Model for l/f Flux Noise in SQUIDs and Qubits* Introduction Roger - - PowerPoint PPT Presentation

Model for l/f Flux Noise in SQUIDs and Qubits*

• Introduction
• 1/f flux noise in SQUIDs and Qubits
• Model for l/f flux noise
• Concluding remarks

Roger Koch† David DiVincenzo IBM Yorktown Heights

†Deceased

*PRL 98, 267003 (2007) DiSQ Berkeley 7 December 2007 Support Army Research Office (DDV) Department of Energy (JC)

The Ubiquitous 1/f Noise

log f log Sx(f) time (t) X(t)

Spectral density: Sx(f) ∝ 1/fβ, β ~1

• Vacuum tubes
• Carbon resistors
• Semiconductor devices
• Metal films
• Superconducting devices

Intensity Fluctuations in Music and Speech

Voss and Clarke Nature 1976

log10[Sintensity(f)] log10(f)

Spectra have been

• ffset vertically

1/f Noise in the Flood Level of the River Nile

Lowest frequency ≈ 2 x 10-11 Hz ≈ 1/2000 years

Voss and Clarke 1976 (unpublished)

Random Telegraph Signals and l/f Noise

time (t) X(t)

• For a single characteristic time τ:

SRTS(f) ∝ τ/[1 + (2πfτ)2]

• The superposition of Lorentzians

from uncorrelated processes with a broad distribution of τ yields 1/f noise (Machlup 1954)

log f log Sx(f) 1/f 1/f2

1/f Noise in Superconducting Devices

Examples: SQUIDS, single electron transistors (SETs), charge qubits, flux qubits, phase qubits Three kinds of noise: Critical current noise: Trapping and release of electrons in tunnel barriers modify the transparency of the junction, causing its resistance and critical current to fluctuate. This is the best understood, and has been widely studied experimentally (Van Harlingen, Buhrman, Martinis…) Charge noise: Hopping of electrons between traps induces fluctuating charges onto nearby films and junctions Flux noise: Flux-sensitive devices (SQUIDs, flux qubits….) exhibit flux noise of unknown origin

1/f Flux Noise in SQUIDs and Qubits

l/f Flux Noise in SQUIDs

• Measurements of noise in dc SQUIDs in a flux-locked loop yield

the spectral density of the equivalent 1/f flux noise SΦ(f) ∝ l/f

• By using two different bias reversal schemes, Koch et al. (1983)

showed there were two independent contributions:

• Critical current noise

– can be removed by bias current reversal

• Flux noise

– cannot be removed by bias current reversal SΦ

1/2 (1 Hz) ≈ 5 – 15 µΦ0 Hz-1/2 for SQUID areas

6 x 10-6 – 7 mm2 Quartz, glass and Si substrates I V

DC SQUIDs: 1/fβ Flux Noise

Nb washer T = 90 mK

• Wellstood, Urbina and Clarke (1987) investigated 12 SQUIDs, using a second

SQUID to measure the 1/f noise.

• Flux noise and critical current noise separated using SΦ(f) ∝ (∂I/∂Φ)2
• For Nb, Pb, PbIn washers: 0.58 < β < 0.80

DC SQUIDs: SΦ

1/2(1Hz) vs. T

Loop materials: Nb (A1-4, F1) Pb (C2) PbIn (A5, B1, C1, E1,2) Loop effective areas: 1,400 µm2– 200,000 µm2

(

Large SQUID > 400 µm

At low T: SΦ

1/2(1Hz) ≈ 7 ± 3 µΦ0Hz-1/2

Small SQUID < 400 µm

l/f Flux Noise in Flux Qubits

(Φa/Φ0) – 1/2 FID dephasing rate 106 s-1

• Decoherence due to flux noise enters via ∂ν/∂Φa,

where ν = (∆2 + ε2)1/2 and ε = 2Iq(Φa – Φ0/2)

• Hence decoherence due to flux noise vanishes

at Φa = Φ0/2 Iq Φa Yoshihara et al. (2006)

• Measured 5 flux qubits:

SΦ

1/2 (1 Hz) = 0.9 – 2µΦ0 Hz-1/2

• Showed that decoherence was not

due to critical current fluctuations

• Loop area of qubit ≈ 3 µm2

l/f Flux Noise in Phase Qubits

Bialczak et al. (2007) SΦ

1/2(1 Hz) = 4µΦ0 Hz-1/2

Area ≈ 40,000 µm2 Showed that noise was not due to critical current fluctuations

Area Dependence of l/f Flux Noise

Area SΦ

1/2 (1 Hz)

(µm2) (µΦ0 Hz-1/2) Wellstood et al. (SQUIDs) ~2 x 105 5 – 10 (Max Area) Bialczak et al. (phase qubit) ~4 x 104 4 Yoshihara et al. (flux qubits) ~3 1 – 3

• Measurements performed at millikelvin temperatures
• Experiments heavily shielded against environmental magnetic

field noise

• Very weak tendency for flux noise to increase with area
• Data rule out a “universal magnetic field noise”
• However: Agreement may not be as good at other frequencies

However: Some Outliers

• Two sets of IBM dc SQUIDs which were passivated showed

somewhat lower 1/f flux noise: – Foglietti et al. (1986) SΦ

1/2(1 Hz) ≈ 0.5 x 10-6 Φ0 Hz-1/2

– Tesche et al. (1985) SΦ

1/2(1 Hz) ≈ 0.2 x 10-6 Φ0 Hz-1/2

• l/f flux noise in Quantronium at Saclay:

– Ithier et al. (2005) SΦ

1/2(1 Hz) ≈ 100 x 10-6 Φ0 Hz-1/2

• Thus, one should perhaps not be too focused on a “universal”

value for flux noise

l/f Flux Noise in High-Tc SQUIDs

• At 77 K large levels of l/f noise can be produced by thermal activation of

vortices among pinning sites.

• This noise can be eliminated by reducing the linewidth w of the film below

(Φ0/B)1/2, where B is cooling field. (John Clem, Dantsker et al., H-M Cho et al.)

• For B = 100 µT, w ≈ 4 µm.
• For low-Tc SQUIDs: pinning energies are much higher

temperature is much lower for qubits, linewidth is typically << 4 µm B is typically << 100 µT.

• Vortex motion believed not to be source of l/f flux noise in low-Tc SQUIDs.

Model for 1/f Flux Noise

Other Recent Models for 1/f Flux Noise

• Rogerio de Sousa (arXiv:0705.4088v2)

Ascribes noise to defects at the Si/SiO2 interface

• Lara Faoro and Lev Ioffe (Preprint)

Ascribes noise to diffusion of spins at the superconductor-substrate interface

Model for 1/f Flux Noise

• Electrons hop on and off defect centers by thermal activation
• r tunnel between def ect centers
• The spin of an electron is locked in direction while the electron
• ccupies a given trap. This direction varies randomly from

trap to trap.

• Two issues:
• What is the mechanism for “spin locking”? This must

persist for times at least as long as the inverse of the lowest frequency at which 1/f noise is observed.

• What is the magnitude of the 1/f flux noise generated by an

ensemble of these RTSs?

Spin Locking I: The Kramers Degeneracy*

• For an odd number of fermions with spin ½ in zero magnetic

field, Kramers showed that the ground state is doubly

• degenerate. It was shown subsequently that this result is a

consequence of symmetry. The two states have oppositely directed momenta.

• For an electron with nonzero orbital momentum (L > 0), the

magnetic moment M = µB(L + 2S) produced by spin-orbit coupling is locked to the direction of the crystal field (electric).

• If there is no orbital angular momentum (for example, if it is

quenched), the remaining angular momentum is due solely to the electron, which does not couple to the crystal field and will thus not be locked ^ ^ ^

*H.A. Kramers, Koninkl. Ned. Akad. Wetenschap., Proc. 33, 959 (1930)

Spin Locking II: Van Vleck Cancellation

• Van Vleck showed that in zero magnetic field matrix elements

for direct transitions between the two states vanish. Thus, spin flip processes are forbidden, and the electron spin remains locked in the state it occupies. The 6 states of a p-electron split by a crystal field

×

Spin Locking III: Second-Order Processes

Second-order spin- flip processes

Second-order processes are in principle

• allowed. However, for the Raman process

Abrahams showed that the lifetime is

τ ∝ 1/T13

Thus, transition rates for such processes are utterly negligible at low temperatures

• Abrahams wrote this paper to explain the absence of electron spin

resonance in donors in Si at low temperatures. In this context, spin locking is well established.

Calculations of 1/f Noise

x y D L d Exterior Hole SQUID/Qubit loop W

Configuration for Simulations Areal density

• f defects

is n

Average radius = (D + d)/2

x y D L d Exterior Hole SQUID/Qubit loop W

Configuration for Simulations Areal density

• f defects

is n

Boundary at which flux coupled from current loop is less than 1% of that from loop at center

Simulation Scheme for One Spin

Loop Hole Exterior

Perpendicular moment

Thickness of superconductor = 0.1µm

1µm In-plane moments

Test loop

Area A Magnetic moment Ai = µB = 9.27 × 10-24 J/T Mutual inductance to SQUID loop M Calculate M using superconducting FastHenry Flux coupled to SQUID loop Φs = M(x,y)µB/A Plot Φs /µB= M(x,y)/A versus position i

0.1 µm

Flux Coupled to SQUID Loop from Current Loop

Flux from perpendicular moment

• Local minimum at the middle
• Peaks at edges of film
• Vanishes at midpoint of the film

(by symmetry) Flux from in-plane moment

• Peaks at midpoint of film
• Falls off rapidly away from film

(and by symmetry is zero exactly in the plane of the film)

Inplane moment

Perp. moment

Position (µm) Magnitude of flux in loop (nΦ0/µB)

Mean Square Flux Noise Coupled to SQUID Loop by Ensemble of Spins

For one spin at (x,y), the flux coupled to the SQUID loop is Φs = M(x,y) µB /A Mean square value <M2(x,y)> = (Mx

2 + My 2 + Mz 2)/3

Mean square flux noise is <(δΦs)2> = 8n µB

2 dx

dy <M2(x,y)>/A2

∫

(D+L)

∫

x

Spectral Density of Flux Noise

Set the spectral density SΦ(f) = α/f, where α is to be determined Introduce lower and upper cut-off frequencies f1 and f2 Then <(δΦs)2> = α df/f = α ln(f2/f1) Assuming f1 = 10-4 Hz and f2 = 109 Hz We find SΦ(f)/Φ0

2 ≈ <(δΦs/Φ0)2>/30f

Note that this result is only very weakly dependent on f1 and f2

∫

f1 f2

Fit of Areal Density of Defects n

We regard n as a fitting parameter to produce values of SΦ(1 Hz) comparable to those observed experimentally. For a SQUID with

• uter dimension 2D = 12 µm and linewidth W = 2 µm:

n = 5 × 1017 m-2. We assume these are surface defects on the substrate and/or oxidized surface of the superconductor, and especially in contaminants produced by exposure to chemicals and the atmosphere. For a 10-nm layer, this value corresponds to about 1 defect per 104 atoms. Koch and Hamers (1987): Performed STM measurements on ultra clean Si surface exposed to 3 × 10-7 torr of O2 for 20 s in UHV

• chamber. They found about 8 two-level systems in a 6.5 × 6.5 nm2

area in a bandwidth of 10-500 Hz, corresponding to an areal density

• f 2 × 1017 m-2. This represents an areal defect density of

21 × 1017 m-2 over 13 decades of frequency.

Remarks on the Defect Density

• Martinis et al. reported a density of two-level systems of 1012 m-2

in the tunnel barriers of Josephson junctions, based on their measurements of 1/f critical current noise.

• In tunnel barriers, however, the barrier is protected with a metal

film immediately after its formation, before it is exposed to any contaminants.

• In contrast, the surfaces of the substrate and films are exposed to

both the atmosphere and a variety of contaminants from processing. It is notewothy that the IBM passivated SQUIDs have lower 1/f flux noise.

• Chris Wilson commented yesterday that a 15-second plasma

cleaning reduced the Q of a superconducting cavity from 104 to 20.

Flux Noise Versus Loop Size for Fixed Aspect Ratio

Loop size D + d (µm) 10 100 1000 SΦ

1/2(1 Hz) (µΦ0/Ηz1/2)

• The dominant noise is from

inplane moments under the superconductor.

• This noise could equally well

arise from inplane moments

• n the superconducting film.
• Inplane noise is negligible for

hole and exterior.

• Noise increases by a factor
• f about 4 for increase in

area of about 400.

Flux Noise Versus Loop Size for Fixed Width

Loop size D + d (µm) SΦ

1/2(1 Hz) (µΦ0/Ηz1/2)

• Hole noise vanishes as the

hole dimension tends to zero.

• For D + d > 50 µm, slope

tends to ½.

• This implies that SΦ(1 Hz)

scales linearly with the loop size, that is with the perimeter, rather than the area.

Checks on the Current Loop

• The results were unaffected when the current loop area A was

varied from 0.1 A to 10 A.

• Dependence of the noise on SQUID-test loop separation:

SΦ

1/2(1 Hz) (µΦ0/Ηz1/2)

• Perpendicular moments
• Noise constant for

separation z < 2 µm.

• Inplane moments
• Loop: Noise constant for

z < 2 µm

• Hole/exterior: Noise tends

to zero as z tends to zero (symmetry)

Comparison with Bialczak et al.

SΦ ∝ (d/W)[(1/2π)ln(2bW/λ2) + 0.28] for d >> W >> b Neglecting the ln term, Faoro & Ioffe have the same scaling

+ + + + + +

10 100 1000 Loop size D + d (µm) d = 2W may not satisfy d >> W

+ [ln(2bW/λ2)]1/2

Comparison with Bialczak et al.

SΦ ∝ (d/W)[(1/2π)ln(2bW/λ2) + 0.28] for d >> W >> b Loop size D + d (µm) When d >> W SΦ ∝ d at fixed W

Could the Noise be Generated by Nuclear Spins?

Model predicts that noise power scales as µ2n Superconducting film

• 52Nb: 5.56 × 1028 m-3, abundance 100%, µ = 0.00336 µΒ
• 207Pb: 3.30 × 1028 m-3, abundance 22%, µ = 0.00032 µB
• Nb noise power/Pb noise power ≈ 850
• Wellstood et al. measured essentially the same 1/f noise in

SQUIDs with Nb and Pb loops Substrate

• 29Si: 5.0 × 1028 m-3, abundance 5%, µ = 0.00030 µB
• Sapphire 27Al: ≈ 3 × 1028 m-3, abundance 100%, µ = 0.0020 µB
• Sapphire noise power/Si noise power ≈ 500
• Phase qubit of Bialczak et al. is on sapphire, and shows similar

levels of 1/f flux noise to devices on silicon.

We have no model for 1/f noise from nuclei

Concluding Remarks

• Crucial underlying physics of “spin locking” is the Kramers

two-fold degeneracy, and that transitions between these states do not occur at low temperature.

• A trap areal density of 1017 - 1018 m-2 is required to account for

the observed levels of 1/f noise. – Can this be explained in terms of surface contamination? – For a 10 nm thick layer: only 1 defect per 104 atoms – Can one reduce the defect density significantly by passivation? Some other treatment?

• The model does not discriminate between defects on the

substrate and on the superconductor. However, for a given defect density, defects on (or underneath) the superconductor make the largest contribution.

• In general terms, the predicted scaling agrees with the data in

Fred Wellstood’s thesis.

• Scaling needs to be investigated for devices of different

dimensions fabricated simultaneously.