Bose-Hubbard Models in Confining Potentials: An Inhomogeneous Mean-Field Theory

4.2 MB·English

by  Ramesh V. Pai

08 January 2012
Bose-Hubbard Models in Confining Potentials: An Inhomogeneous Mean-Field
Theory
Ramesh V. Pai
∗
Department of Physics, Goa University, Taleigao Plateau, Goa 403 206, India
Jamshid Moradi Kurdestany
†
Centre for Condensed Matter Theory, Department of Physics,
Indian Institute of Science, Bangalore 560 012, India
2
1
K. Sheshadri
0 ‡
Bagalur, Bangalore North Taluk, India 562 149, India
2

n
Rahul Pandit
a §
J Centre for Condensed Matter Theory, Department of Physics,
Indian Institute of Science, Bangalore 560012, India.
8
(Dated: January 10, 2012)

] WepresentanextensivestudyofMottinsulator(MI)andsuperfluid(SF)shellsinBose-Hubbard
s
a (BH) models for bosons in optical lattices with harmonic traps. For this we develop an inhomo-
g geneous mean-field theory. Our results for the BH model with one type of spinless bosons agrees
- quantitatively with quantum Monte Carlo (QMC) simulations. Our approach is numerically less
t
n intensive than such simulations, so we are able to perform calculation on experimentally realistic,
a large3Dsystems,exploreawiderangeofparametervalues,andmakedirectcontactwith avariety
u of experimental measurements. We also generalize our inhomogeneous mean-field theory to study
q BH models with harmonic traps and (a) two species of bosons or (b) spin-1 bosons. With two
t. speciesofbosonsweobtainrich phasediagrams withavarietyofSFandMIphasesandassociated
a shells, when we include a quadratic confining potential. For the spin-1 BH model we show, in a
m
representative case, that the system can display alternating shells of polar SF and MI phases; and
- we makeinteresting predictions for experiments in such systems.
d
n PACSnumbers: 05.30Jp,67.40Db,73.43Nq
o
c
[ I. INTRODUCTION ory19, which has been used for the Bose-glass phase in
1 the disordered BH model.
v
In addition to the optical-lattice potential, a confin-
2 High-precision experiments on cold atoms, such as
ing potential, which is typically quadratic, is present in
4 spin-polarized 87Rb, in traps have provided powerful
all experiments. This inhomogeneous potential leads to
6 methods for the study of quantum phase transitions1,
1 e.g., the transition from a superfluid (SF) to a bosonic inhomogeneities in the phases that are obtained: simu-
1. Mott-insulator (MI) in an optical lattice2,3. This transi- lations20,21 of the Bose-Hubbardmodel with this confin-
ing potential and experiments22,23 on interacting bosons
0 tion was predicted by mean-field studies4,5 and obtained
2 inMonte-Carlosimulations6 ofthe Bose-Hubbardmodel in optical lattices with a confining potential have both
1 before it was seen in experiments1–3. Recent experi- seenalternatingshellsofSFandMIregionsinthesingle-
: species, spinless case. We explore such shells via the in-
v ments7,8 have investigated a heteronuclear degenerate
homogeneous mean-field theory, first for single-species,
i mixture of two bosonic species, e.g., 87Rb and 41K, in
X spinless bosons and then for the two-species and spin-1
a three-dimensional optical lattice; such mixtures have
ar also been studied theoretically9–13 and by Monte Carlo generalizations mentioned above.
simulations14. Systems of alkali atoms with nuclear spin Mean-field theories for the Bose-Hubbard model were
I = 3/2 have hyperfine spin F = 1; examples include first developed for the homogeneous case4,5; these the-
23Na, 39K, and 87Rb; these spins are frozen in magnetic ories were then extended to the inhomogeneous case19
traps,sotheseatomsaretreatedasspinlessbosons;how- to develop an understanding of the Bose-glass phase in
ever, in purely optical traps, such spins can form spinor the disordered Bose-Hubbard model. We show that BH
condensates15–18. Thus, we consider the following three models with confining potential can be treated, at the
typesofBose-Hubbard(BH)models: (1)aBHmodelfor level of mean-field theory, as was done in the Bose-glass
spinless interacting bosons of one type; (2) a generaliza- case19; inparticular,we providea naturalframeworkfor
tion of the spinless BH model with two types of bosons; understanding alternating SF and MI shells, which are
and (3) a spin-1 generalization of the spinless BH model seen in simulations20,21 and experiments22,23 on inter-
with bosons of one type. We study these models by de- acting bosons, trapped in a confining potential, and in
velopingextensionsofaninhomogeneousmean-fieldthe- an optical lattice. Though other groups24–29 have stud-
2
ied such shell structure theoretically, they have not ob- II. MODELS AND INHOMOGENEOUS
tained the quantitative agreement with quantum Monte MEAN-FIELD THEORY
Carlo(QMC)simulations20thatweobtain,exceptinone
dimension30. Furthermore, our theory yields results in We begin by defining the three Bose-Hubbard models
good agreement with a variety of experiments; and it that we study. We then develop inhomogeneous mean-
canbegeneralizedeasilyto(a)twospeciesofinteracting fieldtheoriesthatarewellsuitedforstudyingthe spatial
bosons and(b) the spin-S case, as we show explicitly for organizationofphasesinthesemodelswithconfiningpo-
S = 1; in both these cases we provide interesting pre- tentials.
dictions that will, we hope, stimulate new experiments.
Our inhomogeneous mean-field calculations can be car-
ried out with experimentally realistic parameters, so we A. Models
can make direct comparison with experiments. In par-
ticular, we obtain in-trap density distributions of alter- The simplest Bose-Hubbard model describes a single
nating SF and MI shells; these show plateaux in certain species of spinless bosons in an optical lattice by the fol-
regions, which can be understood on the basis of sim- lowing Hamiltonian:
ple geometrical arguments. Furthermore, we obtain the
radii of SF and MI shells from in-trap density distribu- 1 1U 1
tions anddemonstrate howthe phasediagramofthe ho- Hzt =−z X (a†iaj+h.c)+2ztXnˆi(nˆi−1)−ztXµinˆi;
<i,j> i i
mogeneous Bose-Hubbard model can be obtained from
(1)
these radii. We also obtain results that are of direct rel-
here spinless bosons hop between the z nearest-neighbor
evance to recent atomic-clock-shift experiments23. With
two species of bosons we obtain phase diagrams in the pairs of sites < i,j > with amplitude t, a†i, ai, and
homogeneous case over a far wider range of parameters nˆi ≡ a†iai are, respectively, boson creation, annihilation,
and number operators at the sites i of a d-dimensional
than has been reported hitherto. We find rich phase di-
hypercubic lattice (we study d = 2 and 3), U the onsite
agram with phases that include ones in which (a) both
Hubbardrepulsion, µ µ V R2, µ the uniformchem-
typesofbosonsareinSFstates,(b)bothtypesofbosons i ≡ − T i
ical potential that controls the total number of bosons,
areinMIphaseswithdifferentorthesamedensities,and
V the strengthofthe harmonicconfiningpotential,and
(c)onetypeofbosonisinanSFphasewhereastheother T
typeisinanMIphase. Weshowthateachofthesephases Ri2 ≡ Pdn=1Xn2(i), where Xn(i), 1 ≤ n ≤ d, are the
appear in shells when we include a quadratic confining Cartesian coordinates of the site i (in d = 3 X1 = X,
potential; and we also obtain in-trap density distribu- X2 =Y, and X3 =Z) ; the origin is chosen to be at the
tions that shows plateaux as in the single-species case. center of the lattice. In terms of experimental parame-
In the case of the spin-1 Bose-Hubbard model we show, ters1 zUt = √48zπaase2qEV0r, where Er is the recoil energy,
in a representative case, that the system can display al- V the strength of the lattice potential, a (= 5.45nm
0 s
ternating shells of polar SF18 and MI phases; the latter for 87Rb) the s-wave scattering coefficient, a = λ/2 the
have integral values for the boson density. Our inhomo- optical lattice constant, and λ = 825nm the wavelength
geneousmean-fieldtheoryleadstointerestingpredictions of the laser used to create the optical lattice; typically
foratomic-clock-shiftexperimentsinsystemswithspin-1 0 V 22E . We set the scale of energies by using
0 r
bosons in an optical lattice with a confining potential. zt≤= 1 i≤n the Bose-Hubbard model 1; for comparisons
with experimental systems we should scale all energies
by E .
r
For a mixture with two types of bosons, we use the
following Bose-Hubbard Hamiltonian:
t t
a b
Hz = −z X (a†iaj +h.c)− z X (b†ibj +h.c) (2)
The remaining part of this paper is organized as fol- <i,j> <i,j>
lows. InSec. 2wedescribethemodelsweuseandhowwe 1U 1U
a b
develop an inhomogeneous mean-field theory for them. +2 z Xnˆai(nˆai−1)+ 2 z Xnˆbi(nˆbi−1)
Section 3 is devoted to our results; subsection 3A con- i i
U 1 1
tains the results of our inhomogeneous mean-field the- + ab Xnˆainˆbi Xµainˆai Xµbinˆbi;
ory for the single-species, spinless Bose-Hubbard model; z − z − z
i i i
subsection 3B is devoted to the results, for both homo-
thefirstandsecondtermrepresent,respectively,thehop-
geneous and inhomogeneous cases, for the spinless BH
model with two types of bosons; subsection 3C is de- ping of bosons of types a and b between the nearest-
neighbor pairs of sites <i,j > with hopping amplitudes
voted to our results for the single-species BH model for
spin-1bosons. Section4containsourconclusions,acom- ta and tb; here a†i, ai, and nˆai ≡ a†iai and b†i, bi, and
parisonof our work with earlier studies in this area, and nˆbi ≡b†ibi are, respectively, boson creation, annihilation,
the experimental implications of our results. and number operators at the sites i of a d-dimensional
3
hypercubic lattice (we study d = 2 and 3) for the two introduces the superfluid order parameter ψ a for
i i
≡ h i
bosonic species. For simplicity we restrict ourselves to the site i, and thence expresses the Hamiltonian (1) as
thecaset =t =t,and,tosetthescaleofenergy,weuse MF = MF, where the superscript MF denotes
a b H PiHi
zt=1. Thethirdandfourthtermsaccountfortheonsite mean field and the single-site Hamiltonian is
interactions of bosons of a given type, with energies U
a MF 1U µ
aUnd,Uabr,isreesspbeecctiavueslye,owfhtehreeasonthsietefifitnhtetrearcmti,ownsithbeetnweergeny Hzit = 2ztnˆi(nˆi−1)−ztinˆi−(φia†i+φ∗iai)+ψi∗φi. (5)
ab
bosonsoftypes a andb. We havetwochemicalpotential
Here φ 1 ψ and δ labels the z nearest neigh-
terms: µ µ V R2andµ µ V R2,whereµ i ≡ z Pδ i+δ
ai ≡ a− Ta i bi ≡ b− Tb i a borsof the site i. If VT =0,the effective onsite chemical
and µ the uniform chemical potentials that control the
b potential µ = µ, for all i, so the local density and su-
i
total number of bosons, of species a and b, respectively,
perfluid order parameters are independent of i: ρ = ρ
i
V andV arethestrengthsoftheirharmonicconfining
Ta Tb and ψ = ψ. If V > 0 we first obtain the matrix el-
i T
potentials (we restrict ourselves to the case V = V ),
Ta Tb ements of MF in the onsite, occupation-number basis
CanadrtResi2ia≡n Pcoodnr=d1inXan2t(eis),owf htheeresXiten(ii;),th1e≤ornig≤indi,saarte tthhee {n|mniaix},,ttrhuenHtcoaittaeldninumprbaecrtiocef bbyoscohnosospienrgsaitfien,iftoervaalugeivfeonr
center of the lattice. initial set of values for ψ . [For small values of U we
i
The spin-1 Bose-Hubbard Hamiltonian18 that we con- must use large values o{f nm} ax; for the values of U we
sider is consider nmax = 6 suffices.] We then diagonalize this
matrix, which depends on ψ and ψ , to obtain the
1 1U i i+δ
zHt = −z X (a†i,σaj,σ+h.c)+ 2 zt0 Xnˆi(nˆi−1) lnoowteedst, erenseprgecytiavnedly,thbeycEoir(rψesp,ψondin)ganwdavΨe f(unψctio);nf,rdome-
<i,j>,σ i g i i+δ g { i}
these we obtain the new superfluid order parameters
1U 1
+2 zt2 Xi (F~i2−2nˆi)− ztXi µinˆi. (3) oψfiψ=ihaΨsgin({pψuit}s)t|oarie|coΨngs(t{rψuci}t)Hi;iMwFe uasnedtrheepseeantetwhevadliuaegs-
onalizationprocedureuntilweachieveselfconsistencyof
Here spin-1 bosons can occupy the sites i of a
input and output values to obtain the equilibrium value
d dimensional, hypercubic lattice and hop between the ψeq (henceforth we suppress the superscript eq for nota-
− i
z nearest-neighbor pairs of sites < i,j > with ampli-
tionalconvenience). [Thisisequivalenttoaminimization
atunddeati,,σσaries, trheespsepctiniveilnyd,esxitet-haatndcasnpinb-ede1p,e0n,d−e1n,tab†io,σ- ospfetchtettooψtai;liefnmerogryetEhga(n{ψoni}e)s≡oluPtioinEgiis(ψobi,tψaiin+eδd),wwiethpircek-
son creation and annihilation operators, and the num- theonethatyieldstheglobalminimum.] Theonsiteden-
ber operator nˆiσ ≡ a†i,σai,σ; the total number operator sity is obtained from ρi =hΨg({ψi})|nˆi |Ψg({ψi})i. In
awtitshitFe~σi,σi′ssnˆtain≡daPrdσsnˆpii,nσ-,1amndatFr~iice=s.PThσ,eσ′ma†io,σdFe~lσ,(σ3′)aii,nσ-′ rvreeapalulreevsaoelfnutψeasitiiovsferψeca.als;esso,,wheenhcaevfeorftohu,nwdetrheasttrtihcteoeuqrusielilbvreisutmo
cludes, in addition to the onsite repulsion U , an energy
0 For the two-species Hamiltonian (3) our mean-field
U fornonzerospinconfigurationsonasite. Suchaspin-
2 theory obtains an effective one-site problem by decou-
dependent term arises from the difference between the
pling the two hopping terms as follows (cf., Eq. 4):
scattering lengths for S = 0 and S = 2 channels31. The
inhomogeneous chemical potential µ is related to the
uniform chemical potential µ and theiquadratic, confin- a†iaj ≃ ha†iiaj +a†ihaji−ha†iihaji;
ing potential as in the spinless case 1. We set the scale b†ibj ≃ hb†iibj +b†ihbji−hb†iihbji; (6)
of energies by choosing zt=1.
here the superfluid order parameters for the site i for
bosons of types a and b are ψ a and ψ b ,
ai i bi i
≡ h i ≡ h i
respectively. The approximation (6) can now be used
B. Inhomogeneous mean-field theory to write the Hamiltonian (3) as a sum over single-site,
mean-field Hamiltonians MF (cf., Eq. 5) given below:
Hi
The mean-fieldtheory we use has been very successful
MF 1U µ
in obtaining the phase diagrams for models (1) and (3), Hi = anˆ (nˆ 1) ainˆ (7)
ai ai ai
with V =0, i.e., in the absence of the harmonic confin- zt 2 zt − − zt
T
tinhgispthoeteonryti,adl5e,v1e8l.opTehdefirinsthofomrothgeenBeoouses-ggleansesrpahliazsaet1i9onanodf −(φaia†i +φ∗aiai)+ψa∗iφai
1U µ
b bi
spinlessbosons,decouplesthehoppingtermtoobtainan + nˆbi(nˆbi 1) nˆbi
2 zt − − zt
effective one-site problem, neglects quadratic deviations
U
ab
from equilibrium values (denoted by angular brackets), −(φbib†i +φ∗bibi)+ψb∗iφbi+ zt nˆainˆbi.
uses the approximation
Here φ 1 ψ and φ 1 ψ , where δ
ai ≡ z Pδ ai+δ bi ≡ z Pδ bi+δ
a†iaj ≃ha†iiaj +a†ihaji−ha†iihaji, (4) labels the nearest neighbors of the site i. If VT = 0, the
4
effective onsite chemical potentials µ = µ and µ = Theself-consistencyprocedurethatweusenowissim-
ai a bi
µ ,foralli,soρ =ρ ,ρ =ρ ,ψ =ψ ,andψ =ψ ilarto,butmorecomplicatedthan,theonewehaveused
b ai a bi b ai a bi b
are independent of i. for the spinless BH model. If V > 0 we first obtain,
T
If V >0, we first obtain, for a given initial set of val- for a given initial set of values for ψ , the matrix el-
T i,σ
ues for ψ and ψ , the matrix elements of MF ements of MF in the onsite, occu{patio}n-number basis
{ ai} { bi} Hi Hi
in the onsite, occupation-number basis n , n , n ,n ,n , truncated in a practical calculation
ai bi i, 1 i,0 i,1
which we truncate in a practical calculat{i|on biy c|hooi}s- b{|y c−hoosing a fiin}ite value for nmax, the total number
ing a finite value nmax for the total number of bosons of bosons per site, [For small values of U and U2 we
per site. [The smaller the values of the interaction pa- must use large values of nmax; for the values we use
rameters Ua,Ub, and Uab the larger must be the value here, nmax = 4 suffices.] We then diagonalize this ma-
of nmax; for the values of Ua, Ub, Uab, µa, and µb trix, which depends on ψi,σ and ψ(i+δ),σ, to obtain the
that we consider, nmax = 6 suffices.] We then di- lowest energy and the corresponding wave function, de-
agonalize this matrix, which depends on ψ , ψ , noted,respectively,byEi(ψ ,ψ )andΨ ( ψ );
ai bi g i,σ (i+δ),σ g { i,σ}
ψ , and ψ to obtain the lowest energy and the from these we obtain the new superfluid order parame-
a(i+δ) b(i+δ)
corresponding wave function, denoted, respectively, by ters ψ = Ψ ( ψ ) a Ψ ( ψ ) ; we use these
i,σ g i,σ i,σ g i,σ
Ei(ψ ,ψ ;ψ ,ψ ) and Ψ ( ψ ,ψ ), whence new valueshof ψ{ to}re|const|ruct{ MF}aind repeat the
g ai a(i+δ) bi b(i+δ) g { ai bi} i,σ Hi
we obtain the new superfluid order parameters ψ = diagonalization procedure until input and output values
ai
Ψ ( ψ ,ψ ) a Ψ ( ψ ,ψ ) and ψ = are self consistent; thus we obtain the equilibrium value
g ai bi i g ai bi bi
hΨ ({ψ ,ψ }) |b Ψ|( ψ ,{ψ ) ; w}eiuse these new ψeq. We suppress eq as above and recall that this self-
vhalgue{s oafiψbi}and| ψi | asgin{puatis tobi}reiconstruct MF and coi,nσsistent procedure is equivalent to a minimization of
repeatthedaiiagonalibziationprocedureuntilweacHhiieveself the total energy in the spin-1 case18. Here too, we fol-
consistency of input and output values to obtain the low our discussion of the mean-field theory of the BH
equilibrium value ψeq and ψeq; again we suppress the model (1) and restrict ourselves to real values of ψ .
ai bi i,σ
superscript eq for notational convenience. [As we have We have noted in an earlier study18 that, at the level
mentionedinthesingle-speciescase,thisself-consistency of our mean-field theory, the superfluid density in the
procedure is equivalent to a minimization, with respect spin-1 case is
to ψ and ψ , of the total energy E ( ψ ,ψ )
ai bi g ai bi
PiEgi(ψai,ψai+δ;ψbi,ψbi+δ); we pickthe o{nethat y}ield≡s ρs =X|ψσeq |2; (11)
the global minimum.] The onsite densities are obtained σ
from ρai = Ψg( ψai,ψbi ) nˆai Ψg( ψai,ψbi ) and and the magnetic properties of the SF phases follow
h { } | | { } i
ρbi = Ψg( ψai,ψbi ) nˆbi Ψg( ψai,ψbi ) ,respectively. from16,17
h { } | | { } i
We follow our discussion of the mean-field theory of the
BH model (1) and restrict ourselves to real values of ψai F~ = Pσ,σ′ψσeqF~σ,σ′ψσeq′. (12)
and ψbi. h i Pσ|ψσeq|2
The inhomogeneous mean-field theory for the spin-1
If we substitute the explicit forms of the spin-1 matrices
BHmodelfollowsalongsimilarlines. Thespin-1analogs
we obtain
of Eqs. 4 and 5 are respectively,
(ψ ψ +ψ ψ ) (ψ2 ψ2 )
a†i,σaj,σ ≃ ha†i,σiaj,σ +a†i,σhaj,σi−ha†i,σihaj,σi (8) hF~i = √2 1P0σ|ψσ−|21 0 xˆ+ P1σ−|ψσ−|21 zˆ,
and F~ 2 = 2(ψ1ψ0+ψ−1ψ0)2 + (ψ12−ψ−21)2, (13)
h i ( ψ 2)2 ( ψ 2)2
MF 1U 1U µ Pσ| σ| Pσ| σ|
Hi = 0nˆ (nˆ 1)+ 2(F~2 2nˆ ) inˆ
zt 2 zt i i− 2 zt i − i − zt i where xˆ and zˆ are unit vectors in spin space; SF phases
with F~ = 0 and F~ 2 = 1 are referred to as polar and
−X(φi,σa†i,σ+φ∗i,σai,σ)+Xψi∗,σφi,σ.(9) ferromhaginetic, resphecitively. The order-parameter mani-
σ σ
folds of these phases can be found in earlier studies17,18.
Here we use the following superfluid order parameters:
ψi,σ ai,σ ; (10) III. RESULTS
≡h i
and φ 1 ψ , where and δ labels the z near-
i,σ ≡ z Pδ (i+δ),σ Giventhe formalismwe have developedabove,we can
est neighbors of the site i; recall, furthermore, that σ
obtain severalresults for quantities that have been mea-
can assume the values 1, 0, −1, and nˆi,σ ≡ a†i,σai,σ, suredin quantum Monte Carlo (QMC) simulations or in
nˆi ≡ Pσnˆi,σ, and F~i = Pσ,σ′a†i,σF~σ,σ′ai,σ′ with F~σ,σ′ experiments for the spinless case. We cover this in Sub-
standard spin-1 matrices. With these order parame- section3A.Subsection3Bisdevotedtotheresultsofour
ters (cf., Eq. 10) we have developed an inhomogeneous inhomogeneousMFtheoryforthecasewithtwotypesof
version of the homogeneous mean-field theory18 for the bosons. Subsection 3C is devoted to the results of our
spin-1 BH model with V =0. inhomogeneous MF theory for the spin-1 case.
T
5
κlocal =∂ρ /∂µ . For this set ofparameterswe calculate
i i i
ρ and thence κlocal; we also obtain the local superfluid
i i
density ρs ψ2 (the last formula is valid at the level
i ≡ i
of our MF theory). In Fig. 1 we plot versus R our MF
i
1.0 0.20 results for ρ , κlocal, and ρs along with data from QMC
i i i
simulations20. For this set of parameters, the central re-
gion near the origin of the lattice is in the MI phase,
0.8
0.15 i.e., the local density ρ =1 and both ρs and κlocal van-
i i i
ish. This central core is enveloped by an SF shell, with
0.6 nonzero values for ρs and κlocal. As we move radially
ρ i i
ρ (QMC) 0.10 outward from the center, ρi decreases monotonically till
ρ (MFT) it goes to zero, as do ρs and κlocal, in the region where
0.4 i i
κS (MFT) µeff < 0. The quantitative agreement between our MF
κ (QMC) i
(MFT) 0.05 results and those from QMC is shown in Fig. 1; there
0.2
is only a slight discrepancy between the MF ρ and its
i
QMC analog atthe MI-SF interface; our result for κlocal
i
0.0 0.00 alsoseemstomiss,atthisinterface,theshoulderthatap-
0 2 4 6 8 10 12 14 16 18 20 pearsinthe QMC κlocal perhapsbecause ourMFtheory
i
Ri overestimates the stability of the SF phase.
ThisgoodagreementbetweenourMFresultsandthose
of QMC simulations has encouraged us to use our MF
FIG. 1: (Color online) Plots with a comparison of our MF
theory in cases where such simulations pose a significant
results for thelocal densityni (bluefilled circles), local com-
pressibility κlocal = ∂n/∂µeff (blue open circles), and local numerical challenge. In particular, we use our theory to
i
superfluid density ρs = ψ2 (green filled triangles) of spin- make direct comparisons with experiments22 that have
i i
lessbosonsinatwo-dimensionalparabolictrapwithVT/U = observed alternating MI and SF shells in 3D optical lat-
0.002, µ/U = 0.37 and U/t = 25; we have obtained QMC tices by recordingin-trapdensity distributions of bosons
data by digitizing plots in figures in simulation studies20 for at different filling fractions. We use a simple-cubic lat-
ni (red filled squares) and κliocal (red open squares). tice with 1213 sites, µ/Er = 1, VT/Er = 0.0003, and
the optical potential V /E in the range 12 16 so that
0 r
the number of bosons N 106, which is co−mparable to
the number of atoms in t≃he experiments22 we consider.
Thischoiceofparametersleadstotwowell-developedMI
60 2.5
(a) (b) shells (ρ=1 and 2, respectively). The MI and SF shells
40 2.0 appear as annuli22 in a 2D planar section through
z
P
20 the 3D lattice, at a vertical distance z from the center
Y 1.5
0 [see, e.g., Fig. 2(a) for V0/Er = 15 and z = 0 where the
1.0 core region is in the SF phase]. Figure 2(b) shows that,
-20 S
as we moveradially outward,ρ decreasesmonotonically
i
-40 0.5 and ρs is zero in the two MI regimes (16 < R < 34
i i
-60 0.0 and 44 < Ri < 52) in which ρi is pinned at 2 and 1,
-60 -40 -20 0 20 40 60 0 10 20 30 40 50 60
X X respectively. SF and MI shells alternate and the outer-
most one is always in the SF phase; their positions and
radii depend on µ, which also controls the total number
FIG. 2: (Color online) (a) SF (white) and MI regions [ρ=2
N ofatomsinthe system,asillustratedbythe sec-
(red)andρ=1(black)]annuliformedina2Dplanarsection Pz=0
tions in Figs.3(a) and (b) for V /E = 15 and µ = 0.8
Pz through the 3D lattice, at a vertical distance z from the 0 r
center(herez=0)and(b)thecorrespondingradialvariation (N =7.1 105)andµ=0.9(N =8.9 105),respectively.
× ×
osqfudaerness)ityforρiµ/(rEerd=cir1c,leVsT)/aEnrd=su0p.0e0rfl03uidanddeVn0sit=y 1ρ5siE(rb;lathcke Fnuormabneyr2oDf bpolasonnasrsinectthioenρP=z wmeMcaInacnanlcuululast,eaNndm(Nzm)r,(tzh)e,
outermost gray regions in (a) contain no bosons. the remaining number of bosons; the total number of
bosons in this planar section is N(z)=N (z)+Nr(z),
m m
which does not depend on m. In Figs.3 (c) and (d) we
A. Results for the spinless Bose-Hubbard model show, for m= 2 and µ =0.8 and 0.9, respectively, plots
versusz ofN (z)(fullredsquares),Nr(z)(fullbluetri-
m m
First we compare our mean-field (MF) results with angles),andtheirsumN(z)(fullblackcircles). Figures2
those obtained by quantum-Monte-Carlo (QMC) sim- (c)and(d) are remarkablysimilar to the density profiles
ulations in two dimensions20. These simulations use obtained in experiments22 [cf., their Figs. 3(c) and (d)].
U/t = 25, µ/U = 0.37, and V /U = 0.002, and ob- The radii of MI shells follow from such in-trap den-
T
tain the local density ρ and the local compressibility sity profiles: In Figs. 4(a) and (b) we plot N (z) and
i m
6
the 2D planar section has no MI shell with density m,
thus, N (z) = 0, which is also apparent in these fig-
m
ures. For z <R (m), the central parts of the 2D planar
60 60 I
sections show SF shells and Nr =N(z) N (z) de-
40 (a) 40 (b) Pz m − m
creases as we increase z. For R (m) < z < R (m), the
I O
20 20 centralpartsofthe2Dplanarsections showMIshells;
Y Y Pz
0 0 the number of bosons in such MI shells is Nm(z) and it
-20 -20 is proportional to the area of this central shell, namely,
mπ(R2(m) z2); thus, N (z) decreases as we increase
-40 -40 z hereO; howe−ver, Nr = N(mz) N (z) remains indepen-
-60 -60 m − m
-60 -40 -20 0 20 40 60 -60 -40 -20 0 20 40 60 dent of z, because of the simple geometrical arguments
x103 X x103 X gRiv(emn)ab<ovze;<i.eR.,w(emh)a.veFpinlaatlleya,ufxorinzN>mr(Rz)i(nmt)h,etrheegi2oDn
12 r 14 r I O O
10 N2+N2 (c) 12 N2+N2 (d) planar section Pz has no MI shell with density m, from
which it follows that N (z)=0.
10 m
8 r r
6 N2 8 N2
6
4 4 N2
2 N2 2 3 3
x10 r x10
0-60 -40 -20 0 20 40 60 0-60 -40 -20 0 20 40 60 12 (a) N2 12 (c)
z z 8 8 RI(2)
FIG.3: (Coloronline)SF(white)andMIregions[ρ=2(red) 4 N2 4 RO(2)
and ρ = 1(black)] annuli formed in the 2D Pz=0 sections
in Figs.3(a) and (b) for V0/Er = 15, VT/Er = 0.0003,
and (a) µ/Er = 0.8 (N = 7.1×105) and (b) µ/Er = 0.9 0 -60 -30 0 30 60 00 10 20 30 40 50 60 70
(N = 8.9×105). The corresponding integrated in-trap den- 3 Z 3 Z
sity profiles N (z) (red squares), Nr(z) (blue squares) and x10 x10
2 2
N +Nr (black circles) are shown, respectively, in (c) and 12 r 12
(d2); the2se figures are qualitatively similar to Figs. 3(c) and (b) N1 (d)
(d) in recent experimental study22. 8 8 RI(1)
Nr(z) versus z for m=2 and m=1, respectively, with 4 N1 4 RO(1)
m R1
µ/Er = 1. The curves Nm(z) show nearly flat plateaux R2
for RI(m) z RI(m); similar plateaux occur in 0 0
Nr(−z) for R≤(m) ≤ z R (m) [Figs. 4(a), (c) and -60 -30 0 30 60 0 10 20 30 40 50 60 70
(bm), (d) for mI=2 a≤n|dm|≤=1,Orespectively]. Here R (m) Z Z
I
and R (m) are the inner and outer radii of the MI shell
O
with integer density m. Elementary geometry can be FIG.4: (Coloronline)(a)and(b)PlotsofNm(z)(redsquares)
used to surmise the existence of these plateaux from the and Nr (bluecircles) versusz for m=2 and 1, respectively,
m
MI-SF shell structure18 as we show below. with µ/Er =1; the plots in (c) and (d), which are the same
N (z) is m times the total number of sites inside the as the right halves of (a) and (b), respectively, show how
m
ρ = m MI annulus; this number of sites is well approxi- we determine the inner and outer radii (RI(m) and RO(m),
mated by the area A(z,m) of this annulus. Thus, respectively) from theplateaux in Nm(z) and Nmr(z).
N (z)=mA(z,m)=mπ[R2(z,m) R2(z,m)], (14)
m O − I In Fig. 5(a) we plot, for m = 1 and 2, RO(m) and
R (m), which we have determined from plots such as
whereR (z,m)andR (z,m)are,respectively,theouter I
O I
those in Figs. 4(c) and (d), versus V (E ); the MI phase
andinner radiiofthe MI annulus withdensity ρ=m,in 0 r
with ρ = m lies between the curves R (m) and R (m).
the2Dplanarsection . Ifz <R (m),simplegeometry O I
z I
yields R2(z,m)=R2(Pm) z2 and R2(z,m)=R2(m) Figure 5(a) can be used to obtain the phase diagram
z2; thereIfore, I − O O − of the homogeneous Bose-Hubbard model as follows:
µ =µ V R2, soR (m) andR (m) canbe usedto ob-
Nm(z)=mπ[RO2(m)−RI2(m)], (15) wtaihinichµ−a−(rme,)Tr=esipµe−ctVivTeORlyO2, (tmhe)laonwderIµ+an(md )up=pµer−bVoTuRndI2a(mrie)s,
whenceweconcludethatN (z)isindependentofzwhen oftheMottlobewithdensityρ=m. TheresultingMott
m
z <R (m);thisresultyieldstheplateauxinthein-trap lobe(obtainedbytheconversionV (E ) U/zt)isgiven
I 0 r
| | →
density profiles shown in Figs. 4 (a)-(d); if z >R (m), in Fig. 5(b) along with its counterpart for the homo-
O
| |
7

5
us50 (a) MI ( = 1) 4.0x10 V0=12
Radi3400 RRRiOi (( (===121))) SF MI ( = 2) ()b2.0x105 VV00==1146 (a)
20 RO (=2) N
13.0 13.5 14.0 14.5 15.0 15.5 16.0
V0 (E r) 0.0
(b) 0.8 0.9 1.0 1.1 1.2
20 MI ( = 2)
/zt SF 5
10 2x10
MI ( = 1)
0 ) (b)
6 8 U/zt10 12 14 5
( 1x10
b
3.5x105 N
(c)
5
3.0x10N 0
5
2.5x10 1.8 1.9 2.0 2.1 2.2
5 V0/Er=15
2.0x10
15 20 25 30 35 40 45
zt
3
(c)
FIG. 5: (Color online) (a) Plots, for ρ = m = 1 and 2, of
RO(m) and RI(m) versus V0(Er); the MI phase with ρ=m 2
liesbetweenthecurvesRO(m)andRI(m);redcirclesandred
invertedtrianglesdenoteRO(1)andRO(2),respectively;and 1
black squares and black triangles denote RI(1) and RI(2),
respectively; (b) Mott-insulating lobes, in plots of µ+(m) = 0
µ−VTRI2(m) and µ−(m) = µ−VTRO2(m) for m = 2 and 0 10 20 30 40 50 60
m = 1 versus U/(zt) (obtained by the conversion V0(Er) →
X
U/zt); we use the same symbols as in (a); and we show, for
comparison, the boundaries of the MI lobes for m = 1 (blue
diamonds) and m = 2 (blue triangles) that follow from our
mean-fieldtheoryforthehomogeneous Bose-Hubbardmodel5. FIG. 6: (Color online) Representative plots of Nb(ρ), the
(c)AnillustrativeplotofthetotalnumberofbosonsN inthe numberofbosonsinthesystemwithagivendensityρ,versus
system versusthechemical potential µ. ρnear(a)ρ=1and(b)ρ=2forV0 =12Er (blacksquares),
14Er (reddiamonds)and16Er (bluetriangles). (c)theradial
variation of density ρi for V0 = 12Er (black squares), 14Er
geneous Bose-Hubbard model, which we have obtained (red diamonds) and 16Er (bluetriangles).
fromthehomogeneousmean-fieldtheory5;theagreement
between these lobes is striking; and it encourages us to
suggestthatthephasediagramofthehomogeneousBose-
dependent transition-frequency shifts, sites with differ-
Hubbardmodelcanbeobtainedfromtheinnerandouter
entdensities of bosonscanbe distinguishedspectroscop-
radiioftheMIshells. Thus,experimentsoncoldatomsin
ically; and, therefore, MI shells, with different values of
opticallatticeswithaquadraticconfiningpotential22,can
the integer density m, are revealed as peaks in the oc-
beuseddirectlytoobtainthephasediagramofthehomo-
cupation number at the corresponding frequencies. This
geneous Bose-Hubbard model from R (m) and R (m),
O I experimentgivesN (ρ),thenumberofbosonsinthesys-
b
which can be determined for an MI shell with density
tem at a given density ρ. We use our inhomogeneous
ρ = m as described above. Note that (a) µ (m) and
− mean-field theory to obtain N (ρ) and in Figs. 6(a)-(b)
µ+(m) are fixed for a given V (E ) and (b) the total b
0 r we plot N (ρ) (with ρ close to m = 1 and 2, respec-
b
number of bosons N increases linearly with the chemi-
tively) for V /E =12, 14 and 16, with V /E =0.0003
0 r T r
cal potential µ (see Fig. 5(c)). Therefore, the inner and
and µ/E =1. The SF and MI shell structure is evident
r
outer radii of the MI shell RO,I(m) = p(µ−µ−,+/VT) fromtheradialvariationofthelocaldensitygiveninFig.
areproportionalto√N,forfixedVT andV0;thispropor- 6(c). For V0 =12Er, no Mott shells is developed; this is
tionality has been reported in the recent experiments22 reflected in a flat variation of N (ρ) for all ρ. However,
b
[cf., their Fig. 3]. if V = 14E , there is a well-formed MI shell ρ = 1; this
0 r
Images of MI shells have been obtained recently from can be inferred from the peak in N (ρ) at ρ=1; and, as
b
atomic-clock-shift experiments23. By using the density- V increases,more Mott shells, with higher, integral val-
0
8
uesofρ,appear. ThisbehaviorofN (ρ)isinaccordance phase diagram shows the following: an SF phase; blue
b
with recent experiments23 [cf., their Fig. 1]. MI1,brownMI 1,MI 1,andpinkMI 2MI 2lobes;green
a b a b
MI 1 and dark-green MI 2 regions; these are like their
a a
counterparts in Figs. 7 (a) and (e). In addition we have
B. Results for the Bose-Hubbard model with two a red MIb2MIa1 lobe in which ρb =2 and ρa =1.
species of bosons Next we consider the case Uab < Ua, Ub = Ua, and
µ = 0.75µ . Specifically, in the first row of Fig. 8 we
b a
showthe phase diagram(Fig.8 (a)), andplotsversusµ
Webeginwithaninvestigationofrepresentativephase a
oftheorder-parametersψ (redline)andψ (bluedashed
diagrams of the Bose-Hubbard model (3), with two a b
line) and the densities ρ (green full line) and ρ (pink
species of bosons, in the homogeneous case, i.e., with a b
dashed line) for U =0.1U , U =U , and µ =0.75µ
V =V =V =0. These have been explored to some ab a a b b a
Ta Tb T
extent in earlier theoretical studies9–13 and Monte Carlo andUa =9(Fig.8(b)),Ua =11(Fig.8(c)),andUa =13
simulations14, but not over as wide a range of parame- (Fig. 8 (d)). (We do not divide explicitly by zt because
we set zt = 1). The phase diagram shows an SF phase
tersasweconsiderhere. Nextweusetheinhomogeneous
andbrownMI 1MI 1,pinkMI 2MI 2andredMI 2MI 1
mean-field theory that we have developed above to ex- a b a b a b
lobes; green MI 1, dark-green MI 2 and a green-ochre
plore order-parameter profiles and a variety of MI and a a
MI 2 regions; these are like their counterparts in Figs. 7
SF shells that are obtained when we have a quadratic b
(a)and(e). Inadditionwehavealight-blueregionMI 1
trap potential. We also present Fourier transforms of b
inwhichbosonsoftypebareinanMIphasewithρ =1
one-dimensional sections of these profiles. b
and bosons of type a are superfluid.
First we consider the case U < U = U and µ =
ab a b a
µ = µ in which the order parameters and densities for InthesecondrowofFig.8weshowthephasediagram
b
both types of bosons show the same dependence on µ. (Fig. 8 (e)), and plots versus µa of the order-parameters
In the first row of Fig. 7 we show the phase diagram ψa and ψb and the densities ρa and ρb for Uab = 0.3Ua,
(Fig. 7 (a)), and plots versus µ of the order-parameters Ub = Ua, and µb = 0.75µa and Ua = 9 (Fig. 8 (f)),
ψa (red line) and ψb (blue dashed line) and the densities Ua = 11 (Fig. 8 (g)), and Ua = 13 (Fig. 8 (h)). This
ρ (green dashed line) and ρ (pink full line) for U = phase diagram shows the following: an SF phase; brown
a b ab
0.5Ua, Ua = Ub, and µa = µb = µ and Ua = 9 (Fig. 7 MIa1,MIb1 and red MIa2MIb1 lobes; green MIa1, dark-
(b)), Ua =11(Fig.7(c)), andUa =13(Fig.7(d)). (We green MIa2 and light-blue MIb1 regions; these are like
do not divide explicitly by zt because we set zt = 1). their counterparts in Figs. 8 (a). In addition we have a
The phase diagram shows an SF phase in which both dark-graySFa phase in which bosons of type a are in an
speciesaresuperfluid;theblueMI1lobedenotesaMott- SF phase and the bosons of type b have vanished.
insulating phase in which the density ρ = 1 is attained In the third row of Fig. 8 we show the phase diagram
by having ρa = ρb = 1/2; the brown MIa1MIb1 lobe (Fig. 8 (i)), and plots versus µa of the order-parameters
denotes a Mott-insulating phase in which the densities ψa and ψb and the densities ρa and ρb for Uab = 0.7Ua,
ρa = ρb = 1; the pink MIa2MIb2 lobe denotes a Mott- Ub = Ua, and µb = 0.75µa and Ua = 9 (Fig. 8 (j)),
insulatingphaseinwhichthedensitiesρa =ρb =2. Such Ua = 11 (Fig. 8 (k)), and Ua = 13 (Fig. 8 (l)). The
phase diagrams can be obtained from plots like those in phase diagram shows the following: an SFa phase; MIa1
Figs. 7 (b)-(d). and MIa2 regions; these are like their counterparts in
InthesecondrowofFig.7weshowthephasediagram Figs. 8 (a) and (e) in which the bosons density for type
(Fig. 7 (e)), and plots versus µ of the order-parameters b has vanished.
ψa and ψb and the densities ρa and ρb for Uab = 0.2Ua, We now consider the case Uab < Ua, Ub = 0.9Ua, and
Ub = 0.9Ua, and µa = µb = µ and Ua = 9 (Fig. 7 (f)), µb =0.9µa. Specifically,inthefirstrowofFig.9weshow
Ua =11(Fig.7(g)),andUa =13(Fig.7(h)). Thephase thephasediagram(Fig.9(a)),andplotsversusµa ofthe
diagram shows an SF phase and brown MIa1MIb1 and order-parametersψa (redline) andψb (blue dashedline)
pinkMIa2MIb2lobes;theseareliketheircounterpartsin andthe densitiesρa (greenfullline)andρb (pink dashed
Fig.7(a). Inadditionwehavethefollowingphases: (i)a line) for Uab = 0.2Ua, Ua = 0.9Ub, and µb = 0.9µa and
greensliver MIa1 inwhich bosonsoftype a are inanMI Ua = 9 (Fig. 9 (b)), Ua = 11 (Fig. 9 ( c)), and Ua = 13
phase with ρ = 1 and bosons of type b are superfluid; (Fig. 9 (d)). The phase diagram shows an SF phase and
a
(ii) a green-ochre region MIb2 in which bosons of type b brown MIa1MIb1 and pink MIa2MIb2 lobes; green MIa1
are in an MI phase with ρb =2 and bosons of type a are anddark-greenMIa2;these areliketheircounterpartsin
superfluid; and (iii) a dark-green region MI 2 in which Figs. 7 and Figs. 8 (a) and (e).
a
bosons of type a are in an MI phase with ρa = 2 and InthesecondrowofFig.9weshowthephasediagram
bosons of type b are superfluid. (Fig. 9 (e)), and plots versus µ of the order-parameters
a
In the third row of Fig. 7 we show the phase diagram ψ and ψ and the densities ρ and ρ for U = 0.5U ,
a b a b ab a
(Fig. 7 (i)), and plots versus µ of the order-parameters U = 0.9U , and µ = 0.9µ and U = 9 (Fig. 9 (f)),
b a b a a
ψ and ψ and the densities ρ and ρ for U = 0.6U , U = 11 (Fig. 9 (g)), and U = 13 (Fig. 9 (h)). The
a b a b ab a a a
U = 0.9U , and µ = µ = µ and U = 9 (Fig. 7 (j)), phasediagramshowsthefollowing: anSFphaseandSF ;
b a a b a a
U = 11 (Fig. 7 (k)), and U = 13 (Fig. 7 (l)). The brown MI 1,MI 1, red MI 2MI 1 and pink MI 2MI 2
a a a b a b a b
9
30 3 2.5 2
a
25 b c d
b) 2 1.5
20 MI 2 , MI 2 a,2
a b ψ( 1.5
µ 1105 SF MIa 1 , MIb 1 , a,b)1 1 1
5 ρ( 0.5 0.5
MI 1
6 8 U 10 12 14 00 10µ 20 30 00 5 10 15 20 25 30 00 5 10 15 20 25 30
30 MI 2 4 3 3
e a
MI 2 , MI 2 f g h
a b b)3
20 a, 2 2
µ SF MIb2 MIa 1 ψ , b)(2
10 MI 1 , MI 1 a, 1 1
a b ρ(1
0 0 µ 0 0
6 8 U 10 12 14 0 5 10 15 20 25 30 0 5 10 15 20 25 30 0 5 10 15 20 25 30
30 a 3 2.5 2
25 i MIaM 2I a, M2Ib 2 MIb 2 , MIa 1 b)2.5 j 2 k 1.5 l
20 a, 2
µ 15 SF MIa 1 MIa 1 , MIb 1ψ , b)(1.5 1.15 1
10 a, 1
ρ( 0.5 0.5
5 MI 1 0.5
0 6 8 Ua 10 12 14 00 10 µ 20 30 00 5 10 15 20 25 30 00 5 10 15 20 25 30
MI 1
FIG. 7: (Color online) Phase diagram (a), and plots versus µ of the order-parameters ψa (red line) and ψb (blue dashed line)
and the densities ρa (green dashed line) and ρb (pink full line) for Uab = 0.5Ua, Ua = Ub, and µa = µb = µ and Ua = 9 (b),
Ua =11(c),andUa =13(d). (Wedonotdivideexplicitlybyztbecausewesetzt=1). ThephasediagramshowsanSFphase
in which bothspecies aresuperfluid;theblueMI1lobe denotesaMott-insulating phasein which thedensityρ=1is attained
by having ρa =ρb = 1/2; the brown MIa1MIb1 lobe denotes a Mott-insulating phase in which the densities ρa =ρb = 1; the
pinkMIa2MIb2lobedenotesaMott-insulating phasein whichthedensitiesρa =ρb=2. Inthesecond rowweshowthephase
diagram (e), and plots versus µ of the order-parameters ψa and ψb and the densities ρa and ρb for Uab =0.2Ua, Ub = 0.9Ua,
and µa =µb =µ and Ua =9 (f), Ua =11 (g), and Ua =13 (h). The phase diagram shows an SFphase and brown MIa1MIb1
and pink MIa2MIb2 lobes; these are like their counterparts in (a). In addition we have the following phases: (i) a green sliver
MIa1 in which bosons of type a are in an MI phase with ρa =1 and bosons of type b are superfluid; (ii) a green-ochre region
MI 2 in which bosons of type b are in an MI phase with ρ = 2 and bosons of type a are superfluid; and (iii) a dark-green
b b
region MIa2inwhichbosonsoftypeaareinanMIphasewithρa =2andbosonsoftypebaresuperfluid. Inthethirdrowwe
showthephasediagram(i),andplotsversusµoftheorder-parametersψa andψb andthedensitiesρa andρb forUab =0.6Ua,
Ub =0.9Ua, and µa = µb =µ and Ua =9 (j), Ua = 11 (k), and Ua =13 (l). The phase diagram shows the following: an SF
phase; blue MI1, brown MIa1,MIb1, and pink MIa2MIb2 lobes; green MIa1 and dark-green MIa2 regions; these are like their
counterparts in (a) and (e). In addition we havea red MIb2MIa1 lobe in which ρb=2 and ρa =1.
lobes; green MI 1, dark-green MI 2 and light-blue MI 1 the total number of bosons N 106, which is compa-
a a b T
≃
regions; these are like their counterparts in Figs. 8 (a) rable to experimental values. Furthermore, this choice
and (e). of parameters leads not only to SF shells but also two
We now consider the effect of a parabolic potential well-developed MI shells (MI1 and MI2) .
anduse theinhomogeneousmean-fieldtheory,developed Weshowrepresentativeplotsofthedensitiesρ (green
a
in the previous Section, to obtain alternating spherical dashed line) and ρ (pink line) versus the position X
b
shells of the variety of MI and SF phases, shown in along the line Y = Z = 0 are shown in Figs. 10 for
the phase diagrams in Figs. 7, 8, and 9, for the two- µ =µ =30, V /(zt)=0.008,andthe followingsix pa-
a b T
speciesBHmodel(3). Wedothisbyobtainingtheorder- rameter sets, respectively: (a) U /(zt)= 8, U = 0.9U ,
a b a
parameter profiles ψ ,ρ ;ψ ,ρ and also by obtain- and U = 0.6U (b) U /(zt) = 10, U = 0.9U , and
ai ai bi bi ab a a b a
{ }
ing in-trap density distributions of bosons at represen- U = 0.6U , (c) U /(zt) = 11, U = 0.9U , and U =
ab a a b a ab
tative values of U , U , U , and µ = µ . In particu- 0.6U , (d) U /(zt) = 12, U = 0.9U , and U = 0.6U ,
ab a b a b a a b a ab a
lar, we use a 3D simple-cubic lattice with 1283 sites and (e) U /(zt)= 13, U = 0.9U , and U = 0.6U , and (f)
a b a ab a
V /(zt) = 0.008; and we study the following represen- U /(zt) = 14, U = 0.9U , and U = 0.6U . It is also
T a b a ab a
tative case: µ /(zt) = µ /(zt) = 30 , U = 0.6U , usefultoobtainacomplementary,Fourier-representation
a b ab a
U =0.9U , when U /(zt)=13. With these parameters pictureofthe profilesinFigs.10(a)-(f)becauseitmight
b a a
10
30 4 4 3
a MI 2
20 MIa 2 , MIb 2 MbIa 2 ψ(a,b)3 b 3 c 2 d
µa MIa 2 , MIb 1 MIb 1 , b)2 2
10 SF MIa 1 , MIb 1 ρ(a,1 1 1
MI1
0a 6 8 U 10 12 14 00 5 10 µ15 20 25 30 00 5 10 15 20 25 3000 5 10 15 20 25 30
30 a
20 e MIa 2 , MIb 1 MIa 2 ψ(a,b)34 f 23 g 12..255 h
µa10 SF MIb 1 MIa 1 , MIb 1 , a,b)2 1 1
ρ(1 0.5
MI1
a SF
0 6 8 U 10 12 14a 00 5 10µa15 20 25 3000 5 10 15 20 25 30 00 5 10 15 20 25 30
30 4 4 3
20 i MI 2 a,b)3 j 3 k 2 l
SF a ψ(
µa a , b)2 2
10 a, 1
MI 1 ρ(1 1
a
0 0 µ 0 0
6 8 U 10 12 14 0 5 10 a15 20 25 30 0 5 10 15 20 25 30 0 5 10 15 20 25 30
FIG.8: (Color online) Phasediagram (a),and plotsversusµa of theorder-parametersψa (red line) and ψb (bluedashed line)
and the densities ρa (green full line) and ρb (pink dashed line) for Uab = 0.1Ua, Ua = Ub, and µb = 0.75µa and Ua = 9 (b),
Ua = 11 (c), and Ua = 13 (d). (We do not divide explicitly by zt because we set zt = 1). The phase diagram shows an SF
phase and brown MIa1MIb1, pink MIa2MIb2 and red MIa2MIb1 lobes; green MIa1, dark-green MIa2 and a green-ochre MIb2
regions; theseareliketheircounterpartsinFigs.7(a)and(e). Inadditionwehavealight-blueregionMI 1inwhichbosonsof
b
typeb are in an MI phasewith ρ =1 and bosons of typea are superfluid. In thesecond row we show thephase diagram (e),
b
andplotsversusµa of theorder-parametersψa andψb andthedensitiesρa andρb forUab =0.3Ua,Ub =Ua,andµb =0.75µa
and Ua = 9 (f), Ua =11 (g), and Ua =13 (h). The phase diagram shows the following: an SF phase; brown MIa1,MIb1 and
red MIa2MIb1 lobes; green MIa1, dark-green MIa2 and light-blue MIb1 regions; these are like their counterparts in (a). In
additionwehaveanSFa phaseinwhichbosonsoftypeaareinan SFphaseandthebosonsdensityoftypebarevanished. In
thethird row we show thephase diagram (i), and plots versusµa of the order-parameters ψa and ψb and thedensities ρa and
ρb for Uab = 0.7Ua, Ub = Ua, and µb = 0.75µa and Ua = 9 (j), Ua = 11 (k), and Ua = 13 (l). The phase diagram shows the
following: an SFa phase; MIa1 and MIa2 regions; these are like their counterparts in (a) and (e) in which the bosons density
of typeb are vanished.
be possible to obtain them in time-of-flight measure- areshowninFigs.12(a)-(f)forthesameparameterval-
ments1. Three-dimensionaltransformsofthe shellstruc- uesasinFigs.10(a)-(f),respectively;intheseplotsψ is
a
ture can be obtained, but they are not easy to visualize; indicatedby a reddashedline and ψ by a blue line. We
b
therefore, we present the one-dimensional Fourier trans- can also obtain the one-dimensional Fourier transforms
forms of ρ (X,Y = 0,Z = 0) and ρ (X,Y = 0,Z = 0) of ψ (X,Y = 0,Z = 0) and ψ (X,Y = 0,Z = 0) with
a b a b
with respect to X. The moduli of these transforms, respect to X. The moduli of these transforms, namely,
namely, ρ (green lines), and ρ (pink lines) of ψ (red lines) and ψ (blue lines) of the profiles
| a,kX| | b,kX| | a,kX| | b,kX|
theprofilesinFigs.10(a)-(f)areplotted,respectively,in Figs. 12(a) - (f) are plotted, respectively, in Figs. 13(a)
Figs. 11 (a)-(f) versus the wave vector k /π. The prin- - (f) versus the wave vector k /π . Again, the princi-
X X
cipal peaks in these transforms occur at k =0 (or 2π); pal peaks in these transforms occur at k = 0 (or 2π);
X X
these are associated with the spatially uniform MI and butsubsidiarypeaksoccurbecauseofthe shellstructure
SF phases . In an infinite system with no confining po- imposed by the confining potential.
tential, these are the only peaks; however, the quadratic In Fig. 14(a) - (d) we show representative density and
confining potential leads to shells of MI and SF phases order-parameter profiles, for ρ and ψ , and the mod-
a a
(see below); this shell structure leads to the subsidiary uli of their Fourier transforms for U = 2.22U , U =
ab a b
peaksthatappearinFigs.11(a)-(f)awayfromkX =0, 0.65Ua, Ua = 13, µb = 0.8µa,VT = 0.008, and L = 64.
and 2π. With this set of parameter values ρ and ψ vanish.
b b
Wecanalsoobtainorder-parameter-profileplots;these The density and order-parameter profiles of Figs. 10