It is shown that a broad class of cavity quantum electrodynamics (QED) problems---which consider the resonant propagation of a single photon interacting with quantum emitters (QEs), such as atoms, quantum dots, or vacancy centers---can be solved directly without application of the second quantization formalism. In the developed approach, the Hamiltonian is expressed through the ket-bra products of collective ($\mathrm{photon}+\mathrm{cavities}+\mathrm{QEs}$) states. Consequently, the $S$ matrix of input-output problems is determined exactly by the Mahaux-Weidenm\"uller formula, which dramatically simplifies the analysis of complex cavity QED systems. First, this approach is illustrated for the problem of propagation of a photon resonantly interacting with $N$ two-level QEs arbitrary distributed inside the optical cavity. Solution of this problem manifests the effect of cumulative action of QEs previously known for special cases. Can a similar cumulative action of QEs enhance the inelastic resonant transmission of a single photon? We solve this problem for the case of an optical cavity having two modes resonantly coupled to electronic transitions of $N$ three-level QEs. It is shown that the described structure is the simplest realistic structure which enables the down-conversion of the single-photon frequency with the amplitude approaching unity in the absence of the external driving field and sufficiently small cavity losses and QE dissipation. Overall, the simplicity and generality of the developed approach suggest a practical way to identify and describe new phenomena in cavity QED.
It is shown that a broad class of cavity quantum electrodynamics (QED) problems---which consider the resonant propagation of a single photon interacting with quantum emitters (QEs), such as atoms, quantum dots, or vacancy centers---can be solved directly without application of the second quantization formalism. In the developed approach, the Hamiltonian is expressed through the ket-bra products of collective (photon+cavities+QEs) states. Consequently, the S matrix of input-output problems is determined exactly by the Mahaux-Weidenm"uller formula, which dramatically simplifies the analysis of complex cavity QED systems. First, this approach is illustrated for the problem of propagation of a photon resonantly interacting with N two-level QEs arbitrary distributed inside the optical cavity. Solution of this problem manifests the effect of cumulative action of QEs previously known for special cases. Can a similar cumulative action of QEs enhance the inelastic resonant transmission of a single photon? We solve this problem for the case of an optical cavity having two modes resonantly coupled to electronic transitions of N three-level QEs. It is shown that the described structure is the simplest realistic structure which enables the down-conversion of the single-photon frequency with the amplitude approaching unity in the absence of the external driving field and sufficiently small cavity losses and QE dissipation. Overall, the simplicity and generality of the developed approach suggest a practical way to identify and describe new phenomena in cavity QED.
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.
We investigate the multiphoton quantum dynamics of a leaking single-mode quantized cavity field coupled with a resonantly driven two-level system possessing permanent dipoles. The frequencies of the interacting subsystems being considered are very different, e.g., the microwave range for the cavity and the optical domain for the frequency of the two-level emitter, respectively. In this way, the emitter couples to the resonator mode only via its diagonal dipole moments. Furthermore, the generalized Rabi frequency resulting from the external coherent driving of the two-level subsystem is also assumed to be different from the resonator's frequency or its multiples. As a consequence, this highly dispersive interaction regime is responsible for the cavity multiphoton quantum dynamics and photon conversion from the optical to microwave range, respectively.
Devices involving bound or propagating photons are among the basic building blocks for modern quantum technologies. For example, the bound modes of open optical resonators, coupled to quantum emitters such as atoms, molecules, or quantum dots, are used for an enhanced emission in lasing applications, for increased sensitivity in quantum sensing, and as qubits in quantum computing. Meanwhile, propagating photons are crucial for the transmission of quantum states between spatially separated systems. In this thesis, we develop a rigorous theory of multiple, spatially separated, and individually addressable cavities (or plasmonic resonators) for applications in modern quantum devices. Losses are modeled via the use of quasinormal modes (QNMs), which are solutions to the source-free Helmholtz equation under open boundary conditions, making them the natural modes of open resonators. Quantum emitters are modeled as two-level systems (TLSs) which couple to the electric field via dipole coupling. The multi-cavity theory is developed in three steps: 1. We present a classical scattering theory for a system of multiple QNM cavities, where the multi-cavity Green's function is formulated in terms of the single-cavity modes. The inter-cavity scattering is frequency-dependent to account for retardation delays, and is efficiently calculated numerically, using common (commercial) solvers. 2. We formulate a quantization scheme for QNMs of spatially separated open resonators. We introduce the \textit{cavity separation parameter} as a simple measure for when QNM cavities can be quantized separately and when they have to be treated as a single system with a joint set of operators. We derive a general system-bath Hamiltonian that contains coupling between bound modes, quantum emitters, and a bath of propagating photons, expressed via rigorous, numerically calculable coupling parameters. 3. We discuss the time-dependent theory of QNMs and TLSs coupled to propagating photons. The system-bath interaction is formulated in terms of correlation functions that are again rigorously defined in terms of numerically calculable QNM parameters. The correlation functions are immediately applicable to common methods for calculating the dynamics of open quantum systems. Steps 1-3 are all formulated for general cavity shapes and materials, as well as (in principle) arbitrary background structures. Hence, the theory is applicable to a wide range of devices, including cavities or plasmonic resonators in free space, on a photonic crystal, or on a substrate. For each step, we show example calculations for one-dimensional or three-dimensional cavities.
A scheme was proposed to construct bilinear and quadratic Hamiltonians for frequency up-and down-conversions in cavity quantum electrodynamics(QED).Generally,in nonlinear optics,the interaction that the energe swaps between different optic modes without atomic transition is named frequency conversion.The proposed scheme was based on the interactions of a single four-level atom simultaneously with two classical driving fields and a two-mode cavity field,which is in the domain of four-wave mixing.By initially preparing the atom in a suitable state,each pump light was resonant with its transition,and two quantum modes were large tune to the other two transition,respectively,In the strong laser regime,the atomic degrees of freedom could be decoupled from the cavity degrees of freedom and the frequency conversion could be realized for the cavity modes.Due to the different initial states and interactions between optic fields and atom,frequency up-and down-conversions arose,respectively.After the preparation of squeezed operation respect to Frequency down-conversion,a discussion on the feasibility of experiment was given and the theoretical value was obtained.The advantage of this proposal is to realize the transition using two cascade photons,which is dipole-forbidden in a cascade structure atom,with high efficiency.The proposal will be useful for optical quantum control and fundamental tests of quantum theory.
Cavity quantum electrodynamics (cavity QED) describes the interactions of single atoms and single photons in a resonator. In the laboratory, state-of-the-art experiments are able to observe the dynamics of these simple yet fundamental quantum systems. The recent development of quantum information science has also introduced a new role for cavity QED systems as quantum interfaces, in which information can be coherently mapped between photonic and atomic quantum bits. Groundbreaking optical cavity QED experiments in the past twenty years would not have been possible without the technological development of mirror coatings with extremely low scatter and absorption losses. Thermal noise has typically played a smaller role in cavity QED experiments but in certain instances is nevertheless important to consider. In this chapter, we begin with an overview of optical cavity QED experiments and some of the central questions that they address. It is hoped that this description, while by no means exhaustive, will provide a window for the nonspecialist into basic techniques and motivations in the field. We then turn to focus on the roles played by scattering and absorption loss and mechanical loss. We discuss how these losses have set limits on feasible experiments and the extent to which improvements may be possible. Experimental realizations of cavity QED systems An idealized cavity QED system consists of one atom placed between two highly reflective mirrors (Figure 17.1). In this model, a single atomic transition interacts with photons in a single cavity mode; the atom is often pictured at an antinode of the cavity standing wave, where it has a maximal dipole interaction with the cavity field.