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4 Figures# Higher-order quantum bright solitons in Bose-Einstein condensates show truly quantum emergent behavior

**Abstract**

When an interaction quench by a factor of four is applied to an attractive Bose-Einstein condensate, a higher-order quantum bright soliton exhibiting robust oscillations is predicted in the semiclassical limit by the Gross-Pitaevskii equation. Combining matrix-product state simulations of the Bose-Hubbard Hamiltonian with analytical treatment via the Lieb-Liniger model and the eigenstate thermalization hypothesis, we show these oscillations are absent. Instead, one obtains a large stationary soliton core with a small thermal cloud, a smoking-gun signal for non-semiclassical behavior on macroscopic scales and therefore a fully quantum emergent phenomenon.

Figures

Higher-order quantum bright solitons in Bose-Einstein condensates show truly quantum emergent

behavior

Christoph Weiss1, ∗and Lincoln D. Carr2 , †

1Joint Quantum Centre (JQC) Durham–Newcastle, Department of Physics,

Durham University, Durham DH1 3LE, United Kingdom

2Department of Physics, Colorado School of Mines, Golden, Colorado 80401, USA

(Dated: December 19, 2016)

When an interaction quench by a factor of four is applied to an attractive Bose-Einstein condensate, a higher-

order quantum bright soliton exhibiting robust oscillations is predicted in the semiclassical limit by the Gross-

Pitaevskii equation. Combining matrix-product state simulations of the Bose-Hubbard Hamiltonian with ana-

lytical treatment via the Lieb-Liniger model and the eigenstate thermalization hypothesis, we show these oscil-

lations are absent. Instead, one obtains a large stationary soliton core with a small thermal cloud, a smoking-

gun signal for non-semiclassical behavior on macroscopic scales and therefore a fully quantum emergent phe-

nomenon.

PACS numbers: 05.60.Gg, 03.75.Lm, 03.75.Gg

Keywords: Bright soliton, Bose-Einstein condensation, Quantum many-body physics, Far-from-equilibrium quantum dynam-

ics, Semiclassical breakdown

The quantum-classical correspondence is well-established

for single-particle quantum mechanics but is known to be

problematic for some many-body quantum problems such as

strongly correlated systems and even materials as simple as

the antiferromagnet. A key macroscopic prediction of Bose-

Einstein condensates (BECs) is the bright soliton, appear-

ing as a localized robust ground state “lump” for attractive

BECs. Based on the ubiquity of semiclassical limits for non-

interacting and weakly interacting bosons, such as lasers and

BECs, one might expect a well-deﬁned emergent macroscopic

classical behavior generically from such systems. To date,

most aspects of matter-wave bright soliton experiments [1–

11] seem to be explained on the semiclassical mean-ﬁeld level

via the Gross-Pitaevskii equation (GPE): thus they display

quantum behavior on a single-particle level matching classi-

cal wave experiments such as nonlinear photonic crystals [12]

and spin-waves in ferromagnetic ﬁlms [13,14]. This state-

ment is supported by the fact that quantum-quantum bright

solitons [15–22] — matter-wave bright solitons that display

quantum behavior beyond the single-particle mean-ﬁeld level

— for many practical purposes show mean-ﬁeld behavior

predicted by the GPE emerging already for particle num-

bers as low as N'3 [23]. So far, beyond-mean ﬁeld ef-

fects only seem to play a role if two or more distinct bright

solitons are involved: two matter-wave quantum bright soli-

tons can behave quite diﬀerently from matter-wave mean-

ﬁeld bright solitons. Only the latter necessarily have a well-

deﬁned relative phase [24]. Both the limit of well-deﬁned

phase [7] and the limit involving a superposition of many

phases [25,26] are experimentally relevant for matter-wave

bright solitons [7,26]. In this Letter we show that truly quan-

tum many-body eﬀects are responsible for the dynamics of a

single quantum-quantum bright soliton, a smoking-gun signal

for quantum emergence in BEC experiments.

For far-from equilibrium dynamics of beyond-ground state

quantum bright solitons, we are only at the beginning of a

journey similar to the case of quantum dark solitons. That

scientiﬁc voyage required multiple lines of investigations [27–

31] to arrive at the state-of-the art explanation that atom losses

are necessary to obtain mean-ﬁeld properties from many-body

quantum solutions [32]. Dark solitons were also realized ex-

perimentally in BECs [33] and have been further explored

in detail over the years in comparison to such predictions,

e.g. [34]. In contrast, bright solitons to-date lack for instance

a phase coherence measurement, let alone the kind of far-

from-equilibrium dynamics we are predicting here. Thus we

focus on a quantum bright soliton experiment easily acces-

sible in current platforms. Speciﬁcally, one ﬁrst prepares a

single ground-state bright soliton and then rapidly changes

the interaction, an “interaction quench” via a Feshbach res-

onance, a well-established experimental technique. For one-

dimensional Bose gases recent work related to quenches in-

cludes positive-to-negative quenches [21,35], and zero-to-

positive quenches [36]. Quenches involving dark-bright soli-

tons [37,38], quenched dynamics of two-dimensional solitary

waves [39], and breathers in discrete nonlinear Schrödinger

equations [40–42] were also investigated, as well as the

breathing motion after a quench of the strength of a harmonic

trap [43].

For attractive BECs, there are very speciﬁc mean-ﬁeld pre-

dictions [44]: in particular, for an interaction quench by a fac-

tor of four there are exact analytical mean-ﬁeld results avail-

able that predict robust perfectly oscillatory behavior for all

times [45]. However, how quantum bright solitons would be-

have in such a situation is an open question which we address

in the current Letter. One GPE interpretation of a higher or-

der soliton is Nsbound bright solitons, here Ns=2, a kind of

diatomic solitonic molecule in a nonlinear vibrational mode.

One might therefore expect quantum ﬂuctuations to cause

the two solitons to unbind via e.g quantum tunneling out of

a many-body potential, resulting in two equal-sized solitons

moving away from each other [46]. This is not at all what we

arXiv:1612.05545v1 [cond-mat.quant-gas] 16 Dec 2016

2

ﬁnd, and is inconsistent with exact results for the center-of-

mass wave function [47]. Moreover, our beyond-mean-ﬁeld

results are distinct from the GPE failing for strongly correlated

systems like Mott insulators [48–50]; as well as from many-

body systems on short timescales with diﬀerences disappear-

ing for typical experimental parameters and large BECs [51].

We will show that an interaction quench leaves a large soliton

core with small emissions of single particles. Experimentally

these dynamics will appear as a “ﬁzzled” higher order bright

soliton, a stationary soliton core with a small thermal cloud.

Thus we establish a new kind of quantum macroscopicity in

weakly interacting bosonic systems.

The mean-ﬁeld approach via the GPE is a powerful ap-

proximation which provides physical insight into weakly in-

teracting ultracold atoms. In a quasi-one-dimensional wave

guide [1–11,52] the GPE reads

i~∂tϕ(x,t)=−(~2/2m)∂xx ϕ(x,t)+(N−1)g1D|ϕ(x,t)|2ϕ(x,t),

where ϕ(x,t) is a complex wave function normalized to unity

and Nis the number of atoms of mass m. The attractive inter-

action

g1D =2~ω⊥a<0

is proportional to the s-wave scattering length aand the per-

pendicular angular trapping-frequency, ω⊥[53]. Some GPE

predictions for repulsive BECs even become exact [54,55] in

the mean-ﬁeld limit

g1D →0,N→ ∞,(N−1)g1D =const.(1)

While quantum bright solitons in their internal ground

state in addition have a center-of-mass wavefunction (see

Refs. [56,57] and references therein), for measurements both

many-body quantum physics [17,58] and the GPE [59] pre-

dict bright solitons localized at X0with a single-particle den-

sity proﬁle of form

%(x)≡ |ϕ(x)|2=(2ξN{cosh[(x−X0)/(2ξN)]}2)−1,

where the soliton length ξNand the related soliton time τN

ξN≡~2[m(N−1) |g1D|]−1;τN≡mξ2

N/~.(2)

remain constant when approaching the mean-ﬁeld limit (1).

In this Letter we use an interaction quench

g1D(t)=(g0:t≤0

ηg0:t>0;η≥1,g0<0.

After an interaction quench by a factor of η=4, the GPE

yields the analytical result [45, p 300]

%(x,t)=

cosh 3x/(2ξN)+3 e−it/τNcosh x/(2ξN)

3 cos (t/τN)+4 cosh (x/ξN)+cosh (2x/ξN)

2

,

(3)

which is depicted in Fig. 1. For 3/2< η1/2<5/2 mean-ﬁeld

predictions also are very speciﬁc: after losing a few atoms the

0 0.5 1 1.5 2 -4

-2

0

2

0

0.5

1

ρ(x,t) / ρmax

t / (2 π τN)

x / ξN

ρ(x,t) / ρmax

(a)

-2

0

2

0 0.5 1 1.5 2

t / (2 π τN)

0

0.2

0.4

0.6

0.8

1

(b)

FIG. 1: Semiclassical emergent dynamics. After an interaction

quench by a factor of four, the GPE mean-ﬁeld theory predicts per-

fect oscillatory behavior, Eq. (3) [45]. Is it realistic to expect BEC ex-

periments to reproduce these oscillations? (a) GPE density, normal-

ized to its maximum for the ﬁrst two oscillation periods as a function

of space and time in soliton units. (b) 2D projection of (a).

system self-cools to the robust higher-order bright soliton of

Eq. (3) [44].

Within the text book derivation [59] of the GPE, the many-

body wave function corresponding to the GPE is a Hartree-

product state

ψGPE(x1,x2,...,xN)=QN

j=1ϕ(xj).(4)

The mean-square diﬀerence between the position of two par-

ticles

h(x1−x2)2iGPE(t)≡Rd x1Rd x2%(x1,t)%(x2,t)(x1−x2)2,

a measure that is distinct from and independent of the expan-

sion of the center-of-mass wave function, can thus be calcu-

lated form the above analytical result to obtain

∆1,2≡ h(x1−x2)2i(t)/h(x1−x2)2i(0).(5)

While the mean-ﬁeld prediction thus is a perfectly periodic

function of period 2πτN, the question is what we expect to

ﬁnd on the many-body quantum level .

For fundamental considerations on the many-body level

corresponding to the GPE, a very useful tool is the Lieb-

Liniger Model (LLM) with the Hamiltonian [60,61]

ˆ

H=−(~2/2m)PN

j=1(∂2/∂x2

j)+PN−1

j=1PN

ν=j+1g1Dδ(xj−xν),(6)

where xjdenotes the position of particle jof mass m. The

ground-state energy is given by

E0(N)=−mg2

1DN(N2−1)/(24~2).

The energy eigenstates of excited states can be written as

E=PNS

r=1E0(Nr)+Nr~2k2

r/2m,N=PNS

r=1Nr,Nr>0 (7)

corresponding to the intuitive interpretation of NSsolitonlets

— solitons that contain a fraction of the total number of par-

ticles — of size Nr(r=1,2,...,NS) and their individual

center-of-mass kinetic energy. Equation (7) is valid if the sys-

tem size Lis large compared to even a two-particle soliton —

3

this can be included by adding a diverging system size to the

mean-ﬁeld limit (1) to get [62,63]

g1D →0,N→ ∞,L→ ∞, ξN=const., N/L=const.

Reaching such a limit is a diﬃcult numerical problem [64].

However, by replacing the Hamiltonian (6) by the Bose-

Hubbard model (BHM) used to model quantum bright solitons

by e.g. [56,57,65,66], we introduce thermalization mecha-

nisms present in real experiments such as a weak imperfectly

harmonic trap, or a one-dimensional waveguide embedded in

a 3D geometry; for the BHM thermalization is due speciﬁcally

to a lattice, in our case in the limit of very weak discretization.

The BHM takes the form

ˆ

HBHM =−JPjˆ

b†

jˆ

bj+1+ˆ

b†

j+1ˆ

bj+1

2UPjˆnjˆnj−1(8)

where ˆ

b†

j(ˆ

bj) creates (annihilates) a particle on lattice site j,

U<0 quantiﬁes the interaction energy of a pair of atoms and

ˆnjcounts the number of atoms on lattice site j. In order to use

this in a way we can directly use the physical insight gained

from the LLM (6), we choose for the hopping matrix element

(cf. [56, Eq. (17)])

J=~2/(2mδL2)

such that both models have the same single-particle disper-

sion in the long wavelength limit kδLπ, with δLthe lattice

constant. The interaction

U=g1D(32 J~2m+m2g2

1D)1/2/(4~2)

is chosen such that the two-particle ground state has the same

ground-state energy as Eq. (6) compared to the free gas [56].

While both the weak lattice introduced by Eq. (8)] and

a weak harmonic trap [67,68] break the integrability of

the LLM, we can still approximately describe eigenstates by

Eq. (7). Furthermore, from a modeling point of view, by

choosing the lattice we avoid the divergence of the energy

ﬂuctuations of the initial state immediately after the interac-

tion quench, caused by delta functions squared, in h∆ˆ

H2

newi0=

(η−1)2hhˆ

H2

int oldi0− h ˆ

Hint oldi2

0i. While in physics distribu-

tions with well-deﬁned mean and diverging variance are well-

known [69], a more severe reason for avoiding the LLM limit

is that this limit seems to be mathematically ill-deﬁned – an

initial wave function with the wrong boundary conditions at

xj=x`(j,`) has to be expressed in terms of eigenfunc-

tions with the correct boundary conditions [17]. Summariz-

ing, we note that these energy ﬂuctuations are consistent with

the LLM predicting the presence of quantum superpositions

involving many solitonlets in the initial state, but inconsistent

with simple pictures predicting two large solitonlets that ei-

ther oscillate [45] around each other or separate from each

other [46] as the latter cannot happen rapidly [47].

We use time-evolving block decimation (TEBD) [70] —

a numerical method based on matrix product states [71,72]

— to solve the BHM (8). In order to exclude both bound-

ary eﬀects and eﬀects introduced by additional traps, we start

1

2

3

4

0 0.5 1 1.5 2

η

t / (2 π τN)

<n0> / N

0

0.2

0.4

0.6

0.8

(c)

1

2

3

4

0 0.5 1 1.5 2

η

t / (2 π τN)

Σj λj

0.97

0.98

0.99

1

(d)

1

2

3

4

0 0.5 1 1.5 2

η

t / (2 π τN)

∆1,2

0

1

2

3

(b)

-4

-2

0

2

4

0 0.5 1 1.5 2

x / ξN

t / (2 π τN)

ρ(x,t) / ρmax

0

0.2

0.4

0.6

0.8

(a)

FIG. 2: Many-body quantum emergent dynamics. Following an in-

teraction quench by a factor of η, the soliton shows no indication

of special values for the strength of the quench η(initial parameters

used for BHM: N=3, J=0.5, U' −0.5). (a) Single particle

density as a function of both position and time; the mean-ﬁeld oscil-

lations displayed in Fig. 2for η=4 are absent in the TEBD data. (b)

Square root of relative width p∆1,2as a function of both time and

quench strength ηshows that the absence of mean-ﬁeld oscillations

are generic. (c) The condensate fraction provides a further indica-

tor of beyond-mean ﬁeld behavior. (d) The sum over the largest 11

eigenvalues λjof the single-particle density matrix (normalized to 1)

shows that the system becomes even less mean-ﬁeld for longer times.

with a very weak harmonic trap, such that opening it hardly

introduces atom losses [73] and thus the analytical result (3)

remains valid. In our simulations, we switch this trap oﬀat

the same time as we introduce the interaction quench.

If for N'3 quantum bright solitons indeed already show

mean-ﬁeld behavior [23], we should be able to see the mean-

ﬁeld oscillations depicted in Fig. 1already for N=3. Fig-

ures 2and 3show that the mean-ﬁeld oscillations are absent

from the many-body TEBD data for N=3 and Nup to 16,

respectively. For the BHM (8), we note that the relative parti-

cle measurement of Eq. (5) can be rewritten in second quan-

tized form appropriate to TEBD by replacing h(x1−x2)2iwith

Pj,`(j−`)2hˆ

b†

jˆ

bjˆ

b†

`ˆ

b`i.

Within the LLM, for relative distances large compared to

the soliton length ξN, the leading-order contributions to ex-

cited states consists of NSsolitonlets of terms correspond-

ing to NSindividual solitonlets moving apart [17] In order to

obtain a physical understanding on the time scales on which

these solitonlets can move apart if they initially sit on top of

each other, we recall the text book result for the variance of an

initially Gaussian single-particle wave function [74],

∆X2= ∆X2

0{1+[~t/(2M∆X2

0)]2}.

For the relative motion the mass M=Nm has to be replaced

by the relative mass mrel ≡m N1N2/(N1+N2). If initially lo-

calized to ∆X0∝ξN(much stronger localization leads to too

high kinetic energies while much weaker localization leads to

a too wide initial wave function) and for a relative mass inde-

pendent of N, the relative wave function will expand to a size

4

0

0.5

1

1.5

2

2.5

3

0 0.2 0.4 0.6 0.8 1

∆1,2

t / (2 π τN)

(a)

-0.2

0

0.2

0 0.2 0.4 0.6 0.8 1

∆1,2 - q(t)

t / (2 π τN)

(b)

10-6

10-4

10-2

0 0.2 0.4 0.6 0.8 1

δrel

t / (2 π τN)

(c)

FIG. 3: Relative width measures for up to 16 particles. While the

GPE predicts perfect oscillations, the predominant behavior in the

matrix-product state numerics is that the relative distance of particles

grows. (a) Relative width ∆1,2as a function of time: mean-ﬁeld os-

cillations (lowest black dotted curve); TEBD results (N=3,4,8,16

blue, fuscia, brown, red curves from top to bottom); overall behav-

ior is quadratic (ﬁts shown as light blue curves). Here J=0.5 and

U' −0.5,−0.33,−0.14,−0.07. (b) Residual oscillations in panel

(a) after subtracting the quadratic ﬁt. (c) Relative error comparing

TEBD data with distinct convergence parameters. From top to bot-

tom: N=16: χmax =60 versus χmax =80, N=8: χma x =60 vs.

χmax =80, N=4: χmax =30 vs. χma x =40, N=3 (too small to be

visible): χmax =30 vs. χmax =40.

larger than the initial wave function on time scales [cf. Eq. (2)]

t∝mrξ2

N/~=[(N1N2)/(N1+N2)]τN.(9)

The Hartree product states (4) are also ideal to cal-

culate mean energies which in the mean-ﬁeld limit (1)

become identical to the exact many-body quantum re-

sults [17]. The kinetic energy prior to the interaction quench

is hEkiniold =N3mg2

1D/(24~2)=−Eold

0, and the interaction

energy hEintiold =2Eold

0. Immediately after the interaction

quench, the kinetic energy remains unchanged and the inter-

action energy is increased to hEintinew =ηhEintiold =2ηEold

0.

In units of the new ground state energy Enew

0=η2Eold

0we have

a total average energy after the interaction quench of

hEi=[(2η−1)/η2]Enew

0(10)

where 0 <(2η−1)/η2<1 for η > 0.5 and η,1.

If ultracold attractive atoms are initially prepared in their

ground state, by using the eigenstate thermalisation hypoth-

esis [75,76] we conjecture that an interaction quench by a

factor of ηwill on short time scales lead to a ﬁnal state con-

sisting of a single bright soliton containing N1atoms, as given

by thermodynamic predictions, and N−N1free atoms. In the

mean-ﬁeld limit (1), for bright solitons thermodynamic pre-

dictions read [77]

N1= 2η−1

η2!1/3

N, ξN1=1

[η(2η−1)]1/3ξN, η > 1

2,(11)

i.e., one large soliton with reduced particle number N1and

reduced size ξN1; and N−N1single atoms which are not bound

in molecules.

0

0.2

0.4

0.6

0.8

1

1 2 3 4

1 - N1 / N, N1 / N

η

(b)

0

0.2

0.4

0.6

0.8

1

1 2 3 4

ξ / ξN, old

η

(a)

FIG. 4: Extrapolation to large particle number. By applying the

eigenstate thermalisation hypothesis [75,76] we conjecture that after

an interaction quench to more negative interactions by a factor of η,

the attractive BEC approaches the equilibrium predictions for a ther-

mally isolated gas of Ref. [77]. (a) New soliton length, in units of the

original soliton length ξN,old, for a soliton containing all atoms (lower

curve) versus N1emitted free atoms (upper curve) as a function of

the quench η. (b) Fraction of atoms in the soliton (upper curve) and

in the free gas (lower curve) [see also Eq. (11)].

To suggest that this might indeed be what happens seems

counterintuitive at best, since following “thermalization” ac-

cording to the eigenstate thermalization hypothesis, all ener-

getically accessible eigenfunctions will be involved [75,76];

violating both the Landau hypothesis [78, very end], which

at ﬁrst glance seems to prevent co-existence of a large soli-

ton and a free gas; argued against also by mean-ﬁeld predic-

tions [44,45] as well as thermodynamic predictions for ultra-

cold atoms in contact with a heat bath [62,63]. However, the

Landau hypothesis is based on assumptions that are not ful-

ﬁlled for bright solitons [77] and thermally isolated ultracold

atoms, arguably realised in state-of-the-art experiments with

bright solitons [1–11], behave quite diﬀerently from those

in contact with a heat bath [77]. Furthermore, contrary to

rumours stating otherwise, one-dimensional Bose gases do

thermalise, for example in the presence of a weak harmonic

trap [67,68].

As depicted in Fig. 4, we conjecture that after an in-

teraction quench to more negative interactions, the system

will relax towards the situation predicted in thermal equilib-

rium: the co-existence between one large soliton and a free

gas [77]. The application of the eigenstate thermalisation hy-

pothesis [75,76] is supported by the fact that single atoms

will move the initial cloud much faster than larger solitonlets

[Eq. (9)] that can continue to thermalize. Two macroscopic

solitons sitting on top of each other would have to remain at

the same position [47]; a freely expending gas passes the con-

vergence test of Ref. [47] but energy conservation and Eq. (10)

would require at least one soliton(let) to be present.

To conclude, we have combined evidence from three dis-

tinct models (GPE, LLM and BHM) to show that truly quan-

tum emergent behaviour for attractive Bosons happens after

an interaction quench to more attractive interactions. Combin-

ing the numerical evidence with general considerations based

on the eigenstate thermalization hypothesis for larger parti-

cle numbers, we conjecture that the ﬁnal many-body quan-

tum state consists of one smaller bright soliton and lots of sin-

5

gle atoms, thus yielding an ultimate example of a mean-ﬁeld

breakdown on time scales that remain experimentally relevant

even in the mean-ﬁeld limit (1). Our predictions are accessible

to state-of-the-art experiments with thousands of atoms [1–

11]. Furthermore, the above conjecture oﬀers an explanation

as to why experiments that quasi-instantaneously switch from

repulsive to attractive interactions (see for example Ref. [5])

while avoiding the “Bose-nova” collapse or modulational in-

stability can nevertheless lead to one large matter-wave bright

soliton (and a thermal cloud) being formed.

We thank T. P. Billam, J. Brand, J. Cosme, S. A. Gar-

diner, B. Gertjerenken, and M. L. Wall for discussions. We

thank the Engineering and Physical Sciences Research Coun-

cil UK for funding (Grant No. EP/L010844/). This material

is based in part upon work supported by the US National Sci-

ence Foundation under grant numbers PHY-1306638, PHY-

1207881, and PHY-1520915, and the US Air Force Oﬃce of

Scientiﬁc Research grant number FA9550-14-1-0287. L.D.C.

thanks Durham University and C.W. the Colorado School of

Mines for hosting visits to support this research.

Data will be available online soon [79].

∗Electronic address: christoph.weiss@durham.ac.uk

†Electronic address: lcarr@mines.edu

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“Higher-order quantum bright solitons in Bose-Einstein con-

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