Dynamical instability and loss of p-band bosons in optical lattices

0.19 MB·English

by  J. -P. Martikainen

Dynamical instability and loss of p-band bosons in optical lattices
J.-P. Martikainen1,∗
1NORDITA, 106 91 Stockholm, Sweden
(Dated: January 24, 2011)
We study how the bosonic atoms on the excited p-band of an optical lattice are coupled to the
lowest s-bandandthe2ndexcitedd-band. Wefindthatinsomeparameterregimestheatom-atom
interactions can cause a dynamical instability of the p-band atoms towards decay to the s and d-
1 bands. Furthermore, even when dynamical instability is not expected s- and d-bands can become
1 substantially populated. We also find that, the stability properties of the excited bands can be
0
improved in superlattices.
2

n PACSnumbers: 03.75.-b,03.75.Lm,03.75.Mn
a
J
1 I. INTRODUCTION can induce important corrections to the simplest lowest
2 band Hubbard model [15, 16].
The purpose if this article is to point out the exis-
The study of cold atoms in optical lattices has seen a
] tence of dynamical instability which can adversely af-
s dramatic experimental progress in the recent past [1, 2].
a fect the stability of the p-band atoms in the superfluid
Due to realization of optical lattices, low densities, and
g phase. This instability is caused by the scattering of p-
low temperatures, a fantastic degree of control has been
- band atoms into s- and d-band atoms. While the anhar-
t obtained which has made detailed studies of strongly
n monicitiesinanopticallattice canmakethis processoff-
a correlated quantum systems possible. For example, the
resonant for all quasi-momenta, interplay between band
u Mott-superfluidtransition[3,4]hasbeensuccessfullyob-
structureandatom-atominteractionscanstillinducein-
q served in optical lattices.
. stability. We also find that even in the absence of dy-
at The early experiments were confined to the lowest namical instability, the s- and d-bands can become sub-
m band and while increasing interactions can make excited stantiallypopulated. Furthermore,dynamicalinstabities
bandpopulationsnon-negligible[5],the lowestbandstill can be removed in superlattices [8].
-
d dominates. Experimentally, however, atomic population The paper is organized as follows. In Sec. II we out-
n residing on the excited bands can be obtained by cou- line a theory which captures the essential physics of the
o pling atoms from the lowest band to the excited bands. loss of p-band atoms to s and d-bands Furthermore, in
c
[ Thishasbeenexperimentallydemonstratedbyaccelerat- this section we discuss realistic magnitude of various pa-
ingthelatticeforashortperiod[6]orbycouplingatoms rameters occurring in our theory. In Sec. III we derive
2 from the lowest band Mott insulator into the first ex- and solve the Bogoliubov-de Gennes equations for the
v citedp-bandofthelatticeviaRamantransitionsbetween collectivemodesandfindthe possibilityofdynamicalin-
5
bands [7]. In this latter study, it was in particular found stability. In Sec. IIIA we point out the importance of
6
4 that the lifetimes of p-band atoms can be considerably condensationfor the occurrenceof dynamicalinstability,
5 longer than the tunneling time-scale in the lattice and but find that s- and d-bands can be substantially pop-
. it was also possible to explore how coherence of bosons ulated even in the absence of dynamical instability. In
9
on the excited band was established. Very recently p-
0 Sec. IIIB we analyze similar processes in superlattices
0 band bosons and their orbital ordering in the superfluid andfind thattherethe dynamicalinstabilitiescanbe re-
1 phase were studied experimentally in a bipartite optical moved. We end with few concluding remarks in Sec. IV.
: lattice [8]. In this experiment atoms were transferred to
v
thep-orbitalsbydeformingthesuperlatticeinsuchaway
i
X thatatomsoriginallyinthelowestorbitalofcertainsites,
II. FORMALISM
r could tunnel to the p-orbital of the neighboring site.
a
The exciting possibilities in the physics with higher
Let us consider bosonic atoms at zero temperature in
band atoms has naturally attracted also theoretical at-
an optical lattice
tention. P-band bosons have been studied in equilib-
rium[9]andasaplatformtorealizeinterestingcorrelated πx πy πz
V(r)=V sin2 +V sin2 +sin2 ,
quantum states [10–13]. Atoms on the p-band have also L (cid:16) d (cid:17) Dh (cid:16) d (cid:17) (cid:16) d (cid:17)i
been found to have interestingrotationalproperties[14].
Furthermore, there are theoretical studies where multi- where d = λ/2 is the lattice constant. To simplify the
orbital description of bosons is found necessary since it formalismwe assume a similar lattice potential as in the
experimentby Mu¨ller et al. [7] sothat the lattice is deep
in y- and z-directions with depth V = 55E , where
D R
E = ~2k2/2m is the recoil energy, m the atomic mass,
R
∗Electronicaddress: [email protected] and the wavevector k is related to the laser wavelength
2
find a generic interactiontermrelevantto our discussion
18
g
H = ppψˆ†ψˆ†ψ ψ +2n (g n +g n ) (2)
16 I 2 p p p p p sp s pd d
+ 2g n ψˆ†ψˆ +h.c +g ψˆ†ψˆ†ψˆ ψˆ +h.c ,
ppsd p(cid:16) s d (cid:17) ppsd(cid:16) s d p p (cid:17)
14
q) 12 where nα = ψˆα†ψˆα. We further assumed that s- and d-
(n bands are almost unpopulated by dropping terms which
E
10 were of higher order in ψˆ and ψˆ and dropped the
s d
far detuned processes 2p 2s and 2p 2d. The
↔ ↔
8 form of the interaction Hamiltonian here is quite gen-
eral and only assumes the validity of the expansion
6 into 3 modes and the weak population of 2 of these
modes. The various interaction parameters are defined
4 by the integrals g = drφ (x)2 φ (x)2 and g =
αβ α β ppsd
−3 −2 −1 0 1 2 3 drφ (x)2φ (x)∗φ (x)∗R. I|t shou|l|d be |noted that in
q (units of 1/d) p s d
R
estimating the interaction strengths, rather than using
Bloch-wavefunctions for φ (x), we could also have used
FIG. 1: A band structure in units of ER for the three lowest α
thebetterlocalizedWannierstates,ortheirharmonicap-
bandsfor a lattice depth of VL=17ER.
proximations without affecting our results qualitatively.
Even the quantitative differences are not dramatic since
theestimatedinteractionstrengthsarenotverydifferent
(λ = 843nm) through k = 2π/λ. Throughout our cal- in the parameter regimes we consider.
culation we will further assume that we are dealing with The non-interacting part of the Hamiltonian is simply
87Rb atoms. Since the lattice is deep is two directions given by H0 = Esns +Epnp +Ednd so that our total
the dynamics in these directions is frozen and the sys- Hamiltonian is the sum H = H0 +HI. Note that here
tem is effectively one-dimensional. For the atomic wave- we also make an approximation that the atoms are suf-
functions along y- and z-directions we use the Gaussian ficiently localized so that tunneling dynamics plays only
ground states corresponding to the harmonic oscillators a minor role at the timescales of interest. However, tun-
along y and z. Using this effectively one-dimensional neling effects are to some extent still included since they
potential together with the usual kinetic energy oper- influence the band structure and are consequently incor-
ator Kˆ = ~2 2/2m, we get an ideal Hamiltonian porated into the energy levels En. This approximation
H =Kˆ+V−(r)∇fromwhichwenumericallycalculatethe is worst for the d-band (which is initially unoccupied)
lat
whose bandwidth is by far the largest. However, tunnel-
dispersions [17] E (q),E (q), and E (q) for the 3-lowest
s p d
ing processesin the d-band would mainly give rise to (in
(s, p, and d) bands as a function of quasi-momentum q.
practice) irreversible atoms loss from the p-band since
Typical band structure is shown in Fig. 1.
the more mobile d-band atoms are free to relax towards
Atoms interact through the two-body interaction [18] theirzeroquasi-momentumminimum, wheretheprocess
g/2 drψˆ†(r)ψˆ†(r)ψˆ(r)ψˆ(r), where ψˆ(r) (ψˆ†(r)) is the 2p s+d is very far detuned.
R ↔
bosonic annihilation (creation) operator and g = The detuning δ(q,k)=2E (q) E (q k) E (q+k)
p s d
4π~2a/m. For 87Rb atoms the scattering length a which is related to the process−2p −s + d−described
110a0. Our main interest is to explore the role of th≈e by the terms ψˆs†ψˆd†ψˆpψˆp+h.c. in the↔Hamiltonian, plays
processesthattransferatomsfromthecondensedp-band an important role for the stability of the system. If one
tothe lowests-andthe secondexcitedd-bands. Forthis applies Fermi’s golden rule to study the life-time of p-
purpose we expand the field-operators in terms of the band atoms [9], the lowest order processes are no longer
field-operators ψˆ for the three different bands. energetically allowed if the lattice is deeper than about
α
V = 18E , since then δ(q,k) is non-vanishing for all q
L R
and k. If the process is always detuned for the p-band
ψˆ(r)=φ (x)ψˆ +φ (x)ψˆ +φ (x)ψˆ . (1) atoms at q = π/d the detuning is minimized with the
p p s s d d
choice k = 0. Even if the detuning is zero somewhere,
for the lattice depths considered here, the resonance is
For the pure Bose-Einstein condensate in the p-band still located close to k = 0 so we choose k = 0 without
at quasi-momentum q, φ (x) corresponds to the Bloch- introducing large additional uncertainties when comput-
p
wavefunction up(x) at quasi-momentum q. The energy ing the coupling coefficients of the interaction Hamilto-
q
ofthep-bandatomisminimizedwiththechoiceq =π/d. nian. For very deep lattices the detuning approaches a
Likewisethefunctionsφ (x)andφ (x)arerelatedtothe constant value and in this sense the system differs from
s d
Bloch-wavefunctionsus,d(x)whichdominatethedynam- a harmonic oscillator even in this limit [13]. However,
q
ics at short times. Substituting the above expansion, we relative to the onsite trap energy scale ~ω the detuning
3
∂ψ
(a) 0.8 (b) i p = (Ep+gppnp+2gspns+2gpdnd)ψp
∂t
2 + 2g (ψ ψ∗+c.c)ψ +2g ψ∗ψ ψ , (4)
Bα gαβ 0.6 ppsd d s p ppsd p d s
1 and
0.4
∂ψ
0 i d =(E +2g n )ψ +2g n ψ +g ψ2ψ∗. (5)
15 20 25 30 15 20 25 30 ∂t d pd p d ppsd p s ppsd p s
Inordertostudythestabilityofthesystemwenowderive
(c) 3 (d) the relevant Bogoliubov-de Gennes equations. Since the
0.18
max2 system is initially prepared in the p-band we have (at
gppsd00..1167 δδ & min01 siψnhpdo(irtct)att=iema(cid:2)e√ss)mnTψals+l(qtδ)uψa=pn(cid:3)tδeiψt−ysi.eE−Fp′tuiE,rs′twth,heψermrde(otsr)ye=m,wbδeoψlddδeefi−insiEeudd′ste,Eda′nt=do
s
E +2g n , E′ =E +2g n , andE′ =E +2g n .
15 20 25 30 15 20 25 30 s sp T d d pd T p p pp T
V (units of E ) V (units of E ) By keeping only terms which are lowest order in δψ we
L R L R α
get a pair of equations
FfuInGc.tio2n: oTfylpaitctaiclebdeehpatvhioVrLf.orInth(ae)rweleevsahnotwptahreambaentedrwsidasthas iδψ˙s =2gppsdnTe−i(Ed′−Es′)tδψd+gppsdnTe−iδ′tδψd∗
(solid line for the d-band, dashed for the p-band, and dot-
and
dashedforthes-band). Thebandwidthisdefinedasadiffer-
emnicneimbuetmweeennertghyewmitahxiinmtuhmebeannedrg.yInw(ibth)iwnethsheobwangdppa(nsdolitdh)e, iδψ˙d =2gppsdnTe+i(Ed′−Es′)tδψs+gppsdnTe−iδ′tδψs∗,
gsp (dashed),andgpd (dot-dashed),and(c)showsthebehav- where δ′ = 2E′ E′ E′. The first terms on the right
ior of the coupling gppsd responsible for population transfer p− s− d
of these equations can be dropped since they are oscil-
away from the p-band. Finally (d) shows the minimum and
maximumvaluesofthedetuningδ(q,k). Whentheminimum lating rapidly and their time average vanishes over the
value of δ(q,k) > 0 the process 2p ↔ s+d is detuned for timescales of interest. By defining
allquasi-momenta. Allenergiesinthey-axisareexpressedin
terms of the recoil energies ER. δψs(s)=(cid:2)use−iωt+vs∗e+iωt(cid:3)e−iδ′/2t (6)
and
does approach zero since ω ∝√VL. δψ (s)= u e−iωt+v∗e+iωt e−iδ′/2t (7)
InFig.2wesummarizethebehaviorofvariousparam- d (cid:2) d d (cid:3)
eters of our model as the lattice depth is varied. It is to
wefindtheBogoliubov-deGenneseigenvalueproblemfor
be noticedthatcoupling coefficientsappearinginH are
I collective modes
typicallyofthe sameorderofmagnitudeasthe detuning
2Ep(q) Es(q) Ed(q). For this reason the system can ωηˆλ=Mˆλ (8)
− −
be poorly described both in the narrow band as well as
in the wide band limit. described by their frequency ω as well as amplitudes uα
and v . Here λT =(u ,v ,u ,v ),
α s s d d
1 0 0 0
III. NONLINEAR CONDENSATE DYNAMICS
 0 1 0 0 
ηˆ= − , (9)
0 0 1 0
 
In the previous section we derived a minimal theory  0 0 0 1
to describe how p-band atoms are coupled to the s- −
and d-bands. Let us now investigate what this model and
implies for the dynamics of pure Bose-Einstein conden-
δ′/2 0 0 g n
sates. In this limit we can describe the system in terms  −0 δ′/2 g n pps0d T 
of complex amplitudes ψs(t), ψp(t) , and ψd(t) which Mˆ = 0 g− n ppδsd′/2T 0 . (10)
are normalized to the number of particles per site i.e.  g n pps0d T −0 δ′/2 
ψs 2+ ψp 2+ ψd 2 =nT. This description also follows ppsd T −
| | | | | |
when the onsite wavefunctions are described by coher-
This eigenvalue problem has solutions
entstatesandprovidesanaccuratepictureofthesystem
deep in the superfluid regime [13]. The equations of mo- 1
ω = δ′2 4(g n )2. (11)
tions for the order parameters are given by ±2q − ppsd T
∂ψ Importantly, these solutions are imaginary and indicate
i ∂ts =(Es+2gspnp)ψs+2gppsdnpψd+gppsdψp2ψd∗, (3) dynamical instability if δ′ < 2gppsdnT. In a parameter
| |
4
regime where dynamical instability exists a system ini- 0.9
tially prepared on the p-band will lose atoms to the s-
0.8
and d-bands at a rate given by
0.7
Γ= δ′2 4(g n )2 . (12)
q ppsd T
| − | 0.6
Itisalsoeasytoseewhenthisinstabilityismorelikelyto 0.5
occur. Forrelativelydeeplatticesδ =2E E E >0 Γ
p s d
since the anharmonicities of the lattice p−otent−ial shifts 0.4
thed-stateslowermostrelativetotheharmonicoscillator
0.3
energy levels. On the other hand the effective detuning
is given by δ′ = δ+2n (g g g ) and will move 0.2
T pp sp pd
− −
closertozeroifg g g <0,aconditionwhichusu-
pp sp pd
− − 0.1
ally holds since all the coupling coefficients have similar
magnitudes. Thenwheneverδ′ becomessmallrelativeto
0
g n (which increases with lattice depth) dynamical 15 20 25 30
ppsd T V (units of E )
instabilities can be expected. L R
InaFig.3weshowthelossrateofthep-bandatomsas
FIG. 3: Loss rate Γ of the p-band atoms as a function of
a function of lattice depth for two different onsite atom
the lattice depth VL for nT = 1 (dashed) and nT = 2 atom
numbers. This figuresuggeststhatforsmallonsite atom
per site. For comparison, with a dot-dashed line, we show
numbers a dynamical instability can be present with re-
the result where onsite wavefunctions were assumed to be
aVlList=ic 2t5raEpRpaatrawmheitcehrsthaebosvyestVemL i≈s e1x9pEecRtedantdobbeeloiwn hisagrmivoennicinouscnililtastoofr1w/aτve=fuEncRt/io~n.sFaonrd87nRTb=at2o.mTshτe liossasbroautet
the superfluid regime [7]. For higher atom numbers the 49µs.
regionofinstability is greatlyincreasedandextends into
the p-bandMott-insulatingregionwhereourassumption
of coherent states is clearly invalid. Furthermore, we 0.12
find that the timescale for the instability is substantially
less than the timescale for the tunneling of the p-band 0.1
atomsjustifyingourearlierapproximationtoignoretun-
neling dynamics. Interestingly,the dynamical instability
0.08
is expected for lattice depths at which the lowest order
Fermi’s golden rule becomes inapplicable [9].
d
P
We have also solved the Eqs. (3)-(5) numerically even + 0.06
Ps
by modifying them to include interactions between s-
and d-band atoms. We compare the Bogoliubov- de
0.04
GennesapproachwiththefullGross-Pitaevskiiequations
in Fig. 4. As is clear the agreement between the nu-
0.02
merical and analytical Bogoliubov- de Gennes solution
is very good when the p-band population is dominant
and the small difference is mainly due to the facts that 0
0 5 10 15 20 25 30 35 40
short time dynamics has some sensitivity to phases of t [ms]
theamplitudesandthatnumericsalsoincludedmanyfar
off-resonantprocesses giving rise to fast oscillations on a FIG. 4: Comparison between the Bogoliubov- de Gennes re-
much shorter timescale. These processeswere ignoredin sult(dashedline)forthes-andd-bandpopulationsasafunc-
the Bogoliubov- de Gennes approach derived earlier. If tionoftimeandthesolutiontothefullGross-Pitaevskiiequa-
in the dynamically unstable region the initial state has tions(solidline). ThelatticedepthwaschosenasVL =20ER
a substantial s-band population the time-evolution be- andnT =1sothatthesystemwaspredictedtobeintheun-
comes more sinusoidal, but even then the maximum d- stable regime. Initial state had a 10−6 population on s- and
d-bands(to seed thedynamics) and the phases of the ampli-
band population is substantial and cannot be ignored.
tudeswere taken to be zero.
The numerical solutions also reveal a revival in the p-
band population after the initial exponential loss. How-
ever, we believe that this revival is unlikely to persist in
arealisticsystem,duetothehighmobilityofthe d-band density distribution in a harmonic trap is proportional
1/4
atoms. It is more likely that those atoms coupled to the tothe B ,whereB isthe d-bandbandwidth. Forthis
d d
d-bandwillbe irreversiblylostfromthe p-band. Also,in reason, many d-band atoms end up spatially separated
atrapthedensitydistributionofthemoremobiled-band from the s- and p-band atoms residing closer to the trap
atomswillbemuchbroadersincethelengthscaleforthe minimum.
5
A. Role of number fluctuations 0.3
In the previous section we described the system by
assumingpurecondensatesinallbandsandfoundapos-
sibilityofdynamicalinstability. Thequestionthenarises
that what role did the assumption of condensation actu-
ally play in our result? Alternatively, deep in the Mott-
insulating regime with nT = 2 atoms per site we can at sdPmax0.25
short times use an ansatz
ψ(t) =ψ (t)1,0,1 +ψ (t)0,2,0 , (13)
sd p
| i | i | i
where n ,n ,n is the Fock state. To the lowest or-
s p d
| i
der in tunneling the many-body wavefunction factorizes
into single site solutions, so the solution in a single site
0.2
amounts to a solutions throughout the insulator. Due 25 30 35 40 45 50
V (units of E )
to the presence ofthe 1,0,1 -statewe mustfurthermore L R
| i
includethedensity-densityinteraction2g n n between
sd s d FIG. 5: The maximum population of the |1,0,1i-state when
s- and d-bands into our model. In the earlier mean-field
the number of atoms nT = 2 as a function of lattice depth.
theory this term was a small quantity which could be
Theinteractionparameterswereestimatedbyapproximating
safely ignored. We then get (at short times)
thelattice site by aharmonic oscillator while theenergy lev-
els were computed with a band-structure calculation with a
H ψ(t) = (E +E +2g )ψ +g √2ψ 1,0,1 quasi-momentum q=π/d.
| i h s d sd sd ppsd pi| i
+ (2E +g )ψ +g √2ψ 0,2,0 . (14)
h p pp p ppsd sdi| i
fluctuations in feeding the dynamical instability. What
In this case we can easily solve the equations of motion happens if the system is neither a Mott-insulator nor a
forψ (t)andψ (t)andfindasimpleRabi-problemwith pure condensate, is unclear. Presumably, the system is
p sd
purely real eigenenergies more stable than our calculations assuming a pure con-
densate suggest, but the stability cannot be taken for
2E +E +E +g +2g
p s d pp sd granted.
E (n =2) = (15)
± T
2
1
(2E E E +g 2g )2+8g2
±2q p− s− d pp− sd ppsd B. Dynamical instabilities in superlattices
and an initial state ψ(t = 0) = 0,2,0 evolves in such
| i | i Since p-band bosons have also been studied in a bi-
a way that the maximumpopulation ofthe 1,0,1 state
| i partite optical lattice [8, 19], let us briefly discuss the
is given by
stability propertiesofp-bandatoms insuperlattices. We
8g2 will again assume the same deep optical lattice along y-
Psd = ppsd . (16) and z directions, but for the lattice along x-direction we
max (2E E E +g 2g )2+8g2
p− s− d pp− sd ppsd use a potential [20]
InFig.5weshowthemaximumpopulationofthe 1,0,1 πx π
state which gives in indication of how reliable |a purei V(x)=VSsin2h d i−VLsin2h(cid:16)2d(cid:17)(x+d)i (17)
p-band theory can be. Since our ansatz with a fixed
atom number is expected to be reasonable only for deep which is characterized by the depths V and V of the
S L
lattices, in this figure we estimated the interaction pa- short and long lattices respectively.
rameters by approximating the lattice site with a har- In Fig. 6 we show examples of the energy levels to-
monic oscillator. As is clear from this figure, substan- gether with the three lowest eigenstates of this potential
tial fraction of the 87Rb population can reside outside while in Fig. 7 we show an estimate of when dynamical
the p-band. In fact, Psd can even reach unity when instabilities might be expected. As is clear from these
max
2E E E +g 2g =0 which formally happens figures dynamical instabilities might occur only when
p s d pp sd
− − −
somewhat below the lattice depth of V = 20E . How- V /V is fairly large and the eigenstates are well local-
L R L S
ever, in this regime stronger tunneling invalidates the ized to the deep sites. When V /V becomes smaller
L S
simple three-state description as well as the assumption instability is quickly suppressed. The reason for this is
of fixed onsite atom number. twofold. First, as V /V becomes smaller the energy
L S
Withexactlyn =1atomspersiteinteractionsdonot levels become very different from the energy levels of a
T
contribute and instability is not expected. These sim- harmonic oscillator at deep sites and this implies an in-
ple exercisesindicate animportantroleofonsite number crease in the detuning 2E E E which reduces the
p s d
− −
6
30 2 cussions to an effectively one-dimensional system. If the
(a) (b)
dimensionality is increased we have several degenerate
1
20 flavorsin the p- and d-bands andmany more interaction
E (n)u(x)0 0 cohgaonunsetlos.EHqso.w(e3v)e-(r5,)thweouGlrdosstsi-lPlliotaoekvvsekriiyesqimuailtaiornasndantahle-
10
−1 stabilityanalysisislikelytorevealsimilarinstabilitiesas
0 −2
−0.5 0 0.5 1 −1 0 1
0.6
x/d x/d
2
(c) (d) 0.5
20
1
15
x) 0.4
E10 (n)u(0 0
Γ
5 −1 0.3
0 −2
−0.5 0 0.5 1 −1 0 1
0.2
x/d x/d
FIG.6: EnergylevelsinunitsofER(togetherwiththelattice 0.1
potential) and wavefunctions in a superlattice. We fixed the
short lattice depth to VS = 15ER while VL = 1.5VS in (a) 0
and(b)orVL=0.8VS in(c)and(d). In(b)and(d)thedot- 0.4 0.6 0.8 1 V1/.V2 1.4 1.6 1.8 2
dashedlineisthewavefunctionforthes-orbital,solidlinefor L S
thep-orbital, and thedashed line for thed-orbital.
FIG. 7: Loss rate of the p-band atoms in a superlattice due
tothedynamicalinstabilityasafunctionofVL/VS whenthe
likelihood of dynamical instabilities. Second, while for short lattice depth was fixed to VS = 15ER. The coupling
coefficients and band energies were estimated from the q=0
large V /V all the wavefunctions are well localized to
deepsitLesaSndtheiroverlapsarelarge,forsmallerV /V eigenstates and theloss rateis given inunitsof 1/τ =ER/~.
L S
the d-orbitalbecomes peakedin the shallow sites. When
this occurs, the overlap between the d-orbital and the
s- and p-orbitals is drastically reduced and g is re-
ppsd we found here for one-dimensional systems.
ducedbymorethananorderofmagnitude. Fromthiswe
We found that while p-band leaks into s- and d-bands
can conclude that superlattices can enhance the stabil-
even in the Mott-insulating regime, for the dynamical
ity properties of p-band bosons. However, this happens
instabilities to be present it was important that the sys-
at the cost of greater hybridization of the p-orbitals in
tem is not a Mott-insulator. However, even in a Mott-
deepsiteswith the s-orbitalsinthe shallowsites. Infact
insulatingregimeforp-bandatoms,theseatomsarecou-
this hybridization played an important role also in the
experiment by Wirth et al. [8]. pled to s and d-bands. Since the bandwidth for the d-
band atoms is dramatically larger than for lower bands,
d-band atoms would typically be deep in the superfluid
IV. CONCLUSIONS regime. Under such circumstances the p-band atoms are
coupled to a coherent d-band and this coupling will in-
duce number fluctuations also on the p-band. How this
Wehavepointedoutadynamicalinstabilitywhichcan
effect affects the superfluid Mott-insulator transition for
affect the stability of p-band bosons in the broken sym-
p-band atoms has not yet been explored.
metry phase. All our numerical examples assumed 87Rb
atoms and consequently some details are expected to be Finally it should be noted that the stability proper-
different for different atoms. In particular, the scatter- ties of fermionic atoms are likely to be better than those
ing length will be different for different atoms or it can of bosonic atoms. This is because with fermions one
be tuned with Feshbach resonance and this can have a can populate the p-band by first filling the lowest band.
dramatic effect on how big a role the coupling to the s- When the lowest band is filled, the coupling between p,
andd-bandsplays. Forconveniencewerestrictedourdis- s, and d-bands becomes Pauli blocked.
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