Superconducting Quantum Interference Devices

John Clarke University of California, Berkeley

Wallenberg Centre for Quantum Technology Summer School Säröhus, Sweden 22 – 26 August 2019 Superconducting Quantum Interference Devices

• History • The Josephson Tunnel Junction: Characteristics and Noise • The dc SQUID: Characteristics and Noise

• Practical Low-Tc dc SQUIDs and SQUID Amplifiers • The Ubiquitous 1/f Noise • Epilogue SQUID Applications

• Brief Topics • Cosmology • Shedding Light on Dark Energy

• Cold Dark Matter: The Hunt for the Axion • Ultra Low Field Magnetic Resonance Imaging • Epilogue Discussion Superconducting Quantum Interference Devices

• History • The Josephson Tunnel Junction: Characteristics and Noise • The dc SQUID: Characteristics and Noise

• Practical Low-Tc dc SQUIDs and SQUID Amplifiers • The Ubiquitous 1/f Noise • Epilogue A Little Personal History: How Did I Get Into SQUIDs?

King’s College Chapel, Cambridge

English Gothic 1446 - 1515 St. Bene’t’s Church

Anglo-Saxon 1000 – 1050 AD

The Perse School

The Perse School was founded in 1615 by Dr Stephen Perse who left money in his will to educate 100 boys from Cambridge and nearby villages at no cost.

The school was originally located on “Free School Lane”. Perse Outside the shop that was once my grandfather’s picture-framing shop Grandad’s shop

Perse Entrance to the Cavendish Laboratory Through the gate…

The Royal Society Mond Laboratory Grandad’s shop

Mond Perse 1 October 1964

Eric Gill 1933 Thesis advisor: Brian Pippard Royal Society Mond Laboratory Thesis research required Cavendish Laboratory measuring voltages of Cambridge University 10−12 to10−13 volt State-of-the-art 10−9 volt

Brian Josephson Explains Josephson Tunneling

November 1964 Courtesy Brian Josephson The Discovery of Superconductivity (1911)

Hg ) Ω Resistance ( Resistance

Heike Kamerlingh Onnes 4.0 4.2 4.4 Temperature (K) Leiden, The Netherlands

Resistance vanishes below the transition (or critical) temperature Tc The Theory of Superconductivity (1957)

Bardeen-Cooper-Schrieffer Theory (1957) Cooper pairs condensed into a single, macrosopic quantum state

I I Supercurrent is carried by Cooper pairs of electrons with charge -2e BCS theory: electron-phonon interaction

water bed The Birth of Superconducting Electronics: 1961 - 1964

Flux Quantization

Φ = nΦ0

Φ = nΦ0 (n = 0, ±1, ±2, ...) -15 2 where Φ0 ≡ h/2e ≈ 2 x 10 Tm J is the flux quantum

Experimental observation 1961: Deaver and Fairbank Doll and Näbauer

Josephson Tunneling

Insulating barrier

Superconductor 1 Superconductor 2 I I

~ 20 Å

I = I0sinδ δ = φ1 – φ2 dδ/dt = 2eV/ħ = 2πV/Φ0

Josephson 1962 Josephson Tunneling

Insulating barrier Superconductor 1 Superconductor 2 Sn-SnOx-Pb I I Tunnel junction

~ 20 Å 1.5 K

I = I0sinδ δ = φ1 – φ2 0.006 G dδ/dt = 2eV/ħ 0.4 G = 2πV/Φ0

Josephson 1962 Anderson and Rowell 1963 Bell Labs

Birth of the Superconducting Quantum Interference Device (SQUID) I I0 Φ0 Φ V

Φ Sn-SnOx-Sn junctions

• Critical current versus applied magnetic field for two different junction spacings • Rapid oscillations due to interference, slow oscillations due to diffraction • Essential physics analogous to two-slit interference in optics • Sensitive detector of magnetic field Jaklevic, Lambe, Silver and Mercereau (Ford Motor Company Laboratory) 1964 Brian Josephson Explains Josephson Tunneling

November 1964 Courtesy Brian Josephson The Very Next Day

Brian: “John, How would you like a voltmeter with a resolution of 2 × 10−15 V in 1 second?”

Brian Pippard

November 1964 Brian’s Idea

R I

M = L V L L V in out τ = L/R M

IΦ0

Digital voltmeter: IΦ0 = Φ0/M = Φ0/L

-15 Voltage resolution: Vin = IΦ0R = (Φ0/L)R = Φ0/τ = 2 × 10 V for τ = 1 s

Six order of magnitude improvement over the state of the art! Sir Brian Pippard Serves Tea to Lady Bragg

Courtesy Cavendish Laboratory At Tea

• Paul Wraight, a fellow research student, pointed out that Nb has a surface oxide layer and PbSn solder is a superconductor. At Tea • Paul Wraight, a fellow research student, pointed out that Nb has a surface oxide layer and PbSn solder is a superconductor. Why don’t you put a blob of solder on a piece of Nb wire?

Niobium Before dinner

4.2 K I Copper

SnPb solder V 5 5mm mm The Day After

Niobium

I Brian Pippard: “It looks as though a slug crawled through the window last night and I Copper expired on your desk!”

SnPb solder V 5 mm SLUG The SLUG Superconducting Low-Inductance Undulatory Galvanometer

Niobium

IB I

I Voltage Copper

SnPb solder Current IB in niobium wire V 5 mm

I B February 1965

The SLUG as a Voltmeter

Niobium δV

1 µΩ δΦ Φ 0 1 2 Φo Flux bias where V vs. Copper V Φ is steepest

Solder Voltage noise 10 fVHz-1/2 (10-14 VHz-1/2) 5 mm Analog voltmeter

Why Does IB Modulate the Critical Current?

IB IB

• Blue region indicates penetration depths in Nb wire and solder • Current IB couples flux into this region • Why are there typically only 2 or 3 junctions? Other Early SQUIDs

Zimmerman and Silver 1966 John Wires Up a SLUG One has to be lucky! In January 1968 I moved to the University of California, Berkeley for a 1-year postdoctoral appointment. Apart from sabbatical leaves at various institutions in Europe, I have been there ever since. Thin-Film Cylindrical SQUID Shadow masks

5 mm

10-14 THz-1/2 (10 fTHz-1/2) JC, Goubau, Ketchen (1974) Geophysical Prospecting Superconducting Quantum Interference Devices

• History • The Josephson Tunnel Junction: Characteristics and Noise • The dc SQUID: Characteristics and Noise

• Practical Low-Tc dc SQUIDs and SQUID Amplifiers • The Ubiquitous 1/f Noise • Epilogue Josephson Tunnel Junction

Superconducting ﬁlm Superconducting ﬁlm + oxide

• Junction has intrinsic capacitance C Resistively-Shunted Junction (RSJ) Model

U(δ)

I I N 0 I < I . 0 < δ! > = 0 δ V/R + I0sin δ + C V ! = I + IN(t) ! Neglect IN(t), set V = "δ/2e: !C ! 2e ∂U U(δ) "δ"+ δ" + = 0, < δ! > = 0 2e 2eR ! ∂δ δ δ Φ0 where U(δ) = − (Iδ + I0 cosδ). 2π δ! > 0 • This differential equation δ represents the motion of a particle of mass ∝ C in a tilted washboard Stewart (1968) potential, U, with damping ∝ 1/R McCumber (1968) Effects of Damping in the RSJ Model !C ! 2e ∂U "δ"+ δ" + = 0, 2e 2eR ! ∂δ Underdamped Case: !δ! term dominates δ! term

βc≡ (2πI0R/Φ0)(RC) = ωJRC >> 1

• When I is reduced to below I0 the kinetic energy of the particle keeps it rolling: Hence hysteresis in the I-V characteristic.

Overdamped Case: δ! term dominates !δ! term I

βc≡ (2πI0R/Φ0)(RC) = ωJRC << 1

• When I is reduced to below I0 the V = R(I2-I 2)1/2 damping stops the motion of the 0 particle: Hence no hysteresis in the I-V characteristic. V We are concerned only with the Overdamped Case Thermal Noise in the Overdamped RSJ (βc << 1)

"C " 4k BT Langevin Equation !δ! + δ! + I0 sin δ = I + I N (t), SI (f ) = 2e 2eR R U I > I0

Noise rounding (Ambegaokar and Halperin 1969)

I I < I0 V(t)

t V Voltage Noise in the Overdamped RSJ: Thermal (Likharev and Semenov 1972)

2 ⎡ 4k T 1 I 4k T ⎤ I B ⎛ 0 ⎞ B 2 (I > I ) S v (f m ) = ⎢ + ⎜ ⎟ ⎥ R D 0 ⎣⎢ R 2⎝ I ⎠ R ⎦⎥ I0 “Straight through” “Mixed down” Dynamic noise noise resistance V

Measurement frequency fm << Josephson frequency fJ = 2eV/h Voltage Noise in the Overdamped RSJ: Quantum Quantum Langevin equation

(ħC/2e)! δ! + (ħ/2eR) δ! + I0sinδ = I + IN(t)

SI(f) = (2hf/R)coth(hf/2kBT) −1 = (4hf/R){[exp(hf/kBT) − 1] + ½} Planck Zero distribution point fluctuations

Setting hfJ = 2eV: 2 2 Sv(fm) = [4kBT/R + (2eV/R)(I0/I) coth(eV/kBT)]RD (I > I0) “Straight “Mixed through” down” noise noise

2 In the quantum limit eV(I0/I) >> 2kBT:

2 2 Sv(fm) = (2eV/R)(I0/I) RD

Likharev and Semenov (1972) Koch, Van Harlingen & JC (1980) Quantum Noise: Experiment

fm = 70, 106, 183 kHz Immersed in liquid helium helium liquid in Immersed

Rc Nyquist noise calibration resistor Voltage noise across tank circuit: 2 2 2 2 Q Sv(fm) = ω Lt [Sv(fm)/RD ] (Q = ωLt/RD)

2 2 Here, Sv(fm)/RD = [4kBT/R + (2eV/R)(I0/I) coth(eV/kBT)] “Straight “Mixed through” down” noise noise Koch, Van Harlingen & JC (1981) Current Noise in Shunt Resistor

No fitted parameters • Quantum fluctuations With zero must dominate point term to achieve a quantum limited SQUID amplifier

Planck

−1 (4hν/R){[exp(hν/kBT) − 1] + ½}

(4hν/R){[exp(hν/k T) − 1]−1} Koch, Van Harlingen B & JC (1981) Superconducting Quantum Interference Devices

• History • The Josephson Tunnel Junction: Characteristics and Noise • The dc SQUID: Characteristics and Noise

• Practical Low-Tc dc SQUIDs and SQUID Amplifiers • The Ubiquitous 1/f Noise • Epilogue The DC Superconducting Quantum Interference Device • dc SQUID Two Josephson junctions on a superconducting ring I

• Current-voltage (I-V) characteristic modulated by magnetic flux Φ: -15 2 Period one flux quantum Φo = h/2e = 2 x 10 T m I V nΦo Ib δV (n+1/2)Φo

V δΦ Φ ΔV 0 1 2 Φo The DC SQUID: Important Parameters

Flux-to-voltage transfer coefficient: VΦ = (∂V/∂Φ)I

Voltage noise VN(t)

Equivalent flux noise ΦN(t) = VN(t)/VΦ

Voltage noise spectral density SV(f) 2 Equivalent flux noise spectral density SΦ(f) = SV(f)/VΦ Thermal Noise Theory for the dc SQUID

1 2

( C/ 2e)!δ! ( / 2eR)δ! I sin δ I (I/ 2) J " 1 + " 1 + 0 1 + N1 = − SI (f ) = 4k BT/ R

( C/ 2e) !δ! ( / 2eR)δ! I sin δ I (I/ 2) J " 2 + " 2 + 0 2 + N 2 = +

δ1 − δ2 = (2π / Φ0 )(Φ + LJ)

δ"1 + δ" 2 = 4eV/ ! Tesche & Clarke 1976 Thermal Fluctuations: Constraints on dc SQUID

• Josephson coupling energy much greater than thermal energy:

I0Φ0/2π >> kBT (Γ ≡ 2πkBT/I0Φ0 << 1) ⎧0.17µA at 4.2K or I0 >> 2πkBT/Φ0 ≈ ⎨ ⎩3.3µA at 77K

• Energy of a flux quantum in the SQUID loop much greater than thermal energy:

2 Φ 0 / 2L >> k BT 2 ⎧5.6nH at 4.2K or L << Φ 0 / 2 k BT ≈ ⎨ ⎩0.33nH at 77K • At least a factor of 5 greater is required for the inequalities. Computed Current-Voltage Characteristics

β ≡ 2LΙ0/Φ0

Γ ≡ 2πkBT/I0Φ0

Φ Computed Transfer Function, Voltage Noise and Equivalent Flux Noise

Transfer function Voltage noise Equivalent flux noise

1/2

T/R]

R)V

/[16k

(0)/2

1/2

Φ S

( S

0 1 2 3 0 1 2 3 0 1 2 3 I/I0 I/I0 I/I0 1/2 1/2 Equivalent flux noise SΦ (f) = SV (f)/VΦ

Thermal Noise in the dc SQUID

• Originates in Nyquist noise in shunt resistors R. 2 • Optimized for βL ≡ 2LI0/Φ0 = 1, βc ≡ 2πΙ0R C/Φ0 ≤ 1 • For SQUID inductance L and shunt resistance R:

Flux-to-voltage transfer coefficient VΦ ≈ R/L 2 2 Spectral density of flux noise SΦ(f) ≈ 16 kBTR/(VΦ) ≈ 16kBTL /R • For typical parameters L = 200 pH, R = 6 Ω, T = 4.2 K:

1/2 −6 −1/2 SΦ (f )≈1.2x10 Φ0Hz • Noise energy:

−32 −1 ε(f) = SΦ(f)/2L ≈ 10 JHz (≈ 100 ħ)

• Note: since SΦ(f) ∝ T, for given L and R high-Tc SQUIDs

at 77 K will have higher noise than low-Tc SQUIDs at 4.2 K How Big is 10-32 J (~10-13 eV)?

This is the energy required to raise 1 electron

through 1 mm in the earth’s gravitational field

mgx ≈ 10-30 kg.10 ms-2.10-3 m ≈ 10-32 J

10-14 x the ground state energy of one hydrogen atom

• Note: These values are for typical SQUIDs operated at 4.2 K. Specially designed SQUIDs operated at mK temperatures can have noise energies of 10-34 J. Superconducting Quantum Interference Devices

• History • The Josephson Tunnel Junction: Characteristics and Noise • The dc SQUID: Characteristics and Noise

• Practical Low-Tc dc SQUIDs and SQUID Amplifiers • The Ubiquitous 1/f Noise • Epilogue Josephson Tunnel Junction

Nb ﬁlm

Nb ﬁlm + Al + Al2O3 I V

• Trilayer process (John Rowell and coworkers) In a single pumpdown: Deposit Nb base electrode Deposit thin Al film Oxidize Al film (part way through) under carefully controlled conditions Deposit Nb counter electrode Pattern using photolithography

• This highly reproducible process is the workhorse of Nb SQUID technology Thin-Film, Square Washer DC SQUID Ketchen and Jaycox (1981) SQUID with input coil Josephson junctions

500 µm 20 µm

• Wafer scale process • Photolithographic patterning • Nb-AlOx-Nb Josephson junctions The Flux Locked Loop

δΦ lock-in ∫

δV0

• Linear response: δV0 ∝ δΦ

• Large dynamic range: δΦ >> Φ0

• Can detect minute flux changes: δΦ << Φ0 Also: Directly coupled readout (Drung) Flux Modulation Scheme

VLock-in

• Lock-in detection at modulation frequency. • This technique eliminates drift and l/f noise in amplifier and bias current. Superconducting Flux Transformer Superconducting Flux Transformer: Magnetometer

Flux-locked SQUID Flux Transformer: Magnetometer • Approximate treatment: Neglects current noise in SQUID, stray inductance, interaction of SQUID and transformer 2 2 M = α LL δΦ Mi i i

Lp δΦs A p L L δJ i

Flux quantization: δΦ + (Lp + Li)δJ = 0

− M i δΦ Flux applied to SQUID: δΦ s = M i δJ = L p + L i

− (L p + Li )δΦ sn Hence in pickup loop: δΦ n = Mi

⎛ L p ⎞ δΦ = −⎜ + L1/ 2 ⎟ sn ⎜ 1/ 2 i ⎟ 1/ 2 ⎝ Li ⎠ αL

• Assuming Lp is fixed, one finds the equivalent flux noise in the pickup loop is minimized when Li = Lp. Flux Transformer: Magnetometer

Convert to spectral densities:

1/ 2 1/ 2 S ⎛ ε ⎞ S1/ 2 (min) = 2L1/ 2 Φ = 2 2 L1/ 2 ⎜ ⎟ [ε(f) = S (f)/2L] Φ p 1/ 2 p ⎜ 2 ⎟ Φ αL ⎝ α ⎠

1/ 2 1/ 2 SΦ (min) SB (min) = A p

1/ 2 1/ 2 2 2(5µ 0 rp ) ⎛ ε ⎞ ≈ ⎜ ⎟ [L 5µ r ] 2 ⎜ 2 ⎟ p ≈ 0 p πrp ⎝ α ⎠

2 -32 For ε/α = 10 J/Hz, rp = 10 mm

1/ 2 −15-16 −1-1/2/ 2 -1/2 SB ≈ 22x 10x 10 THz THz = 0.2 fTHz Lowest White Noise in SQUID Magnetometers

Storm et al.

0.1 fTHz-1/2 M. Schmelz et al., Leibnitz Institute of Photonic Technology, Jena

0.15 fTHz-1/2 J.-H. Storm et al., Physikalisch-Technische Bundesanstalt (PTB) Berlin tesla

Magnetic Fields 1 Conventional MRI

10-2

10-4 Earth’s field

10-6 Urban noise 10-8 Car at 50 m

10-10 Human heart Fetal heart 10-12

Human brain response 10-14 1 femtotesla 10-16 SQUID magnetometer Superconducting Flux Transformer: Gradiometer

Flux-locked SQUID • Sensitivity of magnetometer to magnetic field ~ 1/r3

• Sensitivity of 1st derivative gradiometer to magnetic field ~ 1/r4

• Sensitivity of 2nd derivative gradiometer to magnetic field ~ 1/r5

• Gradiometers used to reject distant background noise, for example in measurements of signals from the human brain. SQUID Femtovoltmeter

r I

v V

For r = 10-8 Ω, T = 4.2 K Johnson noise ≈ 10-15 V Hz-1/2

SQUID Amplifiers Calculation of Amplifier Noise Temperature

εn

R in -A Vo εn, in assumed uncorrelated

Vo = −Aε n − Ai n R o 2 2 2 2 Sv (f ) = A Sε (f ) + A Si (f )R = 4k BT N RA

1 ⎛ Sε (f ) ⎞ TN (f ) = ⎜ + Si (f )R ⎟ 4k B ⎝ R ⎠

∂TN 1 ⎛ Sε (f ) ⎞ = ⎜ − + Si (f ) ⎟ = 0 (Minimum ) Example: FET at 100 kHz, 300 K R 4k ⎜ R 2 ⎟ ∂ B ⎝ ⎠ 1/ 2 −1/ 2 Sε (f ) ≈1nV Hz 1/ 2 R opt = [Sε (f )/Si (f )] 1/ 2 −1/ 2 Si (f ) ≈100fA Hz 1/ 2 opt opt [S (f )S (f )] (Well known result!) T ≈ 3.5K T = ε i N N opt 2k B R ≈1nV /100fA ≈10kΩ SQUID Amplifier

r I v V DC SQUID: Voltage, Current and Cross-Spectral Noise

1, 2 k T/I 0.05, (2n + 1) /4 IB βL ≈ Γ ≡ π B 0Φ0 ≈ Φ ≈ Φ0

JN (t) VΦ ≈ R/L R V (t) N SV(f) ≈ 16 kBTR S (f) 11 k T/R L J ≈ B

SVJ(f) ≈ 12 kBT

Origin of cross-spectral noise: 0 VN ≈ VN + LJNVΦ 0 VN = voltage noise when VΦ = 0 . Equivalent Noise Sources Referred to Input Coil

JN (t) R VN (t)

dJ N e N (t) = −M i dt

iN is a “virtual” current source

VN (t) e is a “real” voltage source i (t) = N N M V i Φ = f)M J (f) is in quadrature with J (f) L eN(f) -j(2π i N N Mi

Calculate TN from en(t) and in(t) using standard method JC, Tesche & Giffard 1979, Martinis & JC 1985 DC SQUID: Tuned Amplifier on Resonance Ci IB

Ri JN (t) Li Vo (t) eN(t)

Impedance of input circuit is Ri (real)

Current noise eN(f)/Ri in input circuit is in quadrature with JN(f)

Additional voltage noise across SQUID

VNʹ = [e N (f )/ R i ]Mi VΦ

is uncorrelated with intrinsic SQUID noise VN

Thus, total output voltage spectral density includes no correlation term SVJ 1/ 2 2 2 1/ 2 2 2 1/ 2 TN = [Sε (f )Si (f )] / 2k B = [M i (2πf ) SJ (f )] [SV (f )/ M i VΦ ] / 2k B 1/ 2 = πf [SJ (f )SV (f )] / VΦ k B

≈ 4242Tf/VfT/ VΦΦ DC SQUID: Optimized Tuned Amplifier Ci IB

Ri Li Vo (t) eN(t

Mi Off-resonance, impedance of input circuit is complex

Induced current noise is no longer in quadrature with eN(f) V Consequently, the additional output voltage Nʹ is partially correlated with VN

Detailed calculation for the optimized noise temperature yields opt πf 2 1/ 2 18Tf18fT T = (S S − S ) ≈ (~ 0.4 TN on resonance) N V J VJ k BVΦ VΦ at a frequency given by opt 2 1/ωCi = ωLi (1+ α SVJ LVΦ /SV )

Simulations of the quantum Langevin equation as T → 0 yield T↓N↑opt ≈ ħω/ kB Gain and Noise Temperature of Tuned SQUID-Amplifier

T A TN = TN + Ti + TN / G

Measured Predicted G(dB) 18.6 ± 0.5 17

TN(K) 1.7 ± 0.5 1.1

• Slope yields gain G A • Intercept at Ti = 0 yields TN + TN /G Hilbert & Clarke (1985) Practical Quantum Limited SQUID Amplifiers SQUID Amplifiers

To achieve quantum limit we need hf >> kBT For T = 20 mK: f >> 400 GHz

Conventional SQUID Amplifier Microstrip SQUID Amplifier (MSA)

• Source connected to both ends of coil • Source connected to one end of the 25 coil and SQUID washer; the other end 20 of the coil is left open • On resonance when length of coil is a 15 half-wavelength

Gain (dB) 10 5 50 100 150 200 M. Mück, J. Gail, C. Heiden†, Frequency (MHz) M-O André , JC 1998 MSA Gain Measurements

Gain versus coil length

30 600 400 (MHz) (MHz) res 25 res ν ν 200 20 40 60 20 Coil Length (mm) 71 mm 7 mm Gain (dB) Gain (dB) 15 33 mm

10 15 mm

100 200 300 400 500 600 700 Frequency (MHz)

M. Mück, et al.1998 Gain and Noise Temperature 24 140 Tbath = 50 mK 120 20 Gmax = 20 dB 100 16 80 res 12 TN

Gain(dB) 60

8 T↓N↑opt 40 SQLTQ

4 20 Noise Temperature (mK) 590 600 610 620 630 640 Frequency (MHz)

• Measured optimum noise temperature T↓N↑opt = 48 ± 5 mK • Occurs slightly below resonance, as predicted • Quantum limit T = 30 mK Q Darin Kinion, JC 2011

Minimum Measured TN vs. Bath Temperature

615 MHz

At Tbath = 50 mK Noise temperature: opt TN = 48 ± 5 mK

Quantum limit TQ = 30 mK

Darin Kinion, JC 2011 Cooling Fins: Reducing Nyquist Noise

Cooling Fin Reduction of Nyquist Noise

No cooling fins

Cooling fins

Wellstood, Urbina, JC 1989 Varactor Diode Tuning of MSA

Input Output 220 Ω Varactor diode As reverse voltage 56 Ω 56 Ω Varactor bias is increased, 1 - 9 pF capacitance decreases for (depletion layer width 20 dB, 50 Ω IΦ 20 - 0 V Attenuator SQUID increases)

30 -1 V 0 1 2 3 4 6 9 22 V

25 No Varactor 20 Gain (dB)

Gain (dB) 15

5 100 120 140 160 180 200 220 Frequency (MHz) M. Mück, et al.1999 Superconducting Quantum Interference Devices

• History • The Josephson Tunnel Junction: Characteristics and Noise • The dc SQUID: Characteristics and Noise

• Practical Low-Tc dc SQUIDs and SQUID Amplifiers • The Ubiquitous 1/f Noise • Epilogue The Ubiquitous 1/f Noise

(f) x X(t) log S log

time (t) log f Spectral density: • Vacuum tubes α Sx(f) ∝ 1/f , α ~1 • Carbon resistors • Semiconductor devices • Metal films • Superconducting devices Random Telegraph Signals (RTS) and l/f Noise X(t)

time (t) For example, electron hops between traps in • For a single characteristic time τ: a semiconductor 2 SRTS(f) ∝ τ/[1 + (2πfτ) ] (f)

• The superposition of Lorentzians from x uncorrelated processes with a broad

log S log 1/f distribution of τ yields 1/f noise (Machlup 1/f2 1954) log f • To generate 1/f noise at frequency f0, the particle must reside in a well for time 1/f0 −4 • For example, for f0 = 10 Hz, 1/f0 ~ 3 h Intensity Fluctuations in Music and Speech

Spectra have been offset vertically

(f)] intensity [S 10 log

Richard Voss and JC Nature Cover 1975

log10(f) 1/f Noise in the Flood Level of the River Nile

Lowest frequency ≈ 2 x 10-11 Hz ≈ 1/1300 years Voss and Clarke 1976 (unpublished) 1/f Noise in SQUIDs Two basic mechanisms

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. At low temperatures, the process may involve quantum tunneling and atomic rearrangement.

Flux noise: Flux-sensitive devices (SQUIDs, flux qubits….) exhibit “flux noise”. This noise behaves as though there were an external source of magnetic flux noise applied to the SQUID.

How can we distinguish these two sources of 1/f noise?

Removal of Critical Current l/f Noise

δI δI J1/f 01 02 ≡ +

“in-phase noise” “out-of-phase noise”

• Removal of “in-phase noise”: Flux modulation (at, say, 100 kHz) • Removal of “out-of-phase noise”: Reverse bias current [at, say, 3.125 kHz = (100/32) kHz]:

Reverses direction of Jl/f which thus averages to zero

Also reverses sign of VΦ:

Restore sign of VΦ by applying Φ0/2 flux shift. R. H. Koch et al. 1983

High-Tc SQUID: Reduction of Critical Current 1/f Noise with Bias Reversal

Β ≈ 0

frequency (Hz)

All high-Tc dc SQUIDs are operated with bias reversal DC SQUIDs: Flux 1/fα Noise

Nb washer T = 90 mK Φ V

α ≈ 0.8 (spectral density)

First observed 1982 Origin of Flux Noise in Low-Tc SQUIDs

• It required a quarter of a century to invent the first credible theory for 1/f flux noise.

• March Meeting 2006, Roger Koch and I had breakfast, and discussed the problem yet again. This time we had an idea. Spin Model for 1/f Noise

• The near-independence of the noise magnitude on SQUID area rules out a “universal magnetic field 1/f noise” that pervades all of space.

• Rather, the noise must be generated locally.

• Original Model: Postulate that the noise is generated by electron spins at the substrate surface and/or a surface oxide on the metal.

• Assume that electrons are randomly distributed spatially and in direction, and randomly reverse their spin direction with a broad distribution of time constants.

† Roger Koch , David DiVincenzo, JC (2007) Areal Density of Spins

• Calculate 1/f noise for several SQUID configurations • Compare results with measured noise

Required areal density of spins to match data: n ≈ 5 × 1017 m-2

One spin per 2 nm2 But: Later Research Showed Spins Are Not Electrons

• P. Kumar, S. Sendelbach,,….R. McDermott (2016) showed that the spins are NOT electrons but O2 molecules adsorbed on

the surface. (The magnetic moment of O2 is 2µB).

• Furthermore, they showed that the flux noise could be reduced

significantly by removing O2 before cooling the SQUID. The SQUID was not exposed to the atmosphere before cooling.

So Spins Are Probably O2

Many Questions Remain

• Is the oxygen O2 or O2(minus)? Or both?

• Flux noise has been observed at frequencies as low as 10-4 Hz in SQUIDs and as high as 109 Hz in flux qubits.

• Frequency span of 13 decades requires a range in characteristic time τ also of 13 decades

• How can this be? Formation of clusters?

• How do the spins remain oriented without any rotation between switching events?

• Is there a way to remove surface spins permanently? Epilogue • Today’s SQUIDs are a commodity, available both commercially and from various institutions.

• Fabricated on Si wafers using a trilayer process, SQUIDs have high yield and reproducibility, and are very robust.

• The theory for the signal and noise properties of SQUIDs is highly developed, and provides excellent predictions of the performance of actual devices.

• At low frequencies, SQUIDs are generally operated in a flux-locked-loop that provides excellent performance and can be highly automated.

• SQUIDs are generally coupled to superconducting input circuits to give optimum performance as magnetometers, gradiometers, voltmeters or amplifiers.

• As T → 0, optimized SQUID amplifiers become near quantum limited: TN ≈ ħω.

• These properties enable one to use SQUIDs in a wide variety of applications.