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arXiv:2002.01912

Phonon-induced dephasing in quantum dot-cavity QED: Limitations of the polaron master equation

A. Morreau, Chaitanya S. Joshi, Egor A. Muljarov

Published Feb 5, 2020• 0 citations
arXivHTMLPDF

Abstract

A semiconductor quantum dot (QD) embedded within an optical microcavity is a system of fundamental importance within quantum information processing. The optimization of quantum coherence is crucial in such applications, requiring an in-depth understanding of the relevant decoherence mechanisms. We provide herein a critical review of prevalent theoretical treatments of the QD-cavity system coupled to longitudinal acoustic phonons, comparing predictions against a recently obtained exact solution. Within this review we consider a range of temperatures and exciton-cavity coupling strengths. Predictions of the polaron Nakajima-Zwanzig (NZ) and time-convolutionless (TCL) master equations, as well as a variation of the former adapted for adiabatic continuous wave excitation (CWE), are compared against an asymptotically exact solution based upon Trotter's decomposition (TD) theorem. The NZ and TCL implementations, which apply a polaron transformation to the Hamiltonian and subsequently treat the exciton-cavity coupling to second order, do not offer a significant improvement accuracy relative to the polaron transformation alone. The CWE adaptation provides a marked improvement, capturing the broadband features of the absorption spectrum (not present in NZ and TCL implementations). We attribute this difference to the effect of the Markov approximation, and particularly its unsuitability in pulsed excitation regime. Even the CWE adaptation, however, breaks down in the regime of high temperature ($50K$) and strong exciton-cavity coupling ($g \gtrsim 0.2$ meV). The TD solution is of comparable computational complexity to the above-mentioned master equation approaches, yet remains accurate at higher temperatures and across a broad range of exciton-cavity coupling strengths (at least up to $g=1.5$ meV).

DOI
10.48550/arxiv.2002.01912

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Journal of Physics Condensed Matter•Dec 12, 2016•9 citations
A semiconductor quantum dot (QD) embedded within an optical microcavity is a system of fundamental importance within quantum information processing. The optimization of quantum coherence is crucial in such applications, requiring an in-depth understanding of the relevant decoherence mechanisms. We provide herein a critical review of prevalent theoretical treatments of the QD-cavity system coupled to longitudinal acoustic phonons, comparing predictions against a recently obtained exact solution. Within this review we consider a range of temperatures and exciton-cavity coupling strengths. Predictions of the polaron Nakajima-Zwanzig (NZ) and time-convolutionless (TCL) master equations, as well as a variation of the former adapted for adiabatic continuous wave excitation (CWE), are compared against an asymptotically exact solution based upon Trotter's decomposition (TD) theorem. The NZ and TCL implementations, which apply a polaron transformation to the Hamiltonian and subsequently treat the exciton-cavity coupling to second order, do not offer a significant improvement accuracy relative to the polaron transformation alone. The CWE adaptation provides a marked improvement, capturing the broadband features of the absorption spectrum (not present in NZ and TCL implementations). We attribute this difference to the effect of the Markov approximation, and particularly its unsuitability in pulsed excitation regime. Even the CWE adaptation, however, breaks down in the regime of high temperature (50K50K50K) and strong exciton-cavity coupling (g≳0.2g \gtrsim 0.2g≳0.2 meV). The TD solution is of comparable computational complexity to the above-mentioned master equation approaches, yet remains accurate at higher temperatures and across a broad range of exciton-cavity coupling strengths (at least up to g=1.5g=1.5g=1.5 meV).
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.
The effect of phonons on a nonlinear optical response of a quantum dot-cavity system in quantum strong coupling regime can be accounted for by a fully analytical treatment, provided that the exciton-phonon dynamics is much faster than the exciton-cavity dynamics. Modern experiments involving semiconductor quantum dots embedded in optical microcavities typically meet this criterion. We find that, for a relatively small exciton-cavity coupling, the effect of phonons is concentrated mainly in the polaron shift of the exciton frequency and reduction of exciton-cavity coupling by the Huang-Rhys factor. We have generalized this result to an arbitrary optical nonlinearity and demonstrated a good agreement with the exact solution in a wide range of temperatures. This generalization provides access to higher rungs of the Jaynes-Cummings ladder, where exact numerical approaches are impractical or even impossible. At larger coupling strengths and low temperatures, our approximation is also in good agreement with the exact solution, which makes it a very useful tool for addressing the phonon contribution to the coherent dynamics of a nonlinear optical system. We demonstrate our results for third-order optical polarization with varying observation time and delay time between excitation pulses in the form of two-dimensional spectra. These spectra provide useful information about the coherent coupling between the exciton and the cavity modes. The presented analytical approach is also compared with another useful approximation, having the form of a matrix product, which is a special case of the asymptotically exact solution in the limit of a short phonon memory time.
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 demonstrate a strong influence of the phonon environment on the coherent dynamics of the quantum dot (QD)-cavity system in the quantum strong coupling regime. This regime is implemented in the nonlinear QD-cavity QED and can be reliably measured by heterodyne spectral interferometry. We present a semianalytic asymptotically exact path integral-based approach to the nonlinear optical response of this system, which includes two key ingredients: Trotter's decomposition and linked-cluster expansion. Applied to the four-wave-mixing optical polarization, this approach provides access to different excitation and measurement channels, as well as to higher-order optical nonlinearities and quantum correlators. Furthermore, it allows us to extract useful analytic approximations and analyze the nonlinear optical response in terms of quantum transitions between phonon-dressed states of the anharmonic Jaynes-Cummings (JC) ladder. Being well described by these approximations at low temperatures and small exciton-cavity coupling, the exact solution deviates from them for stronger couplings and higher temperatures, demonstrating remarkable non-Markovian effects, spectral asymmetry, and strong phonon renormalization of the JC ladder.
In the first part of this dissertation, a dissipative quantum dot (QD) -cavity system coupled to a longitudinal acoustic (LA) phonon reservoir is studied using a numerically exact real-time path-integral approach. Three distinct dynamical regimes of weak (WC), strong (SC), and coherent coupling (CC) are discussed and more accurate conditions identifying them are presented. Our results show that to have the CC regime, which is characterized by clear vacuum Rabi oscillation (VRO), vacuum Rabi splitting (VRS) should be larger than the sum of the widths of the corresponding peaks. In order to distinguish between contributions of population decay and impure dephasing, induced by LA phonons and the dissipations, on the quantum dynamics of the QD-cavity system, we propose a two-part phenomenological expression which fits the QD-cavity decay curves perfectly and is used to calculate the corresponding spectra. The emission rate increases from the carrier recombination rate to a maximum value, which is the mean of the QD and cavity dissipation rates, with QD-cavity coupling strength. We introduce a quantity that can be applied in determining the distinct coupling regimes; This quantity enables us to identify the onset of the SC regime as the point where the impure dephasing term begins to contribute to the central band of the spectrum significantly, as a result of the existence of VRO with a very small frequency (unclear VRO) at the corresponding decay curve. Its contribution to the width of the central band increases with the coupling strength up to the onset of the CC regime, then reduces as a result of the appearance of sidebands in the spectra. The effective population decay and impure dephasing rate contribute solely to the width—of the central and sideband peaks of the triplet spectra respectively—only beyond a very large coupling strength which is the same across the considered temperature range. In the second part, we demonstrate that all the available experimental data of temperature (T)-dependent shift of photoluminescence (PL) peak of In(Ga)As QD samiii ples can be modeled successfully by using a two-oscillator model if and only if the whole temperature interval (0–300 K) is divided into a few parts (at most four parts), depending on dispersion degree of the PL peak from a monotonic behavior. Analysis of the numerical results show that excitons mostly interact (inelastically) with acoustic (AC) or optical (OP) phonons separately. Increasing QDs uniformity, by using some improved growth techniques, results in decreasing or removing the sigmoidal behavior, enhancing total AC phonon contribution to the redshift of the PL peak. Elevation of the zero bandgap (ZBG) energy up to a critical value about 1.4 eV, for In(Ga)As QDs grown using molecularbeam epitaxy, results in enhancement of QD symmetry and total OP phonon contribution and reduction of QDs uniformity and total AC phonon contribution, while a rollover happens for further increase of the ZBG. Therefore we find that the highest QD symmetry and the lowest exciton fine structure splitting correspond to this critical value of ZBG, in accordance with previous experimental results.
A dissipative quantum dot (QD)-cavity system, where the QD is initially prepared in the excited state with no photon in the cavity, coupled to a longitudinal acoustic (LA) phonon reservoir is studied using a numerically exact real-time path-integral approach. Three distinct dynamical regimes of weak (WC), strong (SC), and coherent coupling (CC) are discussed and more accurate conditions identifying them are presented. Our results show that to have the CC regime, which is characterized by clear vacuum Rabi oscillation (VRO), vacuum Rabi splitting (VRS) should be larger than the sum of the widths of the corresponding peaks. In order to distinguish between contributions of population decay and impure dephasing, induced by LA phonon bath and the dissipations, we propose a two-part phenomenological expression, corresponding to the population decay and impure dephasing, which fits the QD-cavity decay curves perfectly and is used to calculate the corresponding spectra. We demonstrate that the effective population decay rate (the emission rate) increases from the carrier recombination rate to a maximum value, which is the mean of the QD and cavity dissipation rates, with QD-cavity coupling strength. To study the role of the effective impure dephasing rate on the width of the central peak of the spectra we introduce a quantity that can also be applied in determining the distinct coupling regimes. This quantity enables us to identify the onset of the SC regime as the point where the impure dephasing term begins to contribute to the central band of the spectrum significantly, as a result of the existence of VRO with a very small frequency (unclear VRO) at the corresponding decay curve. Its contribution to the width of the central peak increases with the coupling strength up to the onset of the CC regime, then reduces as a result of the appearance of sidebands in the spectra, which originates from clear VRO. The effective population decay and impure dephasing rate contribute solely to the width-of the central and sideband peaks of the triplet spectra respectively-only beyond a very large coupling strength which is the same across the considered temperature range. For higher temperatures, the maximum achievable emission rate can be obtained at larger coupling strengths.