1. Institut fur Theoretische Physik
Seminar in theoretical physics: Non-linear and non-hermitian quantum mechanics
Bose-Einstein Condensation: Basics,Gross-Pitaevskii equation and Interactions
Kathrin KleinbachSupervisor: Apl. Prof. Dr. rer. nat. Jorg Main
25.04.2012
Introduction
First experimental Observation in 1995. Typically the velocitydistribution is shown before the condensate appears (A), shortlyafterwards (B) and very pure condensate (C). from [2]
Ideal Bose gas
Bosons
I spin s is integer
I bosons may occupy the same single-particle state(symmetric wave function → no Pauli-principle)
I for T = 0 all bosons are in the ground state N = N0 and thetotal wave function is a product of the single-particle wavefunctions: ψ(r1, r2, ..., rN) =
∏Ni=1 φ0(ri )
I for T > 0 bosons are in the thermal component as wellN = N0 + NT
I Bose-Einstein condensate when ground state is occupiedmacroscopically
Ideal Bose gas
neglect interaction between particles
grand canonical ensemble:
I chemical potential µ and temperature T are fixed
I εi energies of single-particle states in the system
Itotal energy: ε =
∑i εiNi
total number of particles: N =∑
i Ni
I the chemical potential µ can be seen as:
µ = ε(n)− ε(n − 1)
it can never exceed the energy of the ground state: µ ≤ ε0
Finding Tc and fraction of particles in the condensate
I grand canonical partition function Z for bosons
Z =∏
i
1
1− exp [(µ− εi )/kT ]
I thermal occupation of the ith energy eigenstate
Ni =kT∂
∂µln Zi
=1
exp [(εi − µ)/kT ]− 1
Finding Tc and fraction of particles in the condensate
I total number of particles
N = N0 +∑i 6=0
1
exp [(εi − µ)/kT ]− 1
∑i 6=0 →
∫∞>0 dε with the density of states D(ε) = Cαε
α−1
I density of states:free particle:
D(ε) =V
2π
(2m
~2
)3/2
ε1/2 = C3/2ε3/2−1
harmonic trap:
D(ε) =ε2
2~3ωxωyωz= C3ε
3−1
Finding Tc and fraction of particles in the condensate
I total number of particles
N = N0 + Cα
∫ ∞>0
εα−1
exp[(ε− µ)/kT ]− 1dε
I At Tc all particles are in the thermal component and µc = 0
NT (Tc , µ = 0) =Cα
∫ ∞>0
εα−1
exp [ε/kTc ]− 1dε
=Cα(kTc )αΓ(α)ζ(α) = N
with Γ(α) =∫∞0 xα−1 exp(−x)dx and ζ(α) =
∑∞n=1 n−α
Finding Tc and fraction of particles in the condensate
I Therefore Tc is for a free particle given by:
kTc =2π~2
m
(N/V
ζ(3/2)
)2/3
I compare to the thermal de-Broglie wavelength:
λc =
√2π~2
mkTc= ζ(3/2)1/3 1
(N/V )1/3
I Bose-Einstein condensation takes place when the mean freepath and the de-Broglie wavelength are of the same order ofmagnitude
Finding Tc and fraction of particles in the condensate
I Tc is for a harmonic trap given by:
kTc = ~(ωxωyωz )1/3
(N
ζ(3)
)1/3
typical critical temperatures in harmonic traps are 10−7K
I For T < Tc the chemical potential stays µ = 0
NT (T , µ = 0) = Cα(kT )αΓ(α)ζ(α)
= N
(T
Tc
)α
Finding Tc and fraction of particles in the condensateThe number of particles in the condensate is:
N0 = N
(1−
(T
Tc
)α)
Figure: Total number N and ground state fraction N0 as a function ofscaled temperature T/Tc . from [3]
Derivation of the Gross-Pitaevskii equation
many-particle Schrodinger equation
N∑i=1
− ~2
2m∆i + Vext(ri) +
1
2
N∑j=1j 6=i
W (ri, rj)
ψ(r1, .., rN) = Eψ(r1, .., rN)
Not solvable exactly! Assume that all particles are in the groundstate and use mean-field theory:
I ground state wave function is product of identicalsingle-particle wave functions:
ψ(r1, .., rN) =N∏
i=1
φ(ri)
I neglect correlations between particles
Derivation of the Gross-Pitaevskii equation
I vary single-particle wave function to minimize the total energy
Emf = 〈ψ(r1, ..., rN)|H |ψ(r1, ..., rN)〉
=
∫dr1...
∫drN
(N∏
i=1
φ∗(ri)
)
N∑i=1
− ~2
2m∆i + Vext(ri) +
1
2
N∑j=1j 6=i
W (ri, rj)
(
N∏i=1
φ(ri)
)
= −N~2
2m
∫drφ∗(r)∆φ(r) + N
∫drVext(r)|φ(r)|2
+1
2N(N − 1)
∫dr
∫dr′W (r, r′)|φ(r)|2|φ(r′)|2
Derivation of the Gross-Pitaevskii equation
I vary φ(r) but respect normalization∫drφ∗(r)φ(r) = 1
by using a Lagrange parameter µN
I the first variation with respect to φ∗(r) should be zero
I first variation of a functional:
δF (y)(f ) =d
dεF (y + εf )
∣∣∣∣ε=0
Derivation of the Gross-Pitaevskii equation
δ
(Emf − µN
(∫drφ∗(r)φ(r)− 1
))(f )
=
∫dr f
(−N
~2
2m∆φ(r) + NVext(r)φ(r)
+N(N − 1)
∫dr′|φ(r′)|2W (r, r′)φ(r)− µNφ(r)
)= 0
has to be zero for every test function f
I for a big number of particles N ≈ N − 1
I Gross-Pitaevskii equation[− ~2
2m∆ + Vext(r) + N
∫dr′|φ(r′)|2W (r, r′)
]φ(r) = µφ(r)
The Gross-Pitaevskii equation is a non-linear differentialequation
I existence and uniqueness of solutions of non-linear differentialequations are hard to show
I the superposition of two solutions of a non-linear differentialequation is not necessarily a solution to it as well
I usually symmetries of the problems have to be used to solve anon-linear differential equation
I two solutions φa, φb corresponding to different values µa, µb
are not orthogonal:∫
drφ∗aφb can be different from zero.
I for large values of N the many-particle wave functions becomeorthogonal: (∫
drφ∗aφb
)NN→∞−→ 0
Interactions in the condensate
I Gross-Pitaevskii equation:[− ~2
2m∆ + Vext(r) + N
∫dr′|φ(r′)|2W (r, r′)
]φ(r) = µφ(r)
I up to here: W (r, r′) is any arbitrary interaction between twoparticles
I in the condensate both long-range and short-rangeinteractions take place
I scattering (short-range interaction) will be discussed
Scattering Theory
I Scattering between two particles is described in relativecoordinates with the reduced mass µ
I Schrodinger-equation for the relative motion[− ~2
2µ∆ + V (r)
]ψ(r) = Eψ(r)
I Two particle interaction is described by Lennard-Jonespotential:
V (r) =C12
r12− C6
r6
Second part describes Van-der-Waals interaction
I Schrodinger equation for the relative motion is solved by:
ψ(r) = e ikr + f (θ)e ikr
r
incoming plane wave and a scattered spheric wavef (θ) :scattering amplitude depending on the potential
Scattering Theory
The scattering cross section is given by:
σ =4π
k2
∞∑l=0
(2l + 1) sin2 δl
with the phase shifts δl
I for short-range interaction (here 1/r6) phase shifts becomesmall for small k
I s-wave scattering (l = 0) becomes dominant
in this limit:
tan δ0 = −ak
σ = 4πδ20k2
= 4πa2
The constant a is called the scattering length.
Scattering Theory
the scattering amplitude is given by:
f (θ) =1
2ik
∞∑l=0
(2l + 1)Pl (cosθ)(e2iδl − 1)
with the Legendre polynoms Pl
I consider s-wave scattering (l = 0) → P0 = 1
I use tan δ0 = −ak
in this limit:
f (θ) =1
2ik(e2iδ0 − 1) = − a
1 + iak
Effective Potential
I Use an effective potential Veff which leads to the same f (θ)and σ section as the original potential:
Veff = V0δ(r)
I Solving the Schrodinger equation shows:
V0 =2π~2a
µ=
4π~2a
m
I For the Gross-Pitaevskii equation we do not use relativecoordinates:
Weff (r, r′) =4π~2a
mδ(r − r′)
Trapped Bose-Einstein Condensate with interactions
I The Gross-Pitaevskii equation including a harmonic trappingpotential and the effective scattering potential:[
− ~2
2m∆ +
1
2mω2
0r2 + N
4π~2a
m|φ(r)|2
]φ(r) = µφ(r)
can not easily be solved.
I For a = 0 the ground state of the harmonic oscillator is:
φ0(r) =1
π3/4a1/2osc
exp
[− r2
2a2osc
]with aosc =
√~/mω0.
I We assume that the interatomic interactions change thedimensions of the cloud: replace aosc by b
Trapped Bose-Einstein Condensate with interactions
I To find b: Same approach as derivation of Gross-Pitaevskiiequation. Minimize mean-field energy with respect to b.
I The total energy of the system is given by:
E (b) = 3N~ω0
(a2
osc
b2+
b2
a2osc
)+
N2U0
2(2π)3/2b3
I For large N: interaction energy per particle large compared to~ω0 → neglect kinetic energy term
Trapped Bose-Einstein Condensate with interactions
Figure: Mean-field energy of the Bose-Einstein Condensate including aharmonic trapping potential and scattering. from [4]
Trapped Bose-Einstein Condensate with interactions
I Solution is given by ground state wave function of theharmonic potential:
ψ(r1, r2, ..., rN) =N∏
i=1
1
π3/4b1/2exp
[−
r2i
2b2
]with the new oscillator length:
b =
(2
π
)1/10( Na
aosc
)1/5
aosc
I And the total energy is given by:
E =5N
4
(2
π
)1/5( Na
aosc
)2/5
~ω0
I but no solution for Naaosc≤ −0.671
Conclusions
I Below a critical temperature Tc Bose-Einstein condensationtakes place: The ground state is occupied macroscopically.
I A mean-field approach leads to the Gross-Pitaevskii equationwhich has to be fulfilled by the single-particle wave functionsin order to minimize the total energy in the system.
I Using an effective potential to describe scattering the energyand wave function of the system can be found.
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