To characterize the statistics and indistinguishability of a source, it is common to measure the correlation functions of the emitted field using various interferometers. Here, we present a theoretical framework for the computation of the correlation functions of a two-level system that is resonantly driven by a realistic noisy cw excitation laser. Analytic expressions of the first- and second-order autocorrelation functions are obtained where the various contributions of the noisy excitation source are correctly taken into account. We predict that, even in the low power regime, the noise source has a strong influence on the two-level system dynamics, which is not anticipated by simpler models. The characterization of photon indistinguishability in the pulsed excitation regime is usually done by measuring the value of the zero-delay intensity correlation obtained with a Hong-Ou-Mandel interferometer. We show that this figure is irrelevant in the cw excitation regime and we introduce the coalescence time window, a figure of merit based on a probabilistic interpretation of the notion of photon indistinguishability. We finally use the coalescence time window to quantify how noisy cw excitation influences photon indistinguishability.
To characterize the statistics and indistinguishability of a source, it is common to measure the correlation functions of the emitted field using various interferometers. Here, we present a theoretical framework for the computation of the correlation functions of a two-level system that is resonantly driven by a realistic noisy cw excitation laser. Analytic expressions of the first- and second-order autocorrelation functions are obtained where the various contributions of the noisy excitation source are correctly taken into account. We predict that, even in the low power regime, the noise source has a strong influence on the two-level system dynamics, which is not anticipated by simpler models. The characterization of photon indistinguishability in the pulsed excitation regime is usually done by measuring the value of the zero-delay intensity correlation obtained with a Hong-Ou-Mandel interferometer. We show that this figure is irrelevant in the cw excitation regime and we introduce the coalescence time window, a figure of merit based on a probabilistic interpretation of the notion of photon indistinguishability. We finally use the coalescence time window to quantify how noisy cw excitation influences photon indistinguishability.
The indistinguishability of successively generated photons from a single quantum emitter is most commonly measured using two-photon interference at a beam splitter. Whilst for sources excited in the pulsed regime the measured bunching of photons reflects the full wave-packet indistinguishability of the emitted photons, for continuous wave (cw) excitation, the inevitable dependence on detector timing resolution and driving strength obscures the underlying photon interference process. Here we derive a method to extract full photon wave-packet indistinguishability from cw measurements by considering the relevant correlation functions. The equivalence of both methods is experimentally verified through a comparison of cw and pulsed excitation measurements on an archetypal source of photons, a single molecule.
We revisit Mandel's notion that the degree of coherence equals the degree of indistinguishability by performing Hong-Ou-Mandel- (HOM-)type interferometry with single photons elastically scattered by a cw resonantly driven excitonic transition of an InAs/GaAs epitaxial quantum dot. We present a comprehensive study of the temporal profile of the photon coalescence phenomenon which shows that photon indistinguishability can be tuned by the excitation laser source, in the same way as their coherence time. A new figure of merit, the coalescence time window, is introduced to quantify the delay below which two photons are indistinguishable. This criterion sheds new light on the interpretation of HOM experiments under cw excitation, particularly when photon coherence times are longer than the temporal resolution of the detectors. The photon indistinguishability is extended over unprecedented time scales beyond the detectors' response time, thus opening new perspectives to conducting quantum optics with single photons and conventional detectors.
We revisit Mandel's notion that the degree of coherence equals the degree of indistinguishability by performing Hong-Ou-Mandel-(HOM-)type interferometry with single photons elastically scattered by a cw resonantly driven excitonic transition of an InAs/GaAs epitaxial quantum dot. We present a comprehensive study of the temporal profile of the photon coalescence phenomenon which shows that photon indistinguishability can be tuned by the excitation laser source, in the same way as their coherence time. A new figure of merit, the coa-lescence time window, is introduced to quantify the delay below which two photons are indistinguishable. This criterion sheds new light on the interpretation of HOM experiments under cw excitation, particularly when pho-ton coherence times are longer than the temporal resolution of the detectors. The photon indistinguishability is extended over unprecedented time scales beyond the detectors' response time, thus opening new perspectives to conducting quantum optics with single photons and conventional detectors. Indistinguishable photons are one of the keys for the implementation of scalable quantum information systems [1, 2]. Indistinguishability is investigated using the coalescence phenomenon: two photons with similar spectral, spatial and polarization properties will bunch when arriving simultaneously on two opposite sides of a beam splitter. One of the pioneers of photon coalescence, Mandel, stated in 1991 that the degree of coherence equals the degree of indistinguishability [3, 4] by investigating theoretically the interference of two light sources, thus underlining the fundamental link of the wave-particle duality of light. In a two-photon interference Hong-Ou-Mandel (HOM) experiment [5], the photons from two sources are combined at the two inputs of a beam splitter and the coalescence will be detected through a drop of the coincidence rate at the outputs—the HOM dip. Under pulsed excitation, perfect temporal matching between the arrival times of the photons at the beam splitter will allow for the observation of the HOM dip. When working with a two-level system, its coherence time T 2 and its lifetime T 1 are tightly linked to the photon indistin-guishability. For example, perfect coalescence giving rise to a zero value in the HOM dip is observed only if the radiative limit T 2 = 2T 1 is reached. The figure of merit under pulsed excitation is thus given by the ratio T 2 /2T 1 which constitutes a fundamental limit to the coalescence efficiency [6]. Under continuous wave (cw) excitation, with two ideal ultrafast detectors , the coincidence rate always vanishes at zero time delay , even for deviations in the properties of the photons [7]. In the case of real detectors, the indistinguishability is thus properly resolved only if the temporal resolution of the detectors T R is shorter than the coherence time of the photons [8]. If T R ∼ T 1 , T 2 , the HOM dip is strongly affected and will disappear completely in the limit of very slow detectors. With a cw source, the value at zero delay of the coincidence rate is thus very sensitive to T R and does not accurately characterize the intrinsic properties of the source with regard to photon indis-tinguishability. A new figure of merit has to be considered. Single semiconductor quantum dots (QD) [9], along with other systems under extensive study including atoms [7, 10], molecules [11–13], trapped ions [14, 15], and colored centers in diamond [16], are promising candidates for sources of single indistinguishable photons. In the case of semiconductor QDs, photon indistinguishability is either limited by the QD dynamics under pulsed excitation [6], or by the detec-tors' temporal resolution under cw excitation [17, 18]. Recent experimental studies focused on the regime of resonant Rayleigh scattering (RRS) under low power cw excitation, where the incoming photons are elastically scattered. This is a well-known phenomenon described by the two-level system resonance fluorescence theory [23], which has been observed with QDs [24–28]. As predicted by theory and shown by ho-modyne and heterodyne detection experiments [26–28], the scattered photons inherit the coherence time of the excitation laser T L , which can be much longer than T 2 and T R , while still exhibiting sub-Poissonian statistics [24, 25]. The resulting QD emission spectrum can then be much narrower than the natural linewidth imposed by the radiative limit, even if the percentage of elastically scattered photons remains limited to T 2 /2T 1 [Fig. 1(b)]. Considering that under such conditions the inherited coherence time surpasses T R , along with Man-del's notion that coherence equals indistinguishability [3], the RRS regime constitutes the ideal ground for the generation of highly indistinguishable photons.
We study the single-photon purity of a resonantly driven single-photon source in the realistic scenario where the excitation laser pulse can leak into the detection path. We find that the duration of the excitation pulse strongly influences the quality of the single-photon source. We calculate the influence of this on the effective parameters describing the most relevant properties, including the two-photon component and Hong-Ou-Mandel visibility. Furthermore, we analyze how these properties can be strongly affected by frequency filtering of the outgoing field. Our results highlight that the relation between the two-photon component of the emission and the Hong-Ou-Mandel visibility is more complicated than typically assumed in the literature, and depends on the specific details of the source.
Single-photon sources are at the heart of quantum-optical networks, with their uniquely quantum emission and phenomenon of two-photon interference allowing for the generation and transfer of nonclassical states. Although a few analytical methods have been briefly investigated for describing pulsed single-photon sources, these methods apply only to either perfectly ideal or at least extremely idealized sources. Here, we present the first complete picture of pulsed single-photon sources by elaborating how to numerically and fully characterize non-ideal single-photon sources operating in a pulsed regime. In order to achieve this result, we make the connection between quantum Monte-Carlo simulations, experimental characterizations, and an extended form of the quantum regression theorem. We elaborate on how an ideal pulsed single-photon source is connected to its photocount distribution and its measured degree of second- and first-order optical coherence. By doing so, we provide a description of the relationship between instantaneous source correlations and the typical experimental interferometers (Hanbury-Brown and Twiss, Hong–Ou–Mandel, and Mach–Zehnder) used to characterize such sources. Then, we use these techniques to explore several prototypical quantum systems and their non-ideal behaviors. As an example numerical result, we show that for the most popular single-photon source—a resonantly excited two-level system—its error probability is directly related to its excitation pulse length. We believe that the intuition gained from these representative systems and characters can be used to interpret future results with more complicated source Hamiltonians and behaviors. Finally, we have thoroughly documented our simulation methods with contributions to the Quantum Optics Toolbox in Python in order to make our work easily accessible to other scientists and engineers.
This thesis intends to familiarize the reader with the concepts of photon statistics and correlations in quantum optics. Developing light sources that emit quantum states is central for the realization of quantum technologies. One important step in characterizing these sources is the measurement of field fluctuations and correlations, by coincidence measurements. The expectation value of a coincidence measurement, a simultaneous measurement of two intensities (or, more general, four fields), is represented by the fourth-order correlation function. The value of the correlation function, at zero delay between the detection of two photons, reveals important properties of the state to which they belonged, for example the fluctuations of the photon number. Since predictability is important for many applications, light sources emitting single photons are also characterized by the indistinguishability of consecutively emitted photons, or of two photons from separate emitters. In paper I we investigate blinking behaviour in quantum emitters, and its effect on the interference pattern and photon statistics with photons from two separate emitters. Blinking refers to an emitters transition into a non-emitting state, and subsequent transition back to an emitting state. We show that blinking can not be treated as linear loss, when measuring the fourth-order correlation function for two emitters in a Hong-Ou-Mandel setup. In general, a measurement of the fourth-order correlation function is robust to loss, which makes it a very practical tool. However, the relation between recorded coincidence counts and the correlation function is only direct in the limit of zero detection efficiency, and depends on the detection system. In paper II, we show that by adding a variable attenuation in the beam path, we can trace back to the ''true'' value of the correlation function at zero quantum efficiency. This method improves accuracy in correlation measurements by decreasing a systematic error at the expense of an increased statistical error, which is easier to handle, extending the use of coincidence methods to classical and non-classical multi-photon states.