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Optics Express

Semiclassical analysis of two-level collective population inversion using photonic crystals in three-dimensional systems

Hiroyuki Takeda

Published Jul 12, 2012• 1 citation
Publisher

Abstract

I theoretically demonstrate the population inversion of collective two-level atoms using photonic crystals in three-dimensional (3D) systems by self-consistent solution of the semiclassical Maxwell-Bloch equations. In the semiclassical theory, while electrons are quantized to ground and excited states, electromagnetic fields are treated classically. For control of spontaneous emission and steady-state population inversion of two-level atoms driven by an external laser which is generally considered impossible, large contrasts of electromagnetic local densities of states (EM LDOS’s) are necessary. When a large number of two-level atoms are coherently excited (Dicke model), the above properties can be recaptured by the Maxwell-Bloch equations based on the first-principle calculation. In this paper, I focus on the realistic 1D PC’s with finite structures perpendicular to periodic directions in 3D systems. In such structures, there appear pseudo photonic band gaps (PBG’s) in which light leaks into air regions, unlike complete PBG’s. Nevertheless, these pseudo PBG’s provide large contrasts of EM LDOS’s in the vicinity of the upper photonic band edges. I show that the realistic 1D PC’s in 3D systems enable the control of spontaneous emission and population inversion of collective two-level atoms driven by an external laser. This finding facilitates experimental fabrication and realization.

DOI
10.1364/oe.20.017201

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•Oct 1, 2014•0 citations
I theoretically demonstrate the population inversion of collective two-level atoms using photonic crystals in three-dimensional (3D) systems by self-consistent solution of the semiclassical Maxwell-Bloch equations. In the semiclassical theory, while electrons are quantized to ground and excited states, electromagnetic fields are treated classically. For control of spontaneous emission and steady-state population inversion of two-level atoms driven by an external laser which is generally considered impossible, large contrasts of electromagnetic local densities of states (EM LDOS’s) are necessary. When a large number of two-level atoms are coherently excited (Dicke model), the above properties can be recaptured by the Maxwell-Bloch equations based on the first-principle calculation. In this paper, I focus on the realistic 1D PC’s with finite structures perpendicular to periodic directions in 3D systems. In such structures, there appear pseudo photonic band gaps (PBG’s) in which light leaks into air regions, unlike complete PBG’s. Nevertheless, these pseudo PBG’s provide large contrasts of EM LDOS’s in the vicinity of the upper photonic band edges. I show that the realistic 1D PC’s in 3D systems enable the control of spontaneous emission and population inversion of collective two-level atoms driven by an external laser. This finding facilitates experimental fabrication and realization.
We theoretically demonstrate the population switching of quantum dots (QD's), modeled as two-level atoms in idealized one-dimensional (1D) and two-dimensional (2D) photonic crystals (PC's) by self-consistent solution of the Maxwell-Bloch equations. In our semiclassical theory, energy states of the electron are quantized, and electron dynamics is described by the atomic Bloch equation, while electromagnetic waves satisfy the classical Maxwell equations. Near a waveguide cutoff in a photonic band gap, the local electromagnetic density of states (LDOS) and spontaneous emission rates exhibit abrupt changes with frequency, enabling large QD population inversion driven by both continuous and pulsed optical fields. We recapture and generalize this ultrafast population switching using the Maxwell-Bloch equations. Radiative emission from the QD is obtained directly from the surrounding PC geometry using finite-difference time-domain simulation of the electromagnetic field. The atomic Bloch equations provide a source term for the electromagnetic field. The total electromagnetic field, consisting of the external input and radiated field, drives the polarization components of the atomic Bloch vector. We also include a microscopic model for phonon dephasing of the atomic polarization and nonradiative decay caused by damped phonons. Our self-consistent theory captures stimulated emission and coherent feedback effects of the atomic Mollow sidebands, neglected in earlier treatments. This leads to remarkable high-contrast QD-population switching with relatively modest (factor of 10) jump discontinuities in the electromagnetic LDOS. Switching is demonstrated in three separate models of QD's placed (i) in the vicinity of a band edge of a 1D PC, (ii) near a cutoff frequency in a bimodal waveguide channel of a 2D PC, and (iii) in the vicinity of a localized defect mode side coupled to a single-mode waveguide channel in a 2D PC.
The coherent control of cooperative spontaneous emission from two identical non-overlapping three-level atoms in the V-configuration located within a photonic band gap (PBG) material with two resonant frequencies near the upper band edge of the PBG and confined to a region small in comparison to their radiation wavelengths but still greater than their atomic sizes is investigated. The dependencies of cooperative effects in which a photon emitted by one atom is reabsorbed by the other atom on the inter-atomic separation, on the initial state of the two-atom system, on the strength of the driving control laser field, and on the detuning of the atomic resonant frequencies from the upper band edge frequency is analyzed so as to identify the conditions for which these cooperative effects are enhanced or inhibited. Cooperative effects between atoms are shown to be influenced more by the PBG than by the nature of the atomic transitions involved. Excited state populations as well as coherences between excited levels are expressed in terms of time-dependent amplitudes which are shown to satisfy coupled integro-differential equations for which analytic solutions are derived under special conditions. Unlike for the case of one atom in a PBG where the fractional non-zero steady state populations on the excited levels as well as the coherence between the excited levels are constants independent of time, in the case of two atoms in PBG these quantities continuously oscillate as a manifestation of beating due to the continuous exchange between the two atoms of the photon trapped by the PBG. The values of these quantities as well as the amplitudes and frequencies of their oscillations depend of the parameters of the system, providing different ways of manipulating the system. The general formalism presented here is shown to recapture the special results of investigations of similar systems in free space when the non-Markovian memory kernels of the PBG are replaced by delta function dependent Markovian memory kernels.
We demonstrate the coherent control of spontaneous emission for a three-level atom located within a photonic band gap (PBG) material, with one resonant frequency near the edge of the PBG. Spontaneous emission from the three-level atom can be totally suppressed or strongly enhanced depending on the relative phase between the steady-state control laser coupling the two upper levels and the pump laser pulse used to create an excited state of the atom in the form of a coherent superposition of the two upper levels. Unlike the free-space case, the steady-state inversion of the atomic system is strongly dependent on the externally prescribed initial conditions. This non-zero steady-state population is achieved by virtue of the localization of light in the vicinity of the emitting atom. It is robust to decoherence effects provided that the Rabi frequency of the control laser field exceeds the rate of dephasing interactions. As a result, such a system may be relevant for a single-atom, phase-sensitive optical memory device on the atomic scale. The protected electric dipole within the PBG provides a basis for a qubit to encode information for quantum computations. A detailed literature survey on the nature, fabrication and applications of PBG materials is presented to provide context for this research.
A theoretical model that takes into full account the vectorial nature of electromagnetic (EM) fields is developed to investigate various types of quantum electrodynamics (QED) in a three-dimensional (3D) photonic crystal. The EM fields are quantized via solving the eigenproblem of photonic crystals with the use of a plane-wave expansion method. It is found that the light-atom coupling coefficients strongly depend on the Bloch states, and the key physical function concerning the atomic QED is the photon local density of states (LDOS) instead of the DOS. Both the DOS and the LDOS vary slowly near the band edge and no singularity takes place. This vectorial model show that the spontaneous emission from a two-level atom can be solved via the conventional Weisskopf-Wigner approximation theory, which exhibits a pure exponential decay behavior with a rate proportional to the LDOS. The quantum interference effect from a three-level atom is greatly weakened at the band edge, instead of enhanced. Neglecting this vectorial nature will lead to many discrepancies in understanding quantum optics in 3D photonic crystals and other inhomogeneous media.
Quantum electrodynamics of atom spontaneous emission from a three-dimensional photonic crystal is studied in a full vectorial framework. The electromagnetic fields are quantized via solving the eigenproblem of photonic crystals with use of a plane-wave expansion method. It is found that the photon density of states and local density of states (LDOS) with a full band gap vary slowly near the edge of band gap, in significant contrast to the singular character predicted by the previous isotropic model. Therefore, the spontaneous emission can be solved by conventional Weisskopf-Wigner approximate theory, which yields a pure exponentially decaying behavior with a rate proportional to the LDOS.
In certain three-dimensional crystals, a frequency range exist for all polarizations for which light is not allowed to propagate in any direction, called the 3D photonic band gap: a frequency range where the density of vacuum fluctuations vanishes in an ideal infinitely large and perfect system. The complete absence of vacuum fluctuations can be interpreted as zero density of states and vice versa. It is well known that the characteristics of spontaneous emission of light depend strongly on the environment of the light source. According to quantum electrodynamics, the emission rate of a two-level quantum emitter as described by Fermi’s “golden rule”, is factorized into one part describing the emitter’s properties and a second part describing the effect of the environment on the light field via the local density of optical state (LDOS). The radiative decay rate of the emitters is proportional to the LDOS. The absence of vacuum fluctuations or the zero LDOS leads to complete inhibition of the spontaneous emission. While many great advances towards a band gap have been made, no complete inhibition has ever been observed to date. In this thesis, for the first time, we have studied the spontaneous emission of PbS quantum dots inside real 3D photonic band gap crystal as a function of frequency throughout the band gap. In our experiments we have observed an inhibited emission of 18x in the band gap. The time-resolved decay curves show evidence for intriguing finite-size effects whereby vacuum fluctuations tunnel into the band gap. Unfortunately, most theories assume crystals of infinite extent, where the inhibition in the band gap is also infinite, as vacuum fluctuations don’t exist inside the crystal. Here we propose an original model to calculate the LDOS in the band gap of a finite photonic crystal based on tunneling of the vacuum fluctuations into the crystal. We validate our model for the one-dimensional situation with a direct comparison to the analytic calculations which exist for one-dimensional photonic crystals. For three dimensional crystals, the theory shows an excellent match to the experiment.