Noise induced transition in Josephson junction with second harmonic

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by  Nivedita Bhadra

Noise induced transition in Josephson junction with second harmonic
Nivedita Bhadra∗1
1Department of Physical Sciences, IISER Kolkata, Nadia, West Bengal 741246, India
We show a noise induced transition in Josephson junction with first as well as second harmonic.
A periodically modulated multiplicative coloured noise can stabilise an unstable configuration in
such a system. The stabilisation of the unstable configuration has been captured in the effective
potential of the system obtained by integrating out the high frequency components of the noise.
Numerical analysis has been done to confirm the prediction from the analytical calculation.
7 PACSnumbers:
1
0
2 I. INTRODUCTION comemoreandmoreimportantandtherelationbecomes
n I = f1sinφ+f2sin2φ, where, f1 and f2 are amplitude
of first and second harmonic of Josephson current. At
a Presence of noise can induce several interesting
J the0−π transition,thefirstharmonicbecomeszeroand
phenomena in nonlinear system, such as, stochastic thesecond-harmonictermdominates21–24. Thesestudies
4 resonance1–3, dynamical stabilisation4,5, noise induced
2 show the higher harmonics to be extremely sensitive to
order6, noise induced chaos7, broadly known as noise in-
changes in barrier thickness, temperature etc. For this
duced transition8–10. We have studied a noise induced
] two harmonic potential, i.e., f (cid:54)=0, we have studied the
n transitioninanonlinearsystem,namely,Josephsonjunc- 2
effect of a periodically modulated coloured noise.
o tion(JJ)whichconsistsoffirstaswellassecondharmonic
c Recently, a robust second harmonic CPR has been es-
in its current phase relation(CPR).
- tablishedinaJosephsonjunctionoftwosuperconductors
r
p In a Josephson junction, a supercurrent flows between separated by a ferromagnetic layer25. This experiment
u twoproximatelycoupledsuperconductorsthroughathin reports for JJs(NbN/GdN/NbN) with highly spin polar-
s insulatinglayerduetoCooperpairtunneling. Thevalue
ized barriers and reveals CPR which is purely second
.
at ofthesupercurrentflowingthroughtheinterlayerispro- harmonic in nature and insensitive to barrier thicknes.
portional to the sine of the difference of the phases of
m The probable origin of such robust second harmonic is
the superconductor order parameter11. In general, the
thegenerationoftwotripletpairsfromhigher-ordertun-
d- current can be summation of all harmonics12,13. In con- nelling processes. We have proposed a theoretical model
n ventional JJs at equilibrium, the phase difference of the forJosephson junctionwherea stochasticdrive hasbeen
o superconducting order parameter on the two sides is
appliedandithasbeenshownthattheunstableconfigu-
c zero14,15. These are also known as 0-junctions. Gener-
rationcanbestabilizedwiththiskindofexternalrandom
[
allythesingleharmonicisadequateforthedescriptionof drive.

1 theJJproperties.Theroleofthesecondandotherhigher
Earlier studies has shown appearance of noiselike so-
v harmonics is negligible and hardly observable. However,
9 in recent years, there has been lot of interesting stud- lution in rf-biased JJ due to the emergence of chaotic
7 dynamics26. If in a noise-free rf-biased JJ chaos ap-
ies in Josephson junction with unconventional current
8 pears, due to addition of small thermal noise, the dy-
phase relations(CPR)16–19. Second harmonic in the cur-
6 namics remains unchanged unless the noise level exceeds
rent phase relation has become noticeable in these stud-
0 thechaoticstate. Smallamountofthermalnoisecanlead
. ies. Severalmechanismofgenerationofsecondharmonic
1 has also been discussed16. For example, in JJs with to a hopping between two dynamical states of the junc-
0 tion. The hopping can be between two different periodic
a ferromagnetic interlayer i.e. S/F/S, where for some
7 solutions or it can be between a periodic and a chaotic
1 intervals of the exchange field h and F-layer thickness
solution. Alittleraiseintemperaturecandestabilizethe
: d, the ground state corresponds to the phase difference
v system from the periodic solution and the system can
equal to π, also known as “π” junctions. Experimen-
i spend most of the time in the chaotic state leading to
X tal studies has shown 0 − π transition in S/F/S from
noiseinducedtransition. Thermalnoisehasbeenrealised
ar tchrietimcaelacsuurrreemnte2n0t.sHofowtheevetre,mthpiesrasetcuornedd-ehpaernmdoenniccecoofnttrhie- in terms of additive delta correlated white noise7,27,28.
bution decreases strongly with the increase of the thick- Another study has shown the effect of multiplicative
ness of the F layer and its exchange field. The CPR for periodic drive in a bosonic Josephson junction. Para-
JJs is sinusoidal only near the critical temperature T , metrically driven pendulum is an mechanical analog for
c
I = f sinφ,f being the amplitude of first harmonic in suchsystem29. Thiskindofperiodicdrivecanleadtody-
1 1
CPR.Atlowtemperature,thehigherharmonictermsbe- namicstabilizationofunstableφ=π configuration30. In
this work we have investigated the effect of periodically
modulated multiplicative coloured noise in JJ with both
first and second harmonic present in CPR. It has been
∗email: [email protected] observed that the unstable state for JJ can be stabilized
2
in presence of such noise. It has been known for a long for the system is unstable at φ = π. This work of ours
time that multiplicative noise can enhance the stability shows how this unstable can be stabilised in presence
of nonlinear systems. Several studies have shown noise of multiplicative periodically modulated coloured noise.
induced phase transitions far from equilibrium10. In our Our study is in the other regime, i.e., f > 2f . The
1 2
present work we have observed the effect of noise, power equation governing the stochastically driven system is as
spectrum of which has a peak at high frequency. Due to follows,
the presence of this high frequency “separation of vari-
able” method is applicable, an effect known as “Kapitza φ¨+(1+ξ(t)) (f sinφ+f sin2φ)=0, (4)
1 2
effect”29,31,32. Kapitza effect was first adopted to under-
stand the dynamic stabilization of inverted position of a where,ξ(t)isthenoisewhosepowerspectrumhasapeak
pendulum periodically driven at the point of suspension. at high frequency compared to the natural frequency of
The idea is to separate the “fast” and “slow” variable the system when no drive is present. This choice of ξ
and integrating out the “fast” variable”. An effective ensures that the deviations of the variable φ caused by
potential V for the “slow” component can be obtained thestochasticvibrationofthesuspensionpointaresmall.
eff
thisway. Thiseliminationofrapidcomponentsproduces Hence, a separation of “slow” and “fast” variable is pos-
higher harmonic terms in V whose coefficient can be sible. ThenoiseistakentobeGaussiandistributedwith
eff
adjusted to generate new local minima. We adopted zero mean, (cid:104)ξ(t)(cid:105)=0, and temporal correlation goes as,
this method of separation of variable to obtain the ef-
fective potential for the system. The stable phase for (cid:104)ξ(t)ξ(t+τ)(cid:105)=D(τ)=σ2e−ατcos(ντ). (5)
the system has been achieved from this effective poten-
tial. Henceforth, angular bracket (cid:104)...(cid:105) will denote averaging
over the noise, where σ2 is the variance of the stochastic
process,andν isthemodulationfrequencywhichistaken
II. MODEL to be very high. α is the attenuation parameter.
We would study the problem under the limit 1<α(cid:28)
We consider a Josephson junction model with first ν.
as well as second harmonic in the current phase rela-
tion(CPR). We would consider the case when bias cur-
rent is zero. The governing equation for the system is as III. CALCULATION OF EFFECTIVE
follows, POTENTIAL Veff
φ¨+2βφ˙+ (f1 sinφ+f2 sin2φ)=0, (1) In this section we would calculate the effective poten-
tial for this sytem. This effective potential was for the
where, f and f are amplitude of first and second har-
1 2 first time implemented in the context of dynamic stabi-
monicandβ isthedampingparameterofJosephsoncur-
lization of a pivot driven pendulum. Dynamical stabili-
rent. We would consider β ≈ 0. Hence, We can write
tion of φ=π i.e.,the inverted position of the system was
Eq.(1) in terms of Josephson potential V as,
theoretically explained on the basis of this effective po-
tential. The idea is to obtain the effective potential by
∂V
φ¨+ =0, (2) splitting the motion into “fast” and “slow” variable and
∂φ
averaging over the “fast” component. In such cases, new
harmonicsappearintheeffectivepotential. Nowthesta-
where,
blephasescanbeobtainedbyfindingouttheextremaof
f the effective potential. We adopt the same method for
V =−f cosφ− 2 cos2φ. (3)
1 2 this system.
ThevariablegoverningthedynamicsofthisJosephson
The condition for stabilization of φ = π position is ob- junction is the angular displacement φ. We can separate
tained by putting ∂∂2φV2|φ=π >0, i.e., 2f2 >f1. It can be itinto“fast”φf and“slow”partφs. Writingφ=φs+φf,
observed from the potential that f > 2f configuration where, φ (cid:29)φ , Eq.(4) becomes,
1 2 s f
φ¨ +φ¨ =−(1+ξ(t))(f (sinφ +φ cosφ )+f (sin2φ +2φ cos2φ )), (6)
s f 1 s f s 2 s f s
by keeping terms upto O(φ ). Eq.(6) involves both the which should have its own separate equation of motion.
f
fluctuatingpart(fast)andthesmooth(slow)part,eachof Retaining terms linear in φ or ξ, and replacing φ by
f s
3
the noise averaged value (cid:104)φ(cid:105) for the fluctuation part, the pendulum. We can consider cos(cid:104)φ(cid:105) ≈ −1 as we are in-
equation of motion for the fluctuating part is given by, terested in the regime φ≈π. We would solve Eq. (7) by
Green function G. The effect of noise can be cast into
the form of unit force. Eq. (7) can be written as,
φ¨ +(f cos(cid:104)φ(cid:105)+2f cos2(cid:104)φ(cid:105))φ =
f 1 2 f
−ξ(t)(f sin(cid:104)φ(cid:105)+f sin2(cid:104)φ(cid:105)), (7)
1 2 φ¨ −(f −2f )φ =δ(t−t(cid:48)), (9)
f 1 2 f
and that of the “slow” part is,
¨ where, δ(t−t(cid:48)) is unit force at t=t(cid:48). In terms of Green
(cid:104)φ(cid:105)=−(f sin(cid:104)φ(cid:105)+f sin2(cid:104)φ(cid:105))−
1 2
function Eq. (9) becomes,
(cid:104)ξ(t)φ (cid:105)(f cos(cid:104)φ(cid:105)+2f cos2(cid:104)φ(cid:105)) (8)
f 1 2
Thenaturalinitialconditionfor δφistoset δφ=δφ˙ =0 G¨−γ2G=δ(t−t(cid:48)), (10)
at the initial time which we may choose as t = −∞.
Withthisinitialcondition, Eq.(7)admits(cid:104)δφ(t)(cid:105)=0for
all time, as expected. where, G satisfies the initial condition for t < t(cid:48) with G
Note that the cross-correlation, as we show below, is continuous at t=t(cid:48) and γ2 =f −2f . Solving Eq. (10)
1 2
O(1), and it contributes to the average motion of the and substituting the solution in Eq.(8) we find,
σ2 1
(cid:104)φ¨(cid:105)=−(f sin(cid:104)φ(cid:105)+f sin2(cid:104)φ(cid:105))− √ (f sin(cid:104)φ(cid:105)+f sin2(cid:104)φ(cid:105))(f cos(cid:104)φ(cid:105)+2f cos2(cid:104)φ(cid:105)), (11)
1 2 ν2 f −2f 1 2 1 2
1 2
From now we would opt out (cid:104)..(cid:105) for the simplification of where,
symbol. In terms of effective potential V we can write,
eff
∂V
φ¨+ eff =0, (12)
∂φ
where,
(cid:16) f f σ2(cid:17) (cid:16)f f2σ2(cid:17) f f σ2 f2σ2
V =− f − 1 2 cosφ− 2 + 1 cos2φ− 1 2 cos3φ− 2 cos4φ, (13)
eff 1 2 ν2 2 4 ν2 2 ν2 4 ν2
Now, tofindtheconditionforstabilityofφ=π, wesim- Thestabilityconditionatφ=π showsadivergenceat
plyputthecondition ∂∂2φV2|φ=π <0. Itgivesthecondition f1 = 2f2. Now we study the stable and unstable phase
as, after expanding Veff around φ = π. Expanding around
φ = π we obtain the effective potential in the following
σ2 1 form,
> √ (14)
ν2 f −2f
1 2
1 1 σ2 1 1 1 σ2 1
V(φ) = V + (2f −f )η2+ √ (f −2f )2η2+ (f −8f ))η4− ((2f −f )(8f −f ))η4
0 2 2 1 2!ν2 f −2f 1 2 4! 1 2 3!ν2 f −2f 2 1 2 1
1 2 ! 2
1 1 σ2 1
+ (−f +32f )η6+ √ (16f2−364f f +1024f2)η6, (15)
6! 1 2 6!ν2 f −2f 1 1 2 2
1 2
where, η = φ−π, V0 = f1− f22 − f124∆ − f224∆.We ignore higher order terms since the highest order term consid-
4
ered here(η6) is always positive for the range of values ofperiodicallymodulatedcolourenoiseweplottheeffec-
we are interested in f <8f . tive potential obtained from the analytical calculation.
1 2
Let us now consider a few special situations. Theplotshowsφ=π isamaximumpotentialconfigura-
1. Intheabsenceofnoisei.e. σ2 =0andf =2f ,the tion in absence of noise i.e,σν22 = 0,whereas, for nonzero
ν2 1 2 valuesofnoiseitgraduallybecomesaminimumatφ=π.
series(Eq.(15)) starts with the quartic term, with
the coefficient of η4 as negative (−f2). The coeffi-
4
cient of η6 is positive ( 1 f ), providing stability to
24 2 4
the system. Therefore, ∂2V /∂φ2| < 0. Hence,
eff π V)
η =0orφ=π is an unstable configuration for the al(
system. nti 0
e
2. Icnonpdriteisoenncσe2o>f n√ois1e .i.e.i,swshaetnis,fieσνd22, t(cid:54)=he0c,oaenffidcitehnet ve pot -4
ν2 f1−2f2 cti
of η2 is positive. In this case, ∂2V /∂φ2| > 0, e ∆=0.0
hence, η =0 is stable configuration feoffr the sπystem. Eff -8 ∆∆==01..50
3. At f = 2f , both η2, η4 terms vanish. Effective
1 2 0 π/2 π 3π/2 2π
potential becomes,
Phase fluctuation(φ)
3 1 1
V = f − f η4+ f η6. (16)
eff 2 2 4 2 4! 2 FIG. 2: This figure shows effective potential in presence of
Hence, it is an unstable configuration at φ=π. periodically modulated coloured noise(Eq.(13)). ∆ is defined
as σ2. When ∆ = 0, i.e., in absence of noise φ = π is a
ν2
maximum. Gradual increase of the parameter ∆ shows that
IV. NUMERICAL RESULTS φ=π gradually becomes a minimum of the potential.
The theoretical results obtained are approximate. To Now to observe the effect of coloured noise we have
observe the shape of oscillation we have performed some simulated Eq.4 by Euler algorithm. We have taken the
numerical analysis. parameter values as, α = 1.5,ν = 10,f1 = 2,f2 = 0.5.
According to our analytical expression for potential of Timestep was chosen to be 0.001. We have taken ξ(t)
the system we have found that φ = π is an unstable as sufficiently wideband noise. Fig.3 shows the power
configuration in the regime f > 2f . Fig.1 shows ef- spectral density for the periodically modulated coloured
1 2
fective potential in absence of any drive. The solid line noise. Thespectrumshowsasharppeakatthefrequency
1 50
V) 0.5
al(
enti 0 sity)
ot en 0
p d
e -0.5 al
Effectiv -1-.15 fff111>=<222fff222 g(powerspectr −50
0 π/2 π 3π/2 2π lo
0
1
Phase fluctuation(φ)
FIG. 1: This firgure shows Josephson potential(Eq.(3)). For
f1 > 2f2, φ = π is a maximum in the potential whereas, for −100
f < 2f ,the system achieves a minimum in the potential. 0 5 10 15 20 2ν5 30 35 40 45 50
1 2
φ=0 is always a stable configuration for the system.
FIG.3: Thepowerspectraldensityofthecolourednoiseξ(t).
shows the shape of the potential for f = 2f . f > 2f The spetrum is sharply peaked at the frequency ν =10.
1 2 1 2
shows a maximum whereas, f < 2f shows a minimum
1 2
at φ = π. Curvature of the potential changes from pos- ν =10. The coloured noise ξ(t) is generated by a forced
itive to negative gradually undergoing a zero value indi- oscillatorwithfrequencyν whichhasadampingα=1.5.
catingatransitionatf =2f . Inordertofindtheeffect The applied force is a delta correlated ramdom force.
1 2
5
Thesolutionofthisrandomlyforceddampedoscillator V. CONCLUSION
has been fed into the JJ equation(Eq.(4)). The variance
σ is kept higher than the frequency ν as is required for
In this work we have shown how periodically modu-
the stability of φ = π. The initial condition for φ has
lated coloured noise can stabilize unstable configuration
been kept π+0.001 as we are interested to observe the
in Josephson junction with first and second harmonic.
motion of φ around π. φ˙ has been kept as 0.01. The
The stable state has been detected by analysing the ef-
potential shown in Fig.2 is also consistent with the na-
fective potential we obtained after separation of “first”
ture of the trajectory. In absence of noise the potential
and “slow” variable. This approximate analysis shows
shows only one minimum at φ=π, whereas, in presence
hownewharmonicappearsintheeffectivepotential. We
of noise another minimum appears at φ = π in the ef-
also observe the stability of φ=π in the numerical anal-
fective potential. All these plots shown are average of 40
ysis of the stochastic equation governing the system.
different realisations.
In general for a JJ system with first harmonic the
phase fluctuation for ground state is φ = 0. In pres-
1
ence of second harmonic it depends on the amplitude of
3.1445 τ)i 0.5 the harmonic. In our current study we have shown how
+
ttξξh()(−0.05 sthtoechφa=sticπdcrainveb.Ienmthaedecoantgerxotunodf JsJta,tteheinvaplrueeseonfcephoafsae
3.1435 fluctuationi.,e.,φisactuallyameasureofJosephsoncur-
φ 0 0.5 τ 1 1.5 rent. Since the Josephson current is proportional to the
sine argument of the phase fluctuation, φ being negative
indicates inversion of the direction of the current. The
3.1425
condition to make the φ = π as a ground state depends
of the parameter of the applied periodically modulated
coloured noise.
3.1415
0 100 200 300 400 500
time
FIG. 4: The main panel shows time series of the variable
Acknowledgements
φ. Here, φ represents phase fluctuation. Due to presence of
damping the time series initially shows oscillation and grad-
uallydampsoutandremainsfixedatφ=π. Theinsetfigure The author would like to thank Dr. Siddhartha Lal
showsautocorrelationoftheperiodicallymodulatedcoloured and Dr. Anandamohan Ghosh for valuable discussions
noise ξ(t).
and IISER Kolkata for financial support.
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