Absence of quasiclassical coherence in mean-field dynamics of bosons in a kinetically frustrated regime

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by  Akos Rapp

Absence of quasiclassical coherence in mean-field dynamics of bosons in a kinetically
frustrated regime
Akos Rapp
Institut fu¨r Theoretische Physik, Leibniz Universita¨t, 30167 Hannover, Germany
(Dated: November 11, 2014)
We study numerically the dynamics of bosons on a triangular lattice after quenching both the
on-siteinteractionsandtheexternaltrappingpotentialtonegativevalues. Inasimilarsituationon
thesquarelattice,thedynamicscanbeunderstoodintermsofaneffectivelyreversedHamiltonian.
On the triangular lattice, however, the kinetic part of the reversed Hamiltonian is frustrated and
4
whethercoherencecandevelopisanopenquestion. Thestrengthofthefrustrationcanbechanged
1
bytuningtheratioofthehoppingratesalongdifferentdirections. Wecalculatetime-of-flightimages
0
atdifferenttimesafterthequenchfordifferentvaluesofthehoppinganisotropy. Weobservepeaks
2
at the maxima of the noninteracting dispersion relation both in the isotropic case and also in the

v rhombic limit of high hopping anisotropy showing quasiclassical coherence. For an intermediate
o value, however, nocoherencedevelopsuptothelongestsimulationtimes. Theseresultsimplythat
N experiments along similar lines could study unconventional superfluidity of bosons and aspects of
the conjectured spin-liquid behavior in the hard-core limit.
0
1
PACSnumbers: 03.75.Nt,67.85.-d,

]
s
a I. INTRODUCTION arerelativelyhigh[8],andspontaneousantiferromagnetic
g long-rangeorderhasnotyetbeenestablishedevenonbi-
- partite lattices. The reason behind this is well known:
t A broadly observed fact is that most physical systems
n due to Pauli blocking, fermionic clouds cannot be cooled
undergo transitions to phases of matter which display
a
efficiently while the typical temperature to be reached is
u some kind of order as the system is cooled down. In
relatively low, of the order of the Heisenberg exchange
q geometricallyfrustratedsystems, however, suchordering
. may not be possible down to the lowest temperatures energy for the Mott insulator. We will concentrate on
t
a since the spatial arrangement is incompatible with cer- bosonic atoms in optical lattices in the following.
m
tain order types. For classical Ising spins on the trian- Whilefrustrationoriginatesfromtheinteractionterms
- gular lattice with antiferromagnetic coupling, this prob- of the Hamiltonian in triangular magnets or fermionic
d
lem was first discussed by Wannier [1], who found finite Mott insulators, in spinless bosonic systems it is related
n
o ground-stateentropy, aconsequenceofahighlydegener- to the kinetic energy. Nevertheless, the realization of
c ate ground state. In three dimensions, frustration plays kinetic frustration is not straightforward in optical lat-
[ an important role in spin ice materials, which display tices. Onereasonbehindthisisthatthenearest-neighbor

magnetic monopoles [2], the microscopic version of the hopping amplitude J between lattice sites in the low-
2
v hypothetical cosmic counterpart proposed in the famous est Bloch band has a definite sign, usually defined with
6 paper by Dirac [3]. Strong quantum fluctuations present the convention J > 0. This is implied physically by the
6 for lower spin lengths can give rise to elusive spin-liquid fact that the lowest energy usually implies zero momen-
9 phases [4]. tum and mathematically by the solution of the Mathieu
3 equation describing the one-body problem in the one-
Theidentificationofspinliquidsinsolidstatesystems
.
1 is very challenging, on one hand, due to the featureless dimensionalstanding-waveopticallatticepotential. Sim-
0 nature of the spin-liquid phase, but also because of the ple square or cubic lattices built from this potential nat-
4 urally share this property; moreover, the non-interacting
interplay of additional degrees of freedom, phonons, dis-
1 diagonal (next-nearest-neighbor) hopping is exactly zero
order,etc. Incontrast,ultracoldatomsinopticallattices
:
v (see the review in Ref. [5]) present exceptionally clean due to the separability of the one-body problem. There-
i fore,opticallatticesetupsneedtoovercometwoproblems
X systems where microscopic parameters can be tuned ex-
to realize frustration: the lack of triangular graphs and
perimentally in a broad range with great control. In ad-
r the sign of the hopping. The former can be solved by
a dition, therelativelylargetypicalspatialandtimescales
using a different lattice geometry.
allow for tracking physical processes more easily. Ultra-
coldatomsinopticallatticesarethereforeidealquantum While certain hopping amplitudes in higher Bloch
simulators for many-body systems. bands have opposite signs, such systems are not espe-
Fermionic atoms in a lattice with two spin compo- cially suitable for quantum simulations. One issue is the
nents at half-filling with strong on-site repulsion are de- high instability with respect to decay to other bands,
scribed by the antiferromagnetic Heisenberg model at which is not easily circumvented [6, 7].
low energies. This implies that on a triangular lattice, A very successful idea to effectively change the sign
suchfermionscouldnaturallyexhibitfrustratedquantum of the hopping amplitude J → J is based on a pe-
eff
magnetism. However, the entropies of fermionic clouds riodic shaking of the optical lattice [9]. This idea lead
2
to a proposal for bosons on the triangular lattice with
an elliptical lattice shaking [10], which allows for a con-
tinuous tuning of the effective hopping anisotropically.
Bosons can be mapped to XY spin models in two limits
of the interaction strength. For weak interactions and
sufficiently high boson filling, each site can be described
as an individual “superfluid” droplet with a well-defined
phase and the bosonic Hamiltonian can be mapped to
a classical XY model [11]. In the other limit, at half-
filling and infinitely strong repulsion, the Hamiltonian
can be mapped to a quantum XY model [10]. Both of
FIG.1: (Coloronline.)Triangularlatticeinrealspace(left)
these models on a triangular lattice are frustrated with
andthecorrespondingreciprocallatticevectorswiththeBril-
antiferromagnetic couplings, given by a negative effec-
louinzone(right). Theexperimentalsetupproposedherehas
tive hopping, J < 0. In the case of isotropic nearest-
eff a fixed geometry, and the anisotropy of the hopping rates is
neighbor spin couplings, it is believed that both ground
realizedbydifferentintensitiesofthelaserbeamscreatingthe
statesexhibit120◦spirallong-rangemagneticorder[U(1) optical lattice potential. Note that in comparison to optical
rotational symmetry breaking], with a non-zero chirality lattices created using standing waves, the lattice spacing is
(Z symmetry breaking) [12, 13]. As the anisotropy larger, |a |=2λ /3.
2 i L
of the couplings is increased, the chirality decreases and
vanishes. In the classical model this happens at the so-
called rhombic transition point, beyond which only the Here J >0 describes anisotropic nearest-neighbor hop-
ij
U(1)spinsymmetryisbroken. Mostinterestingly, itwas ping between sites i and j on an equiangular triangular
proposed that in the hard-core limit, instead of a single lattice (cf. Fig. 1), U is the on-site interaction strength,
phase transition, a gapped spin-liquid phase emerges be- V gives the strength of the external harmonic potential,
0
tween two quasi-classically ordered phases [13, 14]. The andthecentralchemicalpotentialµ setsthetotalnum-
0
conjectured phase diagram for the bosons is displayed ber of particles.
in Ref. [10]. While the lattice shaking technique suc-
Using numerically robust methods [exact diagonaliza-
ceeded experimentally in simulating frustrated classical
tion(ED),projectedentangledpairstates(PEPS)[24]or
magnetism [15, 16], no signature of the quantum mag-
quantum Monte Carlo (QMC) [25]], it is very challeng-
netism has been reported so far. To study the quantum
ing to describe the Bose-Hubbard model in Eq. (1) with
aspects of frustration with ultracold bosons, a new ap-
kinetic frustration due to either the size of the Hilbert
proach is required.
space (ED) or the frustration (PEPS, QMC). We use a
An alternative route to reversed hopping in the low- low-entanglement (mean-field) approach. We do not ex-
est Bloch band is employing negative absolute temper- pect that it can describe a spin-liquid phase. However,
atures, T < 0. Negative absolute temperatures can be the estimates presented here can be used as a starting
reachedinclosedsystemswithHamiltonianswithanup- point to initiate experimental quantum simulations.
per bound [17]. With ultracold atoms in optical lattices,
This paper is organized as follows. In Sec. II we dis-
such a Hamiltonian H can be engineered basically by
cuss a specific experimental setup to realize the trian-
switching the sign of the external harmonic trapping po-
gular lattice and outline the band-structure calculation.
tentialV >0→V <0[18–23]. Energyconservationre-
0 0 SectionIIIisdevotedtothediscussionoftheexperimen-
strictsthedynamicsandtheatomiccloudcannotexplode
talparametersbasedonthesetupandthecorresponding
aslongasthekineticenergyisbounded. Thisimportant
microscopic parameters. We also discuss the procedure
condition is provided by a sufficiently deep optical lat-
ofreversingtheinteractionU andharmonicpotentialV .
0
tice. In equilibrium at T < 0, the partition function of
In Sec. IV we outline the numerical simulation method.
the system is equivalent to the partition function of a
Theresultsofthesimulationsareshownandtheirimpli-
system at an effective temperature |T| governed by the
cations are discussed in Sec. V.
reversed Hamiltonian −H. In Ref. [20] it was discussed
that this mapping can be used to simulate Hamiltonians
thathavecouplingswithsignsthatarehardtoreachex-
perimentally. In this work we apply this idea to bosons II. TRIANGULAR OPTICAL LATTICE
onthetriangularlattice, which, onlyatnegativeT, have
“frustrated” kinetic energy.
Following Refs. [15, 26] we consider three phase sta-
For concreteness, we consider bosons described by the bilized running waves at blue detuning with some wave-
Bose-Hubbard model lengthλ in120◦ angles. Thecorrespondingelectricfield
L
is given by
(cid:88) U (cid:88) (cid:88)
H =− J b†b + nˆ (nˆ −1)+ (V r2−µ )nˆ .
ij i j 2 j j 0 j 0 j (cid:88)
<ij> j j E(r,t)= Ei(cid:15)ˆicos(kir−ωLt), (2)
(1)
i=1,2,3
3
where E are the strengths of the electric field in each for the parameter values of V and α relevant to this
i L
plane wave, ω is the laser frequency, the wave vectors work. Due to the partial potential anisotropy, there is
L
are partial hopping anisotropy J =J >J for α>0.
1 3 2
(cid:32) √ (cid:33) Theminimumofthenoninteractingdispersionrelation
1 3 (cid:15) is always at k = 0 momentum, while the maxima lie
k =k (1,0),k =k − ,± , (3) k
1 L 2,3 L 2 2 at
(cid:32) √3 (cid:33) k∗ 3 (cid:18) J (cid:19)
w(cid:15)ˆiitlihetihnetwheavpelannuemobferprkoLpa=ga2tπio/nλ,L,andthepolarizations kA,B = ±k∗, 2 kL ;kL = 2πarccos 2J12 . (8)
(cid:15)ˆ =k−1(zˆ×k ). (4) These points coincide with the corners of the Brillouin
i L i
zone in the isotropic case, J = J = J (α = 0). The
1 2 3
The time averaged laser intensity can be reparametrized momenta are not equivalent in terms of modulo recipro-
conveniently as the optical lattice potential cal lattice vectors, kA (cid:29) kB ∼= −kA, leading to the pos-
sibility of the Z (chirality) symmetry breaking [15, 16].
2
(cid:88)
V(r)=Voffset+VL (1+αi)sin2(bir), (5) The vectors kA,B are incommensurate with the lattice
for a general α. The value of k∗ decreases for α>0 and
i=1,2,3
vanishes at the rhombic transition point α = α when
c
where b1 = 12(k2 − k3), etc., and αi characterize the J1 = 2J2 , thus k∗ serves also as a measure for frustra-
anisotropyoftheopticallatticepotential. Forsimplicity, tion. For stronger anisotropy, the lattice links with the
we will consider the case of partial anisotropy α = −α stronger hopping define a rhombic lattice. Similar to the
1
and α =α/2 corresponding to E ≥E =E [27]. square lattice, the bare rhombic lattice is bipartite and
2,3 1 2 3
therefore there is no frustration.
The optical lattice setup discussed above has a fixed
lattice geometry. This allows for a direct comparison The joint set of elements of all eigenvectors vk˜ of
of the time-of-flight images with different values of α. Eq.(6)ofthelowestbanddefinetheFourier components
Rhombic optical lattices can also be realized with two of the Wannier function up to a phase factor,
standing waves by varying the angle between the beams.
However, it is harder to reach the isotropic case (which w(k˜)∼vk˜ →wj(r)=(cid:88)eik˜(r−rj)w(k˜), (9)
follows trivially from symmetry with the three-beam k˜
setup) and the comparison of TOF images is not so
which is used on one hand to calculate the envelope for
straightforward as the reciprocal lattice vectors change.
thetime-of-flight(TOF)imagesandtodefinethedimen-
An additional experimental advantage of the setup pro-
sionless interaction overlap integral
posedhereoveratwo-beamsetupisthatthelaserinten-
sitiescanbechangedmoreeasilythantheanglebetween (cid:90)
the beams. u=k−2 d2r|w (r)|4 . (10)
L 0
TheperiodicpotentialinEq.(5)defines(possiblyover-
lapping) Bloch bands for a triangular lattice. The lat-
√
tice vectors a = (−1/3,1/ 3)λ , a = (2/3,0)λ , and
1 L 2 L III. EXPERIMENTAL AND MODEL
a = a +a are shown in Fig. 1. The band-structure
3 1 2 PARAMETERS
parameters for Eq. (1) are calculated using the solution
of the two-dimensional one-body problem in the opti-
We consider blue detuned laser beams at wavelength
cal lattice potential (5): for each fixed momentum k of
λ = 736.65 nm for 39K atoms, which was also used in
theBrillouinzone(BZ),wecalculatetheeigenvaluesand L
the experiment described in Ref. [21]. This implies that
eigenvectors of the block matrix (note rescaling in terms
the recoil energy is
of the recoil energy E =(cid:126)2k2/2M)
R L
(cid:126)2k2
h =δ k˜2 − VL (cid:88)3 (1+α )[δ +δ ] ER = 2ML ≈kB450nK≈2π(cid:126)9.4kHz. (11)
k˜,k˜(cid:48) k˜(cid:48),k˜ kL2 4ER j=1 j k˜(cid:48),k˜+Gj k˜(cid:48),k˜−Gj Hopping amplitudes and other band parameters in the
(6)
isotropiccaseareshowninTableIandfortheanisotropic
where the extended momentum k˜ = k+g1G1 +g2G2 case in Tables II and III.
can be indexed by the integers g1,2 ∈ [−gc,gc −1] and The external harmonic potential has a bare strength
Gj =2bj. ThelowesteigenvaluesofEq.(6)asafunction V¯,
ofkdefinethelowestBlochband,whichisapproximately
a nearest-neighbor dispersion relation, V /E ≡±V¯ν2 ≈±2.78×10−8 ν2, (12)
0 R
(cid:88)
(cid:15) =−2 J cos(k·a ), (7) where the value of the trapping frequency ν is in units
k j j
of Hz. The upper sign corresponds to the usual trapping
j=1,2,3
4
Bloch gap between the lowest band and the next band
TABLE I: Band parameters in the isotropic case (α = 0)
may not be large enough (cf. Tables II and III). This
for different values of the lattice depth V . The bandwidth
L is unfavorable since the protocol strongly relies on the
of the lowest band (cid:15) is W, and ∆ gives the gap between
k bounded kinetic energy, which is violated if the Landau-
the lowest Bloch band and the next one (or the bottom of
Zener tunneling rate to other Bloch bands is not negli-
the continuum). The next-nearest-neighbor hopping is J .
nnn
Note that the hopping rates vanish faster with increasing V gible. In weaker lattices the overlap integral for the in-
L
thaninastandingwaveopticallattice;cf. TableIinRef.[5]. teraction is also reduced and therefore larger scattering
The main reason is the larger lattice spacing. lengths are needed to compensate. This usually implies
V /E W/E ∆/E J /E ∆/W J /J gettingclosertoaFeshbachresonance[21], wheremany-
L R R R 1 R nnn 1
1. 0.529 0.572 0.0627024 1.08 -0.08297 body losses are enhanced. Last, the value of the lattice
potential anisotropy is bounded, α ≤ 1, since the wave
2. 0.257 1.412 0.0295701 5.59 -0.03648
intensities cannot be negative; cf. Eq. (5). Additionally,
3. 0.116 2.233 0.0133173 19.28 -0.01476
forV =3E ,therhombictransitionhappensatalower
4. 0.0548 2.975 0.00625151 54.26 -0.00627 L R
value of α than for V = 2E (cf. Tables II and III),
L R
5.5 0.0195 3.903 0.00221681 199.9 -0.00197
which might be favored experimentally.
Taking these considerations into account, we will
mainlyfocusontheparametersV =3E ,ascattering
potentials, the lower sign is active for the antitrapping L,f R
length a = −50a , and anti-trapping ν = 30 Hz
situation. s,f Bohr f
(V /E ≈−0.000025). For these parameters the system
The on-site interaction is given by [5] 0 R
iswellapproximatedbytheone-bandHubbardHamilto-
U/E =8π (a k )uw , (13) nian in Eq. (1).
R s L z
where, for simplicity, we input the value of the scat-
tering length a directly [28]. The interaction overlap
s IV. TIME-DEPENDENT GUTZWILLER
u = u(V ,α) is calculated from the Wannier function
L ANSATZ
in Eq. (10). We consider a layered system similar to
Ref. [22] with a vertical optical lattice depth V =
L,ver We apply the time-dependent Gutzwiller ansatz (GA)
25E [29], which corresponds to a vertical hopping J ≈
R z [22, 31–36] to study the dynamics of the cloud after the
0.00104E and Wannier overlap w ≈0.848035.
R z quench defined in the previous section. In this approx-
To reach negative absolute temperatures in Ref. [21],
imation, the probability amplitudes of finding precisely
theexperimentalparameters(horizontalandverticalop-
m bosons at site j and time t are given by the following
ticallatticeintensities,magneticfield,etc.) werechanged
set of differential equations:
via an involved protocol. However, most of these steps
emergeasatechnicalnecessity. Furthermore,thecloudis m−1
i∂ f (j,t) = [U(t) +V (t)r2−µ ]mf (j,t)
initially trapped in a very deep optical lattice where the t m 2 0 j 0 m
√
atomic density distribution is essentially frozen. From −Φ∗(j,t) m+1f (j,t)
thisperspective, moststepsoftheexperimentalprotocol √ m+1
−Φ(j,t) mf (j,t), (14)
are almost instantaneous. m−1
To simplify the numerical simulations and to improve (cid:80)
where we introduced Φ(j,t)= J (t)(cid:104)b (cid:105), the index
the transparency of the text, we consider an instanta- δ δ j+δ
δ running over the six nearest-neighbor sites, and
neous quench in the system: for time t < 0, we take
anisotropic(α =0)triangularlatticewithdepthV = (cid:88)√
i L,i (cid:104)b (cid:105) = m+1f∗(j,t)f (j,t). (15)
5.5E ,ascatteringlengtha =+400a (U/J ≈582) j m m+1
R s,i Bohr 1
m
and ν =60 Hz horizontal trapping frequency (V /E ≈
i 0 R
0.0001) for a strongly compressed Mott insulator initial In the GA, quantum correlations beyond the mean-
groundstateinequilibrium. AccordingtoRef.[19], such field Φ between the lattice sites are neglected. In higher
an initial state is necessary to optimize the final conden- dimensions, or more precisely, for higher coordination
sate fraction. At t = 0, we instantaneously change to numbers z, the approximation is expected to improve.
a shallower optical lattice V < V , a negative scat- For example, for the cubic lattice with z = 6, the GA
L,f L,i
tering length a < 0 and an anti-trapping harmonic variational wave function gives a good estimate for the
s,f
potential V <0 [30]. quantum phase transition between the Mott insulator
0,f
The optimal regime of the final lattice depth for the and the superfluid phase [5]. In Ref. [22] we studied
numerical and the experimental setups depends on vari- numerically a setup corresponding to the experiments
ous aspects. in Ref. [21] on the square lattice, z = 4. Based on
Fast enough dynamics certainly requires weak enough these findings, deep in the rhombic regime J (cid:29)J , the
1 2
V . Avoiding technical heating from the blue detuned time-dependent GA should work reasonably well. The
L,f
lattice lasers also favors weaker lattice potentials. isotropic triangular lattice with z = 6 is closer to the
On the other hand, there are more arguments in favor mean-field limit z → ∞ than the square or rhombic lat-
of a relatively deep lattice. If the lattice is too weak, the tice; however, frustration is expected to enhance quan-
5
TABLE II: Band parameters for different values of the lattice potential anisotropy α for V =2E .
L R
α W/E ∆/E J /E =J /E J /E ∆/W J /J k∗/k u(∼U)
R R 1 R 3 R 2 R 1 2 L
0. 0.257 1.412 0.0295701 0.0295701 5.586 1. 0.5 0.167994
0.25 0.262 1.277 0.0320253 0.0254546 4.871 1.25813 0.425 0.167238
0.5 0.275 1.145 0.0347427 0.022295 4.166 1.55832 0.323 0.165063
0.75 0.296 1.01 0.0377333 0.0199259 3.408 1.89368 0.156 0.161633
1. 0.325 0.873 0.0409952 0.0182272 2.689 2.24912 0. 0.157101
TABLEIII: BandparametersfordifferentvaluesofthelatticepotentialanisotropyαforV =3E . Thefirstcolumndefines
L R
the identifiers for the different “protocols”.
α W/E ∆/E J /E =J /E J /E ∆/W J /J k∗/k u(∼U)
R R 1 R 3 R 2 R 1 2 L
a) 0. 0.116 2.233 0.0133173 0.0133173 19.28 1. 0.5 0.226388
b) 0.1 0.118 2.156 0.0139504 0.0121748 18.28 1.14585 0.459 0.226222
c) 0.2 0.121 2.079 0.014631 0.0111764 17.24 1.3091 0.409 0.225722
d) 0.3 0.124 2.003 0.015362 0.0103051 16.16 1.49072 0.348 0.224902
e) 0.4 0.128 1.926 0.0161468 0.0095464 15.08 1.6914 0.269 0.223772
f) 0.5 0.134 1.847 0.0169886 0.008888 13.74 1.91141 0.143 0.222343
g) 0.6 0.142 1.768 0.0178908 0.00831952 12.48 2.15046 0. 0.220625
h) 0.7 0.150 1.689 0.0188569 0.00783222 11.30 2.40761 0. 0.218627
tumfluctuationswhicharecapturedpoorlyinthemean- density is given by
field approximation. Nevertheless, we will confirm later
(cid:88)
that the dynamics in the GA gives the expected behav- D = d (t),with
tot j
ior in the isotropic limit. Similar to Ref. [22], we focus j
only on a single layer and entirely neglect the hopping (cid:88) (cid:18)m(cid:19)
between layers. dj(t) = 2 |fm(j,t)|2.
Thelatticeconsistsof192×192latticesites. Theinitial m
state is a strongly compressed Mott insulator, which is
The time evolution of the condensate occupation
determined as the ground state of Eq. (1) in the equilib-
(cid:88)
riumvariationalGAfortheinitialparameters. Thetotal N (t) = |(cid:104)b (cid:105)|2, (17)
0 j
atom number is N ≈ 2260. To numerically integrate
tot j
Eq. (14) we use the fourth-order Runge-Kutta method
and the (conserved) total atom number and nearest-neighbor coherences
Ntot = (cid:88)nj(t),with (16) C(t)=(cid:88)(cid:104)b†jbj+δ(cid:105)G=A(cid:88)(cid:104)b†j(cid:105)(cid:104)bj+δ(cid:105) (18)
j j,δ j,δ
(cid:88)
n (t) = m|f (j,t)|2, follow a qualitatively similar behavior as on the square
j m
m lattice [22], C becoming negative; see Fig. 2. Here and
below GA means that the expectation values are evalu-
serves as a primary measure of numerical accuracy.
ated using the bosonic Gutzwiller wave function.
We note that after the quench the total energy E =
tot Longer-range coherences (l (cid:54)= 0) are calculated in the
(cid:104){f (j,t)}|H |{f (j,t)}(cid:105)+µ N isalsoconserved. The
m f m 0 tot a (x) direction,
2
N and E determine a unique grand-canonical den-
tot tot
sity matrix for the Hamiltonian Eq. (1), and in principle C(l,t) = (cid:88)(cid:104)b† b (cid:105)G=A(cid:88)(cid:104)b† (cid:105)(cid:104)b (cid:105), (19)
thelong-timeaveragesof(macroscopic)quantitiesinthe rj rj+la2 rj rj+la2
j j
GA should approximate the corresponding expectation
values. However, at the moment it is unclear how the C˜(l,t) = (cid:88) (cid:104)b†rjbrj+la2(cid:105)
latter could be computed. (cid:112)n(r )n(r +la )
j j j 2
G=A (cid:88) (cid:104)b†rj(cid:105) (cid:104)brj+la2(cid:105) , (20)
V. NUMERICAL RESULTS (cid:112)n(r )(cid:112)n(r +la )
j j j 2
We calculate various (macroscopic) quantities as a the latter being normalized so that it is less sensitive to
function of time, as defined in Ref. [22]. The total pair spatial inhomogeneities.
6
We calculate two-dimensional TOF images using the
formula (following Ref. [5])
0.5 0.5
I (k˜,t)=|w(k˜)|2 G(k˜,t) (21)
TOF
wheretheenvelope|w(k˜)|2istheFouriertransformofthe Ntot 0.0 0.0 Ntot
Wannierfunction(c.f. Eq.(9))andtheFouriertransform C/ NC/0N/Nto,Vt,VL==2E2ER C/
of the single-particle density matrix at a time t is given N,tot 0.5 NC/0N/Ntotto,Vt,LVL==3E3ERR 0.5N,tot
in the GA by / tot L R /
N0 N0
G(k,t) G=A |(cid:104)b (t)(cid:105)|2+L−2(N −N (t)) 1.0 1.0
k tot 0
(cid:88)
(cid:104)b (cid:105) = L−1 eikrj(cid:104)b (cid:105),L=192. (22)
k j
1.5 1.5
j
(cid:80) 0 20 40 60 80 0 2 4 6 8 10 12
This normalization implies k∈BZG(k,t) = Ntot, and t (ms) J1t/
makes direct comparisons of the absolute TOF intensi-
ties possible. While the different expectation values in FIG. 2: (Color online.)Left: condensate occupation N (t)
0
Eqns. (17)-(20) provide valuable insight regarding cer- andnearest-neighborcoherencesC(t)asafunctionoftimetin
tain quantities and the structure of correlations at var- the isotropic case α=0 for two different final lattice depths.
ious times on the mean-field level, the TOF intensities Right: the same quantities with the time axis rescaled by
calculated using Eq. (21) can be compared directly to J1/(cid:126). While for VL = 3ER the hopping amplitude is lower
experiments. (cf. Table I), and therefore the dynamics is slower in real
time, the initial evolution in the natural time unit ∼ J t is
1
thesameasforV =2E . Thedashedlinesontheleftpanel
L R
correspond to J t/(cid:126)=12.
1
A. Numerical results for the isotropic lattice
We compare macroscopic quantities for two differ- The condensate occupations N , the nearest-neighbor
0
ent sets of the final parameters, V = 2E ,a = coherences C and the total pair density D is shown in
L,f R s,f tot
−100a and V = 3E ,a = −50a in Fig. Fig. 5 as a function of time t for the different values of
Bohr L,f R s,f Bohr
2. The interaction strengths are U/J ≈ −5.5 and α.
1
U/J ≈ −8.2, respectively. The total energies after We show the relevant part of the TOF images for the
1
the quench are E (V = 2E )/E ≈ −46.31 and different values of α in Fig. 6. For weak anisotropies
tot L,f R R
E (V =3E )/E ≈−38.15. TOFimagesatt=200 α≤0.4,weobservecoherencemanifestinginpeaksatthe
tot L,f R R
ms are shown for comparisons in Fig. 3. The main con- correspondingquasiclassicalmaximak ork ofthefree
A B
tribution of the TOF intensities is concentrated around dispersion. TheasymmetryintheTOFintensitiesatthe
the corners of the Brillouin zone. These peaks persist two momenta k and k is related to the chirality when
A B
over time in contrast to the noisy features representing k∗ (cid:54)= 0. Beyond the rhombic transition point, 2J < J
2 1
spatialvariations,whichalsochangeasafunctionoftime correspondingtoα>0.55,thereisalsoap√parentlycoher-
andthuswouldcanceloutoveraveraging. Thenoisyfea- ence at the “N´eel” momentum Q = (0, 3/2)k . How-
L
tures in the TOF images are likely the results of the fact ever, for the intermediate value α=0.5 [ case (f)], there
that the system is in a non-equilibrium state after the are no peaks at the quasiclassical maxima; furthermore,
quench. How exactly these noisy features develop is an no dominant coherent peak is found up to t = 600 ms
open question. of the numerical simulation. Note that the macroscopic
Since in the case of the deeper lattice VL,f = 3ER quantities N0,C, or Dtot show a weak monotonous be-
the system is closer to the hard-core limit and yet the havior as a function of α, cf. Fig. 5.
corresponding TOF images show more enhanced peaks TocomplementtheTOFimages,wecalculatedlonger-
with a pronounced chirality, we will consider this lattice ranged coherences, shown in Fig. 7. While the density
depth in the following. normalizedcoherencesC˜aremorenoisy,theyfollowqual-
itatively the course of the corresponding unnormalized
coherences, which in turn are well approximated in most
cases by
B. Numerical results for the anisotropic lattice
C(x)≈c(x)eiκx ,κ=k∗+iξ−1 , (23)
In this subsection we display macroscopic quantities
and TOF images for different values of the potential and c(x) varies weakly. As a sidenote, ξ determines the
anisotropy α for the final lattice depth V = 3E . approximate size of the coherent “domains,” each con-
L,f R
The microscopic parameters used for the simulations are tributing with a given chirality ±. The asymmerty be-
listed in Table III and shown in Fig. 4. tween k and k becomes suppressed as ξ decreases.
A B
7
FIG. 3: (Color online.)TOF images at t = 200 ms for a weaker (V /E = 2, U/J ≈ −5.5) and a stronger final lattice
L R 1
(V /E =3,U/J ≈−8.2)intheisotropiccase. Notethatthelatterimageshowsastrongchirality. Thearrowsrepresentthe
L R 1
reciprocal lattice vectors G and G , the hexagon represents the border of the first BZ. The rectangular area is displayed in
1 2
Fig. 6.
the x-direction, ∼ 1/k∗, becomes longer than the coher-
00..45○a) ○b) ○c) d) //UJUJ,121248◆● ◆● ◆● ◆● ◆● ◆● ◆● ◆● evrhnaorciemoublesincpgttarhraanξms.ietStioeinnrs,ceiatnkids∗i=cmlopkuo∗dr(tαsa)inz∼teha√orwoαucξn−ddeααpe→≈ndα0scoa.ntUtthhnee-
kL0.3 3 ○ e) 0 0.1 0.3 0.5 0.7 fuosritnugntahteelyp,retsheinstqaupepstriooanchc.annot be addressed properly
*/k 2 ◇ ◇ ◇ ○ α
0.2/JJ121◇ ◇ ◇ ◇ ◇ f) (f)Acnouolbdviboeusanreaasnoonmfaolrouthsleylalocnkgof“creolhaexraetniocen”intitmheecdausee
○
0.1 tothevariousapproximations,andtheexperimentalsys-
0
0.1 0.3 0.5 0.7 g) h) tem [or even the “true” dynamics under Eq. (1)] could
α
0.0 ○ ○ display coherence in shorter times. However, this claim
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
isonlypartiallyjustified. Itistruethatwithinthetime-
α
dependent GA defined by Eq. (14) the “relaxation” is
slow as it is driven by dephasing, i.e., by the mismatch
FIG.4: (Coloronline.)Valueofk∗ asdefinedbyEq.(8)asa
betweenlocalmean-fieldHamiltonians. Thetruedynam-
function of the optical potential anisotropy α for V =3E .
L R ics governed by Eq. (1) should lead to a faster equilibra-
Insets: microscopic parameters for the different values of α
tion in general. However, the main candidate state for
listed in Table III. The rhombic transition point corresponds
to α=α ≈0.55. stronger on-site interactions has simply N´eel-type coher-
c
ence [10, 37], which is found to develop for J > 2J .
1 2
Therefore,itisquitepuzzlingwhyGAfailstofindeither
This can be observed in the case e) or at V /E = 2 in of the mean-field type solutions (spiral or N´eel) in the
L R
theisotropiccase. Thenotableexceptiontothebehavior case (f).
given by Eq. (23) is the case (f), without any apparent
Regardingtimescales,oneshouldalsonotforgetabout
oscillating component and a coherence length ξ of the
lossesanddecoherence intheexperimentalsystem,which
order of one lattice spacing. From Eq. (23) it is obvi-
includesallprocessesthatareleftoutfromEq.(1): tech-
ous that coherence cannot be defined if the condition for
nical heating from the lasers, multiband contributions,
coherent behavior
many-body losses driven by three-body recombination,
k∗ξ (cid:29)1 (24) etc. These determine the experimental lifetime and pro-
vide an upper bound to the coherence lifetime. For the
breaksdown,i.e.,whenthe“pitchlength”ofthespiralin optimal square-lattice setup it was found to be on the
8
t(ms) ative absolute temperature. We also found the expected
0 50 100 150 200 N´eel-typeantiferromagneticorderingintherhombiclimit
0.8
[cases (g) and (h)]. Based on the qualitative agreement
/NN0tot 0000....0246★◆●★◆◇○●▽△★◻◆◇○●▽△★◻◆◇○●▽△★◻◆◇○●▽△★◻◆◇○●▽△★◻◆◇○●▽△★◻◆◇○●▽△★◻◆◇○●▽△★◻◆◇○●▽△★◻◆◇○●▽△★◻◆◇○●▽△★◻◆◇○●▽△★◻◆◇○●▽△★◻◆◇○●▽△★◻◆◇○●▽△★◻◆◇○●▽△★◻◆◇○●▽△★◻◆◇○●▽△★◻◆◇○●▽△★◻◆◇○●▽△★◻◆◇○●▽△★◻◆◇○●▽△★◻◆◇○●▽△★◻◆◇○★●▽△◆◻●◇○▽△◇○◻▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○△▽◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○△▽◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△/◻DN★◆●◇○▽△◻★◆●tottot◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻000000★◆●◇○▽△◻★◆●◇○▽△◻......★◆●◇○▽△◻001122★◆●◇○▽△◻★◆●◇○▽△◻050505★◆●◇○△▽◻★◆●◇○▽△★0◆★◻●◆★●◇▽△◆★●◇○◻▽△◆★●◇○◻▽△◆★●◇○◻▽△★◆★●◇○◻▽△◆◆★●●◇○◻▽△◆★●◇○◻▽△◇○◆▽△★●◇○◻▽△◆★●◻◇○◻▽△◆★●◇○◻▽△◆★●◇○◻▽△◆★●◇○◻▽△◆★●◇○◻▽△★◆★●◇○◻▽△◆◆★●●◇○◻▽△◆★●◇○◻▽△◇○◆▽△★●◇○◻▽△◆★●◻◇○◻▽△◆★●◇○◻△▽◆★●◇○◻△▽◆★●◇○◻△▽◆★●◇○◻△▽★◆★●◇○◻△▽★◆◆●●◇○◻△▽◆●◇○◻△▽◇○▽△◇△▽○◻◻○◻★◆●◇△▽★○◻◆●★◆◇○●▽△◇◻△▽○◻★◆●★◇△▽◆○●◻◇★○▽△◆●◻◇△▽○◻★◆★●◆◇●△▽○◻◇○▽△★◻◆●◇△▽○◻★★◆●◆●◇△▽◇○▽△○◻◻★◆●◇△▽○◻★★◆●◆●◇○◇▽△△▽○◻◻★◆●◇△▽★○◻◆●★◆◇●○▽△◇△▽◻○◻★◆●★◇△▽◆○◻●★◇○▽△◆●◻◇△▽○◻★◆●★◇◆△▽●○◻◇○▽△★◻◆●◇△▽○◻★★◆●◆●◇△▽◇○○◻▽△◻★◆●◇△▽○◻★★◆5●◆●◇◇○△▽▽△○◻◻★◆●◇△▽★○◻◆●★◆●◇○▽△◇△▽◻○◻★◆●◇★△▽○◆◻●0★◇○▽△◆●◻◇△▽○◻★◆●★◇◆△▽●○◻◇○▽△★◆◻●◇△▽○◻★★◆●◆●◇△▽◇○○◻▽△★◻◆●◇△▽○◻★★◆◆●●◇◇△▽○▽△○◻◻★◆●◇△▽○◻★◆★●◆●◇○▽△◇△▽◻○◻★◆●◇△▽★○◻◆●★◇○◆▽△●◻◇△▽○◻★◆●★◇△▽◆●○◻◇○▽△★◆●◻◇△▽○◻★★◆●◆●◇△▽○◻◇○▽△★◻◆●◇△▽○◻★★◆◆●●◇△▽◇○▽△○◻◻t★◆●◇△▽○◻★◆★●◆●◇○▽△◇△▽◻○◻★◆●◇△▽★○◻◆●★◇○◆▽△●◇◻△▽○◻★◆1●(★◇△▽◆●○◻◇○★▽△◆●◻◇△▽○◻★◆★●◆◇●△▽○◻◇○▽△m★◻◆●◇△▽○◻★★0◆◆●●◇△▽◇○▽△○◻◻★◆●◇△▽○◻★★◆●◆●◇○▽△◇△▽◻○◻★◆●◇△▽★○◻◆●0★◆◇○●▽△◇◻△▽○◻s★◆●★◇△▽◆○●◻◇★○▽△◆●◻◇△▽○◻★◆★●◆◇●△▽○◻◇○▽△★◻◆●◇△▽○◻★★◆●◆●◇△▽)◇○▽△○◻◻★◆●◇△▽○◻★★◆●◆●◇○◇▽△△▽○◻◻★◆●◇△▽★○◻◆●★◆◇●○▽△◇△▽◻○◻★◆●★◇△▽◆○◻●★◇○▽△◆●◻◇△▽○◻★◆●★◇◆△▽●○◻◇○▽△★◻◆●◇△▽○◻★★◆●◆●◇△▽◇○○◻▽△◻★◆●◇△▽○◻★★◆●◆●◇◇○△▽▽△○◻◻★◆1●◇△▽★○◻◆●★◆●◇○▽△◇△▽◻○◻★◆●◇★△▽◆○◻●★◇○▽△◆●◻◇△▽○◻★5◆●★◇◆△▽●○◻◇○▽△★◆◻●◇△▽○◻★★◆●◆●◇△▽◇○○◻▽△◻★◆●◇△▽○◻★0★◆◆●●◇◇○△▽▽△○◻◻★◆●◇△▽○★◻◆★●◆●◇○▽△◇△▽◻○◻★◆●◇△▽★○◻◆●★◇○◆▽△●◻◇△▽○◻★◆●★◇△▽◆●○◻◇○▽△★◆◻●◇△▽○◻★★◆●◆●◇△▽○◻◇○▽△★◻◆●◇△▽○◻★★◆◆●●◇△▽◇○▽△○◻◻★◆●◇△▽○◻★◆★●◆●◇○▽△◇△▽◻○◻★◆●◇△▽★○◻◆●★◇○◆▽△●◇◻△▽○◻★◆●★◇△▽◆●○◻◇○★▽△◆●◻◇△▽○◻★◆★●◆◇●△▽○◻◇○▽△★◻◆2●◇△▽○◻★★◆◆●●◇△▽◇○▽△○◻◻★◆●◇△▽○◻★★◆●◆●◇○▽△◇▽△◻○◻★0◆●◇▽△★○◻◆●◇○▽△◻★◆●◇○▽△◻0★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻ bicrrpfi(nfhaorre)mhoosCtt]tw,maehuerccneetbrehoeeoinitonrnciocu(mcrnelqesleiuxelumogty-spabhid,iimesotseefirenpcoeiitrmmlrervaocniepseafaadsdnnlntiietccheghnsbeasiutesrle[lt)rt2aGwheaa1rcaicoA]obntlcrhaaoek[tvtnets2.htahrsd2ieeelc]unidtenrfech.eobeeltroTeyhifvntsehhtautihfneihsmoseteosuoeq“tenbrarxunidsoncpaeepaigrreuslivearocpaisttlmiiatratmvionoteoeptudnn-i6yTltca0iaein”st0[l,ciopatttamhhhnpsaeeeess--
/NCtot---1100....5050★◆●★○○○○○○○○◆◇○▽△●★◻◆◇○▽△●★◻◆◇○▽△●★○○◻◆○○○○△●★◻◆◇○▽△●★◻◆◇○▽△●★◻◆◇○▽△●★◻◆◇○▽△●★◻◆◇○▽△●★◻◆◇○▽△●★◻◆◇○▽△●★◻◆◇○▽△●★◻◆◇○▽△●★◻◆◇○▽△●★◻◆◇○▽△●★◻◆◇○▽△●★◻◆◇○▽△●★◻◆◇○▽△●★◻◆◇○▽△●★◻◆◇○▽△●★◻◆◇○▽△●★◻◆◇○▽△●★◻◆◇○★▽△●◆◻●◇○▽△◇○◻▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○★★▽△◆◆◻●●◇◇○○▽▽△△◻◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆abdegh●c◇○▽△◻f★◆●◇○▽△◻)★◆●◇○▽△)◻))))))★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻★◆●◇○▽△◻ atmtttwttchhiohhofoitonneeeetnedhicracsamrel.obeunultosnssnhS[umeij(1beoienl10nitenc)ca,scqnteseu1utiweos4oredafiie,fntttddthnh3ccdhiuh7oussref]hm.pser.sneuiseetnoWrsTshrei-ttnnilanherciupgtqacaimwsoteuleoeseimeqoinddsrsrubileeibyabcishtllneanheieoebrolhopxvatrtadspapiorviuastitmenibiimqrolfoguiliaurnkemlcaseaiotttsenmtfoaQntioqottitafrMtehnusyuastesilhCtct.eobieqaevfeoonueTacurtalphrhyhvtnPepaeeetlwvlErraulHaoisehtPmfexlneyeaicSiddgtmmoX,htiannhatieatYldogeert--
phase diagram put forward in Ref. [10].
0 50 100 150 200 We did not study the chirality of the spiral order in
t(ms) detail, as in a layered setup with independent layers and
TOF images taken vertically, this feature cannot be ac-
FIG.5: (Coloronline.)Macroscopicparameters(condensate cessed easily.
occupation N , total pair density D , and nearest-neighbor
0 tot An interesting future direction would be the general-
coherences C) as a function of time t for the different values
ization of the Feynman relation [38] to unconventional
of α listed in Table III. Note that the condensate occupation
N varies very weakly with α. superfluids on the triangular lattice. This relation gives
0
a variational estimation of the dispersion relation of the
low-energyexcitationsE asaratioofthenoninteracting
q
order of 700 ms [21]. kineticenergy(cid:15)kandtheformfactorS(q)(Fouriertrans-
formofthedensity-densitycorrelationfunction). Inpar-
ticular, Feynman was able to reproduce approximately
VI. CONCLUSIONS the phonon-roton spectrum for superfluid He using the
form factor measured by neutron scattering. For ultra-
We proposed a specific experimental setup for inter- coldatoms,theformfactorcouldbeextractedfromnoise
acting bosons on an anisotropic triangular lattice and correlations of TOF images [5, 39], which could be used
calculated microscopic parameters for the corresponding torevealthelow-energydispersionrelationoftheexcita-
Bose-Hubbard model. We studied numerically the dy- tions, in particular, the dynamical exponent z from the
namics of the atoms in a time-dependent mean-field ap- relation Eq ∼qz at low momenta.
proximation after instantaneously reversing the signs of Acknowledgements. I am grateful for discussions
the on-site interaction to U < 0 and the external po- with Hendrik Weimer, Ricardo Doretto, Temo Vekua,
f
tential to V <0. We found that quasi-classical coher- Luis Santos, and Ulrich Schneider. I thank especially
0,f
encewith120◦ spiralorderdevelopsintheisotropiccase. Stephan Mandt for a critical reading of the manuscript.
This can be interpreted as a manifestation of the “frus- This research was supported financially by the cluster of
trated”kinetictermoftheBose-Hubbardmodelataneg- excellence QUEST.
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FIG. 6: (Color online.)Parts of the TOF images at V = 3E for different values of α. The rows (a) - (h) correspond to
L,f R
different values of the lattice potential anisotropy α (cf. Table III and Fig. 4); the columns correspond to different times after
thequench. ThecolorcodingissimilartoFig.3. ThegraydashedlinecorrespondstotheborderoftheBZ.Theredcontours
are defined to enclose 128 points with the highest kinetic energy (cid:15) in reciprocal space for the lattice size of 192×192.
k
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10
◻ ◻
0.6 ○○ a) 0.6 f)
○○
0.4 ◻ ○ ○ 0.4
oherences-000...202◻◻◻◻◻◻○◻◻○◻○◻◻○◻○◻○◻○◻○◻◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻ oherences-000...202◻○◻◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻
c ○○ c
-0.4 ○○ -0.4
-0.6 -0.6
0 20 40 60 80 0 20 40 60 80
l l
◻ ◻○
0.6 0.6
c) g)
○ ○○
oherences-0000....2024◻○◻◻◻◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻ oherences-0000....2024◻○◻◻◻◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻
c-0.4 ◻○○ c-0.4
-0.6 -0.6
0 20 40 60 80 0 20 40 60 80
l l
◻ ◻
0.6 0.6 ○
e) h)
○
herences-0000....2024○◻◻○◻○◻○◻○◻◻◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻ herences-0000....2024◻○◻◻◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻○◻
o ◻ ○ o
c ○ c
-0.4 ○ -0.4
-0.6 ○ -0.6
0 20 40 60 80 0 20 40 60 80
l l
FIG. 7: (Color online.)Longer-range coherences in the a (x-) direction as a function of the site index l at time t=400 ms.
2
Red squares: ReC(l); blue circles: ReC˜(l). The grid lines represent the quasiclassical “pitch” length 2π/|a |k∗ of the spiral
2
ordering vector.
man, T. Rom, I. Bloch, and U. Schneider, Science 339, can be inverted for the magnetic field value B using the
52 (2013). parameters of the corresponding Feshbach resonance.
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s