Quantum Gases in Traps

A trapped quantum gas is a gas of bosons or fermions confined by an external potential rather than by a translation-invariant box. The trap changes the one-particle spectrum, makes the equilibrium density position dependent, and introduces global energy scales that are not obtained by inserting one average density into a uniform-gas formula.

For a particle of mass $m$ in a smooth trap,

$$H_1 = \frac{\mathbf p^2}{2m} + V_{\mathrm{ext}}(\mathbf r).$$

Ideal bosons and fermions occupy the eigenmodes of this Hamiltonian with Bose–Einstein or Fermi–Dirac statistics. At sufficiently large particle number or temperature, the same problem can often be described semiclassically in phase space.

The central organizing relation is

$$\mu_{\mathrm{loc}}(\mathbf r) = \mu_{\mathrm{global}} - V_{\mathrm{ext}}(\mathbf r).$$

It says that moving away from the trap minimum lowers the chemical potential available to the local gas. A single trapped cloud can therefore contain a strongly degenerate center and dilute, nearly classical wings.

We first count exact harmonic levels, then derive the continuum density and distinguish a cloud profile from its measured column image. Ultracold atoms explains experimental preparation and calibration.

Helpful background. Ideal Bose and Fermi gases supply occupation laws; the harmonic oscillator supplies the exact levels.

Why Trapping Changes the Problem

In a uniform box, translation invariance makes momentum a good quantum number and the density is spatially constant in equilibrium. A trap removes both simplifications.

The consequences are structural:

• one-particle modes are trap orbitals rather than plane waves;
• energy levels are discrete at finite trap frequency;
• the density of states depends on the confinement geometry;
• local density and local degeneracy vary across the cloud;
• the thermodynamic limit must scale the trap as particle number grows;
• finite trapped clouds exhibit rounded crossovers rather than exact nonanalytic transitions;
• images usually record a column density or an expanded cloud, not the three-dimensional density directly.

Near a stable minimum, many magnetic, optical-dipole, and hybrid traps are approximately harmonic. Anharmonicity, gravity, finite trap depth, species dependence, and optical lattices can matter in precision work, but the harmonic trap is the canonical starting point.

Harmonic Confinement

In $d$ dimensions, an anisotropic harmonic trap has

$$V_{\mathrm{ext}}(\mathbf r) = \frac{m}{2} \sum_{a=1}^{d} \omega_a^2x_a^2.$$

The one-particle energies are

$$\epsilon_{\mathbf n} = \epsilon_0 + \sum_{a=1}^{d} \hbar\omega_a n_a, \qquad n_a=0,1,2,\ldots,$$

with zero-point energy

$$\epsilon_0 = \frac{\hbar}{2} \sum_{a=1}^{d} \omega_a.$$

It is often convenient to measure excitation energy from the trap ground state:

$$E_{\mathbf n} \equiv \epsilon_{\mathbf n}-\epsilon_0.$$

The corresponding shifted chemical potential is

$$\widetilde\mu \equiv \mu-\epsilon_0.$$

For a finite positive-temperature grand-canonical Bose gas, $\widetilde\mu<0$ is strict; zero is approached as a limiting saturation value. Forgetting which energy origin is being used is a common source of apparently contradictory chemical-potential formulas.

Define the geometric mean frequency

$$\overline\omega = \left( \prod_{a=1}^{d} \omega_a \right)^{1/d}.$$

In three dimensions,

$$\overline\omega = (\omega_x\omega_y\omega_z)^{1/3}.$$

The oscillator lengths

$$a_a = \sqrt{ \frac{\hbar}{m\omega_a} }$$

set the spatial widths of the ideal ground orbital.

Exact Mode Occupations

For ideal particles, the mean occupation of trap orbital $\mathbf n$ is

$$\overline n_{\mathbf n}^{\mathrm B} = \frac{1}{ e^{\beta(E_{\mathbf n}-\widetilde\mu)}-1 }$$

for bosons, and

$$\overline n_{\mathbf n}^{\mathrm F} = \frac{1}{ e^{\beta(E_{\mathbf n}-\widetilde\mu)}+1 }$$

for fermions.

These formulas are exact grand-canonical means for the ideal trapped gas when every complete mode, including internal labels, is counted. A fixed-number canonical state generally has different finite-system occupations; here $N$ in the grand-canonical number equation is a mean. Internal degeneracy $g$ can be implemented either by multiplying the mode count or by summing separately over components with their own chemical potentials.

The exact number equation is

$$N = \sum_{\mathbf n,\sigma} \overline n_{\mathbf n\sigma}.$$

At low occupation, both statistics reduce to the Maxwell–Boltzmann form. At high occupation, the lowest bosonic mode must be retained explicitly, while fermions fill a ladder of distinct trap states.

Isotropic Shell Counting

For an isotropic three-dimensional trap with frequency $\omega$, define

$$q = n_x+n_y+n_z.$$

The excitation energy of shell $q$ is

$$E_q = q\hbar\omega,$$

and its orbital degeneracy per internal component is

$$d_q = \frac{(q+1)(q+2)}{2}.$$

The number of orbitals through shell $q$ is

$$\mathcal N_q = \sum_{j=0}^{q}d_j = \frac{(q+1)(q+2)(q+3)}{6}.$$

This exact counting reveals shell effects that a smooth density of states cannot reproduce. In an anisotropic trap, rational frequency ratios can produce different degeneracy patterns, while incommensurate frequencies generally lift most shell degeneracies.

Semiclassical State Counting

When the relevant energy scale resolves many oscillator levels, replace the mode sum by a phase-space integral:

$$\sum_i \longrightarrow g \int \frac{ d^d r\,d^d p }{(2\pi\hbar)^d}.$$

For thermal observables in every active direction, a common condition is

$$k_{\mathrm B}T \gg \hbar\omega_a.$$

For leading ground-state energy and coarse density profiles of a degenerate Fermi gas, the occupied Fermi energy should span many levels:

$$E_{\mathrm F}^{\mathrm{trap}} \gg \hbar\omega_a.$$

This does not guarantee smooth low-temperature heat capacities: resolving the thermal window also requires the relevant discrete shell spacing to be small compared with $k_{\mathrm B}T$. The condensate ground mode of a Bose gas is never replaced by the continuum integral; it must be separated before the semiclassical excited-state approximation is taken.

Cumulative state count

The number of oscillator states per internal component with excitation energy below $E$ is the volume of a simplex in the nonnegative $n_a$ coordinates:

$$\mathcal N(E) \simeq \frac{E^d}{ d!\,\hbar^d \prod_a\omega_a }.$$

Differentiation gives the density of states per internal component:

$$D_{\mathrm{trap}}^{(d)}(E) \simeq \frac{E^{d-1}}{ (d-1)!\,\hbar^d \prod_a\omega_a }.$$

In three dimensions,

$$D_{\mathrm{trap}}(E) \simeq \frac{E^2}{ 2(\hbar\overline\omega)^3 }.$$

Multiply by $g$ only when $g$ independent internal components share the same spectrum and chemical potential.

The $E^2$ scaling differs from the $E^{1/2}$ density of states of a uniform three-dimensional quadratic gas. That difference changes condensation, low-temperature thermodynamics, and particle-number scaling.

Leading continuum number equations and energy zero

There are two related approximations here, and their energy zeros must agree. The exact spectrum has ground energy $\epsilon_0=\frac12\sum_a\hbar\omega_a$ above the minimum of $V$. Its excitation energies $E_{\mathbf n}$ and shifted chemical potential $\widetilde\mu$ give the exact occupations above.

In the leading semiclassical model, level spacing and zero-point offsets are neglected together. Write its chemical potential as $\mu_{\mathrm{sc}}$, measured from the minimum of $V$, and $z=e^{\beta\mu_{\mathrm{sc}}}$. To this accuracy the leading density of states can be written using either the excitation energy or the classical energy $p^2/(2m)+V$. This equivalence does not determine corrections of order $\hbar\omega_a$.

For the three-dimensional harmonic trap,

$$N_{\mathrm{ex}}^{\mathrm B}\simeq g \left(\frac{k_{\mathrm B}T}{\hbar\overline\omega}\right)^3\operatorname{Li}_3(z), \qquad 0<z\le1,$$

and

$$N^{\mathrm F}\simeq g \left(\frac{k_{\mathrm B}T}{\hbar\overline\omega}\right)^3 [-\operatorname{Li}_3(-z)].$$

For fermions the real Fermi-integral branch applies also at $z>1$. Both reduce to $N\simeq gz(k_{\mathrm B}T/\hbar\overline\omega)^3$ when $z\ll1$.

In particular, leading semiclassical Bose saturation sets $\mu_{\mathrm{sc}}=0$, whereas the exact finite-trap chemical potential approaches $\epsilon_0$ from below. Inserting that positive exact value into an uncorrected classical phase-space Bose denominator would create spurious negative occupations near the trap center. Use the discrete spectrum or consistent finite-size corrections if the zero-point offset matters.

Local-Density Approximation

The local-density approximation (LDA) replaces a slowly varying inhomogeneous gas by a homogeneous equation of state evaluated at

$$\mu_{\mathrm{loc}}(\mathbf r) = \mu_{\mathrm{sc}} - V_{\mathrm{ext}}(\mathbf r).$$

Thus

$$n(\mathbf r) \simeq n_{\mathrm{hom}} \left[ \mu_{\mathrm{loc}}(\mathbf r),T \right],$$

and similarly

$$P(\mathbf r) \simeq P_{\mathrm{hom}} \left[ \mu_{\mathrm{loc}}(\mathbf r),T \right].$$

This is not a new definition of chemical potential and not an exact statement for every trapped system. It is a controlled approximation when the potential varies little over the microscopic and correlation lengths governing the local state.

At zero temperature, the local energy available in a harmonic trap is $E_{\mathrm F}^{\mathrm{trap}}-V(x)$. The lower panel compares the normalized central density slice $(1-x^2/R_x^2)^{3/2}$ with the normalized column density $(1-x^2/R_x^2)^2$. Integrating along the imaging direction changes the shape; it does not merely multiply by a constant. These are leading ideal-gas LDA profiles.

LDA conditions

LDA requires more than a large particle number. Check that:

• the trap varies slowly over the local interparticle spacing or Fermi wavelength;
• it varies slowly over the healing or correlation length when interactions matter;
• the gas has reached local thermal and chemical equilibrium;
• spatial averaging resolves a region large compared with microscopic lengths but small compared with the trap variation; spectral measurements must also account for resolved levels;
• no tight direction that should remain quantized has been replaced by a continuum;
• critical correlations have not grown to the trap-variation scale.

The approximation can fail near a cloud edge, a sharp interface, a very small trap, a critical region, or a rapidly varying optical structure.

Semiclassical Approximation Versus LDA

These terms are related but not identical.

The semiclassical trap approximation replaces discrete one-particle levels by a phase-space integral. It is a statement about spectral resolution.

The local-density approximation applies a homogeneous many-body equation of state at a position-dependent chemical potential. It is a statement about spatial scale separation.

For an ideal gas in a smooth potential, the two lead to the same phase-space expression:

$$n(\mathbf r) = g \int \frac{d^d p}{(2\pi\hbar)^d} \frac{1}{ e^{\beta[p^2/(2m)+V(\mathbf r)-\mu_{\mathrm{sc}}]} \mp1 },$$

where the upper minus sign is for bosons and the lower plus sign is for fermions. This is a leading phase-space approximation, not an exact finite-trap density.

For an interacting gas, LDA can remain useful even though the local equation of state is far from ideal. Conversely, a discrete noninteracting trap can be solved exactly without invoking LDA.

Three-Dimensional Local Profiles

Define the thermal wavelength

$$\lambda_T = \sqrt{ \frac{2\pi\hbar^2}{mk_{\mathrm B}T} }.$$

The local fugacity is

$$z(\mathbf r) = z_0e^{-\beta V(\mathbf r)},$$

where $z_0=e^{\beta\mu_{\mathrm{sc}}}$ in the leading semiclassical convention just defined. The local Bose formula requires $z(\mathbf r)\le1$; it describes the thermal cloud, with the condensate added separately.

For ideal bosons outside the condensate mode,

$$n_{\mathrm{th}}^{\mathrm B}(\mathbf r) = \frac{g}{\lambda_T^3} \operatorname{Li}_{3/2} \left[z(\mathbf r)\right].$$

For ideal fermions,

$$n^{\mathrm F}(\mathbf r) = \frac{g}{\lambda_T^3} \left[ -\operatorname{Li}_{3/2} \left[-z(\mathbf r)\right] \right].$$

In the dilute wings, $z(\mathbf r)\ll1$, so both become

$$n(\mathbf r) \simeq \frac{g z_0}{\lambda_T^3} e^{-\beta V(\mathbf r)}.$$

For a harmonic trap, this is a Gaussian profile. The same cloud can therefore require quantum statistics near its center and Maxwell–Boltzmann statistics in its outer wings.

Trapped Ideal Bose Gas

For one ideal bosonic component, the ground orbital is

$$\phi_0(\mathbf r) = \prod_{a=1}^{3} \frac{1}{ \pi^{1/4}a_a^{1/2} } \exp \left( -\frac{x_a^2}{2a_a^2} \right).$$

If $N_0$ particles occupy that orbital, the condensate contribution is

$$n_0(\mathbf r) = N_0 \prod_{a=1}^{3} \frac{1}{\sqrt\pi a_a} \exp \left( -\frac{x_a^2}{a_a^2} \right).$$

The full ideal-gas density below the condensation crossover is

$$n(\mathbf r) = n_0(\mathbf r) + n_{\mathrm{th}}(\mathbf r).$$

This produces a bimodal spatial or velocity profile when the condensate is sufficiently resolved from the thermal cloud.

Condensation scale

At saturation, $z\to1$ and the semiclassical excited-state capacity is

$$N_{\mathrm{ex,max}} = \zeta(3) \left( \frac{k_{\mathrm B}T}{ \hbar\overline\omega } \right)^3$$

for a single component. The leading ideal critical scale is

$$k_{\mathrm B}T_c^{\mathrm{trap}} = \hbar\overline\omega \left[ \frac{N}{\zeta(3)} \right]^{1/3}.$$

Below this scale,

$$\frac{N_0}{N} \simeq 1- \left( \frac{T}{T_c^{\mathrm{trap}}} \right)^3$$

in the ideal semiclassical limit. The canonical derivation, finite-size qualifications, and dimensionality criterion are on Bose–Einstein Condensation.

Interaction warning

Repulsive interactions broaden the condensate beyond the oscillator ground state and can produce an interaction-dominated profile. The resulting Thomas–Fermi approximation is derived in Gross–Pitaevskii Equation; it is unrelated to Fermi–Dirac statistics despite the shared name.

Finite-temperature interactions also shift density profiles and the transition region. The ideal trapped gas is a benchmark, not a precision model for every condensate.

Trapped Ideal Fermi Gas at Zero Temperature

For $g$ equally populated, noninteracting internal components, fill trap levels up to a global Fermi excitation energy $E_{\mathrm F}^{\mathrm{trap}}$. Semiclassical state counting gives

$$N = \frac{g}{6} \left( \frac{E_{\mathrm F}^{\mathrm{trap}}}{ \hbar\overline\omega } \right)^3.$$

Therefore

$$E_{\mathrm F}^{\mathrm{trap}} = \hbar\overline\omega \left( \frac{6N}{g} \right)^{1/3}.$$

Define the global trap Fermi temperature by

$$k_{\mathrm B}T_{\mathrm F}^{\mathrm{trap}} = E_{\mathrm F}^{\mathrm{trap}}.$$

This is not obtained from a volume-averaged density. It approximates the occupied energy boundary in the large-$N$ continuum model. It is not an exact finite-shell energy or a prescription for its zero-point correction.

Local Fermi scales

At $T=0$, LDA gives the local Fermi energy

$$\epsilon_{\mathrm F}(\mathbf r) = E_{\mathrm F}^{\mathrm{trap}} - V_{\mathrm{ext}}(\mathbf r).$$

Where this quantity is positive,

$$k_{\mathrm F}(\mathbf r) = \frac{ \sqrt{2m\epsilon_{\mathrm F}(\mathbf r)} }{\hbar}$$

and

$$n(\mathbf r) = \frac{g}{6\pi^2} \left[ \frac{2m}{\hbar^2} \epsilon_{\mathrm F}(\mathbf r) \right]^{3/2}.$$

Outside the classically allowed ellipsoid, $n(\mathbf r)=0$ in the zero-temperature LDA.

Thomas–Fermi radii

Define

$$R_a = \sqrt{ \frac{ 2E_{\mathrm F}^{\mathrm{trap}} }{m\omega_a^2} }.$$

Then

$$n(\mathbf r) = n(\mathbf0) \left( 1- \sum_{a=1}^{3} \frac{x_a^2}{R_a^2} \right)^{3/2}$$

inside the ellipsoid, with

$$n(\mathbf0) = \frac{g}{6\pi^2} \left( \frac{ 2mE_{\mathrm F}^{\mathrm{trap}} }{\hbar^2} \right)^{3/2}.$$

Here Thomas–Fermi profile refers to the semiclassical filling of many trap states. It should not be confused with the interaction-dominated Thomas–Fermi condensate approximation.

Fermi-gas energy and the harmonic virial theorem

The leading density of states is proportional to $E^2$, so

$$U_0\simeq\frac34 NE_{\mathrm F}^{\mathrm{trap}}.$$

Here $U_0$ includes the physical kinetic and trapping energies, but only its leading large-$N$ term is retained. The exact harmonic virial theorem is $\langle K\rangle=\langle V\rangle=U_0/2$ for a stationary ideal gas, with $V$ measured from its minimum. Consequently

$$\langle K\rangle\simeq\langle V\rangle\simeq\frac38 NE_{\mathrm F}^{\mathrm{trap}}.$$

If exact excitation energies are used instead, write $U_0=N\epsilon_0+U_{\mathrm{exc}}$ and retain $N\epsilon_0/2$ in each virial contribution. Adding only a zero-point term to the leading continuum count is not a complete finite-size correction; the discrete shell count also changes.

For the low-temperature continuum calculation, the total density of states at the Fermi boundary is $D(E_{\mathrm F}^{\mathrm{trap}})=3N/E_{\mathrm F}^{\mathrm{trap}}$. At fixed mean number, the Sommerfeld expansion yields

$$\mu_{\mathrm{sc}}(T)=E_{\mathrm F}^{\mathrm{trap}} \left[1-\frac{\pi^2}{3}\theta^2+O(\theta^4)\right], \qquad\theta=\frac{T}{T_{\mathrm F}^{\mathrm{trap}}},$$ $$C_{N,\boldsymbol\omega}=\pi^2Nk_{\mathrm B}\theta+O(Nk_{\mathrm B}\theta^3).$$

Trap frequencies, rather than a uniform-box volume, are held fixed. The smooth thermal window must still contain many relevant levels. The coefficient differs from the uniform gas because the trap density of states has a different energy dependence.

Hydrostatic Balance and Equation-of-State Mapping

At uniform temperature, LDA implies

$$dP = n\,d\mu.$$

Taking a spatial gradient and using $\nabla\mu_{\mathrm{loc}}=-\nabla V$ gives

$$\nabla P(\mathbf r) = -n(\mathbf r) \nabla V_{\mathrm{ext}}(\mathbf r).$$

This is hydrostatic force balance. It makes trapped profiles useful rather than merely inconvenient: a single image samples the homogeneous equation of state over a range of $\mu_{\mathrm{loc}}$.

If $V(\mathbf r)$ and the imaging geometry are calibrated, measured density profiles can be transformed into pressure, compressibility, and other local thermodynamic quantities. Such inversions require symmetry assumptions, finite-resolution corrections, and a reliable relationship between optical signal and atom number.

Trap Thermodynamic Limit

At fixed finite particle number in this confining ideal model, the partition function is analytic for positive temperature. The grand-canonical Bose trace instead has a boundary of convergence; that boundary alone is not a finite-cloud phase transition. A sharp thermodynamic transition requires a trap thermodynamic limit.

In $d$ dimensions, take

$$N\to\infty, \qquad \overline\omega\to0,$$

with fixed frequency ratios and with

$$N\overline\omega^d$$

fixed. This keeps characteristic density and degeneracy scales finite while every frequency vanishes. Merely sending their geometric mean to zero does not ensure this if anisotropy changes without bound.

For a real finite cloud:

• condensation and degeneracy onsets are rounded;
• shell effects can survive at small $N$ or low $T$;
• critical temperatures acquire finite-size and interaction shifts;
• the precise crossover criterion depends on the observable.

Finite-size effects explains how shell resolution and correlation lengths limit a continuum description.

Anisotropy and Dimensional Crossover

Strong confinement can freeze one or two oscillator directions. If

$$k_{\mathrm B}T, \quad \mu_{\mathrm{int}}, \quad E_{\mathrm F} \ll \hbar\omega_{\perp},$$

then excitations in the tight direction are suppressed and the gas becomes effectively lower dimensional.

One must retain the discrete ground state in each frozen direction and integrate only over the weakly confined directions. Replacing every direction by a three-dimensional continuum can produce wrong densities of states, couplings, and transition criteria.

Low-Dimensional Quantum Gases explains why quasi-one-dimensional and quasi-two-dimensional interaction parameters require confinement-induced renormalization beyond ideal state counting.

From Density to Images

Cold-atom experiments commonly use two broad measurement geometries.

In-situ imaging

An absorption or phase-contrast image usually yields a line-of-sight column density:

$$n_{\mathrm{col}}(x,z) = \int dy\, n(x,y,z),$$

after optical calibration. Reconstructing the three-dimensional density may require cylindrical or spherical symmetry, tomographic data, or a model fit.

Time-of-flight expansion

After the trap is released, a sufficiently long collisionless ballistic expansion approximately maps initial momentum to position:

$$\mathbf r(t) \simeq \frac{\mathbf p}{m}t.$$

This relation is not automatic. Mean-field release energy, collisions, hydrodynamic flow, finite expansion time, gravity, and switching dynamics can alter the mapping.

For a condensate, anisotropic expansion can reflect interaction and confinement energy. For a degenerate Fermi gas, cloud size and shape can reveal Fermi pressure, but an interacting cloud requires the appropriate dynamical model.

Experimental Signatures

Bosons

The 1995 dilute-gas Bose–Einstein condensation experiments used trapped alkali atoms, evaporative cooling, and spatial or velocity-distribution signatures. A characteristic observation was the emergence of a narrow, nonthermal component on a broad thermal distribution.

That bimodality is strong evidence when the imaging response, expansion dynamics, and interaction model are controlled. It should not be reduced to the statement that every atom occupies one point in momentum space; the condensate occupies a trap orbital with finite spatial and momentum widths.

Fermions

The onset of Fermi degeneracy was observed in trapped potassium-40 in 1999. Unlike a Bose condensate, an ideal Fermi gas does not develop a macroscopically occupied one-particle orbital. Signatures include:

• saturation of low-energy state occupation;
• a cloud size supported by Fermi pressure;
• deviations from a classical Gaussian profile;
• Pauli suppression of collisions and fluctuations;
• thermometry expressed through $T/T_{\mathrm F}^{\mathrm{trap}}$.

Mixtures of internal states are often needed for efficient $s$-wave thermalization, because identical spin-polarized fermions suppress that channel at ultralow energy. The component populations then enter $g$, the Fermi scales, and interaction physics explicitly.

Exercises

Count isotropic oscillator shells

For a three-dimensional isotropic harmonic oscillator, show that shell $q=n_x+n_y+n_z$ has degeneracy

$$d_q = \frac{(q+1)(q+2)}{2}$$

and that the number of orbitals through shell $q$ is

$$\mathcal N_q = \frac{(q+1)(q+2)(q+3)}{6}.$$

Solution

The degeneracy is the number of nonnegative integer solutions of

$$n_x+n_y+n_z = q.$$

By stars and bars,

$$d_q = \binom{q+2}{2} = \frac{(q+1)(q+2)}{2}.$$

Summing the shells and using the hockey-stick identity gives

$$\begin{aligned} \mathcal N_q &= \sum_{j=0}^{q} \binom{j+2}{2} \\ &= \binom{q+3}{3} \\ &= \frac{(q+1)(q+2)(q+3)}{6}. \end{aligned}$$

Recover the classical Gaussian cloud

Use the Maxwell–Boltzmann phase-space distribution in a harmonic trap to derive the spatial density and its root-mean-square width in direction $a$.

Solution

In the dilute limit,

$$n(\mathbf r) \propto e^{-\beta V(\mathbf r)}.$$

For the harmonic potential,

$$n(\mathbf r) = n(\mathbf0) \prod_a \exp \left( -\frac{ m\omega_a^2x_a^2 }{2k_{\mathrm B}T} \right).$$

Comparing each factor with a normalized Gaussian gives

$$\langle x_a^2\rangle = \frac{k_{\mathrm B}T}{m\omega_a^2}.$$

Thus the root-mean-square width is

$$\sigma_a = \sqrt{ \frac{k_{\mathrm B}T}{m\omega_a^2} }.$$

Normalize the zero-temperature Fermi profile

Show that

$$n(\mathbf r) = n(\mathbf0) \left( 1- \sum_a\frac{x_a^2}{R_a^2} \right)^{3/2}$$

integrates to

$$N = \frac{g}{6} \left( \frac{E_{\mathrm F}^{\mathrm{trap}}}{ \hbar\overline\omega } \right)^3.$$

Solution

Set $u_a=x_a/R_a$. Then

$$d^3r = R_xR_yR_z\,d^3u,$$

and the support is the unit ball. The dimensionless integral is

$$\int_{u<1} d^3u\, (1-u^2)^{3/2} = \frac{\pi^2}{8}.$$

Also,

$$R_xR_yR_z = \frac{1}{ \omega_x\omega_y\omega_z } \left( \frac{2E_{\mathrm F}^{\mathrm{trap}}}{m} \right)^{3/2}.$$

Multiplying by

$$n(\mathbf0) = \frac{g}{6\pi^2} \left( \frac{2mE_{\mathrm F}^{\mathrm{trap}}}{\hbar^2} \right)^{3/2}$$

gives

$$N = \frac{g}{6} \frac{(E_{\mathrm F}^{\mathrm{trap}})^3}{ \hbar^3\omega_x\omega_y\omega_z }.$$

Since $\overline\omega^3=\omega_x\omega_y\omega_z$, this is the required result.

Project a Fermi cloud into a column density

For the zero-temperature profile, integrate along $y$ and show that the column density has exponent $2$ rather than $3/2$.

Solution

At fixed $x,z$, define

$$A = 1- \frac{x^2}{R_x^2} - \frac{z^2}{R_z^2}.$$

For $A>0$, the allowed line of sight satisfies

$$|y| \le R_y\sqrt A.$$

Then

$$\begin{aligned} n_{\mathrm{col}}(x,z) &= n(\mathbf0) \int_{-R_y\sqrt A}^{R_y\sqrt A} dy\, \left( A- \frac{y^2}{R_y^2} \right)^{3/2} \\ &= n(\mathbf0)R_yA^2 \int_{-1}^{1} dt\, (1-t^2)^{3/2}. \end{aligned}$$

The remaining integral is $3\pi/8$, so

$$n_{\mathrm{col}}(x,z) = \frac{3\pi}{8} n(\mathbf0)R_y \left( 1- \frac{x^2}{R_x^2} - \frac{z^2}{R_z^2} \right)^2.$$

For $A\le0$ the profile is zero. A recorded column profile therefore has a different exponent from the underlying three-dimensional density.

References

• M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose–Einstein Condensation in a Dilute Atomic Vapor”, Science 269, 198–201 (1995) — trapped rubidium-87 condensation and its velocity-distribution signatures.
• F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose–Einstein Condensation in Trapped Gases”, Reviews of Modern Physics 71, 463–512 (1999) — authoritative review of trapped dilute Bose gases.
• K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Bose–Einstein Condensation in a Gas of Sodium Atoms”, Physical Review Letters 75, 3969–3973 (1995) — trapped sodium condensation and bimodal expansion profiles.
• B. DeMarco and D. S. Jin, “Onset of Fermi Degeneracy in a Trapped Atomic Gas”, Science 285, 1703–1706 (1999) — first degenerate trapped atomic Fermi gas.
• S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Ultracold Atomic Fermi Gases”, Reviews of Modern Physics 80, 1215–1274 (2008) — trapped and uniform Fermi gases, interactions, profiles, and thermodynamics.
• W. Ketterle and N. J. van Druten, “Bose–Einstein Condensation of a Finite Number of Particles Trapped in One or Three Dimensions”, Physical Review A 54, 656–660 (1996) — finite-size and trap-dimensionality analysis.
• C. J. Pethick and H. Smith, Bose–Einstein Condensation in Dilute Gases, 2nd ed., Cambridge University Press (2008) — trapped Bose gases, semiclassical profiles, interactions, and experiments.
• L. Pitaevskii and S. Stringari, Bose–Einstein Condensation and Superfluidity, Oxford University Press (2016) — equilibrium and dynamical theory of trapped quantum gases.
• A. G. Truscott, K. E. Strecker, W. I. McAlexander, G. B. Partridge, and R. G. Hulet, “Observation of Fermi Pressure in a Gas of Trapped Atoms”, Science 291, 2570–2572 (2001) — trap-size manifestation of Fermi pressure.