Fast dynamics for atoms in optical lattices

0.33 MB·English

by  Mateusz Lacki

Fast dynamics for atoms in optical lattices
1 2
Mateusz Łącki and Jakub Zakrzewski
1 Instytut Fizyki imienia Mariana Smoluchowskiego,
Uniwersytet Jagielloński, ulica Reymonta 4, PL-30-059 Kraków, Poland
2 Instytut Fizyki imienia Mariana Smoluchowskiego and Mark Kac Complex Systems Research Center,
Uniwersytet Jagielloński, ulica Reymonta 4, PL-30-059 Kraków, Poland
(Dated: October31, 2012)
2
Cold atoms in optical lattices allow to study in a detailed way a many body dynamics. Rapid
1
time-dependent modifications of optical lattice potentials may result in significant excitations in
0
atomicsystems. Thedynamicsinsuchacaseisfrequentlyquiteincompletelydescribedbystandard
2
applications of tight-binding models (such as e.g. Bose-Hubbard model or its extensions) that

t typically neglect the effect of the dynamics on the transformation between the real space and the
c
O tight-bindingbasis. We develop a quasi-exact multi-band approach using time-dependent Wannier
functions and apply it to model cases, directly related to experiments.

0
3 PACSnumbers: 67.85.Hj,03.75.Kk,03.75.Lm

]
s
Ultra-cold quantum gases in optical lattice potentials forfermions[25]ortheso-calledBose-Hubbardmodelfor
a
g allow for precise studies of standard models known from bosons. One may consider 3D cubic lattices realized by
- otherbranchesofphysics(e.g. thecondensedmatterthe- three orthogonally polarized laser standing waves. Of-
t
n ory) as well as proposing novel situations with intrigu- ten the reduced, two- (2D) and one-dimensional (1D)
a ing properties. The latter utilize rich atomic internal geometries are interesting [14, 15] for which very deep
u
structures, a versatility and an extreme controllability lattices in remaining directions cut the atomic sample
q
. of atomic systems [1–3]. Many body physics often ad- into2Dslicesor1Dtubes(withtheconfineddegree(s)of
t
a dresses stationary properties such as phase diagrams – freedom effectively described by the harmonic oscillator
m cold atoms enable also a controlled study of dynamics. groundstate). Explicitly, for the simplest quasi-1Dsitu-
d- That is especially interesting in the vicinity of quantum ation[26]V1(~r)=ssin2(kx)+m2ω2(y2+z2), Parameters
phasetransitions[4]whereoneoftheintriguingproblems s,ω are tunable in the experiment.
n
is the adiabaticity and quantitative analysis of deviation ColdinteractingBosegasdescribedbya secondquan-
o
c from it for slow quenches [5–8]. tized Hamiltonian:
[ Other interesting aspects of the dynamics concern ef-
~2
1 fects resulting from rapid changes of system parameters. H = d3~rΨ†(~r) − ∇2+V (~r) Ψ(~r)
1
v The typical example of such a situation is a well known Z (cid:18) 2m (cid:19)
7 revival experiment [9, 10] where the sample prepared in 1
5 + d3~rd3~r′Ψ†(~r)Ψ†(~r′)V(~r,~r′)Ψ(~r)Ψ(~r′), (1)
asuperfluidstateisplacedintheinsulatingenvironment 2Z
9
by a fast increase of optical potential depth. Other pos-
7
. sibleexamplesincludefastquenches[11–13]andperiodic whereV(~r,~r′)isanisotropicshort-rangepseudopotential
0
modulationsofopticallatticepotentialsoftenusedeither modelling s-wave interactions [3]
1
for measuringthe state ofthe system [14,15] or evenfor
2
1 modifying its effective parameters and thus its proper- 4π~2a ∂
V(~r,~r′)= δ(~r−~r′) |~r−~r′| (2)
v: ties [16–20]. Even faster modulations were suggested to m ∂|~r−~r′|
i efficiently populate excited bands [21].
X
The aim of this letter is to show that for rapid mod- with a being the scattering length.
r
ifications of the optical lattice potential (e.g. its depth) The field, Ψ(~r), is expanded in basis functions,
a
a standardapplication of tight-binding models is incom- Wα(~r,s) = wα(x,s)H(y)H(z) [built as a product of
i i
plete. Wedevelopaquasi-exactmulti-bandtheorywhich Wannier functions in the direction of the lattice with
uses time-dependent Wannier functions and show its ap- harmonic oscillator functions in transverse direction:
plicability on few chosen model examples. H(z)=(κ/π)1/4exp(−κz2/2)], κ=mω/~:
For weak optical potentials, in the deep superfluid
regime,thestandardapproachistouseaGross-Pitaevski Ψ(~r)= aαWα(~r,s), (3)
i i
mean-field approach [22, 23]. In deeper lattices, the de- Xi,α
pletion of the condensate becomes significant and an-
other approach is necessary. The seminal work [24] uses i numbers the sites and α Bloch bands of the lattice.
a tight-binding approach mapping the real space system Performing the integrations in (1) using orthogonality
onto the lattice, resulting in a famous Hubbard model ofWannierfunctionstheextendedBose-Hubbard(EBH)
2
Hamiltonian is obtained s =s(t) (see e.g. [9, 14, 31, 32]).
0
Such an approach neglects the time-dependence of
H=− Jiα−j((aαi)†aαj +h.c.)+ Eiαnαi + Wannier functions. The situation is similar to a ba-
iX6=j,α Xi,α sic textbook unitary transformation case. Recall that
1 Uαβγδ(aα)†(aβ)†aγaδ, (4) if ψ(t) = U(t)χ(t) and the evolution of χ is governed
2 ijkl i j k l by the Hamiltonian H then the proper Hamiltonian for
α,Xβ,γ,δXijkl time evolutionofψ is UHU†(t)+i~d/dt(U(t))U†(t). Us-
with J terms describing tunneling between sites while ing time dependent Wannier functions is equivalent to
U terms 2-body collisions. Explicitly, Jα = performing a similar transformation on our system. A
i−j
Wα(~r) −~2 ∇2+V (~r) Wα(~r)d3~r, while Uαβγδ = straightforwardcalculation [26] yields the proper Hamil-
R d3i~rd3~r′(cid:16)Wiα2(m~r)Wjβ(~r′1)V(~r(cid:17),~r′)jWkγ(~r)Wlδ(~r′). Tihjkel on- ftoornmian HW in the instantaneous Wannier basis in the
sRite energies Eα = Jα do not depend on site for trans-
i 0 d
lationallyinvariantsystems(we leavethis dependence to H =H+W =H+i~ (U(t))U†(t). (5)
W
signify that onemay incorporateeasily additionalslowly dt
varying inhomogeneous term).
Specifying further on that we are interested in changes
Thetightbindingrepresentationisexact. Theapprox- of lattice depth in time we may write d/dtU(t) =
imations emerge when we limit the number of bands as
∂ (U)ds/dt and W =Tds/dt. Then we obtain [26]
s
well as put restrictions on the Hamiltonian parameters.
For sufficiently deep lattices (say s ≥ 3) we may re-
strict the tunneling to nearest neighbors only (see [27] T = −i Tαβ(s)(aα)†aβ
i−j i j
for a shallow lattice case when next nearest neighbor i,Xj,α,β
tunnelings also play a role). Consistently, for interac- d
Tαβ(s) = wα(x,s) wβ(x,s)dx. (6)
tions, we include terms such that i = j = k = l or i−j Z i ds j
i = j = k, and l being nearest neighbor of i up to
a permutation (the so called density dependent tunnel- Transition integrals Tαβ(s) obey relations: ∀i,j,α :
i−j
ings [28–30] are taken into account). From now on by Tαα ≡ 0, Tαβ = (−1)α+βTαβ = −Tβα, Tαβ = 0 for
EBH we shall denote the Hamiltonian (4) with the fi- i−j i−j j−i j−i 0
α−β odd. In particular, T-term correction to a single
nite, low number of bands: α = 0,1,..,B. The corre-
band model (e.g. a standard BH) is zero. Thus the in-
sponding Wannier functions are smooth and the action
fluence of the W term is expected only whencoupling to
of the pseudo-potential is equivalent to a standard con-
higher bands is appreciable. The T term contains both
tact term V(~r,~r′) = 4π~2aδ(~r −~r′) = gδ(~r −~r′). Re-
m on- and off-site terms. Practically, most significant cou-
stricting α to the lowest band only (α = 0) and taking
pling occurs on-site between bands α = 0 and α = 2.
solely i = j = k = l on-site interactions gives the stan- Tαβ(s) decrease rapidly as |d| grows. In Fig. 1 most
d
dardBose-Hubbard(BH)model[24]. Inthefollowingwe
prominent of these parameters are shown as a function
adopt the recoil energy ER = k22m~2,k = 2λπ as an energy of s. In numerical examples below terms up to nearest
unit (λ is a wave of the laser). We take k−1 = λ/2π as neighbors are taken only.
the unit of length. As mentioned above the W term is usually omitted in
It is vital to note that Wannier functions depend on numerical simulations. Its relative importance depends
the lattice parameters, in particular s. While EBH gen- onthe product T ds. Clearly rapidchangesof the lattice
dt
uinely describes the dynamics in such a lattice, we run intime arenecessaryfor the effects ofW to be apprecia-
into problems when, e.g. the lattice depth s varies in ble. Thusforslow(e.g. 100ms)quasi-adiabaticquenches
time. There are two options: either we keep the basis from shallow to deep optical lattices leading to Mott in-
fixed in time determining it once at a given, say initial, sulatorphaseformations[9,14, 15,33],the Wtermmay
s=s0 valueorwemakethebasistime dependentsothat be ignored safely.
Wannier functions change with time [i.e. we use s(t) in- Situation becomes different for fast changes of s. We
stead of s0 in Eq. (3)]. The former, while conceptually shall consider the influence of W term in model situa-
simpler, leads to difficulties: once s in the Hamiltonian tions. We shall discuss first a simple model of a linear
(1)is different thans0 chosenforWannier functions, the quench. Then we briefly mention the famous revival ex-
resultingHamiltonianisnolongerinthe formofEBHas periment [10]. Finally we analyze the recent proposition
definedin(4)-inparticulartunneling-typetermsappear for efficient higher bands excitations [21].
between different bands. Therefore most of the authors A linear quench. Itisrealizedassumings(t)=s t/τ+
1
use the latter approach (at least implicitly) using the (1−t/τ)s , where τ is the duration of the quench. Con-
0
EBH (or BH just for the lowestband). Then changesof, sider N =5 87Rb atoms placed in a 1D lattice of length
e.g., lattice depth s(t) are just translated into changes L = 4 under periodic boundary conditions (PBC). The
ofHamiltonianparametersJα andUαβγδ evaluatedfor exactdiagonalizationgivesthe groundstateats =12–
i−j ijkl 0
3
2 a) b) 1 U(s1)U†(s0) operation obtained directly from Eq.(6).
1 U0000 U0002 0.8 In the revival experiment [10] a rapid quench is real-
U0011 U0022 0.6 ized for bosons in the optical lattice by a rapid increase
0
0.4 ofthe latticedepth. Theauthorswereofcourseawareof
-1 J01 J21 0.2 the factthattoofastaquenchwouldhavepopulatedex-
J11 J31 cited bands thus they chose the duration of the quench,
0
c) d) T10 τ = 50µs, sufficiently long so that the higher bands ex-
0.01 1 -T120 T21 00..011 citations were negligible. As it turns out already for
1
τ = 20µs effects due to Wannier function dynamics are
appreciable [26].
1e-4
T40 T200 1e-5 Higher bands excitations. RecentlySowińskisuggested
0.0001 T301 0 [21] to use periodic modulations of the lattice depth, say
0 10 20 30 0 10 20 30 40
in the x-direction,
Latice depth s [E ]
R
s (t)=s0 +s sin(ωt), (7)
x x m
Figure1: (coloronline)Relevanceofdifferenttransitionam-
plitudes: (a) – nearest neighbor tunnelings J1α for different for excitedorbitalquantum state preparation. Largeen-
Bloch bands (b) – interaction integrals for g = 1,κ = 2π; ergy gap between Bloch bands requires high frequency
the term Ui0i0ii00 term present in the Bose-Hubbard Hamil- driving to couple different Bloch band states [21]. This
tonian is compared with interaction terms involving excited
translates into significant values of ds/dt and contribu-
bands.Panels(c)and(d)showadditionalamplitudesTαβ [see
i−j tionsfromtheW terminthedynamicsmaybesignificant
Eq. (6)] comingfrom thetime-dependenceof Wannierfunc-
- these terms were not taken into account in [21].
tions (i.e. theso called W term).
To see how the important the W part is we have re-
calculated the numerical simulation [21] using (4) with
the initialstate. The quenchis performeduptos =40, and without the W term, Eq. (6). The studied system
1
for different values of the quench time, τ. We find that is a 2D lattice with the tight harmonic confinement in
(see Figure 2) as soon as τ < ~/E the excitation en- the third direction [26]. The system is assumed to be in
R
ergy becomes significantly larger in the presence of the a deep Mott insulator regime (sx = 32,sy = 20,κ = 8).
W termthanwithoutit. Thatreflectsasignificantdiffer- Due to a deep lattice potential, inter-site hopping can
encesintheoccupationofthesecondexcitedband. Thus be neglected, and the whole system decouples into in-
a simple treatment of higher bands via the EBH model dependent 2D sites with the (assumed) integer filling
is insufficient to explain the dynamics; time-variation of ν = 2. We have prepared the system in the ground
Wannier functions has to be taken into account. state |ψ(0)i with energy E0. Following [21] we restrict
the numerical simulation to first three bands (while this
10
may not be sufficient for a simulation of a real situa-
a) b) 0.004
tion since W terms efficiently populate higher bands we
p band 0.002 consider the same model as in [21] to isolate the influ-
] 0.1
R ence of Wannier functions’ dynamics). We have per-
E
E [ c) formed numerical evolution of a system for a time cor-
∆ 0.1
d band responding to 10ms with varying frequencies ω . As in
x
0.001 0.01 [21] we measure the maximal ground state depletion:
0.001 δ(ω )=1−sup |hψ (t)|ψ(0)i|asafunctionofdriv-
0.01 1 100 0.01 1 100 x t∈[0,T] ωx
Quench Time τ [h_ /E ] ing frequency ωx. We find that the presence of W term
R changes significantly the depletion function in the fre-
quency range considered (compare Fig.3). The W term
Figure2: (coloronline)∆E,theenergygain (i.e. theexcess
leads to several additional excitations accompanied also
energy over the corresponding ground state) during a linear
quenchofamodel1Dsystemfroms0 =12ER tos1 =40ER,. bybroadeningandshiftingtheexcitationpeaksobtained
For the adiabatic process ∆E =0. Red (black, dashed) lines without the W term.
correspond to the simulation with (without) the W term in A key result of [21] is a possibility of efficient popula-
time-dependent Hamiltonian. Panels b) and c) show time tion of higher Bloch bands. Population of excited state
variationofaveragesofbandoccupationoperatorshnˆ1i,hnˆ2i. was found to oscillate with period of a few ms, making
the whole process quite efficient. We have found that
In the limit τ →0 the quench becomes instantaneous, including the W term in the analysis makes populating
the evolutiondoes not changethe wavefunction. Yet the the excited bands even faster. The oscillation period is
Wanner function basis changes from B(s ) to B(s ) and decreased usually several times with similar excitation
0 1
so does the field operator, (3) representation in Wan- efficiency. Therefore, while confirming the possibility of
nier function basis. The basis change is realized via directresonanttransferofpopulationtoexcitedbandsby
4
on 1 U0000 may be expanded as [compare (7)]
pleti0.8
e
d0.6 ∂U0000
e U0000(s)≈U0000(s0)+ s sinωt≈1.45+0.050sinωt
stat0.4 x ∂sx m
d (8)
n0.2
u for s0 = 32 and s = 4 (in recoil units). The dominant
Gro 05 10 15 20 25 intraxsite hopping mterm coming from the W term
_
Modulation Frequency [E /h]
R
ds ∂T20
Figure 3: (color online) Excitation via modulation of the T020(s) dtx ≈ (cid:18)T020(s0x)+ ∂s0 smsinωt(cid:19)smωcosωt
latticeheightwith andwithoutW term. Thedepletionfunc- x
≈ 0.79cosωt−0.026sin2ωt
tionduringthefirst10ms,without(black,solid)andwiththe
W term (red, dashed). The broadening of the peak around
ω = 18.5 is a power broadening effect, see discussion in the for ω = 18.5. Thus the driving term coming from
text.
W contribution is an order of magnitude stronger than
driving induced by a modulation of interactions! The
present of an “in phase” and “in quadrature” driving
breaks the time-reversalinvariance [34] that strongly af-
lattice depthmodulation,ouranalysissuggeststhattak-
fectsthestructureofavoidedcrossings[35,36]. Moreover
ing the time variation of Wannier functions into account
W term brings strong second harmonic which addition-
is crucial for controlling the process and for selective ex-
ally strongly modifies Floquet spectrum as well as the
citation of desired bands.
dynamics. For parametersof [21] the W term dominates
the dynamics.
0 0
Ees [ ]R 1 1 inLtheteuesvomluentitoionnaaplpsoeatrhsaatltshoefiomrpsuorffitacniecnetloyftfahsetWoscteilrlma-
gi tory movement of the lattice sites with a fixed lattice
ner 2 2 depth [16, 37]. The selection rules for excitations are
e
asi 3 then modified with respect to the simple, translationally
u
Q 3 invariant,symmetricsituationdiscussedhere[26]. Other
18.0 18.5 19.0 18.0 18.5 19.0
Modulation Frequency [E R ] possible applications of the presented formalism may be
thoughtofduetothegeneralityoftheargumentsleading
Figure4: (coloronline)FloquetspectrumwithoutWcontri- to Eq. (5). In particular, with appropriate modifications
bution (left) and with quasi-exact study,with W term (right
it can be formulated for fermions as well.
panel). Broad avoided crossings for the latter are due to the
strengthoftermsomittedwithinBHdescriptionaswellasthe In conclusionwe have shown that greatcare has to be
influence of higher harmonics - see text for discussion. The taken when using discretized tight-binding approxima-
region of avoided crossings (large curvatures) is highlighted tionsforatomicsystemsinopticallatticesinthepresence
with red (slightly lighter) color. of fast time variations of optical lattice parameters. A
transformation to Wannier functions basis involves time
dependent terms reflecting the time dependence ofWan-
The presence of the W term in the Hamiltonian may nierfunctionsthemselves. Thesetermsmodifythe tight-
be also analyzed using the Floquet approach [16]. Ex- binding Hamiltonian and are responsible for appreciable
emplary spectra inspecting the broad structure around and directly experimentally measurable effects whenever
ω = 18.5 in the depletion function are shown in Fig. 4. the change of lattice parameters is sufficiently fast.
ThespectrumwithoutW contributionshowsasingleiso- It is worth mentioning that in a series of pa-
latedavoidedcrossingindicatingasimpleresonance(cor- pers, Heidelberg group [38–40] developed alternative
respondingtotheisolatedpeakinFig.3)inacontrastto approach for studying non-equilibrium dynamics using
broad structure of avoided crossings for the quasi-exact time-dependentvariationalbasiswhichimportantlytakes
evolution. Thatstructurecorrelateswellwiththebroad- interactions into account. The detailed comparison of
ened peaks observed in the depletion function. both approaches is left for a subsequent study.
To understand this striking difference it is enough to We acknowledge discussions with Dominique De-
consider the magnitude of different oscillatory terms in lande and Krzysztof Sacha on the final form of
the Hamiltonian. For a deep lattice, when tunneling in this work. Projects of M.Ł. and J.Z. have been
negligible and a single site is considered only [21] the financed by Polish National Science Center un-
driving comes from time dependence interaction term as der contract DEC-2011/01/N/ST2/02549 and DEC-
well as from W term. The dominant interaction term 2012/04/A/ST2/00088respectively.
5
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6
SUPPLEMENTARY MATERIAL TO: FAST the potential depths s = (s ,s ,s )) that are products
x y z
DYNAMICS FOR ATOMS IN OPTICAL of the corresponding one-dimensional Wannier functions
LATTICES wα(s ,t)
i t
Mateusz Łącki and Jakub Zakrzewski Ψ(~r)= aαWα(~r,s), (13)
i i
Abstract: Xi,α
We provide a detailed derivation of the tight binding
with
Hamiltonian which takes into account time-dependence
of Wannier functions. We provide expressions valid for
Wα(~s,~r)=wαx(sx,x)wαy(s ,y)wαz(s ,z). (14)
differentdimensionalitiesoftheopticallatticepotentials. i i i y i z
In the expressionsabovethe multiindex α=(α ,α ,α )
x y z
numbers different Bloch bands of the lattice while aα
Multiband Bose-Hubbard Hamiltonian i
is an annihilation operator for bosons at site i and in
α band. Index i numbering the sites is, also really a
To make this material self-contained we repeat below
multiindex i=(i ,i ,i ).
some of the formulae from the letter extending it at the x y z
Usingthe orthogonalityofWannierfunctions andper-
same time by pedagogical comments.
forming integrations in (9) the extended Bose -Hubbard
ColdinteractingBosegasinanexternalpotentialV(~r)
(EBH) Hamiltonian is obtained
maybedescribedbyafollowingsecondquantizedHamil-
tonian:
H = Z d3~rΨ†(~r)(cid:18)−2~m2 ∇2+V(~r)(cid:19)Ψ(~r) H=−t=Xx,y,zXi→tjXα Jiα−tj(st)(aαi)†aαj +
+ 21Z d3~rd3~r′Ψ†(~r)Ψ†(~r′)V(~r,~r′)Ψ(~r)Ψ(~r′), (9) +Xα Eαnα+ 21αXβγδXijklUiαjβklγδ(aαi)†(aβj)†aγkaδl.(15)
whereV(~r,~r′)isaninteractionpotentialbetweenbosons.
t
For a very low temperature, the s-wave scattering dom- Thenotationi→j,following[21]inthesumindicates
inates and the real potential may be represented as an summation over all sites i and j shifted in t direction by
isotropic short-range pseudopotential [3] distance i−j. The J terms describe tunneling between
sites
4π~2a ∂
V(~r,~r′)= δ(~r−~r′) |~r−~r′| (10) ~2
m ∂|~r−~r′| Jαt = Wα(~r) − ∇2+V(~r) Wα(~r)d3~r, (16)
i−j Z i (cid:18) 2m (cid:19) j
with a being the s-wave scattering length.
We shall consider an external potential corresponding while U terms 2-body collisions
to a cubic optical lattice (generalizations to other lattice
configurations are quite straightforward): Uαβγδ = d3~rd3~r′Wα(~r)Wβ(~r′)V(~r,~r′)Wγ(~r)Wδ(~r′).
ijkl Z i j k l
(17)
V(~r)= s sin2(k~e ·~r) (11)
t t The on-site energies Eα = Jα do not depend on site
t=Xx,y,z i 0
for translationally invariant systems (we leave this de-
where s is the depth of the periodic standing wave po- pendence to signify that one may incorporate easily ad-
t
tentialinthet-direction,~e -versorinthecorresponding ditional slowly varying inhomogeneous term).
t
direction, while k = 2π/λ the wavevector for light with Formally this representation of the Hamiltonian is ex-
wavelength λ. Consider x only and recall that Wannier act. Theapproximationsemergewhenwelimitthenum-
functions, wα(x,s), localized at lattice sites, are linear ber of bands as well as put restrictions on the Hamilto-
i
combinations of Bloch functions φα, eigenfunctions of nianparameters. Inparticular,restrictingthe expansion
p
the single particle problem with potential ssin2(kx) and tothelowestbandonly,assumingthatthetunnelingam-
quasimomentum p∈[−k,k) [41] : plitudes are non-zero to nearest neigbors while interac-
tions are entirely on-site we recover the standard Bose-
Hubbard Hamiltonian [24]. Since we will be interested
k
1 inprocessespopulatingexcitedbandsweshalltakethem
wiα(x,s)=r4πk Z φαp(x)dp. (12) intoaccount,limitingourselvestoseveralofthemineach
−k direction.
As long as the number of bands included is limited
Now the field operator, Ψ(~r), is expanded in terms of to a finite number, any function built as a linear com-
three-dimensional (3D) Wannier functions Wα(~r,s) (for bination of Wannier functions will be smooth. Then a
i
7
pseudopotential, Eq.(10), may be replaced by a Fermi Contribution of the time dependence of Wannier
delta-potential functions to the tight-binding Hamiltonian
4π~2a
V(~r,~r′)= δ(~r−~r′). (18) Suppose thatwe areinterestedin the dynamicalprob-
m
lem with time dependent lattice depth s = s(t),s(0) =
From now on we shall assume a convenient units in s . TheinitialwavefunctionisexpressedinWannierbasis
0
which energy is expressed in recoil units (with recoil en- {Wα(·,s )}. Similarly the EBH Hamiltonian is obtained
ergy ER = ~22mk2,k = 2λπ), and 2λπ = 1/k is the unit of forithat0value. As discussed in the letter if s changes
length. The potential, (18) is then in time so does both the tunneling and interactions pa-
rametersoftheEBHHamiltonianaswellastheWannier
V(~r,~r′)=gδ(~r−~r′), (19)
functions themselves (an alternative approach of keep-
with dimensionless coupling constant g =8πak. The in- ing the basis fixedin time wouldrequire a differenttight
teractionintegralsseparateintoaproductcorresponding basisHamiltonian). Therefore,when s=s(t) is time de-
to each coordinate taking the form pendent so is the isometric basis transformation U(s(t))
fromthepositionrepresentationtothe lattice(Wannier)
Uiαjβklγδ = g uiαjtkβltγtδt representation. Of course H = U(s(t))HXU†(s(t)). De-
t=Yx,y,z fine the transformation via ψ(t) = U(s(t))ψ (t) (where
X
= g dtwαt(t)wβt(t)wγt(t)wδt(t). (20) ψX(t) is the wavefunction in the position representation
Z i j k l while ψ(t) corresponds to the lattice). Then a standard
Y
Frequentlydeepopticallatticesareusedtotoseparate textbook derivation gives the proper new Hamiltonian
atomic cloud into parts, a single retroreflected beam in HW in the form:
onedirectioncutsacloudinto2Dslices,twosuchperpen- d
dicularstandingwavesproducea setofveryweaklycou- HW =H+i~ U(s(t)) U†(s(t))=H+W, (25)
(cid:18)dt (cid:19)
pledtubes. Separatingdifferentenergyscalesonemayin
those situations assume, that in the tightly confined di- and the TDSE for ψ(t):i~∂tψ =HWψ.
rection (directions for tubes) the system remains in the For s(t) changing in time slowly enough, the second
lowest band and neglect the tunnelings between slices term may be neglected (this corresponds physically to
(tubes). This has to be done with care, virtual effect the assumption that the system has time to adapt to a
of high lying excited bands may be not negligible [28– given change of basis). For quick variations of s(t) this
30]. In this limit one may often approximate the Wan- term leads to appreciable effects, we shall now evaluate
nier function in that lowest band, say in z direction by its form.
a ground state of an appropriate harmonic oscillator, i.e The basis for the Hilbert space for a gas of N bosons
w0z ≈ H(z) = (κ/π)1/4exp(−κz2/2), where dimension- consists of symmetrized (tensor) products of N single
i
less κ = ~ωz/2ER = sz/ER where sz corresponds to particle basis functions Wiα. In restricted lattice geome-
the lattice depth in thpat direction. Thus the basis func- tries,westudyofmultiparticlestateswhichintransverse
tions for 2D system with the effective potential of the direction(s) contain a harmonic oscillator ground states
form — we take W functions as in (22) or (24). These wave-
m functions are effectively 1D or 2D with finite, effective
V2(~r)=sxsin2(kx)+sysin2(ky)+ 2 ωz2z2, (21) “width”, set by the curvature of transverse confinement.
WedenotesuchabasisforN particleproblembyWN(s),
are
the basis depends on the lattice height by the value of
Wiα(~r,s)=wiαx(x,sx)wiαy(y,sy)H(z). (22) s parameter through the set of single particle Wannier
functions W(s)={Wα(·,s)} .
For 1D tube, the corresponding formulae read i i,α
The Hilbert space for the lattice system (in which
m
V (~r)=ssin2(kx)+ ω2(y2+z2), (23) the Bose-Hubbard hamiltonian is usually expressed) has
1
2
time-independent basis (the Fock basis) F.
and Let us define a shortened notation. The lattice Fock
statewithoccupationn ofmodeιwillbedenotedas|~n i
Wα(~r,s)=wα(x,s)H(y)H(z). (24) ι L
i i Thecorrespondingstateinthepositionrepresentationfor
Observethatinthereducedgeometriestunnelingtakes thelattice withheightswillbe abbreviated: |~n ,si. We
X
place along the periodic lattice potential only. This always assume that n +...n =N.
1 L
simplifies the notation, in particular in 1D geometry, Action ofthe mapU(s) fromthe continous space with
the index α numbering the bands is simply an inte- base WN(s), to the Fock space with base F is rather
ger, α = 0,1,..,B, the tunneling is unidimensional and trivial: inthe chosenorthonormalbasesitis the identity
Hamiltonian (4) of the paper is obtained as an extended matrix — it maps state |~n ,s(t)i to a state |~n ,s(t)i.
X L
Bose Hubbard system . The map is thus always an isometry (note that for 1D
8
and 2D lattice it is only a partial isometry from a full correspond to changing the mode of only one particle
continuous space). from configuration~n — the mode ι to κ. Therefore:
SingleparticlestatesdefinedbyWannierfunctionsand
in the discrete lattice are enumerated by two indices α mi =ni i6=ι,κ
andi,fromthis pointupto theendofthe derivation,we  mι =nι−1 (29)
introduce the multiindex ι=(α,i) to simplify notation.  mκ =nκ +1
Using this notation we can express the map U as: 
As Tι = 0 due to norm preservation, only ι 6= κ terms
ι
contribute. Change of occupation is compatible with ac-
U(s)= |m~ ihm~ ,s|
L X tionofa†a operator. Wewillshowthatalsothenumer-
X~n κ ι
ical factor agrees. A mode to be differentiated (mode ι)
Now we expand the derivative of U isometry. We may be chosen in (27) in n ways, and:
ι
use the fact that basis of the Fock space F is time-
independent: nι nι(nκ +1)
=
d d n1!n2!...nι′! n1!...(nιp−1)!...(nκ +1)!...nι′!
U(s(t)) |ψi = |m~ i hm~ ,s(t)| |ψi . p p (30)
X L X X
(cid:18)dt (cid:19) Xm~ (cid:18)dt (cid:19) Thus from (26), (27) and (28):
Thus the W term is just: d ∞
κ
dt|~nX,s(t)i= Tι nι(nκ +1)|m~X,s(t)i (31)
W = i~ |m~ ih~n | dhm~ ,s(t)| |~n ,s(t)i= κX,ι=1 p
L L X X
(cid:18)dt (cid:19)
~nX,m~ Above m~ is assumed to satisfy relations (29). All in all,
d we obtain
= −i~ |m~ ih~n |hm~ ,s(t)| |~n ,s(t)i (26)
L L X X
(cid:18)dt (cid:19)
n~X,m~
The relation between ~n and m~ that may give nonzero U˙(t)U†(t)=− Tικ|m~Lih~nL| nι(nκ +1)=
Xι,κ p
contributionto the abovesum remains to be workedout
as well as exact values of the coefficient. To do so, we =− Tικa†κaι. (32)
expand the time derivative, by inserting exact action of Xι,κ
the symmetrization operator:
We now go back to the original labeling by Bloch band
d number: ι=(α,i),κ =(β,j). To obtain the form of the
N!n !n !...n ! |~n ,s(t)i=
1 2 ι0 dt X W term from the
d p Now Tκ = Tβα = −Tαβ. We obtain the form of W
W (x )W (x )...W (x )· ι j−i i−j
dt 1 π(1) 1 π(2) 1 π(n1) term used in the main article:
π∈XS(N) (cid:2)
·W (x )...W (x )...W (x )
2 π(n1+1) 2 π(n1+n2) ι0 π(N)
N (cid:3) W = −i~ Tiα−βj(aαi)†aβj,
= W (x )W (x )...W (x )· Xι,κ
1 π(1) 1 π(2) 1 π(n1)
Xk=1π∈XS(N)(cid:2) Tαβ = W˙ β(x)Wα(x)d3x. (33)
·W (x )...W˙ (x )...W (x ) (27) i−j Z j i
2 π(n1+1) ι π(k) ι0 π(N)
i κ
The term T has to be worked out for the basis
ι
In the above line each of Wannier functions Wι depends functions for the lattice in the appropriate dimension.
on t through s(t). The formula is well-stated, because In the 1D lattice, Wannier functions are of form (24),
only finite number of modes has nonzerooccupation: for then: Tκ = dydzH(y)2H(z)2 dxwβ(x)w˙α(x).
ι j i
ι>ι0 we have nι!=1 and no factors Wι. Due to normalizaRtion: Tκ = R dxwβ(x)w˙α(x).
Now we use the partition of unity |Wκ >< Wκ|, ι j κi
For 2D lattice, from (22), we Rget: T =
appying it on W˙ι we get: P dzH(z)2 dxdywβx(x)wβy(y)d wαx(x)wαy(y)ι .
ddtWι(x)= TικWκ(x) (28) RoNronizero=vaRljues∧mαayjxbe=obtβjayi.neTddhotun(cid:16)slytiixfheix =Wijyxt∧erαm(cid:17)x =peβrx-
Xκ y y y y
form hopping of a particle in only one direction
where Tικ = Wκ(x)W˙ι(x)dx. Therefore, by combining (including the Bloch band change). The corre-
together (26)R, (27) and (28) one obtains that the only sponding amplitude for hopping in y direction are:
(~n,m~) givingnonzerocontributionin (28) arethose that dzH(z)2 dxwαx(y)2 dywβy(y)w˙αy(y), which after
ix jy iy
R R R
9
normalization becomes just: dywβy(y)w˙αy(y). Simi- tially prepared in a relatively low lattice, on the super-
jy iy
larly for the x direction we obRtain: dxwβx(x)w˙αx(x). fluid side. Then a rapid quench is realized by a sudden
jx ix increase of the lattice depth. The subsequent evolution
Analogouslyforthe 3Dcase(using14R)the amplitudefor
hopping in direction t∈{x,y,z} is: dtwβt(t)w˙αt(t). of the system in a deep lattice is monitored by measur-
jt it ing the time dependence of the contrast of interference
R
fringesinthe momentumdistribution. The initial coher-
Analysis of the revival experiment entstatelikeoccupationofsitesevolvesdifferentlyinthe
deep insulating-type lattice showing the decay and par-
tialrevivalsofcoherence. Inthesimulationtheevolution
0 20 50 [µs] 1 2 3 4 5 of coherence may be monitored by the time dependence
oftheorderparameterφ=ha0iwherethesuperscriptin-
2 (a)
dicates that the lowest Bloch band is taken into account
only(wedropthesiteindex,asweshallconsiderasingle
φ 1 ramp site only - in a very deep lattice the tunneling may be
r neglected and a single site evolution is considered - see
e
met [10]).
ra 0 ThecontributionofW term(seetheletter)dependson
r pa 2 (b) the speed of the quench, it becomes important when ex-
e
d citationsofhigherBlochbandsbecomeappreciable. The
r
O
ramp authors[10]wantedtoavoidpopulationofexcitedbands
1
(that could be controlled experimentally) thus they ex-
perimentallychosethedurationofthequench,τ =50µs,
sufficientlylongsothatthehigherbandsexcitationswere
0
0 2 5 [µs] 1 2 3 4 5 negligible. Consequently a contribution of Wigner func-
Time [ms]
tions dynamics (the W term) is for experimentalparam-
eters quite small (see the upper panel in Fig. 5. As it
Figure 5: (color online) Time dependence of the order pa-
rameterinthelowestband,φ=hψ(t)|a0|ψ(t)ifor50µsinitial turns out already for τ = 20µs effects due to Wannier
linear quench as used in the experiment [10] (top panel) and functiondynamicsareappreciablewhileastillfaster5µs
for a faster quench of 5µs (bottom panel). Red full line - ex- quench leads to strong Wannier functions dynamics ef-
actevolutionusingHW Hamiltonian,Eq.(25),black(dashed) fects (compare Fig. 5b). Let us point out that to make
line the result obtained neglecting W term. The difference calculations less computer demanding we used a 2D lat-
between the two is negligible for the top panel, showing that
tice, the initial state, ψ(0) was prepared as a coherent
thehigherbandsplayaminorroleintheexperiment. Forthe
state with hn i= 2 at s = 8 and a linear quench up to
faster quench (bottom panel) the W term describing time- 0 0
s = 40 was realized. That roughly corresponds to the
dependenceofWannierfunctionsiscrucialforthedescription 1
of dynamics. Note that the time scale during the quench is revival plot in Fig. 2 of [10].
expandedincomparisontothesubsequentevolutiontomake It seems, therefore, that making the quench time in
theplot more readable. the experiment [10] shorter by an order of magnitude
would allow for a direct experimental verification of the
Intherevivalexperiment[10]theatomicsystemisini- approachpresented here.