squid amplifier

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Journal of Low Tem perature P hysics, Vo l. 61, Nos. 3/4, 1985

S i gn a l an d N o i s e T h eory f or a D C S QU I D A mp l i f i e rJ o h n M . M a r t in i s a n d J o h n C l a r k e

Departm ent of Physics, University o f California, and M aterials and M olecularResearch Division,Lawrence Berkeley Laboratory, Berkeley, C alifornia(Received May 7, 1985)

A the ory is p re se n te d fo r t he ga in an d no i se i n an am p l i f ie r base d on a dcS Q U I D . I n t h e l u m p e d c i rc u it a p p r o x i m a t io n , t h e t o ta l in d u c t a n c e o f th e in p u tc i r c u i t L r i s c o u p l e d t o t h e S Q U I D i n d u c t a n c e L v i a a m u t u a l i n d u c t a n c eM i = c~e(LLT) 1 /2 an d i s in serie s w i th a vo l ta ge so urce wi th a res i s tance R ia n d a c a p a c i t a n c e Ci. The re su lt s a re e x pre s se d in t e rm s o f para m e te r s f o r aS Q U I D w i th r ed u ce d in d u c ta n c e ( I - a ~ ) L . T h e v ol ta g e g a i n o f t he am p l if ie ra t f r e qu e nc y to / 2~r i s M iV~ ,/Z *T (to ), wh i l e t he t o ta l v o l tage n o i s e a t t he ou tpu to f th e S Q U I D is V ~ ( t o ) 2 r rM i V , J N ( t o ) ( R i + 1 / j w C i ) / L r Z * (w ). H er e,Z * ( to ) = ZT ( tO) - - J ~ M ~ ( R i + I / j to C i ) / L r , w h e r e Z T ( t o ) i s t he t o ta l im pe da nc eo f the un load e d inpu t c ir cu it, V~ , and J~ , a re t he f l ux - to - v o l tage an d f l ux - to -c ir c u la t in g c u r r en t tr a n s fe r f u n c t i o n s o f th e r ed u c e d S Q U I D , a n d V ~ ( t o ) a n dJ ~ ( t o ) a r e t h e n o i se v o l t a g e a n d n o i se c u r re n t o f t he r e d u c e d S Q U I D .

1 . I N T R O D U C T I O NS e v e r a l y e a rs a g o , C l a r k e , T e s c h e , a n d G i f f a r d I ( C T G ) d i s c u s s e d t h e

o p t i m i z a t i o n o f v o l t m e t e r a n d m a g n e t o m e t e r c ir cu i ts i n v o lv i n g d c S Q U I D s .T h i s p a p e r t o o k i n t o a c c o u n t t h e t w o n o i s e s o u rc e s o f t h e S Q U I D , n a m e l yt h e v o l ta g e n o i s e ac r o ss t h e S Q U I D a n d t h e c u r r e n t n o i s e c i rc u l a ti n g a r o u n dt h e l o o p , a n d t h e p a r t i a l c o r r e l a t i o n o f t h e s e t w o n o i s e s . E x p r e s s i o n s w e r ed e r i v e d f o r t h e o p t i m i z e d n o i s e t e m p e r a t u r e o f v o l t m e t e rs a n d t h e m i n i m u md e t e c t a b l e si g na l e n e r g y o f m a g n e t o m e t e r s f o r b o t h t h e t u n e d a n d u n t u n e dc a se s. F o r S Q U I D s i n th e t h e r m a l l i m it , c o m p u t e d v a l u e s o f t h e n o i s es o u r c e s 2'3 w e r e u s e d t o m a k e e s t im a t e s o f r e a l iz a b l e n o i s e t e m p e r a t u r e s a n ds i g n a l e n e r g i e s . T h e e s s e n t i a l a p p r o x i m a t i o n i n t h e s e c a l c u l a t i o n s w a s t h a tt h e S Q U I D w a s r e l a ti v e ly w e a k l y c o u p l e d t o t h e i n p u t c i rc u i t a n d t h a t t h ep a r a m e t e r s o f t h e S Q U I D w e r e n o t s i gn i fi c an t ly a ff e c te d b y t h e p r e s e n c eo f t h a t c i r c u i t.

S u b s e q u e n t l y , p l a n a r , th i n -f il m d c S Q U I D S h a v e b e e n d e v e l o p e d th a ta r e s t r o n g l y c o u p l e d t o t h e i r i n p u t c o il s , 4-11 s o t h a t t h e a s s u m p t i o n o f w e a k

2270022-2291/85/1100-0227504.50/0 1985 Plen um Pub lishing Corp oratio n


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228 John M. M a r t i n i s an d J o h n C l a r k e

c o u p l i n g i s n o t n e c e s s a r i l y v al id . I n s t e a d , o n e n e e d s t o d e v e l o p a t h e o r yt h a t t a k e s i n to a c c o u n t p o s s ib l e m o d i f i c a t i o n s o f t h e p a r a m e t e r s o f t h eS Q U I D d u e t o t h e p r e s e n c e o f t h e in p u t c ir cu i t, a n d a t t h e s a m e t i m ep o s s i b l e c h a n g e s i n t h e i m p e d a n c e o f t h e i n p u t c i rc u i t d u e t o t h e p r e s e n c eo f t h e S Q U I D . T o o u r k n o w l e d g e , t h e re h a v e b e e n t w o a p p r o a c h e s t o t hi sp r o b l e m , a t h e o r y b y K o c h 12 a n d a s e r ie s o f th r e e p a p e r s b y Te sc he . 13-15T h e l a t t e r t h e o r y , i n o u r v i e w , c o r r e c t l y c a l c u l a t e s t h e e f fe c t o f t h e i n p u tc ir cu i t o n t h e S Q U I D , b u t d o e s n o t a c c o u n t fo r t h e S Q U I D i m p e d a n c er e f l e c t e d i n t o t h e i n p u t c i r c u i t i n a s e l f - c o n s i s t e n t w a y . T h e p r e s e n t w o r km a k e s u s e o f a m e t h o d t h a t d i f f er s f r o m t h o s e u s e d i n t h e p r e v i o u s t h e o r i e s ,a l t h o u g h i t m a k e s c o n t a c t w i t h b o t h , a s w i l l b e m a d e c l e a r a t a p p r o p r i a t ep l a c e s i n t h e p a p e r .

E a c h o f t h e s e t h e o r i e s a t t e m p t s t o c o n s t r u c t a n e q u i v a l e n t c i r c u it m o d e lf o r t h e a m p l i f ic a t io n a n d n o i s e p r o p e r t i es o f a d c S Q U I D c o u p l e d t o a ni n p u t c i r c u i t , a n d , i n t h e p r e s e n t w o r k , t o a n o u t p u t c i r c u i t . S u c h a m o d e lis d e s i r a b l e b e c a u s e t h e e x a c t s o l u t i o n o f t h e n o n l i n e a r c i r c u i t e q u a t i o n so n a c o m p u t e r f o r g i v en i n p u t a n d o u t p u t c i rc u it s is ti m e - c o n s u m i n g a n dm u s t b e r e p e a t e d f o r a r an g e o f ci rc u it p a r a m e t e r s t o e n a b l e o n e t o o p t i m i z et h e p a r a m e t e r v a lu e s. T h e a p p r o a c h w e h a v e a d o p t e d , d e s c r i b e d in S e c t i o n2 , is t o b r e a k u p t h e f u l l e q u a t i o n s o f m o t i o n i n t o l i n e a r a n d n o n l i n e a rp a rt s . T h e n o n l i n e a r p a r t i s a s s u m e d t o b e s o l v e d o n a c o m p u t e r f o r th et r a n s fe r f u n c t io n s a n d n o i s e p a r a m e t e r s t h a t a r e t h e n u s e d i n t h e e q u i v a l e n tc ir c u it m o d e l o b t a i n e d f r o m t h e l i n ea r p a r t o f th e e q u a t io n s . T h e e q u i v a l e n tc i rc u i t m o d e l a n d t h e v a l u e s o f t h e t r a n s f e r f u n c t i o n s a n d n o i s e p a r a m e t e r sd e p e n d o n h o w t h e e q u a t i o n s a r e d i v i d e d i n t o l i n e a r a n d n o n l i n e a r p a r t s .W i t h i n th e a p p r o x i m a t i o n s m a d e i n S e c t i o n 2 , t h is d i v i s i o n is m a d e s o t h a tt h e n o n l i n e a r p a r t d e s c r i b e s a " r e d u c e d S Q U I D ''12-16 w i t h a r e d u c e d i n d u c t -a n c e t h a t d e p e n d s o n l y o n t h e e f fe c ti v e c o u p l i n g c o e f fi c ie n t t o t h e i n p u tc i rc u i t a n d is i n d e p e n d e n t o f f r e q u e n c y . A s a r es u l t, o n e i s re q u i r e d t oc o m p u t e t h e t r a n s f e r f u n c t i o n s a n d n o i s e p a r a m e t e r s f o r a s i n g l e v a l u e o fi n d u c t a n c e o n l y ; t h e s e v a l u e s c a n b e u s e d i n t h e e q u i v a l e n t c i r c u i t m o d e lt h a t e n a b l e s o n e t o o p t i m i z e th e c i r c u it p a r a m e t e r s i n a s t r a i g h t f o r w a r d w a y .

I n S e c t i o n 3 w e c o m m e n t b r i ef ly o n a f o r m a l m e t h o d o f i n c o r p o r a t i n gt h e e f f ec t s o f p a r a s i t i c c a p a c i t a n c e i n t o t h e t h e o r y , w h i l e S e c t i o n 4 c o n t a i n sa c o n c l u d i n g d i s cu s s io n . A b r i e f p r e s e n t a t i o n o f t h e w o r k h a s b e e n m a d ee l s e w he r e . 17

T h i s p a p e r is th e f ir st o f a s e ri e s o f t h r e e p a p e r s t h a t a r e c l o s e l y l i n k e d .A s w e s h a ll s e e , o u r t h e o r y r eq u i r es a k n o w l e d g e o f t h e d y n a m i c i n p u ti m p e d a n c e o f t h e S Q U I D l o o p , t h at is , t h e c u r r e n t r e s p o n s e t o a sm a l la p p l i e d f l ux . S in c e th e r e h a s b e e n n o p r e v i o u s d e t a i l e d s t u d y o f t h is a s p e c to f S Q U I D b e h a v i o r , t h e s e c o n d p a p e r 18 p r e s e n t s a c o m p r e h e n s i v e s t u d y o ft h e d y n a m i c i m p e d a n c e b o t h o n a n a n a l o g c o m p u t e r a n d o n re a l S Q U I D S .


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Signal and Noise Theory for a DC SQU ID Amplifier 229

In add i t i on , t h is pap e r i nc ludes an exper im en ta l s tudy o f t he e f fect s o fpa ras i t i c capac i t a nce a nd a d i scuss ion o f t he va l id i ty o f t he lum ped c i rcu i tmo de l t ha t i s u sed in the p rese n t pape r . The th i rd pap e r a9 d raws on thework o f t he f ir s t two papers t o d i scuss t he op t imiza t ion o f t une d a nd u n tu nedSQ U ID v o l t a g e a mp l i f i e r s , a n d t h e n d e s c r i b e s t h e p e r fo rma n c e t h a t h a sbeen ach ieved wi th p rac t i ca l ampl i f i e r s .

2. T H E O R Y O F T H E C O U P L E D D C S Q U I D2 . 1. M o d e l C i r c u i t

Ou r mo de l f o r an ampl i f i e r i nvo lv ing a dc S Q UI D is show n in F ig . 1 .T h e SQ U I D c o n si s ts o f t w o i d e n t ic a l J o s e p h s o n t u n n e l j u n c t i o n s , e a c h w i tha cr i t ica l current Io , a se l f-capaci tance C, and a shunt res i s tance R. TheSQUID i s b i a sed wi th a cons t an t cu r ren t I and deve lops a vo l t age V ( t )a c ro s s t h e i n p u t i mp e d a n c e ZA of the n ex t s t age, wh ich migh t be a t rans -fo rmer , a t uned c i rcu i t , o r a p reampl i f i e r . The SQUID loop , wh ich has anind uct an ce L, is co up led to the indu ctan ce L,~ of the inpu t c i rcu i t v ia amu t u a l i n d u c t a n c e Mi = a (LL i) u2. The inpu t c i rcu it con ta ins a p i ckup loopw i t h i n d u c t a n c e Lp, a s t ray ind ucta nce Ls, a capa ci tan ce Ci, and a res i s tanceRi. Thus , t he to t a l im ped anc e o f t he inpu t c i rcu it is

ZT(tO) = R i + 1/jtoC, +j~o (L, + L . + L~)= R, + 1/jtoC~ +jtOLT (1)

Th e inpu t vol tage V~ cou ld ar i se f rom a vol tage sour ce ( the res i s tance o fwh ich is i nc lud ed in R~) o r f rom a t ime-va ry ing m agne t i c f l ux in the p i ckupcoi l . By choosing Lp, Ci, and R~ appropria te ly , one can use th is c i rcu i t tor e p re s e n t a v o l t a ge a mp l i f ie r o r a ma g n e t o m e t e r , t u n e d o r u n t u n e d .

The c i rcu i t show n in F ig . 1 neg lec t s pa ras i t i c capac i t ance , fo r examp le ,b e t w e e n t u rn s o f t h e i n p u t c o il o r b e t w e e n t h e i n p u t c o il a n d t h e S Q U ID ,

o

Lt R

IF j_CiFig. 1. Mode l circu it for a voltage amplifier ba sed on a deSQUID.


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230 John M. M artinis and John Clarke

w h i c h m a y h a v e a s i g n i f ic a n t e f fe c t o n t h e d y n a m i c s o f t h e S Q U I D a t th eJ o s e p h s o n f r e q u e n c y . T h e s e e ff ec ts m a y b e p a r t i c u l a r l y i m p o r t a n t f o r t h i n -f il m d ev i c e s w i t h ti g h t l y c o u p l e d i n p u t c o il s. T h e l u m p e d c i rc u i t m o d e l ise x p e c t e d a l s o to b r e a k d o w n i f t h e w a v e l e n g t h a t t h e J o s e p h s o n f r e q u e n c yb e c o m e s c o m p a r a b l e w i t h t h e d i m e n s i o n s o f t h e i n p u t c ir c ui t. A t t h e s i g n a lf r e q u e n c y , w h i c h w e a s s u m e to b e m u c h l es s t h a n t h e J o s e p h s o n f r e q u e n c y ,t h e p a r a s i ti c c a p a c i t a n c e s s h o u l d b e r e l a ti v e ly u n i m p o r t a n t . F o r t h e m o m e n t ,w e n e g l e c t t h e s e p a r a s i t i c e f f e c t s , a n d c o n c e r n o u r s e l v e s w i t h d e v e l o p i n g as o l u t i o n to t h e e q u a t i o n s o f m o t i o n t h a t d e s c ri b e F i g . 1.

2 .2 . E q u a t io n s o f M o t i o n f or t he C o u p le d D C S Q U I DI n t h e ti m e d o m a i n , th e e q u a t io n s o f m o t i o n f o r t h e S Q U I D c o u p l e dt o a n i n p u t c i r c u i t a r e

V (t ) = [ 81( t ) + t%(t ) ] h / 4 e (2)hC ,2 e 8 , ( 0 + 2 - - ~ 8 , ( t ) = I , ( t ) - J ( t ) - Io s in 8 , ( t ) + I N l ( t ) (3)

a n d

h C . . 2 - ~ "2 e 8 2 ( t ) + 8 2 ( t ) = I 2 ( t ) + J ( t ) - I o s i n 8 2 ( t ) + I N 2 ( t ) (4)I i ( t ) + I2 ( t ) = I - ZA ' ( to ) * V (5 )

[8~( t ) - 8 2 ( t) ]d g o /2 7 r = dp + L J ( t ) + M i I ~ ( t ) (6)w h e r e 8 - = 0 8 / O t , 8 " - - 0 2 8 / 0 t 2 , f d e n o t e s th e F o u r i e r t r a n s f o r m o f f a n d( f * g ) ( t ) d e n o t e s t h e c o n v o l u t io n

(27r) -1/2 I ~ _ o o f ( t ') g ( t - t ') d t 'I n t h e s e e q u a t i o n s , V i s t h e v o l t a g e a c r o s s th e S Q U I D , J i s t h e c u r r e n ta r o u n d t h e S Q U I D l o o p , is t h e e x t e rn a l l y a p p l i e d m a g n e t i c f lu x , a n d I ii s t h e c u r r e n t i n t h e i n p u t c i r c u it . T h e p h a s e d i f f e r e n c e s a c r o s s t h e j u n c t i o n sa r e 8 1 a n d 8 2 , t h e c u r r e n t s t h r o u g h t h e s e j u n c t i o n s a r e 1 1 a n d / 2 , a n d t h en o i s e c u r r e n t s p r o d u c e d b y t h e s h u n t r e si s to r s a re IN ~ a n d I N 2 . T h e e q u a t i o n sa r e i d e n t i c a l to t h o s e u s e d p r e v i o u s l y t o d e s c r i b e t h e b a r e d c S Q U I D , 2 w i t ht h e a d d i t i o n o f t h e l a s t (l i n e a r ) t e r m s o n t h e r i g h t - h a n d s i d e o f E q s . ( 5)a n d ( 6 ), w h i c h a c c o u n t f o r t h e l o a d i n g o f t h e S Q U I D b y th e n e x t s ta g e a n dfo r t he i n f lue nce o f t h e i np u t c i rcu i t , 12-16 re spec t ive ly .


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Signal and Noise Theory for a DC SQ UI D Amplifier 231

In the f requency dom ain , the cur ren t J (~o) in the SQ UID loop inducesa voltage* - j o J M i J ( o ~ ) in to the input c ircui t , so that

I, ( to ) = [ V~( w ) -j w M , J ( ~o ) ]/ Z T ( to ) (7 )In o rder to reso lve the equa t ions in to l inear and non l inear par t s , we

no w int rodu ce low- and high-fr equen cy f i l ter funct ions [ ]LF an d [ ]HF intothe terms tha t represen t the coup l ing o f the SQ UID to the inpu t an d o u tpu tcircui ts . The cutoff f requencies of these f i l ter funct ions are chosen to l iebe tween the s igna l and Joseph son f requenc ies , in o rder to separa te e f fec tsa t the s igna l f requency f rom those a t the Josephson f requency . Thus , werewri te Eqs. (5)-(7) in the form

l l ( t )+I2( t ) = I - - [ z a l ( o g ) * V ] L F - - [ Z'- A' (O ~ )* V ] H F ( 8 )a n d

[t~l (t ) -- 82(t)]aPo/2~r= + L J + { - [floMz~ /ZT(~O )] * J + [M , IZT (~O) ] * V~}LF

- - { [ f loM~/ZT(O J)] * J}HF (9)

TO simplify Eqs. (8) and (9) , we in troduce two assumptions. The f i rs ti s tha t the ou tpu t load ing o f the SQUID is neg l ig ib le a t the Josephsonfrequ ency oJ j /2m so tha t the high-freq uency term in Eq. (8) is negl ig ible .The second a ssum pt ion i s tha t R~, I1/jogj Ci[


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2 3 2 J o h n M . M a r t i n i s a n d J o h n C l a r k e

a n d

w h e r e

a n d

[ (~1 t ) -- 82( t)]qb o/2 ~" = qb + ( 1 -- a 2) L J + Aqb( t ) ( 13 )

A I ( t ) = - - [ Z a l ( t o ) ] * V ]L F ( 14)

A ( t ) = { [ M i / Z r ( o J ) ] * V ~ - [ j o J M ~ / Z T ( o J ) ] * J + a ~ L J } L F (15)W i t h A I = Aqb = 0, E q s . ( 3 ) , ( 4 ) , ( 1 2 ), a n d ( 1 3 ) a r e t h e e q u a t i o n s o f m o t i o nf o r a reduced S Q U I D 12-16 w i t h a n i n d u c t a n c e (1 - a2)L: t h e i n d u c t a n c e i sr e d u c e d b y t h e s h i e l d i n g e f f e ct o f t h e i n p u t c i rc u i t, w h i c h w e h a v e a s s u m e dt o b e p u r e l y i n d u c t i v e a t t h e J o s e p h s o n f r e q u e n c y .

I n t h e n e x t s e c t io n , w e d e s c r ib e a m e t h o d f o r s o l v i n g t h e e q u a t i o n s o fm o t i o n .

2 . 3 . T h e L i n e a r A p p r o x i m a t i o nT o s o l v e f o r t h e e f f e c ts o f t h e i n p u t a n d o u t p u t c i rc u i ts u s i n g t h e l i n e a r

a p p r o x i m a t i o n , w e i n i ti a l ly s et A I = A q b = 0 a n d s o l v e E q s . ( 2 ) - ( 4 ) , ( 1 2 ),a n d ( 13 ) f o r t h e s e t o f l o w - f r e q u e n c y tr a n s f e r f u n c t io n s t h a t r e p r e s e n t t im ea v e r a g e s o v e r th e v o l t a g e a n d c u r r e n t a t t h e J o s e p h s o n f r e q u e n c y , W ~,-=( O V / O ( ~ ) r, W x=-(OV /OI) r , Jr~=-(OJ/O ~)r, a n d Jr1=--(OJ/OI)r, a n d f o r t h ev o l t a g e a n d c u r r e n t n o i s e t e r m s W N ( t) a n d J ~ v (t ). T h e s u p e r s c r i p t r im p l i e st h a t t h e se q u a n t it i es a r e c o m p u t e d f o r a S Q U I D w i t h r e d u c e d i n d u c t a n c e(1 - a2)L. W e n o t e t h a t , i n g e n e r a l , t h e t r a n s f e r f u n c t i o n s w i l l c o n t a i n b o t hr e a l a n d i m a g i n a r y c o m p o n e n t s . 18'2 T h e n o i s e t e r m s a r e c h a r a c t e r i z e d b yt h e l o w - f r e q u e n c y s p e c t r a l d e n s i ti e s S ~, a n d S ~ a n d c r o s s - s p e c t r al d e n s i t yS ~ j .

W e n o w r e i n tr o d u c e th e low-frequency t e r m s A I a n d A~P b y r e g a r d i n gt h e m a s s m a l l c h a n g e s i n t h e b i a s p a r a m e t e r s . W e u s e t h e s m a l l - s i g n a l ,l i n e ar a p p r o x i m a t i o n t o f in d th e n e w o u t p u t v o l t a g e a n d c i r c u l at in g c u r r e n ta t l o w f r e q u e n c y w i t h t h e s e t e r m s i n c l u d e d :

a n dV ( t ) = W N ( t )+ V ~ , A ~ ( t ) + W~ A I ( t )

J ( t ) = J ~ ( t ) + J o A q b ( t) + J ~ A I ( t )

( 16 )

( 17 )I n s e r t i n g E q s . ( 1 4 ) a n d ( 1 5 ) i n t o E q s . ( 1 6 ) a n d ( 1 7 ) , w e o b t a i n , a t l o w


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Signal and Noise Theory for a DC SQUID Amplifier 233

f requency,

=J (~) / Za(~o)-J~ZA(~o)

F-j,, ,M~ 2 1\

+ v (0,)4 zT(o,)\./,,,(o)) H zT(~o)

(18)

The linear equations (18) can be solved in a straightforward manner.To focus a t tention on the interact ion of the input c i rcuit and the S Q UI Dand to simplify the solution, we consider the l imit ZA(tO)~ 0O in which theloading of the S Q UI D b y the following circuitry is negligible.* In this l imit ,we solve Eqs. (18) to find

V~(to) + MJ~N( ~o ( R, + 1/jtoCi) / L r]V(~)= WN(oJ)+M,W~,, " ~ ' ( - r ( o ~ ~ ~ - ~ "/ (19)Equa t ion (19) expresses the ou tput vol tage in terms of reduced SQ UI Dparameters that are dependent only on the effective coupling coefficient aeand are otherwise indepen dent of the input c ircui t. Thus, one can readi lyuse Eq. (19) to optimize the c om pon ent values fo r any type of input circuit .The second term in square brackets represents the noise current f lowing inthe input circuit in the equivalent circuit model: this current produces af lux in the SQUID and thus an addi t ional vol tage noise a t the output . Theterm MiJ~( to) (Ri+l / j toCi) /Lr in the numerator is identical to thatobtained by Tesche13-15 (with L p = L s= 0 ) . On the other hand, the

r 2denom inator of Eq. (19) contains the addi t ional term -Ja, Mi(R~+!Z}~Q)/Lr , representing t lae aodification o f the inp ut circuit by the pres-enbe of the SQ UID , which was not include d in the wo rk o f Tes che J 3-15 Ingeneral, this term modifies both the signal and noise measured at the outputo f t he SQ U ID .

*This is an excellent approximation for the amplifiers studied in the two papers that follow.We note, however, that for many low-frequency applications--for example, when the SQUIDis followed by a transformer-- this approximation is not valid, and one should retain theterms in 1/Za(tO).


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3. E F F E C T S O F P A R A S I T IC C A P A C I T A N C EP r a c t i c a l~ S Q U I D a m p l i fi e rs g e n e r a l l y c o n t a i n p a r a s i t ic c a p a c i t a n c e s ,

f o r e x a m p l e , a m o n g t h e t u r n s o f th e i n p u t c o il o r b e tw e e n t h e i n p u t c o i la n d t h e S Q U I D . A s a r e s u l t , e v e n w h e n t h e i n p u t c o i l i s o p e n - c i r c u i t e d ,t h e h i g h - f r e q u e n c y d y n a m i c s , a n d t h e r e f o r e t h e l o w - f r e q u e n c y t r a n s f e rc o e ff ic ie n ts a n d n o i se , m a y d i f fe r f r o m t h o s e f o r a b a r e S Q U I D . O n e c o m m o nm a n i f e s t a t i o n o f t h e se p a r a s i t ic e f fe c ts is th e a p p e a r a n c e o f s t ru c t u r e o nt h e c u r r e n t - v o l t a g e c h a r a c t e r i s t i c s . 18'21 B e c a u s e t h e s e c a p a c i t a n c e s a r e d i s-t r i b u t e d , i t is d i f f ic u l t t o i n c o r p o r a t e t h e i r e f fe c ts i n t o t h e e q u a t i o n o f m o t i o n .H o w e v e r , i n a f o r m a l s e n s e o n e c a n a ll o w f o r t h e s e e f fe c ts , t o g e t h e r w i t ha n y o t h e r p a r a s it ic e f fe c ts d u e t o th e b r e a k d o w n o f t h e s im p l e l u m p e dc i r c u i t o f F i g . 1 , b y r e p l a c i n g Z A I ( t o ) a n d j ~M 2 /Z T (C O ) i n th e h i g h - f r e q u e n c yp a r t s o f E q s . ( 8 ) a n d ( 9) w i t h th e f u n c t i o n s U A ( W ) an d U~(to ), r e spe c t i ve ly ,t o o b t a i n

I ~ ( t ) + I 2 ( t ) = 1 - [ Z A l ( t o ) * V ] L F - - [ U A ( 0 3 ) * V]HF (20)a n d

[8 1( t) -- t~2( )]qbo/2"rr= Op + L J + { - [ j t o ~ ) ] * J 4- [ ' ~ / Z r ( ~ ] * V/}LF

- [ Ui( t.o)* J]H F (21)I f U A ( t O ) a n d U ~ (to ) a r e k n o w n , o n e c a n p r o c e e d a s i n S e c t io n 2 , a n d , a tl e a s t i n p r i n c i p l e , t h e p r o b l e m c a n b e s o l v e d .

A s a s i m p l e e x a m p l e , w e c o n s i d e r t h e l i m i t i n g c a s e U A(CO j)= Ui(taj) = 0,t h a t i s, t h a t n o c u r r e n t s a t t h e J o s e p h s o n f r e q u e n c y f lo w in t h e i n p u t c i r c u ito r in t h e s ta g e f o l l o w i n g t h e S Q U I D . T h e r e s u l t c a n b e f o u n d e a s i l y u s i n gt h e a p p r o a c h o f S e ct io n 2 a n d b y n o t i n g t h a t th e r e d u c e d S Q U I D p a r a m e t e r sa r e r e p l a c e d w i t h t h e b a r e S Q U I D p a r a m e t e r s. F o r th e c a s e Z A ( t O ) . . .9 00,o n e f i n d s

. . . E - j t o M i J N ( t o )V ( t o ) = V N ( w ) + & li V~, Z - r ~ (22)T h e f o r m o f th e e q u i v a l e n t c i r c u it m o d e l o f E q . ( 2 2) d i f f er s f r o m t h a t o fE q . ( 19 ) b e c a u s e o f t h e w a y i n w h i c h th e h i g h - f r e q u e n c y p a rt s o f th e S Q U I Di n d u c t a n c e w e r e t re a t e d i n t h e t w o a p p r o x i m a t i o n s . W e n o t e t h a t i f o n ew r i t e s H J ~ = - j t o / ~ , w h e r e ~ ( t o ) is t he d y n a m i c im p e d a n c e o f t h e S Q U I Dl o o p , t h e re f le c te d S Q U I D i m p e d a n c e f i o M 2 j , b e c o m e s o j 2 M ~ / ~ . T h u s , a se x p e c t e d , E q . ( 22 ) is t h e C T G r e s u lt [E q . ( 7 ) ] a p a r t f r o m t h e i n t r o d u c t i o no f t h e ( c o r r e c t ) n e g a t i v e s ig n a n d t h e s u b s t i t u t i o n o f ~ ( t o ) f o r 4V t.


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Signal and Noise Theory for a D C SQUID Amplifier 23 5

4. DISCUSSIONWe have presented a theory for the gain and noise properties of an

amplifier based on a dc SQUID. Our central result using the lumped circuitmodel, Eq. (19), differs from the CTG result that was obtained under theassumption that the SQUID parameters were unaffected by the presenceof the input circuit; the CTG result is essentially Eq. (22). The first, obviousdifference is the replacement of the bare SQUID parameters in the CTGresult with the reduced parameters in Eq. (19). The second difference is thereplacement of the CTG expression for the noise voltage induced into theinput circuit, -jtoMJN(~o), with the expression M~J~(to)(Ri + 1/jtoC~)/Lrthat was first obtained by Tesche. 14 The third difference is in the form ofthe denominator of Eqs. (19) and (22).In a real SQUID, it is likely that neither Eq. (19) nor Eq. (22) is exactlyapplicable. Which of the two results is the more appropriate is likely todepend on the details of the structure of the SQUID and its input coil andon the conditions under which the SQUID is operated. Thus, for the moment,we regard the question of whether to use Eq. (19) or Eq. (22) to optimizethe input circuit and noise temperature of the SQUID amplifier as one tobe answered experimentally. For this reason, we have deferred a discussionof the optimization of amplifiers until the third paper 19 in this sequence.The second paper 18 is devoted to the effects of parasitic capacitance, thevalidity of the lumped circuit model, and measurements of the impedanceinduced into the input circuit by the presence of the SQUID.

ACKNOWLEDGMENTSThis research was stimulated by the work of Roger Koch, to whom we

express our appreciation. We also thank Michel Devoret and Claude Hilbertfor countless helpful discussions and Claude Hilbert for a critical readingof the manuscript. J. M. M. gratefully acknowledges the receipt of an NSFgraduate fellowship and an IBM predoctoral fellowship during the courseof this research. This work was supported by the Director, Office of EnergyResearch, Office of Basic Energy Sciences, Materials Sciences Division ofthe U. S. Department of Energy under contract DE-AC03-76SF00098.

REFERENCES

1. J . Clarke, C. D. Tesche, and R. P . Giffard , J . Low Temp. Phys. 37, 405 (1979).2 . C. D. Tesche and J . Clarke, J. Low Temp. Phys. 29, 301 (1977).3 . C. D. Tesche and J . Clarke, J. Low Temp. Phys. 37, 397 (1979).4 . F . De t tman , W. Rich te r , G . Alb rech t , and W. Zahn , Phys. Stat. Sol. (a) 51, K185 (1979).5 . M. B. Ketchen and J . M. Jaycox, Appl. Phy s. Lett. 40, 736 (1982).


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6 . J . M. Mar t i n i s and J . C l a rke , IEE E Trans . Mag . MA G -19 , 446 (1983 ) .7 . V . J . d e W a a l , T . M . K l a p w i jk , a n d P . v a n d e n H a m e r , Z Low Temp. Phys . 53, 287 (1983).8 . C . M . P e g r u m , D . H u t s o n , G . B . D o n a l d s o n , a n d A . T u g w e l l, IEE E Trans. Mag. M A G - 2 1 ,1036 (1985).9 . B . M u h f e l d e r , J . A . B e a ll , M . W . C r o m a r , R . H . O n o , a n d W . W . J o h n s o n , IEEE Trans .Mag. MA G -21 , 427 (1985 ) .10 . P. Care l l i and V. Fogl ie t t i , IEE E Trans . Mag . MA G -21 , 424 (1985 ) .11 . C . D . Tesche , K . H . Brow n , A . C . Ca l l ega r i , M. M. Chen , J . H . G re ine r , H . C . Jones ,M. B . K e t chen , K . K , K im, A . W . K le in sasse r , H . A . N o ta ry s , G . Pro to , R . H . W ang , andT. Y og i , IEEE Trans . Mag . MAG-21, 1032 (1985).12. R . H . K och , P h .D . T hes i s , U n ive rs i t y o f Ca l i fo rn i a , Be rke l ey (1982) .13 . C . D . Tesche , Appl. Phys. Lett. 41, 490 (1982).14 . C. D, Tesche , IEEE Trans . Mag . MA G -19 , 458 (1983 ) .15 . C. D. Tesche , in Noise in Physical Sys tems and 1/ fNoise , M. S ave l l i, G . Lecoy , a nd J . -P .N o n g i e r , e d s . ( N o r t h - H o l l a n d , A m s t e r d a m , 1 9 8 3 ) , p . 1 3 7 .

1 6 . J . E . Z i m m e r m a n , J . Appl. Phys. 42, 4483 (1971).1 7. J . M . M a r t i n i s , C . H i l b e r t , a n d J . C l ar k e , p r e s e n t e d a t A p p l i e d S u p e r c o n d u c t i v i t y C o n f e r -e n c e , S a n D i e g o . S e p t e m b e r 9 -1 3 , 1 9 84 ( u n p u b l i s h e d ) .18 . C . H i lbe r t and J . C l a rke , J . Low Temp. Phys ., 61 ,237 (1985 ) .19 . C . H i lbe r t and J . C l a rke , J . Low Temp. Phys . , 61, 263 (1985).2 0. D . G . M a c D o n a l d , Appl. Ph ys. Lett. 44, 556 (1984).2 1. B . M u h l f e l d e r a n d W . W . J o h n s o n , in Optical Bistability 2 , C . M . B o w d e n , H . M . G i b b s ,and S . L . Mc Ca l l , ed s . (P l e num Press , N ew Y ork , 1984 ), p . 375.

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