Inhomogeneous Bose Gas

File:Superfluid to Mott insulator transition-bec3-first-expt

The physics of a BEC happens not just in momentum space, but also in position space, and it is useful to analyze it accordingly. With a trapping potential applied, the Hamiltonian is

$H=\int d^{3}r\psi ^{\dagger }(r)[{\frac {-\hbar ^{2}}{2m}}+V_{trap}]\psi (r)+{\frac {1}{2}}\int d^{3}r\int d^{3}r'+....$

This must be approximated, in the spirit of Bogolubov's momentum space approximation, to obtain a useful solution. We thus replace

${\hat {\psi }}(r,t)=\psi (r,t)+{\tilde {\psi }}(r,t)\,,$

where $\psi (r,t)$ is an expectation, and ${\tilde {\psi }}(r,t)$ captures the quantum (+ thermal) fluctuations. The resulting equation is a nonlinear Schr\"odinger equation (also known as a Gross-Pitaevskii equation):

$i\hbar {\frac {\partial \psi }{\partial t}}=\left[{-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V_{trap}+U_{0}N|\psi (r,t)|^{2}}\right]\psi (r,t)$

The $|\psi (r,t)|^{2}$ term captures a potential proportional to the density. In the mean field approximation, it is determined by the trapping potential. This equation can now be solved. In the Thomas-Fermi approximation, with positive (repulsive) interactions, there is a characteristic length $\xi$ which arises, known as the healing length,

$\xi =(8\pi an)^{-1/2}\,,$

arising from

${\frac {\hbar ^{2}}{2m\xi ^{2}}}=U_{0}h\,.$

File:Superfluid to Mott insulator transition-bec3-healing

If the interactions are really strong, the kinetic energy term can be neglected, because the interactions will keep the density constant in its spatial distribution. Such an approximation is the Thomas-Fermi approximation, giving an equation for the wavefunction,

$[V_{ext}+U_{0}|\psi (r,t)|^{2}-\mu ]\psi (r)=0\,,$

giving the solution

$|\psi (r,t)|^{2}={\frac {1}{U_{0}}}(\mu -V(r)\,.$

The wavefunction is essentially just the potential filled up to the chemical potential level, inverted. For a quadaratic potential, $V(r)=m\omega ^{2}r^{2}/2$, the chemical potential is

$\mu ={\frac {\hbar \omega }{2}}(15N_{a}/a_{osc})^{2/5}\,,$

where $x=15N_{a}/a_{osc}$ is a common term worth identifying, and $a_{osc}={\sqrt {\hbar ^{2}/m\omega }}$ is a characteristic length scale of the oscillator, its zero point motion. Defining $R=a_{osc}x^{1/5}$, we may find $\mu =m\omega ^{2}R^{2}/2$. This explains the profile of the condensate data obtained in experiments:

File:Superfluid to Mott insulator transition-bec3-bec-peak

\noindent Note that the size of the ground stat BEC is much larger than the zero-point motion of the harmonic oscillator. This is due to the pressure of the repulsive interactions. The Gross-Pitaevskii interaction gives not only the ground state wavefunction, but also the dynamics of the system. For example, it predicts soliton formation: stable wavefunctions with a size scale determined by a balance of the kinetic energy and the internal interactions. This requires, however, an attractive potential. Such soliton formation can nevertheless be seen in BEC's, with tight traps (see recent Paris experiments).

Length and energy scales in BEC

\begin{itemize}

• Size of atom: $a=3$ nm
• Separation between atoms $n^{1/3}=200$ nm
• Matter wavelength $\lambda _{dB}=1$ $\mu$m
• Size of confinement $a_{osc}=30$ $\mu$m

Note that

$a\ll n^{1/3}\leq \lambda _{dB}<2\pi \xi <a_{osc}\,.$

For a gas, $a\ll n^{1/3}$. For a BEC, $\lambda _{dB}\geq n^{1/3}$ in addition. The corresponding energy scales are also useful to identify. Let $\hbar \omega =a_{osc}$. Then:

$k_{B}T_{s-wave}\gg k_{B}T_{c}\geq k_{B}T>U_{int}>\hbar \omega$

The interaction energy scale $U_{int}\sim (h^{2}/m)na$, corresponding to the healing length.

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