1 Dynamics & Entanglement in Synthetic UQM

Victor Gurarie University of Colorado Boulder Synthetic quantum matter 2 New field in quantum many body physics: arose over the last 20 years

Can be loosely split into:

1. Bottom-up engineering: using elements 2. Hamiltonian engineering: a gas of cold designed to perform as quantum degrees atoms with adjustable interactions, placed of freedom to build up a quantum many in external “trap” or optical lattices, with body systems. Superconducting qubits, random potential. cold ions.

Advantage over more conventional solid state systems:

1. Hamiltonian is known precisely. Can be used as “quantum simulators”.

2. Precise accounting over degrees of freedom. Can be fully isolated from the environment. Can easily be driven out of equilibrium or prepared in an arbitrary state.

3. Detailed probes: measures of momentum distribution, spatial density distribution; harder to measure transport articles

depth of around 13 Er the interference maxima no longer increase in which is of the order of 2 ms for a lattice with a potential depth of 9 strength (see Fig. 2e): instead, an incoherent background of atoms Er. A signi®cant degree of phase coherence is thus already restored gains more and more strength until at a potential depth of 22 Er no on the timescale of a tunnelling time. interference pattern is visible at all. Phase coherence has obviously It is interesting to compare the rapid restoration of coherence been completely lost at this lattice potential depth. A remarkable coming from a Mott insulator state to that of a phase incoherent feature during the evolution from the coherent to the incoherent state, where random phases are present between neighbouring state is that when the interference pattern is still visible no broad- lattice sites and for which the interference pattern also vanishes. ening of the interference peaks can be detected until they completely This is shown in Fig. 3b, where such a phase incoherent state is vanish in the incoherent background. This behaviour can be created during the ramp-up time of the lattice potential (see Fig. 3 articles explained on the basis of the super¯uid±Mott insulator phase legend) and where an otherwise identical experimental sequence is diagram. After the system has crossed the quantum critical point used. Such phase incoherent states can be clearly identi®ed by U=J ˆ z 3 5:8, it will evolve in the inhomogeneous case into adiabatically mapping the population of the energy bands onto alternating regions of incoherent Mott insulator phases and coher- the Brillouin zones19,21. When we turn off the lattice potential Quantum phase transition from a ent super¯uid phases2, where the super¯uid fraction continuously adiabatically, we ®nd that a statistical mixture of states has been decreases for increasing ratios U/J. created, which homogeneously populates the ®rst Brillouin zone of

Restoring coherence super¯uid to a Mott insulator in A notable property of the Mott insulator state is that phase coherence can be restored very rapidly when the optical potential is lowered again to a value where the ground state of the many-body a system is completely super¯uid. This is shown in Fig. 3. After only 22 )

agasofultracoldatoms 4 ms of ramp-down time, the interference pattern is fully visible r E

again, and after 14 ms of ramp-down time the interference peaks ( 0 9 have narrowed to their steady-state value, proving that phase V È τ = 20 ms Markus Greiner*, Olaf Mandel*, Tilman Esslinger², Theodor W. Hansch* & Immanuel Bloch* coherence has been restored over the entire lattice. The timescale for the restoration of coherence is comparable to the tunnelling time 0 * Sektion Physik, Ludwig-Maximilians-UniversitaÈt, Schellingstrasse 4/III, D-80799 Munich, Germany, and Max-Planck-Institut fuÈr Quantenoptik, D-85748 Garching, 80 ms 20 ms t t ˆ =J between two neighbouring lattice sites in the system, 3 Time Germany tunnel ~ articles b 125 ² Quantenelektronik, ETH ZuÈrich, 8093 Zurich, Switzerland First quantum-simulated state (2002)

...... m) for the local occupation ni of atoms on a single lattice site is period of 500 ms to nrad ˆ 24 Hz such that a spherically symmetric µ 100 poissonian, that is, its variance is given by Var n †ˆ nà . Further- Bose±Einstein condensate with a Thomas±Fermi diameter of For a system at a temperature of absolute zero,i all thermali ¯uctuations are frozen out, while quantum ¯uctuations prevail. These articlesz more, this state is well described by a macroscopic wavefunctionh i 26 mm is present in the magnetic trapping potential. microscopic quantum ¯uctuations can induce a macroscopic phase transition in the ground state of a many-body system when the 75 relativewith long-range strength of phase two coherence competing throughout energy terms the lattice.is varied across a criticalIn order value. to Here form we the observe three-dimensional such a quantum lattice phase potential, transition three If interactions dominate the hamiltonian, the ¯uctuations in optical standing waves are aligned orthogonal to each other, with x y in a Bose±Einstein condensate with repulsive interactions, held in a three-dimensional optical lattice potential. As the potential atom number of a Poisson distribution become energetically very their crossingQuantum point positioned at phase the centre of the transition Bose±Einstein from a 2 ᐜk 50 depth of the lattice is increased, a transition is observed from a super¯uid to a Mott insulator phase. In the super¯uid phase, each costly and the ground state of the system will instead consist of condensate. Each standing wave laser ®eld is created by focusing a atom is spread out over the entire lattice, with long-range phase coherence. But in the insulating phase, exact numbers of atoms localized atomic wavefunctions with a ®xed number of atoms per laser beamsuper¯uid to a waist of 125 mm at the to position a of Mott the condensate. insulator A in Width of central peak ( 25 aresite localized that minimize at individual the interaction lattice sites, energy. with The no many-body phase coherence ground acrosssecond the lens lattice; and a mirror this phase are then is characterized used to re¯ect the by laser a gap beam in the back excitationstate is then spectrum. a product We of local can Fock induce states reversible for each changes lattice site. between In this theonto two itself, ground creating states the of standing the system. wave interference pattern. The agasofultracoldatoms 0 limit, the ground state of the many-body system for a commensu- lattice beams are derived from an injection seeded tapered ampli®er 0 2 4 6 8 10 12 14 rate ®lling of n atoms per lattice site in the homogeneous case is and a laser diode operating at a wavelength of l ˆ 852 nm. All t (ms) Markus Greiner*, Olaf Mandel*, Tilman Esslinger², Theodor W. HaÈnsch* & Immanuel Bloch* Agiven physical by: system that crosses the boundary between two phases (herebeams between are spatially kinetic ®ltered and interaction and guided energy) to the is experiment fundamental using to 4 cd e changes its properties in a fundamentalM way. It may, for example, quantumoptical ®bres.* phaseSektion Acousto-optical Physik, transitions Ludwig-Maximilians-Universitaand modulators inherentlyÈt, Schellingstrasse different are used 4/III, tofrom D-80799 control normal Munich, the Germany, and Max-Planck-Institut fuÈr Quantenoptik, D-85748 Garching, ² n Germany melt or freeze. This macroscopicjªMI Jˆ0 ~ change aÃi † isj0 driven by microscopic 3† phase transitions, which are usually driven by the competition Figure 1 Schematic three-dimensional interference pattern with measured absorption i & i intensity² ofQuantenelektronik, the lattice beams ETH ZuÈrich, and 8093 introduce Zurich, Switzerland a frequency difference of ¯uctuations. When the temperatureiˆ of1 the system approaches zero, between inner energy and entropy. images taken along two orthogonal directions. The absorption images were obtained after about 30...... MHz between different standing wave laser ®elds...... The ...... Hamiltonian engineering: ballistic expansion from a lattice with a potential depth of V ˆ 10E and a time of ¯ight of all thermalThis Mott ¯uctuations insulator die state out. cannot This be prohibits described phase by transitionsa macroscopic in polarizationThe physics of of a standing the above-described wave laser ®eld system is chosen is captured to be linear by the and 0 r 1 15 ms. classicalwavefunction systems like at zero in temperature, a Bose condensed as their phase, opportunity and thus to change is not Bose±HubbardorthogonalFor a polarized system model at a, which totemperature all describes other of absolute standing an interacting zero, waves. all thermal boson Due ¯uctuations gasto the in are frozen out, while quantum ¯uctuations prevail. These has vanished. However, their quantum mechanical counterparts can a lattice potential.microscopic The quantum hamiltonian ¯uctuations in cansecond induce quantized a macroscopic form phase reads: transition in the ground state of a many-body system when the amenable to a treatment via the Gross-Pitaevskii equation or different1.relative frequencies Bose-Hubbard strength in of twoeach competing standing model energy wave, terms any is varied residual across interfer- a critical value. Here we3. observe Probing such a quantum(quasi)momentum phase transition distribution Figure 3 Restoring coherence. a, Experimental sequence used to measure the restoration showBogoliubov's fundamentally theory different of weakly behaviour. interacting In bosons. a quantum In this system, state no ence betweenin a Bose±Einstein beams propagating condensate with along repulsive orthogonal interactions, directions held in a three-dimensional is optical lattice potential. As the potential 1 of coherence after bringing the system into the Mott insulator phase at V 0 ˆ 22E r and ¯uctuations are present even at zero temperature, due to Heisen- depth of the lattice² is increased, a transition is observed from a super¯uid to a Mott insulatorab phase. In the super¯uid phase,cd each phase coherence is prevalent in the system, but perfect correlations time-averagedH ˆ 2 toJ zeroaÃi andaÃj ‡ thereforeeinÃi ‡ notU seennÃi bynÃi the2 1† atoms. The1† lowering the potential afterwards to V ˆ 9E ; where the system is super¯uid again. The berg's uncertainty relation. These quantum ¯uctuations may be atom is spread^ out over the^ entire lattice,2 with^ long-range phase coherence. But in the insulating phase, exact numbers of atoms 0 r in the atom number exist between lattice sites. resulting three-dimensionali;j opticali potentiali (see ref. 20 and atoms are ®rst held at the maximum potential depth V0 for 20 ms, and then the lattice strong enough to drive a transition from one phase to another, are localizedh ati individual lattice sites, with no phase coherence across the lattice; this phase is characterized by a gap in the 1 potential is decreased to a potential depth of 9 Er in a time t after which the interference As the strength of the interaction term relative to the tunnelling referencesexcitation therein)² spectrum. for the We atoms can induce is then reversible proportional changes between to the the sum two ground states of the system. bringingterm in about the Bose±Hubbard a macroscopic change. hamiltonian is changed, the system Hereof theaà intensitiesi and aÃi ofcorrespond the three standing to the waves, bosonic which annihilation leads to a simple and pattern of the atoms is measured by suddenly releasing them from the trapping potential. 2. Optical lattice ˆ ² reachesA prominent a quantum example critical of such point a quantumin the ratio phase of U/J transition, for which is the the creationcubic type operators geometry of of atoms the lattice: on the ith lattice site, nÃi aÃi aÃi is b, Width of the central interference peak for different ramp-down times t, based on a change from the super¯uid phase to the Mott insulator phase in a the atomicA physical number system operator that crosses counting the boundary the numberbetween two of phases atoms(here on between kinetic andefgh interaction energy) is fundamental to lorentzian ®t. In case of a Mott insulator state (®lled circles) coherence is rapidly system will undergo a quantum phase transition from the super¯uid 4 changesV x; y its; z properties†ˆV insin a2 fundamental kx†‡sin way.2 ky It†‡ may,sin for2 kz example,††quantum 4† phase transitions and inherently different from normal restored already after 4 ms. The solid line is a ®t using a double exponential decay system consisting of bosonic particles with repulsive interactions the ith latticemelt or site, freeze. and This macroscopic0ei denotes change the is energy driven by offset microscopic of thephaseith transitions, which are usually driven by the competition 0 state to the Mott insulator state. In three dimensions, the phase (t ˆ 0:94 7 † ms, t ˆ 10 5† ms). For a phase incoherent state (open circles) using the hoppingtransition through for an a average lattice potential. number of This one system atom was per ®rst lattice studied site is latticeHere sitek¯uctuations.ˆ due2p=l todenotes When an the external the temperature wavevector harmonic of the systemof the con®nement approaches laser light zero, and ofbetweenV theis inner energy and entropy. 1 2 theoretically in the context of super¯uid-to-insulator transitions in atoms2. Theall thermal strength ¯uctuations of the die tunnelling out. This prohibits term phase in thetransitions hamiltonian in The0 physics of the above-described system is captured by the same experimental sequence, no interference pattern reappears again, even for ramp- = ˆ 3 : 1 expected to occur1 at U J z 5 8 (see2 refs 1, 5, 6, 7), with z being the maximumclassical potential systems at zero depth temperature, of a single as their standing opportunity wave to change laser ®eld.Bose±Hubbard model , which describes an interacting boson gas in down times t of up to 400 ms. We ®nd that phase incoherent states are formed by applying liquid helium . Recently, Jaksch et al. have proposed that such a is characterizedhas vanished. by However, the hopping their quantum matrix mechanical element counterparts between can adja-a lattice potential. The hamiltonian in second quantized form reads: the number of next neighbours of a lattice site. The qualitative This depth V0 is conveniently3 measured2 in2 units of the recoil energy a magnetic ®eld gradient over a time of 10 ms during the ramp-up period, when the transition might be observable when an ultracold gas of atoms with cent sites2 i,j2show J ˆ fundamentally2ed xw x different2 xi† behaviour.2~ = = In2m a‡ quantumV lat x†† system,w x 2 xj†, change in the ground-state con®guration below and above the Er ˆ k =2m. The con®ning potential for an atom on a single Figure 2 Absorption images of multiple matter wave interference patterns. These were system is still super¯uid. This leads to a dephasing of the condensate wavefunction due to ~ ¯uctuations are present even at zero temperature, due to Heisen- ² 1 repulsive interactions is trapped in a periodic potential. To illustrate where w x 2 xi† is a single particle Wannier function localized to H ˆ 2 J obtainedaÃi aÃj ‡ afterein suddenlyÃi ‡ U releasingnÃi nÃi 2 the1 atoms† from1† an optical lattice potential with different quantum critical point is also accompanied by a marked change lattice site due to the4. optical Observing lattice can phase be approximated transition by abetween^ synthetic^ 2states^ of matter the nonlinear interactions in the system. c±e, Absorption images of the interference berg's uncertainty relation. These quantum ¯uctuations may be i;j i i this idea, we consider an atomic gas of bosons at low enough the ith lattice site (as long as ni < O 1†), Vlat(x) indicates the optical h potentiali depths V after a time of ¯ight of 15 ms. Values of V were: a,0E ; b,3 E ; c,7 E ; patterns coming from a Mott insulator phase after ramp-down times t of 0.1 ms (c), 4 ms in the excitation spectrum of the system. In the super¯uid regime, harmonicstrong potential enough with to drive trapping a transition frequencies from one phasenr on to another, the order of 0 0 r r r ² temperatures that a Bose±Einstein condensate is formed. The lattice potential2bringing and aboutm a macroscopicis the mass change. of a single atom. The repulsionHere aà and aà correspondd, 10 Er; e to, 13 theEr; f bosonic, 14 Er; g, annihilation 16 E r; and h, 20andE r. (d), and 14 ms (e). the excitation spectrum is gapless whereas the Mott insulator phase n < k =2pm† V =E . In our set-up potential depths of up to i i r ~ 0 r ˆ ² condensate is a super¯uid, and is described5±8 by a wavefunction between twoA atoms prominent on example a single of lattice such a quantum site is quanti®ed phase transition by theis the on-sitecreation operators of atoms on the ith lattice site, nÃi aÃi aÃi is exhibits a gap in the excitation spectrum 3 . An essential feature of a 22 Er canchange be reached, from the resultingsuper¯uid phase in trapping to the2 Mott frequencies insulator phase4 3 of in approxi- a the atomic number operator counting the number of atoms on that exhibits long-range phase coherence . An intriguing situation interaction matrixp element U ˆ 4p~ a=m†ejw x†j d x, with a NATURE | VOL 415 | 3JANUARY2002| www.nature.com © 2002 Macmillan Magazines Ltd 41 quantum phase transition is that this energy gap ¢ opens up as the mately nsystemr < 30 consisting kHz. The of bosonic gaussian particles intensity with repulsive pro®le interactions of thethe laserith lattice site, and ei denotes the energy offset of the ith appearsquantum when critical the pointcondensate is crossed. is subjected to a lattice potential in beingbeams the athopping scattering the position through length a lattice of of the anpotential. condensateatom. This In system our creates case was ®rst the an studied interaction additionallattice site due to an external harmonic con®nement of the which the bosons can move from one lattice site to the next only by energy is verytheoretically well described in the context by of thesuper¯uid-to-insulator single parameter transitionsU, due in toatoms the 2. The strength of the tunnelling term in the hamiltonian Studies of the Bose±Hubbard hamiltonian have so far included weak isotropicliquid helium harmonic1. Recently, con®nement Jaksch et al.2 have over proposed the lattice, that such with a trap-is characterized by the hopping matrix element between adja- tunnel coupling. If the lattice9,10 potential is turned on smoothly, the short range of the interactions, which is much smaller than the 3 2 2 granular superconductors and one- and two-dimensional ping frequenciestransition might of 65 be Hz observable for a potential when an ultracold depth gas of of 22 atomsEr. with cent sites i,j J ˆ 2ed xw x 2 xi† 2~ = =2m ‡ V lat x††w x 2 xj†, system remains in the super¯uid11±16 phase as long as the atom±atom lattice spacing. † Josephson junction arrays . In the context of ultracold atoms, The magneticallyrepulsive interactions trapped is trapped condensate in a periodic potential. is transferred To illustrate intowhere the w x 2 xi is a single particle Wannier function localized to interactions are small compared to the tunnel coupling. In this In the limitthis idea, where we consider the tunnelling an atomic termgas of dominatesbosons at low the enough hamilto-the ith lattice site (as long as ni < O 1†), Vlat(x) indicates the optical atom number squeezing has very recently been demonstrated with a optical latticetemperatures potential that a by Bose±Einstein slowly increasing condensate the is formed. intensity The oflattice the potential and m is the mass of a single atom. The repulsion 17 regimeBose±Einstein a delocalized condensate wavefunction in a one-dimensional minimizes the dominant optical latticekinetic . nian,lattice the lasercondensate ground-state beams tois a their super¯uid, energy ®nal is value and minimized is over described a period by if thea wavefunction of single-particle 80 ms usingbetween an two atoms on a single lattice site is quanti®ed by the on-site energy,The above and experiments therefore also were minimizes mainly carriedthe total out energy in the of limit the many- of large wavefunctionsexponentialthat ramp exhibits of N withatoms long-range a time are phase spread constant coherence out of over3.t Anˆ intriguing the20 ms. entire The situation lattice slow withrampinteraction matrix element U ˆ 4p~2a=m†ejw x†j4d3x, with a body system. In the opposite limit, when the repulsive atom±atom M latticeappears sites. The when many-body the condensate groundis subjected state to a lattice for a potential homogeneous in being the scattering length of an atom. In our case the interaction boson occupancies ni per lattice site, for which the problem can be speed ensures that the condensate always remains in the many-body interactions are large compared to the tunnel coupling, the total system (e whichˆ const the bosons:) is then can move given from by: one lattice site to the next only by energy is very well described by the single parameter U, due to the well described by a chain of Josephson junctions. groundi statetunnel coupling. of the If combined the lattice potential magnetic is turned and on smoothly, optical the trappingshort range of the interactions, which is much smaller than the 87 energyIn our is minimized present experiment when each we lattice load siteRb is atoms®lled with from the a Bose±same potential.system After remains raising in the the super¯uid lattice phase potential asN long as the the atom±atom condensatelattice has spacing. interactions are small comparedM to the tunnel coupling. In this In the limit where the tunnelling term dominates the hamilto- numberEinstein of condensate atoms. The into reduction a three-dimensional of ¯uctuations optical in lattice the poten- atom ² been distributedregime a delocalized overjª more wavefunction than~ 150,000 minimizesaà j0 lattice the dominant sites kinetic (,65 latticenian, 2† the ground-state energy is minimized if the single-particle number on each site leads to increased ¯uctuations in the phase. SFiUˆ0 ^ i i tial. This system is characterized by a low atom occupancy per sites in aenergy, single and direction) therefore also with minimizes aniˆ average1 the! total atom energy ofnumber the many- of upwavefunctions to of N atoms are spread out over the entire lattice with Thus in the state with a ®xed atom number per site phase coherence lattice site of the order of ni < 1 2 3, and thus provides a unique 2.5 atomsbody per system. lattice In site the opposite in the limit, centre. when the repulsive atom±atom M lattice sites. The many-body ground state for a homogeneous h i e ˆ : istesting lost. In ground addition, for a the gap Bose±Hubbard in the excitation model. spectrum As we appears. increase The the HereIn all order atomsinteractions to test occupy are whether large the compared there identical tois the still extended tunnel phase coupling, coherence Bloch the total state. betweensystem An ( i const ) is then given by: energy is minimized when each lattice site is ®lled with the same N competition between two terms in the underlying hamiltonian important feature of this state is that the probability distribution M lattice potential depth, the hopping matrix element J decreases differentnumber lattice of sites atoms. after The ramping reduction of up ¯uctuations the lattice in the potential, atom we jª ~ aò j0 2† exponentially but the on-site interaction matrix element U suddenlynumber turn offon each the site combined leads to increased trapping ¯uctuations potential. in the phase. The atomic SFiUˆ0 ^ i i iˆ1 ! NATUREincreases.| VOL We 415 | are3 JANUARY thereby 2002 able| www.nature.com to bring the system across© 2002 the Macmillan critical MagazineswavefunctionsThus Ltd in arethe state then with allowed a ®xed atom to expand number per freely site phase and coherence interfere with39 is lost. In addition, a gap in the excitation spectrum appears. The Here all atoms occupy the identical extended Bloch state. An ratio in U/J, such that the transition to the Mott insulator state is each other.competition In the super¯uid between two regime, terms in where the underlying all atoms hamiltonian are delocalizedimportant feature of this state is that the probability distribution induced. over the entire lattice with equal relative phases between different lattice sites,NATURE we| VOL obtain 415 | 3 JANUARY a high-contrast 2002 | www.nature.com three-dimensional© 2002 Macmillan interfer- Magazines Ltd 39 Experimental technique ence pattern as expected for a periodic array of phase coherent The experimental set-up and procedure to create 87Rb Bose± matter wave sources (see Fig. 1). It is important to note that the Einstein condensates are similar to those in our previous experi- sharp interference maxima directly re¯ect the high degree of phase mental work18,19. In brief, spin-polarized samples of laser-cooled coherence in the system for these experimental values. atoms in the (F ˆ 2, mF ˆ 2) state are transferred into a cigar- shaped magnetic trapping potential with trapping frequencies of Entering the Mott insulator phase nradial ˆ 240 Hz and naxial ˆ 24 Hz. Here F denotes the total angular As we increase the lattice potential depth, the resulting interference momentum and mF the magnetic quantum number of the state. pattern changes markedly (see Fig. 2). Initially the strength of Forced radio-frequency evaporation is used to create Bose±Einstein higher-order interference maxima increases as we raise the potential condensates with up to 2 3 105 atoms and no discernible thermal height, due to the tighter localization of the atomic wavefunctions at component. The radial trapping frequencies are then relaxed over a a single lattice site. Quite unexpectedly, however, at a potential

40 © 2002 Macmillan Magazines Ltd NATURE | VOL 415 | 3 JANUARY 2002 | www.nature.com 17 years later 4

Progress isn’t as fast as initially anticipated.

Very slow progress on creating equilibrium quantum matter. Cooling in optical lattice remains a challenge.

Progress comes from just a few key labs.

More progress on quenches (sudden changes of the Hamiltonian) and out of equilibrium evolution.

At the same time, entanglement as a characteristics of quantum many body states received wider recognition.

Synthetic quantum matter provides an access to entanglement inaccessible to other methods. Experimental platforms 5

• Quantum gases. Bose, Fermi, Bose-Fermi. Spin-0, 1/2, 1, …

Spinor condensates BCS-BEC Crossover Polarons, impurity in gases • Hubbard model.

Bose, Fermi, higher spin, SU(N). Disorder: Anderson and MB localization • Long-range interacting transverse field Ising model.

Quantum phase transitions. Many body localization

iHt New observables: tr e entanglement entropy

Anderson localization & Many-body localization 6 MBL: Originally had to do with adding AL: Quantum motion in a random potential interactions to the Anderson-localized system and asking if transport is still 1 inhibited. n + V (r) n = En n 2m In this formulation, turned out to be an All wave functions below 3D and extremely difficult question because of sufficiently low energy wave functions at initial lack of technical tools to access 3D and above are localized interesting observables.

n Basko, Aleiner and Altshuler (2006) were | | the first to meaningfully address this (n) r r0 question Annalsusing of Physics suitably 321 (2006) 1126–1205resumed exp www.elsevier.com/locate/aop ⇠ 0 `n 1 perturbation theory. @ A Metal–insulator transition in a weakly r interacting many-electron system with localized single-particle states Manifests itself in inhibited transport D.M. Basko a,b,*, I.L. Aleiner b, B.L. Altshuler a,b,c

a Department of Physics, Princeton University, Princeton, NJ 08544, USA Was successfully studied in the 70s-90s using b Physics Department, Columbia University, New York, NY 10027, USA conventional methods of quantum many body c NEC-Laboratories America, 4 Independence Way, Princeton, NJ 085540, USA Received 14 August 2005; accepted 30 November 2005 theory. Available online 23 January 2006

Abstract

We consider low-temperature behavior of weakly interacting electrons in disordered conductors in the regime when all single-particle eigenstates are localized by the quenched disorder. We prove that in the absence of coupling of the electrons to any external bath dc electrical conductivity exactly vanishes as long as the temperature T does not exceed some finite value Tc. At the same time, it can be also proven that at high enough T the conductivity is finite. These two statements imply that the system undergoes a finite temperature metal-to-insulator transition, which can be viewed as Ander- son-like localization of many-body wave functions in the Fock space. Metallic and insulating states are not different from each other by any spatial or discrete symmetries. We formulate the effective Hamiltonian description of the system at low energies (of the order of the level spacing in the single-particle localization volume). In the metallic phase quantum Boltzmann equation is valid, allowing to find the kinetic coefficients. In the insulating phase, T < Tc, we use Feynmann diagram technique to determine the probability distribution function for quantum-mechanical transition rates. The probability of an escape rate from a given quantum state to be finite turns out to vanish in every order of the perturbation theory in electron–electron interaction. Thus, electron–electron interaction alone is unable to cause the relaxation and establish the thermal equilibrium. As soon as some weak coupling to a bath is turned on, conductivity becomes finite even in the insulating phase. Moreover, in the vicinity of the transition temperature it is much larger than phonon-induced hopping conductivity of non-interacting electrons. The reason for this enhancement is that the

* Corresponding author. E-mail address: [email protected] (D.M. Basko).

0003-4916/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2005.11.014 MBL & Thermalization 7

Progress occurred after D. Huse and collaborators recognized that the signature of MBL was lack of thermalization.

high energy eigenstate | i ⇢ =tr S = trA [⇢A ln ⇢A] A B | ih |

ETH (eigenstate thermalization hypothesis): ⇢A is thermal, S obeys volume law

A MBL: ⇢A is not thermal, S obeys area law B

MBL: every eigenstate looks like a ground state for some Hamiltonian MBL & Thermalization 8

Occurs in isolated quantum systems. Manifests itself in the inhibited transport and absence of thermalization. Hard to see in solid state systems because no solid state system is isolated.

Cold atoms a natural playground to see many body localization.

Cold atoms also provide unique tools which allow to measure signatures of MBL unavailable in solid state. In particular, the ability to see entanglement directly.

high energy density state | i ⇢ =tr S = trA [⇢A ln ⇢A] A B | ih |

ETH: ⇢A is thermal, S obeys volume law A B MBL: ⇢A is not thermal, S obeys area law PRANJAL BORDIA et al. PHYS. REV. X 7, 041047 (2017)

the appearance of a nonergodic many-body phase in two (a) eoeoo e dimensions by directly tuning the strength of a quasiperi- 9 Observationodic potential. By quantifying the dynamicalof MBL relaxation of in Fermi-Hubbard an imprinted striped density pattern, we find evidence for three dynamical regimes: a regime of fast relaxation at weak disorders consistent with thermalization;PHYSICAL a regime of REVIEW X 7, 041047J (2017) slow relaxation at intermediate disorders, resembling the y relaxation expected in a Griffiths regime [23]; and, finally, x a strong-disorder regimeProbing with Slow negligible Relaxation relaxation, and con- Many-Body Localization in Two-Dimensional sistent with the appearance of a MBL phase. TheQuasiperiodic slow Systems J relaxation regime begins only once the single-particle states U are already strongly localized,Pranjal highlighting Bordia,1,2 Henrik that the Lüschen, slow 1,2 Sebastian2 Scherg,1,2 Sarang Gopalakrishnan,3 4 1,2,5 1,2 2 dynamics is an inherent interaction effect.Michael Compared Knap, toUlrich one Schneider,(b) and Immanuel Bloch 1 dimension [27], the slowFakultät relaxation für Physik, regime Ludwig-Maximilians-Universität is observed to München, Schellingstr.Rare 4, regions 80799 Munich, Germany 2 PROBINGThermal SLOW RELAXATION AND MBL MANY-BODY … PHYS. REV. X 7, 041047 (2017) PRANJAL BORDIA et al. much stronger disordersMax-Planck-Institut inPHYS. two REV. dimensions, X 7, 041047 für revealing Quantenoptik, (2017) an Hans-Kopfermann-Straße 1,slow 85748 relaxation Garching, Germany important role of dimensionality.3Department Furthermore, of Engineering tracking Science the and Physics, CUNY College of Staten Island, Staten Island, the appearance of a nonergodic many-body phaserelaxation in two (a) dynamics appearseoe to beoo useful e in locating theNew York 10314, USA dimensions by directly tuning the strength of a quasiperi- 4 many-body localization transition,Department even in of the Physics, presence Walter of Schottky Institute, and Institute for Advanced Study, odic potential. By quantifying the dynamical relaxation of Technical University of Munich, 85748 Garching, Germany Disorder strength, an imprinted striped density pattern, we find evidenceweak for couplings to the environment [27,35]. 5Cavendish Laboratory, Cambridge University, J.J. Thomson Avenue, Cambridge CB3 0HE, three dynamical regimes: a regime of fast relaxation at FIG. 1. Schematic of the experiment. (a) The system is weak disorders consistent with thermalization; a regime of J United Kingdom initialized in a striped density-wave pattern of fermionic 40K slow relaxation at intermediate disorders, resembling the A. Experiment(Receivedy 25 and April model 2017; revised manuscript received 5 October 2017; published 28 November 2017) relaxation expected in a Griffiths regime [23]; and, finally, x 40 atoms in a random mixture of two spin states (red and blue) in a a strong-disorder regime with negligible relaxation, con-Our system is composedIn a many-body of a degenerate localizedK (MBL) Fermi quantum gas system,square the lattice ergodic with hypothesis tunneling breaks matrix down, element givingJ, rise quasiperiodic to a sistent with the appearance of a MBL phase. Theprepared slow in an equal two-componentfundamentally new spin many-body mixture of phase. its two Whether and under which conditions MBL can occur in higher J potential of strength Δ, and tunable on-site interactions U relaxation regime begins only once the single-particle states U lowest hyperfine2 states.dimensions The spinful remains fermions an outstanding hop challenge on a bothbetween for experiments the different and spins. theory. The Here, largest we realized experimentally 2D system is are already strongly localized, highlighting that the slow square lattice, andexplore the two the species relaxation interact dynamics via2 of on-site an interactingcomposed gas of of fermionic approximately potassium200 × atoms100 sites loaded with in aseveral two- thou- dynamics is an inherent interaction effect. Compared to one (b) dimension [27], the slow relaxation regime is observedinteractions to that are tunabledimensionalRare by regions a optical Feshbach lattice resonance. with different Two quasiperiodicsand atoms. potentials (b) For along weak the disorder two directions. strength, We the observesystem thermal- a Thermal MBL much stronger disorders in two dimensions, revealingquasiperiodic an potentialsdramatic withslow slowingrelaxation different down incommensurabili- of the relaxation for intermediateizes quickly disorder (green strengths. area), whereas Furthermore, at strong beyond disorder a critical it is likely to important role of dimensionality. Furthermore, trackingties the are created alongdisorder the x and strength,y directions we see negligible of the lattice relaxationexhibit on experimentally a many-body accessible localized time regime scales, (blue). indicating Close a to the relaxation dynamics appears to be useful in locating the and form a quasiperiodicpossible two-dimensional transition into a disorder two-dimensional poten- MBLtransition phase. (red Our dot, experimentsΔc), a regime reveal of a slow distinct relaxation interplay is observed of many-body localization transition, even in the presence of Disorder strength, weak couplings to the environment [27,35]. tial; see Fig. 1. Ourinteractions, system is described disorder, and by dimensionality the following and(red provide area), insights potentially into regimes caused where by locally controlled insulating theoretical regions. Hamiltonian:FIG. 1. Schematicapproaches of the experiment. are scarce. (a) The system is A. Experiment and model initialized in a striped density-wave pattern of fermionic 40K atoms in a random mixture of two spin states (red and blue) in a In anFIG. ergodic 2. timeTime evolution, evolution this of an density-wave imprinted density-wave pattern will pattern in the interacting, two-dimensional Aubry-Andr´e model. We measure the Our system is composed of a degenerate 40K Fermi gas DOI: 10.1103/PhysRevX.7.041047 Subject Areas: Atomic and Molecular Physics, ˆ square lattice with† tunneling matrix element J, quasiperiodic prepared in an equal two-component spin mixture of its two H −J cˆj;σcˆi;σ H:c: U nˆ i;↑nˆ i;↓ quicklytime vanish evolution as of theCondensed the dynamics imbalance Matter erase Physics between the microscopic atom numbers on even and odd stripes for intermediate interactions U 5J and varying ¼potential of strengthð Δ,þ and tunableÞþ on-site interactions U ¼ lowest hyperfine states. The spinful fermions hop on a i;j ;σ i detailsdisorder of the strength initialΔ conditions.. (a) At weak In disorder contrast, (Δ a persistent1J), the imbalance vanishes quickly within a couple of tunneling times, signaling ergodic betweenhX thei different spins. The largest realizedX 2D system is square lattice, and the two species interact via on-site composed of approximately 200 × 100 sites with several thou- dynamics. For intermediate disorder ( 4J¼), we observe a markedly slow relaxation. At even stronger disorders ( 10J), relaxation cos 2πβ m cos 2πβ n nˆ : 1 pattern indicates a memory of the initialNeΔ state andN henceo Δ interactions that are tunable by a Feshbach resonance. Two sand atoms.Δ (b) For weak disorderx strength, the systemy thermal-i;σ is absent up to a weak, previously measured¼ [40] coupling to the environment. (b) The same time evolution is shown¼ on a double quasiperiodic potentials with different incommensurabili- izesþ quicklyi;σ (green½ area),ð whereasÞþ at strongð disorder itފ is likely to ð Þ nonergodic behavior. This can= be captured by the normal- exhibit a many-body localizedI. INTRODUCTION regime (blue). Close to the MBL can occur in higher-dimensional systems remains a ties are created along the x and y directions of the lattice X izedlogarithmic atom number plot difference for additional between values the of the even disorderNe and strength. The solid lines denote fits to the model described in the main text [Eq. (2)]. transition (red dot, ), a regime of slow relaxation is observed I N + N and form a quasiperiodic two-dimensional disorder poten- Δc challenging question for both theorye and experiment.o While ˆ†(redTheˆ area), ergodic potentially caused hypothesis by locally underlies insulating regions. quantum statisticaloddInN botho plots,stripes, error defined bars denote as the the error imbalance of the meanI from six individual experimental realizations. All times are in units of the tunneling tial; see Fig. 1. Our system is described by the followingHere, ci;σ ci;σFermi-Hubbardis the creation (annihilation) model operator of the initial theoretical work in Ref. [5] on MBL¼ does not mechanics,ð Þ linking reversible microscopic dynamicsN to time,− N τ= Nh= 2NπJ ,. which serves as our dynamical Hamiltonian: a fermion with spin σ ∈ ↑ ; ↓ on a lattice site ð e dependoÞ ¼ð e stronglyþð oÞÞ on dimensionality, it was recently argued i m;irreversible nIn, an characterizedergodic time macroscopic evolution, byfj this thei behavior. density-wavej Cartesianig In pattern an coordinates ergodicwill system,order parameter. Such an observable has several key Hˆ −J cˆ† cˆ H:c: U nˆ nˆ quickly vanish as the dynamics erase the microscopic that rare, locally thermal regions [15] in systems with true ¼ ð j;σ i;σ þ Þþ i;↑ i;↓ ¼ð localÞ degrees† of freedom get rapidly entangled with oneadvantages. Whereas mass transport is a slow process even i;j ;σ i m; n , anddetailsnˆ i; ofσ thecˆ initialcˆi;σ conditions.is the particle In contrast, number a persistent operator. random disorder can destabilize the MBL phase in two hXi X another, andi; localσ quantum correlations are rapidly erasedin cleanfor varying ergodic systems disorder[38] strengths, the imbalanceΔ; see relaxes Fig. within2. In the initial (2) characterized by a nonvanishing imbalance and clearly ð Þ pattern indicates¼ a memory of the initial state and hence Δ cos 2πβxm cos 2πβyn nˆ i;σ: In the1 first term, the angle brackets ; restrict the sum over dimensions. It is presently unclear if such arguments also þ ½ ð Þþ ð ފ ð Þ [1nonergodic–4]. Nonergodic behavior. This many-body can be captured localized by the normal- (MBL) [5–10]a fewstate, hopping almost times all[10,39] the atoms. Since occupy all experiments even stripes, are such that visible differences between the short and long time closed- i;σ nearest-neighbor sites. The tunnelingh i matrix element is set hold for systems with deterministic disorder such as X systems,ized atom however, number difference defy this between ubiquitous the even behaviorNe and and showlimitedthe to imbalance finite observation at zero times, evolution such a time local measure- is close to unity system imbalances. This indicates that, in this regime, the quasiperiodic potentials. At the same time, initial experi- † to J ≈ h odd× 300NoHzstripes, (h is Planck defined’s as constant), the imbalance and UIdenotes Here, cˆi;σ cˆi;σ is the creation (annihilation) operator of persistent local quantum correlations [11–14]. Furthermore,ment allows us to clearly identify any longer relaxation ð Þ N N = N N , which serves as our dynamical¼ [seements Fig. provided2(a)]. For evidence low disorder for a MBL strength phase (Δ in higher1J), we system relaxes much slower than the microscopic time a fermion with spin σ ∈ ↑ ; ↓ on a latticethe site on-sitee interspecies− o e o interaction strength. The disorder ¼ fj i j ig MBLðorder parameter. systemsÞ ð þ Such areÞ believed an observable to behas robust several key to small, localtimeobserve scales emerging a quick because relaxation, of the disorder. and the Furthermore, imbalance vanishes scales. For > 2J, all single-particle states are localized, i m; n , characterized by the Cartesian coordinatespotential is characterized by the strength Δ and the dimensions by measuring global transport [16,17]. Δ ¼ð Þ † perturbationsadvantages. Whereas and massform transport a distinct, is a slow nonergodic process even phase of matter.the dynamical time evolution of the imbalance could m; n , and nˆ i;σ cˆ cˆi;σ is the particle number operator. withinMoreover, a few tunneling the nature times. of a possibleHowever, MBL upon transition increasing in the but in many regions of the system, interactions with nearby ð Þ ¼ i;σ incommensurableThein clean phase ergodic wavelength transition systems [38] from, ratios the imbalance theβx ergodic≈ relaxes0.721 phase withinand toβy the≈ MBL In the first term, the angle brackets ; restrict the sum over capturehigher eventual dimensions microscopic might Griffiths-type itself be very effects, different even compared h i 0.693 [36]phasea few. In hopping appears the absence times to be[10,39] a of highly interactions,. Since unusual all experiments this critical system are phenomenon; is disorder, relaxation slows down dramatically (Δ 4J) atoms can still result in local thermal equilibrium. However, nearest-neighbor sites. The tunneling matrix element is set limited to finite observation times, such a local measure- in higherto dimensions,the one-dimensional where mass transition transport[18 might24]; for not example, be ¼ a to J ≈ h × 300 Hz (h is Planck’s constant), and U denotesseparableas along ergodicity the two breaks directions down in and the MBL admits phase, an Aubry- its description and essentially comes to a full stop– for strong disorder in some rare regions with anomalously low density or large ment allows us to clearly identify any longer relaxation sensitivesubdiffusive to them [23] phase. as a precursor to localization in one the on-site interspecies interaction strength. The disorderAndr´e-typeliestime metal-insulator beyondscales emerging the scope because transition of of the thermodynamics disorder. at a critical Furthermore, disorder and traditional (Δ 10J). spin imbalances (see below), this thermalization mecha- potential is characterized by the strength Δ and the the dynamicalU 0 time evolution of the imbalance could dimension¼ [18,24–28] might not exist in higher dimensions strengthstatistical of Δc ¼ physics2J [37][8,9]. . To quantitatively analyze this slow relaxation, the nism could be largely ineffective. Such regions can be incommensurable wavelength ratios βx ≈ 0.721 and βy ≈ capture eventual¼ microscopic Griffiths-type effects, even [29] (butB. Identifying see Ref. [30] slow). Given relaxation the apparent conflict of 0.693 [36]. In the absence of interactions, this systemTo is probeBecause the many-body of limitations dynamics of the of available this system, numerical we meth- in higher dimensions, where mass transport might not be timeavailable dependence theoretical of results imbalance[15,29,31,32] is modeledand infeasibility as I oft thermalized only by their greater surroundings, which are separable along the two directions and admits an Aubry-prepareods,sensitive a far-from-equilibrium most to them theoretical[23]. explorations initial state of MBL where concentrate atoms onWe~ choose a fixed intermediate~ interaction strength of ð Þ¼ thermal, but to which they couple in a significantly weaker Andr´e-type metal-insulator transition at a critical disorderare selectively loaded only on the even stripes; see Fig. 1(a). U I5Jtreliableand× f monitort numerical. Here, the simulations, timeI t evolutionis the experiments closed-system of the imbalance stand to playimbalance an U 0 one dimension. Whether, and under which conditions, ð Þ ð Þ ð Þ strength of Δc ¼ 2J [37]. ¼describingimportant the role dynamics in elucidating of a these perfectly regimes isolated[10,16,17,33,34] system,. and fashion. Thus, the overall thermalization time scale grows. To probe the many-body¼ dynamics of this system, we B. Identifying slow relaxation prepare a far-from-equilibrium initial state where atoms We choose a fixed intermediate interaction strength of f t representsUltracold atoms a weak in optical coupling lattices to provide the environment. a particularly Such As the disorder is increased, these surroundings themselves are selectively loaded only on the even stripes; see Fig. 1(a). U 5J and monitor the time evolution of the imbalance couplingsð well-suitedÞ are platform present to in explore all real these systems phenomena, and will as they always gradually become more localized and less effective thermal Published¼ by the American Physical Society under the041047-2 terms of combine almost ideal isolation from the environment with the Creative Commons Attribution 4.0 International license. thermalize any system at long enough times [40,41]. In our baths. For strong disorder Δ 9J, we identify regime (3), Further distribution of this work must maintain attribution to individual experimental control of all microscopic param- ≳ 041047-2 experiment, this weak coupling is dominated by a small but where the values of I~ t at short and long times are both the author(s) and the published article’s title, journal citation, eters. In this work, we employ ultracold fermions in a ð Þ and DOI. nonzeroquasiperiodic hopping optical rate lattice between to experimentally multiple two-dimensional investigate large and, within the experimental uncertainty, identical. 3 planes along the z direction, with a rate Jz ≈ J=10 [40]. This is consistent with the system being many-body We model the resulting imbalance relaxation due to this localized. − Γt β 2160-3308=17=7(4)=041047(8) 041047-1weak coupling Published with a stretched by the American exponential Physicalf t Societye ð Þ , −3 −1 ð Þ¼ with the decay rate Γ Γexp 10 τ and the stretching C. Relaxation exponents and noninteracting ¼ ¼ exponent β 0.6 measured independently in a previous inclusions experiment [40]¼ . Identifying a suitable model to analyze the slow relax- The resulting I~ t is shown in Fig. 3 (a) for short (10τ) ation in regime (2) is challenging, as the underlying ð Þ and long (100τ) evolution times and fixed interaction dynamics in two dimensions at intermediate disorder is strength U 5J as a function of the disorder strength. theoretically unknown. In one-dimensional models with We can identify¼ three dynamical regimes. For weak random potentials, anomalously strongly disordered disorders Δ 2J (1), we observe vanishing values of regions have been argued to give rise to a subdiffusive short and long≲ time imbalances, signaling the presence phase via Griffiths effects [18,20,21,23]. Because our of a rapidly thermalizing system. Upon increasing the system contains quasiperiodic rather than random poten- disorder strength, for 2J Δ 9J, we find a regime tials, it should not contain such anomalously disordered ≲ ≲

041047-3 RESEARCH

◥ tunneling across the boundary between the sub- RESEARCH ARTICLE systems. (ii) Configurational entanglement im- plies that the configuration of the particles in one subsystem is correlated with the configuration of QUANTUM SIMULATION the particles in the other. It therefore requires the presence of at least one particle in each subsys- tem. Tunneling alone does not generate config- Probing entanglement in a urational entanglement, as it acts individually on each particle. Interactions, in contrast, can en- – tangle pairs of particles. As a result, the combi- many-body localized system nation of tunneling and interactions can lead to configurational entanglement at long distances. * Alexander Lukin, Matthew Rispoli, Robert Schittko, M. Eric Tai, Adam M. Kaufman , The formation of particle and configurational Soonwon Choi†, Vedika Khemani, Julian Léonard, Markus Greiner‡ entanglement changes in the presence or absence of interactions and disorder in the system (Fig. 1B). An interacting quantum system that is subject to disorder may cease to thermalize owing In thermal systems without disorder, interact- to localization of its constituents, thereby marking the breakdown of thermodynamics. ing particles delocalize and rapidly create both The key to understanding this phenomenon lies in the system’s entanglement, which types of entanglement throughout the entire is experimentally challenging to measure. We realize such a many-body–localized system in system. In contrast, for Anderson localization, a disordered Bose-Hubbard chain and characterize its entanglement properties through number entanglement builds up only locally at

particle fluctuations and correlations. We observe that the particles become localized, Downloaded from RESEARCH | RESEARCH ARTICLE the boundary between the two subsystems. Here, suppressing transport and preventing the thermalization of subsystems. Notably, we the lack of interactions prevents the formation of measure the development of nonlocal correlations, whose evolution is consistent with a a substantial amount of configurational entan- qualitatively distinguishes our system from a high-resolution imaging system through which insulator. The result is a highly pure state, in logarithmic growth of entanglement entropy, the hallmark of many-body localization. glement. In MBL systems, number entanglement noninteracting, localized state. we project site-resolvedOur optical work potentials experimentally (37). which establishes all correlations many-body are expected localization to stem as a qualitatively distinct We first isolate a single,RESEARCH one-dimensional chain from entanglement in the system. builds up in a similarly local way as for Anderson Experimental system from the Mott insulatorphenomenon and then add the from site- localization in noninteracting, disordered systems. Breakdown of thermalization localization. However, notably, the presence of 10 In our experiments, we study MBL in the in- resolved potential offsetsRESEARCHWi with the incommen- | RESEARCH ARTICLE

interactions additionally enables the slow forma- http://science.sciencemag.org/ teracting Aubry-André model for bosons in one surate lattice. AtEntanglement this point, the system remains We first investigate◥ the breakdown& MBL of thermal- (2019) tunneling across the boundary between the sub- dimension (35, 36), which is described by the in a product state of onesolated atom per quantum lattice site. many-bodyization in a systems subsystem main- that consistshowever, of a single has remained elusive because it requires tion of configurational entanglement through- RESEARCH ARTICLE systems. (ii) Configurational entanglement im- Hamiltonian We abruptly switch on thetain tunneling their initial by re- globallattice purity site. The while conserved under- total atomexquisite number control over the system’s coherence. out the entire system. ducing the lattice depth within a fraction of enforces a one-to-one correspondence between plies that the configuration of the particles in one ^ † going unitary time evolution. However, the We study these many-bodydynamicsbyprobing In this work, we realize an MBL system and J a^i a^i 1 h:c: the tunneling time (Fig. 2C). This quench brings the particle number outcome on a single site subsystem is correlated with the configuration of H¼À ð þ þ Þþ qualitatively distinguishes our system from a high-resolution imaging system through which insulator. The result is a highly pure state, in i QUANTUMpresence of SIMULATION interactions drives local ther- the entanglement properties of an MBL system characterize its key properties: breakdown of U X the system to a nonequilibrium state and ini- and the number in the remainder of the sys- the particles in the other. It therefore requires the n^ i n^ i 1 W hin^ i 1 tializes the unitary time dynamicsmalization: correspond- The couplingtem, entangling between the any two sub- during tunnelingwith a dy- fixed particle number (30–34). We dis- quantum thermalization, finite localization length 2 ð À Þþ ð Þ noninteracting, localized state. we project site-resolvedpresence of at optical least one particle potentials in each subsys- (37). which all correlations are expected to stem i i ing to the above Hamiltonian.I The tunneling namics. Ignoring information about the remaining X X system and its complement mimics the contact tinguish two types of entanglement that cantem. Tunnelingof the particles, alone does area-law not generate scaling config- of the number time t ℏ=J 4:3 1 ms and the interaction system puts the subsystem into a mixed state of We first isolate a single, one-dimensional chain from entanglement in the system. † ¼ ¼ ðwithÞ a bath. This causes the subsystem’sdegrees exist between a subsystem and its complement entanglement, and slow growth of the config- where a^ (a^i) is the creation (annihilation) op- strength U 2 87 3 JProbingremain constant in all entanglementdifferent number states. The associated number in a urational entanglement, as it acts individually i : † ¼ ðExperimentalÞ system erator for a boson on site i, and n^ i a^i a^i is the our experiments. Afterof a freedom variable evolution to be time, ultimately described by1 a ther- (Fig. 1A): (i) Number entanglementfrom the implies Mott that insulatoron eachurational particle. and Interactions, entanglement then add in contrast, that the ultimately can site- en- results ¼ entropy is given by Snð Þ pn log pn ,where particle number operator on that site. The first we abruptly increase themal lattice ensemble, depth and even image if the full system is in¼À a puren theð Þ particle number in one subsystem is cor-tanglein pairs a volume-law of particles. scaling. As a result, The first the threecombi- propertiesBreakdown of thermalization Inmany-body our experiments,– welocalized studyX MBL in system the in- resolved potential offsets Wi with the incommen- term describes the tunneling between neighbor- the system in an atomstate number (1–3–).sensitive A consequence way pn is of the thermalization probability of finding is n atomsrelated in the with the particle number in the other.nationare of tunneling also present and for interactions an Anderson can lead localized to state; ing lattice sites with the rate J=ℏ, where ℏ is the with single-site resolution (38). This projects the subsystem (38). Because the atom number is the

teracting Aubry-André model for bosons in1 one surate lattice. Atconfigurational this point, entanglement the system at long remains distances. We first investigate the breakdown of thermal- that local information about the initial state of This type of entanglement is generated through the slowly growing configurational entanglement on September 7, 2019 reduced Planck constant. The second term repre- many-body state onto theAlexander number basis, Lukin, which Matthewonly degree Rispoli, of freedom Robert of a single Schittko, lattice site, M.Snð EricÞ Tai, Adam M. Kaufman*, sents the energy shift U when multiple particles consists of all possiblethe distributions subsystem of the gets par- scrambledcaptures all and of thetransferred entanglement between the Downloaded from The formation of particle and configurational dimensionSoonwon Choi (†35, Vedika, 36), Khemani, which Julian is described Léonard, Markus by the Greiner‡ in a product state of one atom per lattice site. ization in a subsystem that consists of a single occupy the same site. The last term introduces a ticles within the chain.into nonlocal correlationssubsystem that andare its only complement acces- and is equivalent entanglement changes in the presence or absence site-resolved potential offset, which is created with In some realizations,Hamiltonian particle loss during the to the single-site von Neumann entanglement We abruptly switchof interactions on and the disorder tunneling in the system by (Fig. re- 1B). lattice site. The conserved total atom number h i sibleAn interacting through global quantum observables system1 (4 that–6). is subject toFig. disorder 1. Entangle- may cease to thermalize owing an incommensurate lattice i cos 2pb f of time evolution and imperfect readout reduce entropy SvNð Þ. In thermal systems without disorder, interact- ¼ ð þ Þ Disordered systems (7–18) can provide an ment dynamics in period 1=b ≈ 1:618 lattice sites, phase f, and am- the number of detectedto atoms localization compared of with its constituents,Counting the atomthereby number marking on an individual the breakdown of thermodynamics.ducing the latticeing particles depth delocalize within and a rapidly fraction create both of enforces a one-to-one correspondence between plitude W. In our system, we achieve indepen- the initial state, therebyexception injecting to classical this paradigm en- †lattice of site quantum in different thermal- experimentalnonequilibrium realizations The^ key toJ understandinga^ a^i this1 phenomenonh:c: lies in the system’s entanglement, which types of entanglement throughout the entire dent control over J, W , and f (Fig. 2A). tropy into the system. Weis experimentallyeliminate this entropy challengingiallows us to to measure. obtain the Weprobabilities realize suchpn and a many-bodyhttp://science.sciencemag.org/ thelocalized tunneling system in time (Fig. 2C). This quench brings the particle number outcome on a single site ization.H¼À In such systems,ð particlesþ þ 1 canÞþ localize quantum systems. – system. In contrast, for Anderson localization, Our experiments begin with a Mott-insulating by postselecting the data on the intended atomi compute SvNð Þ . We perform such measurements 87 a disordered Bose-Hubbard chain and characterize its entanglement properties through state in the atomic limit with one Rb atom on number, thereby reachingand atransport fidelity of 99 ceases,X:1 2 % whichfor various prevents evolution thermal- times. At low(A) disorder Subsystems the system to anumber nonequilibrium entanglement builds state up only and locally ini- at and the number in the remainder of the sys- U ð Þ Downloaded from each site of a two-dimensional optical lattice unity filling in the initialization.particle state, Thiswhich fluctuations phenomenon is limited anddepth iscorrelations.W called1:0 many-body1 J , We the entropy observe growsA that and over B the of a particlesan isolated become localized, the boundary between the two subsystems. Here, n^ i n^ i ½ 1 ¼ ðWÞ Š hin^ i 1 tializes the unitary time dynamics correspond- tem, entangling the two during tunneling dy- (Fig. 2B). The system is placed in the focus of a by the fraction of doublon-holelocalizationsuppressing pairs2 in (MBL) the transport Mott (6ð, 7few, and19À tunneling–23 preventing).Þþ Experimental times and the then thermalization reachessystem a stationary out of subsystems. ofð Þ Notably, we the lack of interactions prevents the formation of studiesmeasure have the identifiedi development MBL of through nonlocal the correlations, per-i equilibrium whose evolution entangle ising consistent to the with above a a substantial Hamiltonian. amount of configurational The tunneling entan- namics. Ignoring information about the remaining sistencelogarithmic of theX growth initial density of entanglement distribution entropy, (24X–29) thein hallmark two different of many-body localization. glement. In MBL systems, number entanglement Our work experimentally establishes many-body localization as a qualitativelytime t distinctℏ=J 4:3 1 ms and the interaction system puts the subsystem into a mixed state of and two-point correlation functions during tran- ways: Number entan- builds up in a similarly local way as for Anderson phenomenon† from localization in noninteracting, disordered systems. ¼ ¼ ð Þ wheresient dynamicsa^Bose-Hubbardi (a^ (i25) is). However, the creation while model particle (annihilation)glement stems op- from strength U 2:localization.87 3 J remain However, notably, constant the presence in all of different number states. The associated number

transport is frozen, the presence of interactions a superposition† of ¼ interactionsð Þ additionally enables the slow forma- http://science.sciencemag.org/

erator for a boson on site i, and n^ i a^i a^i is theon September 7, 2019 our experiments. After a variable evolution time, 1 givessolated rise to quantum slow coherent many-body many-body systems dynam- main- however,states has with remained different elusive because it requires tion of configurational entanglement through- entropy is given by Sð Þ p log p ,where ¼ n n n n particleics thattain generate their number initial nonlocal global operator purity correlations, while on under- which that site.exquisiteparticle The control numbers first over thewe system abruptly’s coherence. increaseout the the entire lattice system. depth and image ¼À ð Þ going unitary time evolution. However, the We study these many-bodydynamicsbyprobing In this work, we realize an MBL system and are inaccessible to local observables (30–32). in the subsystems and X termpresence describes of interactions the tunnel drives localing ther-betweenthe entanglement neighbor- propertiesthe of an system MBL system in ancharacterize atom number its key properties:–sensitive breakdown way of pn is the probability of finding n atoms in the These dynamics are considered to be the is generated through ing latticemalization: sites The coupling with between the rate anyJ sub-=ℏ, wherewith a fixedℏ is particle the numberwith (30– single-site34). We dis- resolutionquantum thermalization, (38). This finite localization projects length the subsystem (38). Because the atom number is the hallmarkI of MBL and distinguish it from its particle motion across system and its complement mimics the contact tinguish two types of entanglement that can of the particles, area-law scaling of the number 1 reducednoninteractingwith a bath. Planck This counterpart, causes constant. the subsystem called The’sdegrees Anderson secondexistthe term between boundary; repre- a subsystem config- many-body and its complement stateentanglement, onto theand number slow growth basis, of the which config- only degree of freedom of a single lattice site, Snð Þ sentslocalizationof freedom the (to7 energy– be11, ultimately14, 15 shift, 18). described TheirU when observation, by a ther- multiple(Fig.urational 1A): particles (i) entangle- Number entanglementconsists implies of thatall possibleurational entanglement distributions that ultimately of the results par- captures all of the entanglement between the Downloaded from mal ensemble, even if the full system is in a pure thement particle stems number from a in superposition one subsystem of is states cor- within a different volume-law particle scaling. arrangement The first three in properties the subsystems occupy the same site. The last term introducesand requires both a particle motion and interactions. (B) In the absence of disorder, both types subsystem and its complement and is equivalent Departmentstate (1–3 of). Physics, A consequence Harvard University, of thermalization Cambridge, MA is related with the particle numberticles in within the other. theare chain. also present for an Anderson localized state;

that local information about the initial state of Thisof typeentanglement of entanglement rapidly is spread generated across through the entirethe slowlysystem growing owing configurational to delocalization entanglement of particles on September 7, 2019 site-resolved02138, USA. potential offset, which is created with In some realizations, particle(1) loss during the to the single-site von Neumann entanglement *Presentthe subsystem address: JILA, gets National scrambled Institute of and Standards transferred and (left). The degreeSingle of entanglement site entanglement: and the time scales change S drastically= whenp applyingn ln p disordern 1 anTechnologyinto incommensurate nonlocal and University correlations of Colorado, that latticeand are Department onlyhi acces- of cos 2(center):pbi Particlef of localizationtime spatially evolution restricts and number imperfect entanglement, readout yet interactions reduce allow entropy SvNð Þ. Physics, University of Colorado, Boulder, CO 80309, USA. n sible through global observables (4–6). ¼ Fig.ð configurational 1. Entangle-þ Þ entanglement to form very slowly across the entire system. A disordered system period†Present address: 1=b Department1:618 of Physics, lattice University sites, of California, phase f, and am- the number of detected atoms compared with Counting the atom number on an individual Disordered≈ systems (7–18) can provide an ment dynamics in X Berkeley, CA 94720, USA. without interactions showsprobability only local number of entanglement,seeing n whereas bosons the slow on growth the of leftmost site plitude‡CorrespondingexceptionW to author. this. In paradigm Email: our [email protected] system,of quantum thermal- we achievenonequilibriumconfigurational indepen- entanglementpn the initial is completely state, absent thereby (right). injecting classical en- lattice site in different experimental realizations ization. In such systems,1 particles can localize quantum systems. Fig. 2. Site-resolved measurement of thermalization breakdown. von Neumann entropy SvNð Þ computed from the site-resolved atom number

J W http://science.sciencemag.org/ dentand transport control ceases, over which, prevents, and thermal-f (Fig.(A) 2A). Subsystems tropy into the system. We eliminate this entropy allows us to obtain the probabilities pn and (A) One-dimensional Aubry-André model with particle tunneling rate J=ℏ, statistics (inset) after different evolution times (scaled with tunneling time 1 on-site interaction energy U,andquasi-periodicpotentialwith Lukinization.t etℏ= alJ This)., inScience the phenomenon presence364, of256 weak–260 is andcalled (2019) strong many-body disorder. 19 April (E) 2019 Inset:A and Probability B of an isolated Our¼ experiments begin with1 a Mott-insulating by postselecting the data on the intended atom 1 ofcompute 5 Sð Þ . We perform such measurements amplitude W.(B) We prepare the initial state of eight unentangled atoms by p of retrieving the initial state; main panel: Sð Þ as a function of W vN localization1 (MBL) (6, 7, 19–23). ExperimentalvN 87system out of and measured after 100 . The deviation from the thermal-ensemble projecting tailored optical potentials onto a two-dimensional Mott insulatorstatestudies in have the identified atomict MBL limit through with the per- one equilibriumRb atom entangle on number, thereby reaching a fidelity of 99:1 2 % for various evolution times. At low disorder at 45Er lattice depth, where Er h 1:24 kHz is the recoil energy. prediction for strong disorder signals the breakdown of thermalization ¼  24 29 ð Þ (C) We create a nonequilibrium system by abruptly enabling tunneling eachsistencein the site system. of the of initial All lines a density two-dimensional in (C) and distribution (D) show the ( prediction– ) opticalin of two exact different lattice unity filling in the initial state, which is limited depth W 1:0 1 J , the entropy grows over a dynamics. After a variable evolution time, we project the many-body state anddiagonalization two-point correlation calculations functions without any during free parameters. tran- Eachways: data Number entan- ½ ¼ ð Þ Š back onto the number basis by increasing the lattice depth and obtain the(Fig.sientpoint 2B). dynamics is sampled The (from25 system). 197 However, disorder is realizations while placed particle (38). in Error theglement bars focus denote stems of from a by the fraction of doublon-hole pairs in the Mott few tunneling times and then reaches a stationary site-resolved atom number from a fluorescence image (38). (D) Single-site transportthe SEM. is frozen, the presence of interactions a superposition of gives rise to slow coherent many-body dynam- states with different Lukin et al., Science 364, 256–260 (2019) 19 April 2019 ics that generate nonlocal correlations, which particle2 numbers of 5 are inaccessible to local observables (30–32). in the subsystems and These dynamics are considered to be the is generated through hallmark of MBL and distinguish it from its particle motion across noninteracting counterpart, called Anderson the boundary; config- localization (7–11, 14, 15, 18). Their observation, urational entangle- ment stems from a superposition of states with different particle arrangement in the subsystems and requires both particle motion and interactions. (B) In the absence of disorder, both types

Department of Physics, Harvard University, Cambridge, MA 02138, USA. of entanglement rapidly spread across the entire system owing to delocalization of particles on September 7, 2019 *Present address: JILA, National Institute of Standards and (left). The degree of entanglement and the time scales change drastically when applying disorder Technology and University of Colorado, and Department of (center): Particle localization spatially restricts number entanglement, yet interactions allow Physics, University of Colorado, Boulder, CO 80309, USA. configurational entanglement to form very slowly across the entire system. A disordered system †Present address: Department of Physics, University of California, Berkeley, CA 94720, USA. without interactions shows only local number entanglement, whereas the slow growth of ‡Corresponding author. Email: [email protected] configurational entanglement is completely absent (right).

Lukin et al., Science 364, 256–260 (2019) 19 April 2019 1 of 5

1 Fig. 2. Site-resolved measurement of thermalization breakdown. von Neumann entropy SvNð Þ computed from the site-resolved atom number (A) One-dimensional Aubry-André model with particle tunneling rate J=ℏ, statistics (inset) after different evolution times (scaled with tunneling time on-site interaction energy U,andquasi-periodicpotentialwith t ℏ=J) in the presence of weak and strong disorder. (E) Inset: Probability ¼ 1 amplitude W.(B) We prepare the initial state of eight unentangled atoms by p1 of retrieving the initial state; main panel: SvNð Þ as a function of W projecting tailored optical potentials onto a two-dimensional Mott insulator and measured after 100t. The deviation from the thermal-ensemble

at 45Er lattice depth, where Er h 1:24 kHz is the recoil energy. prediction for strong disorder signals the breakdown of thermalization ¼ Â (C) We create a nonequilibrium system by abruptly enabling tunneling in the system. All lines in (C) and (D) show the prediction of exact dynamics. After a variable evolution time, we project the many-body state diagonalization calculations without any free parameters. Each data back onto the number basis by increasing the lattice depth and obtain the point is sampled from 197 disorder realizations (38). Error bars denote site-resolved atom number from a fluorescence image (38). (D) Single-site the SEM.

Lukin et al., Science 364, 256–260 (2019) 19 April 2019 2 of 5 MBL as ultra quantum states 11

• Every eigenstate looks like a ground state for some Hamiltonian

• Topologically ordered states at finite energy density

• A method to prevent heating in driven systems, such as Floquet quantum matter Role of measurement: Quantum Zeno effect 12 Misra, Sudarshan (1977) Measurements inhibit quantum evolution z x Spin rotating in the Zeeman field B. B H = JBxSx with frequency !

Suppose we take measurements of the spin’s z-component with the rate ! Probability that spin the still points up at the time of the measurement cos2 (!/) 1 !2/2 ⇡ Probability that spin the still points up after time t or N = t measurements cos2N (!/) 1 N!2/2 =1 t!2/ 1 ⇡ ! !1 For frequent measurements the spin stops rotating Evolution of entanglement in quantum circuits 13

time

random unitaries

spins

A product state evolves into a highly entangled state quickly, saturating in a volume law

S V (volume of the smaller of the two subsystems) ⇠ A Quantum circuits with projective measurements 14

Quantum Zeno E↵ect and the Many-body Entanglement Transition (also Skinner, Yaodong Li,1 Xiao Chen,2 and Matthew P. A. Fisher1 Ruhman, Nahum) 1Department of Physics, University of California, Santa Barbara, CA 93106,2 USA 2Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA (Dated: November 2, 2018) increasing the rate of the measurement p, this “weak Wet introduce and explore a one-dimensional “hybrid” quantum circuit model consisting of both measurement phase” with volume-law entanglement, un- unitary gates and projective measurements. While the unitary gates are drawn from a random distribution and act uniformly in the circuit, the measurements are made at random positions and dergoes a continuous dynamical quantum phase transi- times throughout the system. By varying the measurement rate we can tune between the volume tion into an area-law entangled “quantum Zeno phase”. law entangled5 phase for the random unitary circuit model (no measurements) and a “quantum Zeno While we do not have analytic access to this transition, phase” where strong measurements suppress the entanglement growth to saturate in an area-law. Extensive numerical simulations of the quantum trajectories ofU the many-particle wavefunctionsunitary evolution our data can be collapsed into a standard finite-size scal- (exploiting4 Cli↵ord circuitry to access systems up to 512 qubits) provide evidence for a stable “weak ing form: the entanglement entropy, SA(p, LA), with sub- measurement phase” that exhibits volume-law entanglement entropy, with a coecient decreasing system size L and p p the deviation of the mea- with3 increasing measurement rate. We also present evidence forUP a novel continuous quantumunitary dy- plus projective A c namical phase transition between the “weak measurement phase” and the “quantum Zeno phase”, surement rate from criticality, fits a form SA(p, LA)= driven by a competition between the entangling tendencies of unitary evolution and themeasurement disentan- 1/⌫ gling2 tendencies of projective measurements. Detailed steady-state and dynamic critical properties L F (p p )L , with scaling function F . Right at of this novel quantum entanglement transition are accessed. A c A criticality,⇣ p = pc,⌘ the entanglement grows with system 1 I. INTRODUCTION ume law entanglement is not present. size as a sub-linear power law, SA(pc,LA) LA,with 1/3 - intermediate between volume and area⇠ law scal- 0 In this work we explore the many-body dynamics of ing.⇡ Moreover, we investigate the entanglement entropyEntanglement provides a convenient characterization a “hybrid” one-dimensional quantum circuit model con- for both stationary states and unitary dynamics of quan- sisting of both (projective) measurements and unitary dynamics starting from an initial product statetum with systems, no especially those with no symmetry. For gates, depicted in Fig. 1. The dynamics of quantum en- entanglement. We find that when p = pc the entangle-many-body systems, the entanglement entropy is use- tanglement can be accessed by following quantum tra- ment grows as a sub-linear power law in time. ful in classifying both ground and excited states [1–7]. jectories of the many-body wavefunctions. We primarily focus on circuits that consist of measurements that are The rest of this article is organized as follows.Entanglement In Sec- dynamicsFIG. 1. under The unitary structure time evolution of the hybrid in circuit model. In this a drivenNovel system ortype after of a quencha transition also exhibits between univer- areamade atlaw random and positions volume and law times as throughout a function the sys- of the relative tion II, we define the circuit model. In SectionsalIII behavior.,wenumber Frompaper anof we unentangledprojective will focus state, onmeasurements. 1D the circuits entangle- withtem nearest(Large and have neighbor number unitary gates. gates, of measurements chosen from a random = dis- area law = Each site has a spin-1/2 degree of freedom,tribution, and that act each uniformly block in the circuit. While being discuss dynamics of the entanglement entropy andment pro- entropy“Zeno grows effect”). linearly inPerhaps, time, saturating with with new a experimental capability, we will see it observed soon? vide numerical evidence for the entanglement transition.volume-law. Forrepresents integrable systems a gate the operation emission on of en-two qubits.a discrete-time See main generalization text for of the model in Ref. [21], tangled pairs ofdetails. quasiparticles [8, 9] o↵ers a convenient this model is non-integrable. By varying the measure- We conclude with discussions in Section IV. picture, but chaotic systems also exhibit similar entan- ment rate we can tune between the volume law entan- glement growth [10, 11], as illustrated in random unitary gled phase for the random unitary circuit model (no mea- circuit models [12–14]. surements) [13, 14] and a strong measurement “quantum Zeno regime” where entanglement growth is suppressed, II. THE CIRCUIT MODEL Quantum systemschosen subjected to be to conditioned both (continuous) on mea- the outcome of the measure- surements and unitaryment ↵ dynamics, which o is↵er approriate another class for of thesaturating binding in of an symmetric area-law. We analyze the quantum tra- jectories numerically both for random Haar and random quantum dynamicalPosner behavior, molecules described [23], in and terms plays of a prominent role in the We consider a setup with one-dimensional geometry,“quantum trajectories”, and well explored in the context Cli↵ord unitaries, which are minimal models describing quantum brain scenario [24–26]. Inchaotic that non-integrable context, a simi- systems. In the latter case, by re- where the qubits are arranged on a chain of L sites,of few with qubit systems [15], quantum spin systems [16], and trapped ultracoldlar circuit atoms for [17, quantum18] . With stronginformation and stricting processing measurements in the Pos- to the Pauli group, we can access one qubit on each site. The dynamics of the systemcontinuous is measurements the state vector can become the long-time quantum dynamics of very large systems, ner model appeared in [27], whereup unitaries to 512 qubits. conditioned Our numerics supports several striking governed by the quantum circuit with “brick-layer”arXiv:1808.06134v2 [quant-ph] 19 Nov 2018 localized struc- in the Hilbert space, an example of a quantum on the measurement outcomes areconclusions. utilized in preparing ture, see Fig. 1. The circuit is composed of quantumZeno e↵ect [19]. While the related quantum dynamics has been exploredresource in many-body states “open” for universal systems [20 measurement-based], Firstly, we find that quan- volume law entanglement of the gates on pairs of neighboring qubits, whose patternwhich focus of on thetum mixed computation. state density matrix and can random unitary model survives “weak” measurements, but with the coecient of the volume law decreasing arrangement is periodic in the time direction. Eachbe described dis- by LindbladIn Fig. equations,1, we thisdraw formalism the circuit does in such a way that the crete time period of the circuit contains two layers,not o↵ ander access to quantum entanglement. Very recently with increasing measurement strength - a stable “weak occurrences of the unitary-projectivemeasurement gates in phase”. space The and absence of this phase in the each layer has L/2 gates, acting on all the odd linksin Ref. in the [21], the entanglement dynamics of quantum state trajectories havetime been explored are random. for (an integrable) For concreteness, model integrable we focus model on of the Ref. sim- [21] is perhaps due to the e↵ec- first layer, and all the even links in the second. Through-of non-interactingple fermions situation subjected in which to continuous we independently mea- tiveness choose of (local) each measurements gate in removing long-ranged surements of local occupancy, performed at a constant entanglement from their highly susceptible EPR pairs. In out the paper we will assume open boundary conditions to be a UP gate with probability p and a U gate with on the circuit. rate throughout the system. Remarkably, authors of contrast, the entanglement in chaotic systems is encoded Ref. [21] concludeprobability that the late 1 timep.Inthelimit entanglement en-p in0, the no sign projections structure of are the (essentially) random wave- In Fig. 1, we take each block to represent a gatetropy opera- growth saturatesmade to in an the area circuit, law for arbitrarily and it weak reduces!function to a unitary [22], evidently circuit. less disturbed by measurement. tion. There are two di↵erent types of gates, as labelledmeasurement. by AIn “weak the measurement limit p phase”1, a projective with vol- measurementSecondly, we present is made numerical evidence that upon di↵erent colors. Each gray block represents one unitary before all of the! unitaries. gate (denoted U), and acts on the states as When simulating these circuit models, we will primar- U , (1) ily follow the evolution of a pure state wavefunction, as | i! | i in Eq. (1) and (2) - that is, a quantum trajectory. Al- while each blue block represents a “unitary-projective” ternatively, these quantum circuit models can be stud- gate, which performs a projective measurement before ied in terms of quantum channels [28], by keeping track the unitary (which we denote UP), and acts on the state of the mixed state due to projections, rather than pure in the following fashion, states given by di↵erent instances of the measurement outcomes individually. Generally, a quantum channel is P U ↵ | i , (2) a trace-preserving, completely positive map that takes | i! P k ↵ | ik one density matrix to another. It has the operator-sum where P is a complete, mutually exclusive set of pro- representation, { ↵} jectors, for which P↵P = ↵P↵ and ↵ P↵ = 1. The m 1 outcome ↵ happens at a probability given by Born’s rule, ⇢ [⇢]= M ⇢M † , (3) P !E ↵ ↵ p↵ = P↵ . We need not specify the ordering of the ↵=0 gatesh within| | eachi layer since they commute with one an- X other. Notice that the unitary transformation U can be where M are Kraus operators which satisfy the con- { ↵} 15 Synthetic matter

• Synthetic matter = an avenue not only to create new types of quantum matter but also to probe it in new ways.

• Provides access to measure entanglement directly.

• Can create matter in new ways, and also allows us to concentrate on those aspects of quantum matter potentially accessible to synthetic matter experiment.