Table Of ContentSuperfluid-Mott-Insulator Transition in a One-Dimensional Optical Lattice with
Double-Well Potentials
H. C. Jiang,1 Z. Y. Weng,1 and T. Xiang2
1Center for Advanced Study, Tsinghua University, Beijing,100084,China
2Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100080, China
(Dated: February 6, 2008)
7
Westudythesuperfluid-Mott-insulatortransitionofultracoldbosonicatomsinaone-dimensional
0
0 optical lattice with a double-well confining trap using the density-matrix renormalization group.
2 At low density, the system behaves similarly as two separated ones inside harmonic traps. At
high density, however, interesting features appear as the consequence of the quantum tunneling
n
between the two wells and the competition between the “superfluid” and Mott regions. They
a
are characterized by a rich step-plateau structure in the visibility and the satellite peaks in the
J
momentum distribution function as a function of the on-site repulsion. These novel properties
3
shed light on the understanding of the phase coherence between two coupled condensates and the
1
off-diagonal correlations between thetwo wells.
]
l PACSnumbers: 03.75.Hh,03.75.Lm,05.30.Jp
e
-
r
t I. INTRODUCTION stronglycorrelatedandarichphenomenonrelatedtothe
s
. evolution of the “superfluid” and Mott insulator regions
t
a as a function of the on-site repulsion is obtained. A de-
The experimental realization of trapped ultracold
m tailed discussion on how to detect these features based
bosonic gases on optical lattices has opened up the pos-
on the visibility, momentum distribution, hopping cor-
- sibility of observing various quantum phases (e.g., su-
d relation function, and other physical quantities is given.
n perfluid and Mott insulator) and studying the quantum
Thesenovelpropertiesshedlightontheunderstandingof
o phasetransitionsbetweentheminawell-controlledman-
thephasecoherenceandoff-diagonalcorrelationbetween
c ner. Withouttheconfiningtrap,thesystemwillundergo
[ a superfluid-Mott-insulator phase transition at integer thecoupledcondensatesinadouble-wellinteractingBose
1 fillings. In the presence of a confining trap, the super- system on optical lattices.
A cold atomic Bose gas on a one-dimensional optical
v fluid and Mott-insulator regions may coexist1,2 due to
latticeinthepresenceofadouble-wellpotentialtrapcan
0 the inhomogeneity induced by the trap.
bedescribedbythefollowingBose-HubbardHamiltonian
9
Numerical calculations have been carried out for the
2
systems without3,4 and with a harmonic trap5,6,7,8, us- H = −t aˆ†aˆ +h.c.
1 i i+1
0 ing the quantum Monte Carlo and density-matrix renor- Xi (cid:16) (cid:17)
7 malization group9,10 (DMRG) methods. Interesting new
+U nˆ (nˆ −1)/2+ V (i)nˆ , (1)
i i T i
0 featuresappearinthesystemwithasingleharmoniccon-
/ finingtrap,whichareassociatedwiththecompetitionbe- Xi Xi
t
a tween the “superfluid” and Mott insulator regimes and where
m
characterizedbythe “kinks”7 inthe visibility ofinterfer- L+1 2 L+1 4
- ence fringes with the increase of the on-site repulsion. VT(i)=VT1 i− +VT2 i− (2)
d (cid:18) 2 (cid:19) (cid:18) 2 (cid:19)
n Recently a bosonic Josephson junction composed of
is the double-well trap with quadratic V and quartic
o two weakly coupled Bose-Einstein condensates in a T1
V coefficients, and L is the number of sites. The hop-
c macroscopic double-well potential has been experimen- T2
v: tally realized11. It raises an interesting question con- tphinegoinn-tseitgerarletpuislssieotn.asaˆt†haenudnaˆitoafreentehregybots=on1ic,acnredaUtioins
i cerning the quantum tunnelling effect in a double-well i i
X potential system in which the ultracold atoms on opti- andannihilationoperators,respectively,andnˆi =aˆ†iaˆi is
r cal lattice are trapped. In such a system, the inter-atom the number operator.
a
interaction and the competition between the “superflu- An important parameter in characterizing the phase
idity” and Mott insulator phases are expected to play a coherenceinasuperfluid-Mott-insulatortransitionisthe
crucial role in shaping the inter-well tunnelling. visibility of interference fringes defined by12,13
Inthispaper,weintendtoclarifythisissuebystudying Smax−Smin
ν = , (3)
a one-dimensional Bose-Hubbard model in the presence S +S
max min
of a double-well potential using the DMRG. At low fill-
whereS andS arethe maximumandminimumof
max min
ing (ρ=1/2), we find the system simply behaves as two
the momentum distribution function
separatedones,eachofwhichisinaharmonictrapwith-
1
out significant coupling between them. At a higher fill- S(k)= L eik·(ri−rj) aˆ†iaˆj . (4)
ing(ρ=1),however,the bosonsinthetwowellsbecome Xi,j D E
2
In the presence of a confining trap, it is useful to intro-
ducethelocaldensityofbosonsn =<nˆ >andthelocal
i i
particle fluctuation6 1.5
U=4.0t
κ ≡[ nˆ2 −n2], (5) U=5.8t
i i i 1.0 U=6.2t
(cid:10) (cid:11) ni U=20 t
to measure the inhomogeneity induced by the confining
trap. To further quantify the spatial correlations within 0.5
the trap potential, we have also calculated the hopping
correlation function
0.0
g(i,j)= aˆ†aˆ (6) 0.4
i j
D E
i
between two lattice sites i and j, the total interaction
energy E , 0.2
U
E =U n (n −1)/2, (7)
U i i
0.0
Xi 0 20 40 60 80
i
the kinetic energy |E |,
k
E =−t aˆ†aˆ +h.c. , (8) FIG. 1: Density profiles ni and the corresponding particle
k Xi D i i+1 E flUu=ctu6a.2ttionneaκrilyfocroiρnc=ide1./2. The profiles for U = 5.8t and
and the trapping energy E
T
inside the two wells atlarge U, due to the low density of
E = V (i)n , (9)
T T i
the system. Fig. 1 shows the density profile and corre-
Xi
spondingparticlefluctuationatvariousvaluesofU/t. At
respectively. small U/t, two separate superfluid regions appear in the
two wells. When U is increased, the superfluid regions
shrinkuntiltheMott-insulatordomainsemergewhenU/t
II. SUPERFLUID-MOTT-INSULATOR is larger than 6.8.
TRANSITION To further quantify this transition, let us define the
integrated density
In the following we present our DMRG calculations
l2
for the Bose-Hubbard model (1). Compared with a har-
N = n (10)
monictrap7,the presentsystemshowsaricherandmore d i
complex structure, due to the coupling between the two iX=l1
wells. When the density is low enough suchthat there is in a range [l ,l ]. In the present system, we choose l =
1 2 1
noparticleinthe middle ofthe system,the systemsplits 11andl =22intheleft-sidewell. AsshowninFig.2(a),
2
into two weakly coupled harmonic ones. However, if the N issaturatedtoaplateauofN =12correspondingto
d d
density is sufficiently high that particles in the two wells a local Mott-insulator state (n = 1) as U/t is increased
i
are not well separated,the interaction between the wells beyond 6.8.
will play a crucial role, which may change the behavior Fig.2(a) shows the visibility ν. Two kinks can be
of the system dramatically. observed. The first one (less evident) occurs around
In the following, we will consider the system with 80 U/t = 5.5, while the second one around U/t = 6.4. As
sites in a double-well trap potential with VT1 =−0.024t U/t is increased between 5.8 and 6.2, there is almost no
and VT2 = 2×10−5t. The calculations are carried out change in the density profile. This is due to presence of
usingtheDMRGwithopenboundaryconditions,fortwo the emergent Mott-insulator regionssurrounding the su-
different filling factors ρ=1/2 and ρ=1. perfluid ones in the two wells. The atoms in the central
superfluidregionscantransfertothe outersuperfluidre-
gions at larger U/t. This will eventually exhaust the su-
A. Filling ρ=1/2 perfluidregionsinthetwowellcenters. AtU/t=20,the
flat Mott-insulator plateaus, with vanishing local parti-
AT the filling ρ = 1/2, the system will not reach a cle fluctuation, centeredaroundthe two wells areclearly
Mott-insulator state, even in the Tonks-Girardeau (TG) seen.
limit U → ∞, without a trap potential V . In the pres- As shownin Fig. 2, the visibility with the correspond-
T
enceofthedouble-wellpotential,however,twoseparated ing E , E and the ratio γ =|E /E |, exhibits a char-
T U U k
Mott-insulator domains with the density n = 1 appear acteristic kink structure. The quick decrease of E in
i U
3
1.0 15 1.0 30
14 Nd 0.8
0.9
25 d
(a) Nd 13 0.6 Nd N
12 0.4 (a)
0.8
20
11 0.2
40 -220 120 -200
ES
E3U 0 (b) ET -230 ET E|k80 -260 ES
|
20 -240 (b) |Ek| -320
EU 40
10 -250
0.6 2.0
0.5 1.5
(c)
1.0
0.4
(c)
0.5
0.3
4 5 6 7 8 9 6 12 18 24
U/t U/t
FIG. 2: (a) Visibility ν and integrated density Nd; (b) FIG.4: (a)VisibilityνandintegrateddensityNdfroml1 =11
Interaction energy EU and trapping energy ET; (c) Ratio to l2 = 30, (b) Kinetic |Ek| and total energy ES, (c) Ratio
γ = |EU/Ek| of interaction to kinetic energy, as functions of γ =|EU/Ek| of interaction to kinetic energy, as functions of
U/t, for thefilling factor ρ=1/2. U/t for ρ=1.
tion function S(k) for the system shown in Fig.1. When
U/tissmall,thesystemisinasuperfluidstateandasin-
glenarrowpeakappearsatzeromomentuminS(k). For
6 U=4.0t large U/t, S(k) is broadened. The quantitative change
U=5.8t in S(k) is best reflected in the kink structure of the
U=6.2t visibility ν. In he Tonks-Girardeau limit, we find that
k)4 U=20 t S = 1.04 and ν = 0.44. The non-zero visibility ν re-
( max
S veals the presence of the superfluid regions surrounding
2 the Mott-insulator plateau regions (Fig. 1).
Finally, we note that at ρ = 1/2, the distribution of
0 bosons remains well separated in the two wells [Fig.1].
-0.6 -0.3 0.0 0.3 0.6
Therefore, it can be approximately treated as two single
k wells with weak coupling. Indeed, the results presented
above are consistent with those obtained in a harmonic
trap7.
FIG.3: MomentumdistributionfunctionS(k)forthesystem
shown in Fig. 1.
B. Filling ρ=1
the interval U/t=6.2−6.8 shows a greatsuppressionof
the double occupancyinthe twowells. Itis a manifesta- At ρ = 1, bosons inside the two wells cannot be sepa-
tion of the transition from superfluid to Mott-insulator rated for sufficiently large U. The system is in a Mott-
in the two wells. This occurs even though the total en- insulator state without the trap in the TG limit.
ergy increases continuously with the increase of U/t. By Fig.4(a) shows the visibility ν and the integratedden-
furtherincreasingU/t,thedensityprofileremainsalmost sity N (l = 11, l = 30) defined in Eq.(10). The evo-
d 1 2
unchanged. lution of these two quantities exhibit a rich step-plateau
Fig. 3 shows the corresponding momentum distribu- structurecomparedwiththe caseofρ=1/2. WhenU is
4
1.5 1.5 18 18
U=7.0t U=7.6t
ni1.0 1.0 ni 12 12
0.5 0.5 6 6
U=9.9t U=22.0t
(a) U =10.0t (b) U= 22.1t
0.0 0.0 0 0
15 15
i0.3 0.3 i
U=8.0t U=8.8t
0.2 0.2 k)10 10
S(
0.1 0.1
5 5
0.0 0.0
0 20 40 60 80 0 20 40 60 80
0 0
i i 10 10
U=9.9t U=10.0t
FIG. 5: Density profiles ni and the corresponding local par-
ticle fluctuation κi for ρ=1. 5 5
0 0
small (about ≤ 10t), the central Mott-insulator domain -0.4 -0.2 0.0 0.2 0.4-0.4 -0.2 0.0 0.2 0.4
has not formed. In this parameter regime, the plateaus
k k
in ν and N are quite narrow, indicating that the Mott-
d
insulator domains surrounding the superfluid ones are
FIG. 6: Momentum distribution function S(k) at different
notrobustandthebosonscanescapefromthesuperfluid
valuesof U/t for ρ=1.
regions in the two wells. This feature is well reflected in
the density profiles and local particle fluctuation around
the step between U/t=9.9 and 10 as shown in Fig.5(a).
satellite peaks disappear.
WhenU/tisfurtherincreased,thecentralregionofthe
More detailed study demonstrates that the satellite
system shows a flat Mott-insulator plateau with n = 1
i peak structure in S(k) is related to the existence of a
[Fig. 5]. This leads to a sudden decrease in ν and N .
d strong off-diagonal long-range correlation between the
Due to the largeon-site repulsionU andthe existence of
two wells. This can be seen from Fig. 7, in which the
the central Mott-insulator regions, bosons can only tun-
hoping correlation function g(16,i) between site 16 and
nel from the inner superfluid region to the outer ones.
site i is plotted. If the system has an off-diagonal long-
Wheneverabosonescapesfromthesuperfluidregion,the
rangecorrelation,the satellite peaksarepresentinS(k).
visibility ν as well as the integrated density N shows a
d If the system does not have the off-diagonal long-range
sharp decrease. When U/t is larger than 22, the super-
correlation,the satellite peaksdisappear. These features
fluid region disappears competely (Fig.5(b)). In the TG
are different from those in a harmonic trap5. All the
limit, the whole system is in a Mott-insulator state and
abovefeaturesappearbeforetheformationofthecentral
κ =0.
i Mott-insulator plateau. In the presence of the central
As shown in Fig. 4(b), the abrupt changes are also
Mott-insulator plateau, however, the satellite structure
present in |E | at the steps of ν, although the total en-
k of S(k) as well as the long-rangeoff-diagonalcorrelation
ergy E increases smoothly with U/t. The steps in |E |
S k between the two wells disappears. This suggests that
can be understood as the abrupt redistribution of the
onecanjudge whetherornotthe off-diagonallong-range
bosondensitybetweenthe superfluidandMott-insulator
correlation exists from the measurement of momentum
regions. During this process, the superfluid region be-
distribution functions.
comes smaller, while the Mott-insulator becomes larger.
Themonotonicdecreaseof|E |withincreasingU/tisdue
k
tothewideningoftheMottinsulatorregionsinwhichthe
kinetic energy is suppressed. In the plateau regions, the III. SUMMARY
increase in γ is also due to the reduction of the kinetic
energy. To summarize, we have explored the superfluid-Mott-
Fig. 6 shows the momentum distribution function insulatortransitioninadouble-welltrappedatomicBose
S(k). S(k) exhibits different features at different U/t gasinaone-dimensionalopticallatticeusingtheDMRG.
regimes. Furthermore, S(k) is correlated with ν. When We find that this transition is well characterized by the
the satellite peaks appear in S(k), ν undergoes a sharp visibility ν, which exhibits a series of kink structures as-
increase; while ν undergoes a sharp decrease when the sociated with the redistribution of bosons between local
5
showsa number of characteristicfeatures in the localin-
tegrateddensity,thedensityprofile,localcompressibility,
2.0 2.0 and the momentum distribution.
U=7.0t U=7.6t
1.5 1.5 At low density (e.g. ρ = 1/2), the system behaves
like two weakly coupled harmonic traps. With the in-
1.0 1.0
crease of the density, the system begins to develop novel
0.5 0.5 propertiesunique to the double wellpotential associated
with a strong quantum tunneling effect between the two
0.0 0.0
wells. At high density (e.g. ρ = 1), a series of steps
2.0 2.0
and plateaus appear in the visibility and other physical
U=8.0t U=8.8t
6,i)1.5 1.5 quantities. Thesefeaturesarerelatedtotheabruptredis-
11.0 1.0 tribution of the bosons in the local superfluid and Mott-
(
g insulator regions. In addition, the presence of the satel-
0.5 0.5
lite peaks in the momentum distribution function indi-
0.0 0.0 cates that the strong off-diagonal long-range correlation
2.0 2.0 betweenthesuperfluidregionsareseparatedbytheMott
U=9.9t U=10.0t insulator regions. Therefore, it is an effective probe for
1.5 1.5
experiment to identify the off-diagonal long-range corre-
1.0 1.0 lationandphasecoherenceduetothetunnellingbetween
the two condensates on the optical lattice.
0.5 0.5
0.0 0.0
0 20 40 60 80 0 20 40 60 80
i i IV. ACKNOWLEDGEMENT
FIG.7: Hoppingcorrelation function g(16,i) betweensite 16 We are grateful for helpful discussions with D.N.
and i at different values of U/t for ρ=1. Sheng, F. Ye, Y.C. Wen, X.L. Qi, Y. Cao, Z.C. Gu.
Support from the NSFC grants and the Basic Research
Programof China is acknowledged.
superfluid and Mott-insulator regions, as a function of
theon-siterepulsionU. Theevolutionofthesystemalso
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