Optical Lattices 1 Greiner Lab Winter School 2010 Florian Huber 02/01/2010.
Transcript of Optical Lattices 1 Greiner Lab Winter School 2010 Florian Huber 02/01/2010.
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?