**Part I – Abstract**

**I.1. INTRODUCTION**

**I.1.1. Basic description of a BEC**

One of the biggest goals in the domain of cold atom physics has been the observation of the Bose- Einstein condensation (BEC), as its accomplishment led to the observation of a variety of exciting phenomena [Deng99, Den00, Grein02, Abo01, Gret06, Billy08, Corn02], as well as to the development of new theoretical tool work for it’s description, to the extend that it is considered today an almost separate, and very widespread field of scientific research.

As predicted in 1925, the fundamental differences between the two different groups of particles, the bosons which have integer spin, and the fermions which have semi-integer spin, lead to very different behavior in low temperatures. Bose-Einstein condensation is actually a separate phase of matter, additional to the gas, the liquid, the solid and the plasma phase, that can be observed when a sample of bosons is cooled bellow a ‘critical temperature’, the value of which depends on the cooled species. It is a phase of matter that allows for the observation of various

exotic phenomena that can be explained only in the context of quantum mechanics^{1}.

Quantum mechanics points out the existence of ‘wave- matter’ duality in nature. For example, light that classical mechanics and electromagnetism consider as a wave, is viewed as a particle in quantum mechanics, the ‘photon’, without which, commonly observed phenomena like the photoelectric phenomena, find no explanation. For what is considered by classic mechanics as

‘matter’, quantum mechanics appoints wave properties. For a collection of particles the ‘wave’

qualities are embodied in the so-called de Broglie wavelength, λ*dB*

* _{dB}*=

###

^{m k}^{2}

^{ ℏ}

^{B}

^{T}^{2}

^{(1.1)}

This de Broglie wavelength is inversely proportional to the temperature and the mass, a fact which
explains why the wave qualities of matter are not evident in everyday life. In a sample consisted of
bosonic particles, when the temperature is very low, and the density is very high, the de Broglie
wavelength can become comparable to the inter-atomic distances. The product of λ*dB* with the
density of the atomic sample, the so-called ‘phase-space density’ increases with decreasing
temperature and increasing density. In conditions of ultra low temperature the sample is best
described as a wave, rather than a sum of many particles, and many phenomena related to wave
properties have been observed [And97].

The picture that is presented in Fig.I.1.1 gives the impression that the BEC is a phenomenon that occurs gradually as phase-space density increases, where actually it appears as a distinct phase transition. To understand why this happens we have to take a look to boson statistics and how these are transformed to describe BECs.

For a sample of bosonic atoms in a trapping potential, with a mean occupation number N*v* of
a single particle v with energy ε*v*, the maximum numbers of atoms that lie in any excited state is
given by

*N*_{}* _{v}*=

### ∑

*v=1*

∞ 1

e^{(}^{}^{v}^{−)/k}^{B}* ^{T}*−1 (1.2)

where µ is the chemical potential that is calculated upon the normalization condition *N*=

### ∑

*v= 0*

*N*_{}* _{v}* .
As the temperature lowers, µ increases in order to preserve the normalization condition. The
chemical potential cannot exceed ε

*0*the energy of the ground state since in this case the expression 1.2 would take an infinite value for v = 0. Thus, equation’s 1.2 limiting form for low temperature is 1The behavior of fermions in ultra-low temperatures, although similarly exotic and interesting, is not related

Fig.I.1.1: The transition from a classical gas to a BEC. Initially (A) the sample behaves as a collection of particles, but as the temperature decreases and the density increases or remains constant (B), the de Broglie wavelength increases, and gradually becomes comparable to the inter-atomic distances (C). Finally, the sample is best described as by a singe wave function as all the atoms participate to the condensate (D).

Figure adapted from [Web03b]

*N*_{ex.max}=

### ∑

*v=1*

∞ 1

e^{(}^{v}^{−}^{0}^{)/k}^{B}* ^{T}*−1 (1.3)

giving the maximum number of the particles that can occupy all excited states. Relation 1.3 is
useful in order to derive a qualitative criterion for the BEC phase transition, and for the critical
temperature Tc for which this occurs. BEC occurs when the number of particles that occupy all
excited states, N*ex,max* becomes comparable to the number of particles in the ground state N*0*, and the
onset of this transition can be defined as the point where N*ex,max**= N**0*. The critical temperature can by
calculated according to this criterion and is found to be [Peth02]

*T** _{C}*=

*N*

^{1}

^{/a}

*k*

*B*

[C* _{a}* a a]

^{1/a}(1.4)

where C*α* is a constant, ζ(α) is the Riemann zeta function, Γ(α) is the gamma function and a is a
constant that depends on the trapping potential geometry, and takes the value 3 for harmonic traps.

We note that k*B**T**c* is usually much higher than the energy difference between the ground and the
first excited state; for this reason BEC does not result to a gradual ‘washing out’ of motion in the
trapping potential, but as a distinct phase transition.

In the BEC experiments that are realized with neutral atoms, this phase transition has an experimental ‘signature’, which is the anisotropic expansion of the BEC when it is released from the trapping potential and falls under the influence of gravity. To explain this anisotropic expansion, we need to take a look of the thermodynamic properties of the BEC in the trapping potential. The mathematical description necessary for the derivation of the basic thermodynamic properties of a BEC, can be done in the context of many-particle theory. However, this complicated process is usually avoided and most basic properties of condensates are derived with the use of the Gross-Pitaevskii (GP) equation:

*i*ℏ ∂

∂*t* r=−ℏ^{2}∇^{2}

2*m* U* _{ext}*

*r*

*g*∣

*r*∣

^{2}

*r* (1.5)

The GP equation resembles to the Schrödinger equation. Nevertheless, an important point is that it is an equation written not for an atomic wavefunction, but for the so-called “order parameter”

*Φ(r), which is a complex quantity which if risen in square gives the condensate's density*^{1}. Apart
from the order parameter, U*ext**(r) is the external trapping potential and * *g*= 4*h*2*a*/*m* with a
being the scattering length. The scattering length is a very important quantity, since in low
temperatures, all collision properties can be parametrized with respect to this single parameter
[Kohl07]. Its value controls the nature and magnitude of the inter-atomic interactions, since when it
takes positive values, the interactions of the sample are repulsive and when it takes negative values
the interactions are attractive. Moreover, the value of the scattering length depends on the magnetic
field, a fact that gives rise to a very important ultra-cold atom phenomenon, the Feshbach resonance
in which we will refer to in the following paragraphs.

Equation 1.5 can be solved to give the density profile of the condensate. In the case of repulsive interactions, the interactions creates a virtual potential, the mean-field potential, which cancels out the trapping potential and stabilizes temporally the BEC’s density profile. If the kinetic energy term in the GP equation is neglected, (this is widely known as the Tomas-Fermi approximation), the cancellation of these potentials gives rise to a flat effective potential and results to an inverted parabola shape for the density of the condensate, which reflects the shape of the trapping potential.

1 However, the order parameter is often refereed to as 'condensate' wavefunction.

After the trapping potential is turned off, the mean field energy, that results from the interaction and that follows the same inverted parabola shape as the density of the BEC, is converted to kinetic energy and leads to an asymmetric ballistic expansion for the condensate. This asymmetric ballistic expansion is an experimental ‘signature’ for condensation and helps to distinguish BECs from dense but not condensed atomic samples.

The creation of BECs with negative scattering length and thus attractive inter-atomic interactions is also possible. In such condensates the attractive interaction is usually balanced by the kinetic energy a fact that can lead to a dynamic stabilization of the BEC. That puts an upper limit to the number of atoms that can participate to negative scattering length BECs [Peth02].