Padé approximations of quantized-vortex solutions of the Gross–Pitaevskii equation

Weiru Chen,Shanquan Lan,Xiyi Liu,Jiexiong Mo,Xiaobao Xu and Guqiang Li

Institute of Theoretical Physics,Lingnan Normal University,Zhanjiang,524048,Guangdong,China

Abstract Quantized vortices are important topological excitations in Bose–Einstein condensates.The Gross–Pitaevskii equation is a widely accepted theoretical tool.High accuracy quantized-vortex solutions are desirable in many numerical and analytical studies.We successfully derive the Padé approximate solutions for quantized vortices with winding numbers ω=1,2,3,4,5,6 in the context of the Gross–Pitaevskii equation for a uniform condensate.Compared with the numerical solutions,we find that(1)they approximate the entire solutions quite well from the core to infinity;(2)higher-order Padé approximate solutions have higher accuracy;(3)Padé approximate solutions for larger winding numbers have lower accuracy.The healing lengths of the quantized vortices are calculated and found to increase almost linearly with the winding number.Based on experiments performed with 87Rb cold atoms,the healing lengths of quantized vortices and the number of particles within the healing lengths are calculated,and they may be checked by experiment.Our results show that the Gross–Pitaevskii equation is capable of describing the structure of quantized vortices and physics at length scales smaller than the healing length.

Keywords:Gross–Pitaevskii equation,Padé approximation,quantized-vortex solution

1.Introduction

Bose–Einstein condensates(BECs)have been extensively studied,both theoretically and experimentally,since their experimental realization in trapped atomic gases at ultralow temperatures[1–3].The Gross–Pitaevskii(GP)equation,which is a mean-field quantum field theory,is a widely accepted theoretical tool[4–6].The GP equation has the merit of providing an effective description of atomic condensates and their dynamics,such as the reconnection of quantized vortices[7–12],the splitting of multiply quantized vortices[13–18],and quantum turbulence[19–21],etc.Before an investigation of its interesting dynamic features,one should know its elementary structures,such as sound waves,solitons,and quantized vortices,which are also interesting and important in themselves.In this paper,we will concentrate on the quantized vortices of different winding numbers in a uniform two-dimensional condensate.

Quantized vortices,the winding of order parameter,are one of the most fundamental topological excitations in BECs.In spite of the seeming simplicity of the GP equation,there is no analytic solution for quantized vortices.Typically,one resorts to numerical methods to obtain quantized-vortex solutions[22–24].Meanwhile,there are many situations in which asymptotic or approximate vortex solutions are desired.For example,Demircan investigated singly quantized vortex dynamics in superfluids[25]based on an approximate vortex solution given by Fetter[26]

where φ is the wave function(see equation(10)).This semianalytical expression is simple.However,there is a big deviation from the numerical solution:the maximum disagreement is about δφ≈0.06(see the blue line in figure 1).Bradley applied a similar expression(where the constant 2 was replaced by a length scale parameter)to investigate the energy spectra of vortex distributions in quantum turbulence[27].Also,in many numerical studies,a specific vortex configuration is required as an initial starting condition[7,8,28–30].Therefore,finding good approximate vortex solutions is important.The Thomas–Fermi approximation method is valid in regions far from the vortex core[31].The Adomian decomposition method achieves good agreement between the semi-analytical solution and the numerical solution for small radii[24],but it fails in regions far from the vortex core.Other methods involve asymptotic expansions in several different regions and asymptotic matching between them,but the resulting approximations of the solutions are complicated[32,33].This problem was greatly abated when a much more accurate expression for singly and doubly quantized vortices was derived by Berloff,who applied the Padé approximation method.

The Padé approximate solution is a rational function formed by the ratio of two power series.The numerator and denominator coefficients are determined by equating the approximate solution’s power series with the power series of the function it is approximating[34,35].The Padé approximate solution is more accurate than the Taylor series in approximating functions with poles.Therefore,as we will see below,this method works well for the quantized vortices that have poles.In practice,the standard Padé approximation method will be modified here.The general idea is that,if a function F(x)has power series expansions of the format x=0 andat x=∞,then the Padé approximate solution is

where n=1,2,3,…,N=0,1,2,…,andwith Q0=F(∞),have the same asymptotes at zero and infinity as function F(x)does[36].

For a singly quantized vortex,Berloff derived a secondorder(N=1)Padé approximate solution[36]

This expression is much more accurate than equation(1).There is only a small deviation from the numerical solution,the maximum disagreement is about δφ≈0.01(see the green line in figure 2).Later,Rorai and others derived a third-order(N=2)Padé approximate solution[23]

This expression is much more accurate than equation(3).Its maximum disagreement with the numerical solution is only δφ≈0.0016(see the blue line in figure 3).Beyond the above solutions,we have derived other solutions which have same order and accuracy(see more below),after a systematic analysis of the Padé approximation method.For a doubly quantized vortex,Berloff derived a second-order(N=1)Padé approximate solution[36]

This has a maximum disagreement of δφ≈0.06 with the numerical result(see the green line in figure 4).The requirement for accuracy is not satisfied,and we have derived other solutions which are more accurate.What is more,we have derived Padé approximate solutions for other multiply quantized vortices and calculated the healing lengths of quantized vortices.

The outline of this paper is as follows.In section 2,the asymptotic behaviors of quantized vortices are analyzed at zero and infinity.In section 3,different orders of Padé approximate solutions of quantized vortices with different winding numbers are presented and then compared with numerical solutions.In section 4,the healing lengths of the quantized vortices are discussed.In the last section,we end with conclusions.

2.Asymptotic behavior of quantized vortices at zero and infinity

We start with the three dimensional Gross–Pitaevskii equation[4–6],which is also known as the nonlinear Schr?dinger equation,

where ψ(x,y,z,t)is the wave function and n3D=|ψ(x,y,z,t)|2is the number density of particles,M is the mass of the bosons,g3D=4π?2as/M is the interaction parameter for an s-wave scattering length as,and μ is the chemical potential.

In this paper,we explore the structures of quantized vortices in a uniform two-dimensional Bose–Einstein condensate.Assuming that the thickness of the thin condensate is d,then the two-dimensional number density of particles can be written as n2D=n3Dd.Thus,the corresponding GP equation becomes

By applying a rescale transformation

the above GP equation can be rewritten in the following dimensionless form,

As quantized vortices are static and have cylindrical symmetry,we will work in cylindrical coordinates(ρ,θ).The complex function φ(x,y)can be rewritten as φ(ρ)eiωθ,where ω is the winding number of the vortex.Here,ω have to be integers(without loss of generality,only positive ω cases are considered throughout this paper)to guarantee that φ(x,y)is a single-valued function.The GP equation is rewritten as

Assuming φ∝ραwhen ρ→0,equation(10)gives α=ω.However,as ρ→∞,equation(10)gives φ=1.To apply the Padé approximation,one needs to check the detail of the asymptotic behavior of quantized vortices at zero and at infinity.

When ρ→0,the asymptotic behavior of quantized vortices can be written as

Here,the first ten terms are sufficient to show the behavior of the quantized vortex function φ(ρ)with winding numbers ω≤6 when ρ→0.We then substitute the above equation into equation(10)and expand the resulting expression in series of ρ.The coefficients at equal powers of ρ should be equal to zero.The first ten equations are listed below.

For a quantized vortex with a winding number ω=1,the ten equations above give

For a quantized vortex with a winding number ω=2,the ten equations above give

For a quantized vortex with a winding number ω=3,the ten equations above give

For a quantized vortex with a winding number ω=4,the ten equations above give

For a quantized vortex with a winding number ω=5,the ten equations above give

For a quantized vortex with a winding number ω=6,the ten equations above give

One can find that the function φ(ρ)of quantized vortices has power series expansions of the format ρ=0.

At ρ→∞,the asymptotic behavior of quantized vortices can be written as

Then we substitute the above equation into equation(10)and expand the resulting expression in series of ρ?1.The coefficients at equal powers of ρ?1should be equal to zero.The first 8 equations are listed below.

The above 8 equations give

One can find that the function φ(ρ)of quantized vortices has power series expansions of the form(here,Q0=1)at ρ=∞.

It is natural to ask that what kind of function has the same power series expansions of the above forms at ρ=0 and at ρ=∞.It is the Padé approximate solution,

where b0=1 and aN=bN+ω.In the following section,we will find the Padé approximate solutions of quantized vortices with winding numbers ω=1,2,3,4,5,6 and then compare them to the numerical solutions.

3.Quantized-vortex solutions in the Padé approximation

3.1.The ω=1 case

For N=0,the approximate solution is

Note that there is only one parameter,which can be determined by just one equation.There are two ways to find the equation.One is to expand the above equation at ρ=0 or at ρ=∞and then compare them to equation(22)or equation(37).The other way is to substitute the above equation into equation(10)and then expand the resulting expression in series of ρ at ρ=0 or in series of ρ?1at ρ=∞.The coefficients of the series are set to zeros.In this paper,we will use the second way.The coefficients at O(ρ)and O(ρ-2)give

Fetter improved the approximate solution by choosing b0=2[26].Figure 1 shows the vortex profiles and the difference between the approximate solutions and the numerical solution.In this paper,the numerical vortex solutions are obtained by the Chebyshev pseudo-spectral method[37].

Figure 1.(a)Vortex profiles:the red line represents the numerical solution,and the green,black,and blue lines represent the solutions of the Padé approximation for ω=1 and N=0.(b)The difference between the approximate solutions and the numerical solution.

Figure 2.(a)Vortex profiles:the red line represents the numerical solution and the green,black,blue,and yellow lines represent the solutions of the Padé approximation for ω=1 and N=1.(b)The difference between the approximate solutions and the numerical solution.

Figure 3.(a)Vortex profiles:the red line represents the numerical solution and the green,black,blue,and yellow lines represent the solutions of the Padé approximation for ω=1 and N=2.(b)The difference between the approximate solutions and the numerical solution.

Figure 4.(a)Vortex profiles:the red line represents the numerical solution and the green,black,blue,and yellow lines represent the solutions of the Padé approximation for ω=2 and N=1,2.(b)The difference between the approximate solutions and the numerical solution.

For N=1,the approximate solution is

Note that there are three parameters which can be determined by three equations.In the same way,we substitute the above equation into equation(10)and then expand the resulting expression in series of ρ at ρ=0 and in series of ρ?1at ρ=∞.The coefficients of the series are set to zeros.The coefficients at O(ρ),O(ρ3),and O(ρ5)give three equations and their only reasonable solution is[36]

There are other choices for the three equations.For example,coefficients at O(ρ),O(ρ3),and O(ρ-2)also give three equations and their only reasonable solution is

The coefficients at O(ρ),O(ρ-2),and O(ρ-4)give three equations and their only reasonable solution is

Although the coefficients at O(ρ-2),O(ρ-4),and O(ρ-6)give three equations,there is no reasonable solution.Figure 2 shows the vortex profiles and the difference between the approximate solutions and the numerical solution for ω=1 and N=1.We can see that when more equations are chosen on the ρ=0 side,the approximate solution is more accurate there.Conversely,when more equations are chosen for ρ=∞,the approximate solution is more accurate there.To find the best approximate solutions,one needs to choose the right equations.

For N=2,the approximate solution is

Note that there are five parameters,which can be determined by five equations.In the same way,the coefficients at O(ρ),O (ρ3),O(ρ5),O(ρ7),and O(ρ-2)give five equations,and their only reasonable solution is

There are other choices.For example,coefficients at O(ρ),O (ρ3),O(ρ5),O(ρ-2)and O(ρ-4)give five equations and their only reasonable solution is

Based on numerical(by shooting)results p1=0.58278,Rorai et al used the first way to expand equation(45)at zero and at infinity,and then compared the results with{pi,qi}in equation(37)and equation(37)to derive[23]

Figure 1,figure 2,and figure 3 show that the Padé approximations of vortex solutions are in good agreement with the numerical solution,especially for the larger N case.

3.2.The ω=2 case

For N=1,the approximate solution is

In the same way,the coeffciients at O(ρ2),O(ρ4),O(ρ6),and O (ρ8)give four equations and their only reasonable solution is[36]

There are,however,other choices.For example,the coefficients at O(ρ2),O(ρ4),O(ρ6),and O(ρ-2)give four equations and their only reasonable solution is

For N=2,the approximate solution is

In the same way,the coeffciients at O(ρ2),O(ρ4),O(ρ6),O(ρ8),O (ρ10),and O(ρ-2)give six equations and their only reasonable solution is

If we choose the equations given by the coeffciients at O(ρ2),O (ρ4),O(ρ6),O(ρ8),O(ρ-2),and O(ρ-4),then the only reasonable solution is

Figure 4 shows the Padé approximation of the quantizedvortex solutions of ω=2 for N=1,2,and the difference between the approximate solutions and the numerical solution.The Padé approximate solutions for N=2 are more accurate than those for N=1.

3.3.The ω=3 case

For N=1,the approximate solution is

In the same way,the coefficients at O(ρ3),O(ρ5),O(ρ7),O (ρ9),and O(ρ-2)give five equations and their only reasonable solution is

There are other choices:for example,the coefficients at O (ρ3),O(ρ5),O(ρ7),O(ρ-2),and O(ρ-4)give five equations.Unfortunately,there is no reasonable solution.However,the coefficients at O(ρ3),O(ρ5),O(ρ-2),O(ρ-4),and O(ρ-6)give five equations and their only reasonable solution is

For N=2,the approximate solution is

In the same way,the coefficients at O(ρ3),O(ρ5),O(ρ7),O (ρ9),O(ρ11),O(ρ13),and O(ρ-2)give seven equations and their only reasonable solution is

If we choose equations given by the coefficients at O(ρ3),O (ρ5),O(ρ7),O(ρ9),O(ρ11),O(ρ-2),and O(ρ-4),then the only reasonable solution is

Figure 5 shows the Padé approximation of the vortex solutions for ω=3 and the difference between the approximate solutions and the numerical solution.

Figure 5.(a)Vortex profiles:The red line represents the numerical solution,and the green,black,blue,and yellow lines represent the solutions of the Padé approximation for ω=3 and N=1,2.(b)The difference between the approximate solutions and the numerical solution.

3.4.The ω=4 case

For N=1,the approximate solution is

In the same way,the coefficients at O(ρ4),O(ρ6),O(ρ8),O (ρ10),O(ρ12),and O(ρ-2)give six equations and their only reasonable solution is

If we choose equations given by the coefficients at O(ρ4),O (ρ6),O(ρ8),O(ρ10),O(ρ-2),and O(ρ-4),unfortunately,there is no reasonable solution.However,the coefficients at O (ρ4),O(ρ6),O(ρ8),O(ρ-2),O(ρ-4),and O(ρ-6)give six equations and their only reasonable solution is

For N=2,the approximate solution is

In the same way,the coefficients at O(ρ4),O(ρ6),O(ρ8),O (ρ10),O(ρ12),O(ρ14),O(ρ16),and O(ρ-2)give eight equations and their only reasonable solution is

If we choose equations given by the coefficients at O(ρ4),O (ρ6),O(ρ8),O(ρ10),O(ρ12),O(ρ14),O(ρ-2),and O(ρ-4),then the only reasonable solution is

Figure 6 shows the Padé approximation for the vortex solutions of ω=4 and the difference between the approximate solutions and the numerical solution.

Figure 6.(a)Vortex profiles:the red line represents the numerical solution and the green,black,blue,and yellow lines represent the solutions of the Padé approximation for ω=4 and N=1,2.(b)The difference between the approximate solutions and the numerical solution.

3.5.The ω=5 case

For N=1,the approximate solution is

In the same way,the coefficients at O(ρ5),O(ρ7),O(ρ9),O (ρ11),O(ρ13),O(ρ15),and O(ρ-2)give seven equations and their only reasonable solution is

If we choose equations given by the coefficients at O(ρ5),O (ρ7),O(ρ9),O(ρ11),O(ρ13),O(ρ-2),and O(ρ-4),unfortunately,there is no reasonable solution.However,the coefficients at O(ρ5),O(ρ7),O(ρ9),O(ρ11),O(ρ-2),O(ρ-4),and O(ρ-6)give seven equations and their only reasonable solution is

For N=2,the approximate solution is

In the same way,the coefficients at O(ρ5),O(ρ7),O(ρ9),O (ρ11),O(ρ13),O(ρ15),O(ρ17),O(ρ-2),and O(ρ-4)give nine equations and their only reasonable solution is

If we choose equations given by the coefficients at O(ρ5),O (ρ7),O(ρ9),O(ρ11),O(ρ13),O(ρ15),O(ρ-2),O(ρ-4),and O(ρ-6),then the only reasonable solution is

Figure 7 shows the Padé approximation of the vortex solutions of ω=5 and the difference between the approximate solutions and the numerical solution.

Figure 7.(a)Vortex profiles:the red line represents the numerical solution and the green,black,blue,and yellow lines represent the solutions of the Padé approximation for ω=5 and N=1,2.(b)The difference between the approximate solutions and the numerical solution.

3.6.The ω=6 case

For N=1,the approximate solution is

In the same way,the coefficients at O(ρ6),O(ρ8),O(ρ10),O (ρ12),O(ρ14),O(ρ16),O(ρ-2),and O(ρ-4)or the coefficients at O(ρ6),O(ρ8),O(ρ10),O(ρ12),O(ρ14),O(ρ-2),O(ρ-4),and O(ρ-6)give eight equations.Unfortunately,there are no reasonable solutions.If we choose equations given by the coefficients at O(ρ6),O(ρ8),O(ρ10),O(ρ12),O(ρ-2),O(ρ-4),O(ρ-6),and O(ρ-8),then the only reasonable solution is

For N=2,the approximate solution is

In the same way,the coefficients at O(ρ6),O(ρ8),O(ρ10),O (ρ12),O(ρ14),O(ρ16),O(ρ18),O(ρ20),O(ρ-2),and O(ρ-4)give ten equations and their only reasonable solution is

If we choose equations given by the coefficients at O(ρ6),O (ρ8),O(ρ10),O(ρ12),O(ρ14),O(ρ16),O(ρ18),O(ρ-2),O(ρ-4),and O(ρ-6),then the only reasonable solution is

Figure 8 shows the Padé approximation of the vortex solutions of ω=6 and the difference between the approximate solutions and the numerical solution.

Figure 8.(a)Vortex profiles:the red line represents the numerical solution and the green,black,blue,and yellow lines represent the solutions of the Padé approximation for ω=6 and N=1,2.(b)The difference between the approximate solutions and the numerical solution.

Figure 9.(a)Density profiles of quantized vortices with winding numbers ω=1,2,3,4,5,6.(b)The healing lengths?defined as φ2(?)/φ2(∞)=0.9 for the six quantized vortices on the left.

4.Healing lengths of quantized vortices with different winding numbers

The healing length?is the scale on which the density rises to the uniform background value n2D(ρ=∞).Thus,it can be used to characterize the sizes of quantized vortices.The healing length of a quantized vortex is usually defined as[38,39]

In our dimensionless GP equation,the above definition gives?=1,which is an important length scale.However,this definition is stereotypical,since the healing lengths of quantized vortices should be different for different winding numbers.In addition,even for the quantized vortex of ω=1,equation(46)gives the result that n2D(ρ=1)≈0.27,which is much less than n2D(ρ=∞)=1.0 and which does not heal the depletion of the quantized vortex.

Therefore,we define a new healing length?as

where the value of the constant c chosen is c=0.9.Thus,equation(46),equation(53),equation(60),equation(66),equation(71),and equation(76)give density profiles n2D=φ2(ρ)of the number of particles and the healing lengths?of quantized vortices with different winding numbers(see figure 9).It is somewhat surprising that the healing lengths increase almost linearly with the winding number according to the fitting function?=0.41+3.13ω.This feature still holds if one chooses the constant c to have values from 0.05 to 0.95.

In experiments with87Rb cold atoms(M=1.44×10?25kg),the density n3Dis about 1019/m3[40,41]and the s-wave scattering length asis about 5nm[42,43].The chemical potential is the derived from the GP equation,yielding μ≈4.8×10?32J.The healing lengths of quantized vortices with different winding numbers become

which are much larger than the average spacing of the particles.For a thin condensate with a thickness d=1um,one can derive the number W of particles within the healing lengths of quantized vortices of different winding numbers as

which are big numbers.The above calculated healing lengths of quantized vortices and the numbers of particles inside the healing lengths could be checked by experiment.If the healing lengths?are small compared with the average spacing of the particles and the quantized vortices possess only a few particles,then the quantized-vortex solutions of the Gross–Pitaevskii equation are unreliable.In other words,we would show that the Gross–Pitaevskii equation is capable of describing the structure of the quantized vortices and the physics at length scales smaller than the healing length.

5.Conclusions

The main work of this paper is to use the Padé approximation method to find the approximate solutions for quantized vortices of winding numbers ω=1,2,3,4,5,6 in the Gross–Pitaevskii equation.First,the asymptotic behaviors of quantized vortices at zero and at infinity were analysed in section 2.At ρ→0,At ρ→∞,

Second,Padé approximate solutions of different orders for quantized vortices with winding numbers ω=1,2,3,4,5,6 were derived and compared with numerical solutions in section 3.These solutions have the same asymptotic behaviours at zero and at infinity of the quantized-vortex solutions.In addition,they approximate the entire solutions quite well elsewhere.The higher-order Padé approximate solutions have higher accuracy,and the Padé approximate solutions of larger winding numbers have lower accuracy.Thus,for numerical or explicit analytic studies involving quantized vortices,we provide good approximate solutions for different orders and different winding numbers.

Thirdly,the healing lengths of the quantized-vortex solutions are calculated in section 4.They increase almost linearly with the winding number.Based on experiments on87Rb condensates,the healing lengths of the quantized vortices and the number of particles inside the healing lengths are derived,which could be checked experimentally.The data show that the Gross–Pitaevskii equation is capable of describing the structure of the quantized vortices and the physics at length scales smaller than the healing length.

In this paper,we only show Padé approximate solutions for quantized vortices with winding numbers 1?6.In fact,we have also derived the Padé approximate solutions for cases with winding numbers of 7 and 8,and the accuracy of the Padé approximate solutions was good.If needed,Padé approximate solutions for cases with even higher winding numbers could also be derived using the same method,according to equation(38).However,on the one hand,we find that the accuracy of the Padé approximate solution decreases as the winding number increases;on the other hand,from an energetic point of view,the higher the winding number,the more unstable the quantized vortex;and it is difficult to obtain quantized vortices with higher winding numbers in the laboratory.

Acknowledgments

This work is supported by Undergraduate Innovation and Entrepreneurship Program Grant No.S201910579797,National Natural Science Foundation of China with Grant No.12005088,11 847 001,11747017,Guangdong Basic and Applied Basic Research Foundation with Grant No.2021A1515010246.S.L.is supported by the Lingnan Normal University Project with Grant No.YL20200203,ZL1930.