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Probing dynamical properties of Fermi-Hubbard systems with a quantum gas microscope Peter Brown Bakr Lab Solvay workshop, February 19 th 2019 Outline 1. Studying strongly-interacting quantum matter with ultracold atoms 2. The Fermi-Hubbard

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SLIDE 1

Probing dynamical properties

  • f Fermi-Hubbard systems with

a quantum gas microscope

Peter Brown Bakr Lab Solvay workshop, February 19th 2019

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SLIDE 2

Outline

  • 1. Studying strongly-interacting quantum

matter with ultracold atoms

  • 2. The Fermi-Hubbard Model
  • 3. Measuring diffusion and conductivity in the

repulsive Hubbard model

  • 4. Measuring spectral functions and the

pseudogap in the attractive Hubbard model

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SLIDE 3

Strongly correlated quantum matter

Spintronic materials Heavy fermion metals

Topological phases

High temperature superconductors

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SLIDE 4

High temperature superconductors

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SLIDE 5

High temperature superconductors Yttrium Barium Copper Oxide

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High temperature superconductors Yttrium Barium Copper Oxide Hubbard model

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SLIDE 7

Use a synthetic quantum system of ultracold atoms

  • Feynman (paraphrased)
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SLIDE 8

Interacting systems of ultracold atoms – enlarged model for condensed matter physics

Why ultracold atoms?

  • Understood from first principles
  • Complete control of microscopic parameters
  • Clean systems, no impurities
  • Dynamics on observable timescales
  • Large interparticle spacing makes optical

imaging/manipulation possible

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SLIDE 9

Microscopy of ultracold atoms in optical lattices

Similar fermion microscopes at: Harvard, MIT, MPQ, Toronto, Strathclyde

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Outline

  • 1. Studying strongly-interacting quantum

matter with ultracold atoms

  • 2. The Fermi-Hubbard Model
  • 3. Measuring diffusion and conductivity in the

repulsive Hubbard model

  • 4. Measuring spectral functions and the

pseudogap in the attractive Hubbard model

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SLIDE 11

The Fermi-Hubbard model

U t How much of the phenomenology of the cuprates does the Hubbard model reproduce?

Hopping (kinetic energy) On-site interaction

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SLIDE 12

Cuprate phase diagram temperature doping

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Cuprate phase diagram temperature doping

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SLIDE 14

Antiferromagnetic spin correlations

Detection of AFMs with microscopes: Parsons … Greiner, Science 353, 1253 (2016) Boll ... Bloch, Gross, Science 353, 1257 (2016) Cheuk … Zwierlein, Science 353, 1260 (2016) Previous work without microscopes: Grief … Esslinger, Science 340, 1307 (2013) Hart … Hulet, Nature 519, 211 (2015) Drewes … Köhl, PRL 118, 170401 (2017)

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SLIDE 15

Cuprate phase diagram temperature doping

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SLIDE 16

Outline

  • 1. Studying strongly-interacting quantum

matter with ultracold atoms

  • 2. The Fermi-Hubbard Model
  • 3. Measuring diffusion and conductivity in the

repulsive Hubbard model

  • 4. Measuring spectral functions and the

pseudogap in the attractive Hubbard model

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SLIDE 17

Cuprate phase diagram temperature doping

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SLIDE 18

Conventional (weakly interacting) Unconventional (strongly correlated)

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SLIDE 19

Previous Work

Mass transport experiments with Fermions Mesoscopic systems: Brantut et al. Science 337, 1069 (2012) (ETH Zurich) … Lebrat et al. PRX 8, 011053 (2018) (ETH Zurich) Valtolina et al. Science 350, 1505 (2015) (Florence) Bulk systems: Ott et al. PRL 92, 160601 (2004) (Florence) Strohmaier et al. PRL 99, 220601 (2007) (ETH Zurich) Schnedier et al. Nat. Phys 8, 213 (2012) (Munich) Xu et al. arXiv:1606.06669 (2016) (UIUC) Anderson et al. arXiv:1712.09965 (2017) (Toronto)

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SLIDE 20

Measurement Protocol

Brown et. al., Science 363, 379 (2019) Spin transport: Nichols … Zwierlein, Science 363, 383 (2019)

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SLIDE 21

Measurement Protocol

Brown et. al., Science 363, 379 (2019)

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Measurement Protocol

Brown et. al., Science 363, 379 (2019)

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Measurement Protocol

Brown et. al., Science 363, 379 (2019)

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Measurement Protocol

Brown et. al., Science 363, 379 (2019)

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Decaying density modulation

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Decaying density modulation Not explained by diffusion alone!

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Decaying density modulation

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Decaying density modulation

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Decaying density modulation

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Hydrodynamic Model

Fick’s Law “charge” conservation

  • Diffusion (Fick’s Law)

neglects finite time to establish current.

  • D, diffusion constant
  • Γ, current relaxation.

rate.

  • Crossover from

diffusive mode to sound mode.

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SLIDE 31

Hydrodynamic Parameters

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SLIDE 32

Compressibility

DQMC High-T limit

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SLIDE 33

Resistivity Versus Temperature

Brown et. al., Science 363, 379 (2019)

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SLIDE 34

Resistivity Versus Temperature

FTLM Brown et. al., Science 363, 379 (2019) DMFT

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SLIDE 35

Outline

  • 1. Studying strongly-interacting quantum

matter with ultracold atoms

  • 2. The Fermi-Hubbard Model
  • 3. Measuring diffusion and conductivity in the

repulsive Hubbard model

  • 4. Measuring spectral functions and the

pseudogap in the attractive Hubbard model

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SLIDE 36
  • Using a photon,

excite a particle from an interacting system

  • Measure the energy

and momentum of the ejected particle

  • single-particle

excitations of a many-body system

  • Rev. Mod. Phys. 75, 473 (2003)

Photoemission spectroscopy

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SLIDE 37

What does ARPES measure?

  • How does an excitation

propagate in a many- body system?

  • Momentum resolved

density of states

  • ARPES particle current

gives access to emission

Remove hole (injection) Remove particle (emission) Emission only Emission + injection

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SLIDE 38

What does ARPES measure?

  • How does an excitation

propagate in a many- body system?

  • Momentum resolved

density of states

  • ARPES particle current

gives access to emission

Remove hole (injection) Remove particle (emission) Emission only Emission + injection

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SLIDE 39

What does ARPES measure?

  • How does an excitation

propagate in a many- body system?

  • Momentum resolved

density of states

  • ARPES particle current

gives access to emission

Remove hole (injection) Remove particle (emission) Emission only Emission + injection

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SLIDE 40
  • Fermi gas,

excitations have definite momentum and energy

  • BCS, pairing

appears as a gap

  • Dispersion exhibits

“backbending”

The BCS limit

Fermi gas

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SLIDE 41

BCS

The BCS limit

  • Fermi gas,

excitations have definite momentum and energy

  • BCS, pairing

appears as a gap

  • Dispersion exhibits

“backbending”

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SLIDE 42

Pseudogap reviews: Low Temp. Phys. 41, 319 (2015)

  • Rep. Prog. Phys. 80, 104401 (2017)

3D Fermi Gas

Stewart … Jin, Nature 454, 744 (2008) Gaebler … Jin, Nature Phys. 6, 569 (2010) Feld … Kohl, Nature 480, 75-78 (2011)

2D Fermi Gas

Pseudogaps

  • Depression in the spectral

function at the Fermi energy.

  • Cold atom experiments:

backbending in dispersion above 𝑈

𝑑.

  • Observed in High-𝑈

𝑑

superconductors and unitary Fermi gas

  • HTSC, PG origin controversial:

precursor to SC or indicative of a competing order.

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SLIDE 43
  • Accessible model: on a

lattice and no DQMC sign problem.

  • BEC-BCS crossover

with interaction strength.

  • Temperatures near

state-of-the-art for experiment

  • Eur. Phys. J. B. 2, 30 (1998)

Pseudogap in the attractive Hubbard model

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SLIDE 44

ARPES with a QGM

rf probe Hubbard system

  • Radiofrequency photon transfers to non-interacting state

but preserves momentum

  • Band mapping transforms quasimomentum to real

momentum

  • 𝑈 4

expansion in harmonic trap maps momentum space to real space (similar to time-of-flight measurement)

  • Freeze atoms in deep lattice and detect

𝑔 ↑

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SLIDE 45

ARPES with a QGM

  • Radiofrequency photon transfers to non-interacting state

but preserves momentum

  • Band mapping transforms quasimomentum to real

momentum

  • 𝑈 4

expansion in harmonic trap maps momentum space to real space (similar to time-of-flight measurement)

  • Freeze atoms in deep lattice and detect

𝑔 ↑

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SLIDE 46

ARPES with a QGM

  • Radiofrequency photon transfers to non-interacting state

but preserves momentum

  • Band mapping transforms quasimomentum to real

momentum

  • 𝑈 4

expansion in harmonic trap maps momentum space to real space (similar to time-of-flight measurement)

  • Freeze atoms in deep lattice and detect

𝑔 ↑

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SLIDE 47

ARPES with a QGM

  • Radiofrequency photon transfers to non-interacting state

but preserves momentum

  • Band mapping transforms quasimomentum to real

momentum

  • 𝑈 4

expansion in harmonic trap maps momentum space to real space (similar to time-of-flight measurement)

  • Freeze atoms in deep lattice and detect

𝑔 ↑

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SLIDE 48
  • Eur. Phys. J. B. 2, 30 (1998)

ARPES data: increasing interaction strength

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SLIDE 49

𝜁𝑙

ARPES data: increasing interaction strength

Expt 𝑉 𝑢 = −4 𝑈 𝑢 = 0.5 DQMC

  • Determine 𝑉 𝑢

, 𝑈 𝑢 , and 𝜈 𝑢 from fitting correlators to equilibrium DQMC

  • Spectral weight shifts to

lower energy (𝑉 < 0)

  • Spectral peak shifts away

from 𝜈 at stronger interaction

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SLIDE 50

𝜁𝑙

ARPES data: increasing interaction strength

Expt 𝑉 𝑢 = −6 𝑈 𝑢 = 0.5 DQMC

  • Determine 𝑉 𝑢

, 𝑈 𝑢 , and 𝜈 𝑢 from fitting correlators to equilibrium DQMC

  • Spectral weight shifts to

lower energy (𝑉 < 0)

  • Spectral peak shifts away

from 𝜈 at stronger interaction

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SLIDE 51

𝜁𝑙

ARPES data: increasing interaction strength

Expt 𝑉 𝑢 = −8 𝑈 𝑢 = 0.5 DQMC

  • Determine 𝑉 𝑢

, 𝑈 𝑢 , and 𝜈 𝑢 from fitting correlators to equilibrium DQMC

  • Spectral weight shifts to

lower energy (𝑉 < 0)

  • Spectral peak shifts away

from 𝜈 at stronger interaction

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SLIDE 52
  • Eur. Phys. J. B. 2, 30 (1998)

ARPES data: increasing interaction strength

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SLIDE 53

𝜁𝑙

ARPES data: increasing temperature

Expt 𝑉 𝑢 = −8 𝑈 𝑢 = 0.44 DQMC

  • Determine 𝑉 𝑢

, 𝑈 𝑢 , and 𝜈 𝑢 from fitting correlators to equilibrium DQMC

  • Second branch emerges with

increasing temperature

  • Lower branch: doublons
  • Upper branch: singles
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SLIDE 54

𝜁𝑙

ARPES data: increasing temperature

Expt 𝑉 𝑢 = −8 𝑈 𝑢 = 1.0 DQMC

  • Determine 𝑉 𝑢

, 𝑈 𝑢 , and 𝜈 𝑢 from fitting correlators to equilibrium DQMC

  • Second branch emerges with

increasing temperature

  • Lower branch: doublons
  • Upper branch: singles
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SLIDE 55

𝜁𝑙

ARPES data: increasing temperature

Expt 𝑉 𝑢 = −8 𝑈 𝑢 = 2.8 DQMC

  • Determine 𝑉 𝑢

, 𝑈 𝑢 , and 𝜈 𝑢 from fitting correlators to equilibrium DQMC

  • Second branch emerges with

increasing temperature

  • Lower branch: doublons
  • Upper branch: singles
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SLIDE 56

𝜁𝑙

ARPES data: increasing temperature

Expt 𝑉 𝑢 = −8 𝑈 𝑢 = 5.0 DQMC

  • Determine 𝑉 𝑢

, 𝑈 𝑢 , and 𝜈 𝑢 from fitting correlators to equilibrium DQMC

  • Second branch emerges with

increasing temperature

  • Lower branch: doublons
  • Upper branch: singles
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SLIDE 57
  • ARPES of repulsive model: further

cooling is a key challenge.

  • Entropy redistribution.
  • Immersion in bosonic baths.
  • Floquet engineering of t-J

models.

  • Dynamical observables
  • More challenging for theory
  • Test approximations
  • Toolkit small compared with

materials

Challenges and opportunities

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SLIDE 58

Thank you!

Funding

PI Waseem Bakr Graduate students Debayan Mitra Peter Brown Elmer Guardado-Sanchez Lysander Christakis Jason Rosenberg Benjamin Spar Post-docs Stanimir Kondov Peter Schauss Undergraduates Joseph Scherrer Adam Bowman Mark Stone Joe Zhang Maria-Claudia Popescu Nathan Agmon Arka Adhikari Theory support David Huse (Princeton) Andre-Marie Tremblay (Sherbrooke) Jure Kokalj (U. of Ljubljana) Edwin Huang (Stanford) Tom Devereaux (Stanford)