# Aligned dipolar Bose-Einstein condensate in a double-well potential: From cigar-shaped to pancake-shaped

**ABSTRACT** We consider a Bose-Einstein condensate (BEC), which is characterized by long-range and anisotropic dipole-dipole interactions and vanishing s-wave scattering length, in a double-well potential. The properties of this system are investigated as functions of the height of the barrier that splits the harmonic trap into two halves, the number of particles (or dipole-dipole strength) and the aspect ratio $\lambda$, which is defined as the ratio between the axial and longitudinal trapping frequencies $\omega_z$ and $\omega_{\rho}$. The phase diagram is determined by analyzing the stationary mean-field solutions. Three distinct regions are found: a region where the energetically lowest lying stationary solution is symmetric, a region where the energetically lowest lying stationary solution is located asymmetrically in one of the wells, and a region where the system is mechanically unstable. For sufficiently large aspect ratio $\lambda$ and sufficiently high barrier height, the system consists of two connected quasi-two-dimensional sheets with density profiles whose maxima are located either at $\rho=0$ or away from $\rho=0$. The stability of the stationary solutions is investigated by analyzing the Bogoliubov de Gennes excitation spectrum and the dynamical response to small perturbations. These studies reveal unique oscillation frequencies and distinct collapse mechanisms. The results derived within the mean-field framework are complemented by an analysis based on a two-mode model. Comment: 21 pages, 16 figures

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arXiv:0908.4313v2 [cond-mat.quant-gas] 25 Nov 2009

Aligned dipolar Bose-Einstein condensate in a double-well potential: From

cigar-shaped to pancake-shaped

M. Asad-uz-Zaman1and D. Blume1

1Department of Physics and Astronomy, Washington State University, Pullman, Washington 99164-2814, USA

(Dated: November 25, 2009)

We consider a Bose-Einstein condensate (BEC), which is characterized by long-range and

anisotropic dipole-dipole interactions and vanishing s-wave scattering length, in a double-well po-

tential. The properties of this system are investigated as functions of the height of the barrier

that splits the harmonic trap into two halves, the number of particles (or dipole-dipole strength)

and the aspect ratio λ, which is defined as the ratio between the axial and longitudinal trapping

frequencies ωz and ωρ. The phase diagram is determined by analyzing the stationary mean-field

solutions. Three distinct regions are found: a region where the energetically lowest lying stationary

solution is symmetric, a region where the energetically lowest lying stationary solution is located

asymmetrically in one of the wells, and a region where the system is mechanically unstable. For

sufficiently large aspect ratio λ and sufficiently high barrier height, the system consists of two con-

nected quasi-two-dimensional sheets with density profiles whose maxima are located either at ρ = 0

or away from ρ = 0. The stability of the stationary solutions is investigated by analyzing the Bo-

goliubov de Gennes excitation spectrum and the dynamical response to small perturbations. These

studies reveal unique oscillation frequencies and distinct collapse mechanisms. The results derived

within the mean-field framework are complemented by an analysis based on a two-mode model.

PACS numbers:

I.INTRODUCTION

Dipolar BECs have recently attracted a lot of attention both theoretically and experimentally [1, 2]. The experi-

mental realization of a Cr BEC just a few years ago constitutes an important milestone [3]. Compared to alkali atoms,

Cr has a comparatively large magnetic dipole moment of 6µB, which leads to an enhancement of the dipole-dipole

interactions by a factor of 36 compared to alkali atoms. The anisotropy of the dipole-dipole interactions has been

observed experimentally by analyzing time of flight expansion images of Cr BECs released from a cylindrically sym-

metric external confining potential [4]. If combined with theoretical calculations, the time of flight images reveal the

initial density distribution of the dipolar gas and depend, e.g., on whether the magnetic dipole moments are aligned

along the axial or longitudinal confining directions, respectively. Furthermore, by taking advantage of the tunability

of the s-wave scattering length near a magnetic Fano-Feshbach resonance, the relative importance of the dipole-dipole

interactions can be changed [5, 6, 7], paving the way for a variety of interesting experimental studies. Dipolar BECs

are characterized by intriguing collapse mechanisms [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], and unique excitation

spectra [15, 18, 19] and vortex structures [20, 21, 22, 23, 24]. In addition, dipolar gases loaded into optical lattices

may allow for the realization of novel phases [25, 26, 27].

Although the dipole-dipole interactions in alkali gases are too weak to result in observable effects in most experi-

ments, it is believed that they play a decisive role in the formation of spin textures in87Rb condensates [28, 29], in

the dynamics of Bloch oscillations of39K BECs loaded into an optical lattice [30] and in7Li BECs [31]. Furthermore,

BECs and degenerate Fermi gases that consist of polar molecules may be realized in the near future [32, 33, 34]. This

prospect adds a new intriguing twist since the interactions between two polar molecules can be tuned by an external

electric field [35]. This opens the possibility to enter the strongly-correlated regime and thus to realize a variety of

condensed matter analogs [25, 36, 37].

This paper considers an aligned dipolar BEC in a double well geometry. Double well potentials play an important

role in chemical and condensed matter physics, among other areas. In the context of cold atom physics, s-wave

dominated alkali systems in a double-well have, e.g., been used to study Josephson-type oscillations [38, 39, 40, 41,

42, 43, 44, 45, 46]. The density oscillations of the Bose gas can be interpreted as corresponding to the charge current

that characterizes “standard” condensed matter Josephson junctions. Related to this, the macroscopic quantum

self-trapping of atoms in one of the wells has been demonstrated experimentally and has been interpreted within a

two-mode model that can be derived from the Gross-Pitaevskii (GP) equation [44, 45]. The double well system has

also been used to experimentally study spin-squeezing [47]. In this context, the left and the right wells of the system

serve as the two arms of an interferometer [48]. The number difference and relative phase of the double well system

are conjugate variables, whose combined measurements has revealed that the system is entangled [47].

Here, we investigate the behaviors of aligned dipoles under cylindrically symmetric harmonic confinement with

a repulsive Gaussian potential centered at z = 0. We limit ourselves to situations where the dipoles are aligned

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along one of the symmetry axis of the external harmonic confining potential. This restriction reduces the parameter

space and also significantly reduces the numerical efforts. Arguably, it may be the conceptually simplest case. We

are particularly interested in determining the phase or stability diagram for both cigar-shaped and pancake-shaped

harmonic confinement. The boundaries of these phase diagrams are governed by the non-trivial interplay of the dipole-

dipole interactions, the energy due to the external harmonic confinement, the energy due to the Gaussian potential

and the kinetic energy. The interplay of these energy contributions leads to density profiles unique to anisotropic

interactions. In agreement with Ref. [49], we observe Josephson oscillations as well as macroscopic quantum self-

trapping of the system for appropriately chosen parameters. We characterize the transition between these two regimes

by analyzing the excitation spectrum and the real time response of the system to a small perturbation. Our study

of two neighboring pancake-shaped dipolar gases can be viewed as a first step towards understanding a multi-layer

system of dipolar pancakes. For a single pancake, an angular roton instability has been predicted to occur [15]. For a

layer of two-dimensional dipolar BECs, a new length scale is given by the interlayer distance and the roton instability

has been predicted to be enhanced compared to the single layer case [50]. Other multi-layer studies can be found in

Refs. [51, 52].

The remainder of this paper is organized as follows. Section II introduces the mean-field GP equation and discusses

how we determine the stationary and time-dependent solutions.

employed to determine the excitation spectrum of the dipolar gas are introduced. Section III reviews the two-mode

model which provides an intuitive understanding of the time-independent and time-dependent GP solutions in the

small λ regime. Section IV presents our stationary solutions. We discuss the phase diagram as functions of the

number of particles (or equivalently, the mean-field strength), the aspect ratio and, in selected cases, the barrier

height. Section V presents our time-dependent studies.

deduce distinct collapse mechanisms from the response of the system to a small perturbation. In addition, selected

Bogoliubov de Gennes eigenmodes are discussed. Lastly, Sec. VI summarizes our main findings and discusses possible

future studies.

The Bogoliubov de Gennes equations that are

We investigate certain dynamically stable regimes and

II.MEAN-FIELD DESCRIPTION OF DIPOLAR BECS

Section IIA introduces the time-dependent mean-field GP equation for a dipolar BEC and discusses the numerical

techniques employed to determine stationary and time-dependent solutions. Section IIB reviews the Bogoliubov de

Gennes equations for the dipolar BEC.

A. Gross-Pitaevskii equation

The time-dependent GP equation for a dipolar BEC consisting of N identical point dipoles is given by [9, 10, 53]

i?∂ψ(? r,t)

∂t

= Hψ(? r,t), (1)

where the mean-field Hamiltonian H reads

H = −?2

2m∇2+ Vext(? r) +

(N − 1)

?

Vdd(? r −? r′)|ψ(? r′,t)|2d3? r′. (2)

Here, m denotes the mass of the dipoles. We interpret ψ(? r,t) as a single-particle wave function and correspondingly

use the normalization?|ψ(? r,t)|2d3? r = 1. The external cylindrically-symmetric confining potential Vextconsists of a

harmonic trapping potential Vhowith angular frequencies ωρand ωzand a Gaussian barrier Vgwith height A (A > 0)

and width b,

Vext(? r) = Vho(ρ,z) + Vg(z), (3)

where

Vho(ρ,z) =1

2m(ω2

ρρ2+ ω2

zz2) (4)

and

Vg(z) = Aexp

?

−z2

2b2

?

. (5)

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3

We define the aspect ratio λ of the harmonic confining potential as

λ =ωz

ωρ. (6)

Throughout, we employ cylindrical coordinates and write ? r = (ρ,ϕ,z).

The third term on the right hand side of Eq. (2) represents the mean-field potential, which depends on both the

density of the system and the dipole-dipole potential Vdd. Throughout, we assume that the dipoles are aligned along

the z-axis,

Vdd(? r −? r′) = d21 − 3cos2ϑ

|? r −? r′|3

,(7)

where d denotes the strength of the dipole moment of the dipolar atom or molecule under study and ϑ the angle

between the relative distance vector ? r −? r′and the z-axis. Throughout, we assume that the s-wave scattering length

asvanishes, implying the absense of the usual s-wave contact interaction term in Eq. (2). For dipolar Cr BECs, e.g.,

this can be achieved by varying an external magnetic field in the vicinity of a Fano-Feshbach resonance [5, 6, 7].

Rewriting the integro-differential equation [Eq. (1) with Eqs. (2)-(7)] in harmonic oscillator units azand Ez, where

az=

?

?

mωz

(8)

and

Ez= ?ωz, (9)

shows that the GP equation depends on four dimensionless parameters: i) d2(N − 1)/(Eza3

the strength of the mean-field potential; ii) the aspect ratio λ; iii) the scaled barrier height A/Ez; and iv) the scaled

barrier width b/az. To reduce the parameter space, we consider a fixed barrier width b, i.e., b = az/5. While most of

our calculations are performed for A = 12Ez, we consider smaller barrier heights in selected cases. The aspect ratio

is varied from λ = 0.1 (cigar-shaped external harmonic confinement) to λ = 12 (pancake-shaped external harmonic

confinement). Lastly, the dimensionless mean-field strength D,

z), which characterizes

D =d2(N − 1)

Eza3

z

, (10)

is, for a given A and λ, varied from 0 to the value Dcrat which collapse occurs.

In practice, the mean-field strength D can be adjusted by loading the double-well potential with condensates of

varying particle number N. More conveniently, one might envision tuning the electric dipole moment of a molecular

BEC through the application of an external electric field [35] or, in the case of magnetic Cr BECs, by changing the

ratio between the dipole-dipole and the s-wave interactions through the application of an external magnetic field in

the vicinity of a Fano-Feshbach resonance [5, 6, 7]. Although our study considers as= 0 and varying D, the latter

scenario should allow for the observation of a number of features predicted in this study. Experimentally, the Gaussian

barrier potential of varying height and width can be realized by a repulsive dipole beam with adjustable intensity and

waist.

The solutions to the integro-differential mean-field equations have to be determined self-consistently since the

density |ψ|2, which is part of the solution sought, also enters into the mean-field potential. The stationary solutions

can be written as ψ(? r) = Ψ(ρ,z)h(ϕ) with h(ϕ) = exp(ikϕ)/√2π. In the following, we seek stationary solutions with

azimuthal quantum number k = 0. Our calculation of the excitation spectrum do, however, include k > 0 modes

(see Sec. IIB). The evaluation of the integral contained in the mean-field potential can be performed most readily

by transforming to momentum space via a combined Fourier-Hankel transform [54]. To determine the stationary

solutions, we implemented two different approaches: i) We minimize the total energy of the system following the

conjugate gradient approach [55]. In this approach, the solution is expanded in terms of harmonic oscillator basis

functions in ρ and z, and the expansion coefficients are optimized so as to minimize the total energy per particle. ii)

We propagate an initial state in imaginary time till the stationary solution has been projected out.

The basis functions and the initial state are both represented on a grid in the ρ and z directions. The grid along ρ

is chosen according to the zeroes of the Bessel functions (see Ref. [54]), which are distributed roughly linearly. Along

the z-direction, we use a linear grid. For most calculations, a grid of Nρ× Nz= 64 × 128 is sufficient. We employ a

rectangular simulation box of lengths [0,ρmax] and [−zmax,zmax]. For pancake-shaped systems (i.e., λ > 1), a “cutoff”

is used for the dipolar potential (i.e., the interaction is truncated for |z| > zmax), which reduces the interaction of the

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true BEC with an “artificial image BEC” and thus allows for the usage of a smaller zmax[54]. For λ < 1, no cutoff is

employed. Typical values for ρmaxand zmaxare around 15aρand 12az, respectively, where aρ=??/(mωρ).

We have checked that the conjugate gradient and imaginary time evolution approaches result, within our numerical

accuracy, in identical energies and densities. Furthermore, for vanishing barrier height, i.e., for A = 0, our solutions

for cylindrically symmetric harmonic traps agree with those reported in the literature [15, 54]. For non-vanishing

barrier height, we compared our solutions with those reported in Ref. [49]. Our energies and chemical potentials are

in reasonable agreement with those reported in Ref. [49]. For A = 4Ez, b = 0.2az, λ = 0.1 and D = 0.6, e.g., we

find E/N = 10.69Ezand µ = 10.00Ezwhile the values reported in Fig. 2 of Ref. [49] are smaller by about 3 and 4%,

respectively. These deviations are somewhat larger than our estimated numerical uncertainty.

The time dynamics of the system is determined by evolving a given initial state in real time. The initial state

is chosen according to the variational two-mode model wave function (see Sec. III) or by adding a small random or

smooth perturbation to the stationary GP wave function of the energetically lowest lying state. If the system collapses

to a high density state in response to the application of a small perturbation, then our simulations are only able to

follow the real time evolution for a limited time period. Eventually, our grid becomes too coarse to accurately present

the time evolved state. Since our main aim is directed at identifying the stability and the collapse mechanisms, this

artefact does not pose any true limitations on our analysis. In fact, once the density becomes sufficiently high, the

mean-field GP description breaks down anyways and beyond mean-field corrections need to be included. Such a

treatment is, however, beyond the scope of the present work.

B.Bogoliubov de Gennes equations

In addition to time-evolving a given initial state, we analyze the stability of the dipolar BEC by seeking solutions

to the time-dependent GP equation of the form [56]

ψ(? r,t) = exp(−iµt/?)[ψ0(? r) + δψ(? r,t)], (11)

where ψ0(? r) denotes the energetically lowest lying solution of the time-independent GP equation with k = 0 and µ

the corresponding chemical potential. We seek “perturbations” δψ(? r,t) that oscillate with frequency ω,

δψ(? r,t) = u(? r)exp(−iωt) + v∗(? r)exp(iωt), (12)

where u(? r) and v(? r) denote the Bogoliubov de Gennes “particle” and “hole” functions [57]. Plugging Eq. (11) with

δψ given by Eq. (12) into Eq. (1), keeping terms up to first order in δψ(? r,t) and its complex conjugate, and equating

the coefficients of the terms oscillating with exp(−iωt) and exp(iωt), respectively, we find the Bogoliubov de Gennes

equations [54]

?ωu(? r) = A(? r)u(? r) +

0(? r′)u(? r′)d3? r′ψ0(? r) +(N − 1)

?

Vdd(? r −? r′)ψ∗

?

(N − 1)Vdd(? r −? r′)ψ0(? r′)v(? r′)d3? r′ψ0(? r) (13)

and

− ?ωv∗(? r) = A(? r)v∗(? r) +

Vdd(? r −? r′)ψ∗

?

(N − 1)

?

0(? r′)v∗(? r′)d3? r′ψ0(? r) +

(N − 1)Vdd(? r −? r′)ψ0(? r′)u∗(? r′)d3? r′ψ0(? r).(14)

In Eqs. (13) and (14), the operator A(? r) is defined as

A(? r) = H0− µ +

Vdd(? r −? r′)|ψ0(? r′)|2d3? r′,(N − 1)

?

(15)

where H0denotes the Hamiltonian of the non-interacting system,

H0= −?2

2m∇2+ Vext(? r). (16)

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Equations (13) and (14) can be decoupled by introducing two new functions f and g, f(? r) = u(? r) + v(? r) and

g(? r) = −u(? r) + v(? r). Assuming, without loss of generality, that ψ0(? r) is real, we find

?2ω2f(? r) = A(? r)[A(? r)f(? r)] +

2(N − 1)A(? r)

??

f(? r′)Vdd(? r −? r′)ψ0(? r′)d3? r′ψ0(? r)

?

(17)

and

?2ω2g(? r) = A(? r)[A(? r)g(? r)] +

Vdd(? r −? r′)ψ0(? r′)A(? r′)g(? r′)d3? r′ψ0(? r). 2(N − 1)

?

(18)

Following Ref. [54], we solve Eq. (18) for the square of the Bogoliubov de Gennes excitation frequency ω and the

corresponding eigenvector g(? r) iteratively using the Arnoldi method. Once g(? r) is determined, the eigenvector f(? r)

can be obtained from the identity

f(? r) = −1

?ωA(? r)g(? r). (19)

The physical meaning of f is elucidated by calculating the density |ψ(? r,t)|2up to first order in δψ and its complex

conjugate. For real u and v, this gives

|ψ(? r,t)|2≈ |ψ0(? r)|2+ 2cos(ωt)ψ0(? r)f(? r), (20)

which shows that f(? r), together with ψ0(? r) and ω, determines the time-dependent density. Due to the cylindrical

symmetry of the system, the ϕ dependence of f(? r) separates, f(? r) =¯f(ρ,z)h(ϕ). Section V discusses the behavior of

¯f(ρ,z), which we refer to as the Bogoliubov de Gennes eigenmode, for different k and various (D,λ) combinations.

The outlined approach allows for the determination of a sequence of excitation frequencies for a given azimuthal

quantum number k at a time. It can be seen from Eq. (12) that a negative ω2and thus a purely imaginary ω

corresponds to a situation where the stationary ground state solution is dynamically unstable.

III. TWO-MODE MODEL

Atomic BECs, coupled through non-vanishing potential barriers, have been used extensively to model coupled

condensed matter systems such as3He-B reservoirs [39, 41, 42, 45, 58]. Although neutral, the study of weakly-

coupled atomic BECs allows, e.g., for the realization of a variety of typical dc and ac effects that characterize charged

Cooper pair superconducting junctions [42, 59]. The connection between weakly-coupled atomic BECs and more

traditional condensed matter systems becomes most apparent if the former is approximated by a two-mode model

and mapped to a Josephson like Hamiltonian. Here, our primary motivation for employing the two-mode model is to

develop an intuitive understanding of some of the phenomena observed in our time-independent and time-dependent

mean-field studies.

Let ψS(? r) and ψA(? r) denote the energetically lowest lying stationary GP solutions that are respectively symmetric

and anti-symmetric with respect to z = 0. If the symmetric function ψS(? r) is the energetically lowest lying solution of

the stationary GP equation, we calculate it by employing the conjugate gradient method or by evolving in imaginay

time (see Sec. II). The anti-symmetric solution ψA(? r) is obtained by restricting the basis functions employed in the

conjugate gradient method to functions that are anti-symmetric with respect to z = 0. Without loss of generality, we

assume in the following that ψSand ψAare real. In the two-mode model, the solutions ψSand ψAare treated as a

basis that defines the two “modes” ΦL(? r) and ΦR(? r),

ΦL,R(? r) =ψS(? r) ± ψA(? r)

√2

.(21)

By construction, ΦL(? r) and ΦR(? r) are normalized to one and orthogonal to each other. The functions ΦLand ΦR

are, for appropriately chosen parameters, located predominantly in the left well and in the right well, respectively.

Within the two-mode model, the time-dependent wave function is approximated by (see, e.g., Ref. [56])

ψ(? r,t) = cL(t)ΦL(? r) + cR(t)ΦR(? r), (22)

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where the complex-valued time-dependent expansion coefficients cL(t) and cR(t) are related through the normalization

condition |cL(t)|2+ |cR(t)|2= 1. Defining cL,R(t) = |cL,R(t)|exp[iθL,R(t)], the time evolution within the two-mode

model is governed by two variables: the fractional difference Z(t) of the population located in the left and in the right

well,

Z(t) = |cL(t)|2− |cR(t)|2,(23)

and the phase difference or relative phase φ(t),

φ(t) = θR(t) − θL(t).(24)

Plugging Eq. (22) into Eq. (1), multiplying by ΦL(? r) and ΦR(? r), respectively, and integrating out the spatial degrees

of freedom, we obtain two coupled equations that govern the time dynamics:

i?dcL(t)

dt

=

?E0+ B + (U − B)|cL(t)|2?cL(t) − TcR(t) (25)

and

i?dcR(t)

dt

=

?E0+ B + (U − B)|cR(t)|2?cR(t) − TcL(t). (26)

In deriving Eqs. (25) and (26), we neglegted terms of the form

(N − 1) ×

? ?

Φi(? r)Φj(? r)Vdd(? r −? r′)Φk(? r′)Φl(? r′)d3? r′d3? r(27)

with i ?= j or k ?= l, where i,j,k and l can take the values L and R. These terms are small as long as ΦLand ΦRare

located predominantly in the left well and in the right well, respectively. In Eqs. (25) and (26), the onsite, offsite (or

interaction tunneling) and tunneling matrix elements U, B and T are defined as

U = (N − 1) ×

? ?

(ΦL(? r))2Vdd(? r −? r′)(ΦL(? r′))2d3? r′d3? r, (28)

B = (N − 1) ×

? ?

(ΦL(? r))2Vdd(? r −? r′)(ΦR(? r′))2d3? r′d3? r,(29)

and

T =

? ?−?2

2m∇ΦL(? r) · ∇ΦR(? r)−

ΦL(? r)Vext(? r)ΦR(? r)]d3? r,(30)

and the “zero point energy” E0is defined as

E0=

? ??2

2m|∇ΦL(? r)|2+ Vext(? r)(ΦL(? r))2

?

d3? r. (31)

Usage of ΦR(? r) instead of ΦL(? r) in Eqs. (28) and (31) gives the same result.

Rewriting the coupled equations (25) and (26) in terms of Z(t) and φ(t) leads to the classical Hamiltonian HTM

(using ? = 1),

HTM= 2T

?

ΛZ2(t)

2

−

?

1 − Z2(t)cos(φ(t))

?

,(32)

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0.11 10100

D

0.1

1

10

λ

S

SB

U

FIG. 1:

dash-dot-dotted, dash-dotted and dashed lines indicate those λ and D values at which the character of the energetically lowest

lying stationary GP solution with k = 0 changes from symmetric (S) to symmetry-broken (SB), from symmetry-broken to

unstable (U), and from symmetric to unstable, respectively. The “solid (red) islands” in the upper right corner of the phase

diagram (comparatively large D and λ) indicate two regions of the phase diagram where the solutions are symmetric but where

the density maximum is located at ρ > 0; these islands are discussed in more detail in the context of Fig. 5. For comparison,

circles show the boundary between the symmetric and symmetry-broken regions predicted by the two mode model. Note the

log scale of both axes.

(Color online) Character of the energetically lowest lying stationary GP solution for b = 0.2az and A = 12Ez: The

where

Λ =U − B

2T

. (33)

Notably, Z(t) and φ(t) are conjugate variables of the classical Hamiltonian. The energy of HTM is conserved and

can, e.g., be obtained by inserting Z(0) and φ(0) into Eq. (32). The properties of HTM have been discussed in detail

in the literature [39, 41, 42, 45]. Here, we review a few points that will aid in the understanding of our GP solutions.

The two-mode model immediately leads to three different classes of stationary solutions, i.e., solutions with constant

Z(t) and φ(t): A symmetric solution for φ(t) = 2πn (n integer) and Z(t) = 0; its energy is −2T. An anti-symmetric

solution for φ(t) = (2n+1)π (n integer) and Z(t) = 0; its energy is 2T. A symmetry-brokensolution for φ(t) = (2n+1)π

(n integer) and Z(t) = ±√1 − Λ−2; this solution exists only if |Λ| > 1 and its energy is T(Λ + Λ−1). Section IV

compares these stationary two-mode model solutions with those obtained from the stationary GP solutions.

It turns out that Hamilton’s equations of motion can be solved analytically for HTM[41]. Of particular interest for

our study is the so-called Josephson oscillation frequency ωJ, which—for small amplitude motion—can be expressed

in terms of Λ (with ? “restored”),

?ωJ,TM= 2T√1 + Λ. (34)

Section V compares the two-mode model frequency ωJ,TMwith the frequency obtained from the real time dynamics

and by solving the Bogoliubov de Gennes equations. For the real time dynamics, we prepare an initial state at time

t = 0 according to Eq. (22) and then time evolve this state according to the time dependent mean-field Hamiltonian.

A Fourier analysis of the expectation value of z(t) then reveals the predominant excitation frequency.

IV.DISCUSSION OF STATIONARY SOLUTIONS

This section discusses our solutions to the stationary GP equation. In particular, we present the phase diagram as

a function of the aspect ratio λ and the mean-field strength D for a fixed barrier height A and discuss selected density

profiles. Furthermore, we discuss how the phase diagram changes with varying barrier height and explain some of the

GP results within the two-mode model.

Figure 1 summarizes the character of the energetically lowest lying solutions with k = 0 of the stationary GP

equation as functions of the aspect ratio λ and the mean-field strength D for A = 12Ez. The parameter combination

(λ,D) = (0.1,1) corresponds, e.g., to a Cr condensate with vanishing s-wave scattering length, ωz = 2π × 10Hz,

ωρ = 2π × 100Hz and N ≈ 1835. The “phase diagram” consists of three regions: Firstly, a region where the

energetically lowest lying state with k = 0 of the stationary GP equation is symmetric with respect to the z-axis; we

refer to this solution as symmetric (“S”) throughout this paper. Examplary density profiles are shown in Figs. 2(a),

Page 8

8

0 0.40.8

-2

0

2

z / az

0 0.4 0.8

-2

0

2

z / az

0

5

10

15

-3

0

3

z / az

0

5

10

15

ρ / az

-3

0

3

z / az

(a)

(b)

(c)

(d)

FIG. 2: (Color online) Density plots of the energetically lowest lying stationary GP solution with k = 0 for b = 0.2az, A = 12Ez

and four different (D,λ) combinations: (a) (D,λ) = (0.5477, 0.3) (symmetric solution), (b) (D,λ) = (1.643,0.3) (symmetry-

broken solution), (c) (D,λ) = (316.2,10) (symmetric solution with density maximum at ρ = 0) and (d) (D,λ) = (474.3,10)

(symmetric solution with density maximum at ρ > 0). The contour lines are chosen equidistant in all four panels. The dashed

contours correspond to (a) 0.05a−3

z

0.35a−3

z

and (d) 0.0007a−3

z , (b) 0.1a−3

z , (c) 0.0001a−3

z .

and (d) 0.0001a−3

z , while the solid contours correspond to (a)

z , (b) 0.7a−3

z , (c) 0.0007a−3

2(c) and 2(d) (see below for more details). Secondly, a region where the energetically lowest lying state of the

stationary GP equation is neither symmetric nor anti-symmetric with respect to the z-axis; we refer to this solution

as symmetry-broken (“SB”) or asymmetric. An examplary density profile is shown in Fig. 2(b). And thirdly, a region

where the stationary GP equation supports a high-density or collapsed solution but no gas-like solution. We refer to

this solution as mechanically unstable (“U”). Sections IVA and IVB discuss the properties of the phase diagram in

more detail for λ ? 1 and λ ? 1, respectively.

A.“Small” aspect ratio (λ ? 1)

Figure 3(a) shows the energy contributions to the total energy per particle Etot/N for λ = 0.3 and A = 12Ezas a

function of D: the kinetic energy per particle Ekin(dashed line); the harmonic trap energy per particle Eho(dotted

line), which is defined as the expectation value of Vho; the Gaussian energy per particle Eg(dash-dotted line), which

is defined as the expectation value of Vg; and the mean-field dipole-dipole energy per particle Edip(dash-dash-dotted

line), which is defined as the expectation value of the mean-field term [third term on the right hand side of Eq. (2)].

A solid line shows the total energy per particle Etot/N. The energies terminate at the critical value Dcrat which the

stationary GP equation first supports a negative energy solution.

Page 9

9

0

1

2

-2

0

2

4

Energy/Ez

012

D

0.2

0.3

Eg / Ez

(a)

(b)

FIG. 3: (Color online) (a) Energy contributions of the energetically lowest lying stationary GP solution with k = 0 as a function

of D for λ = 0.3, A = 12Ez and b = 0.2az. The solid, dashed, dotted, dash-dotted and dash-dash-dotted lines show Etot/N,

Ekin, Eho, Eg and Edip, respectively. (b) Blow-up of the Gaussian energy Eg. Eg exhibits a kink at D ≈ 0.75, indicating the

symmetry change (symmetric to symmetry-broken) of the energetically lowest lying stationary GP solution.

The Gaussian energy Eg is shown on an enlarged scale in Fig. 3(b). It can be seen that Eg shows a “kink” at

D ≈ 0.75. We find that the other energy contributions (i.e., Ekin, Ehoand Edip) and Etot/N exhibit kinks at the same

D value. These kinks are, however, less pronounced and not (or hardly) visible on the scale shown in Fig. 3(a). Our

analysis shows that the D values at which the kinks occur coincide with the D values at which the density profiles

of the energetically lowest lying stationary GP solutions change from symmetric to symmetry-broken. In most of

our calculations for fixed b, A and λ but varying D, we use the kink in Eg to determine the D value at which the

character of the energetically lowest lying stationary GP solution changes and thus to obtain the dash-dot-dotted line

in Fig. 1. The stability of the solutions around the symmetry to symmetry-broken transition is discussed in Sec. V in

the context of Figs. 7 through 10.

The dash-dot-dotted lines in Fig. 1 can be reproduced qualitatively by the two-mode model (see circles in Fig. 1).

To illustrate some aspects of the two-mode model, solid and dashed lines in Fig. 4 show Λ [see Eq. (33)] as a function

of D for λ = 0.3 and 0.4, respectively, and A = 12Ezand b = 0.2az. For |Λ| ≤ 1 and positive T, the two-mode model

predicts a symmetric stationary ground state. For |Λ| > 1, a symmetry-broken solution is supported; if T > 0 and

Λ < −1, the symmetry-broken state has a lower energy than the symmetric state. Vertical arrows in Fig. 4 mark the

D values, D ≈ 1.31 and 2.89, at which the transition from symmetric to symmetry-broken occurs for λ = 0.3 and

λ = 0.4, respectively. These two-mode model predictions (also shown as circles in Fig. 1) are slightly larger than the

results obtained by solving the GP equation but predict the symmetric to symmetry-broken transition qualitatively

correctly.

It is interesting to compare the behavior of Λ, which can be interpreted as the ratio between an effective interaction

energy and twice the tunneling energy, for λ = 0.3 and 0.4 (solid and dashed lines in Fig. 4). For λ = 0.3, an increase

of the mean-field strength D leads to a monotonic decrease of Λ. For λ = 0.4, in contrast, Λ first increases, reaches

a maximum at D ≈ 0.87 and then decreases monotonically. We find that the onsite energy U and the offsite energy

B are both negative for all D shown in Fig. 4. A change of the aspect ratio λ effectively changes the strength of the

dipole-dipole interaction, leading to a less attractive U than B, and thus to a positive Λ, for small D and λ = 0.4. For

λ = 0.3, in contrast, the onsite energy U is always more negative than the offsite energy B, resulting in a negative Λ

for all D.

In addition to the barrier height A = 12Ez, we considered smaller barrier heights A, in particular A = 4Ezand 8Ez,

Page 10

10

012

D

34

-1

0

Λ

FIG. 4: (Color online) Two-mode model parameter Λ for A = 12Ez and b = 0.2az as a function of D for λ = 0.3 (solid line)

and λ = 0.4 (dashed line). Vertical arrows mark the D values at which |Λ| equals 1; for |Λ| < 1 and > 1, the two-mode model

predicts that the energetically lowest lying stationary state is symmetric and symmetry-broken, respectively.

for a few selected λ values. Our calculations suggest that the dash-dot-dotted line in Fig. 1 (i.e., the line that marks the

symmetric to symmetry-broken transition) moves to larger D values with decreasing A while the dash-dotted line (i.e.,

the line that marks the symmetry-broken to unstable transition) remains approximately unchanged with decreasing A.

The dependence of the dash-dot-dotted line on A for fixed λ and b can be explained by applying the two-mode model.

As A decreases, the tunneling energy T becomes more important compared to the absolute value of the effective

interaction energy U − B. This implies that |Λ| decreases with decreasing A (for fixed λ and b). Correspondingly,

a larger D is required for the two-mode model condition |Λ| = 1, which signals the symmetric to symmetry-broken

transition, to be fulfilled. The fact that the dash-dotted line in Fig. 1 remains to first order unchanged with decreasing

A is due to the fact that the density of the system prior to collapse is located predominantly in one of the wells. This

implies that the density prior to collapse is only weakly dependent on A, thus explaining the comparatively small

dependence of the dash-dotted line on A for the parameter combinations investigated.

We note at this point that the linear stationary Schr¨ odinger equation permits only symmetric and anti-symmetric

solutions but no symmetry-broken solutions. This fact emphasizes that the transition from symmetric to symmetry-

broken is driven by mean-field interactions. Furthermore, this fact implies that the symmetry-broken solution should

disappear if sufficiently many higher order corrections to the mean-field GP equation are taken into account (see,

e.g., Ref. [41]). In this sense, the appearance of the symmetry-broken region in the phase diagram is an artefact of

the mean-field formalism. It is, however, intimately related to the dynamical phenomena of Josephson oscillation and

macroscopic quantum self-trapping, both of which have been observed experimentally for s-wave interacting BECs.

We return to these considerations in Sec. V in the context of the discussion of Figs. 7 through 10.

B. “Large” aspect ratio (λ ? 1)

Figure 5 shows an enlargement of the large λ region of Fig. 1 using a linear scale for both λ and D. The S0region of

the phase diagram is characterized by GP solutions whose density maxima are located at ρ = 0 [see Figs. 2(a) and 2(c)

for examples] while the S>0region of the phase diagram is charactericed by GP solutions whose density maxima are

located at ρ > 0 [see Fig. 2(d) for an example]. The latter class of density profiles only exists in a narrow parameter

region of the phase diagram; in particular, these solutions only arise for pancake-shaped confining potentials and not

for cigar-shaped confining potentials. Furthermore, the solutions with S>0character are unique to dipolar gases, i.e.,

they are not observed for purely s-wave interacting gases, and thus directly reflect the anisostropic long-range nature

of the dipole-dipole interactions.

The two different classes of symmetric solutions have previously been characterized for A = 0, i.e., for a pancake-

shaped trapping geometry without barrier [15, 60]. In those studies, a dipolar BEC with density maximum at ρ > 0

was termed “red blood cell”, as its isodensity surface is reminiscent of the shape of a red blood cell. The S>0regions

in Fig. 5 are characterized by the formation of two staggered red blood cells. Section V shows that the dynamical

instability near Dcrof the stationary k = 0 ground state solutions of types S0and S>0is distinctly different.

Figure 5 shows that, generally speaking, the D value at which the dipolar gas becomes unstable increases with

increasing λ. This trend can be understood by realizing that an increase of λ leads to a “flattening” of the system so

that the dipoles interact effectively more repulsively. The boundary near the stable and unstable regions shows a rich

structure: i) As already noted above, S>0islands in which the density profiles are structured exist. ii) The boundary

Page 11

11

0 200400

D

600

6

8

10

12

λ

S0

S>0

U

S>0

FIG. 5:

region where the energetically lowest lying symmetric stationary GP solution has its density maximum at ρ = 0 is labeled by

“S0” and that where the energetically lowest lying symmetric stationary GP solution has its density maximum at ρ > 0 by

“S>0”.

(Color online) Blow-up (with more detail) of the phase diagram for b = 0.2az and A = 12Ez shown in Fig. 1: The

between the S0and the U regions of the phase diagram changes non-monotonically. For D ≈ 240, e.g., the system is

mechanically stable for λ ? 8.13, mechanically unstable for 8.13 ? λ ? 7.72, and then again mechanically stable for

a small λ regime (7.72 ? λ ? 7.42).

We find that some, though not all, of the features of the phase diagram can be reproduced qualitatively by a simple

variational wave function ψvar(ρ,z),

ψvar(ρ,z) =

?

exp

?

−ρ2

2b2

?

1

?

+ b2exp

?

−ρ2

2b2

?

3

??

×

?

exp

?

−z2

2b2

4

?

+ b5

1 −

z2

λa2

z

?

exp−z2

2b2

6

??

, (35)

where b1−b6denote variational parameters that are optimized by minimizing the energy per particle. For b2= b5= 0,

ψvarreduces to the commonly used variational wave function of purely Gaussian shape. The second term in the first

square bracket on the right hand side of Eq. (35) has been added to allow for the description of densities of S>0

character while the second term in the second square bracket on the right hand side of Eq. (35) has been added to

account for the Gaussian barrier along the z-direction. Figure 6 compares the total energy per particle from our

variational calculation (dashed line) with that from the full numerical calculation (solid line) for A = 12Ez, λ = 7 and

b = 0.2az. The variational energy is less than 2 % higher than the energy obtained from the full numerical calculation.

We find that the density of the dipolar gas changes from S0 to S>0 character at D ≈ 35, compared to D = 80.03

obtained from the full calculation. For both sets of calculations, the energy and its derivative change smoothly as

the system undergoes the structural change from S0to S>0. For comparison, a dash-dotted line in Fig. 6 shows the

energy per particle for ψvarwith b2= 0 (we refer to this variational wave function as four-parameter wave function),

i.e., for a wave function that is not sufficiently flexible to describe structured ground state densities of red blood cell

shape. As expected, this variational wave function results in somewhat higher energies.

The variational wave function ψvarpredicts S0to S>0transitions for all aspect ratios λ between 5 and 12, indicating

that it is not flexible enough to describe the island character of the S>0regions of the phase diagram and, furthermore,

that the S0to S>0transition is driven by a delicate balance between the different energy contributions. Motivated

by calculations presented in Ref. [15], we expect that the variational four-parameter wave function can qualitatively

reproduce the existence of alternating stable and unstable regions of the phase diagram as λ is changed for fixed D

and A (see our discussion above for D ≈ 240 and A = 12Ez); we have, however, not checked this explicitly. Lastly,

we note that the variational six-parameter wave function predicts a stable dipolar gas even for fairly large D (i.e., D

values larger than those shown in Fig. 6) while the full numerical calculation predicts collapse at D ≈ 190.49.

V. DISCUSSION OF DYNAMICAL STUDIES

This section presents Bogoliubov de Gennes excitation spectra and discusses the corresponding eigenmodes. For

small λ and appropriate D (see Sec. VA), the lowest non-vanishing Bogoliubov de Gennes excitation frequency is

identified as the Josephson oscillation frequency ωJ. Comparisons with results obtained by time-evolving a properly

Page 12

12

0

50

100

D

150

200

2

2.5

(Etot / N) / Ez

FIG. 6: (Color online) A solid line shows the total energy per particle Etot/N of the energetically lowest lying stationary GP

solution with k = 0 as a function of D for λ = 7, A = 12Ez and b = 0.2az obtained numerically. For comparison, dashed and

dash-dotted lines show Etot/N obtained using the variational six- and four-parameter variational wave functions (see text for

details). The density of the system changes from S0 to S>0 character at D ≈ 35 and 80.03 for the six-parameter variational

and the full numerical calculations, respectively.

012

D

0

2

4

Re(ω) / ωz

FIG. 7:

A = 12Ez, b = 0.2az and λ = 0.3. The real parts of the frequencies for k = 0 and 1 are shown by solid and dashed lines,

respectively. The vertical arrow indicates the D value, D ≈ 0.68, at which the real part of the lowest non-vanishing k = 0

Bogoliubov de Gennes frequency vanishes.

(Color online) Excitation spectrum obtained by solving the Bogoliubov de Gennes equations as a function of D for

prepared initial state and by applying the two-mode model equations are presented. In the regime where the sym-

metric solutions are of types S0and S>0, respectively (large λ, see Sec. VB), the decay mechanisms are identified.

Furthermore, the character of various (avoided) crossings of the excitation frequencies is revealed.

A. “Small” aspect ratio (λ ? 1)

Figure 7 shows the excitation spectrum as a function of D obtained by solving the Bogoliubov de Gennes equations

for A = 12Ez, b = 0.2azand λ = 0.3. The spectrum is characterized by three distinct features that will be elaborated

on in the following paragraphs: i) The real part of the lowest k = 0 frequency vanishes at D ≈ 0.68, and “reappears”

at D ≈ 0.75. ii) The k = 0 frequencies show a series of crossings (or avoided crossings) at D ≈ 2.42. iii) At slightly

larger D values, i.e., near D ≈ 2.45, the real part of several k = 0 excitation frequencies vanishes.

We first discuss the regime i) around D ≈ 0.68−0.75. Figure 8(a) shows the Bogoliubov de Gennes eigenmode¯f(ρ,z)

that corresponds to the lowest non-vanishing k = 0 frequency for D = 0.6573, A = 12Ez, λ = 0.3 and b = 0.2az.

For these parameters, the energetically lowest lying stationary GP solution is symmetric and the corresponding

eigenfrequency has a finite real part and vanishing imaginary part (see Fig. 7). Since Bogoliubov de Gennes functions

with k = 0 have no explicit ϕ dependence, the eigenmode shown in Fig. 8(a) corresponds to a situation where the

population oscillates with frequency ω between the left and the right well as a function of time. The lowest non-

vanishing k = 0 frequency can thus be identified as the Josephson oscillation frequency ωJ (see also below). For

comparison, Fig. 8(b) shows the Bogoliubov de Gennes eigenmode¯f(ρ,z) corresponding to the lowest non-vanishing

Page 13

13

0 0.40.8 1.2

-3

0

3

z / az

0 0.4 0.8 1.2

ρ / az

-3

0

3

z / az

(a)

(b)

FIG. 8: (Color online) Bogoliubov de Gennes eigenmodes¯f(ρ,z) corresponding to the lowest non-vanishing k = 0 frequency

for λ = 0.3, A = 12Ez, b = 0.2az and (a) D = 0.6573 and (b) D = 0.7668. The contours are chosen equidistant, with solid and

dashed lines corresponding to positive and negative values of¯f. The dash-dotted lines indicate the nodal lines of¯f.

k = 0 frequency for D = 0.7668 (i.e., in the regime where the frequency has “reappeared” and where the energetically

lowest lying stationary GP solution with k = 0 is symmetry-broken) and the same A, Ezand b values as before. In

this case, the asymmetry of the eigenmode indicates that there is population transfer between the left and the right

wells but that there is, on average, more population in the right than in the left well. This behavior is identified as

macroscopic quantum self-trapping. Our interpretation of the Bogoliubov de Genne eigenmodes is supported by our

time-dependent calculations.

In our time-dependent studies near the symmetric to symmetry-broken transition, we prepare an initial state and

time evolve it according to the mean-field Hamiltonian H, Eq. (2). As for s-wave interacting BECs, the system

dynamics can be divided into two categories (see also above): A regime where the population is transferred back and

forth between the left well and the right well (this is the Josephson oscillation regime) and a regime where the time

averaged population is asymmetrically divided among the two wells (this is the macroscopic quantum self-trapping

regime). Figures 9(a) and 9(b) show the time evolution of the expectation value ?z(t)? for D = 0.6573 and D = 0.7668,

respectively. Here, ?z(t)? is obtained by calculating the expectation value of z with respect to the GP density at each

time step. The expectation value ?z(t)? is related to but not identical to the population difference Z(t) introduced in

Sec. III. For these D values, the energetically lowest lying stationary GP solution is symmetric and symmetry-broken,

respectively. For D = 0.6573, the initial state is prepared according to Eq. (22) with φ(0) = 0 and Z(0) = 0.002.

Figure 9(a) shows that ?z(t)? oscillates between positive and negative values of equal magnitude and that the time

average of ?z(t)? over a period gives zero. For D = 0.7668, the initial state is prepared by adding a small amount of

random noise to the energetically lowest lying stationary GP solution. Figure 9(b) shows that ?z(t)? oscillates about

a negative value and that the time average of ?z(t)? gives a non-zero value. An analysis of the time evolution of the

density profiles confirms that the system is in the Josephson regime and in the macroscopic quantum self-trapping

regime, respectively.

To determine the oscillation frequency from the time evolution of ?z(t)?, we Fourier transform ?z(t)? and record

the center of the dominant peak for various parameter combinations. Circles in Fig. 10 show the resulting Josephson

oscillation frequency ωJfor A = 12Ez, λ = 0.3 and b = 0.2az. The agreement between the frequency obtained from the

Fourier analysis (circles in Fig. 10) and the lowest non-vanishing k = 0 Bogoliubov de Gennes excitation frequency

(solid line in Fig. 10) is excellent. The D value at which the Josephson oscillation frequency obtained by Fourier

transforming ?z(t)? vanishes, coincides, within our numerical accuracy, with that at which the lowest non-vanishing

k = 0 Bogoliubov de Gennes excitation frequency becomes imaginary. Notably, this D value, D ≈ 0.68, is slightly

Page 14

14

0 400 800

-2

0

2

<z(t)> * 103 / az

0 400

t * ωz

800

-0.8

-0.6

<z(t)> / az

(a)

(b)

FIG. 9: (Color online) Solid lines show the expectation value ?z(t)? (see text)—calculated by time evolving a given initial state

according to the mean-field Hamiltonian, Eq. (2)—as a function of time t for A = 12Ez, λ = 0.3, b = 0.2az and (a) D = 0.6573

(Josephson oscillation regime) and (b) D = 0.7668 (macroscopic quantum self-trapping regime). In panel (a), the initial state

is prepared according to Eq. (22) with Z(0) = 0.002 and φ(0) = 0. In panel (b), the initial state is prepared by adding a small

amount of random noise to the energetically lowest lying stationary GP solution.

01

D

0

0.05

0.1

ωJ / ωz

FIG. 10: (Color online) Josephson oscillation frquency ωJ as a function of D for λ = 0.3, A = 12Ez and b = 0.2az. The circles

show the Josephson oscillation frequency ωJ obtained from our time-dependent study, in which the initial state is prepared

according to Eq. (22) with Z(0) = 0.002 and φ(0) = 0 and then time evolved according to the mean-field Hamiltonian H,

Eq. (2). The solid line shows the lowest non-vanishing k = 0 Bogoliubov de Gennes excitation frequency. For comparison, a

dash-dotted line shows the two-mode model prediction ωJ,TM.

smaller than the D value at which the energetically lowest lying stationary GP solution changes from symmetric to

symmetry-broken (D ≈ 0.75).

We find that the lowest non-vanishing k = 0 Bogoliubov de Gennes frequency for D = 0.7668 and λ = 0.3 is about

30% smaller than the oscillation frequency extracted from Fig. 9(b), i.e., ω = 0.059ωz. The fact that the Bogoliubov

de Gennes excitation frequency differs notably from the frequency obtained by Fourier-transforming ?z(t)? might be

due to the approximate nature of the Bogoliubov de Gennes equations.

For comparison, a dash-dotted line in Fig. 10 shows the Josephson oscillation frequency ωJ,TM, Eq. (34), predicted

Page 15

15

0 0.40.8 1.2

-3

0

3

z / az

0 0.40.8 1.2

-3

0

3

z / az

0 0.40.8 1.2

-3

0

3

z / az

0 0.4 0.81.2

ρ / az

-3

0

3

z / az

0 0.40.81.2

-3

0

3

0 0.40.81.2

-3

0

3

0 0.4 0.8 1.2

-3

0

3

0 0.40.8 1.2

ρ / az

-3

0

3

(a)

(b)

(c)

(d)

(h)

(g)

(e)

(f)

FIG. 11:

frequencies for λ = 0.3, A = 12Ez, b = 0.2az. Panels (a)-(d) show¯f corresponding to the lowest, second lowest, third lowest

and fourth lowest frequencies for D = 2.410 (i.e., just before the crossing) while panels (e)-(h) show¯f corresponding to the

lowest, second lowest, third lowest and fourth lowest frequencies for D = 2.443 (i.e., just after the crossing). The contours are

chosen equidistant, with solid and dashed lines corresponding respectively to positive and negative values of¯f.

(Color online) Bogoliubov de Gennes eigenmodes¯f(ρ,z) corresponding to the four lowest non-vanishing k = 0

by the two-mode model. Figure 10 shows that the two-mode model provides a qualitatively but not quantitatively

correct description of the Josephson oscillation frequency. The fact that the two-mode model does not allow for

quantitative predictions for all D is likely due to the fact that the modes ΦLand ΦRare not entirely located in the

left well and in the right well, respectively, but that the left mode “leaks” into the right well and the right mode into

the left well. This has been discussed in some detail in Ref. [49], which employs a slightly modified version of the

two-mode model. In an attempt to obtain a better simple quantitative description of the system dynamics, we applied

the improved two-mode model proposed in Ref. [45]. For the cases considered, we find that this model leads only to

small changes compared to the simple two-mode model applied above and does not provide a significantly improved

description. In the future, it may be interesting to apply a multi-mode model.

We now discuss the regime ii) near D ≈ 2.42, where the k = 0 frequencies show (avoided) crossings. To shed light

on these (avoided) crossings, Figs. 11(a)-(d) show the Bogoliubov de Gennes eigenmodes¯f(ρ,z) corresponding to the

four lowest non-vanishing k = 0 frequencies just before the crossing (i.e., for D = 2.410), while Figs. 11(e)-(h) show

those corresponding to the four lowest k = 0 frequencies just after the crossing (i.e., for D = 2.443). Dash-dotted

lines in Fig. 11 indicate the nodal lines of the Bogoliubov de Gennes eigenmodes. While some of these nodal lines are

to first order only dependent on z, others depend in a non-trivial manner on ρ and z. In the following we discuss a

few key features of the eigenmodes shown in Fig. 11. The eigenmode corresponding to the lowest frequency extends

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