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SQUIDs: Then and Now 161

Serious theory emerged in the ’70s, and reﬁnements continue today. I cannot

possibly do justice to all the work that has been done, but will summarize

the key results.

The dynamics of the Josephson junction are well explained by the

McCumber

–Stewart

model, in which the Josephson element is in parallel

with a resistance R (which may be an external shunt) and a capacitance C.

The I–V characteristic is hysteretic when

≡ 2πI

C/Φ

≡ ω

RC > 1 , (5)

and nonhysteretic for β

< 1. Here, ω

/2π ≡ I

R/Φ

is the Josephson

frequency at voltage I

R. In the absence of noise and for β

 1, the I–V

characteristic reduces to V = R(I

− I

)

1/2

, which tends asymptotically to

V = IR for I  I

. An external shunt resistance, which is linear, has an

associated Nyquist noise current with a spectral density S

(f) = 4k

T/R

in the classical limit; k

is the Boltzmann constant. Noise has two eﬀects.

First, it rounds the I–V characteristic at low voltages, reducing the apparent

critical current.

To maintain a useful degree of Josephson coupling, one

requires the coupling energy I

/2π  k

T ; at 4.2 K, I

 0.2 µA. The

second eﬀect of the noise current is to induce a voltage noise across the

junction at nonzero voltages.

All computer and analytical calculations of

noise in SQUIDs are based on this model.

6.1. dc SQUID

Claudia Tesche and I worked out the theory for the dc SQUID using this

model.

46,47

That is to say, after we had agreed on the equations of mo-

tion, Claudia spent roughly a year carrying out the computer simulations

including the eﬀects of noise. The inclusion of noise requires a great deal of

averaging to obtain an accurate result, and in those days, computers were

of course orders of magnitude slower than they are today. To speed up the

calculation, we neglected the junction capacitance, and, for given values of

and junction area, chose the value of R that set β

= 1 so that the I–V

characteristic is just nonhysteretic. There is a second limitation imposed by

Nyquist noise in the shunt resistors, namely Φ

/2L  2πk

T . At 4.2 K,

this implies L  6 nH.

For a typical SQUID operating at 4.2 K, we can summarize our results

as follows. First, the noise energy is optimized when

≡ 2LI

/Φ

= 1 . (6)

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162 J. Clarke

The maximum response to a small change in ﬂux δΦ

 Φ

occurs when

≈ (2n + 1)Ω

/4, where the ﬂux-to-voltage transfer coeﬃcient

≡ |∂V /∂Φ

≈ R/L ≈ 1/(πLC)

1/2

; (7)

we have assumed β

= β

= 1. Nyquist noise in the shunt resistors intro-

duces a white voltage noise across the SQUID with spectral density S

(f),

and a current noise around the SQUID loop with spectral density S

(f).

Both spectral densities depend on the current and ﬂux biases. At frequen-

cies much less than the Josephson frequency and under optimum conditions,

the voltage noise leads to a ﬂux noise spectral density

(f) = S

(f)/V

≈ 16k

T L

/R . (8)

The optimized current noise spectral density is

(f) ≈ 11k

T/R . (9)

Furthermore, the voltage and current noises are partially correlated, with a

cross-spectral density

V J

(f) ≈ 12k

T . (10)

The noise energy is

ε(f) = S

(f)/2L ≈ 9k

T L/R ≈ 16k

T (LC)

1/2

. (11)

We note that ε(f) is not a complete characterization of the noise in the dc

SQUID since it does not take into account the noise current in the loop,

which induces a noise voltage into an input circuit coupled to the SQUID.

This theory does not, of course, account for 1/f noise, which arises from

ﬂuctuations in the junction critical currents and from “ﬂux noise,” a subject

that is only now beginning to be understood. We shall not delve into these

subjects.

For a representative SQUID with L = 200 pH, R = 6 Ω, β

= 1 and

= 1, we estimate S

1/2

(f) ≈ 10

−6

−1/2

and ε(f) ≈ 10

−32

JHz

−1

≈

100~, in good agreement with measured values. Equation (11) makes it clear

that “smaller is better” and that ε(f) decreases linearly with temperature,

in good agreement with experiment.

6.2. rf SQUID

The theory of the rf SQUID begins by imposing ﬂuxoid quantization on a

superconducting loop containing a single Josephson junction [Fig. 6(a)]:

δ + 2πΦ

/Φ

= 2πn . (12)

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SQUIDs: Then and Now 163

0 1 2

Φ/Φ

/Φ

=0.5

′

=2.0

′

Fig. 10. Total ﬂux Φ

in the rf SQUID loop versus applied ﬂux Φ

for the disper-

sive (β

= 0.5) and hysteretic (β

= 2) regimes.

Here, Φ

is the sum of the applied ﬂux Φ and the ﬂux generated by the

current J = −I

sin(2πΦ

/Φ

) ﬂowing around the loop. Thus,

= Φ

− LI

sin(2πΦ

/Φ

) . (13)

Inspection of Eq. (13) reveals two distinct kinds of behavior, plotted

in Fig. 10. For β

≡ 2πLI

/Φ

< 1, the slope dΦ

/dΦ

= [1 +

cos(2πΦ

/Φ

)]

−1

is always positive, and the Φ

versus Φ

curve is non-

hysteretic. For β

> 1, on the other hand, there are regions in which

dΦ

/dΦ

is positive, negative or divergent, and the Φ

versus Φ

plot

becomes hysteretic.

In the hysteretic mode, the applied rf ﬂux [Fig. 6(a)] causes the SQUID

to make transitions between quantum states and to dissipate energy at a

rate that is periodic in Φ

. This periodic dissipation in turn modulates the

Q of the tank circuit, so that when it is driven on resonance with a current

of constant amplitude, the rf voltage is periodic in Φ

. This is the origin of

the behavior ﬁrst observed by Zimmerman and Silver [Fig. 6(b)]. A detailed

analysis

–

shows that the SQUID is optimized when k

Q ≈ 1, where k

is the coupling coeﬃcient between the SQUID loop and the inductor of the

resonant circuit. Under this condition, at ﬁxed rf current I

the transfer

coeﬃcient becomes

|∂V

/∂Φ

≈ ω

/L)

1/2

/k . (14)

The intrinsic noise energy is given by

ε(f) ≈ LI

(2πk

T/I

)

4/3

/2ω

. (15)

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164 J. Clarke

For SQUIDs at 4.2 K coupled to a room-temperature preampliﬁer, however,

extrinsic noise contributions — notably preampliﬁer noise and loss in the

line coupling the SQUID and preampliﬁer — may far exceed the intrinsic

noise. For this reason, dissipative rf SQUIDs at 4.2 K are rarely used today.

On the other hand, increasing the temperature of the SQUID from 4.2 K to

77 K has little impact on the system noise energy, whereas the noise energy

of the dc SQUID scales with T [Eq. (11)]. Thus, the noise advantage of the

dc SQUID over the hysteretic rf SQUID is diminished at 77 K.

In the nonhysteretic regime β

< 1, the rf SQUID behaves as a ﬂux-

sensitive inductor. The underlying physics is that the inductance of a Joseph-

son junction in the zero voltage regime is given by L

= Φ

/2π(I

− I

)

1/2

for I < I

. Thus, an applied ﬂux produces a circulating current and hence

a change in L

. When the tank circuit is driven oﬀ-resonance at constant

amplitude and frequency, a ﬂux change in the SQUID changes the resonant

frequency, so that the rf voltage is periodic in the applied ﬂux. In the limit

/2π  k

T , the intrinsic noise energy is

51,53

ε(f) ≈ 3k

T L/β

R . (16)

A noise energy of 23 ± 7~ was reported for an rf SQUID operating in this

regime at microwave frequencies.

7. High-T

SQUIDs

The advent of high-T

superconductors in 1986 generated a furor of

activity in the SQUID community. A successful thin-ﬁlm technology for

YBa

7−x

was developed that included both single-layer SQUIDs and

SQUIDs with integrated coils, very much like the low-T

device of Fig. 8(b),

and planar magnetometers and gradiometers. For reasons of space, I have

decided not to discuss high-T

SQUIDs; a broad overview and copious refer-

ences appear in Koelle et al.

8. Applications of SQUIDs: Overview

The major reason I still work on SQUIDs after 46 years in the ﬁeld is the

sheer breadth of their applications. This diversity is driven by the extraor-

dinary low noise energy of the SQUID, which has opened up a myriad of

applications that would otherwise not exist. Many of these are one-of-a-kind

experiments in basic science, and I cannot possibly hope to review them. In

this section, I touch on some of the areas in which SQUIDs are used, and

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SQUIDs: Then and Now 165

(a) (b)

Fig. 11. Commercial SQUID systems. (a) System for magnetoencephalography

(courtesy Elekta). (b) Cross-section of system for measuring magnetic properties

(courtesy Quantum Design).

in Secs. 9 and 10, I describe in more detail selected applications in which I

have been involved personally.

Systems for magnetoencephalography (MEG)

involve about 300 dc

SQUIDs, coupled to gradiometers, arranged in a helmet into which the sub-

ject inserts his or her head. The system — a typical version is shown in

Fig. 11(a) — is generally installed in a magnetically-shielded room to at-

tenuate external magnetic disturbances. The SQUIDs detect tiny magnetic

signals produced by neuronal electrical currents in the brain. Magnetoen-

cephalography is used in neurological and psychological research, and in

clinical applications such as pre-surgical mapping of brain tumors. A second

medical application is magnetocardiography (MCG),

for example, in the

noninvasive diagnosis of heart disease such as ischemia. Other biomagnetic

applications

include liver susceptometry to monitor the accumulation of

excess iron (thalassemia) and immunoassay — the detection and identiﬁca-

tion of antibodies or antigens in blood specimens.

Almost certainly, the most commercially successful SQUID system

[Fig. 11(b)] is used to measure magnetic properties such as magnetiza-

tion and susceptibility at temperatures ranging from a few kelvin to above

room temperature in magnetic ﬁelds up to several tesla.

This system is

available with a cryocooler, obviating the need to supply liquid helium. A

September 8, 2010 10:53 World Scientiﬁc Review Volume - 9.75in x 6.5in ch9

166 J. Clarke

related system,

also highly successful, is a “rock magnetometer,” with an

open horizontal access to characterize core samples taken from the Earth.

The “scanning SQUID microscope” enables one to move samples at room

temperature in two-dimensions close to a SQUID to map their magnetic

properties.

59,60

In some approaches, one measures intrinsic magnetization,

while in others, one applies a low-frequency magnetic ﬁeld and detects the

eddy-current response. Applications include detecting ﬂaws in aircraft wheel

hubs and under the rivets of aircraft fuselages, ﬁnding tantalum inclusions

in niobium sheets destined for superconducting cavities for particle acceler-

ators, tracking corrosion by detecting the magnetic ﬁelds generated by the

ion currents responsible for the oxidation, and locating defects in computer

multichip modules and other semiconductor circuits. A SQUID microscope

was used to map the swimming of a magnetotactic bacterium.

A quite diﬀerent application is geophysical exploration

— in which

interest has been rekindled using liquid-nitrogen-cooled SQUID magnetome-

ters. Of particular interest is “transient electromagnetics” (TEM), in which

one applies a large current pulse to the ground and measures the ensuing

magnetic ﬁeld. In one version of TEM, a SQUID package is towed behind

an aircraft.

In the examples I have given above, the primary goal is to detect magnetic

ﬁeld. There is another class of applications in which the aim is to measure

other physical quantities. One example, is the SLUG voltmeter. Another is

in transducers for gravity wave detectors and gravity gradiometers.

Pre-

cision gyroscopes with SQUID readout were the basis of Gravity Probe B,

intended to measure the geodetic eﬀect (curved space-time due to the pres-

ence of the Earth) and Lense-Thirring eﬀect (dragging of the local space-time

frame due to rotation) predicted by general relativity. SQUIDs ﬁnd impor-

tant applications in standards and metrology,

for example, in verifying the

universality of the Josephson voltage-frequency relation, in linking the volt

to mechanical SI units, and in the cryogenic current comparator. A fasci-

nating application of SQUIDs that came to prominence in the last decade

is to superconducting qubits.

SQUIDs are used to detect the quantum

state of ﬂux and phase qubits, and — through the ﬂux dependence of their

inductance — to tune thin-ﬁlm, microwave resonators.

In the next two sections, I brieﬂy describe three relatively new applica-

tions of SQUIDs that illustrate their diversity. The ﬁrst two are in cosmol-

ogy

— searching for galaxy clusters and for cold dark matter (Sec. 9) — and

the third (Sec. 10) is the use of SQUID gradiometers in low-frequency nuclear

magnetic resonance (NMR) and magnetic resonance imaging (MRI).

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SQUIDs: Then and Now 167

9. Cosmology

Measurements of the cosmic microwave background (CMB) have taught us

almost everything we know about the origin of the universe — so much so

that the CMB

is sometimes referred to as “the Cosmic Rosetta Stone”. The

CMB, which has a Planck distribution with a characteristic temperature of

2.726 K, originated 379,000 years after the Big Bang and has been traveling

for 13.7 billion years since then. We have learned that the Universe is a

rather bizarre place, consisting of 0.6% neutrinos, 4.6% baryons, 73% dark

energy (DE) and 22% cold dark matter (CDM). Thus, some 95% of the

universe is of unknown origin. SQUIDs are being used to investigate both

DE and CDM, as I brieﬂy describe in the next two sections.

9.1. Dark energy: Searching for galaxy clusters

Galaxy clusters contain typically several hundred galaxies and, with a mass

of 10

to 10

solar masses, are the largest objects in the universe.

Mea-

suring the density of galaxy clusters as a function of red-shift enables one

to determine the parameters of the equation of state for DE. To make

a deﬁnite test, however, one needs to ﬁnd and characterize thousands of

clusters. The red-shift is available from optical astronomy, but the signal is

weak, making it diﬃcult to locate large numbers of clusters. A newly imple-

mented search technique capable of much more rapid searching involves the

Sunyaev–Zel’dovich Eﬀect (SZE).

This technique is based on the fact that

the space between galaxies in a cluster contains a hot electron gas. When a

microwave background photon passes through this gas, it has a small (1–2%)

chance of being scattered to a higher energy, so that a fraction of the CMB

spectrum is shifted to slightly higher frequencies. Thus, as the telescope

scans across the sky, one expects to ﬁnd regions in which there is a shifted

Planck spectrum superimposed on the unshifted spectrum — the signature

of a galaxy cluster.

Since the CMB spectrum peaks at a frequency of about 150 GHz, one

needs ultrasensitive detectors in the far infrared. The detector of choice is the

transition-edge sensor (TES) bolometer.

This consists of three elements:

an optical absorber, a sensitive thermometer and a thermal conductance to

a heat bath. It is convenient ﬁrst to describe the thermometer (TES), which,

in the groups of Bill Holzapfel, Adrian Lee and Paul Richards at Berkeley,

is a thin ﬁlm bilayer of Al-Ti

with a superconducting transition tempera-

ture of about 0.5 K. The TES is connected in series with a voltage source

and the input coil of a SQUID, and operated at a temperature within its

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168 J. Clarke

(a) (b)

Fig. 12. Far infrared bolometer. (a) Spider-web absorber, 3 mm in diameter.

(b) Expanded view. The TES (lower edge of the gold ring) is electrically connected

to the readout circuit by two aluminum lines (vertical in this photograph) and

thermally connected to the heat bath by the gold line to the left of the left-hand

aluminum line. (Reproduced with permission from Ref. 68.)

resistive transition. When a photon is absorbed, thermal energy is trans-

ferred to the TES, initially raising its temperature and increasing its resis-

tance. As the resistance begins to increase, however, the dissipation gen-

erated by the voltage bias drops, so that the power dissipation in the TES

remains constant. The change in the current ﬂowing through the SQUID

input coil is related to the absorbed energy. This electrothermal feedback

results in a fast, linear response. Furthermore, the power dissipation is

low, typically 1 nW. The TES is mounted at the center of the absorber,

which consists of a “spiderweb” of silicon nitride coated with a thin gold

ﬁlm (Fig. 12). The noise in an optimized sensor is limited by ﬂuctuations in

the arrival rate of photons.

To obtain an image of a galaxy cluster, one requires hundreds or prefer-

ably thousands of such sensors mounted at the focal plane of a telescope.

Needless to say, this is a challenging prospect. Although it is perfectly feasi-

ble to fabricate hundreds or thousands of SQUIDs, a serious concern is the

thermal heat load generated by the leads when each TES requires its own

readout SQUID. This issue caused Paul Richards to walk into my oﬃce one

day in the spring of 1999 to discuss the feasibility of using a single SQUID

to read out an array of TESs simultaneously with a frequency-domain

multiplexer. I found this a very interesting problem, and the eventual out-

come was the eight-channel multiplexer

shown in Fig. 13. Each TES is

connected in series with a capacitor and an inductor to form a resonant

circuit, each with a diﬀerent frequency. An oscillator provides a comb of

voltages V

) at frequencies f

corresponding to the resonant frequencies of

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SQUIDs: Then and Now 169

Demodulator

Nulling

comb

Σ-V

)

Bias

comb

ΣV

)

Fig. 13. Circuit for a TES frequency-domain multiplexer. A tuned circuit is con-

nected in series with each TES. The readout SQUID consists of 100 SQUIDs in

series. (Reproduced with permission from Ref. 69.)

the tuned circuit. The currents from the eight tuned circuits are summed at

one end of the input coil of a SQUID (actually a 100-SQUID series array);

the other end is grounded. The oscillator applies a nulling comb of voltages

−V

) to the same summing point so that, in the absence of any photon

ﬂux, the total current injected into the input coil is zero. When a photon

is absorbed by one of the sensors, the resulting change in current is injected

into the input coil of the SQUID, which is operated in a ﬂux-locked loop.

Since the eight frequencies are distinct, all eight sensors can be read out

simultaneously with the aid of a demodulator.

The Berkeley group, in collaboration with several other groups, ﬁrst

implemented this scheme on APEX (Atacama Pathﬁnder EXperiment), a

12-m telescope at about 5100 m on the Atacama plateau in Northern Chile

[Fig. 14(a)]. The resolution is one minute of arc and the ﬁeld of view is

22 minutes of arc. The receiver contained about 280 working channels

with multiplexed readout. As a ﬁrst demonstration of the system opera-

tion, Fig. 14(b) shows an SZE image of a known galaxy cluster, the “Bullet

Cluster” — actually two merging clusters.

The signal-to-noise ratio is 20

in the central one minute of arc beam. Subsequently, the Berkeley group

and others began a cluster search with SPT (South Pole Telescope), a 10-m

telescope at Antarctica at about 2900 m. The resolution is one minute of arc

and the ﬁeld of view 60 minutes of arc; there about 660 TESs with multi-

plexed readout. In the next two years, SPT will survey 4000 square degrees

of sky and — if the theory is correct — is expected to ﬁnd thousands of

galaxy clusters.

September 8, 2010 10:53 World Scientiﬁc Review Volume - 9.75in x 6.5in ch9

170 J. Clarke

00′

Right Ascension (J2000)

µK CMB

Declination (J2000)

58′

56′

54′

−55°52′

−600

−500

−400

−300

−200

−100

(a) (b)

Fig. 14. APEX telescope and ﬁrst results. (a) Telescope at Atacama. (b) SZE

image of the “Bullet Custer”; the contour interval is 100 µK

CMB

. Disk at lower left

represents the full width at half-maximum (FWHM) resolution. (Reproduced with

permission from Ref. 70.)

9.2. Cold dark matter: The hunt for the axion

Since we do not know what CDM is, we have to postulate some kind of

particle and then search for it. One of the two leading candidates is the

WIMP (Weakly Interacting Massive Particle).

Supersymmetry theories

predict that the WIMP has a mass of 10–100 GeVc

−2

. A WIMP search

currently under way involves a 250 g block of Ge. An impinging WIMP would

scatter oﬀ the Ge nuclei, producing phonons, which in turn would break

Cooper pairs in superconducting Al absorbers on the surface, producing

quasiparticles. The absorbers are in electrical contact with 1000 parallel-

connected tungsten TESs. The quasiparticles diﬀuse into the TESs, where

they are trapped, raising the temperature. The TES array is biased with a

static current, and read out with a 100-SQUID array. The second candidate

is the axion, ﬁrst proposed in 1978 to explain the absence of a measurable

electric dipole moment on the neutron.

–

The axion mass m

is predicted

to be 1 µeVc

−2

to 1 meVc

−2

(corresponding to 0.24 to 240 GHz). Since the

measured CMB density ρ

is 0.45 GeV cm

−3

, a mass of 1 µeV corresponds

to an axion density of 4.5 × 10

−3

How does one search for this light, spinless, chargeless particle? In 1983,

Pierre Sikivie

75,76

suggested that it could be found via Primakoﬀ conver-

sion. In this scheme, in the presence of a large magnetic ﬁeld B, the axion

converts to a photon (together with a virtual photon) with energy given

by m

. The conversion takes place in a cooled cavity of volume V with a