Magnetic Gradient Fluctuations from Quadrupolar 73 Ge in Si/SiGe Exchange-Only Qubits
J. Kerckhoff, B. Sun, B. H. Fong, C. Jones, A. A. Kiselev, D. W. Barnes, R. S. Noah, E. Acuna, M. Akmal, S. D. Ha, J. A. Wright, B. J. Thomas, C. A. C. Jackson, L. F. Edge, K. Eng, R. S. Ross, T. D. Ladd
MMagnetic Gradient Fluctuations from Quadrupolar Ge inSi/SiGe Exchange-Only Qubits
J. Kerckhoff, B. Sun, B. H. Fong, C. Jones, A. A. Kiselev, D. W. Barnes, R. S. Noah, E. Acuna, M. Akmal,S. D. Ha, J. A. Wright, B. J. Thomas, C. A. C. Jackson, L. F. Edge, K. Eng, R. S. Ross, and T. D. Ladd
HRL Laboratories, LLC, 3011 Malibu Canyon Rd., Malibu, CA 90265 (Dated: September 18, 2020)We study the time-fluctuating magnetic gradient noise mechanisms in pairs of Si/SiGe quantumdots using exchange echo noise spectroscopy. We find through a combination of spectral inversionand correspondence to theoretical modeling that quadrupolar precession of the Ge nuclei play akey role in the spin-echo decay time T , with a characteristic dependence on magnetic field and thewidth of the Si quantum well. The Ge noise peaks appear at the fundamental and first harmonicof the Ge Larmor resonance, superimposed over 1 /f noise due to Si dipole-dipole dynamics,and are dependent on material epitaxy and applied magnetic field. These results may inform theneeds of dynamical decoupling when using Si/SiGe quantum dots as qubits in quantum informationprocessing devices.
I. INTRODUCTION
Qubits made of silicon are promising candidates forquantum computation for a variety of reasons [1]. Oneof these reasons is the long coherence times observedfor spins in bulk silicon crystals, going back manydecades [2]. These long spin coherence times are dueto the quiet magnetic environment of the surroundingcrystal and allow spin qubits in silicon to hold quantuminformation for timescales long relative to typical control-pulse timescales. This advantage is especially evidentwhen comparing to spin-qubits based on III-V semicon-ductors such as GaAs, in which every substrate nucleushas a nuclear magnetic moment. Critically, the best co-herence times are observed when silicon is isotopicallyenhanced; that is, when the silicon source has a depletedpopulation of the spin-1/2 Si isotope, leaving behindonly the natural spin-0 isotopes of Si and Si [3–11].Silicon spin qubits may host electrons in a variety ofways, including binding them to single donors [4], trap-ping them at a metal-oxide-semiconductor (MOS) inter-face [5, 7, 8], or trapping them in a strained silicon quan-tum well in a Si/SiGe heterostructure [6, 9–16]. Here, wefocus on the SiGe-based quantum well approach. Quan-tum dots in SiGe are lithographically defined and benefitfrom the low disorder of the Si-SiGe interface, advan-tages relative to donor bound spins or MOS, but sufferfrom the disadvantages of lower valley splittings and re-duced compatibility with many commercial complemen-tary MOS (CMOS) fabrication processes [1].One of the key questions about SiGe-based dots notencountered for MOS or donor qubits is whether andhow the only stable nonzero-spin isotope of Ge, Ge,impacts magnetic noise [17]. In the present work, we ex-amine devices in which the Si content is depleted fromthe natural levels of 4.7% to 800 ppm in the Si well, andaddress the critical question of whether the remainingmagnetic noise will be dominated by these residual Sinuclei, by naturally-abundant Ge nuclei, or by othernoise sources. In particular, we focus on magnetic lim- its to coherence in spin-echo experiments (i.e. T times)and find through both modeling and dependence on ma-terial epitaxy that these coherence limits are driven bythe distribution of Ge nuclei in the device when ap-plied magnetic fields are low. This question informs whatengineering or material modifications may be needed inthe future to allow high fidelity quantum gates on qubitsbased on these materials.Magnetic noise in silicon qubits has been studied pre-viously in different types of devices and qubit encod-ings, which we now briefly review. Single qubit gatesmay employ exchange only [6, 10, 18], electron spin res-onance (ESR) [4, 5], or electron dipole spin resonance(EDSR) [9, 11, 12, 15]; two qubit gates may either in-volve exchange only [18] or a combination of exchangeand radio-frequency control [7, 15, 16, 19, 20]. In allcases, fluctuations of local magnetic fields from Si nu-clei or other sources will impact gate fidelity, determin-ing the eventual overhead of quantum error correctionfor allowing quantum information processing [21, 22].For isotopically enhanced donor and MOS-based single-spin qubits at high magnetic field, long static dephas-ing times, exceeding 100 µ s, have been observed, andspin-echo studies indicate low-frequency magnetic noisedominated by fluctuations in the applied magnetic field[4, 5, 23]. Higher-frequency ( >
100 Hz) magnetic noise inthese devices may be dominated by the effect of chargemotion on the position of the electron, which translatesto magnetic noise due the hyperfine coupling to the Pnucleus in the case of the donor or to a spin-orbit-inducedStark shift in the case of MOS. In SiGe, existing stud-ies to date employ large magnetic field gradients inducedby a microferromagnet, allowing single-spin control viaEDSR [9, 11, 12, 15, 24]. In these studies, in bothisotopically natural and isotopically enhanced samples,magnetic noise appears to be dominated by electric-fieldnoise transduced by the deliberately applied gradient.Here, we study the nuclear environment of SiGe qubitswithout introducing any deliberate magnetic gradients,and in a low-field environment where spin-orbit and para- a r X i v : . [ qu a n t - ph ] S e p FIG. 1. a) False-color scanning electron micrograph of a six dot device. A 2DEG is accumulated under the bath gates (Bath),and loaded through the tunnel barriers from by gates (T). The plunger gates (P) form the quantum dots and exchange gates(X) controls the exchange interaction between neighboring dots. The measure dot (M) acts as an electrometer, sensing thecharge state in each dot through capacitive shift. This work focuses on data from double quantum dots on devices featuringarrays of up to six dots, as highlighted. b) Cartoon of the Si/SiGe heterostructure, as labelled; gate stack, color-matched tothe white-dashed-line-indicated region of (a), and a depiction of a double-dot potential, which is depressed due to P gates andlifted due to X gates; electrons with spin are depicted as white. c) Illustration of the Si/SiGe barrier and electron wavefunctionof a single dot with population of spin-carrying nuclear spins. The plot places a green circle for each Ge nucleus and a redcircle for each Si nucleus in a 50 × ×
14 nm slab of material (dot size and shade indicate distance into the out-of-page y dimension). The dots are randomly placed at each silicon lattice site with probability 800 ppm for Si in both well and barrierand 7.76% for Ge multiplied by the alloy content. The sidewall content of Si is enriched in this figure, but is natural in theactual devices. The alloy content for Ge is shown as horizontal bars on the right, for a profile with 1 atomic layer of smearing atthe Si/SiGe interface. This Ge percentage is used to determine the vertical wavefunction | ψ ( z ) | , shown in black and featuringvalley oscillations with phase determined by Ge placement. A 30-nm-diameter Gaussian wavefunction diameter is assumed inthe x and y directions. magnetic or diamagnetic gradients from local screeningeffects are minimized. This is the magnetic environmentof exchange only qubits [6, 10, 18], and in this regard,we examine a simpler magnetic environment than previ-ous studies. In this simple environment, higher frequencymagnetic fluctuations appear to be entirely dominated byresidual nuclear magnetic moments, in contrast to previ-ous studies. Using exchange echo noise spectroscopy, wefind a rich noise spectrum including components originat-ing from quadrupolar Ge nuclear moments and providea detailed model for these effects.
II. THE SiGe QUANTUM DOT DEVICE
We study accumulation mode devices in thismanuscript predominantly employing a gate stack similarto that described in Ref. 25, except with aluminum gatesto control quantum dot loading and a large aluminumscreening gate to prevent electrons from accumulatingunder the gate leads, leading overall to a gate-stack verysimilar to that in Ref. 13. There are several variations inthe gate stack across different samples, but the aspectsvaried are not expected to cause significant variation in T ∗ or T as is our focus here.All devices were fabricated on [100]-orientedSi . Ge . /Si/Si . Ge . buried channel heterostruc-tures grown on thick graded buffers using a variety ofmethods as described in Ref. 26. Silicon well widthsvaried from 3 to 10 nm, set between a 170 nm SiGebuffer below and a 60 nm Si . Ge . barrier above,typically also including a 1.5 nm Si cap. An SEM imageof an example six-dot device is shown in Fig. 1a, and crossection schematic in Fig. 1b. For most samples,the Si wells were isotopically enriched to 800 ppm Si,while the Si . Ge . uses a natural abundance of Ge.The samples in this paper use natural abundance Si inthe SiGe sidewalls, although we will argue in Sec. V thatthe silicon isotopic abundance of the barriers plays littlerole in qubit behavior due to the dominant contributionof isotopically natural barrier Ge.Some samples have been characterized by atomic probetomography, as discussed in Ref. [27], but in most casesthe detailed shape of the Si/SiGe barrier is not known.Figure 1c shows a model for the barrier and wavefunc-tion for a 5 nm well, discussed further in Sec. V, with anillustration of the density of Si and Ge nuclei. Thisfigure illustrates that the wavefunction appreciably over-laps hundreds of Si nuclei in the well and hundreds of Ge nuclei at the Si/SiGe barrier; a more quantitativeevaluation of hyperfine overlap is discussed in Sec. V.In contrast with the actual devices, the figure depictsenriched levels of Si in the sidewalls to show the loca-tion of the Ge nuclei more clearly. One open questionfor the present study is whether magnetic gradient noisespectroscopy may help determine the degree of nuclearoverlap, and hence details of the Si/SiGe barrier. This isespecially plausible if the effect of Ge can be isolatedfrom that of Si, which we will see is enabled by boththe use of isotopically enriched Si in the quantum welland the use of noise spectroscopy.While the SiGe-Si-SiGe heterostructure confines elec-trons to the Si quantum well along the growth direction,voltages applied to the plunger (P) gates create an at-tractive potential which accumulates and confines singleelectron spins inside the quantum well plane. The tunnelbarriers (T) control the loading rate of single electronsfrom the accumulated baths. Applying a voltage to theexchange (X) gates changes the potential barrier betweendots and is used to control the exchange interaction be-tween neighboring dot pairs. The device shown in Fig. 1features six dots, or two triple-quantum-dot exchange-only qubits [6, 10, 18], however all experiments describedin this paper utilize only two dots at a time. The experi-ments described here may be performed on any dot-pairof the device, with similar results. In addition to thedots formed under the P gates, two additional dots areformed under the two measure (M) dot gates, which areused as electrometers to sense the occupation of each dotthrough capacitive shifts.All experiments use a standard energy-selective initial-ization procedure [6, 14, 28] to prepare a singlet state, | S (cid:105) ≡ [ |↑↓(cid:105) − |↓↑(cid:105) ] / √
2, which has total spin amplitude S = 0, in the first two dots. Spin-state measurement isperformed using a Pauli-spin blockade as sensed by theM-dot electrometers which can distinguish between sin-glet states, with S = 0, and triplet states, S = 1, butcannot discriminate between the triplet projection sub-states, m = − , ,
1. Coherent spin swaps are performedby “symmetric” control of the exchange [14]. Calibra-tion of such pulses use all three dots operating as a sin-gle qubit and the calibration procedure is described inRef. 10. All three operations are depicted schematicallyin Fig. 2a.While the spin singlet and m = 0 triplet states are in-sensitive to global magnetic fields, local magnetic fieldssuch as hyperfine-coupled nuclear spins create fluctua-tions of magnetic gradients between quantum dots, driv-ing transitions between singlet and triplet states. We caneffectively homogenize the magnetic fluctuations by co-herently swapping the spins, so that each spin is equallysubjected to the same averaged local magnetic environ-ment. This is the basis for exchange-based spin echo.Exchange echo sequences can use an arbitrary number ofexchange pulses and in the following section we will cal-culate the effect of such a pulse sequence on a double-dotinitialized to a spin singlet. III. EXCHANGE-BASED SINGLET-TRIPLETMAGNETIC NOISE SPECTROSCOPY
Noise spectroscopy is a well-studied suite of techniques,especially for qubits, for deducing noise spectral charac-teristics from an ensemble of measurements using dif-ferent noise-compensating sequences [29–31]. Versions ofnoise spectroscopy have been applied to a variety of spin-qubit systems, and in particular to singlet-triplet qubitsin GaAs [32, 33]. We provide here a brief theoreticalsummary of singlet-triplet magnetic noise spectroscopyas pertinent to the experiments we describe in this paper;we detail the theory further in Appendix A. For this, wewill neglect charge noise, initialization errors, and mea-surement errors and assume a spin singlet is perfectly prepared and measured, and spin-swaps occur perfectlyand instantaneously. While this approximation will in-evitably fail for many-pulse experiments, we will discussno more than 10-pulse experiments in the present workand we find no deleterious effects of charge noise on ourability to extract magnetic noise phenomena.We first note that for singlet-triplet qubits, we are im-mune to global magnetic fields, since the singlet state | S (cid:105) is an invariant eigenstate relative to any global magneticfield, including dynamically fluctuating global magneticfields. However, if the magnetic field on dot 1 and themagnetic field on dot 2 vary in magnitude, direction, orboth, the singlet state will evolve into orthogonal tripletstates, which is the effect we seek to characterize. Thetime-dependent, noisy Hamiltonian of the system is sim-ply H ( t ) = J ( t ) S · S − gµ B [ B z ( S z + S z ) + δ B ( t ) · S + δ B ( t ) · S ] , (1)where S k is the vector spin-operator for electron k with z -component S zk , g ≈ g -factor, µ B theBohr magneton, B z is a global applied magnetic fieldtaken to be in the z direction, and J ( t ) describes voltage-modulated exchange energy. For the present work, wewill consider only instantaneous π pulses, and hence take J ( t ) as (cid:80) j (cid:126) πδ ( t − t j ) for pulses arriving at prescribedtimes t j . Differences in the local magnetic field variationsat each dot, δ B k ( t ), represent the noisy fluctuation whosespectral character we seek to deduce via control of thespin-swap-pulse arrival times t j . We can summarize thefree evolution for a pair of spins via a unitary operator U ( t ) = e − i b ( t ) · S e − i b ( t ) · S , (2)in which b k ( t ) represents the effective axis and angleabout which spin k rotates, which includes a combina-tion of noisy magnetic vector fields and spin-swaps. Ingeneral, the relationship between b k and B k is complex;only if all effective magnetic field vectors are parallel andno exchange is applied can we make the simplifying as-sumption b zk = (cid:82) t dtgµ B ( B z + δB zk ( t )) / (cid:126) .In all experiments, we initialize | S (cid:105) and measure the re-turn probability to | S (cid:105) , and average over a time-ensembleof measurements. Appealing to an ergodic ensemble av-erage, which we notate (cid:104)·(cid:105) and discuss further at the endof this section, the singlet probability measured in eachexperiment can then be approximately summarized (seeAppendix A) as P S ( t ) ≈ (cid:10) |(cid:104) S | U ( t ) | S (cid:105)| (cid:11) ≈
12 + 12 cos (cid:18)(cid:10) | b − b | t (cid:11)(cid:19) exp (cid:18) − σ ( t )2 (cid:19) . (3)Our focus in the present work is the second moment σ ( t ), which captures dynamic fluctuations around ran-dom mean magnetic field gradients, which may occur at avariety of timescales. We decompose this second momentusing a filter function formalism as σ ( t )2 = g µ (cid:126) (cid:88) k =1 (cid:90) ∞ df S k ( f ) F ( f, t ) , (4)where S k ( f ) the power spectral density for each com-ponent of the magnetic field noise in dot k . The filterfunction F ( f, t ) depends on the experiment being per-formed. Experimentally, we only have access to a totalnoise power (cid:80) k S k ( f ), however we derive filter functions F ( f, t ) scaled for one dot in Eq. (4) and assume indepen-dent, identical noise distributions across the pair. Detailsof the calculation method for any number of spin-swapsare discussed in Appendix A. There, we show that thefilter function can be decomposed into three terms; a“central lobe” which we notate as F ( f, t ) and replicas ofthat central lobe at positive and negative electron Lar-mor frequencies ω = gµ B B z / (cid:126) . These side-bands appearwhenever the noisy magnetic field gradients in the systemhave a “transverse” component, meaning that their vec-tor directions include components orthogonal to the ap-plied magnetic field. This in turn means that the experi-ment must somehow drive the system to compensate forthe electron Zeeman energy, rendering these side-lobesnegligible at sufficiently high magnetic fields. In the ex-periments described here, a field of 10 mT is sufficientlylarge to completely suppress these sidelobes.In some experiments we may be sensitive to a first mo-ment (cid:104)| b − b | (cid:11) ; this will depend on any static magneticfield gradients across the pair of dots. Static magneticfield gradients may arise from the screening of the appliedmagnetic field by the superconducting aluminum gates,at fields beneath the critical field, or by spin-orbit effectsat higher magnetic fields. We observe both phenomenain our devices, which we elaborate in Appendix B.A primary role of exchange echo experiments is tocancel the low-frequency noise leading to a finite first-moment by undoing static or nearly static phase evo-lution. This is accomplished using one or more cali-brated spin swaps via voltage-induced exchange in be-tween equal-time durations of evolution in the (1,1)charge state, which alternate which spin sees which dot’smagnetic field. If using only a single swap, we refer tothe experiment as a Hahn echo (HE) experiment, and itscentral-lobe filter function following Appendix A isHE : F ( f, τ ) = sin (2 πf τ ) tan ( πf τ )( πf ) . (5)Here τ is the amount of time of free evolution before andafter the spin swap. If some number n swaps are used,with n >
1, we refer to the experiment as a Carr-Purcell- n (CP n ) experiment [34]. For even n , the filter functionis altered toCP n : F ( f, τ ) = 4 sec (2 πf τ ) sin ( πf τ ) sin (2 nπf τ )( πf ) . (6) In this case, τ is the amount of free evolution time beforefirst and after the last swap; between swaps spins evolvefor the time 2 τ . The protocol for these experiments isdepicted in Fig. 2b.Both Eq. (5) and Eq. (6) vanish at f = 0, indicatingtheir capability to decouple low frequencies [35]. Intu-itively, we may think of these experiments as “homoge-nizing” the magnetic field by swapping the two spins suf-ficiently rapidly between the dots, converting static localgradient fields into global fields, to which singlet-tripletsubspaces are immune. The associated filter functionsare depicted in Fig. 3, which shows F ( f, τ ) for CP10and HE. These two filter functions act as narrow- andbroad-band filter functions, respectively (plus additionalpeaks at odd integer multiples).The key concept of noise spectroscopy is to scan theseband-pass filters across the noise spectrum by varying τ in a series of experiments. The resulting observed de-cay may then be inverted to estimate the underlyingspectrum. Our approach for this inversion, followingRefs. 29 and 30, is to first correct the data using thesignal at τ → τ → ∞ to extract a decay functionexp( − χ ( t )), where t = 2 nτ , from the singlet probabil-ity of form [1 + exp( − χ ( t ))] /
2. We then treat the noiseunder study as roughly constant within the bandwidthof the main lobe of the filter function, which for CP n extends from (1 − /n ) / (4 τ ) to (1 + 2 /n ) / (4 τ ), centeredat f = 1 / (4 τ ). The integral under this lobe is 0 . nτ ,giving a summed spectral density across both dots of (cid:88) k S k ( f ) ≈ f χ ( n/ f )0 . n . (7)While more sophisticated noise spectroscopy inversiontechniques certainly exist [31], we find this simple in-version method sufficient for deducing the sources andmagnitudes of underlying fluctuating gradient fields.We may also analyze the decay by extracting T , whichwe define as the total sequence time (i.e., 2 nτ ) at whichcoherence has decayed to its 1 /e point as τ is increasedand n fixed in HE and CP n experiments. This 1 /e def-inition has limited predictive value, especially since dif-ferent noise spectra can lead to a significant variation ofdecay shapes. For example, if integrated against a sim-ple 1 /f spectrum, both filter functions for CP10 and HEresult in a decay going as exp[ − ( t/T ) ] (i.e., one that isGaussian in time). If integrated against white noise, ex-ponential decay would result, and significantly different T values could be chosen for comparable noise powers.Our chosen definition of T therefore is insufficient topredict dephasing at different timescales; the underlyingdeduced spectrum is necessary to project the impact ofdephasing.Finally, also included in Fig. 3 is the filter functionfor a so-called Free Induction Decay (FID) experiment,which has no spin swap pulses and is thus sensitive tonoise components at low frequencies, as discussed morein Appendix B. The 1 /e point of this decay provides theexperimental definition of T ∗ , but we caution that T ∗ FIG. 2. a) Schematic representation of double dot intialization, single/triplet measurement, and pulsed exchange swaps. b)Representation of the experimental protocol for Hahn echo (n=1) and Carr-Purcell-n experiments in our double dot system.FIG. 3. Center lobe filter functions for Carr-Purcell-10(CP10), Hahn Echo (HE), and Free Induction Decay (FID)experiments. depends on slow fluctuations that are more challengingto spectroscopically invert.We now briefly address ergodicity: in the expressionsabove, we have used an average over an ensemble of in-dependent dot-pairs, but in our experiments we studyonly a single dot-pair evolving over the long timescale ofa time-averaged ensemble of experiments. The ergodicassumption is violated if the averaging timescale is tooshort relative to the lowest noise frequency under study.This discrepancy is particularly acute for power-law noisespectra, S ( f ) ∝ /f α , which diverge as f →
0, andexperiments highly sensitive to low-frequency noise, inparticular FID, are particularly sensitive to insufficientaveraging. The noise sources we will study in this pa-per, however, all appear to achieve an “ergodic” limit atsome finite averaging time, indicative of a low-frequencydeviation from true 1 /f α behavior, and we present datathroughout with sufficient averaging to have seen thislimit reached. See Ref. [6] as an example characterizationof the ergodic limit in a double-dot magnetic noise exper-iment; similar characterizations have been performed forthe measurements in this study. FIG. 4. CP10 echo decay curves vs. τ measured with exter-nal magnetic fields ranging from 10 mT to 150 mT. Singletsurvival times increase dramatically as the field is increasedand complex decay curves appear at intermediate fields. Thegreen and black curves at 40 mT are two separate measure-ments, showing that the observed structure is reproducible.Inset: 10 mT and 150 mT data on linear τ scales. IV. FLUCTUATING MAGNETIC FIELDGRADIENT MEASUREMENTS
In this section we summarize singlet-triplet noise spec-troscopy measurements from HE and CP n experimentsand a variety of devices to help illuminate the magneticdynamics of Ge.
A. Time-domain Decay
Figure 4 shows the decay observed from CP10 echoexperiments performed with an applied magnetic fieldranging from 10 mT to 150 mT (always applied in thein-plane direction in this work, approximately along the[1¯10] crystal axis), from a device with a 5 nm Si quan-tum well enriched to 800 ppm Si. For applied fields
FIG. 5. Hahn Echo T as determined by 1 /e decay time,versus well width and applied magnetic field. Thick lines andsymbols are experimental measurements across six devices;thin lines are results of simulations for 10 devices at each wellwidth with varying random Ge placement. between 10 and 20 mT, the coherence decay as τ is in-creased falls off roughly as exp[( τ /T ) − ], with T timesof about 20 τ = 60 µ s. Between 30 and 80 mT, the coher-ence persists out to progressively higher τ with appliedfield, and the decay curves acquire irregular bumps. Al-though irregular, each bump is highly reproducible, atleast over the many-minute timescales over which thesemeasurements are taken. Demonstrating this are the twooverlapping datasets at 40 mT, labeled T times of almost 1 ms.The first hint that Ge underlies these phenomena isthe observation that T times from both CP10 and HEexperiments appear to have a strong dependence on thewidth of the Si quantum well, which primarily impactsthe amount of hyperfine coupling to Ge nuclei in theSiGe barriers. In Figure 5, we summarize the T timesextracted from HE experiments conducted on several de-vices with varying growths of the Si well (the 5 nm dataalso appeared in the Supplement of [6]). At low fields,the T times increase monotonically with the width ofthe Si well from between 3 nm to 10 nm. The one ex-ception to this is the low field T for an 8 nm well devicewith isotopically natural silicon, whose growth conditionslikely resulted in a more diffuse Si/SiGe interface than allthe others, putting its characteristics similar to a devicewith a 3 nm well. As the applied magnetic field increasesinto the 10s of mT, the T times increase abruptly to ap-proach 1 ms, with narrower well devices requiring largerapplied fields before increasing.Also shown in Fig. 5 are curves for Monte-Carlo sim-ulations capturing the effects of quadrupole split Genuclei, which we discuss in more detail in Sec. V. Qual-
FIG. 6. T ∗ measured at a few mT for the same devices asin Figure 5. The two devices with natural isotopic Si contenthave T ∗ values of 0.51 µ s and 0.66 µ s. The colored bandsshow the µ ± σ range, where µ is the mean value of T ∗ de-fined for each nuclear species and σ is the standard deviationunder random isotopic placement, calculated from Eq. (10).The purple band shows the anticipated contribution of Si,while the cyan band shows the contribution of Ge, whichdominates the amount of variation with well-width. The graybands are the total anticipated T ∗ . The purple dashed linesshow the small reduction of T ∗ due to isotopically natural Siin the barriers; this reduction is negligible in comparison tothe wells for wide well widths and in comparison to the Gecontribution for narrow well widths. itatively, they behave similarly to the data at low mag-netic field, showing a B-field-independent T value de-pending on the well-width, and then diverging at somecritical field. Lacking from these simulations but evidentin the data is a high-field “saturation” T value. Theexperimentally observed T saturation with field is likelylimited by Si dipole-dipole dynamics, which are lack-ing in the simulations. We note that samples of widelyvarying well widths but all with the same Si isotopic en-richment have similar T times at high field, and this T increases gradually with field. Key evidence for this sat-uration being the Si limit is the two 8 nm well deviceswith natural abundance Si in their wells, but differingSi/SiGe interfaces: both have much shorter high-field T times of 100 µ s.The low-field T ’s vary over nearly two orders of mag-nitude with well-width. This contrasts sharply with themuch weaker variation in T ∗ with well width, as mea-sured by FID: Fig. 6 shows the T ∗ from the same devicesas Fig. 5 at a few mT. Unlike T , we see relatively minorvariation of T ∗ with applied magnetic field [6]. The varia-tion we do see is discussed in Appendix B. The T ∗ values(Fig. 6) correspond closely to predictions based on theanticipated overlap with nuclear spins of both species.These predicted values of T ∗ are dominated by Si atboth natural abundance and at 800 ppm for wide wells,while Ge’s contribution increases with narrower wells.As the figure shows, the isotopic content of the silicon inthe barrier plays a negligible role in comparison to eitherthe natural Ge spins in the barrier or the silicon in thewell for all samples. The ratio of magnitudes for the T ’sas measured by HE at high field match the ratio of T ∗ ’sfor each device for wide wells, pointing again to T athigh field being dominated by the wavefunction overlapof Si.
B. Spectroscopic analysis
Both T ∗ and T indicate a strong role for Ge inrelaxation at low field; experimental signatures of thequadrupole splittings of Ge are more evident by con-verting to the frequency domain using Eq. (7), hence es-timating the noise spectral density of magnetic gradientfluctuations from the CP10 data. Using the time-domaindata in Figure 4, Fig. 7a shows the estimated magneticnoise spectra in this 5 nm enriched device [36]. Spectralestimation only achieves a high signal-to-noise ratio forbandwidths over which the time domain signal has sig-nificant variation. Since the time domain data varies onthe time scale τ , the spectral estimation bandwidth isnarrow and changes as T changes.We find that as the field increases, the overall noisemagnitude decreases, while irregular peaks emerge,which flatten and increase in frequency. At sufficientlyhigh field (above 100 mT, where HE saturates to the Silimit) the spectra are much more flat. Especially visibleat fields 30 to 50 mT, a pair of spectral peaks appearto occur at the characteristic Larmor frequency of Ge,1.5 MHz/T, and its first harmonic. This intriguing fea-ture signals the influence of quadrupole splitting, as weelaborate in the next section.The estimated spectra in Fig. 7a may be contrastedwith that from CP10 data from devices with differentmaterial epitaxy, such as the 8-nm-well natural-Si devicewith diffuse sidewalls, Fig. 7b. Here, we observe similarqualitative evolution, but with different magnitudes: theoverall noise magnitude is higher and the noise peaks areless suppressed at the same applied fields.The difference between the CP10-deduced spectra fromdifferent devices may be corroborated with the low-frequency noise estimates deduced from FID experi-ments. To illustrate, Figs. 8a and b show time-domaincomparisons of FID and CP10 experiments, respectively,for three samples. The T ∗ of the device with a 5 nmwell with enriched- Si is longest at 2.9 µ s, while the two8 nm natural devices with sharper and diffuse interfaceshave very similar T ∗ values of 0.51 µ s and 0.66 µ s, re-spectively. The underlying low-frequency ( < /f power laws, as observed in all sam-ples, and have total magnitudes consistent with the ob- served decays shown in Fig. 8a. We may compare these1 /f spectra to the higher-frequency estimated spectra,shown in the lower right of panel c, extracted from theCP10 decay shown in panel b. While an extrapolationof the low frequency fit for the enriched 5 nm devicematches the inverted high-frequency spectral band quitewell, those of the 8 nm devices are much further off, againsuggesting different mechanisms determine gradient noisein these two regimes. This may be understood by notingthat Ge plays a substantial role in low-frequency noise(and thus T ∗ ) for an enriched 5-nm well, but a negligiblerole in a natural 8-nm well; however, it plays a domi-nant role in the low-field T spectrum in all three cases.This difference also points to why the two 8 nm devicesmay behave so differently: they have nearly the same Si character, which effects their low-frequency noise,but highly different Ge overlap, affecting their high fre-quency noise.
V. THEORETICAL MODEL FORQUADRUPOLAR COUPLING DETERMINING T The experiments we have described already pointstrongly to Ge as a source of magnetic gradients limit-ing T and T ∗ in narrow wells, but they also show thatthe often uncertain details of the Si/SiGe interface acrosssamples has significant effect on the measured values ofdecoherence times in FID, HE, and CP10 experiments.This may be understood from the form of the Fermi con-tact hyperfine interaction between the electron spin andthe Si and Ge spins within the electron wavefunc-tion in each dot. The contact hyperfine Hamiltonian fortwo well-separated quantum dots in the effective massapproach is H hf = (cid:88) j =1 (cid:88) k (cid:126) A jk S j · I k , (8)where A jk = 2 µ g µ B γ k η k | ψ j ( r k ) | . (9)The sum over index j is over the pair of quantum dots,the sum over index k is over all nuclear spins in the crys-tal, ψ j ( r k ) is the effective-mass envelope wavefunction forthe electron in dot j at the location of nucleus k , S j isthe spin of electron j with vacuum electron gyromagneticratio g µ B / (cid:126) , and I k is the spin of nucleus k with gyro-magnetic ratio γ k . Note that A k is in units of rad/s here.The overlap of the periodic Bloch wavefunction on eachnuclear site, the “bunching factors,” are taken as unitlessconstants η k . The total spin for each nuclear species are I k = 1 / Si nuclei and I k = 9 / Ge nuclei.The values of A k may be approximately calibrated byassuming that FID measurements measure the high-fieldergodic value of T ∗ , in which nuclear spins give a fairsample of their full distribution of configurations and the FIG. 7. a) Inferred magnetic gradient noise from the CP10 data in Figure 4 using Eq. (7). b) Same as (a), but from a devicewith an 8 nm well with natural Si content and a more diffuse Si/SiGe barrier.FIG. 8. a) FID experiments of the same devices at a few mT, showing that the Si isotopic content has the largest contributionto T ∗ . b) CP10 decay experiments from three devices of differing material epitaxy, with 20 mT magnetic field, showing thatthe Si/SiGe interface has the largest contribution to CP10 decay. c) Comparison of spectral estimates from FID and CP10experiments in 20 mT fields, as well as extrapolation of power law fits to the 5 nm and 8 nm CVD low frequency data. Thenarrow, isotopically enhanced quantum well sample (red) has a clear connection between low and high frequency noise, as bothhave substantial Ge contribution; the isotopically natural sample (green) has more low-frequency noise due to increased Sicontent but comparable high-frequency noise due to comparable Ge content. applied magnetic field causes electron Zeeman splittingssignificantly higher than hyperfine values. By increasingthe averaging time and magnetic field in our experiments,we assure that our measured T ∗ values are close to thislimit. Under these assumptions, T ∗ is estimated simplyas (cid:18) T ∗ (cid:19) = 12 (cid:88) j (cid:88) k I k ( I k + 1)3 A jk . (10)To numerically estimate T ∗ and use it to develop a bar-rier model, we require numeric values of the bunchingfactors η k . For Si, we use an η value of 178, as this iswell corroborated by calculation and magnetic resonanceexperiments [37]. For Ge, the correct value to use is lessclear [17]; our aggregated data across multiple deviceswith multiple quantum wells are consistent with a valueof 570 which we assume throughout.With these values assumed we now note that the char-acter of the Si/SiGe interface impacts the results in twoways. First, a diffuse interface allows the envelope wave-function ψ j ( r k ) to broaden into the barriers more than itwould for a hard-wall interface. Second, a diffuse inter-face places more straggling Ge in the quantum wellregion. Reference 27 show some example microscopiccharacterizations for some Si/SiGe wafers, and indicateclearly that interface diffusion is a significant consider-ation. Here, we find that our T ∗ and T data taken inaggregrate are decently supported by a model interface inwhich we simply convolve a perfect square interface witha Gaussian smearing function with standard deviation of1 silicon mono-atomic-layer, 0.136 nm. Confounding ourability to estimate this interface from existing data is theuncertainty in the vertical electric field and the in-planewavefunction diameter for each device. We assume zerovertical electric field and a 30 nm 1 /e diameter of a Gaus-sian | ψ j ( r ) | , as these are consistent with electrostaticsimulations, but these numbers are undoubtedly depen-dent on disorder and gate tuning variances. The resultingmodel for the barrier and wavefunction is illustrated inFig. 1c for a 5 nm well. We emphasize that our estimatedbunching factors, barrier model, wavefunction diameter,and vertical electric field are all rough estimates, enablingreasonably close postdictions for T ∗ and T ; in reality, thedifferent samples under study may have different barrierand wavefunction parameters, so the assumption that allsamples follow the same profile necessarily limits the ac-curacy of our models.With the hyperfine hamiltonian so determined, weare now prepared to develop the theoretical model toexplain not only the observed field and well-width de-pendences, but also the characteristic decay or spectralshapes of HE and CP10 experiments, arising from theelectric quadrupole effects from the Ge nuclei.The quadrupole moment of Ge makes it quite dif-ferent from Si which is spin-1/2 and therefore onlyhas a magnetic dipole moment. The result of the nu-clear quadrupole moment of Ge is to split the spin- states by an amount proportional to electric field gradi-ents across the nucleus. The quadrupolar hamiltonian,Eq. VII.II.A.23 from Ref. 38, is H Q = (cid:88) j (cid:126) ξ j (cid:26) θ j −
12 [3 I zj − I ( I + 1)]+ 3 sin θ j cos θ j { I zj , I xj } + 3 sin θ j I xj − I yj ] (cid:27) . (11)Here, ξ j = e Q/ [4 I (2 I − ∂ V /∂z j / (cid:126) , where θ j is theangle between the z -axis (i.e. the applied magnetic field),and the principle axis of the electric field gradient tensor ∂ V /∂x j ∂x k at the location of the nucleus. The z direc-tion is explicitly that of the magnetic field (not the crys-tal growth direction; in all experiments we have reportedthe magnetic field is in the plane of the quantum well).The electric quadrupole moment of the Ge nucleus isdefined in terms of eQ = (cid:104) (cid:80) j e j (3 z j − | r j | ) (cid:105) , where thesum is over nucleons and the expectation is over the max-imal spin state m z = I . While the quadrupole moment Q for Ge is known (about −
200 millibarn), the electricfield gradients contributing to quadrupolar splittings arefar more uncertain.We know of no direct measure or calculation of Gequadrupole splittings in SiGe alloys. We note that thesymmetry of a bulk silicon crystal means that an isolated Ge in a perfect silicon host would see no quadrupolesplitting. However, in the SiGe barrier or at its inter-face, the relevant nuclei see an electric environment withsubstantially reduced symmetry, due in majority part toalloy effects (i.e. the neighbors of a given Ge are ran-dom combinations of Ge and Si) and to a much lesserdegree due to strain. For alloy effects, Ref. 39 reportscalculations of quadrupole splittings for a substitutionaliron atom in a Si crystal, in the presence of a pertur-bation due to the Ge atom nearby; these calculationsinform about plausible magnitudes of quadrupole split-tings, and tabulate the effect of first, second, and thirdnearest neighbor Ge atoms. In the barrier, with 30% Ge,most (76%) of lattice sites have at least one 1st nearestneighbor Ge. While the effect of a 2nd nearest neighborGe is reported to be about four times weaker than a firstnearest neighbor, there are 3 times more of them, result-ing in a significant possible range of quadrupole splittingsin the alloy. We estimate that these alloy effects, usingRef. 39 as proxy, overwhelm the effects of macroscopicstrain present in the device due to the lattice mismatchof the Si/SiGe heterostack, as well as internal stresses dueto thermal expansion mismatches of the gatestack duringcool-down; using the similar strain-to-quadrupole cali-brations available from studies of As [40] and
Sb [41].Existing data on quadrupole splittings in other semicon-ductors may also inform estimates for the magnitude ofthe ξ j constants. Reference 42 reports quadrupole split-tings from nuclear magnetic resonance (NMR) studies ofsingle-crystal Ge with varying isotopic contents, attribut-ing electric field gradients to isotopic disorder. In thiscase, quadrupole splitting parameters ξ j are on the order0of 100 Hz. A closer estimate results from noting thatAlGaAs/GaAs quantum wells have similar alloy fluctua-tions to Si/SiGe quantum wells and the Ga, Ga, and As nuclei have similar quadrupole moments Q as Ge.Since NMR spectra are available from these quantumwells via optical detection [43], quadrupole splittings inthe 1 to 30 kHz are observable and suggest a comparablerange for SiGe.A characteristic feature of H Q is the existence of ma-trix elements allowing ∆ m z = 0 , ± , and ±
2. Thesetransitions correspond to shifts of the nuclear Zeemanenergy, described by hamiltonian H Z = (cid:88) k (cid:126) γ k B z I zk , (12)where the Ge nuclei each have γ k = γ =(2 π ) × .
49 MHz/T. In the interaction picture for thisHamiltonian, the quadrupole Hamiltonian ˜ H Q ( t ) =exp( iH Z t/ (cid:126) ) H Q exp( − iH Z t/ (cid:126) ) features terms at zero-frequency, the nuclear Larmor frequency γ B z , andtwice the nuclear Larmor frequency 2 γ B z ; these termsalso provide matrix elements to vary the coupling to elec-trons H hf . Roughly speaking, then, we anticipate this“bath” to provide “noise” at these characteristic frequen-cies. This qualitative observation is somewhat evident inFig. 7, where two peaks are visible at 30-50 mT rangecoincident with γ B z and 2 γ B z .To make a more quantitative assessment, we simu-late the dynamics of randomly precessing and nutatingquadrupolar Ge and track their hyperfine shifts. Thenoise source in question is highly non-Markovian, sincein the regime that Ge hyperfine coupling constants arecomparable to the Ge Larmor frequency, the evolu-tion of quadrupolar nuclei will be highly influenced bythe electron spin-flips which occur in each dot due toexchange pulsing. The act of swapping electron-spins,which amounts to flipping the hyperfine field seen by each Ge nucleus, may be regarded as a periodic variation ofthe effective Larmor frequency, and hence modifies thenoise dynamics as translated through the hyperfine inter-action. For this reason, in addition to the complicationthat all terms of the Hamiltonian are of comparable orderin those regime where the most prominent spectral fea-tures occur, perturbative expansions are challenging foranalysis. Fortunately for calculation of quadrupolar ef-fects, much of the structure of the CP n observations maybe understood considering only non-interacting Ge nu-clei. The resulting motion may therefore be simulated byexponentiating the 10-dimensional spin Hamiltonian foreach Ge and for each electron spin orientation, com-posing unitary operators by alternating the sign of theelectron spin, and summing the contributions of each nu-cleus on the electrons. The detailed equations describingthe simulations of Ge precession amongst spin swaps ispresented in Appendix C.An example simulation of a CP10 experiment for a5-nm-well device, including only Ge spins which are randomly placed in a simulated crystal, is shown in Fig-ure 9(a). While the detailed structure of the decay willdepend on the specific placement of Ge nuclei in thebarrier of the device, we find this non-interacting simula-tion well-simulates the initial structure of the CP n decay,as evident from a qualitative comparison to Fig. 4. Forthis simulation, we find that drawing the hyperfine split-ting constants ξ j from a Lorentzian distribution with zeromean and root-mean-square width of 10 krad/sec givesdecay characteristics comparable to experiment; varyingthis width by up to an order of magnitude around thisestimate gives insignificant variation relative to the vari-ation resulting from random isotopic placement. Fig-ure 9(a) fails to show the smooth Gaussian decay at highfield or at high values of τ , since it results from a non-interacting, ultimately fully coherent model. To capturethis additional decay, we would have to include the 1 /f noise believed to result from nuclear dipole-dipole inter-actions. As this results from an extended many-body sys-tem, comparable simulations would be more complex, re-quiring methods such as coupled-cluster expansion tech-niques [44, 45] not employed here.The resemblence of this simple quadrupolar model toexperiment is perhaps more evident using the noise spec-trum inferred from data and now from simulation, shownin Fig. 9(b). In this case, we simulate CP10 experimentsfor 20 independent simulated 5-nm-well samples (with Ge isotopic placement in the crystal randomized foreach sample) and plot the average and standard deviationof the inverted noise spectrum, for comparison to Fig. 7.Here, we see that our simple model of non-interacting,quadrupole-split Ge nuclei result in broad noise peaksat γ B z and 2 γ B z with noise power at comparable val-ues to that in Fig. 7(a), with similar reduction in powerwith applied field B z . Again, the low-frequency noisecontent varies from experiment due to the lack of nu-clear dipole-dipole interactions; Fig. 8 suggests a modelin which this additional noise source can be measured(or estimated) at low-frequency and simply added to the Ge noise peak.This same simulation, now using only a singleexchange-swap, may be used to understand the magneticfield variation of Hahn-echo experiments as well. Thethin lines in Fig. 5 are the result of Hahn echo simula-tions from 10 isotopically randomized crystals at each ofthe 5 well widths considered, again determining T asthat particular value of 2 τ where the singlet probabilityfalls to 1 /e . Comparable to the data, the simulationsindicate that T due to non-interacting quadrupolar nu-clei is roughly constant at low field and then diverges atsome critical field B . The location of this “critical” fieldcorresponds to where the Ge Larmor frequency beginsto significantly exceed both the broad hyperfine couplingparameters A k and the quadrupole splittings ξ j , whichare of comparable magnitude [46]. At much higher fields,non-interacting Ge spin projections freeze, and remain-ing noise properties would have to result from spin flip-flops, especially Si dipole-dipole dynamics as we have1
FIG. 9. a) Simulated decay of a CP10 experiment vs. magnetic field under the effects of Ge quadrupolar dynamics only, fora single simulated device with a 5-nm-wide, isotopically enriched Si well. Nuclear dipole-dipole dynamics are not included inthis simulation. b) Inferred magnetic noise spectrum, via Eq. (7), from an ensemble of 20 time-domain simulations such as thatin (a), each using a different random configuration of Ge nuclei. The lines show the average spectrum across the ensemble,and the bands the standard deviation. The circles on each trace show the location of the Ge Larmor peak and the squaresits first harmonic. previously discussed, and are excluded from the simula-tions shown in Fig. 5. As with the CP10 simulations,the quantitative results of T vary weakly with the un-known quadrupole coupling parameters ξ k ; the more crit-ical variation is the microscopic character of the Si/SiGebarrier profile. An interesting open question is whethermore detailed quantitative fitting of data such as we haveshown, or the results of more complex noise spectroscopysequences beyond what we have shown here, may allowmore precise determination of barrier profiles. VI. DISCUSSION
The magnetic noise effects reported here are more ap-parent relative to prior quantum dot studies in part dueto the use of isotopically enhanced Si (i.e. 800 ppm Si).In nuclear-rich materials such as GaAs, spin-orbit andquadrupolar effects are also incidentally larger and somay be observed using Landau-Zener transitions [47, 48]and dynamical decoupling [32, 33, 49].We emphasize, though, that a key reason the noisesources are evident in the devices under discussion inthe present study is that these devices are designed to minimize magnetic gradients, either from nuclear spinsor micromagnets, in order to achieve exchange-only con-trol, whose construction relies on magnetic field homo-geneity [6, 10, 18]. Many Si/SiGe devices presently un-der study include magnetic field gradients, and in these,quadrupolar effects may be difficult to observe due to theinterference of charge-noise [9, 11]. The singlet-tripletmeasurement scheme further allows us to study a widerange of low magnetic fields, not available in single-spinreadout schemes [19, 20].Exchange-only operation is particularly sensitive tomagnetic noise due to the possibility of leakage, even while pulsing [10, 14, 50]. The application of correc-tion sequences may still allow very high fidelity oper-ation, only insofar as the operation time well exceedshigh-frequency cut-offs of the magnetic noise [51, 52]. Forthis, the observation of higher frequency Ge quadrupo-lar noise peaks reported here is most pertinent, butoptimistically we may be confident that dipolar noisecannot extend to arbitrarily high frequencies and Gequadrupolar peaks may easily be circumvented. We findthat the effects of Ge quadrupolar dynamics diminishsignificantly with field, yielding the long, Si-limited T times at easily operational magnetic fields.Recently, quadrupolar spins have shown surprising res-onant response to gate voltage in donor devices [41]. Weneither see nor expect evidence of such effects in thepresent results, but under different designs, we specu-late that the electric quadrupole effects of Ge in SiGemight serve as an interesting lever rather than a decoher-ence source in future silicon spin qubit devices.In summary, we have argued that low-field T inexchange-only qubits in the absence of magnetic field gra-dients results from electrical quadrupole effects from the Ge nuclei, as corroborated by the correct well-width, B -field, and decay-shape dependence relative to theoret-ical models. The strong correspondence is reassuring forthe future of exchange-only qubits in this material, as itsuggests this noise can be circumvented by reasonable ap-plied magnetic fields, by noise compensation techniquesperformed at moderate speeds, or by depletion of the Ge spin isotope in the SiGe material. Nonetheless, ad-ditional research remains in the modeling and detectionof the magnetic noise features studied here to solidifypredictions for the ultimate performance of SiGe qubits.2
ACKNOWLEDGMENTS
We acknowledge valuable experimental contributionsfrom Chris Bohn, Devin Underwood, Laura De Lorenzo,Ed Chen, Cathie Erickson, Mitch Jones, Ari Wein-stein, Matt Reed, Reed Andrews, Matthew Rakher,Matt Borselli, and Andy Hunter; valuable theoreti-cal discussions with Matthew Grace, Wayne Witzel,Seth Merkel, and Chris Schnaible; and assistance withfigures from John B. Carpenter.
Appendix A: Filter Function Formalism
In all experiments, a singlet is prepared in the well-separated double-dot system, and the measurement sam-ples the probability of ending in this same state. In be-tween, for each ensemble instance of an experiment, eachspin is presumed to evolve independently in each dot,resulting in the unitary where S j is the vector spin oper-ator for electron j , and b j ( t ) is a time-dependent, noisyvector. This b j ( t ) tracks the fluctuating magnetic field,for which we assume there may be a static component,notated as B , and vector fluctuating components δ B ( t ).The filter-function formalism is ultimately a linear expan-sion with respect to this fluctuation, and hence employsa first order Magnus expansion for b j : b αj ( t ) ≈ gµ B (cid:126) (cid:90) t ds h jk ( s ) R ( t ) αβ B βk ( s ) , (A1)where repeated indices are summed, superscripts corre-spond to the three spatial vector components, and thesubscripts refer to which dot the electron occupies dur-ing the experiment. Here g is electron g -factor, µ B theBohr magneton, and B k ( t ) is the time-dependent mag-netic field in dot k . The matrix R ( t ) rotates vectorsabout the magnetic field axis at the angular Larmor fre-quency ω = gµ B B / (cid:126) . The time-dependent “switchingmatrix” h jk ( t ) tracks how spins are swapped betweendots during the evolution. For the three experiments in this paper, the switching matrices areFID : h ( s ) = (cid:18) (cid:19) (A2)HE : h ( s ) = (cid:32) (cid:33) , ≤ s < τ (cid:32) (cid:33) , τ ≤ s < τ (A3)CP n : h ( s ) = (cid:32) (cid:33) , mτ ≤ s < (4 m + 1) τ (cid:32) (cid:33) , (4 m + 1) τ ≤ s < (4 m + 3) τ (cid:32) (cid:33) , (4 m + 3) τ ≤ s < m + 1) τ , (A4)where in the case of CP n ( n even), m is an integer rang-ing from 0 to n/ −
1, capturing the repetition of this ex-periment. Note that these matrices are necessarily cen-trosymmetric. As a result of the unitary evolution foreach experiment, the probability of measuring the sin-glet in a single experiment is |(cid:104) S | U ( t ) | S (cid:105)| = (cid:18) cos | b | | b | b · b | b || b | sin | b | | b | (cid:19) . (A5)Note that in the case that all magnetic fields are paral-lel, in which case b j may be treated as a scalar b j , thisexpression simplifies to |(cid:104) S | U ( t ) | S (cid:105)| →
12 + 12 cos[ b ( t ) − b ( t )] , B || B . (A6)We now average these expressions over noisy magneticfields, assuming the noise is stationary and Gaussian.Again consistent with the first-order filter-function ex-pansion, we limit our derivation to the first and secondmoments, allowing the estimated evolution of Eq. (3),which would be exact for the all-parallel-field case understrict Gaussian assumptions. In the case of non-parallelfields with non-zero mean, the constant field B is takenalong the difference vector B − B . To evaluate the sec-ond moment, we make the highly simplifying assumptionthat different vector components of noise within each dotare uncorrelated and identical, and that noise in each dotis uncorrelated and identical: (cid:104) B αj ( t ) B βk ( t ) (cid:105) = δ αβ δ jk (cid:90) ∞ df S j ( f ) cos[2 πf ( t − t )] , (A7)where S j ( f ) is the magnetic noise spectral density for dot j . Under this simplification, we find Eq. (4), where thefilter function F ( f, t ) may be divided into a central lobe F ( f, t ) and two side-lobes F ± ( f, t ) = F ( f ± f , t ); i.e.3the side-lobes are separated from the central lobe by theLarmor frequency ω / (2 π ) = gµ B B /h : F ( f, t ) = F ( f, t ) + F ( f + f , t ) + F ( f − f , t ) . (A8)The central lobe is derived by dividing an experimentinto N intervals, such that t = N τ , and writing F ( f, t ) = N (cid:88) p,q =1 C pq cos[2( p − q ) πf τ ] sin ( πf τ )( πf ) , (A9)where the matrix C pq is given by C pq = 12 ( h k,p h k,q − h k,p h k,q − h k,p h k,q + h k,p h k,q ) , (A10)where h jk,p = h jk ( s ) for ( p − τ ≤ s < pτ . For thethree experiments in this manuscript, the matrix C pq isvery simple. For FID, N = 1 and C FID = 1. For HE, N = 2 and C HE is one for both diagonal components and − n , N = 2 n and the matrix is full of n/ × n/ × C HE ⊗ C HE . We note that the presentformalism may be easily extended to more dots and morecomplex sequences, in which case the C matrices becomemore complex.For FID, that central band has the filter function F ( f, t ) = sin ( πf t )( πf ) (A11)while HE and CP n experiments have the central band de-scribed as Eqs. (5) and and (6), respectively. A key obser-vation from our approximations is that the filter function F ( f, t ) contains not only the central lobe F ( f, t ) above,which is independent of B , but also two equivalent side-lobes separated by the Larmor frequency f . We can keeponly the central band if either (1) all magnetic fields areparallel or (2) if the Larmor frequency is much higherthan all constituent noise frequencies of the underlyingmagnetic field fluctuation. For finite external field andlow frequency magnetic noise, the FID decay includingthe Larmor sidelobes takes the form P S ( t ) ≈
12 + 12 exp (cid:18) − t ω + 4(1 − cos ω t ) T ∗ ω (cid:19) , (A12)reducing to the standard expression in the B → ∞ limit. Appendix B: Static and Low-fluctuation-frequencyField Gradients
If no exchange echo pulses are applied, the simple de-cay experiment is referred to as Free Induction Decay(FID), in analogy to the common experiment in NMR.Since this experiment has a central-lobe filter function,Eq. (A11), that is finite at f = 0, it is sensitive to veryslow changes in magnetic gradients. Those magnetic field FIG. 10. Free induction decay taken at very small externalmagnetic field. The dashed green line is a fit to a Gaussiandecay curve, resulting in a T ∗ = 2.04 µ s. The solid blue lineis a fit to Eq. (A12) and results in a T ∗ = 1.98 µ s with amagnetic field of 29.8 µ T. gradients, which arise due to fluctuating nuclear spins inboth isotopically natural and enhanced samples, are welldescribed as approximately 1 /f Gaussian noise, result-ing predominantly in a Gaussian decay, as in Fig. 8(b).However, we also observe structure in the FID decay insome magnetic field regimes, which we discuss in this ap-pendix.Some structure is observed at very low magnetic field,such as the Earth’s magnetic field, due to the influenceof transverse magnetic field fluctuations. In Fig. 10, weshow a well-averaged FID curve in a device with a 3-nmquantum well taken at Earth’s magnetic field. A fit ofthe oscillations superimposed on the Gaussian decay ishighly consistent with the field-dependent filter functionas derived in Appendix A, fitting to a total applied mag-netic field of 29 . ± . µ T, consistent with the Earth’smagnetic field.At high magnetic fields, static gradients appear, asevident from singlet-triplet oscillations indicated as the (cid:104)| b ( t ) − b ( t ) |(cid:105) term of Eq. (3). An example such curvewas presented in Ref. 6, Fig. 6A, and a series of suchcurves is shown in Fig. 11(a). In both cases, we haveobserved that singlet-triplet oscillations have a frequencyincreasing linearly with applied magnetic field. The sam-ples in the present study, however, see larger gradients atsmaller fields. Figure 11(b) shows the Fourier transformof the singlet-triplet oscillations, whose peaks are compa-rable to Fig. 6B of Ref. 6. In the present study we clearlysee two regimes; at a critical field of about 18 mT, we seethe oscillations suddenly reduce in frequency. The higherfrequency, low-field paramagnetic singlet-triplet oscilla-tions were not observed in the sample of Ref. 6, whichemployed Ti/Au gates. The devices in the present studyuse aluminum gates, suggesting the superconductivity ofaluminum at the 50 mK base temperature of these exper-iments is at play. Moreover, the critical field is found tobe mildly hysteretic, characteristic of near-critical super-conducting effects in aluminum thin-films. In a study on4 FIG. 11. Free induction decay versus magnetic field. (a) FID decay in the time domain, showing sinusoidal variation undera Gaussian envelope as in Eq. (3). (b) Fourier transform of (a), showing that the frequency of the singlet-tripet oscillationsincrease with applied field in two varying regimes. Note that the applied field is taken from a calibration of the current in thepower supply of the superconducting solenoid, which typically carries a few-mT offset from true zero field, evident in this data. another series of samples, we have validated that for themethod and thickness of aluminum deposition we employ,the critical field for superconductivity is highly consistentwith the observed field at which the large singlet-tripletoscillations reduce. We therefore posit that the staticmagnetic field gradient at low-field occurs due to screen-ing of the applied magnetic field from the Meissner effectin the aluminum gates. We have further performed elec-trostatic modeling of the field profile resulting from thescreening expected from perfect Meissner gates in ourgate geometry and found it consistent with the magni-tude of the gradients observed.At higher magnetic fields (e.g. >
18 mT in Fig. 11),and in previous devices with Ti/Au gates, the singlet-triplet oscillations are similar to those also observed insilicon MOS devices [8, 53]. These oscillations are nowattributed to spin-orbit coupling at the barrier Si/SiO orthe Si/SiGe barrier, as evidenced by their characteristicvariation with magnetic field orientation [54].Due to the broad filter function of the time-averagedFID experiment, it is not possible to reconstruct the noisespectrum which gives rise to the Gaussian decay usingonly FID decay curves. Instead, we use the sinusoidalshape of the singlet-triplet oscillations due to Meissnerscreening or spin-orbit effects to perform peak-tracking.By fitting a time-series of FID experiments to Eq. 3, wemay track the location of the phase of the fixed-frequencyoscillation and use this to extract the low-frequency char-acter of gradient noise. This technique was previouslydescribed in Ref. 6 (See Fig. S1 there), but in that work,larger magnetic field magnitudes had to be used to in-duce oscillations, whereas the data presented in Fig. 8was taken at an applied field of 20 mT.As shown in Fig. S1 in Ref. 6 and in Fig. 8 of thepresent work, all silicon samples show a nearly 1 /f spec-trum at fields 10s of mT. This contrasts the most basicmodels of spin diffusion which indicate a 1 /f spectrum.Some recent evidence for 1 /f -like behavior is also seenin the modeling of P:Si donor devices [23]. We positthat in both isotopically natural and 800 ppm Si de-vices, the 1 /f noise results from the large distribution of dipole-dipole timescales resulting from random Siplacement. Subdiffusive behavior has similarly and re-cently also been reported in GaAs dipolar dynamics in asingle quantum dot [49]. Detailed time-domain simula-tions of Si dynamics in quantum dots [45] are neededto verify the underlying reasons for this particular powerlaw. The present study at least verifies that the origin ofthis 1 /f noise is indeed nuclear in origin (in contrast to1 /f magnetic noise arising from charge noise in samplesincluding micromagnets [9, 11]), as its amplitude variesas expected with isotopic content and with quantum wellwidth. Appendix C: Simulation of Ge Quadrupole Effectsunder Exchange Echo Experiments
Here we derive the model used to simulate Ge dy-namics during multiple-exchange-echo experiments for asinglet-triplet qubit in a double quantum dot. For thesinglet-triplet subspace of the two electrons in the dou-ble dot, we may neglect hamiltonian components pro-portional to the total spin-projection operator S z + S z since at fields > σ x operator σ x = S z − S z , since itis magnetic field differences between the two dots thatdrive transitions between the singlet-triplet qubit states.We may therefore write the hyperfine hamiltonian as H hf ≈ (cid:80) k A k I zk − (cid:80) k (cid:48) A k (cid:48) I zk (cid:48) σ x , (C1)where the sums over k and k (cid:48) are the sums over the nucleiin the two dots, and σ x drives singlet-triplet oscillations.(We have hence altered our notation relative to Eq. (8)to drop the j (dot) subscript on A jk , and instead track“which dot” of A k by the location of nucleus k .)Our basic model is that each nucleus, indexed by k and k (cid:48) in Eq. (C1), precesses in its individual hyperfine envi-ronment and interacts only with the common electrons ofthe quantum dot. Nuclear polarization must reconfigure5due to some interaction (dipole-dipole, quadrupole relax-ation, hyperfine-mediated interactions, etc.) but we treatthese reconfigurations as quasistatic; i.e. slow in compari-son to a single spin-echo measurement, albeit fast enoughto observe on the averaging timescale of a time-ensembleof measurements. We also treat the nuclear spin bathas having effectively infinite spin temperature; γ B z issmaller than thermal energies at our applied fields, butmore critically nuclear T at mK would not allow ther-malization to a finite temperature even at the longestaveraging times in the experiments described. These as-sumptions, common in analyses of nuclear baths, enableus to simply trace over the nuclear degree of freedomwhen predicting a measurement result.To analyze decoherence, we consider evolution in thetwo subspaces which are eigenstates of σ x , i.e. | + (cid:105) and |−(cid:105) . (These states in fact correspond to |↑↓(cid:105) and |↓↑(cid:105) forelectron-spins in the double dot). Hence we may decom-pose the relevant terms of our total hamiltonian as H = (cid:88) k H + k | + (cid:105)(cid:104) + | + H − k |−(cid:105)(cid:104)−| , (C2)where H ± k = H Z + H Q ± s k A k I zk , (C3)acts on only nucleus k . Here s k = ± k is from dot 1 or dot 2.Evolution for a time τ under one of these Hamiltoniansis calculated via simple exponentiation, as U ± k = e − iH ± k τ/ (cid:126) . (C4)In an exchange-echo experiment, we begin with a singletstate | S (cid:105) = ( | + (cid:105)−|−(cid:105) ) / √
2. We evolve for time τ to state | ψ ( τ ) (cid:105) = 1 √ (cid:20)(cid:89) k U + k | + (cid:105) − (cid:89) k U − k |−(cid:105) (cid:21) . (C5)The exchange echo then swaps | + (cid:105) and |−(cid:105) , which fol-lowed by another evolution time of τ gives | ψ (2 τ ) (cid:105) = 1 √ (cid:20)(cid:89) k U − k U + k |−(cid:105) − (cid:89) k U + k U − k | + (cid:105) (cid:21) . (C6)For the simple echo, we ask the overlap of this state withthe singlet | S (cid:105) : (cid:104) S | ψ (2 τ ) (cid:105) = 12 (cid:20)(cid:89) k U − k U + k + (cid:89) k U + k U − k (cid:21) . (C7)This is an operator over all the nuclei, for which wenow trace against a maximally mixed density matrix ρ I = 1 / (2 I + 1). As we do not simulate initializationor measurement errors, the predicted singlet-triplet de-cay curve goes exactly as 1 / − χ ( t )] /
2; the decay function is therefore given for HE asexp[ − χ HE ( t )] = 2Tr (cid:40) (cid:20)(cid:89) k U − k U + k + (cid:89) k U + k U − k (cid:21) ρ ⊗ kI (cid:20)(cid:89) k U − k U + k + (cid:89) k U + k U − k (cid:21) † (cid:41) − (cid:89) k Tr (cid:110) U − k U + k U − k † U + k † (cid:111) I + 1 , (C8)where t = 2 τ . For a CP n sequence, which contains n π pulses where n is even, the primitive described above issimply repeated, yieldingexp[ − χ CPn ( t )] = (cid:89) k Tr (cid:110) [ U + k U − k U − k U + k ] n/ [ U − k † U + k † U + k † U − k † ] n/ (cid:111) I + 1 , (C9)where now t = 2 nτ .In practice, we perform simulations by constructingrandom crystals of nuclei where the probability of find-ing a Ge nucleus at a given site is given by the prod-uct of the Ge alloy content (30% Ge in the SiGe bar-rier, 0% in the Si quantum well, and following a single-monoatomic-layer-smeared profile between the two asshown in Fig. 1c) multiplied by the isotopic content (the7.76% natural abundance for Ge). We then constructelectron wavefunctions by solving the Schr¨odinger equa-tion in the vertical dimension for the given Ge-contentprofile, assuming a linear potential offset with Ge-contentand a value of 180 meV at 25%. We find the phaseof vertical valley oscillations by minimizing the overlapof the charge density with the random Ge nuclei place-ment. For the transverse wavefunction, we simply employa Gaussian profile with 1/e diameter of | ψ | at 30 nm,which is consistent with electrostatic calculations for ourgate geometry. With the wavefunction calculated, wefind the hyperfine shifts A k for every nearby Ge nu-cleus; we localize the Ge around each dot by using the400 highest values of A k for Ge nuclei in the randomcrystal. Choosing more nuclei at smaller values of A k has negligible effect on results. For each Ge nucleus,we also draw its ξ k quadrupole-splitting value and elec-tric field gradient angle θ k values from random distri-butions; ξ k is drawn from a Lorentzian distribution ofwidth 10 rad/sec, while θ k values are drawn randomlyas tan − ( x/y ), where x and y are both drawn from zero-mean normal distributions with unity variance. Onceeach nucleus has its parameters randomly drawn as de-scribed, then at the values of τ and B z of interest, wediagonalize H ± k for all 400 nuclei k and both signs ± .We then evaluate χ ( t ) according to Eqs. (C8-C9). Wethen plot 1 / − χ ( t )] /
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