Download Bloch Oscillations in cold atoms

Guillermina Ramirez San Juan
Bloch wave in silicon
Optical Lattice
PART I:
Brief description of Bloch
Oscillation and Optical
Lattices
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)
e ik R  1
Particle in a periodic potential
Solve Schrodinger’s eq. for a particle in a periodic potential:
Hˆ   E
pˆ 2
ˆ
H
 V (x)
2M
V (x  d)  V (x)
Solution proposed
by Bloch: a wave function with a periodic

squared modulus
nq (x)  nq (x  d)  e iqxunq (x)
Bloch State
The corresponding eigenvalues for this equation are:

n (q)  n (q  k)
Energy eigenvalues are periodic with periodicity k (reciprocal
lattice vector)

• Periodicity of lattice leads to band structure of energy
spectrum of the particle
(b)
(a)
2
Band structure for a particle in the periodic potential U(z)  U 0 sin ( z /d)
and mean velocity v 0 q : a) Free particle ( U0  0 ) , b) U 0  E 0 




 2 /2md 2
2
Bloch Oscillations
• Particles in a periodic potentials subjected to an external
force undergo oscillations instead of linear acceleration
• Eigenenergies En (q) and eigenstates n,q are Bloch states
• Under the influence of a constant external force, n,q(0)
evolves into the state n,q(t) according to q(t)  q(0)  Ft /


2

T

• The evolution is periodic and has a period of B dF

• The mean velocity in n,q(t) is

1 dE n Rq(t)
v n qt  

dq
• A wave packet with a well defined q in the nth band
 n /2 F where  n is the energy
 an amplitude

oscillates with
width of the nth band
•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
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?
Optical Potential
Consider a 2 level atom in a standing plane wave. The
temporal evolution if the system is given by:
i

 Hˆ (t)
t
The Hamiltonian of this system is:
E e
H  
 0
2 
0 1
0  p
1 0


 2sin(  L t)sin( kL x)

E g  2M 0 1
1 0
M
Eg, Ee

wL, kL
atomic mass
ground and excited electronic states
Rabi frequency
frequency and wave vector
of the standing wave
The wavefunction of this system is:
(x,t)  exp(iL t)e (x,t) e  g (x,t) g
ˆ2
 e (x,t)
p
i
  e (x,t) 
 e (x,t)  sin( kL x) g (x,t)
t
2M
2

(x,t)
ˆ
p
g
i

 g (x,t)  sin( kL x) e (x,t)
t
2M
where  
E
e
 Eg 
  L is the detuning
We consider  , then:
e (x,t)  /sin( kL x)g (x,t)
Then the problem reduces to solving a Schrodinger equation:


 g (x,t)  pˆ 2
i
 
 V (x) g (x,t)
t
2M

V (x)  V0 sin 2 (kL x),
V0   2 /
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
ER 
2
2
kL /2M
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
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]
1. Cooling the atoms
Using laser cooling prepare a gas of free electrons with a
momentum spread p  k / 4 in the direction of the
standing wave
• Precool Cs( 6K )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
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
2
U(z)

U
sin
kz
coupling to the atom leads to the potential:
0
• Spontaneous emission can be neglected because
interaction time is much shorter than the emission rate

Atoms re-arrange and form optical lattice
3. Applying external force
Mimic external force by Introducing a tunable frequency
difference  (t) between 2 counterpropagating laser
waves. So atom feels a force:
F  ma  m

d  (t) 


dt  2 
This is done by applying a frequency ramp of duration


Schematic representation of
Counterpropagating laser waves
ta
4. Measuring the oscillation
• At a given acceleration time ta 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 -mata

Source: See [1]
Momentum distributions in the accelerated frame
for different values of t a between t a  0 and t a   B = 8.2ms
Potential depth is U
0
= 2.3E R and a = -0.85m/s 2
Source: See [1]
Mean atomic velocity
v as a function of t
a
for values
of the potential depth : (a)U 0 =1.4E R , (b)U 0 = 2.3E R , (c)U 0 = 4.4E R
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]
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)