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Physical Review A

Optical pulse dynamics for quantum-dot logic operations in a photonic-crystal waveguide

Xun Ma, Sajeev John

Published Nov 28, 2011• 8 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.84.053848

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Chinese Physics Letters•Aug 1, 2009•3 citations
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\ensuremathμ1.5\phantom{\rule{0.222222em}{0ex}}\ensuremath{\mu}1.5\ensuremathμm) 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\ensuremathμ10\phantom{\rule{0.222222em}{0ex}}\ensuremath{\mu}10\ensuremathμm in a photonic-band-gap (PBG) optical microchip.
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 demonstrate multiwavelength channel optical logic operations on the Bloch vector of a quantum two-level system in the structured electromagnetic vacuum of a bimodal photonic crystal waveguide. This arises through a bichromatic strong-coupling effect that enables unprecedented control over single quantum-dot (QD) excitation through two beams of ultrashort femtojoule pulses. The second driving pulse (signal) with slightly different frequency and weaker strength than the first (holding) pulse leads to controllable strong modulation of the QD Bloch vector evolution path. This occurs through resonant coupling of the signal pulse with the Mollow sideband transitions created by the holding pulse. The movement of the Mollow sidebands during the passage of the holding pulse leads to an effective chirping in transition frequency seen by the signal. Bloch vector dynamics in the rotating frame of the signal pulse and within the dressed-state basis created by the holding pulse reveals that this chirped coupling between the signal pulse and the Mollow sidebands leads to either augmentation or negation of the final quantum-dot population (after pulse passage) compared to the outcome of the holding pulse alone and depending on the relative frequencies of the pulses. By making use of this extra degree of freedom for ultrafast control of QD excitations, applications in ultrafast all-optical logic and, or, and not gates are proposed in the presence of significant (0.10.10.1) THz nonradiative dephasing and (about 1%) inhomogeneous broadening.
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.
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.
In this thesis, we demonstrate a new type of resonant, nonlinear, light-matter interaction facilitated by the unique electromagnetic vacuum density-of-state (DOS) structure of Photonic Band Gap (PBG) materials. Strong light localization inside PBG waveguides allows extremely strong coupling between laser fields and embedded two-level quantum dots (QD). The resulting Mollow splitting is large enough to traverse the precipitous DOS jump created by a waveguide mode cutoff. This allows the QD Bloch vector to sense the non-smoothness of the vacuum structure and evolve in novel ways that are forbidden in free space. These unusual strong-coupling effects are described using a "vacuum structure term" of the Bloch equation, combined with field-dependent relaxation rates experienced by the QD Bloch vector. This leads to alternation between coherent evolution and enhanced relaxation. As a result, dynamic high-contrast switching of QD populations can be realized with a single beam of picosecond pulses. During enhanced relaxation to a slightly inverted steady state at the pulse peak, the Bloch vector rapidly switches from anti-parallel to parallel alignment with the pulse torque vector. This then leads to a highly inverted state through subsequent coherent "adiabatic following" near the pulse tail, providing a robust mechanism for picosecond, femto-Joule all-optical switching. The simultaneous input of a second, weaker (signal) driving beam at a different frequency on top of the stronger (holding) beam enables rich modulation effects and unprecedented coherent control over the QD population. This occurs through resonant coupling of the signal pulse with the Mollow sideband transitions created by the holding pulse, leading to either augmentation or negation of the final QD population achieved by the holding pulse alone. This effect is applied to ultrafast all-optical logic AND, OR and NOT gates in the presence of significant (0.1 THz) nonradiative dephasing and (about 1%) inhomogeneous broadening. Further numerical studies of pulse evolutions inside the proposed devices demonstrate satisfactory population contrast within a PBG waveguide length of about 10 micro meter. These results provide the building blocks for low-power, ultrafast, multi-wavelength channel, on-chip, all-optical computing.
We theoretically investigate controlled tunable all-optical filtering and buffering of optical pulses in a hybrid nano-photonic structure, where a single quantum dot (QD) embedded in a photonic crystal nanocavity is side-coupled between a bare nanocavity and a photonic crystal waveguide. We demonstrate that there is a sharp low-loss transmission peak in the transmission spectrum under even low QD-nanocavity coupling strength and the input optical pulses can be delayed up to several hundred picoseconds within the dephasing time of the QD. The filtering regime can be shifted readily by manipulating the detuning between the QD excitonic transition frequency and resonant frequency of the nanocavity mode, which can be explored in future for on-chip all-optical logic and signal processing.