Kármán vortex street in a two-component Bose–Einstein condensate

Vortex shedding from a moving obstacle potential in a two-component Bose–Einstein condensate is investigated numerically. For a miscible two-component condensate composed of 23Na and 87Rb atoms, in the wake of obstacle, the Kármán vortex street is discovered in one component, while the Kármán-like vortex street named ‘half-quantum vortex street’ is formed in another component. The other patterns of vortex shedding, such as the vortex dipoles, V-shaped vortex pairs and corresponding ‘half-quantum vortex shedding’, can also be found. The drag force acting on obstacle potential is calculated and discussed. The parameter region for various vortex patterns and critical velocity for vortex emission are presented. In addition, a 85Rb–87Rb mixture is also considered, where the Kármán vortex street and other typical patterns exist in both components. Finally, we provide an experimental protocol for the above realization and observation.


Introduction
The phenomenon of periodic vortex shedding from a symmetrical bluff body and the formation of vortices in a street are of vital significance to both theoreticians and experimenters [1]. In the classical physics, this work has been greatly triggered by von Kármán in 1912, who not only analyzed the stability of vortex street configuration, but also established a theoretical link between the vortex street structure and the drag on the body, so the periodic anti-symmetric double-row vortex street was named as Kármán vortex street [1,2]. Looking for this quantum vortex pattern in the superfluids is one way that physicists can gain a deeper understanding of the relationship between classical and quantum fluids.
Nowadays, Bose-Einstein condensate (BEC) has provided a clean testing ground for the microscopic physics of vortex shedding. Numerical simulations based on the Gross-Pitaevskii equation (GPE) have shown that the vortex dipoles or vortex-antivortex pairs can be nucleated when a superfluid moves past an obstacle faster than the critical velocity, above which vortex shedding induces the drag force [3][4][5][6][7][8]. Moreover, the vortex dipoles can also be generated by the moving Gaussian potential [9,10], the oscillating Gaussian potential [11] in a condensate, the circular motion of a Gaussian potential stirring the condensate [12], or the moving circular potential in the plane-wave state of a pseudospin-1/2 BEC with Rashba spin-orbit coupling [13]. In the reported experiment [14], the vortex dipoles have been nucleated by causing a highly oblate BEC to move past a repulsive Gaussian obstacle, and the critical velocity for vortex dipole shedding has been measured.
In 2010, the Kármán vortex street in a single-component BEC, in which an obstacle potential moves, has been first investigated via numerical calculations by Sasaki et al [15]. Subsequently, for a strong phase-separated two-component BEC, in which a bubble of one component moves through the other component, the vortex street is also formed in the wake of moving bubble [16]. Soon after, an elliptical obstacle moving through the BEC also generates the wake of quantum vortices, which resembles the Kármán vortex street, and the role of ellipticity is to facilitate the interaction of vortices nucleated by obstacle [17,18]. In particular, in a strongly correlated superfluid system, i.e. superfluid 4 He at zero temperature, when two-dimensional flow moves past an infinitely long cylinder of nanoscopic radius, the onset of dissipation is characterized by vortex-antivortex pair shedding from the periphery of moving cylinder, while vortex pairs with the same circulation are occasionally emitted in the form of dimers, which constitute the building blocks for the Kármán vortex street structure [19]. On the other hand, through a series of experiments [20][21][22], in a highly oblate BEC, the vortex shedding from a moving obstacle, which is formed by focusing a repulsive Gaussian laser beam, is investigated by Kwon et al Finally, in 2016, they have reported on the experimental observation for the first time, where the shedding of vortex clusters each consisting of the two like-sign vortices is generated to form a regular configuration in a periodic and stagger manner, like the Kármán vortex street [23].
In the two-component BEC, a variety of dynamical properties have been observed and reported [24][25][26][27][28][29], and the tunable inter-component interaction has been realized experimentally by using the Feshbach resonance technology [30,31]. Especially, the quantized vortices have been created and studied [32][33][34][35][36][37][38][39]. For instance, a vortex dipole can not only penetrate the interface between the two components, but also disappear or disintegrate at the interface, depending on its velocity and the interaction parameters [40]. However, as far as we know, there has been little work on studying the vortex shedding from a moving obstacle in a twocomponent BEC.
In this paper, we investigate the vortex shedding from a moving obstacle potential in a two-component BEC. Here, we systematically perform the numerical simulations to solve the coupled GPEs. First, we consider a miscible two-component condensate which is composed of 23 Na and 87 Rb atoms at length. The results show that, in the 87 Rb component, the Kármán vortex street is discovered in the wake of obstacle potential; in the 23 Na component, the Kármán-like vortex street named 'half-quantum vortex street', which is formed for 'halfquantum vortices', is also discovered. The various patterns of vortex shedding, such as no vortex, vortex dipoles, V-shaped vortex pairs, irregular vortices, and corresponding 'half-quantum vortex shedding', can also be found. The drag force acting on the moving obstacle potential is calculated, which depends upon the radius and velocity of obstacle. Also shown is the contribution to the drag force from vortex shedding. The parameter region for all kinds of vortex patterns and the fitting curve of critical velocity for vortex emission are presented. The critical velocity is related to the geometry of obstacle and the speed of sound waves. In addition, a dual-species 85 Rb-87 Rb condensate is also considered, and there exist the Kármán vortex street and other typical patterns in both components. The drag force, parameter region and critical velocity are also presented. In particular, in the 23 Na-87 Rb mixture, the absence of vortices in one component could be related to the different sound velocity with respect to the 85 Rb-87 Rb mixture. Finally, we offer a protocol to realize and observe the above dynamics in future experiments.

Formulation of the problem
Considering a two-component BEC within the framework of zero-temperature mean-field theory, the dynamics of system can be well described by the coupled GPEs [16,41,42]: where ψ j denotes the macroscopic wave function, m j is the atomic mass of the jth component ( j=1, 2), and V is the external potential. The number of the jth atoms is N r d ) being the reduced mass and a jj¢ being the s-wave scattering length between atoms in the jth and j¢ th components, which can be controlled by Feshbach resonance technology [30,31] ) with ω x , ω y , and ω z being the x-, y-, and z-directions trapping frequencies [43]. Here a Gaussian obstacle potential with peak strength V 0 and radius d is employed, which moves in the −x direction at a velocity υ [15] V Ve , 2 obs 0 where (x 0 , y 0 ) is the center position of obstacle potential at t=0. In the experiments, this moving obstacle potential can be produced by a repulsive Gaussian laser beam along the z direction [9,14,15,[20][21][22][23]. Here, the system can be reduced to quasi-two-dimensional by considering a very strong confinement along z axis, i.e. ÿω z being much larger than any other energy scale [42][43][44][45][46], which is easily realized in the experiments, e.g. the experiments of Kwon et al [20][21][22][23]. It is reasonable that the wave function can be expressed as r t x y t z , , , e , and the motion of atoms along the z-direction is frozen into the ground to be the temporal and spatial characteristic quantities, where ω 0 =min[ω x , ω y ]. One can now obtain the dimensionless coupled GPEs as is the 2D Laplacian operator and μ m =m 2 /m 1 . The corresponding dimensionless interaction parameters read g a n l 2 where n 0 is a given particle density. There is an assumption that the moving obstacle potential V obs is the same for both components. In general, one can take the maximum particle density as n 0 (but this is not necessary). The number of the jth atoms is expressed by N n x y d d The speed of sound waves can be written as m The various physical quantities in equation (3) are normalized according to

Vortex shedding from the obstacle
In order to study the vortex shedding from obstacle, we consider a two-component BEC composed of 23 Na and 87 Rb atoms. Here, in order to simplify the problem without loss of generality, the interaction parameters can be taken as g 11 =g 22 =1.0 and g 12 =g 21 =0.5 to satisfy the condition g g g 11 22 12 2 > and g 0 jj > ¢ , so that the two components are miscible and stable [38,39]. This situation can be easily realized in the experiments and it will be discussed later in detail. When the numerical calculations are performed, [−256, 256]⊗[−64, 64] spatial domain will be discretized into 2048×512 grids. First, we find the ground state of equation (3) by the imaginary-time | | , and phase distributions, arg j 1 , arg j 2 , of the vortex shedding from a moving obstacle potential in the 23 Na-87 Rb mixture. The potential radius and velocity are d=1.0 and υ=0.25. The dotted arrow indicates the direction in which the obstacle potential moves and the solid curved arrow indicates the direction in which a pair of two point quantized vortices or two point 'half-quantum vortices' rotates. The symbols + and − denote the clockwise and counterclockwise circulations of quantized vortices. The density is normalized by n 0 . The field of view is 64×32 in dimensionless units.
propagation approach [43] with υ=0 and initial condition j 1 =j 2 =1. Then, the time-splitting spectral method [47][48][49] is employed to make the nonlinear dynamical evolution with periodic boundary condition, where the initial condition is adopted as the ground state obtained above plus a small amount of noise to break the symmetry of system. The Gaussian obstacle potential V obs with V 0 =100 and (x 0 , y 0 )=(128, 0) will begin to move along the −x direction at a velocity υ for t>0. In component 2, it is significant that the two point quantized vortices in a pair, which shed from the moving obstacle potential at a time, have the same circulation. The quantized vortex is the topological defect of order parameter and the circulation is quantized to h/m j [9,40] with h being the Planck's constant. Here, the symbols + and − denote the clockwise and counterclockwise circulations of quantized vortices. Hence, the circulation of top row is +h/m, while that of bottom row is −h/m. Note that, the two point quantized vortices having the same circulation in a pair rotate around their center without changing distance between vortex and vortex and between j | | , and phase distributions, arg j 1 , arg j 2 , of the vortex shedding from a moving obstacle potential in the 23 Na-87 Rb mixture at different time. The used parameters are the same as those in figure 1, the radius and velocity (d, υ)=(1.0, 0.25). The dotted arrow indicates the direction in which the obstacle potential moves and the solid curved arrow indicates the direction in which a pair of two point quantized vortices or two point 'half-quantum vortices' rotates. The symbols + and − denote the clockwise and counterclockwise circulations of quantized vortices. The density is normalized by n 0 . The time interval is 40/ω 0 . The field of view is 32×32 in dimensionless units. pair and pair. As a result, the vortex shedding will form a periodic anti-symmetric double-row vortex street in the wake of moving obstacle potential, this is the signature of quantum analog of the Kármán vortex street. Here we have also measured the distance between two vortex rows being b≈4.38a 0 and that between two vortex pairs in the one vortex row being l≈15.88a 0 on average, and hence b/l≈0.28. This ratio is in good agreement with the stability condition of Kármán vortex street arrangement b l cosh 2 0.28 2,15]. Correspondingly, in component 1, there exists the Kármán-like vortex street formed for the 'half-quantum vortices'. One can see from figures 1(e) and (f) that, 23 Na atoms fill into the 'pits' in component 2 but the 'pits' are not full-filled, and the position of a density peak in component 1 is the center position of a vortex in component 2. The density peak is a part of vortex structure in the two-component BEC, so that this density distribution structure in component 1 can be named as 'half-quantum vortex street'. Figure 2 shows the dynamics of the Kármán vortex street and 'half-quantum vortex street' formation from the moving obstacle potential in detail. The used parameters are the same as those in figure 1. It is obvious that the two point quantized vortices having the same circulation and corresponding 'half-quantum vortices' are periodically and alternately shed from the moving obstacle potential. The pairs of quantized vortices or 'half-quantum vortices' are released obliquely backward left and right with alternate circulations. Eventually, in the wake of moving obstacle potential, the Kármán vortex street and 'half-quantum vortex street' are formed in the component 2 and component 1, respectively. Figures 3 and 4 show the other typical patterns of vortex shedding shed from a moving obstacle potential with the different radius d and velocity υ. As shown in figures 3(a)-(d), when the velocity υ exceeds the critical velocity, that we shall characterize later, the vortex dipoles and 'half-quantum vortex dipoles' will begin to be periodically and symmetrically shed from the obstacle potential. The vortex dipole consists of two point quantized vortices having the opposite circulation, +h/m and −h/m. If the radius d or velocity υ are further increased, the vortex-antivortex pairs and 'half-quantum vortex-antivortex pairs' will be periodically and alternately generated. Since a symmetric double row of vortices is unstable, the vortex-antivortex pairs and 'half-quantum vortex-antivortex pairs' will drift along the +y or −y direction, eventually forming the vortex pattern of V-shaped vortex pairs and 'half-quantum vortex pairs', as shown in figures 3(e)-(h). Especially, as indicated by the solid arrows in figures 3(a), (c), (e) and (g), a pair of two point quantized vortices or 'halfquantum vortices' moves in the direction perpendicular to a line between the pair. For a sufficiently small radius d or velocity υ, the flow around the obstacle is a steady laminar flow and there is no quantized vortex and 'halfquantum vortex' for two components, as shown in figures 4(a)-(d). Conversely, if the radius d or velocity υ is too large, the shedding of quantized vortices and 'half-quantum vortices' will no longer have obvious periodicity and begin to be irregular, as shown in figures 4(e)-(h). Figure 5 shows the normalized drag force acting on the moving obstacle potential, which is given by with Φ=(j 1 , j 2 ) T , where T denotes transpose. Corresponding to the situation of no quantized vortex and 'half-quantum vortex' (figures 4(a)-(d)), it is obvious that both f x and f y are about zero, as shown in figure 5(a). However, when the vortex dipoles and 'half-quantum vortex dipoles' are created periodically and symmetrically (figures 3(a)-(d)), f x begin to have an clear periodicity, as shown in figure 5(b). For the vortex patterns of V-shaped vortex pairs and 'half-quantum vortex pairs' (figures 3(e)-(h)) or Kármán vortex street and 'half-quantum vortex street' (figures 1(a)-(d)), both f x and f y are almost periodic, while the periodicity of f x is only half that of f y , as shown in figures 5(c) and (d). It is due to that the vortex emission are not only periodic but also alternate, just as the quantized vortex and 'half-quantum vortex' are released obliquely backward left and right with alternate circulations (see in figure 2). When the shedding of quantized vortices and 'half-quantum vortices' are irregular (figures 4(e)-(h)), both f x and f y will have no obvious periodicity, as shown in figure 5(e). To deeply understand the dependence of drag force acting on the moving obstacle potential, we calculate the normalized time-averaged drag force which can be given by f f T t T 0 = å = | | with T being total time. As shown in figure 6, it is obvious that the drag force depends upon the radius d and velocity υ of the moving obstacle potential. With the increase of radius and velocity, the drag force f x and f ȳ raises gradually while there are some fluctuations. In particular, when velocity is more than the critical velocity, the drag force f x will increase substantially. Also shown is the contribution to the drag force from vortex shedding, namely the vortex shedding induces drag force.   1(a) and (b)), irregular 'half-quantum vortices' (figures 4(e) and (f)), respectively. Obviously, with the variation of radius d or velocity υ, the pattern of vortex shedding changes from one to another. In particular, compared with the other vortex patterns, the region of vortex street is rather restricted. In figure 7, the white curve is the fitting curve of dimensionless critical velocity for vortex emission c d 1 p c s u u = ( ) with c≈1/4.79 and p≈1/3.71 in the 23 Na-87 Rb mixture. It is obvious that the dimensionless critical velocity υ c /υ s depends on the geometry of moving obstacle potential and shows a 1/d p dependence with p<1, which is in good agreement with the result in the experiments which the vortex shedding is produced by a repulsive Gaussian laser beam moving through a highly oblate BEC [21] and is also the case in a strongly correlated superfluid 4 He system which a two-dimensional flow past an infinitely long cylinder of nanoscopic radius [19]. Besides, the critical velocity υ c is also related to the speed of sound waves υ s in the BEC.
In addition to the 23 Na-87 Rb mixture discussed before, we consider another two-component BEC composed of 85 Rb and 87 Rb atoms. The interaction parameters are taken as g 11 =g 22 =1.0, g 12 =g 21 =0.9 to satisfy the miscible condition. Figure 8 displays the main results, unlike the situation of 23 Na-87 Rb mixture, it is in the both components of 85 Rb-87 Rb mixture that the quantized vortices exist. It is obvious that the Kármán vortex street has been formed in the wake of a moving obstacle potential in both components, as shown in figures 8(a)-(d). Here the ratio is also b/l≈0.28 with b≈5.63a 0 and l≈20.13a 0 on average, which satisfies the stability condition. The other typical patterns of vortex shedding are also created by a moving obstacle potential with the different radius d and velocity υ. Only the density distribution  Finally, we provide an experimental protocol for the realization of above Kármán vortex street and the dynamics of quantized vortices can be observed by the destructive imaging technique [14,42]. To do so, we consider a miscible two-component BEC composed of 23 (3). Then, the moving obstacle potential is produced by a repulsive Gaussian laser beam [9,14,15,[20][21][22][23] along the z direction with peak strength V 0 =2.65×10 −31 J and radius d=10.46 μm, which is initially located at (x 0 , y 0 )=(1.34,0) mm at t=0, and moves in the −x direction at a velocity 0.066 mm s 1 u = for t>0, corresponding to the parameters used in figure 1.

Conclusion
In summary, we have numerically investigated the vortex shedding from a moving obstacle potential in a twocomponent BEC by solving a coupled GPEs. We consider two kinds of miscible two-component condensate, one is composed of 23 Na and 87 Rb atoms , the other is composed of 85 Rb and 87 Rb atoms. In the 23 Na-87 Rb mixture, the Kármán vortex street is discovered in the 87 Rb component, while the Kármán-like vortex street named 'half-quantum vortex street', which is formed for 'half-quantum vortices', is discovered in the 23 Na component. The various patterns of vortex shedding, such as no vortex, vortex dipoles, V-shaped vortex pairs, irregular vortices, and corresponding 'half-quantum vortex shedding', can also be found. In the 85 Rb-87 Rb mixture, there exist the Kármán vortex street and other typical patterns in both components. The drag force acting on the moving obstacle potential is calculated, which depends upon the radius and velocity of obstacle. Also shown is the contribution to the drag force from vortex shedding. The parameter region for all kinds of vortex patterns and the fitting curve of critical velocity for vortex emission are presented. The critical velocity is related to the geometry of obstacle and the speed of sound waves. In particular, in the 23 Na-87 Rb mixture, the absence of vortices in the 23 Na component could be related to the different sound velocity with respect to the 85 Rb-87 Rb mixture. We also provide a protocol for the above realization and observation in future experiments. Finally, we hope that these results can stimulate the further investigation of the vortex shedding in BEC.