When studying the correlations between two sources using two detectors, the Hanbury-Brown & Twiss effect arises from two-particle interferences between direct and exchange paths, pictured on Fig. 10a). As discovered in 1956 when observing distant stars , intensity correlations offer a powerful way to study the emission statistics of sources. In particular, two particle interferences lead to different possible outcomes depending on the fermionic or bosonic character of the two indistinguishable particles that would impinge on a beamsplitter (Fig. 10b)). On one hand, indistinguishable electrons (fermions) antibunch: the only possible outcome is to measure one electron in each output arm. On the other hand, indistinguishable photons (bosons) bunch: two photons are then measured in one of the outputs. Thus, when such particles collide and bunch/antibunch on the beam-splitter, the fluctuations and correlations of output currents encode information on the single particle content of the incoming beams. First observed with light sources , the HBT effect has since then been observed for electrons propagating in a two dimensional electron gas [19-21].
A convenient way to implement the interference between the two exchanged paths on two detectors is to use the geometry described on Fig. 11. The two sources are placed at the two inputs of a beam-splitter and the two detectors at the two outputs. A coincidence detection event on the detectors has then two exchanged contributions. Particles emitted by source 1 and 2 can be reflected to 3 and 4 or transmitted to 4 and 3. These two paths lead to two-particle interferences in the coincidence counts of the two detectors. Using electron sources, a quantum point contact can be used as a tunable electronic beam-splitter with energy-independent reflexion and transmission coefficients R and T () relating incoming to outgoing modes. As single particle detection is not available yet for electrons (at least for subnanosecond time scales), coincidence counts are replaced in electronics by current correlations. The output current operators and the output current correlations can be expressed in terms of input currents and correlations:
where and are the current fluctuations in inputs 1 and 2 and denotes the quantum Hanbury-Brown & Twiss contribution to outcoming current correlations. It encodes the aforementioned two-particle interferences and involves the coherence functions of incoming electrons and holes:
This quantum two-particle interference can be unveiled through the measurement of zero-frequency correlations. Namely, standard low-frequency noise measurement setup gives access to the averaged quantities . Thus it is possible to access the averaged HBT contribution
which is nothing but the overlap between the single electron and hole coherences of channels 1 and 2, and plays a key role in the various experiments one can perform in the Hanbury-Brown & Twiss geometry. In the following, we will study the three situations described on Fig. 11. In the first one, a single source is used and partitioned on the splitter while the second input is kept 'empty'. Contrary to the true vacuum obtained in the optical experiment, in electronics, this second input is always connected to a Fermi sea which is a source at equilibrium. This leads to important differences in the electronic version of this experiment. In the second experiment, each input is connected to a triggered single electron emitter. Two single electrons collide synchronously on the splitter realizing the electronic analog of the Hong-Ou-Mandel experiment in optics [23, 75-77]. Finally, using a reference state in one input, an unknown input state can be reconstructed and imaged by measuring its overlap with the known reference state. The principle of such a single electron state tomography will be described in the last section.
5.1 Single source partitioning
Let us first consider the electronic analog of the seminal experiment performed by Hanbury-Brown & Twiss to characterize optical sources , in which a light source is placed in input 1 whereas the second arm is empty and described by the vacuum. In the electronic analog, the single electron source described previously is used, while the empty arm now consists of a Fermi sea at equilibrium, with fixed temperature and chemical potential. The purpose of this experiment is not here to obtain the charge statistics of the source, that is accessed via high-frequency autocorrelations described in the previous section. It in fact reveals the number of elementary excitations (electron/hole pairs) produced by the electron source, which has no optical counterpart and stems from the fact that particles with opposite charges contribute with opposite signs to the current. The total number of elementary excitations emitted from the source is hard to access through a direct measurement of the current or its correlations (that is without partitioning). Indeed, the emission of one additional spurious electron/hole pair in one driving period, as represented on Fig. 6 (sixth period of the drive on the figure) is a neutral process and cannot be revealed in the current if the time resolution of the current measurement is longer that the temporal separation between the electron and the hole. This temporal resolution is estimated to be a few tens of picoseconds in the high frequency noise measurement presented previously. Spurious electron/hole pairs emitted by the source on a shorter time scale thus cannot be detected. However, the random and independent partitioning of electrons and holes on the splitter can be used to deduce their number from the low frequency current fluctuations of the output currents. Using Eqs. (30)–(33), the excess output current correlations and their low frequency spectrum are given by:
Where is the excess HBT contribution with respect to equilibrium. As can be seen in Eq. (36) and contrary to optics, the single source partitioning experiment involves two sources, the triggered emitter and the Fermi sea at finite temperature, through the overlap between their first order coherence and . This overlap is more easily expressed in Fourier space:
Where is the excess number of electrons (at energy above the Fermi energy) emitted per unit energy in the long measurement time . Similarly, is the energy density of the number of holes emitted at energy (corresponding to a missing electron at energy below the Fermi energy) in the measurement time . For a periodic emitter of frequency f, it is more convenient to use the energy density of the number of excitations emitted in one period. To avoid defining too many notations, in the rest of the manuscript, (resp. ) will refer to the energy density of electrons (resp. holes) emitted in one period. Defining as the number of electron/hole pairs counted per period by the partition noise measurement, Eq. (37) then becomes:
Considering first the limit of zero temperature, equals the average number of electrons/holes emitted in one period. This result can be understood by a simple classical reasoning: electrons and holes are independently partitioned on the beam-splitter following a binomial law. As a consequence, the low-frequency output noise is proportional to the number of elementary excitations arriving on the splitter. Consequently, measuring the HBT contribution directly gives access to the total number of excitations generated per emission cycle. However, large deviations to this classical result can be observed due to finite temperature. Indeed, input arms are populated with thermal electron/hole excitations that can interfere with the ones generated by the source, thus affecting their partitioning. As seen in Eq. (37), is corrected by , corresponding to the energy overlap of thermal excitations and the particles triggered by the source. The minus sign reflects the fermionic nature of particles colliding on the QPC. For vanishing temperatures, classical partitioning is recovered. For non-vanishing temperature, a fraction of the triggered excitations reaching the beamsplitter find thermal ones at the same energy. In virtue of Fermi-Dirac statistics, these indistinguishable excitations antibunch (see Fig. 10): the only possible outcome consists of one excitation in each output, so that no fluctuations are expected in that case, thus reducing the amplitude of the HBT correlations.
Figure 13 presents the HBT low frequency correlations as a function of the beam-splitter transmission T. For all three curves, the dependence is observed, but the noise magnitude notably differ. In particular, , invalidating the classical partitioning of a single electron/hole pair. This discrepancy is attributed to the non-zero overlap between triggered excitations and thermal ones, whose exact value strongly depends on the driving parameters. An intuitive picture can be proposed. The highest value of is observed with a square drive. In this case, a single energy level in the dot is rapidly raised from below to above the Fermi level of the reservoir, and the quasiparticle is emitted at an energy well separated from thermal excitations. Therefore, we expect the outcoming noise to be maximum. For a sine wave, the rise of the energy level in the dot is slower and the electron is emitted at lower energies and thus more prone to antibunch with thermal excitations. This tends to reduce . As the transmission D is lowered, the escape time increases and electron emission occurs at later times, corresponding to higher levels of the sine drive. The quasiparticle is then emitted at higher energies and are less sensitive to thermal excitations. is then increased, as seen by comparing the black and red traces of Fig. 13. This intuitive picture can be confronted to numerical calculations within the Floquet scattering theory [61, 54] which can be used to calculate and for any type of excitation drive (sine or square) and any value of the dot parameters. The resulting curves for the energy distributions can be found on ref , they confirm the intuitive picture discussed above.
These differences in energy distributions can be revealed by the Hanbury-Brown & Twiss interferometry, as shown on Fig. 14 that presents measurements of as a function of the dot transmission D for two different drives, sine or square. Floquet calculations for square and sine drives at are presented in red dashed line: they are almost identical and reach for , as expected for an ideal source that does not emit additional electron-hole pairs. For , the shot noise regime is recovered whereas quantization effects in the dot are progressively lost for . The effect of temperature in arm 2 ( mK, ) is shown in blue line. As already discussed, the presence of thermal excitations reduces . This effect decreases when lowering the transmission, and is more pronounced for sine wave than for square drive. Remarkably, the effect of temperature in arm 1 (blue dashes) is identical to the one in arm 2. When a temperature of 150 mK (extracted from noise thermometry) is introduced in both arms, a good agreement is found with the experimental data (black dashes). This confirms the tendency to produce low energy excitations when using a sine drive, and energy-resolved excitations using a square drive. Note that the Floquet calculations do not take into account the energy relaxation  along the 3 microns propagation towards the splitter that will be discussed in the last section of this article. It only provides the energy distribution at the output of the source, 3 microns away from the splitter where the collision with thermal excitations occur. The good agreement with Floquet calculation implies that energy relaxation has a small effect on the total number of excitations and would require a direct measurement of the energy distribution (and not of its integral on all energies) to be characterized.
5.2 Hong-Ou-Mandel experiment
The previously discussed antibunching effect bears strong analogies with the photon coalescence observed in the Hong-Ou-Mandel experiment . While quasiparticles are generated on-demand in the first input, thermal excitations are however randomly emitted in the second input. To recreate the electronic analog of the seminal Hong-Ou-Mandel experiment [80-82], two identical but independent single electron sources can be placed in the two input arms of the beamsplitter, as pictured in Fig. 15.
As in the seminal HOM experiment, the antibunching of the on-demand quasiparticles provides a direct measurement of the overlap of the two mono-electronic wavefunctions, i.e. their degree of indistinguishability. Indeed, for two sources generating periodically (period ) a single electron described by the wavefunctions and above the Fermi sea (well separated from thermal excitations), as seen in Sec. 'Stationary source versus single particle emission', the coherence function for source i reads such that we have:
For perfectly distinguishable electrons, and the classical random partitioning of two electrons is recovered. However, for perfectly indistinguishable electrons, and the random partitioning is fully suppressed. The overlap between the two particles can be modulated by varying the delay τ between the excitations drives. Dividing by the total partition noise of both sources ( for each source neglecting temperature effects) one then gets the normalized HOM correlations as:
When working at finite temperature, the partition noise in the HOM and HBT configurations is reduced from their overlap with thermal excitations (see previous section). However, if the generated quantum states in sources 1 and 2 remain indistinguishable, the antibunching effect remains total and numerical simulations using the Floquet scattering formalism show that is only marginally modified.
The resulting HOM correlations are presented in Fig. 16 as a function of delay τ. A dip in the correlations is clearly observed around . The measured noise is normalized by its value on the plateaus observed at large delays, and matches as expected the sum of the HBT contributions of each source, that are measured independently by alternatively turning one of the sources off. As seen in Sec. 'Single electron emitters', for a square wave excitation, single electron emission is described by an exponentially decaying wavepacket, with decay time and energy ε0 that depends on the amplitude of the square excitation: . then takes the following simple form:
Taking into account a loss in the visibility η and an error on synchronization τ0, fitting with then gives ps, ps and . The extracted value of is consistent with independent measurements via the average current. Though effects of the partial indistinguishability of the generated excitations are indubitable, the visibility η is far from unity. This may be the result of parameter mismatch between the two sources, resulting in reduced overlap of the wavepackets, but also from decoherence effects due to interaction with the environment. Such effects will be discussed in Sec. 'Interactions in electron quantum optics'.
Figure 16. Excess noise as a function of time delay τ and normalized by the value on the plateau observed for long delays. The sum of both partition noises (in the HBT configuration) is depicted by the blue blurry line, while the red trace is obtained with a fit by
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5.3 Electron-hole correlations in the Hong-Ou-Mandel setup
A unique property of electron optics compared to photon optics is the ability to manipulate hole excitations in addition to electron excitations. Performing the HOM experiment with identical single hole excitations in the two input arms of the beamsplitter will produce results similar to those of electrons (with hole wavefunctions replacing electron wavefunctions in Eq. (41)). But performing the HOM experiment while injecting a single electron excitation in one input arm of the beam-splitter, and a single hole excitation in the other arm will produce results which have no counterpart in optics .
In order to get useful analytical formulas, we first consider theoretically states where one electron charge has been added (removed) from the Fermi sea
where is the Fermi sea at temperature , and , the electron and the hole wavefunctions in real space. Taking the electron-hole symmetric case for simplicity (), the normalized HOM correlation becomes:
Comparing this with Eq. (41), we notice important changes. First, the interferences contribute now with a positive sign to the HOM correlations, that is, the opposite of the electron-electron case. Electron-hole interferences produce a “HOM peak” rather than a dip. Second, the value of this peak depends on the overlap of the electron and the hole wave packets times the Fermi product . This peak thus vanishes as since it requires a significant overlap between electron and hole wave packets, a situation which only happens in an energy range around , where electronic states are neither fully occupied nor empty.
To simulate the electron-hole HOM peak with the real electron emitters, we have used the Floquet scattering matrix formalism. We have computed the correlations when the two single electron sources in the two input arms of the beam splitter are submitted to a square drive. As these sources periodically emit an electron and then (after half a period) a hole, the correlations obtained for a time delay close to a half-period correspond to the correlations between an electron and a hole. The results for as a function of the time-delay τ are shown on Fig. 17, for a drive period of 400 (in units of ). As the correlations are proportional to the overlap in energy of the electron and the hole wavefunctions (see Eq. (44)), in order to observe a peak the electron emission and the hole emission need to happen at energies not too far apart. This can be controlled by the dot level position of the single electron source with respect to the Fermi energy: when a dot level is close to resonance with the Fermi energy ( on Fig. 17), the energy overlap between the emitted electron and the emitted hole is important, and a large peak in the correlations is observed. On the other hand, when the dot levels of the single electron sources are far from resonance ( on Fig. 17), there is no overlap in energy between the emitted electron and the emitted hole, and no peak is visible in the correlations, as observed on the experimental data of Fig. 16 where electron/hole correlations are below experimental resolution. The temperature used in these simulations is , which is similar to the experimental value.
5.4 Tomography of a periodic electron source
In the previous experiments, properties of the source can be inferred by measuring, through current correlations, the resemblance between the state in input arm 1 and its counterpart in input arm 2. Indeed, HBT correlations yield information on the energy distribution of the source, by taking the Fermi sea as a reference, whereas HOM correlation demonstrate the indistinguishability of two quantum states generated by two independent sources. In fact, the complete coherence function in energy domain of a source of electrons and holes can be obtained in the HBT geometry by placing in input arm 2 different reference sources and measuring the corresponding current correlations. These spectroscopy  and tomography processes , inspired by the optics equivalent [85-87] could provide a direct image of electron wavepackets propagating in quantum Hall edge channels through the determination of the first order coherence in the ε, plane. For a periodic source, the definition of the first order coherence in the energy domain needs to be slightly modified. Indeed, has a T-periodicity in the time , and no periodicity along . Using these two variables in time, the Fourier transform is defined in the following way:
From the above definition, and are related through:
Due to the periodicity in time, takes discrete values along the energy difference while the sum of energies takes continuous values, . The population in energy domain thus corresponds to the component of while the coherences correspond to .
The source contribution of the coherence function can be fully reconstructed in the , plane by applying as a reference state on input 2 a voltage sum of a dc bias and an ac excitation at angular frequency . The complete description of this tomography protocol lies beyond the scope of this article and can be found in Ref.. However an intuitive understanding can be drawn, that mainly relies on the two-particle interference between the electron source under study and the reference source. Let us first focus on the reconstruction of the component of the coherence function, associated with the energy distribution, that is on the spectroscopy of the electron source. A sketch supporting this discussion is presented Fig. 18a). In the case , only the dc part of the voltage applied on input 2 is kept: that shifts the chemical potential of the connected edge by the value . As already mentioned, a two-particle interference can only occur between states of same energy. An electron at a well defined energy ε0 finds a symmetric partner in input 2 only if (in the limit of vanishing temperature). Under this threshold, antibunching occurs with unit probability and partition noise is reduced to zero. Otherwise, for , the random partitioning takes place, regardless of the presence of the DC bias. Accordingly, by sweeping the bias , one can then reconstruct the probability of finding a particle at energy ε, namely from the dependence of the partition noise due to antibunching effects. Due to thermal smearing effects, the resolution of such a spectroscopy is in fact limited to in the presence of a finite temperature . In the same manner (Fig. 18b)), dynamical modulations of the noise with a reference voltage enables to gain access to harmonics for , that is the off diagonal elements in the plane.
Using once again the Floquet scattering formalism, simulations of the coherence function of a periodic source have been realized, in the case of the single electron/hole source: three cases with different sets of parameters illustrate the key features on Fig. 19. For clarity, odd and even harmonics of are plotted on separate graphs as they have different parity with respect to ω: is odd while is even. First, these graphs clearly highlight the four quadrants identified in Fig. 2. For a transmission , the parameters are close to the optimal values: every charge is emitted during the dedicated emission cycle and the excitations are highly energy-resolved, around energies . Only weak coherences are detected: the emission probability is very close to one, so that the emission of an electron is decorrelated from the emission of the previous hole as the emission probability is close to one. Going towards higher transmission () yields excitations that lie mostly at low energy, and spread over a wide range of energies. Since the transmission is high, the two emission events of electron and holes are once again decorrelated. On the opposite, for lower transmissions (), strong coherences appear as the emission probability is much smaller than 1. Production of holes and electrons are correlated as the emission of an electron is subject to the emission of the preceding hole, which does not take place in each cycle.
Note that, as suggested in refs [88, 89], the coherence function of the source could also be measured in time domain, , measuring the current at time t at the output of a Mach-Zehnder interferometer as a function of the difference τ in the propagation time between the two arms of the interferometer. This method implies a simpler measurement (average current instead of current fluctuations) but a more complicated sample. Also, decoherence effects during the propagation in the interferometer [90-93] would have to be taken care of.