FLUXON-BREATHER-PLASMA OSCILLATION DECAY IN LONG JOSEPHSON JUNCTIONS

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FLUXON-BREATHER-PLASMA OSCILLATION

DECAY IN LONG JOSEPHSON JUNCTIONS

G. Costabile, R. Parmentier, B. Savo

To cite this version:

JOURNAL DE PHYSIQUE

_{Colloque C6, suppliment}

_{au no 8, Tome 39, aoiif 1978, page}

_C6-567

FLUXON-BREATHER-PLASMA OSC

I

L L A T I O N DECAY

I N

LONG JOSEPHSON J U N C T I O N S

G. Costabile,

R.D.

Parmentier, and B. Savo

Istituto di Fisica

-

Universita'

di

Salerno

-

1-84200 SaZerno, Italy

RGsum6.- Les solutions exactes de l'lquation de sine-Gordon, qui dgcrivent les oscillations sur

une jonction Josephson longue et uni-dimensionnelle avec oonditions au bord de circuit ouvert,

suggsrent que les tourbillons qui ralentissent jusqu'ii une vitesse critique de propagation se

transforment en "breathers", qui

1

leur tour, lorsque leur amplitude diminue jusqu'9 la valeur

critique, se transforment en oscillations du plasma.

Abstract.- Exact solutions of the sine-Gordon equation describing oscillations on a long, one-

dimensional Josephson junction with open cicuit boundary conditions suggest that fluxons that

slow to a critical propagation velocity decay into breathers, which in turn, when their ampli-

tude diminishes to a critical value. decay into plasma oscillations.

The solutions of the sine-Gordon equation with

@ =

4 tan-ID cn(Bx;kf) cn(nt;kgn

finite boundary conditions recently reported by Cos-

(3)

where

tabile et al. /I/, which furnish exact analytical

expressions for the three fundamental types of os-

k

f

- -

A~B~(~+A~)+I

B2(1+A2)2

l1

;k2

g

=

n2(l+A2)2

132(1+~2)-Cl

(4a,b)

cillations on long Josephson junctions, viz., flu-

and

n,

and

A are

related by the nonlinear disper-

xons, breathers, and plasma oscillatio~,suggest

the

sion

equation

existence of excitationldecay mechanisms between

1-A2

n 2 - V

=

these modes of oscillation. The analysis of such me-

(5)

chanislns

is of considerable importance in the study

Imposition of the boundary conditions (2) fixes the

of applications of the Josephson junction as a gene-

spatial periodicity as

2n

rator of high frequency radiation121 and as an ele-

B

=

i;

K (kf)

n

( 6 )

ment for digital computation/3/.

Neglecting dissipative effects, a long, one-

dimensional Josephson junction is described by the

equation

qxx

-

Ptt

=

sin

@

(1)

where

$

is the magnetic flux normalized respect to

h/4se,

x

is distance normalized with respect to the

Josephson penetration length

XJ,

and t is time nor-

malized with respect to the inverse of the Josephson

plasma frequency

wJ. The boundary conditions

@,iO,t)

= 0 =

$x(L,t>

(2)

correspond to imposing open-circuit terminations at

where n

=

1,2,

...

is the number of nodes in the

standing wave and

K(k)

is the complete elliptic in-

tegral of the first kind. For n

=

0, the entire

length of the line oscillates in phase as

@ =

4 tan-'

sn(~2t;kg; k

=

A2 and a = l/(l+A2) (7)

2) Breather oscillation (see figure 28 of

Fulton 1.51)

:

@ =

4 tan-'

{A

dnB(x-xO);kf]sn(f2t;k

g

) }

where

p e l -

1-B2(l+~2)/~2

; k; a

A'O-Q~

(l+A2jl (9a,b)

f

62

(l+A2)

n 2

(1+A2)

and the nonlinear dispersion equation is

the two ends of such a junction having normalized

B

= OA.

(10)

length L. Physically, such boundary conditions can

The boundary conditions (2) now require

n

be closely approximated by using an "overlap" geo-

8,

=

K(kf)

(11)

metry 141. Fulton 151, in a series of remarkably

_{with two ~ossible}

_{values for xo}

_:

_{a) Bxo}

₌

_K(kf),

detailed observations on a mechanical analog, has

and b) xo

= 0.

For n even,

a)

corresponds to brea-

recently reviewed the qualitative nature of the so-

_{thers located near the center of the line, and b)}

lutions of (1) under the boundary conditions (2).We

corresponds to fluxons bound to virtual antifluxons

repeat here, for convenience, the analytic solutions

at both ends of the line. For n odd, a) and b) are

obtained by Costabile, et al.

111.

equivalent.

1) Plasma oscillation (see figure 20 of

Fulton 151)

:

3) Fluxon o s c i l l a t i o n ( s e e f i g u r e s 25 and 26 of F u l t o n /5/ :

4

= 4 t a d l [ A dn(f3x;kf) t n ( Q t ; k

)I

g (12) where

8

- ( A ~ - ~ ) - I (13a,b) k 2 = 1 - A2 f The n o n l i n e a r d i s p e r s i o n e q u a t i o n i s a g a i n a s i n (10) and t h e boundary c o n d i t i o n s (2) a g a i n r e q u i r e (11). The e x i s t e n c e of e x c i t a t i o n / d e c a y mechanisms between t h e s e fundamental modes of o s c i l l a t i o n i s sug- g e s t e d by t h e f o l l o w i n g o b s e r v a t i o n s . 1 ) Breather-plasma o s c i l l a t i o n e x c i t a t i o n / decay : S e t t i n g

f3

< A / ( ~ + A ~ ) i n (8) and (9) y i e l d s s o l u t i o n s t h a t map o n t o t h o s e w i t h

6

>

A/(1+A2). S e t t i n g

B

= A/ (1+A2) i n (9) y i e l d s k i = 0 and k 2 = A 4 . R e c a l l i n g g

t h a t dn(X;O) = 1, and assuming A

<

1, t h i s s u b s t i t u - t i o n reduces (8) t o ( 7 ) . S e t t i n g

B=

A/ (1+A2) i n (11) and r e c a l l i n g t h a t K(0) = ~ / 2 t h u s y i e l d s a minimum v a l u e f o r A i n (81 a s

The n = 0 plasma o s c i l l a t i b n , on t h e o t h e r hand, can e x i s t f o r a l l A i n t h e range 0 < A < 1. These f a c t s s u g g e s t t h a t f o r A

<

Amin, a b r e a t h e r must n e c e s s a r i - l y decay i n t o a n n = 0 plasma o s c i l l a t i o n provided t h a t , from (14), L/n

2

?T, and t h a t a n n = 0 plasma o s c i l l a t i o n w i t h A

2

Amin c a n e x c i t e a b r e a t h e r .

2) Fluxon-breather e x c i t a t i o n / d e c a y :

S e t t i n g

f3

> A2/A2-1) i n (12) and (13) y i e l d s solutions t h a t map o n t o t h o s e w i t h

6

< A2/A2-1). Thus, s e t t i n g

B

= A2/A2-1) i n (11) and (13a) y i e l d s a maximum v a l u e f o r A i n (12) a s

S e t t i n g

f3=

A/(A2-l)Ihin (13) y i e l d s

k;

= 1

and k2 = 1. R e c a l l i n g t h a t dn(X;l) = sech(X) and g

t n ( ~ ; l ) = s i n h ( X ) , t h i s s u b s t i t u t i o n , w i t h t h e i d e n t i - f i c a t i o n A E l / u , where u i s t h e v e l o c i t y i n t h e cen- t e r of mass r e f e r e n c e frame, reduces (12) t o t h e form of t h e f l u x o n - a n t i f l u x o n c o l l i s i o n on t h e i n f i n i t e

l i n e r e p o r t e d by S c o t t , e t a l . 161. R e c a l l i n g t h a t K ( l ) + m

,

t h e s e r e s u l t s imply t h a t s o l u t i o n s of t h e

type (12) can be found on t h e f i n i t e l i n e f o r a l l A such t h a t s o l u t i o n s of t h e t y p e (12) can be found on t h e f i n i t e l i n e f o r a l l A such t h a t 1 < A

5

Amax,£.

B r e a t h e r s , a c c o r d i n g l y , can e x i s t f o r A < A

<_

min- *max,b'

From (15) and ( 1 6 ) , Amax,b < Amax,£; however, f o r L/n >> 1, t h e two maxima tend toward e q u a l i t y . R e c a l l i n g t h a t sn(iX;k) = i t n ( X ; k ' ) , t h e t r a n s f o r - mation A + - i A , Q -+

in,

f3

+ f3

,

which p r e s e r v e s

( l o ) , t r a n s f o r m s (8) and (9) i n t o (12) and (13). R e c a l l i n g t h a t a b r e a t h e r r e p r e s e n t s a bound s t a ' t e of a f l u x o n and a n a n t i f l u x o n , t h e s e f a c t s suggest t h a t a f l u x o n t h a t slows t o near t h e c r i t i c a l pro- p a g a t i o n v e l o c i t y (A + Amax ,£) w h i l e encountering a n a n t i f l u x o n c a n decay i n t o a b r e a t h e r , and t h a t a b r e a t h e r t h a t i n c r e a s e s i n amplitude t o n e a r Amax, b c a n e x c i t e a f l u x o n - a n t i f l u x o n p a i r . References /1/ C o s t a b i l e , G., P a r m e n t i e r , R.D., Savo, B., McLaughlin, D.W., and S c o t t , A.C., Appl-Phys. L e t t . ( i n p r e s s ) .

/ 2 / F u l t o n , T.A. and Dunkleberger, L.N., Revue Phys. Appl.

2

(1974) 299.

/3/ F u l t o n , T.A., Dynes, R.C., and Anderson, P.W., Proc. IBEE

61

(1973) 28.

141 Barone, A., Johnson, W . J . , and ~ a g l i o , R., J .

Appl. P h y s . 5 (1975) 3628.

/ 5 / F u l t o n , T.A., i n Superconductor A p p l i c a t i o n s :

SQUIDS and Machines, B.B. Schwartz and S.Foner eds. (Plenum P r e s s , New York) 1977, p.125. 161 S c o t t , A . C . , Chu, F.Y.F., and McLaughlin, D.W.,

Proc. IEEE

5

(1973) 1443.