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arXiv:2306.17722 · Physical review. B./Physical review. B

Impact of the phonon environment on the nonlinear quantum-dot–cavity QED: Path-integral approach

L. S. Sirkina, Egor A. Muljarov

Published Sep 28, 2023• 5 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.108.115312

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arXiv:1704.03347•Physical review. B./Physical review. B•Nov 22, 2017•42 citations
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.
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.
Interaction between light and matter generates optical nonlinearities, which are particularly pronounced in the quantum strong coupling regime. When a single bosonic mode couples to a single fermionic mode, a Jaynes–Cummings (JC) ladder is formed, which we realize here using cavity photons and quantum dot excitons. We measure and model the coherent anharmonic response of this strongly coupled exciton–cavity system at resonance. Injecting two photons into the cavity, we demonstrate a larger polariton splitting with respect to the vacuum Rabi splitting. This is achieved using coherent nonlinear spectroscopy, specifically four-wave mixing, where the coherence between the ground state and the first (second) rung of the JC ladder can be interrogated for positive (negative) delays. With increasing excitation intensity and thus rising average number of injected photons, we observe spectral signatures of the quantum-to-classical crossover of the strong coupling regime.
We develop an exact analytical approach to four-wave mixing and higher-order optical nonlinearities of a quantum dot-microcavity system. This approach allows us \nto analyse the non-linear response in terms of coherent transitions between rungs \nof the Jaynes-Cummings ladder, finding their complex amplitudes and frequencies. \nIncreasing the pulse area of the excitation field, a transition from quantum strong \ncoupling to coherent classical regime is observed, which manifests itself in a formation of the famous Mollow triplet, with the observation in the three-level model \nof a quantum dot, including a biexciton, to predict two new effects: One is the \nexistence of a Mollow-like quintuplet, the other is a complete elimination of the \nfour-wave-mixing optical non-linearity under certain parameter constraints. We further develop a simple and accurate approximation in the low-damping limit of the \nJaynes-Cummings model of the quantum dot-microcavity system, for nonlinear response of any order in a closed analytic form, predicting the positions and widths \nof the spectral lines. For higher damping, we demonstrate a strong relaxation down \nthe ladder, shown by both numerical and analytic calculation.
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 semiconductor master equation technique to study the input/output characteristics of coherent photon transport in a semiconductor waveguide-cavity system containing a single quantum dot. We use this approach to investigate the effects of photon propagation and anharmonic cavity-QED for various dot-cavity interaction strengths, including weakly-coupled, intermediately-coupled, and strongly-coupled regimes. We demonstrate that for mean photon numbers much less than 0.1, the commonly adopted weak excitation (single quantum) approximation breaks down, even in the weak coupling regime. As a measure of the multiphoton correlations, we compute the Fano factor and the correlation error associated with making a semiclassical approximation. We also explore the role of electron--acoustic-phonon scattering and find that phonon-mediated scattering plays a qualitatively important role on the light propagation characteristics. As an application of the theory, we simulate a conditional phase gate at a phonon bath temperature of 20 K in the strong coupling regime.
The generation of photons in a microcavity coupled to a laser-driven quantum dot interacting with longitudinal acoustic (LA) phonons is studied in the regime of simultaneously strong driving and strong dot-cavity coupling. The stationary cavity photon number is found to depend in a nontrivial way on the detuning between the laser and the exciton transition in the dot. In particular, the maximal efficiency of the cavity feeding is obtained for detunings corresponding to transition energies between cavity-dressed states with excitation numbers larger than one. Phonons significantly enhance the cavity feeding at large detunings. In the strong-driving, strong-coupling limit, the photon statistics is highly non-Poissonian. While without phonons a double-peaked structure in the photon distribution is predicted, phonons make the photon statistics thermal-like with very high effective temperatures $\ensuremath{\sim}{10}^{5}$ K, even for low phonon temperatures $\ensuremath{\sim}4$ K. These results were obtained by numerical calculations where the driving, the dot-cavity coupling, and the dot-phonon interactions are taken into account without approximations. This is achieved by a reformulation of an exact iterative path-integral scheme which is applicable to a large class of quantum-dissipative systems and which in our case reduces the numerical demands by 15 orders of magnitude.