Stability and chaos of a driven nano-electromechanical Josephson junction

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by  P. Berggren

Stability and chaos of a driven nano-electromechanical Josephson junction
P. Berggren∗ and J. Fransson
Department of Physics and Astronomy, Uppsala University, Box 530, SE-751 21 Uppsala
(Dated: January 13, 2012)
We consider the motion of and Josephson current through a mechanically oscillating supercon-
ductingislandasymmetricallyembeddedinaJosephsonjunction. Theelectromechanicalcouplingis
providedbydistancedependenttunnelingratesbetweentheelectrodesandtheisland. Thesystem
asymmetry, resulting from the geometrical configuration, leads, for weak coupling, to an equation
of the mechanical motion that reduces to the well-known Duffing equation. At zero bias voltage
the island motion is determined by the homogenous Duffing equation that opens up two separate
2 regionsofsolutionsdependingonthesuperconductingphases. Theislandeithermovesunderinflu-
1 ence of an anharmonic single well potential, or is governed by a double well potential that allows
0 for off-center oscillations. Under applied bias voltage the island equation of motion turns into a
2 modified Duffing equation, with time dependent coefficients, that demonstrate both quasi periodic
and chaotic behavior.
n
a
PACSnumbers: 85.25.Cp,05.45.Ac,85.85.+j,73.40.Gk
J

2
1 I. INTRODUCTION equation, thoroughly studied in mathematics41–44, has

received a lot of attention in NEMS research since non-
]
l linear restoring forces act on small scale resonators, see
l Nano electromechanical system (NEMS) resonators
a Ref. 45forareview. TheDuffingequationalsoshowsup
may now be micro fabricated precise enough that the
h in driven macro scale resonators that are geometrically
s- esyffsetcetms oafrteunmneealisnugraeblleec1t.ronSuccohupdliynngamtoictahleinmteecrhacatniiocnasl similar to our setup46,47.
e
between the charge carriers and vibrational modes in a At finite bias voltages the equation of motion is mod-
m
mesoscopic system have been observed in single electron ified to include harmonically time dependent coefficients
t. tunneling to suspended carbon nano tubes2,3. Similar to both the linear and cubic term as well as a harmonic
a
effects of vibron–electron coupling possibly explain dif- driving force. No studies have been published on this
m
ferential conductance dips and peaks in molecular elec- Duffing equation variant to our knowledge. At Joseph-
- tronics devices4–17 and differential conductance steps in son frequencies above and below the eigen frequency of
d
n STM based inelastic tunneling spectroscopy on local vi- the oscillating island regular and stable quasi periodic
o bration modes on surfaces18,19. motion is found, wheres more resonant frequencies yield
c Nanoscaleresonatorsetupsareinterestingasfasthigh chaotic solutions. Chaos is an inherent property of the
[ sensitivity detection devices20 for mass21–26, charge27, driven Duffing equation48.

1 force28 and displacement29 and as mechanical systems TheimportanceofnonlinearitiesandtheDuffingequa-
v reach the quantum limit implications for quantum infor- tion in a NEMS aspect comes from a number of sug-
7 mation technology may be tremendous30–33. gestedandinvestigatedapplications. Weaksignalampli-
1
The field has further evolved to include and explore fication with low noise levels based on system sensitivity
5
2 superconducting NEMS. One investigative direction has near bifurcation points is one active subject49–52. Other
. beentocouplenanomechanicalresonatorstoasupercon- novel experiments utilize buckled nano resonator beams
1
ducting Cooper pair box34, or a superconducting quan- that oscillate within the confines of a double well poten-
0
2 tum interference device (SQUID)35, in order to probe tial, typical to the Duffing equation, either to produce
1 andcontrolsuperconductingqubits,aswellasdetectdis- mechanical quantized qubit states in the resonator by
: placements near the quantum limit. Another course has cooling53 or to construct mechanical memory bits that
v
beentostudyJosephsoncurrentscoupledtomechanical, work under room temperature by controlling transitions
i
X or molecular, oscillators situated within the tunneling between the potential wells54.
r junction36–40. The oscillator then acts to shuttle Cooper This paper is outlined as follows. In Sec. II a detailed
a
pairs at resonant levels. description of the mechanical system is given. Here we
InthispaperweintroduceadoubleJosephsonjunction also derive the Josephson tunneling currents dependent
thatisasymmetricwithrespecttotheoscillatorymotion on island position as well as the equation of motion for
ofasuperconductingisland. Thedynamicsofthesystem theislandasitcouplestothetunnelingCooperpairs. In
is captured as the mechanical motion is coupled to the Sec. III the solutions to the island equation of motion in
electrontunneling. Apartfromreproducingtheexpected absence of bias voltage are presented together with nu-
equationofmotiontermsfoundinRef. 39theasymmetry merical results for the tunneling currents under different
adds a nonlinear cubic term. At zero bias voltage the is- conditions including zero bias voltage with varying su-
landmotionisconsequentlycontrolledbytherelativesu- perconducting phases and finite bias voltage. In Sec. IV
perconductingphasesthroughtheDuffingequation. This we summarize our findings.
2
II. THEORY The first three terms of H separately give the electronic
structure of the leads in terms of BCS Hamiltonians, c.f.
A. Description of model Eq. (2). The superconductors couple through tunnel-
ing term H . Here, c and c† annihilate and create an
T κ κ
electron in electrode χ=L,I,R with momentum κ and
Thenanomechanicalsystemconsideredcomprisethree
spin σ. Electrons in the left, island, and right leads are
superconducting electrodes out of which two are fixed
denoted by momentum p, k, and q, respectively. ∆ is
perpendicular to each other. At the intersection where χ
the superconducting pairing potential in lead χ.
fixedleft(L)andright(R)electrodespointathirdmov-
The tunneling response to island vibrations is mod-
able island is suspended by a cantilever. In absence of
eled by distance dependent tunneling matrix elements
electromechanical coupling the island is allowed to vi-
T and T . For small vibrations we use the linear ap-
brate in the direction of the left lead with a restoring pk qk
proximation
force proportional to the distance from equilibrium. The
setup is inherently asymmetric with respect to the mo-
T =T(0)(1−αu), (4)
tion of the island and an illustration of the system is pk pk
shown in Fig. 1.
where α is a positive coupling constant and T(0) is the
tunneling rate to the island at its equilibrium position.
k Thematrixelementfortunnelingbetweentheislandand
I
the right lead is given by
T =T(0)(cid:16)1−α(cid:104)(cid:112)R2+u2−R(cid:105)(cid:17), (5)
qk qk
SC L SC I
where R is the equilibrium distance from the island to
m
I the right lead.
Byassuminglowtemperatures,T ∼0.01−1K,wecan
workwithvibrationalenergiesoftheuncoupledislandin
therangeω ∼10−6−10−3 eV,whichissmallinrelation
0
to the typical electron energy of 1 eV.
V
SC R B. Josephson current modulated by the island
oscillation
We derive the tunneling current, defined by I (t) =
χ
FIG. 1: Schematic picture of the mechanically and electron- −e(cid:104)N˙ (t)(cid:105),whereN isthenumberoperator,atjunction
χ χ
ically coupled system. The superconducting island (SC I) is χ(=L,R) to the island by following Ref. 40 and obtain,
free to move as indicated by the arrows, while the left (SC
L) and right (SC R) superconductors are rigid. mI and kI (cid:90) t
denote the island mass and spring constant respectively. A I (t)=2eRe e−iωχ(t+t(cid:48))(cid:104)[A(t),A(t(cid:48))](cid:105)
χ
bias voltage V can be applied. −∞ (6)
+e−iωχ(t−t(cid:48))(cid:104)(cid:2)A(t),A†(t(cid:48))(cid:3)(cid:105)dt(cid:48),
ThesystemformsadoubleJosephsonjunctionandwe
assume the island displacement u to be small compared whereω =µ −µ definesthevoltagedropbetweenlead
χ χ I
to the distance between the superconducting parts. Our χ and the island (µ and µ are the chemical potentials
χ I
aim is to describe the tunneling current as the electronic of lead χ and the island, respectively). The operators
process couples to the mechanical motion of the island A(t)=(cid:80) T c† (t)c (t),forκ∈L,Randthetime-
κkσ κk κσ kσ
and we begin by addressing the electron Hamiltonian of dependence is defined by
the system,
c (t)=eiKχtc e−iKχt
κσ κσ
H =H +H +H +H , (1) (7)
L R I T c (t)=eiKItc e−iKIt,
kσ kσ
where,
where K =H −µ N and K =H −µ N .
(cid:88) (cid:88)(cid:104) (cid:105) χ χ χ χ I I I I
H = (cid:15) c† c + ∆ c† c† +H.c. In Eq. (6) the junction current is divided into two
L,R,I κ κσ κσ L,R,I κ↑ −κ↓
terms which describe different tunneling mechanisms.
κσ κ
(2) The second term accounts for the single electron tunnel-
ing and will not be addressed further in this text. Our
(cid:88) (cid:88)
HT = Tpkc†pσckσ+ Tqkc†qσckσ+H.c. (3) focus is here devoted to the first term, which describes
pkσ qkσ the Josephson tunneling current.
3
We make use of the Bogoliubov–Valatin transforma- of motion we construct a Hamiltonian that include en-
tion cκσ = uκγκσ − ηνκ∗γκ†σ¯, where η = ±1 differs in ergy terms HJ,L and HJ,R originating from coupling in
sign for spin up or spin down electrons, whereas u and each junction,
κ
ν are the coherence factors satisfying |u |2+|ν |2 = 1
aκnd u∗κνκ = |∆χ|eiφχ/(2Eκ), where φχ isκthe supκercon- Hosc =Ho(0sc)+HJ,L+HJ,R, (13)
ductingphaseinleadχ. Throughthetransformation,we
define the quasi-particle energies where Ho(0sc) = 2pm2I + kI2u2. p is the island momentum,
whilem denotesitsmass,andk isthecantileverspring
(cid:113) I I
E = ((cid:15) −µ )2+|∆ |2. (8) constant.
κ κ χ χ
H andH areconstructedoutoftherequirement,
J,L J,R
We can, thus, write
∂H
(cid:104)[A(t),A(t(cid:48))](cid:105)= (cid:88) |∆4Eχ||E∆I|Tκk(t)Tκk(t(cid:48)) 2e ∂φJ =IJ, (14)
κkσ κ k (9)
(cid:16) (cid:17) which is fulfilled if,
× ei(Eκ+Ek)τ −e−i(Eκ+Ek)τ e−iφχ,
J
where τ = t−t(cid:48) and φχ = φI −φχ. Our assumptions HJ,L =2Le[1−αu]2(1−cos(ωJ,Lt+φL))
of small vibrational energies justifies the approximation (15)
Γ
T (t(cid:48))=T (t)−τT˙ (t),whichleadstotheJosephson − L [1−αu]αpsin(ω t+φ )
pk pk pk 2em J,L R
current I (t) from the left lead to the island I
L
I (t)=J [1−αu]2sin(ω t+φ ) and
L L J,L L
(10)
+ΓL[1−αu]αu˙cos(ωJ,Lt+φL). H =JR (cid:16)1−α(cid:104)(cid:112)R2+u2−R(cid:105)(cid:17)2
J,R 2e
Here, the amplitudes
×(1−cos(ω t+φ ))
J,R R
Jχ(eV)=e(cid:88)|Tκ(0k)|2|∆2Eχ||E∆I| + ΓR (cid:16)1−α(cid:104)(cid:112)R2+u2−R(cid:105)(cid:17) (16)
κk κ k 2emI
(cid:18) 1 1 (cid:19) (cid:18) αpu (cid:19)
× − , × √ sin(ω t+φ ).
eV +Eκ+Ek eV −Eκ−Ek R2+u2 J,R R
(11a)
WithacompleteHamiltonianthefullislandmotion,u,
Γ (eV)=e(cid:88)|T(0)|2|∆χ||∆I| isobtainedbysolvingtheHamiltonequationsofmotion,
χ κk 2E E
κ k
κk
∂H ∂H
(cid:18) 1 1 (cid:19) u˙ = osc, p˙ =− osc. (17)
× − , ∂p ∂u
(eV +E +E )2 (eV −E −E )2
κ k κ k
(11b) In doing so we arrive at the following differential equa-
tion,
definethetunnelingbetweenthefixedelectrodeχ=L,R
andtheislandinabsenceandpresenceofthecouplingto
m u¨+(γ +γ )u˙ +(k +γ )u=F , (18)
I L R,2 I R,1 L
the vibrational mode, respectively.
Thetunnelingfromtherightleadtotheislandisgiven where,
byEq. (10)afterreplacingthetunnelingmatrixelement
Eq. (4) with Eq. (5), i.e. γ =− ΓLα2 sin(ω t+φ ),
(cid:16) (cid:104)(cid:112) (cid:105)(cid:17)2 L e J,L L
IR(t)=JR 1−α R2+u2−R sin(ωJ,Rt+φR) γ =ΓRα(cid:20)(1+αR)(cid:18)√ 1 − u2 (cid:19)−α(cid:21)
(cid:16) (cid:104)(cid:112) (cid:105)(cid:17) R,2 e R2+u2 [R2+u2]3/2
+Γ 1−α R2+u2−R
R
×sin(ω t+φ ),
(cid:18) (cid:19) J,R R
αu˙u (cid:18) (cid:19)
× √R2+u2 cos(ωJ,Rt+φR). γR,1 =JReα √1R+2+αRu2 −α
(12)
(cid:20)(cid:18) (cid:19) (cid:21)
Γ ω
× R J,R +1 cos(ω t+φ )−1 ,
2J J,R R
R
C. Island motion modulated by the electron
coupling and
(cid:20)(cid:18) (cid:19) (cid:21)
In addition to the cantilever spring force acting on the F =−JLα(1+αu) ΓLωJ,L +1 cos(ω t+φ )−1 .
island,electromechanicalcouplingcontributeswithady- L e 2J J,L L
L
namical force. To model this more complicated equation (19)
4
One may note that equation (18) lacks a driving force A
term, F , that will be present if the angle between the
R
right lead and the island motion differs from 90◦.
The central island equation of motion contains both
time and nontrivial position dependence in its coeffi-
cients. Amoretransparentequationisfoundintheweak
coupling and low bias voltage limit. Under such condi-
tions α is small and Γ (cid:28) J , so terms proportional to
χ χ
either Γ , αΓ or α2 are dropped to enlighten the terms
χ χ
of greatest physical relevance. B
We also bear in mind that u/R(cid:28)1 and keep only the
second order Taylor expansions, u
1 1 u2
√ (cid:39) − (20)
R2+u2 R 2R3 0
etc.
The coefficients (19) approximate to,
u
0
γ ≈0,
L
γ ≈0,
R,2
J α (cid:18) u2 (cid:19) FIG.2: Zerobiasvoltagephasediagramsoffourcharacteristic
γ ≈ R (cos(ω t+φ )−1) 1− , (21) solution regions to the equation of motion (23) as it depends
R,1 eR J,R R 2R2 onthecoefficientsAandBwhenF =0. Bisalwaysnegative
J α and only solutions from leftmost quadrants are physical.
F (t)≈− L (cos(ω t+φ )−1),
L e J,L L
and in defining, nonzero. Depending on the coefficients sign in equation
(23) this Josephson effect opens up two distinct regions
1
of solutions, depicted in the phase diagrams of Fig. 2.
A(t)= [k +k (cos(ω t+φ )−1)]
mI I D J,R R (22) Whilefourregionsofsolutionsareobtainablemathemat-
k ically, only two are physical since B is non-positive.
B(t)= D (cos(ω t+φ )−1)
2m R2 J,R R The force term F is non-negative and acts to shift the
I
island motion away from the left lead, consequently low-
where,kD =(JRα)/(eR),actasadynamicalspringcon- ering the DC tunneling rate. The following analytic so-
stant, we end up at the equation of motion, lutionsapplytotheφ =0casewhereF =0. Notethat
L
the current I = 0 at all times under such conditions
L
u¨+A(t)u−B(t)u3 =FL(t)/mI. (23) within the approximations made above.
Also note that Γ and Γ are zero when no bias volt-
L R
This is a Duffing equation modified by time dependent
age is applied which means that the second term in the
coefficients. It is only analytically solvable for zero bias
current expressions vanish.
voltage,throughseriesexpansions56,orbyJacobi’sellip-
In terms of energy the island is confined by the poten-
tic functions57. tial V(u) = Au2 − Bu4 which for A > 0, B < 0 is a
2 4
single well. Under these conditions the equation of mo-
tion (23) has one singular point of center type and all
III. RESULTS AND DISCUSSION
phase trajectories are closed. The equation is satisfied
by the solution,
Thedynamicsofequation(23)directlyeffectstheover-
all Josephson current through its solutions. Due to the u=u cn(Ωt,k), (24)
0
equations nonlinear nature we approach these numeri-
cally in the general case and analytically for zero bias where u is the amplitude, cn(x,y) is the Jacobi elliptic
0
voltage. cosine, Ω = (cid:112)A−Bu2, and k = (cid:112)−B/2·u /Ω. See
0 0
upper left quadrant of Fig. 2.
If A < 0, B < 0 the island motion is governed by a
A. System under zero bias voltage double well potential which give rise to three singular
points of which two are centers, corresponding to the
Evenatzerobiasvoltage,A(t),B(t),F (t)→A,B,F, double well bottoms, while the third is of saddle type
L
the island equation of motion has a rich variety of centered between the two wells. All phase trajectories
solutions if the superconducting phase differences are are closed and as is clear from Fig. 2 solutions exist that
5
circumfere either one of the two singular points of center A>0, B<0 A<0, B<0
type as well as solutions that enclose all three singular F
points. For A < 0,B < 0 solutions to equation (23) can
be written,
0
 (cid:113) (cid:113)
±u0dn(ω1t,k1) for (−|AB|) <u0 < (2−|AB|)
u=
(cid:113)
u cn(ω t,k ) for u > 2|A| ,
0 2 2 0 (−B)
(25)
whereω =(cid:112)−B/2u ,ω =(cid:112)−Bu2−|A|,k =ω /ω , 0
1 0 2 0 1 2 1
k =ω2/ω2 and dn is a Jacobi elliptic function.
2 1 2
The upper solutions above correspond to trajectories
u
enclosing either one of the two singular points of center
type. Asignchangeontheinitialconditionu ,withinthe
0
limit, gives rise to oscillations of equal frequency whose
(cid:112)
origin is separated by a distance 2 |A|/(−B) in real 0
space. A schematic picture of the two solutions are de-
picted in Fig. 3. The different solutions will not change
u
the tunneling current between the island and right lead 0 0
but the DC component of the tunneling current between
theleftleadandtheislandisclearlyaffected. AtφL =0, FIG. 4: Phase portraits of the two possible solution regions
orinotherwordsF =0,thiseffectisabsentsinceequa- as they depend on the size of the force term, F. The centre
L
tion (10) yields zero current. For a small phase shift in the A > 0,B < 0 case shifts toward the right while the
φ (cid:54)=0, on the other hand, both solutions exist together phase trajectories become more elliptical with a major axis
L
with a non zero tunneling current I . The magnitude paralleltothevelocity. IntheA<0,B <0casetheleftmost
L
twocentersvanishwithgrowingF whiletherightmostcenter
of the DC tunneling current difference between solutions
slowly shift towards the right.
confined to the two separate potential wells is less than
(cid:112)
∆I <J 4α |A|/(−B)sinφ .
L,DC L L
Figure 4 illustrates how phase shifts φ distorts solu-
L tion. The two points eventually merge, leaving only the
tiontrajectoriesaswellasmovesthesingularpoints. On
singular point of centre type on the positive side. This
thenegativesideoftheoriginthesingularpointofcenter
point, on the other hand, slowly shifts to more positive
typemovesinpositivedirection,asF growslarger,while
values as φ increases.
thesingularpointofsaddletypemovesinnegativedirec- L
For A > 0, B < 0 the singular point of center type
shiftstowardspositivevaluesasφ increasesatthesame
L
k timeasthephasetrajectoriesdistortstowardanelliptical
I
shape with major axis along the u˙ direction.
SC I
A<0,B<0 m One may also note that while the island oscillations
I
SC L
pass the origin the frequency of the motion induced I
R
2|A| is double that of IL due to the system geometry. As
0< u0< -B 0 soonasoscillationsarerestrictedtoeitherthepositiveor
negativesidethetunnelingcurrentshaveequalfrequency.
V
To analyze the Josephson tunneling under coupling to
SC R
themechanicalmotionoftheisland,atzerobiasvoltage,
k the Fourier transform is taken for a fine mesh of varying
I
A<0,B<0 SC I phase shifts φR and φL. For φL =0 the analytical solu-
SC L mI tions are used while a numerical solver is utilized when
φ (cid:54)=0. As far as values goes the results are to be taken
L
2|A| qualitatively even though realistic input parameters are
- -B < u0<0 0 used.
The current parameters and the coupling constant are
V
SC R set to58 J = J = 0.1mA and α = 0.01˚A−1. As equi-
L R
librium distance between the leads R = 10˚A is taken,
FIG. 3: Schematic image of the island trapped in one of two while an island mass of mI =1fg is used.
potential wells present in the A<0,B <0 case, at zero bias Figure5depictstheFouriertransformofthetunneling
voltage and small phase shifts φL. For the indicated limits current IR as a function of its frequency and the phase
on u0 the island oscillates with its center either to the right shift φR for a given initial value u0 = 0.1nm and me-
(above) or to the left (below).
chanical spring constant k = 0.01N/m. Unless stated
I
6
[μA] the initial value.
6 The inset of Fig. 5 depicts a small area around φr =
7/8
5 π/2 and indicates that the transition between the two
arc structures is not direct. As A goes from positive to
6/8 4
negativetheislandmotionphaseportraitinthelowerleft
3
quadrant of Fig. 2 builds up from the origin. The two
5/8 2
singular points of center type divide from the the single
1.579 1
φπ/R1/2 φR1.571 [μA] caesnatedriasncodnlteianvueessastseapddilnepthoeinltobweehrinadrc. Tofhtishecainnsbeetsaenend
0.08 solutiontrajectoriesnowencloseallthreesingularpoints.
3/8 1.563
0 0.1 0.2 0.3 0.06 WithbiggerφR valuesAbecomesmorenegativeandthe
lyingeightshapeofthephaseportraitgrows. Eventually
1/4
0.04 the separatrix curve reaches the initial value u where
0
the island nears the origin infinitesimally slow and the
1/8
0.02
frequencydropstozero. Shortlyafterthispointispassed
0 0 theoscillationamplitudedropstozeroastheinitialvalue
0 0.5 1 1.5 2 2.5 3
u and the bottom of the double well potential coincide.
ω/ω 0
I 0
The differences in amplitude between solution regions
even out when input values are changed to u = 0.4nm
FIG.5: FouriertransformoftheJosephsontunnelingcurrent 0
andk =0.017N/m,inaccordwiththediscussionabove.
IR as a function of its frequency ωI and the phase shift φR. I
ω =6.6·10−8 eVistheislandeigenfrequencyandφ =0at For these values the transitions of the Fig. 5 inset hap-
0 L
alltimes. Thefigureiscomposedoftwoimagesdividedasthe pens over a larger phase shift φR range as well. All por-
color bars indicate. Both color bars are graded in µA. The trayed in Fig. 6.
lowerhalfcorrespondtosinglewellsolutionswhereA>0and Withnonzerophaseshifts,φ (cid:54)=0,theforcetermF in
L
B <0 while the upper half stems from solutions confined to equation (23) becomes finite positive which changes the
oneofthethedoublewellsintheA<0andB <0case. The
island vibration signature in the tunneling current. Fig-
inset is an enlargement of a small area around φ /π=1/2.
R ure 7 illustrates this with four consecutive images where
φ increases for each image from left to right.
L
First off the current amplitude from solutions in the
otherwise these are the input values used in calculations A > 0,B < 0 region varies with growing φ and sec-
L
throughout the remainder of this paper. As a result the ondly the distinctive features of each solution region be-
energy associated with the eigenfrequency of the uncou- come less pronounced. The changes can be understood
pled island is ω = 6.6·10−8 eV. All figures of Fourier
0
transformed currents are given without DC component.
As φ increases from 0 to π/2 solutions to the island [μA]
R
equationofmotionarerestrictedtotheA>0,B <0re- 1.4
7/8
gion, while phase values between φ =π/2 and π result
R
in solutions within the A < 0, B < 0 region having tra- 1.2
6/8
jectoriesthatenclosethepositivesingularpointofcenter
type. Tunneling currents are close to singly harmonic in 1
5/8
the 0<φ <π/2 region and with increasing phase shift
R
thefrequencydropsuntilitapproacheszeroasAvanishes π 0.8
/R1/2
whenφ =π/2. Thevalueφ =π/2hasnosignificance φ
R R
in it self but stems from the parameter input kI =kD. 3/8 0.6
Above φ = π/2 the island motion has a more com-
R
plicated shape which is reflected in the larger number of 1/4 0.4
harmonicsneededforitsdescription. Thecurrentampli-
tude, given in µA, is noticeably two orders of magnitude 1/8 0.2
largercomparedtotheamplitudeofthebottomhalfarc.
This is expected for solutions to the island motion with 0 0
0 0.4 0.8 1.2 1.6 2
trajectories enclosing one of the singular points of center ω/ω
I 0
type when the initial value lies close to the origin for the
A < 0,B < 0 case in comparison to the A > 0,B < 0 FIG. 6: Fourier transform of I as in Fig. 5 with initial
R
case. The AC current lies superimposed on top of the condition and spring constant changed to u = 0.4nm and
0
DC current and while solution trajectories depicted in k =0.017N/m,whichleadstoω =8.6·10−8 eV. Currents
I 0
the bottom left quadrant of Fig. 2 moves away from the from the additional set of island motion solutions, enclosing
origin all the way around the singular point to the right, all three singular points in the A < 0,B < 0 region, clearly
trajectories in the upper left quadrant never go beyond show up above the discontinues step in the lower arc.
7
φ=0.1 φ=0.4 φ=0.8 φ=1.2
L L L L
3
1/2 2.5
2
3/8
π
φ/R 1.5
1/4
1
1/8
0.5
0 0
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3
ω/ω ω/ω ω/ω ω/ω
I 0 I 0 I 0 I 0
FIG. 7: The Fourier transform of the Josephson tunneling current I as a function of its frequency ω and phase shift φ
R I R
depictedwithincreasingphaseshiftsφ fromlefttoright. Theamplitudescaleislogarithmicontheformlog (1+I ∗108).
L 10 R
with figure 4 in mind. In the first two images of Fig. 7 order of 10−3. All figures presented below are obtained
the stable center in the A > 0,B < 0 region moves in with φ =φ =0.
l R
the positive direction toward the initial value u and the Atlowbiasvoltage,ω /ω <0.24(ω <1.6·10−8 eV),
0 J 0 J
amplitude diminishes. In the following two images the the island motion follows a regular pattern where it is
stablecenterhaspassedu andamplitudegetsbigger. A quasi periodic with a high frequency, low amplitude, os-
0
nonzeroφ complicatesthepicturefurthersinceitcauses cillation superimposed on a higher amplitude vibration
L
φ to also shift the stable center towards more positive offrequencyequaltoω . Thiscausessmallrippleson,as
R J
values. This is evident in the φ = 0.4 image of Fig. 7 well as distorts, the tunneling current dominated by the
L
where the stable center passes the initial value u just Josephson factor. The typical case situation is shown in
0
below φ =π/4. the bottom four images of Fig. 9.
R
The tunneling current frequency never goes to zero in Thecurrentinducedcoefficientsintheislandequation
eitherimageofFig. 7eventhoughthedoublewellpoten- of motion are zero at
tial governs the island motion shortly above φ = π/2,
R
and upwards, in the φL = 0.1 case. The positive force t=0+n·2π/ωJ n=0,1,2,... (26)
termF shiftstheφ valueatwhichseparationofthesin-
R
gular points occur. When this happens the center point where the island vibration is harmonic. The cantilever
of the single well potential has already passed the ini- spring dictates the motion for a few periods until the
tial value u0 and the separatrix curve is never crossed. nonlinearanddrivingforcecontributionsrapidlygrowin
In the remaining three images with higher φL values the sync with the Josephson AC. In such time intervals of
doublewellpotentialneverformandnoseparatrixcurve strong nonlinearity the phase portrait implies a stable
appears. center markedly shifted away from the left SC lead.
AllimagesinFig. 7seemtoindicatethatthetunneling Between 0.24 < ω /ω < 6.1 (1.6·10−8 eV < ω <
J 0 J
currentfrequenciesperfectlymatchatthetransitionfrom 4.0·10−7 eV) the harmonic driving force and island mo-
region A > 0,B < 0 to A < 0,B < 0 in contrast to the tion frequencies are comparable, but instead of simply
φL = 0 case. The discontinuity in Fig. 5 is very small resonating, the island vibrates in a chaotic fashion. This
on the other hand and numerical noise makes it hard to behavior comes as no surprise for such a strongly non-
distinguish such fine features. linear driven system. Chaotic solutions are a well stud-
ied property of the ordinary driven Duffing equation but
here we only go as far as to compare Poincar´e maps
B. System under finite bias voltage taken at ω /ω = 0.48 and ω /ω = 24 to conclude
J 0 J 0
that the low voltage map is compliant with a charac-
A bias voltage ω = −ω > 0 over the junction teristic chaotic map while the higher voltage map has a
J,L J,R
setup significantly changes the character of the tunnel- clearly quasi periodic structure. The middle four images
ing current. Most noticeably the island motion is no of Fig. 9 illustrates the island motion and Josephson
longerstrictlyperiodic,butratherquasiperiodicoreven tunneling current in the chaotic region at ω /ω = 0.48
J 0
chaotic. The current amplitude also becomes some three (ω =3.2·10−8 eV).
J
orders of magnitude greater than in the zero bias case. Above ω /ω > 6.1 (ω > 4.0 · 10−7 eV) regularity
J 0 J
ThisiseasilyunderstoodastheusualJosephsonfactorin in the island motion reappears, as the top four images
expressions (10) and (12) varies between −1 and 1 while of Fig. 9, taken at ω /ω = 24 (ω = 1.6·10−6 eV),
J 0 J
the factor associated with island motion changes in the indicate. InthisregiontheJosephsonfrequencyishigher
8
0.15 0.2 pling adds a linear, cubic and force term to the equation
of motion - all time dependent at finite bias voltage. In
0.15
0.1
the zero bias voltage limit the differential equation re-
m/s] 0.05 0.1 duces to the Duffing equation if the super conducting
y [ 0.05 phases differ.
cit
o
el 0 0 ThehomogenousDuffingequationhastwosetsofphys-
V
−0.05 ical solution regions in our setup. One with a single well
−0.05
potentialifthelineartermispositiveandonewithadou-
−0.1
−0.1 blewellpotentialifthelineartermisnegative. Whichof
−0.15
the double wells the oscillator is vibrating in is indistin-
−0.15 −0.2 guishablebylookingatthealternatingtunnelingcurrent.
0 0.5 1 1.5 2 −3 −2 −1 0 1 2
Position [nm] Position [nm] TheDCcontributionswillhowevervarybetweenthetwo.
Superconductingphaseshiftsassociatedwiththejunc-
FIG. 8: Poincar´e maps of the island motion taken at the
tion in line with the direction of oscillations govern the
t = 0 + n · 2π/ω n = 0,1,2,... intersection and bias
J force term and can be manipulated to shift the potential
voltages ω /ω =24 (left) and ω /ω =0.48 (right).
J 0 J 0 well bottoms away from the rigid SC lead. The phase
shiftassociatedwiththesecondjunctioncontrolsthelin-
ear and cubic term and a sweep through 0 to π reveal
than that of the major oscillatory island motion and the
thatsinglewellsolutionsareobtainedatlowphaseshifts
tunneling current is subject to a slow modulation.
while larger values produce a double well potential if the
In contrast to the low bias voltage case, where the
mechanical spring constant is chosen properly.
momentary phase portraits vary adiabatically with re-
spect to the island vibrations, high bias voltage causes At nonzero bias voltage we find three domains of so-
rapid changes in the time dependent equation of motion lutions to the island equation of motion with their own
coefficients. The comparatively slow island is subject characteristics. For very low bias voltage, such that the
to a quickly deforming single well whose bottom shifts Josephsonfrequencyislowcomparedtothevibrationfre-
from the origin to a finite positive value with period quency,theislandmotionisquasiperiodicwhichdistorts
T =2π/ω . With increasing bias voltages the fine wave the tunneling current by superimposing small ripples on
J
pattern in the solution trajectories seen in the top left the current. In the intermediate voltage span, where the
image of Fig. 9 diminishes. Josephson and island frequencies are of the same order,
the system turns chaotic and the tunneling current gets
irregular distortions. For larger voltages, such that the
IV. SUMMARY Josephson frequency is much larger than the vibration
frequency,theislandmotionisagainquasi-periodic. The
currentisroughlyharmonic,however,withaslowmodu-
We have studied electron tunneling coupled to a me-
lation superimposed arising from the mechanical motion
chanical oscillator in a double Josephson junction. The
of the island.
geometry of the setup is asymmetric with respect to the
mechanical motion which introduces a nonlinear term in The present study is based on theoretical assumption
the oscillator equation of motion. that should be within the realms of the state-of-the-art
The Josephson tunneling current over one junction is experimental capabilities. Detection of chaotic dynam-
modeled as linearly dependent on the oscillator displace- ics in nanoscale systems would be interesting from many
mentdirectlywhilethegapwidthofthesecondjunction perspectivesanditiswithgreatconfidenceweanticipate
changesasahypothenusetothedisplacement. Animme- experimental verification of our proposal.
diate consequence is that the mechanically induced cur-
rent frequency over the first junction is half that of the
secondjunctioniftheoscillatorpassesitsequilibriumpo-
V. ACKNOWLEDGEMENTS
sitionandthatthefrequenciesareequalifthevibrations
are restricted to either the positive or negative side.
In the uncoupled system the oscillator is taken to vi- We like to acknowledge the support of the Swedish
brate harmonically with a linear restoring force. Cou- Research Council.
∗ Electronic address: [email protected] 3 B.Lassagne,Y.Tarakanov,J.Kinaret,D.Garcia-Sanchez,
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9
Island motion Island motion I FFT of I
2 R 1 R
0.15 0.1
0.1 1.6 0.8
s] m] A] e
Velocity [m/-00..00550 Position [n 01..82 Current [m 0 Magnitud 00..46
-0.1 0.4 0.2
-0.15 −0.1
0 0
-2 -1 0 1 2 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 10 20 30 40 50
0.1
2 0.8
0.2
Velocity [m/s] -00..110 Position [nm] 01 Current [mA] 0 Magnitude 00..46
-1 0.2
-0.2
−0.1
-2 0
-2 -1 0 1 2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4
1.6
0.015 0.08 0.8
m/s]0.00.0051 nm] 1.2 mA] 0.04 de 0.6
Velocity [-0.0050 Position [ 00..48 Current [−0.040 Magnitu 0.4
-0.01 0.2
-0.015 0 −0.08
0
-1.5 -1 -0.5 0 0.5 1 1.5 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4
Postition [nm] Time [μs] Time [μs] ωI/ω0
FIG. 9: Images depicting the phase portrait of the island motion, the island position as a function of time, the Josephson
tunneling current as a function of time, and the fast Fourier transform of the current from left to right at bias voltages
ω /ω =24 (top), ω /ω =0.48 (middle), and ω /ω =0.048 (bottom).
J 0 J 0 J 0
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