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arXiv:1702.02784 · Physical review. B./Physical review. B

Time-resolved pump-probe signals of a continuously driven quantum dot affected by phonons

Doris E. Reiter

Published Mar 10, 2017• 20 citations
arXivHTMLPDF

Abstract

The interaction of a light field with a quantum-mechanical system can be studied in an optically controlled semiconductor quantum dot. When driven by a continuous light field switched on instantaneously, the quantum dot occupation performs Rabi oscillations. Unlike an atomic system, the quantum dot is coupled to phonons, which leads to a damping of the Rabi oscillations. Here we model the time-resolved probe spectra to monitor these dynamics and study the influence of phonons on the spectra. The spectra consist of up to three peaks, similar to the Mollow triplet known from quantum optics. We develop analytical equations within a rate equation model and show that they agree excellently with a numerical solution using a well established correlation expansion approach.

DOI
10.1103/physrevb.95.125308

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The interaction of a light field with a quantum-mechanical system can be studied in an optically controlled semiconductor quantum dot. When driven by a continuous light field switched on instantaneously, the quantum dot occupation performs Rabi oscillations. Unlike an atomic system, the quantum dot is coupled to phonons, which leads to a damping of the Rabi oscillations. Here we model the time-resolved probe spectra to monitor these dynamics and study the influence of phonons on the spectra. The spectra consist of up to three peaks, similar to the Mollow triplet known from quantum optics. We develop analytical equations within a rate equation model and show that they agree excellently with a numerical solution using a well established correlation expansion approach.
Abstract Semiconductor quantum dots are solid state few‐level systems which can interact strongly with light. As such, they can be used as a single photon source or to perform solid‐state quantum‐optics experiments. A major difference to atoms is the interaction with the lattice vibrations, that is, the phonons, which strongly affect the optical signals from a quantum dot. In this paper, we calculate the time‐resolved pump‐probe signals from a continuously driven quantum dot accounting for detuned excitation and electron–phonon interaction. We provide analytical equations giving insight into the different phenomena observed in the spectra like an oscillation between absorptive and dispersive line shapes. For detuned excitation, the electron–phonon interaction can lead to phonon‐assisted occupation of the exciton, giving rise to a strong asymmetry in the dynamical behavior depending on the sign of the detuning. The time‐resolved spectra monitor this relaxation path, which helps understanding the effect of electron–phonon interaction in quantum dots.
A model of semiconductor quantum dots coupled to phonons and laser light, which is relevant in the limit of strong electronic confinement, is investigated. For an arbitrary sequence of excitations by ultrafast pulses, analytical solutions are obtained for all pertinent density-matrix elements. The results are nonperturbative with respect to the coupling to both phonons and laser light. The derivation depends neither on a particular choice of electronic wave functions nor on a particular wave-vector dependence of the phonon interaction or phonon dispersion. All components of the density matrix exhibit oscillatory dependences on the pulse areas of the exciting pulses analogously to the Rabi oscillations known from simple two-level systems. Three typical applications of the general results are worked out: (a) the influence of phonons on four-wave-mixing spectra, (b) the coherent control of electronic dot occupations in the presence of phonons, and (c) the dynamics of phonon correlations including the formation of a stable polaron spatially localized in the dot and the emission of a pulsed phonon wave packet that leaves the dot.
We present a semiconductor quantum optics formalism to study the dynamics of a coherently-driven semiconductor quantum dot interacting with an acoustic phonon bath and a high Q microcavity. A quantum master equation is derived in the polaron frame, where multiphoton and multiphoton effects are included to all orders. As applications of the theory, we study the Mollow triplet of a driven quantum dot in the regime of semiconductor cavity-QED. Pronounced signatures of electron-phonon-photon scattering are observed through excitation-induced dephasing and off-resonant cavity coupling. We also present an effective phonon master in Lindblad form and show example quantum trajectory simulations that help one to understand the features in the Mollow triplet spectra.
We study the resonance fluorescence spectra of a driven quantum dot placed inside a high-Q semiconductor cavity and interacting with an acoustic phonon bath. The dynamics is calculated using a time-convolutionless master equation in the polaron frame. We predict pronounced spectral broadening of the Mollow sidebands through off-resonant cavity emission which, for small cavity-coupling rates, increases quadratically with the Rabi frequency in direct agreement with recent experiments using semiconductor micropillars [S. M. Ulrich et al., preceding Letter, Phys. Rev. Lett. 106, 247402 (2011)]. We also demonstrate that, surprisingly, phonon coupling actually helps resolve signatures of the elusive second rungs of the Jaynes-Cummings ladder states via off-resonant cavity feeding. Both multiphonon and multiphoton effects are shown to play a qualitatively important role on the fluorescence spectra.
While the advanced coherent control of qubits is now routinely carried out in low frequency (GHz) systems like single spins, it is far more challenging to achieve for two-level systems in the optical domain. This is because the latter evolve typically in the THz range, calling for tools of ultrafast, coherent, nonlinear optics. Using four-wave mixing micro-spectroscopy, we here measure the optically driven dynamics of a single exciton quantum state confined in a semiconductor quantum dot. In a combined experimental and theoretical approach, we reveal the intrinsic Rabi oscillation dynamics by monitoring both central exciton quantities, i.e., its occupation and the microscopic coherence, as resolved by the four-wave mixing technique. In the frequency domain this oscillation generates the Autler-Townes splitting of the light-exciton dressed states, directly seen in the four-wave mixing spectra. We further demonstrate that the coupling to acoustic phonons strongly influences the FWM dynamics on the picosecond timescale, because it leads to transitions between the dressed states.
We present a comprehensive theoretical study of the resonance fluorescence spectra of an coherently-driven quantum dot (QD) placed inside a high-QQQ semiconductor cavity and interacting with an acoustic-phonon bath. We derive a quantum master equation (ME) in the polaron frame, which includes exciton-phonon and exciton-cavity coupling to all orders. This work details and extends the theory used in a recent paper [C. Roy and S. Hughes, Phys. Rev. Lett. 106, 247403 (2011)] to describe the QD Mollow triplet in the regime of semiconductor cavity QED. Here, we introduce two ME forms, Nakajima-Zwanzig and time convolutionless (TC), both to second order in the system--phonon-reservoir perturbation. In the polaron frame, these two ME forms are shown to yield equivalent population dynamics and fluorescence spectra for a continuous wave (cw) driving field. We also demonstrate that a Markov approximation is valid for computing the incoherent scattering processes, and we subsequently exploit the TC ME to explore the resonance fluorescence spectra of an exciton-driven QD. Both cavity-emitted and exciton-emitted spectra are studied, and these are found to have qualitatively different spectral features. Using a coherent driving field, the well-known characteristics of the atomic Mollow triplet are shown to be considerably modified with electron--acoustic-phonon scattering, and we highlight the key effects arising from both cavity coupling and electron-phonon coupling. Regimes of pronounced cavity feeding and anharmonic cavity QED are exemplified, and we find that the cavity coupling depends sensitively on the exciton-cavity detuning and the temperature of the phonon bath. We show how the full width at half maximum (linewidth) of the Mollow-triplet sidebands varies as a function of the square of the Rabi frequency of the cw pump. Phonon-mediated cavity coupling also contributes to the spectral broadening of the Mollow triplet, depending upon the exciton-cavity detuning and the strength of the exciton-cavity coupling rate. Finally, we calculate the fluorescence spectra for off-resonance cw driving and investigate the resulting Mollow-triplet linewidths.