Solutions of the GGPE vs further approximations

The transformed GGPE

− 1

2m∇²φ( r) +V_trap( r)φ( r) +g|φ( r)|²φ( r) = µφ( r) . (2.8) is more convenient to study because the parameter

g = 4πa_sc

m N (2.9)

incorporates both interaction and number of particles. Then g is the only variable parameter for the system in a given trap. In the absence of an analytical solution, Equation (2.8) is solved numerically in the cylindrical coordinate system. The wave function φ( r) is obtained by expressing the radial and axial components as linear products of B splines (Appendix B) and an exponential angular part with quantum number m,

φ(r) =

nρ

X

i nz

X

j

C_ijB_i,kρ(ρ)B_j,kz(z)e^imφ. (2.10) The GGPE is solved numerically using the following iterative procedure. The solution of Equation (2.7) obtained for no interaction situation is used for the construction of the interaction term. Then the GGPE with this interaction term is solved, yielding the value for the first iteration. This solution is used to construct the new interaction term. The procedure is repeated until convergence is reached.

Numerical solutions of the GGPE can be used in order to check avaliable approximations. One of these approximations is the Thomas-Fermi approxi-mation (TFA). It implies that the interaction is so strong or that the number of particles is so large, that the kinetic energy term in the GGPE can be ignored. The solution is then trivial and is given as

φ(r) =

s 1

U₀N [µ−V_trap(r)] (2.11)

2.1 Mean-field approach

where the chemical potential is µ= 1 2 Another sometimes invoked approximation is based on the variational prin-ciple (VP). It gives an upper bound for the ground state energy. The as-sumption about the ground state is

φ(r) = treating the effective frequencies ˜ω_ρ and ˜ω_z as the variable parameters. The substitution of (2.12) into the GGP energy functional yields the ground state energy

Figure 2.1 shows the wave functions of the GGPE against TFA and VP.

This figure clearly shows the correctness of the numerical results. As is also

0 1 2 3 4 5 6

Wave function φ (units of aρ-3/2)

-6 -4 -2 0 2 4 6

Wave function φ (units of az-3/2)

Figure 2.1: Numerical solution of the GGPE (black and green solids) together with the TFA for 20000 atoms (blue dashes) and the VP with Gaussians for 10 atoms (red dashes).

The calculations are done for⁸⁷Rb atoms in an isotropic trap ofωρ=ωz = 2π×100kHz interacting repulsively witha_sc= 1a₀.

evident from Figure 2.1, as the number of particles increases, the repulsion

Chapter 2. Investigation of ultracold many-body systems

between atoms tends to lower the central density, which expands the cloud of atoms towards the regions where the trapping potential is higher.

An important feature is the difference between systems interacting with either repulsive or with attractive forces. If interaction between particles is attractive (a_sc <0), then the solution of GGPE is metastable. In this case, if the number of particles in the condensate is sufficiently large, it becomes unstable and collapses. However, this case will not be considered in this work.

Using the GGPE, it is possible to discuss various ground-state properties of the system: the form of the atomic cloud, the role of the interatomic potential, and the velocity distribution. An important question is the role of the interatomic potential. At first sight it is expected to be negligible for such a dilute system like the BEC. However, the interaction has a deep influence on how the GGPE is solved.

0 1 2 3 4

Wave function φ (units of aρ-3/2) ^0.01

0.05

Wave function φ (units of az-3/2)

0 5 10 15

Wave function φ (units of aρ-3/2)

10

Wave function φ (units of az-3/2)

Figure 2.2: The ground state wave functions of the radial and the transverse motion of 20000⁸⁷Rb atoms in the pancake-shaped harmonic trap of frequenciesωρ= 2π×100Hz andωz= 2π×10kHz. The variable parameter (2.9) is indicated in the figure.

In order to understand how the behavior of the condensate changes as

2.1 Mean-field approach

the interaction strength varies, further solutions of the GGPE are considered here. Figure 2.2 shows the wave functions of the radial and longitudinal motion for the pancake-shaped geometry of the trap. Figure 2.3 shows the chemical potential as a function of g-factor (2.9). Figure 2.4 shows the wave

0 0.2 0.4 0.6 0.8 1

Figure 2.3: The chemical potential µ as a function of the g-factor for the solutions of Figure 2.2

functions of the radial and longitudinal motion for the cigar-shaped geome-try of the trap. Figure 2.5 shows the chemical potential as a function of the g-factor (2.9). As is evident from Figure 2.2 and Figure 2.4, an increase

0 1 2 3 4

Wave function φ (units of aρ-3/2) ^0.01

0.05

Wave function φ (units of az-3/2)

Figure 2.4: The ground state wave functions of the radial and the transverse motion of 20000⁸⁷Rb atoms in the cigar-shaped harmonic trap of frequenciesω_ρ= 2π×10kHz and ωz= 2π×100Hz. The variation parameter (2.9) is indicated in the figure.

in interaction between particles lowers the central density (it becomes rather flat) expanding the cloud of atoms towards regions where the trapping poten-tial is higher. The final result is that the system is still fully condensed, but

Chapter 2. Investigation of ultracold many-body systems

0 0.2 0.4 0.6 0.8 1

g factor 1.05

1.1 1.15

µ (units of ωρ)

Figure 2.5: The chemical potential µ as a function of the g-factor for the solutions of Figure 2.4

the structure of its wave function can be strongly affected by the interatomic forces.

The GGPE is valid if the gas of atoms is dilute. BEC satisfies this condi-tion. The diluteness parameter is n a³_sc where n is the density of the sample.

As long as this parameter small, the mean-field description should be ac-curate. The GGPE is formulated in the limit of zero temperature and so corrections are expected as the temperature of the gas increases. In order to investigate the effects of the density and finite temperature the beyond-mean-field (BMF) description is required.