Model for l/f Flux Noise in SQUIDs and Qubits* • Introduction Roger Koch † David DiVincenzo • 1/f flux noise in SQUIDs and Qubits IBM Yorktown Heights • Model for l/f flux noise † Deceased • Concluding remarks *PRL 98, 267003 (2007) DiSQ Support Berkeley Army Research Office (DDV) 7 December 2007 Department of Energy (JC)
The Ubiquitous 1/f Noise log S x (f) X(t) log f time (t) Spectral density: • Vacuum tubes S x (f) ∝ 1/f β , β ~1 • Carbon resistors • Semiconductor devices • Metal films • Superconducting devices
Intensity Fluctuations in Music and Speech Spectra have been offset vertically log 10 [S intensity (f)] Voss and Clarke log 10 (f) Nature 1976
1/f Noise in the Flood Level of the River Nile Lowest frequency ≈ 2 x 10 -11 Hz ≈ 1/2000 years Voss and Clarke 1976 (unpublished)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 X(t) time (t) • For a single characteristic time τ : S RTS (f) ∝ τ /[1 + (2 π f τ ) 2 ] log S x (f) • The superposition of Lorentzians from uncorrelated processes with a 1/f 1/f 2 broad distribution of τ yields 1/f noise (Machlup 1954) log f
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 S Φ (f) ∝ l/f • By using two different bias reversal schemes, Koch et al . (1983) showed there were two independent contributions: I • Critical current noise V – can be removed by bias current reversal • Flux noise – cannot be removed by bias current reversal 1/2 (1 Hz) ≈ 5 – 15 µΦ 0 Hz -1/2 for SQUID areas S Φ 6 x 10 -6 – 7 mm 2 Quartz, glass and Si substrates
DC SQUIDs: 1/f β 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 S Φ ( f ) ∝ ( ∂ I/ ∂ Φ) 2 • For Nb, Pb, PbIn washers: 0.58 < β < 0.80
1/2 (1Hz) vs. T DC SQUIDs: S Φ Loop materials: Nb (A1-4, F1) ( Pb (C2) PbIn (A5, B1, C1, E1,2) Loop effective areas: 1,400 µ m 2 – 200,000 µ m 2 Large SQUID > 400 µ m 1/2 (1Hz) ≈ 7 ± 3 µ Φ 0 Hz -1/2 At low T: S Φ Small SQUID < 400 µ m
l/f Flux Noise in Flux Qubits • Decoherence due to flux noise enters via ∂ν / ∂Φ a , where ν = ( ∆ 2 + ε 2 ) 1/2 and ε = 2I q ( Φ a – Φ 0 /2) • Hence decoherence due to flux noise vanishes I q at Φ a = Φ 0 /2 Φ a FID dephasing rate 10 6 s -1 Yoshihara et al. (2006) • Measured 5 flux qubits: 1/2 (1 Hz) = 0.9 – 2 µΦ 0 Hz -1/2 S Φ • Showed that decoherence was not ( Φ a / Φ 0 ) – 1/2 due to critical current fluctuations • Loop area of qubit ≈ 3 µ m 2
l/f Flux Noise in Phase Qubits Bialczak et al . (2007) 1/2 (1 Hz) = 4 µΦ 0 Hz -1/2 S Φ Area ≈ 40,000 µ m 2 Showed that noise was not due to critical current fluctuations
Area Dependence of l/f Flux Noise 1/2 (1 Hz) S Φ Area ( µ m 2 ) ( µΦ 0 Hz -1/2 ) Wellstood et al. (SQUIDs) ~2 x 10 5 5 – 10 (Max Area) Bialczak et al . (phase qubit) ~4 x 10 4 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: 1/2 (1 Hz) ≈ 0.5 x 10 -6 Φ 0 Hz -1/2 – Foglietti et al. (1986) S Φ 1/2 (1 Hz) ≈ 0.2 x 10 -6 Φ 0 Hz -1/2 – Tesche et al. (1985) S Φ • l/f flux noise in Quantronium at Saclay: 1/2 (1 Hz) ≈ 100 x 10 -6 Φ 0 Hz -1/2 – Ithier et al . (2005) S Φ • Thus, one should perhaps not be too focused on a “universal” value for flux noise
l/f Flux Noise in High-T c 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 ( Φ 0 /B) 1/2 , where B is cooling field. (John Clem, Dantsker et al ., H-M Cho et al .) • For B = 100 µ T, w ≈ 4 µ m. • For low-T c SQUIDs: pinning energies are much higher temperature is much lower for qubits, linewidth is typically << 4 µ m B is typically << 100 µ T. • Vortex motion believed not to be source of l/f flux noise in low-T c 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/SiO 2 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 or tunnel between def ect centers • The spin of an electron is locked in direction while the electron occupies 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 = µ B ( L + 2 S ) 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 processes are in principle allowed. However, for the Raman process Abrahams showed that the lifetime is τ ∝ 1/T 13 Thus, transition rates for such processes are Second-order spin- utterly negligible at low temperatures flip processes • 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
Configuration for Simulations y Areal density Exterior of defects is n SQUID / Qubit loop x Hole W d L D Average radius = (D + d)/2
Configuration for Simulations y Areal density Boundary at which Exterior flux coupled from of defects current loop is less is n than 1% of that SQUID / Qubit loop from loop at center x Hole W d L D
Simulation Scheme for One Spin Loop Hole Exterior Thickness of 1µ m superconductor = 0.1 µ m Perpendicular In-plane moment moments Test loop Area A Magnetic moment Ai = µ B = 9.27 × 10 -24 J/T i Mutual inductance to SQUID loop M Calculate M using superconducting FastHenry Flux coupled to SQUID loop Φ s = M(x,y) µ B /A 0.1 µ m Plot Φ s / µ B = M(x,y)/A versus position
Flux Coupled to SQUID Loop from Current Loop Magnitude of flux in loop (n Φ 0 / µ B ) Flux from perpendicular moment • Local minimum at the middle • Peaks at edges of film • Vanishes at midpoint of the film Inplane (by symmetry) moment Flux from in-plane moment • Peaks at midpoint of film Perp. moment • Falls off rapidly away from film (and by symmetry is zero exactly in the plane of the film) Position ( µ m)
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 Φ s = M(x,y) µ B / A 2 + M y 2 + M z Mean square value < M 2 (x,y) > = (M x 2 ) / 3 Mean square flux noise is (D+L) x 2 dx < ( δΦ s ) 2 > = 8 n µ B ∫ ∫ dy < M 2 (x,y) >/ A 2 0 0
Recommend
More recommend
