Self-consistent Maxwell-Bloch theory of quantum-dot-population switching in photonic crystals

Abstract

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.where q is the phonon wave vector, ω q is the phonon frequency, η q is the electron-phonon interaction coefficient, and b q and b † q are the annihilation and creation operators of phonons, respectively ([b q ,b † q ] = 1). b q (t) satisfies the Heisenberg equation of motion,(Conventionally, the second term is neglected for simplicity, although η q is larger than hω q at a certain |q| [28].) Then, b q (t) = b q e −iω q t (independent boson model). Using the polaron transformation H = e X H e −X , where X = qthe polaron shift and 053811-3 HIROYUKI TAKEDA AND SAJEEV JOHN PHYSICAL REVIEW A 83, 053811 (2011) D ± = exp[± q η q hω q (b † q − b q )] are lattice displacement operators. Then, H = e −XĤ e X is rewritten as

Collective population evolution of two-level atoms based on mean-field theory

“…5(a). These behaviors are almost the same as those obtained by the semiclassical Maxwell-Bloch equations [17]. On the other hand, I consider the truncation of the input electric field, and then (t) = (0 < t < t 0 ) and e −(t−t 0 )/τ (t t 0 ).…”

“…Although in three-level and four-level atoms ω L > ω A is necessary for population inversion, in two-level atoms this condition is not always necessary. [17]. This means that collective spontaneous emission can be interpreted as classical dipole radiation.…”

“…In the case of a large number of two-level atoms, the steadystate population evolution discontinuously changes near the threshold value of input electric fields [2,7,8], and this property can be recaptured by the semiclassical Maxwell-Bloch theory [17]. In the semiclassical theory, while electrons are quantized to the ground and excited states, electromagnetic fields are treated classically.…”

Self-consistent Maxwell-Bloch model of quantum-dot photonic-crystal-cavity lasers

“…This simple model of an artificial atom, implemented with Maxwell's equations, has the distinct benefit of allowing one to study general light-matter interactions without using either of the rotating-wave approximation or the slowly-varyingenvelope approximation, which have already led to a wide range of new effects such as the dynamic nonlinear skin effect [38] and carrier-wave Rabi flopping [39,40], even with simple 1D equations of motion. Moreover, when studying quantum information systems that are dominated by radiative decay, it is critical to preserve the coherent radiative contributions that a MB analysis provides [41], without recourse to adding in phenomenological damping constants. Otherwise, if nonradiative processes dominate, a more straightforward MB formalism may be used, in which phenomenological damping terms like pure dephasing are implemented [42,43].…”

Optical bistability in one-dimensional doped photonic crystals with spontaneously generated coherence

“…They can be tailored for efficient optical switching by using embedded atoms in PCs [17][18][19][20][21][22]. Conventional OB in PCs utilizes dynamic shifting of the band edge while the doped PCs allow for dispersive OB via dynamic shifting of the defect mode [23].…”