Josephson radiation from gapless Andreev bound states in HgTe-based topological junctions

Frequency analysis of the rf emission of oscillating Josephson supercurrent is a powerful passive way of probing properties of topological Josephson junctions. In particular, measurements of the Josephson emission enables to detect the expected presence of topological gapless Andreev bound states that give rise to emission at half the Josephson frequency $f_J$, rather than conventional emission at $f_J$. Here we report direct measurement of rf emission spectra on Josephson junctions made of HgTe-based gate-tunable topological weak links. The emission spectra exhibit a clear signal at half the Josephson frequency $f_{\rm J}/2$. The linewidths of emission lines indicate a coherence time of $0.3-\SI{4}{ns}$ for the $f_{\rm J}/2$ line, much shorter than for the $f_{\rm J}$ line ($3-\SI{4}{ns}$). These observations strongly point towards the presence of topological gapless Andreev bound states, and pave the way for a future HgTe-based platform for topological quantum computation.

In recent years, schemes for fault-tolerant quantum computation have been theoretically developed on the premises of non-abelian particle statistics [1]. Such statistics can arise in condensed matter systems for so-called Majorana quasiparticles, that may be braided around one another to execute quantum information protocols. Majorana zero-modes can be conveniently engineered by inducing p-wave superconductivity in a two-dimensional topological insulator [2,3] (2D TI). Coupling the topological edge channels of a 2D TI to a nearby conventional s-wave superconductor leads to the appearance of an induced p-wave superconducting phase [4]. In a topological Josephson junction, a doublet of p-wave Andreev bound states is predicted to have a topologically protected level crossing for a superconducting phase difference ϕ = π, 3π, ... across the junction. Such states can in principle be detected via the resulting energy dispersion that is 4π-periodic in ϕ, with, in the simplest case of a short junction, E = E J cos ϕ/2 [4,5]. However in the thermodynamic limit of a time-independent phase ϕ, the current is 2π-periodic as only the lower branch at E ≤ 0 is populated. Experiments relying on out-of-equilibrium dynamics in the GHz range are thus useful to provide evidence for the existence of gapless 4π-periodic Andreev bound states on time scales shorter than equilibration time. Equilibration occurs through various relaxation processes such as coupling to the continuum, to other Andreev bound states, or quasiparticle poisoning [5][6][7][8], see Fig.1a. On such short time scales, Josephson emission at half the Josephson frequency f J /2 is then predicted [4,6,8].
In previous works [9,10], we reported a doubling of the Shapiro step size (hf /e) in Josephson weak links based on thick strained HgTe layers (3D TI) and HgTe quantum wells (2D TI) which clearly indicates the presence of a 4π-periodic component in the supercurrent. Though experimentally easily accessible, detailed interpretation of such experiments is hindered by the strongly non-linear nature of the Josephson response to an rf excitation. In contrast, Josephson emission under a dc voltage bias provides a passive and direct probe of supercurrents in topological junctions, but the radiated power is low and difficult to measure. Moreover, the linewidths of the emission lines reflect the lifetime of the Andreev bound states. In this article, we report on the study of Josephson emission in a range from 2 to 10 GHz using cryogenic microwave measurements. Besides conventional emission at f J and 2f J , we observe clear emission at f J /2 in Josephson junctions based on inverted HgTe quantum wells, which are 2D TIs and exhibit the quantum spin Hall effect [11]. These emission measurements provide very direct evidence of the presence of a 4π-periodic supercurrent. Additionally, the coherence time of the unconventional emission line at f J /2 is observed to be up to an order of magnitude shorter than that at f J , indicating its sensitivity to relaxation processes. This set of experimental signatures is attributed to the presence of gapless Andreev bound states. In a reference experiment, a non-topological HgTe-based superconducting weak link exhibits only conventional emission at f J .
Due to a band inversion, HgTe quantum wells become 2D topological insulators for a thickness larger than a critical thickness d c 6.3 nm. In this regime, they exhibit a pair of counter-propagating edge channels when tuned into the band gap (quantum spin Hall effect [11] surements of emission spectra are nowadays accessible via microwave cryogenic amplifiers [14]. To this end, the junction is connected to a coaxial line and decoupled from the dc measurement line via a bias-T (see Fig.1b and Fig.1c, and SI for details). The rf signal is then amplified at both cryogenic and room temperatures before being measured with a spectrum analyzer. The commercial rf components used in the readout line limit the frequency range of detection to 2-10 GHz. An essential requirement to successfully perform such measurements is a stable bias. Under current bias, instabilities and hysteretic behavior may occur at low voltages [9,10]. We therefore employ a small resistive shunt R S (between 1 and 50 Ω) to enable a stable voltage bias (though residual switching below a few microvolts is sometimes seen), while a small resistance R I in series with the junction yields a measurement of the current I through the junction (Fig.1c).
In the absence of any drive on the junction, a background noise is observed, probably originating from black body radiation and parasitic stray noise from the environment. It is taken as a reference and subtracted from all measurements to isolate the contribution of the junction. When the junction is biased and a finite voltage V develops, the contribution of the junction appears. In Fig.2  (non-topological) microbridges [12,13]. In strong contrast, the topological Josephson junction unveils a new and strong feature at half the Josephson frequency f J /2 (sometimes concomitant with emission at f J ). This observation, illustrated for two different electron densities in Fig.2b and 2c, is a direct manifestation of the presence of a 4π-periodic supercurrent flowing in our topological junctions [6] and constitutes our main finding. In   When the detection frequency is swept, one can verify that the emission lines follow the linear relation f J = 2eV h . In the weak link with a trivial band structure (Fig.2d), the conventional 2π-periodic line is visible over the range 2 − 10 GHz (for each value of f d , the reference at I = 0 is subtracted, and the data is normalized to its maximum to correct for frequency-dependent coupling and amplification). In the topological device and for V g = −0.55 V (Fig.2e), the colormap shows that the emission is entirely dominated by the 4π-periodic supercurrent below f = 5.5 GHz, before the conventional line is recovered.
At higher frequencies, the emission spectrum is influenced by resonant modes within the electromagnetic environment. These can be easily identified by a characterization of the electromagnetic environment of the junction (see SI). When V g −1.4 V (Fig.2f), the colormap reveals that the 4π-periodic component at f J /2 is visible only up to f d 4.5 GHz, while the conventional emission line at f J is seen in the whole range of frequencies.
We now analyze the measurements of Fig the dynamics of the junction is highly non-linear, the 2π-periodic component of the voltage is effectively suppressed, resulting in 4π-periodic oscillating voltages [15] and emission at f J /2. Consequently, for the corresponding low voltages, the junction is expected to emit mainly at f J /2. For higher biases, the dynamics of the system is ruled by a single time scale, and resembles that of a 2π-periodic junction. Computations for increasing voltages V and detection frequency f d yield a good qualitative agreement with the I-V characteristic (Fig.3a) as well as the emission features (Fig.3b) for a contribution of 4π-periodic modes amounting to around 40% of the critical current, in agreement with previous estimates [10].
We now detail the dependence of the emitted power as a function of the gate voltage. In the non-topological device (see SI), we observe that the amplitude of the collected signal reflects the amplitude of the critical current, and verify that A ∝ I c with a good agreement [16]. This confirms the conventional behavior of the device in the conduction and valence bands of the quantum well, as well as close to the gap. In Fig.4a and 4b, we present two sets of measurements of the collected rf amplitude A on the topological weak link, taken at a low (f d = 2.98 GHz) and high (f d = 5.5 GHz) frequency. We observe three clear regimes in the emitted power that correlate with the expected band structure. When the gate voltage is above V g > −0.4 V, we observe that emission occurs (Fig.4a), and for both f J and for f J /2 at high frequency of f d = 5.5 GHz (Fig.4b). These observations suggest transport in the conduction band of the quantum well, where gapless Andreev bound states have been seen to coexist with n-type conventional states, in agreement with previous observations and predictions dc   c) The ratio of the intensity of the 4π-to the 2π-periodic amplitudes r = A 4π A 2π (averaged over frequency in 2-10 GHz range) (as a red line) and the critical current (as a blue line) are plotted as function of gate voltage V g . After rescaling of the critical current measured in Ref. [10], the emission features can be compared with previous observations on the Shapiro response. [10,17]. The overall gate voltage dependence is consistent with the expected band structure of a quantum spin Hall insulator. Finally, we compute from the measurement data the amplitude A 2π , A 4π extracted along the emission line at f J and f J /2 respectively. For each gate voltage V g , we calculate the ratio r = A 4π A 2π (which is independent of the frequency response of the amplification scheme). We plot this ratio r averaged over frequency f d as a red line in Fig.4c and compare our results with information inferred from the Shapiro response in our previous work [10]. As expected from Fig.4a and 4b, r shows a maximum around V g = −0.7 V, where the conventional line is strongly suppressed.
When correcting for density differences (offset on the gate voltage) and the strength of induced superconductivity (multiplication factor on the current), the critical current in the present experiment can be mapped to the one measured in Ref. [10]. We observe that a good agreement is found between the observation of edge transport (SQUID-like response to a magnetic field), the 4π-periodic Shapiro response, and the emission at the half the Josephson frequency f J /2.
Beyond the direct detection of a 4π-periodic supercurrent, the measurement of Josephson emission in these devices provides other new insights. First, we detect emission at f J /2 as low as 1.5 GHz. If responsible for the 4π-periodic supercurrent [15,18], potential Landau-Zener transitions would be activated for a voltage V LZ 6 µV. This sets an upper bound on the existence of a residual avoided crossing δ [7,19], and tend to rule this mechanism as origin for the 4π-periodic emission. Furthermore, the linewidth of both emission lines can be examined. For conventional Josephson radiation, the linewidth is in principle related to fluctuations in the pair or quasiparticle currents [20,21] or can be dominated by the noise in the environment [16].   Fig.7a), and a current bias I is applied. When combined with the Josephson equation, one then readily obtains a first order differential equation [24,25] on the superconducting phase difference ϕ(t) that can be solved and consequently yields the voltage V (t) = 2e dϕ dt : In previous works [9,15], we have studied an extended RSJ model which simulated the effect a 4π-periodic contribution in the CPR (simply written as I S (ϕ) = I 2π sin ϕ + I 4π sin ϕ/2). This differential equation is highly non-linear, and it is of importance in determining the parameter space for which a 4π-periodic contribution can be observed even in the case where it is accompanied by a larger 2π-periodic contribution.
For bias currents slightly exceeding the critical current, I I 2π , the time-dependent voltage V (t) is highly non-sinusoidal (see Fig.6a), as a result of the non-linearity of the RSJ equation. This plays an important role in the case of a 4π-periodic contribution to the supercurrent, as discussed below. In contrast, the response is almost harmonic for higher biases, and V (t) becomes sinusoidal for I I 2π .
Case of an additional 4π-periodic contribution to the supercurrent Following the same reasoning as for 2π-periodic supercurrent, we can define f 4π = 2/T 4π with the period Simulations computed for I = 1.05I c , I 4π = 0.2I 2π . c) A mostly 2π-periodic dynamics is recovered at higher voltages, and T 1 T 2 T 2π . The voltage becomes closer to a 2π-periodic sine wave. Simulations computed for I = 8I c , I 4π = 0.2I 2π defined by the following integral" Since the term I 4π sin(ϕ/2) has contributions of opposite signs in T 1 and T 2 , these two time scales can be very different (in particular when I ∼ I c + I 4π ), thus affecting the periodicity of the dynamics. When the current I slightly exceeds the critical current, such that I ∼ I c + I 4π , one observes that T 1 T 2 , and the system exhibits a non-sinusoidal 4π-periodic behavior (Fig.6b). In contrast, for high currents (I I c + I 4π ),

Extended RSJ model with shunt circuit
We first need to derive and adapt the equations of motion to the experimental shunt circuit, shown in Fig.7b. The junction is as before represented by its CPR I S (ϕ) and its resistance R n , but resistors R I and R S are added. Applying Kirchhoff laws, one extracts the modified equations of motion for the experimental setup, that read: One sees that Eq. (7) is identical to the standard RSJ equation (Eq.(1)) with substitutions R n →R n and I c = max ϕ I S (ϕ) → I c 1 + R I R S . Simulations performed in the standard RSJ model can be readily adapted to this new setup. Besides, the experimental data is more naturally presented as a function of I 1 rather than I, which is obtained from

Simulations of Josephson emission
To compare the simulations with our experimental data, it is more convenient to set I c directly (rather than I 2π and I 4π independently). We use the following parametrization : With these notations, the critical current is set to I c and the ratio of 4π-to 2π-periodic supercurrents is tuned by x.
Fitting the I-V curve We first optimize the fitting of the I-V curve (experimental data of Fig.2b of the main text) by choosing the value of I c and R n . R I and R S are set to the value used in the setup, i.e. R I = R S = 25 Ω. We observe that x has a marginal effect on the I-V characteristic, and can be ignored in this part. As depicted in Fig.8, we find the best agreement for I c (240 ± 10) nA and for two different values of R n . For R n (130 ± 15) Ω (presented in the main text, here as a red line), we obtain a good agreement for low voltages only. For R n (220 ± 15) Ω (blue line) the agreement is not as good for low voltages, but remains decent also for higher voltages.
Estimating the 4π-periodic supercurrent The last parameter that needs to be evalu- voltage V 4π that increases with x. In the experimental data, V 4π 12 µV. In our simulations, the crossover is not as abrupt, but reaches this value for x 0.6 ± 0.1 (similar for both values of R n ), so that I 4π 0.4I c . From these simulations, we find the estimate I 4π 80 − 120 nA which exceeds the expected contribution of two edge modes ( 50 nA).
However, though the physics previously described is quite universal, this estimate of I 4π has to be taken cautiously as it is more strongly dependent on the model and choice of parameters.

Inductive effects and period doubling
Our experimental setup comprises bond wires between the resistors and the device chip. Bond wires add an inductive contribution to the shunt circuit, estimated around a few nH (typically 1 nH mm −1 ), Inductance in shunt circuits strongly modify the RSJ equation (by adding higher order derivatives) and lead to complex dynamical behavior of Josephson junctions [26][27][28][29]. Of particular interest, it can lead to period doubling and could be responsible for a Josephson radiation at f J /2. Several observations in our experiments show that this explanation is in fact unlikely.
First, the presence of these phenomena is governed by the dimensionless inductance and capacitance parameters in the shunt branch [29], β L = 2eIcL , β C = 2eR 2 S IcC , where L and C are respectively the inductance in the shunt branch and the capacitance of the junction. We find that, indeed, the inductive parameter is rather large (around β L 5 for the parameters at V g = −0.55 V). However the shunt resistance and the capacitance of the junction are small and yield β C 0.01. In these conditions, it has been predicted [28,29] that inductive effects have limited consequence, and the junction should not be subject

Possible influence on emission measurements
Self-induced steps The first consequence of unintentional resonant structures near the device is the appearance of Shapiro steps in the device : at resonance, the emitted Josephson radiation is fed back onto the junction resulting in self-induced Shapiro steps. The differential resistance of our device dV /dI exc exhibits a series of equidistant peaks. This appear to be in very good agreement with the Shapiro steps associated to the resonance at f d = 1.9 GHz (dashed line in Fig.9b). Though the differential resistance can modify the amplitude of the emitted radiation [16], no strong connection to our measurements of the power has been established. Similar effects have been theoretically and experimentally investigated in the regime of dynamical Coulomb blockade [30,31]. When the junction is embedded in a cavity resonating at f r , two-photon processes can give rise to replicas of the main emission line (at f J ) shifted by the energy of a photon, namely f J + f r . Given the impedance of our device R s , R n R K = h e 2 , our device is not in the appropriate regime to observe dynamical Coulomb blockade effects. In particular, two-photon processes are second-order in R n /R K and should always be much less visible as standard emission at f J . As such they cannot solely explain the observation of radiation at f J /2. Besides, when no radiation is detected at f J /2, the complex pattern of Fig.9d is absent or barely visible (see Fig.2d and f in the main text). However, we speculate that the interplay of resonant two-photon processes and anomalous emission at f J /2 are a possible explanation for the observed high-frequency features.

Josephson emission
Alternatively, embedding of a Josephson junction described by RSJ equations into a resonator has been theoretically investigated in Refs. [32,33]. Self-induced Shapiro steps are then predicted to occur, as well as radiation at the resonator frequency f r , but no explicit prediction has been made concerning emission at f J + f r and further investigation is required.

ADDITIONAL EXPERIMENTAL RESULTS
Gate voltage dependence of the non-topological weak link In this section, we show that the non-topological weak link follows to a good accuracy the expected behavior A ∝ I c between the collected rf amplitude A and the critical current I c [16]. The latter is tuned via the gate voltage V g . In Fig.10a, we present as a colormap the amplitude A as function of voltage V and gate voltage V g . The amplitude of the measured emission line at f J scales exactly with the amplitude of the dc supercurrent, as shown in Fig.10b. Moreover, it is additionally shown in the 2D map that radiation at f J /2 is completely absent in this device.  FIG. 11. Emission at f J /2 in HgTe-based 3D topological insulatorsa) I-V curve (red line) and normalized emission magnitude A (blue line) for a 3D topological insulator weak link.
The radiation is collected at fixed detection frequency f d = 5 GHz. b) 2D map of the collected rf amplitude A as function of bias voltage V and detection frequency f d . For better visibility, the data is normalized to its maximum for each frequency.
In this section, we show measurements of the Josephson emission on weak links made of 70 nm-thick strained layers of HgTe. Such layers have been demonstrated to be 3D topological insulators [34,35]. In a previous work [9], we have detected the presence of a weak 4π-periodic supercurrent flowing in such Josephson junctions, as a signature of a single topologically protected Andreev doublet in agreement with theoretical predictions [36]. The measurement of anomalous emission at f J /2 confirms the presence of this 4πperiodic supercurrent.
Though the shunt circuit is similar to that of Fig.1b (main text), the measurements have been performed slightly differently. Here the detection frequency f d is swept at a fixed value of the bias current. The disadvantages of this method are an increased time consumption and the difficulty to correct for slow drifts of the background. The measurements are shown in Fig.11. In Fig.11a, the I-V curve of the device is presented together with the normalized rf amplitude A at a detection frequency f d = 5 GHz. As previously, a clear peak is observed at f J /2. In Fig.11b, the 2D map of the normalized amplitude A as function of voltage V and detection frequency f d is shown. As in the main text, the 4π-periodic supercurrent is seen to dominate the low-frequency/voltage regime, while the conventional 2π-periodic supercurrent is recovered at f d = 7.5 GHz.
In this plot with high frequency resolution, it is seen that the frequency of the radiated signal is slightly shifted from the expected value of f J /2. This shift seems to follow oscillations in the transmitted signal and could reflect again the influence of resonances in the surrounding electromagnetic environment, as discussed in Section II. The study of line widths yields here δV 2 − 8 µV at half-width, corresponding to τ 4π 0.25 − 1 ns. * All three authors contributed equally to this work, email: erwann.bocquillon@physik.uniwuerzburg.de