Part III:
Dynamics of cold atoms in an optical lattice
Outline of part 3
3.1 The effective Hamiltonian
3.2 Bloch oscillations of a cold-atom wave packet
3.3 Wannier-Stark resonances
3.4 Berry phase in 2D optical lattices
3.5 Bloch oscillations with self-interaction
54
3.1 Gross-Pitaevskii equation in an optical lattice
• Optical lattice: Periodic optical potential Vp ( x )
[20,28]
Wave function
• Combined with an ‘external’ potential Vx ( x )
The many body Hamiltonian
�
H = dx
�
h2
†
†
2m ∂ x ψ ∂ x ψ + (Vp ( x ) + Vx ( x ))�ψ ψ
+ g ( ψ † )2 ψ2
• Assumption: Self-interaction energy is
smaller than hopping energy. Condensate
wave function equation:
ih̄∂t χ =
h2 2
2m ∂ x χ + (Vp ( x ) + Vx ( x )) χ +
Optical potential
Lattice constant a
gN |χ|2 χ
• Linear approximation:
ih̄∂t χ =
h2 2
2m ∂ x χ + (Vp ( x ) + Vx ( x )) χ
55
3.1 Wave packet in a periodic environment:
canonical and internal degrees of freedom
[5,21,23]
• Idea: use the Zak representation to average
over the fast oscillation of the potential
Position in the unit cell
� p, x |ψ � = ∑ e
Quasi-momentum
p( x +na)
ih̄
n
� x + na |ψ �
• Periodic in x , � p, x + a |ψ � = � p, x |ψ � ,
quasi-periodic
in p
�
�
p+
2πh̄
a , x |ψ
• Fourier transforms
Wave function
x
= e2πi a � p, x |ψ �
Optical potential
Lattice constant a
Integer band index
� p, x |ψ � −→ �q, n |ψ �
Position in the lattice
• q̂, p̂ canonically conjugate observables: [q̂, p̂] = ih̄
subject only to external potential
56
3.1 Matrix-valued phase space representation
• Hilbert space operators in the Zak representation have an integer index
[9,10,12,21]
• Weyl representation is (infinite)-matrix valued: Ô ↔ O(q, p)
�n| O(q, p) |m� = �n| Trc Ô∆ˆ (q, p) |m� =
• Properties
�
ˆ (q, p) |qm�
dq �qn| Ô∆
1. Inner product: trace over band index TrÔ1 Ô2 =
�
dqdp
2πh̄
tr O1 (q, p)O2 (q, p)
Matrix trace
2. Product representation: Phase space functions are matrices—respect ordering
←
−−
→ ←
−−
→
Ô1 Ô2 ↔ O1 (q, p) exp( ih2 ( ∂q ∂ p − ∂ p ∂ p )O2 (q, p) = O1 O2 + ih2 {O1 , O2 } + O(h̄2 )
57
3.1 The effective Hamiltonian: Peierls substitution
and beyond
• Zak representation of the 1-body
Hamiltonian
�m| V̂p |n� = Ṽm−n
H=
1 2πh̄
2m ( a n̂ +
[9,10,21]
Wave function
p̂)2 + V̂p + Vx (q̂)
• Without Vx , Ĥ is diagonalized by
U ( p̂)† ĤU ( p̂) = ε( p̂)
ε n ( p ), 0 ≤ p <
2πh̄
a
diagonal
= nth Bloch band
Optical potential
Lattice constant a
• With Vx , Ĥ is diagonalized U ( p) for
commuting q, p
• [q̂, p̂] = ih̄ � 1
diagonalize
semiclassically. Actual small parameter =
scale of Vx / lattice constant
58
3.1 The effective Hamiltonian: Peierls substitution
and beyond
[9,10,21]
• General problem: diagonalize operatorvalued matrix Hamiltonian
U † H (q̂, p̂)U = ε̂
Wave function
diagonal
• where U (q, p)† H (q, p)U (q, p) = ε(q, p)
c-numbers
• Idea: Let
U = U0 + h̄U1 + · · ·
ε̂ = ε̂ 0 + h̄ε̂ 1 + · · ·
Optical potential
Effective potential
• Results: nth band
1. ε̂ 0 = ε n (q̂, p̂) — Peierls substition
Band dispersion
• In a periodic environment: ε̂ 0 = ε n ( p̂) + Vx (q̂)
• No lattice potential, band kinetic energy
External potential
59
3.1 The effective Hamiltonian: Peierls substitution
and beyond
• General problem: diagonalize operatorvalued matrix Hamiltonian
U † H (q̂, p̂)U = ε̂
[9,10,21]
Wave function
diagonal
• where U (q, p)† H (q, p)U (q, p) = ε(q, p)
c-numbers
• Idea: Let
U = U0 + h̄U1 + · · ·
ε̂ = ε̂ 0 + h̄ε̂ 1 + · · ·
Optical potential
Lattice constant a
• Results: nth band
1. ε̂ 0 = ε n (q̂, p̂) — Peierls substition
2. ε̂ 1 = i (�un | ∂q ε n ∂ p |un � − q ↔ p) − 2i (∂q �un | ( H − ε n )∂ p |un � − q ↔ p)
Berry term
Wilkinson-Rammal term
60
3.2 Bloch oscillations of a forced wavepacket
• Wavepacket supported in ground band
[9,10,21]
Wave function
1. Uniform forcing: ε̂ 0 = ε g ( p̂) − q̂F
Force
Wavepacket dynamics:
• ∂t p = F
p = Ft
qt =
• E = ε g ( p) − qF = const
ε g ( Ft)−ε g (0)
F
p
• Band dispersion is periodic
Effective potential
phase space trajectory
• Constant force but periodic motion
ε g ( p)
Br
illo
v<0
v>0
ui
n
p
zo
ne
q
3.2 Bloch oscillations of a forced wavepacket
[9,10,21]
• Wavepacket supported in ground band
Wave function
1. Uniform forcing: ε̂ 0 = ε g ( p̂) − q̂F
Force
Wavepacket dynamics:
• ∂t p = F
p = Ft
• E = ε g ( p) − qF = const
qt =
ε g ( Ft)−ε g (0)
F
• Band dispersion is periodic
p
Effective potential
phase space trajectory
• Constant force but periodic motion
• Example: Deep potential, tight-binding limit
ε g ( p) =
pa
2u cos( h̄ )
Br
illo
ui
n
zo
ne
q
3.2 Bloch oscillations of a forced wavepacket
• Wavepacket supported in ground band
[14,15,19]
Wave function
1. Uniform forcing: ε̂ = ε g ( p̂) − q̂F
Force
Wavepacket dynamics:
•
2πh̄
Evolved wavepacket after a period t = aF
iA
|ψ�t = e h̄ T (qt , pt ) M(St ) |ψ�0
•
Overall action:
�
A = − qdp + Hdt =
�
dtε g ( Ft)dt = 0
Effective potential
p
phase space trajectory
• Phase space shift: T (qt , pt ) = T (0, 2πh̄
a )=1
• Squeezing M(St ) = M(1) = 1
• Periodic wavepacket dynamics
Brillouin zone
q
3.2 Bloch oscillations of a forced wavepacket
• Wavepacket supported in ground band
2. Harmonic trap:
[14,15,19]
Wave function
ε̂ = ε g ( p̂) − 12 kq2
• Pendulum-like Hamiltonian with q ↔ p
• Libration for low energies < ε g ( πh̄
a )
Effective potential
• Bloch oscillations for high energies > ε g ( πh̄
a )
displacement bounded by
• Maximal
�
(`2)
2
πh̄
ε
(
g
a )
k
phase space trajectories
below/above the separatrix
• Aperiodic dynamics of wavepacket shape
— squeezing
p
Libration
Brillouin zone
q
3.3 Wannier-Stark resonances
[29]
• Energy states supported in ground band
1. Uniform forcing: momentum representation
(−ih̄F∂ p + ε g ( p)) � p | E � = E � p | E �
� p |E � = e
Force
�
i ( Ep − p ε ( p� ) dp� )
g
h̄F
boundary conditions:
• Quasi-momentum
�
�
p+
2πh̄
a
|E = � p |E �
�
�
�
1
En = 2πa
n + 0 dxε g ( 2πh̄
F
a x)
Mean band dispersion
(Landau-Zener) tunnelling to excited bands
`
States are in effect resonances
65
3.3 Wannier-Stark resonances
[29]
• Energy states supported in ground band
2. Harmonic trap:
ε̂ = ε g ( p̂) − 12 kq2
• Pendulum levels:
�
• Below separatrix: qdp = 2π (n + 12 )h̄
�
• Above separatrix: qdp = 2πnh̄
• Density of state diverges at separatrix
p
Libration
q
66
3.4 Berry phase in wavepacket dynamics
• Next-order term in the effective Hamiltonian
Berry term
[19,21,22]
Wave function
ε̂ 1 = i (�un | ∂q ε 0 ∂ p |un � − q ↔ p)
− 2i (∂q �un | ( H − ε n )∂ p |un � − q ↔ p)
Wilkinson-Rammal term
• Bloch eigenvectors depend only on quasimomentum p
Wilkinson-Rammal term = 0
• ε 0 = ε g ( p) + U (q)
Effective potential
ε 1 = U � (q) · i �u| ∂ p |u� = F (q) · A( p)
• ε̂ 1 generates new dynamics. Berry potential A( p) = −i �u| ∂ p |u�
∂pε g
• Semiclassical equations of motion: ∂t q = v + h̄F · ∂ p A
−∂q U
∂t p = F − h̄∂q F · A
• Must remember: ε̂ acts on transformed space U † H (q̂, p̂)U = ε̂
q̂ → U q̂U † = q̂ + h̄A( p̂) p̂ → U p̂U † = p̂
3.4 Berry phase in wavepacket dynamics
• Next-order term in the effective Hamiltonian
[19,21,22]
Wave function
ε̂ 1 = i �un | ∂q ε 0 ∂ p |un � = F (q) · A( p)
• Semiclassical equations of motion:
∂t q = v + h̄F · ∂ p A ∂t p = F − h̄∂q F · A
• Transformed observables
q̂ → U q̂U † = q̂ + h̄A( p̂) p̂ → U p̂U † = p̂
Effective potential
�
∂t p = F + h̄ ( A · ∂) F − A · (∂F )) = F
�
�
∂t q = v + h̄ F · (∂A) − ( F · ∂) A = v( p) + h̄Ω( p) × ∂t p
• Berry curvature Ω = ∇ × A acts like a magnetic field in momentum space
68
3.4 Berry phase in wavepacket dynamics
• Semiclassical equations of motion:
∂t q = v( p) + h̄Ω( p) × ∂t p
[19,21,22]
Wave function
∂t p = F
• Berry curvature Ω = ∇ × A acts like a
magnetic field in momentum space
1. Generates a drift velocity — Hall effect.
Effective potential
2. Can be nonzero only in 2D and 3D
3. Curvature Ω is gauge invariant, while the
potential A depends on gauge, i.e. choice
of phase of Bloch eigenvectors |u�
4. Symmetries: time reversal Ω( p) ↔ −Ω(− p) • Must break space reflection or
time reversal symmetry for Ω �= 0
space reflection Ω( p) ↔ Ω(− p)
69
3.4 Berry phase of a Bloch oscillating wavepacket
[19,21,22]
• Effective Hamiltonian
Wave function
ε = ε g ( p) + U (q) + h̄F (q) A( p)
• Overall phase in a full period |ψ� → eiφZB |ψ�
φZB =
− 1h̄
�
�
qdp + Hdt = − F (q) A(q)dt
�
�
0
libration
�
= A( p)dp =
2π
0 A ( p ) dp Bloch osc.
• Topological Zak-Berry phase
Effective potential
p
phase space trajectories
Libration
q
3.5 Dynamics of a self-interacting wavepacket
Condensate wave function
• Gross-Pitaevskii equation
ih̄∂t χ =
h2 2
2m ∂ x χ + (Vp ( x ) + Vx ( x )) χ +
[27,37,38]
gN |χ|2 χ
• Slowly varying wavepacket: Peierls
substitution
ih̄∂t χ = (ε g ( h̄i ∂q )∂2x χ + Vx (q) + gN |χ|2 )χ
pa
• Tight-binding limit: ε g ( p) = 2u cos( h̄ )
Effective potential
• Discrete nonlinear Schrödinger equation
ih̄∂t χn = u(χn+1 + χn−1 ) + (Vx (qn ) + gN |χn |2 )χn
na
χ(qn )
71
3.5 Dynamics of a self-interacting wavepacket
1. No external potential
Condensate wave function
[27,37,38]
ih̄∂t χn = u(χn+1 + χn−1 ) + gN |χn |2 χn
• Wavepacket with position q̄ , momentum p̄ ,
width w, and chirp b
χn = s(
qn −q̄ i p̄(qn −q̄)+b(qn −q̄)2
w )e
Localized envelope
• Center evolves classically: ∂t q̄ = −u sin p̄ ∂t p̄ = 0
• Weak nonlinearity gN |s|2 � u
Wavepacket broadens in real space w → ∞
• Strong nonlinearity gN |s|2 � u
Wavepacket broadens in momentum
space b → ∞ : Self-trapping
72
3.5 Dynamics of a self-interacting wavepacket
2. Forced wavepacket
[27,37,38]
Condensate wave function
Harmonic potential
ih̄∂t χn = u(χn+1 + χn−1 ) + 2k q2n χn + gN |χn |2 χn
• Wavepacket with position q̄ , momentum p̄ ,
width w, and chirp b
χn = s(
qn −q̄ i p̄(qn −q̄)+b(qn −q̄)2
w )e
Localized envelope
ROLE OF THE MEAN FIELD IN BLOCH OSCILLATIONS…
(c)
(d)
• Weak nonlinearity
t (ms)
• Center evolves classically: ∂t q̄ = −u sin p̄
gN |s|2
|χ|2 0.2 (a)
∂t p̄ = −kqn
�u
Wavepacket broadens in momentum space b → ∞
• Strong nonlinearity:
Self-trapping
gN |s|2
� up (units of p
-1
+1 -1
z
+1 -1
+1
rec
-1
0.2
FIG. 2. Momentum distributions of the BEC as a function of
time for different atom numbers N. !a"–!d" correspond to N = 0,
2000, 25 000, and 60 000, respectively. The brightness represents
the atomic density on a linear gray scale.
the overall motion resembles that of a Bloch oscillation.
Comparing the Bloch oscillations for different atom num-
! t "# = 0.154
! t " 15 =0.803
(b)
t=0 ms
t=15 ms
! t "# = 0.130
! t " 15 =0.248
0.0
0.2
(c)
t=0 ms
t=15 ms
! t "# = 0.100
! t " 15 =0.115
0.0
+1
)
t=0 ms
t=15 ms
0.0
Atomic density
(b)
(a)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
PHYSICAL REVIEW A 78, 053607 !2008"
0.2
(d)
t=0 ms
t=15 ms
0.0
-1
0
! t "# = 0.088
! t " 15 =0.219
p+1 73
p z (units of p rec )
FIG. 3. !Color online" Probability distribution of z momentum at
Summary of part 3
• Adiabatic regime in a periodic environment — wavepacket extent much larger
than lattice unit. Average over fast oscillations of lattice potential.
• Zak transform in the linear limit separates the adiabatic degrees of freedom,
from the internal lattice degrees of freedom represented by the band index.
The phase space representation becomes matrix valued.
• Projection on a simple band leads to en effective Hamiltonian, whose leading
term, Peierls substitution, obtains by replacing the kinetic energy by the band
dispersion; for a strong lattice potential this is the tight-binding Hamiltonian.
• Under a constant external force, wavepackets undergo Bloch oscillations that
are strictly periodic in the leading approximation. Harmonic forcing leads to a
pendulum-like Hamiltonian, with libration and Bloch oscillations.
74
Summary of part 3 (continued)
• Energy states with external forcing, Wannier-Stark resonances, form a unifrom
ladder for constant force. The pendulum spectrum for harmonic forcing has
typical libration and rotation level spacing, that diverge toward the separatrix.
• The subleading effective Hamiltonian is a Berry-potential Hamiltonian, that acts
like a magnetic field in momentum space. It may lead to Hall effect-like drift
velocity in two or more dimensions. In one dimension it generated a topological
Zak-Berry dynamical and quantization phase.
• Self interactions in the tight-binding regime lead to the discrete nonlinear
Schrödinger equation. The interactions inhibit broadening in real/momentum
space; strong nonlinearity seeds dark solitons and other defects.
75
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