Experiments in Tilman Esslinger’s group, ETH Zurich

Leticia Tarruell

Les Houches – 02/07/2015

Simulating the Fermi-Hubbard model with ultracold atoms in optical lattices

« with a suitable class of quantum machines you could imitate any quantum system »

Quantum simulation

Many-body system

R. P. Feynman, 1981

Quantum simulator

Model

t U ?

Quantum simulation

t U ? Strongly correlated

materials Model

« with a suitable class of quantum machines you could imitate any quantum system »

R. P. Feynman, 1981

Quantum simulator

Quantum simulation

t U ? Strongly correlated

materials Fermi-Hubbard

model

« with a suitable class of quantum machines you could imitate any quantum system »

R. P. Feynman, 1981

Quantum simulator

Quantum simulation

t U ? Strongly correlated

materials Fermi-Hubbard

model

Ultracold fermions in optical lattices

The Fermi-Hubbard model

tunneling interaction

t U

Repulsive phase diagram

Metal Mott insulator

Magnetic order Doping: high-Tc?

Building the Fermi-Hubbard model Metal-Mott insulator transition Short-range magnetic correlations

Outline

Building the Fermi-Hubbard model Metal-Mott insulator transition Short-range magnetic correlations

Outline

100000 trapped fermionic atoms T<0.1TF

TF

(fermionic isotopes of K, Li, Yb, Sr, Dy, Er, Cr)

Creating a periodic potential

Quantum gases in optical lattices

x

y

z

100x100x20 sites

Brillouin zones

(k=2π/λ)

quasi momentum

Realizing tight-binding models

Time-of-flight absorption imaging

CCD

Atoms

Glass cell (ultra-high vacuum)

Lens

Band mapping

Deep lattice Weak lattice Free atoms

lattice switch-off time

Metal – band insulator transition

Metal Band insulator filling

M. Köhl, H. Moritz, T. Stöferle, K. Günter and T. Esslinger, Phys. Rev. Lett. 94, 080403 (2005)

Observing the Fermi surface

Other non-standard lattices: aligned dimers, plaquettes, Kagome, Lieb (NIST, Munich, Bonn, Hamburg, Berkeley, Kyoto)

Many lattice geometries possible

Chequerboard

Triangular

Dimer 1D chains

Square Honeycomb

L. Tarruell et al., Nature 483, 302 (2012)

Studying strongly correlated systems

The Fermi-Hubbard model

t U

Control: tunneling (lattice depth) interaction (Feshbach resonances) filling (atom number) spin (internal states) lattice geometry gauge fields …

Correlated materials vs. cold atoms

x

y

z

High-temperature superconductor Ultra-cold fermions + optical lattice

Lattice spacing: ~ 500 nm Density: ~ 1013 atoms/cm3

Temperature: ~ nK

Lattice spacing: ~ nm Density: ~ 1022 electrons/cm3 Temperature: ~ K

40K

Some specificities of cold atom systems Trap: varying chemical potential over the cloud

U Vtrap

t

Varying filling: several phases in one cloud

Some specificities of cold atom systems

Isolated: total entropy of the system is fixed

Building the Fermi-Hubbard model Metal-Mott insulator transition Short-range magnetic correlations

Outline

The metal – Mott insulator transition

kinetic energy interaction energy

Bosons : superfluid – Mott insulator

Fermions : metal – Mott insulator

M. Greiner et al. Nature 415, 39 (2002)

U/t

Delocalization vs. interactions

R. Jördens et al., Nature 455, 204 (2008); U. Schneider et al., Science 322, 1520 (2008)

Temperature scales

U>>t

ener

gy

T > U: metallic behaviour

T < U: Mott insulator

T

Which observable? Mainz: cloud compressibility (Mott insulator = incompressible)

U. Schneider et al., Science 322, 1520 (2008)

Which observable? Zurich: occupation of lattice sites

Mott insulator: reduced number fluctuations

Metal Mott insulator

In-situ fluorescence imaging

Harvard, Munich, Tokyo (bosons)

Harvard, Glasgow, MIT (fermions)

Which observable?

Double occupancy

Zurich: occupation of lattice sites

Mott insulator: reduced number fluctuations

Metal Mott insulator

4. Expansion and Stern-Gerlach separation

Measuring double occupancy

2. Induce energy shift

1. Suppress tunneling 3. RF transfer

Doubly occupied sites

Measuring double occupancy

mF=-9/2 mF=-5/2 mF=-7/2

Doubly occupied sites

Measure D for values as low as 1%

Qualitative results

Qualitative results

U/6t = 0 Kinetic energy and trap dominated U/6t = 4.8 Interaction energy dominated

R. Jördens, N. Strohmaier, K. Günter, H. Moritz and T. Esslinger, Nature 455, 204 (2008)

U/6t=4.8

U/6t=0 Non interacting

Mott insulator

Quantitative comparison

R. Jördens, L. Tarruell, D. Greif, T. Uehlinger, N. Strohmaier, H. Moritz, T. Esslinger, L. De Leo, C. Kollath, A. Georges, V. Scarola, L. Pollet, E. Burovski, E. Kozik, and M. Troyer

Phys. Rev. Lett. 104, 180401 (2010)

Weakly interacting Mott insulator Intermediate

U/6t

DMFT and high-temperature series expansions

Metal Mott insulator

Next challenge

Quantum magnetism

Building the Fermi-Hubbard model Metal-Mott insulator transition Short-range magnetic correlations

Outline

Magnetism: a temperature challenge

U>>t

ener

gy

T > U: metallic behavior

T < U: Mott insulator

T

T < J: spin ordering

T

R. Jördens et al., Phys. Rev. Lett. 104, 180401 (2010) P. Duarte et al., Phys. Rev. Lett. 114, 070403 (2015)

J=4t2/U

Superexchange J

Approaches to magnetism

S. Trotzky et al., Science 319, 295 (2008) S. Nascimbène et al., Phys. Rev. Lett. 108, 205301 (2012) S. Murmann et al., Phys. Rev. Lett. 114, 080402 (2015)

Isolated double-wells or plaquettes (Munich, Heidelberg)

Approaches to magnetism

J. Simon et al., Nature 472, 307 (2011)

Ising spin chains (Harvard)

J. Struck et al., Science 333, 996 (2011) J. Struck et al., Nature Phys. 9, 738 (2013)

Classical magnetism, Ising XY (Hamburg)

Mappings

Approaches to magnetism

Dipolar interactions (JILA, Paris)

B. Yan et al., Nature 501, 521-525 (2013) A. de Paz et al., Phys. Rev. Lett. 111, 185305 (2013)

D. Greif et al., Science 340, 1307 (2013) R. A. Hart et al., Nature 519, 211 (2015)

Approaches to magnetism

Short-range quantum magnetism in the Fermi-Hubbard model (ETH, Rice)

J < T < Jd,s

T J

Jd

Jd > J

Dimerized lattice

ener

gy

Enhancing magnetic correlations

Magnetic correlations T < J

Js > J

Anisotropic cubic lattice

Jd,s

Magnetic correlations in dimerized lattice

singlet

triplet

Jd

Spin correlations on neighboring sites

T < Jd : NS > NT

Local spin correlations in cubic lattice Nearest-neighbor spin correlations vs. temperature

antiferromagnetic transition

DCA simulation 3D Fermi-Hubbard model

S. Fuchs, E. Gull, L. Pollet, E. Burovski, E. Kozik, T. Pruschke, and M. Troyer, Phys. Rev. Lett. 106, 030401 (2011)

Merging lattice sites

Chequerboard

Dimer

Square

Detecting magnetic correlations

singlet

or

triplet t0

singlet triplet t0

Dimerized lattice

Singlet-Triplet Imbalance

Measuring singlets and triplets

𝑝𝑆

𝑝𝑡𝑡 Sin

glet

s Tr

iple

ts

Merging neighboring sites Singlet-triplet oscillations

Singlet-triplet oscillations: S. Trotzky et al., Phys. Rev. Lett. 105, 265303 (2010)

Theory: second order high-temperature series expansion of coupled dimers

Dependence on dimerization

s=1.7 kB

Jd

T J

isotropic strongly dimerized

D. Greif, T. Uehlinger, G. Jotzu, L. Tarruell, and T. Esslinger, Science 340, 1307 (2013)

Dependence on entropy

Theory: second order high-temperature series expansion of coupled dimers

U/t = 11.0(8) td/t = 22(2) t/h = 67(3) Hz

Jd T

J

D. Greif, T. Uehlinger, G. Jotzu, L. Tarruell, and T. Esslinger, Science 340, 1307 (2013)

Anisotropic simple cubic lattice

transverse spin correlator ⟺ population difference

AFM correlations along x

Effective 1D chains

Dependence on anisotropy

isotropic strongly anisotropic

VY,Z = 11.0(3) ER s = 1.8 kB

normalized spin correlator

D. Greif, T. Uehlinger, G. Jotzu, L. Tarruell, and T. Esslinger, Science 340, 1307 (2013)

Dependence on entropy

tS /t=7.3

D. Greif, T. Uehlinger, G. Jotzu, L. Tarruell, and T. Esslinger, Science 340, 1307 (2013)

Comparison with theory

Theory: DCA+LDA for anisotropic simple cubic lattice J. Imriška, M. Iazzi, L. Wang, E. Gull, D. Greif, T. Uehlinger, G. Jotzu, L. Tarruell, T. Esslinger, and M. Troyer, Phys. Rev. Lett. 112, 115301 (2014)

Correlations over 2 sites

T<t

Analogous results with DMRG: B. Sciolla et al., Phys. Rev. A 88, 063629 (2013)

Conclusion

Metal-Mott insulator transition

Nearest-neighbor magnetic correlations

Building the Fermi-Hubbard model

With a bit more cooling…

High-T phase diagram of cuprates

QCP

Dimers AFM

T/J

J/Jd

Geometry-induced quantum phase transitions

Frustration

Gregor Jotzu Daniel Greif L. T. Thomas Uehlinger Tilman Esslinger

Robert Jördens

Henning Moritz

Thomas Uehlinger

Niels Strohmaier

Daniel Greif

L. T. Tilman Esslinger

Zurich: V. Scarola, L. Pollet, E. Burovski, E. Kozik, J. Imriška, M. Iazzi, L. Wang, E. Gull, M. Troyer Paris: L. De Leo, C. Kollath, A. Georges Geneva/Bonn: B. Sciolla, A. Tokuno, S. Uchino, P. Bartmettler, T. Giamarchi, C. Kollath

THEORY

Ultracold Quantum Gases group @

L. T.

Pierrick Cheiney

César Cabrera

www.qge.icfo.es

Luca Tanzi

Jordi Sastre

Julio Sanz

Manel Bosch (now at Laboratoire Kastler Brossel, Paris) Vincent Lienhard (now student at ENS Cachan) Lisa Saemisch (now at ICFO’s Molecular Nanophotonics group)