Higher-order quantum bright solitons in Bose-Einstein condensates show truly quantum emergent behavior
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.
Higher-order quantum bright solitons in Bose-Einstein condensates show truly quantum emergent
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-
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 
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 . 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 . Both the limit of well-deﬁned
phase  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 . Dark solitons were also realized ex-
perimentally in BECs  and have been further explored
in detail over the years in comparison to such predictions,
e.g. . 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 . Quenches involving dark-bright soli-
tons [37,38], quenched dynamics of two-dimensional solitary
waves , 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
For attractive BECs, there are very speciﬁc mean-ﬁeld pre-
dictions : 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 . 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 . This is not at all what we
arXiv:1612.05545v1 [cond-mat.quant-gas] 16 Dec 2016
ﬁnd, and is inconsistent with exact results for the center-of-
mass wave function . 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 .
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
where ϕ(x,t) is a complex wave function normalized to unity
and Nis the number of atoms of mass m. The attractive inter-
is proportional to the s-wave scattering length aand the per-
pendicular angular trapping-frequency, ω⊥. 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  pre-
dict bright solitons localized at X0with a single-particle den-
sity proﬁle of form
where the soliton length ξNand the related soliton time τN
remain constant when approaching the mean-ﬁeld limit (1).
In this Letter we use an interaction quench
After an interaction quench by a factor of η=4, the GPE
yields the analytical result [45, p 300]
cosh 3x/(2ξN)+3 e−it/τNcosh x/(2ξN)
3 cos (t/τN)+4 cosh (x/ξN)+cosh (2x/ξN)
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
ρ(x,t) / ρmax
t / (2 π τN)
x / ξN
ρ(x,t) / ρmax
0 0.5 1 1.5 2
t / (2 π τN)
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) . 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) .
Within the text book derivation  of the GPE, the many-
body wave function corresponding to the GPE is a Hartree-
The mean-square diﬀerence between the position of two par-
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
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]
where xjdenotes the position of particle jof mass m. The
ground-state energy is given by
The energy eigenstates of excited states can be written as
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 —
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 .
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
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)])
such that both models have the same single-particle disper-
sion in the long wavelength limit kδLπ, with δLthe lattice
constant. The interaction
is chosen such that the two-particle ground state has the same
ground-state energy as Eq. (6) compared to the free gas .
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
int oldi0− h ˆ
0i. While in physics distribu-
tions with well-deﬁned mean and diverging variance are well-
known , 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 . 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  around each other or separate from each
other  as the latter cannot happen rapidly .
We use time-evolving block decimation (TEBD)  —
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
0 0.5 1 1.5 2
t / (2 π τN)
<n0> / N
0 0.5 1 1.5 2
t / (2 π τN)
0 0.5 1 1.5 2
t / (2 π τN)
0 0.5 1 1.5 2
x / ξN
t / (2 π τN)
ρ(x,t) / ρmax
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  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 , 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
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  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 ,
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
0 0.2 0.4 0.6 0.8 1
t / (2 π τN)
0 0.2 0.4 0.6 0.8 1
∆1,2 - q(t)
t / (2 π τN)
0 0.2 0.4 0.6 0.8 1
t / (2 π τN)
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)]
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 . The kinetic energy prior to the interaction quench
is hEkiniold =N3mg2
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
In units of the new ground state energy Enew
a total average energy after the interaction quench of
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 
[η(2η−1)]1/3ξN, η > 1
i.e., one large soliton with reduced particle number N1and
reduced size ξN1; and N−N1single atoms which are not bound
1 2 3 4
1 - N1 / N, N1 / N
1 2 3 4
ξ / ξN, old
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. . (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  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 . Furthermore, contrary to
rumours stating otherwise, one-dimensional Bose gases do
thermalise, for example in the presence of a weak harmonic
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 . 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 ; a freely expending gas passes the con-
vergence test of Ref.  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-
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. )
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 .
∗Electronic address: email@example.com
†Electronic address: firstname.lastname@example.org
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“Higher-order quantum bright solitons in Bose-Einstein con-
densates show truly quantum emergent behavior: Supporting