1. Introduction

Few problems in physics have received more attention than the complex nature of light [1,2]. Our modern understanding of light was triggered by Planck’s empirical blackbody radiation formula [3,4]. Einstein can be credited with many significant contributions, especially his explanation of the photoelectric effect [5] and the derivation of Planck’s blackbody law by use of the concept of light quanta [6,7]. It is this concept that also defines what we refer to as “light” in this paper, owing to its detection through the photoelectric effect. The founders of quantum theory furthermore left us with key insights, such as Heisenberg’s uncertainty principle [8], which underlies the important diffraction limit and Schrödinger’s puzzling concept of entanglement [9], famously criticized by Einstein as “spooky action at a distance” [10]. The existence of entanglement, arguably the most nonclassical manifestation of the quantum formalism, has by now been convincingly proven and underlies the evolving field of quantum information science [11–14].

1.1. From Single- to Multi-Photon Interference

The name photon was coined by Lewis in 1926 [15] and was formally defined a year later through Dirac’s quantization of the electromagnetic field [16,17], where the photon corresponds to an elementary excitation of a single mode of the quantized field. Dirac’s field quantization into monochromatic modes, also referred to as number or Fock states, defined by their wavevector $\overrightarrow{k}$ and polarization $p$, was the basis for his famous and bold statement that photons interfere only with themselves [18].

In his statement, which has led to much discussion and confusion over the years [19], he loosely used the word “photon” instead of “photon probability amplitude.” The statement was meant to explain the formation of Young-type diffraction patterns in the detector plane through one-photon counts at a time [20], with the final pattern forming with time as the mosaic of the number of single-photon counts deposited at different positions (see Section 2.6). The crucial point Dirac tried to convey is that the conventional diffraction pattern, today referred to as the first-order pattern, consists of a binary assembly of 0 and 1 photon counts, corresponding to the destructive and constructive interference of a single photon (wavefunction = field) with itself.

This picture of diffraction pattern formation was famously discussed for electrons by Feynman et al. [21] and has been beautifully verified experimentally [22–24] as discussed in Section 2.6. Feynman clearly recognized and most eloquently articulated the essence of quantum interference in his space–time formulation of quantum mechanics [25–27]. What interferes in quantum mechanics are not particles or photons but their complex space–time probability amplitudes between birth and destruction (detection). We shall introduce and extensively utilize the important concept of probability amplitudes in this paper for the discussion of both one-photon and two-photon diffraction.

For independent single photons, the complex probability amplitudes assemble the same way as complex fields through superposition (addition) to a single complex amplitude. In both cases, its absolute value squared gives the real detection probability at a given point. Remarkably, Feynman showed [27] that the conventional wave description of light emerges from quantum electrodynamics (QED), the fundamental theory of light, in first order. This resolves the schizophrenic “wave–particle duality” through the equivalence of the classical wave concept and the QED first-order probability amplitude concept associated with independent photons.

The interference of single-photon probability amplitudes considered by Dirac and Feynman was extended in the mid-1960s by Glauber [28–30], who developed the modern quantum theory of coherence based on the photon detection process. His concept of higher order degrees of coherence may also be understood in terms of a perturbation series description of QED, where first order coherence is described by single photon probability amplitudes, second order by two photon amplitudes, and so on [27]. Glauber also showed how the time-independent Heisenberg picture used in Dirac’s field quantization, resulting in monochromatic number or Fock states that are delocalized in time, may be phrased in the time-dependent Schrödinger picture, where through superposition of Fock states one may create polychromatic wavepackets [31], sometimes referred to as wavefunctions [32]. This naturally allows the definition of a coherence time in terms of the energy width of the wavepacket. Interference between wavepackets exists when the wavepackets overlap, i.e., if there is no possibility of “welcher Weg” (which path) distinction from the creation (birth) to the destruction (detection) events [33–35].

The two most fundamental quantum optics experiments that proved the indivisibility of single photons and the destructive interference of the probability amplitudes of two photons were carried out in the mid-1980s by Grangier, Roger, and Aspect [36] and Hong, Ou, and Mandel [37], respectively. They are discussed in Section 4. The two-photon case naturally extends Dirac’s first order to Glauber’s second-order quantum description. In first order, 0 and 1 counts mean destructive or constructive interference of fields, or in Dirac’s language, 1 count represents the interference of a single-photon probability amplitude with itself. From a detection point of view, a single photon is an excitation of the electromagnetic field whose photon-number statistics has a mean value of one photon and a variance of zero. In second order, two photons can only lead to 0, 1, and 2 counts, and they correspond respectively to destructive, no, and constructive interference, respectively, of two-photon probability amplitudes at a single point in the detector plane.

Today, the term “photon–photon interference” or “multi-photon interference” has assumed common usage in the quantum optics community [33–35], as reflected by the title of this paper. Like Dirac’s statement, it needs to be understood as an abbreviation for “interference of two or more photon probability amplitudes.” The shorter expression simply reflects the fact that in $n$th-order coherence theory, the diffraction pattern is defined through the coincident detection of $n$ photons. Similar to the $n=1$ single-photon case, the $n$th-order diffraction patterns are formed with time as the mosaic of $n$ coincident counts.

In the optical and x-ray range, photons are detected through the created photoelectric charge. In the process the photon gets destroyed. This is mathematically treated through the action of a quantum mechanical operator. If $n$ photons are created simultaneously, their destruction is treated to occur simultaneously or in coincidence through the actions of $n$ destruction operators. This corresponds to the instantaneous collapse of the multi-photon wavefunction. Remarkably, in quantum mechanics, superposition and interference of photon wavepackets can exist even when their paths to two or more detector positions cannot be distinguished. In this case, interference is defined as the coincident count rate between detectors at different positions, and it is a purely quantum mechanical phenomenon. For this reason, the treatment of multi-photon detection through the photoelectric effect is a central part of Glauber’s theory.

Glauber’s and Feynman’s approaches explain the seminal Hanbury Brown–Twiss (HBT) experiment [38–42] in terms of the interference of two-photon probability amplitudes [2,27] rather than classical statistical intensity fluctuations [43,44]. Since intensity fluctuations naturally arise in QED through the photon-based graininess of light, the HBT result must naturally follow from the more fundamental quantum picture, and we shall derive it here using the probability amplitude concept. It just happens that for a chaotic source, the second-order quantum and corresponding statistical optics formalisms give the same answer. However, the quantum formulation includes experimental observations when two or more photons are involved that cannot be explained classically [33–35,45,46]. Here we will specifically review the evolution of diffraction from single independent photons to two correlated photons and highlight how the conventional diffraction limit can be overcome by two-photon interference and further reduced by multi-photon interference.

1.2. Objectives of the Paper

The present paper elucidates the concepts of coherence and diffraction by comparing the well-established wave or statistical optics [43,44] predictions to those of quantum optics [30,34,35,46]. In first-order coherence theory, the two approaches are shown to yield identical results, first pointed out by Mandel and Wolf in 1965 [47]. In second order, the classical field-based and quantum photon-based approaches are known to differ [48]. This is here directly shown for the cases of cloned and entangled biphoton diffraction. The concepts are elucidated by calculations of the conventional and biphoton diffraction patterns for the fundamental case of a circular source with a uniform intensity distribution, which historically has been the basis for the definition of the conventional diffraction limit [49].

The quantum approach, based on the interference of photon probability amplitudes, also reveals a remarkable duality between partial coherence in first and partial entanglement in second order, which was first recognized by Saleh et al. [50] and experimentally verified by the same group a year later [51]. While in first order the concept of chaoticity is the lack of first-order coherence, in second order the concept of entanglement is the lack of second-order coherence. Thus, the first-order concept of partial coherence with its chaotic and coherent limits has the second-order analogue of partial entanglement with its limits of entangled and cloned biphotons.

We furthermore explore the evolution of the far-field diffraction pattern with increasing order $n$ of coherence, following Glauber’s theory. The area of the far-field diffraction pattern, which determines the diffraction limit, is shown to decrease as $1/n$ with a concomitant increase of the central intensity by $n$ due to power conservation. For very large $n$, diffraction effects disappear and the pattern in the far field exhibits the same area as the source. This reveals the remarkable equivalence of the 0th-order ray-optics and the $n$th-order quantum-optics picture, where photons within the collective state may be envisioned to travel on particle-like trajectories. It supports a general view of coherence based on photon density, which in first order is equivalent to the conventional wave-based picture.

Another goal of the present paper is to introduce quantum optics and higher order coherence concepts to the broader x-ray community, which only recently has ventured into the exploration of effects outside the conventional first-order description of light. A broader understanding of the concepts of coherence is necessitated by the advent of x-ray free electron lasers (XFELs) [52–56]. It is envisioned that the exploration of higher order coherence in the x-ray regime may reveal new aspects of the complex nature of light, and it may well become a new paradigm for obtaining unprecedented structural resolution.

1.3. Formal Structure of the Paper

We start in Section 2 by reviewing the conventional concepts of “light,” defined in this paper as extending from the visible to the x-ray range, and the characteristic time scales associated with light and its detection. We then review the conventional concepts of coherence, photon degeneracy parameter, source brightness, and their link to the existence of a diffraction pattern. We point out that the conventional concepts provide only a first-order description and correspond to the detection of independent, noninteracting photons. We close the section by defining the diffraction limit as the central paradigm of optics and show that it follows from both the classical wave-based and quantum mechanical photon-based formalisms.

Section 3 presents an overview of the evolution of the study of two-photon phenomena. We then discuss the controlled generation of two photons, so-called biphotons, whose correlation in space–time is created by their simultaneous birth through a nonlinear process. We review biphoton creation by spontaneous parametric downconversion (entangled biphotons) and stimulated resonance scattering (cloned biphotons). We point out that stimulated emission and the cloned biphoton concept may also be viewed classically as a parametric amplification process, while the concept of entangled biphotons is truly quantum mechanical.

In Section 4 we review fundamental one- and two-photon experiments and the associated one- and two-photon detection schemes. In particular, we use Feynman’s two-photon probability amplitude concept to explain the HBT experiment. We then outline seminal experiments with entangled and cloned biphotons that reveal some of the unique interference and diffraction features relative to conventional experiments with independent photons.

In Section 5 we formally introduce the concept of first-order coherence in the frameworks of statistical and quantum optics. We review Zernike’s powerful statistical optics theorem for the propagation of coherence, and derive it quantum mechanically by use of one-photon probability amplitudes. This formally proves the equivalence of the field and photon descriptions of coherence in first order. We then apply Zernike’s theorem to the pedagogical case of a Schell model source, and derive the Fraunhofer diffraction formula for a coherent source and the van Cittert–Zernike formula for the coherent part of the intensity distribution for a chaotic source. The chaotic and coherent diffraction patterns are directly compared through the concept of a first-order coherence matrix. The equivalence of the wave and particle pictures is then established through the introduction of a single-photon probability amplitude or wavefunction. Partial coherence, i.e., the evolution between the chaotic and coherent limits, is elucidated through the example of a circular Schell model source.

Second-order coherence is introduced in Section 6. We start by formally extending the description of first-order coherence to second order and discussing their link. We then discuss the essence of two-photon probability amplitudes and how they are formed from one-photon amplitudes using Feynman’s rules. The general second-order propagation law for two correlated photons (biphotons) is derived and compared to Zernike’s first-order law for independent photons. The difference between the statistical and quantum optics formulations is emphasized. We illustrate the quantum approach based on the interference of two-photon probability amplitudes through the example of two photons that are correlated by their simultaneous birth and emitted either as an indistinguishable cloned pair or an entangled pair. We show how the concept of partial coherence in first order has its analogue in partial entanglement in second order. Partial entanglement is shown to reduce to the limits of entangled and cloned (i.e., second-order coherent) biphotons. The evolution of the two-photon diffraction pattern for the general case of partial entanglement is discussed for the case of a circular Schell model source, leading to the entangled and cloned limits.

In Section 7 we discuss how multi-photon interference overcomes the conventional diffraction limit. We start with a simple picture of entangled and cloned biphotons that reflects their diffraction behavior. We then generalize the case of cloned biphotons to $n>2$ cloned photons, and show that the one-photon diffraction limit is lowered by $1/n$. In the collective $n$th-order coherent case, the individual photons are shown to propagate on particle-like parallel trajectories. The reduced diffraction limit is discussed in terms of Heisenberg’s uncertainty principle. Finally, we present a remarkably simple picture of lateral coherence based on the creation and propagation of spatial photon density. The paper is summarized in Section 8.

2. Conventional Concepts of Light and Diffraction

What we refer to as “light” in this paper is distinguished from longer wavelength electromagnetic (EM) radiation through its detection process. It covers the range illustrated in Fig. 1(a).

 

Figure 1. (a) Definition of “light” within this paper, defined through its photoelectric detection process. The cycle time is given by ${\tau }_0=\lambda /c$. (b) The number of photons generated in a birth volume of order ${\lambda }^3$ by different sources [57–59], the so-called photon degeneracy parameter. (c) Coherence time ${\tau }_{\mathrm{coh}}$ associated with electronic decays in atoms. Here the lifetime of the upper electronic state defines the coherence time according to ${\tau }_{\mathrm{coh}}=\hslash /\mathrm{\Gamma }$, where $\mathrm{\Gamma }$ is the energy FWHM of the emitted Lorentzian line. (d) Coherence time ${\tau }_{\mathrm{coh}}$ of a thermal source, where oscillating electrons emit wavetrains that are interrupted by atomic collisions, indicated by vertical lines. The average time of the uninterrupted wavetrains defines the coherence time. (e) Typical range of cycle times (red) and coherence times (blue) encountered for optical and x-ray sources. In green we show the shortest detector time of interest, the time jitter ${\tau }_{\mathrm{jit}}$, as discussed in Section 2.2.

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The frequency of light is sufficiently high that direct detection of any induced charge oscillation, utilized for example in the microwave regime, is too fast to be observable. Light detection is accomplished through the photoelectric effect, and this process must be taken into account in the description of the measured intensity of the light signal. We start with a brief overview of typical parameters that characterize light sources.

2.1. Time Scales of Light and Photon Density of Sources

In classical electromagnetism, EM radiation consists of self-propagating fields that have separated from an accelerated charge. Such “acceleration fields” are distinguished from the Coulomb or “velocity fields” that remain attached to charges moving with a constant velocity [60]. EM radiation needs to be defined at least over one wavelength $\lambda$, and one may associate the volume ${\lambda }^3$ with the minimum birth volume of photons. Figures 1(a) and 1(b) summarize some key properties of the light spectrum emitted by different sources, such as the wavelength range, the quality of a source defined by the photon density it produces, called the photon degeneracy parameter, given here as the number of photons generated in a birth volume of $\simeq {\lambda }^3$ [59,61], and the fundamental oscillation time of the EM field, ${\tau }_0=\lambda /c$.

The second time scale is set by the (longitudinal) coherence time ${\tau }_{\mathrm{coh}}$, illustrated in Figs. 1(c) and 1(d) for two different processes of light generation. In Fig. 1(c) we illustrate the case of characteristic atomic emission due to electronic decays from an upper to a lower electronic state. The coherence time is defined by the 1/e lifetime width of the electronic transition according to ${\tau }_{\mathrm{coh}}=\hslash /\mathrm{\Gamma }$, where $\mathrm{\Gamma }$ is the FWHM energy-width of the emitted Lorentzian line shape. After a time ${\tau }_{\mathrm{coh}}$, the probability of finding the atom in the upper state has reduced to 1/e or to $\simeq 37\%$ of the start value. In Fig. 1(d) the coherence time of thermal light sources is illustrated. At a given temperature, an oscillating electron emits an EM wave until its emission is interrupted during a collision, as schematically indicated by vertical lines in Fig. 1(d). If the collision is elastic, the emission will resume with the same frequency but a different phase until another collision occurs. In general, the coherence time is then given by the average time interval between collisions. The existence of collision intervals adds frequency Fourier components that cause a finite energy bandwidth of Gaussian shape.

The range of coherence times is shown in blue in Fig. 1(e). For optical sources, the shown range extends from conventional sources to optical lasers with coherence times as long as $\sim 1\mathrm{ms}$. The dark blue region indicates the range for biphotons created by parametric downconversion discussed in this paper. For x-ray sources, the light blue range covers the inherent source coherence times up to the time scales reached by insertion of a monochromator shown in dark blue.

The temporal properties of modern synchrotron radiation (SR) sources are illustrated in Fig. 2. The electron bunches in a storage ring that generate SR have a typical length of about 3 cm or $\sim 100\mathrm{ps}$ and consist of $\sim {10}^{10}$ electrons. The temporal x-ray pulse length ${\tau }_{\mathrm{p}}$ is about the same as the electron bunch length. The individual electrons radiate coherent wavetrains over a time $\sim 1$ as for a bending magnet source and $\sim 100$ as for an undulator source owing to their longer path through about 100 undulator periods. In both cases, however, the wavetrains emitted by individual electrons are not correlated, so that the total $\sim 100\mathrm{ps}$ SR pulses are globally chaotic [62,63].

 

Figure 2. (a) Schematic of the electron bunches in a storage ring. (b) The temporal x-ray pulse length ${\tau }_{\mathrm{p}}$ is about the same as the electron bunch length. The wavetrains emitted by the individual electrons are coherent with coherence time ${\tau }_{\mathrm{coh}}\sim 1$ as for a bending magnet source and ${\tau }_{\mathrm{coh}}\sim 100$ as for an undulator source.

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The temporal nature of x-ray pulses in a self-amplified spontaneous emission (SASE) XFEL is schematically illustrated in Fig. 3. In this case a linear accelerator is used to compress the electron bunches to micrometer lengths, and the x-ray pulse length ${\tau }_{\mathrm{p}}$ is shortened to femtoseconds. Most importantly, the interaction of the emitted radiation with the electrons leads to a self-ordering effect in the electron bunch. The ordered micro-regions within a bunch emit coherently so that the x-ray pulse contains coherent intensity spikes of ${\tau }_{\mathrm{coh}}\sim 1\mathrm{fs}$ width. As a rule of thumb, the number of micro-regions scales with the total bunch length, and for a 10 fs pulse there are about 10 spikes, as shown in Fig. 3(b). There is no correlation between the spikes and the individual single pulses are globally partially coherent. The structure and intensity of the spikes varies from pulse to pulse, with an average over many pulses assuming a Gaussian distribution, as shown by the dashed black curve. The temporal average renders XFEL sources, despite their name, to be statistically chaotic [56]. It has recently been shown that external seeding of the FEL process, which is presently only possible in the XUV range, indeed generates pulses that statistically behave like a laser [64]. By use of ultrashort electron bunches of low charge or passing a bunch through a narrow aperture [52,65], it is possible to produce individual near transform-limited pulses of about 1 fs duration as shown in Fig. 3(c). Note, however, that the statistical average of such pulses is still chaotic.

 

Figure 3. (a) Electron bunches in a linear accelerator may consist of $\sim {10}^9$ electrons and through compression may have a length of $\sim 3\mathrm{\mu m}$ or $\sim 10\mathrm{fs}$. This comes at a price of lower repetition rate than for storage rings. (b) The temporal duration of the bunch and the x-ray pulse length are about the same (${\tau }_{\mathrm{p}}\sim 10\mathrm{fs}$). The wavetrains emitted by small regions of electrons ordered with a periodicity of the wavelength form coherent spikes with ${\tau }_{\mathrm{coh}}\sim 1\mathrm{fs}$. There is no correlation between the individual spikes and the total single pulse is globally spatially and temporally partially coherent. (c) A very short electron bunch of $\sim 300\mathrm{nm}$ may produce a single spike that is nearly transform-limited in the time-energy domain.

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2.2. Detector Time Scales

The third important time scale in modern quantum optics experiments is defined by the detector. In general, the most relevant of the various time scales associated with the detection process itself and the electronics depends on the experiment. Detector time scales also need to be considered in conjunction with other detector characteristics, such as the detection efficiency, dark count background, linearity of response, and the ability to provide photon energy and photon number resolution.

Photon detectors involve the conversion of a photon into an electrical signal. The timing characteristics of a detector are determined not only by the photon/electron conversion process but also by the associated electronics. Today, the two most-used detectors in quantum optics are semiconducting avalanche photodiodes (APDs) and superconducting single-photon detectors (SSPDs) [66–69].

APDs are typically made of Si, Ge, or InGaAs and offer low dark count rates, high detection efficiencies, and high count rates. Most APDs can operate at room temperature and, due to their low expense and common availability, have increasingly replaced photomultiplier tubes and microchannel plates [66]. In SSPDs, the detector element (typically a NbN nanowire) has to be kept well below the superconducting critical temperature, adding cost and experimental complexity at the benefit of the fastest response times (see below). Since the energy gap across which electrons are excited is about 2 to 3 orders of magnitude lower in a superconductor than in a semiconductor, for the same incident photon energy the avalanche electron charge is considerable larger in SSPDs than APDs.

For the one- and two-photon diffraction experiments discussed in this paper, the most important detector time scales may be summarized as follows.

Typically, APDs and SSPDs give only a binary response of 0 and 1 so that a multiphoton pulse triggers the same output signal as a single photon. This limitation is overcome by photon-number resolving detectors, which can distinguish between the arrival of single and two photons within a time window of about a nanosecond [67,73,74]. They have been less used for quantum optics experiments relative to APDs, but we will encounter an interesting application of such a detector in Section 4.4 below.

According to Fig. 1(e), the shortest time jitter, ${\tau }_{\mathrm{jit}}$, falls into the range of optical coherence times, but it is orders of magnitude longer than x-ray coherence times. For SR and XFEL sources one may, however, utilize their pulsed nature to correlate signals observed at different positions within the duration of each pulse, with a temporal average obtained by summing over many pulses, as pioneered by Vartanyants and collaborators [53,62,63].

2.3. Fields, Intensities, and Photons

In the conventional framework based on the wave–particle duality, light is born locally as a photon, propagates as a wavefield, and is detected in a local detector pixel as a photon that generates charge through the photoelectric effect. From a wave point of view, in the detection process, energy equivalent to that of a single photon is extracted locally from the total electromagnetic field. The process is governed by the electric field $\overrightarrow{E}$ of the EM wave, and the measured intensity of dimension [energy / (area × time)] is linked to the field strength, number of photons, and their energy through the relation

2.4. Nature of Incoherent Sources

A source or secondary source is called “incoherent” or “chaotic” when the intensity distribution in a distant detector plane shows no structure, i.e., no distinct diffraction pattern. Classically, the mathematical description of a chaotic source is based on the fundamental notion that radiation can propagate away from the generating electric charge only if it is defined at least over the dimension of the mean wavelength [75,76].

In practice, the description of an incoherent source is treated as the limiting case of a partially coherent source whose coherence area shrinks to the size of the wavelength. The description of this limit is facilitated by Wolf’s concept of quasi-homogeneity [61,77]. In quasi-homogeneous sources, the intensity distribution varies slowly while the coherence may vary fast from region to region. In practice one assumes that such a source has a well-defined macroscopic intensity distribution while it contains uncorrelated microscopic coherence regions. Hence, from a global perspective, the source is spatially chaotic.

The concept of quasi-homogeneity is typically used together with the assumption that the correlation (coherence) function of all microscopic regions has the same functional form. For example, a blackbody source with a broad bandwidth spectrum actually contains small regions with a sinc-function $\sin (kr)/(kr)$ field–field correlation between two points separated by $r$ [43,78,79]. The best description of a quasi-monochromatic source that emits a characteristic line is through an Airy correlation function of the form $2J_1(kr)/(kr)$ [75,78,79].

A SR or XFEL source, on the other hand, contains microscopic Gaussian electron distributions, $\exp [-k^2{r}^2]$, that emit x-ray pulses with no discernible internal coherence structure for a storage ring source but coherent “spikes” for an XFEL source, as shown in Figs. 2 and 3. The radiation is also emitted preferentially in the forward direction due to relativistic effects, as addressed later in Section 7.6.

Mathematically, the description of lateral incoherence is facilitated by treating the small-area limit of any peaked field correlation function as a cylindrically symmetric 2D Dirac $\delta$-function. We adopt the notation that the dimensionality of the $\delta$-function is defined by the dimensionality of its argument, which in our case is a 2D vector ${\overrightarrow{r}}^{\prime }={\overrightarrow{r}}_2-{\overrightarrow{r}}_1$. Denoting the microscopic coherence area of any peaked function as $A_{\mathrm{coh}}^{\mu }$, we may write the following explicit expressions for 2D Gaussian, flat-top, and Airy distribution functions:

2.5. Diffraction of a Sample Illuminated by an Incoherent Source

The existence of diffraction when a sample is inserted into a beam emitted by an incoherent source is a consequence of coherence propagation, as illustrated in Fig. 4. While the longitudinal coherence time remains unchanged upon propagation in vacuum, the lateral micro-coherence areas in the source expand upon propagation, and their overlap defines the size of the coherence areas at the sample position shown as a red circle in Fig. 4(a). For a crystalline sample, the circle typically covers several lattice unit cells. The periodicity of the lattice then leads to the same diffraction patterns for micro-regions that are individually illuminated coherently, without the requirement of longer range x-ray coherence between micro-regions [80].

 

Figure 4. Coherence requirements for x-ray diffraction. Because of the expansion of the small coherence areas in a quasi-homogeneous globally incoherent source, there will be coherence areas in the distant sample plane due to overlap of coherence cones from different source areas, indicated by red circles in a distant plane. (a) For a single crystal sample, the beam will be coherent over regions that cover several atomic unit cells. The periodicity of the lattice then leads to the same diffraction patterns for all coherently illuminated micro-regions. (b) For a sample with non-periodic nanoscale order, the micro-regions have random structure and the coherent patterns from different regions will superimpose incoherently. The reciprocal space diffraction pattern consists of a fuzzy ring of average diameter $Q=4\pi /d$, where $d$ is the statistical correlation length of the domains. (c) If the same sample is moved away from the source it can be entirely illuminated coherently. The resulting speckle pattern encodes the real space structure of the entire sample. For a circular aperture, the central bright spot has a diameter determined by the diffraction limit, as discussed in Section 2.9.

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When the sample is non-periodic, as shown in Fig. 4(b), the interference patterns from small domains in the sample are no longer identical and therefore the diffraction pattern is blurred. The conventional small angle scattering pattern corresponds to the incoherent superposition of the coherent patterns from different domains. The intensity distribution consists of a ring-like pattern located at a momentum transfer $Q=2\pi /d$, where $d$ is the statistical correlation length of the domains.

By moving the sample away from the source, as shown in Fig. 4(c), one increases the effective coherence area at the cost of coherent intensity. If the entire sample is coherently illuminated, the scattered intensity recorded by an imaging detector is “speckled.” The speckle pattern contains the complete information on the real space structure within the illuminated area. For a circular aperture, the central bright spot has a diameter determined by the diffraction limit, as discussed in Section 2.9. In practice, the inversion of the reciprocal space diffraction pattern into real space requires the use of holographic reference beams [81,82], the exploitation of the strong dependence of the optical constants near a resonance [83,84], or computer-based phase retrieval algorithms [85].

2.6. Diffraction and the Wave–Particle Duality

To account for interference and diffraction, we still use the magical Huygens–Fresnel principle that each point may be viewed as the origin of a spherical wave of uniform amplitude and an outward traveling phase front, referred to as a wavelet. The concept of spherical waves around all possible source points, e.g., atoms in a sample, has been a great aid since it produces the experimentally observed diffraction pattern and it underlies the first Born approximation [86].

Remarkably, one can use the Huygens–Fresnel principle of spherical waves emanating from points even when calculating the diffraction pattern of an illuminated hole where there are no real atoms that can scatter! In this case, one may invoke Babinet’s principle as an explanation, which states that the far-field diffraction pattern is the same for an opaque object surrounded by “nothing” and for a hole (“nothing”) replica of the object surrounded by opaque matter. From a fundamental point of view, both the Huygens–Fresnel and Babinet principles are unsatisfactory, since the complete intensity distribution in the detector plane is simply calculated by forming the absolute value squared of the field interference pattern without the need to consider photons, which after all are the building blocks of QED, the complete theory of light.

The first demonstration that the diffraction pattern accumulates through the addition of single photon counts was carried out in 1909 by Taylor [87], who recorded the diffraction pattern of the tip of a needle by using an extremely weak gas flame source that was so weak that it required three months to record the pattern. In 1961, Jönsson [22] repeated Young’s double-slit experiments with electrons. It appears that this experiment was not known to Feynman, who prominently discussed it in his 1962 lectures as a “Gedanken” experiment, published later in Volume III of his lecture series [21]. Jönsson’s experiment showed that the use of electrons gave the same pattern as photons. Improvements of the experiment clearly revealed how the diffraction pattern assembles with an increasing number of electrons [23,24]. It has also been repeated with increasing number of photons [88], neutrons [89], and large organic molecules [90] with similar results. As shown in Fig. 5, the diffraction pattern builds up probabilistically through single “particle” processes.

 

Figure 5. Assembly of the double-slit diffraction pattern, one photon at a time. The granular patterns on the right were actually recorded with single electrons [24], but the photon case is the same [88]. The yellow curve is the intensity distribution in the limit of recording a large number of counts.

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These experiments have provided clear evidence that the diffraction pattern emerges as a point-by-point mosaic of single particle detector clicks. Young’s double-slit experiment therefore proceeds by single-photon processes whose probabilistic repetition leads to the buildup of the diffraction pattern as shown in Fig. 5. The yellow curve is the intensity distribution in the limit of recording a large number of counts. The experimental results support Dirac’s famous statement: “…each photon interferes only with itself. Interference between different photons never occurs” [18]. It is likely that Dirac knew about Taylor’s experiment.

In 1909 Einstein derived a fluctuation formula for Planck’s blackbody radiation that contained a wave and a particle term and suggested an inherent wave–particle duality in the nature of light that combines two seemingly different concepts [91]. By using such a semiclassical combination of the wave and particle concepts, one may naively envision three steps leading to a diffraction pattern, as indicated at the bottom of Fig. 5. A photon is created by an electronic transition in an atom. It then expands into space with the speed of light and acts as a wave that may be diffracted by a sample put in its path. The shape of the diffraction pattern is determined by wave interference. However, a single photon is detected at only a single point in the detector plane, where an energy packet $\hslash \omega$ is locally taken from the total wave field. The wave collapses into a photon again. Local photon to electron conversion in a detector pixel then leads to a “bright spot.” While one can envision the build-up of the diffraction pattern by such a naive semiclassical model, it has the flavor of science fiction where “phase shifting” between particle and wave is allowed. Below we will discuss an entirely quantum mechanical derivation of diffraction.

2.7. Diffraction and Quantum Electrodynamics

Today, quantum electrodynamics (QED) is regarded as the fundamental theory of light and its interactions with electronic matter. It can account for the most sophisticated experimental results, such as the Lamb shift of the energy states of hydrogen and the anomalous magnetic moment of the electron. In principle, it must therefore be able to account for all conventional laws of optics. In practice, the optical laws governing absorption, reflection, refraction, and diffraction are typically derived from a classical wave picture. While this is partly due to the historical development of optics, the general theory of QED also appears too complicated and intimidating. Many of the QED complications may be avoided, however, by Feynman’s space–time formulation of quantum mechanics [25]. In lowest order perturbation theory, the space–time formulation leads to simple rules regarding the addition and multiplication of single-photon probability amplitudes [26], which are used in Feynman’s 1985 book [27] to derive the conventional laws of optics. In the same book, Feynman also reveals the amazing fact that within first order, individual photons and electrons are both described in QED by the same single particle probability amplitudes, which explains the experimental observations discussed in the previous section.

Feynman’s treatment is remarkable in that it exposes the very origin of the schizophrenic wave–particle duality, which suggests that light acts sometimes like a wave and sometimes like a particle. In QED, this ambiguity does not exist since light is fundamentally granular or quantized and composed of photon building blocks. The remarkably simple solution of the wave–particle dilemma is that the complex QED theory may be approximated in first order by a wave theory with its own governing principles. Examples of these principles are the magical Huygens–Fresnel principle and the concept of spherical waves. Although light is fundamentally composed of photons, the wave picture with its own associated physical principles or postulates may then simply be used as a convenient approximation.

We will use Feynman’s quantum approach to diffraction for different cases below. In Section 2.9 we will derive the fundamental diffraction limit by associating independent probability amplitudes with each photon. In this case, the individual complex probability amplitudes for each alternative space–time path between the birth and destruction of photons are added at a given detector point. The detection probability at a given point is then calculated as the absolute value squared of all superposed (added) probability amplitudes. This quantum approach is obviously analogous to the superposition of complex fields in wave optics, with the intensity calculated as the absolute value squared of all superposed fields.

The equivalence of the photon and wave calculations in first-order QED breaks down in second and higher order, which is a key topic discussed in Section 6. Historically, this became apparent through Glauber’s treatment of higher order coherence, which was largely motivated by the HBT experiment discussed in Section 4.1. In second order, one accounts for a quantum correlation between two photons that are born simultaneously in space–time. In this case, single-photon probability amplitudes are first multiplied to obtain a two-photon probability amplitude, reflecting a concomitant propagation and simultaneous (coincident) detection. As for the first-order single-photon case, the correlated (multiplied) two-photon amplitudes for alternativetwo-photon space–time paths are then added. The absolute value squared of all summed two-photon amplitudes constitutes the two-photon detection probability. We will show in Section 4.2 that this quantitatively accounts for the HBT effect.

Today, diffraction calculations based on particle concepts are sometimes based on alternative formulations of quantum mechanics by Madelung [92], de Broglie [93], and Bohm [94–96], referred to as the pilot-wave or quantum trajectory method [97–100]. A key ingredient of the theory is Bohmian mechanics [95,96], which through a deterministic or “causal” interpretation of quantum mechanics can provide information about particle trajectories. The particles perform quantum mechanical fluctuations around the trajectories that give rise to Heisenberg’s position–momentum uncertainty. The fluctuations may be viewed as arising from the interaction of the particles with the stochastic zero-point field and also lead to a finite coherence length. The trajectories of single photons in a double-slit experiment have recently been reconstructed from experiment [101].

It has been pointed out, however [102,103], that the quantum trajectory calculations for the double-slit experiment [97] disagree with experimental data recorded with different types of particles and photons, which all give the same diffraction pattern. In the quantum trajectory calculations, the intensity of the central peak is too large and of the side peaks too small. This suggests that the quantum trajectory method overestimates the particle-like behavior. This may be explained, as discussed in Section 7.5, by higher order QED contributions to the quantum trajectory method.

2.8. Link of Diffraction Pattern, Degeneracy Parameter, Brightness, and Coherence

Figure 1(b) reveals the radical change in the photon degeneracy parameter in the optical regime with the advent of the laser and in the x-ray regime from x-ray tubes to XFELs. Before lasers and XFELs, the degeneracy parameter was less than unity, so all early diffraction experiments with conventional optical sources and even the best SR sources were carried out one photon at a time. This naturally supports the assembly of diffraction patterns illustrated in Fig. 5. It raises the question, however, whether the diffraction pattern changes with the increase of the photon degeneracy parameter.

Experimentally, one finds that Young’s experiment carried out with the slits illuminated by a distant monochromatic conventional optical source or a laser give the same pattern. Similarly, x-ray tubes, synchrotron, and XFEL radiation with degeneracy parameters differing by a factor of $\sim {10}^{25}$ all give the same Bragg diffraction patterns! It therefore does not matter whether the photons arrive in the detector plane one after the other or all together. In both cases, their energy is deposited probabilistically to create the final pattern, as shown in Fig. 5. This agrees with Dirac’s statement that photons do not interfere with each other, and we can therefore state that the conventional diffraction pattern does not depend on the photon degeneracy parameter.

The physics behind this statement, however, is non-trivial and we need to explore it in more detail since the title of our paper indicates that “multi-photon interference” exists and may indeed change the diffraction pattern. Dirac’s quantization of the EM field [16,17] resulted in so-called number or Fock states [45]. Photons in these states occupy the same wavevector $\overrightarrow{k}$ and polarization $p$ modes, and the photon degeneracy parameter is nothing but the number of photons $n_{p\overrightarrow{k}}$ in the same mode $p$ and $\overrightarrow{k}$. They can be envisioned to be born in a source volume of order ${\lambda }^3$. As the name implies, number states $|n_{p,\overrightarrow{k}}⟩$ have a well-defined number and corresponding field amplitude, but their quantum phase is random [45,104]. While for a small number $n$ a single observation may still produce interference [105,106], it increasingly washes out when one averages over many observations, corresponding to forming a statistical ensemble average or a quantum mechanical expectation value. We shall see below that the interference washout does not exist in other forms of light called coherent states, as discussed in conjunction with cloned biphotons in Section 7.2.

The photon degeneracy parameter is directly linked with the concept of spectral brightness or brilliance of a source, which was introduced for the quantitative description of the quality of SR sources by Kim [107], following an earlier paper that focused on x-ray coherence [108]. The brightness is defined as [109–112]

The so-defined brightness also defines the coherence of the source. At first sight this connection is puzzling since coherence is usually defined by assuming that light behaves as waves and then demanding that the waves are in-phase. In contrast, the brightness expression does not contain any phase information, and on the right side of Eq. (4) is expressed simply as the density of photons. This indicates that coherence may be phrased either in a wave or a photon picture. In the latter, the coherence of a source is simply determined by the number of photons it produces in the coherence volume $\simeq {\lambda }^3$. We will see later in Section 7.5 that the coherence concept based on photon density or energy density of the EM field is the most general description of coherence.

It turns out that the concepts of coherence, brightness, degeneracy parameter, and diffraction limit constitute only a first-order description of the properties of light. “First order” is most elegantly defined by the quantum theory of light, where it means non-interacting and non-interfering photons. In quantum optics, the classical field $\overrightarrow{E}$ and its complex conjugate ${\overrightarrow{E}}^{*}$ are replaced by field operators ${\mathbf{E}}^{+}$ and ${\mathbf{E}}^{-}$(see Section 5), which act as raising or creation and lowering or destruction operators. In quantum mechanics, photons are created by these operators out of the zero-point quantum vacuum and destroyed back into it. In first order, treated by Dirac, the detection process consists of the destruction of a single photon.

Glauber, in a series of papers [28–31,113], went a step further and considered the possibility that two or more photons were simultaneously destroyed, leading to a substructure of light where $n$th order of coherence means $n$ indistinguishable photons or clones. In second order, one considers the simultaneous creation of two photons and their simultaneous destruction at a later time. At a given point in the detector plane, there are only three possible outcomes within a time interval defined by the two-photon creation process in the source, corresponding to the arrival of 0, 1, or 2 photons. If we detect zero photons (0 counts), the two photons must arrive elsewhere; if we detect one photon (1 count), the other photon must arrive elsewhere; and the last possibility is that we indeed detect both photons (2 counts). The three cases correspond to destructive interference, no interference, and constructive interference of the two-photon probability amplitudes at a given point in the detector plane, respectively. More generally, in $n$th order, $n$ photons are simultaneously born and simultaneous destroyed at the same or different points in the detector plane.

Before going into the details of multi-photon phenomena, we need to briefly review a key concept of conventional optics, the definition of the conventional diffraction limit. It arises in a quantum picture by treating all photons as being independent.

2.9. Central Paradigm of Conventional Optics: the Diffraction Limit

The central dilemma of wave optics or first-order diffraction theory is the finite width of the central Airy disk in a distant detector plane produced by a uniformly illuminated circular aperture, as illustrated and defined in Fig. 6. This problem has haunted astronomers for centuries, since the finite size of the aperture of a telescope necessarily limits the resolution due to diffraction. Through the years, opticians have tried to reduce the width of the detector plane Airy disk through modification of the intensity distribution across the aperture, e.g., through non-uniform coatings. It was found that because of power conservation, the narrowing of the width of the central Airy disk always comes at the expense of an increase of the intensity of the outer rings [114], which reduces the sharpness of the total image in a telescope.

 

Figure 6. (a) Assumed geometry of a flat-top secondary source of area $\pi R^2$ that is coherently illuminated. The inset above the aperture shows the 1D projection of the flat-top field and intensity distribution. (b) 1D projection of the Airy intensity distribution (black curve) plotted on a logarithmic scale. The blue curve is for an inserted film that transmits $1/e$ of the intensity. (c) Enlarged central part of the Airy pattern for a hole (black) and effective flat-top distribution (red), plotted on a linear scale. The areas under the two curves, given by $B_{\mathrm{eff}}=\pi {\rho }_{\mathrm{eff}}^2$, contain the same power. The Rayleigh diffraction limit ${\rho }_0=0.61\lambda z_0/R$ is defined by the first zero of the Airy intensity.

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The diffraction pattern of a uniformly illuminated circular aperture of area $\pi R^2$ in a detector plane at a large distance $z_0$ is the shown Airy diffraction pattern. It is typically derived by use of the Huygens–Fresnel principle, where each point in the circular aperture emits a spherical wave [49]. The diffracted intensity is obtained by considering the interference of the different waves at a given detector point. The diffracted intensity is found to change with the distance $\rho$ from the optical axis as (see Section 5.6.a)

2.10. Photon Probability Amplitude Formulation of the Diffraction Limit

To show the complete equivalence of the classical wave-based and the quantum mechanical photon-based derivation of the conventional diffraction pattern, we use Feynman’s space–time formulation of quantum mechanics [25–27,115] and calculate the diffraction limit for the example of a circular source of uniform average intensity. This is done as illustrated in Fig. 7 by use of the concept of single-photon space–time probability amplitudes $\varphi (\overrightarrow{r},\overrightarrow{\rho })$, which are associated with the birth of a photon at position $\overrightarrow{r}$ and the destruction of a photon at position $\overrightarrow{\rho }$ in a distant detector plane.

 

Figure 7. Quantum mechanical description of diffraction by means of one-photon space–time probability amplitudes ${\varphi }_{\mathrm{AX}}=\varphi (\overrightarrow{r},\overrightarrow{\rho })$. Each probability amplitude is defined as the product of a phase factor associated with a point of birth A (coordinate $\overrightarrow{r}$) and with propagation along the path $\overline{\mathrm{AX}}$ of length $r_{\mathrm{AX}}$ to the point of detection X (coordinate $\overrightarrow{\rho }$), according to Eq. (6). For a finite size source, the one-photon amplitudes from all source points ${\overrightarrow{r}}_i$ to a general detection point $\overrightarrow{\rho }$ are added, which in practice is done by integration over a general source point coordinate, yielding the total one-photon amplitude at a detection point $\psi (\overrightarrow{\rho })$, given by Eq. (8). The diffraction pattern consists of the expectation value (many single-photon events) of the detection probabilities ${|\psi (\overrightarrow{\rho })|}^2$ at different points, which accumulate through binary counts of either 0 or 1.

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The single-photon amplitude for the path length $r_{\mathrm{AX}}$ may be written as

The birth is defined through a phase constant ${\alpha }_A=\alpha (\overrightarrow{r})$ that depends on the place of birth $\overrightarrow{r}$, and the propagation phase $k{r}_{\mathrm{AX}}=2\pi r_{\mathrm{AX}}/\lambda$ depends on the propagation path length $r_{\mathrm{AX}}$. The fact that the source and propagation amplitudes in Eq. (6) are multiplied follows from Feynman’s rule that the amplitudes of consecutive events are multiplied, corresponding to the addition of associated phases [27]. Owing to the constant speed of light, the propagation may also be expressed by the travel time $\omega t_{\mathrm{AX}}=\omega r_{\mathrm{AX}}/c$, hence the term “space–time.” Each one-photon probability amplitude is normalized to reflect a single photon according to ${|\varphi (\overrightarrow{r},\overrightarrow{\rho })|}^2=1$.

Different source points $\overrightarrow{r}$ and a given detection point $\overrightarrow{\rho }$ are linked by alternative space–time paths. The final one-photon amplitude $\mathrm{\varphi }(\overrightarrow{\rho })$, which determines the detection or destruction probability at point $\overrightarrow{\rho }$ according to ${|\mathrm{\varphi }(\overrightarrow{\rho })|}^2$, is therefore determined by alternative places of birth and propagation paths according to

At large average source–detector separation $z_0$, the so-called Fraunhofer diffraction limit, we can express the path length $r_{AX}$ in Fig. 7 by assuming $r,\rho \ll z_0$, the so-called paraxial approximation, and obtain

As discussed in Section 2.4, for a chaotic source there are only microscopic coherence regions of average size $A_{\mathrm{coh}}^{\mu }$, so that the integration over the source in Eq. (10) can be conveniently expressed by a Dirac $\delta$-function. The cases of a chaotic and coherent source then can be distinguished by averaging over many emission events, corresponding to the expectation values:

The diffraction limit has been overcome in the past in the imaging of biological specimens by utilizing the nonlinear response of fluorophores emitters. In 2014, the Nobel Prize in Chemistry was awarded to Betzig, Moerner, and Hell for “the development of super-resolved fluorescence microscopy” [117]. These developments are examples of the general principle developed in this paper that the diffraction limit can be overcome by use of nonlinear electronic response in the object whose diffraction pattern or real space image is of interest. As the name implies, nonlinear interactions with the sample induce photon correlation effects that appear only in second and higher order.

In this paper we will specifically address the fundamental case illustrated in Fig. 6, which defines the diffraction limit as a result of first-order coherence, and demonstrate how the conventional diffraction limit, defined by the area of the central Airy peak, can be overcome with a concomitant disappearance of the weak outer rings. As an introduction of the formal treatment, we shall review the case where instead of two photons being born independently at two points, their birth process is correlated in space–time.

3. Overview of Two-Photon Interference and Diffraction

While conventional diffraction is based on individual non-interacting photons, there are interesting cases, where two photons are born simultaneously in space–time and maintain some kind of correlation upon propagation to the detector, where they arrive simultaneously at the same point or two different points. In contrast to the one-photon case in which the photon probabilistically deposits its energy at a single point in the detector plane, in two-photon diffraction, the two photons may deposit their energy simultaneously at one or two points. The correlation of the two-photon destruction events contains cases that have no classical analogue. In this section we first give a short overview of the historical development of two-photon experiments and the controlled generation of biphoton probability amplitudes or wavepackets.

3.1. Historical Overview of Two-Photon Experiments

The two-photon (boson) correlation case is of similar fundamental importance as the two-electron (fermion) correlation case, with the two cases being distinguished by the symmetry of the total wavefunction according to the symmetrization postulate [86]. The case of two electrons, first studied for the two-electron He atom and ${\mathrm{H}}_2$ molecule, famously laid the foundation for Pauli’s exclusion principle [80]. Similarly, the case of two photons has allowed fundamental tests of the nature of light and quantum theory. The two-photon case also exhibits the most dramatic departures from the classical field-based description of light.

Schrödinger already recognized in 1935 that two electrons (fermions) or photons (bosons) may form states that he referred to as verschränkt or entangled [9]. Entanglement can exist in different physical properties, such as position, momentum, spin, energy, and polarization. For example, spatial entanglement between two particles described by one-particle wavefunctions $\varphi ({\overrightarrow{x}}_1)$ and $\varphi ({\overrightarrow{x}}_2)$ exists if the total two-particle wavefunction can be written only in the form $\mathrm{\Psi }({\overrightarrow{x}}_1,{\overrightarrow{x}}_2)\propto \varphi ({\overrightarrow{x}}_1){\varphi }^{*}({\overrightarrow{x}}_2)+\varphi ({\overrightarrow{x}}_2){\varphi }^{*}({\overrightarrow{x}}_1)$, but not in the position-factored form $\mathrm{\Psi }({\overrightarrow{x}}_1,{\overrightarrow{x}}_2)\propto \varphi ({\overrightarrow{x}}_1){\varphi }^{*}({\overrightarrow{x}}_1)+\varphi ({\overrightarrow{x}}_2){\varphi }^{*}({\overrightarrow{x}}_2)$.

The existence of entanglement was famously related to the question of the completeness of quantum mechanics by Einstein, Podolsky, and Rosen (EPR) [10]. The first investigations of the concept of entanglement and the EPR paradox formulated in terms of Bell’s inequalities [118] were carried out by using two photons created in atomic cascade emission by Clauser and co-workers [119–121]. Many studies of entanglement have since followed, as reviewed in Refs.  [13,14,33–35,46,122]. They convincingly demonstrated the existence of entanglement, with the latest study showing spatial entanglement over ground–satellite distances of up to 1200 km [123].

Two-photon physics was experimentally pioneered in the late 1950s by Hanbury Brown and Twiss, who extended a radio-astronomy method of correlating the signals in two antennas to the optical range where the correlation was measured between two photodetectors [38,39,41,42]. A nice review of the experiment from a classical point of view has been given by Wolf [61], and we will discuss the experiment in Section 4.1 and give its quantum derivation in Section 4.2.

3.1a. Spontaneous Parametric Downconversion—Entangled Biphotons

Two-photon experiments were revolutionized by the discovery of spontaneous parametric downconversion (SPDC) in 1967 [124–126]. In SPDC, discussed in more detail below, a nonlinear process in a crystal without inversion symmetry is used to simultaneously generate two lower energy photons through fission of an incident photon [127,128]. For example, an incident photon of energy $\hslash \omega$ may be split into two photons of energy $\hslash \omega /2$. Quantum correlations between SPDC photon pairs were first observed in 1970 by Burnham and Weinberg [129]. In the 1970s and ’80s, SPDC was utilized by Mandel et al. [130–134] to investigate how the interference of two photons changes as a function of their temporal separation. These studies, reviewed by Ou [33], culminated in the study by Hong, Ou, and Mandel [37] that demonstrated the destructive interference of two temporally overlapping photon probability amplitudes, as discussed in Section 4.4.

From a spatial interference or diffraction point of view, two-photon experiments with SPDC photons were pioneered by Klyshko, who in 1982 gave the first description of their transverse correlation [135], and in 1988 suggested the concept of coincidence imaging and coined the term “biphotons” [127]. Coincidence imaging was experimentally demonstrated by Shih and collaborators in 1995 [136], and the related technique of “ghost imaging” was also introduced in the same year [137]. It was later shown to be possible with chaotic sources, as well [138,139].

In ghost imaging, photons are separated by a beam splitter and propagate to different detectors, where they are detected in coincidence. Remarkably, the image of a sample located in one beam is recorded by scanning the detector in the other beam that does not contain a sample. Ghost “images” can be recorded in either real or reciprocal space [140,141], and the technique has recently been extended to x rays [142,143]. The potential of ghost imaging in the x-ray regime lies in the fact that sample damage may be avoidable by subjecting the sample to the weaker part of an entangled beam while recording its image with a stronger one that does not traverse the sample [144].

In practice, experiments with entangled photon pairs, here referred to as entangled biphotons, are typically performed by coincidence correlation of the signals of two single photon detectors, although detectors exist today that can discriminate between the arrival of single and two photons [73,74], as discussed in Section 4.4. The coincidence method assures that the wavetrains of the two photons overlap in time. As illustrated in Section 4.5, the so-measured diffraction pattern of entangled biphotons is characteristic of the wavelength $\lambda /2$ of the incident photon that is fractured, and not of the individual wavelength $\lambda$ of the two downconverted photons. This was first demonstrated by Fonseca et al. in 1999 [145] and led to the proposal of “quantum lithography” below the conventional diffraction limit [146,147]. The diffraction behavior has been attributed to the entangled biphotons forming a collective new “quanton” [148,149] state of wavelength $\lambda /2$, that interferes with itself. Owing to the close link of multi-photon states to Bose condensates or “de Broglie matter waves” [150], the entangled biphoton has also been called a “photonic de Broglie wave” [151].

3.1b. Stimulated Emission—Cloned Biphotons

Stimulated emission, conjectured by Einstein in 1916 [6,7], constitutes another nonlinear process where two photons are born correlated in space–time. It may also be explained classically as the complement to absorption, with the two processes respectively describing the decrease and increase of energy of a driven system [152]. In modern optics, stimulated emission is a consequence of a nonlinear interaction, which may be pictured semiclassically as a parametric amplification process [153,154] and treated in terms of a third-order nonlinear susceptibility [155]. The stimulation process has been studied in the time domain by superimposing a coherent laser beam onto one of the beams generated by SPDC [156]. With increasing laser beam intensity, the effect of stimulation could then be directly observed by an increase of interference fringe visibility of the SPDC beam.

From a quantum perspective, stimulation leads to the simultaneous creation of two indistinguishable photons or cloned biphotons in an atomic decay [see Fig. 1(c)]. The maximum achievable cloning was quantitatively determined by a beautiful experiment carried out by Lamas-Linares et al. [157] in 2002. It proved the so-called “no cloning” theorem [133,158–160], which states that no quantum operation exists that can duplicate perfectly an arbitrary quantum state. The optimal cloning fidelity is 5/6 = 0.833, and the experiment yielded a value of $0.81\pm 0.01$.

The cloning limitation arises from consideration of emission by an atom in an excited state, whose decay probability scales with the well-known factor $1+n$, where 1 represents the relative spontaneous decay probability induced by the zero-point field and $n$ expresses the relative stimulated decay probability driven by the presence of $n$ photons in the same mode. If only a single incident photon, $n=1$, is available to stimulate the excited state decay, there is equal probability of spontaneous and stimulated emission. While the incident polarization is preserved in stimulated emission, it is arbitrary in spontaneous emission. Hence the presence of spontaneous emission negates perfect cloning of the incident photon. More generally, the essence of the no-cloning theorem is founded in the linearity of quantum theory.

The nature of cloned biphotons has been difficult to probe since they are indistinguishable in space–time. The question arises whether they behave differently than two photons in the same wavevector and polarization mode, i.e., the photons emitted on average by a source with degeneracy parameter 2? We have seen earlier that the diffraction pattern does not depend on the degeneracy parameter of the source, and therefore a source of degeneracy parameter 2 will produce the same pattern as a source of lower degeneracy parameter. Hence, the answer to our question depends on whether the diffraction pattern of a cloned biphoton source is different.

We will show below that the diffraction pattern indeed changes because photons in the same mode are only first-order coherent, while cloned biphotons are second-order coherent. The cloned biphoton diffraction pattern has the same spatial periodicity as the single photon one, reflecting the common wavelength $\lambda$, but the central intensity of the cloned biphoton image is increased by a factor of 2 and the image area is reduced by a factor of 2 below the conventional diffraction limit [59].

3.2. Creation and Properties of Biphotons

Biphotons are created simultaneously in space–time through a nonlinear fusion or fission process, which induces a “correlation” between them that remains after birth and manifests itself in the diffraction pattern. The creation of secondary biphoton sources for the entangled and cloned cases is illustrated in Fig. 8.

 

Figure 8. Illustration of the production of photon pairs. (a) Spontaneous parametric downconversion produces two photons that are entangled in either wavevector, photon energy, or polarization, called entangled biphotons. One distinguishes Type I and Type II biphotons by their relative polarizations. Not shown is the case where Type I photons are made to propagate collinearly by suitably cut crystals. Detection of the correlation requires coincident two-photon detection. (b) Stimulated resonance scattering produces two photons that are completely indistinguishable and second-order coherent, called cloned biphotons. They are not entangled, and a single point two-photon measurement in the detector plane produces a pattern that is the same as the multiplication of one-photon patterns recorded with two symmetrically arranged detectors.

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3.2a. Entangled Biphotons

In the SPDC process, illustrated in Fig. 8(a), entangled biphotons are produced by interaction of a pump laser with a non-centrosymmetric crystal. In the optical regime, atomic oscillators are driven into the nonlinear regime, and the downconverted photons created in the nonlinear process usually have a broad bandwidth corresponding to coherence times of the order of 10 fs that can be lengthened by use of energy filters. For an in-depth discussion of entangled biphoton physics, the reader is referred to the books by Ou [33,35] and Shih [34].

In the optical regime, the frequency and wavevector directions of the biphotons are determined by the nonlinear crystal response and momentum conservation. One distinguishes two types of SPDC processes. In Type I, the photons have the same polarization, which is opposite to that of the pump beam, and are typically emitted into different $\overrightarrow{k}$ directions, although collinear emission can also be achieved by appropriately cut crystals. In Type II, the photons have opposite polarizations and, in practice, are most often created with parallel wavevectors so that they are emitted collinearly [33,128], as shown in Fig. 8(a). In all cases the two photons are mutually incoherent. The cross section of SPDC is linear in the incident intensity and may be described by the second-order nonlinear susceptibility ${\chi }^{(2)}$ [127,155]. Although the biphoton creation probability is very low (of order ${10}^{-10}$), one may nevertheless eliminate the high-intensity pump beam by a polarization or energy filter, so that one can study the correlations between the two entangled photons emitted by the secondary source.

SPDC has also been demonstrated in the x-ray regime. Suggested by Freund and Levine in 1970 [161], the process was first demonstrated by Eisenberger and McCall in 1971 [162]. This experiment, performed with a 17 keV Mo x-ray tube source and about two weeks of accumulation of coincidence events, constituted the observation of the first nonlinear x-ray effect, about 40 years before the advent of XFELs! Later experiments used brighter 8 keV x rays from a rotating Cu anode [163] and SR [164–167]. Following Yoda et al. [164], today diamond single crystals are typically used as the nonlinear medium.

Two-color experiments have also been performed using entangled x rays of very different energy [166] with a primary beam of 11.107 keV that produced 11.007 keV signal and 100 eV idler photons, and with 760 and 840 nm optical photons [168].

3.2b. Cloned Biphotons Produced in Stimulated X-Ray Resonant Scattering

Cloned biphotons are naturally created in stimulated emission, as discussed above. Spontaneous and stimulated emission may occur independent of the excitation process. For example, in the x-ray regime a core hole may be created by electron or ion beam impact on a sample. The hole may then be filled by decays of electrons from outer shells. They may be driven either by the Coulomb field of the hole, resulting in Auger decay, or by the EM zero-point fluctuations, resulting in spontaneous emission of a photon. In the presence of a photon of the right energy, the decay may be stimulated by the photon, which clones itself in the process, constituting stimulated emission.

In the soft x-ray region, one often uses resonant excitations where the incident photon energy is tuned to an atomic resonance and a core electron is excited to a rather localized valence state (orbital). The resonant process maximizes the excitation (absorption) cross section and provides atomic, chemical [169], and magnetic [80] specificity. In addition to polarization dependent x-ray absorption measurements, which provide bonding and magnetic information, one can also obtain structural information through diffraction experiments. They are carried out as illustrated in Fig. 4(c) and yield information on the nanoscale domain structure [81,85]. In diffraction experiments, one employs position sensitive CCD detectors to record the entire pattern, and with XFELs the diffraction pattern may even be obtained with a single shot [170].

Before the advent of XFELs, the incident beam, even at the most advanced SR facilities, had a degeneracy parameter below 1, as shown in Fig. 1(b). The probability of stimulating decays with the incident beam itself was therefore negligible. With XFELs, however, the large degeneracy parameter may readily lead to stimulated decays. If the excitation and emission process are both driven by the incident beam one speaks of impulsive stimulation. The resonant x-ray scattering process then combines the first absorption step with the second decay step, and with increasing incident intensity one may observe how the conventional diffraction pattern changes with the onset of stimulation [171,172]. This will be discussed in Section 4.5 after we have introduced the concepts of two-photon detection.

Stimulated resonant scattering is a second-order nonlinear process. The atom is modeled as a two-level electronic system consisting of a core and a valence state, separated for soft x rays by an energy of the order of 1 keV. At lower intensities the process may still be described in second-order Kramers–Heisenberg–Dirac perturbation theory [17,173], where any population changes of the atomic ground and excited states, averaged over all illuminated atoms, are neglected. At higher intensities, population changes in the coherently illuminated number of atoms need to be included by use of the optical Bloch equations [45,174]. In elastic resonant scattering, the temporal degree of second-order coherence $g^{(2)}(t)$ is related to the relative upper state population ${\rho }_{22}(t)$ of the collective atomic two-level system by [45,48,175]

Equilibrium is consistently reached only if the coherence time of the incident x rays is longer than the total spontaneous decay time $\mathrm{\Gamma }/\hslash$ of the upper state. In the soft x-ray region, $\mathrm{\Gamma }\simeq 100–500\mathrm{meV}$ is the total (Auger plus radiative) decay energy width [172,174], corresponding to excited state decay times $\hslash /\mathrm{\Gamma }\simeq 7–1.4\mathrm{fs}$. The strength and coherence time of the field ensure that it controls the electronic transitions for a sufficient time to equalize the upper and lower state populations with the process becoming quasi-stationary. Absorption is then compensated by stimulated emission and the sample becomes transparent [172]. Stimulated resonant scattering scales with the square of the incident intensity and may be described semiclassically by a third-order nonlinear susceptibility ${\chi }^{(3)}$ [155].

The production of cloned biphotons in impulsive stimulated resonant forward scattering does not violate the famous “no-cloning theorem” [133,158–160] because the system evolves into an equilibrium state where stimulated emission completely dominates. The situation is similar to transportation and amplification of telecom signals in optical fibers, where stimulated emission dominates over spontaneous emission. The no-cloning theorem then does not apply [176]. The inapplicability of the theorem in these cases may also be explained by the violation of the basic premise of an arbitrary input state. Since all incident photons are in the same mode, prior information about a state to be cloned is available and it can be cloned with higher fidelity [177].

Stimulated elastic resonant scattering entails the interaction of two independent photons in the same mode (first-order coherent) with an atom. It requires only a single beam of sufficient degeneracy parameter tuned to an atomic resonance to control the stimulated decay. In the atomic interaction, two photons are converted into indistinguishable clones, as illustrated in Fig. 8(b). This process needs to be distinguished from other stimulated processes. Examples are amplified spontaneous emission (ASE) utilized in a conventional laser, where the process is started by a spontaneous atomic decay that then self-amplifies. This process has also been observed in the x-ray range [178–181]. Another process is the impulsive stimulation of the inelastic channel by the incident beam, referred to as stimulated x-ray Raman scattering [182]. It has been observed by use of broadband incident radiation from a SASE XFEL [183]. The broad incident spectrum contained photons above the ionization energy to create core holes and below the ionization energy to stimulate inelastic decays into the created core holes.

4. Fundamental One- and Two-Photon Experiments

For many quantum optics experiments one would like the detection time uncertainty ${\tau }_{\mathrm{jit}}$ to be the same as the coherence time given by the length of an uninterrupted wavetrain (wavepacket). For flat-top distributions, the coherence time is given by ${\tau }_{\mathrm{coh}}={\lambda }^2/(c\mathrm{\Delta }\lambda )=\hslash /\mathrm{\Delta }\mathcal{E}$, which may be lengthened through a reduction of the wavelength bandwidth $\mathrm{\Delta }\lambda$ or energy bandwidth $\mathrm{\Delta }\mathcal{E}$ by a filter or monochromator inserted into the beam. As shown in Fig. 1(e), one may achieve ${\tau }_{\mathrm{jit}}\simeq {\tau }_{\mathrm{coh}}$ in the optical regime, but this is impossible at x-ray energies where ${\tau }_{\mathrm{jit}}\gg {\tau }_{\mathrm{coh}}$.

Some of the most fundamental experiments in quantum optics have dealt with the spatial or temporal correlations between two photons. In such experiments, the existence of photon–photon correlations is typically revealed by detecting the two-photon arrival probability at the same position as a function of the difference in their arrival time $\tau =t_2-t_1$, or detecting the probability of two photons arriving at the same time as a function of the difference in their position ${\rho }^{\prime }=|{\overrightarrow{\rho }}_2-{\overrightarrow{\rho }}_1|$. The two types of measurements yield information of their temporal and spatial correlations, respectively. Below we focus on spatial correlation, interference, and diffraction.

4.1. Hanbury Brown–Twiss Experiment

Together with the development of the maser and laser, it was the Hanbury Brown–Twiss (HBT) experiment conducted with chaotic sources in the mid1950s that played an important role in the development of statistical optics as outlined in a recent book by Wolf [61] and especially quantum optics, as recalled by Glauber in his Nobel lecture [2].

HBT performed two experiments, illustrated in Fig. 9, one for the star Sirius [39,42] and another with a laboratory mercury arc lamp [38,41]. In both cases, the thermal sources were quasi-monochromatic, with the photons emitted by chaotic processes. The coincident arrival of two photons was detected in coincidence at two spatial points as a function of their separation. The correlated arrival of two photons was detected with one-photon detectors whose response time was much longer than the coherence time of the photons, determined by their spectral bandwidth. Since the detector response time significantly exceeded the coherence time, the identification of the true coincidence signal required excellent signal-to-noise data as reviewed by Wolf [61].

 

Figure 9. (a) Illustration of the HBT stellar intensity interferometer, used to measure the correlation of light emitted by the star Sirius ($\lambda \simeq 500\mathrm{nm}$) [39,42]. The coincident counts in two one-photon detectors were measured as a function of the detector separation ${\rho }^{\prime }=|{\overrightarrow{\rho }}_2-{\overrightarrow{\rho }}_1|$, which was varied up to about 10 m. (b) Experimental arrangement for a laboratory-based experiment with a mercury-arc lamp ($\lambda =436\mathrm{nm}$). In this case, the spatial or lateral correlation was measured by scanning one of the detectors perpendicular to its incident beam, as shown [38,41]. The source–detector distances were about 2.5 m, and ${\rho }^{\prime }$ was changed up to about 10 mm. (c) Schematic result of both experiments, reflected by the normalized coincident counts, consisting of an accidental and a true contribution, as a function of the relative change in path length ${\rho }^{\prime }=|{\overrightarrow{\rho }}_2-{\overrightarrow{\rho }}_1|$. The shown blue curve, representing preferential “photon bunching” at small detector distances, is taken to be a Gaussian but, in general, it depends on the macroscopic shape of the source.

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For their experiment with light emitted by a star, HBT used the schematic arrangement shown in Fig. 9(a). The measurement allowed the determination of the angular size of Sirius. The experiment with a mercury arc lamp used the scheme in Fig. 9(b), where the two single-photon detectors, joined by a coincidence circuit, are placed at the exit ports of a beam splitter. For convenience, only one of the detectors was scanned perpendicular to its incident beam to determine the relative spatial correlation of the two photons. The two-photon detection schemes introduced by HBT have provided the foundation for many modern quantum optics experiments, as will become apparent below.

The results of the HBT experiment are sketched in Fig. 9(c), and the two contributions to the coincidence signal are indicated. The blue curve represents true correlations between photons at small detector distances, referred to as “photon bunching.” Its shape depends on the macroscopic shape of the source. It is taken here to be a Gaussian, corresponding to a source with a Gaussian intensity distribution. For a chaotic circular flat-top source, the blue curve would have the shape of an Airy function of the form ${|g^{(1)}({\rho }^{\prime })|}^2={[2J_1(kR{\rho }^{\prime }/z_0)/(kR{\rho }^{\prime }/z_0)]}^2$ [184] [see Eq. (24)]. For chaotic sources, the random emission of photons also leads to a background, shown in red, that arises from accidental coincident arrival of two photons that were born at different points and different times in a completely uncorrelated way, but their path lengths to the detectors just happen to compensate for their different birth times. This contribution does not depend on the detector separation and is constant.

As indicated by the label of the ordinate in Fig. 9(c), the measured normalized coincidence rate reflects the degree of second-order coherence, which as discussed below is given by Eq. (66). Under the condition of spatio-temporal separability, it is given by

For the measurement of the temporal correlation $g^{(2)}(\tau )$, the light is made spatially coherent, $|g^{(1)}({\rho }^{\prime })|=1$, by a small aperture. However, the temporal coherence measurement is more difficult because it requires a time resolution of the order of the coherence time ${\tau }_{\mathrm{coh}}$. Such a measurement was only accomplished about 10 years after the original HBT spatial correlation measurement by Morgan and Mandel [185], using the blue line of a mercury discharge lamp with a 5 ns temporal coherence time.

In the x-ray region, a temporal correlation measurement is much more difficult due to the much shorter coherence time, which even with a monochromator typically does not exceed ${\tau }_{\mathrm{coh}}\simeq 10\mathrm{fs}$ (see Fig. 1). Therefore, only the spatial coherence $g^{(2)}({\rho }^{\prime })$ has been measured so far [186,187]. For these measurements, one can overcome the slow detector response time by gating the detector to the arrival time of the x-ray pulses, so that the counting window is effectively determined by the shorter x-ray pulse length. Since the pulse repetition time is typically 2 ns for synchrotron sources and $>10\mathrm{\mu s}$for XFELs (see Figs. 2 and 3), the detector may be read out pulse-by-pulse and a statistical average is obtained by averaging over many pulses [63].

4.2. Quantum Derivation of the HBT Effect

The HBT result may be accounted for by intensity fluctuations within the framework of statistical optics in conjunction with a semiclassical model for photon detection [44,61,188]. Since intensity fluctuations are nothing but fluctuations in the number of emitted photons, we might as well derive the effect in the more fundamental QED formulation. In QED, the HBT effect originates from the interference of the probability amplitudes associated with two photons. This was first pointed out by Fano [189] and supported by Glauber’s treatment of two-photon coherence based on their detection [28–30].

The original HBT effect was observed with sources of finite extent that obviously contain a large number of independent emitters. The shape of the bunching peak in Fig. 9(c) is then defined by a Fourier transform of the shape of the source, as will become apparent later. Theoretical studies of the effect, particularly of its quantum mechanical origin, have typically used a simple two-point source model, first considered by Fano [189] and also treated in the book by Scully and Zubairy [46]. In the following section we shall first explore this model system using Feynman’s perturbation treatment of QED and then treat the more complicated realistic source case for the example of a uniform circular source.

4.2a. Fano’s Simple Four-Atom Model

For independent single photons, probability amplitudes for alternative paths are added, similar to the superposition of fields, before the detection probability is calculated as the absolute value squared. For two or more photons, the multi-photon probability amplitudes are constructed from those of single photons by multiplication and addition [27]. This extension consists of remarkably simple rules that we shall illustrate in the following through treatment of the HBT effect.

To illustrate the quantum mechanical origin of the HBT effect, we start with the four atom case shown in Fig. 10 following Fano [189], where photons are created at two source atoms, separated by a distance $r^{\prime }=|{\overrightarrow{r}}_2-{\overrightarrow{r}}_1|$ and detected by two atoms separated by ${\rho }^{\prime }=|{\overrightarrow{\rho }}_2-{\overrightarrow{\rho }}_1|$.

 

Figure 10. Quantum mechanical description of the HBT experiment following Fano [189]. We assume that two photons are emitted by source atoms A and B and consider the probability for their coincident detection by detector atoms X and Y. For single-photon emission and detection per atom, there are only four single-photon probability amplitudes connecting the four atoms. Concomitant two-photon amplitudes ${\varphi }_{\mathrm{AX}},{\varphi }_{\mathrm{BY}}$ (green) and ${\varphi }_{\mathrm{AY}},{\varphi }_{\mathrm{BX}}$ (orange) are shown in the same color. Alternatives of concomitant paths are shown in different colors. According to Feynman’s rules, concomitant one-photon amplitudes combine by multiplication into two-photon amplitudes, while alternatives of the concomitant scenarios are then added to form the final two-photon amplitude $\mathrm{\Psi }({\overrightarrow{\rho }}_1,{\overrightarrow{\rho }}_2)$ given by Eq. (17). The two-photon detection probability is given by ${|\mathrm{\Psi }({\overrightarrow{\rho }}_1,{\overrightarrow{\rho }}_2)|}^2$.

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In Fig. 10 we depict one-photon probability amplitudes ${\varphi }_{\mathrm{IJ}}$ with the birth and propagation of a photon from source point I to detection point J. We note that the atomic model assumes that only single photons can originate from a given source atom. It therefore does not include the case where biphotons are birthed at the same point or within an indistinguishably small source area. However, as will be shown in Section 6.5, the atomic model remains valid in a more rigorous treatment since all other possible interference processes average to zero. This detailed discussion is postponed until we have properly introduced the concept of second-order coherence in Section 6.1.

According to Feynman’s rules, concomitant two-photon amplitudes, shown in the same color in Fig. 10 are formed by multiplication of one-photon amplitudes. The resulting two-photon amplitudes are then ${\varphi }_{\mathrm{AX}}{\varphi }_{\mathrm{BY}}$ (green) and ${\varphi }_{\mathrm{AY}}{\varphi }_{\mathrm{BX}}$ (orange). Alternatives of the concomitant amplitudes, shown in different colors, are then added to form the total two-photon amplitude $\mathrm{\Psi }({\overrightarrow{\rho }}_1,{\overrightarrow{\rho }}_2)$, which is given by

We now express the distances in Eq. (17) by assuming $r^{\prime },{\rho }^{\prime }\ll z_0$ (see Fig. 10) in the paraxial approximation of Eq. (9) and obtain

When the two detectors are scanned in opposite directions parallel to the line defined by the two source points (${\overrightarrow{\rho }}_i\parallel {\overrightarrow{r}}_i$), then the dot product in Eq. (20) is simply the product of the absolute values $r^{\prime }=|{\overrightarrow{r}}_2-{\overrightarrow{r}}_1|$ and ${\rho }^{\prime }=|{\overrightarrow{\rho }}_2-{\overrightarrow{\rho }}_1|$ (see Fig. 10), and we have

4.2b. Finite Size Sources

For the considered case of two source points of infinitesimal area, the cosine modulation has a constant unit amplitude and does not resemble the peak-shaped blue bunching curve in Fig. 9(c). If the source has a finite size, however, a bunching peak appears since the interference-modulation averages to zero with increasing ${\rho }^{\prime }$. For example, the simplest case is a continuous line of source points, so that ${\overrightarrow{r}}_1$ and ${\overrightarrow{r}}_2$ lie within an infinitesimally thin slit of length ${\ell }_s$. For such a chaotic line source, we can assume that there are only microscopic coherence regions of length ${\ell }_{\mathrm{coh}}^{\mu }$, where photons are born without a relative phase shift, corresponding to a 1D version of the 2D case treated in Section 2.4. If the detectors are scanned so that ${\overrightarrow{\rho }}_i\parallel {\overrightarrow{r}}_i$, then Eq. (21) changes to

The normalized value of Eq. (23) is the degree of second-order coherence (see Section 6.1) given with ${\rho }^{\prime }=|{\overrightarrow{\rho }}_2-{\overrightarrow{\rho }}_1|$ by

4.2c. The Physics behind the HBT Effect

Our quantum mechanical derivation of the HBT effect reveals its origin within the fundamental QED-based photon picture of light. While it can also be accounted for in the framework of statistical optics, there is really no argument that the quantum formalism is more fundamental. It just turns out that for a chaotic source the treatments of second-order correlations give the same answer in both cases.

In Feynman’s space–time probability amplitude formulation of QED, the HBT bunching peak arises in second-order perturbation theory from the interference between alternatives of concomitant two-photon probability amplitudes, formed by products of one-photon amplitudes. The constant background is due to the arrival of independent single photons at the two detectors at the same time. These photons were born independently at different points and different times, but the distances covered from birth to detection just accidentally compensate for their different birth times.

As will be formally shown in Section 6.5, the birth of biphotons at the same point, not considered in the above derivations based on Fig. 10, make no contribution to the HBT effect. This is expected from the fact that in thermal sources the births of cloned biphotons by stimulated emission and entangled biphotons by spontaneous parametric downconversion are considerably less probable than the birth of single photons.

4.3. Single Photons, Detection, and Self-Interference

The quantum theory of light is based on the existence of photons, which is often said to be proven by the photoelectric and Compton effects. However, both effects can be explained by assuming classical EM waves with quantum properties associated only with the atomic and electronic structure of matter in the photoelectron generation process [190]. The experiment that convincingly proved the existence of photons as elementary single units was performed in 1986 by Grangier, Roger, and Aspect [36] and is illustrated in Fig. 11.

 

Figure 11. Single-photon detection experiment of Grangier, Roger, and Aspect [36], demonstrating the quantum concept of indivisible single photons. Two photons are produced in cascade emission, with the first acting as the trigger for detection of the second one, emitted 4.7 ns later. The second photon is incident on a beam splitter, and the coincidence counts are observed at the two output ports of the beam splitter. Only single-photon counts are observed in the two detectors with zero cross-correlation coincidence counts.

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The experiment utilized two photons emitted consecutively in an atomic cascade decay, with the first photon serving as a trigger to mark the arrival of the second photon, emitted about 4.7 ns later. The second photon entered the shown HBT-like detection scheme, where it could choose between two paths through a beam splitter, leading to two single-photon detectors, gated to the arrival of the first trigger photon. The second photon was found to register in only one of the two detectors with zero cross-correlation coincident counts. This confirmed the quantum mechanical prediction of indivisible single photons.

In a wave picture, it takes the constructive interference of two fields, represented by the product $E{E}^{*}$, to account for photon creation and destruction (detection). For a single photon, only detector counts 0 and 1 are possible, corresponding to the classical picture of destructive or constructive interference between two fields. From a quantum point of view, a detector count of 1 corresponds to interference of a single photon with itself, as stated by Dirac.

4.4. Observation of Two-Photon Interference

One of the most important experiments in quantum optics was carried out in 1987 by Hong, Ou, and Mandel (HOM) [37] using entangled biphotons. It is the basis for what we today refer to as two-photon interference. In the following we will discuss two versions of the experiment that illustrate its complementary aspects.

The original HOM experiment, shown schematically in Fig. 12(a), utilized a Type I SPDC source whose two beams of the same polarization were inserted into two input ports of a loss-less beam splitter, and the arrival of the two photons at the two output ports was measured with single-photon detectors as a function of their relative arrival time.

 

Figure 12. (a) Schematic of the HOM experiment using Type I biphotons is shown on the left. When there is no difference in the paths from the SPDC source to the two single-photon detectors, the probability for cross-coincidence counts at the detectors was observed to be zero, $P(1,1)=0$. On the right we show the possible two-photon concomitant paths and alternative concomitant paths, whose probability amplitudes are treated by Feynman’s multiplication and addition rules. In a classical picture, the photons are distinguishable, as indicated by the two colors. The case where two photons arrive at the same output port was not measured. The experimental result $P(1,1)=0$ is explained in the text. (b) Experiment of Di Giuseppe et al. [191] using Type II collinear orthogonally polarized biphotons. The combination of the non-polarizing beam splitter and the 45° polarizers in each detector arm resulted in $P(1,1)=0$, as in (a). The use of two-photon detectors (blue) in the two output ports allowed the direct comparison of single- and two-photon arrivals at the exit ports, resulting in the simultaneously measured probabilities $P(1,1)=0$, $P(2,0)=0.5$, and $P(0,2)=0.5$, as explained in the text.

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Femtosecond time resolution was achieved through precision movement of the beam splitter to change the relative path length to the two detectors. With decreasing delay between the arrival of the two photons, a “dip” in the coincidence counts was observed, with zero counts or a coincident arrival probability $P(1,1)=0$ for no delay, as indicated in the figure. The situation encountered in the experiment was independently shown in 1987 to exist by Fearn and Loudon, who treated the effect of a loss-less beam splitter quantum mechanically [192].

The missing one-photon cross-coincidence counts, labeled $P(1,1)=0$, cannot be explained classically. On the right of Fig. 12(a), we show the four different possibilities of how photons, viewed as distinguishable classical particles, can propagate through the beam splitter. To indicate their classical distinguishability, we have colored their respective paths black and red and used a notation $P(n_1,n_2)$ to denote their arrival probabilities, where due to photon conservation, $n_1+n_2=2$ and the sum of all probabilities is unity. Two photons arriving at the same output port can happen two ways—either both are reflected with a probability $P(2,0)=0.25$ or both are transmitted $P(0,2)=0.25$. Two photons exiting different output ports can also happen two ways—either one photon is reflected and the other is transmitted $P(1,1)=0.25$ and vice versa $P(1,1)=0.25$. Hence, from a classical point of view, cross-coincidence counts are expected half of the time, i.e., a total of $P(1,1)=0.5$, in contrast to the measured result.

From a quantum perspective, the situation is different. The quantum theory of a beam splitter is presented in Refs.  [33,193–195], and here we only need to utilize the fact that there is no phase shift for the transmitted beam but the reflected beam is phase shifted by $\pi /2$. The probability amplitudes of the reflected beams (see right side of Fig. 12) are therefore both changed by a phase factor of $\exp (\mathrm{i}\pi /2)=\mathrm{i}$. The two-photon probability amplitudes follow the rules established in Section 4.2. The amplitudes of concomitant paths are multiplied, and those of alternative concomitant paths are added. We therefore have the detection probabilities $P(2,0)+P(0,2)={|(\mathrm{i}*1+1*\mathrm{i})/2|}^2=1$ and $P(1,1)={|1*1+\mathrm{i}*\mathrm{i}|}^2=0$. One may also say that the result $P(1,1)=0$ is due to destructive interference of biphoton probability amplitudes at the two detectors caused by their relative phase shift of $\pi$.

A more complete version of the HOM experiment carried out in 2003 by Di Giuseppe et al. [191], which reveals the complementarity of the two photons exiting the beam splitter together versus separately, is schematically shown in Fig. 12(b). The real novelty of the experiment was the use of special detectors [73] that could discriminate one (gray) from two (blue) photon arrivals within the detector response time ${\tau }_{\det }=1\mathrm{ns}$. The experiment also differed from that of HOM by utilizing collinear biphotons with orthogonal polarizations created in a Type II SPDC process, following an earlier scheme introduced by Shih and Sergienko [196].

The collinear biphotons were sent into the same beam-splitter input port, and the combination of a non-polarizing beam splitter and two 45° polarizers in each detector arm reproduced the HOM dip. The two polarizers behind the beam splitter served to establish polarization indistinguishability and also added to the $\pi /2$ beam-splitter phase shift between reflected and transmitted photons, so that the total phase shift between the wavepackets arriving at the two one-photon coincidence detectors was again $\pi$, as in the HOM experiment. The coherence time of the biphotons was not increased by use of energy filters, but accidental coincidences were avoided by using a low-power pump beam so that the time between biphoton arrivals exceeded the coincidence detection window of $\simeq 1\mathrm{ns}$.

The experiment thus allowed simultaneous measurement of the coincident cross correlation of single-photon arrivals at two exit ports and the complementary arrival of two photons at a single exit port, which was not measured by HOM. The results are schematically indicated in gray and blue in Fig. 12(b). The three arrival probabilities were found to be complementary, as required by the conservation of photons, adding the missing piece of the original HOM experiment.

If two photons (2 counts) are detected at one beam-splitter output, there are zero photons (0 counts) at the other beam-splitter output, as required by conservation of photons. This may be interpreted as constructive interference of the two-photon probability amplitudes at one port and destructive interference at the other. The other possibility of single photons exiting simultaneously at the two exit ports is not observed. It has zero probability because of the opposite phase of their wavepackets at the two exit ports. The implied destructive interference of coincident two probability amplitudes at two positions is a pure quantum phenomenon.

Comparison of the entangled biphoton results in Fig. 12(b) with the single photon results in Fig. 11 reveals a striking similarity. The single-photon counts are simply replaced by two-photon counts. In Dirac’s formulation, one might say that similar to individual photons, entangled biphotons form a collective new quanton state that may interfere only with itself. Then constructive biphoton self-interference corresponds to 2 counts and destructive biphoton self-interference to 0 counts.

4.5. Two-Photon Diffraction Experiments

Two-photon diffraction experiments are based on the detection of two rather than single photons. This is typically accomplished in the optical regime by use of a “two-photon detector” consisting of a beam splitter and two single-photon detectors, as in the experiments discussed above. In principle, a true two-photon detector could also be used as done by Di Giuseppe et al. [191], but the use of two single-photon APD detectors is more convenient. The HOM experiment, however, reveals that care has to be taken that the single-photon cross-coincidence probability $P(1,1)$ does not vanish. This can always be achieved by suitable choices of beam splitters and phase shifters.

We note that the manipulation of photon beams by beam splitters and polarizers is significantly more difficult in the x-ray regime [197]. The following discussion of optical concepts and experiments is envisioned to be fully extendable in the future to the x-ray regime through the development of a complete “x-ray optics toolbox.”

Two-photon diffraction experiments have typically been carried out by introducing a slit or transmission gratings right after the entangled biphoton source, essentially repeating Young’s experiment with biphotons. In this case the diffraction pattern is one dimensional, which facilitates its interpretation. Below we first illustrate typical results through two examples obtained with Type I and Type II entangled biphotons. We then discuss a soft x-ray experiment with cloned biphotons where the slit was replaced by a circular aperture. This experiment used the geometry of Fig. 6 and thus directly revealed how the diffraction limit can be overcome.

4.5a. Diffraction of Entangled Biphotons with Orthogonal Polarizations

Our first example of biphoton diffraction is shown in Fig. 13. The experiment was performed by D’Angelo et al. [147] utilizing collinearly Type II biphotons of $\lambda =916\mathrm{nm}$ wavelength with orthogonal polarizations produced by shining a 458 nm Ar ion laser beam onto a $\beta \text{-}{\mathrm{BaB}}_2{\mathrm{O}}_4$ (BBO) crystal. The diffracting object was a double slit positioned immediately behind the biphoton source, as shown in Fig. 13(a).

 

Figure 13. Biphoton diffraction experiment by D’Angelo et al. [147]. (a) Collinear biphotons with orthogonal polarizations were generated in a Type II process by an argon ion laser of 458 nm wavelength incident on a BBO crystal. The biphotons of 916 nm wavelength were diffracted by a double slit placed as close as possible to the BBO output surface. The pump photons were eliminated by a coated mirror and cut-off filter, and the biphotons were guided by two mirrors onto the entry port of a two-photon detector, indicated by a blue box. It consisted of a polarizing beam splitter and two single-photon counters linked by a coincidence circuit. Instead of scanning the detector, the beam was scanned by rotation of the first mirror as indicated. (b) Biphoton diffraction pattern recorded in coincidence with the two-photon detector. (c) The conventional one-photon double-slit diffraction pattern recorded in the same setup with first-order coherent classical light of 916 nm.

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The two-photon diffraction pattern was recorded as a function of the angle from the optical axis by use of a two-photon detector, represented by the entire blue box, consisting of a polarizing beam splitter and two one-photon coincidence detectors. The polarizing beam splitter optimized the coincident one-photon counting rate at the expense of the two-photon counting rate behind each output port. For experimental convenience, the detector was kept fixed in position, and rotation of a mirror was used to sweep the beam across the entrance pinhole of the detector. The experiment was also repeated by use of a conventional first-order coherent source of the same 916 nm wavelength using one-photon detection. This pattern constitutes the conventional double-slit diffraction pattern for reference.

The diffraction patterns for the two cases are shown in Figs. 13(b) and 13(c). Remarkably, the pattern for the biphotons of 916 nm wavelength corresponds to the conventional one-photon pattern of photons with half the wavelength, i.e., 458 nm. It thus beats the conventional diffraction limit by a factor of 2. From a diffraction point of view, the biphotons act as if they were a collective quanton that still has the wavelength of the 458 nm Ar ion pump beam that created them through a fission process. This remarkable observation cannot be explained classically, and arises as a consequence of quantum mechanical entanglement of the two photons.

4.5b. Diffraction of Entangled Biphotons with Identical Polarizations

A similar biphoton diffraction experiment performed with collinear Type I entangled biphotons with the same polarization by Shimizu et al. [198] is shown in Fig. 14(a). The biphotons of $\lambda =862\mathrm{nm}$ wavelength were created in a potassium niobate (${\mathrm{KNbO}}_3$) crystal by a Ti:sapphire pump laser (862 nm) that was frequency doubled to $\lambda =431\mathrm{nm}$. The biphotons were diffracted by a transmission grating placed immediately behind the crystal. The pump beam of opposite polarization was separated from the biphoton beam by a polarizing beam splitter. The experiment used the same scanning scheme as that by D’Angelo et al. and a similar two-photon detector, but a simple rather than polarizing beam splitter provided the required non-vanishing cross-coincidence two-photon detection probability $P(1,1)$.

 

Figure 14. Biphoton diffraction experiment by Shimizu et al. [198]. (a) Collinear biphotons with the same polarization were produced by a Type I process by a frequency-doubled Ti:sapphire laser operating at 431 nm in a ${\mathrm{KNbO}}_3$ crystal. The biphotons of 862 nm wavelength were diffracted by a transmission grating placed as close as possible to the ${\mathrm{KNbO}}_3$ output surface. They were separated from the pump beam of opposite polarization by a polarizing beam splitter and cut-off filter. The biphotons were then guided by two mirrors onto the entry port of a two-photon detector, indicated by a blue box, consisting of a regular (non-polarizing) beam splitter and two single-photon counters linked by a coincidence circuit. Instead of scanning the detector, the beam was scanned by rotation of the first mirror, as indicated. (b) Detected biphoton signal using a one-photon detector. (c) Biphoton diffraction pattern recorded in coincidence with the two-photon detector. (d) The conventional one-photon double-slit diffraction pattern recorded in the same setup without the ${\mathrm{KNbO}}_3$ crystal, when the grating is illuminated with coherent 862 nm light from the Ti:sapphire laser. (e) Diffraction pattern recorded in coincidence with the blue-box two-photon detector with the same Ti:sapphire laser light as in (d).

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For comparison, the one-photon counting rate was recorded concomitantly with the two-photon counting rate. Another experiment was performed after removal of the ${\mathrm{KNbO}}_3$ crystal with the grating directly illuminated by the 862 nm light from the Ti:sapphire laser. In this case, the light incident of the grating was coherent and not entangled in contrast to the biphoton case. The diffraction pattern was then recorded in the same experimental configuration by using either one of the green one-photon detectors or the entire blue-box two-photon detector. The results are shown in Figs. 14(b)–14(e).

The constant one-photon intensity in the detector plane in Fig. 14(b) is due to the fact that the biphoton source is mutually incoherent so the one-photon intensity is constant similar to that of a chaotic source. A similar one-photon measurement was not performed by D’Angelo et al. [147] but it would have yielded the same result. The diffraction pattern of the entangled biphotons of wavelength 862 nm, shown in Fig. 14(c), is again representative of a conventional pattern of single photons of 431 nm wavelength, i.e., half of the biphoton wavelength. This is revealed by comparison of the one-photon reference pattern in Fig. 14(d) recorded with $\lambda =862\mathrm{nm}$ light from a conventional source.

Comparison of the patterns in Fig. 14(d) recorded by one-photon detection and in Fig. 14(e) by two photon detection contains some altogether new information. Note that the two cases correspond to photon detection at the same point determined by the entrance slit of the detector, as a function of momentum transfer changed by rotation of the mirror. The peaks of both patterns are in the same positions but their relative intensities are different. The solid line fit through the data points reveals that the shape of the two-photon pattern in Fig. 14(e) is simply the square of the shape of the one-photon pattern in Fig. 14(d). Remarkably, the conventional one-photon diffraction pattern of the coherently illuminated grating can be changed to its square by picking out those events where two photons arrive simultaneously at a given position in the detector plane!

Closer inspection also shows that the widths of the peaks in the two-photon pattern in Fig. 14(e) are narrower than those in the one-photon pattern in Fig. 14(d). This is an indication that two-photon diffraction may reduce the diffraction limit. The reason for this behavior will become clear through the formal derivation of the one- and two-photon diffraction patterns below. It is a manifestation of the evolution of coherence in quantum optics from first to second order.

4.5c. Diffraction of Cloned Biphotons

Our last example is a cloned biphoton experiment performed by Wu et al. [171], illustrated in Fig. 15. The experiment was conducted at the Linac Coherent Light Source (LCLS) XFEL using 50 fs pulses of linearly polarized laterally coherent SASE pulses.

 

Figure 15. (a) Coherent imaging experiment of a magnetic Co/Pd film with magnetic worm domains in a circular Au aperture illuminated by laterally coherent 50 fs XFEL pulses tuned to the $\mathrm{Co}{\mathrm{L}}_3$ resonance at 778 eV with a 0.2 eV bandpass [171]. A beam stop in front of the CCD detector eliminated the intense central part of the diffraction pattern. The linearly polarized incident radiation allows separation of the diffraction pattern from the Au aperture and the magnetic domains due to the polarization dependence of charge and magnetic scattering, as shown. (b) Experimentally recorded pattern by averaging over many low-power shots ($0.6\mathrm{mJ}/{\mathrm{cm}}^2/50\mathrm{fs}=0.012\mathrm{TW}/{\mathrm{cm}}^2$). (c) Pattern for a single high-power shot ($272\mathrm{mJ}/50\mathrm{fs}=5.4\mathrm{TW}/{\mathrm{cm}}^2$). (d) Radial intensity average of the patterns in (b) and (c) for the two power values. (e) Simulation of the change in the Airy pattern of the aperture, assuming the onset of stimulated resonant scattering [59].

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The x rays were transmitted through a monochromator, which defined the photon energy to $778\pm 0.1\mathrm{eV}$, matching the resonant energy of the $\mathrm{Co}\hspace{0.17em}{\mathrm{L}}_3$ transition from the $2p_{3/2}$ core to the $3d$ valence shell. The sample was a Co/Pd multilayer with a nanoscale magnetic worm domain structure as shown. The Co/Pd film plus a circular gold mask of 1.45 μm diameter, deposited directly onto the film on the side of the incident beam, constituted the diffracting object. The diffraction pattern was recorded with a position sensitive CCD detector, with the total accumulated charge per pixel forming the diffraction pattern. No temporal gating of the detector and no electronic coincidence circuit was employed.

As indicated in Fig. 15(a), the diffracted signal consists of a magnetic part, determined by the domains, and a charge part due to the aperture and the homogeneous charge distribution in the film, which according to Fig. 6 only attenuates the intensity of the first-order diffraction pattern. For the employed linear polarization, the magnetic and charge patterns are separable, as indicated in the figure  [199].

When the incident intensity was increased, a change of the diffracted intensity from both the magnetic domains and the circular aperture was observed [171]. This is shown by comparison of the low- and high-intensity patterns shown in Figs. 15(b) and 15(c). The decrease of the diffracted intensity was attributed to intrinsic interference of the two cloned photons in the forward direction [171,174], enhancing the “in-beam” transmitted at the expense of the “out-of-beam” diffracted intensity.

The decrease of the outer Airy rings, in particular, is indicative of a stimulation effect that redirects the outer ring intensity into the central part of the Airy disk [59]. The intensity enhancement in the forward direction, the hallmark of stimulation, could not be observed in the original experiment [171] due to a beamstop, but it was directly observed in a recent experiment [172].

Figures 15(d) and 15(e) compare the experimentally observed change in the outer Airy rings with that predicted by the theory of Ref.  [59]. It predicts that the conventional one-photon diffraction pattern changes to its square when the stimulation effect saturates. This behavior parallels the change of the one- to two-photon pattern in Figs. 14(d) and 14(e) recorded with the same coherent illumination of the grating but changing one- to two-photon detection. The two observations are indeed related since the roles of two-photon creation and detection are reversed.

While the two-photon pattern in Fig. 14(e) depends on coincidence detection of the two photons at the same point in the detector plane, the cloned biphoton pattern does not require coincidence detection. Rather, the correlation of the cloned biphotons is intrinsically created in the source and is maintained upon propagation to the detector plane where the two photons are naturally deposited at the same time. There is no need of inducing two-photon interference extrinsically by multiplication of the single-photon patterns by a coincidence circuit. Comparison of the two cases demonstrates the conjugate nature of two-photon creation in the source and destruction in the detector plane.

In the following sections, we shall formally treat one- and two-photon diffraction, comparing the predictions based on the wave and particle nature of light. We shall see that the experimental observations discussed above naturally follow from the quantum theory of light, which emphasizes the photon behavior. In contrast, statistical optics, which emphasizes the wave nature, will be seen to fail in some cases.

5. First-Order Coherence

Our starting point for the formal treatment of coherence is the description of light propagation from two points in a source to two others in a distant detector plane, as illustrated in Fig. 16. It was first treated by van Cittert in 1934 [200] and Zernike in 1938 [201]. What we are interested in is how the fields created within the source area propagate to a distant “detector” plane, where the quotation marks indicate that, in practice, the fields cannot be detected directly. In an experiment one would place mirrors at the points ${\overrightarrow{\rho }}_1$ and ${\overrightarrow{\rho }}_2$ and reflect the fields to a detector that measures the interference intensity.

 

Figure 16. Schematic of coordinates of fields at two points ${\overrightarrow{r}}_1$ and ${\overrightarrow{r}}_2$ in a source plane and at two points ${\overrightarrow{\rho }}_1$ and ${\overrightarrow{\rho }}_2$ in a “detector” plane at large distance $z_0$.

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Classically, we distinguish the two-point correlation of fields and intensities. Field propagation is formally expressed by a correlation function between two fields and is referred to as a “second-order” correlation in statistical optics [43,44,61]. In contrast, quantum optics is built on the notion of indivisible photons, and following Glauber, it is customary to refer to field–field or one-photon probability amplitude correlations (see Section 2.10) as first-order coherence [45,46,193]. In Section 4.2 we have already introduced the correlation between two-photon probability amplitudes corresponding to the classical correlation of two intensities or four fields. This case, which is the main topic of Section 6 below, constitutes second-order coherence in quantum optics and fourth-order coherence in statistical optics. We shall use the quantum optics terminology in this paper.

5.1. Field Quantization and Definition of First-Order Coherence Function ${\mathsf{G}}^{(\mathsf{1})}$

In the quantum theory of light, the electric field is an operator that contains both creation and annihilation operators associated with the positive and negative frequency parts of the field according to [46]

To briefly outline the quantum formalism, we use the general coordinate label $\overrightarrow{x}$ to designate either the coordinates $\overrightarrow{r}$ in the source plane or $\overrightarrow{\rho }$ in the detector plane, as shown in Fig. 16. The quantum-optical spatio-temporal first-order correlation function is defined as [46]

The statistical optics notation is similarly given by replacing ${\mathbf{E}}^{-}\leftrightarrow {\overrightarrow{E}}^{*}$ and ${\mathbf{E}}^{+}\leftrightarrow \overrightarrow{E}$, with the triangular brackets in Eq. (27) now referring to a statistical ensemble average. However, in principle, one needs to distinguish the quantum and statistical optics formulations. In statistical optics, $\overrightarrow{E}$ and ${\overrightarrow{E}}^{*}$ correspond to complex numbers that convey equivalent information, with ${|\overrightarrow{E}|}^2=\overrightarrow{E}{\overrightarrow{E}}^{*}={\overrightarrow{E}}^{*}\overrightarrow{E}$ defining the detected intensity, given by $I={ϵ}_{0}c{|\overrightarrow{E}|}^2$ in SI units. In contrast, the quantum operators ${\mathbf{E}}^{-}$ and ${\mathbf{E}}^{+}$ are intimately linked with the concept of the photon and describe the different processes of creation and destruction. We shall see that within first order, the statistical and quantum approaches give the same results but that this is not necessarily true in second order.