Phase-fluctuating condensates are fragmented: An experimental benchmark for self-consistent quantum many-body calculations

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by  Oleksandr V. Marchukov

Phase-fluctuating condensates are fragmented: An experimental benchmark for
self-consistent quantum many-body calculations
Oleksandr V. Marchukov and Uwe R. Fischer
Center for Theoretical Physics, Department of Physics and Astronomy,
Seoul National University, 08826 Seoul, Republic of Korea
(Dated: March 17, 2017)
Westudyzero-temperaturequantumphasefluctuationsinharmonicallytrappedone-dimensional
interacting Bose gases, using the self-consistent multiconfigurational time-dependent Hartree
7 method. In a regime of mesoscopic particle numbers and moderate contact couplings, it is shown
1 thatthephase-fluctuatingcondensateisproperlydescribedasafragmentedcondensate. Inaddition,
0 wedemonstratethatthespatialdependenceoftheamplitudeofphasefluctuationssignificantlyde-
2 viatesfromwhatisobtainedinBogoliubovtheory. Ourresultscanbeverifiedincurrentlyavailable
experiments. They therefore provide an opportunity both to experimentally benchmark the mul-
r
a ticonfigurational time-dependent Hartree method, as well as to directly observe, for the first time,
M the quantum many-body phenomenon of fragmentation in single traps.

5
1 ThestandardparadigmoftheweaklyinteractingBose of the order of percent, and thus for moderate coupling
gas is Bogoliubov theory [1]. A single orbital is occupied constants and densities (particle numbers N = 10–30).

]
macroscopically, with a continuum of excitations exist- In particular, we show that, on the self-consistent level,
s
a ing on top of this condensate. On the other hand, in the phase fluctuations develop a peak in position space.
g
spatialdimensionsmallerthanthree,theabsenceofsuch This is akin to what is observed for a pair of bosons in
-
t a Bose-Einstein condensate (BEC) in infinitely extended Monte Carlo simulations at infinitely large interactions
n
homogeneous systems directly follows from the funda- andlowdensitiesintheTonks-Giradeaulimit[24]aswell
a
u mental quantum statistical Bogoliubov inequality [1], as as for an analytical N = 2 solution at finite couplings,
q derived in [2–4]. More recently, the advances in modern employing the approach of [25]. This peak is absent in
t. precision experiments using ultracold atomic gases have Bogoliubov mean-field theory and its extension to larger
a
rekindled the interest in the coherence properties of low- coupling in Luttinger liquid theory [12]. The fluctuation
m
dimensional quantum gases [5, 6]. peak height is essentially proportional to the degree of
-
d For finite systems, which are those actually realized fragmentation. In addition, the mean-field phase fluc-
n in experiments with trapped Bose gases, the question of tuations saturate at large distances, while we find that
o theexistenceofBECinlowdimensionsismoreintricate. they decrease again towards the condensate border. We
c For example, a one-dimensional (1D) BEC, at given in- takethesefactsasconcreteevidencethatinhomogeneous
[
teraction coupling and density, can exist up to a critical phase-fluctuating fragmented condensates in a harmonic

2 length of the condensate [7, 8]. Within the realm of Bo- trap must be described by self-consistent many-body so-
v goliubovtheory[9,10],itsextensiontoquasicondensates lutions that go beyond hydrodynamic, local density, and
1
2 [11],andtheLuttingerliquidapproach[12],phasefluctu- Hartree-Fock type approximations.
8 ationshavebeenshowntograduallydestroyoff-diagonal- Theimportanceofmany-bodycorrelationsbetweenall
6 long-range-order (ODLRO) in finite 1D Bose-Einstein modes in the phase-fluctuating regime which we reveal
0 condensates, and to lead to a characteristic power law can be directly verified by current experiments [16–20].
1. decay of correlation functions. These phase-fluctuating Thisfacilityofexperimentalaccessisinmarkedcontrast
0 condensates have been probed in numerous experiments, to the large degrees of fragmentation necessary to ob-
7 initially in [13–15], and with increasing sophistication in serve significant density-density correlations [26]. Our
1
recent years cf., e.g., [16–20]. self-consistent many-body results therefore pave the way
:
v to study experimentally the many-body phenomenon of
Ourprimaryaiminwhatfollowsistoshowthatphase-
i fragmentation in a single harmonic trap [27]. They pro-
X fluctuating 1D condensates are properly to be described
vide a benchmark to correlate theory and experiment in
r as fragmented condensates [21–23] in a self-consistent
a quantum many-body physics, and in principle to arbi-
many-body approach. The many-body correlations cor-
trarily high order in the correlation functions [18].
responding to fragmentation are not captured by mean-
In the interacting Bose gas, the (basis invariant) def-
field theories, for which all excited modes are uncorre-
inition of BEC is due to Penrose and Onsager [21]. It
lated with each other and with the condensate. Frag-
employsthepositionspacesingle-particledensitymatrix
mented condensates in 1D harmonic traps, on the other
in its eigenbasis,
hand,mustbedescribedbyaself-consistenttheorywhich
accommodates the correlations between all significantly
M
occupied orbitals. Importantly, we reveal that this re- ρ(1)(x,x(cid:48))=(cid:104)ψˆ†(x)ψˆ(x(cid:48))(cid:105)=(cid:88)N ϕ∗(x)ϕ (x(cid:48)), (1)
i i i
mains true even at very small degrees of fragmentation
i=1
2
where ψˆ†(x) and ψˆ(x) are bosonic creation and annihi- correlation properties of fragmented BECs cf., e.g., [38–
lation field operators, respectively. The angled brack- 47]. Briefly, see for details [34, 35, 48], the MCTDH
ets indicate quantum-statistical average over states; we method constructs the many-body wavefunction as a
work at zero temperature. Furthermore, ϕ (x) are the sum of all possible configurations of distributing N par-
i
single-particle wavefunctions (called natural orbitals in ticles over M time-dependent single-particle wavefunc-
thiseigenbasisofρ(1)),N aretheiroccupationnumbers, tions. Applying the Dirac-Frenkel variational principle,
i
andM isthenumberoforbitals(inpracticefixedtobea one obtains a system of coupled equations for both or-
finite number by the available computational resources). bitals and their occupation number distributions, which
The definition [21] states that if the eigenvalues N are aresolvedself-consistentlytoyieldthemany-bodywave-
i
“macroscopic,” i.e. N = O(N), then the many-body function and the corresponding correlation functions.
i
state of the Bose gas is a simple or fragmented BEC
when there are one or several such eigenvalues, respec- Using the representation of the field operators
tively [22, 23]. We note that this formal definition must ψˆ(x) = eiφˆ(x)(cid:112)ρˆ(x),ψˆ†(x) = (cid:112)ρˆ(x)e−iφˆ(x), where
obviouslyremainsomewhatvaguewherethereisnother- ρˆ(x) is the particle density operator and φˆ(x) is a
modynamic limit of very large N, but this does not usu- (hermitian) phase operator, the single-particle den-
ally lead to practical difficulties. sity matrix can be written in the form ρ(1)(x,x(cid:48)) =
The Hamiltonian that we consider to address the (cid:68)(cid:112)ρˆ(x)e−i(φˆ(x)−φˆ(x(cid:48)))(cid:112)ρˆ(x(cid:48))(cid:69). It is well known that
many-body problem describes bosons interacting by a
one should exercise care when defining the number and
contact pseudopotential and placed in a harmonic trap:
phase operators in this way, as thoroughly reviewed in
Refs. [49, 50]; also see the coarse-graining procedure ap-
H =(cid:88)N (cid:18) p2i + 1mω2x2(cid:19)+g (cid:88)δ(x −x ). (2) plied in [11]. We will use the definition above which co-
2m 2 i 1D i j incides with the traditional Dirac approach [50], since
i=1 i>j
we consider that particle density fluctuations are small,
Here, x and p are 1D position and momentum opera- andtodirectlyrelateourresultswithpreviousworkper-
i i
tors of a given atom (or molecule) i, respectively, ω is formed in [9, 51]. One then has
the frequency of the trapping harmonic potential, N is
the number of particles, m is their mass, and g1D is the (cid:104)ψˆ†(x)ψˆ(x(cid:48))(cid:105)=(cid:112)ρ(x)ρ(x(cid:48))exp(cid:20)−1(cid:68)δˆφ2 (cid:69)(cid:21), (3)
contact coupling. In order to establish the connection 2 xx(cid:48)
with experiment, one considers a quasi-1D gas, i.e. the
particles are trapped in a three-dimensional (3D) har- where δˆφxx(cid:48) =φˆ(x)−φˆ(x(cid:48)) is the phase difference oper-
monic potential that is strongly anisotropic, with the ator and ρ(x) = (cid:104)ψˆ†(x)ψˆ(x)(cid:105) is the mean local density;
transverse frequency being much larger than the axial weneglecteddensityfluctuations. Solvingfor(cid:104)δˆφ2 (cid:105),we
xx(cid:48)
one, ω/ω⊥ (cid:28) 1, such that transverse motion is con- obtainadirectrelationofmean-squarephasefluctuations
fined (frozen) to the ground state. The 1D coupling and single-particle density matrix in position space:
strength, g , is related to the 3D scattering length, a ,
1D sc
tbeyrign1gDr=eso4nπa(cid:126)n2caessc,/w(πhmerle⊥2l)⊥fa=r(cid:112)aw(cid:126)a/ymfrωo⊥misgethometertarnicsvsecraste- (cid:104)δˆφ2xx(cid:48)(cid:105)=−2ln(cid:34)(cid:104)(cid:112)ψˆ†ρ((xx))ψˆρ((xx(cid:48)(cid:48)))(cid:105)(cid:35). (4)
oscillator length [28].
Thehomogeneousquantummany-bodyproblemcorre- Theaboverelationrepresentsthedefinitionofphasefluc-
sponding to the Hamiltonian (2) without harmonic trap tuations within our analysis. Below, we aim to demon-
(ω = 0) can be solved by Bethe ansatz, as was shown strate that at mesoscopic numbers of particles trapped
longagobyLiebandLiniger[29]. Ontheotherhand,for in a 1D harmonic oscillator potential with contact inter-
trapped (spatially confined) and thus generally inhomo- action, the mean-square phase fluctuations obtained by
geneoussystems,themathematicalcasemostrelevantto using self-consistent calculations differ significantly from
actual experiments, the exact solution is not known (ex- mean-field results (cf., e.g., Refs. [9, 51]). Using the di-
cept for the hard-core limit [30]). There are several nu- agonalized form of the single-particle density matrix in
mericalmethods thatareapplicable tothepresentprob- Eq.(1) one can rewrite Eq. (4) as
lem,suchas,forexample,densitymatrixrenormalization
group [31, 32] and quantum Monte Carlo methods [33]. (cid:104)δˆφ2 (cid:105)=−2ln(cid:34)(cid:80)Mi=1Niϕ∗i(x(cid:48))ϕi(x)(cid:35). (5)
Here, we used the multiconfigurational time-dependent xx(cid:48) (cid:112)ρ(x)ρ(x(cid:48))
Hartree (MCTDH) method for bosons [34] and, in par-
ticular, its implementation MCTDH-X [35]. This power- Weevaluate(5)afterfindingthemany-bodygroundstate
ful method, long known in physical chemistry for distin- ofthesystemusingMCTDH-X[35]. Forourcalculations
guishableparticles[36,37],hassincetheadventofultra- in the number range N = 10–30, we used five available
cold quantum gases proven its value for the study of the orbitals, M = 5, to ensure that there is no significant
3
g n n n n n
0 1 2 3 4 5 a)
0.1 0.99875 0.00095 0.00022 0.00004 0.00002
0.5 0.98068 0.01338 0.00425 0.00119 0.00049
0.75 0.96611 0.02241 0.00788 0.00254 0.00105
1.0 0.95152 0.03087 0.01168 0.00418 0.00175
TABLE I. Table of relative occupation numbers, n =N /N
i i
for N =10 and various couplings g [Eq.(6)].
0
occupation in the highest orbitals. It is convenient to
introduce a dimensionless, scaled interaction parameter
g =mg l/(cid:126)2 =4a l/l2, (6) b)
0 1D sc ⊥
wherel=(cid:112)(cid:126)/mωistheaxialharmonicoscillatorlength.
We considered the range g = 0.1–1.0, which even for
0
our relatively small number of particles corresponds to
the weakly-interacting Bose gas in the Thomas-Fermi
regime [9]. This range of couplings is well within cur-
rent experimental possibilities [16–20].
The occupation numbers we obtain after convergence
of the self-conistent equations has been reached, for
N = 10 and varying interaction strength g , are given
0
in Table I (for N = 30, cf. [48]). We see that the degree
of fragmentation, defined as the fraction of particles not
c)
being in the energetically lowest orbital, 1−n , grows
1
rapidly with interaction strength. However, for the g
0
rangesweconsiderthefragmentationisstillintherange
of a few percent only. This, in turn, also represents a
condition for our calculations, with a fixed number of
orbitals M =5, to be reliable for the chosen range of g .
0
In Fig.1, we display surface plots of the mean-square
of quantum phase fluctuations (cid:104)δˆφ2 (cid:105) in the x–x(cid:48) plane
xx(cid:48)
for three different values of dimensionless interaction
strength. We see two very distinct bulges that emerge
even for small interaction, which grow in size with in-
creasing interaction strength. The detailed shape and
fine structure of the bulges corresponds to the shape FIG. 1. Mean-square quantum phase fluctuations (cid:104)δˆφ2 (cid:105) for
and weight of the different orbitals in the self-consistent xx(cid:48)
N = 10 and different interaction strengths: a) g = 0.1,
0
solution for the quantum field. The emergence of the
b) g = 0.5, and c) g = 1.0. The maxima along the off-
0 0
bulges has the direct interpretation of the loss of phase diagonals, x(cid:48) = −x, correspond to the fact that the gas be-
coherence between distant parts of the cloud by phase comes phase-uncorrelated in distant regions of the cloud.
fluctuations. This phase-phase-correlations induced phe-
nomenon is conjugate to the coherence loss indicated by
density-density correlations which was discussed in [26]. tions contributing to the degree of fragmentation when
Fig.2 shows the dependence of the maximum value of one increases the coupling towards the boson-localized
phase fluctuations Φ2 := max[(cid:104)δˆφ2 (cid:105)] (measured at the Tonks-Girardeau regime. The smooth dependence on g0
xx(cid:48)
we get is expected from the fact that we observed in our
top of the bulges in Fig.1), and of the degree of frag-
simulationsthattheshapeoftheorbitalsdoesnotchange
mentation 1−n , as functions of the dimensionless in-
1
drastically by increasing the interaction strength.
teraction strength g . We recognize a very smooth, al-
0
most linear dependence of both quantities on g . The Hence, we obtain as an important result that in our
0
inset shows that their ratio slightly decreases with cou- regimeofmoderatecouplingsthedegreeoffragmentation
pling strength (by about 15 % over the range of g0 in- 1−n1 isameasureofthemaximalstrengthofphasefluc-
vestigated). We conjecture that this slight decrease is tuations. To gain further insight, in Fig.3, we compare
related to the increasing importance of density fluctua- thephasefluctuationsrelativetothecloudcenter,(cid:104)δˆφ2 (cid:105),
x0
4
FIG. 2. Maximum value of mean-square phase fluctuations
of (5) Φ2 := max[(cid:104)δˆφ2 (cid:105)] (peak height in Fig.1), and frag-
mentationdegree,1−xnx1(cid:48),wheren1 istheoccupationnumber FIG. 3. The comparison of phase fluctuations (cid:104)δˆφ2x0(cid:105), cal-
of the energetically lowest orbital, as function of interaction culated in the hydrodynamic mean-field approach [51] (black
strength g , for N = 10. The inset shows the ratio of the solidline)andself-consistentlywithMCDTHusing (5)(blue
0
maximum fluctuation Φ2 and fragmentation degree 1−n . dashed line) for N = 10 and different interaction strengths:
1
a) g = 0.1, b) g = 0.5, c) g = 0.75 and d) g = 1.0. The
0 0 0 0
axial coordinate x is scaled by the size of the gas cloud R.
with those obtained from the hydrodynamical limit of
Bogoliubov theory, which read [51]
construction, do not describe.
√
(cid:104)δˆφ2 (cid:105) = 2g0l(cid:18)2ln(cid:20)R(cid:21)−Ci(cid:20)2R|x|(cid:21)+Ci(cid:20)2|x|(cid:21)(cid:19). To summarize, our primary result is twofold. First,
x0 mf π R l l2 R that self-consistent many-body calculations for a 1D
(7) trapped Bose gas, performed with MCTDH-X, reveal
thatphase-fluctuatingBECsareproperlytobedescribed
Here, R is the total length of the cloud, calculated from as fragmented condensates containing many-body corre-
thedomaininwhichthedensityofthegasisnotzero,and lations between all significantly occupied orbitals. And,
Ci(x)=−(cid:82)∞cos(t)dt/t is the Cosine integral function. second,thatthespatialdependenceofphasefluctuations
x
The difference which manifests itself in Fig.3 is evi- obtained via mean-field theory is both qualitatively and
dent: The mean-field result does, in particular, not cap- quantitativelydifferentfromtheself-consistentone. Asa
ture the distinct phase fluctuation maximum in the mid- corollary, we conclude that self-consistency is crucial for
dle of the cloud which is due to the self-consistently ob- theaccuracyandpredictivepowerofmany-bodycalcula-
tainedpreciseshapeoftheorbitals,andhasatendencyto tions, even when the degree of fragmentation is as small
overestimatethemagnitudeofphasefluctuations. More- asonthelevelofpercent. Thisisbecause1Dmean-field,
over, the mean-field phase fluctuations saturate to an despiteitsbeingreliableinthecaseofverylargeparticle
asymptotic value at the edge of the cloud for mean-field. numbers and very weak couplings, does not describe the
On the other hand, we obtain that, rather, the fluctua- more subtle many-body correlations which develop for
tionsdecreaseagaintowardstheboundariesofthecloud. evenmoderateinteractionstrengths. Futureworkween-
As a result, self-consistent many-body physics predicts visageincludesexploringtherelevanceofself-consistency
potentially larger, more stable 1D condensates, because to the large fragmentation regime, and thus its potential
distant regions of the cloud tend to remain more phase- impact on the crossover location to the Tonks-Girardeau
correlated than in mean field. gas in a single harmonic trap [19, 20, 52].
The emergence of the local phase-fluctuation maxi- Our results suggest a pathway to implement an ex-
mumweobservewithinMCDTHisakintothefirst-order perimental benchmark for MCTDH, by verifying its pre-
correlation function dips obtained in Monte Carlo calcu- dictive power through readily accessible experimental
lations for the Tonks-Girardeau limit of very strong in- means, versus the purely theoretical arguments put for-
teractions [24] and in the N = 2 analytics of [25], as ward, e.g., in[53–55], whichquestiontheaccuracyofthe
demonstrated in the supplemental material [48]. The convergenceofMCDTHandthusalso,ultimately,itsva-
self-consistent approach MCDTH, similarly, therefore lidity. Further possible extensions in this respect, which
contains many-body correlations between field-operator probethelatteraspectsofconvergencetothetruemany-
modes with spatially inhomogeneous modulus, which body solution even more deeply, concern nonequilibrium
mean field and local density approximations, by their setups created by starting from phase-fluctuating con-
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1
SUPPLEMENTAL MATERIAL
Multiconfigurational time-dependent Hartree method
Here we present a brief description of the multiconfigurational time-dependent Hartree (MCTDH) method that we
useforourself-consistentmany-bodycalculations. Themethodhasbeengeneralizedtoapplyforallindistinguishable
particles [35]; here. we focus on bosons. For more detailed reviews and examples see Refs. [34, 36, 37].
A system of N interacting bosons is described by using the time-dependent Schr¨odinger equation
 
N
(cid:88)hˆ(xi;t)+(cid:88)Wˆ(xi−xj)Ψ(x1,...,xN;t)=i(cid:126)∂∂Ψt , (S1)
i=1 i>j
where hˆ(xi;t)= 2pm2i +V(xi) is the one-body Hamiltonian, with m as mass of the particles, xi and pi as position and
momentum operators of a given boson i, and V(x ) as the potential energy. The term Wˆ(x −x ) is the pairwise
i i j
particleinteractionoperator. Themany-bodywavefunction,Ψ(x ,...,x ;t),intheMCTDHformulationisexpressed
1 N
by the following ansatz
(cid:88) →−
Ψ(x1,...,xN;t)= C→−N(t)|N;t(cid:105), (S2)
→−
{N}
→−
where the basis |N;t(cid:105) consists of all possible symmetrized wavefunction products of N particles (permanents) dis-
→−
tributed over M single-particle functions (orbitals), where N =(N ,N ,...,N ) and N +N +···+N =N, i.e.
1 2 M 1 2 M
Nj represents the occupation of the orbital j, and C→−N(t) are the time-dependent expansion coefficients. The basis
sizeequals (N+M−1)!,i.e. thetotalnumberofdistributionsofN particlesamongM states. Inthelanguageofsecond
N!(M−1)!
quantization, the permanents can be written as
→− 1
|N;t(cid:105)= √ (b†(t))N1(b†(t))N2...(b† (t))NM|vac(cid:105). (S3)
N !N !...N ! 1 2 M
1 2 M
Theoperatorb†(t)isthetime-dependentbosoniccreationoperator,and|vac(cid:105)isthevacuumstate. Clearly,theansatz
j
isexactifM →∞,thatisifweconsiderthefullHilbertspaceoftheproblem. Inpractice,thisishowevernotpossible
due to the unavoidable computational constraints for any nontrivial (that is, two-body interacting) problem, however
for large enough M, i.e., when the occupation of the highest orbitals is negligible, the many-body function (S2)
represents a numerically exact solution of time-dependent many-body Schr¨odinger equation.
In order to calculate the expansion coefficients C→−N(t) and orbitals {φj(xi;t), j = 1,...,M}, one applies the
Dirac-Frenkel variational principle to the action
 
S[{C→−N(t)},{φj(xi;t)}]=(cid:90) dt(cid:104)Ψ|Hˆ −i(cid:126)∂∂t|Ψ(cid:105)− (cid:88)M µjk(t)((cid:104)φj|φk(cid:105)−δjk), (S4)
j,k=1
whereµ (t)aretime-dependentLagrangemultipliers,ensuringthattheorbitalsremainorthonormal. Thevariational
jk
procedure gives the equations of motion
∂C(t)
i(cid:126) =H(t)C(t),
∂t
 
M
i(cid:126)∂|∂φtj(cid:105) =Pˆhˆ|φj(cid:105)+ (cid:88) ρ−jk1ρksqlWˆsl|φq(cid:105), (S5)
k,s,q,l=1
that can be solved numerically in order to obtain the many-body wavefunction Ψ(x ,...,x ;t). Here, C(t) is the
1 N
columnvectorthatconsistsofallpossibleexpansioncoefficientsC→−(t),H(t)isamatrixcomposedofmatrixelements
of the time-dependent Hamiltonian in the corresponding basis N|→−N;t(cid:105), Wˆ = (cid:82) dx(cid:48)φ∗φ are the local interaction
sl s l
potentials,Pˆ =1−(cid:80)M |φ(cid:48)(cid:105)(cid:104)φ(cid:48)|isaprojectionoperator,andρ andρ arethematrixelementsoftheone-body
k(cid:48)=1 k k jk ksql
and two-body density matrices, respectively.
2
g n n n n n
0 1 2 3 4 5
0.1 0.99694 0.00220 0.00064 0.00015 0.00006
0.5 0.97256 0.01633 0.00698 0.00277 0.00136
0.75 0.95917 0.02303 0.01067 0.00473 0.00239
1.0 0.94785 0.02843 0.01381 0.00653 0.00337
TABLE I. Table of relative occupation numbers, n = N /N for N = 30, and various dimensionless couplings g , which are
i i 0
defined in Eq.(6) of the main paper.
ThesystemofcoupledequationsinEqs.(S5)istobesolvedbothfortheorbitalsandexpansioncoefficientstogether.
This constitutes the notion of self-consistency in inhomogenous quantum many-body systems of interacting bosons
we employ throughout the paper.
MCDTH Results for N =30
Inthissubsection,wepresenttheresultsofourcalculationsthatwerediscussedinthebulkofthepaperforN =10,
butnowwithanincreasednumberofparticles,N =30. ThemainfeaturesarequalitativelyverysimilartotheN =10
case. Table I shows the relative occupation numbers of the orbitals. The degree of fragmentation 1−n is again not
1
large, but also not negligible even for relatively weak interaction.
In Fig. S1 we show surface plots of the mean-square of quantum fluctuations (cid:104)δˆφ2 (cid:105) in the x−x(cid:48) plane for various
xx(cid:48)
values of the dimensionless interaction strength g . Bulges of a similar geometric shape to the N = 10 case emerge,
0
cf. Fig.1 in the main text, since the shape of the correlations is dictated by the geometry of the trapping potential
and single-particle orbitals, also cf. the discussion in the subsection below. Fig. S2 shows the dependence of the
maximumvalueofmean-squarephasefluctuationsΦ2 :=max[(cid:104)δˆφ2 (cid:105)]andfragmentationdegree1−n , asafunction
xx(cid:48) 1
ofdimensionlesscouplingg . WeseethatΦ2 growsslightlyslowerforN =30thanforN =10,butremainsasmooth
0
and almost linear function of g , cf. Fig. 2 in the main text. Finally, Fig. S3 displays the comparison of the MCTDH
0
calculations with the hydrodynamic limit of Bogoliubov theory [51]. As for N = 10, cf. Fig. 3 in the main text, the
results are qualitatively different for the mean-field and MCTDH calculations. This is to be expected, since we still
consider a relatively small number of particles.
Comparison with Monte Carlo for hard-core bosons and N =2 analytics
Inthissubsection,wecomparetheresultsofourMCTDHcalculationswiththeMonteCarlocalculationsperformed
by Minguzzi et al. in Ref. [24] for the one-body density matrix of a pair (N =2) of hard-core bosons. In Fig. S4 we
(cid:112)
plot the normalized one-body density matrix ρ(1)(x,x(cid:48))/ ρ(x)ρ(x(cid:48)) as function of x, scaled with the axial harmonic
oscillator length l, and at fixed x(cid:48) =0, for different values of dimensionless coupling strength g . The gray circles are
0
taken from Ref. [24] for comparison purposes. Specifically, the gray circles represent the results of the Monte Carlo
calculation for N =2 bosons.
Clearly, the qualitative behavior of the MCTDH results is well in accord with the Monte Carlo calculations – the
dip in the correlation at approximately x = l is consistently visible for both approaches, although MCDTH has a
tendency to overestimate the depth of the dip. Note that a dip in the correlation function ρ(1)(x,x(cid:48)) corresponds to
a peak in the phase fluctuations according to Eq. (4) in the main text. The dip is clearly due to the geometry of
the trap and, consequently, due to the shape of the occupied orbitals and, exists even for small interactions. This
comparisonconfirmsthattheintrinsicself-consistencyoftheMCTDHmethodiscrucialinordertocorrectlydescribe
the correlation phenomena in trapped many-body systems.
The solid lines in Fig. S4 show the comparison of MCTDH results with the analytic solution for N =2 bosons in a
harmonic trap [25]. We note a sizable quantitative difference to the analytic solutions even for interaction strengths
that are far below the hard-core limit of g → ∞. However, for small couplings the agreement between the exact
0
results and MCTDH is very satisfactory, see Fig. S5. Thus, the dip in the correlation function indeed appears for any
interaction strength and is correctly reproduced by the MCTDH method. For these calculations, we increased the
number of oribitals to M = 10, to (partially) compensate for the known deficiencies of the MCTDH method when
dealing with small particle numbers [58].
3
FIG.S1. Mean-squarequantumphasefluctuations(cid:104)δˆφ2 (cid:105)forN =30andvaryinginteractionstrength: a)g =0.1,b)g =0.5,
xx(cid:48) 0 0
andc)g =1.0. Themaximaalongtheoff-diagonals,x(cid:48) =−x,correspondtothefactthatthegasbecomesphase-uncorrelated
0
in distant regions of the cloud.
4
FIG. S2. Maximum value of mean-square phase fluctuations of (5) Φ2 :=max[(cid:104)δˆφ2 (cid:105)] (peak height in Fig.S1), and fragmen-
xx(cid:48)
tationdegree,1−n ,wheren istheoccupationnumberoftheenergeticallylowestorbital,asfunctionofinteractionstrength
1 1
g , for N =30. The inset shows the ratio of the maximum fluctuation Φ2 and fragmentation degree 1−n .
0 1
FIG. S3. The comparison of phase fluctuations (cid:104)δˆφ2 (cid:105), calculated in the hydrodynamic mean-field approach [51] (black solid
x0
line)andself-consistentlywithMCDTHusing(5)(bluedashedline)forN =30anddifferentinteractionstrengths: a)g =0.1,
0
b) g =0.5, c) g =0.75 and d) g =1.0. The axial coordinate x is scaled by the size of the gas cloud R.
0 0 0