II. Applications of the Josephson Effect

Chapter 4 

Applications 

of the 

Josephson Effect

R. Gross, A. Marx, and F. Deppe © Walther-Meißner-Institut (2001 - 2013) 

II. Applications of the Josephson Effect 

AS-Chap. 4 - 2 

large number of analog and digital applications: 

• I s m = I s m B : 

magnetic field sensors (SQUIDs) (ch. 4) 

• β C ≫ 1: 

bistability: zero/voltage state 

switching devices, Josephson computer (ch. 5) 

• 2 nd Josephson equation dφ/dt = 2eV/ħ 

VCO, voltage standard (ch. 6) 

• nonlinear IVC 

mixers up to THz, oscillators (ch. 7) 

• macroscopic quantum behavior 

superconducting qubits (ch. 9)


4. Superconducting Quantum Interference Devices 

Superconducting Quantum Interference Devices = SQUID 

single Josephson junction as can be used as 

magnetic field sensor: I s m = I s m B 

- sensitivity: dI s /dΦ ≈ I s m /Φ 0 = I s m /B 0 t B L 

increase t B L to increase sensitivity 

• superconducting loop with one or more Josephson junctions 

relevant area: loop area 

• dual or multi-beam interference: 

Superconducting Quantum Interference Devices (SQUIDs) 

• relevant physics: 

flux quantization in superconducting loops and Josephson effect 

• SQUIDs = most sensitive detectors for magnetic flux 

• can detect every quantity that can be converted into magnetic flux: 

magnetic field, field gradient, current, voltage, displacement, … 

• most relevant types of SQUIDs: 

direct current (dc) and radio frequency (rf) SQUIDs 

• dc-SQUIDs: highest energy sensitivity at low temperatures 

AS-Chap. 4 - 3


Die Kalmare (Teuthida) stellen mit rund 300 Arten die größte Gruppe innerhalb der Kopffüßer 

dar. Gemeinsam mit den Echten Tintenfischen (Sepiida), den Zwergtintenfischen (Sepiolida) und 

dem Posthörnchen (Spirula spirula), das allein eine eigene Gruppe Spirulida darstellt, bilden sie 

die Gruppe der Zehnarmigen Tintenfische (Decapoda). 

AS-Chap. 4 - 4



4.1 The dc-SQUID 

4.1.1 The Zero Voltage State 

2 JJs (lumped elements) connected in parallel, identical I c




total phase change along closed contour = 2¼n: 

use:




• substitution: 

• integration of A: 

closed loop total flux Φ 0 threading the loop 

• integration for J s : 

J s vanishes deep inside electrode material ∫ ΛJ s ⋅ dl vanishes 

• result: phase differences are not independent but linked via the fluxoid quantization: 

• if Φ = Φ ext : maximum possible supercurrent is obtained for sin(…) = 1 as 

problem solved!




• problem: finite inductance L of the loop 

• for a symmetrical loop we can write: 

average supercurrent 

circulating current 

equations have to be solved 

self-consistently 

maximum I s m : maximize I s with 

respect to φ 1 with constraint




• relevance of finite inductance is characterized by 

screening parameter: 

• negligible screening: β L ≪ 1 

flux due to circulating current can be neglected 

maximum supercurrent for given Φ ext : 

from dI s /dφ 1 = 0 

A = 2 mm 2 , 

B ext = 1 nT 

Φ ext = 

Φ 0



AS-Chap. 4 - 10 

• large screening: β L ≫ 1 

LI c ≫ Φ 0 circulating current compensates external field 

total flux in the loop tends to be quantized: 

transport current: 

circulating current: 

constraint: 

for given I and Φ: 2 equations for φ 1,2 ⇒ Φ cir = LI cir and Φ ext 

e.g.: for I ≃ 0: sin φ 1 ≃ − sin φ 2 and: 

in summary we get: Φ Φ ext 

• how does Φ Φ ext look like in detail ?



• finite screening: 

β L > 0 

• small β L : single valued/non-hysteretic Φ Φ ext 

• maximum value of Φ cir ≃ LI c 

• single valued Φ Φ ext curve for: |Φ cir | ≤ Φ 0 

⇒ LI 2 c ≤ Φ 0 

2 

(more precisely: β L ≤ 2/π) 

⇒ 

β L ≤ 1 

AS-Chap. 4 - 11



AS-Chap. 4 - 12 

• how does I m s Φ ext look like? 

- qualitative discussion for large β L 

I cir → 0 for large I → I s m ≃ 2I c 

start with n = 0: 

small screening current I cir ≃ −Φ ext /L screens field 

I 1 decreases, I 2 increases with increasing Φ ext I 2 ≤ I c 

modulation depth of I s m Φ ext is strongly reduced with increasing β L (∝ 1/β L )


4.1.2 The Voltage State 

• negligible screening: ¯L ¿ 1, strong damping: ¯C ¿ 1 

total flux = applied flux 

displacement current can be neglected: 

total current = Josephson + resistive current 

• identical junctions: 

• define new phase: 

with F ' F ext = const 

IVC ´ IVC of single junction with flux dependent I c 

mechanical analogue: two pendula, for ¯L ¿ 1 rigid coupling of the pendula 

AS-Chap. 4 - 13



AS-Chap. 4 - 14 

• single junction IVC for ¯C ¿ 1: 

periodic IVCs, period: © 0 

V(© ext ): periodic 

minima and maxima at same positions 

maximum modulation for I ' 2I c



AS-Chap. 4 - 15 

• finite screening: ¯L » 1, intermediate damping: ¯C » 1 

- increase flux threading loop: larger loops large L 

- consider displacement + noise current 

numerical solution required ! 

• basic equations: 

relates voltage to phase change 

fluxoid quantization 

coupling



AS-Chap. 4 - 16 

• mechanical analogue: 2 pendula with mass M and length l coupled via twistable bar 

- for negligible screening (¯L ¿ 1): rigid bar 

relative angle 1 - 2 = 2¼© ext /© 0 fixed by external flux 

single combined pendulum with mass 2M, distance from pivot point: l cos(( 1 - 2 )/2) 

without torque: pendula reside at ( 1 - 2 )/2 

with torque (bias current): combined pendula rotate 

I = 0 I > 0


4.1.3 Operation and Performance of dc-SQUIDs 

¯C < 1 non-hysteretic IVCs 

I > 2I c 

AS-Chap. 4 - 17



¯C < 1 non-hysteretic IVCs 

I > 2I c 

AS-Chap. 4 - 18



AS-Chap. 4 - 19 

• important parameters for practical applications of SQUIDs 

flux-to-voltage transfer coefficient (measure of sensitivity to flux): 

maximum at steepest point of hVi(F ext ) 

flux to voltage transducer 

equivalent flux noise (measure of flux resolution) 

S V (f): power spectral density of voltage 

noise at fixed bias current 

noise energy (measure of energy resolution, good for comparison of different SQUID geometries) 

sets energy resolution 

should be as small as possible



for given S V (f): maximize H and L: 

1. bias current: I ¸ I c : largest modulation of hVi(F ext ) 

2. flux bias: close to (2n+1) F 0 /4 

3. junction critical current: coupling energy E J À k B T 

simulations show: I c ¸ 5¢I th ´ 5¢2¼k B T/F 0 

@ 4.2 K: I c ¸ 1 ¹A 

4. loop inductance: should be large 

but: thermal noise flux (k B TL) 1/2 ¿ F 0 

define thermal inductance: L th I th = F 0 /2 

simulations: L · 0.2¢L th 

@ 4.2 K: L · 1 nH 

define: 

5. screening parameter: ¯L = 2I c L/© 0 < 1 no hysteresis in hVi(F ext ) 

small L, but large area choose ¯L » 1 

small I c ≈ 1 ¹A @ 4.2 K L · 1 nH 

6. Stewart-McCumber parameter: ¯c < 1 no hysteresis in IVC (external shunt) 

choose ¯c ' 1 for large voltage output 

AS-Chap. 4 - 20



AS-Chap. 4 - 21 

• detailed numerical simulations: 

²(f) minimum for ¯L » 1, ¯c ' 1 at (2n+1)F 0 /4, at max. voltage modulation (≈I c R N ) 

then: 

• small signal analysis in white noise regime (@ optimal point): 

T, C 

J c , ! p



• improve performance od dc-SQUID: decrease T and C, increase J c 

- ²(f) is given in units of ~ ' 10 -34 Js 

• optimized SQUIDs: ²(f) approaches quantum limit, practical SQUIDs: ²(f) ≈ 10 ~ 

• dimensionless parameter: 

• reduced noise energy: 

rapid increase for 

°¯L > 0.2 

thermal noise rounding 

of IVC 

D. Kölle et al., Rev. Mod. Phys. 71, 631 (1999) 

AS-Chap. 4 - 22


4.1.4 Practical dc-SQUIDs 

AS-Chap. 4 - 23 

we need: 

- antenna, 

- SQUID, 

- room temperature electronics



AS-Chap. 4 - 24 

• today: 

fabrication: 

SQUIDs and antenna consist of thin film structures 

by optical and electron beam lithography 

• washer type SQUID: 

- large sensitivity large area 

- but large L deterioration of performance 

large effective area & 

small inductance: 

perfect diamagnetism 

washer type 

easy coupling to 

antenna: 

planar spiral coil



AS-Chap. 4 - 25 

• loop currents around inner opening defines L = 1.25 ¹ 0 D (for W À D) 

effective area: A eff / D¢W 

• limit: flux trapping in washer area flux noise by thermally activated motion 

• flux focusing effect in washer-type YBCO grain boundary junction dc SQUID 

M.B. Ketchen et al., 

Appl. Phys. Lett. 40, 736 (1982) 

R. Gross et al., 

Appl. Phys. Lett. 57, 727 (1990)



spiral input coil: 

coil inductance: 

mutual inductance: 

coupling coefficient: 

L S : stripline inductance 

n: # of turns 

n 2 L: coil geometric self inductance 

example: D = 20 ¹m L ' 30 pH 

n = 50 L i ' 75 nH and M i ' 1.5 nH (® ex ' 0.6 – 0.8) 

AS-Chap. 4 - 26



low-T c dc-SQUIDs: 

- multilayer thin film technology 

- high plasma frequency (high J c ) 

- small junctions 

( small capacitance) 

- Nb/AlO x /Nb technology 

specific problem: capacitance between spiral input coil and square washer 

LC resonances excess noise 

AS-Chap. 4 - 27


AS-Chap. 4 - 28 


high-T c dc-SQUIDs: 

- heteroepitaxial growth of the different SC layers 

- low reproducibility of junctions 

- alternatives: flip-chip design, directly coupled SQUID (only 1 SC layer required)


AS-Chap. 4 - 29 

4.1.5 Read-Out Schemes 

• flux-locked loop operation 

- hVi(F ext ) curve is nonlinear 

- linearization by feedback circuit 

linear input – output relation 

- oscillating flux, amplitude F 0 /2 

- f mod ≈ 100 kHz – several MHz 

• quasistatic flux = nF 0 : rectified input 

2f mod component 

lock-in detection at f mod zero 

• quasistatic flux = (n+1/4) F 0 : 

maximum lock-in output signal 

• detection of ac voltage: 

cooled transformer: 

SQUID impedance matching: 

R d N 2 R d 

low-noise preamplifier



flux-locked loop: 

L f : 

feedback coil 

small flux change ±F to SQUID positive output / ±F 

increase of integrator output voltage 

increase of current through feedback coil 

compensates ±F 

SQUID voltage (and integrator output) stay constant null detector 

±V in / ±F ±I f = ±V in /R f ±F f = k 2 L f ±I f 

then from |±F f | = |±F |: 

f mod » 100 kHz – several MHz: bandwidth up to 100 kHz, dynamic range ¸ 140 dB 

“slew rate”: up to 10 7 © 0 /s 

AS-Chap. 4 - 30


AS-Chap. 4 - 31 


• bias current reversal: 

- bias current is modulated (to suppress low-frequency noise of Josephson junctions) 

- asymmetric part of critical current fluctuations can be eliminated 

double modulation technique 

• additional positive feedback: 

- asymmetric hVi(F ext ) (part of the bias current is used) 

- steeper slope larger V/F 

direct read-out with RT electronics 

• digital read-out schemes: 

- cryogenic digital feedback schemes compact, wide bandwidth 

digitized output signals for transmission to RT 

• relaxation oscillation schemes: 

- hysteretic junctions 

- SQUID shunted by series LR 

- frequency of relaxation oscillations depends on flux 

large SQUID output voltage


4.2 The rf-SQUID 

AS-Chap. 4 - 32 

requires only 1 Josephson junction: 

- rf current is applied via tank circuit inductively coupled to SQUID loop 

- measure time-averaged tank circuit voltage 

advantage: only 1 JJ, no dc-current to SQUID 

disadvantage: energy resolution (limited by read-out electronics) 

4.2.1 The zero voltage state 

total phase change: 

using: 

and:


4.2.1 The zero voltage state 

substitution: 

integration of A total flux 

large thickness: 2 nd integral can 

be omitted 

supercurrent: 

total flux: 

screening parameter: 

¯L,rf >1: 

unstable regions 

negative slope 

practically relevant 

at © ext = © ext,c : 

junction critical current reached 

flux quantum enters SQUID ring 

hysteresis loop 

AS-Chap. 4 - 33


4.2.2 Operation and Performance of rf SQUIDs 

AS-Chap. 4 - 34 

operation of rf-SQUIDs: 

- SQUID loop inductively coupled to LC tank circuit 

- quality factor Q = R T /! rf L T 

! rf2 = 1/L T C T 

- mutual inductance M = ®(L T L) 0.5 

- rf current in tank circuit: I T = QI rf 

for ¯L,rf < 1: 

©(© ext ) non-hysteretic 

rf-SQUID as nonlinear inductor 

periodic modulation of resonance frequency (L T,eff C T ) -1/2 

operation close to resonance frequency: strong change in rf-voltage



AS-Chap. 4 - 35 

for ¯L,rf > 1: 

©(© ext ) hysteretic 

total flux through SQUID: 

for © S + © rf > © ext,c hysteresis loop 

energy loss / area 

damping of tank circuit: 

minimum close to n¢© 0 

maximum close to (2n+1)/2¢© 0 

tank voltage is periodic in applied flux


dependence of V T on © S and © rf : 

start at © S = n¢© 0 , n = 0: 

V T increases linearly with I rf as long as © rf = MQI rf < © ext,c 

critical rf-current: I rf,c = © ext,c /M 

tank voltage: 

for I rf > I rf,c : jump to k = +1 or k = -1 branch 

hysteresis loop, energy loss, decrease of rf-current 

no hysteresis loops until tank circuit has recovered 

further increase of I rf (A B) 

transitions at higher rate 

at I rf,r : 1 transition in each rf cycle 

start at © S = (n+1/2)¢© 0 , n = 0: 

transition to k = +1 branch at: © ext,c - © 0 /2 

transition to k = -1 branch at:- (© ext,c +© 0 )/2 

first horizontal branch at: 

V T (I rf ) curves for intermediate values: 

between © S = © 0 /2 and © S = 0 

AS-Chap. 4 - 36



AS-Chap. 4 - 37 

dependence of V T on © S and © rf : 

- start at © S = n¢© 0 , n = 0: 

- V T increases linearly with I rf as long as © rf = MQI rf < © ext,c 

- critical rf-current: I rf,c = © ext,c /M 

- tank voltage: 

for I rf > I rf,c : jump to k = +1 or k = -1 branch 

hysteresis loop, energy loss, decrease of rf-current 

no hysteresis loops until tank circuit has recovered 

further increase of I rf (A B) 

transitions at higher rate 

at I rf,r : 1 transition in each rf cycle 

start at © S = (n+1/2)¢© 0 , n = 0: 

transition to k = +1 branch at: © ext,c - © 0 /2 

transition to k = -1 branch at:- (© ext,c +© 0 )/2 

first horizontal branch at:



change of V T from © S = 0 to © 0 /2: ! rf L T © 0 /2M 

flux-to-voltage transfer function near © S = © 0 /4: 

lower bound for M / ®: there must be I rf that intersects the first step of V T (I rf ) 

point F has to be to the right of E 

operation: stay on step for all © S , flux locked loop with flux modulation 

AS-Chap. 4 - 38



AS-Chap. 4 - 39 

additional topic: noise in rf-SQUIDs: 

- switching k = 0 k = 1: stochastic fluctuations (thermal activation) 

noise in step voltage V T flux noise: 

noise causes finite slope of horizontal branches 

extrinsic noise source: preamplifier, lines T amp 

eff 

energy resolution: 

high frequencies (few GHz), cold amplifiers: ² ≈ 3£10 -32 J/Hz 

comparison: 

rf-SQUID: ² ≈ k B T/! rf 

dc-SQUID: ² ≈ k B T/! c (! c = 2¼IcR N /© 0 ≈ 100 GHz) 

better energy resolution of dc-SQUID because: ! c À ! rf


4.2.3 Practical rf-SQUIDs 

AS-Chap. 4 - 40 

low T c rf SQUIDs: 

- early versions: toroidal configuration machined from Nb 

- operated at 10 MHz, ² ≈ 5£ 10 -29 J/Hz 

- now: thin film technology, ² ≈ 10 -32 J/Hz 

high T c rf-SQUIDs: 

- operating at 77 K 

- washer-type rf-SQUIDs incorporated in ¸/2 microstrip resonator


4.3 additional topic: Other SQUID Configurations 

4.3.1 The DROS (Double Relaxation Oscillation SQUID) 

AS-Chap. 4 - 41 

large flux-to-voltage transfer coefficient, large modulation voltage 

direct read-out by RT amplifier 

hysteretic dc-SQUID (¯C > 1), hysteretic junction in series + L-R shunt 

DROS as comparator of the two critical currents 

voltage output behaves like square wave 

flux-to-voltage coefficients up to 3 mV/© 0


4.3.2 The SQIF 

AS-Chap. 4 - 42 

for ¯L ¿ 1: dc-SQUID double slit configuration 

steeper I sm (© ext ) for optical grid analogue: N junctions in parallel 

experimental problem: uniformity of junctions and loops 

irregular parallel array of JJ: Superconducting Quantum Interference Filters 

I sm (© ext ) shows sharp peak at zero flux


4.3.3 Cartwheel SQUID 

AS-Chap. 4 - 43 

SQUID loop: several loops in parallel reducing total SQUID inductance


Summary: dc Squid 

AS-Chap. 4 - 44 

flux-to-voltage transfer coefficient 

equivalent flux noise 

noise energy


4.4 Instruments Based on SQUIDs 

• SQUIDs sense any signal that can be converted to flux 

• SQUID based instrument: antenna determines quantity to be measured: 

• input circuit influences signal and noise properties of SQUID 

• reduction of SQUID inductance 

® 2 : coupling coefficient 

mutual inductance: 

AS-Chap. 4 - 46


4.4.1 Magnetometers 

AS-Chap. 4 - 51


4.4.1 Magnetometers 

AS-Chap. 4 - 52


4.4.2 Gradiometers 

AS-Chap. 4 - 53 

for biomagnetic measurements: 

reduce perturbing environmental magnetic fields 

- non-magnetic materials 

- ¹-metal shields 

- magnetically shielded rooms 

and/or: n th -order gradiometers suppression of remote signals 

PTB Berlin: 

7 ¹-metal shields 

1 Al layer 

active field reduction 

shielding factor: 

2£ 10 6 @ 0.01 Hz 

2£ 10 8 @ 5 Hz


4.4.2 Gradiometers 

AS-Chap. 4 - 54 

axial gradiometer 

planar gradiometer 

1 st order 2 nd order 1 st order, dB z /dx 1 st order, dB z /dy


4.4.3 Susceptometers 

measurement of magnetic properties of materials 

- 1 st or 2 nd order gradiometer & sample in static B field 

- sample in one of the pick-up loops 

non-magnetic sample 

sample with susceptibilty Â 

no output signal 

additional flux detected by SQUID 

commercial systems: 2 nd order gradiometer, sample axially moving 

Â with resolution of 10 -8 emu @ 1.8 – 400 K up to 7 T 

AS-Chap. 4 - 55


4.4.4 Voltmeters 

AS-Chap. 4 - 57 

voltage transformed into current via input resistor 

- feedback of SQUID output to input resistor 

flux locked loop (null-balancing of voltage) 

resolution: 

≈ 10 -12 V/(Hz) 1/2 for R i = 0.01 

≈ 10 -10 V/(Hz) 1/2 for R i = 100 

superior for low impedance samples 

applications: 

transport, noise, thermoelectric properties of metallic/superconducting samples


4.4.5 Radiofrequency Amplifiers 

AS-Chap. 4 - 58 

tuned amplifier: 

- input circuit: R i , C i , and L i 

- for frequencies up to 100 MHz 

- noise temperature close to the quantum limit T N ≈ hn / k B


4.5 Applications of SQUIDs 

4.5.1 Biomagnetism 

AS-Chap. 4 - 60 

biomagnetic method: 

- non-invasive detection of magnetic signals from human body 

biomagnetic imaging: 

- field map of heart / brain activity source location 

(only simple volume conductor models are needed) 

MEG: magnetoencephalography 

MCG: magnetocardiography 

extremely small fields: 

brain: 100 fT (one neuron: 0.1 fT), heart: 10 pT 

highly sensitive SQUID magnetometers 

low 1/f noise 

spatial and temporal magnetic 

field distribution: 

array of SQUID sensors (multichannel systems) 

(or scan single SQUID)



AS-Chap. 4 - 61 

signal reconstruction 

- current distribution cannot be calculated from 

measured field distribution 

inverse problem has no unique solution 

model assumptions 

based on elementary current dipoles 

(= short localized conductor segments & 

volume backflow)


AS-Chap. 4 - 62 


Magnetocardiography (MCG): 

heart signals ≈ 100 pT, fine structure (≈ pT) clinically relevant 

source of pathological signals with high precision in 3 dimensions 

• planar sensor loops B z component 

positive and negative components 

instantaneous field maps 

eg.: false color maps 

• each patient: own fingerprint 

• use noise suppression techniques 

MFMS: magnetic field map sequences 

healthy 

heart 

ventricular 

tachycardia



AS-Chap. 4 - 63 

11 SQUIDs: 

vector components B x , B y , B z on two planes: 

B x ~ (B x1 + B x2 )/2 

gradient: 

B x /x ~ (B x1 - B x2 ) 

magnetic field during R-peak above thorax of healthy heart:



AS-Chap. 4 - 64 

Magnetoencephalography (MEG) 

functional brain mapping (in contrast to CT/MRI) 

intercellular neuronal currents 

spatial resolution 2 mm, temporal resolution: 1 ms 

optimum SQUID resolution, noise suppression 

completely non-invasive, undistorted biosignals 

epilepsy, localizing eloquent cortex


4.5.2 Nondestructive Evaluation (NDE) 

non-invasive identification of structural or material defects 

(sub-) surface cracks in aircrafts 

reinforcing rods in concrete strutures 

short distance between inner 

cold and out warm wall 

(spatial resolution) 

HTS SQUIDs (@ 77 K) 

advantageous 

eddy-current techniques: 

- alternating field 

- eddy currents disturbed by material defects 

AS-Chap. 4 - 65


4.5.2 Nondestructive Evaluation (NDE) 

AS-Chap. 4 - 66


4.5.3 SQUID Microscopy 

AS-Chap. 4 - 67 

image magnetic field distribution, high sensitivity, modest spatial resolution 

- initially: low-T c dc SQUIDs, now: high-T c SQUIDs @ 77 K 

- frequency: dc up to 1 GHz 

spatial resolution: 

- cold samples: 5 ¹m (with soft magnetic focusing tip: 0.1 ¹m) 

- RT-samples: 30 – 50 ¹m 

applications. 

- diagnostics of SC devices 

- properties of ultra-thin magnetic films 

- analysis of semiconducting devices 

SQUID – sample separation: 

75 ¹m Sapphire window: 150 ¹m 

3 ¹m Si x N y window: 15 ¹m


4.5.3 SQUID Microscopy 

AS-Chap. 4 - 68 

alignment track of 5.25” 

floppy disk 

magnetic ink in 1 $ note 

flux quanta trapped in 

SQUID washer


4.5.4 Gravity Wave Antennas and Gravity Gradiometers 

topics: inertial navigation, general relativity, deviations from 1/r 2 , and gravitational waves: 

e.g. collapsing stars, rotating double stars 

expansion and contraction oscillations 

expected length change ¢l/l ~ 10 -19 

required resolution ~ 10 21 

resonant mass transducer 

displacement current 

antenna mK regime 

resolution: zero point motion 

quantum limited antenna 

typical resonance frequency: 1 kHz T< ~! ant /k B ≈ 50 nK 

but: increase of Q increase of effective noise temperature T eff 

assume: gravitational pulse, length ¿, antenna decay time Q/! ant : 

quantum limit (bar energy > k B T eff ) T < Q~/k B ¿ 

for Q = 2£10 6 and ¿ = 1 ms T ≈ 20 mK 

quantum limited sensor is required 

presently: sensitivities ¢l/l ≈ 10 -18 , no gravity waves reported yet 

AS-Chap. 4 - 69


AS-Chap. 4 - 70 

4.5.4 Gravity Wave Antennas and Gravity Gradiometers 

gravity gradiometer: 

gravity gradient separation of test masses coil induction 

mapping earth’s gravity gradient, test of 1/r 2 law 

4.5.5 Geophysics 

SQUIDs used to probe the magnetic properties of earth 

• rock magnetometry 

• mapping earth magnetic field / em impedance 

• geophysical surveying


Summary 

negligible screening ¯L = 2LI c /© 0 

¿ 1 © » © ext 

strong damping: 

large screening ¯L = 2LI c /© 0 À 1 © = © ext + LI cir » n© 0 

intermediate ¯L: I sm (© ext ) self-consistently from ©(© ext ) and I sm (©) 

performance: 

optimum operation (¯L » 1, ¯C » 1): 

operation in flux-locked-loop as null detector 

AS-Chap. 4 - 71


Summary (rf-SQUID) 

operation: inductive coupling to tank circuit 

performance: 

operation in flux-locked-loop as null detector 

Summary (SQUID based instruments) 

• antenna, SQUID (flux-to-voltage transformer), read-out electronics 

• magnetometer, gradiometer, voltmeter, susceptometer, … 

• magnetic field resolution: few fT/(Hz) 1/2 

• application: magnetocardiography/-encephalography, NDE, microscopy, geophysics 

AS-Chap. 4 - 72

II. Applications of the Josephson Effect

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