Microwave sensing of Andreev bound states in a gate-defined superconducting quantum point contact

We use a superconducting microresonator as a cavity to sense absorption of microwaves by a superconducting quantum point contact deﬁned by surface gates over a proximitized two-dimensional electron gas. Renormal-ization of the cavity frequency with phase difference across the point contact is consistent with coupling to Andreev bound states. Near π phase difference, we observe random ﬂuctuations in absorption with gate voltage, related to quantum interference-induced modulations in the electron transmission. Close to pinch-off, we identify features consistent with the presence of a single Andreev bound state and describe the Andreev-cavity interaction using a Jaynes-Cummings model. By ﬁtting the weak Andreev-cavity coupling, we extract ∼ GHz decoherence consistent with charge noise and the transmission

The flow of supercurrent across a Josephson junction (JJ) is described by Andreev bound states (ABSs), coherent superpositions of electrons and holes that transport Cooper pairs by Andreev reflection at the junction interfaces. 1 By setting the Josephson inductance LJ, ABSs play a central role in superconducting transmon qubits 2 and also form a two-level basis for spin-based quasiparticle qubits. 3In the short-junction limit, ABSs are spatially confined to the JJ with energy   = ±Δ√1 −   sin 2 ( 2) ⁄ , where Δ is the superconducting gap, δ is the phase across the JJ, and   the transmission probability of the n-th electron channel in the JJ material. 4Conventional qubits based on aluminium either host a fixed high number of low-τ ABSs in oxide JJs, yielding the familiar Josephson energy   =  0 cos(δ), or a few mechanically tuneable high-τ ABSs used for quasiparticle qubits in atomic point contacts. 5Rapid improvement in the growth of hybrid superconductor-semiconductor interfaces over the last decade has unlocked semiconductor JJs that can span these operating regimes, using local capacitively-coupled gate electrodes to tune the number 6,7 and   distribution 8 of ABSs.
Gatemons, the voltage-controlled variant of a transmon, 9 have so far been realised with semiconductor nanowires, 10,11 carbon, 12,13,14,15 and topological insulator 16 JJs.Single channels in InAs nanowire gatemons were recently associated with suppressed relaxation 17 and charge dispersion. 18,19Splitting of microwave transitions in quasiparticle qubits 20 due to strong spin-orbit interaction 21 has enabled spin-to-supercurrent conversion 22 and paves the way towards time-domain analysis of Majorana bound states. 23Qubit states can be probed by coupling the dipole transition to a transmission line via a microwave cavity resonator.
Capacitive coupling is used for transmons and inductive coupling for quasiparticle qubits, generating a state-dependent shift of the cavity frequency described by circuit quantum electrodynamics (cQED). 24oximitized two-dimensional electron gases (2DEGs) are an attractive scalable and versatile platform for realising voltage-controlled JJs, with ABS confinement potentials tailored by the outline of lithographically-patterned surface gates.Gatemons with 2DEG JJs have microsecond lifetimes, limited by microwave losses in the InP substrate. 25Here we establish the 2DEG variant for quasiparticle qubits by studying ABSs in a gate-defined superconducting quantum point contact (SQPC), shown schematically in Fig. 1(a).Our 2DEG comprises a trilayer InGaAs/InAs/InGaAs stack grown on a buffered InP substrate.A 50 nm-thick epitaxial Al capping layer proximitizes the 2DEG via Andreev reflection while preserving high (~ 1 m 2 V −1 s −1 ) charge carrier mobilities. 26ABSs are lithographically confined by laterally etching a mesa with width  1 ~4 μm down to the InP buffer layer and removing a  ~ 100 nm strip of Al.To form a SQPC the ABSs are further electrostatically confined to a narrow constriction, by depleting carriers from regions under the split-gate electrodes, which are spaced by width  2 ~100 nm.Fig.   ~1 μA , however, corresponds to an order of magnitude lower number of channels (~20), a feature also reported in an earlier study on wide JJs. 27We speculate that channels with higher momentum parallel to the interface have path lengths Λ >  that exceed the superconducting coherence length , and thus do not contribute to the transfer of Cooper pairs.Using the conductivity  =  2  and  =  * /ℏ 2 for the constant 2D density of states, we obtain ~0.07  2  −1 for the diffusion coefficient,   =  * /~2.5 ps for the scattering time, and mean free path   = √  ~ 70 nm, placing our JJs between the diffusive and ballistic regimes. * is between the clean (~ℏ  /Δ * ~1.4 μm) and dirty (~√ 0   ~300 nm) limits, where   ~1.2 × 10 6 is the Fermi velocity, and Δ * ~180 μeV is the induced gap measured by bias spectroscopy.In either case our devices satisfy  <  * and a critical current contribution of  0 per channel is not expected in region I since  2 ~1 . 28During definition (region II),   drops over a ∆  ~ 0.5 V interval as channels contributing to   either side of the SQPC are closed.Once the SQPC is defined,   ~0 and reduces gradually over a ∆  ~ 1.5 V as  1 decreases (region III).The reduced   /  in this region results from the reduced capacitive coupling from the split gates to the constriction.Having observed the correlation between ∆  and   , it is perhaps suprising that the fluctuations observed in ∆  (  , 0.5) are typically absent from   (  , 0) in region I.To elucidate this we measure S21(Φ ′ , ) for   values shown by dashed lines in Fig. 2(d).The series of plots in Fig. 3(a) show the decrease in overall push ∆  (  , 0.5) with increasing negative   , and also reveals a variety of avoided crossings symmetric about Φ′ = 0.5.Avoided crossings are a sign of virtual cavity-ABS photon exchange, inducing a push on the cavity frequency described by a Jaynes-Cummings (JC) interaction.In the JC picture, avoided crossings result from a divergence in the cavity push   () =   2 /2(  −   ), where   is the coupling rate, when the detuning from the  →  ABS transition approaches   ~ . 28nerally   comprises spin-orbit and subband kinetic energy contributions from different parity manifolds. 21For transitions within the same manifold,   =   () (  ,  ± ∆  ), channels with sufficient   meet this condition at different phase offset ∆  , giving rise to the different patterns of crossings in Fig. 3(a).
In order to capture these features quantitatively we simulate the coupled ABS-cavity system using a master equation for the JC model (see Appendix). 32Note the dispersive JC shift ∑   (𝐽𝐶) is valid in the range of flux shown in Fig. 3(a).Finally, since the ABS-cavity interaction is sensitive to quasiparticle dynamics, we can estimate the lifetime   via the decoherence rate  = 1/  in the JC model.We fix   and measure S21(Φ ′ , ) where a single ABS transition hybridises with the cavity [Fig.5(a)] and extract the cavity frequency to compare with the JC result and the simple dispersive shift  2 /2∆ [Fig.5(b)].Figure 5(c) and (d) also shows the equivalent plots for the avoided crossing when the ABS transition is tuned with   .The JC model, unlike the dispersive shift, shows good agreement with both data sets using  ≈ 1 MHz (estimates from a 6.163 GHz resonator with internal ~7000) and /2~16 MHz, corresponding to a ~10 pH, yielding /2~1 GHz.The fact  ≪  accounts for the weak coupling and absence of clear Rabi splitting, with similar behaviour observed in multiple devices.Note that  is ~3 times lower than extracted from the experimental data in Fig. 2 taken in the many-channel regime.This could be due to an underestimate of the channel number in the many-channel regime, which would imply lower , or to using the JC approximation in the single-mode regime.The   ~ 1 ns is shorter by several orders of magnitude than atomic point contacts 5 and recent work with InAs nanowire JJs. 19We speculate that the main mechanism for inelastic quasiparticle relaxation are emission or absorption of phonons 36 and electromagnetic coupling to the environment. 37Charge noise in 2DEGs is expected to generate root-mean-squared (RMS) fluctuations of 〈  〉  ~0.5 mV. 25 The slope   /  ~120 GHz/V [Fig.4(c)] yields  2 * ~4 ns, 38 comparable to our measured   .Nanowire gatemons with RT showed off-resonant   /   ~500 GHz/V, 19 but the lower noise 〈  〉  ~10 μV 35 and suppressed charge dispersion improves  2 * ~50 ns. 19Note that odd-parity states in which a single quasiparticle occupies the ABS would limit coherence on the ~μs timescale and thus strongly limit coherent manipulation.We anticipate further improvements by using thicker buffer layers to increase the carrier mobility, decreasing the gate capacitance, and reducing the maximum frequency in longer channels.
In summary we have used a superconducting microresonator to probe the microwave response of a 2DEG SQPC as a function of phase difference and carrier density.A monotonic shift in cavity frequency as a function of phase is consistent with coupling to ABSs channels.In the dispersive regime close to  phase across the JJ, we observed random shifts in the cavity frequency as a function of gate voltage related to

2(c).
A good fit is obtained for   = 4 eV, with 3 pairs of highly-transmitting channels with    = {0.97,0.92, 0.71} and    = {0.95,0.90, 0.81}) and  = 30 pH.This is larger than the ~10 pH deduced from comparison of the master equation simulation to data in the single-channel regime (Fig. 5).This discrepancy could be due to the precise QPC potential deviating from the PB or HW induced by the splitgate, as this would modify the number of modes and precise -distribution in the QPC at a given   .
To understand the measured Δ  (  ) shown in Fig.

Figure 1 .
Figure 1.(a) Stack structure of an InAs 2DEG with split-gates (red) used to define a SQPC.ABSs comprise right/left-(red) and left/right-(blue) propagating electron/hole pairs Andreev reflected at the interface.(b) Differential resistance ~/  of a SQPC as a function of   and   .Regions: open (I), definition (II), depletion (III), and tunnelling (IV).Lower panel shows  over a narrower range of   in region IV, with   resonances approaching ~45 nA indicated by arrows.(c) Schematic ABS energy spectrum of the junction reconstructed as a function of phase difference  =  2 −  1 (lower panels) and   at  =  (upper).In each region, the energy detuning between bare cavity photons (  ) and ABS transitions (  = 2  /ℎ) determines the effective shift in the observed cavity frequency.

Figure 2 .
Figure 2. (a) Optical and scanning electron micrographs of the device.A serpentine microwave cavity (blue) is inductively coupled to a loop of Al/InAs (orange) interrupted by a SQPC (red).(b) Feedline transmission coefficient S21 as a function of feedline frequency f and flux Φ.  0 ~6.163GHz is the observed resonance frequency at zero flux at zero gate voltage.(c) Resonant frequency shift extracted from (b).(d) S21 as a function of frequency and   at Φ/Φ 0 = 0.5.(e) Number of cavity fluctuations per volt (black points) and resonant frequency (red line) extracted from (d).

Figure 3 .
Figure 3. (a) S21 as a function of  and Φ ′ for   values indicated by the dashed lines in Fig. 2(d).Measured and simulated S21 as a function of (b,c) Φ ′ and (d,e)   and readout difference frequency.Lower panels show the transition frequency for 3 Andreev channels used in the simulations. ̃ is the simulation sweep parameter scaled to the same range as   .

Figure 3 (
b) shows the raw S21(Φ ′ , ) data at   = −3 V and corresponding simulation in Fig.3(c).The discrete crossings and overall cavity shift ~− 2 MHz are well reproduced assuming  = 3 spin-degenerate channels, consistent with modelling of Fig.2(b).We also confirmed that the model reproduces the dependence on   at Φ ′ = 0.5 [Figs.3(d)and (e)] using the same model parameters.Note that the  ̃ of the simulation is a randomly generated value of   , related to the underlying statistics of universal conductance fluctuations (UCFs).30Using Δ  =  1 Δ  ℏ 2 / * 2 , where  1 =  HfO / is the capacitance to the 2DEG,  = 20 nm, and  HfO = 11, and equating the number of avoided crossings 1/Δ  ~30 −1  , where Δ  is the correlation voltage, yields Δ  ~10 meV.This is in agreement with the intermediate limit ( >   ),33 where fluctuations of ~0 are expected when ∆  ~∆( √  *   /)~15 meV (for  * ~1 μm).In this picture the factor of ~3 increase in the correlation voltage [Fig.2(e)] in region IV is due to the weaker capacitive coupling between the split gate and the constriction.UCFs therefore appear in the cavity push as it only couples to the lowest modes, while in transport they are averaged out from region I in   [Fig.1(b)].A notable feature below   ~− 6 V is the presence of paired crossings [see arrows in Fig.3(d)].We focus on the pair highlighted by the dashed box in Fig.3(d) and shown in detail in Fig.4(a).Paired crossings are naturally explained by peaks in (  ), as illustrated in region IV of Fig.1(c).Resonant passage at the the conduction band edge is possible along periodically spaced impurity configurations,34 or a quantum

Figure 4 .
Figure 4. (a) S21 measured as a function of frequency and V  at the paired crossings highlighted by the dashed box in Fig. 3(d).(b) Schematic of a quantum dot formed in the junction, coupled to the leads with tunneling rates  1 and  2 .The energy level sketch (middle panel) shows a quantum dot state with energy detuning  from the chemical potential μ modulated by V  .(Lower panel) Plot of Breit-Wigner transmission ( 1 =  2 = ) and ABS frequency.(c) Simulated S21 as a function of V ෩  using the   () shown in lower inset.

Figure 5 .
Figure 5. (a) S21 measured as a function of Δ and  in the single-channel regime (  ~− 6 V).(b) Shift in cavity frequency extracted from the S21 data in (a) and plotted as a function of  (green points).Fit to the experimental data using the J-C model (orange dashed) and dispersive shift (blue dashed).(c) S21 measured as a function of frequency and   at  = .(d) Shift in cavity frequency shift with   (green points), J-C model (orange dashed), and dispersive shift (blue dashed).Insets in (b) and (d) illustrate the ABS-cavity hybridisation.
quantum interference-induced fluctuations in transmission.We reproduced the phase and gate-voltage dependence using a JC model.In the single-channel regime we observe paired crossings as a function of gate voltage, suggesting the presence of resonant tunnelling in a quantum dot.The absence of Rabi splitting is due to the ~GHz decoherence induced by gate noise and level dispersion.With optimisation of materials our study paves the way for quantum control of gate-defined quasiparticle qubits and provides insight into how the gate geometry and material of a JJ relates to qubit performance.a function of   in Fig.6(a) and (b) for both HW and PB potentials.Note that the channels are spindegenerate, so each line represents two channels.As   increases, the transmission of each channel drops to near-zero.One or two pairs of channels remain highly transmitting and undergo fluctuations in transmission before finally being backscattered as the QPC is pinched off.To visualise the overall number

Fig. 6 .
Fig. 6.Mode transmissions {  } for the hardwall (HW) potential in (a) and the parabolic (PB) potential in (b) as a function of chemical potential   .(c) The number of highly-transmitting channels, defined with   > 0.5, showing a progressive decrease with increasing   as each channel is backscattered.

Fig. 7 .
Fig. 7. (a) Plot showing the Andreev transition frequency as a function of phase difference for hardwall (HW, blue solid) and parabolic (PB, red dashed) confinement at   = 4 .(b) Shift in cavity frequency as a function of phase difference for HW (blue solid) and PB (orange dashed), and the experimental data (black crosses).
Fig. 8(a).As expected, at low   the number of channels drops rapidly while the lowest energy mode remains highly transmitting.The corresponding Δ  (  , ) in Fig. 8(b) shows ∼ −5 MHz average shift

Fig. 8 . 2 .
Fig. 8. (a) Plot showing the number of transmitting channels as a function of chemical potential for hardwall (HW) and parabolic (PB) confinement.Also shown is the transmission   of the lowest energy mode.(b) Calculated frequency shift of the cavity as a function of chemical potential parameter   .(c) Experimental frequency shift with gate voltage   along with a running average to show the overall frequency shift excluding fluctuations.Regions II-IV are indicative only and correspond to those indicated in Fig. 1(b).
1(b)shows a plot of the differential resistance ~/  as a function of gate voltage   and bias current   of a typical SQPC. was determined by conventional AC lock-in measurements in a fourterminal configuration, using an external 1 MΩ resistor to implement current-biased measurements.A dissipationless state ( = 0) is maintained up to a maximum critical current   .In principle   = ∑