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Lecture 1: sine-Gordon equation and solutions

· Equivalent circuit · Derivation of sine-Gordon equation · The most important solutions plasma waves a soliton! chain of solitons resistive state breather and friends · Mechanical analog: the chain of pendula · Penetration of magnetic field

Introduction to the fluxon dynamics in LJJ

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Why LJJ?

· Almost ideal system to study soliton dynamics (simple measurable quantities e.g. V u) · Applications as oscillators (FF, ZFS, FS, Cherenkov, FF transistors) · Physics of layered HTS (dynamics+losses) · Studying "fine" properties: fluxon in a potential, energy level quantization, etc. · Some JJ are just long · It is nice non-linear physical system ;-)

Introduction to the fluxon dynamics in LJJ

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Equation of long Josephson junction
I ILk-1
e

Ie ILk ILk+1

Ik-1

Ik

k+1 - k k IL + Ie Ik,e Jk,e dx, x =
k IL dx

= 2 k /0 = k = IL+1 + Ik L ldx,

2 0

k e - LIL

e H dx

2 0

k H - lIL

= Je - Jk
0 2l



k @ x = 0,L IL = 0 x = 2H 0

k We get rid of IL :

xx = J (x) - Je
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Introduction to the fluxon dynamics in LJJ


Sine-Gordon Equation
RSJ model & l = µ0 d /w, J (x) = j (x)w, Je = je w: 0 V xx = jc sin + + CVt - je R 2µ0 d 0 0 C je 0 t + tt - xx = sin + jc 2Rjc 2jc 2µ0 d jc
2 J
- c 1 - p 2



~ Normalized units: x = x/J , t = tp . ~ xx - tt - sin = t - ~~ ~ ~~ perturbed sine-Gordon equation 1 p = = = c c Other normalized quantities: v = t , ~ h = x ~ u = v/c0 ¯
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1 2 jc R2 C 0

Characteristic velocity: c0 = J p , ¯

Introduction to the fluxon dynamics in LJJ


Boundary conditions

Boundary conditions for linear LJJ: x |x=0, = h ~ Boundary conditions for annular LJJ: |x=0 = |x= +2N ~ ~ x |x=0 = x |x= ~~ ~~ Perturbations are small: 1, 1. For Nb-Al-AlOx -Nb junctions at T = 4.2K 10 The typical value of 0.1. Taking = = 0 we get: xx - tt - sin = 0 ~~ ~~ unperturbed sine-Gordon equation
-2

.

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1: Josephson plasma waves
Consider small amplitude waves: (x, t) = A sin(kx - t), A 1

Substituting into xx - tt - sin = 0 and using approximation sin [A sin(kx - t)] A sin(kx - t) we get the dispersion relation for EM waves in the LJJ: (k ) = 1+ k
2

Picture. Non-Josephson strip-line. c0 = 1 is the Swi¯ hart velocity. Plasma gap. Phase velocity: u
ph

=

= k

1+

1 >1 2 k

i.e. uph > c0 Swihart velocity in the LJJ is the ¯ minimum phase velocity and maximum group velocity of linear EM waves.

Introduction to the fluxon dynamics in LJJ

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Disp ersion of linear waves

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Mechanical analog of LJJ

are generated pendula This twisting program 1999 by PostScript Edward Goldobin, by Dr. Coded

Josephson phase bias current damping coefficient Josephson voltage t

angle of pendulum torque friction in the axis angular frequency

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2: Soliton
Unperturbed sine-Gordon equation has exact solution: (x, t) = 4 arctan exp ±
x -ut 1-u2

This is a solitary wave or soliton. It can move with velocity 0 u < 1 (i.e. c0 !). Picture. Soliton is a ¯ kink which changes the Josephson phase from 0 to 2 (soliton) or from 2 to 0 (anti-soliton). The field of soliton is h = x = cosh(
x -ut 1-u2

2

)

,

h|x=0 = 2

Existence of the soliton

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Fluxon shap e & contraction


Introduction to the fluxon dynamics in LJJ







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Lorentz invariance
Sine-Gordon equation is invariant with respect to the Lorentz transformation: x - ut xx = , 2 1-u t - x/u tt = 1 - u2

Thus, soliton behaves as relativistic object and contracts when approaching the velocity of (our!) light -- Swihart velocity! Picture. In spite of contraction, soliton always carries one quantum of magnetic flux:
-

x dx = () - (-) = 2

Since = 20 , = 0 . Therefore, the soliton in LJJ is called fluxon. An antifluxon carries -0 . The energy (mass) of the soliton (next slide):

E (u) = m(u) =

8 1-u

2

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Mechanical analog of LJJ

are generated pendula This twisting program 1999 by PostScript Edward Goldobin, by Dr. Coded

Josephson phase bias current damping coefficient Josephson voltage t

angle of pendulum torque friction in the axis angular frequency

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Fluxon interaction
Hamiltonian (energy) of the LJJ: H=
+ -

2 2 t + x +(1 - cos ) dx 2 2
K U

Substituting two solitons with the distance x between them






· two fluxons repel each other. · fluxon and anti-fluxon attract.

Introduction to the fluxon dynamics in LJJ



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3: Chain of fluxons
Fluxons can form a dense chain (x, t) = 2 am(x - ut, k )+ For h 1: (x, t) h(x - ut) - sin [h(x - ut)] h2 (1 - u2 )

are generated pendula This twisting program 1999 by PostScript Edward Goldobin, by Dr. Coded

(-: mincing machine :-) Intuitive explanation of repelling.

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4: Resistive (McCumb er) state

(x, t) = (Hx + t) -

sin (Hx + t) 2 - H 2

are generated pendula This twisting program 1999 by PostScript Edward Goldobin, by Dr. Coded

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5. Breather
Since fluxon and antifluxon attract each other, they can form a bound state which oscillates around common center of mass: sin(t cos ) (x, t) = 4 arctan tan cosh(x sin ) where = 0 ... /2. A breather with the moving center of mass can be obtained using Lorentz transformations. Fluxon-antifluxon collision: ut sinh 1-u2 (x, t) = 4 arctan u cosh 1x u2 - There is a positive phase-shift ! Fluxon-fluxon collision: sinh (x, t) = 4 arctan u cosh There is a negative phase-shift !
Introduction to the fluxon dynamics in LJJ Nr. 17

x 1-u


2


2

ut 1-u


Penetration of magnetic field into LJJ

Introduction to the fluxon dynamics in LJJ









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Penetration of magnetic field into LJJ
When h exceeds 2, the fluxons enter the junction and fill it with some density, forming a dense fluxon chain.

Example: h = 4, = hx, so (L) - (0) = h , 50 h N = 2 = 4â28 31.8. Looking at picture, we 6. see 30 fluxons. For smaller fields the correspondence is worse, since the dense fluxon chain approximation works not so good, and at h < 2 does not work at all. e.g. for h = 2.1, N = 16.7, but we see only 10 fluxons.

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Lecture 2: Dynamics of fluxon

· Perturbation theory · Fluxon steps in annular LJJ · ZFS in linear LJJ · Flux-Flow and FFS (Eck peak) · Fiske Steps

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Perturbation theory of McLoughlin and Scott
All solutions of s-G equation (except resistive state) considered during the previous lection are solutions of the unperturbed sG equation: xx - tt - sin = 0 We also have seen that: H=
+ -

2 2 t + x +(1 - cos ) dx 2 2
K U

(1)

The real equation which governs the Josephson phase dynamics in the system is perturbed s-G equation: xx - tt - sin = t - (2)

The Hamiltonian (1) corresponds only to the l.h.s. of (2) while r.h.s. describes the energy dissipation and injection. Let us write down the change of energy with time.

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Energy balance equations
dH dt
+

=
- +

2 d 2 t + x +(1 - cos ) dx 2 dt 2

=
-

(t tt + x xt + t sin ) dx
+

=

x t |- zero if localized
+

+

+
-

(t tt - xx t + t sin ) dx

=
-

-t (xx - tt - sin ) dx l.h.s. of sine-Gordon -t
- + - +

=

(t - ) r.h.s. of sine-Gordon

dx

=

t - 2 dx = 0 t

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Energy balance for fluxon
+ +

t dx =
- F u -

2 dx t
F u

x - ut (x, t) = 4 arctan exp 1 - u2 2 -u t (x, t) = x-ut 1 - u2 cosh 1-u2 8u2 - 2u = 1-u |u| =
1+ 1 2

4 for

2

, |u| 4 |u| 1,

1

for 1

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I ­V Characteristic

Example: annular LJJ V= n0 n0 u = = t L/u L

I -V characteristic ­u characteristic V
max



=

¯ n0 c0 L

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Collision with the edge


Collision with edge fluxon-antifluxon collision

Introduction to the fluxon dynamics in LJJ











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Fluxon trajectories

animations see at http://christo.pit.physik.uni-tuebingen.de:88/FluxonDynamics/

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Zero Field Steps

V=

0 - (-0 ) 0 u = = t 2L/u L

But frequency of collisions is f = u/2L, i.e., two times lower!



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How the fluxons get into the junction?

Moving down along McCumber branch the rotation frequency becomes lower resulting in instability due to thermal fluctuations.

are generated pendula This twisting program 1999 by PostScript Edward Goldobin, by Dr. Coded

ZFS are better visible at T > 4.2K

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Flux-flow

Let us suppose that LJJ is filled with fluxons e.g. some field H > Hc1 is applied to the linear LJJ.

V

FF

=

t

=

H L L/u

= H u

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Flux-flow IVC
t H L L/u

V

FF

=

=

= H u

The maximum on the IVC at u = c0 is called a flux¯ flow resonance or Eck peak. Application: tunable oscillators for the frequencies 50­ 800 GHz.



Introduction to the fluxon dynamics in LJJ

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Interaction with edges


ILHOG1[























FRRUGLQDWH[





· The boundary conditions x = h are not satisfied if we take running solutions (x - ut). · we have to add "reflected wave" which propagates towards the middle of LJJ. · This wave decays on the distances 1/. · L 1 results in the formation of the standing wave. · moving fluxons synchronize with this standing wave, resulting in geometrical resonances on the c0 ¯ FS IVC at: Vn = 0 n 2L
Introduction to the fluxon dynamics in LJJ Nr. 13


Fiske Steps

· Linear theory (H

2,L

1) is developed.

· Non-linear theory (any H , any L) in the present state gives only the amplitude of resonances in 1-harmonic approximation. · General nonlinear theory is not developed yet. · Experimental IVC contains some features (shift or sub-families, fine structure of FSs) which are not explained.

Introduction to the fluxon dynamics in LJJ

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