Dilute Bose gas in 3D

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Dilute Bose gas in 3D

• Wei Bobo
• Jan 8, 2009

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Outline

• Dilute Bose gas.
• Mean field treatment.
• Beyond mean field.
• Pair distribution function.
• Summary and references.

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Ideal Bose gas

• There are two characteristic lengths
• The mean particle distance
• Thermal de Broglie wavelength

The number of atoms in the lowest quantum state
is proportional to the so called phase-space
density, and has to exceed a critical number of
2.612 to achieve Bose-Einstein condensation.
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Dilute Bose gas

• Accessible in experiments.
• Weakly interacting Bose gas
• There are one more characteristic length range
  of interacting potential
• At very low temperature, the effective
  interaction can be described by the s wave
  scattering length. which can be considered as
  hard-sphere diameter
• Dilute condition

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Effective Hamiltonian
The commonly used effective interaction is
Effective Hamiltonian Where
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Mean field treatment
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Mean field
It can not describe the correlations between the
N atoms. The best state given by this variational
approach will describe how the state of each atom
is modified by the mean field of the N-1 other
atoms. So we can use Lagrange multiplier to
get the best single particle state.
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Mean field
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Mean field treatment

• By Minimization procedure
• This equation describes the evolution of each
  atoms in the trapping potential and in the mean
  field created at its position by the remaining
  N-1 atoms.

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Mean field treatment
For dilute gases, the potential takes the form
Finally, we have the Gross-Pitaevskii equation
which plays a central role in the study of the
properties of Bose-Einstein condensation.
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Mean field energy

• The spatial density
• The Kinetic energy
• The Kinetic energy comes from the
  confinement.
• Trapping energy
• Interaction energy

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Mean field ground state energy

• For the homogenous system we are considering,
  there is no trapping potential, then
• Total energy
• Finally, we have mean field ground state energy

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Beyond mean field
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Beyond mean field

• How far does the system differ from the product
  of single particle state?
• T0, all the particles are not stay at one single
  particle state.
• What is the number of particles occupy the other
  states?
• Quantum depletion.
• Thermal depletion.
• What is the elementary excitations?

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Second quantized Hamiltonian
Expand the field operators into free particle
fields
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Bogoliubov approximation

• 1. In the dilute gases near T0, the population
  of k0 state is macroscopic compared to the total
  population of the other states.
• Thus the matrix element of are on
  the order of . We will neglect the non
  commutation of them and make a c-number
  substitution

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Bogliubov approximation

• 2. In the interaction terms, there are terms
  including 4,2,0 of , and no terms
  with odd number of due to the
  conservation of momentum.
• For large value of N we are considering, they
  become three categories in magnitude

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Effective Hamiltonian

• By considering terms of categories (1) and (2),
  we would obtain the dominant term of the energy
  of the system
• This is a quadratic expression of creation and
  annihilation operators, which can be diagonalized
  by Bogoliubov transformation.

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Bogoliubov transformation

• Introducing
• Constrain
• One have
• New expressions of the Hamiltonian

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Bogoliubov transformation

• In order to diagonalize the Hamiltonian, we must
  have

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Ground state energy

• When one evaluate E0, it diverges. This is
  because the potential we have used. When one do
  Fourier transform, it is a constant.
• To solve this problem, LHY proposed
  pseudo-potential

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Ground state energy
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Ground state and excited state
The ground state is that all the oscillators
stay in their ground state, which satisfies
The first excited states with momentum K is
So the quasi-particle appears as a linear
superposition of ordinary particles with momentum
K and ordinary hole with momentum K.
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Low energy excitations
At low energy excitations, K is very small
So we have obtained the phonon dispersion
relations for the excitations of a weakly
interacting Bose gas. The repulsive interactions
change the elementary excitations at long wave
lengths from free particles to Phonons.
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Quantum depletion
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Physical picture of ground state

• The ground state turns to be a collection of n
  pairs of particles, each pair consists of two
  particles with momentum k and k.
• Average number of pairs
• The pairs has a correlation length
• Within one correlation length, the number of
  particles is, on the average

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Physical picture of ground state

• Within one correlations, the number of particles
  have k not equal to 0 is
• In other words, within one correlation length,
  there are many particles, but only few of them
  with k not equal to 0.

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Pair distribution function

• Pair distribution function is defined as
• Physical meaning of pair distribution function
• Given a particle at a point, the average
  number of particles in dr at a distance r is
  rouD(r)dr.
• Thus D(r) extends to 1 as r goes to infinity.
• It is measurable in experiment.
• We evaluate D(r) based on the wave function of
  the system.

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Pair distribution function
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Summary

• The mean field gives a good description if the
  system are sufficiently dilute.
• Beyond mean field can be obtained based on
  Bogliubov approximation.
• We have evaluated the pair distribution function
  of dilute Bose gas based on the ground state wave
  function.

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References

• Lecture notes of BEC by Prof.Cohen-Tannoudji at
  CityU.
• Bose-Einstein condensation in the alkali gases
  Some fundamental concepts, REVIEWS OF MODERN
  PHYSICS, 73,307 (2001).
• Statistical mechanics, K. Huang,1987.