Optical Lattices 1

Greiner Lab Winter School 2010

Florian Huber02/01/2010

Outline

• Solid State Physics• How to make optical lattices• How to diagnose optical lattices

Solid State Physics

Phonons• Lattice vibrations• Thermal properties

(isolators)• Mediating electron-electron

interaction in type 1 (=BCS) superconductors

• Acoustic and optical phonons

Electrons• Electrical and thermal

properties• Semiconductors• Magnetism

Quantum Simulation

Lattice

• Atoms in solids arranged in regular pattern

• In general: 3D = 14 Bravais lattices…• … but we usually only have to deal with simple cubic (s.c.) lattices

a=𝜆2

x

y

Free Electrons

• High School Physics:

MetalReduction of

energy by delocalizing

outer electrons

(more or less) free electron gas

Bloch’s Theorem

• Delocalized electrons “feel” periodic potential, thus their wave function has to inherit periodicity

aFree Electrons Electrons in Per. Pot.

Wave function , where

Dispersion , where

Band Gap

2 1 0 1 2qG

2

1

2

3

4

1

2 m

1st Brillouin Zone

Restrict to 1st BZ

x

Standing wave 2 has higher probability near the ion cores higher energy than 1 band gap

Both waves with Standing waves

From free to tightly boundLattice Depth

ℏ𝜔

Harmonic oscillator energies

(Solid state systems: Atomic energy levels)

Comment from Markus’ thesis: (b/c J Tunneling Eff. Mass, see later)

Bloch VS. Wannier

• Bloch waves:– Delocalized – Plane-wave-like

• Deeper lattices better described by Wannier functions:– Localized on each lattice site– Closer to QHO Eigenstates– Intuitive picture for J in Bose-Hubbard

∑N , q

❑

∑lattice sites

❑

Bloch Oscillations

• Group velocity: (slope of dispersion)• Effective mass

(inverse curvature)• Apply external force: Direction of acceleration

changes, when mass changes sign! Oscillations• Not observable in “real” s.s. systems, scattering

rate with impurities to high• Note: Effective mass large for deep lattices

suppressed tunneling

Optical Dipole Force

• Reason: Position dependent AC Stark shift

• Focused Laser Beam:– Red detuned: Optical dipole trap

– Blue detuned: Plug beam

One Dimension

• Interfere with counter-propagation beam: Standing Wave:

Rayleigh Range

Inte

nsity

𝜆2≈500 nm

Here: red detuned

More dimensions

1D 2D 3D

“Pancakes” “Tubes”

D>1: Typically orthogonal beams are not interfering.different frequency or orthogonal polarizationsOtherwise: Relative phase matters!

Simple-cubic

Harmonic Confinement

• In D>1 configuration: – Additional (anti-) confinement due to Gaussian

profile of orthogonal laser beams.• Red detuned:

• Blue detuned:

Pote

ntial

Realization

• Non-interfering orthogonal beams:– Different frequency and/or polarizations– Separable lattice: +…

• Mirror to create standing wave– “Easy” to implement

• Cavity enhanced– Deeper lattice for given power– Cleaner potential (cavity is filtering the modes)

Recycling

• Recycle a beam to make lattice along another axis– Beams are interfering! Different lattice pattern

• Adiabatic loading of superfluid (slower than what? Tunneling?)

• Sudden release and TOF: Matter wave point sources on each lattice site

BEC in Lattice

q=0pcm=0

1/ext. confinement

1/lattice spacing

1/f(Tunneling)

Lattice Pulsing

• Depth measurement• Cycle:– BEC (superfluid)– Lattice suddenly pulsed on– Lattice suddenly switched of again– Image diffraction pattern in TOF– Repeat and vary intensity

Lattice Pulsing: Grating Picture

• Position dependent AC Stark shift of lattice imprints a phase pattern into BEC depending of the intensity/duration of the pulse (thin-grating)

Lattice Pulsing: Band Picture

• Free space WF: Thomas-Fermi gets projected onto Bloch waves Superposition of

• Different energies: Time evolution produces different accumulated phase while lattice is on.

𝜓

Projection

Time

evolution

TOF

LatticeOn

LatticeOff

Lattice Pulsing: Raman Picture

• Length of the pulse short broad spectrum

• So called Raman-Nath Regime

• Different bands (=vibrational levels) are not resolved

ℏ𝜔S

P

Bragg Scattering

• Here: • Bands are resolved Single diffraction order

(like AOM)• “Kick” the BEC using

Raman Transition• Angles of beams have

to be such that energy and momentum is conserved

Lattice Pulsing: Math

• Projected states in lattice evolve with )• Some identities later one gets for the

population in the Nth order:

Bessel Proportional to lattice depth

Lattice Pulsing: Pictures

Parametric Heating

• Small modulation of lattice to cause heating into higher bands

• Symmetry only allows excitations into bands with even wave function for – Initial state: symmetric– Perturbation: symmetricFinal state: symmetric

• Measurement gives twice the trap frequency

Band Mapping

Adiabatic Ramp Down of Lattice Depth preserves the quasi-momentum

1st BZ

Band Mapping

• Problem: BEC is in • One has to scramble the phases first (B-Field

gradient) to populate the whole 1st band• Fermions: Pauli-Exclusion Principle already

populating many momentum states• Imaging of Fermi surface

Increase lattice depth

Band Mapping: Higher Bands

Why only every other band?