lecture 8 : optical lattices and band structurechevy/atomesfroids/lecture8.pdf · plane wave...
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Lecture 8 : optical lattices and band structure
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Atoms are trapped near the intensity maxima (red detuning) or minima (blue detuning)
One dimensional lattice : stacks of 2D gases
Additional (weaker) trapping potentials provide confinement in the plane perpendicular to the lattice
One-dimensional optical lattices
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Invariance by translation of a basis vector
Common eigenstates of Td and H have the generic form
Bloch theorem
Invariance by changing the quasi-momentum by a multiple of 2π/d (a reciprocal lattice vector)
Quasi-momentum can be restricted to the first Brillouin zone BZ1 =
Bloch wave Bloch functionperiodic with period d
1D sinusoidal potential :
d
Neil Ashcroft and David Mermin. Solid State PhysicsJean Dalibard, Cours au Collège de France 2013 http://www.phys.ens.fr/~dalibard/
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Plane wave expansion
Fourier series expansion of the potential :
Fourier series expansion of the eigenstates :
Schrödinger equation reduces to a matrix equation for the coefficients in the plane wave basis.
For a sinusoidal potential, only the terms with m=0,+1,-1 remain: tridiagonal matrix
Length scale :Momentum / quasi-momentum scale :
Energy scale :
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Band structure for V0=2 ER
−0.5 0 0.50
2
4
6
8
10
12
14
16
18
quasi−momentum (2//d)
Ener
gy (E
r)
V0=2ER
−2 −1.5 −1 −0.5 0 0.5 1 1.5 20
0.02
0.04
Bloch wave functions
position (d)
un=
0,k=
0−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
−0.1
−0.05
0
0.05
0.1
position (d)
un=
1,k=
0n=1
n=2
n=3
n=4
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Band structure for V0=4 ER
−0.5 0 0.50
2
4
6
8
10
12
14
16
18
20
quasi−momentum (2//d)
Ener
gy (E
r)
V0=4ER
−2 −1.5 −1 −0.5 0 0.5 1 1.5 20
0.02
0.04
0.06
Bloch wave functions
position (d)
un=
0,k=
0
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.1
−0.05
0
0.05
0.1
position (d)
un=
1,k=
0n=1
n=2
n=3
n=4
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Band structure for V0=10 ER
−0.5 0 0.52
4
6
8
10
12
14
16
18
20
22
quasi−momentum (2//d)
Ener
gy (E
r)
V0=10ER
−2 −1.5 −1 −0.5 0 0.5 1 1.5 20
0.02
0.04
0.06
0.08Bloch wave functions
position (d)
un=
0,k=
0
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.1
−0.05
0
0.05
0.1
position (d)
un=
1,k=
0n=1
n=2
n=3
n=4
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Band structure for V0=20 ER
−0.5 0 0.50
5
10
15
20
25
30
quasi−momentum (2//d)
Ener
gy (E
r)
V0=20ER
−2 −1.5 −1 −0.5 0 0.5 1 1.5 20
0.05
0.1Bloch wave functions
position (d)
un=
0,k=
0
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.1
−0.05
0
0.05
0.1
position (d)
un=
1,k=
0n=1
n=2
n=3
n=4
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Wannier functions
−2 −1 0 1 2
0
1
2
3
4
position x (d)
Wan
nier
func
tions
V0=4ER
−2 −1 0 1 2−1
0
1
2
3
4
5
position x (d)
Wan
nier
func
tions
V0=10ER
−2 −1 0 1 2−1
0
1
2
3
4
5
position x (d)
Wan
nier
func
tions
V0=20ER
wn(r� ri) =1pNs
X
k
un,k(r)eik·ri
un,k(r)
Wannier functions form an orthogonal basis :
For large lattice depths, they become more and more localized around site ri
Analog of the localized states introduced for the double-well case
Solid: Wannier functions
Dots: harmonic oscillator approximation
Instead of working in the Bloch basis , it is often convenient to use the so-called Wannier functions defined as
(some subtleties for higher bands, see W. Kohn Phys. Rev. 1959)
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Tight-binding limit
In the Wannier basis, we can express the hamiltonian as
Single-band approximation :
At low temperatures/chemical potential, only the lowest energy band n=0 is occupied appreciably : higher energy bands can be neglected.
Tight-binding approximation :
This leads to the simplest non-interacting lattice model describing particles in the ground band tunneling from sites to sites:
0 5 10 15 20
10−6
10−4
10−2
100
V0 [ER]
Tunn
el E
nerg
ies
[ER
]
0 5 10 15 2010−3
10−2
10−1
100
V0 [ER]
Tunn
el E
nerg
ies
[ER
]
J0(1)J0(2)J0(3)
J0(1)> J0W0/4J0 (approx)
When the lattice depth is large (roughly V0>>5 ER) Jn(i-j) decreases very fast with site distance : one can neglect all terms but the ones connecting nearest neighbors.
In the Bloch basis, we can express the hamiltonian as
lundi 19 mai 14