Cooper L.N., Feldman D. (Eds.) BCS: 50 Years

Подождите немного. Документ загружается.

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

SQUIDs: Then and Now 171

Fig. 15. Axion detector installed at Lawrence Livermore National Laboratory.

(Courtesy Darin Kinion.)

tunable resonance frequency. An antenna couples the signal in the cavity to

a cooled, low-noise ampliﬁer. When the cavity frequency corresponds to the

frequency of photons from the putative axions, the ampliﬁer detects a peak;

oﬀ-resonance, it detects only blackbody noise. Clearly, one has to scan the

cavity frequency to search for the axion. The axion-to-photon conversion

power is given by

∝ B

V g

, (17)

where g

is a coupling coeﬃcient. There are two contending theories

for g

, the KSVZ (Kim–Shifman–Vainshtein–Zakharov) model

78,79

and the

DFSZ (Dine–Fischler–Srednicki–Zhitnitsky) model,

80,81

with g

=+0.97 and

−0.36, respectively. Evidently, the DFSZ model predicts the smaller density

of photons in the cavity.

Following the ﬁrst-generation detectors based on this scheme at

Brookhaven National Laboratory and the University of Florida,

a second-

generation detector was constructed at Lawrence Livermore National

Laboratory (Fig. 15), and began operation in 1996. The cavity and the

surrounding 7-T persistent-current magnet were cooled to 1.5 K. The cooled

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

172 J. Clarke

Nb coil isolated

from washer

Nb washer

Nb counter

electrode

Josephson

junctions

resistive shunt

out

(a) (b)

Fig. 16. Microstrip SQUID ampliﬁer. (a) Conﬁguration (courtesy Darin Kinion).

(b) Gain versus frequency for four lengths of input coil.

HEMT (high electron mobility transistor) ampliﬁer had a noise temperature

= 1.7 K, giving a system noise temperature T

= T + T

= 3.2 K. For

the parameters of the experiment and assuming g

= −0.36, the time to

scan from f

= 0.24 GHz to f

= 0.48 GHz is

τ(f

, f

) ≈ 4 × 10

/1K)

(1/f

− 1/f

) sec ≈ 270 years . (18)

Evidently, this experiment was not likely to ﬁnd an axion very quickly!

In 1994, Leslie Rosenberg and Karl van Bibber approached me to ask

whether it would be possible to make a SQUID ampliﬁer with a lower noise

temperature than a HEMT at frequencies around 1 GHz. At the time, this

was not so simple: the essential problem with the usual SQUID washer

SQUID (Fig. 8) is that the parasitic capacitance between the input coil

and the washer rolls oﬀ the gain at frequencies above typically 100 MHz.

Needless to say, this was a challenging problem, and, Michael M¨uck, Marc-

Olivier Andr´e, Jost Gail, the late Christoph Heiden and I set about ﬁnding

a solution.

Michael made the crucial breakthrough: instead of applying the rf signal

between the two ends of the input coil, he moved one wire and applied it

between one end of the coil and the SQUID washer. In this conﬁguration,

the washer serves as a groundplane for the coil, forming a microstrip or

transmission line — thus making a virtue of the coil-washer capacitance

[Fig. 16(a)].

When the length of the microstrip corresponds to a half-

wavelength of the signal, the resonance produces a substantial amount of

gain — typically 20 dB — as shown in Fig. 16(b). As the coil is progressively

shortened, the resonance moves to higher frequency. In fact, the dependence

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

SQUIDs: Then and Now 173

of the resonance frequency on the coil length is quite complicated because

the microstrip inductance is dominated by the inductance coupled in from

the SQUID loop.

Suﬃce to say, the microstrip SQUID ampliﬁer (MSA)

achieves useful levels of gain over the frequency range

0.2–2 GHz.

An important requirement for the axion detector is the ability to tune the

resonant frequency of the cavity and hence of the ampliﬁer. The resonant

frequency of the MSA can be readily tuned by connecting a varactor diode

between the previously open end of the coil and the SQUID washer.

The

capacitance of the varactor — and hence the phase change of the reﬂected

signal — is a strong function of the reverse voltage bias. One can change the

resonant frequency by a factor of nearly two by adjusting the reverse bias

voltage.

The next step was to measure the noise temperature of the MSA,

ultimately in a dilution refrigerator since, from Eq. (11), we expect T

to scale with T in the classical regime. In these experiments, we used a

second MSA to amplify signals from the ﬁrst. We found

that T

was

indeed proportional to T at temperatures down to about 100 mK, below

which it ﬂattened out. Separate experiments showed that this ﬂattening

was due to hot electrons

in the resistors shunting the SQUID junctions.

At the lowest temperatures and f = 519 MHz, the optimized noise temper-

ature was 47 ± 10 mK, about a factor of two above the quantum limited

value T

= hf/k

≈ 25 mK. Subsequently, using a single MSA with shunts

equipped with cooling ﬁns, Darin Kinion and I measured

= 48 ±5 mK

at 612 MHz, with a bath temperature of about 50 mK. At these frequencies,

≈ 29 mK. This noise temperature is about 40 times lower than that of a

state-of-the-art HEMT.

The potential impact on the scan rate of the axion detector is dramatic.

As a scenario, suppose that the cavity is cooled to T = 50 mK with a

dilution refrigerator, and that the readout ampliﬁer has a noise temperature

= 50 mK. Thus, the system noise temperature T

= 100 mK. Since the

scan rate scales as T

, the scan time in Eq. (18) becomes 270(0.1/3.2)

years

≈100 days!

The very low noise temperature of the MSA spurred a ﬁrst-step upgrade

of the axion detector in which the HEMT was replaced with an MSA while

the temperature was maintained at about 2 K. Since blackbody noise from

the cavity is not reduced, the decrease in scan time is modest. Rather, the

object of the upgrade was to demonstrate that the MSA could indeed operate

as expected on the axion detector; a major challenge was eﬀective screening

of the 7-T ﬁeld. In fact, the system worked extremely well at a frequency of

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

174 J. Clarke

about 842 MHz. As of writing, 88,732, 80 sec data sets have been acquired,

corresponding to a net 82 days of data.

Given the success of these experiments, one can be optimistic that the full

upgrade that includes a dilution refrigerator to cool the cavity to (say) 50 mK

will be funded. The combination of the MSA and a millikelvin-temperature

cavity turns a rather impractical axion search into an extremely viable one.

10. SQUID-Detected Microtesla NMR and MRI

Magnetic resonance imaging is wonderfully successful, with perhaps 30,000

machines deployed worldwide. These machines can image most parts of the

human body with a spatial resolution of about 1 mm. Probably few patients

are aware of the fact that they are surrounded by roughly 100 km of NbTi

wire wound in a persistent-current solenoid and immersed in about 1000

liters of liquid helium! Most machines operate at a magnetic ﬁeld of 1.5 T

but there is a trend towards 3-T machines. These machines are expensive,

however, roughly $2M for a 1.5-T system, have a large footprint, and are

heavy: it is not unknown for the cost of reinforcing the ﬂoor to exceed the

cost of the machine.

MRI

is based on NMR,

91,92

almost always of protons, which have

spin

/2 and magnetic moment µ

. In the presence of an applied mag-

netic ﬁeld B

, the protons align either parallel or antiparallel to B

, the

energy level splitting being given (in angular frequency units) by the Lar-

mor frequency ω

= γB

; γ is the gyromagnetic ratio. The NMR fre-

quency is 42.58 MHzT

−1

. The magnetic moment of N protons in the limit

 k

T is M

= Nµ

T (Curie’s Law). At room temperature in

achievable magnetic ﬁelds, the magnetic moment is very small. For example,

for T = 300 K and B

= 1 T, M

/N µ

= µ

T ≈ 3.6 × 10

−6

In NMR, M

is initially in thermal equilibrium aligned with B

along the

z-axis. One applies a “π/2 pulse” along the x-axis at the Larmor frequency

that tips M

into the x-y plane in which it precesses about the z-axis at fre-

quency ω

/2π. As it precesses, M

undergoes two relaxation processes. First,

it relaxes its direction towards the z-axis, where it eventually regains ther-

mal equilibrium in a characteristic longitudinal (spin-lattice) relaxation time

. Second, individual spins dephase via local ﬁeld ﬂuctuations produced at

each site by neighboring spins in the transverse (spin-spin) relaxation time

. In water at room temperature, T

and T

are about 1 s; T

≥ T

al-

ways. The FWHM linewidth of the NMR line is ∆f = 1/πT

(“homogeneous

broadening”). Inhomogeneities in the magnetic ﬁeld, however, can substan-

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

SQUIDs: Then and Now 175

tially broaden the linewidth (“inhomogeneous broadening”), and reduce T

to a value T

∗

given by 1/T

∗

= 1/T

+ 1/T

; T

is the inhomogeneous life-

time. The very important spin echo technique invented by Erwin Hahn

eliminates inhomogeneous broadening but not homogeneous broadening.

In MRI, one uses NMR to determine spatial structure by means of three

orthogonal magnetic ﬁeld gradients that deﬁne a “voxel”. These gradients

establish B

in a small volume that produces a speciﬁc NMR frequency.

Gradient switching translates the voxel through the patient to construct the

magnetic resonance image.

Given the success of high-ﬁeld MRI, why would one consider low-ﬁeld

imaging and SQUID detection? To address this issue, we note ﬁrst that in

conventional NMR the oscillating magnetic signal induces a voltage across

an inductor shunted with a capacitor to form a resonant circuit. The volt-

age across the tank circuit, by Faraday’s Law, scales as ω

, that is, as

. Thus, at ﬁrst sight, reducing B

would seem to be exactly the wrong

thing to do. Two factors counter this thinking. First, consider replacing the

tank circuit with an (untuned) ﬂux transformer coupled to a SQUID. Since

this detector responds to ﬂux, rather than rate of change of ﬂux, one factor

of B

is eliminated and the output voltage scales as B

. Second, one can

prepolarize

the spins in a magnetic ﬁeld B

much greater than B

. After

has been turned oﬀ, the spins retain a corresponding magnetic moment

 M

that decays in a time T

, so that, although the spins precess at

frequency γB

/2π, they produce a signal amplitude proportional to B

Thus, the amplitude of the SQUID-detected NMR signal becomes indepen-

dent of B

; one can choose B

at will provided that B

 B

There is an immediate advantage of low-ﬁeld NMR and MRI. For an

inhomogeneity ∆B

in B

, the inhomogeneous linewidth ∆f

scales as

(∆B

, that is, as B

for a ﬁxed relative inhomogeneity ∆B

Thus, narrow linewidths — and high spatial resolution — can be achieved

in relatively inhomogeneous ﬁelds. For example, to obtain a 1 Hz linewidth

in a 900 MHz NMR system, it is necessary to shim the magnetic ﬁeld

homogeneity to about one part in 10

over the volume of the sample.

Although achievable, this is extremely challenging. Furthermore, spatial

variations in magnetic susceptibility across a sample produce a linewidth

broadening — and a loss of spatial resolution in MRI — that cannot be

compensated. In contrast, at an NMR frequency of 2 kHz (for protons,

corresponding roughly to the Earth’s ﬁeld), the ﬁeld homogeneity required

for a 1 Hz linewidth is only one part in 2000, which is easily obtainable; in

addition, the eﬀects of susceptibility variation are negligible.

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

176 J. Clarke

60 80

Frequency (kHz)

100 60 80

Frequency (Hz)

100

∼1 kHz

∼1 Hz

(a) (b)

Fig. 17. NMR spectra of mineral oil. (a) B

= 1.8 mT, 10,000 acquisitions.

(b) B

= 1.8 mT, B

= 1.8 µT, 100 acquisitions.

Following these ideas, Robert McDermott, Andreas Trabesinger, Michael

M¨uck, Erwin Hahn, Alex Pines and I studied NMR in protons in min-

eral oil.

The sample, maintained near room temperature, was placed in

one loop of a ﬁrst-derivative gradiometer coupled to a ﬂux-locked SQUID.

Separate coils supplied B

, an oscillating ﬁeld to tip the spins through

π/2, and B

— which was deliberately made relatively inhomogeneous.

Figure 17(a) shows the results of a conventional spin echo measurement

with B

= 1.8 mT. The signal-to-noise ratio is poor, despite the 10,000 av-

erages, and the linewidth is broad, about 1 kHz. In contrast, the spectrum

in Fig. 17(b) was obtained at 1.8 µT following prepolarization of the spins

at 1.8 mT. As expected, the linewidth was reduced by a factor of 1000, and,

even with only 100 averages, the signal-to-noise ratio of the spectrum was

greatly increased.

We exploited the high spectral resolution achievable in microtesla NMR

to detect scalar coupling (J-coupling) in heteronuclear spin systems, for

example, trimethyl phosphate, which contains a single

P atom and nine

equivalent protons.

The

P nuclear spin interacts with the proton spins

via the chemical bonds, producing a 10.4 ± 0.6 Hz splitting of the proton

levels. This splitting is easily resolved in ﬁelds of a few microtesla. Since

such splittings are unique to a particular chemical bond, this technique could

be used as a “bond detector”.

Subsequently, we constructed an MRI system,

–

made from wood with

copper wire coils (Fig. 18). The system is generally operated at 132 µT,

corresponding to a proton Larmor frequency of 5.6 kHz. Magnetic ﬁeld

gradients are typically 200 µT m

−1

. The NMR signal is detected by a second-

derivative gradiometer in a low-noise ﬁberglass dewar. A novel feature is the

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

SQUIDs: Then and Now 177

(a)

(b)

Fig. 18. MRI in microtesla ﬁelds. (a) Wooden coil frame. The “cube” carries coils

to cancel the Earth’s magnetic ﬁeld, and supports coils to provide the Larmor ﬁeld,

three sets of ﬁeld gradients, the polarizing ﬁeld and the pulsed ﬁeld. It also supports

the helium dewar containing the SQUID. (b) MRI slices of a forearm obtained with

= 40 mT, B

= 132 µT and gradients of 150 µT/m. The slice thickness is

10 mm and the in-plane resolution about 2 mm.

array of 25 unshunted Josephson junctions in series with the gradiometer

coils and the input coil of the SQUID. When the polarizing coil or gradient

coils are being switched, the current in the superconducting circuit causes

the junctions to switch to the normal state, restricting the current to a low

value. Once the ﬁelds stabilize, the junctions rapidly revert to the super-

conducting state, enabling us to acquire the data. The magnetic ﬁeld noise

referred to the lowest loop of the gradiometer is typically 0.3–0.5 fT Hz

−1/2

Experiments on “phantoms” with B

= 85 mT, B

= 132 µT and gradients

of 150 µT m

−1

demonstrated an inplane resolution of 0.7 mm. Figure 18

shows six MRI slices of a forearm acquired with B

= 40 mT. The inplane

resolution is about 2 mm.

An important issue in MRI is distinguishing diﬀerent tissue types — for

example, healthy and cancerous tissues — even though the proton densities

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

178 J. Clarke

(a) (b) (c)

Fig. 19. T

-weighted contrast imaging. (a) Phantom consisting of nine water-ﬁlled

drinking straws, with diameters ranging from 1 to 6 mm, standing vertically in a

tube containing 0.5% agarose gel solution in water. (b) Image with B

= 100 mT;

the 6-mm straw is barely discernible. (c) Image with B

= 132 µT; all straws are

highly visible.

may be identical. This distinction can often be made using T

-weighted

contrast imaging if the T

-values in the tissues are diﬀerent. Low-ﬁeld MRI

is readily adapted to this modality, and has the important virtue of oﬀering

much higher contrast than high-ﬁeld MRI in many instances.

Our low-ﬁeld technique is as follows. Suppose two regions, A and B, have

relaxation times T

and T

, with T

> T

. After polarizing the proton

spins, we turn oﬀ B

, and, following a delay of a few tens of milliseconds

during which the spins partially relax, we apply the imaging sequence. The

image of region A will be brighter than that of region B since the magne-

tization at the beginning of the imaging sequence is higher. To illustrate

this principle, we studied samples with 0.25% and 0.5% concentrations of

agarose gel in water.

At ﬁelds below 500 µT, the T

value for the 0.25%

concentration is a factor of two longer than for the 0.50% concentration,

whereas above about 10 mT there is essentially no contrast. The impact on

imaging capability is illustrated in Fig. 19, which shows images of a phantom

consisting of plastic straws containing water standing vertically in a 0.5%

agarose gel solution. At 100 mT, the image shows virtually no contrast,

whereas at 132 µT, the contrast is stark. The last two ﬁgures provide a

vivid demonstration of the enhanced T

-contrast at very low ﬁelds.

Encouraged by these results, Sarah Busch, Michael Hatridge, Michael

M¨ossle, Jeﬀ Simko (a pathologist at the University of California, San Fran-

cisco) and I embarked on a study of ex vivo prostate tissue. After a cancerous

prostate was removed surgically, it was taken immediately to the pathology

laboratory where two small specimens were excised. One was judged (by

eye) to be cancerous and the other normal. The specimens were enclosed in

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

SQUIDs: Then and Now 179

a biohazard bag, placed on ice, and rushed to our MRI machine where their

-values were measured simultaneously in a gradient ﬁeld (which separates

the NMR frequencies of the two specimens). Subsequently, the specimens

were returned to UCSF, where Jeﬀ characterized the nature of the tissues.

In preliminary measurements on specimens from eight patients, we found

that T

was signiﬁcantly higher for healthy tissue than for cancerous tissue:

the average values were 77±2 ms and 47±5 ms respectively. Further studies

are in progress.

These ex vivo results suggest that microtesla MRI with T

-contrast may

be a viable technique for in vivo imaging of prostate cancer. Only in vivo

studies, however, can determine the sensitivity and speciﬁcity. If indeed the

method proves successful, what might its role be? It is unlikely to be adopted

as a screening technique, since prostate cancer is generally diagnosed by

means of a PSA (prostate speciﬁc antigen) test. Managing prostate cancer,

however, poses a dilemma because of the wide range of malignancy; treat-

ments include “watchful waiting”, radiation therapy and surgery. Thus,

reliable, cost-eﬀective imaging would greatly beneﬁt staging of prostate

cancer. In fact, there are MRI-based techniques that can image prostate

cancer,

100

but these techniques require a 3 T system, and their complexity

unfortunately results in a cost that is too high for routine clinical use.

Thus, if microtesla MRI were able to image prostate cancer, it might

well play a key role in its diagnosis and treatment. In addition to assessing

the severity of the disease, microtesla MRI could provide a reference image

to guide subsequent biopsy. Because of its potentially low cost per image,

it could be used routinely to monitor the progression of prostate cancer

during watchful waiting or radiation therapy. Radiation therapy involves

the accurate placement of metallic radioactive seeds. Microtesla MRI —

with its insensitivity to the presence of metallic objects — could be used

to monitor the insertion of the seeds with the aid of a previously acquired

reference image. A broader issue concerns the applicability of microtesla

MRI to imaging other cancers, for example, breast, brain or abdominal.

In the meantime, other groups have begun research on microtesla NMR

and MRI. A proton linewidth of 0.034 Hz was measured for benzene in

the Earth’s ﬁeld using a high-T

SQUID,

101

and a linewidth of 0.34 Hz in

a J-coupling spectrum.

102

Volegov and co-workers

103

made simultaneous

measurements of a somatosensory response and microtesla NMR from the

brain (no claim is made that the two are correlated). Zotev et al.

104

used

a seven-channel SQUID system to acquire three-dimensional images of the

brain at 46 µT using a polarizing ﬁeld of 30 mT. A European-Union-funded

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

180 J. Clarke

consortium led by Aalto University is focused on combining microtesla MRI

with a 300-channel system for MEG. Such a system would allow one to per-

form MRI and MEG with the same system, eliminating co-registration prob-

lems and potentially reducing costs. The use of 300-SQUID gradiometers to

acquire the MRI signal could greatly increase the signal-to-noise ratio.

11. Epilogue

SQUIDs have come a long way since those blobs of solder and machined

pieces of niobium. In those early days, none of us could have imagined that

SQUIDs would have such diversity of applications — in physics, chemistry,

materials science, biology, medicine, geophysics, cosmology, quantum com-

puting — and that in each case they are the best device for the job. SQUIDs

are remarkably broadband — ranging from 10

−4

Hz in geophysical surveying

to 10

Hz in the axion detector. SQUID ampliﬁers approach the quantum

limit. In the three applications I have described in some detail — searching

for galaxy clusters, the hunt for the axion and microtesla MRI — the SQUID

plays an essential role. In each case, the application would simply not exist

without the extraordinarily low noise energy of the SQUID.

Acknowledgments

I thank the members of my research group and numerous other collaborators

for their countless contributions to the research described in this chapter.

I am grateful to Mark Ketchen, Adrian Lee, Bernard Sadoulet and Paul

Richards for insightful discussions during the preparation of this manuscript.

The writing of this article was supported by the Director, Oﬃce of Science,

Oﬃce of Basic Energy Sciences, Materials Sciences and Engineering Division,

of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.

References

1. L. N. Cooper, Phys. Rev. 104, 1189 (1956).

2. J. Bardeen, L. N. Cooper and J. R. Schrieﬀer, Phys. Rev. 108, 1175 (1957).

3. F. London, Superﬂuids (Wiley, New York, 1950).

4. B. S. Deaver and W. M. Fairbank, Phys. Rev. Lett. 7 (1961).

5. R. Doll and M. N¨abauer, Phys. Rev. Lett. 7, 51 (1961).

6. I. Giaever, Phys. Rev. Lett. 5, 147 (1960).

7. I. Giaever, Phys. Rev. Lett. 5, 464 (1960).

8. M. H. Cohen, L. M. Falicov and J. C. Phillips, Phys. Rev. Lett. 8, 316 (1962).

9. B. D. Josephson, Phys. Lett. 1, 251 (1962).