Guillermina Ramirez San Juan

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Bloch Oscillations of cold atoms in Optical
Lattices

• Guillermina Ramirez San Juan

Bloch wave in silicon
Optical Lattice
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PART I Brief description of Bloch Oscillation
and Optical Lattices
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Background

• 1rst formulated in the context of condensed
  matter physics. Pure quantum effect
• It was predicted that a homogeneous static
  electric field induced oscillatory motion of
  electrons in lattice
• Every crystal structure has 2 lattices associated
  with it, the real lattice and the reciprocal
  lattice

Crystal lattices
Reciprocal space unit cell (B-Z)
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Particle in a periodic potential

• Solve Schrodingers eq. for a particle in a
  periodic potential
• Solution proposed by Bloch a wave function with
  a periodic squared modulus
• The corresponding eigenvalues for this equation
  are
• Energy eigenvalues are periodic with periodicity
  k (reciprocal lattice vector)

Bloch State
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• Periodicity of lattice leads to band structure of
  energy spectrum of the particle

(b)
(a)
Band structure for a particle in the periodic
potential and mean velocity
a) Free particle , b)
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Bloch Oscillations

• Particles in a periodic potentials subjected to
  an external force undergo oscillations instead of
  linear acceleration

• Eigenenergies and eigenstates
  are Bloch states
• Under the influence of a constant external force,
  evolves into the state
  according to
• The evolution is periodic and has a period of
• The mean velocity in is
• A wave packet with a well defined q in the nth
  band oscillates with an amplitude
  is the energy width of the nth band

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• Electrons acted on by a static electric field
  oscillate
• Oscillations have never been observed in nature
• Studied in semiconductor super lattices, but
  oscillations are still dominated by relaxation
  process.
• Solution Optical lattices

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Motivation to study optical lattices

• Studying Bloch oscillations, properties of
  condensed matter systems
• Study superfluid behaviour in the lattice
• Can be used for laser cooling atoms (lattice
  potential increases efficiency of some optical
  cooling methods)
• Study of many body quantum mechanics
• Atomic clocks
• Quantum computers?

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Optical Potential

• Consider a 2 level atom in a standing plane wave.
  The temporal evolution if the system is given by

The Hamiltonian of this system is

• M atomic mass
• Eg, Ee ground and excited electronic states
• Rabi frequency
• wL, kL frequency and wave vector
• of the standing wave

The wavefunction of this system is
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We consider ??? ?, then Then the problem
reduces to solving a Schrodinger equation
This eq. describes the motion of the atom
along the standing wave. The potential is the
optical lattice and has a spatial period of
d?/2 The dept of the lattice is measured in
units of recoil energy
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Optical Lattices

• Create a Bose-Einstein condensate or a cold gas
  of fermionic atoms with a well defined momentum
  spread
• Slowly ramp up lasers to create a lattice
  potential
• Put the lattice into the atoms and the atoms
  reorder to adapt to their new environment

Standing laser waves and cold neutral atoms play
the role of the crystal lattices and electrons
respectively
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PART II How to measure Bloch oscillations

• Description of the experiment performed by
• M.B Dahan,E.Peik, J.Richel, Y. Castin, C.Salomon.
  See 1

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1. Cooling the atoms

• Using laser cooling prepare a gas of free
  electrons with a momentum spread
  in the direction of the standing wave
• Precool Cs ( )using a MOT . Turn off
  magnetic field and 1D Raman cooling with
  horizontal beams

Cloud of cold atoms
MOT with cloud of cold atoms visible in red
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2. Setting the Potential

• Adiabatically switch on light potential, initial
  momentum distribution is turned into a mixture of
  Bloch states
• Laser is split in 2 beams with the same
  polarization and intensity. Beams are
  superimposed in counterpropagating directions
• Initially beams have the same frequency, their
  dipole coupling to the atom leads to the
  potential
• Spontaneous emission can be neglected because
  interaction time is much shorter than the
  emission rate

Atoms re-arrange and form optical lattice
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3. Applying external force

• Mimic external force by Introducing a tunable
  frequency difference between 2
  counterpropagating laser waves. So atom feels a
  force
• This is done by applying a frequency ramp of
  duration

Schematic representation of Counterpropagating
laser waves
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4. Measuring the oscillation

• At a given acceleration time the standing
  wave is turned off fast
• Obtain atomic momentum distribution in the lab
  frame
• The distribution in the accelerated frame is
  obtained by a translation -ma

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Source See 1
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Source See 1
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Advantages of this Method

• Initial momentum distribution is well defined and
  can be tailored at will
• Periodic potential can be turned on and off
  easily
• There is virtually? no dissipation or scattering
  from defects in the periodic potential
• We observe Bloch periods in the millisecond
  range, i.e. 10 orders of magnitude longer than in
  semiconductors.

Source See 2
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References

• 1 M.B Dahan et al., Phys. Rev. Lett 76,24
  (1996)
• 2 M.Greiner S.Folling, Nature 5, 736-738
  (2008)
• 3 D.Budker, D. Kimball D.P DeMille, Atomic
  physics An Exploration through Problems and
  Solutions (Oxford University Press, 2008)
• 4 I.Bloch, Nature Phys. 1, 23-30 (2005)
• 5 O.Morsch et al., Phys. Rev. Lett. 87,14
  (2001)