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Composite vortex rings in ferromagnetic spin-1 Bose–Einstein condensates
Chaos, Solitons and Fractals 132 (2020) 109546
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Composite vortex rings in ferromagnetic spin-1 Bose–Einstein condensates Yong-Kai Liu a,∗, Shi-Jie Yang b a b
College of Physics and Information Engineering, Shanxi Normal University, Linfen 041004, China Department of Physics, Beijing Normal University, Beijing 100875, China
a r t i c l e
i n f o
Article history: Received 26 July 2019 Revised 18 November 2019 Accepted 22 November 2019
a b s t r a c t The recent experimental creation of Shankar skyrmion in ferromagnetic spin-1 Bose–Einstein condensates opens an interesting pilot for exploring the stability and dynamics of the three-dimensional skyrmion. We find similar vortex rings can be accommodated in the spin-orbit coupled ferromagnetic spin-1 Bose– Einstein condensates. However, the composite vortex rings has mixing manifolds of ferromagnetic and antiferromagnetic states, where the Shankar skyrmion is hidden. Our results suggest an alternative way of creating skyrmion-like vortex rings in spin-1 Bose–Einstein condensates. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction Vortex rings present in various systems with length scales ranging from nanometer to interstellar plasmas[1–3]. In recent years, the study of vortex ring in Bose–Einstein condensates (BECs) has become a hot topic due to the BEC systems are highly controllable [4–8]. The mean-field description is simple and the topological stability of the vortex ring is guaranteed by the quantization of circulation [3]. In addition to vortex rings, spinor BEC can support more complex topological structures such as the three-dimensional (3D) skyrmion [5]. In its simplest realization which is originally proposed for two-component BECs, the 3D skyrmion consists of a vortex ring in one component coupling to a vortex line in the second component [9], or the 3D dimeron consists of a vortex ring in one component interlocking a vortex ring in the second component [10]. One of the increasing interests is the single-component counterpart of the 3D skyrmion, the so-called hopfion [11,12], which consists of a vortex ring and a vortex line in the same component. The 3D skyrmion that extends over the entire threedimensional space are characterized by the third homotopy group [13]. A prime example called Shankar skyrmion with the topology π 3 (SO(3))∼ =Z[13], which is supported by the ferromagnetic phase of the spin-1 BECs [13]. Four decades ago, Shankar skyrmion was theoretically predicted [14], and many attempts were made to observe experimentally. Although the skyrmion is topologically protected, the biggest difficulty is energetic instability against collapse to zero
∗
Corresponding author.. E-mail addresses: [email protected] (Y.-K. Liu), [email protected] (S.-J. Yang). https://doi.org/10.1016/j.chaos.2019.109546 0960-0779/© 2019 Elsevier Ltd. All rights reserved.
size. Recently, this question is becoming experimentally accessible within ferromagnetic 87 Rb BECs exposed to a time-dependent magnetic field [15]. In this paper we propose an alternative way of creating Shankar skyrmion in spin-1 BEC with spin-orbit coupling (SOC). Recently, synthetic SOC is generated in ultracold atoms by utilizing atom-light interactions [16,17]. In the past few years, the study of spin-orbit coupled quantum gases has become one of the hottest topics in cold atom physics. It opens an interesting avenue for exploring new quantum states and novel quantum phenomena in large-spin systems [17]. The evolution of the atomic wave function likes a charged particle effected by the gauge potentials. Many interesting properties have been predicted for spin-orbit coupled Bose gases due to the interplay between SOC and the unique properties of dilute atomic gases [18–20]. All these effects make the cold atom gases with the synthetic gauge field a rapidly developing field. In this paper, we investigate the vortex rings in the ferromagnetic spin-1 Bose–Einstein condensates with SOC. We display the stable structure by numerically solving the coupled Gross– Pitaevskii (GP) equation in imaginary-time simulation and demonstrate that they can be naturally generated from a initial vortexfree Gaussian wave packet(uniform spin), as shown in Fig. 2. The numerical results show that the vortex rings is energy stable, and has the same shape of a Shankar skyrmion. However, it is not exactly consistent with Shankar skyrmion from the boundary. We find the vortex rings is a mixture of the ferromagnetic and the antiferromagnetic manifolds, where the Shankar skyrmion confined in ferromagnetic manifolds is hidden in the composite vortex rings
2
Y.-K. Liu and S.-J. Yang / Chaos, Solitons and Fractals 132 (2020) 109546
structure. We also check the dynamical stability of the vortex rings by real-time simulation. The paper is organized as follows. In Section 2, we describe the model of spin-1 BECs with SOC and Shankar skyrmion in spin-1 BECs. In Section 3, we present numerical results and describe the structure in comparison to the Shankar skyrmion. In Section 4, we give a theoretical analysis of this composite structure. A summary is included in Section 5. 2. Model We consider a ferromagnetic spin-1 BEC with SOC in a harmonic trap. The mean-field order parameter describing the BEC is expressed as ψ (r ) = n(r )ξ (r ), where n(r ) is the local atomic density and ξ (r ) = (ξ+1 (r ), ξ0 (r ), ξ−1 (r ))T is a three-component normalized spinor [21,22]. In the mean-field approximation, we theoretically describe the condensates with the coupled Gross– Pitaevskii equation
ih ¯
∂ (r, t ) = {h(r, t ) + n(r, t )[c0 + c2 F(r, t ) · f]}(r, t ), ∂t
Fig. 1. Upper panels: The density profiles for (a) ψ+1 , (b) ψ 0 and (c)ψ−1 in the x-y plane. Lower panels: The density profiles for (d) ψ+1 , (e) ψ 0 and (f)ψ−1 in the x-z plane. All quantities in this figure and the following other figures are dimensionless.
(1)
where h ( r, t ) denotes the single-particle Hamiltonian, f = ( fx , fy , fz ) the vector of spin-1 matrices, and F(r, t ) = ξ (r, t )† fξ (r, t ) the spin density vector. The density-density and spin-spin coupling are described by c0 = (g0 + 2g2 )/3 and c2 = (g2 − g0 )/3, with gF = 4π h ¯ 2 aF /m [13,15]. The single-particle Hamiltonian is
h ( r, t ) = − h ¯ 2 ∇ 2 /(2m ) + V (r ) + γ f · p,
(2)
where γ is the strength of SOC and the harmonic potential V = m[ωρ2 (x2 + y2 ) + ωz2 z2 ]/2. In harmonic oscillator units, we can get the dimensionless form of GP equations. The length, time, en ergy and SO coupling strength are scaled by h ¯ /mw, w−1 , h ¯w
Fig. 2. Isosurfaces of the density distributions: (a) a vortex ring in ψ 1 ; (b) a vortex ring and a vortex line in ψ 0 ; (c) a vortex line in ψ−1 . The isosurfaces correspond to |ψi |2 = 10−4 . These numerical results are obtained with the parameter γ = 2.8, which are the same as Shankar skyrmion at first glance.
and h ¯ w/m, respectively. The subsequent results are simulated by solving the coupled equations with the Fourier pseudospectral method. As we know, without the spin-orbit coupling, the system becomes ferromagnetic (FM) when the spin-spin interaction c2 < 0 and antiferromagnetic or polar when c2 > 0. One can construct the general spinor wavefunction of the ferromagnetic phase by applying a spin rotation to the representative spinor ξF = (1, 0, 0 )T [13,23]. The general spinor in the ferromagnetic phase is
ξ
1 = Uˆ (τ , α , β ) 0 0
=
(3)
(cos τ2 − i cos β sin τ2 )2 √ iα − 2e sin β sin τ2 (i cos τ2 + cos β sin τ2 ) , 2 2 −e2iα sin τ2 sin β
where matrix Uˆ (τ , α , β ) = e−iτ n·f and n= (cos α sin β , sin α sin β , cos β ). A Shankar skyrmion with the winding number 1 can be realized by rotating about the direction n through an angle τ (r). The radial profile function τ (r) is a monotonically decreasing function of r, subject to the boundary conditions τ (0 ) = 2π and τ (∞ ) = 0. The boundary condition ensures that the three-dimensional space is a compacted S3 manifold [15]. The 3D order parameter of the ferromagnetic BEC may be characterized by a field of rigid triads F = t × u, the local spin F and the superfluid phase parameterized in terms of two unit vectors t and u [13]. For the Shankar skyrmion, the local spin F shows a knotted spin texture with each tube is linked. The knotted spin texture in the Shankar skyrmion is reminiscent of the field d knot in polar state of spin-1 BEC. The Shankar skyrmion(SO(3)) is distinguished from knot(S2 )by an additional 4π twist(S1 )in the triad
orientation about its spin axis as one surround a closed tube [15]. Fig. 1 shows schematic picture of the Shankar skyrmion for the analytical function with the Gaussian profile of density distribution of Eq.(3) in spin-1 BEC. For the Shankar skyrmion, the order parameter continuously changes from the core (1, 0, 0)T via (0, 0, 1)T to the boundary (1, 0, 0)T , which differs to the spin texture of the Anderson–Toulouse coreless vortex at the boundary (0, 0, 1)T [13]. 3. Numerical results Despite of a topological object, the skyrmion solution in several physical systems is energetically unstable against shrinking to zero size if there are no additional stabilizing measures [24–26]. We numerically study the stable configurations by minimizing the energy of the full 3D mean-field theory of coupled GP equations. The stationary states (Fig. 2) are obtained by using the imaginary-time propagation method (time-splitting Fourier pseudo-spectral method). In our numerical simulations we use space and time steps of 0.1 and 0.005, respectively. We have chosen the parameters c0 = 9, c1 = −9/216, which correspond to the parameters of 87 Rb. The initial state was prepared in the ferromagbetic phase ξF = (1, 0, 0 )T with the density profile yields a Gaussian-like shape. Fig. 2 illustrates the isosurfaces of density for each of the three components, respectively. It shows that the ψ+1 component forms a density ring in the center surrounded with a 2π phase changing. This indicates a ring vortex. The ψ 0 component forms a density torus with a inner ring. The inner ring changes a 2π phase along the circle. The big torus has a vortex line with one unit of circulation around the core oriented along the z axis. This indicates a line vortex and a ring vortex coexist in the component ψ 0 . In twocomponent BECs, a line vortex in one component and a ring vortex
Y.-K. Liu and S.-J. Yang / Chaos, Solitons and Fractals 132 (2020) 109546
Fig. 3. (Color online) (a) The preimages of Fx = 0.9 and Fy = 0.9. (b) The preimage of Fz = −0.5. (c) The preimages of Fx = 0.9 and Fy = 0.9 of the Shankar skyrmion.
3
romagnetic ingredient. The F reflects ferromagnetic characteristics of the skyrmion. When the magnetization is relatively weak, the antiferromagnetic state dominates. From the Fig. 4(a) and (b), we can see that the vortex-ring has the boundary (0, 1, 0)T instead of (1, 0, 0)T of a Shankar skyrmion. The component ψ 0 extends to the boundary of the finite-size atomic cloud, altering the boundary conditions and the density of other component. The emergence of the antiferromagnetic parts looks like the three components pass though each other, causing local enhancement and local attenuation and indicating a composite vortex ring.
4. Theoretical analysis
Fig. 4. (Color online) (a) The cross sections of |ψ 1 |2 , |ψ 0 |2 and |ψ−1 |2 along the x axis for the Shankar skyrmion. (b) The cross sections of |ψ 1 |2 , |ψ 0 |2 and |ψ−1 |2 along the x axis for our numerical result. (c) The magnitude of |F|2 along the x axis for our numerical result.(d)The plot of Fi i = (x, y, z ) along the x axis for the Shankar skyrmion. (e)The plot of Fi i = (x, y, z ) along the x axis for our numerical result.
in other component form a 3D skyrmion. Some authors also call this structure as Hopfion type, i.e., twisted toroidal tubes with two independent winding numbers in single-component model [11,12]. The second component in our numerical result has the winding number m = 1 in the (x,y) plane and the second winding number (twist) s = 1 in the (r,z) plane. ψ−1 shows a torus isodensity contour. It shows a vortex line in its core along the z-axis. We have noticed this is different from the vortex line in ψ 0 with 4π phase change around the z-axis. From the perspective of overall density distribution, the line component of ψ−1 and ψ 0 is confined in a toroidal region inside the ring component of ψ 1 and ψ 0 in the neighborhood of the ring core. The three components can repel each other and prevent the vortex ring from shrinking to zero radius due to the toroidal filling of the line component. Compared to an empty vortex ring, a vortex ring filled with line component, can be more stable against collapse. Although the F texture(S2 ) cannot constitute the entire Shankar skyrmion (SO(3)). Knotted F texture with a Hopf charge π 3 (S2 )∼ =Z is an intuitive embodiment of Shankar skyrmion. In a knotted field configuration, any given two preimages interlock Q times. The linking number provides an alternative perspective on its physical significance. Fig. 3(c) shows the preimages of the Shankar skyrmion where any two tubes link once. However, F is not a good physical quantity for our numerical result. We find that the vortex-ring contains both the ferromagnetic and the antiferromagnetic states in the condensates. The value of |F| gradually becomes smaller as one moves away from the center, it then becomes larger and then smaller. The F texture is shown in Fig. 3(a) and (b). For Fx and Fy , the point on them does not correspond to a closed tube. For Fz , a negative value still correspond to a closed tube. We analyze the magnetization from the x-axis as shown in Fig. 4(c)(e). The reason that the spin F is not a linked ring can been seen from the amplitude of F and the boundary of F. When the magnetization is relatively strong, there is less antifer-
The energy stability of the 2D and 3D skyrmion in twocomponent BEC with SOC is physically understandable with the concept of the helical modulation of the order parameter (OP) [27,28]. This concept is also successfully applied to the twodimensional structure of spin-1 and spin-2 with SOC [29,30]. However, the energy stability of the 3D structure in the ferromagnetic phase of an F = 1 spinor BEC remains nontrivial. We extend this theory to a ferromagnetic spin-1 BEC with the OP under influence of SOC. The order parameter manifold in the ferromagnetic phase is SO(3). The favorable OP in the ferromagnetic state is ψ = R(k · r, nR )ξF , where R(θ , n ) denotes the rotation matrix with the angle θ about n. The 3D helical modulation propagating with all the directions of k along r fulfills the OP manifold within 0 ≤ k · r ≤ 2π . We can draw a qualitative picture of the Shankar skyrmion texture correspond to Eq.(3), which is the candidate of the ground state. But our numerical calculations reveal that it is actually not the ground state. Even though the Shankar skyrmion is a topological object. An isolate shankar skyrmion is energy unstable in the spin-1 BEC without SOC, where the Shankar skyrmion is energy unstable to shrink to zero [13]. In the ferromagnetic spin-1 Bose–Einstein condensates with SOC, a composite vortex-ring where a Shankar skyrmion can be energetically stable. In the stable vortex-rings, the magnetization leads to a mixing of the polar and FM phases. The √ √ structure can be described as ψ = nF ψShankarskyrmion + n p ψpolar , which means the shankar skyrmion is hidden in the structure. If we exclude the antiferromagnetic parts, the rest of the structure is a shankar skyrmion. We can simply analyze this conclusion by exhibiting the density distribution on the x-axis. The asymptotic boundary conditions is (0, 1, 0)T , which is the antiferromagnetic or polar state with nF → 0. If we remove ψ 0 in the nearby region, the asymptotic boundary conditions change to (1, 0, 0)T as shown in the Fig. 4(a) and (b). From the perspective of internal distribution, the most obvious difference for the composite structure and Shankar skyrmion is that the density zeros of ψ 0 and ψ 1 do not coincide. If we extract ψ 0 in the density zeros of ψ 1 , the zeros of the two densities will coincide and form (0, 0, 1)T . The reason of the manifold mixing lies in that, in the presence of SOC, the spin-dependent interaction prefers the pure ferromagnetic state, while the total energy functional prefer the manifold mixing. The wave functions can also be seen as the superposition of different orbital angular momentum states. The high orbital angular momentum states cost more kinetic energy, SOC energy and are less populated. For example, fix ratio of Y21 ~ eiα sin β cos β and Y11 ~ eiα sin β of ψ 0 in Eq.(3) is the ideal combination to form the Shankar skyrmion. The proportion of the radial portion of the Y21 and Y11 in ψ 0 is not limited to sin τ (2r ) / cos τ (2r ) due to the aforementioned reason. In comparison to Eq.(3), some antiferromagnetic states are added and form the composite vortex-rings. Composite defects also exist in other systems[31,32]. Spinor BECs may therefore shed light on topological features of composite vortex.
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From our analysis, the type of Shankar skyrmion has a higher energy than the type of the componsite vortex ring. Our numerical results also confirm that the componsite vortex ring instead of Shankar skyrmion is energy stable. We also performed real-time simulation of the imaginary-time profile as the initial state over a long interval of time. The real-time evolution establishes the dynamical stability of the composite vortex rings with the initial rings undecayed. 5. Summary We have shown that the SO coupling leads to composite vortexrings in the ferromagnetic spin-1 BECs. The structure shows a manifold mixing of ferromagnetic and antiferromagnetic states. We find the Shankar skyrmion is hidden in the structure. The emerged structures are physically understandable with the concept of the helical modulation and manifold mixing. Our work provides an alternative way of creating composite vortex-rings and enriches the field of 3D spin-orbit coupling research in the spinor BECs. This work is supported by the NSF of China under grant No.11604193 and No.11774034. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] Shariff K, Leonard A. Vortex rings. Annu Rev Fluid Mech 1992;24:235. [2] Barenghi CF, Donnelly RJ. Fluid Dyn Res 2009;41:051401.
[3] Zhu T, Evans ML, Brown RA, Walmsley PM, I Golov A. Phys Revfluids 2016;1:044502. [4] Crasovan L-C, Pérez-García VM, Danaila I, Mihalache D, Torner L. Phys Rev A 2004;70:033605. [5] Bisset RN, Wang W, Ticknor C, Carretero-González R, Frantzeskakis DJ, Collins LA, Kevrekidis PG. Phys Rev A 2015;92:063611. [6] Wang W, Bisset RN, Ticknor C, Carretero-González R, Frantzeskakis DJ, Collins LA, Kevrekidis PG. Phys Rev A 2017;95:043638. [7] Yakimenko AI, Bidasyuk YM, Prikhodko OO, Vilchinskii SI, Ostrovskaya EA, Kivshar YS. Phys Rev A 2013;88:043637. [8] Oliinyk A, Yatsuta I, Malomed B, Yakimenko A. Symmetry 2019;11:1312. [9] Ruostekoski J, Anglin JR. Phys Rev Lett 2001;86:3934. [10] Liu Y-K, Yang SJ. Phys Rev A 2015;91:043616. [11] Kartashov YV, Malomed BA, Shnir Y, Torner L. Phys Rev Lett 2014;113:264101. [12] Bidasyuk YM, Chumachenko AV, Prikhodko OO, Vilchinskii SI, Weyrauch M, Yakimenko AI. Phys Rev A 2015;92:053603. [13] Kawaguchi Y, Ueda M. Phys Rep 2012;520:253. [14] Shankar R. J Phys France 1977;38:1405–12. [15] Lee W, Gheorghe AH, Tiurev K, Ollikainen T, Möttönen M, Hall DS. Sci Adv eaao 2018;4:3820. [16] Lin Y-J, Jiménez-García K, Spielman IB. Nature 2011;471:83. [17] Campbell DL, Price RM, Putra A, Valdés-Curiel A, Trypogeorgos D, Spielman IB. Nature. Commun 2016;7:10897. [18] Wang C, Gao C, Jian C-M, Zhai H. Phys Rev Lett 2010;105:160403. [19] Sinha S, Nath R, Santos L. Phys Rev Lett 2011;107:270401. [20] Hu H, Ramachandhran B, Pu H, Liu XJ. Phys Rev Lett 2012;108:010402. [21] Ho TL. Phys Rev Lett 1998;81:742. [22] Ohmi T, Machida K. J Phys Soc Jpn 1998;67:1822. [23] Mäkelä H. J Phys A: Math Gen 2006;39:7423. [24] Savage CM, Ruostekoski J. Phys Rev Lett 2003;91:010403. [25] Ruostekoski J, Anglin JR. Phys Rev Lett 2001;86:3934. [26] Zhai H, Chen WQ, Xu Z, Chang L. Phys Rev A 2003;68:043602. [27] Kawakami T, Mizushima T, Nitta M, Machida K. Phys Rev Lett 2012;109:015301. [28] Liu Y-K, Yue H-X, Xu L-L, Yang SJ. Front Phys 2018;13:130316. [29] Liu Y-K, Yang SJ. New J Phys 2017;16:053046. [30] Kawakami T, Mizushima T, Machida K. Phys Rev A (R) 2011;84:011607. [31] Eltsov VB, Kibble TWB, Krusius M, Ruutu VMH, Volovik GE. Phys Rev Lett 20 0 0;85:4739. [32] Dossou KB, Poulton CG, Botten LC, Mahmoodian S, McPhedran RC, deSterke CM. Phys Rev A 2009;80:013826.
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Composite vortex rings in ferromagnetic spin-1 Bose–Einstein condensates Yong-Kai Liu a,∗, Shi-Jie Yang b a b
College of Physics and Information Engineering, Shanxi Normal University, Linfen 041004, China Department of Physics, Beijing Normal University, Beijing 100875, China
a r t i c l e
i n f o
Article history: Received 26 July 2019 Revised 18 November 2019 Accepted 22 November 2019
a b s t r a c t The recent experimental creation of Shankar skyrmion in ferromagnetic spin-1 Bose–Einstein condensates opens an interesting pilot for exploring the stability and dynamics of the three-dimensional skyrmion. We find similar vortex rings can be accommodated in the spin-orbit coupled ferromagnetic spin-1 Bose– Einstein condensates. However, the composite vortex rings has mixing manifolds of ferromagnetic and antiferromagnetic states, where the Shankar skyrmion is hidden. Our results suggest an alternative way of creating skyrmion-like vortex rings in spin-1 Bose–Einstein condensates. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction Vortex rings present in various systems with length scales ranging from nanometer to interstellar plasmas[1–3]. In recent years, the study of vortex ring in Bose–Einstein condensates (BECs) has become a hot topic due to the BEC systems are highly controllable [4–8]. The mean-field description is simple and the topological stability of the vortex ring is guaranteed by the quantization of circulation [3]. In addition to vortex rings, spinor BEC can support more complex topological structures such as the three-dimensional (3D) skyrmion [5]. In its simplest realization which is originally proposed for two-component BECs, the 3D skyrmion consists of a vortex ring in one component coupling to a vortex line in the second component [9], or the 3D dimeron consists of a vortex ring in one component interlocking a vortex ring in the second component [10]. One of the increasing interests is the single-component counterpart of the 3D skyrmion, the so-called hopfion [11,12], which consists of a vortex ring and a vortex line in the same component. The 3D skyrmion that extends over the entire threedimensional space are characterized by the third homotopy group [13]. A prime example called Shankar skyrmion with the topology π 3 (SO(3))∼ =Z[13], which is supported by the ferromagnetic phase of the spin-1 BECs [13]. Four decades ago, Shankar skyrmion was theoretically predicted [14], and many attempts were made to observe experimentally. Although the skyrmion is topologically protected, the biggest difficulty is energetic instability against collapse to zero
∗
Corresponding author.. E-mail addresses: [email protected] (Y.-K. Liu), [email protected] (S.-J. Yang). https://doi.org/10.1016/j.chaos.2019.109546 0960-0779/© 2019 Elsevier Ltd. All rights reserved.
size. Recently, this question is becoming experimentally accessible within ferromagnetic 87 Rb BECs exposed to a time-dependent magnetic field [15]. In this paper we propose an alternative way of creating Shankar skyrmion in spin-1 BEC with spin-orbit coupling (SOC). Recently, synthetic SOC is generated in ultracold atoms by utilizing atom-light interactions [16,17]. In the past few years, the study of spin-orbit coupled quantum gases has become one of the hottest topics in cold atom physics. It opens an interesting avenue for exploring new quantum states and novel quantum phenomena in large-spin systems [17]. The evolution of the atomic wave function likes a charged particle effected by the gauge potentials. Many interesting properties have been predicted for spin-orbit coupled Bose gases due to the interplay between SOC and the unique properties of dilute atomic gases [18–20]. All these effects make the cold atom gases with the synthetic gauge field a rapidly developing field. In this paper, we investigate the vortex rings in the ferromagnetic spin-1 Bose–Einstein condensates with SOC. We display the stable structure by numerically solving the coupled Gross– Pitaevskii (GP) equation in imaginary-time simulation and demonstrate that they can be naturally generated from a initial vortexfree Gaussian wave packet(uniform spin), as shown in Fig. 2. The numerical results show that the vortex rings is energy stable, and has the same shape of a Shankar skyrmion. However, it is not exactly consistent with Shankar skyrmion from the boundary. We find the vortex rings is a mixture of the ferromagnetic and the antiferromagnetic manifolds, where the Shankar skyrmion confined in ferromagnetic manifolds is hidden in the composite vortex rings
2
Y.-K. Liu and S.-J. Yang / Chaos, Solitons and Fractals 132 (2020) 109546
structure. We also check the dynamical stability of the vortex rings by real-time simulation. The paper is organized as follows. In Section 2, we describe the model of spin-1 BECs with SOC and Shankar skyrmion in spin-1 BECs. In Section 3, we present numerical results and describe the structure in comparison to the Shankar skyrmion. In Section 4, we give a theoretical analysis of this composite structure. A summary is included in Section 5. 2. Model We consider a ferromagnetic spin-1 BEC with SOC in a harmonic trap. The mean-field order parameter describing the BEC is expressed as ψ (r ) = n(r )ξ (r ), where n(r ) is the local atomic density and ξ (r ) = (ξ+1 (r ), ξ0 (r ), ξ−1 (r ))T is a three-component normalized spinor [21,22]. In the mean-field approximation, we theoretically describe the condensates with the coupled Gross– Pitaevskii equation
ih ¯
∂ (r, t ) = {h(r, t ) + n(r, t )[c0 + c2 F(r, t ) · f]}(r, t ), ∂t
Fig. 1. Upper panels: The density profiles for (a) ψ+1 , (b) ψ 0 and (c)ψ−1 in the x-y plane. Lower panels: The density profiles for (d) ψ+1 , (e) ψ 0 and (f)ψ−1 in the x-z plane. All quantities in this figure and the following other figures are dimensionless.
(1)
where h ( r, t ) denotes the single-particle Hamiltonian, f = ( fx , fy , fz ) the vector of spin-1 matrices, and F(r, t ) = ξ (r, t )† fξ (r, t ) the spin density vector. The density-density and spin-spin coupling are described by c0 = (g0 + 2g2 )/3 and c2 = (g2 − g0 )/3, with gF = 4π h ¯ 2 aF /m [13,15]. The single-particle Hamiltonian is
h ( r, t ) = − h ¯ 2 ∇ 2 /(2m ) + V (r ) + γ f · p,
(2)
where γ is the strength of SOC and the harmonic potential V = m[ωρ2 (x2 + y2 ) + ωz2 z2 ]/2. In harmonic oscillator units, we can get the dimensionless form of GP equations. The length, time, en ergy and SO coupling strength are scaled by h ¯ /mw, w−1 , h ¯w
Fig. 2. Isosurfaces of the density distributions: (a) a vortex ring in ψ 1 ; (b) a vortex ring and a vortex line in ψ 0 ; (c) a vortex line in ψ−1 . The isosurfaces correspond to |ψi |2 = 10−4 . These numerical results are obtained with the parameter γ = 2.8, which are the same as Shankar skyrmion at first glance.
and h ¯ w/m, respectively. The subsequent results are simulated by solving the coupled equations with the Fourier pseudospectral method. As we know, without the spin-orbit coupling, the system becomes ferromagnetic (FM) when the spin-spin interaction c2 < 0 and antiferromagnetic or polar when c2 > 0. One can construct the general spinor wavefunction of the ferromagnetic phase by applying a spin rotation to the representative spinor ξF = (1, 0, 0 )T [13,23]. The general spinor in the ferromagnetic phase is
ξ
1 = Uˆ (τ , α , β ) 0 0
=
(3)
(cos τ2 − i cos β sin τ2 )2 √ iα − 2e sin β sin τ2 (i cos τ2 + cos β sin τ2 ) , 2 2 −e2iα sin τ2 sin β
where matrix Uˆ (τ , α , β ) = e−iτ n·f and n= (cos α sin β , sin α sin β , cos β ). A Shankar skyrmion with the winding number 1 can be realized by rotating about the direction n through an angle τ (r). The radial profile function τ (r) is a monotonically decreasing function of r, subject to the boundary conditions τ (0 ) = 2π and τ (∞ ) = 0. The boundary condition ensures that the three-dimensional space is a compacted S3 manifold [15]. The 3D order parameter of the ferromagnetic BEC may be characterized by a field of rigid triads F = t × u, the local spin F and the superfluid phase parameterized in terms of two unit vectors t and u [13]. For the Shankar skyrmion, the local spin F shows a knotted spin texture with each tube is linked. The knotted spin texture in the Shankar skyrmion is reminiscent of the field d knot in polar state of spin-1 BEC. The Shankar skyrmion(SO(3)) is distinguished from knot(S2 )by an additional 4π twist(S1 )in the triad
orientation about its spin axis as one surround a closed tube [15]. Fig. 1 shows schematic picture of the Shankar skyrmion for the analytical function with the Gaussian profile of density distribution of Eq.(3) in spin-1 BEC. For the Shankar skyrmion, the order parameter continuously changes from the core (1, 0, 0)T via (0, 0, 1)T to the boundary (1, 0, 0)T , which differs to the spin texture of the Anderson–Toulouse coreless vortex at the boundary (0, 0, 1)T [13]. 3. Numerical results Despite of a topological object, the skyrmion solution in several physical systems is energetically unstable against shrinking to zero size if there are no additional stabilizing measures [24–26]. We numerically study the stable configurations by minimizing the energy of the full 3D mean-field theory of coupled GP equations. The stationary states (Fig. 2) are obtained by using the imaginary-time propagation method (time-splitting Fourier pseudo-spectral method). In our numerical simulations we use space and time steps of 0.1 and 0.005, respectively. We have chosen the parameters c0 = 9, c1 = −9/216, which correspond to the parameters of 87 Rb. The initial state was prepared in the ferromagbetic phase ξF = (1, 0, 0 )T with the density profile yields a Gaussian-like shape. Fig. 2 illustrates the isosurfaces of density for each of the three components, respectively. It shows that the ψ+1 component forms a density ring in the center surrounded with a 2π phase changing. This indicates a ring vortex. The ψ 0 component forms a density torus with a inner ring. The inner ring changes a 2π phase along the circle. The big torus has a vortex line with one unit of circulation around the core oriented along the z axis. This indicates a line vortex and a ring vortex coexist in the component ψ 0 . In twocomponent BECs, a line vortex in one component and a ring vortex
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Fig. 3. (Color online) (a) The preimages of Fx = 0.9 and Fy = 0.9. (b) The preimage of Fz = −0.5. (c) The preimages of Fx = 0.9 and Fy = 0.9 of the Shankar skyrmion.
3
romagnetic ingredient. The F reflects ferromagnetic characteristics of the skyrmion. When the magnetization is relatively weak, the antiferromagnetic state dominates. From the Fig. 4(a) and (b), we can see that the vortex-ring has the boundary (0, 1, 0)T instead of (1, 0, 0)T of a Shankar skyrmion. The component ψ 0 extends to the boundary of the finite-size atomic cloud, altering the boundary conditions and the density of other component. The emergence of the antiferromagnetic parts looks like the three components pass though each other, causing local enhancement and local attenuation and indicating a composite vortex ring.
4. Theoretical analysis
Fig. 4. (Color online) (a) The cross sections of |ψ 1 |2 , |ψ 0 |2 and |ψ−1 |2 along the x axis for the Shankar skyrmion. (b) The cross sections of |ψ 1 |2 , |ψ 0 |2 and |ψ−1 |2 along the x axis for our numerical result. (c) The magnitude of |F|2 along the x axis for our numerical result.(d)The plot of Fi i = (x, y, z ) along the x axis for the Shankar skyrmion. (e)The plot of Fi i = (x, y, z ) along the x axis for our numerical result.
in other component form a 3D skyrmion. Some authors also call this structure as Hopfion type, i.e., twisted toroidal tubes with two independent winding numbers in single-component model [11,12]. The second component in our numerical result has the winding number m = 1 in the (x,y) plane and the second winding number (twist) s = 1 in the (r,z) plane. ψ−1 shows a torus isodensity contour. It shows a vortex line in its core along the z-axis. We have noticed this is different from the vortex line in ψ 0 with 4π phase change around the z-axis. From the perspective of overall density distribution, the line component of ψ−1 and ψ 0 is confined in a toroidal region inside the ring component of ψ 1 and ψ 0 in the neighborhood of the ring core. The three components can repel each other and prevent the vortex ring from shrinking to zero radius due to the toroidal filling of the line component. Compared to an empty vortex ring, a vortex ring filled with line component, can be more stable against collapse. Although the F texture(S2 ) cannot constitute the entire Shankar skyrmion (SO(3)). Knotted F texture with a Hopf charge π 3 (S2 )∼ =Z is an intuitive embodiment of Shankar skyrmion. In a knotted field configuration, any given two preimages interlock Q times. The linking number provides an alternative perspective on its physical significance. Fig. 3(c) shows the preimages of the Shankar skyrmion where any two tubes link once. However, F is not a good physical quantity for our numerical result. We find that the vortex-ring contains both the ferromagnetic and the antiferromagnetic states in the condensates. The value of |F| gradually becomes smaller as one moves away from the center, it then becomes larger and then smaller. The F texture is shown in Fig. 3(a) and (b). For Fx and Fy , the point on them does not correspond to a closed tube. For Fz , a negative value still correspond to a closed tube. We analyze the magnetization from the x-axis as shown in Fig. 4(c)(e). The reason that the spin F is not a linked ring can been seen from the amplitude of F and the boundary of F. When the magnetization is relatively strong, there is less antifer-
The energy stability of the 2D and 3D skyrmion in twocomponent BEC with SOC is physically understandable with the concept of the helical modulation of the order parameter (OP) [27,28]. This concept is also successfully applied to the twodimensional structure of spin-1 and spin-2 with SOC [29,30]. However, the energy stability of the 3D structure in the ferromagnetic phase of an F = 1 spinor BEC remains nontrivial. We extend this theory to a ferromagnetic spin-1 BEC with the OP under influence of SOC. The order parameter manifold in the ferromagnetic phase is SO(3). The favorable OP in the ferromagnetic state is ψ = R(k · r, nR )ξF , where R(θ , n ) denotes the rotation matrix with the angle θ about n. The 3D helical modulation propagating with all the directions of k along r fulfills the OP manifold within 0 ≤ k · r ≤ 2π . We can draw a qualitative picture of the Shankar skyrmion texture correspond to Eq.(3), which is the candidate of the ground state. But our numerical calculations reveal that it is actually not the ground state. Even though the Shankar skyrmion is a topological object. An isolate shankar skyrmion is energy unstable in the spin-1 BEC without SOC, where the Shankar skyrmion is energy unstable to shrink to zero [13]. In the ferromagnetic spin-1 Bose–Einstein condensates with SOC, a composite vortex-ring where a Shankar skyrmion can be energetically stable. In the stable vortex-rings, the magnetization leads to a mixing of the polar and FM phases. The √ √ structure can be described as ψ = nF ψShankarskyrmion + n p ψpolar , which means the shankar skyrmion is hidden in the structure. If we exclude the antiferromagnetic parts, the rest of the structure is a shankar skyrmion. We can simply analyze this conclusion by exhibiting the density distribution on the x-axis. The asymptotic boundary conditions is (0, 1, 0)T , which is the antiferromagnetic or polar state with nF → 0. If we remove ψ 0 in the nearby region, the asymptotic boundary conditions change to (1, 0, 0)T as shown in the Fig. 4(a) and (b). From the perspective of internal distribution, the most obvious difference for the composite structure and Shankar skyrmion is that the density zeros of ψ 0 and ψ 1 do not coincide. If we extract ψ 0 in the density zeros of ψ 1 , the zeros of the two densities will coincide and form (0, 0, 1)T . The reason of the manifold mixing lies in that, in the presence of SOC, the spin-dependent interaction prefers the pure ferromagnetic state, while the total energy functional prefer the manifold mixing. The wave functions can also be seen as the superposition of different orbital angular momentum states. The high orbital angular momentum states cost more kinetic energy, SOC energy and are less populated. For example, fix ratio of Y21 ~ eiα sin β cos β and Y11 ~ eiα sin β of ψ 0 in Eq.(3) is the ideal combination to form the Shankar skyrmion. The proportion of the radial portion of the Y21 and Y11 in ψ 0 is not limited to sin τ (2r ) / cos τ (2r ) due to the aforementioned reason. In comparison to Eq.(3), some antiferromagnetic states are added and form the composite vortex-rings. Composite defects also exist in other systems[31,32]. Spinor BECs may therefore shed light on topological features of composite vortex.
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From our analysis, the type of Shankar skyrmion has a higher energy than the type of the componsite vortex ring. Our numerical results also confirm that the componsite vortex ring instead of Shankar skyrmion is energy stable. We also performed real-time simulation of the imaginary-time profile as the initial state over a long interval of time. The real-time evolution establishes the dynamical stability of the composite vortex rings with the initial rings undecayed. 5. Summary We have shown that the SO coupling leads to composite vortexrings in the ferromagnetic spin-1 BECs. The structure shows a manifold mixing of ferromagnetic and antiferromagnetic states. We find the Shankar skyrmion is hidden in the structure. The emerged structures are physically understandable with the concept of the helical modulation and manifold mixing. Our work provides an alternative way of creating composite vortex-rings and enriches the field of 3D spin-orbit coupling research in the spinor BECs. This work is supported by the NSF of China under grant No.11604193 and No.11774034. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] Shariff K, Leonard A. Vortex rings. Annu Rev Fluid Mech 1992;24:235. [2] Barenghi CF, Donnelly RJ. Fluid Dyn Res 2009;41:051401.
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