A Josephson relation for fractionally charged anyons

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Science  22 Feb 2019:
Vol. 363, Issue 6429, pp. 846-849
DOI: 10.1126/science.aau3539

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


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 fJ = 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 fJ. 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 pe2/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/(2k + 1), where k is an integer, are well described by Laughlin states (3). The elementary excitations, or quasiparticles, bear a fraction e* = e/(2k + 1) of the elementary charge e (36) 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/(2kp + 1), where p and k are integers, display e* = e/(2kp + 1) fractionally charged excitations (9); these excitations are composite fermions, i.e., electrons to which –2k flux quanta ϕ0 = 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 (1214); understanding the dynamics of anyons is thus of utmost importance.

Central to this understanding is achieving an experimental observation of the Josephson relation fJ = 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 Vdc, a steady-current oscillation occurs at frequency fJ, providing evidence of the Cooper pair charge e* = 2e. The inverse AC Josephson effect occurs when a superconducting junction is irradiated at frequency f; when the bias voltage Josephson frequency fJ 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 eVdc. For normal metals, described by a Fermi sea, no steady Josephson oscillations are expected, but transient current oscillations at frequency fJ= eVdc/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 = fJ.

First predicted (18) for mesoscopic conductors, PASN is also expected to occur in interacting electronic systems, like the FQHE (1922). 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 g0 = e2/h and a unique scatterer of reflection probability R, the zero-temperature current noise spectral density under DC bias Vdc is given by Embedded Image. When adding an AC voltage to the biased contact: V(t) = Vdc + Vac(t), with Vac(t) = Vac cos(2 πft), the phase of all carriers emitted by the contact gets a time-dependent part Embedded Image, giving rise to energy scattering. The emitted carriers end in a superposition of quantum states with energy shifted by lhf and probability amplitude Embedded Image, where l is an integer and T = 1/f. Using the voltage Vdc in units of the Josephson frequency fJ = e*Vdc/h, the predicted PASN spectral density can be written asEmbedded Image(1)where Embedded Image is the DC shot noise measured when Vac = 0. Equation 1 expresses that the measured observable is the result of the sum of simultaneous measurements with shifted voltage VdcVdc + lhf/e* and weighted by the probability |pl|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, ~|Vdc|~|fJ|, is replicated whenever fJ = e*Vdc/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 fJ = eVdc/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* = 2e). 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 (1922). However, experimentally combining high magnetic fields, sub-fA/Hz1/2 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/(2p + 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 VG 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, It and IB, respectively. These currents are measured at contacts (1) and (2) via the voltages V1(2) = (h/e2νB)It(B), where νB is the filling factor in the lead (far from the QPC). The partitioning of the carrier generates a current noise SI, which is measured by recording the negative cross-correlation of the voltage fluctuations ΔV1(2) = (h/e2νBIt(B), giving Embedded Image. Δf is the bandwidth of the low-frequency resonant detecting circuit as described in (30).

Fig. 1 Schematics for PASN measurements.

(A) A DC voltage Vdc applied to contact (0) injects carriers at bulk filling factor νB = 2/5 into two chiral fractional edge modes (red lines). The carriers are partitioned by a quantum point contact (QPC) into transmitted and reflected currents, which are absorbed at the grounded contacts, giving rise to voltages V1 = IthBe2 and V2 = IBhBe2 at contacts (1) and (2), respectively. The negative voltage fluctuation cross-correlation ΔV1ΔV2 = −(hBe2)2SIΔf is recorded to obtain the noise SI. The voltages are sent to two identical resonant circuits followed by cryogenic amplification and fast digital acquisition. A computer performs the fast Fourier transform cross-correlation. The RF excitation from a microwave photon source is added to Vdc and sent to contact (0). (B) QPC conductance Gt= dIt/dVdc at bulk filling factor νB = 2/5 versus the QPC gate voltage VG. The (2/5)e2/h plateau is followed by a (1/3)e2/h plateau signaling complete reflection of the inner 2/5 fractional edge channel. Points A and B show the weak backscattering conditions for measurements with fractional carriers e/3 and e/5, respectively.

We focus on bulk filling factor νB = 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 VG (Fig. 1B). Starting with a (2/5)e2/h conductance plateau, we observe a second conductance plateau (1/3)e2/h at lower VG. This corresponds to a fully reflected inner channel with conductance g2 = (2/5 − 1/3)e2/h, whereas the outer edge channel with conductance g1 = (1/3)e2/h is fully transmitted. To probe the e/3 charged excitations of the 1/3 − FQHE state locally formed at the QPC, VG 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 Vdc to the injecting contact. The incoming current of the outer edge mode I0 = (1/3)(e2/h)Vdc divides into a backscattering current IB RI0 and a forward current I = I0 IB. In the FQHE, the chiral modes form chiral Luttinger liquids (14). At finite backscattering, transport becomes energy dependent, giving nonlinear variations of IB with voltage Vdc. A complete modeling is difficult, and comparison to experiments is only easy in the WB regime (R << 1). The small backscattered current IB(Vdc) results from rare quasiparticle tunneling events following Poisson statistics. In this limit, the DC shot noise cross-correlation is (29, 31, 32)Embedded Image(2)with e* = e/3 for the 1/3 − FQHE regime considered here and Te 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 Vdc is used as IB(Vdc) is found to be almost linear (fig. S3).

Fig. 2 Josephson relation for 1/3 FQHE state.

(A) The fully reflected 2/5 inner edge state gives rise to a ν = 1/3 FQHE state at the QPC. For VG = −0.090 V (point A of Fig. 1B), the counterpropagating outer edge states are weakly coupled, allowing e/3 backscattered carriers to be probed. (B) Raw shot noise measurements: Black dots show the DC shot noise [i.e., with only Vdc applied at contact (0) and no RF]. The dashed line is Eq. 2 with e* = e/3 and constant R = 0.026 (point A of Fig. 1A). Blue and red open circles are noise measurements for 22 GHz −67dBm and −61dBm RF irradiation. Blue- and red-dashed line curves plot Eq. 1 with fJ = (e/3)Vdc/h and using |p0|2 and |p1|2 deduced from the analysis of Fig. 2C. (C) Excess PASN ΔSI (blue, green, and red dots) for three frequencies 22, 17, and 10 GHz, respectively. The average of measurements at several excitation powers is shown to improve the noise statistics. The blue-, green- and red-dashed lines, computed from Eq. 3 using fJ = (e/3)Vdc/h, compare well to the data. For clarity, the constant ΔSI(Vdc = 0) has been subtracted from the excess PASN and the 17- and 10-GHz data have been offset. (D) Determination of carrier charge from the Josephson relation. A best fit of ΔSI gives, for each frequency, the threshold voltages VJ = hf/e* above which ΔSI rises. They are plotted in units of (e/3)VJ/h versus f (blue points; error bars are SEM); a linear fit gives e* = e/(3.06 ± 0.20). For comparison, the red-dashed line corresponds to e* = e/3 exactly.

Next, to show the Josephson relation using PASN, the AC voltage Vac(t) = Vaccos2πft is superimposed on Vdc with f = 22GHz. The blue and red dots show the measured (PASN) noise for several values of Vac corresponding to −61 and −67dBm nominal radiofrequency (RF) power (disregarding RF lines losses) sent to the contact. At low Vdc, the PASN noise increases with power; for values of Vdc 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 Embedded Image. Finding the condition to cancel the (l = 0) DC shot noise term in ΔSI provides the value of |p0|2 = J0(α)2 for the excitation Vac = αhf/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 |p1|2 values obtained from the average |p0|2 value, Embedded Image, the theoretical excess PASN is

Embedded Image(3)

Equation 3 is plotted using fJ = (e/3)Vdc/h as the blue-dashed line in Fig. 2C. The extracted value of Embedded Image and the choice of fJ 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 Embedded Imageextracted from each experimental curve and fixing e* = e/3 when calculating fJ . Using e* as a free parameter, the quantity VJ = 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 VJ is plotted in Josephson frequency units (e/3)VJ/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)e2/h (Fig. 3A). The backscattered current in this regime is IB = R(1/15)e2/hVdc. We set VG 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 Embedded Image is plotted in Fig. 3C (blue dots). The data compare well with Eq. 3 (red dashed line), using fJ = (e/5)Vdc/h and Embedded Image, including the finite temperature Te = 30 mK in the DC shot noise (Eq. 2). Note that we have no independent way to determine Te. A similar procedure is done for 10-GHz excitation. Then, the voltage VJ = 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, fJ = (e/5)VJ/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 VJ versus f passing through zero gives e* = e/(5.17 ± 0.31).

Fig. 3 Josephson relation for the 2/5 FQHE state.

(A) Chiral edge schematics: The 2/5 inner edge state is weakly reflected; see point B of Fig. 1B. Here, backscattered e* = e/5 carriers contribute to current IB and shot noise SI. (B) Raw shot noise measurements: Black dots show the DC shot noise (i.e., no RF) measured during the 17-GHz PASN measurement run. The black dashed line is Eq. 2 using R = 0.064 (point B of Fig. 1A). Blue and red open circles are the PASN for 17-GHz −58d Bm and −51dBm RF irradiation. Blue- and red-dashed curves plot Eq. 1 with fJ = (e/5)Vdc/h. (C) Excess PASN ΔSI. Green and red dots correspond to 17 and 10 GHz, respectively. Green- and red-dashed lines are computed from Eq. 3 using fJ = (e/5)Vdc/h. For clarity, for each curve the corresponding ΔSI(Vdc = 0) has been subtracted from the excess PASN. (D) Determination of e*: A best fit of ΔSI gives, for each frequency, the threshold voltages VJ = hf/e* above which ΔSI rises. They are plotted in units of (e/5)VJ/h versus f (blue points; errors bars are SEM); a linear fit gives e* = e/(5.17 ± 0.31). For comparison, the red-dashed line corresponds to e* = e/5 exactly.

Measurement of fJ for non-Abelian anyons at νB = 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 pl 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

Materials and Methods

Supplementary Text

Figs. S1 to S8

References (3554)

References and Notes

  1. Supplementary materials are available online.
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).
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