Probing the relaxation towards equilibrium in an isolated strongly correlated 1D Bose gas

1.3 MB·English

by  Stefan Trotzky

Probingtherelaxationtowardsequilibriuminanisolatedstronglycorrelated1DBosegas
S.Trotzky1−3,Y.-A.Chen1−3 ,A.Flesch4,I.P.McCulloch5,U.Schollwo¨ck1,6,J.Eisert6,7 andI.Bloch1−3
1 Fakulta¨t fu¨r Physik, Ludwig-Maximilians-Universita¨t, 80798 Mu¨nchen, Germany
2 Max-Planck Institut fu¨r Quantenoptik, 85748 Garching, Germany
3 Institut fu¨r Physik, Johannes Gutenberg-Universita¨t, 54099 Mainz, Germany
4 Institute for Advanced Simulation, Forschungszentrum Ju¨lich, 52425 Ju¨lich, Germany
5 SchoolofPhysicalSciences, TheUniversityofQueensland, Brisbane, QLD4072, Australia
6 Institute for Advanced Study Berlin, 14193 Berlin, Germany and
7 Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany
The problem of how complex quantum systems eventually come to rest lies at the heart of statistical me-
1
chanics. Themaximumentropyprincipleputforwardin1957byE.T.Jaynessuggestswhatquantumstates
1
0 oneshouldexpectinequilibriumbutdoesnothintastohowclosedquantummany-bodysystemsdynamically
2 equilibrate. Anumberoftheoreticalandnumericalstudiesaccumulateevidencethatunderspecificconditions
quantummany-bodymodelscanrelaxtoasituationthatlocallyorwithrespecttocertainobservablesappears
n
asiftheentiresystemhadrelaxedtoamaximumentropystate.Inthiswork,wereporttheexperimentalobser-
a
vationofthenon-equilibriumdynamicsofadensitywaveofultracoldbosonicatomsinanopticallatticeinthe
J
regimeofstrongcorrelations. Usinganopticalsuperlattice,weareabletopreparethesysteminawell-known
3
initialstatewithhighfidelity.Wethenfollowthedynamicalevolutionofthesystemintermsofquasi-localden-
1
sities,currents,andcoherences.Numericalstudiesbasedonthetime-dependentdensity-matrixrenormalization

groupmethodareinanexcellentquantitativeagreementwiththeexperimentaldata. Forverylongtimes,all
]
s threelocalobservablesshowafastrelaxationtoequilibriumvaluescompatiblewiththoseexpectedforaglobal
a maximumentropystate. Wefindthisrelaxationofthequasi-localdensitiesandcurrentstoinitiallyfollowa
g power-lawwithanexponentbeingsignificantlylargerthanforfreeorhardcorebosons. Forintermediatetimes
- thesystemfulfillsthepromiseofbeingadynamicalquantumsimulator,inthatthecontrolleddynamicsrunsfor
t
n longertimesthanpresentclassicalalgorithmsbasedonmatrixproductstatescanefficientlykeeptrackof.
a
u
q Ultracold atoms in optical lattices provide highly control- Concept of the experiments. We consider a one-
.
t lablequantumsystemsallowingtoexperimentallyprobevar- dimensionalchainoflatticesitescoupledbyatunnelcoupling
a
ious quantum many-body phenomena. In this way, ground J andfilledwithrepulsivelyinteractingbosonicparticles. In
m
statepropertiesofHamiltoniansthatplayafundamentalrole the tight-binding approximation, the Hamiltonian takes the
-
in the condensed matter context have been investigated un- formofaone-dimensionalBose-Hubbardmodel[3,20]
d
on dhearrdperrectoiseplyrotbuenaibnleacctounadlitcioonndse[n1s–e3d].mFaetatteurremsattheartiaalrseoervetno Hˆ =(cid:88)(cid:20)−J(cid:16)aˆ†aˆ +h.c.(cid:17)+ Unˆ (nˆ −1)+ Knˆ j2(cid:21),
c simulate in numerical studies are dynamical ones, including j j+1 2 j j 2 j
j
[
dynamical properties emerging in adiabatic sweeps [4] and
1 farfromequilibrium[5–11]. Inthisrespect,forexample,the where aˆj annihilates a particle on site j, nˆj = aˆ†jaˆj reflects
v quenchfromashallowtoadeepopticallattice[6–8]andthe thenumberofatomsonsitej andU istheon-siteinteraction
9 phase dynamics emerging after splitting a one-dimensional energy. TheparameterK = mω2d2 (mistheparticlemass,
5 Boseliquid[12]havepreviouslybeenstudiedexperimentally. dthelatticespacing)describesanexternalharmonictrapwith
6
In this article, we report on the direct observation of re- trapping frequency ω (cid:39) 2π ×61Hz, present in the experi-
2
laxationdynamicsinaninteractingmany-bodysystemusing ments.
.
1 ultracoldatomsinanopticallattice. Startingwithapatterned Theexperimentalsequencecanbedescribedinthreeparts
0
density with alternating empty and occupied sites in isolated (see Fig. 1a): (i) At t = 0, the system is initialized in a
1
Hubbardchains,wesuddenlyswitchedonthetunnelcoupling density wave represented as a state vector |ψ(t = 0)(cid:105) =
1
: along these chains and measured the emerging dynamics in |··· ,1,0,1,0,1,···(cid:105), such that only lattice sites with an
v termsofquasi-localdensities,currentsandcoherences. Both evensiteindexareoccupiedandnotunnel-couplingispresent
i
X theinitialstatepreparationandthedetectionwasrealizedus- alongthechain. (ii)Afterthequenchtoadistinctsetofpos-
ing a bichromatic optical superlattice [13, 14]. For a wide itive parameters J, U and K, the system follows the non-
r
a range of (repulsive) inter-particle interactions, we find a fast equilibrium dynamics of the above Hamiltonian Hˆ. (iii) Fi-
relaxation of the measured observables to steady state val- nally,thetunnel-couplingissuppressedagainandtheproper-
ueswhichareconsistentwithadynamicalversionofJaynes’ tiesoftheevolvedstatevector|ψ(t)(cid:105)arereadout.
principle [15]. The timescale of the relaxation cannot be at- We started our experiments by loading a BEC of about
tributedtoaclassicalensembleaverage. Forshorttimes, we 45×10387Rbatomsinthe|F =1,m =−1(cid:105)Zeemanlevel
F
compare the experimental results to time-dependent density- intoa3Dopticallatticeformedbyretroreflectedlaserbeams
matrixrenormalizationgroupsimulations(t-DMRG,forare- ofwavelengthλ =1530nmalongonedirection(“longlat-
xl
viewseeRefs.[16,17]andreferencestherein)oftheHamil- tice”)andλ =844nmalongtheothertwo. Inthisloading
y,z
tonian dynamics without free parameters, further developing wewerecrossingthetransitiontoaMott-insulatorwhichre-
theideasofpreviousnumericalstudies[18,19]. sultedinanoccupationofnotmorethanoneparticlepersite.
2
a
0.6 a b
t
. . . 0.4
U/J = 2.44(2) U/J = 3.60(4)
(i) Preparation (ii) Evolution (iii) Readout 0.2
K/J = 5·10-3 K/J = 7·10-3
b
nodd dd 0
o
neven n 0.6 c d
c d
1.0 0.4
hk)4
sition (-202 nodd0.5 0.2 UK/J/J = = 5 9.1·160(7-3) KU/J/ J= =1 59·.190(1-3)
Po-4 0
0.0
0 1 2 3 4 0 1 2 3 8 101214161820 0 1 2 3 4 5 0 1 2 3 4 5
t (ms) t (ms)
4Jt / h
FIG.1. Relaxationofthedensitypattern. (a)Conceptoftheexper- FIG. 2. Relaxation of the local density for different interaction
iment: after having prepared the density wave |ψ(t = 0)(cid:105) (i), the strengths. We plot the measured traces of the odd-site population
lattice depth was rapidly reduced to enable tunneling (ii). Finally, n (t) for four different interaction strengths U/J (circles). The
odd
thepropertiesoftheevolvedstatewerereadoutafteralltunneling solidlinesareensemble-averagedresultsfromt-DMRGsimulations
wassuppressedagain(iii). (b)Even-oddresolveddetection: parti- withoutfreeparameters. Thedashedlinesrepresentsimulationsin-
clesonsiteswithoddindexwerebroughttoahigherBlochband.A cluding next-nearest neighbor hopping with a coupling matrix ele-
subsequentband-mappingsequencewasusedtorevealtheodd-and mentJ /J (cid:39)0.12(a),0.08(b),0.05(c)and0.03(d)calculated
NNN
even-sitepopulations[13,14]. (c)Integratedband-mappingprofiles fromthesingle-particlebandstructure.
versusrelaxationtimetforh/(4J) (cid:39) 0.9ms,U/J = 5.16(7)and
K/J (cid:39)9×10−3. (d)Odd-sitedensityextractedfromtherawdata
shown in c. The shaded area marks the envelope for free Bosons
profilesasafunctionofrelaxationtimeforh/(4J)(cid:39)0.9ms,
(lightgrey)andincludinginhomogeneitiesoftheHubbardparame-
U/J = 5.16(7)andK/J (cid:39) 9×10−3. Weplottheresulting
tersintheexperimentalsystem(darkgrey).
traces n (t) in Fig. 1d. We generally observe oscillations
odd
inn withaperiodT (cid:39)h/(4J)whichrapidlydampenout
odd
within3-4periodstoasteadyvalueof(cid:39)0.5. Thesamequal-
Finallyweaddedtothelonglatticeanotheropticallatticewith itative behavior is found in a wide range of interactions (see
wavelengthλ = 765nm = λ /2(“shortlattice”)withthe Fig.2).
xs xl
relativephasebetweenthetwoadjustedtoloadeverysecond We performed t-DMRG calculations, keeping up to 5000
site of the short lattice [14, 21]. Completely removing the states in the matrix-product state simulations (solid lines in
long lattice gave an array of practically isolated 1D density Fig. 2). The Bose-Hubbard parameters used in these sim-
waves|ψ (cid:105) = |··· ,1,0,1,0,1,···(cid:105)–thusrealizingstep(i) ulations were obtained from the respective set of experi-
N
– with a distribution of particle numbers N and thus lengths mental control parameters. Furthermore, we took into ac-
L = 2N −1givenbytheexternalconfinement. Forourpa- count the geometry of the experimental setup by perform-
rameters,weexpectchainswithamaximalparticlenumberof ing the corresponding ensemble average E over chains
{N}
N (cid:39)43andameanvalueofN¯ (cid:39)31(seeSupplementary with different particle numbers N (see Supplementary Ma-
max
Materialfordetailsontheloadingprocedure). terial). For the times accessible in the simulations, these av-
Toinitializethemany-bodyrelaxationdynamicsofstep(ii), erages differ only slightly from thetraces obtained for a sin-
we quenched the short-lattice depth to a small value within gle chain with the maximal particle number N = 43 of
max
200µs, allowing the atoms to tunnel along the x-direction. the ensemble (see Supplementary Material). For interaction
After a time t, we rapidly ramped up the short lattice to its strengths U/J (cid:46) 6 (Fig. 2a-c), we find a good agreement
original depth, thus suppressing all tunneling. Finally, we oftheexperimentaldataandthesimulations. Inthisregime,
read out the properties of the evolved state in terms of den- only small systematic deviations can be observed, which are
sities, currents and coherences in step (iii). Note that in the strongest for the smallest value of U/J which corresponds
experiments we always measured the full ensemble average to the smallest lattice depth. They can be attributed to the
X(t) = E (cid:104)ψ (t)|Xˆ|ψ (t)(cid:105)ofanobservableXˆ overthe breakdownofthetight-bindingapproximationforshallowlat-
{N} N N
array of chains (denoted by the averaging operator E ), ticeswhichgivesrisetoasignificantamountoflonger-ranged
{N}
rather than the expectation value for a single chain with N hopping. When including a next-nearest neighbor hopping
particles. term−J (cid:80) (aˆ†aˆ +h.c.)inthet-DMRGsimulations
NNN j j j+2
Relaxationofquasi-localdensities. Wefirstdiscussmea- we obtain quantitative agreement with the experimental data
surementsofthedensityonsiteswitheitherevenoroddindex. (dashed line in Fig. 2). For larger values of U/J and corre-
Afterthetimeevolution,wetransferredthepopulationonodd spondinglydeeperlattices,thetight-bindingapproximationis
sitestoahigherBlochbandusingthesuperlatticeanddetected valid. For U/J (cid:38) 10 (Fig. 2d), larger deviations are found.
these excitations employing a band-mapping technique (see Here,thedynamicsbecomemoreandmoreaffectedbyresid-
Fig.1b)[13,14]. Fig.1cshowstheintegratedband-mapping ualinter-chaintunnelingandnon-adiabaticheatingastheab-
3
solutetimescaleoftheintra-chaintunneling∝ 1/J becomes a o e o e o e o - e o o e - o
larger. or
The results of the density measurements can be related to
the expectations for an infinite chain with K = 0. There,
b 1
the time-evolution can be calculated analytically in the case
d
of either non-interacting bosons (U/J = 0) or infinite inter- od0.5 A
actions (U/J → ∞) [18, 19]. These limiting cases can be n
well-understoodthroughthemechanismoflocalrelaxationby 0
0 50 100 150 200 250
ballisticallypropagatingexcitations.Theon-sitedensitiesfol-
low0-thorderBesselfunctionsdescribingoscillationswhich tDW (µs)
are asymptotically dampened by a power law with exponent c 2π
φ
−1/2. The damping we observe in the interacting system,
0
however,ismuchfaster. Thisbehaviorhasalsobeenfoundin α
1
t-DMRG simulations of homogeneous Hubbard chains with 1 2
A
finiteinteractions[18,19]. Theexactoriginofthisenhanced e
relaxation in the presence of strong correlations constitutes d 0.1 1
oneofthemajoropenproblemsposedbytheresultspresented plitu0.5 0.01 0
here. m 1 0 5 10 15
A U/J
Measurements of quasi-local currents. Employing the
0
bichromaticsuperlattice,wewerealsoabletodetectthemag- 0 1 2 3 4
nitudeanddirectionofquasi-localdensitycurrents.Insteadof 4Jt/h
raising the short lattice at the end of step (ii), we ramped up
the long lattice to suppress the tunnel-couplings through ev-
erysecondpotentialbarrierinthechain(seeFig.3a). Atthe FIG.3. Quasi-localcurrentmeasurement. (a)Tomeasurethequasi-
same time, we set the short lattice to a fixed value to obtain local density flow every second tunnel coupling was suppressed,
always the same value of (U/J) (cid:39) 0.2 in the emerging couplingeitherodd-evenoreven-oddpairs. (b)Oscillationsofthe
DW
odd-site population in the double-wells with fitted sine-waves for
double wells. By tuning the relative phase between the long
t = 100µs(solid),200µs(dashed)and400µs(dotted). Thevalue
andshortlatticewewereabletoselectivelycouplesiteswith
ofU/J duringtherelaxationwas5.16(7). (c)Extractedamplitude
index(2j,2j+1)(“even-odd”,jinteger)or(2j−1,2j)(“odd-
A and phase φ of the double-well oscillations for odd-even (filled
even”). We recorded the time-evolution in the now isolated
circles)andeven-odd(opencircles)couplings. Thesolidlinesshow
double-wellsusingthesamefinalread-outschemeasforthe
therespectiveresultsofthet-DMRGsimulations. Thedashedlines
densities(seeFig.3b). Wefindsinusoidaltunneloscillations arefitstoalinearincreaseinthephaseandapower-lawdecayofthe
whichdephaseonlyslowlyanddecreaseinamplitudewithin- amplitude.Theinsetsshowtheamplitudeinalog-logplot(left)and
creasing relaxation time t. The phase φ and amplitude A of theextractedpower-lawcoefficients(right).Thehorizontalgreyline
these oscillations were extracted from a fit of a sine-wave to indicates the power-law coefficient α = 0.5 for free and hardcore
the data and are plotted in Fig. 3c as a function of the relax- bosons.
ationtimeforU/J = 5.16(7). Whilethephasecontainsthe
information about the direction of the mass flow, the ampli-
tude is a combination of the local population imbalance and It is key to the experiment that the observed fast damping
thestrengthofthelocalcurrent. cannot be attributed to a mere classical ensemble averaging
Wefindφtoevolvelinearlyintime,givingstrongevidence duetotheinhomogeneousdistributionoftunnel-couplingsin
thattheexcitationsinthesystemexpandapproximatelyballis- the various chains (var(J)/J (cid:39) 0.4%) or the external trap.
tically as suggested in Refs. [18, 19]. Furthermore, its value Furthermore, we ensure that the transverse tunnel-coupling
does not change when coupling even-odd or odd-even sites, between adjacent chains J⊥ is always one to two orders of
indicating the absence of center-of-mass motion in the sys- magnitude smaller than J. Furthermore, the dynamics of
tem. The amplitude A on the other hand decays to zero on a single site – or of the densities of odd sites – cannot be
the same timescale as the oscillations in the local densities described in terms of simple rate equations, and not even
dampen out – in fact the quantities (1 ± A)/2 provide en- in terms of Markovian quantum master equations reflecting
velopestothetracesn andn (seeSupplementaryMa- damped motion (see Supplementary Material). Similarly, no
odd even
terial). On short timescales, 0 < 4Jt/h < 3, we find the dynamical mean-field description can capture the dynamics
decay of the amplitude to follow an approximate power-law forlargeU [22]. Hence, anyrealisticdescriptionhastonec-
∝t−αwithα=0.86(7). Thisbehaviormightwellchangeat essarily include the many-body and non-Markovian features
longertimes,wherenosignificantamplitudewasmeasurable. ofthedynamics,contributingtothechallengeforanumerical
We extract the power-law coefficients α for a wide range of simulationforintermediatetimes.
U/J (rightinsettoFig.3c). Inallcases, theabsolutevalues Time-evolution of the quasi-momentum distribution. A
ofthecoefficientsarelargerthantheoneexpectedforfreepar- different view on the relaxation can be obtained from the
ticles,whereα=0.5,againindicatingthefasterrelaxationin quasi-momentumdistributionoftheensemble. Wheninstan-
thepresenceofinteractions. taneously switching off all trapping potentials after a relax-
4
a 2 onlythenext-neighborcoherencesremainasalsofoundfrom
) Experiment t-DMRG
k t-DMRG simulations of homogeneous Hubbard chains with
h
(2 0 finiteinteractions[19]. Weextractthevisibilityofthelowest
F
o frequencycomponentasdescribedinRef.[14]bothfromthe
T
x-2 experimentaldataandthet-DMRGcalculations(seeFig.4b).
0 0.5 1 1.5 2 0 0.5 1 1.5 2
Intheabsenceoflocalcurrentsinthesystem,thevisibilityof
4Jt / h
b 1 thelowestFouriercomponentisgivenby4E{N}Re(cid:104)aˆ†jaˆj+1(cid:105).
We find good agreement between experiment and numerics.
y
bilit0.5 T0.h5ecvoirsriebsipliotyndbiunigldtsoutphetofiwrsatrdmsaaxfiimrsutmmainxinmum(ta)t(4FJigt/.h2c(cid:39)),
si odd
Vi followedbydampenedoscillations. From4Jt/h (cid:39) 2on,we
find the visibility to increase slowly with t towards an equi-
0
0 0.5 1 1.5 2 2.5 libriumvaluewhichissignificantlyhigherthantheonefound
4Jt / h
c for a homogeneous system (dashed line in Fig. 4b). In the
1
presenceofthetrap,thesystemcanexpandafterthequantum
0.8 t = 5 x h/4J
bility0.6 ~J/U quench,convertingkineticenergyproportionalto−(cid:104)aˆ†jaˆj+1(cid:105)
si0.4 into potential energy associated with (cid:104)nˆjj2(cid:105). As a conse-
Vi0.2 quence,theabsolutevalueof(cid:104)aˆ†aˆ (cid:105)increases.
j j+1
0 We observe the same qualitative behavior for the whole
0 10 20 30 40
U / J rangeofinteractionstrengthU/Jaccessedintheexperiments,
but with a strength of the next-neighbor correlations which
FIG. 4. Build-up of short-ranged correlations. (a) Plot of the
depends on U/J. In Fig. 4c, we plot the visibility of the in-
integrated density profiles obtained after ToF versus 4Jt/h for
terferencepatternsfoundfromexperimentsatdifferentvalues
U/J = 5.16(7) as obtained in the experiment (left) and recon-
of U/J with at a relaxation time of 4Jt/h = 5, where den-
structed from numerical t-DMRG simulations (right). The images
sities and currents are fully relaxed. We find the visibility
showthecrossoverfromapurelyGaussiandistribution(t = 0)to
a more complex quasi-momentum distribution (0 < 4Jt/h < 2) to have a maximum around U/J = 4 while it decreases to-
to a purely sinusoidal pattern (4Jt/h > 2). (b) Visibility of the wards the analytic limits of U → 0 and U → ∞. In both
interferencepatternsversus4Jt/hobtainedexperimentally(circles) oftheselimits,theproblemcanbemappedontofreeparticles
andfromthesimulations(solidcurve). Thegreylinerepresentsthe where no coherences will survive for long times. In the lat-
measuredvisibilityat4Jt/h(cid:39)5,whilethedashedlinecorresponds ter case of hardcore bosons, the coherences are found to be
tothevalueobtainedfromthesimulationofahomogeneoussystem suppressedwithJ/U intheexperiments. Thesamebehavior
[19].(c)Steady-statevalueofthevisibilitymeasuredat4Jt/h(cid:39)5.
can be found from perturbation theory in J/U for a thermal
Thesolidlineisaguidefortheeye∝J/U.
state in the lattice [19]. Recent theoretical results show that
fornon-degeneratespectra,thetime-averagedstateisalways
identicaltothemaximumentropystategivenbythefullsetof
ation time t and letting the cloud expand freely for a time
constantsofmotion[23]. Ontheotherhand,ifastaterelaxes
t , the density distribution takes the form n (r) ∝
ToF ToF locally,ithastobetotheprojectionofthistime-averagedstate
|w (mr/(cid:126)t)|2S(mr/(cid:126)t). Here, w˜ (k) is the Fourier trans-
(cid:101)0 0 ontherespectivesubsystem. However,itisuptonowunclear
formoftheon-siteWannierorbitalandtheinterferenceterm
how to physically interpret and identify all constants of mo-
for the ensemble of decoupled Hubbard chains in the far-
tion defining the maximum entropy state. Interestingly, the
fieldlimitisS(k) = E{N}(cid:80)j,j(cid:48)eikx(j−j(cid:48))d(cid:104)aˆ†jaˆj(cid:48)(cid:105)withd = findings presented here are compatible with the expectations
λxs/2 being the lattice spacing along the chain direction. In foraGibbsstatedefinedonlybythetotalenergyandthetotal
Fig.4a,weplotthemeasureddensityprofilesintegratedover number of particles. Finally note that the build-up of short-
they-andz-directionasafunctionoftherelaxationtime(left range phase coherence to a finite value complements the ob-
panel)togetherwiththecorrespondingpatternsreconstructed serveddecayofthedensitypatternandthecurrents. Itcannot
from t-DMRG simulations for the full distribution of chains be explained by any classical dephasing mechanism, but is a
(rightpanel)forU/J (cid:39)5.Boththeexperimentaldataandthe resultofgenuinemany-bodydynamicsinthesystem.
numericalcalculationshowarapidbuild-upofshort-rangeco- Forthetime-evolutionofthedensities,currentsandcoher-
herence,notpresentintheinitialstate. ences, t-DMRG simulations and experimental results show
At short relaxation times 4Jt/h (cid:46) 2, the simulation data a remarkable congruence. This emphasizes the clean im-
shows a strong cosinusoidal component with a period of plementation of controlled quantum dynamics in the one-
2(cid:126)k =2ht /(mλ )andweakercontributionsfromhigher dimensional interacting Hubbard-model and the high fidelity
ToF xs
harmonics. While the former correspond to next-neighbor of the initial-state preparation. All parameters are calculated
coherences in the system, the latter are a signature of corre- abinitofromtheexperimentalcontrolparameters. Therefore,
spondingly longer range coherences which rapidly decay in the experiments can be seen as a self-sustained dynamical
the relaxation process [19]. Due to the noise on the experi- quantum simulation where the simulation effort is the same
mental data, the higher frequency components are weak, but for each value of t and each set of parameters. As long as
canstillbeidentified. Forlongerrelaxationtimes4Jt/h(cid:38)2, the time-evolution is not perturbed by experimental imper-
5
fection, this dynamical quantum simulator outperforms any canbeseenasadynamicalversionofJaynes’principlewhich
continuous-time numerical simulation for which the calcula- couldrecentlybesubstantiatedtheoreticallyforlocalobserv-
tionaleffortincreaseswitht. Simulationmethodsonclassical ables or reduced states [18, 28, 29] and for two-periodic ob-
computerssuchasmatrix-productstatebasedtime-dependent servables[19,22]asconsideredhere. Afterafinitetime,the
DMRGusedheresufferfromanextensiveincreaseinentan- closed quantum system cannot be distinguished locally from
glement entropy which limits the relaxation times accessible havingreachedaglobalGibbsstateundertheconstraintofa
inthecalculations[24,25]. setofmacroscopicconstantsofmotionsetbytheinitialstate
ConclusionsandOutlook. Inconclusion,wehavedemon- [15,26,29,30].
strated measurements on the relaxation of a charge-density A direct measurement of global observables to, e.g., iden-
wave of in a strongly correlated one-dimensional Bose gas tify constants of motion of the dynamics is inhibited by the
with varying interactions. Using a bichromatic optical su- ensemble of chains with various particle numbers in the ex-
perlattice, we were able to prepare a patterned density state perimentalrealization. Thislimitationmightbeovercomeby
with high fidelity and to induce non-equilibrium dynamics eitherpreparingasinglechainwithfixedlengthorbyselect-
by rapidly switching on the tunnel-coupling along the chain. ingasinglechainfromtheensemblefordetection. Thiswill
We could follow the dynamical evolution of the initial state openthewaytoanswerwhatglobalstateisafterallreachedin
in terms of even- and odd-site densities, local currents and the evolution, including possible pre-thermalization [31] and
short-range correlations visible in the quasi-momentum dis- experimentalstudiesoftheeigenstatethermalizationhypothe-
tribution. Wehavecomparedourmeasurementstoparameter- sis[32–34].Itisthehopethatthepresentworktriggersfurther
free t-DMRG simulations, finding excellent agreement and experimentalstudieswhichaddresstheoreticallyunsolvedkey
identifyingcontributionsfromnext-nearestneighborhopping. questionsofnon-equilibriumdynamics.
All three observables can be seen as local probes of the sys- We acknowledge stimulating discussions with B. Paredes,
temandshowarapidrelaxationtosteadystatevaluesasitis M. Cramer and C. Gogolin. This work was supported by
predictedforHubbard-typemodelsbyacentrallimittheorem the DFG (FOR 635, FOR 801), the EU (NAMEQUAM,
[26, 27]. These steady state values are compatible with the QESSENCE,MINOS,COMPAS),theEURYI,andDARPA-
systemgloballybeinginamaximumentropystate. Thisidea OLE.
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6

a b 0.6 a b
150 0.10
1000)100 eight00..0068 00..24
( w
N n=1 50 rel. 00..0024 odd 00 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5
n
00 50 100 150 00 10 20 30 40 0.6 c
Ntot (1000) N 0.4
0.2
FIG.5.aMeasuredatomnumberonsiteswithn=1versusthetotal d
atomnumber.ThesolidlinerepresentstheconditionN =N . 0
tot n=1
0 0.5 1 1.5 2 2.5 0 1 2 3
Thearrowindicatesthetotalatomnumberusedinourexperiments.
4Jt / h
bDistributionofparticlenumbersperchaincausedbytheharmonic
confinementcalculatedforourloadingsequence.
FIG.6. DMRGresultsfortherelaxationofthequasi-localdensities
n .Thesolidlinesrepresenttheensembleaveragedt-DMRGdata
odd
plottedinFig.2a-d.Thedashedlinesarethesimulationresultsfora
SUPPLEMENTARYMATERIAL singlechainintheensemblewithN =43particles.Theparameters
usedinthesimulationsarethesameasinFig.2.
I. LOADINGPROCEDUREANDSUDDENQUENCH
DMRGcalculationswereperformedforseveralparticlenum-
We started the loading procedure by ramping up the long
latticetoadepthof30Exlwhichresultsinanarrayofisolated bersrangingfrom5to43andweightedaccordingtothedis-
r tributionshowninFig.5b.InFig.6,weplottheaveragedden-
two-dimensionalquantumgases. Thelatticedepthisgivenin
unitsoftherespectiverecoilenergyEi = h2/(2mλ2). Sub- sities as obtained from DMRG and shown in Fig. 2 together
r i with the results for a single chain with the maximal number
sequently, we ramped up the two transverse optical lattices
withwavelengthsofλ = 844nmto30Ey,z, crossingthe ofparticles. Bothcurvesdifferonlyslightlyonthetimescales
y,z r accessiblebythesimulations.
Mottinsulatortransitionineachofthetwo-dimensionalgases
Intheexperiment,thesmoothexternaltrapleadstoaslow
with a maximal filling of one atom per site (see Supplemen-
expansion of the cloud along the x-direction, rather than a
taryMaterial). Wethenaddedtheshortlatticewithadepthof
30Exs to the long lattice, forming a bichromatic period-two sharp reflection of excitations traveling at a velocity ∝ J in
r
the system. Thus, the length of the one-dimensional lattices
superlattice[14]. Wesettherelativephasebetweentheshort
intheDMRGcalculations(∼121)waschosensuchthatthe
andthelonglatticesuchthateverysecondshort-latticesiteco-
particles do not reach the system boundary during the simu-
incidedwithaminimumofthelong-latticepotential. Finally
lated times. This absence of a sharp reflection also explains,
removingthelonglatticecompletelyyieldedanensembleof
togetherwiththeaveragingoverdifferentchainswithdiffer-
one-dimensionaldensitywaves|··· ,1,0,1,0,1,···(cid:105)[7,14]
ent particle numbers for which recurrences would happen at
withvariousparticlenumbersN.
different times, why no recurrences of the density wave are
We verified the loading of single atoms by bringing each
visibleinourexperiments.
atomintoasuperpositionofthestates|F =1,m =−1(cid:105)and
F
F = 2,m = 1(cid:105), where atom pairs can undergo hyperfine-
F
relaxing collisions which would expel them from the lattice.
We measured the total number of particles with (N ) and III. CURRENTMEASUREMENTS
tot
without this filtering step (N ), finding that up to a total
n=1
number of 60×103 atoms, no pairs were removed from the We measure the quasi-local currents by suppressing every
lattice (see Fig. 5a). For the experiments, we chose a parti- secondtunnelcouplingin200µswiththehelpofthesuperlat-
cle number of 45×103 in order to obtain maximally singly tice,suchastoobtainchainsofsymmetricdouble-wells(see
occupiedsites. Fig. 3a). For all current measurements, we chose the super-
lattice depths to be 40Exl and 4Exs, respectively. In this
r r
configuration, we have a tunnel coupling within the double-
II. DMRGSIMULATIONRESPECTINGTHEGEOMETRY wells of J (cid:39) h × 8kHz and an interaction strength of
DW
OFTHEPROBLEM (U/J) (cid:39) 0.2, such that the particles in the double-well
DW
willpracticallytunnelindependently. InFig.3b, weplotthe
Aftertheloadingprocedure,theoccupiedsitesformanel- relativepopulationontheleftsideofthedoublewellsn asa
L
lipsoid around the center of the trap. Thus, due to the sup- functionoftheholdtimet andforthreedifferentrelaxation
DW
pressed transversal hopping, the experimental setup is de- timesastheywereevaluatedforFig.3c. Here,wehavemea-
scribed by a two-dimensional approximately circular array suredthepopulationontheleftandrightsidesofthedouble
of one-dimensional chains with different particle numbers wells by the same band mapping technique which was used
N with N = 43. For a realistic theoretical description, forthedensitymeasurements. Thephaseφandtheamplitude
max
7
oftheoscillationsshowninFig.3careextractedfromasimple reflects Markovian dynamics. Since this is a bosonic free
sinusoidalfittothedata. model, onecankeeptrackoftheevolutionbyspecifyingthe
Ifweassumetheparticlesinthedouble-wellformedbythe firstandsecondmomentsalone. Thesecondmoment√matrix
siteswithindexjandj+1toevolveunderthesimplesingle- of ρ(t) is given by, with ˆb = ˆb , Xˆ = (ˆb +ˆb†)/ 2 and
√ j
particle Hamiltonian Hˆ = −J (aˆ†aˆ + h.c.), we P =i(ˆb†−ˆb)/ 2,
DW,j DW j j+1
findtheoscillationamplitudetobe
(cid:20) 2(cid:104)Xˆ2(cid:105)(t) (cid:104)XˆPˆ(cid:105)(t)+(cid:104)PˆXˆ(cid:105)(t)(cid:21)
(cid:113) γ(t)= ,
A (t)= (cid:0)(cid:104)nˆ (t)(cid:105)−(cid:104)nˆ (t)(cid:105)(cid:1)2+4Im(cid:0)(cid:104)aˆ (t)†aˆ (t)(cid:105)(cid:1)2, (cid:104)XˆPˆ(cid:105)(t)+(cid:104)PˆXˆ(cid:105)(t) 2(cid:104)Pˆ2(cid:105)(t)
j j j+1 j j+1
(1)
and the evolution is given by the Gaussian channel γ(t) =
where t is the relaxation time. From the simultaneous mea-
A(t)γ(0)AT(t)+B(t). Astraightforwardcalculationgives
surement of the local densities, it is thus possible to recon-
struct the bare mass current Im(cid:104)aˆ (t)†aˆ (t)(cid:105) through the
j j+1 (cid:18)L/2 (cid:19)
barrier between the two sites as a function of the relaxation B(t)= (cid:88)(2|V (t)|2+1) I , A(t)=|V (t)|2I ,
j,2k 2 j,j 2
time t. Furthermore, it is evident from Eq. (1), that when-
k=1
everthemasscurrentvanishestheamplitudeismeasuringthe
populationimbalancebetweenthesitesj andj+1. Atthese withV(t)=e−itH,asamatrixexponential.
points,thequantities(1±A)/2reproducetheeven-andodd- This evolution then has to be compared with a Markovian
site densities, respectively. Therefore they provide two en- timeevolution,onethatsatisfies
velopeston (t).
odd,even
For the phase φ (t), we find from the same calculation as A(s+s(cid:48))=A(s)A(s(cid:48)), B(s+s(cid:48))=A(s)B(s(cid:48))A(s)T+B(s)
j
above
aswouldnecessarilybetrueforadynamicalsemi-groupgen-
(cid:32) 2Im(cid:0)(cid:104)aˆ (t)†aˆ (t)(cid:105)(cid:1)(cid:33) erated by a Gaussian Markovian master equation. A reason-
φj(t)=arctan − (cid:104)nˆ (t)j(cid:105)−(cid:104)nˆj+1(t) . (2) able figure of merit of Markovianity of a Gaussian channel,
j j+1 is, for a given time t ≥ 0, how different the dynamical map
isfromaMarkovianone. Thereexistsmorethanonereason-
Weusethesetworelationstocalculatetheensemble-averaged
amplitude E A(t) = |E (cid:80) A (t)exp(iφ (t))| and able way to quantify such a difference, yet all lead to qual-
phaseE φ{(Nt)}=arg[E {(cid:80)N}A j(t)eixp(iφ (t))j]fromthe itatively identical results. A physically well-defined way of
{N} {N} j i j quantifying this is a norm difference from the Jamiolkowski
DMRGsimulations(solidlinesinFig.3b).
covariancematrixoftheclosestMarkovianGaussianchannel
to the given channel at hand [35]. A more pragmatic—but
in the certification of non-Markovianity equally effective—
IV. DEVIATIONFROMMARKOVIANANDMEAN-FIELD
meansis,foratimet≥0,tosimplycompute
DYNAMICS
inf(cid:107)A(s+s(cid:48))−A(s)A(s(cid:48))(cid:107) , (3)
2
In this section, we show that the time evolution cannot be
describedbyaMarkovianquantummasterequation,signify- withtheminimizationbeingperformedwithrespecttos,s(cid:48) ≥
ing the complex relaxation dynamics beyond a situation that 0 such that s+s(cid:48) = t. This quantity can easily be seen to
can be described in terms of rates. That is to say, each con- take positive values for intermediate times, giving rise to a
stituent does not see the rest of a chain as a mere bath it is boundthatcanbedirectlyrelatedtoobservabledifferenceson
weakly coupled to, but intricate memory effects do play a states. Hence, no Markovian dynamics—amounting to rate
role. To see this, it is sufficient to consider the simple case equations—can model the dynamics encountered here, and
of local relaxation dynamics of a single site and the case of memoryandgenuinequantummany-bodyeffectshavetobe
U =0. Werestrictourselvestotheinfinitetranslationallyin- considered.
variantcaseandaninitialstateofadensitywaveasdescribed The dynamics found here is also not consistent with a
above—anyothersettingisonlymorecomplexandingeneral mean-fieldpictureforvaluesofU significantlydifferentfrom
also non-Markovian. To follow a Markovian time evolution zero. In such a picture, one looks for a time-dependent self-
of the reduced state of some odd site j, denoted as ρ, means consistentsolutionfor
that its time evolution follows a master equation in Lindblad
1 (cid:88)
form x(t)= (−1)j−1(cid:104)nˆ (cid:105)(t),
L j
dρ(t)=i[ρ(t),hˆ]+(cid:88)G (cid:16)Fˆ ρ(t)Fˆ†− 1{Fˆ†Fˆ ,ρ(t)} (cid:17), j
dt α,β α β 2 β α +
α,β forLsitesinamean-fieldHamiltonian
(cid:20) (cid:21)
wherehˆ isaHermitianoperatorthatcanbedifferentfromthe Hˆ(t)=−(cid:88) ˆb† ˆb +ˆb†ˆb + U(cid:0)−1 +(−1)j−1x(t)(cid:1)nˆ ,
freeHamiltonianHˆ,andGandFˆ aresomearbitrarymatri- j+1 j j j+1 2 2 j
α j
ces, reflecting the influence of the rest of the chain. This is
themostgeneralformofamasterequationwhenthedynam- in a variant of the findings of Ref. [22]. Using Runge-Kutta
ical map, mapping the initial state |1(cid:105)(cid:104)1| = ρ(0) (cid:55)→ ρ(t), numerical integration one finds that although for short times
8
andsmallU, theevolutionofdensities(cid:104)nˆ (cid:105)(t)isquitecom- non-trivialityofthedynamics,inthatameanfieldpicturecan-
j
patible with a mean-field picture, one encounters significant notcapturethedynamicsathand,andsophisticatedt-DMRG
deviationsforlargerU andlargertimes. Thisagainshowsthe simulationsarenecessary.