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. We focused on the semi-classical case 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 kT \gg \delta E} . Here $\delta E$ is the scale for enery splitting 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 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 \approx 2000 Hz} 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 E}

It seems to be contradictary 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. This assumption is usually valid in the

Thermodynamics of a Bose Gas

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 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 \langle n_{\nu} \rangle = e^{-(\epsilon_{\nu}-\mu)/kT} } which is much less than unity. This feature is qualitatively captured by the 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 \textit{Phase-space Density} } defined as (3D, homogeneous gas)

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

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 \lambda_T = (2\pi\hbar^2/mkT)^{1/2}} is the thermal de Broglie wavelength.

Some typical parameters for

• Classical thermal gas
  • Atom density 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^{25} \text{m}^{-3}}
  • Interatomic distance 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/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) }
  • 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^{-8} }
• BEC in dilute gas
  • Atom density 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^{20} \text{m}^{-3}}
  • Interatomic distance 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/n)^{1/3} \sim 10^{2} \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^{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

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

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.

Transition Temperature - Semi-classical Picture

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 emrgence 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 critieria 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_{ex} = \int\limits_{0}^{\infty} f(\epsilon, \mu=0, T_c) g(\epsilon) d \epsilon \,. }

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 g(\epsilon) } is the density of states. The number of atoms in the ground state 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 N_0 = N - N_{ex}} . The form of the transition temperature $T_{c}$ 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 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) \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 $x=\epsilon /(kT_{c})$. 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
	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}	$3/2$	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}	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 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 $\omega _{i}\sim 2\pi \times 100Hz$

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 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} 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

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 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 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\propto T^{\delta}}

Thermodynamic Property:	E	$C_{V}=\partial E/\partial T$	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}	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}	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 $N\propto T_{c}^{\alpha }$ we therefore obtain

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 E \propto NkT\gamma^{\alpha}, C_V \propto Nk\gamma^{\alpha}, S \propto Nk\gamma^{\alpha} \,. }

Padagogical Example

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