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Physical Review A

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

Hiroyuki Takeda, Sajeev John

Published May 9, 2011• 19 citations
Publisher

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.

DOI
10.1103/physreva.83.053811

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Quantum Theory of Phonon-Mediated Decoherence and Relaxation of Two-Level Systems in a Structured Electromagnetic Reservoir

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Physical Review A•Dec 17, 2007•28 citations
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.
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 demonstrate dynamical near-complete inversion switching of a two-level quantum dot driven by picosecond optical pulses in a bimodal photonic band gap (PBG) waveguide at microwatt power levels. This is enabled by a sharp steplike discontinuity in the local (electromagnetic) density of states (LDOS) provided by a cutoff in one of the two waveguide modes. The atomic Bloch vector equations in this colored vacuum are derived using dressed state description of the atomic density operator in the presence of phonon-mediated dephasing. Instead of assuming phenomenological decay rates, we derive explicit expressions for decay terms of bare state atomic dipole moments and population in the Born-Markov approximation from a master equation approach. Giant Mollow splitting of the atom level due to subwavelength light localization of strong driving pulse causes the time-dependent Mollow bands to straddle the LDOS jump, leading to different radiative decay rates in the upper and lower Mollow sidebands. This results in remarkable field-dependent spontaneous emission and dipolar dephasing rates, combined with a novel ``vacuum structure'' term in the Bloch equations. Our Bloch equations predict ultrafast high-contrast inversion switching that is activated and deactivated by picosecond pulse trains detuned below and above the atomic resonance, respectively. This dynamic inversion is due to the rapid rise in relaxation rates as the pulse amplitude rises, causing the Bloch vector to switch from antiparallel to parallel alignment with the pulse ``torque vector.'' Subsequent near-complete inversion occurs through an adiabatic following process retained long after the pulse amplitude subsides and the system reverts to slow relaxation. For a 1% inhomogeneously broadened distribution of quantum dot with average of 100 D dipole moment, driven by $1.5\text{ }\ensuremath{\mu}\text{m}$ picosecond pulses and coupled to a cutoff mode LDOS jump with radiative emission rates of ${\ensuremath{\gamma}}_{high}=2.5\text{ }\text{THz}$ and ${\ensuremath{\gamma}}_{low}=5\text{ }\text{GHz}$, a large average population switching contrast of 0.5 is demonstrated with a phonon dephasing rate ${\ensuremath{\gamma}}_{p}=0.5\text{ }\text{THz}$. A 1.6 fJ control pulse is required per switching operation and a $30\text{ }\ensuremath{\mu}\text{W}$ pulse train is sufficient to maintain the inversion. This switchable gain (loss) segment of the PBG waveguide can be used to controllably amplify (absorb) signal pulses conveying optical information. This provides a robust mechanism for ultrafast multiwavelength channel all-optical logic in PBG microchips.
We numerically demonstrate all-optical logic operations with quantum dots (QDs) embedded in a bimodal photonic-crystal waveguide using Maxwell-Bloch equations in a slowly varying envelope approximation (SVEA). The two-level QD excitation level is controlled by one or more femtojoule optical driving pulses passing through the waveguide. Specific logic operations depend on the relative pulse strengths and their detunings from an inhomogeneouslly broadened (about 1% for QD transitions centered at $1.5\phantom{\rule{0.222222em}{0ex}}\ensuremath{\mu}$m) QD transition. This excitation controlled two-level medium then determines passage of subsequent probe optical pulses. Envelope equations for electromagnetic waves in the linear dispersion and cutoff waveguide modes are derived to simplify solution of the coupled Maxwell-Bloch equations in the waveguide. These determine the quantum mechanical evolution of the QD excitation and its polarization, driven by classical electromagnetic (EM) pulses near a sharp discontinuity in the EM density of states of the bimodal waveguide. Different configurations of the driving pulses lead to distinctive relations between driving pulse strength and probe pulse passage, representing all-optical logic and, or, and not operations. Simulation results demonstrate that such operations can be done on picosecond time scales and within a waveguide length of about $10\phantom{\rule{0.222222em}{0ex}}\ensuremath{\mu}$m in a photonic-band-gap (PBG) optical microchip.
We demonstrate picosecond, high-contrast population flipping of two-level atoms interacting with femtojoule optical pulses in a 3D photonic band gap waveguide. This is the result of giant Mollow splitting caused by strong light localization and a remarkable field-dependent relaxation of the atomic Bloch vector as it evolves near an abrupt jump in the electromagnetic density of states. Unlike steady-state switching, population inversion can remain nearly complete after the pulse subsides. This may be used for multiwavelength channel, all-optical, on-chip information processing.
In this thesis we study the role of nonradiative degrees of freedom on quantum optical properties of mesoscopic quantum dots placed in the structured electromagnetic reservoir of a photonic crystal. We derive a quantum theory of the role of acoustic and optical phonons in modifying the optical absorption lineshape, polarization dynamics, and population dynamics of a two-level atom (quantum dot) in the ``colored" electromagnetic vacuum of a photonic band gap (PBG) material. This is based on a microscopic Hamiltonian describing both radiative and vibrational processes quantum mechanically. Phonon sidebands in an ordinary electromagnetic reservoir are recaptured in a simple model of optical phonons using a mean-field factorization of the atomic and lattice displacement operators. Our formalism is then used to treat the non-Markovian dynamics of the same system within the structured electromagnetic density of states of a photonic crystal. We elucidate the extent to which phonon-assisted decay limits the lifetime of a single photon-atom bound state and derive the modified spontaneous emission dynamics due to coupling to various phonon baths. We demonstrate that coherent interaction with undamped phonons can lead to enhanced lifetime of a photon-atom bound state in a PBG by (i) dephasing and reducing the transition electric dipole moment of the atom and (ii) reducing the quantum mechanical overlap of the state vectors of the excited and ground state (polaronic shift). This results in reduction of the steady-state atomic polarization but an increase in the fractionalized upper state population in the photon-atom bound state. We demonstrate, on the other hand, that the lifetime of the photon-atom bound state in a PBG is limited by the lifetime of phonons due to lattice anharmonicities (break-up of phonons into lower energy phonons) and purely nonradiative decay. We demonstrate how these additional damping effects limit the extent of the polaronic (Franck-Condon) shift of the atomic excited state. We also derive the modified polarization decay and dephasing rates in the presence of such damping. This leads to a microscopic, quantum theory of the optical absorption lineshapes. Our model and formalism provide a starting point for describing dephasing and relaxation in the presence of external coherent fields and multiple quantum dot interactions in electromagnetic reservoirs with radiative memory effects.
We study the detailed propagative characteristics of optical pulses in photonic band-gap (PBG) waveguides, coupled near resonantly to inhomogeneously broadened distributions of quantum dots. The line centers of the quantum-dot (QD) distributions are placed near a sharp discontinuity in the local electromagnetic density of states. Using finite-difference time-domain (FDTD) simulations of optical pulse dynamics and independent QD susceptibilities associated with resonance fluorescence, we demonstrate subpicosecond switching from pulse absorption to pulse amplification using steady-state optical holding and gate fields with power levels on the order of 1 milliwatt. In the case of collective response of QDs within the periodic dielectric microstructure, the gate power level is reduced to 200 microwatt for room temperature operation. In principle, this enables 200 Gbits per second optical information processing at wavelengths near 1.5 microns in various wavelength channels. The allowed pulse bandwidth in a given waveguide channel exceeds 0.5 THz allowing switching of subpicosecond laser pulses without pulse distortion. The switching contrast from absorption to gain is governed by the QD oscillator strength and dipole dephasing time scale. We consider dephasing time scales ranging from nanoseconds (low-temperature operation) to one picosecond (room-temperature operation). This all-optical transistor action is based on simple Markovian models of single-dot and collective-dot inversion and switching by coherent resonant pumping near the photon density of states discontinuity. The structured electromagnetic vacuum is provided by two-mode waveguide architectures in which one waveguide mode has a cutoff that occurs, with very large Purcell factor, near the QDs resonance, while the other waveguide mode exhibits nearly linear dispersion for fast optical propagation and modulation. Unlike optical switching based on Kerr nonlinearities in an optical cavity resonator, switching power levels and switching speeds for our QD device are not inversely proportional to cavity quality factors.