Bohmian analysis of dark solutions in interfering Bose-Einstein condensates the dynamical role of underlying velocity fields J. Tounli and A. S. Sanz

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Bohmian analysis of dark solutions in interfering Bose-Einstein condensates:

the dynamical role of underlying velocity ﬁelds

J. Tounli and A. S. Sanz∗

Department of Optics, Faculty of Physical Sciences, Universidad Complutense de Madrid

Pza. Ciencias 1, Ciudad Universitaria – 28040 Madrid, Spain

(Dated: July 12, 2024)

In the last decades, the experimental research on Bose-Einstein interferometry has received much

attention due to promising technological implications. This has thus motivated the development

of numerical simulations aimed at solving the time-dependent Gross-Pitaevskii equation and its re-

duced one-dimensional version to better understand the development of interference-type features

and the subsequent soliton dynamics. In this work, Bohmian mechanics is considered as an addi-

tional tool to further explore and analyze the formation and evolution in real time of the soliton

arrays that follow the merging of two condensates. An alternative explanation is thus provided

in terms of an underlying dynamical velocity ﬁeld, directly linked to the local phase variations

undergone by the condensate along its evolution. Although the reduced one-dimensional model is

considered here, it still captures the essence of the phenomenon, rendering a neat picture of the full

evolution without diminishing the generality of the description. To better appreciate the subtleties

of free versus bound dynamics, two cases are discussed. First, the soliton dynamics exhibited by a

coherent superposition of two freely released condensates is studied, discussing the peculiarities of

the underlying velocity ﬁeld and the corresponding ﬂux trajectories in terms of both the peak-to-

peak distance between the two initial clouds and the addition of a phase diﬀerence between them. In

the latter case, an interesting correspondence with the well-known Aharonov-Bohm eﬀect is found.

Then, the recurrence dynamics displayed by the more general case of two condensates released from

the two opposite turning points of a harmonic trap is considered in terms of the distance between

such turning points. In both cases, it is presumed that the initial superposition state is generated

by splitting adiabatically a single condensate with the aid of an optical lattice, which is then turned

oﬀ. Nonetheless, although the lattice does not play any active role in the simulations, the param-

eters deﬁning the initial states are in compliance with it, which helps in the interpretation and

understanding of the results observed.

I. INTRODUCTION

In the late 1990s, Ketterle and coworkers [1, 2] pro-

duced the ﬁrst experimental realization of interference

with a Bose-Einstein condensate (BEC). As they showed,

when an ultracold atomic cloud is coherently split up

and then the two resulting separate clouds are released

again, the latter interact in such a way that a pattern

of alternating bands of more and less atomic density can

be observed. To some extent, an analogy can be estab-

lished between this behavior and the interference fringes

observed in a typical Young-type two-slit experiment, al-

though the latter case obeys a linear dynamics, while the

former is a consequence of a nonlinear one. A similar

eﬀect can also be observed by splitting the condensate

and then letting one of the parties to drop on top of the

other, as it was shown by Javanainen et al. [3, 4] by the

same time. Since the performance of these crucial ex-

periments, interferometry with BECs [5–8] has become

an important test ground for the understanding of quan-

tum coherence as well as a remarkable source of novel

quantum technology, because of its extraordinary sensi-

tivity, with applications in atom lasers [9] or quantum

metrology [10], where the manipulation and preparation

of BECs in optical lattices plays a major role [11, 12].

∗Corresponding author: a.s.sanz@ﬁs.ucm.es

In order to understand and describe the dynamics ex-

hibited by BECs in real time, the Gross-Pitaevskii equa-

tion (GPE) constitutes an ideal and simple working tool,

without requiring a further many-body description of the

full atomic cloud (involving of the order of 103to 106

atoms, in general terms). This is possible by recasting the

many-body problem in the form of an eﬀective nonlinear

Schr¨odiner equation, which accounts for the dynamics

displayed by a single atom from the cloud acted by an

external potential plus a self-interaction term that rep-

resents the collective action of any other identical atom

from the cloud [5–7]. Because the GPE is not analyt-

ical in general, a number of numerical techniques have

been considered in the literature to solve this equation

and extract useful information from it [13–15]. To some

extent, these techniques are analogous to those earlier on

applied to solve the time-dependent Schr¨odinger equa-

tion, and have been used recently to study, analyze and

describe diﬀerent aspects involved in BEC interferometry

[16–19], the BEC dynamics in periodic and harmonic po-

tentials [20], or the role of the nonlinearity when weakly

harmonic and Gaussian traps are considered [21, 22].

Within this context, a suitable tool to understand the

dynamics displayed in real time by BECs is the Bohmian

formulation of quantum mechanics [23, 24]. The hydro-

dynamic language introduced by this quantum formu-

lation allows us to understand the interference dynam-

ics leading to the appearance of solitons in terms of an

arXiv:2210.13175v2 [quant-ph] 10 Jul 2024

underlying velocity ﬁeld, associated with the phase of

the condensate, and to follow its subsequent evolution in

time by means of swarms of trajectories. It was shown

by Benseny et al. [25] that these trajectories constitute

a convenient tool to determine how each element of the

condensate (not to be confused with each individual atom

itself) moves apart, thus providing some clues on its dy-

namical evolution beyond more conventional-type infor-

mation, such as the density distribution. In other words,

it is possible to determine which portions in the BEC are

going to separate faster or slower at each time by simply

observing local variations in the underlying velocity ﬁeld.

In this work we investigate by means of a series of nu-

merical simulations the formation of dark solitons both

from two freely released condensates and also under the

action of an underlying harmonic trap in order to study

the appearance and persistence of recurrences in time.

The appearance of soliton-type solutions is associated

with the value of the scattering length, as. Speciﬁcally,

if as<0, as it happens in 85Rb [26] or 7Li [27], atomic

interactions are attractive and bright soliton solutions

(spike-type deformations in the density) can appear. On

the contrary, for as>0, as it is the case of 87Rb [28]

or 23Na [29], the interactions are repulsive, which corre-

sponds to a situation where dark solitons (dips in the den-

sity) can be observed. The ﬁrst experimental observation

of dark solitons dates back to 25 years ago, where these

solitons were produced in an elongated (cigar shaped)

BEC by the so-called phase imprinting method [30]. A

year before, though, Scott et al. [31] identiﬁed the for-

mation of persistent dark fringes in the the case of the

collision of two separated condensates under the inﬂu-

ence of a harmonic trap. This work showed evidence of

the relationship between the dark solitons and the phase

change, an idea that is implicit in the phase imprinting

method devised to generate vortices in the condentasate

[32], but that also led to the experimental generation of

dark solitons [30]. An overview of the advances in dark

soliton dynamics both experimentally and theoretically

can be found in [33, 34]. Such phase imprinting methods

are actually strongly connected to the Bohmian trajec-

tories here. As it is shown below, these trajectories will

allow us to monitor in real time some features that are

not evident at a ﬁrst glance from the usual density distri-

butions, and also to elucidate some important diﬀerences

between the nonlinear and linear dynamical regimes that

characterize the BEC dynamics in harmonic traps. This

is by virtue of the relationship between the trajectories

and the local phase of the condensate. Indeed, pioneering

work by Tsuzuki [35] in the early 1970s on the derivation

of soliton-like solutions of the Gross-Pitaevskii equation

are strongly connected to this formulation (Tsuzuki’s is

only a ﬁrst approximation to the exact approach here

considered).

The work has been organized as follows. In Sec. II the

model and computational details are introduced as well

as some elementary notions of the Bohmian formulation.

In Sec. III the results obtained from the numerical simu-

lations carried out for the two scenarios mentioned above,

namely, free propagation and motion inside a harmonic

trap, are presented and discussed. To conclude, some

ﬁnal remarks are summarized in Sec. IV.

II. THEORY

A. The reduced one-dimensional GPE

Within the Hartree-Fock approximation, the dynamics

of a BEC consisting of Nidentical atoms with mass mis

described by the Gross-Pitaevskii equation [5],

iℏ∂Ψ(r, t)

∂t =−ℏ2

2m∇2+Vext(r, t) + gN(r, t)Ψ(r, t),

(1)

where Vext describes any external interaction acting on

the BEC (e.g., the conﬁning optical trap), while the

nonlinear term gN(r, t) accounts for the self-interaction

contribution, i.e., the eﬀective interaction with the re-

maining N−1 atoms in the cloud. The strength of

such interaction is determined by the coupling constant

g= 4πℏ2as/m, with asbeing the scattering length [7].

For convenience, the wave function (order parameter) can

be recast in polar form, as

Ψ(r, t) = pN(r, t)eiθ(r,t),(2)

which enables a description of the condensate in terms

of its density distribution, N(r, t) = |Ψ(r, t)|2, with

R|Ψ(r, t)|2dr=N, and its local phase variations, ac-

counted for by θ(r, t). Here, for numerical convenience,

we have chosen the wave function to be normalized

to unity, which implies that the factor Narising from

N(r, t) will be included as a multiplicative constant in g,

that is, from now on g= 4πℏ2asN/m.

Consider that a prolate conﬁguration for the con-

densate, which arises assuming typical frequency ranges

fz∼20−60 Hz and f⊥∼400−900 Hz. Accordingly, the

transverse harmonic trap values will be much larger than

the longitudinal one, since ω2

⊥≫ω2

z, and hence, for the

times considered, the transverse degrees of freedom can

be assumed to be frozen. This thus allows us to provide

an eﬀective description of the condensate dynamics along

the z-direction by means of a reduced one-dimensional

(1D) GPE [7], which reads as

iℏ∂ψ(z, t)

∂t =−ℏ2

∂2

∂z2+Vext(z) + g1Dn(z, t)ψ(z, t),

(3)

where g1D=g/2πa2

⊥= 2ℏω⊥as, with a⊥=pℏ/mω⊥,

is an eﬀective 1D coupling constant that arises from

the dimensional reduction of Eq. (1) [36, 37] (see fur-

ther details about the potential parameters in Sec. II C).

From now on, for computational convenience we assume

that the wave function ψ(z, t) is normalized to unity,

i.e., R|ψ(z, t)|2dz =Rn(z, t)dz = 1, with n(z, t) being

a linear density distribution (measured in µm−1). Ac-

cordingly, the expression for the coupling constant g1D

here will read as g1D=g′/4πa2

⊥= 2ℏω⊥asN, with

a⊥=pℏ/mω⊥(to further simplify notation, g1Dis used

instead of g′

1D).

Typical single dark soliton solutions for this nonlinear

equation display the functional form [36]

ψ(z, t) = √n0e−i(µ/ℏ)tβtanh β(z−vt)

χ+iv

c(4)

for BEC clouds with a constant density n0, where µ=

n0gis the chemical potential, c=pµ/m is the so-

called Bogoliubov speed of sound, β=p1−(v/c)2and

χ=ℏ/√mn0gis the coherence or healing length char-

acterizing the soliton extension. The solution (4) rep-

resents a soliton propagating towards positive zwith a

speed von a homogeneous background density n0. As

it can be readily inferred from Eq. (4), in the particular

case v=c, the soliton will not be distinguishable from

the background ﬂuid [38]. Here, instead of a constant

density, we analyze the case of space-limited condensates

with an initial Gaussian proﬁle, which somehow mimic

the typical parabolic density proﬁle that corresponds to

a stationary solution inside a harmonic trap [7]. Note

that solutions like (4) can be generated from the Gaus-

sian ansatz by imprinting a phase shift to a part of the

condensate leaving the other part unaﬀected [18]. Fol-

lowing this basic imprinting technique, the appearance

of interference-type features in BECs can be understood

as a sequential phase change, which allows us to identify

the deeps in the cloud with a sequence of dark solitions,

each with a shape analogous to Eq. (4). This will be

made evident by means of the Bohmian analysis.

B. Bohmian description

A central idea to hydrodynamics is that ﬂuid diﬀusion

can be monitored by means of streamlines, which pro-

vide us with information on the expansion, contraction,

or rotation of the ﬂuid. In a similar fashion, the Bohmian

formulation of quantum mechanics or Bohmian mechan-

ics renders a trajectory-based description of the evolution

of quantum systems or, more strictly speaking, the dif-

fusion of their probability density in the corresponding

conﬁguration space [24]. In its original version, either as

formulated by Madelung in 1926 [39] or, later on, as pos-

tulated by Bohm in 1952 [40, 41], the main equations of

motion are obtained after recasting the wave function in

polar form (i.e., considering a nonlinear transformation

from a complex-valued ﬁeld to two real-valued ﬁelds) and

then substituting it into the time-dependent Schr¨odinger

equation. Proceeding the same way, i.e., substituting the

polar ansatz (2) into Eq. (1), leads to the set of coupled

hydrodynamic equations of motion

∂N(r, t)

∂t =−∇ · j(r, t),(5a)

ℏ∂θ(r, t)

∂t +ℏ2

2m(∇θ)2+Vext(r)

+gN(r, t) + Q(r, t) = 0,(5b)

where

j(r, t)≡ℏ

mN(r, t)∇θ

=ℏ

2mi [Ψ∗(r, t)∇Ψ(r, t)−Ψ(r, t)∇Ψ∗(r, t)]

(6)

is the quantum ﬂux [42] associated with the process,

which ensures the conservation of particles in the cloud

by virtue of Eq. (5a). Equation (5b) displays the func-

tional form of a Hamilton-Jacobi equation, although with

the particularity that it contains the term

Q(r, t) = −ℏ2

4m(∇2N(r, t)

N(r, t)−1

2∇N(r, t)

N(r, t)2),(7)

which is the so-called Bohm’s quantum potential [43].

As it can be noticed, unlike the motion described by a

standard Hamilton-Jacobi equation, the presence of this

additional term is going to induce a permanent coupling

between the motion displayed by a single particle of mass

mand a statistical ensemble of identical particles, speci-

ﬁed by the density n.

From Eqs. (5) it is clear that individual trajectories or

streamlines associated with the evolution of the quantum

system described by (1) can be obtained either by postu-

lating an equation of motion from (5b) [40, 43], in direct

analogy with the classical counterpart,

r(r, t) = ℏ

m∇θ(r, t),(8)

or just, in a more natural way, by noting from the con-

tinuity equation (5a) that the quantum ﬂux is typically

associated with a local velocity ﬁeld [24],

j(r, t) = N(r, t)v(r, t),(9)

from which the same equation of motion follows,

v(r, t) = ˙

r(r, t) = j(r, t)

N(r, t)=1

mRe ˆ

pΨ(r, t)

Ψ(r, t).(10)

The last term, emphasizes the direct connection of this

velocity vector ﬁeld with the real part of the local value

of the usual momentum operator ˆ

p=−iℏ∇. Hence, the

trajectories obtained in this manner, i.e., from Eq. (10),

are well-deﬁned and are not in contradiction with stan-

dard quantum mechanics, where, in principle, one cannot

appeal to the concept of trajectory in the conﬁguration

space, because of the uncertainty relation between po-

sition and momentum. Unlike classical trajectories, the

swarms of trajectories obtained from Eq. (10) are con-

strained to obey a phase relation, according to which the

corresponding momenta cannot be independent one an-

other. This phase relation is precisely what we regard

as coherence, which, in the present context, gives rise

to the well-known Bohmian trajectory non-crossing rule:

Bohmian trajectories cannot get across the same point at

the same time, this being a direct consequence from the

single-valuedness of the local phase of quantum systems.

Besides, it is worth stressing the fact that the equation of

motion (10) not only establishes a direct link with usual

transport equations, but also with the hydrodynamical

description of Bose ﬂuids earlier on proposed by Landau

[44, 45].

C. Trapping potential model

The conﬁnement of neutral cold atoms in periodic lat-

tices of reduced dimensions, which enables the investiga-

tion of BECs and their applications [46], including dark

soliton dynamics [47], relies on the generation of periodic

optical traps by counterpropagating laser beams [48, 49].

Thus, let us consider that a single atomic cloud, trapped

in a harmonic potential, is adiabatically split up by ap-

plying an optical lattice a standing laser ﬁeld. If the

steepness of the harmonic trap is relevant in relation to

the period of the optical lattice and the depth of its wells

(see Fig. 1), and we assume an unbiased (or nearly un-

biased) splitting of the cloud, we can assume that the

BEC breaks into two, one half of each occupying a local

minimum of the trap+lattice potential. To better under-

stand the process in terms of the potential generated (for

further details on these optical lattice potentials, see, for

instance, Ref. [48, 49]), and hence the model here con-

sidered, let us consider that the conﬁning potential along

the z-direction consists of two contributions, namely,

Vext(z) = Vtrap(z) + Vlatt(z),(11)

where the harmonic trapping contribution reads as

Vtrap(z) = 1

2mω2

zz2,(12)

and the optical lattice as

Vlatt(z) = V0cos2πz

ℓ.(13)

Here we consider a 85Rb atomic cloud, with m=

1.44 ×10−25 kg, so we have chosen typical experimen-

tal values [37, 50] for the parameters involved in these

two contributions, in particular, fz= 50 Hz (ωz=

2πfz≈314.2 rad/s) and V0/h = 850 Hz (where h=

6.62607 ×10−34 J·s is Planck’s constant). Furthermore,

with the purpose to discuss diﬀerent dynamical behav-

iors, three values of the lattice constant, ℓ, will be con-

sidered below: 5.7 µm, 15 µm, and 26 µm. To illustrate

the overall eﬀect, the two separate contributions as well

as the combined potential (11) are displayed in Fig. 1

for ℓ= 5.7µm. The trap potential is denoted with the

gray dashed line and the lattice potential with the blue

dashed line, which their sum is represented with the black

solid line. As it can be noticed, the activation of the lat-

tice potential generates a barrier that is going to split

up the atomic cloud basically into two parts. In a ﬁrst

approximation, each one of these parts can be assumed

FIG. 1. External potential (solid black line) applied to split

up a single atomic condensate and generate a coherent su-

perposition of two condensates. The conﬁning harmonic trap

(dash-dotted gray line) here has a frequency fz= 50 Hz,

while the lattice standing ﬁeld (dashed blue line) has a pe-

riod ℓ= 5.7µm and a potential barrier V0/h = 850 Hz. Each

site or well on either side of the central barrier can be approx-

imated by a harmonic potential (dotted and solid red lines)

with an eﬀective frequency feﬀ ≈245 Hz (see text for details),

which can be used to determine the initial ansatz in the sim-

ulations.

to be acted by a harmonic well with its frequency being

determined by the lattice properties (barrier height and

period). As it can readily be noticed, in a good approxi-

mation, for the case displayed in Fig. 1, these harmonic

well can be approximated by the functional form

Vha,±(z)≈Vtrap(z±) + 1

2mω2

eﬀ (z−z±)2,(14)

with z±≈ ±ℓ/2 and

ωeﬀ =r2π2V0

mℓ2.(15)

From this expression, we can choose for the initial width

of the Gaussian wave packet the one corresponding to the

ground state of the harmonic oscillator, i.e.,

σeﬀ =rℏ

2mωeﬀ

.(16)

Of course, a more precise (though still approximated)

form can easily be derived. However, for the purpose

here, expression (15) suﬃces, since we are interested in

releasing two coherently separated clouds from a ﬁxed

distance, ℓ, which are assumed to be kept inside harmonic

wells determined by the lattice potential (13) and not

from the combined eﬀect (11). Note that, as ℓincreases,

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BohmiananalysisofdarksolutionsininterferingBose-Einsteincondensates:thedynamicalroleofunderlyingvelocityfieldsJ.TounliandA.S.Sanz∗DepartmentofOptics,FacultyofPhysicalSciences,UniversidadComplutensedeMadridPza.Ciencias1,CiudadUniversitaria–28040Madrid,Spain(Dated:July12,2024)Inthelastdecades,theexper...

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Bohmian analysis of dark solutions in interfering Bose-Einstein condensates the dynamical role of underlying velocity fields J. Tounli and A. S. Sanz

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