Spectroscopy of transverse and longitudinal low-frequency noiseand their temperature dependencies in a superconducting qubit

Fei Yan1,∗ Jonas Bylander2, Simon Gustavsson2, Fumiki Yoshihara3, Khalil Harrabi3,† David G.

Cory4,5, Terry P. Orlando2,6, Yasunobu Nakamura3,7,‡ Jaw-Shen Tsai3,7, and William D. Oliver2,8

1Department of Nuclear Science and Engineering,Massachusetts Institute of Technology (MIT), Cambridge, Massachusetts 02139,

USA 2Research Laboratory of Electronics, MIT, Cambridge, Massachusetts 02139,USA 3Institute of Physical and Chemical Research (RIKEN), Wako, Saitama 351-0198,

Japan 4Institute for Quantum Computing and Department of Chemistry, University of Waterloo,ON, N2L 3G1, Canada 5Perimeter Institute for Theoretical Physics, Waterloo, ON,N2J, 2W9, Canada 6Department of Electrical Engineering and Computer Science,

MIT, Cambridge, Massachusetts 02139, USA 7Green Innovation Research Laboratories,NEC Corporation, Tsukuba, Ibaraki 305-8501, Japan 8MIT Lincoln Laboratory,

244 Wood Street, Lexington, Massachusetts 02420, USA(Dated: January 30, 2012)

We report a direct measurement of the transverse and longitudinal low-frequency noise thatcauses dephasing of superconducting flux qubits. The former is due to flux fluctuations, and itaffects the qubit’s energy detuning; the latter is parameterized as effective critical-current or chargefluctuations, and affects the tunnel coupling between the qubit states. Our method uses repeatedsingle-shot measurements of the qubit state to resolve the noise-power spectral density at frequencieslimited only by the achievable repetition rate. This approach enables an explicit measurementof the noise over a range of frequencies that directly contributes to the dephasing observed inensemble-averaged time-domain measurements. For both noises, we determine that the very same1/f -type power laws measured at considerably higher frequencies (0.2 − 20 MHz) are consistentwith the 0.01 − 100-Hz range. We find no evidence of temperature dependence of the noises over65−200 mK, and also no evidence of time-domain correlations between the two noises. These resultshelp us understand the properties of noise and dephasing pertinent to all superconducting qubits.

PACS numbers: 03.67.Lx, 74.40.-n, 74.25.Sv, 85.25.Cp, 85.25.Dq

A major remaining obstacle to fault-tolerant quantumcomputation with superconducting qubits is the insuffi-cient coherence time, T2, compared to the gate-operationtime. In the Bloch–Redfield picture of two-level systemdynamics, there are two contributions that both can limitT2 in superconducting qubits: energy relaxation (T1),due to transverse noise (in the qubit’s eigenbasis) at thetransition frequency ν01, and dephasing (Tϕ), due to lon-gitudinal low-frequency fluctuations of ν01. In cases whenboth relaxation and dephasing exhibit exponential de-cay laws, their inverse times add to a decoherence rateT−1

2 = (2T1)−1+T−1ϕ . While T1 can now exceed 10µs [1–

3], energy relaxation is irreversible in the absence ofmulti-qubit quantum error-correction protocols. Dephas-ing, on the other hand, can be refocused by dynamical-decoupling techniques [2], at the cost of added overhead.The ultimate goal is to mitigate and eliminate the noiseleading to both types of decoherence. To this end, a moredetailed understanding of the noise processes – such asmagnetic-flux, critical-current, and charge fluctuations –would expedite materials science, device engineering, andthe development of coherent-control methods.

Effective surface spins have recently been identifiedas one dominant source of low-frequency magnetic-fluxnoise [4, 5], detrimental to several types of superconduct-ing qubits; however, open questions remain regarding thedynamics of these spins. Their noise is known to be due

to local fluctuators [6–9] and the spectrum exhibits a1/fα power-law dependence from hertz to tens of mega-hertz [10–14]. This power law’s dependence on the devicegeometry [15, 16] merits further study.

Similarly, for the flux qubit, the noise in the tunnel cou-pling ∆ between the qubit states shows a 1/f -type spec-trum from hertz to hundreds of kilohertz [2]. This canbe due to critical-current fluctuations in the Josephsonjunctions [17–20]. Another contributor may be the fluc-tuating offset charges, thought to be due to an ensembleof charge traps located in the tunnel barriers or surfaceoxides. Charge noise can lead to dephasing even in theflux qubit, despite its relatively high ratio of Josephson-tunneling to Coulomb-charging energies, EJ/EC ∼ 50.

In this Letter we introduce a new measurement tech-nique for low-frequency noise, extensible to any system,and report on a direct characterization of the 1/f -noise-power spectral densities (PSD) S(f) in an aluminum su-perconducting flux qubit [21, 22] [28]. Its Hamiltonianis H = −(h/2) (εσx + ∆σz): with this choice of eigen-basis, ε is transverse and ∆ is longitudinal. We writeeach of the parameters λ = ε, ∆ as a sum of its nom-inal value and a fluctuation, λ(t) = λ(0) + δλ(t) anddistinguish between the two noises δε, which is effec-tive flux noise, and δ∆, which can be parameterized aseffective critical-current noise or effective charge noise.Interestingly, we find that the same 1/fα power laws,

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measured at much higher frequencies [2], extend down tothe 10−2 − 10+2-Hz range nearly unchanged. Over thetemperature range 65− 200 mK, both noises, δε and δ∆,are independent of temperature; any δε − δ∆-noise cor-relations are very small or non-existent. In contrast toprevious works [13, 15, 23], our technique uses single-shotmeasurements to extend the spectrum to frequencies rel-evant to time-domain experiments such as Ramsey free-induction decay: the upper frequency is limited only bythe system-dependent repetition rate [29].

The Hamiltonian’s two parameters are the tunnel cou-pling, ∆ = 5.4 GHz, which depends on the EJ/EC ra-tio, and the energy detuning between the diabatic states,ε = 2IpΦb/h, which is adjusted by the external magneticflux Φ via Φb = Φ − Φ0/2 (here Φ0 is the flux quantumand Ip = 0.18µA is the persistent loop current), see theschematic in Fig. 1(a). We distinguish between the effectsof δε and δ∆ fluctuations by rotating the qubit’s quan-tization axis (eigenbasis), thereby altering the sensitiv-ity of the level splitting ν01 =

√ε2 + ∆2 to fluctuations,

Dλ = ∂ν01/∂λ = λ/ν01, see Fig. 1(b). This means thatδ∆-noise dominates at ε = 0, whereas δε- (flux) noise isstrong enough to dominate at |ε| & 0.1 GHz.

We use a hysteretic SQUID to read out the qubit’sstate. To directly probe the fluctuations of ν01, we re-peatedly let the qubit undergo Ramsey free inductionwith a fixed pulse spacing τ , see Fig. 1(c–e). Then thequbit’s varying detuning from the applied microwaves,∆ν(t) = ν01(t) − νµw, due to δλ fluctuations, translatesinto fluctuations of the SQUID’s switching probability,psw(∆ν, τ) = p0−a0(T ) cos(2π∆ν τ), where a0(T ) is thetemperature-dependent read-out visibility (2 a0 = 79 %at the refrigerator’s 12-mK base temperature) and p0

is the switching probability for the qubit’s 50-% super-position state. By choosing the nominal detuning νµw

such that, on average, ∆ν = 1/4τ (see Figs. 1(d,f); afree-evolution π/2 rotation in the X−Y plane), we canlinearize about a working point at psw = p0 to obtainδpsw/δν01 = 2π a0(T ) τ . A correction factor arises inthe calculation of ensemble-averaged quantities, assum-ing Gaussian statistics, due to the ν01-to-psw transferfunction; see supplementary material [29]. This factoris a(τ, T )/a0(T ), where a(τ, T ) is the amplitude of thefringe at pulse separation τ . The psw-to-δλ conversionfactor then becomes ηλ(τ, T ) = 2π a(τ, T ) τDλ.

The sampling results in a stochastic, binary time series{zn} of length ttot, with a repetition time of ∆t = 2 msto allow for the read-out induced quasiparticles to re-lax between trials; see Fig. 1(g). Each element zn is aBernoulli trial with expectation value psw. The standardmethod to determine the noise-PSD [13, 15, 23], is tocollect switching events during a gate time tacq = NG∆t,with typically NG ∼ 1, 000, then determine the aver-age switching probability psw, which is binomially dis-tributed, and finally take the Fourier transform of thetime series of switching probabilities psw; see Fig. 2(a).

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FIG. 1: (a) Schematic of the qubit device with biasingand read-out circuitry. (b) Qubit energy-level diagram nearΦb = Φ0/2 (ε = 0). (c) Double-π/2 pulse sequence (Ramseyinterference). (d) Bloch sphere representation of dephasingand our measurement scheme: the initial π/2 pulse puts theBloch vector on the equator (gray arrows); during the free-induction time τ , the nominal detuning is ∆ν = 1/4τ , butslightly different at each trial (quasi-static noise), and theBloch vectors therefore acquire different phases; the final π/2pulse rotates the Bloch vectors out of the equatorial plane sothat they can be distinguished by the SQUID read out. (e)Ramsey fringe measured at ε = 0 with 5, 000 averages perpoint. (f) Calibration by scanning νµw with τ = 0.3µs, ε = 0.Red is a sinusoidal fit used to determine the working point formaximal sensitivity (dashed, line at ∆ν = −0.8 MHz). (g)Individual SQUID-switching events measured on an oscillo-scope. The dashed, horizontal line is a software thresholddetector.

It gives us access to the frequency range 1/ttot− 1/2tacq.Alternatively, we can directly calculate the (bilateral)

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nS

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δ∆ noise

Ensemble averaged Single shot

T = 65 mK

(Single-shot method)

FIG. 2: Bilateral noise PSDs. (a) Standard method PSD of the ensemble-averaged time series of switching probabilities psw

(blue data) and of the binary time series of single-shot measurements {zn} (gray), measured at base temperature. The redpoints are averages of hundreds of traces. (Sε measured at ε = 450 MHz.) (b) Cross-PSDs of interleaved time series, Sλ(f)(Eq. 2), vs. device temperature (Sε measured at ε = 640 MHz). The data was smoothed by a sliding average with a triangularweight function of width ∆f = f/4. The dashed, diagonal lines are the 1/f noises Sλ(f) = (2π)2κ2

λAΦ,ic/|f |, derived inRef. [2]; the sensitivities are κ∆,ic ≡ ∂∆/∂ic = 3.2 GHz (ic is normalized critical current) and κε ≡ ∂ε/∂Φ = 1.1 GHz/mΦ0,and the noise strengths are Aic = (4.0 × 10−6)2 and AΦ = (1.7µΦ0)2. The solid, diagonal line is Sε(f) = (2π)2κ2

εA′Φ/|f |0.9,

with A′Φ = (0.8µΦ0)2. See Ref. [2] and its supplementary information for details. The horizontal, green, dashed lines are thesampling-noise levels Sn(τ, T ) at low temperature; the triangles represent an upper cut-off frequency fc for sufficient averaging,above which the data is not dependable.

PSD of the series of single-shot measurements (Bernoullitrials), S(fk) = |Zk|2/(N/∆t), where fk = k/N∆t,k = 0, . . . , N/2, and {Zk} is the discrete Fourier trans-form of {zn} [30], with n typically ranging from 1 toN = 5 × 104 (ttot = N∆t = 100 s). This method in-creases the upper cut-off frequency from 1/2NG∆t to1/2∆t, which may approach 1/τ and is limited by theachievable repetition rate; see Fig. 2(a–b) and the sup-plementary material [29].

Both the PSDs originating from single-shot andfrom ensemble-averaged time series in Fig. 2(a) suf-fer from the statistical-sampling noise Sn(τ, T ) =(2π)2 σ2

s ∆t/η2λ(τ, T ), where the variance is σ2

s = psw(1−psw). We can implicitly eliminate this white backgroundnoise by calculating the (bilateral) cross-PSD of the twointerleaved (single-shot) time series z′n = z2n−1 andz′′n = z2n,

Scross(fk) =Z ′k(Z ′′k )∗

N/∆t, fk =

k

N∆t. (1)

Now k = 0, . . . , N/4. Dividing Eq. (1) by the conversionfactor, we obtain the angular-frequency correlator,

Sλ(fk) = (2π)2 Scross(fk)/η2λ(τ, T ). (2)

We typically average the spectra of 500 time series forimproved statistical accuracy, recalibrating the workingpoint periodically. Note that both the 1/f noise and the

sampling noise dominate all other background noise atthe temperatures considered.

The δε and δ∆ noise-PSDs are plotted in Fig. 2(b)for several temperatures. There is striking agreementwith the 1/fα power laws inferred in Ref. [2], measuredat considerably higher frequencies (0.02 − 20 MHz). Atthe flux-insensitive bias point, ε = 0, the δ∆ noiseis responsible for the observed Ramsey dephasing, assecond-order δε noise is negligible [2]. For δ∆ noise thatis strictly 1/fα=1 over the frequency range relevant toRamsey free-induction (10−1 ∼ 106 Hz), we expect aGaussian decay. Using the approach of Ref. [24], withε = 0, typical pulse spacing τ = 1µs, acquisition timetacq = 10 s per measured point, and noise sensitivity κ∆,ic

and strength Aic as defined in the legend of Fig. 2, wecalculate an inhomogeneously broadened time constant

T ∗ϕ = (2π κ∆,icD∆)−1A−1/2ic

(ln[tacq/2 τ ])−1/2 = 3.2µs, invery good agreement with the observed T ∗ϕ in Fig. 1(e),after taking the 12-µs T1 contribution into account.

We now turn to possible δε − δ∆-noise correlationsin the time domain, to exclude that the two spectra inFig. 2 are due to one and the same mechanism. We candetermine the correlation amplitude γε∆ by repeatedlymeasuring the switching probability at alternating fluxbiases, psw(±ε,∆) = p±, with ±ε chosen such that theeffects of δε and δ∆ noise on the qubit are similar inmagnitude; see Fig. 3. We then get access to the noise

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FIG. 3: δε−δ∆-noise correlations measured at base temper-ature. (a) Top panel: Time series of psw with fixed τ and∆ν, and 1,000 averages per measured point. The flux-biaspolarity was alternated between positive (blue) and negative(red), ±Φb = ±0.058 mΦ0, see inset in (b), with a 2-s rep-etition period. Middle and bottom panels: Differential(δε) and common-mode (δ∆) noise inferred from the data inthe top panel. (b) Correlation amplitude (4). Inset: Ram-sey free-induction decay with τ = 0.3µs: plot of psw vs. Φb

and ∆ν measured over 30 min. White circles indicate the biaspoints used in the top panel of (a).

correlations on time scales slower than about 2 s, the timeit takes to average and to shift the flux bias. Keepingτ and the nominal ∆ν fixed, δ∆ fluctuations lead to acommon-mode fluctuation in the level splitting ν01 for ±εbiases, and consequently a positive correlation betweenthe p± shifts, whereas δε fluctuations lead to a differentialfluctuation, and therefore anticorrelated p± shifts. Weuse the decay function of the Ramsey fringe,

psw(ε,∆) = p0 − a0 exp (−τ/2T1)× (3)

× exp(−[τ/T ∗ϕ(ε)

]2)cos (2π∆ν τ) ,

where [1/T ∗ϕ(ε)]2 = [1/T ∗ϕ(0)]2 +KAΦD2ε and K is a con-

stant. For each point in time, we obtain a system of twonon-linear algebraic equations in two unknowns, fromwhich we determine the δε and δ∆ noises numerically,as shown in Fig. 3(a). We then calculate the cross-PSD

0 100 200Device temperature, T (mK)

10

Π

([r

ad/s

] x

10

)

0

210

∆

Π ([rad/s] x 10 )

214

ε

δ∆ noise

δε noise

5

0

6

FIG. 4: Temperature dependencies of the total noise powersΠλ(T ) = (2π)2κ2

λ

∫Fλ

df Sλ(f ;T ) in the frequency intervals

Fε = 0.02 − 50 Hz and F∆ = 0.02 − 2 Hz, cf. Fig. 2(b). It ispossible to measure the δε noise up to somewhat higher tem-peratures and frequencies than the δ∆ noise. Note that Πλ de-pend on the integration limits although the choice of Fλ doesnot make any significant difference in the trends. (The dou-ble data points for the 165- and 180-mK δ∆ noise were mea-sured with different pulse spacings τ .) We estimated the de-vice temperature from switching-current measurements on theSQUID, which suggest saturation at dilution-refrigerator tem-peratures below about 65 mK. The error bars are derived onlyfrom the fit error of the read-out visibility a(τ, T ), includedin the psw-to-∆λ conversion factor ηλ(τ, T ), see Fig. 1(f).

Sε∆(f) as the Fourier transform of the cross-correlationfunction, and obtain an upper bound on the correlationamplitude, shown in Fig. 3(b),

γε∆(f) =|Sε∆(f)|

[Sε(f)S∆(f)]1/2< 0.2. (4)

Finally, we discuss the temperature dependencies ofthe two different types of noise. Figure 4 shows the inte-grated noise powers Πλ vs. temperature T in the 65−200-mK range, where our read-out visibility is sufficient. Weobserve in essence temperature independence for bothnoises. For δε (flux) noise, this is consistent with previ-ous observations in SQUIDs [5, 10]; we discuss the δ∆noise below.

In order to analyze the δ∆ noise, we parameter-ize it as an effective, normalized critical-current noise,ic = δIc/Ic, with Ic = 0.4µA, in a Josesphsonjunction with area A = (0.2µm)2. Van Harlin-gen et al. [17] found a “canonical” value for the 1/fδIc-noise power at 1 Hz and 4.2 K: it was close toAcanIc

= 144 (pA)2(Ic/µA)2/(A/µm2) in several SQUIDsand qubits of various sizes, made of different materi-als. The authors hypothesized that it would have aquadratic temperature dependence, consistent with cer-tain plausible models for the noise sources below 100 mK(they note that some other models suggest a linear de-pendence). The (bilateral) normalized noise-PSD thenbecomes Scan

ic(f) = Acan

IcI−2c (T/4.2 K)2/|f |, which, for

T = 65 mK, is considerably lower (almost 20 times)

5

than our measured value. On the other hand, Eroms etal. [20] measured resistance fluctuations in tunnel junc-tions of the type our device is made of: they found ∼100times lower noise power at 4.2 K, a linear temperaturedependence, and saturation below 0.8 K, i.e., Sic(f) =(1/100) × Acan

IcI−2c (T/4.2 K)/|f |. With T = 0.8 K, this

gives a value about 2.5 times lower than what we observe.We also note that recently, contrary to these findings,Paik et al. [3] reported no evidence for 1/f ic noise in aJosephson junction.

An alternative source of δ∆ noise is the fluctuatingoffset charges, δQ, known to exhibit 1/f noise [25–27];these charges effectively supply a continuous gate volt-age to each island. The charge-noise power typicallyobserved in single-electron tunneling (SET) devices isproportional to temperature [31] (although quadraticdependencies have also been observed [26]) and satu-rates below ∼200 mK, due to self heating of the SET,at a “canonical” value AQ of about (1 ∼ 10 me)2 at1 Hz. We estimate our qubit’s maximum sensitivityto charge fluctuations, κ∆,Q ≡ ∂∆/∂Q, to be in therange 0.1∼1 MHz/e. We can then parameterize the δ∆noise as charge noise and estimate the dephasing time

T ∗ϕ = (2π κ∆,QD∆)−1A−1/2Q (ln[tacq/2 τ ])−1/2 ≈ 4−400µs.

The lower end of this range is not far from our observedvalue. Moreover, the tunneling of charged quasiparticlesbetween the small islands constituting our device maydisplace offset charges and contribute to dephasing atε = 0.

In conclusion, our spectroscopy of both δε noise (fluxnoise) and δ∆ noise (effective critical-current or chargenoise), facilitated by single-shot measurements and thor-ough data analysis, shows that the very same 1/fα de-pendencies, measured at substantially higher frequencies,extend down to millihertz frequencies. This apparentlyindicates that the same noise mechanisms are active anddominant over some ten orders of magnitude or morefor δε noise and at least eight orders of magnitude forδ∆ noise. The δε noise may extend, with roughly con-stant slope (on a logarithmic scale), up to the qubit’stransition frequency at several gigahertz [2]: there, thisnoise is nearly transverse to the flux qubit’s eigenbasis,and would therefore also contribute to energy relaxation.The small, if not negligible, δε − δ∆-noise correlations(over 7 × 10−4 − 2 × 10−1 Hz) show that the noises aredue to distinct underlying mechanisms. Moreover, bothnoises are temperature independent in the 65− 200-mKrange, which suggests that the microscopic mechanismsare dominated by even lower energy scales than that.This is useful information for the development of noisemodels, and it also calls for further studies of the δ∆noise, in particular, as it limits the coherence time insuperconducting flux and transmon qubits.

We acknowledge discussions with G. Chen, L. DiCarlo,M. Gustafsson, X. Jin, and L. Wang. We thank theLTSE team at MIT Lincoln Laboratory for technical

assistance. This work was sponsored by the U.S. Gov-ernment; the Laboratory for Physical Sciences; the Na-tional Science Foundation; and the Funding Program forWorld-Leading Innovative R&D on Science and Technol-ogy (FIRST), CREST-JST, MEXT kakenhi “QuantumCybernetics.” Opinions, interpretations, conclusions andrecommendations are those of the author(s) and are notnecessarily endorsed by the U.S. Government.

During the preparation of this paper, we became awareof related results on low-frequency, low-temperature fluxnoise by Sank et al. [23].

∗ fyan@mit.edu† Present address: Physics Dept., King Fahd University of

Petroleum & Minerals, Dhahran 31261, Saudi Arabia‡ Present address: Dept. of Applied Physics, University of

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[23] D. Sank, R. Barends, R. C. Bialczak, Y. Chen, J. Kelly,M. Lenander, E. Lucero, M. Mariantoni, M. Neeley,P. J. J. O’Malley, et al. (2011), arXiv:1111.2890.

[24] J. M. Martinis, S. Nam, J. Aumentado, K. M. Lang, andC. Urbina, Phys. Rev. B 67, 094510 (2003).

[25] K. Bladh, D. Gunnarsson, A. Aassime, M. Taslakov,R. Schoelkopf, and P. Delsing, Phys. E 18, 91 (2003).

[26] O. Astafiev, Y. A. Pashkin, Y. Nakamura, T. Yamamoto,and J. S. Tsai, Phys. Rev. Lett. 93, 267007 (2004).

[27] N. M. Zimmerman and W. H. Huber, Phys. Rev. B 80,195304 (2009).

[28] The Al/AlOx/Al device was made by shadow evapora-tion at NEC; the experiments were performed at MIT.

[29] See supplementary material.[30] Contrary to Refs. [2, 11, 12], we use the type-1 Fourier

transform, Sx(f) =∫∞−∞ dt〈x(0)x(t)〉 exp(−i2πft).

[31] M. Gustafsson, private communication (manuscript inpreparation).

7

SUPPLEMENTARY MATERIAL

Spectral density and the statistical noise floor

Here we describe how we calculate the noise-powerspectral density (PSD) from the noisy time series, andeliminate the statistical white-noise floor due to sam-pling.

PSD

The fluctuations of our qubit’s transition frequencyconstitute a zero-mean, wide-sense stationary processδν(t); at our chosen working point, δν(t) = ∆ν(t)−1/4τ .We seek its bilateral noise-PSD (in units of rad/s, i.e., weuse the angular-frequency correlator),

S(f) = (2π)2 limT→∞

⟨|V (f)|2

⟩T

. (5)

Our measurements’ raw data, however, consists of a bi-nary time series {zn} with elements of expectation valueyn = 〈zn〉 = p0 + a0 sin(2π δν(n∆t) τ), where ∆t is thetime step. The statistical properties of {zn} representthose of the underlying process δν(t), up to a conversionfactor and a correction factor (explained in the next sec-tion). We can therefore take the discrete Fourier trans-form {Zk} = F [{zn}], identify Z(fk) = Zk ×∆t for fk =k/N∆t, and compute the discrete, bilateral noise-PSDover the frequency range from 1/ttot = 1/N∆t = 10 mHzto 1/2tacq = 1/2∆t = 250 Hz,

Sk=0 = (2π)2 1

2

Z20 (∆t)2

N∆t, Sk 6=0 = (2π)2 Z

2k (∆t)2

N∆t,

(6)

for k = 0, . . . , (N − 1)/2 with fk =k

N∆t.

We then take the statistical average of M different PSDsobtained from different time series measured in succes-sion,

〈Sk〉stat =1

M

M∑m=1

S(m)k , (7)

and finally smooth the result with a sliding average inthe frequency domain.

With this method (Eqs. 6–7), each element zn is asingle-shot measurement; the sampling time step tacq isthe same as the pulse sequence repetition time ∆t. Thissets it apart from the standard approach of first takingthe ensemble average of typically NG = 1, 000 samplesin the time domain, before calculating the PSD of theresulting N/NG sampled points. The acquisition timeis then tacq = NG∆t, and the upper cut-off frequencybecomes only 1/2tacq ≈ 0.25 Hz.

White-noise floor

The PSD of the single-shot time sequence suffers fromstatistical sampling noise because each time step n consti-tutes a Bernoulli trial (bn): the read-out SQUID switches(bn = 1) with probability p and does not switch (bn = 0)with probability 1− p. This statistical noise has a whitespectrum; it dominates possible white background noisefrom other sources, and dominates also the 1/f noise athigh frequencies. To estimate it, we can treat the stochas-tic variable bn as independent and identically distributed(i.i.d.) with ensemble-averaged mean 〈b〉 = p and vari-ance σ2

b ≡ 〈(∆b)2〉 ≡ 〈b2〉 − 〈b〉2 = p (1 − p). Samplingat a fixed rate 1/∆t, the white-noise floor of the bilateralPSD becomes

Sn(fk) = (2π)2(σ2b + 〈b〉2δk,0

)∆t. (8)

Here we use Kronecker’s delta δk,0 in the discrete PSD.The same expression is valid for the PSD of the

ensemble-averaged time series, whose elements consti-tute a binomially distributed switching probability (c for“counts”) during a gate time tacq = NG∆t : we obtainEq. (8) after substituting 〈c〉 = p and σ2

c = p (1− p)/NGfor 〈b〉 and σ2

b , respectively.Equation (8) is, in fact, a modification of Carson’s

theorem, which is valid for temporally random pulsearrivals. There, one considers a random pulse trainw(t) =

∑Ll=1 bl g(t − tl), in which g(t) is the pulse

envelope, the stochastic variable bl is the (continuous)pulse height, and the stochastic variable tl is the pulse-arrival time. The Fourier transform of w(t) is W (f) =

G(f)∑Ll=1 bl exp(−i2πftl), where G(f) = F [g(t)]. Car-

son’s theorem is then

SCarsonn (fk) = (2π)2

(〈1/∆t〉〈b2〉|G(fk)|2 + 〈b〉2δ(fk)

).

(9)In our case, the pulse height bl is binary and the pulse-arrival rate is fixed at 1/∆t; we can therefore write (withKronecker’s delta) G = F [δm,0] = 1 ×∆t. We just haveto replace the mean-square 〈b2〉 by the variance σ2

b andset δ(fk) = δk,0∆t to obtain Eq. (8).

Parenthetically, one can also derive Eq. (8) by using theWiener–Khintchine theorem. The autocorrelation func-tion is

R(m)bb = (1/N)×

N−1∑n=0

bnbn−m = 〈b〉2 + σ2b δm,0, (10)

and F [1] = δk,0 ×∆t, so that

Sn(fk) = (2π)2F[R

(m)bb

]= Eq. (8). (11)

We find that the PSD resulting from a simulation ofBernoulli- and binomially distributed noise agrees wellwith the measured data and with Eq. (8): we therefore

8

conclude that our experimental noise floor is due to thestatistical sampling.

If the data consisted of a train of pulses of finite lengthin time, the PSD would have a roll-off near the Nyquistfrequency 1/2∆t. For example, the Fourier transformof a boxcar (square) pulse of length ∆t is the func-tion ∆t sinc (πf∆t). In our case, after conversion of theSQUID’s response (the presence or absence of a voltagepulse) to binary form, our data can be seen as representedby a train of delta-functions, and their Fourier transformis frequency independent, i.e., our white noise floor hasno roll-off.

Cross-PSD: implicit white-noise elimination

In order to implicitly eliminate the white-noise floor, atthe expense of a halved Nyquist frequency, we calculatethe discrete cross-PSD of interleaved time series, i.e. bysetting z′n = z2n−1 and z′′n = z2n (with n = 1, . . . , N/2)and computing the cross spectrum of z′n with z′′n. (Weagain assume that the stochastic switching process is un-correlated from sample to sample, at frequency 1/∆t.)The resulting PSD is

Scrossk=0 = (2π)2 1

2

Z ′k (Z ′′k )∗

N/2∆t, Scross

k 6=0 = (2π)2 Z′k (Z ′′k )

∗

N/2∆t,

(12)where k = 0, . . . , N/4 and fk = k/N∆t. These ex-pressions reproduce the noise spectrum, with the use ofcorrection factors, as explained in the following section(Eqs. 18–24).

Compared to the previous section, we have eliminatedthe white noise by circumventing the zero-delay autocor-

relation term R(0)bb in Eq. (10), and are only left with the

delta-function component,

Scrossn (fk) = (2π)2 〈b〉2 δk,0∆t. (13)

In the same way as in Eqs. (10–11), this can be de-rived by applying the Wiener–Khintchine theorem tothe cross correlation function, which this time simply

gives R(m)b′b′′ = 〈b〉2 (the subscript b′b′′ indicates two in-

terleaved sub-series obtained from the original series bn),and therefore

Scrossn (fk) = (2π)2 F

[R

(m)bb′

]= Eq. (13). (14)

When calculating the PSD, we take the statistical aver-age of Scross

k , keeping the averaging coherent throughout(i.e. retaining Scross

k as a complex quantity), and, just asfor Sk, smoothen it with a sliding average before plottingits magnitude |〈〈Scross

k 〉stat〉fq|.The result (12) is equivalent to the explicit subtraction

of the incoherent noise Sn from Sk (Eqs. 6, 8), with theadvantage, however, of drastically reduced uncertainty,in particular at high frequencies where the (1/f -noise)

signal is much smaller than the white noise. This methodis appropriate for the analysis of, e.g., 1/f -type noise.However, it is not applicable in a predominantly white-noise environment: then, the noise under study would beeliminated along with the statistical white noise.

Correction factors: Quasi-static noise and thenon-linear transfer function

In this section, we treat the effects on the PSD causedby quasi-static noise, and by the sine nonlinearity in theconversion from the measured switching events to thevariations of the qubit’s transition frequency.

Decay of the Ramsey fringe – quasi-static noise

Noise in the effective longitudinal field coupled to thequbit results in decoherence of the quantum superposi-tion. We denote a fluctuation as “quasi-static” or “inco-herent” noise, when it can be considered as static duringeach free-induction period, but varying over the longertime span between experimental realizations. Dephasingresults from such uncorrelated fluctuations of the Lar-mor frequency ν01, and therefore of the accrued phase ofthe superposition state, ϕ(τ) = 2π

∫ τ0

dt ν01(t). It leadsto decay of the Ramsey free-induction signal, as eachmeasured point is the incoherent average of many exper-imental realizations. We describe this fluctuation by astandard deviation,

σ2 = 2

∫ 1/τ

1/tacq

df S(f). (15)

The higher integration limit is here the inverse of thefree-induction time, 1/τ ∼ 0.1 − 100 MHz; fluctuationsat even higher frequencies are effectively canceling out.The lower limit is given by the total acquisition time tacq

used to infer the qubit’s population at each fixed free-induction time span τ . Typically averaging over Navg =5, 000 measurements with a repetition time ∆t = 2 ms,we obtain tacq = Navg∆t = 10 s. (If instead the measure-ments were done in the opposite order, stepping over τin the inner loop, with Npts ∼ 100 steps, and averagingover Navg in the outer loop, the total acquisition timewould be NptsNavg∆t = 1, 000 s, and the lower cut-offfrequency would be correspondingly lower.)

Ensemble averaging over all realizations of δϕ(τ), andassuming Gaussian fluctuations resulting from numerousfluctuators, we obtain the dephasing envelope

h(τ) = 〈exp(iδϕ(τ))〉 = exp(−〈(δϕ)2〉/2) = (16)

= exp

(−τ2

∫ fhigh

flow

df S(f) sinc2(πfτ)

)≈

9

≈ exp(−σ2 τ2/2

),

where the sinc-squared function is due to the square timewindow of the Ramsey pulse sequence, and we can ap-proximate it by unity for f < 1/2τ .

As an illustration, we now evaluate h(τ) for the twocases of 1/f noise and white noise. For 1/f noise, S(f) =A/f , Eq. (15) becomes σ2 = 2A log [(1/2τ)/(1/tacq)].The weak, logarithmic sensitivity to the cut-off frequen-cies effectively allows us to treat it as a time-independentconstant, σ2 ≈ 2AC, giving Gaussian decay p1/f (τ) =exp(−σ2 τ2/2). For white noise, S(f) = Sw, on the otherhand, the integral is linearly sensitive to the upper cut-off frequency, so that σ2 = Sw/τ , yielding an exponentialdecay pw(τ) = exp(−Swτ/2). Here the exponent is pro-portional to time; we can therefore identify 1/T ∗ϕ = Sw/2as the dephasing rate.

Repeated fixed-time free-induction

The previous section described how quasi-static noisedetermines the dephasing of the Ramsey-fringe. Now weturn to its effect on Ramsey interference with a fixedfree-induction time τ , repeated numerous times.

With our single-shot measurements, each element ofthe binary time series {zn} is a Bernoulli random vari-able zn with expectation value given by the switchingprobability psw, which we now denote

yn = p0 + a0 sinxn. (17)

This function has a non-linear dependence on xn =2πδνnτ , the phase accrued during τ , where δνn is theaverage fluctuation of the transition frequency at timestep n. This phase xn, in turn, has noise contributionsfrom two distinct frequency intervals, “1” and “2.”

We denote as “interval 1” the frequencies which wecan resolve by taking the Fourier transform of the series{zn}, of total length N∆t and step size ∆t, i.e. from1/ttot = 1/N∆t ∼ 10−2 Hz to 1/2tacq = 1/2∆t ∼ 250 Hz(or with the interleaving method up to 1/4∆t ∼ 125 Hz).The noise within this interval has zero mean and varianceσ2

1 (Eq. 15).In addition, there is a contribution from the quasi-

static noise in “interval 2,” which is the range from1/2tacq to 1/τ ; see Fig. 5. This noise cannot be resolved,but acts in aggregate and leads to dephasing, e.g. in aRamsey-fringe experiment. It has zero mean and vari-ance σ2

2 (Eq. 15).Noise at even higher frequencies than 1/τ averages out

during free induction.At each time step n, the element xn is subject to noise

contributions from both intervals: we write xn = un+vn,where u and v refer to the noise originating in intervals1 and 2, respectively. Here un has correlations betweenthe different time steps n due to the memory effect of the

Range relevant for time-domain meas.

f

S

1 1

σ 12

τt tot

1/2tacq (avg)

Resolved bysingle-shot meas.

Resolved byensemble avg.

Quasi- static noise

σ22Interval 1:

Interval 2

1/2tacq (sing)

FIG. 5: Sketch of the PSD, indicating the frequency inter-vals resolved by the ensemble-averaging and single-shot sam-pling methods. Also indicated are the Gaussian, quasi-staticnoise and the variances σ2

1,2 (Eq. 15). Here ttot is the totallength of the time trace (can be several minutes to hours);tacq(avg) = NG∆t = 1 ∼ 10 s is the acquisition time per mea-sured point in time-domain experiments such as Ramsey andspin-echo decay; tacq(sing) = ∆t = 2 ms is the repetition time(acquisition time of the single-shot samples); and τ ∼ 1µs isthe pulse spacing.

1/f noise; on the other hand, vn is incoherent and canbe taken as a Gaussian i.i.d. random variable.

While it is impossible to unequivocally infer xn fromthe measured zn at each instance n, we can infer statisti-cal properties of {xn}, such as its correlations and spec-tral density, up to the frequency 1/2tacq = 1/2∆t, whichcan approach 1/τ . This is advantageous compared to theensemble-averaging method, which has a longer acquisi-tion time tacq = NG∆t.

We can write the m 6= n autocovariance function for∆zn = zn − 〈zn〉 as

〈∆zm ∆zn〉 = 〈∆ym ∆yn〉 = a20〈sinxm sinxn〉 ' (18)

' a20〈xm xn〉 = a2

0〈um un〉.

The first equality holds because the Bernoulli trials areindependent, and the last equality is the consequence ofvn being i.i.d., which implies 〈um vn〉 = 〈vm vn〉 = 0.The third step is an equality only when |xn| � 1; whenσ2 is large, e.g. at higher temperatures, or when we usea larger free-induction time τ to decrease the statisticalnoise level, the variation of xn can be large, and thenthis is not a good approximation. Instead of approxi-mating, however, we can compensate the result for thesine nonlinearity. Expanding the correlator 〈∆ym∆yn〉,we obtain

〈sinxm sinxn〉 = 〈sin(um + vm) sin(un + vn)〉 = (19)

= 〈(sinum cos vm + cosum sin vm)(m→ n)〉.

10

Since sine is an odd function and vn is a zero-mean, Gaus-sian i.i.d. variable, 〈sin vm,n〉 = 0, and (19) becomes

〈sinum cos vm sinun cos vn〉 = (20)

= 〈cos vm〉〈cos vn〉〈sinum sinun〉.

The cosine factors depend on noise in interval 2, i.e.,above the sampling frequency. This is similar to dephas-ing due to quasi-static noise, which acts uniformly on allthe samples in time (incoherent averaging over a distribu-tion of the noise), and leads to Gaussian decay functions

〈cos vm,n〉 = exp(−σ22τ

2/2). (21)

For the sine factor, the noise is from interval 1, i.e., itis resolved by our sampling, and therefore is not uniform.The process is a combination of ensemble-averaged inco-herent noise and a frequency-dependent filtering due tothe (m − n)∆t time difference in the correlator. Eval-uating this factor, we obtain Gaussian damping of ahyperbolic-sine function of the correlator,

〈sinum sinun〉 = (22)

=

∫∫dumdun sinum sinunN(0, σ) =

= exp(−σ21τ

2) sinh(〈umun〉),

where the integral is taken over a two-dimensional nor-mal distribution N(0, σ) with zero mean and correlationmatrix σ = {σm, σn, σmn}. (The distribution widths areequal, σm = σn, and σmn = 〈umun〉 is the correlationfunction).

The correlator (18) finally becomes

〈∆zm∆zn〉 = (23)

= a20 exp(−σ2

2τ2) exp(−σ2

1τ2) sinh(〈umun〉).

Note that no approximation has been made so far (cf.Eq. 10). If the noise correlation due to 1/f -type noiseis small, as in our case where exp(σ2

1τ2) < 10, we can

neglect the frequency-dependent filtering effect and ap-proximate sinh(〈umun〉) ≈ 〈umun〉.

Now remains only the determination of the correctionfactors, which we know from the calibration measure-ment, exp

[(σ2

1 + σ22)τ2

]= (a0/a(τ))2, where we identify

a(τ)/a0 = h(τ) (Eq. 16), so that, finally,

〈umun〉 ≈ 〈∆zm∆zn〉/ (a(τ))2. (24)

We note that it resembles the signal damping due to de-phasing in a Ramsey fringe. The actual numbers used inour analysis of the data in Figs. 2 and 4 of the main textare presented in Tables I–III.

TABLE I: δε noise (ε = 640 MHz). Data in the main text’sFigs. 2 and 4.

Temp. (mK) τ (ns) exp[(σ2

1 + σ22

]τ2)

65 50 1.5

120 50 1.6

165 50 1.6

210 50 1.8

TABLE II: δ∆ noise (ε = 0). Data in the main text’s Figs. 2and 4.

Temp. (mK) τ (ns) exp(σ2τ2)

65 300 1.2

120 300 1.3

165 1,200 7.4

180 1,000 5.6

TABLE III: δ∆ noise (ε = 0). Data in the main text’s Fig. 4(but not in Fig. 2).

Temp. (mK) τ (ns) exp(σ2τ2)

165 300 1.3

180 300 1.6

Data smoothing and reproducibility of the PSD

The following Figs. 6 and 9 show that our PSD’s powerlaws are independent of the choice of smoothing windows.Figures 7–8 show the reproducibility of our results, withsets of data taken on different days.

11

0.01 0.1 1 10 100

107

108

109

1010

1011

1012

0.01 0.1 1 10 100

-1

- 0.5

0

0.5

1

0.01 0.1 1 10 100Frequency (Hz) Frequency (Hz) Frequency (Hz)

PSD

, S

(ω)

(rad

/s)

Cros

s-PS

D p

hase

(ra

d)

∆f/f = 1/10 ∆f/f = 1/4 ∆f/f = 1/2

65 mK120 mK165 mK180 mK

All plots

∆

FIG. 6: δ∆ noise with different smoothing windows ∆f/f . We choose the upper cut-off frequency fc for the main paper’sFig. 2 as the lowest frequency for which the phase of the cross-PSD deviates from zero by more than 1 rad. In that figure weuse the smoothing window ∆f/f = 1/4. The spectrum displays no significant difference depending on ∆f/f , and the structurecan be attributed to insufficient averaging. The phase deviation is, also, due to insufficient averaging, and becomes larger forincreasing temperature, for fixed pulse separation τ .

10−2

10−1

100

101

10210 7

10 8

10 9

1010

1011

1012

Frequency (Hz)

PSD

, S

(ω)

(rad

/s)

∆f/f = 1/4

T = 165 mK

τ = 0.3 µsτ = 1.2 µs

∆

FIG. 7: δ∆ noise at 165 mK with two different pulse spacings τ , showing reproducibility of the noise-PSD and Tables II–III.

12

10−2

10−1

100

101

102

107

108

109

1010

1011

1012

Frequency (Hz)PS

D,

S (ω

) (r

ad/s

)

∆f/f = 1/4

T = 180 mK

τ = 0.3 µsτ = 1.0 µs

ε

FIG. 8: δ∆ noise at 180 mK with two different pulse spacings τ , showing reproducibility of the noise-PSD, cf. Fig. 7 andTables II–III.

0.01 0.1 1 10 100

1011

1012

1013

1014

1015

0.01 0.1 1 10 100

−2

0

2

0.01 0.1 1 10 100

Frequency (Hz) Frequency (Hz) Frequency (Hz)

PSD

, S

(ω)

(rad

/s)

Cros

s-PS

D p

hase

(ra

d)

∆f/f = 1/10 ∆f/f = 1/4

∆

65 mK120 mK165 mK180 mK

All plots

∆f/f = 1/2

FIG. 9: δε noise with different smoothing windows ∆f/f , cf. Fig. 6.