An effect on the critical current of a DC SQUID by electromagnetic vacuum fluctuations

~

0038-I098/9053.00+.00 Pet@~mon Press plc

Solid State C~,mmications, Vol. 75, NO. 6, PP. 483-484, 1990. Printed in Great Britain.

AN EFFECT ON 1"dE CRITICAL CURRENT OF A DC SQUID BY ELECTROMAGNETIC VACUUM FLUCTUATIONS

Leon Cunther Department

of P h y s i c s

and Astronomy/

Tufts U n i v e r s i t y Hedford, (Received

HA 02155 1990 by H. S u h l )

26 F e b r u a r y

I t i s shown t h a t i f a c o n d u c t i n g s l a b i s b r o u g h t close to a DC SQUID, coupling of electrons with the quantised electromagnetic field increases the critical c u r r e n t o f t h e SQUID. At z e r o t e m p e r a t u r e , a p e r f e c t c o n d u c t o r p r o d u c e s a n i n c r e a s e by a f a c t o r e x p [ a r / 4 s ] , where u i s t h e f i n e structure c o n s t a n t , r i s t h e r a d i u s o f t h e SQUID, and z i s t h e d i s t a n c e o f t h e SQUID f r o m t h e c o n d u c t o r b o u n d a r y , a s s u m e d parallel t o t h e p l a n e o f t h e SQUID.

R e c e n t l y , F o r d end P a t h t n a y a k e l ( F p ) showed that vacuum fluctuations of the electromagnetic field in the presence of a conductor produce a significant e f f e c t on a t w o - s l i t electron interference experiment. Since the electrons n e e d n o t be i n a r e g i o n w h e r e t h e r e i s m a g n e t i c field, we h a v e h e r e a n A h a r o n o v - B o h m - l i k e effect. I n t h i s p a p e r we d i s c u s s a c o u n t e r p a r t o f t h e i r r e s u l t i n t h e b e h a v i o r o f a DC SQUID.

To s t u d y t h e e f f e c t o f t h e s e vacuum fluctuations on a DC SQUID, we s t a r t w i t h t h e macroscopic equations of motion for the voltage V a n d t h e p h a s e O o f t h e t w o - J u n c t i o n SQUID. [See Tinkham, ref.2, for details.] We h a v e d___e_B 2eV(t) dt b

(3a)

and The e x p e r i m e n t a l l a y o u t i s d e p i c t e d i n f i g . 1. A thin layer of insulator - s a y one m i c r o n thick - is deposited upon a thick slab of metal. k SQUID i s t h e n d e p o s i t e d u p o n t h e i n s u l a t o r film. We f o c u s o u r a t t e n t i o n on t h e e f f e c t o f the vacuum fluctuations of the magnetic flux @ t h r o u g h t h e SQUID, d u e t o t h e p r e s e n c e o f t h e conductor. B o t h , t h e mean s q u a r e f l u c t u a t i o n s <@2>o i n t h e a b s e n c e o f t h e c o n d u c t o r a s w e l l a s t h e mean s q u a r e f l u c t u a t i o n s <@ > i n t h e p r e s e n c e o f t h e c o n d u c t o r d i v e r g e . The Duly 2 meanureable quantity is their difference <@ ~ . FP f i n d t h a t when t h e c o n d u c t o r i s p e r f e c t and the distance z from the conductor to the area corresponding t o t h e f l u x i s much l e s s t h a n t h e circumference C of that area (z<
<@2>p.. In units

of the flux

<#2>

q u a n t u m 0o ~ h c / 2 e ,

[ I - 2 I 0 cos(w~) s i n 0 ] ,

8w 3 z

'

$ -

2e

~

+ +

/ A.dl

,

structure

(4)

~here the line integral is along a closed path lying within the superconductors, except across the Junctions. If ~ were a constant and I > 2Io, t h e DC v o l t a g e c o m p o n e n t would be g i v e n by 2

we h a v e

VDC . R~ [ 12 - 41o 2 c o s 2 ( w ~ ) ] l / 2

(s)

(2)

The m a g n e t i c f l u x r e d u c e s t h e c r i t i c a l w h i c h i s h e r e g i v e n by 2I o c o s ( w # ) . w h e r e ~ • @/#o a n d a ~ e21hc- i s

(3b)

where I is the bias current. We a s s u m e , f o r simplicity, t h a t t h e two J u n c t i o n s h a v e t h e same critical c u r r e n t I o and n o r m a l r e s i s t a n c e R. Furthermore, in our case @ is a quantum operator expressible in terms of the vector potential A :

(1)

.

R

aC

.

R

V -~

current,

the fine

coherent.

In addition, we h a v e a f l u x w h i c h v a r i e s rapidly in time. The f r e q u e n c y s p e c t r u m o f t h e flux fluctuations is characterized by ~ e f r e q u e n c y c / z . [ F o r z = 3 m~, c / z - I 0 Hz]. For frequency components of the flux fluctuations which are ~reater than the relaxation rate of the microscopic degrees of freedom of the electrons - s a y - 1016 Hz - t h e

The f a c t t h a t <@2> i s n e g a t i v e r e f l e c t s the reduction of the fluctuations by t h e p r e s e n c e o f the conductor. We w i l l l a t e r s e e t h a t t h i s f a c t results in a surprising increase in the critical c u r r e n t o f t h e SQUID.

483

484

ELECTROMAGNETIC VACUUM FLUCTUATIONS

Vol. 75, NO. 6

VDC- ~R [I 2 - 4Z~ exp ( a C / S , z ) ] 1/2 We s e e t h a t vacuum f l u x f l u c t u a t i o n s i n c r e a s e the critical c u r r e n t . For • c i r c u l a r SQUID of r a d i u s r = I mm and a s p a c i n g z = 3 mU, t h e e x p o n e n t i n e q . ( 7 ) i s a r / 4 z = 0 . 6 and t h e effect is quite significant!

Conductor

Fig.

I.

Schematic drawing of the experimental layout depicting a DC SQUID which is s e p a r a t e d from a t h i c k c o n d u c t o r s l a b by an i n s u l a t o r f i l m of t h i c k n e s s z.

m a c r o s c o p i c e q u a t i o n s of m o t i o n ( 3 a ) and (3b) b r e a k down. We w i l l a s s u m e t h a t we can n e g l e c t t h e e f f e c t of t h e s e c o m p o n e n t s . N e x t , i f t h e d o m i n a n t f r e q u e n c i e s of t h e f l u x f l u c t u a t i o n s exceed the characteristic f r e q u e n c y of v o l t a g e e l o R / h , we can o b t a i n VDC by r e p l a c i n g c o s (1~) i n e q . ( 3 b ) by i t s q u a n t u m a v e r a g e < c o e ( w ~ ) ) . F o l l o w i n g the steps of FP, we o b t a i n = exp [ - x 2 ( t 2 ~ R / 2 ]

.

How can our theoretical prediction be checked experimentally? First of all , one must measure the critical current precisely. One would hope t o s e e t h e d e p e n d e n c e of t h e o b s e r v e d crltlcal current on z. This might require use o f many i d e n t i c a l SQUIDS l a i d down w i t h v a r i e d z . U n f o r t u n a t e l y , i t i s n o t e a s y to m a n u f a c t u r e SQUIDS which a r e i d e n t i c a l to t h e e x t e n t n e e d e d . F u r t h e r m o r e , r e a l c o n d u c t o r s have a l i m i t e d r e s p o n s e r a t e g i v e n a p p r o x i m a t e l y by t h e plasma f r e q u e n c y mpl . As e c o n s e q u e n c e , F P ' s c a l c u l a t i o n b r e a k s down when z i s l e s s t h a n o r a b o u t e q u a l to t h e p l a s m a w a v e l e n g t h ~ 1 = v~ C/mpl = (mr 2 /4wne 2 )s / 2 , where m is the electron mass and n i s t h e d e n s i t y of c o n d u c t i o n e l e c t r o n s . 3 T y p l c a l l y ~ I i s IO0-1000A. (Note t h a t Xpl i s t h e London p e n e t r a t i o n d e p t h of a s u p e r c o n d u c t o r a t a b s o l u t e z e r o . ) Nhen z < < ~ l , we e x p e c t e q s . ( 1 ) , ( 2 ) , and ( 7 ) to s a t u r a t e w i t h z b e i n g r e p l a c e d by ~ 1 . For t h e p u r p o s e of dealing with specific condensed matter e n v i r o n m e n t s , s u c h as r e a l m e t a l and t e m p e r a t u r e effects, detailed calculations are in progress.

(6)

U s i n g e q s . ( 2 ) and ( 5 ) , w i t h cos2(w~) r e p l a c e d by 2, we o b t a i n

Acknowledgment - The a u t h o r i s v e r y g r a t e f u l to L a r r y Ford f o r t h e i n t r o d u c t i o n t o h i s work a n d , a l o n g w i t h Alan Widom, f o r h e l p f u l and insightful discussions.

References

I.

L. H. Ford and Chandra P a t h l n a y a k e , to be p u b l i s h e d i n t h e P r o c e e d i n g s of t h e C o n f e r e n c e on F u n d a m e n t a l A s p e c t s of Quantum T h e o r y , Columbia S . C . Dec. 1989.

2. M i c h a e l Tinkham, I n t r o d u c t i o n t o Superconductivity

( R o b e r t E. K r e i g e r

Publishing Corp., Inc.,

1980). 3. P o i n t n o t e d by L. Ford.

H u n t i n g t o n , N.Y.,