Ideal Bose Gas

A Bose–Einstein condensate (BEC) is a defined as the occurrence of the macroscopic occupation of one-particle states. When a dilute gas of bosons cooled to temperatures very close to 0K (usually ~100nK in experiments), 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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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) }
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu } is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_{min} } . As temperature is lower, the chemical potential rises until it reaches Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta E} is the scale for enery level spacing in the trapping potential (for example, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta E = \hbar \omega} for 3D harmonic trapping. This is usually valid in a real experiment where $kT\approx 2000Hz$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_c } . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N = N_0 + N_{ex} \,. }

where we define

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_{ex} := \int\limits_{0}^{\infty} f(\epsilon, \mu=0, T_c) g(\epsilon) d \epsilon \,. }

Here $g(\epsilon )$ is the density of states. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_0 } is the number of atoms in the ground state. Notice that the chemical potential Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu = 0} is set to 0 (or in fact $\epsilon _{min}$ without any justifications. In fact, by setting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_c } and therefore the condensate atom number depends strongly on the form of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N \propto C_{\alpha}(kT_c)^{\alpha}\int\limits_{0}^{\infty} dx\frac{x^{\alpha-1}}{e^{x} - 1} \,. }

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = \epsilon/(kT_c)} . Straightforwardly we have a simple scaling function

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle kT_c \propto N^{\frac{1}{\alpha}} \,. }

Some common cases are summarized below

cases:	3D box	2D box	3D Harmonic	2D Harmonic
$\alpha$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3/2}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1, diverge}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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)

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N \sim 10^6 }
• harmonic trapping frequencies Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_i \sim 2\pi \times100Hz }

We have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_c \approx 450nK }

Thermodynamic Properties

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_V=\partial E/\partial T}	Entropy S
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha + 1}	$\alpha$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha}

It is also useful to express the relationship with dimensionless parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma = T/T_c} considering Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_i} which is often determined by the trapping potential, and temperature Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} , the constraint allows the calculation of $z(N,T)$ at least numerically. We then immediately get the ground state occupation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_0 = \frac{z}{1-z}} .

Off-Diagonal Long-Range Order

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{r}} and ${\vec {r}}'$ with

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho^{(1)}(\vec{r}, \vec{r}')= \langle \hat{\psi}^{\dagger}(\vec{r})\hat{\psi}(\vec{r}') \rangle \,. }

The term Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{r}} .

In a homogeneous system, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho^{(1)}(\vec{s})= \frac{1}{V}\int d\vec{p} \rho(\vec{p})e^{i\vec{p}\cdot\vec{s}} \,. }

• In a thermal gas, we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho^{(1)}(\vec{s}) \rightarrow 0 } as $|vec{s}|\rightarrow \infty$.
• In a BEC, we have macrosopically populated state ( for example in a box we populate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\vec{p} = 0 \rangle } ), we then have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho^{(1)}(\vec{s}) \rightarrow \sim \frac{N_0}{V} } as $|{\vec {s}}|\rightarrow \infty$.

Some Remarks

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