Quantum Simulations using Ultracold Quantum Matter Outline Introduction

Quantum Gas Microscopy of Fermi Hubbard Model Fundamentals Bilayer Readout Fractionalization and Dynamical Spin-Charge Separation Imaging Magnetic Polarons in 2d From Polaron to Fermi-Liquid

Outlook I.B. Quantum Science Seminar 3 September 2020

Quantum Spin Systems

Particle Systems: Bosons, Fermions, Mixtures

Classically Intractable Computational Regimes

Quantum Gas Microscopy

Few particles up to 1000s of particles ! courtesey: T. Hänsch Single Atoms Measuring a Many-Body Quantum System Single Atoms Experimental Setup

lattice beams single 2D degenerate gas 1064 nm Local occupation measurement d ~ 1000 87Rb atoms (bosons) c

b 4 µm

mirror 1084 nm window 780 nm high-resolution f 2 µm objective e NA = 0.68 z x y Enables access to all position correlation between particles! y

Extendable to other observables (e.g. local currents etc…) x y Subwavelengthx spatial16 µmresolution: 50resolution nm of the imaging system:

J. Sherson, Nature (2010) ~700 nm see also: W. Bakr, Science (2010)

Addressing Arbitrary Light Patterns Potential Shaping Potential Shaping

Example: Hubbard ladders in homogeneous potential

! 2

Measured Light Pattern 1

0 Atoms DMD Pattern Light Field Atoms (Single Shot) (Averaged) (but: fight Laser Speckle) Digital Mirror Device (DMD) Exotic Lattices Quantum Wires Box Potentials Almost Arbitrary Light Patterns Possible!

Single Spin Impurity Dynamics, Domain Walls, Quantum Wires, Novel Exotic Lattice Geometries, ...

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RESEARCH REVIEW Quantum Ladders with Fermi-Hubbard Model B. Keimer et al., Nature 518 2015

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The two-dimensional copper oxide layers (Fig. 3) are ? AFM Heisenberg Model separated by ionic, electronically inert, buffer layers. The stoichiometric (K) (K) Pseudogap J Half filling & strong interaction T 200 ‘parent’ compound (Fig. 2, zero doping) has an odd-integer number of electrons per CuO2 unit cell (Fig. 3). The states formed in the CuO2 unit T SC, onset cells are sufficiently well localized that, as would be the case in a collec- T C, onset tion of well-separated atoms, it takes a large energy (the Hubbard U)to remove an electron from one site and add it to another. This effect pro- Temperature, Charge Tc 14 100 order duces a ‘traffic jam’ of electrons . An insulator produced by this classical Spin jamming effect is referred to as a ‘‘Mott insulator’’15. However, even a order AF TCDW localized electron has a spin whose orientation remains a dynamical degree 2 T S, onset 4t Fermi of freedom. Virtual hopping of these electrons produces, via the Pauli = T d-SC H = J Si Sj J SDW liquid exclusion principle, an antiferromagnetic interaction between neighbour- Â 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 U , · ing spins. This, in turn, leads to a simple (Ne´el) ordered phase below room 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 i j 0 0.1 0.2 h i p p p p temperature, in which there are static magnetic moments on the Cu sites 50x50 atoms min c1 c2 max with a direction that reverses from one Cu to the next16,17. Hole doping, p The Cu-O planes are ‘doped’ by changing the chemical makeup of Fully tuneable coupling strengths Away from half filling: t-J model Figure 2 | Phase diagram. Temperature versus hole doping level for the interleaved ‘charge-reservoir’ layers so that electrons are removed (hole- copper oxides, indicating where various phases occur. The subscript ‘onset’ doped) or added (electron-doped) to the copper oxide planes (see the competition between marks the temperature at which the precursor order or fluctuations become horizontal axis of Fig. 2). In the interest of brevity, we will confine our apparent. TS, onset (dotted green line), TC, onset and TSC, onset (dotted red line for discussion to hole-doped systems. Hole doping rapidly suppresses the hole delocalizationboth) refer to the onset temperaturesmagnetic of spin-, charge order and superconducting antiferromagnetic order. At a critical doping of p , superconductivity fluctuations, while T* indicates the temperature where the crossover to the min sets in, with a transition temperature that grows to a maximum at p , pseudogap regime occurs. The blue and green regions indicate fully developed opt antiferromagnetic order (AF) and d-wave superconducting order (d-SC) then declines for higher dopings and vanishes for pmax (Fig. 2). Materials A. Maruzenko et al. Nature (2017), M. Boll et al. Science (2016), T. Hilker et al. Science (2017), with p , p are referred to as underdoped and those with p , p are setting in at the Ne´el and superconducting transition temperatures TN and Tc, opt opt L. Cheuk et al. Science (2016), P. Brown et al. Science (2017) respectively. The red striped area indicates the presence of fully developed referred to as overdoped. charge order setting in at TCDW. TSDW represents the same for incommensurate It is important to recognize that the strong electron repulsions that spin density wave order. Quantum critical points for superconductivity and cause the undoped system to be an insulator (with an energy gap of 2 eV) charge order are indicated by the arrows. are still the dominant microscopic interactions, even in optimally doped copper oxide superconductors. This has several general consequences. The be the most important open problem in the understanding of quantum resulting electron fluid is ‘highly correlated’, in the sense that for an elec- materials, and it is here that radically new ideas, including those derived tron to move through the crystal, other electrons must shift to get out of from recently developed non-perturbative studies in string theory, may its way. In contrast, in the Fermi liquid description of simple metals, the be useful. quasiparticles (which can be thought of as ‘dressed’ electrons) propagate More unique to the copper oxides is the behaviour observed in a range freely through an effective medium defined by the rest of the electrons. of temperatures immediately above Tc in what is referred to as the The failure of the quasiparticle paradigm is most acute in the ‘strange metal’ ‘pseudogap’ regime. It is characterized by a substantial suppression of the regime, that is, the ‘normal’ state out of which the pseudogap and the electronic density of states at low energies that cannot be simply related to superconducting phases emerge when the temperature is lowered. None- the occurrence of any form of broken symmetry. Although much about theless, in some cases, despite the strong correlations, an emergent Fermi this regime is still unclear, convincing experimental evidence has recently liquid arises at low temperatures. This is especially clear in the overdoped emerged that there are strong and ubiquitous tendencies towards several regime (Fig. 2). But recently it has been shown that even in underdoped sorts of order or incipient order, including various forms of charge- materials, at temperatures low enough to quench superconductivity by Li-Microscope Fermi-Hubbard - 2016-2017 Li-Microscope AFM Orderdensity-wave, spin-density-wave,in the andFermi electron-nematic Hubbard order. There is the applicationModel of a high magnetic field, emergent Fermi liquid behaviour also suggestive, but far from definitive, evidence of several sorts of novel order—that is, never before documented patterns of broken symmetry— including orbital loop current order and a spatially modulated super- O conducting phase referred to as a ‘pair-density wave’. There are many Cu fascinating aspects of these ‘intertwined orders’ that remain to be under- stood, but their existence and many aspects of their general structure were Ba/Ca anticipated by theory7.Superconductingfluctuations also have an important role in part of this regime, although toanextentthatisstillmuchdebated.

The high-temperature superconducting phase itself has a pattern of O 2py

broken symmetry that is distinct from that of conventional superconduc- Cu 3dx2 – y2 tors. Unlike in conventional s-wave superconductors, the superconduct- ing wavefunction in the copper oxides has d-wave symmetry8,9, that is, it O 2p L. Cheuk et al. Science (2016) changes sign upon rotation by 90u. Associated with this ‘unconventional x D. Greif et al. Science (2016) pairing’ is the existence of zero energy (gapless) quasiparticle excitations A. Omran et al. PRL (2015) at the lowest temperatures, which make even the thermodynamic prop- erties entirely distinct from those of conventional superconductors (which Figure 3 | Crystal structure. Layered copper oxides are composed of CuO2 are fully gapped). The reasons for this, and its relation to a proximate anti- planes, typically separated by insulating spacer layers. The electronic structure ferromagnetic phase, are now well understood, and indeed were also anti- of these planes primarily involves hybridization of a 3dx2 { y2 hole on the 10–12 cipated early on by some theories . However, while various attempts copper sites with planar-coordinated 2px and 2py oxygen orbitals.

180 | NATURE | VOL 518 | 12 FEBRUARY 2015 ©2015 Macmillan Publishers Limited. All rights reserved

M. Boll et al. Science (2016)

M. Parsons et al. Science (2016) A. Mazurenko et al., Nature 545, 462 (2017) AFM Correlations (Short, Medium & Long Ranged) now visible! A. Maruzenko et al. Nature (2017), M. Boll et al. Science (2016), T. Hilker et al. Science (2017), L. Cheuk et al. Science (2016), P. Brown et al. Science (2017) The Electron

Charge -e Spin 1/2

Fractionalization of Quasi-particles Deconfinement that make up the elementary particle

Charge -e Spin 1/2 Quasi-Particle Quasi-Particle J. Vijayan et al. Science 367, 186 (2020)

AFM Incommensurate Magnetism The Electron Postselection to Mz = 0 !

z z z z 0.2 C(d)=4Sˆ Sˆ 4 Sˆ Sˆ h i i+di h i ih i+di Charge -e Spin 1/2 0.1

0.0 Fractionalization -0.1 of Quasi-particles -0.2 10-1 Deconfinement C ( d ) d

1 ) that make up the elementary particle

-0.3 Spin correlation C(d) correlation Spin ( 10-2 -0.4 1 2 3 4 5 6 7 8 9 d (sites)

2 4 6 8 10 12 d Distance Charge -e Spin 1/2 x(0, T) Largest possible decays length ! Quasi-Particle Quasi-Particle Fractionalization Previous Work Spin-Charge Separation

X Theory

Experiment

Spinon Spin 1/2 Chargon +e Vs (Holon) Vc Spectroscopic determination: DMRG Simulation: C. Kim, et al. Phys. Rev. Lett. 77, 4054 (1996) C. Kollath, U. Schollwöck, W. Zwerger O.M. Auslaender et al. Science 308, 88 (2005) Phys. Rev. Lett. 95, 176401 (2005)

FHM Dynamics Dynamical Spin Charge Separation FHM Dynamics Spin & Charge Velocities

Hole Dynamics 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

Spin Dynamics 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 (squeezed space) Fractionalization - Hole Shedding Spinon SC Separation Spin-Hole-Spin Correlations

Spin attached to hole

Spin structurehole position! independent of 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 X

Sˆz hˆ Sˆz < 0 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 i 1 i i+1 h i z z Hole got rid of spin Sˆ hˆ Sˆ 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 i 1 i i+1 h i

SC Separation Seeing the Spinon

Qˆ = Sˆz f ( x x ) i  j L | i j|

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 j ˆ

AAAJNHiclVZbb+Q0FM4ut9ly2S77uDxYjJCm0mw1KSuBeFqxVIIHRLeie1EzjRzbM2ONY0e20+ng+oVfwys88F+QeEO88hs4TjLtJClFGynKyXe+c/Hx8UmyQnBjJ5M/7tx96+133n1vcG/n/Q8+/Oj+7oOPXxhVasJOiBJKv8qwYYJLdmK5FexVoRnOM8FeZstnQf/ynGnDlfzRrgs2zfFc8hkn2AKU7n6SCCzngqFLlCywRc9THkRdgenucLI/qS7UF+JGGEbNdZQ+GEQJVaTMmbREYGNO40lhpw5ry4lgficpDSswWeI5OwVR4pyZqauW4dFngFA0UxpuaVGFbls4nJsc28UYgXBhNa4Es86zMQq4VUqYMeJG1SxJSC1cGPDAxmiFzRrorTRclkNklClBKyftFO3sy6njsigtk6TOcFYKZBUKxUSUa0asWIOAieawSEQWWGNioeStKMbmWK819TstuMACNkKqdkohjwL/1EGpCWUJDiibJZnGw9glgcqlZNo1G+mGsb/0viYtmW2TLoO62d0NCTz1eFvOttiSrYjKcyypS3Bm/Gk8dQmTptQsmIIVm9kE2s2iYQxNxOeL+vUqn3NGXB0nm6HlBqYtmG5g3YL1Bj5swYfdxEBr68zCnobuUCKso0NTRU2Cpq+1bTWXlnqXWHZhHe3a5lRZUKL9Ll6KgH8FsGbbimPmQ0SmsVU6NL07rgraZn2Xd1mA9BLj3vFOXHZRqJWs1sPObljrrID4T5KCJ6wwXMC5n3QYR4brkDo8R1UFnfZju9dnyQ0tlbcQS6m9K7coPYKusi3TkO1/sigNwWZwnlxShBmChb+SkO3VhlJ9C193ywLuF8ziNzMpFvxNDIhUEopPUnnmRvGeH/VqxfgSsoZ942i7Y5e+9QqOe3Yru9lyMFU5m+OqnFAXdHV1bPIto8ddq35eQG7ch5ONRrcm+Lj2hmxz8ve6Ho8llOo4dfAY6W4ZXgvo/tepg8coqfZlnJxjDeXuMquCXsabudRW2pUC5cHNSgrt7x1NXXzgzzZnoTvXYCAsfT0XsE83cwaW25uAQKHX1LOE4vmc6dtMJNDl//m88nhTvF64DumwdtAEqD7pvQlproMkhs/zXiSzFalhXIeDM/cNg4+7Zt9DkB+acQWVhy6GG/4Z4u4fQl94cbAff74fP38yfPp18/cwiB5Fn0ajKI6+iJ5G30ZH0UlEop+jX6Jfo98Gvw/+HPw1+Lum3r3T2DyMWtfgn38BFflq3g== Qi Magnitude ofh Magnetization| |i in length L Hidden Order in the Ground State related Qˆ 2 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 i i

Magnetization Fluctuations in Region L AFM Charge & Spin Order around Hole

Minimize Energy Two Conditions

Holes want to delocalise Spins want to align antiferromagnetically

Ground State Excited State

AFM Microscopic Origin of SC —Separation

Hole introduce domain wall “parity” kinks in AFM background!

Hole = Non local topological excitations! AFM AFM around Holes AFM Hidden (Topological) Order in the 1D Hubbard

0.2 5 2 / on s

d 0.1 i t + a l rr e o

0 0 c

n i p s - n i

-0.1 p S P o s i t n h l e -5 -0.2

-5 0 5 Distance spin-spin d Single Hole in AFM

Charge Impurities in 2D

J. Koepsell et al. Nature 572, 358 (2019)

Polaron Competing Energy Costs: Kinetic vs Magnetic Polaron Competing Energy Costs: Kinetic vs Magnetic

Hopping events Hopping events t J t J 0 1 2 0 1 2 energy cost energy cost magnetic kinetic magnetic kinetic magnetic kinetic magnetic kinetic magnetic kinetic magnetic kinetic Polaron Competing Energy Costs: Kinetic vs Polaron Ground State?

Hopping events t J unaffected AFM background 0 1 2 = + unaffected AFM | polaroni | i | background i

Antiferromagnetic + spin disturbance dresses ... hole/doublon locally

reduced AFM background = + reduced AFM | noi polaron | i | background i energy cost +... magnetic kinetic magnetic kinetic magnetic kinetic Ferromagnetic Doping reduces AFM spin order: [Mazurenko et al., Nature 545 2017]

Polaron Polarons in the FHM Polaron C(1) Local Spin Correlations Around

Semiclassical considerations 1 -0.12 t 4 2 R Auerbach, Springer 1994 ' J ✓ ◆ Nagaoka ferromagnetism Large t/J (large U): dressing becomes 1 ferromagnetic -0.14 F. GRUSDT et al. Nagaoka, Phys. Rev. 147 1966 PHYS. REV. X 8, 011046 (2018) 0

(a) (b) Quasiparticle with bandwidth W=2J : Chernyshev and Wood,advanced arXiv:cond-mat/0208541, theoretical 2002 techniques. These include effective model Hamiltonians [17,19],fullyself-consistentGreen’s y - hopping distance (sites) -0.16 function methods [25], nontrivial variational wave func- -1 Attraction between polarons tions [20,23,26], or sophisticated numerical methods such superconductivity, stripes? as Monte Carlo [26,28] and density matrix renormalization group [27,29] calculations. The difficulties in understand- String picture: spinon-holon binding -2 ing the single-hole problem add to the challenges faced by -0.18 0 3 string length theorists trying to unravel the mechanisms of high-T -3 -2 -1 1 2 c x - hopping distance (sites) (c) Grusdt et al., PRX 8 2018 2.5 superconductivity.1 3 1 Heret we study the problem of a single hole moving in an 3 L antiferromagnet from a different perspective, focusing on 2 244 ' J Lee et al., Rev. Mod. Phys. 78 2006 the✓ t-J◆z model for simplicity. In contrast to most earlier Conditioned C(1) Correlator Schrieffer et al., PRB 39 1988 works, we consider the strong-coupling regime, t ≫ Jz. 1.5 8 Starting from first principles, we derive an effective parton theory of magnetic polarons (mp). This approach not only 1 3 provides new conceptual insights to the physics of mag- netic polarons, but it also enables semianalytical deriva- tions of their properties. We benchmark our calculations by 0.5 comparison to the most advanced numerical simulations known in literature. Notably, our approach is not limited to 0 1 low energies but provides an approximate description of the 0 0.2 0.4 0.6 0.8 1 entire energy spectrum. This allows us, for example, to calculate magnetic polaron dynamics far from equilibrium. Note that in the extreme limit when J 0, Nagaoka has FIG. 1. We consider the dynamics of a hole propagating in a z ¼ N´eel-ordered spin environment. (a) The hole creates a distortion shown that the ground state of this model has long-range of the spin state (green). We introduce two types of partons, ferromagnetic order [14]. We work in a regime where spinons and holons, describing the spin and charge quantum this effect does not yet play a role; see Ref. [27] for a numbers, respectively. They are confined by a string of displaced discussion. spins connecting them (blue), similar to mesons which can be understood as bound states of confined quark-antiquark pairs. (b) In the strong-coupling limit the dynamics of the holon is much A. Partons and the t-Jz model faster than the dynamics of the spinon. The holon motion takes Partons have been introduced in high-energy physics to place on a fractal Bethe lattice which is comoving with the describe hadrons [30]. Arguably, the most well-known spinon. Moreover, the holon wave function affects the spinon and example of partons is provided by quarks. In quantum introduces coherent motion of the spinon on the original square chromodynamics (QCD), the quark model elegantly ˆ lattice. The holon states on the Bethe lattice obey a discrete C4 explains mesons (baryons) as composite objects consisting symmetry that corresponds to rotations of the string configuration of two (three) valence quarks. On the other hand, individual around the spinon position. (c) As a result, discrete rotational and vibrational excited states of the magnetic polaron can be formed. quarks have never been observed in nature, and this has been attributed to the strong confining force between a pair They are characterized by quantum numbers nm4 …, where iπm4/2 ˆ of quarks mediated by gauge fields [31]. Even though there e denotes the eigenvalue of C4 and n labels vibrational excitations. We use labels S, P, D, F for m4 0, 1, 2, 3 and is little doubt that quarks are truly confined and can never indicate the multiplicity of the degenerate states¼ by numbers in be separated at large distances, a strict mathematical proof circles. Different colors correspond to the different types of is still lacking, and the quark confinement problem is still rovibrational excitations. For the calculation, we used S 1/2 attracting considerable attention in high-energy physics; ¼ and linear string theory (LST, described later in the text) with see, for example, Ref. [32]. strings of length up to l 100. max ¼ To understand how the physics of holes moving in a spin environment with strong AFM correlations is connected to relevant for high-temperature cuprate superconductors for the quark confinement problem, consider removing a which typically t/J ≈ 3 [3]. While several theoretical particle from a two-dimensional N´eel state. When the hole approaches have been developed, which are reliable in moves around, it distorts the order of the surrounding spins. the weak-to-intermediate coupling regime t J, Jz, to date In the strong-coupling regime, t ≫ J, Jz, these spins have there exist only a few theories describing≲ the strong- little time to react and the hole can distort a large number of coupling limit [17,19] and simple variational wave func- AFM bonds. Assuming for the moment that the hole tions in this regime are rare. Even calculations of qualitative motion is restricted to a straight line, as illustrated in ground state properties of a hole in an antiferromagnet, Fig. 1(a), we notice that a string of displaced spins is such as the renormalized dispersion relation, require formed. At one end, we find a domain wall of two aligned

011046-2 Polaron C(1,1) Local Spin Correlations around Polaron Magnetic Polaron Size

-0.12 0.05 2 2

0.04

0.03 1 1

-0.14 0.02 0.05 2 0 0 0.01

Doublons induce sign flip ! 0.00 y - hopping distance (sites) y - hopping distance (sites) -0.16 0.04 -1 -1 -0.01

-0.02 0.03 -2 -2 1 -0.18 -0.03 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 x - hopping distance (sites) x - hopping distance (sites) Impurity mediated 0.02 interactions!

0 0.01 0.02

2

0.00 0.00 y - hopping distance (sites)

1 -1 -0.01 -0.02

0

-0.04

-0.02 Complete model independenty - hopping distance (sites) characterisation -1 -2 of magnetic environment around charge impurity-0.06 -0.03 -3 -2 -1 0 1 2 3 -2

x - hopping distance (sites) -0.08 -3 -2 -1 0 1 2 3

x - hopping distance (sites)

related: detection of string patterns Ch. S. Chiu et al. Nature (2019)

Polaron to Fermi Liquid Doping 2d Antiferromagnets

- Cuprates: nature of charge carriers changes from (polaron) hole-like to electron-like at around 19% doping [Badoux et al. Nature 531, 2016]

- Pseudogap + Strange Metal behaviour in the < 19% doping regime. What is the link to polarons and antiferromagnetic correlations?

- QMC of Fermi-Hubbard model: single- Polaron Breakdown into particle bandwidth is polaronic (order J) up to around 30% doping, then free-particle like (8t) Fermi-Liquid [Preuss et al. PRL 79, 1997] - Numerical Simulations suggest a Spin- Density-Wave in Fermi-Liquid regime, with New observables from QGM’s spin-structure peak moving (pi,pi) over (0,pi) (with full resolution) to (0,0) (expected from random-phase approximation and noninteracting Fermions) [Moreo et al. PRB 41, 1990] [Furukawa&Imada, Jour. Phy. Soc. J. 61, 1992]

- Magnetic Susceptibility (q=0,w=0) predicted to rise for PG+SM phase dopings, then drop 2-point 3-point 4-point like weakly interacting Fermi-Liquid [Moreo, PRB 48, 1993] check arXiv soon….. check arXiv soon….. Correlations Polaron to Fermi Liquid Connected vs. Disconnected Correlations

Observations

Spin-Spin Correlations (2-point)

Doping 0.07 0.21 0.35 0.49

Polaron to Fermi Liquid Spin Structure Factor + Susceptibility Polaron to Fermi liquid Spin-Spin Correlator

- Most correlations change sign around 20-30% doping - Spin-Density Wave behaviour for extreme dopings - Weakly oscillating magnetism with doping - Incommensurate peak moving from (pi,pi) to (pi,0) - Extreme dopings: Spin-Density Wave expected from Fermi-Liquid/ - Magnetic Susceptibility expected Noninteracting Fermions from weakly interacting Fermi-Liquid: Doping monotonic decrease with doping - Comparison to 4 numerical 0.07 0.21 0.35 0.49 simulations - Pink curve: Noninteracting Fermions (ED: has large finite size offsets RVB states: - Two regimes in experiment: - ED: large finite size (only C(2,2) 0-30% doping: constant/increase shown with scaling factor) >30% doping: monotonic decrease - pi-flux good for low doping (susc. can be computed via - uniform: very good agreement all fluctuation dissipation for q=0) dopings - string: ok for low dopings

uRVB pi-flux String Polaron to Fermi Liquid Hole-Spin-Spin Correlator

- polaronic correlations <20% doping

Doping - polaron breakdown 20-40% 0.08 0.18 0.28 0.50 Hole-Spin-Spin Correlations doping - transformation into Pauli-hole driven correlations for (3-point) dopings >40%

- very good agreement with ED (T=0.4t)

Polaronic Pauli-hole

Doping 0.08 0.18 0.28 0.50 ED uRVB pi-flux String Nonint Polaron to Fermi Liquid Hole-Spin-Spin Correlator

- polaronic correlations <20% doping

- polaron breakdown 20-40% doping Hole-Hole-Spin-Spin Correlations - transformation into Pauli-hole driven correlations for (4-point) dopings >40%

- very good agreement with ED (T=0.4t)

ED uRVB pi-flux String Nonint Doping 0.08 0.18 0.28 0.50

Doping 0.08 0.18 0.28 0.50

Polaron to Fermi Liquid Probing Interactions Polaron to Fermi Liquid Probing Interactions

Doping 0.1 0.3 0.5 tJ-DMRG (A. Bohrdt)

Doping 0.1 0.3 0.5 - Singlet-character of spins around - Singlet-character of spins around hole-pairs seen with DMRG hole-pairs seen with DMRG ED uRVB pi-flux String Nonint

- Experiment sees antiferromagnetic - Experiment sees antiferromagnetic ED uRVB pi-flux String Nonint features features

- Explanation: Pauli-hole driven + dh - Explanation: Pauli-hole driven + dh fluctuations fluctuations

- Key observation: Qualitative features - Key observation: Qualitative features of paired holes already built in of paired holes already built in

- Pairing affects correlations (mostly) - Pairing affects correlations (mostly) quantitatively, not qualitatively quantitatively, not qualitatively

Challenges Scaling Up

813 nm

Next 1-2 years 1 cm High power and robust cavity lattices Ultracold Atoms larger

20 µm 1 3 ~1000 atoms tuneout S0 ( P0) 689 nm (633 nm) 1 3 tuneout S0 ( P0) 689 nm (633 nm) 1 µm Outlook and Development Tweezer Arrays Challenges ~100 atoms Controllability

Ion Traps

~50 atoms Big advantage: all atoms (molecules)(built by are nature) the same! Challenges Programmability Outlook Search for New Phases of Matter Extremely Strong Magnetic Field Physics Novel Quantum Magnets Controlled Quasiparticle Manipulations Non-Equilibrium Dynamics (Universality?)

Programmable Quantum Wires Box Potentials Thermalization in Isolated Quantum Systems Lattices Entanglement Measures in Dynamics Almost Arbitrary Light Patterns Possible! (but: fight Laser Speckle) Supersolids Cosmology - Black Hole Models? High Energy Physics/String Theory New clocks/Navigation Quantitative testbeds

Unique Geometries (reconfigurable on the fly) for theory!

Munich Center for Quantum www.mcqst.de F. Grusdt Science & Technology A. Bohrdt E. Demler Petar Bojovic Thomas Chalopin

Y. Wang

Joannis Guillaume Koepsell Christian Salomon Dominik Jayadev Vijayan Gross Bourgound

Sarah Hirthe Pimponpan Sompet www.quantum-munich.de