Spin Josephson vortices in two tunnel coupled spinor Bose gases
SSpin Josephson vortices in two tunnel coupled spinor Bose gases
T.W.A. Montgomery, W. Li, and T.M. Fromhold
School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, UK (Dated: November 1, 2018)We study topological excitations in spin-1 Bose-Einstein condensates trapped in an elongateddouble- well optical potential. This system hosts a new topological defect, the spin Josephson vortex(SJV), which forms due to the competition between the inter-well atomic tunneling and short-rangeferromagnetic two-body interaction. We identify the spin structure and formation dynamics of theSJV and determine the phase diagram of the system. By exploiting the intrinsic stability of theSJV, we propose a dynamical method to create SJVs under realistic experimental conditions.
PACS numbers: 67.85.Hj 67.85.Fg 67.85.De
Ultracold spinor atomic gases that exhibit both super-fluidity and magnetic order display an abundance of richstatic and dynamical properties. This has attracted con-siderable theoretical and experimental study, with par-ticular focus on the topological excitations of trappedspinor gases [1–6]. Topological phases of single trappedspinor gases, such as spin vortices [6–9], knots [10] andskyrmions [11–13], depend critically on the mean-fieldorder-parameter manifold. A remarkable feature of thesedynamical excitations is that their size is typically largerthan the underlying spin healing length (SHL) [1, 2].When spinor atoms are confined in optical lattice po-tentials, atomic tunneling between adjacent lattice sitescompetes with the spin dependent inter-atomic interac-tion. This competition provides a mechanism for theemergence of topological phases, which have been iden-tified and investigated in several studies [14–21]. It canalso strongly influence the behavior of a simpler systemcomprising atoms confined in double-well (DW) poten-tials [22, 23], which are analogous to Josephson junctionsin solid state devices. Analysis of such systems often usesthe lowest energy mode approximation [24], which allowsthe intra-well spatial motion to be mapped as a functionof time. Even in this limit, spin-dependent populationoscillations between the two potential wells have beenidentified [25–29]. But beyond this limit, it remains un-clear whether quantum fluctuations can trigger the for-mation of extended topological excitations when the sizeof the individual spinor gases in each well exceeds thespin healing length (SHL).In this work, we show that a dynamically stable topo-logical excitation, the so-called spin Josephson vortex(SJV), forms in two weakly coupled spin-1 ferromagneticBose-Einstein condensates (BECs) trapped in an elon-gated DW potential [Fig. 1(a)]. As depicted in Fig. 1(b),a key feature of an SJV is its fixed spin current, facilitatedby the inter-well atomic tunneling, which circulates abouta point mid-way between the two wells. Due to its largesize, on the order of several SHLs, the SJV is a macro-scopic topological object. We determine analytically theparameter space required for SJVs to form in a uniformsystem, where they are the only stable topological exci-
FIG. 1: (Color online) (a) Schematic diagram of the system.The weakly coupled spin-1 BECs are trapped in a double-well potential, which gives strong (weak) confinement in the y − z plane ( x -direction). A uniform magnetic field B ˆ z isapplied along the z -direction. (b,c) show spin vector patternscorresponding to (b) a spin Josephson vortex (SJV) centeredon the blue cross, (c) a ferromagnetic domain wall (FDW).The parameters are κ = κ c / κ = 2 κ c for theFDW. Other parameters are q = 0 and α = π/
2. See text formore details of the parameters and spin patterns. tation. We show that, as a consequence of this stability,the SJV can be created dynamically through the decayof a ferromagnetic domain wall (FDW) [6, 30, 31]. Wedemonstrate that the SJV can be realized by implement-ing this dynamical scheme under conditions that can befully attained with current experimental techniques.Our system comprises two one-dimensional (1D) spin-1 BECs trapped in a symmetric optical DW potential[Fig. 1(a)]. Both atom clouds are strongly confined inthe transverse ( y, z ) directions. The dynamics of weaklycoupled spinor BECs may be described by the spin-1 field operator [32], ˆΨ ( x ) = ˆ Ψ l ( x ) + ˆ Ψ r ( x ), whereˆ Ψ j ( x ) = [ ˆ ψ j ( x ) , ˆ ψ j ( x ) , ˆ ψ j − ( x )] T with j = l ( r ) indicat-ing the left (right) well and m = { , , − } denoting thethree Zeeman levels. A uniform magnetic field B ˆ z is ap-plied along the z -axis. The many-body Hamiltonian is H = H t + H l + H r , (1)where H t = − κ (cid:82) dx [ ˆ Ψ l † ˆ Ψ r + H.c.] denotes the inter-welltunneling with state-independent tunneling strength κ a r X i v : . [ c ond - m a t . qu a n t - g a s ] D ec [24]. The Hamiltonian of atoms in the j th well is H j = (cid:82) dx [ ˆ Ψ j † h j ˆ Ψ j + c (ˆ n j ) + c (cid:126) F j · (cid:126) F j ], where h jm,m (cid:48) = − δ m,m (cid:48) ( ∂ x / − µ j − pm + qm ). Here, µ j , p = − µ B B/ q = ( µ B B ) / E hf denote the chemical potential, lin-ear and quadratic Zeeman energies respectively, where µ B is the Bohr magneton and E hf is the hyperfine en-ergy splitting [31]. The two-body collisional interactionsin the j th spinor BEC enter the expression for H j viathe scalar density ˆ n j = ˆ Ψ j † ˆ Ψ j and the spin-dependentvector density, (cid:126) F j = ˆ Ψ j † (cid:126) f ˆ Ψ j , where (cid:126) f is the Cartesianvector of the spin-1 matrices ( f x , f y , f z ). The effective1D interaction strengths are c = 16 (cid:126) ( a + 2 a ) / M r ⊥ and c = − (cid:126) ( a − a ) / M r ⊥ , where a S is the 3D s-wave scattering length for collisions with total angularmomentum S = 0 , r ⊥ is the width of theBEC in the transverse directions.We study the system using mean field theory. Let usfirst investigate the stationary state of the system. Weuse a simple ansatz to describe the order parameter ψ j ψ j ψ j − = ψ j (cid:113) n j − | ψ j | ( ψ j ) ∗ , (2)where n j = Ψ j † Ψ j is the total density of the j th BEC [31]. For convenience, we scale length, time andenergy by the variables ξ = (cid:126) / ( M c n R ) / , t = (cid:126) /c n R and (cid:15) = c n R , where the reference density, n R , ischosen to ensure correct chemical potentials in the twoatom clouds. Neglecting the spatial dependence of n j , ψ j satisfies the coupled non-linear differential equations, (cid:104) − ∂ x − µ j eff − γ | ψ j | (cid:105) ψ j − κψ j (cid:48) = 0, where γ = c /c and j = l ( r ) when j (cid:48) = r ( l ). Consequently, withinthis approximation, the spinor BECs are described bytwo coupled scalar equations. Each scalar equation ischaracterized by an effective chemical potential µ j eff = µ j − (1 + 2 γ ) n j − q and an interaction strength equalto − γ >
0. The corresponding stationary solution isreadily obtained [33] ψ j = [ C tanh( vx ) ± iA sech( vx )] e iα , (3)where C = (cid:112) (2 γn j + q ) / γ , v and A are constants andthe +( − ) sign corresponds to j = l ( r ). This analyticalsolution allows us to calculate many properties of thesystem. For example, one can directly find the density n j = (1 + κ + γ ) / (1 + γ ) in units of n R and chemicalpotential µ j = q/ γ + 1) in units of (cid:15) .We now discuss the topological excitations of the sys-tem. Depending on the value of A , two distinct solutionscan be obtained from Eq. (3). When A = 0, the solutiondescribes a FDW [31], which has a characteristic spatialwidth of 1 /v = √ ξ s , where ξ s = 1 / (2 | γ | n j − q ) , is thespin healing length. When A = (cid:112) (2 γn j + q + 8 κ ) / γ ,we obtain a totally different topological excitation, the FIG. 2: (Color online) (a) and (b): | ψ l ( x ) | for a FDW and anSJV respectively using the same parameters as in Fig. 1. Thedashed (dotted) curves are calculated with (without) the con-stant atom density approximation and the solid curves shownumerical solutions found by evolving the GPEs in imaginarytime. (c) Phase diagram of the topological excitations. TheSJV is dynamically stable in region I (yellow). The FDW isunstable in both regions I and II. In region III, the system ex-hibits a polar groundstate phase (see text). Red (blue) circlemarks system parameters in region I (II), which are discussedin the text. spin Josephson vortex . The size of an SJV is approx-imately 1 /v = 1 / √ κ and, hence, controlled by theinter-well tunneling strength, κ . To distinguish thetwo distinct topological excitations, we calculate theirspin texture, characterized by the local spin orienta-tion φ j ( x ) = tan − ( F jy /F jx ) and its magnitude | (cid:126) F j | =[( F jx ) + ( F jy ) ] / , from Eq. (3). The spatial variationof the local spin vector along the x axis is shown inFigs. 1(b,c). In an SJV [Fig. 1(b)], the spin current formsa vortex structure in which the local spin vector rotatesbetween the two spinor BECs around a point [blue crossin Fig. 1(b)] mid-way between them. By contrast, thereis no spin current associated with the FDW. Instead, thespin vectors in the two atom clouds are locally alignedfor all x and vanish at x = 0 [Fig. 1(c)].Although the analytical ansatz in Eq. (3) is simple,it produces accurate wavefunctions when compared withfull numerical solutions of the equations of motion, whichwe obtain by propagating the Gross-Pitaevskii equa-tions (GPEs) for the coupled spinor BECs in imaginarytime [34]. Figs. 2(a,b) reveal a small deviation between ψ l ( x ) curves obtained analytically (dashed curves) andnumerically (solid curves) near the center of the SJV andFDW. This deviation is caused by the constant densityassumption used in the above analytical calculation. Toovercome this, we now allow a spatially dependent den- FIG. 3: (Color online) (a) Schematic diagram showing phase imprinting of the coupled spinor BECs by a focused laser beam(red). The phase-imprinting laser is switched on at t = 0 and only affects atoms in the region − r b < x < r b spanned by thebeam (upper panel). The laser beam is switched off at time τ = π/ ( βI ) and coherently flips all atomic spins in the region − r b < x < r b (lower panel). (b) and (c) Color maps showing how the spin vectors (cid:126) F l and (cid:126) F r , respectively, evolve after the phaseimprint when κ = 2 κ c . The orientation ( φ j ) and magnitude ( | (cid:126) F j | ) of the local spin vector in the x − y plane are represented,respectively, by the color and brightness of the images (see scale). The two black stripes (where | (cid:126) F j | = 0) visible for | γ | t < x/r b = ± κ = κ c /
2. Again, two black stripes centered at x/r b = ± | γ | t <
5. In this case, though, the FDW evolves towards a new quasi-static spintexture [green and red stripes in (d) and (e)]. At | γ | t = 50, this spin texture corresponds to the local spin vectors shown in(f). Comparing the region of (f) within the blue dashed box to Fig. 1(b), we see that an SJV with α = π/ x/r b ≈ − x/r b ≈ r x = 1250 and r b = 100 (see text). sity perturbation, n j ( x ) = n j + δn j ( x ), in the ansatz. Theresulting values of ψ l ( x ) [dotted curves in Figs. 2(a,b)]agree much better with the values obtained numerically(solid curves).We are now in a position to obtain the phase diagramof the topological excitations from the above stationarysolutions. Three distinct phase regions are found, whichare summarized in Fig. 2(c). Region I (yellow) shows theparameter space where the SJV exists. The FDW existsboth in region I and region II (green) and a “polar” phasein which all atoms are in the m = 0 spin level occupiesregion III (blue) [1, 35]. We emphasize that the phasediagram can be determined completely analytically. Forexample, the phase boundary between region I and II oc-curs along the red dotted line in Fig. 2(c), whose equationis κ = κ c = (2 | γ | − q )(1 − | γ | ) / (4 − | γ | ).To investigate the dynamical stability of the topo-logical excitations, we use an extended Bogoliubov the- ory [36] in which we evolve a stationary solution, Ψ j s ( x ),to Ψ j ( x, t ) = Ψ j s ( x ) + δ Ψ j ( x, t ) at time t , where δ Ψ j ( x, t ) = u j ( x ) e − iλt − v j ( x ) ∗ e iλ ∗ t is a small pertur-bation. Linearizing these vector equations with respectto u j and v j yields an eigenequation with eigenvectors( u l , v l , u r , v r ) T and eigenvalues λ . A stable solution re-quires that Im( λ ) = 0 [37]. Analysis of the eigenvaluesreveals that the SJV is dynamically stable in region I. Bycontrast, the FDW is unstable in all regions of Fig. 2(c).Guided by this stability analysis, we now explain howto realize the SJV in experiment. First, a FDW is cre-ated in the spinor BECs by a phase-imprinting methodof the type used previously to generate topological exci-tations [9, 38, 39]. Provided the system is in region I ofFig. 2(c) ( κ < κ c ), the unstable FDW can decay into thestable SJV.We now consider the details of the spin-dependentphase-imprinting process. A phase-imprinting laser beampropagating along z is switched on at t = 0. We approx-imate the intensity profile of the beam along the x di-rection by a square wave of the form I ( x ) = I θ ( x ± r b ),where I is the laser intensity, θ is the Heavyside functionand 2 r b determines the width of the laser beam along the x direction. Such a shape can be achieved, for example,by reflecting the laser beam from a spatial light modu-lator [39]. The laser light is circularly polarized ( σ + ) toinduce a linear Zeeman shift through the spin-dependentA.C. Stark energy shift p ( x ) = βI ( x ), where β is a con-stant [40]. Applying the beam for a duration τ = π/ ( βI )coherently flips the atomic spins near the central regionof the atom clouds where | x | < r b . Such a process isshown schematically in Fig. 3(a).As in typical cold atom experiments, we further as-sume that the spinor atoms are confined along the x di-rection by a shallow harmonic trap. The atom densityof the spinor BECs is then given by a Thomas-Fermiprofile, n ( x ) = n R [1 − ( x/r x ) ], where r x is the Thomas-Fermi radius. Initially, the two BECs are prepared inthe ferromagnetic groundstate, whose wavefunction is Ψ l,r = (cid:112) n ( x )[ − / , i/ √ , /
2] when q = 0, in whichthe spin vectors point along the y direction [1]. Afterapplying the phase-imprinting laser, we determine thedynamics by evolving the coupled GPE for the spinorBECs. We include dissipation in our numerical simula-tions following the methods in [8].Let us now analyze the dynamics of the atom cloudsafter the laser illumination. We first consider the evo-lution of a system with parameters located in regionII of Fig. 2(c). Specifically, we choose q/ | γ | = 0 and κ/ | γ | = 2 κ c / | γ | [marked by the blue circle in Fig. 2(c)],for which the evolution of the local spin vectors in theleft and right atom clouds is shown in Fig. 3(b) and (c)respectively. For short times ( | γ | t < x/r b = ± | (cid:126) F j | = 0. The stripesseparate two bright yellow regions, where the atom spinspoint along the y axis, from a bright blue region wherethe atom spins point along the − y direction. Due to theirinstability, the FDWs decay into spreading spin textureswhen | γ | t >
10. This decay appears in Figs. 3(b,c) asmuti-colored bands, which emerge from the black stripesand spread outwards with increasing t .The evolution of the spin texture differs markedlywhen the system parameters are prepared in region Iof Fig. 2(c). To illustrate this, we choose q/ | γ | = 0and κ/ | γ | = κ c / (2 | γ | ) [marked by the red circle inFig. 2(c)], for which the evolution of the local spin vec-tors is shown in Figs. 3(d,e). Comparison of Figs. 3(d,e)with Figs. 3(b,c) shows that in both cases the systeminitially (for | γ | t <
10) forms two FDWs (black stripesat x/r b = ± x/r b = ±
1. To demonstratethat these spin textures correspond to SJV formation, inFig. 3(f) we show the associated spin vector configura-tions in the two spinor BECs at | γ | t = 50. Comparisonof the spin vectors around the point x/r b = − α = π/ x/r b = 1 (within the reddashed box). This corresponds to a solution in Eq. (3)with α = − π/ Rb BECs. If the total number of atoms is 2 × and r x ( r ⊥ ) = 200 µ m (2 . µ m), the characteristic timescale is | γ | t = 16 ms. The atomic tunneling strength, κ , canbe controlled by changing the intensity and/or waist ofthe laser that creates the double-well trap [23]. All of thesystem parameters and procedures required to implementour proposed route to creating SJVs can be attained us-ing current experimental setups [35]. Consequently, weexpect that the dynamical regime that we have identifiedwill be directly accessible to experimental study.In conclusion, we have identified SJVs in spin-1 ferro-magnetic BECs trapped in an elongated DW potential.We have presented a detailed analysis of the stability andformation of the SJVs. In particular, we have shown thatthe SJV can be created from the decay of a FDW. Ouranalysis can be extended to study topological phases inmulti-well optical potentials and for higher atomic spins,which seem certain to reveal further exotic spin textures.This work is funded by EPSRC. WL acknowledgesfunding through an EU Marie Curie Fellowship. [1] T.-L. Ho, Phys. Rev. Lett. , 742 (1998).[2] T. Ohmi and K. Machida, J. Phys. Soc. Jpn. , 1822(1998)[3] H. Schmaljohann, M. Erhard, J. Kronjager, M. Kottke,S. van Staa, L. Cacciapuoti, J. J. Arlt, K. Bongs, andK. Sengstock, Phys. Rev. Lett. , 040402 (2004).[4] N. Bigelow, Nat. Phys , 89 (2005).[5] M.-S. Chang, Q. Qishu, W. Zhang, L. You, and M. S.Chapman, Nat Phys. , 111116 (2005).[6] L. E. Sadler, J. M. Higbie,S. R. Leslie, M. Vengalattore,and D. M. Stamper-Kurn, Nature 443, 312 (2006).[7] T. Mizushima, K. Machida and T. Kita Phys. Rev. Lett.89, 030401 (2002)[8] H. Saito, Y. Kawaguchi, and M. Ueda, Phys. Rev. Lett.96, 065302 (2006).[9] K. C. Wright, L. S. Leslie, A. Hansen, and N. P. Bigelow,Phys. Rev. Lett. , 030405 (2009).[10] Y. Kawaguchi, M. Nitta, and M. Ueda, Phys. Rev. Lett. , 180403 (2008).[11] A. K.Usama, and H. Stoof, Nature , 918 (2001).[12] L. S. Leslie, A. Hansen, K. C. Wright, B. M. Deutsch,and N. P. Bigelow, Phys. Rev. Lett. , 250401 (2009). [13] J.-y. Choi, W. J. Kwon, and Y.-i. Shin, Phys. Rev. Lett. , 035301 (2012).[14] H. Pu,W. Zhang, and P. Meystre, Phys. Rev. Lett. ,140405 (2001).[15] E. Demler, and F. Zhou, Phys. Rev. Lett. , 163001(2002).[16] H. Pu, W. Zhang, and P. Meystre, Phys. Rev. Lett. ,090401 (2002).[17] S. K. Yip, Phys. Rev. Lett. , 250402 (2003).[18] M. Rizzi, D. Rossini,G. De Chiara, S. Montangero andR. Fazio, Phys. Rev. Lett. , 240404 (2005).[19] J. L. Song, G. W. Semenoff, and F. Zhou, Phys. Rev.Lett. , 160408 (2007).[20] G. G. Batrouni, V. G. Rousseau, and R. T. Scalettar,Phys. Rev. Lett. , 140402 (2009).[21] K. Rodriguez,A. Arguelles, A. K. Kolezhuk, L. Santos,and T. Vekua, Phys. Rev. Lett. , 105302 (2011).[22] I. Bloch, Nature Physics , 23 (2005).[23] M. Albiez, R. Gati, J. F¨olling, S. Hunsmann, M. Cris-tiani, and M. K. Oberthaler, Phys. Rev. Lett. , 010402(2005).[24] G. J. Milburn, J. Corney, E. M. Wright, and D. F. Walls,Phys. Rev. A , 4318 (1997).[25] O. E. M¨ustecaplıo ˘glu, M. Zhang, and L. You, Phys. Rev.A , 053616 (2005).[26] O. E. M¨ustecaplıo˘glu, W. Zhang, and L. You, Phys. Rev.A , 023605 (2007).[27] B. Julia-Diaz, M. Mele-Messeguer, M. Guilleumas, and A. Polls, Phys. Rev. A , 043622 (2009).[28] M. Mel-Messeguer, B. Juli-Daz, M. Guilleumas, A. Polls,and A. Sanpera, New J. Phys. , 033012 (2011).[29] Dan-Wei Zhang, Li-Bin Fu, Z. D. Wang, and Shi-LiangZhu, Phys. Rev. A , 043609 (2012)[30] H. Saito and M. Ueda, Phys. Rev. A , 023610 (2005).[31] H. Saito, Y. Kawaguchi, and M. Ueda, Phys. Rev. A ,013621 (2007).[32] W. Zhang and L. You, Phys. Rev. A , 025603 (2005).[33] V. M. Kaurov and A. B. Kuklov, Phys. Rev. A , 013627(2006).[34] Y. Kawaguchi, H. Saito, and M. Ueda, Phys. Rev. Lett. , 130404 (2006).[35] D. M. Stamper-Kurn and M. Ueda, arXiv:1205.1888(2012).[36] C. Pethick and H. Smith, Bose-Einstein condensation indilute gases (Cambridge University Press, 2002).[37] T. W. A. Montgomery, R. G. Scott, I. Lesanovsky, andT. M. Fromhold, Phys. Rev. A , 063611 (2010).[38] W. Li, M. Haque, and S. Komineas, Phys. Rev. A ,053610 (2008).[39] C. Becker, S. Stellmer,P. Soltan-Panahi,S. Dorscher,M. Baumert, E. Richter, J. Kronjager, K. Bongs, andK. Sengstock, Nature Physics , 496 (2008).[40] J. M. Higbie, L. E. Sadler, S. Inouye, A. P. Chikkatur,S. R. Leslie, K. L. Moore, V. Savalli, and D. M. Stamper-Kurn, Phys. Rev. Lett.95