Analysis and E–cient Computation for the Dynamics of Two … · 2019. 5. 2. · condensates...

Contemporary Mathematics

Analysis and Efficient Computation for the Dynamics ofTwo-Component Bose-Einstein Condensates

Weizhu Bao

Abstract. In this paper, we review some recent results on the analysis and ef-

ficient computation for the dynamics of rotating two-component Bose-Einsteincondensates (BECs). We begin with the three-dimensional (3D) coupled Gross-Pitaevskii equations (GPEs) with an angular momentum rotation term and an

external driving field, show how to scale it into dimensionless form, reduce itto a 2D and 1D GPE in the limiting regime of strong anisotropic confinement

and present its semiclassical scaling and geometric optics. Dynamic laws on

the angular momentum expectation, density of each component, condensatewidths and analytical solutions of stationary states with its center shifted from

the trapping center are presented. In addition, efficient and accurate numerical

methods for computing the dynamics of rotating two-component BEC are pre-sented and some numerical results are reported to demonstrate the efficiency

and accuracy of the numerical methods.

1. Introduction

This paper summarizes recent work by the author on analysis and efficientcomputation for the dynamics of rotating two-component Bose-Einstein condensate(BEC) with/without an external driving field [2, 19, 14]. Although the theoriesand numerical methods presented here apply to multi-component BEC with anynumber of components [43, 40, 1, 28, 30], we concentrate here on two-componentBEC for simplicity [28, 29, 42, 33]. At temperatures T much smaller than thecritical temperature Tc [44, 20], in the rotating frame, a two-component BECwith an external driving field can be well described by two self-consistent nonlinearSchrodinger equations (NLSEs), also known as coupled Gross-Pitaevskii equations(CGPEs) [32, 22, 37, 36],(1.1)

i~∂ψj(x, t)

∂t=

[− ~

2

2m∇2 + ~κj + Vj(x)− ΩLz +

2∑

l=1

Ujl|ψl|2]

ψj −λ~ψkj, j = 1, 2,

Key words and phrases. Bose-Einstein condensation, two-component, ground state, dynam-

ics, external driving field, condensate width, time-splitting.The author acknowledges support from the Ministry of Education of Singapore grant No.

R-146-000-083-112.

c©0000 (copyright holder)

1

2 WEIZHU BAO

where ψj(x, t) denotes the macroscopic wave function of the jth (j = 1, 2) compo-nent with x = (x, y, z)T being the Cartesian coordinate vector and t being time, ~is the Planck constant, m is the atomic mass (for simplicity, here we assume thatthe atomic mass of the two components is the same), Ω is the angular velocity ofthe rotating laser beam, Lz = −i~(x∂y − y∂x) is the z-component of the angularmomentum, κj (j = 1, 2) are the Raman transition constants [30], and λ is theeffective Rabi frequency describing the strength of the external driving field. Vj(x)is the external trapping potential acting on the jth component, and if the harmonicpotential is considered, it takes the form

Vj(x) =m

2(ω2

x,j x2 + ω2y,j y2 + ω2

z,j z2), j = 1, 2,(1.2)

with ωx,j , ωy,j and ωz,j the trapping frequencies of the jth component in x-,y- and z-directions, respectively. The interactions of particles are described byUjl = 4π~2ajl/m with ajl = alj (j, l = 1, 2) being the s-wave scattering lengthsbetween the jth and lth component (positive for repulsive interaction and negativefor attractive interaction). The integers kj in (1.1) are chosen as

(1.3) kj =

2, j = 1,1, j = 2.

It is necessary to ensure that the wave functions are properly normalized. Espe-cially, we require∫

R3

(|ψ1(x, t)|2 + |ψ2(x, t)|2) dx = N = N01 + N0

2 , t ≥ 0,(1.4)

where

(1.5) N0j =

∫

R3|ψj(x, 0)|2 dx,

is the particle number of the jth (j = 1, 2) component at time t = 0 and N is thetotal number of particles in the condensate.

Under the normalization (1.4), we introduce the dimensionless variables asfollows: t → t/ωm with ωm = min1≤j≤2ωx,j , ωy,j , ωz,j, Ω → ωm Ω, κj → ωm κj ,λ → ωm λ, x → a0x with a0 =

√~/mωm, and ψj → ψj

√N/a

3/20 , i.e. we choose

1/ωm and a0 as the dimensionless time unit and length unit, respectively. Aftersome computations from (1.1), we obtain the dimensionless CGPEs as(1.6)

i∂ψj(x, t)

∂t=

[−1

2∇2 + κj + Vj(x)− ΩLz +

2∑

l=1

βjl|ψl|2]

ψj − λψkj, j = 1, 2,

where the dimensionless interaction parameters are characterized by βjl = βlj =mUjlN~2a0

= 4πNajl

a0(j, l = 1, 2). The dimensionless angular momentum rotation term

becomes

(1.7) Lz = −i(x∂y − y∂x) = −i ∂θ := −i Lz

with (r, θ) the polar coordinate in 2D, and resp. (r, θ, z) the cylindrical coordinatein 3D. The dimensionless external potentials are

(1.8) Vj(x) =12

(γ2

x,j x2 + γ2y,j y2 + γ2

z,j z2), j = 1, 2

with γx,j = ωx,j/ωm, γy,j = ωy,j/ωm and γz,j = ωz,j/ωm (j = 1, 2).

DYNAMICS OF ROTATING TWO-COMPONENT BEC 3

Theoretical treatment of such systems began in the context of superfluid he-lium mixtures and spin polarized hydrogen, and has now been extended to BECin alkalis [29, 20, 35, 45]. With the realization of BEC in experiments, the theo-retical predications of multi-component condensates, e.g. density profile, dynamicsof interacting BEC [26], motion damping [30] and formation of vortices, can nowbe compared with experimental data [28, 3]. Needless to say that this dramaticprogress on the experimental front has stimulated a wave of activity on both theo-retical and numerical fronts.

The paper is organized as follows. In section 2, we review dimension reductionand geometric optics of coupled GPEs. In section 3, we review the dynamic laws forthe density of each component, angular momentum expectation, condensate widthand a stationary state with its center shifted from the trap center. In section 4,we review efficient and accurate numerical methods for computing the dynamics oftwo-component BEC. In section 5, some numerical results are reported. Finally,some conclusions are drawn in section 6.

2. The coupled Gross-Pitaevskii equations

In this section, we will review dimension reduction of the coupled GPEs, re-duction to one-component BEC, and the semiclassical scaling and geometric opticsin strongly repulsive interaction regimes.

2.1. Dimension reduction. In a disk-shaped condensate, i.e.

ωx,j ≈ ωy,j ≈ ωm, ωz,j À ωm ⇐⇒ γx,j ≈ γy,j ≈ 1, γz,j À 1, j = 1, 2,

the 3D CGPEs (1.6) can be formally reduced to 2D CGPEs with x = (x, y)T

[41, 7, 5, 6]. When Ω = 0 and in a cigar-shaped condensate, i.e.

ωx,j ≈ ωm, ωy,j À ωm, ωz,j À ωm ⇐⇒ γx,j ≈ 1, γy,j À 1, γz,j À 1, j = 1, 2,

the 3D CGPEs (1.6) can be formally reduced to 1D CGPEs with x = x [41, 7, 5, 6].Thus here we consider the following CGPEs in d-dimensions (d = 2, 3 when Ω 6= 0and d = 1, 2, 3 when Ω = 0):

i∂ψj(x, t)

∂t=

[−1

2∇2 + κj + Vj(x)− ΩLz +

2∑

l=1

βj l|ψl|2]

ψj − λψkj , t ≥ 0,(2.1)

ψj(x, 0) = ψ0j (x), x ∈ Rd,(2.2)

where the initial data are normalized as

(2.3) ‖ψ01‖2 + ‖ψ0

2‖2 :=∫

Rd

(|ψ01(x)|2 + |ψ0

2(x)|2) dx =N0

1

N+

N02

N= 1,

and the external potentials are given as

(2.4) Vj(x) =

12γ2

x,j x2, d = 1,12

(γ2

x,j x2 + γ2y,j y2

), d = 2,

12

(γ2

x,j x2 + γ2y,j y2 + γ2

z,j z2), d = 3,

j = 1, 2.

In fact, in (2.1), if Ω = 0, then the equations are for nonrotating two-componentBEC, and if Ω 6= 0, they are for rotating two-component BEC; if λ = 0, there is noexternal driving field, and if λ 6= 0, there is an external driving field.

The dimensionless CGPEs (2.1) conserve the total density:

N(t) = N1(t) + N2(t) ≡ ‖ψ01‖2 + ‖ψ0

2‖2 = 1, t ≥ 0(2.5)

4 WEIZHU BAO

with

Nj(t) = ‖ψj(·, t)‖2 :=∫

Rd

|ψj(x, t)|2 dx, t ≥ 0, j = 1, 2,(2.6)

and the energy

E(ψ1, ψ2) =∫

Rd

[2∑

j=1

(12|∇ψj |2 + (κj + Vj(x)) |ψj |2 − Ω Re

(ψ∗j Lzψj

)

+2∑

l=1

βjl

2|ψj |2|ψl|2

)− 2λ Re(ψ∗1ψ2)

]dx

≡ E(ψ0

1 , ψ02

), t ≥ 0(2.7)

with f∗ and Re(f) denoting the conjugate and real part of a function f , respectively.In addition, if there is no external driving field in (2.1), i.e. λ = 0, the density ofeach component is also conserved, i.e.

Nj(t) =∫

Rd

|ψj(x, t)|2 dx ≡ ‖ψ0j ‖2 =

N0j

N, t ≥ 0, j = 1, 2.(2.8)

2.2. Reduction to single GPE. If there is no external driving field in (2.1),i.e. λ = 0, and the initial particle numbers of the two components N0

1 and N02

(w.l.o.g., assuming that N01 ≥ N0

2 ) in (1.4) satisfy N01 À N0

2 , i.e. N01 = O(N) and

N02 = o(N), when N À 1, we have

N2(t) =∫

Rd

|ψ2(x, t)|2 dx ≡ N02

N:= ε ¿ 1,(2.9)

N1(t) =∫

Rd

|ψ1(x, t)|2 dx ≡ N01

N:= 1− ε ≈ 1, t ≥ 0.(2.10)

These immediately imply that the effect of the second component is insignificantand the original two-component system is mainly dominated by the first component.Formally, we can drop the second component from the two-component BEC andget a single-component BEC, and in this case the CGPEs (2.1) are reduced to

i∂ψ(x, t)

∂t=

[−1

2∇2 + κ + V (x) + β|ψ|2 − ΩLz

]ψ, t ≥ 0(2.11)

by setting ψ(x, t) =√

N/N01 ψ1(x, t), κ = κ1, V (x) = V1(x) and β = N0

1 β11/N ≈β11. The GPE (2.11) conserves the normalization of the wave function(2.12)

‖ψ(·, t)‖2 ≡∫

Rd

|ψ(x, 0)|2 dx =∫

Rd

N

N01

|ψ1(x, 0)|2 dx =N

N01

N01

N= 1, t ≥ 0,

and the energy(2.13)

Es(ψ) =∫

Rd

[12|∇ψ|2 + (κ + V (x)) |ψ|2 − Ω Re (ψ∗Lzψ) +

β

2|ψ|4

]dx, t ≥ 0.


In addition, by setting ψ1(x, t) =√

N01 /Nψ(x, t) and ψ2(x, t) =

√N0

2 /Nφ(x, t) inthe energy of the two-component BEC (2.7) with λ = 0, we obtain

E(ψ1, ψ2) =N0

1

NEs(ψ) +

N02

NEr(ψ, φ) = (1− ε)Es(ψ) + ε Er(ψ, φ)

= Es(ψ) + ε [−Es(ψ) + Er(ψ, φ)] ,(2.14)

where

Er(ψ, φ) =∫

Rd

[12|∇φ|2 + κ2|φ|2 + V2(x)|φ|2 − Ω Re (φ∗Lzφ)

+β21N0

1

N|ψ|2|φ|2 +

β22

2N0

2

N|φ|4

]dx.(2.15)

This formally implies that the relative error between the energy of the two-componentBEC (2.7) and that of the single-component BEC (2.13) converges to 0 linearlywhen ε = N0

2N goes to 0, i.e.

(2.16)|E(ψ1, ψ2)− Es(ψ)|

Es(ψ)= ε

[1− Er(ψ, φ)

Es(ψ)

]= O(ε), when 0 < ε ¿ 1.

2.3. Semiclassical scaling and leading asymptotics in energy. Let βmax =maxβ11, β12, β22. If βmax À 1, i.e. in the strong repulsive interaction regime orthere are many particles in the condensate, under the normalization (2.5), the semi-classical scaling for the CGPEs (2.1) is also very useful in practice by choosing

x = ε−1/2x, ψεj = εd/4ψj , ε = β−2/(d+2)

max .(2.17)

Substituting (2.17) into (2.1), we get the CGPEs in the semiclassical scaling underthe normalization (2.5) with ψj = ψε

j (j = 1, 2):(2.18)

iε∂ψε

j (x, t)∂t

=

[−ε2

2∇2 + εκj + Vj(x)− εΩLz +

2∑

l=1

αjl |ψεl |2

]ψε

j −ελψεkj

, j = 1, 2,

where αjl = βjl/βmax = O(1) (or o(1)). In this case, the associated energy func-tional Eε(ψε

1, ψε2) is defined as

Eε (ψε1, ψ

ε2) =

∫

Rd

[2∑

j=1

(ε2

2|∇ψε

j |2 + (εκj + Vj(x)) |ψεj |2 − εΩ Re

((ψε

j )∗Lzψ

εj

)

+2∑

l=1

αjl

2|ψε

j |2|ψεl |2

)− 2ελ Re ((ψε

1)∗ψε

2)

]dx = O(1), t ≥ 0,(2.19)

by assuming that ψεj is ε-oscillatory (see (2.21)) and ‘sufficiently’ integrable such

that all terms have O(1)-integral. Then the leading asymptotics of the energyfunctional E(ψ1, ψ2) in (2.7) can be given by

E(ψ1, ψ2) = ε−1Eε(ψε1, ψ

ε2) = O(ε−1) = O

(β2/(d+2)

max

).(2.20)

6 WEIZHU BAO

2.4. Geometrical optics of nonrotating two-component BEC. For non-rotating two-component BEC without external driving field and in strongly repul-sive interaction regimes, i.e. Ω = 0, λ = 0 and 0 < ε ¿ 1 in (2.18), we can set, i.e.the WKB ansatz [24]

ψεj (x, t) =

√ρε

j(x, t) exp(

i

εSε

j (x, t))

, j = 1, 2,(2.21)

where ρεj = |ψε

j |2 and Sεj = ε arg

(ψε

j

)are the position density and phase of the

wave function ψεj of jth-component (j = 1, 2), respectively. Inserting (2.21) into

(2.18) and collecting the real and imaginary parts, we get the coupled transportequations for the densities ρε

j and the Hamilton-Jacobi equations for the phases Sεj

(j = 1, 2):

∂tρεj + div

(ρε

j∇Sεj

)= 0,(2.22)

∂tSεj +

12|∇Sε

j |2 + εκj + Vj(x) +2∑

l=1

αjlρεl =

ε2

2√

ρεj

∇2√

ρεj , j = 1, 2.(2.23)

Furthermore, by defining the current densities

Jεj(x, t) = ρε

j∇Sεj = ε Im

[(ψε

j

)∗∇ψεj

], j = 1, 2,(2.24)

where Im(f) is the imaginary part of a function f , we can rewrite (2.22)-(2.23) asa coupled Euler system with third-order dispersion terms

∂tρεj + divJε

j = 0, j = 1, 2,(2.25)

∂tJεj + div

(Jε

j ⊗ Jεj

ρεj

)+ ρε

j∇Vj(x) +∇Pj (ρε1, ρ

ε2) =

ε2

4∇ (

ρεj∇2 ln ρε

j

);(2.26)

where the pressures Pj are defined as

Pj (ρε1, ρ

ε2) =

12

2∑

l=1

αjl ρεj ρε

l , j = 1, 2.

By formally passing to the limit ε → 0+ in (2.22)-(2.23), we get

∂tρ0j + div

(ρ0

j∇S0j

)= 0,(2.27)

∂tS0j +

12|∇S0

j |2 + Vj(x) +2∑

l=1

αjlρ0l = 0, j = 1, 2,(2.28)

with ρ0j = limε→0+ ρε

j and S0j = limε→0+ Sε

j . Similarly, letting ε → 0+ in (2.25)-(2.26), formally we get an Euler system coupling through the pressures

∂tρ0j + divJ0

j = 0, j = 1, 2,(2.29)

∂tJ0j + div

(J0

j ⊗ J0j

ρεj

)+ ρ0

j∇Vj(x) +∇Pj

(ρ01, ρ

02

)= 0,(2.30)

where J0j (x, t) = ρ0

j∇S0j (j = 1, 2). The system (2.29)-(2.30) is a coupled isotropic

Euler system with quadratic pressure-density constitutive relations in the non-rotational frame. The formal asymptotics is supposed to hold up to caustic onsettime [24, 25].


Remark 2.1. When Ω = 0 and λ 6= 0 in (2.18), the WKB analysis for studyingthe semiclassical limit of the nonlinear Schrodinger equation [24] is no longer validfor (2.18). Alternatively, one may need to use the Wigner transform [25] to studythe semiclassical limit of (2.18) when λ 6= 0.

2.5. Geometrical optics of rotating BEC. For rotating two-componentBEC without external driving field and in strongly repulsive interaction regimes,i.e. λ = 0 and 0 < ε ¿ 1 in (2.18), due to the appearance of vortices in the initialdata (2.2), we set [27, 15]

ψεj (x, t) = Aε

j(x, t) exp(

i

εSε

j (x, t))

, j = 1, 2,(2.31)

where Aεj = (aε

j , bεj) ∈ C (with aj ∈ R and bj ∈ R) represents the density and slow

varying phase, and Sεj = ε arg

(ψε

j

)is the fast varying phase of the wave function

ψεj of jth-component (j = 1, 2). Inserting (2.31) into (2.18) and collecting the

O(1) and O(ε) terms, we get the coupled Schrodinger-type equation for Aεj and the

Hamilton-Jacobi equations for the phases Sεj (j = 1, 2):

∂tAεj +∇Sε

j · ∇Aεj +

12Aε

j∆Sεj − Ω LzAε

j + iκjAεj =

iε

2∆Aj ,(2.32)

∂tSεj +

12|∇Sε

j |2 + Vj(x) +2∑

l=1

αjl |Aεl |2 − Ω LzS

εj = 0, j = 1, 2.(2.33)

Furthermore, by defining the quantum hydrodynamic velocity

uεj(x, t) = ∇Sε

j = ε Im[(

ψεj

)∗∇ψεj

]/|ψε

j |2, j = 1, 2,(2.34)

we can rewrite (2.32)-(2.33) as

∂tAεj + uε

j · ∇Aεj +

12Aε

j∇ · uεj − Ω LzAε

j + iκjAεj =

iε

2∆Aj ,(2.35)

∂tuεj + (uε

j · ∇)uεj +∇Vj(x) +

2∑

l=1

αjl∇|Aεl |2 − Ω Lzuε

j = 0, j = 1, 2.(2.36)

By formally passing to the limit ε → 0+ in (2.32)-(2.33), we get

∂tA0j +∇S0

j · ∇A0j +

12A0

j∆S0j − Ω LzA0

j + iκjA0j = 0,(2.37)

∂tS0j +

12|∇S0

j |2 + Vj(x) +2∑

l=1

αjl

∣∣A0l

∣∣2 − Ω LzS0j = 0, j = 1, 2;(2.38)

where A0j = limε→0+ Aε

j and S0j = limε→0+ Sε

j for j = 1, 2. Similarly, lettingε → 0+ in (2.35)-(2.36), formally we get

∂tA0j + u0

j · ∇A0j +

12A0

j∇ · u0j − Ω LzA0

j + iκjA0j = 0,(2.39)

∂tu0j + (u0

j · ∇)u0j +∇Vj(x) +

2∑

l=1

αjl∇∣∣A0

l

∣∣2 − Ω Lzu0j = 0, j = 1, 2;(2.40)

where u0j = limε→0+ uε

j for j = 1, 2. The formal asymptotics is supposed to holdup to caustic onset time [27, 15].

8 WEIZHU BAO

Remark 2.2. Again, when λ 6= 0 and Ω 6= 0 in (2.18), the above formal analysisfor studying the semiclassical limit of the nonlinear Schrodinger equation [27, 15]is no longer valid for (2.18) with Ω 6= 0. Alternatively, one may need to use theWigner transform [25] to study the semiclassical limit of (2.18) when λ 6= 0.

3. Dynamics of two-component BEC

In this section, we review the dynamic laws of two-component BEC with exter-nal driving field including the conservation of the angular momentum expectationin symmetric traps, time evolution of the density of each component and conden-sate widths, and analytical solutions for a stationary state with its center shiftedfrom the trap center in rotating two-component BEC.

3.1. Dynamics of the density of each component. We define

W1(t) = i

∫

Rd

[ψ∗1(x, t)ψ2(x, t)− ψ1(x, t)ψ∗2(x, t)] dx,(3.1)

W2(t) =∫

Rd

[ψ1(x, t)ψ∗2(x, t) + ψ∗1(x, t)ψ2(x, t)] dx, t ≥ 0.(3.2)

For the dynamics of the density of each component, we have the following lemmas

Lemma 3.1. Suppose (ψ1(x, t), ψ2(x, t)) is the solution of the CGPEs (2.1)-(2.2); then we have,

N ′1(t)− λ W1(t) = 0, N ′

2(t) + λ W1(t) = 0,(3.3)W ′

1(t)− 2λ [1− 2N1(t)] + (κ1 − κ2)W2(t) = F1(t), t > 0,(3.4)W ′

2(t) + (κ2 − κ1)W1(t) = F2(t),(3.5)

with initial conditions

N1(0) =∫

Rd

|ψ01(x)|2 dx =

N01

N:= N

(0)1 ,(3.6)

N2(0) =∫

Rd

|ψ02(x)|2 dx =

N02

N:= N

(0)2 ,(3.7)

W1(0) = i

∫

Rd

[(ψ0

1

)∗(x)ψ0

2(x)− ψ01(x)

(ψ0

2

)∗(x)

]dx := W

(0)1 ,(3.8)

W2(0) =∫

Rd

[ψ0

1(x)(ψ0

2

)∗(x) +

(ψ0

1

)∗(x)ψ0

2(x)]

dx := W(0)2 ;(3.9)

where for t ≥ 0,

F1(t) =∫

Rd

(ψ∗1ψ2 + ψ1ψ∗2)

[V2(x)− V1(x) + (β12 − β11)|ψ1|2

+(β22 − β12)|ψ2|2]dx,(3.10)

F2(t) = i

∫

Rd

(ψ∗1ψ2 − ψ1ψ∗2)

[V1(x)− V2(x) + (β11 − β12)|ψ1|2

+(β12 − β22)|ψ2|2]dx.(3.11)


Proof. Differentiating (2.6) with respect to t, noticing (2.1), integrating by parts,we obtain

N ′j(t) =

d

dt‖ψj(·, t)‖2 =

d

dt

∫

Rd

|ψj(x, t)|2 dx =∫

Rd

(ψj∂tψ

∗j + ψ∗j ∂tψj

)dx

= i

∫

Rd

[ψj

((−1

2∇2 + κj + Vj(x)− ΩL∗z +

2∑

l=1

βj l|ψl|2)

ψ∗j − λ ψ∗kj

)

−ψ∗j

((−1

2∇2 + κj + Vj(x)− ΩLz +

2∑

l=1

βj l|ψl|2)

ψj − λ ψkj

)]dx

= i

∫

Rd

[(12|∇ψj |2 + (κj + Vj(x)) |ψj |2 − ψ∗j ΩLzψj + |ψj |2

2∑

l=1

βj l|ψl|2)

−(

12|∇ψj |2 + (κj + Vj(x)) |ψj |2 − ψ∗j ΩLzψj + |ψj |2

2∑

l=1

βj l|ψl|2)

−λ ψjψ∗kj

+ λ ψ∗j ψkj

]dx

= iλ

∫

Rd

(ψ∗j ψkj

− ψjψ∗kj

)dx, t ≥ 0.(3.12)

Plugging (3.1) into (3.12), noticing (1.3), we obtain

N ′1(t) = iλ

∫

Rd

(ψ∗1ψ2 − ψ1ψ∗2) dx = λ W1(t),(3.13)

N ′2(t) = iλ

∫

Rd

(ψ∗2ψ1 − ψ2ψ∗1) dx = −λ W1(t), t ≥ 0.(3.14)

Differentiating (3.1) with respect to t, noticing (2.1) and (2.5), we get

W ′1(t) = i

d

dt

∫

Rd

[ψ∗1ψ2 − ψ1ψ∗2 ] dx

= i

∫

Rd

[ψ∗1 ∂tψ2 + ψ2 ∂tψ∗1 − ψ1 ∂tψ

∗2 − ψ∗2 ∂tψ1] dx

=∫

Rd

ψ∗1

[(−1

2∇2 + κ2 + V2(x)− ΩLz +

2∑

l=1

β2 l|ψl|2)

ψ2 − λψ1

]dx

−∫

Rd

ψ2

[(−1

2∇2 + κ1 + V1(x)− ΩL∗z +

2∑

l=1

β1 l|ψl|2)

ψ∗1 − λψ∗2

]dx

+∫

Rd

ψ1

[(−1

2∇2 + κ2 + V2(x)− ΩL∗z +

2∑

l=1

β2 l|ψl|2)

ψ∗2 − λψ∗1

]dx

−∫

Rd

ψ∗2

[(−1

2∇2 + κ1 + V1(x)− ΩLz +

2∑

l=1

β1 l|ψl|2)

ψ1 − λψ2

]dx.(3.15)

10 WEIZHU BAO

Integrating by parts in (3.15), noticing (3.2), (2.5) and (3.10), we get

W ′1(t) =

∫

Rd

[(ψ∗1ψ2 + ψ1ψ

∗2)

(κ2 − κ1 + V2(x)− V1(x) + (β12 − β11)|ψ1|2

+(β22 − β12)|ψ2|2)

+ 2λ(|ψ2|2 − |ψ1|2

)]dx

= 2λ (N2(t)−N1(t)) + (κ2 − κ1)W2(t) + F1(t)= 2λ (1− 2N1(t)) + (κ2 − κ1)W2(t) + F1(t), t ≥ 0.(3.16)

Similarly, differentiating (3.2) with respect to t, noticing (2.1), (3.1) and (3.11),integrating by parts, we get

W ′2(t) =

d

dt

∫

Rd

[ψ∗1ψ2 + ψ1ψ∗2 ] dx

=∫

Rd

[ψ2 ∂tψ∗1 + ψ∗1 ∂tψ2 + ψ1 ∂tψ

∗2 + ψ∗2 ∂tψ1] dx

= i

∫

Rd

ψ2

[(−1

2∇2 + κ1 + V1(x)− ΩL∗z +

2∑

l=1

β1 l|ψl|2)

ψ∗1 − λψ∗2

]dx

−i

∫

Rd

ψ∗1

[(−1

2∇2 + κ2 + V2(x)− ΩLz +

2∑

l=1

β2 l|ψl|2)

ψ2 − λψ1

]dx

+i

∫

Rd

ψ1

[(−1

2∇2 + κ2 + V2(x)− ΩL∗z +

2∑

l=1

β2 l|ψl|2)

ψ∗2 − λψ∗1

]dx

−i

∫

Rd

ψ∗2

[(−1

2∇2 + κ1 + V1(x)− ΩLz +

2∑

l=1

β1 l|ψl|2)

ψ1 − λψ2

]dx

= i

∫

Rd

[(ψ∗1ψ2 − ψ1ψ

∗2)

(κ1 − κ2 + V1(x)− V2(x) + (β11 − β12)|ψ1|2

+(β12 − β22)|ψ2|2)]

dx

= (κ1 − κ2)W1(t) + F2(t), t ≥ 0.(3.17)

¤Lemma 3.2. Suppose that there is no external driving field in the CGPEs (2.1),

i.e. λ = 0, then the density of each component is conserved, i.e.

(3.18) N1(t) ≡ N1(0) =N0

1

N, N2(t) ≡ N2(0) =

N02

N, t ≥ 0.

In addition, we have(i) If the external trapping potentials are the same and the inter-/intra-component

s-wave scattering lengths in (2.1) are the same, i.e.

(3.19) V1(x) = V2(x), x ∈ Rd, and β11 = β12 = β22 (i.e. a11 = a12 = a22),

then for any initial data (ψ01(x), ψ0

2(x)), we have

W1(t) = W(0)1 cos ((κ1 − κ2)t)−W

(0)2 sin ((κ1 − κ2)t) ,(3.20)

W2(t) = W(0)1 sin ((κ1 − κ2)t) + W

(0)2 cos ((κ1 − κ2)t) , t ≥ 0.(3.21)


This immediately implies that: (i) if κ1 = κ2, then W1(t) ≡ W(0)1 and W2(t) ≡ W

(0)2

are two conserved quantities; and (ii) if κ1 6= κ2, then W1(t) and W2(t) are periodicfunctions with period T = 2π/|κ1 − κ2|.

(ii) For all other cases, we have, for any t ≥ 0,

W1(t) = W(0)1 cos ((κ1 − κ2)t)−W

(0)2 sin ((κ1 − κ2)t) + f1(t),(3.22)

W2(t) = W(0)1 sin ((κ1 − κ2)t) + W

(0)2 cos ((κ1 − κ2)t) + f2(t), t ≥ 0;(3.23)

where (f1(t), f2(t)) is the solution of the following first-order ODE system:

f ′1(t) + (κ1 − κ2)f2(t) = F1(t),(3.24)f ′2(t) + (κ2 − κ1)f1(t) = F2(t), t ≥ 0,(3.25)f1(0) = f2(0) = 0.(3.26)

Proof. When λ = 0, the ODE system (3.3)-(3.5) collapses to

N ′1(t) = 0, N ′

2(t) = 0,(3.27)W ′

1(t) + (κ1 − κ2)W2(t) = F1(t), t > 0,(3.28)W ′

2(t) + (κ2 − κ1)W1(t) = F2(t).(3.29)

Thus (3.18) is a combination of (3.27), (3.6) and (3.7).(i) When the conditions in (3.19) are satisfied, noticing (3.10), (3.11), we im-

mediately obtain

(3.30) F1(t) ≡ 0, F2(t) ≡ 0, t ≥ 0.

Plugging (3.30) into (3.28) and (3.29), we have

W ′1(t) + (κ1 − κ2)W2(t) = 0,(3.31)

W ′2(t) + (κ2 − κ1)W1(t) = 0, t ≥ 0.(3.32)

Thus (3.20)-(3.21) is the unique solution of the first-order ODE system (3.31)-(3.32)with the initial data (3.8) and (3.9).

(ii) From the results in (i) and using the superposition principle, we get that(3.22)-(3.23) is the unique solution of the first-order ODE system (3.28)-(3.29) withthe initial data (3.8) and (3.9). ¤

Lemma 3.3. Suppose that there is an external driving field in the CGPEs (2.1),i.e. λ 6= 0,

(i) If the external trapping potentials are the same and the inter-/intra-components-wave scattering lengths in (2.1) are the same, i.e. the conditions in (3.19) aresatisfied, we have

N1(t) =C0

ω2+

(N

(0)1 − C0

ω2

)cos(ωt) +

λ W(0)1

ωsin(ωt),(3.33)

N2(t) = 1− C0

ω2−

(N

(0)1 − C0

ω2

)cos(ωt)− λ W

(0)1

ωsin(ωt),(3.34)

W1(t) = W(0)1 cos(ωt) +

(C0

λω− ωN

(0)1

λ

)sin(ωt),(3.35)

W2(t) = W(0)2 + C1 [1− cos(ωt)] + C2 sin(ωt), t ≥ 0;(3.36)

12 WEIZHU BAO

where

ω =√

4λ2 + (κ1 − κ2)2, C0 = 2λ2 + λ(κ2 − κ1)W(0)2 + (κ2 − κ1)2N

(0)1 ,(3.37)

C1 =κ2 − κ1

λ

(N

(0)1 − C0

ω2

), C2 =

W(0)1 (κ1 − κ2)

ω.(3.38)

These solutions immediately imply that N1(t), N2(t), W1(t) and W2(t) are periodicfunctions with period T = 2π/ω = 2π/

√4λ2 + (κ1 − κ2)2.

(ii) For all other cases, we have,

N1(t) =C0

ω2+

(N

(0)1 − C0

ω2

)cos(ωt) +

λ W(0)1

ωsin(ωt) + f(t), t ≥ 0,(3.39)

N2(t) = 1− C0

ω2−

(N

(0)1 − C0

ω2

)cos(ωt)− λ W

(0)1

ωsin(ωt)− f(t),(3.40)

W1(t) = W(0)1 cos(ωt) +

(C0

λω− ωN

(0)1

λ

)sin(ωt) +

1λ

f ′(t),(3.41)

W2(t) = W(0)2 + C1 [1− cos(ωt)] + C2 sin(ωt) +

κ1 − κ2

λf(t) +

∫ t

0

F2(τ) dτ ;(3.42)

where f(t) is the solution of the following second-order ODE:

f ′′(t) + ω2f(t) = λ F1(t) + λ(κ2 − κ1)∫ t

0

F2(τ) dτ,(3.43)

f(0) = f ′(0) = 0.(3.44)

Proof. When λ 6= 0, differentiating the first equation in (3.3) with respect to t,noticing (3.4), we obtain

N ′′1 (t) = λ W ′

1(t)= 2λ2[1− 2N1(t)] + λ(κ2 − κ1)W2(t) + λ F1(t), t ≥ 0.(3.45)

Plugging the first equation in (3.3) into (3.5), we have

(3.46) W ′2(t) +

κ2 − κ1

λN ′

1(t) = F2(t), t ≥ 0.

Solving the above first order ODE, noticing the initial conditions (3.6) and (3.9),we get

W2(t) =κ1 − κ2

λN1(t) + W2(0) +

κ2 − κ1

λN1(0) +

∫ t

0

F2(τ) dτ

=κ1 − κ2

λN1(t) + W

(0)2 +

κ2 − κ1

λN

(0)1 +

∫ t

0

F2(τ) dτ, t ≥ 0.(3.47)

Plugging (3.47) into (3.45), noticing (3.37), we have

(3.48) N ′′1 (t) + ω2N1(t) = C0 + λ F1(t) + λ(κ2 − κ1)

∫ t

0

F2(τ) dτ, t ≥ 0.

(i) When the conditions in (3.19) are satisfied, we immediately obtain (3.30).Plugging (3.30) into (3.48), we obtain

(3.49) N ′′1 (t) + ω2N1(t) = C0, t ≥ 0,


together with initial conditions

(3.50) N1(0) = N(0)1 , N ′

1(0) = λ W1(0) = λ W(0)1 .

Thus, (3.33) is the unique solution of the second-order ODE (3.49) with the initialdata (3.50). Then the solution (3.34) is a combination of (3.33) and (2.5). Fi-nally, plugging (3.33) into (3.3) and (3.47) with F2(τ) ≡ 0, we immediately get thesolution (3.35) and (3.36), respectively.

(ii) From the results in (i) and using the superposition principle, we get that(3.39) is the unique solution of the second-order ODE (3.48) with the initial data(3.50). Then the solution (3.40) is a combination of (3.39) and (2.5). Finally, plug-ging (3.39) into (3.3) and (3.47), we immediately get the solution (3.41) and (3.42),respectively. ¤

3.2. Conservation of angular momentum expectation. As a measure ofvortex flux, we define the total angular momentum expectation:

〈Lz〉(t) = 〈Lz〉1(t) + 〈Lz〉2(t), t ≥ 0,(3.51)

where for j = 1, 2

(3.52) 〈Lz〉j(t) =∫

Rd

ψ∗j (x, t)Lzψj(x, t) dx = i

∫

Rd

ψ∗j (x, t)(y∂x−x∂y)ψj(x, t) dx.

In fact, 〈Lz〉j(t) := 〈Lz〉j(t)Nj(t)

is the angular momentum expectation of the jth (j =1, 2) component. Typically when λ = 0, as the density of each component isconserved, then 〈Lz〉j(t) = 〈Lz〉j(t)

Nj(0). For the dynamics of the angular momentum

expectations in rotating two-component BEC, we have the following lemmas.

Lemma 3.4. Suppose (ψ1(x, t), ψ2(x, t)) is the solution of the CGPEs (2.1)-(2.2); then we have,

d〈Lz〉j(t)dt

=(γ2

x,j − γ2y,j

) ∫

Rd

xy|ψj |2 dx− βjkj

∫

Rd

|ψj |2(x∂y − y∂x)|ψkj|2 dx

−2λ Re[∫

Rd

ψ∗kj(x∂y − y∂x)ψj dx

], t ≥ 0, j = 1, 2.(3.53)

Proof: The proof follows the line of the analogous result for the case of κ1 = κ2 = 0in [53]. ¤

From the above lemma, we have

Lemma 3.5. Suppose the traps in (2.4) are radially symmetric in 2D, and resp.cylindrically symmetric in 3D, i.e. γx,1 = γy,1 and γx,2 = γy,2.

(i) For any given initial data(ψ0

1(x), ψ02(x)

)in (2.2), the total angular mo-

mentum expectation is conserved, i.e.

(3.54) 〈Lz〉(t) ≡ 〈Lz〉(0) =2∑

j=1

∫

Rd

(ψ0

j (x))∗

Lzψ0j (x) dx, t ≥ 0.

14 WEIZHU BAO

In addition, the energy for non-rotating part is also conserved, i.e.

En(ψ1, ψ2) :=∫

Rd

[2∑

j=1

(12|∇ψj |2 + (κj + Vj(x)) |ψj |2 +

2∑

l=1

βjl

2|ψj |2|ψl|2

)

−2λ Re(ψ∗1ψ2)

]dx ≡ En

(ψ0

1 , ψ02

), t ≥ 0.(3.55)

(ii) Suppose the initial data (ψ01(x), ψ0

2(x)) in (2.2) is chosen as

(3.56) ψ0j (x) = fj(r)eimjθ with mj ∈ Z and fj(0) = 0 when mj 6= 0,

in 2D, and resp. in 3D,(3.57)

ψ0j (x) = fj(r, z)eimjθ with mj ∈ Z and fj(0, z) = 0 when mj 6= 0.

If λ = 0, then 〈Lz〉1(t) and 〈Lz〉2(t) are conserved, i.e.

(3.58) 〈Lz〉j(t) ≡ 〈Lz〉j(0) =1

Nj(0)

∫

Rd

(ψ0

j (x))∗

Lzψ0j (x) dx, t ≥ 0, j = 1, 2.

On the other hand, if m1 = m2 := m in (3.56) for 2D, and resp. in (3.57) for 3D,then for any given λ, 〈Lz〉1(t) and 〈Lz〉2(t) are conserved, i.e.

〈Lz〉j(t) ≡ 〈Lz〉j(0) = m, t ≥ 0, j = 1, 2.(3.59)

Proof: Again, the proof follows the line of the analogous result for the case ofκ1 = κ2 = 0 in [53]. ¤

3.3. Dynamics of condensate widths. Another important quantity char-acterizing the dynamics of a rotating two-component BEC is the condensate widthdefined as

σα(t) =√

δα(t) =√

δα,1(t) + δα,2(t), α = x, y or z,(3.60)

where

(3.61) δα,j(t) = 〈α2〉j(t) =∫

Rd

α2|ψj(x, t)|2 dx, t ≥ 0, j = 1, 2.

For the dynamics of condensate widths, we have the following lemmas.

Lemma 3.6. Suppose (ψ1(x, t), ψ2(x, t)) is the solution of problem (2.1)-(2.2);then we have

d2δα(t)dt2

=∫

Rd

2∑

j=1

[(∂yα− ∂xα)

(4iΩψ∗j (x∂y + y∂x)ψj + 2Ω2(x2 − y2)|ψj |2

)

+2|∂αψj |2 − 2α|ψj |2∂α(Vj(x)) + |ψj |22∑

l=1

βjl|ψl|2]dx, t ≥ 0,(3.62)

with initial conditions

(3.63) δα(0) = δ(0)α =

∫

Rd

α2(|ψ0

1(x)|2 + |ψ02(x)|2) dx, α = x, y, z,


(3.64) δ′α(0) = δ(1)α = 2

2∑

j=1

∫

Rd

α[−Ω|ψ0

j |2 (x∂y − y∂x) α + Im((ψ0

j )∗∂αψ0j

)]dx.


From the above lemma, we have

Lemma 3.7. In 2D with radially symmetric traps, i.e., d = 2, κ1 = κ2 andγx,1 = γy,1 = γx,2 = γy,2 := γr in (2.1), we have

(i) If there is no external driving field, i.e. λ = 0 in (2.1), for any given initialdata (ψ0

1(x), ψ02(x)) in (2.2), we have, for t ≥ 0,

δr(t) =E(ψ0

1 , ψ02) + Ω〈Lz〉(0)− κ1

γ2r

[1− cos(2γrt)]

+δ(0)r cos(2γrt) +

δ(1)r

2γrsin(2γrt),(3.65)

where δr(t) = δx(t)+ δy(t), δ(0)r := δx(0)+ δy(0) and δ

(1)r := δ′x(0)+ δ′y(0). Further-

more, when the initial data (ψ01(x), ψ0

2(x)) in (2.2) satisfies (3.56) with m1 = m2,we have, for t ≥ 0,

δx(t) = δy(t) =12δr(t)

=E

(ψ0

1 , ψ02

)+ Ω〈Lz〉(0)− κ1

2γ2r

[1− cos(2γrt)]

+δ(0)x cos(2γrt) +

δ(1)x

2γrsin(2γrt).(3.66)

Thus in this case, the condensate widths σr(t), σx(t) and σy(t) are periodic functionswith frequency doubling the trapping frequency.

(ii) If there is an external driving field, i.e. λ 6= 0 in (2.1), we have

δr(t) =E(ψ0

1 , ψ02) + Ω〈Lz〉(0)− κ1

γ2r

+δ(1)r

2γrsin(2γrt) + gr(t)

+(

δ(0)r − E(ψ0

1 , ψ02) + Ω〈Lz〉(0)− κ1

γ2r

)cos(2γrt), t ≥ 0,(3.67)

where gr(t) is the solution of the following second-order ODE:

d2gr(t)dt2

+ 4γ2r gr(t) = Gr(t), gr(0) = g′r(0) = 0,(3.68)

with

Gr(t) = 4∫

Rd

[λ (ψ∗1ψ2 + ψ1ψ

∗2) + (κ1 − κ2)|ψ2|2

]dx.

Proof: Again, the proof follows the line of the analogous result for the case ofκ1 = κ2 = 0 in [53]. ¤

16 WEIZHU BAO

3.4. Dynamics of a stationary state with its centers shifted. Whenλ = 0 in (2.1), let (φe

1(x), φe2(x)) be a stationary state of the CGPEs (2.1) with

chemical potential (µe1, µ

e2), i.e., (µe

1, µe2;φ

e1, φ

e2) satisfying

µej φe

j(x) = −12∇2φe

j + (κj + Vj(x))φej − Ω Lzφ

ej +

2∑

l=1

βjl|φel |2φe

j , x ∈ Rd,(3.69)

‖φej‖2 :=

∫

Rd

|φej(x)|2 dx =

N0j

N, j = 1, 2.(3.70)

If the initial data (ψ01(x), ψ0

2(x)) in (2.2) is chosen as a stationary state with ashift in its center, one can construct an exact solution of the CGPEs (2.1) withharmonic oscillator potentials (2.4). This kind of analytical construction can beused, in particular, in the benchmark and validation of numerical algorithms forthe CGPEs (2.1). For single-component non-rotating and rotating BEC, this kindof analytical construction can be found in the literature [23, 5]. For rotating two-component BEC, we have the following lemma.

Lemma 3.8. If the initial data (ψ01(x), ψ0

2(x)) in (2.2) is chosen as

(3.71) ψ01(x) = φe

1(x− x01), ψ0

2(x) = φe2(x− x0

2), x ∈ Rd,

where x01 and x0

2 are two given points in Rd, when λ = 0, x01 = x0

2 := x0 andV1(x) ≡ V2(x), then the exact solution of the CGPEs (2.1)-(2.2) satisfies

(3.72) ψj(x, t) = φej(x− x(t))e−iµe

j teiwj(x,t), x ∈ Rd, t ≥ 0, j = 1, 2,

where for any t ≥ 0, wj(x, t) is a linear function for x, i.e. for j = 1, 2(3.73)

wj(x, t) = cj(t) · x + gj(t), cj(t) = (cj,1(t), · · · , cj,d(t))T , x ∈ Rd, t ≥ 0,

and x(t) satisfies the following second-order ODE system

x′′(t)− 2Ωy′(t) +(γ2

x,1 − Ω2)x(t) = 0,(3.74)

y′′(t) + 2Ωx′(t) +(γ2

y,1 − Ω2)y(t) = 0, t ≥ 0,(3.75)

x(0) = x0, y(0) = y0, x′(0) = Ωy0, y′(0) = −Ωx0.(3.76)

Moreover, if in 3D, another ODE needs to be added:

z′′(t) + γ2z,1 z(t) = 0, z(0) = z0, z′(0) = 0.(3.77)


The ODE system (3.74)-(3.77) governing the motion of the center of mass x(t)[52] for rotating two-component BEC is the same as that for single-component BEC[5]. This ODE system was solved analytically in [52] and different motion patternsof the center were classified in details based on the parameters Ω, γx,1, γy,1 andγz,1.

4. Numerical methods

In this section, we review efficient and accurate numerical methods for solvingthe CGPEs (2.1)-(2.2) for the dynamics of two-component BEC. The key ideasare: (i) to apply a time-splitting technique for decoupling the nonlinearity; and(ii) to adopt the Cartesian coordinates and the polar coordinates in 2D (and resp.


cylindrical coordinates in 3D) for nonrotating and rotating two-component BEC,respectively. Due to the trapping potentials V1(x) and V2(x) given by (2.4), thesolution (ψ1, ψ2) of (2.1)-(2.2) decays to zero exponentially fast when |x| → ∞.Thus in practical computation, we truncate the problem (2.1)-(2.2) into a boundedcomputational domain Ωx with homogeneous Dirichlet boundary conditions:

i∂ψj

∂t=

[−1

2∇2 + κj + Vj(x)− ΩLz +

2∑

l=1

βjl|ψl|2]

ψj − λψkj , t ≥ 0,(4.1)

ψj(x, t) = 0, x ∈ Γ = ∂Ωx, t ≥ 0,(4.2)

ψj(x, 0) = ψ0j (x), x ∈ Ωx, with

∫

Ωx

(|ψ01(x)|2 + |ψ0

2(x)|2) dx = 1.(4.3)

In practical computation, we use sufficiently large domain so as to make sure thehomogeneous Dirichlet boundary condition (4.2) doesn’t introduce aliasing error.Usually, the radius of the bounded computational domain depends on the problem.In general, it should be larger than the “Thomas-Fermi radius”. Of course, theuse of more sophisticated radiation boundary conditions is an interesting topic thatremains to be examined in the future.

4.1. For nonrotating two-component BEC. In this case, i.e. Ω = 0 in(4.1), we adopt the Cartesian coordinates and use the sine pseudospectral methodfor solving the linear Schrodinger equations.

Time-splitting We choose a time step ∆t > 0. For n = 0, 1, . . . , from timet = tn = n∆t to t = tn+1 = tn + ∆t, the CGPEs (4.1) with Ω = 0 is solved in twosplitting steps [4, 7]. One first solves

i∂ψj

∂t= −1

2∇2ψj − λψkj

, j = 1, 2(4.4)

for the time step of length ∆t, followed by solving

i∂ψj

∂t= (κj + Vj(x))ψj +

2∑

l=1

βjl|ψl|2ψj , j = 1, 2(4.5)

for the same time step. For time t ∈ [tn, tn+1], the ODE system (4.5) leaves|ψ1(x, t)| and |ψ2(x, t)| invariant in t [4, 7], and thus it can be integrated exactlyto obtain [4, 5], for j = 1, 2 and t ∈ [tn, tn+1](4.6)

ψj(x, t) = ψj(x, tn) exp

[−i

(κj + Vj(x) +

2∑

l=1

βjl |ψl(x, tn)|2)

(t− tn)

].

The equations (4.1) with Ω = 0 are now decoupled and thus we need only showhow to discretize (4.4). Various algorithms were introduced in the literature fordiscretizating the GPE (4.4) [4, 9]. For the convenience of the reader, here wereview a method which discretizes the equation (4.10) by using sine pseudospectralmethod. In order to do so, we choose the bounded computational domain Ωx =(a, b) in 1D, Ωx = (a, b)× (c, d) in 2D, and resp. Ωx = (a, b)× (c, d)× (e, f) in 3Dwith |a|, b, |c|, d, |e| and f sufficiently large. For simplicity, we only present themethod in 1D. Extensions to 2D and 3D are straightforward by tensor products.

18 WEIZHU BAO

Numerical algorithm In 1D, the equations (4.4) will be discretized in spaceby the sine-spectral method and integrated in time exactly. We choose the spatialmesh size h = ∆x = (b−a)/M with M an even positive integer and define the gridpoints by

xm = a + m h, m = 0, 1, 2, . . . , M.

Let(ψn

1,m, ψn2,m

)be the approximation of (ψ1(xm, tn), ψ2(xm, tn)) for m = 0, 1, 2, . . . , M

and denote (ψn1 , ψn

2 ) be the solution vector at time t = tn with component(ψn

1,m, ψn2,m

).

From time t = tn to t = tn+1, we combine the splitting steps via the standardsecond-order splitting:

ψ(1)1,m = ψn

1,m e−i∆t(κ1+V1(xm)+β11|ψn1,m|2+β12|ψn

2,m|2)/2,

ψ(1)2,m = ψn

2,m e−i∆t(κ2+V2(xm)+β12|ψn1,m|2+β22|ψn

2,m|2)/2,

ψ(2)1,m =

2M

M−1∑

l=1

e−i∆tµ2l /2

[cos(λ∆t)(ψ(1)

1 )l + i sin(λ∆t)(ψ(1)2 )l

]sin

(ml π

M

),

ψ(2)2,m =

2M

M−1∑

l=1

e−i∆tµ2l /2

[i sin(λ∆t)(ψ(1)

1 )l + cos(λ∆t)(ψ(1)2 )l

]sin

(ml π

M

),

ψn+11,m = ψ

(2)1,m e

−i∆t(

κ1+V1(xm)+β11|ψ(2)1,m|2+β12|ψ(2)

2,m|2)

/2,

ψn+12,m = ψ

(2)2,m e

−i∆t(

κ2+V2(xm)+β12|ψ(2)1,m|2+β22|ψ(2)

2,m|2)

/2, m = 1, 2, . . . , M − 1;

where φl (l = 1, 2, . . . , M − 1), the sine-transform coefficients of the vector Φ =(φ0, φ1, φ2, . . . , φM )T with φ0 = φM = 0, are defined as

(4.7) µl =lπ

b− a, φl =

M−1∑m=1

φm sin(

ml π

M

), l = 1, 2, . . . , M − 1.

The overall time discretization error comes solely from the splitting, which is secondorder in time step ∆t, and the spatial discretization is of spatial order of accuracy.The discretization is time reversible and time transverse invariant. Furthermore,for the stability of the above discretization, we have the following lemma, whichshows that the total mass of the two-component BEC is conserved for any λ ∈ R,and the mass of each component is conserved when there is no external drivingfield, i.e. λ = 0.

Lemma 4.1. The above time-splitting spectral method for nonrotating two-component BEC is unconditionally stable and conserves the total mass of two-component BEC. In fact, for any mesh size h > 0 and time step ∆t > 0, wehave

‖ψn1 ‖2 + ‖ψn

2 ‖2 := hM−1∑m=1

[|ψn1,m|2 + |ψn

2,m|2]

≡ hM−1∑m=1

[|ψ01(xm)|2 + |ψ0

2(xm)|2] , n ≥ 0.(4.8)


Furthermore, when λ = 0 in (4.1), without the external driving field, we have

(4.9) ‖ψnj ‖2 := h

M−1∑m=1

|ψnj,m|2 ≡ h

M−1∑m=1

|ψ0j (xm)|2, n ≥ 0, j = 1, 2.


Remark 4.2. When the external potentials V1(x) and V2(x) are chosen as theharmonic potentials, one can also solve the coupled Gross-Pitaevskii equations (4.1)with Ω = 0 by using the time-splitting Laguerre-Hermite pseudo-spectral methodproposed in [9].

4.2. For rotating two-component BEC. In this case, i.e. Ω 6= 0 in (4.1),we adopt the polar coordinates in 2D, and resp. cylindrical coordinates in 3D,such that the angular momentum rotation term becomes a term with constantcoefficients.

Time-splitting For n = 0, 1, . . . , from time t = tn = n∆t to t = tn+1 =tn + ∆t, the CGPEs (4.1) are solved in three splitting steps [4, 7]. One first solves

i∂ψj

∂t= −1

2∇2ψj − ΩLzψj , j = 1, 2(4.10)

for the time step of length ∆t, followed by solving

i∂ψj

∂t= (κj + Vj(x))ψj +

2∑

l=1

βjl|ψl|2ψj , j = 1, 2(4.11)

for the same time step, and then by solving

i∂ψj

∂t= −λψkj

, j = 1, 2(4.12)

again for the same time step. The solutions for (4.11) are given in (4.6). For theODE system (4.12), we can rewrite it as

i∂Ψ∂t

= −λAΨ, with A =(

0 11 0

)and Ψ =

(ψ1

ψ2

).(4.13)

Since A is a real and symmetric matrix, it can be diagonalized and integratedexactly, and then we obtain [4], for t ∈ [tn, tn+1](4.14)

Ψ(x, t) = eiλA (t−tn)Ψ(x, tn) =(

cos (λ(t− tn)) i sin (λ(t− tn))i sin (λ(t− tn)) cos (λ(t− tn))

)Ψ(x, tn).

The equations (4.1) are now decoupled and thus we need only show how to discretizethe following single GPE in a rotational frame:

i∂ψ

∂t= −1

2∇2ψ − ΩLzψ, x ∈ Ωx, tn ≤ t ≤ tn+1.(4.15)

Various algorithms were introduced in the literature for discretizating the GPE(4.15) [5, 52, 10]. For the convenience of the reader, here we review a method whichdiscretizes the equation (4.15) in the θ-direction by the Fourier pseudospectralmethod, in the r-direction by the fourth-order finite difference method, in the z-direction by the sine pseudospectral method and in time by the Crank-Nicolson(C-N) scheme [5, 53]. In order to do so, we choose the bounded computational

20 WEIZHU BAO

domain Ωx = (x, y), r =√

x2 + y2 < R in 2D, and resp. Ωx = (x, y, z), r =√x2 + y2 < R, a < z < b in 3D with R, |a|, and b sufficiently larger than the

Thomas-Fermi radii.

Discretization in 2D When d = 2, we use the polar coordinate (r, θ) andassume that

ψ(r, θ, t) =L/2−1∑

l=−L/2

ψl(r, t)eilθ,(4.16)

where L is an even positive integer and ψl(r, t) is the Fourier coefficient for the l-thmode. Plugging (4.16) into (4.15) and noticing the orthogonality of the Fourierfunctions, we obtain, for −L

2 ≤ l ≤ L2 − 1 and 0 < r < R:

i∂ψl(r, t)

∂t= − 1

2r

∂

∂r

(r∂ψl(r, t)

∂r

)+

(l2

2r2− lΩ

)ψl(r, t),(4.17)

ψl(R, t) = 0, (for all l), ψl(0, t) = 0, (for l 6= 0).(4.18)

In order to discretize (4.17)-(4.18) in space by the finite difference method, wechoose an integer M > 0, a mesh size ∆r = 2R/(2M +1) and grid points rm = (m−1/2)∆r for 1 ≤ m ≤ M + 1. Let ψl,m(t) be the approximation of ψl(rm, t). Then afourth-order finite difference discretization for (4.17)-(4.18) with t ∈ [tn, tn+1] reads[5, 34]

idψl,m(t)

dt=

(l2

2r2m

− lΩ)

ψl,m(t)

−−ψl,m+2(t) + 16ψl,m+1(t)− 30ψl,m(t) + 16ψl,m−1(t)− ψl,m−2(t)24(∆r)2

−−ψl,m+2(t) + 8ψl,m+1(t)− 8ψl,m−1(t) + ψl,m−2(t)24rm∆r

, 1 ≤ m ≤ M,(4.19)

idψl,M+1(t)

dt=

(l2

2r2M+1

− lΩ)

ψl,M+1(t)

−11ψl,M+2(t)− 20ψl,M+1(t) + 6ψl,M (t) + 4ψl,M−1(t)− ψl,M−2(t)24 (∆r)2

−3ψl,M+2(t) + 10ψl,M+1(t)− 18ψl,M (t) + 6ψl,M−1(t)− ψl,M−2(t)24 rM+1 ∆r

,(4.20)

ψl,−1(t) = (−1)lψl,2(t), ψl,0(t) = (−1)lψl,1(t), ψl,M+1(t) = 0.(4.21)

Finally, the ODE system (4.19)-(4.21) is discretized by the standard C-N schemein time. Although an implicit time discretization is applied for (4.19)-(4.21), theone-dimensional nature of the problem makes the coefficient matrix for the linearsystem pentadiagonal, which can be solved very efficiently, i.e. via O(M) arithmeticoperations.

In practice, we always use the second-order Strang splitting [49]; i.e. from timet = tn to t = tn+1 (i) evolve (4.11) for half time step ∆t/2 with the initial datagiven at t = tn; (ii) evolve (4.12) for half time step ∆t/2 with the new data; (iii)evolve (4.10) for time step ∆t with the new data obtained in (ii); (iv) evolve (4.12)


for half time step ∆t/2 with the new data obtained in (iii), and (v) evolve (4.11)for half time step ∆t/2 with the newer data.

For the discretization considered here, the total memory requirement is O(ML)and the total computational cost per time step is O(ML lnL). The method istime reversible and time transverse invariant when the original CGPEs (2.1) does.Furthermore, following the similar proofs in [4, 7, 5], the total density can beshown to be conserved in the discretized level.

Remark 4.3. When λ = 0 in (4.1), in the above second-order Strang splittingfor the problem, the step (ii) and (iv) can be removed, and then the method willconsist of three steps. In this case, the density of each component is also conservedin the discretized level. In addition, a second-order finite difference discretizationfor (4.17)-(4.18) was proposed in [5].

Discretization in 3D When d = 3, we use the cylindrical coordinate (r, θ, z)and assume that

(4.22) ψ(r, θ, z, t) =L/2−1∑

l=−L/2

K−1∑

k=1

ψl,k(r, t) eilθ sin(µk(z − a)),

where L and K are two even positive integers, µk = πkb−a (k = 1, · · · ,K − 1) and

ψl,k(r, t) is the Fourier-sine coefficient for the (l, k)th mode. Plugging (4.22) into(4.15) with d = 3, noticing the orthogonality of the Fourier-sine modes, we obtain,for −L

2 ≤ l ≤ L2 − 1, 1 ≤ k ≤ K − 1 and 0 < r < R, that

(4.23) i∂ψl,k(r, t)

∂t= − 1

2r

∂

∂r

(r∂ψl,k(r, t)

∂r

)+

(l2

2r2+

µ2k

2− lΩ

)ψl,k(r, t),

with essential boundary conditions

(4.24) ψl,k(R, t) = 0 (for all l), ψl,k(0, t) = 0 (for l 6= 0).

The discretization of (4.23)-(4.24) is similar as that for (4.17)-(4.18) and thus omit-ted here.

For the algorithm in 3D, the total memory requirement is O(MLK) and thetotal computational cost per time step is O(MLK ln(LK)).

Remark 4.4. Another way to discretize the coupled GPEs (4.1) in the rota-tional frame is to use the efficient and accurate numerical method proposed in [10]for rotating single-component BEC. The key ideas are to apply a time-splitting fordecoupling the nonlinearity and to properly use the alternating direction implicit(ADI) technique for the coupling in the angular momentum rotation terms in theGPEs, at each time step, the GPEs in rotational frame is decoupled into a nonlin-ear ordinary differential equations (ODEs) and two systems of partial differentialequations with constant coefficients. For more details, we refer to [10].

5. Numerical results

For the completeness and convenience of the readers, here we also present somenumerical results for the dynamics of rotating two-component BEC [53]. For morenumerical results, one can refer to [4, 9, 53].

22 WEIZHU BAO

Example 1. Dynamics of the density of each component [53], we take d = 2,κ1 = κ2 = 0, λ = 1, Ω = 0.6, γx,j = γy,j = 1 (j = 1, 2) in (2.1). The initial data in(2.2) is chosen as

(5.1) ψ01(x) =

x + iy√π

exp(−x2 + y2

2

), ψ0

2(x) ≡ 0, x ∈ R2.

The problem is solved numerically on a bounded computational domain Ωx =(x, y), r =

√x2 + y2 < R with R = 12 by the numerical method in the previous

section and we choose mesh sizes ∆r = 0.005, ∆θ = π/128 and time step ∆t =0.0001.

Figure 1 [53] shows the time evolution of density of each component for two setsof interaction parameters: (i) β11 = β12 = β22 = 500 (i.e. a11 : a12 : a22 = 1 : 1 : 1);(ii) β11 = 500, β12 = 300 and β22 = 400 (i.e. a11 : a12 : a22 = 1 : 0.6 : 0.8).

(a)0 2 4 6 8 10

0

0.2

0.4

0.6

0.8

1

t (b)0 2 4 6 8 10

0

0.2

0.4

0.6

0.8

1

t

Figure 1. Time evolution of the densities N1(t) = ‖ψ1(·, t)‖2(dash line), N2(t) = ‖ψ2(·, t)‖2 (dot line) and N(t) = N1(t)+N2(t)(solid line) for two sets of interaction parameters: (a) β11 = β12 =β22; (b) β11 6= β12 6= β22.

From Fig. 1 [53], we can see that (i) the total density N(t) is conserved inthe discrete level for both cases; (ii) the densities of both components, i.e. N1(t)and N2(t), are periodic functions of period T = 2π/

√4λ2 + (κ1 − κ2)2 = π when

β11 = β12 = β22 (cf. Fig. 1a); otherwise when β11 6= β12 6= β22 they are periodicfunctions of period T = π with a perturbation (cf. Fig. 1b), which confirms theanalytical results in (3.33) and (3.34).

Example 2. Dynamics of vortex lattices [53], i.e. we d = 2, κ1 = κ2 = 0 andΩ = 0.9 in (2.1). The initial data in (2.2) is taken as the stationary square vortexlattices [53], which are computed numerically by using the above parameters aswell as λ = 0 and γx,j = γy,j = 1 (j = 1, 2) in (2.1) [?]. The problem is solvednumerically on a bounded computational domain Ωx = (x, y), r =

√x2 + y2 < R

with R = 12 by the numerical method in the previous section and we choose meshsizes ∆r = 0.005, ∆θ = π/128 and time step ∆t = 0.0001.

Figures 2 and 3 depict the contour plots of the wave functions |ψ1|2 and |ψ2|2at different times for two cases: (i) adding an external driving field, i.e. at t = 0,


changing λ in (2.1) from 0 to 1, and (ii) changing the trapping frequencies, i.e. att = 0, setting γx,1 = γy,1 = 0.9 and γx,2 = γy,2 = 1.1, respectively.

(a)

t = 0 t = 1.5 t = 3 t = 5

(b)

t = 0 t = 1.5 t = 3 t = 5

Figure 2. Contour plots of the wave functions |ψ1|2 (top row (a))and |ψ2|2 (bottom row (b)) at different times for case (i).

(a)

t = 0 t = 2.5 t = 5 t = 10

(b)

t = 0 t = 2.5 t = 5 t = 10

Figure 3. Contour plots of the wave functions |ψ1|2 (top row (a))and |ψ2|2 (bottom row (b)) at different times for case (ii).

From Figs. 2 and 3 [53], we can see that initially there are two square latticeswith about 16 and 21 quantized vortices in the first and second components, respec-tively (cf. Figs. 2&3 leftmost column). When we add an external driving field att = 0, the two vortex lattices rotate due to the angular momentum term and shifttheir condensate shapes almost periodically due to the external driving field (cf.

24 WEIZHU BAO

Fig. 2). On the other hand, if we change the trapping frequencies at time t = 0,the two vortex lattices rotate again due to the angular momentum term but thecondensate shape of each component keeps almost unchanged and the number ofvortices in each lattice doesn’t change during the dynamics (cf. Fig. 3). Of course,the lattice patterns are changed due to the inter-component interactions (cf. Figs.2&3).

6. Conclusion

We reviewed some recent results on the dynamics of rotating two-componentBose-Einstein condensates (BEC) and their efficient and accurate computation. Webegan with the three-dimensional (3D) coupled Gross-Pitaevskii equations (GPEs)with an angular momentum rotation term and an external driving field, showedhow to scale it into dimensionless form, reduce it to a 2D and 1D GPE in thelimiting regime of strong anisotropic confinement and presented its semiclassicalscaling and geometric optics. Analytical and numerical results for the dynamics oftwo-component BEC were reviewed. Finally, the analytical results and numericalmethods for two-component BEC can be extended to spin-1 BEC [10, 8, 51].

Acknowledgment. I would like to thank my collaborators over the years fortheir help in appreciating the difficulties inherent to the mathematical analysis andnumerical simulation for Bose-Einstein condensation: Peter A. Markowich, DieterJaksch, Qiang Du, Jie Shen, Hailiang Li, Yanzhi Zhang, Hanquan Wang, WeijunTang, Fong Yin Lim, I-Liang Chern, Rada M. Weishaupl, Yunyi Ge and Ming-Huang Chai.

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Department of Mathematics and Center for Computational Science and Engineer-

ing, National University of Singapore, Singapore 11743E-mail address: [email protected]; URL: http://www.math.nus.edu.sg/~bao/

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