The physics of thermal light second-order interference beyond coherence
A novel thermal light interferometer was recently introduced in Ref. . Here, two classically correlated beams, obtained by beam splitting a thermal light beam, propagate through two unbalanced Mach-Zehnder interferometers. Remarkably, second-order interference between the long and the short paths in the two interferometers was predicted independently of how far, in principle, the length difference between the long and short paths is beyond the coherence length of the source. This phenomenon seems to contradict our common understanding of second-order coherence. We provide here a simple description of the physics underlying this effect in terms of two-photon interference.
1 Introduction and motivation
The discover of the Hanbury Brown and Twiss (HBT) effect [2, 3] in triggered an intense debate about the physics behind this effect and even its correctness. To quote Hanbury Brown, the reason behind a lot of this criticism was that “many physicists had failed to grasp how profoundly mysterious light really is”. However, as often happens in science for a new phenomenon which is initially not well understood and received, the HBT effect paved the way to the development of an entire new field, the field of quantum optics [4, 5, 6, 7, 8]. One can consider this discovery as a precursor of fundamental studies and experiments based on multi-photon interference and correlations [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 1, 19], with intriguing applications in imaging [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], quantum information processing [31, 32, 33, 34, 35, 36], and metrology [37, 38, 39, 40, 41, 42, 43, 44].
In the temporal domain , the HBT effect consists of beam splitting a thermal beam and measuring the correlation in time between the photon intensities at the beam splitter output ports (Fig. 1(a)). Two-photon interference is observed for a detection time delay small compared to the coherence time of the source. Indeed, two-fold detection events have twice the chance to occur at equal detection times than with a time delay much larger than the coherence time. This effect arises from the interference between the two possible indistinguishable pairs of paths that two detected photons may have undertaken from the source to the two detectors (Fig. 1(b)). We emphasize that, since these two pairs of path completely overlap in space, the HBT effect is not sensitive to any relative phase delay. Indeed, no sinusoidal fringes can be observed in standard HBT experiments.
Recently, was proposed a novel interferometric scheme, depicted in Fig. 2(a), where, differently from the standard HBT scheme, two Mach-Zehnder interferometers are introduced at the two output ports of the beam splitter before a correlation measurement at approximately equal detection times is performed . Furthermore, the difference between the long (dashed red) paths and the short (dotted blue) paths in each Mach-Zehnder interferometer is assumed to be larger than the coherence length of the source, while the differences in length between paths of the same type, and , are negligible with respect to the coherence length.
This interferometer has some similarities with the famous Franson interferometer , which however differs in two fundamental aspects:
- a source of two classically correlated beams obtained by beam splitting a thermal light beam is replaced in the Franson interferometer by a source of two photons entangled in energy and time;
- the time delay between the long and short paths in each Mach-Zehnder interferometer is required to be within the second-order coherence time of the two-photon source, which is usually given by the coherence time of the pump laser which generates the entangled pair of photons when interacting with an SPDC crystal.
These two conditions ensure the emission of two entangled photons simultaneously but with a “quantum uncertainty” in time (second-order coherence time) larger than the time delay due to the unbalance between the long and short paths in each Mach-Zehnder interferometer, even if the usual first order coherence length of each single photon is much smaller than the path length difference . This is at the very heart of the observation of the well known two-photon Franson interference between the two pairs and of two-photon paths.
Can second order interference between the path configurations and be also observed in the thermal light interferometer in Fig. 2(a) where the path length difference in each Mach-Zehnder interferometer is beyond the coherence length of the thermal source? Does it matter here how far beyond? Interestingly, as shown in Ref. , the answer is yes and it does not matter, in principle, how far beyond. This result is highly counter-intuitive since, differently from the Franson interferometer, no energy-time entanglement occurs to justify any two-path indistinguishability in terms of second-order coherence length. How does such two-path indistinguishability arise? How can we reconcile it with our common understanding of second-order coherence?
Even more interestingly, it was predicted that this second-order interference phenomenon with thermal light, differently from the standard HBT effect, could manifest itself in sinusoidal fringes as a function of the difference between the phase delays in the two Mach-Zehnder interferometer . This differs from the Franson interferometer which is sensitive to the sum of the interferometer relative phases.
But what is the physics behind the observation of these sinusoidal fringes? And why such a fundamental difference with respect to the Franson interferometer?
All these questions have puzzled different scientists who were reluctant in believing that this interference effect was even correct when it was first predicted and no experimental realizations were yet performed. Fortunately, two main features were of substantial help in experimentally verifying this phenomenon: 1) a thermal source can be easily simulated in a laboratory by using laser light impinging on a fast rotating ground glass ; 2) given a coherence time for a thermal source which can range from the order of to , the calibration of the interferometric paths and of the detection times can be easily achieved. Furthermore, the same effect can be realised also in the spatial domain, by substituting the two Mach-Zehnder interferometers with two double slits and performing spatially correlated measurements at the output, as proposed in Ref. . Indeed, it did not take too long until experimental demonstrations of this effect were first reported both in the temporal and spatial domain in Refs. [19, 41], respectively. Furthermore, as predicted in Ref. , interference fringes as a function of the phase difference have been observed with almost constant visibility at the increasing of the path length difference even if of the order of beyond the coherence length of the source . This phenonenon therefore paves the way to applications in high precision metrology and sensing, including the characterization of remote objects as demonstrated first theoretically in Ref.  and than experimentally in Ref. .
In addition, the genuineness of the correlations between the long and the short paths in the observed second order interference is testified by the possibility to exploit them to simulate quantum logic gates in absence of entanglement . Indeed, the simulation of a CNOT gate by using this interference effect, first theoretically proposed in the temporal domain in Ref.  and extended to the spatial domain in Ref. , was experimentally realised in Ref. . This can lead to potential applications in the development of novel optical algorithms with thermal light [47, 48, 49, 50, 51].
In this paper, we provide a deeper understanding of the physics behind this interference phenomenon in terms of interference of “two-photon detection amplitudes” [9, 10]. We will demonstrate how, differently from standard HBT experiments, interference can actually occur in a phase-dependent manner between two-photon detection amplitudes corresponding to the propagation of a pair of detected photons from the source to the detectors through two possible pairs and of two different types of paths and (Fig. 2(c)). Further, standard HBT interfering amplitudes also arise from the propagation through pairs of the same type of path (either interfering with or interfering with (Fig. 2(b)).
It is than the combination of all these interference contributions from all the possible pairs of photons emitted by the source that leads to the observed “collective” interference between the pair of long paths and the pair of short paths. Evidently, this does not mean that each pair of two detected photons have either taken the or the paths in an indistinguishable manner. Indeed, the correlated paths and cannot be interpreted as two-photon paths. On the other hand, their interference is a result of the non trivial collective contribution of all the possible pairs of detected photons associated with different interfering two-photon detection amplitudes within the statistics of the thermal source (Figs. 2(b) and 2(c)).
In Section 2, we will first analyze this interference phenomenon by using the standard description of thermal light in terms of the Glauber-Sudarshan probability distribution (Eq. (2)). In Section 3, we will give a detailed analysis of the fundamental two-photon interference physics underlying this effect. We will finally conclude the paper with conclusions and perspectives in Section 4.
2 Thermal light interference “beyond coherence”
and the average photon number 
at frequency , with the mean photon rate , average frequency and spectral width .
At the interferometer output the intensity fluctuations
at the detection time around the mean value , at each detector , can be measured by using single-photon detectors with integration time [53, 54]. One can than evaluate the correlation in the intensity fluctuations at the two output ports [53, 54, 5]
by subtracting from the correlation in the intensities 
in terms of the field operators and their Hermitian conjugate at the detectors , the background term 
associated with the product of the intensities
where is a constant.
At approximately equally detection times with respect to the coherence time of the source,
and for path delays
the expression in Eq. (LABEL:eq:corrfluc) becomes 
Here, the interference between the two-path contributions 
with , leads to a sinusoidal dependence on the difference between the relative phases , with , in the two Mach-Zehnder interferometers. These sinusoidal oscillations can be measured independently of how much the difference in length between the L-type paths and the S-type path is beyond the coherence length of the source, since each interfering contribution in Eq. (10) depends only on the relative distance between paths of the same type. Indeed, the other interfering terms and from path of different types give a negligible contributions to Eq. (10) in the conditions in Eq. (9).
We point out that the phase difference can be tuned by varying the difference in length between paths of the same type , within the coherence length of the source as in Eq. (9). This was obtained, for example, experimentally in Ref.  by using a thermal source of coherence length of about and central wavelength of by varying the differences in the path lengths with piezoelectric actuators at the rate of . We also emphasize that interference fringes can be observed also by measuring directly the correlation in the photon numbers in Eq. (5) but with a visibility of due to the additional background term in Eq. (6).
How can we interpret these second-order interference fringes in terms of two-photon interference?
3 Two-photon interference physics
We describe here the observed interference effect in terms of interference of two-photon detection amplitudes [9, 10]. Toward this goal, it useful to mimic the thermal field emitted from the point-like source by a very large number of subfields m with frequency , which state can be written in the coherent representation as 
where is an eigenstate of the annihilation operator with eigenvalues which contain a real-positive amplitude and a random phase arising from the thermal nature of the subfields emitted by the source.
We can then evaluate the correlation in the intensity fluctuations
where denotes the ensemble average over all the possible values of . Here, the field operator at the detectors can be expressed as the sum
where the subfield operators
take into account the propagation of each detected photon m from the point-like source where each corresponding subfields m is generated to the point-like detector through either the long path or the short path .
We can now describe the correlation in the intensity fluctuations in Eq. (3) in terms of the interference of all the possible “two photon-detection amplitudes”
with , associated with all the possible contributions of two different subfields m and n to a two-photon detection, in analogy to a HBT experiment. However, since the time delay between the long (dashed red) and the short (dotted blue) paths (Fig. 2(a)) is beyond the coherence time of the source, interference can occur only if each given subfield m,n undertakes the same type of path in the two-interfering two-photon amplitudes. There are therefore only four possible ways for these amplitudes to interfere with each other as in the expression