand, the spontaneous emission of photons is proportional to the number of decay channels, which decrease when the interatomic interaction between excited states breaks the degeneracy in the subspaces associated with the same number of excitations.
It is in accordance with such experiments as single photon self-interference that a photon, conveying one radiation energy quantum " h × frequency", is spatially extensive and stretches an electromagnetic wave train. A wave train, hence an energy quantum, can only be emitted (or absorbed) by its source (or absorber) gradually. In both two processes the wave and "particle" attributes of the radiation field are simultaneously prominent, where an overall satisfactory theory has been lacking; for the latter process no known theoretical description currently exists. This paper presents a first principles treatment, in a unified framework of the classical and quantum mechanics, of the latter process, the emission (similarly absorption) of a single radiation quantum based on the dynamics of the radiation-emitting source, a charged oscillator, which is itself extensive across the potential well in which it oscillates. During the emission of one single radiation quantum, the extensive charged oscillator undergoes a continuous radiation damping and is non-stationary. This process is in this work treated using a quasi stationary approach, whereby the classical equation of motion, which directly facilitates the correspondence principle for a particle oscillator, and the quantum wave equation are established for each sufficiently brief time interval. As an inevitable consequence of the division of the total time for emitting one single quantum, a fractional Planck constant h is introduced. The solutions to the two simultaneous equations yield for the charged oscillator a continuously exponentially decaying Hamiltonian that is at the same time quantised with respect to the fractional- h at any instant of time; and the radiation wave field emitted over time stretches a wave train of finite length. The total system of the source and radiation field maintains at any time (integer n times) one whole energy quantum, ( n × ) h × frequency, in complete accordance with the standard notion of quantum mechanics and experiment.