transmissionsTof the beamsplitter:T= 0.2,
T= 0.27, andT= 0.5 ( 37 ) (Fig. 2). In the three
cases, forI+≥50 pA, negative cross-correlations
varying linearly with the currentI+were ob-
served. We extracted the slopeaof the varia-
tion ofSI 3 I 4 = 2 e by a linear fit (dashed lines)
of the experimental data. The three extracted
values ofaare plotted in the inset of Fig. 2 as
a function of the beamsplitter transmission
T. The observedTdependence agrees with
the binomial lawT(1ÐT) (dashed line) extend-
ing Eq. 7 for transmissions beyond the weak-
backscattering regime ( 32 ). The generalized
Fano factor can be extracted from the fit ofa
with the dependencea=PT(1ÐT), givingP=
Ð2.1 ± 0.1 and demonstrating the fractional
statistics atn=⅓with the predicted exchange
phasef=p/3. In striking contrast, we observed
P≈0 at filling factorn= 2 (fig. S5), correspond-
ing to the expected fermionic behavior for an
integer filling factor.
The fermionic behavior can be restored at
n=⅓by increasing the transmissions of the
input QPCsT 1 andT 2 , thereby deviating from
the weak-backscattering regime suitable for the
emission of anyons. ForTS= 1 (black points
in Fig. 3A), we observed fermionic behavior:
SI 3 I 4 ¼0 for all values ofI+. For intermediate
values ofTS, the current-voltage character-
istics of the input QPCs were strongly non-
linear (Fig. 3B).TSdecreased whenI+increased,
eventually restoring the weak-backscattering
limit at large bias. The measurements ofSI 3 I 4
forTS= 0.14 andTS= 0.2 (forI+=0)plottedin
Fig. 3A reflect this evolution. At low current
I+, fermionic behavior was observed:SI 3 I 4 ¼0.
At higher current, where the weak-backscattering
limit is restored, the linear evolution of the
cross-correlations withI+was recovered, with
a generalized Fano factor almost constant.P
slightly increased fromP=Ð2.00 ± 0.15 for
TS= 0.04 toP=Ð1.94 ± 0.12 forTS= 0.14, and
toP=Ð1.73 ± 0.10 forTS= 0.2. As expected,
the domain where the fermionic behavior was
observed (SI 3 I 4 ¼0) increased when the trans-
missionTSincreased; it varied from |I+|≤
200 pA atTS= 0.14 to |I+|≤400 pA atTS=
0.2. These data confirm thatP=Ð2 is ob-
served only in the regime of anyon emission
and that regular fermionic behaviorP≈ 0
takes place away from the weak-backscattering
limit.
Finally, we checked in more detail the
agreement between our measurements and
the quantum description of anyon collisions
( 20 ) by investigating the dependence of the
Fano factorPon the ratioIÐ/I+. Contrary to
the previous experiments whereIÐ= 0 wasimposed byV 1 =V 2 andT 1 =T 2 , we instead
modified the ratioIÐ/I+by varying the values
of the input voltagesV 1 ≠V 2. Figure 4A presents
the evolution ofSI 3 I 4 as a function of the total
currentI+for four different values of the ratio
IÐ/I+andTS=0.05.Weobservedinthefour
cases a linear evolution withI+, with a slope
Pthat decreases whenIÐ/I+increases. The
different values ofPextracted from a linear fit
of the data (dashed lines in Fig. 4A) are plotted
in Fig. 4B. ForIÐ/I+≤0.2,Pis constant with
P≈Ð2.Pthen decreases linearly towardP≈Ð 3
forIÐ/I+≈1. These experimental results can
be compared with the calculation of ( 20 )
(dashed line). The excellent agreement between
our experimental results and the calculations
further supports the quantum description of
anyons withf=p/3.
Our measurement of the Fano factorP=Ð 2
demonstrates the anyonic statistics of the
charge carriers with an exchange phasef=p/3
in accordance with the predictions for the
Laughlin staten=⅓. Interestingly, the pre-
dictionP=Ð2 forf=p/3 is valid when edge
reconstruction effects can be neglected. Although
neutral modes have been observed even atn=
⅓( 38 ), the agreement with the prediction for a
simple edge structure suggests that their effect
can be neglected in our experiment ( 33 ). Col-
lision experiments similar to ours could be used
to characterize the elementary excitations of
other fractional quantum Hall phases with
different fractional statistics or even more
exotic cases where non-Abelian statistics ( 17 )
are predicted. Additionally, combining collision
experiments with the triggered emission of
fractional quasiparticles ( 39 , 40 ) would allow
one to perform on-demand braiding of single
anyons in a quantum conductor.REFERENCESANDNOTES- J. Leinaas, J. Myrheim,Nuovo Cim. Soc. Ital. Fis. B 37 ,1– 23
(1977). - F. Wilczek,Phys. Rev. Lett. 48 , 1144–1146 (1982).
- F. Wilczek,Phys. Rev. Lett. 49 , 957–959 (1982).
- D. C. Tsui, H. L. Stormer, A. C. Gossard,Phys. Rev. Lett. 48 ,
1559 – 1562 (1982). - R. B. Laughlin,Phys. Rev. Lett. 50 , 1395–1398 (1983).
- B. I. Halperin,Phys. Rev. Lett. 52 , 1583–1586 (1984).
- D. Arovas, J. R. Schrieffer, F. Wilczek,Phys. Rev. Lett. 53 ,
722 – 723 (1984). - C. de C. Chamon, D. E. Freed, S. A. Kivelson, S. L. Sondhi,
X. G. Wen,Phys. Rev. B 55 , 2331–2343 (1997). - K. T. Law, D. E. Feldman, Y. Gefen,Phys. Rev. B 74 , 045319
(2006). - F. E. Camino, W. Zhou, V. J. Goldman,Phys. Rev. Lett. 98 ,
076805 (2007). - N. Ofeket al.,Proc. Natl. Acad. Sci. U.S.A. 107 , 5276–5281 (2010).
- D. T. McClure, W. Chang, C. M. Marcus, L. N. Pfeiffer,
K. W. West,Phys. Rev. Lett. 108 , 256804 (2012). - R. L. Willett, C. Nayak, K. Shtengel, L. N. Pfeiffer, K. W. West,
Phys. Rev. Lett. 111 , 186401 (2013). - J. Nakamuraet al.,Nat. Phys. 15 , 563–569 (2019).
- R. L. Willett, K. Shtengel, C. Nayak, L. N. Pfeiffer, Y. J. Chung,
M. L. Peabody, K. W. Baldwin, K. W. West, arXiv 1905.10248
[cond-mat.mes-hall] (24 May 2019). - B. I. Halperin, A. Stern, I. Neder, B. Rosenow,Phys. Rev. B 83 ,
155440 (2011). - G. Moore, N. Read,Nucl. Phys. B 360 , 362–396 (1991).
- M. Banerjeeet al.,Nature 559 , 205–210 (2018).
- Y. Kasaharaet al.,Nature 559 , 227–231 (2018).
176 10 APRIL 2020•VOL 368 ISSUE 6487 sciencemag.org SCIENCE
Fig. 4. Experimental test
of the quantum
mechanical description of
an anyon collision.(A)SI 3 I 4
as a function ofI+for various
values of the ratioI–/I+. The
dashed lines are linear fits of
SI 3 I 4 .(B) Generalized Fano
factorPextracted from the
slope of the linear fits in (A),
plotted as a function of the
ratioI–/I+. The colors corre-
spond to the ratiosI−/I+
plotted in (A). The horizontal
error bars correspond to
the standard deviation of the
ratioI–/I+of the data. The
vertical error bars are given
by the uncertainties of the
linear fits. The dashed line is
the prediction extracted from
( 20 ) for the quantum
description of anyon colli-
sions withf=p/3.
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