Coherent manipulation of an Andreev spin qubit
M. Hays, V. Fatemi, D. Bouman, J. Cerrillo, S. Diamond, K. Serniak, T. Connolly, P. Krogstrup, J. Nygård, A. Levy Yeyati, A. Geresdi, M. H. Devoret
CCoherent manipulation of an Andreev spin qubit
M. Hays, ∗ V. Fatemi, † D. Bouman,
2, 3
J. Cerrillo,
4, 5
S. Diamond, K. Serniak,
1, 6
T. Connolly, P. Krogstrup, J. Nyg˚ard, A. Levy Yeyati,
5, 8
A. Geresdi,
2, 3, 9 and M. H. Devoret ‡ Department of Applied Physics, Yale University, New Haven, CT 06520, USA QuTech and Delft University of Technology, 2600 GA Delft, The Netherlands Kavli Institute of Nanoscience, Delft University of Technology, 2600 GA Delft, The Netherlands ´Area de F´ısica Aplicada, Universidad Polit´ecnica de Cartagena, E-30202 Cartagena, Spain Departamento de F´ısica Te´orica de la Materia Condensada C-V,Universidad Aut´onoma de Madrid, E-28049 Madrid, Spain Current Affiliation: MIT Lincoln Laboratory, 244 Wood Street, Lexington, MA 02420, USA Center for Quantum Devices, Niels Bohr Institute,University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark Condensed Matter Physics Center (IFIMAC) and Instituto Nicol´as Cabrera,Universidad Aut´onoma de Madrid, E-28049 Madrid, Spain Quantum Device Physics Laboratory, Department of Microtechnology and Nanoscience,Chalmers University of Technology, SE 41296 Gothenburg, Sweden
Two promising architectures for solid-state quantum information processing are electron spins insemiconductor quantum dots and the collective electromagnetic modes of superconducting circuits.In some aspects, these two platforms are dual to one another: superconducting qubits are more eas-ily coupled but are relatively large among quantum devices ( ∼ mm), while electrostatically-confinedelectron spins are spatially compact ( ∼ µ m) but more complex to link. Here we combine beneficialaspects of both platforms in the Andreev spin qubit: the spin degree of freedom of an electronicquasiparticle trapped in the supercurrent-carrying Andreev levels of a Josephson semiconductornanowire. We demonstrate coherent spin manipulation by combining single-shot circuit-QED read-out and spin-flipping Raman transitions, finding a spin-flip time T s = 17 µ s and a spin coherencetime T E = 52 ns. These results herald a new spin qubit with supercurrent-based circuit-QED inte-gration and further our understanding and control of Andreev levels – the parent states of Majoranazero modes – in semiconductor-superconductor heterostructures. A weak link between two superconductors hosts dis-crete, fermionic modes known as Andreev levels [1, 2].They govern the physics of the weak link on the micro-scopic scale, ultimately giving rise to macroscopic phe-nomena such as the Josephson supercurrent. Supercon-ducting quantum circuits crucially rely on the nonlinear-ity of the supercurrent in Josephson tunnel junctions, amanifestation of the ground-state properties of millions ofAndreev levels acting in concert [3–5]. While the vast ma-jority of conduction electrons participate in the nonlinearbosonic oscillations of the superconducting condensate,each Andreev level is itself a fermionic degree of freedom,able to be populated by electronic excitations known asBogoliubov quasiparticles.In 2003, it was proposed to store quantum informa-tion in the spin state of a quasiparticle trapped in aweak link possessing a spin-orbit coupling [6–9]. It waspointed out that this Andreev spin qubit would carrya state-dependent supercurrent, opening new paths forspin manipulation and measurement that are unavail-able to electrostatically-confined spin qubits [10, 11]. Inparticular, such a state-dependent supercurrent could beused to achieve strong coupling with a superconduct-ing microwave resonator, an area of active research inthe spin qubit community [12–18]. This supercurrent-based coupling has been used in such circuit quantumelectrodynamics (cQED) architectures [19, 20] to detectand manipulate pairs of quasiparticles trapped in An-dreev levels [21, 22]. However, because the Andreev levelsof most weak links are paired into spin-degenerate dou-blets, quasiparticle spin manipulation has remained outof reach. The level structure of an Andreev doublet is de-termined by the geometric and material propertiesof the host weak link, as shown in weak linkscomposed of superconductor-proximitized semiconductornanowires, or “Josephson nanowires” for short [23, 24].Thanks to recently-achieved atomic-scale perfection ofthe superconductor-semiconductor interface, it is nowpossible to observe the Andreev spectra of Josephsonnanowires, revealing a rich interplay between electromag-netic field effects, device geometry, and spin-orbit cou-pling [25, 26]. These properties of Andreev levels insuperconductor-semiconductor nanowires have been em-ployed to demonstrate gate-tunable weak links for super-conducting qubits [27, 28], probe non-abelian Andreevlevels known as Majorana zero modes [29–33], and, im-portantly for this experiment, investigate spin-split dou-blets without a Zeeman field [26, 34].In this letter, we demonstrate the first coherent ma-nipulation of the spin of an individual quasiparticle exci-tation of a superconductor. The quasiparticle is trappedin the Andreev levels of a Josephson nanowire, where itresides predominantly in the two spin states of the lowest-energy Andreev doublet with roughly equal probability.First, we initialize this Andreev spin qubit in one of thetwo spin states by post-selecting on a single-shot cQEDspin measurement, which we demonstrated in an earlierwork [34]. We then achieve full coherent control of theAndreev spin qubit by driving Raman transitions in anatural Λ system formed by the two spin states and anexcited state. We observe spin lifetimes up to T s = 17 µ sat the presented gate voltages [see Supplementary Infor-mation Fig. S9]. However, the much shorter spin coher- a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n ence time T = 52 ns appears to be limited by a spinfulbath. FIG. 1: Principle of the Andreev spin qubit. (a) Illustrationof a semiconductor nanowire (white) coated with epitaxial su-perconductor (light blue). A quasiparticle is trapped in theexposed weak link by the pair potential ∆ of the supercon-ducting leads. Due to spin-orbit coupling, if the quasiparticleis in the spin-up state (upper panel) supercurrent flows tothe right near zero phase bias ϕ (cid:54) = 0, while in the spin-downstate (lower panel) supercurrent flows to the left. Applyingnonzero ϕ thus breaks spin degeneracy. (b) Level structureof two Andreev doublets tuned to a Λ configuration. Twomicrowave drives (frequencies f ↓ and f ↑ ) are equally detunedfrom | ↓ q (cid:105) ↔ | ↑ a (cid:105) and | ↑ q (cid:105) ↔ | ↑ a (cid:105) , inducing a coherentRaman process between the qubit states | ↓ q (cid:105) and | ↑ q (cid:105) via avirtual level (black dashed line). (c) Both microwave drivesinduce an rf electric field E d between the superconductingleads. But, for a nanowire symmetric across the plane M (i.e.only nanowire + substrate), drive-induced spin-flips would beforbidden. However, as depicted in the inset, here the mirrorsymmetry is broken by both the partial aluminum shell as wellas the cutter (blue, bias V g , c ) and plunger gates (black, bias V g , p ). (d) The Josephson nanowire (light blue) is embedded ina superconducting loop (gray), which enables phase bias viaan external flux ϕ ≈ π Φ / Φ as well as inductive coupling toa superconducting microwave resonator (maroon). The res-onator reflection coefficient Γ = I + iQ is probed with a tonenear the resonator frequency f r = 9.188 GHz. (e) Repeated1 . µ s measurements of I and Q clustered into three distribu-tions, corresponding to | ↓ q (cid:105) , | ↑ q (cid:105) and | g (cid:105) (standard deviation σ ). The system state was assigned based on thresholds indi-cated by the black dotted lines. Our realization of the Andreev spin qubit hinges onthe interplay between spin-orbit coupling in the semicon- ductor nanowire and the superconducting phase bias ϕ across the weak link [Fig. 1(a)] [6–9, 26, 34]. In a con-ventional weak link, a trapped quasiparticle is restrictedto spin-degenerate Andreev doublets and therefore thespin cannot be coherently manipulated. In a Joseph-son nanowire, however, an inter-subband spin-orbit in-teraction can cause spin to hybridize with translationaldegrees of freedom (this hybridized spin is sometimesknown as pseudospin, though we will continue to referto it as “spin” for simplicity). Due to this interaction be-tween spin and motion, the two spin states of an Andreevdoublet carry equal and opposite supercurrent ± I s / ϕ = 0, with I s doublet-dependent. The doublet degen-eracy can thus be lifted with a nonzero phase bias: per-turbatively near ϕ = 0, the spin splitting is given by (cid:15) s = I s ϕ Φ / π .Microwave quantum optics techniques are well-suitedto achieve quasiparticle spin manipulation, given the fre-quency selectivity brought about by such a flux-inducedspin splitting. In this experiment, the two spin states | ↓ q (cid:105) , | ↑ q (cid:105) of one Andreev doublet form the qubit basis[Fig. 1(b)], while a second, higher-energy doublet pro-vides auxiliary states | ↑ a (cid:105) , | ↓ a (cid:105) critical for both qubitcontrol and measurement [34]. To manipulate the An-dreev spin qubit, we use the qubit states | ↓ q (cid:105) , | ↑ q (cid:105) inconjunction with | ↑ a (cid:105) as a Λ system. We apply simul-taneous microwave drives to both the | ↑ q (cid:105) ↔ | ↑ a (cid:105) tran-sition (drive frequency f ↑ ) and | ↓ q (cid:105) ↔ | ↑ a (cid:105) (drive fre-quency f ↓ ). By equally detuning the two drives fromtheir respective transitions, a Raman process is inducedsuch that the {| ↓ q (cid:105) , | ↑ q (cid:105)} manifold can be coherentlymanipulated while | ↑ a (cid:105) remains minimally populated.The success of the Raman process is contingent onour ability to drive both the spin-conserving transition | ↑ q (cid:105) ↔ | ↑ a (cid:105) and the spin-flipping transition | ↓ q (cid:105) ↔| ↑ a (cid:105) . While spin-orbit hybridization is necessary to en-able electric-field induced spin-flips [35], in our situationa broken spatial symmetry of the Josephson nanowireis also required (see Supplementary Information for de-tails). In this experiment, our hexagonal nanowire wasmade of [001] wurtzite indium arsenide grown by molec-ular beam epitaxy. Such a nanowire lying alone on asubstrate would possess a transverse mirror symmetry[Fig. 1(c)]; this property would then be inherited by thelevels of the nanowire such that one spin state of eachdoublet would be mirror-symmetric and the other anti-symmetric. Since we apply the microwave drive voltageacross the weak link, the rf electric field respects the mir-ror symmetry (it points along the nanowire) and thereforecannot flip spin.In the device used in this work, the mirror symme-try is broken by both the superconducting leads and theelectrostatic gates [Fig. 1(c)], as well as any symmetry-breaking disorder present in the nanowire. The super-conducting leads consist of 10 nm-thick epitaxial alu-minum, of which a 500 nm length was removed to formthe weak link. The aluminum only covers two of sixnanowire facets, thereby breaking the mirror symmetryof the nanowire-substrate system. As the gates are fab-ricated on one side of the nanowire, they also break themirror symmetry. Both the cutter and plunger gates wereused to tune the transparency of the weak link and werebiased to V g , c = − . V g , p = 4 . I + iQ of the reflected tone clustered into three distributions,corresponding to the two spin states | ↓ q (cid:105) , | ↑ q (cid:105) of atrapped quasiparticle as well as the ground state | g (cid:105) of thejunction where no quasiparticle was present. Throughoutthis work, we present data in terms of spin state occupa-tion probabilities P ↑ , P ↓ , which we compute based on thethresholds displayed in Fig. 1(e).The Andreev spin qubit exists exclusively when aquasiparticle is stochastically trapped in the Josephsonnanowire (see Supplementary Information for a quantumjumps trace). For the bias conditions presented in thiswork, we found that a trapped quasiparticle occupiedthe two spin states of the lowest doublet with roughlyequal probability. Thus, under any coherent manipula-tion | ↓ q (cid:105) ↔ | ↑ q (cid:105) the observed spin state populationswould not change. Throughout this work, we overcamethis problem by initializing the quasiparticle in | ↑ q (cid:105) viaan initial readout pulse and post-selection (see Supple-mentary Information for the same measurements with | ↓ q (cid:105) post-selection). Our single-shot spin readout wasthus critical to our observation of coherent populationtransfer between | ↑ q (cid:105) and | ↓ q (cid:105) .The first step in driving the Raman process [Fig. 1(b)]was locating the two transitions that defined the Λ sys-tem: | ↓ q (cid:105) ↔ | ↑ a (cid:105) and | ↑ q (cid:105) ↔ | ↑ a (cid:105) . After breakingspin degeneracy with Φ = − . and tuning the tran-sitions to a local maximum in V g , c to mitigate chargenoise (see Supplementary Information), we measured thespectrum shown in Fig.2(a) using two-tone spectroscopy,without spin initialization. The dip in P ↑ at 13 .
000 GHzcorresponds to the drive coming into resonance with the | ↑ q (cid:105) ↔ | ↑ a (cid:105) transition, resulting in population transferout of | ↑ q (cid:105) and into | ↑ a (cid:105) . Similarly, the dip in P ↓ at13 .
684 GHz corresponds to the | ↓ q (cid:105) ↔ | ↑ a (cid:105) transition(see Supplementary Information for gate voltage and Φdependence). Taking the difference yields the spin split-ting (cid:15) s /h = 684 MHz.Having characterized the Λ system, we then inves-tigated two-photon Raman transitions of the trappedquasiparticle. After initializing the quasiparticle in | ↑ q (cid:105) ,we applied two simultaneous Gaussian pulses with vari-able respective carrier frequencies f ↑ and f ↓ and thenmeasured the final qubit spin state [Fig. 2(b)]. Through-out the main text, we present data with f ↑ and f ↓ blue-detuned from | ↑ q (cid:105) ↔ | ↑ a (cid:105) and | ↓ q (cid:105) ↔ | ↑ a (cid:105) respectively FIG. 2: Raman transitions of a trapped quasiparticle. (a)In a two-tone measurement, the Josephson nanowire was firstdriven by a 1 µ s saturation pulse (gray) of variable carrier fre-quency f d before the quasiparticle state was determined witha readout pulse (maroon). A dip is observed in P ↑ correspond-ing to the | ↑ q (cid:105) ↔ | ↑ a (cid:105) transition and in P ↓ corresponding tothe | ↓ q (cid:105) ↔ | ↑ a (cid:105) transition [see Fig. S5 for Φ-dependence].(b) The quasiparticle was first prepared in | ↑ q (cid:105) via an ini-tial readout pulse and post-selection. Simultaneous Gaussianpulses (235 ns full width at half maximum, 30 dB more powerthan used in (a)) with variable frequencies f ↓ , f ↑ were thenapplied, followed by a final readout pulse. The observed peakin the final | ↓ q (cid:105) population lies along f ↓ = f ↑ + 609 MHz(black dashed line). (c) Full Γ histograms of the final read-out pulse for the two subsets of measurements enclosed by thegray and black solid lines in (b). Data accrued in the regionenclosed by the gray line shows little population transfer fromthe post-selected | ↑ q (cid:105) (left panel), while data in the regionenclosed by the black shows significant population transfer to | ↓ q (cid:105) (right panel). (see Supplementary Information for data over a wider fre-quency range). Along a line given by f ↓ = f ↑ + 609 MHz,we observe increased | ↓ q (cid:105) population that we attributeto the onset of a Raman process. As expected for Ramantransitions, the slope of this line is equal to one, sincea shift of one drive frequency must be compensated byan equal shift of the other. The discrepancy between thespin splitting (cid:15) s /h = 684 MHz and the 609 MHz offsetwas due to an uncontrolled shift of the Andreev spec-trum that occurred in between the measurements shownin Fig. 2(a) and Fig. 2(b) (see Supplementary Informa-tion).To further illustrate the dynamics of the quasiparti-cle under the Raman transitions, we histogram Γ fordata points off/on resonance with the Raman process[Fig. 2(c)]. Off resonance, the quasiparticle was foundpredominantly in | ↑ q (cid:105) , as expected from post-selectionon the initial readout pulse. On resonance, there was sig-nificant population transfer to | ↓ q (cid:105) as desired, as well asa small population transfer to | g (cid:105) . This is due to drive-induced quasiparticle de-trapping, which we comment onfurther below. FIG. 3: Coherent Λ-Rabi oscillations of the quasiparticle spinat f ↓ = 13 .
280 GHz and f ↑ = 13 .
964 GHz. (a) Independentlyvarying the amplitudes A ↑ and A ↓ of the simultaneous Gaus-sian drive pulses (94 ns full width at half maximum) resultedin coherent oscillations between | ↑ q (cid:105) and | ↓ q (cid:105) characteristicof a Raman process. The oscillations are only present awayfrom A ↑ , A ↓ = 0, and are symmetric under sign flips of A ↓ , A ↑ .(b) Simulated dynamics of the quasiparticle under the actionof the drive pulse. The reduced contrast observed in (a) istaken into account using the measured readout fidelities. Finally, having induced spin population transfer usingRaman transitions, we demonstrate the first coherent ma-nipulation of the spin of an individual quasiparticle. Wefirst chose our Raman drive frequencies using the samemeasurement as shown in Fig. 2, but with shorter pulses(94 ns full width at half maximum): we detuned f ↓ by280 MHz from | ↓ q (cid:105) ↔ | ↑ a (cid:105) and varied f ↑ until we ob-served maximum spin population transfer. We then var-ied the amplitudes A ↑ , A ↓ of the two Gaussian pulses be-fore determining the final quasiparticle state [Fig. 3(a)].The observed oscillations in the population difference be-tween the two spin states are characteristic of a coher-ent Raman process. Qualitatively, when either A ↓ = 0or A ↑ = 0 there is no population transfer because bothdrives are required to induce the Raman process. As theamplitudes of both drives are increased (roughly alongthe diagonals | A ↓ | (cid:39) | A ↑ | ), the spin population differenceundergoes coherent oscillations. As expected, the dataare symmetric under A ↓ → − A ↓ and A ↑ → − A ↑ .Quantitatively, the data are well-represented by a sim-ulation of the coherent quasiparticle dynamics under theaction of the drive pulses [Fig. 3(b)]. Details of thisnumerical simulation can be found in the Supplemen-tary Information. Using a Lindblad master equation [36]we calculated the dynamics of the quasiparticle betweenthe two Andreev doublets, with the inter-doublet transi-tion frequencies and dephasing rates, spin dephasing rate,state-dependent readout fidelities, and pulse frequenciesand envelopes fixed to values determined by indepen-dent measurements and instrument settings. We thenfit the simulation to the measured data by varying thefour inter-doublet transition matrix elements [Fig. 1(d)], as well as a slight detuning from the Raman resonancecondition, which we found to be 5 . ± . . ± . | A ↓ | = | A ↑ | = 1)to capture the measured increase of | ↓ q (cid:105) , | ↑ q (cid:105) → | g (cid:105) for larger drive powers. We were thus able to capturethe measured Raman spin dynamics of a quasiparticletrapped in the Andreev levels of a Josephson nanowire. FIG. 4: Coherence decay of the quasiparticle spin ( V g , c = − . , V g , p = − . , Φ = − . ). Ramsey (a)and Hahn-echo (b) experiments reveal T R = 18 ± T E = 52 ± τ to the final Ramanpulse. With the ability to perform coherent spin manipulationin hand, we then characterized the coherence lifetime ofan Andreev spin. A Ramsey measurement [Fig. 4(a)]revealed spin coherence decay with a timescale T R =18 ± T E = 52 ± − ( τ /T ) α ] with α = 0 . ± .
1, indicativeof excess low-frequency components compared to a whitenoise spectrum where α would be zero. The observedoscillations in P ↑ , P ↓ are centered about a lower value inFig. 4(b) as compared to 4(a), which we attribute to addi-tional quasiparticle de-trapping | ↓ q (cid:105) , | ↑ q (cid:105) → | g (cid:105) causedby the echo pulse. Both the observed Ramsey and Hahn-echo coherence times are comparable to that of the spin-orbit qubit [12, 35, 37], the closest cousin of the Andreevspin qubit in that it consists of the spin-orbit hybridizedpseudospin of a single electron. However, because herethe quasiparticle was trapped in Andreev levels, we pos-sessed a different experimental lens with which to investi-gate the effects of the environment on the spin coherence.As the Andreev levels of a Josephson nanowire are tun-able via both electrostatic voltages and flux, we first sus-pected charge or flux noise as the source limiting the An-dreev spin qubit coherence. However, we found that nei-ther T R nor T E varied with V g , c around the sweet-spotbias point (see Supplementary Information). This indi-cated that the spin coherence was not limited by chargenoise, consistent with the observed weak dependence of (cid:15) s on V g , c . By comparing to the charge-noise-limited inter-doublet transitions, we extracted a lower-bound of 4 . µ son the charge-noise-induced dephasing time of the quasi-particle spin (see Supplementary Information). More-over, the spin coherence time was not limited by fluxnoise, as we found no measurable dependence on Φ.To better understand what was limiting the coherenceof the Andreev levels, we additionally measured the co-herence lifetimes of both inter-doublet transitions andso-called “pair transitions” at several gate bias points[Tab. 1]. The latter correspond to the excitation oftwo quasiparticles out of the condensate into both lev-els of a doublet [21, 22, 26, 38]. We found that the pairtransition coherence times were systematically an orderof magnitude longer than inter-doublet transition coher-ence times. To first order, perturbations that couple tospin (such as a Zeeman field) result in equal and oppo-site energy shifts of the two doublet levels [26]. As such,these perturbations do not change the frequency of thedoublet pair transition, and therefore do not cause de-phasing. However, such spin-specific perturbations do in-duce dephasing of both inter-doublet transitions and theAndreev spin qubit. It thus appears that the coherencelifetime of the Andreev spin qubit is limited by a spin-specific noise source such as hyperfine interactions withthe spinful nuclei of indium and arsenic (though nuclearbaths are typically lower frequency than the measuredratio T E /T R = 2 . Transition V g , c (mV) V g , p (mV) T R (ns) T E (ns) | ↑ q (cid:105) ↔ | ↑ a (cid:105) -166.3 -127.6 n.m. 39 ± ± ± | ↓ q (cid:105) ↔ | ↑ a (cid:105) ± ± ± | ↑ q (cid:105) ↔ | ↑ a (cid:105) ± ± ± In this work, we have demonstrated the first coherentmanipulation of an Andreev spin qubit by driving Ramantransitions of a single quasiparticle spin. In future exper-iments, the fidelity of this process may be improved bythe implementation of a resonant STIRAP protocol [40],which would reduce the necessary pulse amplitude andlength, thereby mitigating the effects of dephasing andquasiparticle de-trapping. The parity dynamics may alsobe suppressed by improved filtering of high-frequency ra-diation [41]. In addition, engineering of the nanowire mir-ror symmetry could result in larger spin-flip drive matrixelements without sacrificing the spin lifetime. Criticalto the demonstration of spin manipulation was our abil-ity to perform single-shot, cQED readout, which inher- ently relies on substantial coupling between the Joseph-son nanowire and a microwave resonator. In future exper-iments, this coupling could be used to achieve long-rangeinteraction between separate qubits, a worthy goal in spinqubit research [12–14, 16, 18]. The nature of the spinfulbath limiting the quasiparticle spin coherence may be elu-cidated by the application of magnetic field [10], whichcould also push the Josephson nanowire through a topo-logical phase transition [29–33]. In general, a detailedunderstanding of the effect of magnetic field on quasipar-ticle dynamics will be critical for further progress.We thank Gijs de Lange for assistance with device de-sign, and thank Nick Frattini and Vladimir Sivak forproviding us with a SNAIL parametric amplifier. Weare grateful to Marcelo Goffman, Cyril Metzger, HuguesPothier, Leandro Tosi, and Cristi´an Urbina for sharingtheir experimental results and hypotheses. We acknowl-edge useful discussions with Nick Frattini, Luigi Frunzio,Leonid Glazman, Manuel Houzet, Pavel Kurilovich, VladKurilovich, and Charles Marcus.This research was supported by the US Office of NavalResearch (N00014-16-1-2270) and by the US Army Re-search Office (W911NF-18-1-0020, W911NF-18-1-0212and W911NF-16-1-0349). D.B. acknowledges support byNetherlands Organisation for Scientific Research (NWO)and Microsoft Corporation Station Q. J.C. acknowledgesthe support from MICINN (Spain) (“Beatriz Galindo”Fellowship BEAGAL18/00081). J.N. acknowledges sup-port from the Danish National Research Foundation.Some of the authors acknowledge the European Union’sHorizon 2020 research and innovation programme for fi-nancial support: A.G received funding from the Euro-pean Research Council, grant no. 804988 (SiMS), andA.G., A.L.Y., J.C., and J.N. further acknowledge grantno. 828948 (AndQC) and QuantERA project no. 127900(SuperTOP). A.L.Y. acknowledges support by SpanishMICINN through grants FIS2017-84860-R and throughthe “Mar´ıa de Maeztu” Programme for Units of Excel-lence in R&D (Grant No. MDM-2014-0377).
Contributions
M.H., V.F., K.S., D.B., T.C., A.G., and M.D. designedthe experimental setup. P.K. and J.N. developed thenanowire materials. D.B. and A.G. fabricated the de-vice. M.H. and V.F. performed the measurements. V.F.,M.H., J.C. and A.L.Y. developed the symmetry analysisand microscopic modeling. M.H., J.C., V.F., and A.L.Y.developed and performed the Raman simulations. M.H.,V.F., K.S., S.D., and M.D. analyzed the data. M.H.,V.F., and M.D. wrote the manuscript with feedback fromall authors. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected][1] C. Beenakker and H. Van Houten, Phys. Rev. Lett. ,3056 (1991).[2] A. Furusaki and M. Tsukada, Phys. Rev. B , 10164(1991). [3] J. Clarke and A. I. Braginski, eds., The SQUID Handbook ,vol. 1 (Wiley, Weinheim, 2004).[4] M. H. Devoret and R. J. Schoelkopf, Science , 1169(2013).[5] A. Roy and M. Devoret, Comptes Rendus Physique ,740 (2016).[6] N. M. Chtchelkatchev and Y. V. Nazarov, Phys. Rev.Lett. , 226806 (2003).[7] C. Padurariu and Y. V. Nazarov, Phys. Rev. B , 144519(2010).[8] A. A. Reynoso, G. Usaj, C. A. Balseiro, D. Feinberg, andM. Avignon, Phys. Rev. B , 214519 (2012).[9] S. Park and A. L. Yeyati, Phys. Rev. B , 125416 (2017).[10] R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha,and L. M. K. Vandersypen, Rev. Mod. Phys. , 1217(2007).[11] L. Childress and R. Hanson, MRS Bull. , 134 (2013).[12] K. D. Petersson, L. W. McFaul, M. D. Schroer, M. Jung,J. M. Taylor, A. A. Houck, and J. R. Petta, Nature ,380 (2012).[13] N. Samkharadze, G. Zheng, N. Kalhor, D. Brousse,A. Sammak, U. Mendes, A. Blais, G. Scappucci, andL. Vandersypen, Science , 1123 (2018).[14] X. Mi, M. Benito, S. Putz, D. M. Zajac, J. M. Taylor,G. Burkard, and J. R. Petta, Nature , 599 (2018).[15] S. P. Harvey, C. G. L. Bøttcher, L. A. Orona, S. D.Bartlett, A. C. Doherty, and A. Yacoby, Phys. Rev. B , 235409 (2018).[16] A. J. Landig, J. V. Koski, P. Scarlino, U. Mendes,A. Blais, C. Reichl, W. Wegscheider, A. Wallraff, K. En-sslin, and T. Ihn, Nature , 179 (2018).[17] T. Cubaynes, M. R. Delbecq, M. C. Dartiailh, R. As-souly, M. M. Desjardins, L. C. Contamin, L. E. Bruhat,Z. Leghtas, F. Mallet, A. Cottet, et al., npj Quantum Inf. , 1 (2019).[18] F. Borjans, X. Croot, X. Mi, M. Gullans, and J. Petta,Nature , 195 (2020).[19] A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, andR. J. Schoelkopf, Phys. Rev. A , 062320 (2004).[20] A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S.Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J.Schoelkopf, Nature , 162 (2004).[21] C. Janvier, L. Tosi, L. Bretheau, C¸ . ¨O. Girit, M. Stern,P. Bertet, P. Joyez, D. Vion, D. Esteve, M. F. Goffman,et al., Science , 1199 (2015).[22] M. Hays, G. de Lange, K. Serniak, D. van Woerkom,D. Bouman, P. Krogstrup, J. Nyg˚ard, A. Geresdi, andM. Devoret, Phys. Rev. Lett. , 047001 (2018).[23] P. Krogstrup, N. L. B. Ziino, W. Chang, S. M. Albrecht,M. H. Madsen, E. Johnson, J. Nyg˚ard, C. M. Marcus,and T. S. Jespersen, Nat. Mater. , 400 (2015).[24] W. Chang, S. Albrecht, T. Jespersen, F. Kuemmeth,P. Krogstrup, J. Nyg˚ard, and C. M. Marcus, Nature nan-otechnology , 232 (2015).[25] D. J. van Woerkom, A. Proutski, B. van Heck,D. Bouman, J. I. V¨ayrynen, L. I. Glazman, P. Krogstrup,J. Nyg˚ard, L. P. Kouwenhoven, and A. Geresdi, Nat.Phys. , 876 (2017).[26] L. Tosi, C. Metzger, M. Goffman, C. Urbina, H. Pothier,S. Park, A. L. Yeyati, J. Nyg˚ard, and P. Krogstrup, Phys.Rev. X , 011010 (2019).[27] T. W. Larsen, K. D. Petersson, F. Kuemmeth, T. S.Jespersen, P. Krogstrup, J. Nyg˚ard, and C. M. Marcus,Phys. Rev. Lett. , 127001 (2015).[28] G. De Lange, B. Van Heck, A. Bruno, D. Van Woerkom,A. Geresdi, S. Plissard, E. Bakkers, A. Akhmerov, andL. DiCarlo, Phys. Rev. Lett. , 127002 (2015).[29] L. Fu and C. L. Kane, Phys. Rev. Lett. , 096407(2008). [30] R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Phys. Rev.Lett. , 077001 (2010).[31] Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett. , 177002 (2010).[32] V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P.A. M. Bakkers, and L. P. Kouwenhoven, Science ,1003 (2012).[33] M. Deng, S. Vaitiek˙enas, E. B. Hansen, J. Danon, M. Lei-jnse, K. Flensberg, J. Nyg˚ard, P. Krogstrup, and C. M.Marcus, Science , 1557 (2016).[34] M. Hays, V. Fatemi, K. Serniak, D. Bouman, S. Diamond,G. de Lange, P. Krogstrup, J. Nyg˚ard, A. Geresdi, andM. Devoret, Nat. Phys. , 1103 (2020).[35] S. Nadj-Perge, S. Frolov, E. Bakkers, and L. P. Kouwen-hoven, Nature , 1084 (2010).[36] J. R. Johansson, P. D. Nation, and F. Nori, ComputerPhysics Communications , 1234 (2013).[37] J. Van den Berg, S. Nadj-Perge, V. Pribiag, S. Plissard,E. Bakkers, S. Frolov, and L. Kouwenhoven, Phys. Rev.Lett. , 066806 (2013).[38] L. Bretheau, C¸ . Girit, H. Pothier, D. Esteve, andC. Urbina, Nature , 312 (2013).[39] F. K. Malinowski, F. Martins, P. D. Nissen, E. Barnes,(cid:32)L. Cywi´nski, M. S. Rudner, S. Fallahi, G. C. Gardner,M. J. Manfra, C. M. Marcus, et al., Nature nanotechnol-ogy , 16 (2017).[40] J. Cerrillo, M. Hays, V. Fatemi, and A. Levy Yeyati,arXiv preprint: 2012.07132 (2020).[41] K. Serniak, S. Diamond, M. Hays, V. Fatemi, S. Shankar,L. Frunzio, R. Schoelkopf, and M. Devoret, Phys. Rev.Applied , 014052 (2019).[42] Frattini, N. E., Sivak, V. V., Lingenfelter, A., Shankar, S.& Devoret, M. H., Phys. Rev. Applied , 054020 (2018).[43] M. Governale and U. Z¨ulicke, Phys. Rev. B , 073311(2002).[44] E. Levenson-Falk, F. Kos, R. Vijay, L. Glazman, andI. Siddiqi, Phys. Rev. Lett. , 047002 (2014).[45] J. M. Martinis, S. Nam, J. Aumentado, K. M. Lang, andC. Urbina, Phys. Rev. B , 094510 (2003).[46] A. A. Houck, J. Koch, M. H. Devoret, S. M. Girvin, andR. J. Schoelkopf, Quantum Information Processing , 105(2009).[47] W. Press, Numerical Recipes in C++ : The Art of Sci-entific Computing (Cambridge University Press, 2002).
Supplementary Information
I. SYMMETRY ANALYSIS OF PSEUDOSPIN-FLIP MATRIX ELEMENTS
In this section we motivate a simple physical picture to explain the qualitative features of the driven spin-flippingtransitions. First, note that while an electron spin does in principle couple to the magnetic field of our microwavedrive, this coupling is extremely weak. The electric field of the drive, on the other hand, only couples to motionaldegrees of freedom, so it cannot induce spin-flip transitions if spin is a good quantum number. Spin-orbit coupling isthe conventional invocation to solve this problem since it hybridizes spin and translational wavefunction componentsinto what is often referred to as pseudospin [35]. We will use symmetry considerations to show that this conversion topseudospin is necessary but insufficient to allow electric fields to flip the quasiparticle pseudospin in our system, andthat an additional broken spatial symmetry is required. Following this we further validate these ideas by inspectionof a tight-binding model that specifically incorporates the physics and energy scales of Andreev levels.
A. General Considerations
We are interested in how transitions between the Andreev levels of the Josephson nanowire are induced by ourmicrowave drive voltage V d cos ω d t . Here ω d is the drive frequency and V d the spatial profile. Initially, we will modelthe spatial profile as a purely longitudinal differential voltage along the Josephson nanowire V d ( x ) ≈ − V d ( − x ), whichis a reasonable starting point given our highly symmetric device design [Fig. S3]. The transition rates will dependboth on the matrix element of V d between the initial/final states, as well as on the mismatch between ¯ hω d and theenergy difference between the initial/final states. We first focus on matrix element considerations, initially assumingthat Φ = 0 such that the system is time-reversal symmetric, before generalizing to any value of Φ.Let’s begin by imagining the Josephson nanowire as a quasi-1d system described by a Hamiltonian H . ThisHamiltonian includes both the superconducting leads and the semiconductor nanowire, though for the moment weneglect spin-orbit coupling. Additionally, let us suppose that the system is rotationally invariant about the x -axis.As with any spin-1/2 system with time-reversal symmetry, the energy levels are paired into spin-degenerate doublets(Kramers theorem). In order to achieve the Raman process investigated in this experiment, it was necessary tosimultaneously drive spin-conserving and spin-flipping inter-doublet transitions. Under our current model of thesystem, while V d can induce spin-conserving inter-doublet transitions by coupling to the spatial character of thewavefunctions, it cannot induce spin-flipping transitions because both H and V d are block-diagonal in spin (they arespin-rotation-symmetric).Spin-orbit coupling can help solve this problem by mixing spin and motional degrees of freedom. We can see thisin the form of a Rashba interaction generated by a static electric field in the z direction H R = iE z γ ( σ x ∂ y − σ y ∂ x ),where σ i are Pauli matrices of the spin and γ is a material parameter. Thus, in the presence of spin-orbit interaction,the quasiparticle “spin” is actually a pseudospin. We stress that a hybridization like this must be present in oursystem, as it is critical to break the Andreev-doublet degeneracy [8, 26, 43]. Upon including a Rashba interaction, theHamiltonian H = H + H R is no longer block-diagonal in spin, and one might imagine that V d could flip pseudospin.However, a selection rule prevents this. While the static electric field in the z direction associated with the Rashbaeffect breaks rotational symmetry, a transverse mirror symmetry in the y direction remains [see main text Fig. 1(c)].This mirror symmetry is described by the operator M y = − iσ y δ y, − y , where δ y, − y sends y to − y . Because [ H, M y ] = 0,the energy eigenstates are also mirror eigenstates. Moreover, as spin is flipped under time-reversal T σ y T − = − σ y , themirror eigenvalue is also flipped. Therefore, the pair of pseudospins comprising each doublet have mirror eigenvalues+ i and − i (this remains true in the presence of a time-reversal-symmetry-breaking phase-bias, as explained below).As our longitudinal drive V d ( x ) ≈ − V d ( − x ) also obeys the mirror symmetry [ V d , M y ] = 0, it cannot induce transitionsbetween states of different mirror eigenvalue (pseudospin flips).To induce pseudospin-flip transitions, the mirror symmetry must be broken. In the real device, this symmetry isbroken by the epitaxial aluminum shell which covers two of six nanowire facets, the presence of the side gates andtheir applied voltages [main text Fig. 1(c)], as well as any non-idealities of the device. As such, the symmetry maybe broken in any/all of the Hamiltonian terms: • [ H , M y ] (cid:54) = 0: asymmetric superconducting leads, gate-electrode perturbations to the transverse confinement • [ H R , M y ] (cid:54) = 0: modified Rashba interaction due to non-symmetric electric field profiles • [ V d , M y ] (cid:54) = 0: drive applied via asymmetric leads and perturbed by presence of the metallic gateUnder this broken mirror symmetry, both the pseudospin-conserving transition | ↑ q (cid:105) ↔ | ↑ a (cid:105) and the pseudospin-flipping transition | ↓ q (cid:105) ↔ | ↑ a (cid:105) are allowed and therefore the Raman process can be driven.Thus far in this section, we have assumed Φ = 0 such that T HT − = H . Because the drive also obeys time-reversalsymmetry T V d T − = V d , the direct pseudo-spin transition | ↓ q (cid:105) ↔ | ↑ q (cid:105) is thus forbidden for Φ = 0. While thisfeature is ideal for the use of the Andreev doublets as a Λ system, in this experiment it was necessary to break thedoublet degeneracy such that | ↓ q (cid:105) ↔ | ↑ a (cid:105) and | ↑ q (cid:105) ↔ | ↑ a (cid:105) were frequency-resolved. We therefore performed mostmeasurements at a nonzero flux bias (Φ ≈ − . ) such that we could drive the inter-doublet transitions as well asdetect the pseudospin state of the lower doublet. As we investigate numerically in the next section, we expect thedirect spin-flip transition to remain suppressed even for nonzero Φ. Note that as long as the additional time-reversal-breaking terms in H respect the mirror symmetry, as is the case for a dc phase applied to mirror-symmetric leads, allthe inter-doublet transition considerations presented above for Φ = 0 automatically generalize to the case of nonzeroΦ. In particular, one state of each doublet has mirror eigenvalue + i and the other − i , independent of Φ. B. Numerical Tight-Binding Model
We now explore these symmetry considerations numerically using a tight-binding model. We begin by investigatingspin-orbit hybridization and broken mirror symmetry in the band structure of an un-proximitized nanowire, beforemoving on to the Andreev levels themselves.
1. Unconfined Normal Channels
Our model of the un-proximitized nanowire is graphically represented in Fig. S1(a). It consists of two coupledinfinite parallel chains aligned along the x -direction, and with Rashba spin-orbit coupling: H = (cid:88) i,τ,σ ( (cid:15) i,τ − µ ) c † i,τ,σ c i,τ,σ + t x c † i,τ,σ c i +1 ,τ,σ + σα x c † i,τ,σ c i +1 ,τ, ¯ σ + H.c.+ (cid:88) i,σ t y c † i, ,σ c i, ,σ + iα y c † i, ,σ c i, , ¯ σ + H.c. (1)Here the operators c † i,σ,τ create an electron on the site i , within the channel τ = 1 , σ . The scalar hoppingstrengths { t x , t y } correspond to the { longitudinal, transverse } directions, while the Rashba hopping strengths aregiven by { α x , α y } . The strengths are scaled by the discretization of a continuous Hamiltonian with a longitudinallattice parameter a = 20 nm and width W = 100 nm, giving t x = ¯ h / ( m ∗ a ), α x = α/ (2 a ), t y = ¯ h / ( m ∗ W ) and α y = α/ (2 W ), where m ∗ = 0 . m e is the effective mass for InAs and α = E z γ = 25 meV · nm is the Rashbaparameter. In the mirror-symmetric case, the on-site energies (cid:15) i,τ are given by 2 t x . Momentarily, we will breakthis mirror symmetry by including an energy offset V A,y between the two chains such that (cid:15) ,τ → (cid:15) ,τ + V A,y / (cid:15) ,τ → (cid:15) ,τ − V A,y /
2. Below, we will express all energies as fractions of the superconducting gap of bulk aluminum∆ = 0 .
185 meV.------------------------
FIG. S1: (a) Visual representation of the tight-binding model of the un-proximitized infinite nanowire. (b) Band structureof the tight-binding model for V A,y = 0. Bands split by the longitudinal Rashba term (gray dashed lines) undergo avoidedcrossings due to the transverse Rasbha term to give the final bands of the model (colored lines). Bands colored in blue areanti-symmetric eigenstates of M y , and bands colored in red are symmetric eigenstates. The black star marks the slow, positivemomentum Fermi point, the transverse wavefunction components of which are plotted in (c) for the values of µ indicated by thegray strip. Only the anti-symmetric amplitudes a − and b − are nonzero. As the chemical potential is tuned through the avoidedcrossing between the bands, weight shifts from | ⇐ , S (cid:105) to | ⇒ , A (cid:105) (see Eqn. S3). (d) For V A,y = 2∆, the bands are qualitativelysimilar to (b), but are no longer mirror eigenstates. As shown in (e), the wavefunction of the slow, positive-momentum Fermipoint has both symmetric and anti-symmetric components.
First, we consider the mirror-symmetric case V A,y = 0. The model has four bands [Fig. S1(b)]: the spatial characterof the transverse wavefunction can either be symmetric | S (cid:105) (lower energy) or anti-symmetric | A (cid:105) (higher energy),while the spin can be either | ⇒(cid:105) or | ⇐(cid:105) in the y -direction. The longitudinal Rashba term iα x c † i,τ,σ c i +1 ,τ, ¯ σ + H.c.generates momentum split-bands [gray dashed lines in Fig. S1(b)], but spin in the y -direction remains a good quantumnumber. However, the transverse part of the Rashba term iασ x ∂ y → iα y c † i, ,σ c i, , ¯ σ +H.c. generates an avoided crossingbetween the lower-energy, transverse-symmetric band and the higher-energy, transverse-anti-symmetric band. Thisinter-band mixing results in hybridization between spin and transverse motional degrees of freedom, so that thetransverse character of the resultant low-energy band is given by | + (cid:105) = a + | ⇒ , S (cid:105) + b + | ⇐ , A (cid:105) (2) |−(cid:105) = a − | ⇐ , S (cid:105) + b − | ⇒ , A (cid:105) (3)The numerically-calculated value for the |−(cid:105) coefficients is shown in Fig. S1(c). As µ is swept through the avoidedcrossing, weight shifts from | ⇐ , S (cid:105) to | ⇒ , A (cid:105) , with maximal hybridization occurring around µ ≈ − . M y |±(cid:105) = ± i |±(cid:105) . As such,a mirror-symmetric drive cannot induce transitions between them.To model a broken mirror symmetry, we set a non-zero inter-chain potential difference V A,y = 2∆. The bands forthis case are depicted in Fig. S1(d). While the change in the energy dispersion relation is subtle, we can see the effectof the broken mirror symmetry in the wavefunction character [Fig. S1(e)]. The band near the avoided crossing nowhas character of all four basis states | ⇒ , S (cid:105) , | ⇐ , S (cid:105) , | ⇒ , A (cid:105) , | ⇐ , A (cid:105) . There are thus no selection rules forbiddingtransitions induced by a mirror-symmetric drive.
2. Andreev Levels
Now we confine the normal region between two superconducting leads with a pair potential ∆ and a differentchemical potential µ S [Fig. S2(a)]. The total number of sites in each of the two chains is N . The superconductingphase difference is applied within the gauge of the t x hopping elements in the middle of the chain, i.e. between sites N/ N/ H = (cid:88) i,τ,σ ( (cid:15) i,τ − µ i ) c † i,τ,σ c i,τ,σ + t x c † i,τ,σ c i +1 ,τ,σ − α x c † i,τ,σ c i +1 ,τ, ¯ σ + H.c.+ (cid:88) i,τ ∆ i c i,τ, ↓ c i,τ, ↑ + H.c. + (cid:88) i,σ t y c † i, ,σ c i, ,σ + iα y c † i, ,σ c i, , ¯ σ + H.c. (4)where µ i = µ S for the sites in the superconducting leads and µ i = µ for the normal region as in the previous section.We simulate the Hamiltonian Eqn. S(4) in Nambu space, fixing µ S = 1 . N = 34, and the number of sites in eachlead N leads = 6. We have found that while N leads = 6 is large enough for the qualitative investigation we present here,more lead sites are necessary if quantitative accuracy is desired.We examine both the mirror-symmetric case V A,y = 0 and the non-symmetric case V A,y = ∆. We note that we apply V A,y only to the sites in the normal region N leads < i < N − N leads . For both the mirror-symmetric and non-symmetriccases, we find Andreev levels with qualitatively similar energies [Fig. S2(b/c)] and therefore similar inter-doublettransition frequencies [Fig. S2(d/e)]. For V A,y = 0, the Andreev levels are eigenstates of the mirror operator just asin the states of the infinite nanowire.Next, we check the drive matrix elements. We model the drive voltage profile as a linear potential drop betweenthe leads V d ( x ) = L V x , choosing V = 2∆. For the mirror-symmetric case, only mirror-preserving transitions canbe driven regardless of flux [Fig. S2(f)] and chemical potential [Fig. S2(h)], as expected. For V A,y = ∆, we indeedfind that all transitions are allowed. However, while the inter-doublet transitions are finite for all values of Φ, thedirect spin-flip matrix element goes to zero at Φ → , Φ / FIG. S2: (a) Visual representation of the tight-binding model, now including superconducting leads. (b-h) For V A,y = 0 , ∆, wecompute the Andreev level energies (b,c), all transition frequencies (d,e), all V d matrix elements versus Φ (f,g), and V d matrixelements (drive amplitude V = 2∆) versus µ involving the lowest energy level (h,i). For V A,y = 0, states are colored by theirmirror eigenvalue as before: red for +1 and blue for −
1. For V A,y = ∆, states are colored the same as in main text Fig. 1(a).Inter-doublet transition frequencies and matrix elements take the same color as the participating lower doublet state, whilethose for the intra-doublet transition are grey. Thick dashed lines correspond to pseudospin-conserving transitions, while thindashed lines correspond to pseudospin-flipping transitions. Vertical dashed black lines in (f)/(g) correspond to Φ for (h)/(i),while vertical dashed black lines in (h)/(i) correspond to µ for (f)/(g). II. EXPERIMENT SCHEMATIC
FIG. S3: Cryogenic wiring diagram and device micrographs (see Extended Data of Ref. [34] for original publication). Opticalmicrograph (e) is of the device on which the presented measurements were performed. Optical micrographs (b), (c), (d) andscanning electron micrograph (f) are of an extremely similar (unmeasured) device, the main difference being that the lengthof the weak length is 750 nm instead of 500 nm. The microwave readout and drive tones pass through the depicted circuitry(a) before being routed through the ∆ port of a 180 ◦ hybrid resulting in differential microwave voltages at the device input.After reaching two coupling capacitors (c), the readout tone was reflected off the differential ∼ λ/ f r = 9 . κ c = 2 π × .
23 MHz, internal loss κ i = 2 π × .
00 MHz) and then routedthrough the depicted amplification chain (a), which was comprised of a SNAIL parametric amplifier (SPA) [42], HEMT, androom-temperature amplifiers. In this circuit, the drive tone creates an ac phase drop across the nanowire (f), which is embeddedin the superconducting Φ-bias loop (green) at the end of the resonator (d,e). One edge of the loop connects the two stripsof the resonator and thereby forms the shared inductance with the nanowire. We controlled the electrostatic potential in thenanowire weak link (f) with a dc gate (pink, voltage V g , c ). Gates on the nanowire leads (orange) were used to gain additionalelectrostatic control (voltage V nw ). To reference the resonator/nanowire island to ground, an additional strip runs between theresonator strips, and connects to a large finger capacitor (purple). This strip does not significantly perturb the resonator’smicrowave properties because it resides at the zero voltage point with respect to the resonator’s differential mode. III. TUNING UP THE DEVICE
In this experiment, we possessed three in-situ control knobs of the nanowire Andreev levels: a loop flux Φ [Fig. S3(d,e)], a main gate voltage V g , c acting on the nanowire weak link [Fig. S3(f)], and an additional gate voltage V g , p appliedto two more gates positioned on either side of the main gate [Fig. S3(f)]. Upon cooling down the device, we observedΦ and gate voltage dependence of the readout resonance around V g , c , V g , p = 0, indicating that conduction channelsin the nanowire link were transmitting. With Φ = − . and V g , p = 0, we swept V g , c while performing two-tonespectroscopy [Fig. S4(a)]. We observed several dispersing transitions, with a local maximum at V g , c = − . V g , c at the local maximum to mitigate the effects of electrostatic noise on the Andreev level coherence (seebelow for further data and discussion), we then performed two-tone spectroscopy while sweeping Φ [Fig. S4(b)]. Fourflux-dependent resonances were observed, which cross at Φ = 0. This is characteristic of inter-doublet transitions of aquasiparticle between spin-orbit split Andreev levels [26]. In conjunction with the population transfer measurementsshown in Fig. 2(a) of the main text, this characteristic spectrum allowed us to identify the two lowest-frequencytransitions as | ↑ q (cid:105) ↔ | ↑ a (cid:105) and | ↓ q (cid:105) ↔ | ↑ a (cid:105) .At certain Φ bias points, some of the transitions are not visible, or become significantly dimmer. For example,the | ↑ q (cid:105) ↔ | ↑ a (cid:105) transition is barely visible at Φ = (cid:39) − . . This drop in signal occurs because the quasiparticlepopulation of the relevant level of the lower doublet (either | ↓ q (cid:105) or | ↑ q (cid:105) ) decreases, often below 0.01. We attributethese population drops to evacuation of the quasiparticle into cold, dot-like levels in the nanowire that are broughtinto resonance with the Andreev doublets as Φ, V g , c , and V g , p are varied. While these features are not completelyunderstood, we found they could be easily avoided with an appropriate choice of bias conditions. The effects of thesepopulation drops can be observed in Figs. SS4 and SS5.Having identified the transitions that defined the Λ system, we searched for a local maximum of the transitionsin both gate voltages in order to mitigate electrostatic noise. We found such a bias point at V g , c = − . V g , p = 4 . FIG. S4: Gate and flux dependence of the inter-doublet transitions. (a) A local maximum (“sweet spot”) is observed in both | ↑ q (cid:105) ↔ | ↑ a (cid:105) and | ↓ q (cid:105) ↔ | ↑ a (cid:105) at V g , c = − . V g , c = − . V g , p = 0 . | ↑ q (cid:105) ↔ | ↑ a (cid:105) and | ↓ q (cid:105) ↔ | ↑ a (cid:105) at the bias point for all data presented in the main text, excludingFig. 4 ( V g , c = − . V g , p = 4 . IV. SEARCHING FOR RAMAN TRANSITIONS
With the gate voltages set to the optimum values discussed above and Φ = − . , we applied simultaneousGaussian drive pulses of variable carrier frequency, as explained in the main text and Fig. 2(b). Here we presentall measured transition probabilities, and over a wider frequency range than in the main text [Fig. S6]. The drivepowers used in this measurement were 30 dB larger than in the two-tone spectroscopy measurement of Fig. 2(a). The | ↓ q (cid:105) ↔ | ↑ a (cid:105) transition is visible (though broadened) just above its low-power value, while the | ↑ q (cid:105) ↔ | ↑ a (cid:105) transitionis no longer visible. Other transitions that are constant in one drive frequency or the other are observed, perhaps dueto evaporation of the quasiparticle into the same dot-like levels as discussed above. Several multi-photon transitionsare observed that lie along f ↑ = − f ↓ + c . The − | ↓ q (cid:105) ↔ | ↑ q (cid:105) occurs along the blackdashed line, which has slope 1 and intersects with the crossing point of the two single-photon transitions of the Λsystem (unlike the data shown in Fig. 2(b) of the main text). This measurement was taken between the measurementsdisplayed in Fig. 2(b) and 2(a) of the main text. We thus conclude that there was an uncontrolled shift of the Andreevlevels between the measurement shown Fig. S6 and Fig. 2(b), as referenced in the main text. Such jumps were notuncommon in this experiment, and occurred on a timescale of days to weeks. FIG. S6: All transition probabilities under the action of simultaneous drive pulses of variable carrier frequencies, as in Fig.2(b) of the main text. Measured transition frequencies of | ↑ q (cid:105) ↔ | ↑ a (cid:105) and | ↓ q (cid:105) ↔ | ↑ a (cid:105) are indicated by pink and purpledashed lines, respectively. Black dashed lines have a slope of one, and run through the crossing point of the two transitions. V. COHERENT SPIN DYNAMICS
Using an experiment similar to that shown in Fig. S6, we chose drive frequencies such that the desired Ramanprocess | ↓ q (cid:105) ↔ | ↑ q (cid:105) was being driven, but the drives were maximally detuned from undesired processes. We thenvaried the amplitudes A ↓ , A ↑ to induce coherent oscillations of the quasiparticle spin [main text, Fig. 3(a)]. To betterunderstand these coherent dynamics, we performed a simulation of our system using QuTiP [36] [main text, Fig. 3(b)].Here we present transition probabilities out of the two spin states under the action of these variable amplitude drivepulses, both measured [Fig. S7(a)] and simulated [Fig. S7(b)] (measurements where the system started in | g (cid:105) showedno features).The simulation of the coherent dynamics included all four Andreev levels | ↓ q (cid:105) , | ↑ q (cid:105) , | ↑ a (cid:105) , and | ↓ a (cid:105) [Fig. S7(c)].While there were only two drives applied in this experiment, because each drive could couple to each of the fourinter-doublet transitions we needed to account for a total of eight Hamiltonian terms. Four of these terms producedtwo Raman processes [dashed double-headed arrows in Fig. S7(c)], one of which was via | ↑ a (cid:105) as desired, and theother was via | ↓ a (cid:105) . The | ↑ a (cid:105) Raman process dominated the dynamics, as the detuning of the drives from | ↑ a (cid:105) was∆ R = −
290 MHz, as compared to the detuning to | ↓ a (cid:105) which was ∆ (cid:48) R = 1 .
36 GHz. The other four Hamiltonian termsproduced Stark shifts [thin solid double-headed arrows in Fig. S7(c)].The fixed parameters in the simulation were the four inter-doublet transition frequencies, the measured dephasingrates of both doublets, the pulse length and shape (Gaussian, 40 ns standard deviation), and the detunings ∆ R = −
290 MHz, ∆ (cid:48) R = 1 .
36 GHz. We fit the simulation to the data using six free parameters: the four drive matrixelements associated with the four inter-doublet transitions ( M ↓ , ↑ , M ↑ , ↑ , M ↓ , ↓ , M ↑ , ↓ ), the detuning δ from the Ramanresonance condition, and the ratio α of the f ↑ drive amplitude to the f ↓ drive amplitude. From the fit, we extract thebelow values and associated co-variance matrix: δ/ (2 π ) = 5 . M ↑ , ↑ / (2 π ) = 232 MHz M ↓ , ↑ / (2 π ) = 255 MHz M ↓ , ↓ / (2 π ) = 280 MHz M ↑ , ↓ / (2 π ) = 80 MHz α = 0 . C = +0 .
02 +0 . − .
09 +0 . . . .
01 +70 − −
200 +200 +0 . − . −
50 +30 +100 − − . . −
200 +100 +500 − − . . − −
40 +400 +0 . . . − . − .
05 +0 .
06 +0 . (5)Note that the extracted values of the matrix elements include the drive amplitudes at the device, which we estimateto be ∼
400 nV across the junction. We note that, in the experimental data, the oscillation minima/maxima do notlie perfectly along | A ↑ | = | A ↓ | ; the inclusion of δ in the fit is necessary to reproduce this. The inclusion of the Starkshift terms, on the other hand, were not needed to reproduce the data, but in reality they must be present. FIG. S7: Coherent Λ-Rabi oscillations of the quasiparticle spin, as in Fig. 3 of the main text. (a) Measured final stateprobabilities after the application of simultaneous drive pulses, with the spin initialized in either | ↓ q (cid:105) or | ↑ q (cid:105) . (b) Simulatedprobabilities. (c) Level diagram with simulation parameters. Additionally, as can be seen in Fig. S7, we observed a drive-induced quasiparticle evaporation rate; the | g (cid:105) popu-lation grows as the drive amplitudes are increased. In simulation, we found that this quasiparticle evaporation wascaptured by including two dissipators on the lower doublet of the form (cid:112) Γ de − trap ( | A ↓ | + | A ↑ | + | A ↓ || A ↑ | ) | g (cid:105)(cid:104)↓ q | , (cid:112) Γ de − trap ( | A ↓ | + | A ↑ | + | A ↓ || A ↑ | ) | g (cid:105)(cid:104)↑ q | , with Γ de − trap / (2 π ) = 1 . ± . | g (cid:105) populationdata. While this is certainly an over-simplified model (no frequency dependence, no spin dependence, etc.), the scalingof the rate with the drive amplitudes indicates that the evaporation is likely due to multi-photon transitions of thetrapped quasiparticle to excited states either in the dot-like levels previously discussed or into the continuum above thesuperconducting energy gap of the leads. This differs with previous results on drive-induced quasiparticle evaporationwhere a linear scaling of the rate with power was observed [44], most likely because the Andreev levels studied in thiswork exist at lower energy. These results are also consistent with our observation that the undesired transitions seenin Fig. S7 only occur at high powers.0Finally, we included readout errors in the simulation by extracting the transition probabilities between the outcomesof the first and second readout pulses for the zero-amplitude experimental data and then applying the resultant transfermatrix to the simulated probabilities for all drive amplitudes. While this did not result in a qualitative change ofthe data, it was important for replication of the experimental contrast. Note that this is why there are some Ramanfeatures visible in the simulated | g (cid:105) data: the rate at which the quasiparticle spontaneously evacuates the junctionduring readout is slightly spin-dependent. Although we were unable to measure the upper doublet population directlybecause the dispersive shift of these states was too small at this bias point, the simulation indicates that it was below20%. VI. ANALYSIS OF THE ANDREEV LEVEL COHERENCE TIMES
As discussed in the main text, we observed no dependence of the quasiparticle spin T on any of the in-situ biasknobs (Φ , V g , c , V g , p ). However, we did find that the coherence times of the inter-doublet transitions depended on V g , c [Fig. S8(a)]. In particular, we found that the T of the | ↑ q (cid:105) ↔ | ↑ a (cid:105) was maximum around a V g , c sweet spotat − . T todrop. We model this behavior using the relation for exponential coherence decay T = (cid:0) πV rms dfdV g , c (cid:1) + Γ c , where V rms = 0 . ± .
01 mV is the effective root-mean-square voltage noise and Γ c = 0 . ± .
001 ns − is a V g , c -independentdephasing rate [45]. We note that the T at the sweet spot is given by Γ c , as we found that second-order noise couplingto d fdV , c was negligible [46].To calculate the effect of this electrostatic noise on the quasiparticle spin T , we first extracted the V g , c dependenceof the | ↓ q (cid:105) ↔ | ↑ q (cid:105) splitting (cid:15) s , which is given by the difference between the frequencies of the | ↓ q (cid:105) ↔ | ↑ a (cid:105) and | ↑ q (cid:105) ↔ | ↑ a (cid:105) transitions. We visualize this in Fig. S8(b) by plotting the same two-tone data as shown in Fig. S8(a), butwith the fitted value of the | ↑ q (cid:105) ↔ | ↑ a (cid:105) transition f ↑↔↓ subtracted from the drive frequency f d at every V g , c bias. The | ↑ q (cid:105) ↔ | ↑ a (cid:105) transition thus lies along f d − f ↑↔↓ = 0 (pink horizontal line), while the V g , c dependence of (cid:15) s /h is givenby the behavior of f ↓↔↓ (purple horizontal line). We observe that (cid:15) s /h has no discernible slope with V g , c , consistentwith the lack of a spin T dependence on V g , c . However, using an upper bound on this slope d(cid:15) s /hdV g , c <
32 MHz / mV[black dashed line in Fig. S8(b)] and twice the extracted value of V rms as an upper bound on the electrostatic noise[black dashed line in Fig. S8(a)], we find a lower bound on the spin dephasing time of 4 . µ s. FIG. S8: Extracting a lower-bound on the electrostatic-noise-induced dephasing time of the quasiparticle spin. (a) Samespectroscopy data as shown in Fig. S2(a). A local maximum is observed in both transitions at V g , c = − . | ↑ q (cid:105) ↔ | ↑ a (cid:105) are shown in pink (right axis). White dashed line is a fit to the T data assuming first-order noise in V g , c plus a constant dephasing rate. Black dashed line corresponds to the expected T given this same constantdephasing rate, but twice the V g , c noise. (b) Same data as shown in (a), but with the fitted frequency of the | ↑ q (cid:105) ↔ | ↑ a (cid:105) transition subtracted from f d for each V g , c bias. Both the purple and pink lines have no slope, and lie along the average valuesof the two transitions. The yellow dashed line is an upper bound on the slope of the | ↓ q (cid:105) ↔ | ↑ a (cid:105) transition. VII. ANALYSIS OF THE ANDREEV LEVEL LIFETIMES
To extract the transition rates between | g (cid:105) , | ↓ q (cid:105) , and | ↑ q (cid:105) , we analyzed the quantum jumps of the system using ahidden Markov model algorithm [Fig. S9] [22, 34, 47]. As we reported in Ref. [34], the spin lifetime T s increased with | Φ / Φ | . Unlike in the data presented in Ref. [34], at this gate bias point the parity lifetime did vary with Φ due to1evacuation of the quasiparticle into the cold, fermionic modes discussed above. In Fig. S9, we plot quantum jumps forΦ = − . where T s = 17 ± µ s and T parity = 22 ± µ s. FIG. S9: Quantum jumps of spin and parity at Φ = − . . (a) Γ histogram resulting from 10 consecutive readout pulses.(b) Plotting Q against time reveals quantum jumps between | g (cid:105) , | ↓ q (cid:105) , and | ↑ q (cid:105) (only a subset of the time trace is shown). Ahidden Markov algorithm was employed to perform state assignment (colored rectangles), as well as to extract all six transitionrates between the three states. These are summarized by the parity lifetime T parity = 22 ± µ s and spin lifetime T s = 17 ± µµ