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Physical Review A

Modeling coherence measurements on a spectrally diffusing single-photon emitter

Laurent Coolen, Xavier Brokmann, Jean‐Pierre Hermier

Published Sep 20, 2007• 24 citations
Publisher

Abstract

We examine the possibility of measuring the emission linewidth $2\ensuremath{\hbar}{\ensuremath{\Gamma}}_{2}$ of a single-photon source by Michelson interferometry. Such an emitter is characterized by a limited fluorescence intensity and, when embedded in a solid matrix, a large spectral diffusion. We show that fast spectral diffusion renders standard Fourier spectroscopy irrelevant. We then calculate the correlations of the intensities detected at the two interferometer outputs, point out the existence of two-photon interferences (coalescence) even for a non-single-mode emission, and show that the correlations are not affected by spectral diffusion at correlation times shorter than spectral dynamics. This yields two ways to measure the decoherence rate ${\ensuremath{\Gamma}}_{2}$: the recently proposed photon-correlation Fourier spectroscopy, and the study of the coalescence dip. We examine the feasibility and spectral resolution of each method, depending on spectral-diffusion characteristics, and show that both methods could be applied to typical nanometer-sized emitters such as colloidal quantum dots.

DOI
10.1103/physreva.76.033824

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We examine the possibility of measuring the emission linewidth 2\ensuremathℏ\ensuremathΓ22\ensuremath{\hbar}{\ensuremath{\Gamma}}_{2}2\ensuremathℏ\ensuremathΓ2​ of a single-photon source by Michelson interferometry. Such an emitter is characterized by a limited fluorescence intensity and, when embedded in a solid matrix, a large spectral diffusion. We show that fast spectral diffusion renders standard Fourier spectroscopy irrelevant. We then calculate the correlations of the intensities detected at the two interferometer outputs, point out the existence of two-photon interferences (coalescence) even for a non-single-mode emission, and show that the correlations are not affected by spectral diffusion at correlation times shorter than spectral dynamics. This yields two ways to measure the decoherence rate \ensuremathΓ2{\ensuremath{\Gamma}}_{2}\ensuremathΓ2​: the recently proposed photon-correlation Fourier spectroscopy, and the study of the coalescence dip. We examine the feasibility and spectral resolution of each method, depending on spectral-diffusion characteristics, and show that both methods could be applied to typical nanometer-sized emitters such as colloidal quantum dots.
Solid-state quantum emitters are playing a crucial role in the development of photonic quantum technologies. However, several decoherence mechanisms are known to impact the device performances and scalability. In various quantum emitter materials the fluctuation of the two-level system has been investigated by different methods to capture the evolution of coherence reduction. However, resolving the dynamics of spectral diffusion in the fundamental framework of on-demand single-photon generation remained unsolved. Hereby, the challenge is to observe with high time and energy resolution the impact of fluctuations directly in the properties of the emitted photons. In this Rapid Communication, we demonstrate the use of dispersion in a slow-light medium to map the photons frequency domain into time domain, observed as frequency-dependent time-of-flight. This allows for the measurement of the emission spectrum and the quantification of the spectral diffusion dynamics in one intensity correlation measurement. On exemplary semiconductor quantum dots, the impact of charge and spin noise on the spectral diffusion are revealed to follow an Ornstein-Uhlenbeck process. By a single measurement, broadening from the excitation repetition up to the stationary limit is resolved. This enables one to extract time-dependent two-photon interference visibilities for various timescales, which is a key performance measure for quantum emitters.
Solid-state single-photon emitters (SPEs) are quantum light sources that combine atomlike optical properties with solid-state integration and fabrication capabilities. SPEs are hindered by spectral diffusion, where the emitter's surrounding environment induces random energy fluctuations. Timescales of spectral diffusion span nanoseconds to minutes and require probing single emitters to remove ensemble averaging. Photon correlation Fourier spectroscopy (PCFS) can be used to measure time-resolved single emitter line shapes, but is hindered by poor signal-to-noise ratio in the measured correlation functions at early times due to low photon counts. Here, we develop a framework to simulate PCFS correlation functions directly from diffusing spectra that match well with experimental data for single colloidal quantum dots. We use these simulated datasets to train a deep ensemble autoencoder machine learning model that outputs accurate, noiseless, and probabilistic reconstructions of the noisy correlations. Using this model, we obtain reconstructed time-resolved single dot emission line shapes at timescales as low as 10 ns, which are otherwise completely obscured by noise. This enables PCFS to extract optical coherence times on the same timescales as Hong-Ou-Mandel two-photon interference, but with the advantage of providing spectral information in addition to estimates of photon indistinguishability. Our machine learning approach is broadly applicable to different photon correlation spectroscopy techniques and SPE systems, offering an enhanced tool for probing single emitter line shapes on previously inaccessible timescales.
We describe a method to probe the spectral fluctuations of a transition over broad ranges of frequencies and timescales with the high spectral resolution of Fourier spectroscopy, and a temporal resolution as high as the excited state lifetime, even in the limit of very low photocounting rates. The method derives from a simple relation between the fluorescence spectral dynamics of a single radi-ating dipole and its fluorescence intensity correlations at the outputs of a continuously scanning Michelson interferometer. These findings define an approach to investigate the fast fluorescence spectral dynamics of single molecules and other faint light sources beyond the time-resolution capabilities of standard spectroscopy experiments.
The excited state population of single solid-state emitters is subjected to energy fluctuations around the equilibrium driven by the bath and relaxation through the emission of phonons or photons. Simultaneous measurement of the associated spectral dynamics requires a technique with a high spectral and temporal resolution with an additionally high temporal dynamic range. We propose a pulsed excitation-laser analog of photon-correlation Fourier spectroscopy (PCFS), which extracts the linewidth and spectral diffusion dynamics along the emission lifetime trajectory of the emitter, effectively discriminating spectral dynamics from relaxation and bath fluctuations. This lifetime-resolved PCFS correlates photon-pairs at the output arm of a Michelson interferometer in both their time-delay between laser-excitation and photon-detection T and the time-delay between two photons τ. We propose the utility of the technique for systems with changing relative contributions to the emission from multiple states, for example, quantum emitters exhibiting phonon-mediated exchange between different fine-structure states.
Light emission from solid-state quantum emitters is inherently prone to environmental decoherence, which results in a line broadening and in the deterioration of photon indistinguishability. Here we employ photon correlation Fourier spectroscopy (PCFS) to study the temporal evolution of such a broadening in two prominent systems: GaAs and In(Ga)As quantum dots. Differently from previous experiments, the emitters are driven with short laser pulses as required for the generation of high-purity single photons, the time scales we probe range from a few nanoseconds to milliseconds and, simultaneously, the spectral resolution we achieve can be as small as ∼ 2µeV. We find pronounced differences in the temporal evolution of different optical transition lines, which we attribute to differences in their homogeneous linewidth and sensitivity to charge noise. We analyze the effect of irradiation with additional white light, which reduces blinking at the cost of enhanced charge noise. Due to its robustness against experimental imperfections and its high temporal resolution and bandwidth, PCFS outperforms established spectroscopy techniques, such as Michelson interferometry. We discuss its practical implementation and the possibility to use it to estimate the indistinguishability of consecutively emitted single photons for applications in quantum communication and photonic-based quantum information processing.