## Probing the dynamics of anyons

A two-dimensional electron gas in the fractional quantum Hall regime has unusual excitations called anyons that carry only a fraction of the electron's charge. This fractional charge can be observed through a dynamical response to irradiation by microwaves, but such experiments require a combination of high magnetic fields with sensitive noise measurements and very low temperatures. Kapfer *et al.* observed this dynamical response in a GaAs/AlGaAs heterostructure hosting a high-mobility two-dimensional electron gas with fractional excitations of one-third and one-fifth of the electron's charge. The method may be of interest for use in topological quantum computing.

*Science*, this issue p. 846

## Abstract

Anyons occur in two-dimensional electron systems as excitations with fractional charge in the topologically ordered states of the fractional quantum Hall effect (FQHE). Their dynamics are of utmost importance for topological quantum phases and possible decoherence-free quantum information approaches, but observing these dynamics experimentally is challenging. Here, we report on a dynamical property of anyons: the long-predicted Josephson relation *f*_{J }= *e***V*/*h* for charges *e** = *e*/3 and *e*/5, where *e* is the charge of the electron and *h* is Planck’s constant. The relation manifests itself as marked signatures in the dependence of photo-assisted shot noise (PASN) on voltage *V* when irradiating contacts at microwaves frequency *f*_{J}. The validation of FQHE PASN models indicates a path toward realizing time-resolved anyon sources based on levitons.

The quantum Hall effect (QHE) occurs in two-dimensional electron systems when strong magnetic fields quantize the electron cyclotron energy into Landau levels. For integer Landau level filling factor ν = *p*, the integer QHE (IQHE) shows a topologically protected quantized Hall conductance *pe*^{2}/*h* with zero longitudinal conductance (*1*). For very low-disorder samples, the Coulomb repulsion favors topologically ordered phases at rational ν = *p*/*q* showing a fractional QHE (FQHE) with fractional Hall (*2*) and zero longitudinal conductance. For electrons filling the first Landau level (ν < 1), the states with ν = 1/(2*k* + 1), where *k* is an integer, are well described by Laughlin states (*3*). The elementary excitations, or quasiparticles, bear a fraction *e** = *e*/(2*k* + 1) of the elementary charge *e* (*3*–*6*) and are believed to obey a fractional anyonic (*7*) statistics intermediate between that of bosons and fermions. For ν < 1, the Jain states (*8*) with ν = p/(2*kp *+ 1), where *p* and *k* are integers, display *e** = *e*/(2*kp* + 1) fractionally charged excitations (*9*); these excitations are composite fermions, i.e., electrons to which –2*k* flux quanta ϕ_{0 }= *h*/*e* are attached (here, *h* is Planck’s constant). For higher Landau level filling, even-denominator FQHE states are found, such as the 5/2 state that hosts Majorana excitations and *e** = *e*/4 non-Abelian anyonic quasiparticles (*10*, *11*); these have possible applications to topologically protected quantum computation. An important breakthrough in this context would be the time domain manipulation of anyons, allowing braiding interference using Hong-Ou-Mandel correlations (*12*–*14*); understanding the dynamics of anyons is thus of utmost importance.

Central to this understanding is achieving an experimental observation of the Josephson relation *f*_{J }= *e***V*/*h*, which implies that *e** anyons are elementary excitations that undergo photon-assisted energy transitions while a voltage *V* is applied to the conductor. The Josephson relation has a long history, starting from the discovery of the AC Josephson effects (*15*, *16*) in superconductors. When two tunnel-coupled superconductors are biased by a voltage *V*_{dc}, a steady-current oscillation occurs at frequency *f*_{J}, providing evidence of the Cooper pair charge *e** = 2*e*. The inverse AC Josephson effect occurs when a superconducting junction is irradiated at frequency *f*; when the bias voltage Josephson frequency *f*_{J} matches a multiple of *f*, photon-assisted singularities called Shapiro steps appear in the *I*-*V* characteristics (*16*). The AC Josephson effects arise from the quantum beating between tunnel-coupled Cooper pair condensates at energies separated by *eV*_{dc}. For normal metals, described by a Fermi sea, no steady Josephson oscillations are expected, but transient current oscillations at frequency *f*_{J}* _{}*=

*eV*

_{dc}/

*h*were demonstrated in numerical quantum simulations for two voltage-shifted Fermi seas put in quantum superposition in an electronic interferometer (

*17*). Also, in normal metals, the Josephson frequency manifests itself in the high-frequency shot noise when the bias voltage equals the emission noise frequency. Reciprocally, the low-frequency photo-assisted shot noise (PASN) shows Josephson-like singularities when microwaves irradiate a contact at frequency

*f = f*

_{J}.

First predicted (*18*) for mesoscopic conductors, PASN is also expected to occur in interacting electronic systems, like the FQHE (*19*–*22*). In the absence of microwave irradiation, the (DC) shot noise is the result of the quantum beating of two voltage-shifted Fermi seas when scattering in the conductor mixes the carrier states. For a single-mode normal conductor (*e** = *e*) with conductance *g*_{0 }= *e*^{2}/*h* and a unique scatterer of reflection probability *R*, the zero-temperature current noise spectral density under DC bias *V*_{dc} is given by . When adding an AC voltage to the biased contact: *V*(*t*) = *V*_{dc} + *V*_{ac}(*t*), with *V*_{ac}(*t*) = *V*_{ac} cos(2 πft), the phase of all carriers emitted by the contact gets a time-dependent part , giving rise to energy scattering. The emitted carriers end in a superposition of quantum states with energy shifted by *lhf* and probability amplitude , where *l* is an integer and *T* = 1/*f*. Using the voltage *V*_{dc} in units of the Josephson frequency *f*_{J }= *e***V*_{dc}/*h*, the predicted PASN spectral density can be written as(1)where is the DC shot noise measured when *V*_{ac }= 0. Equation 1 expresses that the measured observable is the result of the sum of simultaneous measurements with shifted voltage *V*_{dc}→*V*_{dc }+ *lhf/e** and weighted by the probability |*p*_{l}|^{2}, as the Fermi sea of the driven contact is in a quantum superposition of states with energy shifted by *lhf*. The zero-bias voltage DC shot noise singularity, ~|*V*_{dc}|~|*f*_{J}|, is replicated whenever *f*_{J} = *e***V*_{dc}/*h* = *lf*, indicating the Josephson frequency and hence the existence of photon-assisted transitions by anyons of charge *e**. This effect parallels the Shapiro steps of superconducting junctions IVC, i.e., the inverse AC Josephson effect (*16*). The PASN singularity for *f*_{J }= *eV*_{dc}/*h* = *f* has been observed in normal conductors (*e** = *e*), such as diffusive metallic wires (*23*), quantum point contacts (*24*), and tunnel junctions (*25*). For interacting systems, Eq. 1 has been derived in (*26*) and observed in (*27*) for superconducting/normal junctions (*e** = 2*e*). In FQHE systems, the concept of fractional Josephson frequency was introduced in (*28*) and (*29*), which discussed photo-assisted processes (*28*) or finite-frequency noise (*29*). The concept and the terminology were later used in FQHE PASN models (*19*). Equation 1 was explicitly shown in (*20*) and is implicit in PASN expressions of (*19*–*22*). However, experimentally combining high magnetic fields, sub-fA/Hz^{1/2} current noise, and >10-GHz microwaves at ultralow temperature (~20 mK) is highly challenging.

In this study, we combine microwave frequency irradiation with low-frequency shot noise measurements to provide evidence of the above Josephson relation and a conclusive test of PASN models.

A schematic view of the setup and the sample is shown in Fig. 1A. In topologically insulating QHE conductors, the current flows along *p* chiral edge modes for filling factor ν = *p*/(2*p* + 1). To inject current or apply a microwave excitation, metallic contacts connect the edges to an external circuit. A narrow tunable constriction called quantum point contact (QPC) is formed by applying a negative voltage *V*_{G} to split gates to induce backscattering by the quantum mixing of counterpropagating edge modes. Carriers incoming from contact 0 and scattered by the QPC contribute to transmitted and backscattered currents, *I _{t}* and

*I*

_{B}, respectively. These currents are measured at contacts (1) and (2) via the voltages

*V*

_{1(2) }= (

*h*/

*e*

^{2}ν

_{B})

*I*

_{t}_{(B)}, where ν

_{B}is the filling factor in the lead (far from the QPC). The partitioning of the carrier generates a current noise

*S*, which is measured by recording the negative cross-correlation of the voltage fluctuations Δ

_{I}*V*

_{1(2) }= (

*h*/

*e*

^{2}ν

_{B})Δ

*I*

_{t}_{(B)}, giving . Δ

*f*is the bandwidth of the low-frequency resonant detecting circuit as described in (

*30*).

We focus on bulk filling factor ν_{B} = 2/5, which conveniently allows us to probe both *e*/3 and *e*/5 anyons. The two copropagating chiral edge modes of the 2/5 Jain state are revealed by sweeping the QPC gate voltage *V*_{G} (Fig. 1B). Starting with a (2/5)*e*^{2}/*h* conductance plateau, we observe a second conductance plateau (1/3)*e*^{2}/*h* at lower *V*_{G}. This corresponds to a fully reflected inner channel with conductance *g*_{2 }= (2/5 − 1/3)*e*^{2}/*h*, whereas the outer edge channel with conductance *g*_{1 }= (1/3)*e*^{2}/*h* is fully transmitted. To probe the *e*/3 charged excitations of the 1/3 − FQHE state locally formed at the QPC, *V*_{G} is set to −0.090V (point A in Fig. 1B) so as to induce a weak backscattering (WB) between counterpropagating outer edge modes with reflection probability *R* = 0.026. Next, we apply a DC voltage *V*_{dc} to the injecting contact. The incoming current of the outer edge mode *I*_{0 }= (1/3)(*e*^{2}/*h*)*V*_{dc} divides into a backscattering current *I*_{B }≈ *RI*_{0} and a forward current *I *= *I*_{0 }− *I*_{B}. In the FQHE, the chiral modes form chiral Luttinger liquids (*14*). At finite backscattering, transport becomes energy dependent, giving nonlinear variations of *I*_{B} with voltage *V*_{dc}. A complete modeling is difficult, and comparison to experiments is only easy in the WB regime (*R* << 1). The small backscattered current *I*_{B}(*V*_{dc}) results from rare quasiparticle tunneling events following Poisson statistics. In this limit, the DC shot noise cross-correlation is (*29*, *31*, *32*)(2)with *e** = *e*/3 for the 1/3 − FQHE regime considered here and *T*_{e} the electronic temperature.

In Fig. 2B, the black dots show DC shot noise data. The black-dashed line, computed using Eq. 2, compares well with the data for *e** = *e*/3. Here, a constant *R* ≈ 0.026 versus *V*_{dc} is used as *I*_{B}(*V*_{dc}) is found to be almost linear (fig. S3).

Next, to show the Josephson relation using PASN, the AC voltage *V*_{ac}(*t*) = *V*_{ac}cos2π*ft* is superimposed on *V*_{dc} with *f* = 22GHz. The blue and red dots show the measured (PASN) noise for several values of *V*_{ac} corresponding to −61 and −67dBm nominal radiofrequency (RF) power (disregarding RF lines losses) sent to the contact. At low *V*_{dc}, the PASN noise increases with power; for values of *V*_{dc} above ≈250 μV, the PASN noise merges into the DC shot noise curve. The change in the slope of the noise variation at this characteristic voltage is suggestive, but not conclusive, of the expected PASN noise singularity. To reveal pure photon-assisted contributions to PASN, guided by the form of Eq. 1, we cancel the *l *= 0 term by subtracting the independently measured DC shot noise data from the raw PASN data. This defines the excess PASN . Finding the condition to cancel the (*l *= 0) DC shot noise term in Δ*S _{I}* provides the value of |

*p*

_{0}|

^{2 }=

*J*

_{0}(α)

^{2}for the excitation

*V*

_{ac }= α

*hf*/

*e** used. This is done for three RF powers: 67, 63, and 61dBm (fig. S4). For clarity and better data statistics, only the calculated average of the three excess PASN curves is shown in Fig. 2C, blue dots. Neglecting |

*l*| > 1 photon process and using the average |

*p*

_{1}|

^{2}values obtained from the average |

*p*

_{0}|

^{2}value, , the theoretical excess PASN is

Equation 3 is plotted using *f*_{J }= (*e*/3)*V*_{dc}/*h* as the blue-dashed line in Fig. 2C. The extracted value of and the choice of *f*_{J} account well for the measured excess PASN variation, strongly supporting the validity of Eq. 1.

A further validation of Eq. 1 is given by changing the excitation frequency. We have repeated similar measurements and analyses for *f* = 17 GHz and *f* = 10 GHz. In Fig. 2C, the green- and red-dotted curves show excess PASN data, and the green- and red-dashed lines provide convincing comparisons to Eq. 3 using the average parameter extracted from each experimental curve and fixing *e** = *e*/3 when calculating *f*_{J} . Using *e** as a free parameter, the quantity *V*_{J} = *hf*/*e** (which signals the onset of excess PASN) is extracted from the best fit of Eq. 3 to the excess PASN data for each frequency *f*. When *V*_{J} is plotted in Josephson frequency units (*e*/3)*V*_{J}/*h* versus *f* (Fig. 2D), a linear fit to the data gives *e** = *e*/(3.06 ± 0.20), yielding the fractional charge of the anyon. For comparison, the red-dashed straight line, with a slope of 1, corresponds to *e** = *e*/3 exactly.

We then confirmed that we were measuring the Josephson frequency by changing the excitation charge. We consider the WB regime of the inner edge of the 2/5 FQHE Jain state, whose nominal conductance is (2/5 − 1/3)*e*^{2}/*h* (Fig. 3A). The backscattered current in this regime is *I*_{B }= *R*(1/15)*e*^{2}/*hV*_{dc}. We set *V*_{G} to −0.03 V for which *R* = 0.064 (point B of Fig. 1B). Figure 3B shows the DC noise data (black dots). A comparison of data to Eq. 2 with *e** = *e*/5 and *R* = 0.064 (black dashed lines) confirms a one-fifth quasiparticle charge (*9*). The PASN for various RF powers at 17 GHz is shown as colored circles. As previously done for *e*/3 charges, we average the PASN data at 17 GHz for three different RF powers (−60, −58, and −55 dBm). The resulting mean excess noise is plotted in Fig. 3C (blue dots). The data compare well with Eq. 3 (red dashed line), using *f*_{J }= (*e*/5)*V*_{dc}/*h* and , including the finite temperature *T*_{e }= 30 mK in the DC shot noise (Eq. 2). Note that we have no independent way to determine *T*_{e}. A similar procedure is done for 10-GHz excitation. Then, the voltage *V*_{J }= *hf*/*e** characterizing the onset of excess PASN is left as a free parameter to fit the excess PASN data and is plotted in units of Josephson frequency, *f*_{J }= (*e*/5)*V*_{J}/*h* versus *f*, in Fig. 3D. The dashed line, with a slope of 1, corresponds to *e** = *e*/5 exactly. A linear fit of the actual *V*_{J} versus *f* passing through zero gives *e** = *e*/(5.17 ± 0.31).

Measurement of *f*_{J} for non-Abelian anyons at ν_{B }= 5/2 will require ultrahigh-mobility samples. A possible route is the realization of a single anyon source based on levitons (*12*) using periodic Lorentzian voltage pulses instead of a sine wave. The PASN caused by periodic levitons is also given by Eq. 1 except that all the *p*_{l} values for *l* < 0 vanish, characterizing a minimal excitation state (*33*) without hole-like excitations [see (*30*) for more details]. A charge *e* Leviton sent to a QPC in the WB regime would provide a convenient time-controlled single anyon source with Poisson statistics (*13*, *22*). Combining two similar sources opens the way for anyon braiding interference through Hong-Ou-Mandel tests of anyonic statistics (*13*) (fig. S8).

## Supplementary Materials

www.sciencemag.org/content/363/6429/846/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S8

This is an article distributed under the terms of the Science Journals Default License.

## References and Notes

**Acknowledgments:**We thank P. Jacques for technical help and P. Pari, P. Forget, and M. de Combarieu for cryogenic support. We acknowledge discussions with M. Freedman, I. Safi, Th. Martin, X. Waintal, H. Saleur, and members of the Saclay Nanoelectronics Group. D.A.R. and I.F. acknowledge the EPSRC.

**Funding:**The ANR FullyQuantum AAP CE30 grant is acknowledged.

**Author contributions:**D.C.G. designed the project. M.K. made the measurements and with P.R. and D.C.G. analyzed the data. M.S. contributed to the early experimental setup and not to the present measurements. All authors, except M.S., discussed the results and contributed to writing the article. Samples were nanolithographed by M.K. on wafers from D.A.R. and I.F.

**Competing interests:**None declared.

**Data and materials availability:**Data shown in the paper are available as a single zip file from Zenodo (

*34*).