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Triplet-blockaded Josephson supercurrent in double quantum dots

Serial double quantum dots created in semiconductor nanostructures provide a versatile platform for investigating two-electron spin quantum states, which can be tuned by electrostatic gating and external magnetic field. In this work, we directly measure the supercurrent reversal between adjacent charge states of an InAs nanowire double quantum dot with superconducting leads, in good agreement with theoretical models. In the even charge parity sector, we observe a supercurrent blockade with increasing magnetic field, corresponding to the spin singlet to triplet transition. Our results demonstrate a direct spin to supercurrent conversion, which can be exploited in hybrid quantum architectures coupling the quantum states of spin systems and superconducting circuits.

Semiconductor quantum dots, where the orbital and spin states of single localized electrons can be controlled [1], are one of the leading platforms for quantum information processing [2]. Specifically, double quantum dots (DQDs) connected in series [3] became the preferred physical implementation of spin [4], and spin-orbit quantum bits [5] in low-dimensional semiconductor nanodevices, such as heterostructures hosting a two-dimensional electron gas or semiconductor nanowires. In these devices, the readout of the spin quantum state relies on spin-dependent single electron tunneling processes, which then enable charge readout via direct electronic transport [1], charge sensing techniques [6], or dipole coupling to a microwave resonator [7,8].
In a superconducting nanodevice, the dissipationless supercurrent I S at zero voltage bias is driven by the quantummechanical phase difference ϕ up to a maximum amplitude, I C , the critical current [9]. In the lowest order of tunneling, the supercurrent-phase relationship (CPR) [10] is sinusoidal, I S (ϕ) = I C sin(ϕ), which describes the coherent transfer of single Cooper pairs through the weak link. When the weak link is a non-magnetic tunnel barrier, a zero phase difference is energetically favorable in the absence of supercurrent, which is described by a positive critical current, I C > 0. In contrast, a negative coupling yields a supercurrent reversal, I C < 0, often denoted a π junction due the π phase shift in the CPR. This negative coupling has been observed in ferromagnetic weak links [11,12], out-of-equilibrium electron systems [13] and semiconductor quantum dot junctions [14,15].
The dependence of the critical current on the spin-and charge state of a DQD has also been addressed theoretically [16][17][18][19][20][21], and the recent progress in materials science of superconductor-semiconductor hybrid nanostructures [22] enabled measurements of the amplitude of the criti-cal current as well [23,24], in correlation with the charge states of the DQD.
In this Letter, we report on direct measurements of the CPR through a DQD weak link formed by an electrostatically gated InAs nanowire. By employing a phasesensitive measurement scheme, where the DQD is embedded in a superconducting quantum interference device (SQUID), we characterize the full CPR of the DQD, enabling a signful measurement of I C . The direct observation of the supercurrent reversal in the total charge number boundaries allowed us to identify the even and odd occupied states. Finally, the magnetic-field dependence of the supercurrent amplitude in the even occupied state reveals the presence of a supercurrent blockade in the spin triplet ground state, in agreement with numerical calculations.
We built our device ( Fig. 1) from an approximately 7 µm long InAs nanowire grown by molecular beam epitaxy, and in-situ partially covered by a 6 nm thick epitaxial aluminum shell with a typical superconducting gap of ∆ ≈ 200 µeV [22,25]. We formed two segments with the aluminum layer selectively removed where the DQD and the reference arm would be defined. Next, we created the SQUID loop from a sputtered NbTiN superconducting film, and covered the device with a 10 nm thick AlO x dielectric by conformal atomic layer deposition. Finally, 40 nm wide and 50 nm thick Ti/Au gates (in yellow in Fig. 1a) were evaporated under three angles to ensure a conformal coverage around the wire (schematically shown in Fig. 1b). Five gates defined the DQD (on the right) and a single gate controlled the reference arm (on the left in Fig. 1c). Details on the device fabrication are shown in the Supplementary Information. All of our measurements were performed in a dilution refrigerator with a base temperature of approximately 30 mK.  We first characterize the DQD with the leads driven to the normal state by a large magnetic field, B ⊥ = 0.5 T. We measure the differential conductance dI/dV of the DQD with the reference arm fully depleted (Fig. 1d). We control the coupling to the leads with the gate voltages V BL and V BR , and the interdot coupling is tuned by V C (Fig. 1a). A characteristic honeycomb diagram is plotted in Fig. 1f, where the charge occupancy of the dots (n L , n R ) is set by the voltages applied on the two plunger gates, V L and V R .
We perform the CPR measurements with the leads being superconducting and with the reference arm of the SQUID opened with its electrostatic gate so that it exhibits a higher critical current than the DQD arm. Due to this asymmetry, the phase drop over the DQD junction is determined by the magnetic flux Φ through the SQUID loop area (Fig. 1e) [14,26], which is proportional to the applied magnetic field B ⊥ . We measure the switching current I SW of the SQUID by ramping a current bias in a sawtooth waveform and recording the bias current value when the junction switches to the resistive state marked by a threshold voltage drop of the order of 10 µV. We show a typical dataset in Fig. 1g, where each pixel in the main panel is an average of 18 measurements. The right sidepanel shows the raw datapoints at two plunger gate settings denoted by the magenta and green lines in the main panel, as well as the fitted sinusoidal curves in the following functional form: where ϕ = 2π(B ⊥ −B o )/B p , with B p ≈ 1.7 mT being the magnetic field periodicity corresponding to a flux change  Fig. 1g, which displays the sign change of I DQD at the charge state boundaries. We note that the change in the environmental impedance [27] causes a slight modulation of I ref as well, despite the lack of any capacitive coupling between the two weak links. However, in our measurements I ref > 5|I DQD | is always fulfilled, enabling a reliable observation of the supercurrent reversal in the DQD. In Fig. 2a, we plot I DQD as a function of the plunger gate voltages V L and V R , resulting in the zero magnetic field charge stability diagram of the DQD mapped by the supercurrent. Remarkably, our phase-sensitive measurement directly shows that the supercurrent reversal is associated with the change in the total charge number, and it is absent in the case of internal charge transfers with (n L , n R ) → (n L ± 1, n R ∓ 1). However, |I DQD | exhibits maxima near all charge boundaries, consistently with earlier experiments [24].
We understand this data using a two-orbital Anderson model, where each dot with an on-site charging energy U i hosts a single spinful level at ε i with the dot index i = L, R. In the experiment, this corresponds to a quantum dot orbital level spacing which is larger than the charging energy [14]. We consider an interdot charging energy term U C n L n R and an effective interdot tunneling amplitude t C . The tunnel coupling energies to the superconducting leads are denoted by Γ L,R . We consider the leading term of the supercurrent in the weak coupling limit where t C , Γ L , Γ R ∆ U i [18,28], and evaluate the current operator I(ϕ) = i e [H, n R ], where H is the Hamiltonian of the system at a phase difference of ϕ between the superconducting leads (see the Supplementary Information). We numerically evaluate I(ϕ) = I C sin ϕ to find the signful I C . We perform a global fit of the calculated sign reversal contours (see dashed lines in Fig. 2a) against the experimental dataset and recover U L = 596.6 µeV, U R = 465.9 µeV, U C = 41.5 µeV and t C = 85 µeV. We match the critical current amplitude scale with the experimental data by setting Γ L = Γ R = 33.2 µeV. The width of the even-odd transitions establishes an upper bound on the electron temperature of the DQD, T < 80 mK. We use these parameters to display I C (V L , V R ) in Fig. 2b and find a good correspondence with the experimental data using a zero bandwidth (ZBW) approximation [24,29], see the Supplementary Information.
The observed supercurrent reversal [14,30] is linked to the number of permutations of fermion operators required to transfer a spin-singlet Cooper pair through the DQD (see the Supplementary Information). In the weak coupling limit, this amounts to counting the number of same-spin dot electrons, which each electron in the Cooper pair crosses. Each such crossing contributes with a factor of −1 to I C , which we illustrate for a DQD with even (Fig. 2c), and odd charge occupations (Fig. 2d). Consequently, the sign of I C is determined by the groundstate charge parity of the DQD.
Next, we focus on the magnetic-field dependence of I DQD (Fig. 3a) along the total energy axis (solid line in Fig. 2a) spanning both even and odd charge states. At B = 0, a finite t C results in a singlet-triplet splitting ∆ ST in the even occupied (1, 1) charge state [1]. We model the DQD with an effective identical g-factor on both dots, which results in a spin-polarized triplet ground state above a threshold magnetic field, B ST = ∆ ST /(g * µ B ). To account for spin-orbit coupling, we refine our interdot tunneling Hamiltonian to include both spin-conserving and spin-flip tunneling amplitudes, t 0 and t x , resulting in an effective t C = t 2 0 + t 2 x (see the Supplementary Information).
With a global fit to the experimental data (Fig. 3a,b), we extract t 0 = 80 µeV, t x = 30 µeV and g * = 15.9. This g-factor is in agreement with earlier experimental values measured on InAs quantum dots [5,[31][32][33] and ballistic channels with superconducting leads [25,34]. We esti- mate the spin-orbit length l SO = l dot t 0 /( √ 2t x ) ≈ 75 nm [35], using the gate pitch as an estimate of the dot length, l dot = 40 nm. This coupling length yields an energy scale E SO = 2 /(2m * l 2 SO ) = 290 µeV with an effective electron mass of m * = 0.023m e , which is similar to earlier experimental results on semiconductor nanowires in the presence of strong electrostatic confinement [36,37].
In Fig. 3c, we plot the calculated expectation value S Z of the total spin z component of the DQD, which visualizes the transition between the spin singlet state S Z = 0 and the spin-polarized triplet state, where S Z = 1, as a function of the magnetic field. This transition point at B ST is accompanied by a drop of the critical current in the (1,1) sector, however this sudden decrease is absent in the odd sector (see blue regions in Fig. 3b). We note that the gradual global decrease in I DQD is consistent with the orbital effect of the magnetic field applied along the nanowire [38].
We analyze this data in Fig. 4, where we first find the charge state boundary at each value of B at I DQD = 0 (blue dots and error bars in Fig. 4a) and overlay the calculated boundary (black solid line, corresponding to Fig. 3b). We quantify B ST ≈ 80 mT, which agrees consistently with the characteristic cutoff magnetic field of I DQD at several plunger gate values (dots in Fig. 4b, colors corresponding to the arrows in Fig. 3a). Furthermore, we find an excellent agreement with the calculated critical current I C (B) (solid lines in Fig. 4b) with a common scaling factor of 0.29, which may stem from the reduced switching current inside the charge state due to thermal activation compared to the corresponding critical current [27]. Cooper pair transfer. Starting from the (1, 1) T state close to the charge boundary with the single occupation sector, we always encounter a virtual state with a double occupation on one of the dots (magenta circle in Fig. 4c).
In the U ∆ limit corresponding to our experiments, this configuration is energetically unfavorable and suppresses Cooper pair tunneling. In contrast, a spin singlet starting condition can avoid this configuration (Fig. 4d). We finally note that the opposite limit, where U ∆, also leads to a triplet supercurrent blockade [19] (see the Supplementary Information), which persists with a finite residual supercurrent in the spin triplet state when U ∼ ∆.
In conclusion, we directly measured the supercurrent reversal associated with the even-odd charge occupation in an InAs DQD, where the large level spacing allows us to use a single orbital for each dot in our quantitative modeling. In the (1, 1) charge sector, we showed that the singlet to triplet transition is accompanied by a supercurrent blockade. This enables a direct spin to supercurrent conversion [34,39] in hybrid semiconductor nanodevices [28] used for quantum information processing.
The authors thank J. Danon  Raw datasets and computer code are available at the Zenodo repository [40].