²~k =¹h2k2

x

2m

²~k =¹h2

2m(k2

x + k2y)

Contours d’énergie constante

D(²) /1

p²

D(²) / Cted = 2

d = 1

D(²)d² = nombre d'¶etats dans une tranche d'¶energie [²;² + d²]

kx

kxky

²~k

kx

kx

ky

2

²~k =¹h2k2

x

2m

²~k =¹h2

2m(k2

x + k2y)

Contours d’énergie constante

D(²) /1

p²

D(²) / Cted = 2

d = 1

D(²)d² = nombre d'¶etats dans une tranche d'¶energie [²;² + d²]

kx

kxky

²~k

kx

kx

ky

2( ) [ ( )]x

kx x

ej v f

Vk k k

2( ) [ ( )]

kxx x k

ej v k k f

V

2( ) [ ( )] 0x x

k

ej v f

Vk k

Fk

F

k

F

k

F

k

F

F

0xx

Ek e

2( ) [ ( )] [ ( )] 0x x x

k

ej v k f k k i f k

V

xE

2Fk

a

BT BT Drude

B

hT

Fa

F

k

F

k

F

k

Fk

2Fk

a

F t

k

BT BT Drude

B

hT

Fa

Oscillations de Bloch

F

k

F

k

F

k

k

B

hT

Fa

710BT s1210 s

Pour des électrons dans les solides

Pas d’oscillations de Bloch

F tk

BT Oscillations de Bloch

F

k

k

Atoms re-arrange and form optical lattice

Bloch Oscillations of Atoms in an Optical Potential, M. Dahan et al.

Phys. Rev. Lett. 76, 4508 (1996)

F tk

BT Oscillations de Bloch

F

k

Atomes dans un réseau optique

k

Periodicity of lattice leads to band structure of energy spectrum of the particle

Band structure for a particle in the periodic potential

and mean velocity : a) Free particle , b)

U(z) U0 sin2( z /d)

v0 q

( U0 0 )

U0 E0 2 2 /2md2

(a) (b)

Mean atomic velocity v as a function of t a for values

of the potential depth : (a)U0 =1.4ER, (b)U0 = 2.3ER, (c)U0 = 4.4ER

k

k

v

Mean atomic velocity v as a function of t a for values

of the potential depth : (a)U0 =1.4ER, (b)U0 = 2.3ER, (c)U0 = 4.4ER

k

k

v