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CRC Press eBooks

Microcavity Lasers and QED

Gunnar G. Bjork

Published Dec 23, 2020• 0 citations
Publisher

Abstract

This chapter gives an overview of the field as it stands at the time of writing in mid 1995. It shows how the picture radically changes when there is more than one atom in the cavity. The simplest light-atom interaction one can think of is the coupling between a single atom and a single mode of the electromagnetic field. This system provides considerable insight into the fundamental physics of the light-atom coupling but the reader is warned that the more general many mode-many atom interactions are much richer in their dynamics. An atom in free space couples not only to one, but to a continuum of modes. The mathematics of the Weisskopf-Wigner theory of spontaneous emission tends to obscure the relatively simple physics of the emission process. An alternative, shorter way of calculating the single atom spontaneous emission rate is to use Fermi’s Golden rule, which is a prescription for calculating general transition rates to first order in perturbation theory.

DOI
10.1201/9781003072836-4

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A single two-level emitter coupled to a single mode of an electromagnetic resonator constitutes the most fundamental instance of light--matter interaction. Realizations of this type of system are investigated in the field of cavity quantum electrodynamics (cavity QED). Particularly important are schemes in which the coupling is strong enough to exceed the spontaneous emission rate as well as cavity damping. In this case, the coherent exchange of excitation between source and photon is the dominating process, providing the unique possibility to deterministically control the quantum evolution of the system. Great progress has recently been accomplished, technologically as well as conceptually, and the focus section in this issue of Journal of Optics B: Quantum and Semiclassical Optics features five articles highlighting important recent developments in the field. 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The most notable example is quantum information processing. This requires the coherent manipulation of single-quantum systems, which is ideally provided by cavity QED interaction in the strong-coupling limit. The development of practical devices will be among the principal future tasks for research in the field. This will include physical systems, which have not been covered in this issue, such as semiconductors, quantum dots as emitters, or photonic crystals as resonator structures. Cavity QED is therefore guaranteed to remain a vigorous field in the years to come.

Journal of Optics B Quantum and Semiclassical Optics•Feb 1, 2004•0 citations
This chapter gives an overview of the field as it stands at the time of writing in mid 1995. It shows how the picture radically changes when there is more than one atom in the cavity. The simplest light-atom interaction one can think of is the coupling between a single atom and a single mode of the electromagnetic field. This system provides considerable insight into the fundamental physics of the light-atom coupling but the reader is warned that the more general many mode-many atom interactions are much richer in their dynamics. An atom in free space couples not only to one, but to a continuum of modes. The mathematics of the Weisskopf-Wigner theory of spontaneous emission tends to obscure the relatively simple physics of the emission process. An alternative, shorter way of calculating the single atom spontaneous emission rate is to use Fermi’s Golden rule, which is a prescription for calculating general transition rates to first order in perturbation theory.
In this chapter, the authors develop a consistent theory of spontaneous emission in an optical micro cavity from a broadened atom. They derive the atomic broadening from first principles by assuming the presence of reservoirs coupled to the atoms. The simplest model of the atom is the so-called two-level atom, which has only two energy levels, the excited and the ground level. However, realistic atoms are surrounded by media that affect the atoms, giving them homogeneous or inhomogeneous broadening. The important parameter of coupling ratio of spontaneous emission to a desired quasi mode cannot be exactly treated by the present mathematical model of the cavity, although it is easily seen qualitatively that the ratio will decrease with increasing atomic broadening. The authors assume that each atom emits independently and ignored the possible interaction among atoms during spontaneous emission. Solving the master equation for the density matrix elements, the readers have derived formulas for spontaneous emission rate and emission spectrum.
Abstract The quantum mechanical description of the interaction of light with atoms treats absorption, gain, and recombination as upward and downward induced transitions of electrons between two states by perturbation due to the optical electric field. This provides the basis for a quantitative theory of diode lasers. The chapter begins with Einstein’s description of absorption, stimulated emission, and spontaneous emission, setting the scene for calculation of the transition probability by solving the time-dependent Schrödinger equation for a coherent superposition of the wavefunctions of two states. The resulting absorption spectrum has a Lorentzian lineshape due to dephasing of these wavefunctions. Expressions for the optical absorption cross section and Einstein coefficients are derived in terms of the dipole matrix element, and the dipole is illustrated using a hypothetical one-dimensional atom. Fermi’s Golden Rule for transitions to a continuum of states (as in quantum wells) is introduced in terms of the momentum matrix element.
It is well known that spontaneous emission is not an intrinsic atomic property, but rather results from the coupling of the atom to the vacuum modes of the electromagnetic field. As such, it can be modified by tailoring the electromagnetic environment into which the atom can radiate. This was already realized by Purcell, who noted that the spontaneous emission rate can be enhanced if the atom placed inside a cavity is resonant with one of the cavity is resonant with one of the cavity modes, and by Kleppner, who discussed the opposite case of inhibited spontaneous emission. It has also been recognized that spontaneous emission need not be an irreversible process. Indeed, a system consisting of a single atom coupled to a single mode of the electromagnetic field undergoes a periodic exchange of excitation between the atom and the field. This periodic exchange remains dominant as long as the strength of the coupling between the atom and a cavity mode is itself dominant. 23 refs., 6 figs.
In Chapter 3 the interaction Hamiltonian which determines the way light interacts with matter is derived. Simple perturbative analysis is applied to see if basic dynamics of atoms can be explained. The partial successes of perturbation theory are compared with predictions of an “exact” method of calculating the occupation probabilities of various atomic states. Even the “exact” method is shown to fail, establishing the need for improved approaches that yield correct results in later chapters. Some elementary results for the transition rates, including Fermi's golden rule are covered. Then the density matrix is introduced as a tool for describing not only the populations of atomic energy levels but coherence that can be created and lost during the dynamic evolution of atoms in time. A vector model based on the Bloch vector is presented as a useful way of picturing coherent atom–field interactions in an optical “spin” space, particularly multiple‐pulse interactions. Mechanisms are described that cause line‐broadening in optical spectroscopy, such as the Doppler effect. The chapter closes by showing that it is usually possible to reduce the complexity of real atoms on an experimental basis to facilitate direct comparisons with theoretical calculations based on simple models with only two or three energy levels.
The spectrum of spontaneously emitted photons from an atom interacting with a single cavity mode depends on the relative sizes of the mode coupling constant and the cavity loss rate. If the coupling constant is not too large, perturbation theory can he used to show that the atomic linewidth is either increased (enhanced spontaneous emission) or decreased (inhibited spontaneous emission) by the presence of the cavity. When the coupling constant is so large that perturbation theory cannot be used, the spontaneous- emission spectrum forms a doublet, with peaks located at the normal-mode frequencies for the coupled-dipole and cavity-mode oscillators (the so-called "vacuum" Rabi splitting).
A single two-level emitter coupled to a single mode of an electromagnetic resonator constitutes the most fundamental instance of light--matter interaction. Realizations of this type of system are investigated in the field of cavity quantum electrodynamics (cavity QED). Particularly important are schemes in which the coupling is strong enough to exceed the spontaneous emission rate as well as cavity damping. In this case, the coherent exchange of excitation between source and photon is the dominating process, providing the unique possibility to deterministically control the quantum evolution of the system. Great progress has recently been accomplished, technologically as well as conceptually, and the focus section in this issue of Journal of Optics B: Quantum and Semiclassical Optics features five articles highlighting important recent developments in the field. Strong coupling between single atoms and the electromagnetic cavity field have been achieved in two distinct regimes, the optical and the microwave region. The latter is represented by the micromaser, a high-Q microwave cavity pumped by a beam of highly excited Rydberg atoms. It has proved to be an almost ideal system for the generation of non-classical field states. A constraint in these experiments is the fact that the field inside the cavity can only be examined indirectly, by analysing the occupation of atoms leaving the cavity. While only two atomic states were previously identified for this purpose, V A Reshetov shows in his article (pages 119--126) that more information on the field may be extracted by taking advantage of the Zeeman substructure of the atomic levels. In the optical domain of cavity QED, by contrast, the state of the resonator field may be probed directly by measuring the light leaking from the cavity using photodetectors. In order to minimize damping, high-quality cavities must be employed. In most optical experiments these are Fabry--Perot resonators with ultra-high reflectivity mirrors, with losses as low as a few parts per million being state of the art. At the same time, the coupling of atoms and photons is enhanced by reducing the length of the cavity, so that strong coupling conditions are reached. A very attractive application of this type of cavity QED setup is the possibility of exerting control over the quantum dynamics of the system. An impressive example is the implementation of quantum feedback, reported in the paper by W P Smith and L A Orozco (pages 127--134). Using information gained from detecting a single photon escaping from the cavity, active feedback is applied to halt the dynamical evolution of the system. This is the first time active feedback has been used to modify the quantum evolution of a system at the single-photon level. The authors discuss how detunings between the subsystems affect the quantum feedback, an issue particularly relevant for ensembles of atoms with a spread in detuning. Perfect quantum feedback in this case requires the use of additional parameters to control the system. The majority of optical cavity QED experiments performed so far have used as a medium atoms traversing the cavity on random trajectories, such as, for example, thermal atomic beams. In this case, the observation of quantum effects is degraded in two ways. The Doppler effect and the finite transit time lead to line broadening, while the spatial structure of the cavity mode results in atoms in different locations experiencing different coupling with the photonic field. J E Reiner et al (pages 135--142) investigate the impact of these effects on quantum fluctuations in their cavity QED system. In order to overcome line broadening, they have developed a continuous cold atomic beam, extracted from a magneto-optical trap. An even better control of broadening effects and fluctuating coupling strength is provided by trapping the atoms inside the cavity and cooling them to temperatures at which their position spread is smaller than the optical wavelength. Much recent activity has been devoted to developing such an ultimate cavity QED system. A Vukics et al (pages 143--153) present a theoretical proposal for dynamically trapping and cooling atoms inside a cavity. By driving the atoms with a standing wave injected from the side of the cavity and by weakly exciting the cavity field itself, they predict three-dimensional trapping of the atom for periods around 1 second, an extremely long time on the typical scale of cavity QED dynamics. The essential component of their scheme is the interference between the two driving fields, determining the temperature of the particle and the stable trapping positions. In the quest for better confinement of radiation, lately, an appealing alternative to the Fabry--Perot cavity has been investigated -- the optical microsphere resonator. A sphere with a diameter of a few tens of microns, it supports so-called whispering gallery modes near its circumference, for which quality factors up to 10 10 have been measured. In order to exploit the correspondingly low field damping rates for cavity QED, a method must be found to couple atomic or solid state emitters to these modes. S Göotzinger et al (pages 154--158) report the coupling of single nanocrystals on the surface of a microsphere through the evanescent field of the mode. Selective excitation of a single nano-emitter was accomplished with a scanning confocal microscope. The system is a promising candidate for cavity QED with single-quantum emitters. The ability to fully control the quantum dynamics of atoms and light is beginning to spawn early applications. The most notable example is quantum information processing. This requires the coherent manipulation of single-quantum systems, which is ideally provided by cavity QED interaction in the strong-coupling limit. The development of practical devices will be among the principal future tasks for research in the field. This will include physical systems, which have not been covered in this issue, such as semiconductors, quantum dots as emitters, or photonic crystals as resonator structures. Cavity QED is therefore guaranteed to remain a vigorous field in the years to come.