Table Of ContentSemifluxons in Superconductivity and Cold Atomic Gases
8
0
0
2 R. Walser1, E. Goldobin2, O.Crasser1,D. Koelle2, R. Kleiner2,
n W.P. Schleich1
a
J 1Institutfu¨rQuantenphysik,Universita¨tUlm,Albert-EinsteinAllee11,D-89069Ulm,
0 Germany
1 2PhysikalischesInstitutII,Universita¨tTu¨bingen,AufderMorgenstelle14,D-72076
Tu¨bingen,Germany
]
l
l E-mail:[email protected]
a
h
-
Abstract. Josephsonjunctionsandjunctionarraysarewellstudieddevicesinsuperconduc-
s
e tivity. Withexternalmagneticfieldsonecanmodulatethephaseinalongjunctionandcreate
m
traveling,solitonicwavesofmagneticflux,calledfluxons. Today,itisalsopossibletodevice
.
t twodifferenttypesofjunctions: dependingonthesignofthecriticalcurrentdensityjc ≷ 0,
a
m theyarecalled0-orπ-junction.Inturn,a0-πjunctionisformedbyjoiningtwoofsuchjunc-
tions. As a result, one obtainsa pinnedJosephson vortex of fractionalmagneticflux, at the
-
d 0-πboundary.Here,weanalyzethisarrangementofsuperconductingjunctionsinthecontext
n
ofanatomicbosonicquantumgas,wheretwo-stateatomsinadoublewelltraparecoupledin
o
ananalogousfashion.There,anall-optical0-πJosephsonjunctioniscreatedbythephaseofa
c
[ complexvaluedRabi-frequencyandweaderiveadiscretefour-modemodelforthissituation,
whichqualitativelyresemblesasemifluxon.
1
v
7
6
5 PACSnumbers: 03.75.Fi,74.50.+r,75.45.+j,85.25.Cp,
1 2 February 2008
.
1
Keywords: Long Josephson junction, sine-Gordon equation, fractional Josephson vortex,
0
8 quantumtunneling,coldatoms,Bose-Einsteincondensates
0
:
v
i
X
r
a
SemifluxonsinSuperconductivityandCold AtomicGases 2
1. Introduction
During the past two decades, the field of cold atomic gases has come a long way starting
from almost lossless trapping and cooling techniques [1] to reaching quantum degeneracy
of Bosons and Fermions [2]. Many phenomena that are the hallmarks of condensed matter
physics,whetherinsuperfluidorsuperconductingmaterials[3],arerevisitedwithinthisnovel
context. Due to the remarkable ease with which it is possible to isolate the key mechanisms
from rogue processes, one can clearly identify phase transitions, for example, Bose-Einstein
condensation, the Mott phase transition or the BEC-BCS crossover. Today however,
degenerategasesarestillatadisadvantageifweconsiderrobustness,portabilityortheability
for a mass production compared to solid-statedevices, which is the great achievement of the
semiconductor industry. Strong attempts to miniaturize cold gas experiments [4, 5] and to
makethemportable[6, 7, 8, 9]are currentlyunderway inmanylaboratories.
Due to the great importance and practical relevance of the Josephson effect in
superconducting systems [10, 11, 12], it has also received immediate attention after the first
realization of Bose-Einstein condensates [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25].
In particular, thecombinationof optical lattices with ultracold gases [26, 27] has boostedthe
possibilities to investigate junction arrays experimentally. Remarkably, even the absence of
phase-coherence between neighboring sites can lead to interference as demonstrated in [28].
The possibility to study atomic Josephson vortices in the mean field description was raised
first inconnectionwiththesine-Gordonequation[29, 30].
Inthepresentarticlewewillreportonsuchatransferofconceptsfromasuperconducting
device [31], i.e., in various realizations of Josephson junction arrays and their unusual
state properties of traveling (fluxons) and pinned (semifluxons) magnetic flux quanta to an
analogous set up for neutral bosonic atoms in a trap. In particular, we will investigatean all-
optical0-π Josephsonjunctionthatcan becreated withajumpingphaseofan opticallaser.
This article organized as follows: in Sec. 2, we will give a brief review of the current
statusofthesuperconductorphysicsofJosephsonjunctions. Inparticular,wewillrefertothe
most relevant publications in this thriving field of fluxon and semifluxon physics; in Sec. 3,
we will discuss a similar setup, which allows to find a pinned semifluxon in an atomic 0-π
Josephsonjunctionandwecomparetheresults. Finally,wewilldiscussfurtheropenquestions
ina Conclusion.
2. Fluxonsand Semifluxons inSuperconductivity
TheJosephsoneffectisawellestablishedphenomenoninthesolidstatephysics. AJosephson
junction (JJ) consists of two weakly coupled superconducting condensates. JJs are usually
fabricated artificially using low- or high-T superconducting electrodes separated by a thin
c
insulating (tunnel), normal metal, or some other (exotic) barrier. JJs can also be present
intrinsically in a anisotropic layered high-T superconductors such as Bi Sr Ca Cu O [32,
c 2 2 1 2 8
33].
ThedcJosephsoneffect(flowofcurrentthroughaJosephsonjunctionwithoutproducing
SemifluxonsinSuperconductivityandCold AtomicGases 3
voltagedrop,i.e. withoutdissipation)isexpressedusingthefirstJosephsonrelation,whichin
thesimplestcasehas theform
I = I sin(φ), (1)
s c
where I is the supercurrent flowing through the junction, I is the critical current, i.e. the
s c
maximum supercurrent which can pass through the JJ, and φ = θ θ is the difference
2 1
−
betweenthephasesofthequantummechanicalmacroscopicwavefunctionsψ = √n eiθ1,2
1,2 s
ofthesuperconductingcondensatesin theelectrodes.
Recent advances in physics and technology allow to fabricate and study the so-called
π-Josephson junctions — junctions which formally have negative critical current I < 0.
c
This can be achieved by using a ferromagnetic barrier, i.e. in Superconductor-Ferromagnet-
Superconductor (SFS) [34, 35, 36, 37, 38] or Superconductor-Insulator-Ferromagnet-
Superconductor (SIFS) [39, 40] structures. One can also achieve the same effect using a
barrier which effectively flips the spin of a tunneling electron, e.g. when the barrier is made
of a ferromagnetic insulator [41], of a carbon nanotube [42] or of a quantum dot created by
gatinga semiconductingnanowire[43].
The change in the sign of a critical current has far going consequences. For example,
analyze the Josephson energy (potential energy related to the supercurrent flow). In a
conventionalJJ withI > 0
c
U(φ) = E (1 cosφ) (2)
J
−
andhasaminimumatφ = 0+2πn(thegroundstate),whereE = Φ I /2π istheJosephson
J 0 c
energy. IfI < 0,wedefine E = Φ I /2π > 0 and
c J 0 c
| |
U(φ) = E (1+cosφ). (3)
J
Obviously,theminimumofenergyisreachedforφ = π+2πn. Thus,inthegroundstate(the
JJ is not connected to a current source, no current flows through it) the phase drop across a
conventionalJJwithI > 0isφ = 0+2πn,whileforajunctionwithI < 0itisφ = π+2πn.
c c
Therefore, onespeaks about“0-JJs”and“π-JJs”.
Further, connecting the two superconducting electrodes of a π-JJ by a not very small
inductor L (superconducting wire), the supercurrent π/L will start circulating in the loop.
∝
Note, that this supercurrent is spontaneous, i.e., it appears by itself, and has a direction
randomly chosen between clockwise and counterclockwise [44]. The magnetic flux created
by this supercurrent inside the loop is equal to Φ /2, where Φ = h/2e 2.07 10−15 Wb
0 0
≈ ·
is the magnetic flux quantum. Thus, the π-JJ works as a phase battery. This phase battery
willworkasdescribed,supplyingasupercurrentthroughtheloopwithinductor,providedthe
inductance L Φ /I . If the inductance L is not that large, the battery will be over loaded,
0 c
≫
providingasmallerphasedropandsupportingsmallercurrent. Forverysmallinductancethe
battery willstopworkingcompletely.
Similar effects can be observed in a π dc superconducting quantum interference device
(SQUID: one 0-JJ, one π-JJ and an inductorL connected in series and closed in a loop)or in
0-π Josephsonjunction. Letus focuson alattercase.
SemifluxonsinSuperconductivityandCold AtomicGases 4
0 part part
1.5
(a)
1.0
x()/ 0.5 semifluxon
e
as 0.0
h
p antisemifluxon
-0.5
-1.0
-1.5
(b) 2
x()x 1.0 semifluxon
d
el
etic fi 0.0
n
g
a
m
-1.0
antisemifluxon
1.0 (c)
antisemifluxon
n
si 0.5
nt
e
curr 0.0
er
p
u
s-0.5
semifluxon
-1.0
-5 -4 -3 -2 -1 0 1 2 3 4 5
coordinate x
Figure 1. Sketch of a 0-π JJ and profiles of (a) the phase φ(x), (b) the magnetic field
dφ(x)/dxand(c)thesupercurrentdensityj (x) sinφ(x)correspondingtoasemifluxon
s
∝ ∝
(black)andantisemifluxon(gray).
Consider a long (along x) Josephson junction (LJJ) one half of which at x < 0 has
the properties of a 0-JJ (critical current density j > 0) and another half at x > 0 has the
c
properties of a π-JJ (critical current density j < 0). Long means that the length is much
c
larger than the so called Josephson length λ , which characterizes the size of a Josephson
J
vortex; typically λ 10 20µm. What will be the ground state of such a 0-π LJJ?
J
∼ −
It turns out that if the junction is long enough (formally infinitely long), then far away
from the 0-π boundary situated at x = 0, i.e., at x the phase φ will have the
→ ±∞
values 0 or π (we omit 2πn here), while in the vicinity of 0-π boundary the phase φ(x)
±
smoothly changes from φ( ) = 0 to φ(+ ) = π, see Fig. 1a. The exact profile can
−∞ ∞ ±
bederived analytically[45, 46, 31]. Sincethe phasebends, thelocal magneticmagneticfield
H dφ/dx will be localized in the vicinity of the 0-π boundary and carry the total flux
∝
equalto Φ /2,seeFig.1b. Thesigndependsonwhetherthephasebendsfromφ( ) = 0
0
± −∞
to φ(+ ) = +π or to φ(+ ) = π. Thus, such an object is called a semifluxon or an
∞ ∞ −
SemifluxonsinSuperconductivityandCold AtomicGases 5
antisemifluxon. IfoneanalyzestheJosephsonsupercurrentdensityflowingthoughthebarrier
j (x) = j (x)sinφ(x), one can see in Fig. 1c that the supercurrent has different directions
s c
on different sides from the 0-π boundary. Since we do not apply any external current, the
flow of current should close in the top and bottom electrodes, i.e. the supercurrent circulates
(counter)clockwise in the case of (anti)semifluxon. Thus, a semifluxon is a Josephson vortex
of supercurrent. It is pinned at the 0-π boundary and has two degenerate ground states with
thelocalizedmagneticfield carryingtheflux Φ /2.
0
±
Semifluxons in various types of JJ has been actively investigated during the last years.
In fact, thefirst experimentsbecame possiblebecause of deeper understandingthe symmetry
of the superconducting order parameter in cuprate superconductors. This order parameter
with the so-called d-wave symmetry is realized in anisotropic superconductors, such as
YBa Cu O or Nd Ce CuO . It allowed to fabricate 0-π grain boundary LJJs [47, 48]
2 3 7 2−x x 4
and, later, more controllable d-wave/s-wave ramp zigzag JJs [49, 50] and directly see and
manipulatesemifluxonsusingaSQUID microscope[51, 52].
Semifluxons are very interesting non-linear objects: they can form a variety of ground
states [53, 54, 55, 56], may flip [51, 47] emitting a fluxon [57, 55, 58], or be rearranged [59]
by a bias current. Huge arrays of semifluxons were realized [51] and predicted to behave as
tunablephotoniccrystals[60]. Semifluxonsarealsopromisingcandidatesforstoragedevices
in the classical or quantum domain and can be used to build qubits [61] as they behave like
macroscopicspin1/2 particles.
Now, an interesting question arises: Can one realize π or even 0-π JJs in an atomic
BEC? In the latter case, the degenerate ground state corresponding to a semifluxon should
haveanon-trivialspatialphaseprofileandsemifluxonphysicscanalsobestudiedusingBEC
implementation.
3. Semifluxons inBose-Einstein Condensates
Here, we will address this question and examine a configuration were the two-state atoms
are trapped in a quasi-one dimensional cigar-shaped trap with an additional superimposed
double well potential in the longitudinal direction. The spatial localization of the two-state
atoms inside the double well potential leads to two internal atomic Josephson junctions that
are drivenviaan optical, complexvalued “0-π”laserfield and they aremotionallyconnected
viatunneling.
First,wewilldescribedetailsofthemodelandintroducetheHamiltonianofthesystem.
Then, we will examine the classical limit of the field theory and study the ground state of
the Gross-Pitaevskii equation. Finally, we will exploit the fact that the spatial wavefunctions
arelocalizedinsideadeepdoublewellandstudyasimplefour-modequantummodelderived
from aWannierbasisstaterepresentation.
SemifluxonsinSuperconductivityandCold AtomicGases 6
3.1. 0-π-junctionina Bose-Einsteincondensate
To model the 0-π-Junction [29, 30] in a BEC, we are guided by the the condensed
matter physics setup depicted in Fig. 2. As in the single atom case, we replace the two
superconductors by an atomic two-level BEC in a cigar shaped trap. The two states of the
atom, i.e., the excited state e and ground state g couple via a position dependent Rabi
| i | i
frequency Ω (x), whichexhibitsaphasejumpat theoriginofthex-axes
0
Ω , x < 0,
Ω (x) = 0 (4)
0
Ω = Ω , x 0.
( 1 0
− ≥
V(x)
a) b)
ψ
e
0 π
Ω0(x) x
2δ
ψ
x g
Figure2. Analogyofa0-π Josephsonjunctionina superconductor(a)with anatomictwo-
level BEC (b) in a double well trap V(x) with a position dependentRabi frequencyΩ0(x).
Theexcitedatomiclevel e isseparatedfromthegroundlevel g bythedetuning2δ.
| i | i
In this quasi one-dimensional scenario, we will represent the two-state atoms by a
spinorialbosonicquantumfield Ψˆ, whichsatisfies thecommutatorrelation
[Ψˆ(x),Ψˆ(y)†] = 1δ(x y). (5)
−
This field can be decomposed in any complete single particle basis σ,l , which resolves the
| i
spatialextentofthefield andtheinternalstructureoftheatoms,i.e.,
∞
Ψˆ = σ,l ˆa (6)
σ,l
| i
σ={e,g} l=0
X X
and we denote the corresponding discrete bosonic field amplitudes by aˆ . Here, σ
σ,l
characterizes the internal states by (e,g) and the external motion in thedouble well potential
by a quantum label l. The dynamical evolution of the atomic field is governed by following
Hamiltonian
∞ δ Ω (x)
Hˆ = dxΨˆ†(x) ∂2 +V(x)+ 0 Ψˆ(x) (7)
− x Ω∗(x) δ
Z−∞ " 0 − !#
+gΨˆ†(x)Ψˆ†(x)Ψˆ (x)Ψˆ (x)+gΨˆ†(x)Ψˆ†(x)Ψˆ (x)Ψˆ (x).
e e e e g g g g
In here, weusedimensionlessunits,in particularwehaveset ~ = 1and themassoftheatom
m = 1. The energy consists of the single particle energy in a trap V(x), which is identical
forbothspecies,theelectricdipoleinteractionofthetwo-stateatom[62],aswellasageneric
SemifluxonsinSuperconductivityandCold AtomicGases 7
collisionenergyproportionaltothecouplingconstantg = g = g . Tosimplifytheanalysis,
ee gg
wehavedeliberatelyset thecross-componentscatteringlengthg = 0. No unaccountedloss
eg
channelsare present. Therefore, wehavenumberconservation
Hˆ,Nˆ = 0, (8)
as asymmetry.hIfwedienoteagenericstateofthemany-particlesystemin Fock-spaceby
∞
ψ(t) = ψn(t) n = (n0,n1,...) , (9)
| i | i
n=0
X
then one can obtain the dynamics of the system most generally from the Lagrangian
formulation
[ψn(t),ψ˙n(t)] = ψ(t) i∂t Hˆ ψ(t) . (10)
L h | − | i
Inthisfieldtheory,thecanonicalmomentumisgivenbyπn = δδψL˙n = iψn∗. FromtheHamilton
equationπ˙n = δL ,onerecovers theconventionalSchro¨dingerequationin Fockspace
−δψn
i∂ ψ = Hˆ ψ . (11)
t
| i | i
The Lagrangian approach is obviously a central concept in the path integral formulation of
quantum mechanics [63]. However, it is also of great utility in the approximate description
of the dynamics if we connect it with concepts of classical mechanics as we will see in the
following.
3.2. Spatiallyextended classicalmodel: theGross-Pitaevskiiequation
The classical limit of the field equations [2] can be recovered quickly by approximating the
stateofthesystembyacoherent state
Ψˆ (x) ψ = ψ (x) ψ . (12)
σ σ
| i | i
Within this approximation, we obtain from the Lagrangian of Eq. (10) the two component
Gross-Pitaevskiiequationψ = (ψ (x,t),ψ (x,t))⊤
e g
δ +2g ψ 2 Ω (x)
i∂ ψ = ∂2 +V(x)+ | e| 0 ψ. (13)
t − x Ω∗(x) δ +2g ψ 2
" 0 − | g| !#
Foramacroscopicallyoccupiedfieldthisequationmodelsthespatialevolutionofthecoupled
Josephsonjunctionsverywell [17, 18, 19, 20, 21, 22,23, 24, 30].
3.3. Discretequantummodel: two coupledJosephsonjunctions
To gain more insight into the quantum properties of the groundstate of the system [61], one
can decompose the field into its principal components and disregard small corrections. A
double well potential can be considered as the limiting case of a periodic lattice. While even
and odd parity modes relate to delocalized Bloch-states in a periodic system, left and right
localized modes i.e., φ and right φ , resemblethe Wannierbasis. These localized modesare
0 1
SemifluxonsinSuperconductivityandCold AtomicGases 8
depicted in Fig. 3. With respect to these basis states, we can approximate the field with four
modes
Ψˆ (x) = eˆ φ (x)+eˆ φ (x)+δΨˆ , (14)
e 0 0 1 1 e
Ψˆ (x) = gˆ φ (x)+gˆ φ (x)+δΨˆ . (15)
g 0 0 1 1 g
The four bosonic amplitudes gˆ ,eˆ ,gˆ ,eˆ satisfy the usual commutation relations and we
{ 0 0 1 1}
willdisregardthesmallcorrections oforder δΨˆ
σ
Figure3. Fromevenandoddparitymodesofadouble-wellpotential,onecanconstructthe
leftφ0 andrightφ1localizedWannierstatesofthesystem.
If this approximate field is substituted in the Hamiltonian, Eq. (7), we can exploit the
orthogonalityofthewavefunctionsandobtainthetwo-bodymatrixelementsφ
ijkl
∞ ∞
δ = dxφ (x)φ (x), φ = dxφ (x)φ (x)φ (x)φ (x). (16)
ij i j ijkl i j k l
Z−∞ Z−∞
Out of the sixteen combinations for φ , we only retain the physically most relevant
ijkl
contributionsanddisregardothersdeliberately. ThisleadstothefollowingmodelHamiltonian
fortwocoupled Josephsonjunctions
Hˆ = Λ(eˆ†eˆ +eˆ†eˆ +gˆ†gˆ +gˆ†gˆ ) (17)
0 1 1 0 0 1 1 0
+ δ(eˆ†eˆ gˆ†gˆ )+δ(eˆ†eˆ gˆ†gˆ )
0 0 − 0 0 1 1 − 1 1
+ Ω (eˆ†gˆ +gˆ†eˆ )+Ω (eˆ†gˆ +gˆ†eˆ ) (18)
0 0 0 0 0 1 1 1 1 1
+ g(eˆ†eˆ†eˆ eˆ +gˆ†gˆ†gˆ gˆ +eˆ†eˆ†eˆ eˆ +gˆ†gˆ†gˆ gˆ )
0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1
where all coupling constants are implicitly rescaled by the corresponding single particle or
two-bodymatrixelementsandthenewparameterΛmeasuresthespatialhoppingratebetween
thesites.
SemifluxonsinSuperconductivityandCold AtomicGases 9
θ
Λ
φ1
φ0 Ω0 Ω1
x
Λ
0 1
Figure 4. Coupling pattern of two coupled Josephson junctions JJ0 and JJ1 with, Ω1 =
Ω0. The classical phase θ refers to a particle current between the spatial sites, while the
−
phasesφ0andφ1arecurrentswithinthetwo-stateatoms.
3.4. Fock-spacerepresentationofthefourmodemodel
In principle, it is possible to solve the four-mode Schro¨dinger equation in Fock space by
projectingiton theN-particle sector
∞
ψ,N = δ(Nˆ N) ψn n = (ne,ng,ne,ng) . (19)
| i − | 0 0 1 1 i
n=0
X
The number constraint on the state remains valid throughout the time-evolution as number
conservation is encoded into our Hamiltonian from the beginning in Eq. (8). This reduces
thediscreted = 4dimensionaleigenvalueproblemtoaneffectivethree-dimensionalproblem
with nontrivialboundaries. We haveillustrated the finitesupport of the amplitudefield ψn in
Fig. 5. It is a (d 1)-dimensional simplex embedded into a d-dimensional Fock space. The
−
fullanalysisofthisproblemisaninterestingprobleminitsownrightandwillbepresentedin
aforthcomingpublication.
3.5. The classicallimitofthefourmodemodel
It is not necessary to solve the four-mode problem in Fock-space to understand the principal
features of the equilibrium configuration. Thus, we will again resort to the classical
approximation and use the number-symmetry broken coherent state approximation for the
quantumstate
ψ = α = (e ,g ,e ,g )⊤ , eˆ ψ = e ψ , gˆ ψ = g ψ . (20)
0 0 1 1 i i i i
| i | i | i | i | i | i
Thedynamicsissimplyobtainedfrom theLagrangian
i d
(α,α˙) = α i∂ Hˆ α = iα∗α˙ (α,α∗) (α,α∗), (21)
t
L h | − | i −H − 2dtN
if we introduce the classical Hamilton functions (α,α∗) = α Hˆ α and number
H h | | i
expectationvalues (α,α∗) = α Nˆ α , as
N h | | i
(α,α∗) = Λ(e∗e +e∗e +g∗g +g∗g ) (22)
H 0 1 1 0 0 1 1 0
SemifluxonsinSuperconductivityandCold AtomicGases 10
30
25
20
n215
10
5
0
0
10
20 30
25
20
15
30 5 10
0
n n
0 1
Figure 5. A three dimensional simplex represents the four-mode Fock space for N = 32
particles. Theaxesare labeledin a genericlexicographicalorder(n0,n1,n2)andimplicitly
n4 =N n0 n1 n2.
− − −
+ δ e 2 g 2 +δ e 2 g 2
0 0 0 1 1 1
| | −| | | | −| |
+ Ω (e∗g +g∗e )+Ω (e∗g +g∗e )
0(cid:0) 0 0 0 0(cid:1) 1(cid:0) 1 1 1 1(cid:1)
+ g( e 4 + g 4 + e 4 + g 4),
0 0 1 1
| | | | | | | |
(α,α∗) = e 2 + g 2 + e 2 + g 2. (23)
0 0 1 1
N | | | | | | | |
Ifthedynamicalcoordinateis α, then wefind thecanonical momentumas
∂
π = L = iα∗, (24)
k ∂α˙ k
k
Consequently, variables and momenta obey the Poisson bracket α ,π = δ . By
j k jk
{ }
construction,numberconservationissatisfied dynamicallyas
d
= , = 0 (25)
dtN {H N}
and theHamiltonequationsofmotionforthecoordinates read
d
α = ,α = i α (26)
dt {H } − K
δ +2g α 2 Ω Λ 0
0 0 0
| |
Ω δ +2g α 2 0 Λ
= 0 − 0 | 1| .
K Λ 0 δ +2g α 2 Ω
1 2 1
| |
0 Λ Ω δ +2g α 2
1 1 3
− | |
In classical mechanics, we can deliberately choose new coordinates and momenta to
accounts for symmetries of the Hamiltonian. If a new variable of a canonical transformation
matches a conserved quantity, it follows that the conjugate variable becomes cyclic. In this
