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Semifluxons 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

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