Ideal Bose Gas

A Bose–Einstein condensate (BEC) is a state of matter of a dilute gas of bosons cooled to temperatures very close to 0K (usually ~100nK in experiments). Under such conditions, a large fraction of bosons occupies the lowest quantum state, at which point macroscopic quantum phenomena become apparent.

Overview

In this section, we summarize some basic and useful thermodynamic results for Bose-Einstein condensation in a uniform, non-interacting gas of bosons. Most of the discussion here will be limited to 3D case. Physics for lower dimensions will be mentioned in the end.

Phase Space Density

The fundamental difference between a BEC and a classical gas is the occupancy of a single-particle state. In a classical gas, the mean occupation number for a single quantum state satisfies the Boltzmann distribution $\langle n_{\nu }\rangle =e^{-(\epsilon _{\nu }-\mu )/kT}$ which is much less than unity. This feature is qualitatively captured by the ${\textit {Phase-spaceDensity}}$ defined as (3D, homogeneous gas)

$\rho _{D}=n\lambda _{T}^{3}\,,$

where $\lambda _{T}=(2\pi \hbar ^{2}/mkT)^{1/2}$ is the thermal de Broglie wavelength.

Some typical parameters for

• Classical thermal gas
  • Atom density $n\sim 10^{25}{\text{m}}^{-3}$
  • Interatomic distance $(1/n)^{1/3}\sim 3{\text{nm}}$
  • Thermal de Broglie wavelength Failed to parse (syntax error): {\displaystyle \lambda_T \sim 10^{-2} \text{nm}（T = 300K) }
  • $\rho _{D}\sim 10^{-8}$
• BEC in dilute gas
  • Atom density $n\sim 10^{20}{\text{m}}^{-3}$
  • Interatomic distance $(1/n)^{1/3}\sim 10^{2}{\text{nm}}$
  • Thermal de Broglie wavelength Failed to parse (syntax error): {\displaystyle \lambda_T \sim 10^{3} \text{nm}（T = 100nK) }
  • $\rho _{D}\sim 10^{2}$

The Bose-Einstein Distribution

For non-interacting bosons in thermodynamic equilibrium, the mean occupation number of the single-particle state $\nu$ is

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

$z=exp(\beta \mu )$ is defined as fugacity. At high temperature, the chemical potential lies below $\epsilon _{min}$. As temperature is lower, the chemical potential rises until it reaches $\epsilon _{min}$ and the mean occupation numbers increase.

Thermodynamics in Semi-classical Limits

We focused on the semi-classical case where $kT\gg \delta E$. Here $\delta E$ is the scale for enery level spacing in the trapping potential (for example, $\delta E=\hbar \omega$ for 3D harmonic trapping. This is usually valid in a real experiment where $kT\approx 2000Hz$ and $\delta E\approx 100Hz$. In this case, the system can be treated as a continuous excited energy spectrum plus a separated ground state. It seems to be contradictory to the nature of BEC when most of the population is found in the single ground state, but this description is a good enough approximation in many situations. The fully quantum description is necessary for some cases as we will see in the Pedagogical Example in end.

Transition Temperature

When $\nu =\epsilon _{min}$ , the occupation number on the ground state can be arbitrarily large, indicating the emergence of a condensate. The corresponding temperature is the transition temperature $T_{c}$. $T_{c}$ can be calculated with the criteria that the maximum number of particles can be held in the excited states is equal to the total particle number $N$. In the semi-classical limit where the sum over all states is replaced by an integral and simple assumption that $\epsilon _{min}=0$ we have

$N=N_{0}+N_{ex}\,.$

where we define

$N_{ex}:=\int \limits _{0}^{\infty }f(\epsilon ,\mu =0,T_{c})g(\epsilon )d\epsilon \,.$

Here $g(\epsilon )$ is the density of states. $N_{0}$ is the number of atoms in the ground state. Notice that the chemical potential $\mu =0$ is set to 0 (or in fact $\epsilon _{min}$ without any justifications. In fact, by setting $\mu =0$ what we are calculating here is the maximum possible number of atoms that can be accommodated by the "excited" states. If the total number of atoms is larger tan that, the rest must go to the ground state. A more rigorous calculation without this assumption can be found in the section Finite number effects.

The form of the transition temperature $T_{c}$ and therefore the condensate atom number depends strongly on the form of $g(\epsilon )$ which is affected by the dimension, trapping potential and the dispersion of the system. Under the most general assumption that $g(\epsilon )\propto \epsilon ^{\alpha -1}$, we reach

$N\propto C_{\alpha }(kT_{c})^{\alpha }\int \limits _{0}^{\infty }dx{\frac {x^{\alpha -1}}{e^{x}-1}}\,.$

where $x=\epsilon /(kT_{c})$. Straightforwardly we have a simple scaling function

$kT_{c}\propto N^{\frac {1}{\alpha }}\,.$

Some common cases are summarized below

cases:	3D box	2D box	3D Harmonic	2D Harmonic
$\alpha$	$3/2$	$1,diverge$	$3$	$2$

We see that the semi-classical picture is already good enough to capture some basic condensate physics. As a quick exaple, for the parameters in a typical AMO experiment (3D, harmonic trapping)

• $N\sim 10^{6}$
• harmonic trapping frequencies $\omega _{i}\sim 2\pi \times 100Hz$

We have $T_{c}\approx 450nK$

Thermodynamic Properties

The thermodynamic properties $H$can be readily calculated from the Bose distributions and sum over all the states.

$H=\int \limits _{0}^{\infty }H(\epsilon )f(\epsilon ,\mu ,T)g(\epsilon )d\epsilon \,.$

For example, the total energy

$E=\int \limits _{0}^{\infty }\epsilon f(\epsilon ,\mu ,T)g(\epsilon )d\epsilon \propto (kT)^{\alpha }\int \limits _{0}^{\infty }{\frac {x^{\alpha }}{e^{x}-1}}dx\,.$

We therefore can obtain a scaling law for all the importnat thermodynamic quantities as listed below assuming that $g(\epsilon )\propto \epsilon ^{\alpha -1}$ and $H\propto T^{\delta }$

Thermodynamic Property:	E	$C_{V}=\partial E/\partial T$	Entropy S
$\delta$	$\alpha +1$	$\alpha$	$\alpha$

It is also useful to express the relationship with dimensionless parameter $\gamma =T/T_{c}$ considering $N\propto T_{c}^{\alpha }$ we therefore obtain

$E\propto NkT\gamma ^{\alpha },C_{V}\propto Nk\gamma ^{\alpha },S\propto Nk\gamma ^{\alpha }\,.$

Beyond Semi-classical Limits

A more quantum way to deal with the system is by treating the energy levels as discrete and replace the integral with summation and also consider the constraint

$\sum _{i=0}^{\infty }f(\epsilon _{i},z,T)=N$

Notice that this time the ground state is not separated from the summation. Given the energy spectrum $\epsilon _{i}$ which is often determined by the trapping potential, and temperature $T$, the constraint allows the calculation of $z(N,T)$ at least numerically. We then immediately get the ground state occupation $N_{0}={\frac {z}{1-z}}$.

Superfluidity and Coherence Length

A Bose-Einstein condensate offers an example of superfluidity. One of the signatures of superfluidity is the emergence of long-range order where atoms far apart are correlated. we shall see that macroscopic occupation of a single particle state produces such order.

Mathematically, we characterise the correlation between atoms at position ${\vec {r}}$ and ${\vec {r}}'$ with

$\rho ^{(1)}({\vec {r}},{\vec {r}}')=\langle {\hat {\psi }}^{\dagger }({\vec {r}}){\hat {\psi }}({\vec {r}}')\rangle$

The term $\rho ^{(1)}({\vec {r}},{\vec {r}}')$ is sometimes also called real space first-order correlation function. We notice that $\rho ^{(1)}({\vec {r}},{\vec {r}})$ is nothing but the atomic density at potision ${\vec {r}}$.

In a homogeneous system, $\rho ^{(1)}({\vec {r}},{\vec {r}}')$ depends only on the relative coordinate ${\vec {s}}={\vec {r}}-{\vec {r}}'$. The correlation function then reveals the distribution in the momenutm space.

$\rho ^{(1)}({\vec {s}})={\frac {1}{V}}\int d{\vec {p}}\rho ({\vec {p}})e^{i{\vec {p}}\cdot {\vec {s}}}$.

We the value $\rho ^{(1)}({\vec {s}})$

Some Remarks

Back to: Quantum gases