The researches on higher-order coherence and quantum statistics of light field are the important researching issues in quantum optics. In 1956, Hanbury-Brown and Twiss (HBT) (Hanbury-Brown R, Twiss R Q 1956 Nature 177 27) revolutionized optical coherence and demonstrated a new form of photon correlation. The landmark experiment has far-reaching influenced and even inspired the quantum theory of optical coherence that Glauber developed to account for the conclusive observation by HBT. Ever since then, the HBT effect has motivated extensive studies of higher-order coherence and quantum statistics in quantum optics, as well as in quantum information science and cryptography. Based on the HBT scheme, the degree of coherence and photon number distribution of light field can be derived from correlation measurement and photon counting technique. With the rapid development of the photoelectric detection technology, single-photon detection, which is the most sensitive and very widespread method of optical measurement, is used to characterize the natures of light sources and indicate their differences. More recently, HBT scheme combined with single-photon detection was used to study spatial interference, ghost imaging, azimuthal interference effect, deterministic manipulation and detection of single-photon source, etc. Due to broadband RF spectrum, noiselike feature, hypersensitivity to the initial conditions and long-term unpredictability, chaotic laser meets the essential requirements for information security and cryptography, and has been developed in many applications such as chaos-based secure communications and physical random number generation, as well as public-channel secure key distribution. But the research mainly focused on macroscopic dynamics of the chaotic laser. Moreover, the precision of measurement has reached a quantum level at present. Quantum statistcs of light field can also uncover profoundly the physical nature of the light. Thus, it is important to exploit the higher-order degree of coherence and photon statistics of chaotic field, which contribute to characterizing the field and distinguishing it from others. In this paper, photon number distribution and second-order degree of coherence of a chaotic laser are analyzed and measured based on HBT scheme. The chaotic laser is composed of a distributed feedback laser diode with optical feedback in fiber external cavity configuration. The bandwidth of the chaotic laser that we obtain experimentally is 6.7 GHz. The photon number distribution of chaotic laser is fitted by Gaussian random distribution, Possionian distribution and Bose-Einstein distribution. With the increase of the mean photon number, the photon number distribution changes from Bose-Einstein distribution into Poissonian distribution and always accords with Gaussian random distribution well. The second-order coherence g(2)(0) drops gradually from 2 to 1. By changing the bias current (I = 1.0Ith-2.0Ith) and feedback strength (010%), we compare and illustrate different chaotic dynamics and g(2)(0). From low frequency fluctuation to coherence collapse, the chaotic laser shows bunching effect and fully chaotic field can be obtained at the broadest bandwidth. Furthermore, the physical explanation for sub-chaotic or weakening of bunching effect is provided. It is concluded that this method can well reveal photon statistics of chaotic laser and will open up an avenue to the research of chaos with quantum optics, which merges two important fields of modern physics and is extremely helpful for the high-speed remote chaotic communication.
The researches on higher-order coherence and quantum statistics of light field are the important researching issues in quantum optics. In 1956, Hanbury-Brown and Twiss (HBT) (Hanbury-Brown R, Twiss R Q 1956 Nature 177 27) revolutionized optical coherence and demonstrated a new form of photon correlation. The landmark experiment has far-reaching influenced and even inspired the quantum theory of optical coherence that Glauber developed to account for the conclusive observation by HBT. Ever since then, the HBT effect has motivated extensive studies of higher-order coherence and quantum statistics in quantum optics, as well as in quantum information science and cryptography. Based on the HBT scheme, the degree of coherence and photon number distribution of light field can be derived from correlation measurement and photon counting technique. With the rapid development of the photoelectric detection technology, single-photon detection, which is the most sensitive and very widespread method of optical measurement, is used to characterize the natures of light sources and indicate their differences. More recently, HBT scheme combined with single-photon detection was used to study spatial interference, ghost imaging, azimuthal interference effect, deterministic manipulation and detection of single-photon source, etc. Due to broadband RF spectrum, noiselike feature, hypersensitivity to the initial conditions and long-term unpredictability, chaotic laser meets the essential requirements for information security and cryptography, and has been developed in many applications such as chaos-based secure communications and physical random number generation, as well as public-channel secure key distribution. But the research mainly focused on macroscopic dynamics of the chaotic laser. Moreover, the precision of measurement has reached a quantum level at present. Quantum statistcs of light field can also uncover profoundly the physical nature of the light. Thus, it is important to exploit the higher-order degree of coherence and photon statistics of chaotic field, which contribute to characterizing the field and distinguishing it from others. In this paper, photon number distribution and second-order degree of coherence of a chaotic laser are analyzed and measured based on HBT scheme. The chaotic laser is composed of a distributed feedback laser diode with optical feedback in fiber external cavity configuration. The bandwidth of the chaotic laser that we obtain experimentally is 6.7 GHz. The photon number distribution of chaotic laser is fitted by Gaussian random distribution, Possionian distribution and Bose-Einstein distribution. With the increase of the mean photon number, the photon number distribution changes from Bose-Einstein distribution into Poissonian distribution and always accords with Gaussian random distribution well. The second-order coherence g(2)(0) drops gradually from 2 to 1. By changing the bias current (I = 1.0Ith-2.0Ith) and feedback strength (010%), we compare and illustrate different chaotic dynamics and g(2)(0). From low frequency fluctuation to coherence collapse, the chaotic laser shows bunching effect and fully chaotic field can be obtained at the broadest bandwidth. Furthermore, the physical explanation for sub-chaotic or weakening of bunching effect is provided. It is concluded that this method can well reveal photon statistics of chaotic laser and will open up an avenue to the research of chaos with quantum optics, which merges two important fields of modern physics and is extremely helpful for the high-speed remote chaotic communication.
High-order photon correlations of light fields are important for characterizing the quantum nature. Since Hanbury Brown and Twiss conducted the pioneering experiments in the 1950s, the HBT effect has inspired extensive research on high-order photon correlation in quantum optics, quantum information, and quantum imaging. The Single-photon counting module is one of the most widely used single-photon detectors. Due to its high detection efficiency and low dark counts in the visible and near-infrared region, it is reasonably chosen for basic research on quantum mechanics. Many researches have demonstrated that the maximum value of second-order photon correlation g(2)(τ) at zero delay (τ = 0) can be used to distinguish different light fields. Therefore, the HBT scheme containing two single photon detectors have been widely used in many advanced studies, such as space interference, ghost imaging, single photon detection with high efficiency, etc. However, higher-order photon correlations g(n) (n > 2) can reveal more measurable characteristics of light fields, such as information about the non-Gaussian scattering process, the skewness and kurtosis of photon number distribution, etc. When the extended HBT scheme is used to measure higher-order photon correlations, the experimental conditions including quantum efficiency and background noise greatly affect the photon correlation measurement. The influences of the counting rate and resolution time of the detection system on the measurements are also very important and cannot be ignored. Therefore, the comprehensive considering of various influence factors is necessary for accurately measuring the high-order photon correlations and also a challenge. In this paper, we present a method based on double Hanbury Brown-Twiss scheme for the accurate measuring of high-order photon correlations g(n) (n > 2). The system consists of four single photon counting modules and is used to detect and analyze the joint distribution probability of temporal photon correlation. Considering the effects of the background noise and overall efficiency, theoretically, we analyze the correlations of the third- and fourth-order photon with the incident light intensity, squeezing parameter and photon number respectively for thermal state, coherent state, squeezed vacuum state, and Fock state. Meanwhile, experimentally we study the influences of resolution time and counting rate on correlations of the coherent state and thermal state with third- and fourth-order photon. On condition that the resolution time is 210 ns and the counting rate is 80 kc/s, the correlations of third and fourth-order photon with the thermal state at zero time delay are accurately measured, and the relative statistical deviations of the measured vales from the theoretical values are 0.3% and 0.8%, respectively. In addition, the third- and fourth-order photon correlations of the thermal state at different delay times are also observed. It is demonstrated that the high-order photon correlations of light fields are measured accurately by comprehensively analyzing various influencing factors. This technique provides a promising and useful tool to investigate quantum correlated imaging and quantum coherence of light fields.
Two-photon correlation measurement provides a promising way to experimentally demonstrate the statistical nature of a light source, which is very significant for the deep understanding of the photon-generating process and the diagnosing of the coherence property. Quantitatively two- photon correlation is described by second-order coherence. Usually the behavior of the second-order coherence against any of the parameters defining the phase volume is different for different photon statistics. The Poisson photon statistics for coherent light gives its second-order coherence as a flat response; The Bose-Einstein photon statistics for chaotic light gives its second-order coherence as a bunching effect; While the Sub-Poisson photon statistics for non-classical light gives its second-order coherence as an anti-bunching effect. Therefore the measurement of two-photon correlation is proved to be a good finger print to check whether light is in coherent state or incoherent state such as thermal state or non-classical state. Historically the measurement of two-photon correlation was first performed by Hanbury-Brown and Twiss (HBT) in 1956. They used a linear mixer to realize the correlation of the two currents from the photoelectric detectors illuminated by a stationary thermal light souece, a mercury arc, and the photo-bunching effect was first successfully observed in the visible region of 435.8 nm. HBT method is no doubt a good way to extract the small excess two-photon correlation for a stationary light because the background, that is the DC components, has been cut off automatically by the broad band amplifiers, which is in fact the key of the success of HBT experiment. However there exists a general problem, to which no attention has ever been paid, in measuring the two-photon correlation of non-stationary light such as synchrotron radiation (SR) by the HBT method. Here the "non-stationary" means a sense of classical mechanics that the observed intensity has some deterministic time structure. The systematic time structure of SR decided by the bunch distribution of the electric current in a storage ring will give rise to a large amount of unexpected accidental correlation, which in fact has nothing to do with the inherent photon statistics of light source and usually l000～l0000 times larger than the true two-photon correlation due to the short bunch separation length (2ns) and the short coherence time (～0.1ps) which is not comparable to the time resolution (1ns) of the measuring system. The existence of the accidental correlation would severely prevent us from observing the bunching effect of the true two-photon correlation. Therefore to suppress the much larger accidental correlation and to extract the small true two-photon correlation, a novel intensity interferometer has been developed for soft X-ray synchrotron radiation. This intensity interferometer consists of an optical vacuum chamber and an electric correlator. All the essential optical elements which includes a wire scanner, a precise diffraction slit, a grating monochromator with a coherence time modulator, a beam divider and two fast-response photon detectors (microchannel plates) are mounted in this high vacuum chamber. The electric correlator completes the multiplication of the two broad band electric currents coming from the photoelectric detectors. The basic idea to suppress the much larger accidental correlation is to modulate the coherence time by modulating the entrance slit width of the monochromator by a piezoclectric translator. The two sets of light intensity are simultaneously modulated too. When the frequency of modulation is f, the third harmonics 3f is detected with a loch-in amplifier because the 3f components include only the true two-photon correlation. Practically it is difficult to modulate with frequency f without any higher order harmonics distortion which might add some false 3f components. To overcome this difficulty we have used a sharp bandpass filter of l00～350 MHz in each branch of the correlator, which is lower than the RF frequency 500 MHz and much higher than 1.6 MHz, the revolution frequency of the stored beam of the 2.5 GeV storage ring. This new apparatus has been operated successfully in the measurement of the horizontal two-photon correlation for the first harmonic of undulator radiation with photon energy of 70 eV at the Photon Factory, KEK. By narrowing the precise slit width which correspondingly changes the spatial coherence of the incident SR, a bunching effect of the normalized excess two-photon correlation has been clearly observed. This explicit bunching effect implies that synchrotron radiation is chaotic radiation. Further investigation shows that although second-order coherence is completely determined by the first-order coherence for the case of chaotic light, the measured information from the light source is essentially different. The two-photon correlation of synchrotron radiation does not depend on the response time of the detectors but gives the information of instantaneous emittance of the stored beam with the time scale of coherence time τC. By fitting the experimental data, the horizontal instantaneous emittance of the stored beam is estimated to be 40nmrad. This intensity interferometer can be utilized to characterize the coherence properties of incomplete FELs, such as SASE, because if they are fully coherent light sources the normalized excess two-photon correlation would have a flat response, but not showing a photon-bunching effect.
Abstract We present an experimental technique that can accurately measure the high-order photon correlations of various light fields based on double Hanbury Brown–Twiss scheme. The system consists of four conventional binary response single-photon counting modules which are non-photon-number-resolving detectors. We theoretically analyze the effects of the experimental conditions on the high-order photon correlations of coherent, chaotic thermal, Fock and squeezed vacuum states with considering the overall efficiency and background noise. In experiment, we determine the proper resolution time from 2 9 = 512 ns to 2 11 = 2048 ns and counting rate from 10 kc s −1 to 100 kc s −1 for characterizing the exact second-, third- and fourth-order photon correlations of chaotic thermal state. At complete time delays, we also observe the third- and fourth-order photon correlations of chaotic thermal state, with the maximum high-order correlation values of 5.99 ± 0.02 and 23.9 ± 0.2 respectively, which are in good agreement with the theoretical values of 6 and 24. The demonstrated experimental technique with high accuracy is a promising tool for investigating quantum optical coherence and ghost imaging.
We investigated the properties of various light fields based on the Hanbury-Brown-Twiss(HBT)experiment consisted by two single-photon-counting modules(SPCMs),one 50/50beam splitter and the Fast-Com P7888card.Having experimentally investigated the second-order degree of coherence g2(τ)for the continuous-wave and the pulse lasers and the single-photon source generated by single atom,we confirmed the experiment results by theoretical analysis.Specially,we have achieved a high quality of single-photon source based on single atom trapped in the magnetic-optical trap and the far-off-resonance trap with g(2)(τ=0)=0.09and g(2)(τ=0)=0.08,respectively.Using the pulse laser and the single atom,we can perform the triggered single photon experiment.
Two-photon correlation phenomena, including the historical experiment of Hanbury Brown and Twiss, may have to be described quantum mechanically, regardless of whether the source of radiation is classical or quantum. Supporting this point, we present a ghost imaging type of second-order spatial correlation experiment of chaotic light to show that the classical understanding based on the concept of statistical intensity fluctuations does not give a correct interpretation for the observation. From a practical point of view, this experiment demonstrates the possibility of having high contrast lensless two-photon imaging with chaotic light, suggesting imaging applications for radiations for which no effective lens is available.
Photon statistics is a branch of physics that involves observing the behavior of single or a small number of photons and then aggregating the results through statistical averaging. This process is essential to obtain meaningful and reliable data, especially in quantum optics research. Photon statistics provides valuable information about correlations in the electromagnetic field and characteristics of the type of light being measured. For example, second-order correlation function obtained via Hanbury Brown and Twiss experiment is the fundamental way to differentiate between coherent, chaotic and quantum light. Moreover, extensive investigations into the quantum properties of light resulted in significant progress that allows for precise control of quantum optical systems and sets the stage for genuine quantum engineering. This progress extends to applications in confirming the generation of entangled photons and exploring their properties for various purposes. The non-classical correlation between the entangled photons has resulted in various quantum optical techniques that enable the practical implementation of conceptual experiments that probe the core principles of quantum theory. This enhanced control over quantum phenomena offers exciting opportunities to explore novel information-processing approaches, with the potential to revolutionize technologies rooted in quantum information science. Driven by the significance and pivotal role of photon statistics, our research pursuits were directed toward two specific domains: the characterization of nanolasers and the characterization of time-bin entangled photons. In the first segment, we delve into the realm of metallic coaxial nanolasers, renowned for their intriguing threshold-less lasing behavior. Through comprehensive photon statistics analysis, we seek to unravel the coherence properties inherent in these nanolasers. Our focus also lies on understanding the elusive threshold-less operation, which has remained a subject of debate and intrigue. To evaluate this unique lasing characteristic, we conduct rigorous second-order coherence measurements alongside time-resolved second-order correlation analyses. The second segment of this study ventures into the generation and characterization of high-dimensional time-bin entangled states, a critical resource for quantum technological applications. We simulate a synthetic photonic lattice by employing a coupled fiber-loop system to perform various quantum walk schemes to generate and process high-dimensional time-bin entangled states. Our investigation focuses on the quantum interference phenomena through detailed photon statistics analysis, with an emphasis on their applications across diverse quantum technologies. Through these two interconnected research endeavors, this thesis demonstrates the profound significance of photon statistics as a versatile tool, bridging classical and quantum realms, and paving the way for advancements in diverse applications, ranging from nanolasers to quantum technologies.