Tensor networks, and coherent and dissipative dynamics of AMO systems

29 Pages·2016·16.74 MB·English

Tensor networks, and coherent and dissipative

dynamics of AMO systems
Andrew Daley
Department of Physics and SUPA

University of Strathclyde
AMO systems for studying many-body physics
• Atoms • Ions • Molecules
e.g., Bose-Hubbard: D. Jaksch et al. PRL ’98
e.g., variable-range Ising model: 

~ 0.5 µm
D. Porras, J. I. Cirac, PRL ‘04
¯
J
U ~ 0 - 10kHz
x x z
H = � � + B �
k l l
k l ↵
kl | � | l
J ~ 100Hz X X
~ 30 nm
U

ˆ ˆ Experiments – cold gases:

H = J b b + nˆ (nˆ 1)
†
j i i
� i 2 � Munich, Zurich, NIST / JQI, MIT, Harvard, 

i,j i Innsbruck, Hamburg, Pisa, Florence, Oxford, 

� �
� ⇥
Cambridge, Austin, Chicago, Penn State, Kyoto, 

• Microscopic understanding / control Toronto, Stony Brook, Paris, Strathclyde, Illinois, 

Cornell, Stanford, Berkeley, Heidelberg ......
• Extendable to many well-controlled models

• Study thermodynamics / quantum phases Experiments – ions and molecules:

Maryland, Innsbruck, NIST, JILA, ……
• Study out of equilibrium dynamics
Non-equilibrium dynamics with quantum simulators:
M. Greiner et al., Nature 419, 51 (2002).
S. Will et al., Nature 465, 197 (2010).
M. Cheneau et al., Nature 481, 484 (2012)

J.-S. Bernier et al., PRL 106, 200601 (2011)

•
Millisecond timescales - track+control in real time Calabrese, Cardy, Essler, Olshanii, Rigol,……
•
Long coherence times; isolated system
Läuchli, Kollath, Heidrich-Meissner, Fazio, Montangero, 

Schollwöck, White, ……
•
Intrinsic / fundamental questions
• Quench Dynamics / correlation spreading / growth of entanglement
• Thermalisation in quantum systems
• Behaviour near critical points
• Emergence of many-body states in driven/dissipative dynamics
• …
Tensor network methods
104 A.J. Daley
Matrix product state methods can be applied to any system for which we can write the Hilbert
space as a product of local Hilbert spaces, . This works well for spin
1 2 L
H = H ⊗ H ⊗ · · · ⊗ H
chains, where each local Hilbert space corresponds to the states of a single spin, as well as
i
Matrix producHt states
for bosons or fermions on a lattice, where each local Hilbert space corresponds to the different
possible occupations of particles on a single lattice site. If we write the d basis states of the local
6 particles, 6 sites: 462 states
space as i , i 1 . . . d, then any state ψ of the complete system can be expressed in the
l l l l
H | ⟩[ ] = | ⟩ 50 particles, 50 sites: 5 x 1028 states
form
d
ψ c i i i . (56)
i i ...i 1 1 2 2 L L
| ⟩ = 1 2 L| ⟩ ⊗ | ⟩ ⊗ · · · ⊗ | ⟩
[ ] [ ] [ ]
i ,i ,...,i 1
1 2!L
=
The key to matrix product state methods is a decomposition of the state in which we write c
i i ...i
1 2 L
from this equation as the multipliVcaetciotonr:of a serieMs aotfrmixa:trices A i , Matrix-Vector product =
l l
[ ]
c A i A i . . . A i , (57)
i i ...i 1 1 2 2 L L
Full 1la2 ttLic=e q[u]an[tu]m sta[ te] (M sites)
 Matrix product state

4
1
0
where the boundary matrices are assumed to be 1D in order to produce coefficients from the matrix
2
[1] [2] [3] [M]

t [1] [2] [3] [M]
s multiplication.
u
g =
This state is represented in the diagram of Figure 8, where we visualise the basic concept of
u
...
A | i | i ⇡
a matrix product state. If the matrices were just complex coefficients (i.e. matrices of dimension

4 . . .
0
D 1), then this state would precisely be the same as the Gutzwiller ansatz state Equation (53)

6
=
3
described in the previous section. It woduMld-dbiemaepnrosdiounctaslt ateten,swoirth a simple physical interpDre-tadtiimone,nsional matrices
:
1
1 and no entanglement between any of the local Hilbert spaces. However, by using matrices of

t
a
dimension D D, where D > 1, we can produce a superposition of different product states, in so

]
3 × • State coefficients expressed as a product of dMD2 coefficients
. doing, allowing entanglement between different local Hilbert spaces. For sufficiently large D, any
6
6
1 state can be represented in this form, but in general D must grow exponentially with the system
.
1
1 size in order to represent an arbitrary state of the Hilbert space D exp(L). InGfa. cVt,idtaol,r Pephryess.e Rnet v. Lett. 91, 147902 (2003)

2 ∝
. an arbitrary state of the form given in Equation (56), and L is even, then we muGs.t Vhidavael, mPhaytrsi.c Resev. Lett. 93, 040502 (2004)

9
0
A. J. Daley et al., J. Stat. Mech. P04005 (2004)

2 at the centre of the system of dimension D dL/2.
[
= S. R. White and A. E. Feiguin, PRL 93, 076401 (2004)

y
b F. Verstraete et al., Adv. in Phys. 57, 143 (2008).

d
e
d
a
o
l
n
w
o
D
Figure 8. Diagram comparing a product state and a matrix product state. Each square represents a state ψ
l l
| ⟩
[ ]
in a local Hilbert space , e.g. a single spin or lattice site, and the vertical leg on each square box indicates an
l
H
index for the states of the local Hilbert space. The Hilbert space representing the whole system is the tensor
product .A product state can be written as ψ ψ ψ , and
1 2 L 1 1 2 2 L L
H = H ⊗ H ⊗ · · · ⊗ H | ⟩ ⊗ | ⟩ ⊗ · · · ⊗ | ⟩
[ ] [ ] [ ]
represents a quantum state with no entanglement between any two local Hilbert spaces. By multiplying out
D D matrices, a matrix product state makes it possible to represent superpositions of product states, and
×
this superposition contains entanglement between the local Hilbert spaces. For sufficiently large D, any state
can be represented in this way, and the ground states of large classes of local 1D many-body Hamiltonians
can be represented faithfully for small D in this fashion. This is the basis of the t-DMRG method, as discussed
in the main text. The maximum amount of entanglement we can represent across any bipartite splitting in the
system is bounded in terms of the von Neumann entropy by S log D. Note that while each square box
vN 2
≤
here is a similar notation to that used in Ref. [226] and similar references, in the notation of those references,
contractions between matrices are represented by connecting boxes with horizontal lines.
Ground state calculations
•
Density Matrix Renormalisation Group S. R. White, PRL 69, 2863 (1992)
•
Direct tensor optimisation
•
Imaginary time evolution
Time evolution
• Time evolution via operations on states, e.g., via Trotter decomposition 

(Time Evolving Block Decimation algorithm / adaptive time-dependent DMRG)
G. Vidal, Phys. Rev. Lett. 93, 040502 (2004)
A. J. Daley et al., J. Stat. Mech. P04005 (2004)

S. R. White and A. E. Feiguin, PRL 93, 076401 (2004)
[1] [2] [3] [M]

...
| i ⇡
•
Also time-dependent variational principle
J. Haegeman et al., PRL 107, 070601 (2011)
•
Quench dynamics lead to entanglement growth, and limit computations
•
Short-time dynamics after a quench, and long time near-adiabatic dynamics computable
Review: U. Schollwöck, Annals of Physics 326, 96 (2011)
10 10
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Spatial Entanglement in Many-body states thelargeapertureimagingbeyondtheparaxiallimit.Here,wecreate minated,suchpairsundergolightassistedcollisionsandleavethe
~100!m solid state cryabnludseadtneotauvnereladlslsGq:auuasrse(ialan3tteinc)eveplooptCeen.Atiaomlsawujoitrhaldaodpitemiroiondalibcaidtyvaanc5tag6re4i0ysnthmse ttsaruafpflicwsieinthtiptnhyoatotpinmsetioocbfetadheetleoclrtdeyder27.oTfbh1e0r0eufmosri,elltohdnegrebmefaoinreintghneyumembeirt
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up in ultra-hiagparhtfromvdiaffracctiounliumitmsandch(ro(m1atic0abe−rrat9ions.in.the.le1nsf0or−11)mbar) and are
largewavelengthchanges.Thisallowsustousespectrallywide‘white’
1 1 1 1 (b) very1w.5ell shi1els.litgr5dhaytwliegithhdtainstheorafretrcegonhceaer.eWnicitnehleansliggtthhttsoodruerdciueccseenuttnrewudaanrrtoedubnddisa7o5rd8nenrmfcr,owmees from the environ-
generateaconservativelatticepotentialwithalatticedepthofupto35
1 1 ment, thus prEworaevc,evwlenhiegrtdhe,Ewrieictnh5mhg2t/h8emlma2oasissntohfe8g7rRecbo.iclenoerghyoefthreeeffenctivceleatticte imes. (4) Coulomb
0.8 0.8 0.8 0.8 Theprojectionmethodalsoenablesustodynamicallychangethe
~100 m crystals featuwraveelengthloafthtetlatiticceleightwcithooutnchasngtinagthnelatttiscegeoomeftry.a few micrometres
0.8 0.8 ! Thisisimportant,aswestronglyincreasethelatticedepthforsite-
al al 0.6 0.6 0.6 0.6 (see fig1ure 3),rsthiets1eeodsllivdgeuhedetitpmionartgeehicneogni2liDnhedolaartdtitneeicrgetnboaysntutdhpepthirmoeesavsgendirnitfigfcualsiltgioshnthta1on3.fdeFtihnoegrattwhtoiasrmv,ewsatbeoesptwnweietaecprnh- ing potential coun-
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lection of fluorescence light and high-resolution imaging of the
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ing of the ions and for individual preparation, control
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© 2009Macmillan Publishers Limited. All rights reserved
and readout of their electronic and motional states. (5)
The Coulomb interaction between the charged ions is not
shielded within the crystal, as in Coulomb crystals the
charge of all ions has the same sign in contrast to ionic
1 1 1 1
crystals in solid state systems.
l l
a a
n n 0.8 0.8 0.8 0.8 1 1 A B It ha1s to be1pointedAout that it is poBssible to deter-
o o ministically achieve phase transitions between different
ti ti 0.6 0.6 0.6 0.6 structures of Coulomb crystals for large numbers of ions
c c
e e [59, 60]. When the ratio of radial to axial confinement is
~20 m
dir dir 0.4 0.4 0.4 0.4 0.5 0.5 == A ! B reduc0e.5d or t0h.e5am ount =of con finAed ions is i ncrBeased, we
i i | i 6 | i ⌦ | iobserve the tran|sitioin f6rom |a lineair c⌦hain|of ionis via a
b b 0.2 0.2 0.2 0.2
two-dimensional zigzag structure to a three-dimensional
Figure 3. Fluorescence images of laser-cooled ions in a com-
0 0 0 0 ⇢ m=on tcronfin(ing pote ntia)l o0f a linPea0Ur RRFEtrap (see TfigRurAeC2)E, structu0re (see0figurMe 3IXb–Ed)D. TRACE
A B
0 10000 21000000 32000000 43000000 54000000 50000 10000 21000000 fo32r00m00i0n0g4d30i0ff00e0r|0en5t4il0y0h00s0t0ru|c5t0u0re00d Co1u0lo00m0b c21r0y00s0t0a0ls.320(0a00)00A 4s3i0n00g00le0 54000000De5sp0i00te0 th1e0u00n0iqu21e000c00o0nd3i2t00i0o00n0s 4in3000C00o0u5lo40m000b00cry5s0t0a0ls in
+
ion (Mg ). (b) A linear string of 40 ions at ω ω . linear RF traps and the high fidelities of operations, cur-
X/Y Z
≫
The axis of the chain coincides with the trap Z-axis, which rent experimental approaches on QS (and QC) are still
is identically orientated in the rest of the images. (c) A lin-
limited to a small number of ions. The approaches in-
FigurFei7g.urDey7n.aDmyincaalmpiucrailfipcuatriiofincaitniotnheincatshceacdaesdcasedteudpesa(er�tsutrpi=ng(0e�m,bfie=drds0itn,grfiaorwtws)to-,drciomhweinr)sia,onlcashleiztriguazalpgss(et�rtuuctp=ure(0�o.f5�=0,.s5e�co,nsdercoownd) aronwd)baidnidrebcitdioirneacltional
L L Von Neumann entLropyL Rclude oRf the order of ten ioRnsénaryrain egnedtrinoaplyinear chain
60 ions for ω > ω . (d) A three-dimensional structure
setupse(t�up=(�� ,=t�hir,dtrhoirwd).roIwn)t.hIenfitrhste cfiorlsutmcnoluthmendtehtuendinetguXnp/iaYntgteprZnatitsercnhoissecnhsousecnh stuhcaht tthheats[t4te3h,ae6d1]ys.tTeshatiads tiyseascdctoiammtpeelirsdhiseidmesbe,yrcwihsoheoissci,nhgwtihhseircahdiailscon-
L LR R of more than 40 ions at ω ! ω . The enhanced signal-
X/Y Z
finement much stronger than the ax1ial one. The linear
signaslliegdnablyleda vbaynaisvhainngishenintrgoepnytroofptyheofretdhuecreeddudceendsidtyteon-mnsoiiastetyrraitmxioaiontf(rdti)xhiseoacfchoiterhveredebscypoeorxrtneendsdipendogenxpsdopsiuninreg. pSstapruiircn-s.pWairhsi.leWinhitlheeincatshceadcaesdcasedteudpstehtuep then
S (⇢ ) = tr [⇢ (log ⇢ )]chain orientates alSong(t⇢he w)e=akest (Z) dliroegctitorn, ⇢where S (⇢ )
tural phase transitionsvcNan beAinduced betwAeen onAe-, two-2andA n A A V N A
� 1 n { }
systesmyspteumrifipeusrfirfioems flreofmt tolefrtigthot,riignhtth, einchthirealchciarsael tchaesesythsteemsysptuemrifipesuraisfieaswashoalew. hIonlet.heInsetchoentsdienyccooosnlcuidllmactinoonlstuhomfetnhdeetchtouoelneddineiotgnuspnaariontutgnedrptnhaetmteinrinmum
three-dimensional crystals, for example by reducing the ratio
�
of the pseudopotential (X and Y ) and thus micromotion
of radial to axial trapping frequencies.
is choissecnhossuecnh stuhcaht tthhaettthheesttheaedsytesatdaytesbtareteakbsreuapksinutop aintteotraamteetrraamnedr tawnod dtiwmoerdsi.mTerhse. lTashtetlwaostctowluomcnoslusmhnows sahnoawlogaonuaslogous
still remain negligible.
L. Amico, R. Fazio, A. Osterloh and V. Vedral, Rev. Mod. Phys. 80, 517 (2008).
situatsiiotunastfioornsdefoturndientgunpiantgteprnatctoerrrnescporornedsipnogntdointgwtoottewtroamteetrrsamanedrsaannodcatonmoecrt,ormesepr,ecretisvpeelcyt.ivOetlyh.erOptahrearmpeatrearms eatreers⌦a=re0⌦.5=� 0,.5� ,
Different laser cooling schemes can be applied to reduce For purposes of a QC and QS, scaling toRa largerRnum-
J. Eisert, M. Cramer and M. B. Plenio, Rev. Mod. Phys. 82, 277 (2010).
� = �0.6=� 0,.6�� =, �0.4=� 0,.4�� =, �0.2=� 0.a2n�d �an=d �0.1=� 0..1�the.total energy of motion of the ion [45]. Doppler cooling ber of spins and more dimensions while keeping sufficient
a a R bR b R cR c R R d d R R
[55–58] of several ions already allows to enter a regime, control over all required degrees of freedom remains the
where the kinetic energy (k T mK) of the ions be- challenge of the field. Using longer linear chains confined
B
∼
comes significantly smaller than the energy related to the in anharmonic axial potentials [62] might provide a way
mutual Coulomb repulsion. Hence, the ions cannot ex- to reach a number of ions in the system that in principle
strensgttrhenigsthrepislarceepdlabcyed�b�y, �an�d, aanadddaitaidondaitlioLnianldbLlianddbladspinss.pIifnst.heIfntuhme bneurmisbeorddis, oinddt,heinctahsceacdaesdcasdeetudpsestoumpesome
change their position anymore. A phase transition from already exceeds capabilities of a classical supercomputer.
termtewrmithwtihthe tshineglesincgollelecctoilvleectjiuvme pjuompperaotpoerrtahcetogarseoocufs (liqouifod)fptlahsmeoafltatohsatecrsylpsataisnltlisnsepastrirnuecstduarreirovece-dnriiAnvnetoontheairnwmtaoyixmaeigdmht,ibnxe eothdne,-udsneaoorfnkR-Fdstarianrgtketr.satpas toffee.ring
R R
curs [59, 60]. On the one hand, the resulting Coulomb periodic boundary conditions for static Coulomb crystals
strensgttrhen2g�tLh:2�L: crystals (see figureC3o) pnrosvCeidqoeunmesanneytqsluiymeitlnahrtiitlisyesswtthiathitsseosliidtsanteo[5t9is,a60n]soatntedaaevdesynt(emsatodarety-deimsotefnasttioehnaeol)fccrhtyhistreaalllcinhe bireaamls of
state crystals already partially explaining why Coulomb ions [63–65]. A microfabricated ring trap is currently de-
pendpanent.daFntu.rthFeurrtnhoetre ntohtaet tthhaist cthonisstcrouncsttiorunctrieoqnuirreeqsuires
crystals appear naturally suited to simulate many-body veloped and fabricated at Sandia National Laboratories
i i �� �� �� ��
physics: (1) The io�ns�resi=d�e o�n0in=dainvidd0ualtalhantutdiscetscihtaeusn.s(2n)coatn[6b6n]e.otexbteenedxetdenndaeidvenlyaivtoelythteo the
⇢˙ = ⇢˙ =H +H +H , ⇢H+, ⇢ + [c ]⇢[+c 2]�⇢+2[�c ]⇢[.c (3]⇢0.) (30)
sys sys R R R R L LR TheRmotion of the ions (6 externa6 l degree of freedom) can The two main limitations for further scaling of the
� ~ � ~ � � � D� D D D bidirebcidtiiornecatliosentatlinsgetotifnSgeco.f 7S.ec*.*7*.A*g*a*inAtghaiisnctohmismcoenmtmisent is
R R R R
be described easiest in terms of common motional modes number of ions in a common potential are, from a practi-
h h i i
for odfodr, omdady,bmeawyebeshwoewsihnocwludinecliut dbeefiotrbe.e*f*o*re.***
As sAhoswsnhoiwnnthine Sthece. SIeVc.AIVthAe dthime edriismederissteadtestiastethies the
EvenEtvheonugthhotuhgehctahsceacdaesdcaadneddtahnedchthirealchmiraasltemraesqtuera-equa-
uniquuenipquureepsutereadsytesatdaytestoafttehoef ctahsecacdaesdcapdaerdt.paBryt. coBny- con-
tion thiaovne htahvee stahmeesasmteeadsytesatdaytess,tatthees, dtyhneamdyincsamofictshoef the
strucsttirouncttihoins dthairskdsatraktestisataeninsihainlantiehdilabtyedthbeystinhgelesincoglle- col-
two stywsotesmyssteinmhs oiwn hthoiws sttheiasdsytesatdaytesstaisteaspipsroaapcphreodacihsed is
lectivleecjtuivmepjuompepraotpoerracto,rscuch, stuhcaht tithaist aitlsios aaldsoarakdsatarktestate
R R ratherratdhieffrerdeinffte.reWnth. ilWe ihnilethien ctahsecacdaesdcasdeetdupsetthuep stphine spin
of thoef atdhdeitaidodnaitliotnearml teinrmthien cthhieraclhsieratul ps.etuTph.usT, haunsy, any
chainchpauirnifipeusrfirfioems flreofmt tloefrtigthotrdiguhettdoutehteoutnhiediurencitdiiornecatlional
uniquuenipquureepsutreeadsyteastdaytestoaftethoef ctahsecacdaesdcamdeadstmeraestqeura-equa-
flow flofowexociftaexticointast,iionnst,hiencthhireacl hciarsael tchaesesythsteesmyspteumrifipeusrifies
tion,tiisona,lsios tahlseoutnhiequuenisqtueaedsytesatdaytesotafttehoefctohrerecsoprornedspinognding
“as a“washaolew”h[oclfe.” F[cifg.. 7F(iag-.h7)(].a-h)].
chiraclhmiraasltmeraesqteuraetiqouna,twiointh, wthitehidthenetiidfiecnattiifiocna�tion � ��. ��.
R R
! !
In paIrntipcaurlatirc,utlhaer,dtihmeedriismedersisteadtest(a2t6e) (is26a)lsios tahlseosttheaedsyteady
statestoafttehoefcthhireaclhmiraasltmeraesqteuraetiqouna.ti*o*n*.M*a*y*bMeatyhbise itshnisotis not
C. CM.ultMipualrttiiptearetnitteanegnlteadngslteedadsytesatdaytesstates
so msuochmaucshuraprsiusrepirfisweeifdwefiendedefi↵nedw↵ithw�it�hb�ef�orbee?f*o*r*e.?***.
j j
NoteNtohtaet tthhaet sttheeadsyteastdaytesstaotfesthoef ctahsecacdaesdcadmeadstmeraster
equaetiqounatairoenoanrley osntelyadsyteaaldsoy ianlstohiencthhireaclhsiertatlinsegtitfintgheiyf they The Tabheoveabdoivsecudssisiocunsssihoonwsshothwast tthhaet stphien scphianinchiasin is
are daarrek.daTrkh.isTrheqisuirreeqsufiorersefxoarmepxlaemapleeveanenvuenmbneurmobfer ofdrivednriivnetno iantpourae pduimreerdiismedersisteedadsytesatdaytesitfatite iisf ditriivsednriven
Entanglement and the matrix product state ansatz
[1] [2] [3] [M]
Matrices D= matrix/bond 


D D
| i ⇡ ... dimension
⇥
max(S ) = 0 = log (1) max(S ) = log (D)
vN 2 vN 2
MPS
H
H
D = 1 “low energy/

slightly entangled states”
max(S ) = M/2
product state

 vN
(for spins)
• Matrix product states can represent a state exactly if entanglement is small

(necessary condition: von Neumann; sufficient condition: Rényi, order < 1)

F. Verstraete and J.I. Cirac Phys. Rev. B 73, 094423 (2006)
N. Schuch, M. M. Wolf, K. G. H. Vollbrecht, and J. I. Cirac, New J. Phys. 10, 033032 (2008)
N. Schuch, M. M. Wolf, F. Verstraete, and J. I. Cirac, Phys. Rev. Lett. 100, 030504 (2008)
•
1D systems with local Hamiltonians have low ground state entanglement
(S bounded in M for gapped systems; grows as log(M) for critical systems)
Entanglement and quench dynamics (local 1D system)
•
Quench experiment for bosons
U /J = 10 U /J = 1
t<0 t>0
J
U
⇢
A
time

A B
• Propagation of quasi-particle pairs, velocity limited
 t
⇤
by Lieb-Robinson bound v
g
�
v
g
E. H. Lieb and D. W. Robinson, Comm. Math. Phys. 28, 251 (1972).
P. Calabrese and J. Cardy, J. Stat. Mech, P04010 (2005)

•
Linear von Neumann entropy increase
S
V N
l = 9
max(S ) = log (D)
vN 2 8
.
.
For exact simulation: D exp [ct]
.
6
/
N
l = 5
In practice:
v
S 4
l = 4
D Runtime:
2
l
D = 2
512 1 day
0 l
D = 2
1024 8 days 0 1 2 3 4 5
1
time [J ]
2048 2 months �
Control over accuracy
• Convergence in the maximum allowed truncation error is used to control 

2
validity of the calculation
a b
0.6
2
0.4 2
S. Trotzky et al., 
 2
e.g., experiment vs. t-DMRG

Nature PhysicUs /J8 ,= 3 22.544 ((220)12) U/J = 3.60(4)
0.2
in a quantum quench 0.6 0.6 a K/J = 5·1a0-3 b K/J = 7b·10-3
0.4 0
d 0.4
d U/J = 2.44(2) U/J = 3.60(4)
0.2
a o K/J = 5·10U-3 /J = 2.44(2)cK/J = 7·10-3 U/J = 3.60(4) d
n 00..26
0 K/J = 5·10-3 K/J = 7·10-3
d a b
d 0.6
no 0d.6 0.04 c d
d
o
0.4 0.6 U/J = 5.16c(7) U/J = 9d.9(1)
n 0.2
t
U/J = 5.16(7) K/J = 9·10-U3/J = 9.9(1) K/J = 15·10-3
. . . 0.2 0.4
0.4 K/J = 9·10-3 K/J = 15·10-3
0
0
0 1 02 13 42U5/J =03 5.1164(72) 53 40 51 2U/J =3 9.9(14) 5
0.2
K/J = 9·10-3 K/J = 15·10-3
4Jt / h 4Jt / h
U/J = 2.44(2) U/J = 3.60(4)
0 0.2
(i) PreparatiFoIGn. 1. Relaxation of the densi(tyiip)at teErnv. (oa)lCuontcieoptnof the exper- (iii) Readout
FIG. 2. Relaxa0tion of1the lo2cal den3sity fo4r diffe5rent 0interac1tion 2 3 4 5
FIG. 1. Relaxation oimfetnht:eafdteernhsaviitnyg ppraetptaererdnt.he(ade)nsCityonwcaveept (otf=th0e) e(xi),ptehre- streFngItGhs.. W2.e ploRt ethleamxaeatsiuorend torafcetshoef thleocodadl-sidteepnospiutlyatiofnor different interaction K/J = 5·10-3 K/J = 7·10-3
| i
lattice depth was rapidly reduced to enable tunneling (ii). Finally, 4Jt / h
b iment: after having prepared the density wave (t = 0) (i), the n (t) for four different interaction strengths U/J (circles). The
odsdtrengths. We plot the measured traces of the odd-site population
the properties of the evolved state were |read out after alil tunneling
Entanglement growth leads to breakdown of simulsaolitdiloinnes avreieans etmebnle-savoerrag nederetsuwltsofrrokmst-DMRG simulations
lattice depth was rapidly reduced to enable tunneling (ii). Finally,
FIG. 1. Relaxatiownasosfutphpreesdseednasgiatiyn (piiai)t.te(br)nE.v(ean-)odCdorenscolevpedt doeftetchtieone: xpapretir-- withFnoIouGtdfd.re(e2t.p)arfaRomreetlefarosx.uaTrthieodndifafsoheefrdeltnihnteesinrleotpecrerasalecntdtiseoimnnusliasttytiroenfsnoignr-thdsiffUer/eJnt(icnirtecrlaecst)i.onThe0
theipmreonpte:rtaifetserohfatvhienclgeesvpoonrlesvipteeasdrwesidtthatothdeedwidndeeenrxeswiteryreeabwdroauovguehttto aaf(thteigrh=earlBl0lto)uchn(bnia)en,dli.tnhAge cludsionglindexlti-nneeasresatrneeieghnbsoenrmhobpplein-gawviethraagceoudplrinegsumlattsrixfreolem- t-DMRG sdimulations
strengths. We plot thoedmdeasured traces of the odd-site population
subsequent band-mapping sequence was u|sed to reveal thie odd- and
waslasttuipceprdeespsethd wagasairnap(iiidil)y. r(ebd)uEcevdent-ooedndabrelesotluvnendeldinegtec(itii)o.n:Fipnaarltliy-, ment JNNN/J 0.12 (a), 0.08 (b), 0.05 (c) and 0.03 (d) calculated d
even-site populations [13, 14]. (c) Integrated band-mapping profiles nwitho(tu)t ff'orerefopuarradmifefeterersn.t TinhteeradcatsiohnedstlriennegsthrespUre/sJen(tcsiricmleusl)a.tioTnhsein-
fromothdedsingle-particle band structure.
clesthoenpsriotepserwtieitshoofdtvdhereisnuesdvreeolxalxvwaetideonrsettimabtereot wfuogrehhre/t(4troJe)aadho0i.ug9thmeasrf,tUBe/rlJoa=cllh5t.ub1n6a(nn7ed)la.innAdg scoluliddinlignenseaxrte-neenaseremsbt lnee-aigvherbaogredhorepspuilntsgfwroimthta-DcMouRpGlinsgimomulaattriioxnsele-
subwseaqsuseunptpbreasnsde-dmaagKpa/piJnin'(gi9isi⇥)e.q10u(�be3)n. cE(dev) Oewndad--ossidtuedsdeerndes'sittoyolevrxeetrdvacetdeadeltfetrohcmetitoohned:rdaw-padaarnttaid- wmiethnotuJt free/pJaramn0et.e1r2s.(aT)h,e0.d0a8sh(ebd),li0n.e0s5r(ecp)reasnednt0s.0im3un(dla)ticoanlscuinl0a-te.d6 c d
shown in c. The shaded area marks the envelope for free Bosons NNN even
evecnl-essitoenpsoipteuslawtiitohnosd[d13in,d1e4x].w(ecr)eIbnrtoeuggrahttetdobaahnidg-hmeraBpploincgh bparonfid.leAs procfilluesdainsga fnunecxtito-nneoaf'rreelsatxantieoinghtimboerfohrohp/(p4iJn)g w0i.t9hmas,coupling matrix ele-
(light grey) and including inhomogeneities of the Hubbard parame- from the single-particle band structure'.
subsequent band-mapping sequence was used to reveal the odd- and U/J = 5.16(7) and K/J 9 10 3. We plot the resulting
versus relaxation timteerstinfthoerehxp/er(im4eJnt)al syst0em.9(dmarksg,reUy)/. J = 5.16(7) and ment J /J 0.12 (a),�0.08 (b), 0.05 (c) and 0.03 (d) calculated
c even-site populations [13, 14]. (c) I'ntdegrated band-mapping profiles trace s n (Nt)NiNn Fig.'1d. 'We ⇥generally observe oscillations
3 fromodtdhe single-particle band structure.
K/J 9 10 . (d) Odd-site density extracted from the raw data
� in n with a period T h/(4J) which rapidly dampen out 0.4
ver'sus r⇥elaxation time t for h/(4J) 0.91m.0s, U/J = 5.16(7) and odd
'
) shoKwn/Jin c.9 Th1e0�sh3a.d(edd) Oadreda-smiteardkesns'tihtye eexntvraeclotepdefrfoomr ftrheeerBawosdoantsa witphirno3fi-4lepesriaodssatofausntecadtyiovnaluoefofr'el0a.x5.aTtihoensamtiemquealf-or h/(4J) 0.9 ms,
k
4(light gre'y) an⇥d inclFuidnainllygwienahdodmedotogtehneelointigelsattoicfetahneothHeruobpbtiacarldlapttaicreawmiteh- itative behavior is found in a wide range of interactions (see
h shown in c. The shaded area marks the envelope for free Bosons 3 '
wavelength � = 765nm = � /2 (“short lattice”) with the FigpU. 2r/o).Jfile=s a5s.1a6f(u7n)ctaionndoKf /reJlaxat9ion t1im0e fo. rWhe/(p4lJo)t the0r.9esmulst,ing
n ( 2ters(liinghthtegreexyp)earnimdeinnrectlalaultidvseiynpsghteaismnexhbs(oedmtwaorekegnegtnhreeediyttwi)e.osaxodljfusthteed tHoulobabdaervderpyasreacomnde- WUtrea/pcJeerfs=ornm5ed.1t6-(D(tM7))RiGanncdFalicKgul.a/ti1Jodns., 'kWe9epein⇥gg1eu0pneto3r�a.50lWl0y0eopblsoetrtvhee'oresscuillltaintig0on.s2 U/J = 5.16(7) U/J = 9.9(1)
o 0 ters in the experimsietentoafltshyesstheomrt (ladtatircke [g1rd4e,y2)10.]..5Completely removing the states in the moadtrdix-product state simu'lations⇥(solid�lines in
o tirnacnes n wit(ht)aipneFriiogd. 1Td. Whe/(g4eJne)rawllhyicohbsraeprvideloysdcialmlatpieonnsout K/J = 9·10-3 K/J = 15·10-3
i long lattice gave an arrany of practically isolated 1D density Fig. 2). oTdhde oBdodse-Hubbard parameters used in these sim-
t-2 '
si waves N = ,1,0,1,0,1, – thus realizing step (i) ulaiwtinoinntshowidnedre3w-o4ibtthapineaerdipoefrdroismotdothTearsestpeehacdt/ivy(e4vJseat)luowfeheoxicpfehri-ra0p.i5d.lyTdhaemspameneoquutal-
| i |··· ···i '
o-4 – with a distribution of particle numbers N and thus lengths mewntailthcoinntr3o-l4papraemreiotedrss. tFourathsetrmeaodrey, wvealtuooekoinfto a'c0-.5. The same qual-
Finally we added to the long lattice another optical lattice with itative behavior is found in a wide range of interactions (see0
P L = 2N 1 given by the external confinement. For our pa- count the geometry of the experimental setup by perf'orm-
Finally we added to the�long lattice ano0th.e0r optical lattice with itative behavior is found in a wide range of interactions (see
wavelength �xs =ra7m6et5ersn, mwe e=xpe�ctxchl/ai2ns (w“itshhaomratxilmatatlipcaert”ic)lewnuimthberthofe ingFtihge.c2or)re.sponding ensemble average E N over chains 0 1 2 3 4 5 0 1 2 3 4 5
0wavel1ength �x2sN= 76354n3 amnd=a m4�eaxnlv/a2lue(“ofsNh¯or0t3l1a(tsteiec1eSu”p)pwle2mitehntatrhy3e witFhi8dgif.fe21re)0n.t p1ar2ticl1e 4num1b6ers1N8(s2ee0Supp{le}mentary Ma-
relative phase betwemeaxn the two adjusted to load every second We performed t-DMRG calculations, keeping up to 5000
' '
relative phta s(embseMt)awteeriealnfotrhdeettawilsoonatdhejulosatdeidngtporolcoedaudree).very second tt e(rimal).WsF)oer tpheertfimoersmaeccdessti-bDleMin RtheGsimcaullactiuolnas,titohenses,avk-eeping up to 5000
site of the short lattice [14, 21]. Completely removing the states in the matrix-product state simulations (solid lines in
site of the short lTaotitniictieali[z1e t4h,e m2a1n]y.-boCdyormelapxlaetitoenldyynraemmicsoovfisntegp (tiih),e erasgteastdeifsferinontlyhesligmhtalytrfiroxm-ptrhoedtruaccets osbttaatineedsifomr ualsaint-ions (solid lines in
4Jt / h
long lattice gave awne qaurernachyedotfhepsrhaorct-tliacttaiclelydepitshotloaatesdma1llDvaludeewnisthiitny gleFcihgai.n 2w)it.h thTe hmeaxiBmaolspea-rtHicluebnbumabredr Nparam= e4t3eorfs used in these sim-
long lattice gave an array of practically isolated 1D density Fig. 2). The Bose-Hubbard parmaamx eters used in these sim-
200µs, allowing the atoms to tunnel along the x-direction. the ensemble (see Supplementary Material). For interaction
wavweasve s = = , 1,,10,,01,,10,,01,,1, ––ththuussrereaalilziziinnggsstteepp ((ii)) uullaattiioonnss wweerree oobbtatainineeddfrformomthteherersepsepceticvteivesetseotf oefxpeexrpi-eri-
| N| iNi | · |·A··f·te·r a time t, we rap·i·d·l··yi·riamped up the short lattice to its strengths U/J . 6 (Fig. 2a-c), we find a good agreement
– w–itwhitahdaisdtrisibtruibtiuootrniigoinnoaflodpfeapprthat,ircttihlceulsensuunpmuprmebsbesirnesgrsaNlNl tuanannndedlintgthh.uusFsinlleaelnlnyg,gtwthhess of tmmheeeennxtptaaelrlimcceonontnatltrdroaoltalpapnadararthamemseitemeturelsar.tsio.nFs.uFIrnuthrthteihsremrergmoimroee,,rew, ewteootokoikntiontaoc-ac-
FIG. 1. Relaxation of the density pattern. (a) Concept of the exper-
L =L =2N2N 1 g1ivgeirvneeadbnyobutyththteehpeerxoeptxeertrteinersanloaflctohceonenfivfinolneveemdmestenantte.t.inFFtoeorrmrosououfrrdpepnaa--- onlccyoosumunanltltstythhsteeemgagetiecoomdmeveietarttiyroynsoocfafnthbtehe eoebsxeepxrveperdei,mrwimheincehtnaaltraeslestueptupbFybIpyGerpfeo.rrfmo2-rm.- Relaxation of the local density for different interaction
�
� sities, currents and coherences in step (iii). Note that in the strongest for the smallest value of U/J which corresponds
ramraemteerste,rws,eweexepxepcetccthcahinaisnws withithaammaxaximimaallppaartritciclleennuummbbeerrooff iinngg tthhee ccoorrrreessppoonnddininggenesnesmemblbeleavaevreargaegeE EN ovoervecrhacihnasins
iment: after havingexpperimreentspwae arlweayds ¯meatshureed thedfuell ennsesmiblteyaverawge avto ethe sm alles(t ltattice=depth. 0Th)ey can(bie)at,tributthed eto the { }N
NmNaxmax '434a3nadnadXma(etm)ae=naEvnaNvlaul euNeo(tfo)NfX¯ˆN N'(t3)31o1(fs(aesneeoebSsSeuruvpappbplleleeXmˆmoeevnnerttaathrreyy brewwakidittohhwnddoiifffftfheeerrteiegnnhttt-bpipnaadritrnitgciclaepleprnonuxiummmabtiebornesfrosNr sNha(lslo(ewseelaeSt-uSpupplep{mles}emtnrteanertyanrMygaMt- ha-s. We plot the measured traces of the odd-site population
Mat'erial for detaarirlasyoonf ct{hhae}inhlso(addeni|ontge|d'pbryoctiheedauvreera)g.ing operator E ), ticetsewriha|icl)h.giFveosrristehteo atismigneifiscaancticaemsosuinbtloef lionngtehr-eransgiemd ulations, these av-
Material for details on the loading procedure). N terial). For the times accessible in the simulations, these av-
lattice depth was rapidly reduced to enab{ l} e tunneling (ii). Finally,
To initialize thraethmerathnayn-tbhoe dexypercetlaatixonatviaolunedfoyrnaasminigclescohfaisntewpith(iNi), hopepriangg.esWdheifnfeinrcloudninlyg aslniegxht-tnleyarefsrtonmeigthhbeorthroapcpeinsgobtained for a sin-
n (t) for four different interaction strengths U/J (circles). The
To initialize the many-body relaxation dynamics of step (ii), erages differ only slightly from the traces obtained for a sin-
particles. term J (aˆ aˆ + h.c.) in the t-DMRG simulations odd
we quenched the short-lattice depth to a small value within gl�e cNhNNainjwi†jthj+t2he maximal particle number Nmax = 43 of
the propewretiqeuesncohefd tthhe eshRoeleratx-valtaiootntiolcfvequdeaseid-plotchasltdotenasaittiesesm. aWwlle fiverastlrudieescuwssrimteheaia-ndweogolbuetaitcnhquaaainnftittawetiviterhagtraheeelmlemntatwxuitihmnthaenlexpepaerlritmiicennlteaglndautamber N = 43 of
200 µs, allowing the atoms to tunnel along the x-direction. the ensePmble (see Supplementary Material). Forsmionatxelriadctiolnines are ensemble-averaged results from t-DMRG simulations
surements ofthe density onsites witheither even orodd index. (dashed line in Fig. 2). For larger values of U/J and corre-
200 µs, 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 6 (Fig. 2a-c), we find a good agreement
After the time evolution, we transferred the population on odd spondingly deeper lattice.s, the tight-binding approximation is
was suppressed again (iii). (b) Even-odd resolved detection: parti-
Aftoerrigaintaiml deetp,thw, estihterusastpoiasduhlipyghperrraeBsmlsoicpnhegbdanadululpsitntughntehneesslhuipnoegrrlt.attliacFettiainncadeldleyttoe,ctwietdse valoisdtf.retFhnoergUteh/xJsp&eUri1/m0J(eFnig.t.a2ld6d),al(taaFrgieagrn.dde2vtiaaht-ieocn)ss,iamwreuefloaufitnidno.dns.a Ignowtohdisiartgehgreiomemeu,etntfree parameters. The dashed lines represent simulations in-
these excitations employing a band-mapping technique (see Here, the dynamics become more and more affected by resid-
origreinadaloduetpththe, ptrhoupserstiuepsporefstshiengevaolllvetudnsntaetleinign. teFrminsalolyf,dewne- oonfltyhesmeaxlplesryimsteemntaatlicdadteaviaantidonthsecasnimbuelaotbiosenrsv.edIn, wthhiischreagrieme,
cles on sites with odFigd. 1b)i[n13,d14e]. xFig. w1c sheowrsethe ibntergroatedubagndh-mtapptinog aualhintierg-chhaineturnnelBinglanod nconh-adiabbataicnheadtin.g asAthe ab-
sities, currents and coherences in step (iii). Note that in the strongest for the smallest value of U/J which corresponds
read out the properties of the evolved state in terms of den- only small systematic deviations can be observcedl,uwdhiicnh gare next-nearest neighbor hopping with a coupling matrix ele-
experiments we always measured the full ensemble average to the smallest lattice depth. They can be attributed to the
sities, currents and coherences in step (iii). Note that in the strongest for the smallest value of U/J which corresponds
subsequent band-mappˆing sequence wˆas used to reveal the odd- and
expXer(itm) e=ntsEwNe a lNw(aty)sXm eaNsu(tr)edotfhaenfuobllseernvsaebmlebXle aovveerratghee btoretahkedoswmnalolfesthtelatitgtihcte-bdinedpitnhg. apTphreoyximcaantiobnefaotrtmrsihbauelltoenwdtltaoJt-the /J 0.12 (a), 0.08 (b), 0.05 (c) and 0.03 (d) calculated
h | | i
{ } NNN
X(atr)ra=y of cha ins((td)eXnˆot ed (bty) thoef aanveorbagseinrgvaobpleerXaˆtoorvEerNth)e, tbirceeaskwdhoiwchngoifvethseritsieghtot-abisnigdninifigcaapnpt raomxoimunattioofnlofnogresrh-arallnogwedlat- '
even-site proatpheur EtlhaaNntithoeNnexspe[ct1ati3Non, v1alu4e]f.or (acsi)ngIlenctheaingwrait{hte}Nd bhaopnpidng-.mWahepnpinicnludgingparonefixt-lneeasrest neighbor hopping
h | | i from the single-particle band structure.
{ }
array of chains (denoted by the averaging operator ), tices which gives rise to a significant amount of longer-ranged
E
particles. N term J (aˆ aˆ + h.c.) in the t-DMRG simulations
{ } NNN j †j j+2
versus relraathxeratthiaon ntheteixmpecetattionfvoalruehfo/r (a 4sinJgl)e chain 0wi.t9h Nms,hoUpp�i/ngJ. W=hen5in.c1lu6din(g7a)neaxtn-ndearest neighbor hopping
Relaxation of quasi-local densities. We first discuss mea- we obtain quantitative agreement with the experimental data
P
parstuicrleems.ents of the density on sites with either even'or odd index. (tedramshedJline in Fig(aˆ. †2aˆ). Fo+r lahr.gce.)r vinaltuheesto-Df UM/RJGansidmcuolrartei-ons
3 NNN j j j+2
K/J 9 10 . (d) Odd-site density extracted�from the raw data
RAefltaexrathtieo�tnimoef qevuoalsuit-iloonc,awl deetrnasnitsifeesr.redWtheefiprosptudlaisticounssonmoedad- swpeonodbitnagilnyqdueaenpteirtalatitvtieceasg,rteheemtiegnhtt-wbiinthdinthgeaepxpproexriimmeatnitoanl idsata
' ⇥ P
surseimteesntotsaohfitghheedr eBnlosicthyboannsdituessinwgitthheesituhpeerrelavtetinceoarnoddddeitnedcetexd. v(daalisdh.eFdolrinUe/iJn F&ig1.02)(.FiFg.or2dla),rglaerrgveraldueevsiaotfioUns/Jareanfoducnodr.re-
shown in c. The shaded area marks the envelope for free Bosons
these excitations employing a band-mapping technique (see Here, the dynamics become more and more affected by resid-
After the time evolution, we transferred the population on odd spondingly deeper lattices, the tight-binding approximation is
profiles as a function of relaxation time for h/(4J ) 0.9 ms,
Fig. 1b) [13, 14]. Fig. 1c shows the integrated band-mapping ual inter-chain tunneling and non-adiabatic heating as the ab-
sites to a higher Bloch band using the superlattice and detected valid. For U/J 10 (Fig. 2d), larger deviations are found.
&
(light grey) and including inhomogeneities of the Hubbard parame-
'
these excitations employing a band-mapping technique (see Here, the dynamics become more and more affected by resid-
3
U/J = 5.16(7) and K/J 9 10 . We plot the resulting
Fig. 1b) [13, 14]. Fig. 1c shows the integrated band-mapping ual inter-chain tunneling and non-adiabatic heating as the ab- �
ters in the experimental system (dark grey).
' ⇥
traces n (t) in Fig. 1d. We generally observe oscillations
odd
in n with a period T h/(4J ) which rapidly dampen out
odd
'
within 3-4 periods to a steady value of 0.5. The same qual-
'
Finally we added to the long lattice another optical lattice with itative behavior is found in a wide range of interactions (see
wavelength � = 765 nm = � /2 (“short lattice”) with the Fig. 2).
xs xl
relative phase between the two adjusted to load every second 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 = , 1, 0, 1, 0, 1, – thus realizing step (i) ulations were obtained from the respective set of experi-
N
| i | · · · · · · i
– with a distribution of particle numbers N and thus lengths mental control parameters. Furthermore, we took into ac-
L = 2N 1 given by the external confinement. For our pa- count the geometry of the experimental setup by perform-
�
rameters, we expect chains with a maximal particle number of ing the corresponding ensemble average over chains
E
N
¯ { }
N 43 and a mean value of N 31 (see Supplementary with different particle numbers N (see Supplementary Ma-
max
' '
Material for details on the loading procedure). terial). For the times accessible in the simulations, these av-
To initialize the many-body relaxation dynamics of step (ii), erages differ only slightly from the traces 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
200 µs, 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 6 (Fig. 2a-c), we find a good agreement
.
original depth, thus suppressing all tunneling. Finally, we of the experimental data and the simulations. In this regime,
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
ˆ ˆ
X (t) = (t) X (t) of an observable X over the breakdown of the tight-binding approximation for shallow lat-
E
N N N
h | | i
{ }
tices which gives rise to a significant amount of longer-ranged
array of chains (denoted by the averaging operator ),
E
N
{ }
rather than the expectation value for a single chain with N hopping. When including a next-nearest neighbor hopping
particles. term J (aˆ aˆ + h.c.) in the t-DMRG simulations
†
NNN j+2
j j
�
Relaxation of quasi-local densities. We first discuss mea- we obtain quantitative agreement with the experimental data
P
surements of the density on sites with either even or odd index. (dashed line in Fig. 2). For larger values of U/J and corre-
After the time evolution, we transferred the population on odd spondingly deeper lattices, the tight-binding approximation is
sites to a higher Bloch band using the superlattice and detected valid. For U/J 10 (Fig. 2d), larger deviations are found.
&
these excitations employing a band-mapping technique (see Here, the dynamics become more and more affected by resid-
Fig. 1b) [13, 14]. Fig. 1c shows the integrated band-mapping ual inter-chain tunneling and non-adiabatic heating as the ab-