1. No category

Institut für Laser-Physik
Universität Hamburg
Andreas Hemmerich
Bose-Einstein Condensation
List of Topics:
Bose-Einstein Statistics and Bose-Einstein Condensation
Cold Collisions, S-Wave Approximation
Evaporative Cooling in Magnetic Traps
Experimental Realization and Observation of Bose-Einstein Condensation
Order Parameter, Gross-Pitaevski-Equation
Thomas-Fermi Limit, Excitations, Superfluidity
Light Scattering in Bose-Einstein Condensates
Hydrodynamic Gross-Pitaevski-Equation, Expansion of Bose-Einstein Condensates, Vortices
Phase and Interference of Bose-Einstein Condensates
Bose-Condensates in Periodic Potentials
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Universität Hamburg, Bose-Einstein Condensation, SoSe 2015
Andreas Hemmerich 2015 ©
Textbooks & Reviews
Bose-Einstein Condensation in Dilute Gases
C. J. Pethick and H. Smith, Cambridge University Press (2002)
Bose-Einstein Condensation
L. P. Pitaevskii and S. Stringari, Oxford University Press (2003)
Ultracold Atomic Gases in Optical Lattices
M. Lewenstein, et al., Oxford University Press (2012)
Theory of Bose-Einstein Condensation in Trapped Gases
F. Dalfovo, S. Giorgini, L. P. Pitaevskii, S. Stringari, Reviews of Modern Physics, Vol. 71, (1999)
Making, Probing and Understanding Bose-Einstein Condensates
W. Ketterle, D.S. Durfee, and D.M. Stamper-Kurn, http://arxiv.org/abs/cond-mat/9904034
Bose-Einstein Condensation of Trapped Atomic Gases
Ph.W. Courteille , V.S. Bagnato , V.I. Yukalov, http://arxiv.org/abs/cond-mat/0109421
Ultracold Atomic Gases in Optical Lattices
M. Lewenstein, et al., Advances in Physics 56 (2), 243 (2007).
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Andreas Hemmerich 2015 ©
Bose-Einstein-Condensation:
cooling @ high density
three-body collisions
yield molecule formation
cooling @ low density
3
2
kBT ≈
1 mV2
2
⇒
Thermal de Broglie Wavelength:
2π
λth = 2π/mV =
3 m kBT
Luis Victor de Broglie 1923
Thermal Ensemble:
Coherence Length = Thermal de Broglie Wavelength
Critical Temperature:
de Broglie Wavelength λth ≈ mean particle separation d ≈ 1/ρ1/3
⇒
T =
Bose-Einstein-Condensate
3
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2 2 2/3
4π  ρ
3m kB
Satyendra Nath Bose, Albert Einstein
S. N. Bose, Z. Phys. 26, 178 (1924)
A. Einstein, Sitzber. Kgl. Akad. Wiss. 3 (1924/25)
Andreas Hemmerich 2015 ©
Quantum Statistics Revisited:
ni = occupation of i-th mode
use Fock-states to describe system of bosons: |n0 ,n1, ....>
εi
E = n0ε0 + n1ε1 + ... energy of |n0 ,n1, ....>
N = n0 + n1 + ... particle number of
ε0
|n0 ,n1, ....>
We do not know the state of the system. Description via probability distribution:
P [n] , n ≡ [n0,n1, ...]
Each possible probability distribution P is characterized by its entropy S(P) which measures the degree of ignorance connected with P.
The entropy S can be evaluated for arbitrary P (Shannon 1948) as :
S(P) = – kB
∑
Shannon, C.E. (1948)
“A Mathematical Theory of Communication”
Bell Syst. Tech. J., 27, 379-423, 623-656
P [n] ln( P [n] )
n
How to find P[n] :
maximize S(P) under boundary conditions 0 = f0(P) = f1(P) = ....
0
Example of complete ignorance:
solution:
4
!
=
∂
∂P[n]
S(P) +
λ0 f0(P)
+
λ1 f1(P)
+ ....
f0(P) ≡ 1 –
∑
P [n]
n
λn = Lagrange Parameters
only a single boundary condition f0
P [n]
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= constant
Andreas Hemmerich 2015 ©
Canonical Ensemble:
∑
0 = f1(P) ≡ E –
system is subject to the constraint:
E [n] P [n]
E =
n
∑
E [n] P [n]
= mean energy
n
find P by maximizing entropy with the constraints f0(P), f1(P): →
Boltzmann-distribution
P [n] =
C e
– β E [n]
β =
1
kBT
, C = Normalization
Example:
thermal photons in a box V with mode separation
mean occupation of i-th mode:
Ω ≡
ni =
πc
V1/3
, ωi = i Ω , εi = i Ω
∑
n0 ,n1, ....
→
e
– β E [n]
=
Π
e
–
βωi ni
i
ni P[n0,n1, ... ]
1
=
e
βωi
–
1
(Plancks black-body radiation law)
mean particle number N is not conserved but rather a function of temperature T :
N =
∑
i=0
ni
=
V
8π γ3/2(1)
Λ3
~
T3
Λ ≡ 2π c β = thermal de Broglie-wavelength
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→
N k BT
~
T4
(Stefan-Boltzmann-law)
Andreas Hemmerich 2015 ©
Grand Canonical Ensemble:
system is subject to two constraints:
known mean energy:
∑
0 = f1(P) ≡ E –
E [n] P [n]
n
known mean particle number:
∑
0 = f2(P) ≡ N –
N [n] P [n]
n
P [n] =
find P by maximizing entropy with the constraints f0(P), f1(P), f2(P):
C e
– β E [n]
e
– λ N [n]
,
C = Normalization
physical significance of Lagrange parameters β and λ:
S(P) = – kB
∑
P [n] ln( P [n] )
= – kB
n
⇒
E
=
∑
P [n] (–β E [n] – λ N [n] )
– kB ln(C)
=
kB β E
+ kB λ N
–
kB ln(C)
n
λ
S
–
N
kB β
β
+
kBT ln(C)
!
=
TS + µN
+
⇒
kBT ln(C)
β =
1
kBT
λ = –
µ
k BT
T = Temperature, µ = Chemical Potential
mean occupation of i-th mode:
ni =
∑
n0 ,n1, ....
Fermions:
Bosons:
6
ni = 0,1
ni =
ni = 0,1,2, ....
ni =
Universität Hamburg, Bose-Einstein Condensation, SoSe 2015
= Ai ∂ ln
∂Ai
ni P[n0,n1, ... ]
∑
ni
n
Ai i
Ai ≡ exp( –β εi– λ)
1
Ai–1 +
1
1
Ai–1 – 1
Andreas Hemmerich 2015 ©
The ideal Bose-Gas
Bose-Einstein-distribution:
εi
ε0
ni =
gi
ξ = exp(β µ) , β = 1/kBT
-1
ξ exp(β εi) – 1
εi , i = 0,1,2....
single particle energies, choose ε0 = 0
gi , i = 0,1,2....
degrees of degeneracy
µ = chemical potential ∈ [-∞,0], ξ = fugacity ∈ [0, 1]
T = temperature
Normalizing to N Particles:
N =
∞
∑
i=0
ni = n0 + G(ξ)
Bose-Condensation:
G(ξ) grows monotonously
and is bounded on [0,1]
ξ = 1, N > Nc ≡ G(1)
g3
g1
µ ε0
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ε1
g2
ε2
energy
ε3
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Example1: N Bosons in a (large) Box with Volume V
εi
∞
G(ξ) =
gi
∑
i=1
=
V
Λ3
≈
ξ-1 exp(εi/kBT) – 1
∞
2
π1/2
Density of States:
dx
0
g(ε) =
∞
dε
0
x1/2
g(ε)
ξ exp(ε/kBT) – 1
=
ξ-1 exp(x) – 1
ε0
-1
V g (ξ)
3/2
Λ3
(2m)3/2 ε1/2 V
43 π2
Thermal deBroglie Wavelength:
2 1/2
Λ ≡ ( 2π ) ,
mkBT
k ≡
2π /Λ
⇒
2k2
= π kB T
2m
The Function g3/2(ξ):
g3/2(ξ)
≡
2
π1/2
∞
0
dx
x1/2
ξ-1 exp(x) – 1
∞
=
∑
n=1
g3/2(ξ) is monotonous on [0,1], g3/2(1) ≈ 2.61
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ξn
2.61
g3/2(ξ)
n3/2
1
ξ
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Bose-Einstein-Condensation in a Box with Volume V:
V
N = n0 + 3 g3/2(ξ) ,
Λ
N > Nc ≡ g3/2(1) V3
Λ
Normalisation for N Particles:
Critical Particle Number Nc :
Critical Temperature Tc , Λc:
k BTc =
n0 = 1/(ξ-1–1) = ξ/(1 – ξ)
ρ
2/3
2π2
(
)
m g3/2(1)
or
(ε0, g0 = 0)
ρΛ3 > g3/2(1) ≈ 2.61
ρ ≡ N/V Particle-Density, ρΛ3 Phase-Space-Density
,
ρΛc3 = g3/2(1)
( Exp: Λ-Point 4He: Tc = 2.177 K, 87Rb at ρ=1014 cm-3 ⇒ Tc = 400 nK )
Bose-Kondensat
Chemical Potential µ(T,V,N):
ξ für N →
1
ξ
1 g (ξ)
1 = 1
+
ρΛ3 3/2
N 1–ξ
⇒
ξ(T,V,N) =
1
ξ für N = 105
⇒ ξ = ξ(T,V,N)
Thermodynamic Limes: V, N → ∞ such that ρΛ3 constant:
g3/2-1 (ρΛ3)
∞
0
g3/2(1)
0
1
2
3
4
5
ρΛ3
if ρΛ3 < g3/2(1)
if ρΛ3 > g3/2(1)
i.e. in Bose-Condensed Regime
µ(T,V,N) = kBT ln( ξ(T,V,N) )
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Occupation of Ground State:
n0
= 1 –
N
1 g (ξ)
ρΛ3 3/2
and
g3/2-1 (ρΛ3)
ξ(T,V,N) =
1
if ρΛ3 < g3/2(1)
ρΛ3 > g3/2(1)
if
yield
n0
N
if
ρΛ3 < g3/2(1)
T 3/2
= 1 –
if
Tc
ρΛ3 > g3/2(1)
0
=
1–
g3/2(1)
ρΛ3
Bose-Condensate
n0
N
1
0
10 Universität Hamburg, Bose-Einstein Condensation, SoSe 2015
0
1
2
T
Tc
Andreas Hemmerich 2015 ©
Internal Energy:
∞
U =
∑
i=0
g5/2(ξ)
∞
n i εi
≈
dε
0
∞
≡ 4
3π1/2
dx
0
g(ε) ε
ξ-1exp(ε/kBT) – 1
V g (ξ)
3
kBT
5/2
2
Λ3
=
∞
x3/2
-1
ξ exp(x) – 1
=
∑
n=1
ξn
1.2
0.8
0.4
0
0
n5/2
monotonous on [0,1], g5/2(1) ≈ 1.34
g5/2(ξ)
1
ξ
Specific Heat:
Cv
N
≡
1 ∂U
N ∂T
V,N
1 g ( ξ(T,V,N) )
15
kB
5/2
4
ρΛ3
=
d g ( ξ(T,V,N) )
=
3/2
dξ
d
dξ
d g ( ξ(T,V,N) )
=
5/2
dξ
d
dξ
∂ξ
∂T
V,N
=
∞
∑
n=1
∞
∑
n=1
ξn
n3/2
ξn
n5/2
d
d
(ρΛ3) =
g3/2-1(ρΛ3)
dT
dy
11 Universität Hamburg, Bose-Einstein Condensation, SoSe 2015
∞
=
ξ-1
=
ξ-1
∑
n=1
∞
∑
n=1
+
ξn
n1/2
ξn
n3/2
d
(ρΛ3)
dT
d g ( ξ)
3/2
dξ
3 kBT
2 ρΛ3
d g ( ξ(T,V,N) )
5/2
dξ
=
ξ-1 g1/2( ξ(T,V,N) )
=
ξ-1 g3/2( ξ(T,V,N) )
∂ξ
∂T
V,N
ξ ρΛ3
= – 3
2 T g1/2(ξ)
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Cv
⇒
1 g (ξ)
15
kB
5/2
4
ρΛ3
=
N
ξ(T,V,N)
–
9 k g3/2(ξ)
4 B g1/2(ξ)
g3/2-1 (ρΛ3)
falls T > Tc
1
falls T < Tc
=
ρΛ3
N 2π2 3/2
)
= (
V mkBT
Limiting Cases:
ρΛ3 = g3/2(ξ)
T > Tc ⇒
1 = ξ, g1/2( 1) = ∞
T < Tc ⇒
T→ ∞
⇒ ρΛ3 → 0 ⇒
Cv
⇒
kBN
⇒
Cv
kBN
ξ = g3/2-1 (ρΛ3) → 0
=
15 g5/2(ξ)
4 g3/2(ξ)
–
=
15 g5/2(1)
4
ρΛ3
15 g5/2(1)
=
4 g3/2(1)
⇒
Cv
kBN
=
9 g3/2(ξ)
4 g1/2(ξ)
15
– 9
4
4
=
T 3/2
Tc
3
2
Cv
kBN
3/2
0
0
1
T
Tc
2
cf. classical ideal Gas:
Cv
kBN
12 Universität Hamburg, Bose-Einstein Condensation, SoSe 2015
= 3/2
Λ-Point of 4He
Andreas Hemmerich 2015 ©
Example2: N Bosons in a 3D harmonic Potential
States: |ix,iy,izi , ix,iy,iz = 0,1,2,....
g(ε) =
Density of States:
∞
G(ξ) =
dε
0
g3(ξ) ≡
1
2
∞
0
ε2
2 (Ω)3
kB T
=
-1
ξ exp(ε/kBT) – 1
ξ-1 exp(x) – 1
=
N = n0 +
Bose-Einstein-Condensation:
N > Nc ≡
Critical Temperature TC and Density ρC:
ξn
∑
n=1
Normalising to N Particles:
3
Ω
∞
x2
Ω
k B Tc ≡
0
dx
x2
ξ-1 exp(x) – 1
3
g3(ξ) ,
Ω
kB T
∞
monotonous on [0,1], g3(1) ≈ 1.202
n3
k BT
1
2
3
Θ > g3(1) , Θ
≡
N
Ω
3
k BT
Θ Phase-Space Density
g3(1)
(N /g3(1))1/3 Ω
(cf. Boltzmann gas:
13 Universität Hamburg, Bose-Einstein Condensation, SoSe 2015
Ω
ε0
g(ε)
dx
ε1
1
(i+1)(i+2)
2
gi =
mΩ2(x2+y2+z2)
ε2
Single Particle Spectrum: εi = i Ω , i = (ix+iy+iz) = 0,1,...
Degeneracies:
1
2
Ω
> k BT
> k T)
B
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Chemical Potential µ(T,Ω,N):
ξ
1
1 =
+
N 1–ξ
1
g (ξ)
Θ 3
Thermodynamic Limes: N → ∞ with Θ
⇒
ξ(T,Ω,N) =
n0
N
ξ = ξ (T,Ω,N)
⇒
constant:
g3-1 (Θ)
falls Θ
< g3(1)
1
falls Θ
> g3(1)
0
=
1–
g3(1)
Θ
T 3
= 1 –
Tc
Bose-Einstein-Condensate of Rubidium atoms
µ(T,Ω,N) = kBT ln( ξ(T,Ω,N) )
falls Θ
< g3(1)
falls Θ
> g3(1)
1
note small deviations due to
collisional interactions
J. Ensher et al. Phys. Rev. Lett. 77, 4984 (1996)
0
0
14 Universität Hamburg, Bose-Einstein Condensation, SoSe 2015
T/Tc
1.8
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Interaction via two-body collisions
U(|r1 – r2|)
R0
|r1 – r2|
Λ >> R0 ⇒ S-Wave Scattering :
Elastic scattering cross section:
eikz → eikz +
Spherical Wave:
phase δ0(k), amplitude sin(δ0(k))
4π sin(δ (k)) e iδ0(k)
0
k
σ ≈ 4πA0(k)2, A0(k)
≡ –
eikr
r
sin(δ0(k))
= Scattering Length
k
Effect of potential is described by a single number "A0" : details of potential are not resolved in S-wave scattering
→ potential U can be replaced by as a contact potential:
 2A 0
U( r1 - r2 ) = g δ ( r1 - r2 ) , g = 4πm
Examples:
No bound states: A0 > 0
Bound states: depending on exact form of potential A0 > 0 or A0 < 0
15 Universität Hamburg, Bose-Einstein Condensation, SoSe 2015
H
7Li
A0(Å) 0.72 –14
23 Na 87 Rb
49
55
Andreas Hemmerich 2015 ©
Physical significance of the sign of the scattering length:
Negative Scattering Length:
Potential provides scattering resonance near dissociation limit
This yields large values of the wave function near
r1 - r2 = 0
Large pobabillity for small distances of collision partners → attractive character of interaction
U(|r1 – r2|)
|r1 – r2|
Positive Scattering Length:
Potential provides bound state near dissociation limit
This yields small values of the wave function near |r1 – r2| = 0
Small distances of collision partners are avoided → repulsive character of interaction
U(|r1 – r2|)
|r1 – r2|
16 Universität Hamburg, Bose-Einstein Condensation, SoSe 2015
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Potential U acts as a contact potential:
 2A 0
U( r1 - r2 ) = g δ ( r1 - r2 ) , g = 4πm
collision energy / particle:
1
2
Ecol =
dr dr’ U(r-r’) ρ(r) ρ(r’)
=
N gρ
2
,
ρ =
N =
1
N
dr ρ2(r)
mean density
particle number
dr ρ(r)
for small ρ the collision energy Ecol is determined by two-body collisions (dashed lines)
for larger ρ three-body collisions may yield molecule formation (solid lines)
A + A + A → 2A + A , τ-1 = α ρ2
E col
N
87Rb
7
Li
5x10-39
cm6 s-1
H↑: α =
α = 4x10-30 cm6 s-1
[Kagan et al., JETP 54, 590(1981)]
87Rb↑:
Example:
ρ=
1014
cm-3
H
ρ
[Moerdijk et al., Phys.Rev.A(1995)]
⇒
τ(87Rb)
17 Universität Hamburg, Bose-Einstein Condensation, SoSe 2015
= 25 s
1gρ
2
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Magnetic Traps
Interaction W = – µB
Magnetic moments µ parallel to B are dragged towards local maxima of |B|: –> high field seakers
Magnetic moments µ antiparallel to B are dragged towards local minima of |B|: –> low field seakers
Local maxima of |B| are not available because ∇B = 0
–> only low field seakers can be trapped.
Magnetic moment follows the field orientation adiabatically, if
v
∂
ω
∂r B
<< ωB2
m=1
m=0
m=-1
Condition cannot be realized in the vicinity of a field zero: Loss of particles by spin flip transitions
(Majorana-transitions)
m=1
m=0
m=-1
1. Quadrupole Geometry
→ linear potential, small trap volume (i.e., high compression), however, B has a zero.
Atoms have to be kept away from the field zero, e.g. by a blue detuned laser beam (optical plug).
2. “TOP”-Trap (quadrupole + rotating offset field)
→
quadratic potential, large trap volume, B has no zero.
3. Ioffe Geometry (linear quadrupole + static offset)
→
quadratic potential, large trap volume, B has no zero.
18 Universität Hamburg, Bose-Einstein Condensation, SoSe 2015
Andreas Hemmerich 2015 ©
Cooling by Evaporation:
H. Hess, Phys. Rev. B, 34, 3476 (1986).
W. Ketterle and N. van Druten, Adv. Phys. 37, 181 (1996).
Evaporative Cooling in Magnetic Traps
Energy
Potential
• Expelling high energy atoms (→ loss of particles)
• Thermalization by elastic two-body collisions
⇒ cooling rate limited by rate for thermalization:
typical elastic collision rate @ ρ = 1010 cm-3 : 1 s-1
µ
Forced Evaporation by RF-Transitions
Energy
m=2
1
νRF
0
–1
Run away effect:
temperature decrease
→ density increase
→ increase of thermalization rate
→ faster cooling
–2
B
19 Universität Hamburg, Bose-Einstein Condensation, SoSe 2015
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Making and Observing a Bose-Einstein Condensate
M. Anderson et al., Science 269,198 (1995)
K. Davis et al., Phys. Rev. Lett. 75,3969 (1995).
C. Bradley, et al.,Phys. Rev. Lett. 75, 1687 (1995)
trap-potential
→ Laser-Cooling
→ Magneto-Optic Trap
uncondensed atoms
→ Magnetic Trap
→ Forced Evaporative Cooling
→ Free Expansion
→ Absorption Image
Lens
CCD Camera
ballistic expansion of a Bose-condensate
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Theoretical description of Bose-Einstein Condensate
Many-Body-Hamiltonian
H
+
2 ∇ 2
dr Ψ (r,t) –
+ Vtrap(r) + 1 dr' Ψ +(r' ,t) U(r' – r)Ψ(r' ,t) Ψ (r,t)
2m
2
=
Vtrap(r ) = trap potential
= two-body interaction potential
U(r' – r)
Ψ (r,t)
= field operator
Ψ (r,t)
=
∑ Ψ ν (r)
+
Ψ(r ,t) , Ψ (r ',t)
a ν (t)
= δ (r' – r)
Ψ ν (r) =
basis-set of functions (e.g., single particle wave functions)
a ν (t) =
corresponding anihilation operators
Heisenberg Equation
use commutation relations
i  ∂ Ψ(r,t) =
∂t
Ψ(r ,t) ,
21 Universität Hamburg, Bose-Einstein Condensation, SoSe 2015
H
=
–
2 ∇ 2
+ Vtrap(r) +
2m
dr' Ψ +(r' ,t) U(r' – r)Ψ(r' ,t) Ψ (r,t)
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Calculation of Heisenberg Equation
22 Universität Hamburg, Bose-Einstein Condensation, SoSe 2015
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Gross-Pitaevskii Equation
Start with Heisenberg equation for many-body system:
i  ∂ Ψ(r,t)
∂t
U(r' – r)
–
=
2 ∇ 2
+ Vtrap(r) +
2m
dr' Ψ +(r' ,t) U(r' – r)Ψ(r' ,t) Ψ (r,t)
= two-body interaction potential
+
Ψ(r ,t) , Ψ(r ',t)
= δ (r' – r)
Apply Bogoliubov approximation:
Ψ(r,t)
=
Φ(r,t) +
Φ (r,t)
=
Ψ (r,t)
Ψ '(r,t)
n0(r,t) = |Φ(r,t)|2
Intoduce S-Wave-limit:
Obtain GP-Equation:
U(r' – r) = g δ (r' – r) , g =
∂
i
Φ(r,t)
∂t
23 Universität Hamburg, Bose-Einstein Condensation, SoSe 2015
=
4 π  2 A0
m
2 ∇ 2
–
+ Vtrap(r) + g Φ(r,t)
2m
2
Φ(r,t)
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Structure of Gross-Pitaevskii Equation
∂
i  ∂t Φ(r,t)
=
2 ∇ 2
–
+ Vtrap(r) + g Φ(r,t)
2m
2
Φ(r,t)
GP-Equation = Schrödinger-Equation + χ(3)-term
g=0
→
GP-Equation = Schrödinger-Equation
→
matter waves experience dispersion in vacuum
→
matter waves self-interact at large densities giving rise to inherent
four-wave mixing similar as photons in a χ(3)-medium at large intensity
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Time-Independent GP Equation
Ansatz:
Φ(r,t) = Φ(r) e
µ
-i  t
Time-independent GP-Equation:
µ Φ(r) =
µN
Ekin
=
Φ(r) –
2 ∇
2m
–
2 ∇ 2
+ Vtrap(r) + g Φ(r)
2m
Epot
Φ(r)
Ekin + Epot + 2Ecol
=
2
Φ(r)
2
=
Φ(r) Vtrap(r) Φ(r)
Ecol = 1 Φ(r) g ρ(r) Φ(r)
2
Recall Thermodynamics:
T = 0 → total energy = chemical potential × particle number:
U ≡ µN
is the internal energy
25 Universität Hamburg, Bose-Einstein Condensation, SoSe 2015
µ is the chemical potential
thermodynamical relations: equation of state etc.
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EXAMPLE
inside box:
Consider box potential for large volume V:
Φ (r)
determine internal energy:
=
N
V
and
Vtrap(r) = 0
µ N = Ekin + Epot + 2Ecol
2 ∇
2m
2
=
0
Φ (r) Vtrap(r) Φ (r)
=
0
Ecol = 1 Φ (r) g ρ(r) Φ (r)
2
=
g N2
2V
Ekin
=
Φ (r) –
Epot
=
Φ (r)
derive thermodynamic quantities with respect to variables N, V (Entropy S = 0):
chemical potential
µ
=
equation of state
P
=
sound velocity
c
=
gN
V
-
dU
dV
1
=
=
=
κs ρmass
4 π  2 A0 N
mV
g N2
2V2
-
, U = Ekin + Epot + Ecol
V2 dP
mN dV
1 dV
compressibility: κs ≡ − V dP
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S
=
gN
mV
mass density:
ρmass ≡
mN
V
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Healing Length ξ : minimal length scale for change of order parameter
condition for locally stable quantum fluid:
collisional pressure εcol(r) must exceed quantum pressure εkin(r)
assume exponentially changing order parameter
εcol(r) ≡ g ρ(r)2
Φ(r) ≈ Φ0 exp(– r )
ξ
Φ0 exp(– r )
ξ
ξ
2
εkin(r) ≡ – Φ* (r)
2
2 ∇
Φ(r) ≈  2 ρ(r)
2m
2m ξ
εcol > εkin
ξ >
ξ0 ≡
1
8π ρ0 A0
typical situation in experiments:
ρ0 ≈ 1014 cm-3, A0 ≈ 10-8 m
→
ξ0 ≈ 200 nm
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significance of collisional interactions
Define Interaction parameter:
EXAMPLE:
η≡
E col
E kin
measures significance of collisional interactions
as compared to quantum pressure
consider harmonic oscillator potential and small interaction
L0 ≡

mω0
ω0 ≡ ωx ωy ωz
η << 1
1/ e radius of single particle ground state
1
3
mean vibrational frequency
L0
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interaction small η << 1 :
→ GP-Equation ≈ Schrödinger-Equation
condensate wave function
Ekin
≈
wave function of single particle ground state
= N  (ωx + ωy + ωz)
2
η ≡
E col
E kin
=
1 Φ (r) g ρ(r) Φ (r)
2
Φ (r) –
2 ∇
2m
2
≈
N
A0
L0
Φ (r)
condensate
thermal cloud
r
L0
Lth
Condensate:
Bose-Condensate corresponds to macroscopic population of single particle ground state
Thermal Cloud:
In a harmonic potential the spatial distribution of a thermal sample is described by a Gaussian with Radius Lth :
Lth =
2 kBT L = 1/ e
0
ω
radius of thermal wavefunction
Lth[T=Tc] = 2 N1/6 L0
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Typical parameter set in experiments:
Thomas-Fermi Limit: η >> 1
N ≈ 106
L0 ≈ 10-5 m
A0 ≈ 10-8 m
η ≈ 103
neglect kinetic energy term in GP-equation
µ Φ(r) =
ρ(r) =
Vtrap(r) + g Φ(r)
µ – Vtrap(r)
g
2
if
Φ(r)
µ > Vtrap(r)
0 otherwise
Energy
Vtrap(r)
µ
r
ρ(r)
r
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Thomas-Fermi-Limit for Harmonic Potential
Vtrap(r) = m ωx2 x2 + ωy2 y2 + ωz2 z2 ,
2
ω0 ≡ ωx ωy ωz
1
3
, L0 ≡

mω0
using
N =
d3 r ρ(r) , µ = Vtrap ( r = Ri , i = x,y,z)
ρ(r) =
µ – Vtrap(r)
g
if
Energy
µ > Vtrap(r)
0 otherwise
Vtrap(r) = m ωx2 x2 + ωy2 y2 + ωz2 z2
2
µ
r
yields
ρ(r)
Ri
ω
= ω0 R0 ,
i
Ecol =
µ
R0 ≡ L 0 15 N A0
L0
1
5
≈
7 L0
R
r
2 µN
7
A
=  ω0 15 N 0
2
L0
2
5
≈
25  ω0
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Harmonic Potential: Thomas-Fermi-Limit versus Exact Solution
0
0
0
radius / L0
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Elementary Excitations
i  ∂ Ψ(r,t) =
∂t
Heisenberg equation for many-body system:
Bogoliubov approximation:
–
2 ∇ 2
+ Vtrap(r) + g Ψ +(r ,t) Ψ(r ,t)
2m
Ψ(r,t) = (Φ(r) + Ψ´(r,t)) e-i
Ψ´(r,t) =
∑
Ψ (r,t)
µ
 t
|
uν(r) aν(t) + vν*(r) aν+(t)
ω
0
uν(r), vν(r) =
aν+(t) =
quasi-particle wave functions
quasi-particle creation operator
[aν(t), aµ+(t) ] = δνµ , aν(t) = aν(0) e-iωνt
Neglect contributions non-linear with respect to u,v :
µ Φ(r) =
2 ∇ 2
–
+ Vtrap(r) + g Φ(r)
2m
2
(µ + ω) u(r) =
–
2∇
2m
(µ - ω) v(r) =
–
2 ∇2
+ Vtrap(r) + 2 g Φ (r)
2m
33 Universität Hamburg, Bose-Einstein Condensation, SoSe 2015
+ Vtrap(r) + 2 g Φ (r)
2
u(r) + g Φ(r)2v(r)
2
v(r) + g Φ(r)2u(r)
2
Φ(r)
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Calculation of Bogoliubov equations
neglect all terms higher order than linear in u,v
group all terms oscillating with same frequency
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Examples of Elementary Excitations:
Uniform Gas, i.e.
Vtrap = 0 :
µ
ρ = Φ(r)2 = g , u(r) = u0 eiqr , v(r) = v0 e-iqr
ground state density is constant, excitations are plane waves
dispersion relation takes Bogoliubov form:
2 q2 2q2 +
2gρ
,
2m 2m
ω =
ω
ω
=
q
c2 +
q
2m
2
q2
ω = 
2m
ω= cq,
c ≡
gρ
m
sound velocity
q
mc

phonons versus particle excitation:
<
mc

q
>
mc

ω <
Ecol
N
ω >
Ecol
N
q
λ
>
8π ξ
phonons
35 Universität Hamburg, Bose-Einstein Condensation, SoSe 2015
2m c

λ
<
Ecol ≡
N
ξ ≡
collision energy per particle
1
8π ρ0 A0
healing length
8π ξ
particles
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Calculation of Bogoliubov dispersion relation
Ansatz:
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→ superfluidity
for low q, system cannot be excited by classical particle
consider classical particle with mass M>>m and momentum q0 and dispersion relation:
energy momentum conservation:
Eclass(q) ≡
2 q2
2M
Eclass(q0) − Eclass(q0 - q) = ω(q)
µ2q2 + 4qc2 = (q - 2q0)2
µ ≡
M
m
,
qc ≡ Mc

q02 ≥ (1 - µ-2 ) qc2
v02 ≥ (1 - µ-2 ) c2
ω
critical velocity
Eclass(q)
classical
particle
mass M > m
qc
q
Eclass(qc + q) − Eclass(qc)
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Calculation of critical momentum
quadratic equation for variable q
dicriminante > 0 :
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Bragg Spectroscopy
momentum space:
resonance condition:
q = 2kx = 2k sin(θ/2)
δω =
phonon like regime:
λ
>
2π ξ
δω
≈
q
gρ
m
particle like regime:
λ
<
2π ξ
0
39 Universität Hamburg, Bose-Einstein Condensation, SoSe 2015
δω
≈
2q2
+ gρ
2m
2 sin(θ/2)
2 sin(θ/2)
δω
2q2 2 q2 +
2g ρ
2m 2m
 2kx
δω
 2kx
0
Andreas Hemmerich 2015 ©
Excitation spectrum in particle-like regime:
box-like trap potential:
inhomogeneous trap:
Δω ≈ g Δρ ≈
excitation
probability
g <ρ>
excitation
probability
frequency
δω
δω
=
≈
2 q2
2m
δω
frequency
+ gρ
2 q2
2m
δω
δω
=
≈
δω
2q2
+ g <ρ>
2m
2q2
2m
J. Stenger et al, PRL 82, 4569 (1999).
Observation for
trapped BEC
BEC after 3 ms
expansion
momentum distribution
of single-particle
ground state
thermal cloud
at T= 1µK
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Preparing selected momenta
position space:
momentum space:
P2
1. pulse
P1
k
P2
2. pulse
P3
P1
2k
P2
frame moving at k:
P1
–k
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P3
+k
Andreas Hemmerich 2015 ©
Four Wave Mixing:
P3
P1
–k
+k
–k
P4
P2 and P1 form a density-grating,
which Bragg-scatters P3 into P4
0 = P2 – P1 + P3 – P4
P3
P1
–k
+k
–k
P4
P2 and P3 form a density-grating,
which Bragg-scatters P1 into P4
0 = P2 – P3 + P1 – P4
P3
P1
P4
Laboratory Frame
Experiment: Deng et al., Nature 398, 218 (1999)
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Rayleigh Scattering
position space:
k
laser beam
MIT Data (Ketterle et al.)
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Self-induced Bragg- Scattering
self-organization process: atoms arrange in optical lattice with every second site occupied
position space:
momentum space:
second order
k
first order
k
k
k
laser beam
laser beam
4 k
2 k
k
MIT Data (Ketterle et al.)
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Short Pulse with High Intensity
q
position space:
δω =
2q2
m
-q
k
laser beam
Photons scattered into laser beam are recoil shifted. Stimulated process only efficient
if band width of laser beam larger than recoil shift → pulse duration < recoil time
momentum space:
2 k
first order
k
k
laser beam
-2k
-k
k
2 k
-k
Schneble et al., Science 300, 475 (2003)
-2k
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Hydrodynamic GP-Equation:
Define:
Ψ(r,t) ≡
∂
ρ + ∇(ρv) = 0
∂t
ρ(r,t) e i S(r,t)
 ∇ S(r,t)
v(r,t) ≡ m

j (r,t) ≡ ρ(r,t) v(r,t) = m
1
2i
(Ψ∗ ∇
Ψ -
Ψ ∇ Ψ∗)
m
2
∂
v + ∇ Vtrap + gρ -  1 ∇2 ρ +
∂t
2m ρ
∂
ρ + ∇(ρv)
∂t
η >> 1 : Collisionless Hydrodynamic Limit (CHL)
(no collisions with thermal atoms)
m
∂
v + ∇ Vtrap + gρ +
∂t
1
2
1
2
mv 2
= 0
= 0
mv 2
= 0
T = 0 case of Landaus theory of superfluidity
v = 0 yields Thomas-Fermi Limit
∇ Vtrap + gρ
⇒
Vtrap + gρ
= 0
= const ≡ µ
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Equivalence of conventional and hydrodynamic GP-equation
evaluate equation for
real and imaginary part
insert definition of velocity
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Collective Excitations ω << µ
Linearization: ρ = ρ0 + δρ , ρ0 ≡ Th-F density
∂2
δρ = ∇ c2(r) ∇ δρ
∂t2
Solutions for Vtrap = 0 :
m c2(r) ≡ µ - Vtrap(r)
,
∝ Sin(ω t - q r + φ )
local sound velocity
ω=cq , c=
µ
m =
g ρ0
m
Bogoliubov dispersion relation for Th-F Limit
Solutions for Vtrap(r) =
1m
2
ω2 r2 :
ω> (nr , l) =
2 nr 2 + 2 nr l + 3nr + l ω
Stringari et.al, PRL 77, 2360 (1996)
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Excitations in a 3D Spherical Harmonic Oscillator Potential:
η << 1 : ω< (nr , l) = ( 2 nr + l ) ω = (nx + ny + nz) ω
η >> 1 : ω> (nr , l) =
2 nr 2 + 2 nr l + 3nr + l ω
surface excitations, nr = 0 :
n=1 :
nr = 0, l = 1, m =
±1,0
n=2 :
nr = 0, l = 2, m =
nr = 1, l = 0, m =
±2,±1,0
0
n=3 :
nr = 0, l = 3, m =
nr = 1, l = 1, m =
±3,±2,±1,0
±1,0
n=4 :
nr = 0, l = 4, m =
nr = 1, l = 2, m =
nr = 2, l = 0, m =
±4,±3,±2,±1,0
±2,±1,0
0
ω<(nr , l) =
ω>(nr , l) =
l
l
ω
ω
compressional modes, nr > 0, l = 0 :
Vibrationen nach Hauptquantenzahl n angeordnet:
n=1 dipol mode, nr = 0, l = 1 :
1
+
m = 0, nz =1:
ω> = ω< = ω
=
n=2
n=2
5
quadrupol mode, nr = 0, l = 2 :
ω< =
2 ω
ω> =
2 ω
ω< =
ω> =
2 ω
5 ω
n=1
n=0
1
monopole mode, nr = 1, l = 0:
+
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=
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Variation of harmonic trap potential (Th-F Limit)
Vtrap(r,t) =
d α + α 2 + ωi2 - 2 g a
i
m i
dt i
m ω (t)2 x2 + ω (t)2 y2 + ω (t)2 z2
x
y
z
2
= 0
(1)
d a - (α + α + α + α ) a = 0
x
y
z
i
i
dt i
(2)
⇒
v(r,t) =
1
2
∇ αx(t) x2 + αy(t) y2 + αz(t) z2
Hydrodynamic
GP-Equation
in CHL
a0 =
ρ(r,t) = a0(t) - ax(t) x2 - ay(t) y2 - az(t) z2
15N
8π
2
5
(axayaz)1/5 normalization
Consider Th-F- radii Ri(t), i = x,y,z
Ri(t):
0 = a0(t) - ai(t) Ri(t)2
Ri(t) =
⇒
Ri(t) = R0(0) bi(t) , Ri(0) =
2µ
m ωi,02
Study excitations with Ansatz:
50 Universität Hamburg, Bose-Einstein Condensation, SoSe 2015
a0(t)
ai(t)
⇒
(1),(2)
ωi,02
d2 b + ω 2 b –
= 0
i
i
dt2 i
bibxbybz
term1:
linear, accounts for instantaneous curvature
of potential
term2:
non-linear, only depends on initial curvature;
accounts for conversion of collisional energy
into kinetic energy and vice versa
ωi(t)2 ≡ ω i,02 + δ i,02 Sin(Ω t)
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EXAMPLE
consider axial symmetric trap:
b⊥ ≡ bx = by , ω⊥ ≡ ωx = ωy
consider non-adiabatic release:
⇒
ω i (t) =
ω i,0 , i = ⊥,z
0
d2 b = 1
⊥
dτ2
b3⊥ bz
d2 b = λ2
z
dτ2
b2⊥ b2z
for t < 0
for t > 0
τ ≡ ω⊥,0 t
ω
λ ≡ ωz,0
⊥,0
solution for λ << 1 (cigar shaped condensate) :
b⊥(τ) =
1 + τ2
bz(τ) = 1 + λ2 τ arctan τ - ln 1 + τ2
aspect ratio :
λ 1 + τ2
R⊥(τ)
=
Rz(τ)
1 + λ2 τ arctan τ - ln 1 + τ2
asymptotic value for τ→∞ :
R⊥(τ)
Rz(τ)
=
2
πλ
1 ms
λ = 0.03
6
λ = 0.1
4
λ = 0.3
2
0
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0
100
200
τ ≡ ω⊥,0 t
300
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Experiment: 25 ms expansion
non-adiabatic expansion of harmonic potential for g = 0 (ideal gas) :
L⊥(τ) = L⊥,0
1 + λ2 τ 2
Lz(τ) = Lz,0
Final RF
644 kHz
1 + τ2
aspect ratio :
L ⊥(τ)
L z(τ)
1/2
=
1 + τ2
1 + λ2 τ 2
λ
604 kHz
L ⊥(τ)
L z(τ)
asymptotic value for τ→∞ :
=
1
λ1/2
595 kHz
λ = 0.1
6
540 kHz
Th-F Limit
4
Ideal Gas
2
0
495 kHz
0
100
200
τ ≡ ω⊥,0 t
300
480 kHz
Fit: Thomas- Fermi for condensed fraction,
Gauß for thermal fraction
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Single Vortex Solutions
y
z
r = (x,y,z)
Ψ(r,t) ≡ Φ(r) e
x
φ
≡
µ
t

v(r)
⇒
ρ( x2 + y2 ,z) e i S(x,y)
y
S(x,y) ≡ κ arctan( x )
Φ(r)
y
-i
≡
1
2
x2
κ
+ y2
2 2
1
mv 2 =  κ 2
2m x + y2
∇
y
arctan( x )

= m
-y
x
0
x
 κ 2π δ(x)δ(y)
= m
v(r)
velocity field diverges at x,y -> 0, thus the density must approach zero at the vortex line.
Estimation of vortex core radius ζ:
 κ =
m ζ
!
v(ζ) = c =
gρ
m
→
ζ =
2 κξ
healing length
v(ζ) = sound velocity
total angular momentum:
L =
r≡
x2 + y2
dz 2π rdr n(r,z) r mv(r)
53 Universität Hamburg, Bose-Einstein Condensation, SoSe 2015
=
dz 2π rdr n(r,z) r mv(r)
=
dz 2π rdr n(r,z) κ
= N κ
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Multi-Vortex Solutions
two vortices (κ1and κ2) with same sense of rotation at distance d:
two vortices (κ1and κ2) with opposite sense of rotation at distance d:
kinetic energy:
kinetic energy:
d >> 0 -> Ekin ∝ κ12 + κ22
d = 0 -> Ekin ∝ (κ1 + κ2)2
d >> 0 -> Ekin ∝ κ12 + κ22
d = 0 -> Ekin = 0
d
energy conserved -> d constant
energy conserved -> d constant
dissipation -> vortices repell
vortices can form lattices
dissipation -> vortices attract & anihilate
54 Universität Hamburg, Bose-Einstein Condensation, SoSe 2015
d
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harmonic potential ωx = ωy :
Hydrodynamic GP-Equation:
1 ∇2 ρ
ρ
κ2
x2 + y2
=
2m
Vtrap - µ + gρ
2
η << 1, κ = 1
right hand side diverges
ρ = 0
for x2 + y2 → 0
thus
+
Lim
η >> 1, κ = 0
x 2+ y 2 → 0
x2 + y2 << ξ2 :
1 ∇2 ρ
ρ
x2
+ y2
0
2
κ
x2 + y2
≈
>>
ξ2,
⇒
ρ ∝
x2
+ y2
κ2
η >> 1 : Th-F density distribution
0 =
Vtrap - µ + gρ
Solutions of hydrodynamic GP-equation exist
only for integer κ
0
1
2
4
3
x / L0
5
6
η >> 1, κ = 1
vortex solution has reduced maximum density as compared to corresponding ground state:
→ for η >>1, excitation of vortices stabilizes BEC with
attractive interaction
excitation of N vortices with κ=1 costs less kinetic
energy than excitation of one κ=N vortex:
→ vortex lattices
Experiment:
vortex lattice NIST
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Interference of two Bose-Condensates
Which state describes BEC ?
Make BEC
Can a BEC interfere with itself ?
Split & Wait
Can two independent BECs interfere ?
Drop
Optical Pumping Laser
(mark a slice of atoms)
Lens
CCD Camera
M. Andrews et al., Science 275, 637 (1997).
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Toy model for bosons in a single mode in a single well
States:
|Σi ≡ C0 |0i + C1 |1i + C2 |2i + ...
|C0|2 + |C1|2 + |C2|2 + ...
Field Operator:
Ψ(x) ≡ α(x) a
= 1
α(x) ≡ |α(x)| exp( iφα(x))
+
Ψ(x) = α(x)* a+
dx |α(x)|2 = 1
[a,a+] = 1
a+|ni = (n+1)1/2 |n+1i
a|ni = n1/2 |n-1i
Particle Number:
Density:
Ψ+(x)Ψ(x) dx = a+a
+
ρ(x) ≡ hΣ| Ψ(x) Ψ(x) |Σi =
|α(x)|2 hΣ| a+a|Σi =
|α(x)|2
∑ n |Cn|2
n
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Hamiltonian for bosons in a single mode of a single well
H =
2
d3r ψ+(r) [ -  ∆ + V(r) ] ψ(r)
2m
single mode expansion
+
g
2
d3r ψ+(r)ψ+(r) ψ(r) ψ(r)
,
g =
4π 2 A
m
ψ(r) = a α(r)
H
≈
Va a+ a
on-site trap energy WV
+
Ua a+a (a+a – 1)
on-site collisions WU
g
2
2
[ -  ∆ + V(r) ] α(r)
2m
Note: the mode geometry is not a dynamical variable. It merely appears via the two numbers Va, Ua
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Interference of two Bose-Einstein Condensates (two particles):
Before Splitting:
Total number of particles = 2
After Splitting:
State: |Σi ≡ C11 |1,1i + C20 |2,0i + C02 |0,2i
|C11|2 + |C02|2 + |C20|2 = 1
|n,mi ≡ |ni ⊗ |mi
Field Operator:
Ψ(x) ≡ α(x) a
+
+
Ψ(x) = α(x)* a+
Particle Number:
β(x) b
+
β(x)* b+
α(x) ≡ |α(x)| exp( iφα(x))
β(x) ≡ |β(x)| exp( iφβ(x))
Ψ+(x)Ψ(x) dx = a+a + b+b
dx α(x) β(x)* = 0
+
ρ(x) ≡ hΣ| Ψ(x) Ψ(x) |Σi
Density:
=
dx |α(x)|2 =
dx |β(x)|2 = 1
|α|2 (|C11|2 + 2 |C20|2) + |β|2 (|C11|2 + 2 |C02|2 )
+ √2 (C20* C11 + C02 C11*) α* β + √2 (C20 C11* + C02* C11) α β*
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=
|α|2 (|C11|2 + 2 |C20|2) + |β|2 (|C11|2 + 2 |C02|2 )
+ √2 (C20* C11 + C02 C11*) α* β + √2 (C20 C11* + C02* C11) α β*
=
(|C11|2 + 2 |C20|2) |α(x)|2 + (|C11|2 + 2 |C02|2 ) |β(x)|2 +
√2 (C20* C11 + C02 C11*)
≡ C exp( iφγ) ,
2 |C| |α| |β| cos(φγ + φβ(x) – φα(x))
C = interference contrast
interference pattern arises only, if interference contrast C ≡ √2 |C20* C11 + C02 C11*|
≠ 0,
i.e. only, if particle number in each condensate is uncertain
Example: mode functions are counterpropagating plane waves φα = –kx, φβ = kx
→
Fock State:
ρ(x) =
→
interference term ~ √8 |α| |β| C cos(φγ + 2kx)
each condensate prepared in Fock-state |1,1i :
→
C20 = C02 = 0 , C11 = 1
|α(x)|2 + |β(x)|2
density ρ(x) independent of φα , φβ, i.e. no interference pattern in density distribution
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Two-Particle Correlation:
ρ(x1,x2) ≡
αi ≡ α(xi) , βi ≡ β(xi) , i =1,2
+
+
hΣ| Ψ(x1) Ψ(x2) Ψ(x2) Ψ(x1) |Σi =
2 |C20|2 |α1α2|2 + 2 |C02|2 |β1β2|2
+
2 C20* C02 α1*α2* β1β2
+
2 C20 C02* α1α2 β1*β2*
+
√2 C20* C11 α1*α2* (α1β2 + α2 β1 )
+
√2 C20 C11* α1 α2 (α1β2 + α2 β1 )*
+
√2 C02* C11 β1*β2* (α1β2 + α2 β1 )
+
√2 C02 C11* β1 β2 (α1β2 + α2 β1 )*
+
|C11|2 | α1β2 + α2 β1 |2
Fock State: each condensate is prepared as a Fock state, i.e., C20= C02 = 0, C11= 1, ρ(x) = |α|2 + |β|2
ρ(x1,x2) = | α(x1)β(x2) + α(x2) β(x1) |2
mode functions are counterpropagating plane waves, i.e., α ≡ |α0| exp( –ikx), β ≡ |β0| exp( ikx)
ρ(x1,x2) = 4 |α0|2 |β0|2 cos2( k(x2–x1) )
→
experiment yields interference pattern in density, however with random spatial phase
J. Javanainen, S. M. Yoo, Phys. Rev. Lett. 76, 161 (1996).
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Split Bose-Einstein condensates with 2 particles and defined relative phase:
Single atom:
1
2
2
Two atoms:
⊗
L
eiφ +
α(x)
R
β(x)
evaluate product
1
2
L
eiφ +
R
φ =
1
2
|Li⊗|Li ei2φ + |Ri⊗|Ri + |Li⊗|Ri eiφ + |Ri⊗|Li eiφ
=
1
2
|Li⊗|Li ei2φ + |Ri⊗|Ri
1
2
|2,0i ei2φ
µ=1
+
1
2
|Li⊗|Ri + |Ri⊗|Li
2
eiφ
symmetrize for bosons
=
+ |1,1i
2
eiφ
+ |0,2i
Phase state |φi maximizes interference contrast :
C11 =
1
2
C02 =
1
2
C20 =
1
2
eiφ
√2 |C20* C11 + C02 C11*| = 1
maximal
→
ei2φ
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ρ(x) ≡ hφ| Ψ+(x) Ψ(x) |φi = | α(x) eiφ + β(x) |2
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Dynamics of split Bose-Einstein condensates:
H =
2
d3r ψ+(r) [ -  ∆ + V(r) ] ψ(r)
2m
Expansion with respect to basis α(r) and β(r) :
H
≈
neglect collisions
involving different
sites
J a+ b + J* a b+
g
2
d3r ψ+(r)ψ+(r) ψ(r) ψ(r)
α(r)
ψ(r) = a α(r) + b β(r)
+
tunnelling WJ
+
Va a + a +
Vb b + b
+
,
2 A
g = 4π m
β(r)
Ua a+a (a+a – 1)
on-site trap energy WV
+
Ub b+b (b+b – 1)
on-site collisions WU
on-site energy WUV
g
2
d3r |α(r)|4
g
2
d3r |β(r)|4
Ua
≡
J
≡
2
d3r α(r)* [ -  ∆ + V(r) ] β(r)
2m
Va
≡
2
d3r α(r)* [ -  ∆ + V(r) ] α(r)
2m
real
Vb
≡
2
d3r β(r)* [ -  ∆ + V(r) ] β(r)
2m
real
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Ub
≡
real
=
κ eiξ
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Two Particles:
States of defined relative phase are stationary states of Tunnelling Operator WJ:
Eigenstates:
Eigenvalues:
=
J a+ b + J* a b+ , J
≡
|e+i =
1
2
|2,0i ei2ξ + |1,1i
2
eiξ + |0,2i
= |ξi
|e–i =
1
2
|2,0i ei2ξ – |1,1i
2
eiξ + |0,2i
= |ξ+πi
|e0i =
1
2
|2,0i ei2ξ – |0,2i
WJ
λ+
=
+2κ
λ-
=
–2κ
λ0
=
0
κ eiξ
Fock-states are stationary states of on-site energy WUV:
WUV =
→
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a+a (Ua a+ a + (Va – Ua) ) +
b+b (Ub b+ b + (Vb – Ub) )
|2,0i , |1,1i , |0,2i
λ20
= 2 (Va + Ua)
λ11
= Va + Vb
λ02
= 2 (Vb + Ub)
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Stationary states of Hamilton Operator:
U ≡ Ua = Ub , V ≡ Va = Vb , J ≡ κ eiξ
H
=
WJ
–
2U a+a b+b
(U
+
|e±i
1
≡
|2,0i ei2ξ + |1,1i
2 + |λ±|2 / 2κ2
|e0i
Eigenvalues:
1
2
≡
total particle number
=
λ±
2κ
eiξ + |0,2i
|2,0i ei2ξ – |0,2i
λ± = - U ±
λ0
≡ a+a + b+b
contributes energy offset: N (U N + (V–U))
Two Particles (V=0)
Eigenstates:
+ (V–U)) ,
U2 + 4κ2
2U
–2|κ|
U/|κ| << -1 → groundstate: |e-i ≈
≈
0 → groundstate:
U/|κ|
0
Limiting cases: (κ < 0)
U
2|κ|
λ+(U) + 2U
1
2
|2,0i ei2ξ + |0,2i
“cat” state
|e–i = |ξi
U/|κ| >> 1 → groundstate: |e–i
λ-(U) + 2U
phase state
C
= |1,1i
Fock state
1
Interference contrast for |e±i:
C = √2 |C20* C11 + C02 C11*| =
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4|κ| |λ±|
4κ2
+
|λ±|2
=
2|κ|
U2
+
-5
5
U/|κ|
4κ2
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Temporal evolution of coherence:
split
α(x)
β(x)
prepare arbitrary |Σ(t=0)i
wait during time t : J=0, H = WUV
merge
On site energy per particle in each condensate:
λ11, λ20, λ02, eigenvalues of WUV:
λ20
= 2 (Va + Ua)
λ11
=
λ02
= 2 (Vb + Ub)
|2,0i
Va + Vb
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|1,1i
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Time Evolution of |Σi:
|Σ(t=0)i ≡ C11 |1,1i + C20 |2,0i + C02 |0,2i
|Σ(t)i = C11 exp(-i λ11t /)
λ20
= 2 (Va + Ua)
λ11
=
λ02
= 2 (Vb + Ub)
|1,1i + C20 exp(-i λ20 t /) |2,0i + C02 exp(-i λ02t /) |0,2i
Va + Vb
U ≡ Ua = Ub , V ≡ Va = Vb
→
λ20 – λ11 = λ02 – λ11 = 2U
exp(i λ11t /) |Σ(t)i = C11 |1,1i + C20 exp(-i 2U t /) |2,0i + C02 exp(-i 2U t /) |0,2i
→ C(t) ≡ √2 |C20(t)* C11(t) + C02(t) C11(t)*| = √2 |C20* exp(i 2U t /) C11 + C02 exp(-i 2U t /) C11*|
with Cij ≡ |Cij| exp(γij )
C(t)
1
for
→ Interference contrast |C(t)| decays and revives periodically.
0
2U
t

π
+ γ0
* For large particle numbers the decay of |C(t)| becomes increasingly fast
* The revival time scales with 1/ U. For large traps U is small and the revival time for |C(t)|
exceeds the typical life time of the condensate due to three-body collisions
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Relative Phase of two Split Bose-Einstein Condensates with N Particles in Total:
Single atom:
1
2
L
eiφ +
α(x)
R
β(x)
Symmetrization
N
N atoms:
⊗
µ=1
1
2
L
eiφ +
ν, N – ν
φ
R
≡
evaluate product
symmetrize for bosons
Two BECs with relative phase φ :
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φ
≡
1
2
N/2
{
L
⊗ ... ⊗
ν - times
N
Σ
ν=0
eiνφ
N
ν
L
⊗
R
⊗ ... ⊗
R
}
(N – ν) - times
ν, N – ν
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Properties of |φi and |n,mi :
|φi is eigenstate of tunneling operator WJ
=
State |φi yields maximal interference contrast:
J a+ b + J* a b+
+
φ Ψ(x) Ψ(x) φ
=
, J ≡ κ eiφ
N
2
| α eiφ + β |2
Example: counterpropagating plane waves, i.e., α ≡ |α| exp(-ikx), β ≡ |β| exp(ikx)
ρ(x) = (|α|– |β|)2 + 4 |α| |β| cos2( kx – φ/2)
|n,mi is eigenstate of on-site energy WUV = a+a (Ua a+a + (Va – Ua)) + b+b (Ub b+b + (Vb – Ub))
Only second order interference for independent BECs (each described by a Fock-state):
+
m,n Ψ(x) Ψ(x) m,n
+
+
m,n Ψ(x1) Ψ(x2) Ψ(x2) Ψ(x ) m,n
1
no interference term
=
m |α|2 + n |β|2
=
m(m-1) |α1α2|2 + n(n-1) |β1β2|2 + m n | α1β2 + α2 β1 |2
Example: counterpropagating plane waves, i.e., α ≡ |α| exp(-ikx), β ≡ |β| exp(ikx)
ρ(x1,x2) = m(m-1) |α|4 + n(n-1) |β|4 + 4 m n |α|2 |β|2 cos2( k(x2–x1) )
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Time evolution of two independent (no tunnelling) BECs with relative phase φ :
t=0:
φ
N
1
≡
2
N/2
Σ
N
ν
eiνφ
ν=0
ν, N – ν
P(ν, N) =
1
2
N
ν
N/2
peaks around ν = N/2
Expand energy of state
ν, N – ν
E(ν, N – ν) =
around ν = N/2 :
E(ν) + E(N – ν)
=
ω0 =
ξ
=
ω0 + ξ (ν – N/2)2
+
O(ν4)
2 E(N/2)
d2E(ν)
dν2
=
ν = N/2
dµ
dν
ν = N/2
State with initial relative phase φ after evolving during a time t:
N
φ(t)
≡
exp(– iω0t)
Σ
exp(– iξ t (ν – N/2)2 + iνφ ) P(ν, N)
ν, N – ν
ν=0
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ψ
States with defined relative phase ψ :
≡
N
1
2
Σ
N/2
N
ν
eiνψ
ν=0
ν, N – ν
phase distribution of |φ(t)> :
N
ψ φ(t) =
Σ
P(ν, N)2 exp(– iξ t (ν – N/2)2 + iν(φ – ψ) )
ν=0
P(ψ) =
ψ φ(t)
2
π
≈
2
N >> 1
Δψ2
exp –
(ψ – φ)2
2
use Laplace formula for P(ν, N)
Δψ(t)
Δψ2
=
1 + N ξ2 t 2
N
J. Javanainen, M. Wilkens, Phys. Rev. Lett. 78, 4675 (1997).
Note:
time evolution yields phase decoherence: width Δψ doubles for ξ t = 3 N–1
width Δψ = π
for ξ t ≈ π N–1/2
complete revival occurs for ξ t = 2 π
Example: Th-F-Limit in harmonic trap:
ξ
=
72
125
1/5
ωV
N
3/5
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A0
L0
2/5
µ
A
=  ωV 15 N 0
2
L0
2
5
N ≈ 106
L0 ≈ 10-5 m
A0 ≈ 10-8 m
ωV
Revival time:
τ ≡ ξ-1 ≈ 100 s
≈ 2π 102 s-1
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Probability to measure the phase ψ for the state φ(t) at time t:
P(ψ, t) =
ψ φ(t)
2
N
Σ
=
2
P(ν, N)2 exp(– iξ t (ν –
N/2)2
+ iν(φ – ψ) )
ν=1
Plots of P(ψ, t) for N = 20, φ = 0
1
0.1
1
1
0
0
0 –1
0
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–1
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Basic Definitions in a Lattice:
unit cell
Bravais-lattice
a = primitive vector
Wigner-Seitz unit cell
Bravais-lattice:
R = {R | R = n1a1 + n2a2 + n3a3 , ni ∈ Z }
reciprocal lattice:
K = {K | eiKR = 1 for all R ∈ R }
basis of K :
K = {K | K = n1b1 + n2b2 + n3b3 , ni ∈ Z }
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bi = 2π
aj x ak
a1(a2 x a3)
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First Brillouin Zone:
Bragg planes
Bravais lattice
Wigner-Seitz unit cell
Bragg Scattering:
reciprocal lattice
Wigner-Seitz unit
cell = 1st Brillouin
zone ( FBZ )
Elastic scattering: | kin | = | kout |
constructive interference: kin - kout ∈ K
kin
kout
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K ∈K
Bragg plane bisects K-vector perpendicularly
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Brillouin zones in square lattice
1
2
3
4
5
6
n-th Brillouin zone:
all k-vectors between n-th order and (n-1)-order Bragg-planes
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Bloch-states, Wannier-states
2
H = -  ∆ + V(r) , V(r + R) = V(r)
2m
Assumption:
Eigen states ψk(r), k ∈ FBZ exist, with
(a)
ψk(r) = eikr uk(r), uk(r) = uk(r+R)
(b)
ψk(r+R) = eikR ψk(r)
(c)
ψk(r) =
∑ eikR
W(r-R)
ψk(r) Bloch functions
δkk’ =
W(r) Wannier function
δRR’ =
d3r W*(r-R) W(r-R’)
R
proof (b):
proof (c) → (a):
Define TRψ(r) ≡ ψ(r+R)
⇒ [TR,TR’]
⇒ ψ(r+R) = TRψ(r) = c(R) ψ(r)
⇒
d3r ψk(r)* ψk’(r)
= [TR,H] = 0
with c(R+R’) = c(R)c(R’)
k ∈ FBZ : c(R) = eikR , normalizability
⇒k
define uk(r)
≡ e-ikr ψk(r) = ∑ e-ik(r-R) W(r-R)
R
real
define: ψk+K(r) ≡ ψk(r)
proof (a) → (b):
straight forward
proof (b) → (c):
ψk(r) = ψk+K(r)
ψR(r) = 1
N
⇒ ψk(r) = ∑
d3k e-ikR ψk(r)
FBZ
R
eikR ψR(r)
= 1
N
(b)
, N ≡ Number of unit cells
d3k ψk(r-R) = ψR=0(r-R)
FBZ
Define W(r) ≡ ψR=0(r)
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Some useful relations
H
≡
–
2
∆ + V(r) , ψk(r) = eikr uk(r), H ψk(r) = εk ψk(r)
2m
2
(∇ + ik)2
2m
(a)
Hk uk(r) = εk uk(r) , Hk
(b)
 ∇ψ (r) = k ψ (r) + eikr  ∇ u (r)
k
k
k
i
i
≡
–
+ V(r)
i.e., crystal momentum ≠ momentum
(c)
1 dεk
=
 dk
proof (a):
H ψk(r) = - eikr
proof (c):
εk+q ≈ εk + q
d3r ψk*(r)
(a)
,
Hk+q = –
dk
1
= Lim q (εq+k – εk)
q→0
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hvi
- eikr (
2
(∇ + ik)2 uk + V uk ) = - eikr Hk uk
2m
2
2
(∇+ik+iq) 2 + V(r) ≈ Hk – i q  (∇ + ik)
m
2m
+ O(q2)
2
εk+q ≈ εk + huk| q  (-i∇ + k) |uki
m
1st order perturbation theory:
dεk
≡
2
(∇2 uk + k(i∇) uk - k2 uk) + V ψk(r) =
2m
dεk
dk

∇ψk(r)
im
=
2
m
d3r uk*(r) (-i∇+k) uk(r)
=
d3r ψk*(r)
2
∇ψk(r)
im
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Band structure (perturbation picture)
H = H0 + V(r) ,
H0 = -
2
2m
∆ ,
V(r)
∑
=
e-iKr VK
K∈K
eigenstates and eigenvalues of H0 :
hk’| V(r) |ki =
∑
|ki = V
VK A(k,k’,K) ,
K∈K
⇒
hk’| V(r) |ki ≈
∑
-1/2
eikr,
εk =
2k2
2m
A(k,k’,K) = V
, V = volume of lattice
-1
d3r ei(k-k’-K)r
≈
δk-k’,K
V
VK δk-k’,K
K∈K
k,k’ ∈ n.BZ ⇒ the only non-vanishing matrix element hk’| V(r) |ki arises for k = -k’ = n K0/2 ,
where K0 ∈K with K0/2 pointing to edge of 1.BZ .
At the edge of the n.BZ the counterpropagating waves eikr and e-ikr with k = n K0/2
are coupled by Bragg-scattering, thus yielding an energy splitting:
2h-nK0/2 | V(r) |nK0/2 i = 2VnK
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0
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Band Structure 1D
εq
2 |V3K|
ε2,k
2 |V2K|
ε1,k
ε0,k
3.BZ
2.BZ
2 |VK|
1.BZ
1.BZ
0
V(r)
=
∑
2.BZ
K/2
q
3.BZ
K
3K/2
e-iKr VK
K∈K
Note: At edge of FBZ
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1 dεk
 dk
parallel to Bragg-plane
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Band Structure of 2D Square Lattice
P-band
S-band
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Semiclassical equations of motion
consider wavepacket
with gk centered around k with a spread Δk << FBZ
d3k gk ψk(r)
ψ(r) =
Δ k << FBZ
 dk = F
dt
1 dεk
=
 dk
d3r ψ*(r)
(A)
 ∇ ψ (r)
im
≡
(B)
v
note: replacement of momentum by crystal momentum in (A) accounts for effect of periodic potential
essential precondition: Force F sufficiently small such that no interband transitions occur
Landau Zener condition (LZ):
1.BZ
 |h1,k| d |0,ki| << 2 |VK|
dt
k=K
ε1,k
(A)
dk
F = 
dt
2 |VK|
ε0,k
0
K/2
q
81 Universität Hamburg, Bose-Einstein Condensation, SoSe 2015
|h1,k| dtd
(LZ)
⇒
|0,ki|
k=K
=
d |0,ki|
|h1,k| dk
k=K
dk
dt
F <<
2 |VK|
d |0,ki|
|h1,k| dk
k=K
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Effective Mass Tensor:
1 dε k
=
 dk
1 dε k
 dk
k=k0
+
M k0 -1  (k - k0) + ...
≡
M k0 -1
1 dεk
 dk
consider k-values e.g. at center and edge of FBZ where
k=k0
1 d2ε k
2 dkidkj
≈
k=k0
M k0 effective mass tensor
0
ε1,k
2 |VK|
ε0,k
0
K/2
q
1.BZ
⇒
note:
at edge of FBZ
M k0 a
≡ M k0
d
v
dt
=

d
k
dt
≡
F
Bragg-scattering yields negative effective mass
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Example : 1D optical lattice
{R|R=n λ ,n∈
2
4π , n ∈
K = {K|K=n
λ
R =
FBZ =
Z
}
Z
}
ω0
2π
,
[ - 2π
λ λ ]
Use Wannier representation of Bloch function:
⇒
ω0
εk
=
d3r ψk*(r)
 ∆
[ - 2m
2
∑ eikR W(r - R)
ψk(r) =
+ Vlatt (r) ] ψk (r)
R
=
∑
d3r W*(r - R)W(r - R’) = δRR’
ik(R-Rʼ)
A RR' e
R' R
A RR'
83 Universität Hamburg, Bose-Einstein Condensation, SoSe 2015
,
≡
2
d3r W *(r - R) [ -  ∆ + Vlatt (r) ] W (r - R ' )
2m
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Special Case: keep only next-neighbour tunneling
⇒
λ
ε(k) = ε0 - 2|J| cos(k 2 - χ)
v
J
≡
ε0
≡
−
=
|J| λ

sin(k λ - χ)
2
2 ∆ + V (r) W (r + λ )
*
d3r W (r) [ ]
latt
2
2m
≡
tunneling rate
- |J| eiχ
2
d3r W *(r) [ -  ∆ + Vlatt (r) ] W (r)
2m
on-site energy
ε(k)
ε0 + 2|J|
ε0 - 2|J|
χ=0
0
2π
λ
k
1.BZ
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Adiabatic Loading of Lattice
εk [ω]
εk [Erec]
25
3
16
2
9
1
Occupation Probability
-5-4 -3 -2 -1
1 2 34 5
p [2π/λ]
4
1
1
0
shallow wells
k [2π/λ]
1
k [2π/λ]
0
deep wells
Adiabatic Cooling
adiabatic reduction of well depth
harmonic oscillator theory
A. Kastberg et al., PRL 74. 1542 (1995)
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Bloch Oscillations:
→
accelerate optical standing wave: ω = ω0 + ω t
standing wave moves with velocity:
F=
in accelerated frame this corresponds to the action of a constant force:
Semiclassical equations of motion
⇒
k(t) =
F

t
=
ω- ω0
ωλ
λ =
t
4π
4π
mω λ
4π
mω λ
t
4π
(*)
F
ω(t)
wavepacket
v
=
ω0
ψ(r) oscillates with Bloch-frequency:
1 d ε - 2|J| cos(k λ - χ)
2
 dk 0
ΩBloch =
λF
2 
=
2
mω λ =
8π
=
|J| λ

λ
sin(k 2 - χ)
=
(*)
|J| λ

sin(ΩBloch t - χ)
π ω
ω rec
Note:
- Bloch frequency does not depend on potential depth
- increase of potential depth
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→
decrease of tunneling rate J
→
decrease of oscillation amplitude
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momentum distribution
in accelerated frame
for different acceleration times
mean velocity in accelerated frame
plotted versus acceleration time
U0 = 1.4 ER
U0 = 2.3 ER
U0 = 4.4 ER
U0 = 2.3 ER
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Bosons in an optical lattice
Vlatt (r)
V trap(r)
(Mean field description)
Heisenberg Equation:
i  ∂ Ψ(r,t) =
∂t
Ψ(r ,t) ,
H
=
–
2 ∇ 2
+ Vext (r) +
2m
dr' Ψ +(r' ,t) U(r' – r)Ψ(r' ,t) Ψ (r,t)
contact potential, Bogoliubov approx. in Wannier basis:
U(r’ - r) = g δ(r’ - r)
ψ(r) =
∑
αR W(r - R) ,
αR = complex amplitude
R
d3r W *(r - R) W (r - R ')
⇒
= δRR'
discrete GP-Equation:
i ∂ α
∂t R
= -
∑
R'
K RR' αR'
KRR' = –
Λ R 1 R2 R 3 R =
88 Universität Hamburg, Bose-Einstein Condensation, SoSe 2015
g
+
∑
R 1,R 2,R 3
Λ R1 R2 R3 R α R1αR2α*R3
2
d3r W *(r - R) [ -  ∆ + Vext (r) ] W (r - R ')
2m
d3r W (r-R 1 )W (r-R 2 ) W*(r-R 3 ) W *(r-R)
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Vlatt(r)
1D optical lattice inside trap potential
Vtrap(r)
ρ(r)
single particle on-site energy of lattice scaled to zero at minimum of Vtrap(r)
approximations: - only on-site collisions
- next-neighbour tunneling
i  ∂ αn
∂t
K
= –
en =
Λ
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=
=
- K α n-1 - K *α n+1 + e n + Λ α n
d3r W *(r)
 ∆
[ - 2m
2
+ Vlatt (r) ] W (r + 2λ )
λ
λ
d3r W *(r - n 2 ) Vtrap(r) W (r - n 2 )
g
d3r |W(r)|
≈
2
≡
αn
κe
Vtrap(n
iχ
,κ>0
λ
)
2
4
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αn
=
π n e i φn
⇒
∂ φ
∂t n
=
κ

πn-1
πn cos(φn-1 − φn + χ)
∂ π
∂t n
=
2κ

πn πn-1 sin(φn − φn-1 − χ)
∑
define center of mass coordinate:
ξ(t)
≡
+
κ

∑ πn
−
1 (e + Λ π )
n
n

2κ π π
n n+1 sin(φn+1 − φn − χ)

–
πn n
n
π n+1
π n cos(φn+1 − φn − χ)
⇒
∂
∂t
∑ πn
= 0
n
n
∂ ξ(t)
∂t
=
2κ

∑
n
πn π n+1 sin(φn+1 − φn − χ)
∑ πn
n
phase locking ansatz:
Δφ = φn+1 − φn independent of n
2
mean crystal momentum of wave packet: k ≡ λ Δφ
90 Universität Hamburg, Bose-Einstein Condensation, SoSe 2015
⇒
( πnπn+1 ≈ πn2 )
∂ ξ(t)
∂t
≈
2κ sin(Δφ − χ)

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∂ φ
∂t n
κ

≈
−1

∂ Δφ(t) ≈
∂t
π n-1
π n cos(Δφ − χ)
∑
n
+
κ

πn (en+1 − en)
∑ πn
−
π n+1
πn cos(Δφ − χ)
Λ

=
−
λ
2
n
π n ∂Vtrap (n λ )
2
∂x
∑ πn
n
∑ πn
n
−
Λ λ
2
∑
n
≈
Note:
−
λ
2
λ
2
h ∂V∂xtrap (n λ2 ) i
πn
∂ρ λ
(n 2 )
∂x
∑ πn
n
n
=
1 (e + Λ π )
n
n

∑ πn (πn+1 − πn)
n
∑
−
−
Λ λ
2
∂ρ
h ∂x
i
hFi
Λ dependence neglected upon assumption of small density gradient
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Connection to Semiclassical Model for Single Particle:
k
≡
2
Δφ
λ
,
a
⇒
λ
2
≡
,
ε(k)
∂ ξ(t)
∂t
≈
∂ k(t)
∂t
≈
≡
2κ (1 - cos(ka− χ))
1
∂ ε(k)
 ∂k
1

hFi
zero force :
∂ ξ(t)
∂t
Δφ
≈
2κ sin(Δφ − χ)

const
=
constant force :
∂ ξ(t)
∂t
≈
Δφ(t) ≈
DC Josephson-effect
hFi
=
f
2κ sin(Δφ − χ)

AC Josephson-effect
= Bloch-oscillation
φ0 + Ω t
Ω
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≡
λf
2
coincides with Bloch-frequency
Andreas Hemmerich 2015 ©
harmonic potential/linear force :
en =
hFi
∂ ξ(t)
∂t
≈
1
2
m ω 2 ( 2λ )2 n2
= - mω2
λ
2
ξ
2κ sin(Δφ − χ)

∂ Δφ(t) ≈
∂t
−
m
ω2 λ 2
( 2) ξ

small amplitude oscillations ( χ = 0 ) :
∂ ξ(t)
∂t
≈
∂ Δφ(t) ≈
∂t
⇒
2κ Δφ

−
m
ω2 λ 2
( 2) ξ

harmonic oscillation with frequency
Ω2 =
2κ

m
ω2 λ 2
( 2)

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⇒
Ω =
π
κ
ω
Erec
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Bosons in an optical lattice
Vlatt (r)
(Beyond mean field)
V trap(r)
Three leading energy contributions expected:
tunneling between adjacent lattice sites (hopping term)
on-site potential energy (trap potential)
on-site collisional energy
General expression of Hamilton-Operator:
H =
d3r ψ+(r) [ -
2
∆ + Vlatt(r) + Vtrap(r)] ψ(r)
2m
Expansion in terms of Wannier basis of lowest band:
+
g
2
d3r ψ+(r)ψ+(r)ψ(r)ψ(r) ,
ψ(r) =
∑
R
g =
4π2A
m
bR W(r - R)
W(r) = lowest band Wannier function, [bR+, bR’] = δRR’
⇒
H =
∑
R,R’
bR+bR’
2
d3r W*(r - R) [ -  ∆ + Vlatt(r)] W(r - R’)
2m
+
g
2
∑
+
bR1+ bR2+ bR3 bR4
∑
bR+bR’
d3r W*(r - R) Vtrap(r) W(r - R’)
R,R’
d3r W*(r - R1) W*(r - R2) W(r - R3) W(r - R4)
R1,R2,R3,R4
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Approximations:
First term:
∑
2
d3r W*(r - R) [ -  ∆ + Vlatt(r)] W(r - R’)
2m
bR+bR’
R,R’
Single well energy scaled to zero, i.e.,
d3r W*(r - R) [ -
0 =
2
∆ + Vlatt(r)] W(r - R)
2m
Only next neighbor tunelling, i.e., sum extends only over R, R' , where a = R-R' is a primitive vector.
Next neighbor tunelling
⇒
Second term:
J
≡ −
First term
2
∆ + Vlatt(r)] W(r - a)
2m
d3r W*(r) [ -
=
∑
–J
a,R
∑
bR+bR’
is chosen real, independent of a.
bR+ bR+a + bR bR+a+
d3r W*(r - R) Vtrap(r) W(r - R’)
R,R’
No tunneling contribution from trap potential, only on-site contribution
⇒
Second term =
∑
R
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eR
≡
d3r Vtrap(r) |W(r - R)|2
≈
Vtrap(R)
eR bR+bR
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g
2
Third term:
∑
bR1+ bR2+ bR3 bR4
d3r W*(r - R1) W*(r - R2) W(r - R3) W(r - R4)
R1,R2,R3,R4
Only on-site collisional energy contribution
⇒
Third term =
1
2
U
≡
U
∑
R
g
d3r |W(r)|4
bR+bR (bR+bR – 1)
In summary: find Bose-Hubbard Hamiltonian
H
=
–J
∑
a,R
bR+ bR+a + bR bR+a+
+
∑
R
eR bR+bR
+
1
2
U
Note: increase of well depth yields more localized W(x) and less overlap with W(x-R)
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∑
R
bR+bR (bR+bR – 1)
→
increase of U, decrease of J
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Bose-Hubbard Hamiltonian coupled to a particle reservoir:
Grand canonical potential:
Hµ
≡
H – µN
=
–J
∑
a,R
bR+ bR+a + bR bR+a+
+
∑
R
(eR – µ) bR+bR
+
1
2
U
∑
R
bR+bR (bR+bR – 1)
Flat trap potential (eR = 0), no tunnelling (J = 0), on-site terms only:
Eigenstates are Fock states |{nR}i ≡ | ...,nR,...i with nR atoms
at nR-th site, M = numer of sites, N = numer of particles:
→
|{nR}i
E{nR}
=
+ nR
(n!)-M/2 ∏ bR
=
∑ nR (nR-1)
R
|4iR
3U
|3iR
U
|2iR
0
|1iR, |0iR
|0i
R
U
2
6U
– nR µ
Ground state:
E0 = M n (n-1) U2
|{n}i =
– n µ , n = N/M
(n!)-M/2 ∏ bR
+n
Minimize E{nν} with constraint ∑ nR = N
R
|0i
R
Energy gap: | ...,n,n,...i → | ...,n-1,n+1,...i , ΔE(n) = [ (n-1)(n-2) + (n+1) n - 2 n(n-1) ] U2 = U
Ground state separated from closest excited state by a gap U →
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Mott-Insulator
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Flat trap potential (eR = 0), tunnelling only, no on-site terms (U = 0, µ = 0):
ψ(r) =
≡
–J
∑
a,R
∑
W(r-R) bR
bR+ creates a single particle in Wannier mode W(r-R)
∑
ψk(r) Bk
Bk+ creates a single particle in Bloch mode ψk(r)
R
ψ(r) =
H
k2 FBZ
Bk = M-1/2 ∑ bR e-ikR ,
R
bR+ bR+a + bR bR+a+
ψk(r) = M-1/2 ∑ eikR W(r-R)
R
single atom wave function of ground state (Bloch state with k=0):
ψ0(r) = M-1/2 ∑ W(r-R)
R
B+ ≡ M-1/2
∑
R
bR +
creates a single particle in Bloch mode ψ0(r)
commutation relations: [B, B+] = 1
N atoms in mode ψk=0(r) comprising M lattice sites:
|ni ≡
(N!)-1/2 B+
N
|0i , n ≡ N/M
n = average number of atoms per site
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Gutzwiller Approximation: | n i is a product of coherent states at each lattice site
+N
B
1
=
N
∑
M R
bR
+
1
=
∑
M N/2
s1,...,sM
s1!
s
+ s1
N!
. . . b+ M
b
1
M
... sM!
N!
=
N
s1+...+sM = N
N/2
N
M
∑
s1
+ s1
b1
...
s1!
s1,...,sM
N
M
sM
+ sM
bM
sM!
s1+...+sM = N
(use generalized binomial formula)
N!
≈
N,M →
|ni =
1,
N
M
(N!)-1/2
B
M
≈
N,M →
e-N/2
1
N!1/2
|0i ≈
1
∏
∑
µ=1
s=0
NN/2
(N/M)s/2
s!1/2
(use Stirling formula: ln(k!) ≈ k ln(k) – k
1
| n iR ≡
|siR
e- n /2
N N/2
= n = const
+N
∑
s=0
→
n s/2
|siR
s!1/2
≡ s!-1/2 bR+ s |0iR
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M
M
1
∏
∑
µ=1
s=0
1
∏
∑
µ=1
s=0
(N/M)s/2
s!
s
b+µ
M
|siµ =
k!/kk ≈ e-k )
∏
µ=1
e-(N/M)/2
N
M
s
+s
bµ
s!
|0i
1
∑
s=0
N!1/2
=
NN/2
(N/M)s/2
s!1/2
M
1
(N/M)s/2
∏
∑
µ=1
s=0
|siµ
=
s!1/2
|siµ
∏ | n iR
R
(Gutzwiller Approximation)
=
coherent state with average number of n atoms at site R
=
Fock state with exactly s atoms at site R
Andreas Hemmerich 2015 ©
M. Fisher et al., Phys. Rev. B 40, 546 (1989)
Phase diagram
U >> J ⇒ Ground state = Fock-state:
relative phases maximally uncertain, no number fluctuations, energy gap
→
Mott insulator
U << J ⇒ Ground state = coherent state:
relative phases maximally certain, Poisson number fluctuations, no energy gap
→
superfluid
µ/U
Mott
phase
2nd order phase transition
Superfluid phase
3
n=3
hni = 3
n=2
hni = 2
hni = 2 – ε
2
1
n=1
hni = 1 + ε
hni = 1
zJ/U
0
zJc /U = 3 - 8
z = number of
next neighbours
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D. van Oosten et al., Phys. Rev. A 63,053601 (2001)
Calculation of Phase Boundary
Hµ
≡
H – µN
=
–J
∑
b i+ b j
1
2
+
U
i
hi,ji
mean field ansatz:
bi = hbii + bi – hbii ,
bi+ bj
ni (ni – 1)
–
∑
i
µ ni
bi – hbii sufficiently small correction
bi+ bj = (bi+ – hbii*) hbji + hbii* (bj – hbji) + hbi+ihbji
⇒
∑
≈ hbi+i bj + bi+hbji – hbi+ihbji
+ (bi+ – hbii*) (bj – hbji)
neglect
= ψ* bj + bi+ψ – |ψ|2
assume hbii = ψ
independent of lattice site
(spatially uniform order parameter ψ)
two possible kinds of solutions: ψ ≠ 0 → defined relative phase for different sites, superfluidity
ψ = 0 → no relative phase for different sites, Mott insulator
insert into Hamiltonian:
⇒
Hµ =
H(0)
H(1)
+
=
∑
H(0)i
H(1)i
+
i
H(0)i
=
H(1)i
=
1
2
U ni (ni – 1)
–
µ ni
– J z ( ψ* bi + ψ bi+ )
+
J z |ψ|2
z ≡ number of next neighbours
(z = 2 in 1D, z = 4 in 2D square lattice, etc.)
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rescaling:
hµ ≡ Hµ / J z =
h(0)
h(1)
+
=
∑
h(0)i
i
h(0)i
=
h(1)i =
1
2
U ni (ni – 1)
–
h(1)i
+
µ ni
+
U ≡ U/J z
|ψ|2
µ ≡ µ/J z
– ( ψ* bi + ψ bi+ )
find ni = g such that
eigenbasis of h(0)i :
Fock states |nii form eigenbasis
ε(0)(ni) =
ground state of h(0)i :
⇒
1
2
U ni (ni – 1)
–
µ ni +
|ψ|2
all ni are equal ni = g ; for given µ, determine g such that ε(0)(g) is minimal:
ε(0)(g+1) – ε(0)(g) = U g – µ > 0
ε(0)(g–1) – ε(0)(g) = µ – U (g-1) > 0
⇒
U (g-1) < µ < U g
ε(0)(g) =
1
2
determines value of g as closest
integer larger than µ/U
U g (g – 1) – µ g
+
|ψ|2
g
µ/U
g-1
|nii = |gi
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Consider h(1)i as a perturbation of h(0)i and apply perturbation theory up to 2nd order to find
ground state energy ε(g) = ε(0)(g) + ε(1)(g) of total Hamiltonian h(0)i + h(1)i :
1st order correction is zero because hg| h(1)i |gi = hg| ψ* bi + ψ bi+|gi = 0
2nd order corection:
ε(1)(g) =
|hg| h(1)i |ni|2
∑
n≠g
=
=
⇒
ε(g) =
=
|ψ*hg| bi |g+1i|2
ε(0)(g)– ε(0)(g+1)
|ψ|2 (g+1)
µ–Ug
U g (g – 1) – µ g
=
ε(0)(g)– ε(0)(n)
+
+
|ψ|2
∑
| ψ*hg| bi|ni + ψhg| bi+|ni |2
ε(0)(g)– ε(0)(n)
n≠g
+
|ψhg| bi+ |g-1i|2
ε(0)(g)– ε(0)(g-1)
|ψ|2 g
U (g-1) – µ
1 +
(g+1)
µ–Ug
+
g
U (g-1) – µ
A(0)(g,U,µ) + A(2)(g,U,µ) |ψ|2
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Landau theory of 2nd order phase transitions:
ε
ε = A(0) +
⇒
⇒
A(2)(g,U,µ)
=
A(2) = 0
phase boundary
1 +
, A(4) > 0
|ψ|
|ψ|
A(2) positive
ψ = 0 → insulator
A(4) |ψ|4 + ...
ε
ε
|ψ|
0 =
A(2) |ψ|2 +
(g+1)
µ–Ug
+
A(2) negative
ψ ≠ 0 → superfluid
g
U (g-1) – µ
0 = U2 g(g-1) + µ2 + Uµ (1-2g) + U + µ
∼2
0 = µ
∼
∼
∼
+ µ (1 – 2g + J) + g(g-1) + J
∼
µ ≡ µ/U
∼
J ≡ zJ/U
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discriminant:
∼
∼
1
2
(2g – 1 – J) ±
0
1
4
(2g – 1 – J)2 – g(g-1) – J
=
1
4
∼
∼
∼
(2g – 1 – J)2 – g(g-1) – J
∼
⇒ µ
=
∼
∼
⇒ J2 + 2J (2g+1) + 1 = 0
∼
⇒ Jc = 2g+1 –
g=1
g=2
∼
⇒ Jc = 3 –
(2g+1)2 – 1
81/2
∼
µ
Mott
phase
Superfluid phase
3
∼
⇒ Jc = 5 – 241/2
g=3
2
g=2
1
g=1
∼
J
0
∼
Jc = 5 - 24
105 Universität Hamburg, Bose-Einstein Condensation, SoSe 2015
∼
Jc = 3 - 8
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Optical Lattice in external trap
Mott Insulator Phase:
Superfluid Phase:
external trap potential Vtrap(x)/U
J=0
chemical potential µ/U
µ = µ0 - V(x) , µ0/U = 3.5
µ/U ∈ [3,4]
n=4
[2,3]
[1,2]
[0,1]
n=3
n=2
n=1
n=3
n=2
n=1
J>0
n=4
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Detection of momentum spectra in an optical lattice
→ Prepare BEC in the microtraps of periodic light shift
potentials (Optical Lattice)
→ Manipulate atoms in lattice during a variable time t
→ Rapidly switch off potential and let BECs ballistically expand
→ Absorption Image
Ballistic
expansion
Absorption Laser
107 Universität Hamburg, Bose-Einstein Condensation, SoSe 2015
Momentum
distribution
CCD Camera
Andreas Hemmerich 2015 ©
Inter-site coherence yields Bragg maxima in momentum spectra
ky
kx
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Momentum distribution
Consider mean number of particles at momentum k: hψ+(k) ψ(k)i
ψ(r) =
∑
R
bR W(r-R)
ψ(k) =
1 3/2
( )
2π
d3r e-ikr ψ(r)
W(k) =
1 3/2
( )
2π
d3r e-ikr W(r)
⇒
hψ+(k) ψ(k)i = |W(k)|2
∑ hbR+bR’i
eik(R-R’)
=
|W(k)|2
hNi +
hNi ≡
|W(k)|2
hNi +
∑ eikR S(R)
R≠0
diffuse background
109 Universität Hamburg, Bose-Einstein Condensation, SoSe 2015
eik(R-R’)
R,R’
R≠R’
R,R’
=
∑ hbR+ bR’i
, S(R) ≡ ∑ hbR’+ bR-R’i
∑ hbR+ bRi
R
R’
can be strongly k-dependent with sharp resonance structure
depending on S(R), which crucially depends on the state used
to evaluate the expectation value.
Andreas Hemmerich 2015 ©
Superfluid state: | n i =
∏ | n iR
R
⇒
⇒
h n |bR+bR’| n i = h n |bR+| n iR h n |bR’| n iR’
S(R) =
∑ hbR’+ bR-R’i
=
=
n
n M = hNi
R’
⇒
hψ+(k)ψ(k)i =
|W(k)|2
hNi +
∑ eikR S(R)
= |W(k)|2
hNi
1 +
∑ eikR
R≠0
R≠0
hψ+(k)ψ(k)i can be large for all reciprocal lattice vectors within support[W(k)]
Mott-Insulator State:
(n!)-M/2 ∏ bR+
|{n}i =
n
|0i
R
⇒
h{n}|bR+bR’|{n}i = n δRR’
⇒
S(R) =
∑ hbR’+ bR-R’i
R’
⇒ hψ+(k)ψ(k)i =
|W(k)|2
= n δR,0
⇒
∑ eikR S(R)
= 0
R≠0
hNi
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Example: simple cubic lattice
configuration space
wave fronts, which can propagate in lattice
K = (2k,2k)
K = (2k,0)
λ
K = (4k,2k)
momentum space (reciprocal lattice)
+2k
2D projection
0
-2k
-2k
0
+2k
4k
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Experiment:
Greiner et al., Nature 419, 51 (2002)
prepare lattice, slowly ramp-up well depth,
ballistic expansion
112 Universität Hamburg, Bose-Einstein Condensation, SoSe 2015
coherence can be restored
Andreas Hemmerich 2015 ©
Phase decoherence for large U
Energy of |siR: Es =
U
2
⇒
e-n/2
=
∑
s
hn(t)|bR|n(t)iR =
∑
s!1/2
s
n s/2
|siR
|siR exp(-iEst/)
s!1/2
e-n
n s/2
n1/2
∏ | n iR , ψ(r) =
R
3U
|3iR
U
|2iR
0
|0iR, |1iR
∑
s
ns
s!
|hn(t)|bR|n(t)iR|2
n = 0.5
exp(-isUt/)
U > 0 ⇒ amplitude of matter wave field ψ(r)
collapses and revives !
|ni =
|4iR
s(s-1)
|n(t=0)iR = | n iR = e-n/2
|n(t)iR
6U
∑
R
bR W(r-R)
mean field: hn(t)|ψ(r)|n(t)i = hn(t)|bR|n(t)iR
mean density: |hn(t)|ψ(r)*ψ(r)|n(t)i|2 =
n
n=3
1
2π
2
U
t

⇒
∑ W(r-R)
R
∑ W(r-R)
R
Greiner et al., Nature 419, 51 (2002).
Experimental steps: slowly ramp-up well depth, stop shortly before phase transition, sudden
further increase of well depth to increase U, wait, ballistic expansion
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Imaging single atoms in an optical lattice
(M. Greiner, Harvard)
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Configuration space image of Mott insulator
(M. Greiner, Harvard)
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Orbital optical lattices
Can we access orbital degrees of freedom in optical lattices?
+ -
lowest band
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higher bands
Andreas Hemmerich 2015 ©
Bipartite optical lattice
Interference of two standing waves
E(x,y) = z I0 (cos(kx) + cos(ky) eiφ)
I(x,y) = I0 ( cos2(kx) + cos2(ky) + 2 cos(φ) cos(kx) cos(ky) )
Laser
cold
atoms
2λ
φ = 0°
φ < 90°
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φ = 90 °
φ > 90°
φ = 180°
Andreas Hemmerich 2015 ©
S-band lattice: dependence on θ
0.46
Lattice
118 Universität Hamburg, Bose-Einstein Condensation, SoSe 2015
1.BZ
θ [π]
0.5
0.54
Lattice
1.BZ
Andreas Hemmerich 2015 ©
Populating higher bands
A
B
A
B
A
B
A
B
A
A
B
A
B
A
B
A
B
A
119 Universität Hamburg, Bose-Einstein Condensation, SoSe 2015
ΔV
Andreas Hemmerich 2015 ©
Mapping of Brillouin zones:
→ Slowly ramp down lattice potential
→ Ballistic expansion
→ Absorption Image
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Energy minima in anisotropic 2nd band
1
-1
1
1
-1
1
-1
1
1
A
λ
-1
1
1
B
1
1
-1
A
1
1
-1
-1
-1
1
-1
-1
-1
1
-1
-1
-1
1
1
2λ
1
1
-1
-1
1
-1
-1
-1
1
-1
0
1
Bloch-function at condensation point
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Condensation at energy minima of 2nd band
1 ms
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3 ms
6 ms
13 ms
Andreas Hemmerich 2015 ©
Higher order Bragg peaks
2λ
e2
e1
±(1,-2)
±(3,-2)
±(1,0)
±(3,2) ±(1,2)
(-3,2)
(-1,0)
(-1,2)
(-3,-2)
(1,2)
(-1,-2)
(3,2)
(1,-2)
(1,0)
(3,-2)
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Energy minima in isotropic 2nd band
→ Deep wells support P-orbits
1
-1
Px ± i Py maximizes volume and hence
minimizes local collision energy
→ Shallow wells support S-orbits
1
-1
-1
1
1
+
+
i
-i
1
-1
-i
=
1
-1
i
1
=
-1
→ Tunneling maximized by matching
phases at tunneling junctions
k
-i
1
1
-1
-i
i
1
i
-1
-1
i
1
i
i
-1
-i
-i
-i
-1
2λ
-1
i
1
-1
-i
1
1
i
-i
-i
1
i
i
i
-i
-i
-1
1
i
-i
1
1
-1
-i
-1
i
i
-i
-1
-1
i
1
i
1
1
-1
-i
1
-i
-1
-i
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-1
1
-1
i
Andreas Hemmerich 2015 ©
Tuning lattice distortion
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Tuning occupation of local p-orbits via change of ΔV
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Interaction stabilizes complex-valued order
10
9
8
7
6
5
4
3
2
1
0
20
2.0
40
1.5
60
1.0
0.5
80
0.0
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