Direct Microwave Measurement of Andreev-Bound-State Dynamicsin a Semiconductor-Nanowire Josephson Junction

M. Hays,1,* G. de Lange,1,2,3 K. Serniak,1 D. J. van Woerkom,2,3 D. Bouman,2,3 P. Krogstrup,4

J. Nygård,4 A. Geresdi,2,3 and M. H. Devoret1,†1Department of Applied Physics, Yale University, New Haven, Connecticut 06520, USA

2QuTech and Delft University of Technology, 2600 GA Delft, Netherlands3Kavli Institute of Nanoscience, Delft University of Technology, 2600 GA Delft, Netherlands

4Center for Quantum Devices and Station Q Copenhagen, Niels Bohr Institute, University of Copenhagen,Universitetsparken 5, 2100 Copenhagen, Denmark

(Received 22 November 2017; published 23 July 2018)

The modern understanding of the Josephson effect in mesosopic devices derives from the physics ofAndreev bound states, fermionic modes that are localized in a superconducting weak link. Recently,Josephson junctions constructed using semiconducting nanowires have led to the realization of super-conducting qubits with gate-tunable Josephson energies. We have used a microwave circuit QEDarchitecture to detect Andreev bound states in such a gate-tunable junction based on an aluminum-proximitized indium arsenide nanowire. We demonstrate coherent manipulation of these bound states, andtrack the bound-state fermion parity in real time. Individual parity-switching events due to nonequilibriumquasiparticles are observed with a characteristic timescale Tparity ¼ 160� 10 μs. The Tparity of atopological nanowire junction sets a lower bound on the bandwidth required for control of Majoranabound states.

DOI: 10.1103/PhysRevLett.121.047001

The fundamental process governing the physics ofmesoscopic superconductors is Andreev reflection, wherebyelectrons are coherently scattered into holes due to spatialvariation of the superconducting order parameter [1]. Withina Josephson junction conduction channel, Andreev reflectionprocesses constructively interfere to form localized fer-mionic modes known as Andreev bound states (ABS).These modes have energies less than the superconductinggap, and are responsible for the flow of the Josephsonsupercurrent [2,3]. While the phenomenological propertiesof Josephson junctions are widely utilized in superconduct-ing circuits [4–6], they can only be understood in detail byconsidering the underlying ABS.Herewe outline the physics of the lowest-energy ABS of a

Josephson junction, which is spin degenerate with energy EAassuming time-reversal invariance [Fig. 1(a)]. The many-body configurations of this level can be separated into twomanifolds indexed by the parity of fermionic excitations. Theeven-parity manifold is spanned by the ground state jgi anddoubly excited state jei, while the odd-parity manifold isspanned by the singly excited spin-degenerate states jo↓iand jo↑i. As the parity-conserving jgi ↔ jei transitioninvolves only discrete subgap levels, the even manifold isamenable to coherent manipulation by microwave fields atfrequency fA ¼ 2EA=h [7–10]. We thus refer to the evenmanifold as the Andreev qubit. Dynamics between the evenand odd manifolds cannot be controlled, as parity-breakingtransitions result from incoherent quasiparticle exchange

with the continuum of modes in the junction leads [11–13].However, it is possible to observe these quasiparticlepoisoning events by tracking the ABS fermion parity inreal time. The ABS can therefore act as a single-particledetector of the nonequilibrium quasiparticles that plaguesuperconducting devices [14–19]. Experiments revealingthese dynamics have been performed on ABS hosted bysuperconducting atomic contacts [10,11].Advances in the fabrication of superconductor-

proximitized semiconducting nanowires [20,21] haveenabled reliable construction of highly transparent nano-wire Josephson junctions (NWJJ). Because of the lowcarrier density of semiconductors, the conduction channelsof NWJJs can be tuned in situ by electrostatic gates,providing convenient control over the ABS [22,23].Such control has been used to create gate-tunableJosephson elements for superconducting quantum circuits[24,25]. Moreover, high-spin-orbit, large-g-factor NWJJscan, in principle, be tuned into a topological phase in whichthe lowest-energy ABS evolves into a Majorana boundstate (MBS) [26,27]. As poisoning by nonequilibriumquasiparticles will hinder efforts to probe the physics ofMBS [28], monitoring the fermion parity switches of theprecursor ABS is a first step towards understanding andmitigating poisoning in a topological NWJJ.In this Letter, we report the microwave detection and

manipulation of ABS in an aluminum-proximitizedindium arsenide (InAs) NWJJ using circuit quantum

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electrodynamics (cQED) [29,30]. We perform microwavespectroscopy of a gate- and flux-tunable Andreev qubit,and we achieve coherent manipulation of this qubit usingpulsed microwave fields. In addition, we monitor the ABSin real time to directly observe transitions between theeven- and odd-parity manifolds, which we attribute toexchange of nonequilibrium quasiparticles between theABS and the junction leads. These parity-switching eventsare observed to occur with a characteristic timescaleTparity ¼ 160� 10 μs.Our cQED detection scheme hinges on the supercurrent

properties of the ABS. While the even manifold supportsthe flow of supercurrent, the odd manifold does not.To probe the ABS of a NWJJ, we therefore inductivelycoupled the junction to a superconducting microwaveresonator with bare frequency fr [Fig. 1(b)] [10]. With thesystem tuned such that the magnitude of Δ ¼ 2πðfA − frÞis much greater than the coupling strength gc, theinteraction between the resonator and the current-carryingAndreev qubit is well described by a dispersive couplingterm in the Hamiltonian ℏχa†aðjeihej − jgihgjÞ, whereχ ¼ g2c=Δ [29]. This results in a qubit-state-dependentshift by �χ of the resonator frequency when the ABS arein the even manifold, while no shift occurs for thecurrentless odd manifold. By monitoring the resonatorresponse to a microwave readout tone, the quantum state

of the ABS can thus be determined. However, thesefrequency shifts must be resolvable with practical meas-urement integration times. This results in two require-ments on the Andreev spectrum. First, the bandwidth ofthe dispersion of EA with the superconducting phasedifference φ should be maximized, as this sets the scaleof the ABS supercurrent operator and thus the value of gc[7,10]. Second, the minimum of fAðφ ¼ πÞ should betuned close to fr, which can be achieved by adjustingthe transparency τ of the conduction channel hosting thelowest-energy ABS [Fig. 1(c)] [31]. In particular, theconduction channel must be quasiballistic, such that τ canbe tuned close to unity [10].To achieve a high-τ NWJJ, we used an MBE-grown

[001] wurtzite InAs nanowire [Fig. 2(a)] with an epitaxialaluminum (Al) shell [21–23]. The device substrate wascomposed of intrinsic silicon capped with a 300 nm layerof silicon dioxide. First, the readout resonator (fr ¼9.066 GHz, line width κ=2π ¼ 9 MHz) and control struc-tures were patterned by electron-beam lithography and

FIG. 1. Model of ABS coupled to a microwave resonator.(a) Many-body configurations of a spin-degenerate Andreev levelin the excitation representation. A microwave transition (purplearrow) links jgi and jei, while quasiparticle poisoning (graydashed arrows) links the even and odd manifolds. (b) A NWJJ(purple) inductively coupled to microwave readout resonator(orange) via a superconducting loop (green). An externallyapplied flux Φ phase biases the NWJJ. (c) A representativespectrum of the system consists of fAðΦÞ (purple) and theresonator transition (orange). The maximum of fA occurs atΦ ¼ 0 and depends on the geometric and material propertiesof the NWJJ. Enlargement: the minimum of fAðΦÞ occurs atΦ ¼ Φ0=2 and is determined by the channel transparency τ. TheAndreev qubit is coupled with strength gc to the resonator mode,while the odd manifold is decoupled leaving only the bareresonator transition (dashed orange line).

(d)

500 µm

5 µm

200 nm

50 µm

(b) (c)(a)

NbTiN

Alshell

Vg

FIG. 2. Color-enhanced device micrographs and simplifiedexperimental setup. (a) Scanning electron micrograph (SEM)of the InAs NWJJ hosting the ABS. The nanowire was partiallycoated with a 10 nm thick epitaxial Al shell (blue) [21], with a200 nm gap forming the junction. A NbTiN gate (pink) was usedfor electrostatic tuning of τ. (b) SEM of the NWJJ contacted byNbTiN leads (green). (c) Optical micrograph of the inductivecoupling (strength gc=2π ¼ 23 MHz) between the NbTiN loop(green) and the λ=4 coplanar stripline resonator (orange). The topof the loop was capacitively coupled to a microwave drive line[see panel (d)]. (d) Optical micrograph of the full chip. Theresonator was measured using the microwave setup depicted insummary on the left of the figure. A readout tone with frequencyfr (orange arrow) was transmitted through a 180° hybrid,differentially driving the resonator through coupling capacitors(see zoom). The reflected tone was routed through a circulatorand amplified before being processed at room temperature. Thegate was biased with an electrostatic voltage Vg, with aninterdigitated capacitor (green) providing a reference to thedevice ground plane. A microwave drive (purple arrow) wasused to induce transitions between jgi and jei.

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reactive ion etching of sputtered niobium titanium nitride(NbTiN). Then, the nanowire was deposited using amicromanipulator and the junction was defined by selec-tively wet etching a 200 nm long section of the Al shell[Fig. 2(a)]. NWJJs of this length have been shown to hostABS with large phase dispersion [22]. The Al leads of theNWJJ were contacted to the rest of the circuit with NbTiN[Fig. 2(b)]. We implemented control over τ via anelectrostatic gate voltage Vg [Figs 2(a), 2(d)], and weapplied an external flux Φ through a NbTiN loop to phasebias the NWJJ [Fig. 2(c)] [22]. Because the inductance ofthe NWJJ was much greater than that of the NbTiN loop,φ ≃ 2πΦ=Φ0. In contrast with dc transport measurements,the NWJJ was galvanically isolated from all off-chipcircuitry. The large critical fields of NbTiN and thin-filmAl make our devices compatible with high magnetic fieldmeasurements, enabling future experiments in the topo-logical regime [21,32]. The measurements presentedhere were performed in a dilution refrigerator with a basetemperature of ∼30 mK.We first investigated the effects of Vg and Φ on the

device properties. WithΦ ¼ Φ0=2, we monitored the phaseθ of the readout tone (frequency fr) while sweeping Vg

[Fig. 3(a)]. For several ranges of Vg, θ exhibits featuresconsistent with a transition crossing fr [inset Fig. 3(a)]. Weattribute this transition to a gate-controlled Andreev qubit

coupled to the readout resonator [Fig. 1(b)]. The abundanceof features observed in Fig. 3(a) may be explained bymesoscopic fluctuations of the nanowire conductance[22,23,33], with fA crossing fr whenever τ approachesunity [see Fig. 1(c)]. With fAðΦ0=2Þ tuned below fr, thefrequency of the microwave drive [Fig. 2(d)] was sweptto pinpoint the qubit transition [inset Fig. 3(b)]. Repeatingthis measurement at various flux biases revealed strongdispersion of fAðΦÞ [Fig. 3(b)], consistent with recent dcand rf measurements of ABS hosted by high-transparencyconduction channels in InAs/Al NWJJs [22,23]. Under thesimplifying assumption that the coherence length is muchgreater than the junction length, we therefore apply thesimplified formula for the Andreev qubit frequencyfAð0Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 − τ sin2ðπΦ=Φ0Þp

[31] and extract τ ≃ 0.98 andfAð0Þ ≃ 60.0 GHz½EAð0Þ ≃ 124 μeV�. Measurements offA over a wider flux range were impeded by drifts in Vg

bias on minute-to-hour timescales, which we attribute tocharging effects in the dielectric surrounding the nanowire.While these instabilities made systematic studies requiringlong measurement times impossible, they did not inhibitour ability to investigate the fast temporal dynamics ofthe ABS.

FIG. 3. Control of the Andreev qubit frequency. (a) The averagephase θ of the readout tone for a range of Vg values (Φ ¼ Φ0=2).Each point was integrated for 1.28 μs. As Vg is varied, transitionsin the nanowire come into proximity with the resonator fre-quency, resulting in avoided crossings. Inset: enlargement of twoof these avoided crossings of fA with the resonator frequency.(b) Inset: Continuous-wave two-tone spectroscopy reveals thequbit transition. The transition frequency fA is extracted from abest fit to a Lorentzian line shape. Main figure: dependence of fAon Φ. Solid line is a fit to the short-junction formula for fA [31].

FIG. 4. Coherent dynamics of the Andreev qubit. Data arerescaled by the standard deviation σ of the jgi distribution.(a) Rabi oscillations of the Andreev qubit (fA ¼ 9.35 GHz,Φ ¼ Φ0=2). A resonant 10 ns square pulse of varying amplitudeA was applied to the qubit, followed by a readout pulse at frwhich was integrated for 640 ns. Nominal π and π=2 qubitrotations were calibrated by fitting the data to a sinusoid (solidline). (b) Histogram of the Im and Qm quadratures of the readouttone following no qubit rotation (left) and a π rotation. (c) Energyrelaxation of the qubit (fA ¼ 6.84 GHz, Φ ¼ Φ0=2). Fitting to adecaying exponential (solid line) yields a time constantT1 ¼ 12.8� 0.2 μs. (d) Coherence of the qubit measured usinga Hahn-echo pulse sequence. The phase of the final π=2 pulseis varied with the delay to introduce oscillations. Solid lineis a best fit to a Gaussian decaying sinusoid with time constantT2E ¼ 390� 10 ns.

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Next we probed the coherent dynamics of the fAtransition. Figure 4(a) displays Rabi oscillations of theAndreev qubit atΦ ¼ Φ0=2. To verify the effect of the Rabidrive on the ABS, this measurement was performed withhigh photon number n ∼ 100 and small detuning Δ=2π ¼280 MHz. In this regime, the integrated quadraturesðIm;QmÞ of the readout pulse clustered into three well-separated Gaussian distributions [Fig. 4(b)] which weattribute to jgi, jei, and the odd manifold [10]. As expected,the populations of jgi and jei are affected by the Rabi drive,while the population of the odd manifold is mainly constant(see Supplemental Material [34], Sec. I). The energy andcoherence decay of the Andreev qubit were measured atincreased Δ to avoid resonator-induced transitions. Themaximum energy relaxation time T1 ¼ 12.8� 0.2 μs wasmeasured with fA ¼ 6.84 GHz [Fig. 4(c)]. At this workingpoint, the Hahn-echo decay time was found to be T2E ¼390� 10 ns [Fig. 4(d)]. Low-frequency fluctuations in fAresulted in an immeasurably short Ramsey decay time,which we attribute to the gate-bias instabilities. We notethat these energy and coherence decay times are of similarmagnitude to those observed in Andreev qubits hosted bysuperconducting atomic contacts [10], indicating that theloss and dephasing mechanisms may be largely indepen-dent of the junction materials.In addition to studying even-manifold coherence, we

observed incoherent transitions between all of the ABSmany-body configurations by continuously monitoring theresonator while at small detuning Δ=2π ¼ −0.5 GHz andhigh photon number n ∼ 100 [Figs. 5(a),5(b)]. The tran-sition rates between configurations were extracted using ahidden Markov model algorithm [38]. This analysisassumes that the system possesses three states (jgi, jei,and the odd manifold), and that each state i emits ðIm;QmÞpairs with different (but potentially overlapping) proba-bility distributions pðIm;QmjiÞ. Importantly, pðIm;QmjiÞdo not need to be known a priori. By analyzing the timeevolution of ðIm;QmÞ, the algorithm yields the most

probable pðIm;QmjiÞ, state assignments [Fig. 5(b)], andtransition rates of the system [Fig. 5(c)]. The extracted ratesyield a reduced qubit lifetime T1 ¼ 3.2� 0.1 μs, which weattribute to off-resonant driving of the jgi ↔ jei transitionby the high-power readout tone. This is consistent with theobserved nonthermal qubit population [Fig. 5(a)]. Inaddition, the extracted rates yield a parity-decay timescaleTparity ¼ 160� 10 μs (see Supplemental Material [34],Sec. II), which we attribute to spontaneous poisoning ofthe ABS by nonequilibrium quasiparticles. We note thatsince Tparity ≫ T1, the lifetime of the Andreev qubit waslimited by direct jgi ↔ jei processes and not by quasi-particle poisoning.Previous measurements of bound-state poisoning in

proximitized semiconducting nanowires have usedCoulomb blockade spectroscopy to estimate the rate ofquasiparticle relaxation from the superconductor into abound state [39,40]. Our Tparity measurement is distinct inthat we directly monitor the parity of the ABS and aretherefore sensitive to all parity-breaking processes. Tolowest order, the readout tone should not induce parity-breaking transitions, which involve energies on the order ofthe superconducting gap. However, recent measurements ofABS in superconducting atomic point contacts have showndependence of Tparity on n [41]. In future experiments, thedependence of Tparity on n will be measured using aJosephson parametric converter [42].In conclusion, we have detected and manipulated the ABS

of an InAs NWJJ using a cQED approach. We realized agate- and flux-tunable Andreev qubit with maximum coher-ence times T1 ¼ 12.8� 0.2 μs and T2E ¼ 390� 10 ns.Moreover, we achieved continuous monitoring of theABS fermion parity in a NWJJ, revealing that quasiparticlepoisoning of the ABS occurred on a timescale Tparity ¼160� 10 μs. The measurement time of experiments aimingto detect the non-Abelian properties of MBS in a topologicalnanowire must fall within a certain range. The upper boundis set by Tparity, as quasiparticle poisoning of MBS will

g

o

e

0 max

110

2102.3

1.5

3.2

1.8

g

o

e

FIG. 5. Dynamics of incoherent transitions between many-body configurations of the ABS. (a) Histogram of the Im and Qm

quadratures of the readout tone (fA ¼ 8.5 GHz, Φ ¼ Φ0=2). Each of the 9.6 × 105 counts corresponds to an integration period of480 ns. The ðIm; QmÞ-pairs cluster into three Gaussian distributions corresponding to the ABS many-body configurations. Data arerescaled by the standard deviation σ of the jgi distribution. (b) Time evolution of Im=σ for a sample of the data in (a). State assignments(blue, gray, and red bars) result from a maximum-likelihood estimation to a hidden Markov model. (c) Transition rates in ms−1 betweenthe ABS many-body configurations extracted from the hidden Markov model analysis.

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decohere superpositions of quantum states with differentparity. Conversely, adiabatic manipulation of MBS restrictsthe lower bound to nanosecond timescales [28]. Therefore,our measured value of Tparity leaves an experimentallyaccessible window for the investigation of Majorana physics.

We acknowledge fruitful discussions with Nick Frattini,Sergey Frolov, Luigi Frunzio, Leonid Glazman, MarceloGoffman, Bernard van Heck, Leo Kouwenhoven, CharlieMarcus, Hugues Pothier, Leandro Tosi, Cristian Urbina,Jukka Väyrynen, Uri Vool, and Shyam Shankar. Facilitiesuse was supported by Yale Institute for Nanoscience andQuantum Engineering (YINQE), the Yale SEAS clean-room, and NSF MRSEC DMR 1119826. This research wassupported by ARO under Grant No. W911NF-14-1-0011,by MURI-ONR under Grant No. N00014-16-1-2270, byMicrosoft Corporation Station Q, by a Synergy Grant of theEuropean Research Council, and by the Danish NationalResearch Foundation (DG-QDev). G. de L. acknowledgessupport from the European Union’s Horizon 2020 researchand innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 656129. A. G. acknowledgesfunding from the Netherlands Organisation for ScientificResearch (NWO) through a VENI grant.

*max.hays@yale.edu†michel.devoret@yale.edu

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