1. Guillermina Ramirez San Juan
Bloch Oscillations of cold atoms in
Optical Lattices
Guillermina Ramirez San Juan
Bloch wave in silicon
Optical Lattice

2. Brief description of Bloch Oscillation and Optical Lattices
PART I:
Brief description of Bloch Oscillation and Optical Lattices

3. 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)

4. Particle in a periodic potential
Solve Schrodinger’s 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

5. Band structure for a particle in the periodic potential
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)

6. 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

7. 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

8. 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?

9. 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:

10. 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

11. 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

12. How to measure Bloch oscillations
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]

13. MOT with cloud of cold atoms
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

14. Atoms re-arrange and form optical lattice
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

15. 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

16. 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

17. Source: See [1]

18. Source: See [1]

19. 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]

20. References [1] M.B Dahan et al., Phys. Rev. Lett 76,24 (1996)
[2] M.Greiner & S.Folling, Nature 5, (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, (2005)
[5] O.Morsch et al., Phys. Rev. Lett. 87,14 (2001)