Ultracold Atoms

An ultracold atomic gas is an ensemble whose center-of-mass motion, collisions, and many-body correlations are governed by energy scales far below ordinary atomic electronic and often hyperfine splittings. In common laboratory usage, temperatures range from the low-microkelvin regime down to nanokelvin scales. The numerical temperature alone is not the definition.

The useful questions are dimensionless:

1. Is the collision wavelength long compared with the interaction range?
2. Do matter waves overlap, so that Bose or Fermi statistics affect the distribution?
3. Is the gas weakly interacting, resonant, or in a crossover between those regimes?
4. Are one or more motional directions energetically frozen?
5. Do elastic collisions establish equilibrium before loss, heating, or control ramps change the state?

These tests distinguish several claims that are often blurred together:

Claim	Representative criterion	What it does not establish
threshold collision regime	$kR\ll1$	quantum degeneracy
dilute gas	$nR^3\ll1$	weak scattering if $\lvert a\rvert\gg R$
quantum-degenerate gas	$n\lambda_{\mathrm{dB}}^3\gtrsim1$ or $T/T_{\mathrm F}\lesssim1$	condensation or superfluidity
weakly interacting Bose gas	$n\lvert a\rvert^3\ll1$, away from instability	ideal-gas behavior at every scale
resonant Fermi gas	$k_{\mathrm F}\lvert a\rvert\gg1$ with small range corrections	absence of loss or finite-temperature effects
lower-dimensional gas	active energies $\ll\hbar\omega_\perp$	validity of unmodified three-dimensional interactions

No single row implies all the others. A cloud can be ultracold but nondegenerate, degenerate but normal, dilute but strongly interacting, or geometrically thin while occupying many transverse modes.

Required background. Evaporative cooling explains how selective loss raises phase-space density; scattering length defines the threshold interaction parameter. Helpful background. Quantum statistics supplies Bose and Fermi occupations, and quantum gases in traps develops trap state counting.

Energy and length scales

An ultracold-gas statement should identify the energies that remain active. Useful scales are

$$\begin{aligned} E_T &= k_{\mathrm B}T, & E_{\mathrm F} &= k_{\mathrm B}T_{\mathrm F}, \\ E_{\mathrm{trap},i} &= \hbar\omega_i, & E_{\mathrm{int}} &\sim gn \quad\text{or}\quad E_{\mathrm F}F(k_{\mathrm F}a), \\ E_{\mathrm{rec}} &= \frac{\hbar^2k_L^2}{2m}, & E_{\mathrm{drive}} &= \hbar\Omega . \end{aligned}$$

They must be compared with internal-state splittings, molecular binding energies, optical linewidths, and the van der Waals energy scale when those degrees of freedom participate.

Cooling usually freezes electronic excitation first. Hyperfine and Zeeman states may remain as controlled internal components, while translational motion becomes quantum mechanical. Calling the atoms “two-level systems” does not remove their motional Hilbert space, and calling the gas “ultracold” does not imply that every internal state is in its ground level.

Three length scales

For a nonrelativistic particle of mass $m$, use the thermal de Broglie wavelength convention

$$\lambda_{\mathrm{dB}} = \frac{h}{ \sqrt{2\pi m k_{\mathrm B}T} }.$$

The mean spacing in a uniform three-dimensional gas is of order

$$d = n^{-1/3}.$$

A short-range interaction has a microscopic range $R$, often represented by a van der Waals length for neutral ground-state atoms. The hierarchy

$$R \ll d \lesssim \lambda_{\mathrm{dB}}$$

describes a dilute, quantum-degenerate threshold gas. It is not automatic: $R\ll d$ separates the interaction range from the mean spacing; it does not guarantee negligible three-body loss, which also depends on correlations and the recombination coefficient. $\lambda_{\mathrm{dB}}\gtrsim d$ marks the importance of exchange statistics.

Degenerate Quantum Gases

The classical Maxwell–Boltzmann approximation fails when exchange cycles and occupation constraints affect observables. For a uniform one-component gas, the phase-space density is

$$\mathcal D = n\lambda_{\mathrm{dB}}^3.$$

The crossover begins around $\mathcal D\sim1$. For an ideal uniform three-dimensional Bose gas, excited states saturate at

$$\mathcal D_c = \zeta\!\left(\frac32\right) \simeq 2.612.$$

That number is not a universal experimental threshold. A harmonic trap has a different density of states, finite clouds round the transition, and interactions shift both density profiles and transition properties. The canonical trap formulas are on Quantum Gases in Traps.

Fermionic degeneracy scale

For one spin component of uniform density $n_\sigma$,

$$k_{\mathrm F,\sigma} = \left( 6\pi^2n_\sigma \right)^{1/3}, \qquad E_{\mathrm F,\sigma} = \frac{\hbar^2k_{\mathrm F,\sigma}^2}{2m}.$$

For a balanced two-component gas with total density $n=2n_\sigma$,

$$k_{\mathrm F} = \left( 3\pi^2n \right)^{1/3}, \qquad T_{\mathrm F} = \frac{E_{\mathrm F}}{k_{\mathrm B}}.$$

The degeneracy parameter is $T/T_{\mathrm F}$. A value much smaller than unity means strong degeneracy. For a normal ideal gas, thermal occupation changes are confined to a shell of width $\sim k_{\mathrm B}T$ around the Fermi surface. Interactions and pairing can change the excitation spectrum; the ratio alone does not demonstrate superfluidity.

In a harmonic trap, one may instead quote a global shell-filling Fermi energy. That scale is not equal to the local $E_{\mathrm F}(\mathbf r)$ inferred from the density everywhere in the cloud. State which definition was used.

Bose and Fermi gases are not mirror images

Bosons can macroscopically occupy one orbital. Fermions fill distinct one-particle states and develop Fermi pressure even without interactions. For collisions:

• identical bosons may scatter in the $s$ wave;
• two distinguishable fermionic components may scatter in the $s$ wave;
• identical spin-polarized fermions cannot use an even spatial partial wave, so their low-energy $s$-wave collision channel is absent;
• higher partial waves are normally threshold-suppressed, although resonances can make them important.

This difference is central to cooling. Pauli blocking reduces available final states for collisions in a deeply degenerate Fermi gas, which can slow rethermalization. Fermions are therefore often evaporated in two-component mixtures or cooled sympathetically.

Worked scale comparison

Consider $^{87}\mathrm{Rb}$ at

$$T = 100\ \mathrm{nK}, \qquad n = 1.0\times10^{19}\ \mathrm{m}^{-3}.$$

Using $m=1.443\times10^{-25}\ \mathrm{kg}$ gives

$$\lambda_{\mathrm{dB}} \simeq 0.592\ \mu\mathrm m, \qquad d \simeq 0.464\ \mu\mathrm m,$$

and therefore

$$\mathcal D \simeq 2.08.$$

Quantum statistics are already important, but the ideal uniform Bose threshold has not quite been reached. Inferring condensation from this one number would still ignore trapping, interactions, finite size, and density calibration.

For a balanced two-state $^6\mathrm{Li}$ gas at the same total density,

$$k_{\mathrm F} \simeq 6.67\times10^6\ \mathrm{m}^{-1},$$

so

$$T_{\mathrm F} \simeq 1.79\ \mu\mathrm K.$$

At $T=200\ \mathrm{nK}$, the gas has $T/T_{\mathrm F}\simeq0.112$. The same numerical temperature can therefore represent very different degeneracy for different masses and densities.

Scattering Length as the Interaction Coordinate

At long wavelength, short-range two-body scattering is organized by the effective-range expansion

$$k\cot\delta_0(k) = - \frac1a + \frac12r_ek^2 + \text{higher-order terms}.$$

The $s$-wave amplitude is

$$f_0(k) = \frac{1}{ -1/a+r_ek^2/2-ik }.$$

For an elastic channel, away from a zero crossing where this expansion can be singular, a sufficient condition for neglecting the range term is $k|r_e|\ll1$, with all remaining range corrections small. Then

$$f_0(k) \simeq - \frac{a}{1+ika}.$$

This compression of microscopic chemistry into $a$ is one reason ultracold gases are controllable. It is a threshold expansion, not permission to discard every other length scale.

Cross sections and statistics

Neglecting effective-range corrections, distinguishable particles have

$$\sigma_{\mathrm{dist}}(k) = \frac{4\pi a^2}{ 1+k^2a^2 }.$$

For identical bosons in the same internal state, symmetrization gives

$$\sigma_{\mathrm B}(k) = \frac{8\pi a^2}{ 1+k^2a^2 },$$

under the standard total-cross-section convention. The factor of two is not an interaction enhancement; it follows from indistinguishable outgoing configurations.

When $k|a|\ll1$, the cross section is proportional to $a^2$. When $k|a|\gg1$, it no longer grows as $a^2$ but approaches the partial-wave unitarity scale proportional to $1/k^2$.

Contact coupling

For equal-mass particles in three dimensions, the dilute weak-coupling parameter commonly used for a Bose gas is

$$g = \frac{4\pi\hbar^2a}{m}.$$

For a nearly pure, weakly depleted condensate, the leading chemical-potential shift is

$$\mu_{\mathrm{int}} \simeq gn.$$

The gas parameter

$$n|a|^3$$

measures diluteness relative to the scattering length. For a stable, weakly repulsive Bose gas, $na^3\ll1$ controls the expansion beyond mean field. A negative $a$ does not merely reverse a perturbative sign in a large homogeneous gas; attraction can cause mechanical instability, with finite traps supporting only limited metastable populations.

For a two-component Fermi gas, a standard interaction coordinate is

$$\frac{1}{ k_{\mathrm F}a }.$$

The regimes

$$\frac1{k_{\mathrm F}a} \ll -1, \qquad \frac1{k_{\mathrm F}a}=0, \qquad \frac1{k_{\mathrm F}a} \gg 1$$

label the BCS side, unitarity, and the molecular BEC side of the broad $s$-wave crossover, respectively. These labels assume a two-component gas, short effective range, and sufficient equilibrium. They do not apply unchanged to a single spin-polarized component.

What the sign of the scattering length says

The sign of $a$ describes the threshold phase shift, not the sign of the microscopic potential at every radius.

• Large positive $a$ commonly accompanies a shallow two-body bound state.
• Large negative $a$ commonly places the corresponding pole on the virtual side of threshold rather than as a physical shallow dimer.
• A potential containing both attraction and repulsion can have either sign of $a$.

For $a>0$ much larger than the interaction range, a universal shallow dimer has approximate binding energy

$$E_b \simeq \frac{\hbar^2}{ 2\mu_{\mathrm r}a^2 },$$

where $\mu_{\mathrm r}$ is the two-body reduced mass. For equal masses,

$$E_b \simeq \frac{\hbar^2}{ ma^2 }.$$

Effective-range, closed-channel, and finite-range corrections matter as $a$ becomes less dominant.

Limits of the one-parameter description

A scattering-length model needs revision when:

1. $k|r_e|$ is not small;
2. higher partial waves contribute;
3. several internal thresholds are nearby;
4. dipole–dipole, Coulomb, or other long-range forces remain active;
5. three-body parameters enter observables;
6. confinement changes the collision boundary conditions;
7. inelastic channels make $a$ complex or introduce separate loss coefficients.

“Universal” always means universal with respect to specified unresolved short-distance details, within a declared scale hierarchy.

Feshbach Resonances

A magnetic Feshbach resonance converts a magnetic field into an interaction control parameter. An incoming pair occupies an open channel. A different internal-state combination supports a closed-channel bound state whose energy shifts relative to the open-channel threshold because the two configurations have different magnetic moments. Coupling between channels shifts the resonance away from the uncoupled crossing. At the scattering pole a dressed molecular state reaches threshold; on the other side the near-threshold pole can continue as a virtual state rather than a bound state.

Near an isolated resonance, a commonly used zero-energy parametrization is

$$a(B) = a_{\mathrm{bg}} \left( 1 - \frac{\Delta}{ B-B_0 } \right).$$

Here:

• $a_{\mathrm{bg}}$ is the background scattering length;
• $B_0$ is the pole position;
• $\Delta$ is the signed field width in this convention;
• the zero crossing occurs at $B=B_0+\Delta$.

Quoting $\Delta$ without the sign convention can reverse which side is called positive. Quoting the magnet power-supply setting without an independent field calibration does not establish $B-B_0$.

Pole, zero, and finite collision energy

The divergence in the zero-energy $a(B)$ formula is not a divergent physical cross section at finite momentum. At large $|a|$,

$$|f_0(k)|^2 \longrightarrow \frac1{k^2}$$

when range corrections are negligible. The scattering length itself still has a zero-energy pole. The finite collision momentum bounds the cross section; a finite-temperature or finite-density experiment probes a range of momenta and need not have a divergent observable.

For the same isolated-resonance model, (a) $a/a_{\rm bg}=1-1/b$ diverges at $b=(B-B_0)/\Delta=0$ and vanishes at $b=1$. Panel (b) shows the distinguishable-particle cross section divided by $4\pi/k^2$ at $ka_{\rm bg}=0.5$, neglecting effective range. It reaches one at the pole and zero at the zero crossing. The horizontal dotted line is the unitarity bound, not another divergence.

The zero crossing is useful for suppressing the leading contact interaction, but it does not eliminate:

• effective-range terms;
• residual interactions in other channels;
• dipolar interactions;
• three-body processes;
• state-dependent light shifts;
• technical heating.

Near the zero, $r_e$ itself may diverge, so a truncated expansion of $k\cot\delta$ must not be interpreted term by term at $a=0$. Use a regular energy-dependent phase shift or a multichannel model there.

Broad and narrow resonances

A resonance’s usefulness is not ranked by $\Delta$ in gauss alone. The physical comparison includes the differential magnetic moment, van der Waals scales, background scattering, and the resonance range parameter. Broad open-channel-dominated resonances can support a large interval where

$$k|r_e| \ll 1$$

and scattering-length universality is accurate. Narrow resonances retain stronger energy dependence and closed-channel character. Their effective range can become an essential many-body parameter.

Control is a trajectory

Changing $B$ can:

1. alter the elastic collision rate;
2. cross a molecular avoided crossing;
3. create or dissociate weakly bound dimers;
4. change three-body recombination;
5. release binding energy into untrapped products;
6. move the gas through a many-body crossover;
7. change equilibration and hydrodynamicity.

The ramp speed must therefore be compared with two-body association, many-body response, trap, collision, and loss timescales. “Adiabatic” is not a property of the waveform alone; it is relative to the gaps and relaxation processes relevant to the intended state.

Field noise maps nonlinearly into interaction noise:

$$\frac{da}{dB}=\frac{a_{\rm bg}\Delta}{(B-B_0)^2}.$$

The linear uncertainty estimate requires fluctuations small compared with $|B-B_0|$. Across the pole, use the full field distribution and finite-energy scattering amplitude. Loss maxima can help locate a resonance, but depend on density, temperature, hold time, and inelastic pathways; they need not mark the elastic pole precisely.

Traps, Inhomogeneity, and Dimensionality

Near a stable minimum, a smooth trap is often approximated by

$$V(\mathbf r) = \frac{m}{2} \left( \omega_x^2x^2 + \omega_y^2y^2 + \omega_z^2z^2 \right).$$

The oscillator lengths are

$$a_i = \sqrt{ \frac{\hbar}{ m\omega_i } }.$$

These lengths determine ground-state wave-packet sizes and compare confinement with scattering scales. They are distinct from thermal cloud radii and from interaction-broadened condensate radii.

A trap creates many local gases

In the local-density approximation,

$$\mu_{\mathrm{loc}}(\mathbf r) = \mu_0 - V(\mathbf r).$$

The center can be strongly degenerate while the wings remain classical. Likewise, a trapped interacting gas can sample several parts of a homogeneous phase diagram in one image. An average density inserted into a uniform-gas formula generally does not reproduce this structure.

LDA requires the potential to vary slowly on the local correlation length. It can fail near small clouds, sharp edges, critical regions, low-density boundaries, or microscopic lattice structure.

Geometry is not dimensionality

Suppose one direction has tight frequency $\omega_\perp$. Subtract the transverse zero-point energy and define an active many-body window

$$E_{\mathrm{active}} = \max \left\{ k_{\mathrm B}T,\, E_{\mathrm F}^{\parallel},\, |\mu_{\mathrm{int}}| \right\}.$$

For equilibrium populations, a useful freeze-out hierarchy is

$$E_{\mathrm{active}} \ll \hbar\omega_\perp.$$

One frozen direction gives a quasi-two-dimensional gas. Two frozen directions give a quasi-one-dimensional gas. The symbol $E_{\mathrm F}^{\parallel}$ denotes the Fermi scale associated with the remaining weak directions; all chemical potentials here are measured relative to the transverse ground level. Drives must also avoid exciting the tight motion: a weak modulation resonant with a transverse transition can populate it even if its amplitude is small. A ramp duration alone is not a complete frequency or selection-rule test.

A cigar-shaped or pancake-shaped density profile is not sufficient evidence. A hot anisotropic classical gas can have the same shape while occupying many tight-direction levels.

Confinement changes interactions

Even when transverse excitations are absent from the real population, virtual transverse excitation affects collisions. Effective one- and two-dimensional couplings depend on both

$$\frac{a}{ a_\perp } \quad\text{and}\quad \hbar\omega_\perp.$$

This produces confinement-induced renormalization and, in one dimension, a confinement-induced resonance. Substituting the three-dimensional $g=4\pi\hbar^2a/m$ directly into a one-dimensional Hamiltonian is generally incorrect.

Low-Dimensional Quantum Gases derives the state counting and effective interactions. In two dimensions, short-range scattering depends logarithmically on collision energy relative to a binding-energy scale. There is no interacting, scale-free counterpart of the three-dimensional $a^{-1}=0$ limit merely obtained by flattening a cloud at a Feshbach pole. See the final exercise and Levinsen–Parish.

Worked confinement check

For $^{87}\mathrm{Rb}$ with

$$\frac{\omega_\perp}{2\pi} = 50\ \mathrm{kHz},$$

the tight-direction level spacing is

$$\frac{\hbar\omega_\perp}{k_{\mathrm B}} = \frac{h(50\ \mathrm{kHz})}{k_{\mathrm B}} \simeq 2.40\ \mu\mathrm K.$$

The oscillator length is

$$a_\perp \simeq 48.2\ \mathrm{nm}.$$

A cloud with $T=100\ \mathrm{nK}$ and $\mu_{\mathrm{int}}/k_{\mathrm B}=300\ \mathrm{nK}$ is plausibly in the transverse ground mode, because both scales lie well below $2.40\ \mu\mathrm K$. That conclusion should still be checked spectroscopically or through a calibrated excited-mode fraction, especially after rapid loading or strong driving.

Harmonic, box, and lattice confinement

Different potentials answer different questions:

• Harmonic traps are smooth and experimentally common, but spatially inhomogeneous.
• Optical box traps approach uniform density over a central region and simplify equation-of-state comparisons, while walls and residual roughness remain finite.
• Optical lattices create bands, tunneling, and on-site interactions; they are not merely tight continuum traps.
• Tweezers emphasize programmable few-body or site-resolved preparation, with array uniformity and transport as additional controls.

Optical lattices develops the implementation; the Bose–Hubbard model explains the band projection and its approximations.

Connection to Many-Body Quantum Mechanics

Once internal electronic excitations and microscopic collision structure are integrated out, a multicomponent continuum gas is often organized by

$$H ={} \sum_\sigma \int d^dr\, \hat\psi_\sigma^\dagger(\mathbf r) \left[ - \frac{\hbar^2\nabla^2}{2m_\sigma} + V_\sigma(\mathbf r,t) \right] \hat\psi_\sigma(\mathbf r) + \frac12 \sum_{\sigma,\sigma'} g_{\sigma\sigma'} \int d^dr\, \hat\psi_\sigma^\dagger \hat\psi_{\sigma'}^\dagger \hat\psi_{\sigma'} \hat\psi_\sigma + H_{\mathrm{long}} + H_{\mathrm{drive}} .$$

All field operators in the interaction term are evaluated at $\mathbf r$. They obey commutation or anticommutation relations according to the isotope and chosen constituents.

The apparent delta interaction in three dimensions is an effective low-energy description. Its bare coupling requires regularization in ultraviolet-sensitive calculations. The experimentally meaningful scattering length fixes the renormalized low-energy amplitude.

Field Operators develops the operator construction. The Gross–Pitaevskii equation follows in the weakly interacting condensate mean-field limit.

Experimental knobs become Hamiltonian terms

AMO control	Effective many-body coordinate
isotope and internal state	particle statistics, mass, component labels
magnetic field near a resonance	$a$, effective range, molecular detuning
atom number and trap shape	density, Fermi scale, inhomogeneity
transverse confinement	dimensionality and effective coupling
optical lattice depth	hopping, bandwidth, on-site interaction
state-dependent light shift	spin-dependent potential or field
Raman coupling	coherent intercomponent coupling, synthetic momentum transfer
controlled disorder	random or quasiperiodic potential
periodic modulation	Floquet drive and heating channels
imaging and loss	measurement backaction and open-system terms

This control is powerful but not unlimited. The realized Hamiltonian includes calibration errors, higher bands, residual confinement, photon scattering, loss, and finite entropy. A quantum simulator is validated by comparing those terms with the target-model scales, not by naming the target Hamiltonian.

The links above develop the many-body descriptions. Experimentally, evaporation or sympathetic cooling prepares a dense, cold sample before interaction and geometry ramps. A ramp that lowers temperature need not remove entropy: reversible trap decompression is one example. Compare the prepared distribution with the assumptions of the effective Hamiltonian.

Observables and What They Infer

Ultracold-gas measurements usually infer a many-body quantity through a forward model.

Measurement	Common inference	Required qualification
absorption or phase-contrast image	column density	optical depth, saturation, detuning, point-spread function
time-of-flight image	momentum distribution	release dynamics, collisions, far-field condition
bimodal fit	condensate fraction	model choice, interactions, finite resolution
in situ equation of state	$T$, $\mu$, pressure, compressibility	LDA, potential calibration, imaging response
radio-frequency spectroscopy	pairing or excitation spectrum	final-state interactions, pulse response
Bragg spectroscopy	dynamic structure factor	momentum resolution and linear-response regime
noise correlations	occupation correlations	finite imaging transfer and ensemble averaging
site-resolved fluorescence	parity or occupation	light-assisted loss, reconstruction fidelity
collective modes	equation of state or hydrodynamics	excitation amplitude, damping model, anisotropy
atom loss	inelastic coefficient or resonance indicator	density calibration and competing channels

Thermometry is model dependent

At high enough temperature, a thermal wing can be fitted to a classical distribution. Near degeneracy, Bose or Fermi occupation factors must be used. Deep in a correlated regime, thermometry may rely on an equation of state, fluctuation relation, impurity, spin gradient, or adiabatic connection to a calibrated reference state.

The colder the gas, the less reliable a naive Gaussian width becomes. Report the thermometer, its calibration range, and whether it measures the state before or after an interaction or trap ramp.

Equilibration before loss

An equilibrium interpretation needs a window such as

$$\tau_{\mathrm{eq}} \ll \tau_{\mathrm{loss}}, \tau_{\mathrm{heat}}, \tau_{\mathrm{drift}}.$$

The hold time must also exceed the relevant equilibration time. Compare that time with $\hbar/E_{\mathrm F}$, trap periods, or tunneling times as appropriate; there is no universal requirement that equilibration be much slower than every microscopic timescale. It can nevertheless be very slow near integrability, with Pauli blocking, or after a large quench.

Holding longer without observing stationarity does not prove equilibrium. A useful test varies the hold time and preparation path while comparing several independent observables.

Exercises

1. Matter-wave overlap

A uniform one-component gas has

$$n = 5.0\times10^{18}\ \mathrm{m}^{-3}, \qquad \lambda_{\mathrm{dB}} = 0.70\ \mu\mathrm m.$$

Compute $\mathcal D$ and the mean spacing $d=n^{-1/3}$. Is the gas safely classical?

Solution

The phase-space density is

$$\begin{aligned} \mathcal D &= n\lambda_{\mathrm{dB}}^3 \\ &= \left( 5.0\times10^{18}\ \mathrm{m}^{-3} \right) \left( 0.70\times10^{-6}\ \mathrm m \right)^3 \\ &\simeq 1.72. \end{aligned}$$

The spacing is

$$d = n^{-1/3} \simeq 0.585\ \mu\mathrm m.$$

Thus $\lambda_{\mathrm{dB}}/d\simeq1.20$. The matter waves overlap and the gas is not safely in the Maxwell–Boltzmann regime. For bosons, the value is still below the ideal uniform condensation threshold $2.612$, but this comparison alone does not diagnose a trapped interacting sample.

2. Balanced Fermi-gas convention

Show that a balanced two-component Fermi gas with total density $n$ has

$$k_{\mathrm F} = \left( 3\pi^2n \right)^{1/3}.$$

Explain why using $(6\pi^2n)^{1/3}$ would be inconsistent if $n$ is the total density.

Solution

Each component has density

$$n_\sigma = \frac n2.$$

For one component,

$$k_{\mathrm F} = \left( 6\pi^2n_\sigma \right)^{1/3}.$$

Substituting $n_\sigma=n/2$ gives

$$k_{\mathrm F} = \left( 3\pi^2n \right)^{1/3}.$$

The expression $(6\pi^2n)^{1/3}$ would treat the total density as though every particle occupied one spin component. It would overestimate $k_{\mathrm F}$ by $2^{1/3}$ and $E_{\mathrm F}$ by $2^{2/3}$.

3. Finite-energy unitarity correction

For distinguishable particles with negligible effective range, compare the finite-energy cross section at $k|a|=1$ with the threshold approximation $4\pi a^2$.

Solution

The finite-energy expression is

$$\sigma(k) = \frac{4\pi a^2}{ 1+k^2a^2 }.$$

At $k|a|=1$,

$$\sigma(k) = \frac{4\pi a^2}{2} = 2\pi a^2.$$

It is one half of the threshold approximation. Thus $4\pi a^2$ is already a poor estimate when $k|a|$ reaches unity.

4. Pole, zero, and sign

An isolated resonance is described by

$$a(B) = a_{\mathrm{bg}} \left( 1 - \frac{\Delta}{ B-B_0 } \right),$$

with $a_{\mathrm{bg}}=100a_0$ and $\Delta=20\ \mathrm G$. Find the zero crossing and evaluate $a$ at $B=B_0+4\ \mathrm G$.

Solution

The zero satisfies

$$1 - \frac{\Delta}{ B-B_0 } = 0,$$

so

$$B_{\mathrm{zero}} = B_0+\Delta = B_0+20\ \mathrm G.$$

At $B=B_0+4\ \mathrm G$,

$$\begin{aligned} a &= 100a_0 \left( 1-\frac{20}{4} \right) \\ &= -400a_0. \end{aligned}$$

The negative result does not mean the microscopic potential is everywhere attractive. It states the sign of the threshold scattering parameter in that channel.

5. Universal dimer scaling

For two equal-mass atoms with large positive $a$, show how the shallow-dimer binding energy changes when $a$ is doubled. What assumption must be checked before trusting the result?

Solution

For equal masses,

$$E_b \simeq \frac{\hbar^2}{ ma^2 }.$$

Replacing $a$ by $2a$ gives

$$E_b(2a) = \frac14E_b(a).$$

The dimer becomes four times less deeply bound. This universal result assumes that $a$ is much larger than the interaction range and that effective-range and closed-channel corrections are small.

6. Dimensional freeze-out

A gas has

$$\frac{\omega_\perp}{2\pi} = 30\ \mathrm{kHz}, \qquad T = 150\ \mathrm{nK}, \qquad \frac{E_{\mathrm{int}}}{k_{\mathrm B}} = 400\ \mathrm{nK}.$$

Compare its largest quoted active energy with $\hbar\omega_\perp$. Is transverse freeze-out plausible?

Solution

The confinement spacing in temperature units is

$$\frac{\hbar\omega_\perp}{k_{\mathrm B}} = \frac{h(30\ \mathrm{kHz})}{k_{\mathrm B}} \simeq 1.44\ \mu\mathrm K.$$

The largest quoted active scale is

$$\frac{E_{\mathrm{active}}}{k_{\mathrm B}} \simeq 0.40\ \mu\mathrm K.$$

Therefore

$$\frac{E_{\mathrm{active}}}{ \hbar\omega_\perp } \simeq 0.28.$$

Freeze-out is plausible but not asymptotically deep. A quantitative claim should measure or bound transverse excited-state occupation and include any Fermi, drive, or quench energy omitted from the question.

7. Local-density boundary

In an isotropic harmonic trap,

$$V(r) = \frac12m\omega^2r^2.$$

Within LDA, a homogeneous phase exists for $\mu_{\mathrm{loc}}>\mu_c$. Find the radius of its boundary in terms of $\mu_0$, $\mu_c$, $m$, and $\omega$.

Solution

The boundary obeys

$$\mu_{\mathrm{loc}}(r_c) = \mu_0 - \frac12m\omega^2r_c^2 = \mu_c.$$

Solving gives

$$r_c = \sqrt{ \frac{ 2(\mu_0-\mu_c) }{ m\omega^2 } },$$

provided $\mu_0>\mu_c$. The trap can therefore display a central phase and surrounding phases at lower local chemical potential in one equilibrium cloud.

8. Why a two-dimensional resonance retains an energy scale

Use the zero-range two-dimensional $T$ matrix convention

$$\mathcal T(E)=\frac{4\pi\hbar^2/m}{\ln(E_b/E)+i\pi},\qquad E>0,$$

where $E_b>0$ is the dimer binding energy. At which energy is $|\mathcal T|$ largest? Does making $E_b$ vanish give a finite, scale-free interaction at fixed $E$? Explain why a three-dimensional Feshbach pole does not alone specify a strongly interacting quasi-two-dimensional gas.

Solution

The squared denominator is $[\ln(E_b/E)]^2+\pi^2$, minimized at $E=E_b$. The maximum is finite, $|\mathcal T|=4\hbar^2/m$, and selects an energy scale. As $E_b\to0$ at fixed positive $E$, the logarithm diverges and $\mathcal T\to0$. Thus this limit is weak scattering, rather than the interacting three-dimensional unitary limit. Quasi-two-dimensional scattering depends on confinement as well as the three-dimensional scattering length; one must determine the effective $E_b$ and compare it with temperature and Fermi energy. The zero-range expression itself requires negligible finite range and transverse excited-channel corrections.

For applications, ultracold-atom quantum simulation connects these controls to lattice, synthetic-dimension, and nonequilibrium models.

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