Guillermina Ramirez San Juan

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

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

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 The Hamiltonian of this system is: Consider a 2 level atom in a standing plane wave. The temporal evolution if the system is given by: • M atomic mass • Eg, Ee ground and excited electronic states • Rabi frequency wL,kLfrequency 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. 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. 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. 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, 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)