bose-einstein condensation: basics, gross-pitaevskii ...€¦ · conclusions i below a critical...

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

Figure: Theoretical prediction of Bose-Einstein condensation in 1924.from [1]

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 GasThe Gross-Pitaevskii equationScattering theory

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


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


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


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


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


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


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


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


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


Figure: Mean-field energy of the Bose-Einstein Condensate including aharmonic trapping potential and scattering. from [4]


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.

A. Einstein.

Quantentheorie des einatomigen gases.SITZUNGSBERICHTE DER PREUSSICHEN AKADEMIE DER WISSENSCHAFTENPHYSIKALISCH-MATHEMATISCHE KLASSE 3, 1925.

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell.

Observation of bose-einstein condensation in a dilute atomic vapor.Science, 269(5221):198–201, 1995.

D. S. Jin, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell.

Collective excitations of a bose-einstein condensate in a dilute gas.Phys. Rev. Lett., 77:420–423, Jul 1996.

C.J. Pethick and H. Smith.

Bose-Einstein Condensation in Dilute Gases.Cambridge University Press, 2nd edition edition, 2008.

L. Pitaevskii and S. Stringari.

Bose-Einstein Condensation.Oxford Science Publications, 2003.

H. Cartarius, P. Christou, A. Eberspacher, R. Eichler, A. Junginger, P. Koberle, J. Main, S. Rau,

T. Schwidder, and M. Zimmer.Bose-einstein-kondensate mit langreichweitiger wechselwirkung, 2010.

Thanks for your attention!

bose-einstein condensation: basics, gross-pitaevskii ...€¦ · conclusions i below a critical...

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