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Optics and Photonics News

CAVITY QUANTUM ELECTRODYNAMICS AT OPTICAL FREQUENCIES

S. Morin, Qilin Wu, THOMAS W. MOSSBERG

Published Aug 1, 1992• 20 citations
Publisher

Abstract

Quantum electrodynamics (QED) tells us that the electromagnetic field is, on a mode-by-mode basis, quantized according to the harmonic oscillator model. Each mode is ascribed a lowest energy (or dark) state possessing one half quanta of non-removable energy plus an infinite ladder of equally spaced excited states accessed through the addition or removal of energy quanta from the mode. In the optical regime and under normal thermal conditions, electromagnetic field modes are typically in their dark state. Nevertheless, a residual dark-state atom-field coupling remains, and it is this coupling that mediates shifts and spontaneous radiative decay of atomic states. The experimental reality of dark-state atom-field coupling is frequently interpreted as proof of the quantum description of light.

DOI
10.1364/opn.3.8.000008

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Quantum electrodynamics (QED) tells us that the electromagnetic field is, on a mode-by-mode basis, quantized according to the harmonic oscillator model. Each mode is ascribed a lowest energy (or dark) state possessing one half quanta of non-removable energy plus an infinite ladder of equally spaced excited states accessed through the addition or removal of energy quanta from the mode. In the optical regime and under normal thermal conditions, electromagnetic field modes are typically in their dark state. Nevertheless, a residual dark-state atom-field coupling remains, and it is this coupling that mediates shifts and spontaneous radiative decay of atomic states. The experimental reality of dark-state atom-field coupling is frequently interpreted as proof of the quantum description of light.
It was first realized by Purcell (1) that the spontaneous emission rate of a quantum system can be enhanced or suppressed by placing it in a resonant radiofrequency cavity. Spontaneous emission as it was first theoretically described by Einstein (2) is not a pure property of matter. It is described by matter coupled to quantized radiation field modes, which can be manipulated in an artificial structure such as a cavity. Cavity quantum electrodynamics (QED) is used to describe such effects that intimately depend on the fact that light is made of photons. Cavity QED has been extensively studied in atoms and the experimental proof was awarded with the Nobel Prize in 2012 (3). It has been applied to cooling of single atoms and for the detection of single atoms and creating and studying states with few photons and few atoms (4). In more recent developments these ideas have been extended to manipulate electronic states (5) and vibrations in molecules (6, 7). In PNAS, Flick et al. (8) introduce recent theoretical developments for the application of cavity QED to molecules and suggest possible novel applications to photochemistry. These include nonadiabatic dynamics of molecules in cavities, the modification of molecular properties, and the integration of QED with density functional theory. A quantized electromagnetic field mode can be described as a harmonic oscillator whose coordinate is the electric field displacement (Fig. 1 A ). The zero point energy translates into a nonvanishing field intensity ⟨ e 2 ⟩ in the ground state | 0 ⟩ (vacuum fluctuations). The electric vacuum field increases for a small cavity volume e c = ℏ ω c V ϵ 0 , [1] Fig. 1. ( A ) Illustration of the standing wave cavity mode and the resulting field quantization (Fock states). ( B ) Combined matter-field states (dressed states, | ± , n ⟩ … [↵][1]1To whom correspondence should be addressed. Email: smukamel{at}uci.edu. [1]: #xref-corresp-1-1
We will begin our journey through the background of quantum optics with an elementary, but quantitative, classical theory of radiative fields.Wewill first briefly outline the electromagnetic (EM) theory of radiation, and describe how the EM radiation may be understood as a wavewhich can be represented by a set of harmonic oscillators. We shall describe how the free or non-interacting EM field may be understood as a collection of harmonic oscillators which is quantized in the standard manner, and whose energy cannot be zero as a consequence of the basic non-commutability of the canonical field variables. This chapter discusses the properties of plane EM waves and normalization of the EM field in one dimension. This is followed by a description of the Hamiltonian and the amplitudes of the EM field in terms of the annihilation and creation operators. Based on this chapter, it is possible to considerably simplify the formulation of the physical basis for the mathematical description of the major problems of quantum optics.
The quantum properties of electromagnetic, mechanical or other harmonic oscillators can be revealed by investigating their strong coherent coupling to a single quantum two level system in an approach known as cavity quantum electrodynamics (QED). At temperatures much lower than the characteristic energy level spacing the observation of vacuum Rabi oscillations or mode splittings with one or a few quanta asserts the quantum nature of the oscillator. Here, we study how the classical response of a cavity QED system emerges from the quantum one when its thermal occupation-or effective temperature-is raised gradually over 5 orders of magnitude. In this way we explore in detail the continuous quantum-to-classical crossover and demonstrate how to extract effective cavity field temperatures from both spectroscopic and time-resolved vacuum Rabi measurements.
A macroscopic, canonical quantization of the EM field and radiating atom system in quantum optics and cavity QED involving classical, linear optical devices, based on expanding the vector potential in terms of quasi mode functions is presented. The quasi mode functions approximate the true mode functions for the device, and are obtained by solving the Helmholtz equation for an idealized spatially dependent electric permittivity function describing the device. The Hamiltonian for the EM field and radiating atom system is obtained in multipolar form and the quantum EM field is found to be equivalent to a set of quantum harmonic oscillators, one oscillator per quasi mode. However, unlike true mode theory where the quantum harmonic oscillators are uncoupled, in the quasi mode theory they are coupled and photon exchange processes can occur. Explicit expressions for the coupling constants are obtained. The interaction energy between the radiative atoms and the quantum EM field depends on the amplitudes of the quasi mode functions at the positions of the radiating atoms, similar to that for the true mode approach. The simpler forms for the quasi mode functions enable the atom-field interaction energy to be written in a form in which the atoms are only coupled to certain types of modes—for example cavity quasi modes, which are large inside the optical cavity. In such cases the escape of energy from excited atoms in the cavity can be pictured in quasi mode theory as a two step process—the atom de-excites and creates a photon in a cavity quasi mode, the photon in the cavity quasi mode is then lost and appears as a photon in an external quasi mode. In this process the first step occurs via the atom-cavity quasi mode interaction, the second through coupling between cavity and external quasi modes. This may be contrasted with the true mode approach, where the excited atom loses its energy and the photon is created in one of the true modes. As all true modes have non-zero amplitudes outside as well as inside the cavity, the escape of energy from excited atoms in the cavity is seen as a one step process. An application of the quasi mode theory to the quantum theory of the beam splitter is outlined. The unitary operator used to describe this device is a scattering operator, relating initial and long time values of annihilation, creation operators for pairs of incident and reflected modes, interpreted here as quasi modes.
Atoms absorb and emit resonant light photon by photon. However, an atom strongly coupled to the light field of a high-finesse optical resonator forms a new system with new characteristics, enabling the emission and absorption of photon pairs. The energy eigenstates of the coupled system form an anharmonic ladder of doublets, the so-called Jaynes-Cummings ladder. The first doublet, consisting of the two so-called normal modes, has been observed in various cavity QED systems. It can be explained classically using Maxwell equations as arising from the coupling of a classical dipole to a classical electromagnetic field. In contrast, the higher-order doublet has so far no classical analogue. It requires both the quantization of the atomic internal state as well as the quantization of light.
We derive the quantum mechanical description of light-matter interactions in optical cavities characterized solely by radiative decay. Unique to radiative decay is the conservation of photon number and coherence, in stark contrast to absorptive losses. This prohibits the description of such cavities by traditional means, e.g., coupling it to a bath of harmonic oscillators into which energy is dissipated. Here, we propose a description of cavities with radiative decay by introducing cavity and noise operators in terms of scattering modes. A multimode input-output formalism to predict measurable far field quantities arises naturally. We apply our general model to the special case of the single excitation regime. We find dynamics reminiscent of the dissipative Jaynes-Cummings model, but with vanishing backaction and a rich temporal and spectral structure of the output modes.