Quantum Theory of Photonic Matter
July 11, 2009 2 Comments
The electromagnetic field inside an inhomogeneous dielectric is different from free space. Guest atoms inside the structure notice this, either because their energies and transition dipole moments change, or because the atom-field interaction changes, or both. Only when the former (electronic) effects can either be neglected or accounted for, can the guest atoms be considered good probes of the latter (photonic) effects of the medium.
We recently described the quantum electrodynamics of an inhomogeneous dielectric in the presence of guest atoms. Such a theory is a prerequisite for calculations of spontaneous-emission rates, line shifts, interatomic interactions, and other observable effects of the medium on embedded atoms.
We describe the medium macroscopically in terms of a position-dependent dielectric function; guest atoms are described microscopically in terms of elementary charges. After choosing a suitable gauge and performing a canonical quantization, we have identified the field operator to which atomic operators couple in an inhomogeneous medium.
Electronic effects of the medium on the guest atom are absent only if the relative dielectric function at the position of the atom equals unity. It is known that even such a tiny modification of the dielectric function will have noticeable effects (see below).
b. Spontaneous Emission
Almost all light around us is produced by spontaneous emission. The reason that spontaneous emission occurs at all can only be understood in a quantum theory of light in dielectrics. Nevertheless, the Einstein coefficient for spontaneous emission can often be calculated in a classical treatment of light.
In general, the spontaneous-emission rate and spatial radiation profile of atoms are different in a medium. A precise knowledge of these modified properties is crucial for applications. One should like to control (enhance or inhibit) emission rates or to engineer preferred emission directions. In lasers for example, it would be advantageous to suppress spontaneous emission, because it is a loss mechanism that determines the pump threshold for lasing.
For guest atoms embedded in the solid part of an inhomogeneous dielectric, dipole orientations will be fixed and spontaneous-emission rates will depend both on the atomic position and dipole orientation. The orientation dependence comes about because light has polarization degrees of freedom. For guest atoms in the gas phase, dipole orientations are not fixed and should be averaged over.
We have shown that the orientation-averaged position-dependent emission rate is proportional to a classical quantity called the “local optical density of states” (LDOS). This is the direct optical analogy of the more familiar electronic local density of states. The electronic LDOS is measured with a scanning-tunnelling microscope (or STM). After recent claims that the signals recorded with a scanning optical near-field microscope (or SNOM) are a probe of the optical LDOS, the latter quantity has gained the attention of a new group of research
c. Photonic Crystals
Spontaneous-emission rates change in a medium. For example, in the close vicinity of a perfect mirror, it is well known that emission rates depend on the atomic transition frequency, its dipole orientation and its distance to the mirror.
One could place instead many infinitely thin mirrors at regular distances from each other on the distance scale of the wavelength of light. The optical properties of such a dielectric structure would be strongly modified compared to free space, even if the mirrors are only partly reflecting. For example, the structure exhibits Bragg scattering of external light sources. We have set up a multiple-scattering theory for such a Bragg mirror of plane scatterers, both for scalar and for polarized waves. We find that the one-dimensional crystal can be an omnidirectional mirror. Still, emission rates of atoms embedded in this omnidirectional mirror are nonzero, because emission into guided modes is still possible.
Recently, three-dimensional inhomogeneous dielectric structures with characteristic length scales matching the wavelength of the luminescence have been put forward as structures in which spontaneous emission can be severely modified. In these three-dimensional photonic crystals even a photonic band gap could appear that would give rise to a vanishing spontaneous-emission rate. From a fundamental perspective, it is fascinating to measure long lifetimes of atoms due to a band gap. Technologically, a suppressed spontaneous emission makes other processes more probable. For example, it would reduce the threshold pump intensity in lasers.
Our group has measured a fivefold reduction of spontaneous-emission rates in air-sphere fcc-crystals. We fabricated the crystals ourselves. Theoretically, we focus on finite-size effects of the crystals, and the effect of disorder in the crystal. (Remember the important role of crystal disorder is in electronics for understanding Ohm’s law.) Furthermore, we are interested in local-field effects (see below).
d. Lorentz Cavity
We investigate the fundamental relation between microscopic properties of individual polarizable building blocks (such as atoms, dipoles, and dielectric spheres) and the macroscopic electromagnetic response of matter. This long-standing problem is particularly important in the interpretation of the dielectric constant (from zero frequency up to the x-ray regime), the Einstein coefficient for spontaneous emission, the absorption of an impurity in a host material, and in all nonlinear optical susceptibilities.
Lorentz made a beautiful contribution by introducing a local field felt by an atom in a medium that is different from the macroscopic field. Up to today the validity of the concept of a local field and the particular value suggested by Lorentz, is still a matter of intense debate. We have recently proved that the original intuitive approach by Lorentz can be given a solid foundation. If an atom couples to a local field different from the macroscopic field, then emission rates in the former case are equal to the macroscopic emission rate, multiplied by the square of a so-called local-field factor. Every local-field model has its own local-field factor.
Experimental and theoretical efforts focused until now on the modification of the atomic emission rates in real or virtual cavities in an otherwise homogeneous dielectric. Photonic crystals are known to have a strong influence on atomic emission rates, when assuming that a dipole couples to a macroscopic field. When considering local fields inside photonic crystals, it is to be expected that local-field effects become position-dependent, but no one has calculated such effects so far, although they will certainly influence the quantum efficiencies of embedded atoms.
e. Spontaneous Emission And The Kramers-Kronig Relations
Photonic crystals are usually described by a nondispersive and real position-dependent dielectric function. In reality, the Kramers-Kronig relations tell that the dielectric function must be a frequency-dependent and complex function. The usual description is therefore only valid in a finite frequency-interval.
When taking the KK-relations into account, the quantum electrodynamics inside a medium becomes more complicated. We have described the QED in absorbing homogeneous dielectrics. We found a simple method to identify the field operators in the medium and showed that the canonical commutation relations are conserved in time. We showed that emission rates become time-dependent quantities in a transient regime when the atomic frequencies are close to resonances of the medium.
We did not yet take local-field effects into account. Recent work done in other groups made us interested in quantum theories of light that predict local-field effects in dielectrics satisfying the Kramers-Kronig relations.