Sample description

The sample is described in Fig. 1b. It is realized in a two dimensional electron gas of nominal density n s =1.9 × 1015m−2 and mobility μ=2.4 × 106cm2V−1s−1 placed a in strong magnetic field B=4T so as to reach a filling factor ν=2 in the bulk. The emitters are two quantum dots synchronously driven by a periodic square excitation applied on the dot top gates with a 40ps risetime. They are placed at a distance l=3.2±0.4μm (corresponding to the interaction region) of a quantum point contact (QPC) used as the electronic beam-splitter. Source 1 is placed at input 1 of the splitter, source 2 at input 2. Changing the voltage V QPC , the QPC can be set to partition either the outer or the inner edge channel. The dots are only coupled to the outer edge channel such that the current pulse is generated on the outer channel only. The dot to edge transmission D is used to tune the dot emission time and the dot charge quantization. Two configurations are studied: at D=1 the dot is perfectly coupled, charge quantization is lost and a classical current pulse (carrying a charge close to e) is generated in the outer channel. This configuration provides the shortest emission time and thus the best time resolution. At D≈0.3, charge is quantized and single quasiparticles are emitted in the outer channel. As we use a periodic square excitation, the electron emission is followed by hole emission5 corresponding to the dot reloading, with a repetition time T=1.10 ns. The HOM noise Δq(τ) normalized by the random partition noise is measured on output 3 of the QPC (see Methods).

HOM interferometry reveals fractionalization

Figure 2 shows Δq(τ) for D=1 (Fig. 2a) and D≈0.3 (Fig. 2b), both when the outer (orange points) or the inner (black points) channels are partitioned. From the outer channel partitioning, we probe the evolution of the generated electron pulse during propagation, inner channel partitioning results from the collective excitations generated by the interaction process. All the traces show a noise reduction (dip) on short times τ, which is reminiscent of two-particle interference. However significant differences are observed in the width of the HOM dips, labelled τ w , which we estimate using an exponential fit. Focusing first on D=1: the outer channel dip is roughly twice larger than the inner one: τ w =80 ps (outer) versus τ w =40 ps (inner). The increased width of the outer channel dip reflects the fractionalization of the current pulse that splits in two pulses of the same sign (see Fig. 2c). The smaller width on the inner channel reflects the dipolar current trace (see Fig. 2c) and equals the temporal extension of the current pulse of a given sign (electron like or hole like), limited by the excitation pulse rise time. For larger time delays (|τ|≈100 ps), the inner channel normalized HOM signal shows an overshoot above unity. As predicted in refs 24, 33, Δq(τ)≥1 occur when an electron-like pulse collides with a hole-like one. It occurs in the inner channel for |τ|≈τ s , the electron part of the inner channel current pulse in input 1 then collides with the hole part of the current pulse in input 2 (see sketch on Fig. 2d). This contrasts with the monotonical increase of Δq(τ) towards 1 for the outer channel. When the dot transmission is decreased to D=0.3±0.05 (D=0.4±0.05 for inner channel partitioning), we observe the expected increase of the HOM dip width compared with D=1, reflecting the increase in the dot emission time: τ w =120 ps (respectively τ w =80 ps) for the outer (respectively inner) channel. Note that the dot to edge transmission are slightly different for outer (D≈0.3) and inner (D≈0.4) channel partitioning. Due to gate coupling, it is hard to tune the dot transmissions to the exact same values when the QPC voltage V qpc is set to partition the outer or the inner channel. This limited accuracy on the dot transmission does not allow for a quantitative comparison between the outer and inner channels dip widths at D≈0.3.

Figure 2: Normalized HOM noise. (a) Δq(τ) at D=1 for outer (orange points) and inner (black points) channel partitioning. Error bars on a and b equal the s.e. of the mean reflecting the statistical dispersion of points. (b) Δq(τ) at D≈0.3 for outer (orange points) and inner (black points) channel partitioning. Encircled c and d refer to the sketches on c and d. The black and orange dashed lines on both the panels represent the fits of the dips using the following exponential dependence: . The extracted values at D=1 are γ=0.73 (both for outer and inner channels) and τ w =40 ps (inner channel) and τ w =80 ps (outer channel). At D≈0.3, we have γ=0.41 and τ w =120 ps (outer channel) and γ=0.31 and τ w =83 ps (inner channel). (c) Sketch of current pulses synchronization at τ=0 for the outer and inner channel partitioning. The outer channels are represented as orange lines, the inner as black lines. Negative (positive) charge pulses are represented by blue (red) colours. Pulses colliding synchronously are emphasized by red circles. (d) Sketch of inner and outer channel current pulses when the time delay between the sources is τ=τ s . The inner channels (black lines) are partitioned while the outer ones (orange dashed lines) are not. Full size image

Decoherence of single-electron states

The contrast γ=1−Δq(0) measures the degree of indistinguishability between the states at inputs 1 and 2, γ=1 corresponding to full partition noise suppression, γ=0 to the absence of interference. The contrasts are much higher for D=1 (γ≈0.73 for both channels) compared with D≈0.3 (γ=0.35 for the outer channel and 0.25 for the inner one). This suppression of the contrast is a consequence of interaction-induced24 decoherence. In principle, the contrast of the classical pulse (D=1) should not be affected by the interactions and we attribute the observed reduction (from 1 to≈0.75) to residual asymmetries in the colliding pulses. As a matter of fact, when the dot is fully open, a classical charge density wave, or edge magnetosplasmon (EMP), carrying current I(t) is generated in the outer channel, as if it was driven selectively by the time-dependent voltage V(t)=h/e2 I(t). The EMP is a collective charge excitation of bosonic nature. It corresponds in the bosonic description, to a product of coherent states: , where the coherent state parameter encodes the outer channel current4 at pulsation ω and |0〉 is the inner edge in the vacuum state (thermal fluctuations are discarded). As a result from interactions, this EMP is partially transferred to the inner channel at the output of the interaction region18,34: (respectively r ω ) is the transmission amplitude to the outer (respectively inner) channel that encodes the interaction parameters. As seen from |Ψ out 〉, the outer channel (conductor) does not get entangled with the inner one (environment)15. A perfect dip γ=1 should be observed both for the outer and inner channels as long as I 1,ω =I 2,ω and t 1,ω =t 2,ω . This can be understood from gauge transformation arguments. Indeed for classical applied voltage pulses V 1 (t) and V 2 (t), all the applied voltage can be brought to one input only (for example, 2) by the overall shift −V 1 (t). Noise is then obviously suppressed (γ=1) for V 1 (t)=V 2 (t). The situation is completely different for the single-particle state produced at D≈0.3. The emission of an electronic excitation with wavefunction φe(x) has no classical counterpart in the bosonic representation and corresponds to a coherent superposition of coherent states4,15: with (ν being the Fermi velocity). It gets entangled with the environment after interaction, each coherent state in the superposition leaving a different imprint in the environment: . After tracing out the environment (inner channel) degrees of freedom, outer channel coherence is suppressed, corresponding to a strong reduction of indistinguishability between the inputs, and thus of the interference contrast (the same argument holds for two-particle interferences in the inner channel by tracing on the outer channel degrees of freedom). This suppression shows that, as Coulomb interaction favours the emergence of collective excitations through the fractionalization process, it is accompanied by the progressive destruction of the quasiparticle that degrades into the collective modes4,15.

Comparison between data and model

Further evidence of fractionalization can be observed on longer time delay |τ|≈T/2 when electron emission for source 1 is synchronized with hole emission for source 2. For |τ|≈T/2, Δq(τ) for D=1 plotted on Fig. 3 exhibits again contrasted behaviours for the outer and inner channels. While it monotonically increases above 1 for the outer channel (see Fig. 3a), as expected for electron/hole collisions, the inner channel shows an additional dip for |τ|≈T/2−τ s (see Fig. 3b). This reveals again the dipolar nature of the inner current: as the dipoles have opposite signs for electron and hole emission sequences, the electron parts of each dipole are synchronized for |τ|=T/2−τ s (see sketch on Fig. 3d). A quantitative description of the HOM traces can be obtained (black and orange lines) by simulating (see Methods) the propagation of the current pulse in the interaction region (see Fig. 4) taking interaction parameters and and measured on a similar sample (ref. 18). The obtained current traces at the output of the interaction region (black and red dashed lines on Fig. 4) reproduce the sketch depicted on Fig. 1a. The good agreement obtained for the HOM trace (Fig. 3b) supports the above qualitative descriptions of the dips observed at τ s and T/2−τ s related to charge fractionalization. Note that an additional spurious modulation of the current resulting from a rebound in our excitation pulse also occur causing an additional dip at |τ|≈350 ps on the outer channel and |τ|≈225 ps on the inner one. Finally, Figure 5 presents Δq(τ) at D≈0.3 for the full range of time shifts −T/2≤τ≤T/2. The qualitative behaviour, although strongly blurred by decoherence, is similar to that of Fig. 3. In particular, the additional dip for |τ|≈T/2−τ s is only observed on the inner channel, which is a hallmark of single-electron fractionalization. Compared with D=1, its position is slightly shifted to lower values of |τ| (|τ|≈430 ps), we attribute this difference to the larger width of the emitted current pulse related to the larger emission time.

Figure 3: Temporal investigation of charge fractionalization. (a) Δq(τ) at perfect dot to edge coupling D=1 for outer channel partitioning (orange points). Error bars on both the panels equal the standard error of the mean reflecting the statistical dispersion of points. (b) Δq(τ) at D=1 for the inner channel partitioning (black points). The orange and black lines on both the panels are simulations for Δq(τ). The vertical red lines correspond to a time delay matching the half-period of the excitation drive: τ=±T/2. Encircled c and d refer to the sketches on c and d. (c) Sketch of current pulses synchronization at τ=τ s for inner channel partitioning. The outer channels are represented as orange lines, the inner as black lines. Negative (positive) charge pulses are represented by blue (red) colours. Pulses colliding synchronously are emphasized by red circles (electron/hole collision in this case). (d) Sketch of current pulses synchronization at τ=T/2−τ s for inner channel partitioning. Pulses colliding synchronously are emphasized by red circles (electron/electron and hole/hole collisions in this case). exp., experimental; theo., theoretical. Full size image

Figure 4: Output current simulation. (a) Simulation of the excitation pulse (black dashed line) applied to the dot. The exact shape of the excitation pulse is not known as it is affected by its propagation in the cryostat. The resulting emitted current at D=1 before interaction is plotted in blue. It shows the electron emission sequence followed by the hole one. (b) Outer (red dashed line) and inner (black line) channel currents obtained using and . The outer channel shows the pulse splitting while the inner channel is a dipolar charge excitation. Full size image