Ideal Fermi Gas

We talk about basics for an ideal Fermi gas.

Fermi-Dirac distribution

The particles in an atom trap are isolated from the surroundings, thus the atom number $N$ and total energy content $E_{\rm {tot}}$ of the atomic cloud is fixed. However, it is convenient to consider the system to be in contact with a reservoir, with which it can exchange particles and energy (grand canonical ensemble). For non-interacting particles with single-particle energies $\epsilon _{i}$, the average occupation of state $i$ is

$f(\epsilon _{i},\mu ,T)={\frac {1}{e^{(\epsilon _{i}-\mu )/kT}+1}}\,.$

These is the Fermi-Dirac distribution. For a fixed number of particles $N$ one chooses the chemical potential $\mu$ such that $N\sum _{i}f(\epsilon _{i},\mu ,T)$.

Trapped Fermi Gas

Harmonic Trap

Applying these distributions to particles confined in a harmonic trap, with trapping potential

$V({\vec {r}})={\frac {1}{2}}m(\omega _{x}^{2}x^{2}+\omega _{y}^{2}y^{2}+\omega _{z}^{2}z^{2})\,.$

We assume that the thermal energy $kT\equiv 1/\beta$ is much larger than the quantum mechanical level spacings $\hbar \omega _{x,y,z}$ (Thomas-Fermi approximation). In this case, the occupation of a phase space cell $\left\{{\vec {r}},{\vec {p}}\right\}$ (which is the phase-space density times $h^3$) is given by

$f({\vec {r}},{\vec {p}})={\frac {1}{e^{({\frac {{\vec {p}}^{2}}{2m}}+V({\vec {r}})-\mu )/kT}+1}}\,.$

The density distribution of the thermal gas is

$n_{th}({\vec {r}})=-{\frac {1}{\lambda _{dB}^{3}}}\,{\rm {Li}}_{3/2}\left(-e^{\beta \left(\mu -V({\vec {r}})\right)}\right)\,.$

where ${\sqrt {\frac {2\pi \hbar ^{2}}{mk_{B}T}}}$ is the de Broglie wavelength. ${\rm {Li}}_{n}(z)$ is the $n^{th}$-order Polylogarithm, defined as

${\rm {Li}}_{n}(z)\;\equiv \;{\frac {1}{\pi ^{n}}}\int {\rm {d}}^{2n}r{\frac {1}{e^{{\vec {r}}^{2}}/z-1}}\;{\stackrel {n\neq 0}{=}}\;{\frac {1}{\Gamma (n)}}\int _{0}^{\infty }{\rm {d}}q{\frac {q^{n-1}}{e^{q}/z-1}}\,.$

where the first integral is over $2n$ dimensions, ${\vec {r}}$ is the radius vector in $2n$ dimensions, $n$ is any positive half-integer or zero and $\Gamma (n)$ is the Gamma-function. The Polylogarithm can be expressed as a sum ${\rm {Li}}_{n}(z)=\sum _{k=1}^{\infty }{\frac {z^{k}}{k^{n}}}$ which is often used as the definition of the Polylogarithm. This expression is valid for all complex numbers $n$ and $z$ where $|z|\leq 1$. The definition given in the text is valid for all $z\leq l$.

Special cases: ${\rm {Li}}_{0}(z)={\frac {1}{1/z-1}}$, ${\rm {Li_{1}}}(z)=-\ln(1-z)$. $f({\vec {r}},{\vec {p}})$ can be written as $\pm {\rm {Li}}_{0}(-\exp[\beta (\mu -{\frac {{\vec {p}}^{2}}{2m}}-V({\vec {r}}))])$. When integrating density distributions to obtain column densities, a useful formula is:

$\int _{-\infty }^{\infty }dx\;{\rm {Li}}_{n}(z\,e^{-x^{2}})={\sqrt {\pi }}\;{\rm {Li}}_{n+1/2}(z)\,.$

Limiting values: ${\rm Li}_n(z) \stackrel{z \ll 1}{\rightarrow} z$ and $-{\rm Li}_n(-z) \stackrel{z\rightarrow\infty}{\rightarrow} \frac{1}{\Gamma(n+1)}\; \ln^n(z)$.}. Note that expression~\ref{e:density} is correct for any potential $V(\vec{r})$. The constraint on the number of thermal particles is \begin{equation} N_{th} = \Int{r} \; n_{th}(\vec{r}) \end{equation} For a harmonic potential (~\ref{e:potential}), we obtain \begin{equation} N_{th} = \pm \left(\frac{k_B T}{\hbar \bar{\omega}}\right)^3 {\rm Li}_3(\pm\,e^{\beta\mu}) \label{e:numberofatoms} \end{equation} with $\bar{\omega} = (\omega_x \omega_y \omega_z)^{1/3}$ the geometric mean of the trapping frequencies.

In the classical limit at high temperature, we recover the Maxwell-Boltzmann result of a gaussian distribution, \begin{equation}

   n_{cl}(\vec{r}) = \frac{N}{\pi^{3/2} \sigma_x \sigma_y \sigma_z} e^{- \sum_i x_i^2/\sigma_{x_i}^2} \qquad {\rm with} \; \sigma_{x,y,z}^2 = \frac{2 k_B T}{m \omega_{x,y,z}^2}

\end{equation}

The regime of quantum degeneracy is reached when $\lambda_{dB} \approx n ^{-1/3}$, or when the temperature $T \approx T_{\rm deg}$. The degeneracy temperature $T_{\rm deg} = \frac{\hbar^2}{2m k_B} n^{2/3}$ is around or below one $\mu \rm K$ for typical experimental conditions.

For {\bf bosons}, it is at this point that the ground state becomes macroscopically occupied and the condensate forms. The density profile of the ideal gas condensate is given by the square of the harmonic oscillator ground state wave function: \begin{equation} n_c(\vec{r}) = \frac{N_0}{\pi^{3/2} d_x d_y d_z} e^{-\sum_i x_i^2/d_{x_i}^2} \end{equation} where $d_{x_i} = \sqrt{\frac{\hbar}{m \omega_{x_i}}}$ are the harmonic oscillator lengths. The density profile of the thermal, non-condensed component can be obtained from Eq.~\ref{e:density} if the chemical potential $\mu$ is known. As the number of condensed bosons $N_0$ grows to be significantly larger than 1, the chemical potential $\mu \approx - \frac{k_B T}{N_0}$ (from Eq.~\ref{e:BoseFermidist} for $E_0 = 0$) will be much closer to the ground state energy than the first excited harmonic oscillator state. Thus we set $\mu = 0$ in the expression for the non-condensed density $n_{th}$ and number $N_{th}$ and obtain \begin{eqnarray} n_{th}(\vec{r}) &=& \frac{1}{\lambda_{dB}^3} {\rm Li}_{3/2}(e^{-V(\vec{r})/k_B T})\\ N_{th} &=& N (T/T_C)^3\qquad \mbox{for $T<T_C$} \end{eqnarray} with the critical temperature for Bose-Einstein condensation in a harmonic trap \begin{equation} T_C \equiv \hbar \bar{\omega}\; (N / \zeta(3))^{1/3} = 0.94 \; \hbar \bar{\omega} N^{1/3} \end{equation} where $\zeta(3) = {\rm Li}_3(1) \approx 1.202$. At $T=T_C$, the condition for Bose condensation is fulfilled in the center of the trap, $n = {\rm Li}_{3/2}(1)/\lambda_{dB}^3 = 2.612/\lambda_{dB}^3$. For lower temperatures, the maximum density of the thermal cloud is ``quantum saturated at the critical value $n_{th} = 2.612/\lambda_{dB}^3 \propto T^{3/2}$. The condensate fraction in a harmonic trap is given by \begin{equation} N_0/N = 1 - (T/T_C)^3 \end{equation} For $T/T_C = 0.5$ the condensate fraction is already about 90\%.

For {\bf fermions}, the occupation of available phase space cells smoothly approaches unity without any sudden transition: \begin{equation}

 f(\vec{r},\vec{p}) = \frac{1}{e^{(\frac{\vec{p}^2}{2m} + V(\vec{r}) - \mu)/k_B T} + 1} \stackrel{T \rightarrow 0} \rightarrow \left\{%

\begin{array}{ll}

   1, & \hbox{$\frac{\vec{p}^2}{2m} + V(\vec{r}) < \mu$} \\
   0, & \hbox{$\frac{\vec{p}^2}{2m} + V(\vec{r}) > \mu$} \\

\end{array}% \right. \label{e:fermiphasespace} \end{equation} Accordingly, also the density profile changes smoothly from its gaussian form at high temperatures to its zero temperature shape: \begin{eqnarray} n_F(\vect{r}) &=& \Intp{p} \, f(\vect{r},\vect{p}) \stackrel{T\rightarrow 0}{\rightarrow} \int_{\left|\vect{p}\right|< \sqrt{2m(\mu-V(\vect{r}))}} \frac{{\rm d}^3\vect{p}}{(2\pi\hbar)^3}\nonumber\\ &=& \frac{1}{6\pi^2} \left(\frac{2m}{\hbar^2}\right)^{3/2} \left(\mu - V(\vect{r})\right)^{3/2}. \end{eqnarray}

From Eq.~\ref{e:fermiphasespace} we observe that at zero temperature, $\mu$ is the energy of the highest occupied state of the non-interacting Fermi gas, also called the Fermi energy $E_F$. The (globally) largest momentum is $p_F \equiv \hbar k_F \equiv \sqrt{2 m E_F}$, the Fermi momentum. {\it Locally}, at position $\vect{r}$ in the trap, it is $p_F(\vect{r}) \equiv \hbar k_F(\vect{r}) \equiv \sqrt{2 m \epsilon_F(\vect{r})} \equiv \hbar (6\pi^2 n_F(\vect{r}))^{1/3}$ with the local Fermi energy $\epsilon_F(\vect{r})$ which equals $\mu(\vect{r},T=0) = E_F - V(\vect{r})$. The value of $E_F$ is fixed by the number of fermions $N$, occupying the $N$ lowest energy states of the trap. For a harmonic trap we obtain \begin{eqnarray}

   N &=& \Int{r} \; n_F(\vect{r}) = \frac{1}{6} \left(\frac{E_F}{\hbar \bar{\omega}}\right)^3\nonumber\\

\Rightarrow E_F &=& \hbar \bar{\omega} (6 N)^{1/3}

   \label{e:Ferminumber}

\end{eqnarray} and for the zero-temperature profile \begin{eqnarray} n_F(\vect{r}) &=& \frac{8}{\pi^2} \frac{N}{R_{Fx} R_{Fy} R_{Fz}} \; \left[\max \left(1 - \sum_i \frac{x_i^2}{R_{Fi}^2},0\right)\right]^{3/2} \label{e:Fermidensity} \end{eqnarray} with the Fermi radii $R_{F{x,y,z}} = \sqrt{\frac{2 E_F}{m \omega_{x,y,z}^2}}$. The profile of the degenerate Fermi gas has a rather flat top compared to the gaussian profile of a thermal cloud, as the occupancy of available phase space cells saturates at unity.

At finite $T \lesssim T_F$, we can understand the shape of the cloud by comparing $k_B T$ with the local Fermi energy $\epsilon_F(\vect{r})$. For the outer regions in the trap where $k_B T \gg \epsilon_F(\vect{r})$, the gas shows a classical (Boltzmann) density distribution $n(\vect{r}) \propto e^{-\beta V(\vect{r})}$. In the inner part of the cloud where $k_B T \ll \epsilon_F(\vect{r})$, the density is of the zero-temperature form $n(\vect{r}) \propto (E_F - V(\vect{r}))^{3/2}$. The Polylogarithm smoothly interpolates between the two regimes. We notice here the difficulty of thermometry for very cold Fermi clouds: Temperature only affects the far wings of the density distribution. While for thermal clouds above $T_F$, the size of the cloud is a direct measure of temperature, for cold Fermi clouds one needs to extract the temperature from the shape of the distribution's wings.

Note that the validity of the above derivation required the Fermi energy $E_F$ to be much larger than the level spacing $\hbar \omega_{x,y,z}$. For example, in very elongated traps and for low atom numbers one can have a situation where this condition is violated in the tightly confining radial dimensions.