Degenerate Fermi Gases Overview

A degenerate atomic Fermi gas is a trapped ensemble in which Fermi statistics, rather than thermal equipartition, controls occupation, pressure, fluctuations, and collision phase space. Its basic reduced temperature is $T/T_{\mathrm F}$, but that symbol is meaningful only after the density, spin-component, or trap convention used to define $T_{\mathrm F}$ has been stated.

Unlike a Bose–Einstein condensate, a Fermi gas cannot put a macroscopic number of atoms into one one-particle mode, even when atoms form pairs. Pauli exclusion permits at most one fermion per spatial orbital per internal component; the condensate of paired fermions is a two-body property.

Degeneracy, pairing, and superfluidity are distinct. A gas can be degenerate without being paired, paired without having established long-range pair coherence, and spectroscopically gapped without a direct superfluid-response measurement. Reliable analysis keeps those claims separate.

Required background. Ultracold atoms introduces the preparation and interaction scales. Degenerate Fermi gases develops ideal occupations and thermodynamics; scattering length defines the threshold interaction parameter.

Fermi Scales

Component-resolved convention

Consider a nonrelativistic gas with internal components labeled by $\sigma$. For one homogeneous component of density $n_\sigma$,

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

and

$$v_{\mathrm F,\sigma} = \frac{\hbar k_{\mathrm F,\sigma}}{m}.$$

For a balanced two-component gas,

$$n_\uparrow = n_\downarrow = \frac n2,$$

where $n=n_\uparrow+n_\downarrow$ is the total density. The common Fermi wavevector is then

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

The factor of two is not cosmetic. Inserting total density into the single-component formula overestimates $k_{\mathrm F}$ by $2^{1/3}$ and $E_{\mathrm F}$ by $2^{2/3}$.

For an imbalanced gas, report both component scales or explicitly define a reference scale from the total density. No single convention removes the physical mismatch between $k_{\mathrm F,\uparrow}$ and $k_{\mathrm F,\downarrow}$.

Reduced temperature

For a noninteracting gas in thermal equilibrium, the mean occupation of a one-particle state with energy $\epsilon$ is

$$f(\epsilon) = \frac{1}{ \exp[ (\epsilon-\mu)/(k_{\mathrm B}T) ] +1 }.$$

At $T\ll T_{\mathrm F}$, states far below $\mu$ remain nearly full, states far above it remain nearly empty, and only an energy shell of width $O(k_{\mathrm B}T)$ near the Fermi surface is thermally active. The fraction of particles in that shell scales as

$$\frac{N_{\mathrm{active}}}{N} \sim \frac{T}{T_{\mathrm F}}.$$

Thus $T/T_{\mathrm F}$ measures more than “coldness.” It controls the available particle–hole phase space, heat capacity, fluctuations, and many collision rates in the normal regime. The interacting atomic momentum distribution is not generally this Fermi function; for example, pairing mixes particle and hole amplitudes, and short-range correlations produce high-momentum tails. Quasiparticle occupations require their own energies and spectral weights.

At fixed positive chemical potential, (a) the ideal-gas Fermi edge broadens with temperature and (b) the mode variance $f(1-f)$ is concentrated near that edge. The curves use $k_{\rm B}T/\mu=0.08$ and $0.25$, not fixed atom number: the chemical potential would also shift in a fixed-density sample. The variance peak always has height $1/4$, while its width and integrated weight shrink as temperature decreases.

Worked homogeneous scale

Take a balanced two-state gas of lithium-6 with total density

$$n = 10^{19}\ \mathrm{m^{-3}}.$$

Using $m=9.99\times10^{-27}\ \mathrm{kg}$ gives

$$\begin{aligned} k_{\mathrm F} &\simeq 6.67\times10^6\ \mathrm{m^{-1}}, & \frac{E_{\mathrm F}}{h} &\simeq 37.3\ \mathrm{kHz}, \\ T_{\mathrm F} &\simeq 1.79\ \mu\mathrm K, & v_{\mathrm F} &\simeq 70.4\ \mathrm{mm\,s^{-1}}. \end{aligned}$$

At $T=180\ \mathrm{nK}$, the reduced temperature is about $0.10$. The gas is degenerate even though its atoms retain momenta up to $p_{\mathrm F}=\hbar k_{\mathrm F}$; degeneracy does not mean that every atom is nearly at rest.

Harmonic-trap convention

For $N_\sigma\gg1$ ideal fermions of one component in a three-dimensional harmonic trap,

$$E_{\mathrm F,\sigma}^{\mathrm{trap}} = \hbar\bar\omega \left( 6N_\sigma \right)^{1/3},$$

where

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

For a balanced mixture with total atom number $N=2N_\sigma$,

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

For $N=2\times10^5$ and $\bar\omega/(2\pi)=150\ \mathrm{Hz}$,

$$\frac{E_{\mathrm F}^{\mathrm{trap}}}{h} \simeq 12.7\ \mathrm{kHz}, \qquad T_{\mathrm F}^{\mathrm{trap}} \simeq 607\ \mathrm{nK}.$$

The trap Fermi energy is a global state-counting scale. A local-density analysis instead uses

$$\mu_\sigma(\mathbf r) = \mu_{\sigma,0} - V_\sigma(\mathbf r)$$

and defines a position-dependent $k_{\mathrm F,\sigma}(\mathbf r)$ from the local density or equation of state. Mixing a central $k_{\mathrm F}$ with a trap-averaged temperature without saying so makes dimensionless interaction and temperature coordinates ambiguous.

Pauli Pressure

Pressure without repulsion

At zero temperature, an ideal balanced three-dimensional gas has

$$\frac EV = \frac35 nE_{\mathrm F}$$

and

$$P = \frac25 nE_{\mathrm F}.$$

Because $E_{\mathrm F}\propto n^{2/3}$,

$$P \propto n^{5/3}.$$

This degeneracy pressure exists even when the interparticle potential vanishes. Compressing the cloud forces atoms into higher momentum states because lower states are already occupied. It is therefore a kinetic and statistical pressure, not a hidden pairwise repulsion.

The zero-temperature isothermal compressibility of the ideal balanced gas is

$$\kappa_T = \frac{3}{2nE_{\mathrm F}}.$$

Low compressibility, a broad momentum distribution, and finite cloud size at very low temperature are all consequences of the filled Fermi sea. Here “low compressibility” means relative to a classical gas at the same density and a temperature much lower than the Fermi scale.

A trapped zero-temperature profile

In the local-density approximation, a balanced ideal gas in an external potential has

$$n(\mathbf r) = \frac{1}{3\pi^2} \left[ \frac{ 2m[ \mu_0-V(\mathbf r) ] }{\hbar^2} \right]^{3/2} \Theta[ \mu_0-V(\mathbf r) ].$$

For a harmonic trap, the cloud edge satisfies

$$\mu_0 = \frac12m\omega_i^2R_{\mathrm F,i}^2$$

along principal axis $i$. The profile is flatter than a classical Gaussian and remains finite in size as $T\to0$. Interactions deform this profile, so extracting a temperature from cloud shape requires the appropriate equation of state.

Density fluctuations

For an ideal mode with mean occupation $f$,

$$\operatorname{Var}(n_{\mathbf k}) = f(1-f).$$

Modes deep inside the Fermi sea have $f\simeq1$ and suppressed occupation fluctuations. Spatially resolved number fluctuations likewise fall below the classical Poisson expectation after imaging resolution, finite cell size, and interactions are accounted for. This is a direct statistical signature of degeneracy, but it is not itself evidence of pairing.

Spin Mixtures and Collisions

Why one component stalls

Two identical fermions in the same internal state have an antisymmetric total wavefunction. At ultralow collision energy, their symmetric $s$-wave spatial channel is forbidden. The leading allowed odd partial wave is usually $p$ wave, whose threshold rate is strongly suppressed away from a $p$-wave resonance.

A single spin-polarized Fermi gas can therefore become collisionally isolated during evaporation: energetic atoms leave, but the remaining cloud cannot rethermalize rapidly enough. This is why experiments commonly use:

• two hyperfine components that can collide in an $s$ wave;
• sympathetic cooling with another internal state or species;
• controlled spin-changing preparation before the final evaporation stage.

The statement “fermions do not scatter” is false. Identical fermions lack the low-energy $s$-wave channel in the same internal state; distinguishable components can have large $s$-wave scattering.

Minimal two-component Hamiltonian

For two equal-mass components with short-range interactions, a continuum model is

$$\hat H = \sum_{\sigma=\uparrow,\downarrow} \int d^3r\, \hat\psi_\sigma^\dagger \left[ - \frac{\hbar^2\nabla^2}{2m} + V_\sigma \right] \hat\psi_\sigma + g \int d^3r\, \hat\psi_\uparrow^\dagger \hat\psi_\downarrow^\dagger \hat\psi_\downarrow \hat\psi_\uparrow.$$

The contact term couples unlike components. Same-component contact terms vanish for fermionic fields at one point. In three dimensions the bare coupling requires ultraviolet matching to the physical scattering length; writing $g=4\pi\hbar^2a/m$ without its regime and regulator is only a low-energy shorthand.

Pauli blocking of final states

A two-body collision integral contains occupation factors schematically of the form

$$f_1f_2 \left( 1-f_3 \right) \left( 1-f_4 \right).$$

The initial states must be occupied and the final states must be empty. Deep in a degenerate gas, many energetically allowed final states are already filled. Near equilibrium, only a thin shell around the Fermi surface participates, and ordinary Fermi-liquid relaxation rates commonly acquire a low-temperature factor proportional to $(T/T_{\mathrm F})^2$.

This is distinct from the absence of same-spin $s$-wave scattering:

• exchange antisymmetry removes a partial wave for identical particles;
• Pauli blocking suppresses final-state phase space for an occupied many-body distribution.

Both can slow evaporation and equilibration.

Population balance

Define total polarization

$$\mathcal P = \frac{ N_\uparrow-N_\downarrow }{ N_\uparrow+N_\downarrow }.$$

For pairing between opposite momenta and spins, imbalance produces mismatched Fermi surfaces. The system may respond through an unpaired majority component, spatial phase separation in a trap, polarized quasiparticles, or more exotic finite-momentum pairing under restricted conditions. A two-component sample is not automatically balanced, and equal preparation pulses do not guarantee equal final atom numbers after state-dependent loss.

Feshbach Tuning

Magnetic-field control

Near an isolated magnetic Feshbach resonance, the zero-energy scattering length is commonly parameterized as

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

Here $B_0$ is the pole, $\Delta$ is the signed width in the stated convention, and the zero crossing occurs at $B_0+\Delta$. The pole is not the same field as the zero crossing or a maximum in loss.

For a single elastic channel at small relative wavevector $k$, the effective-range approximation is

$$f_0(k) = \frac{1}{ - a^{-1} + \tfrac12r_{\mathrm e}k^2 - ik }.$$

In the zero-range limit, the distinguishable-particle elastic cross section is

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

As $\lvert a\rvert\to\infty$, the cross section approaches $4\pi/k^2$ rather than diverging. Collision energy, effective range, inelastic channels, and density remain essential near resonance.

The many-body interaction coordinate

For a homogeneous balanced gas, define

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

The broad-resonance crossover is organized as:

$$\begin{array}{ccl} x\ll-1 &:& \text{weak BCS side}, \\ x=0 &:& \text{unitarity}, \\ x\gg+1 &:& \text{molecular-BEC side}. \end{array}$$

This coordinate is meaningful only with the quoted $k_{\mathrm F}$ convention. A complete regime label also includes

$$\frac{T}{T_{\mathrm F}}, \qquad k_{\mathrm F}r_{\mathrm e}, \qquad \mathcal P, \qquad \frac{\hbar\omega_i}{E_{\mathrm F}},$$

and relevant loss or ramp times. The single number $1/(k_{\mathrm F}a)$ does not identify a state by itself.

Broad and narrow resonances

For a broad, open-channel-dominated resonance with $k_{\mathrm F}\lvert r_{\mathrm e}\rvert\ll1$, low-energy equilibrium properties can approach a universal zero-range description. A narrow resonance introduces an additional range or closed-channel scale. Then two gases at the same $1/(k_{\mathrm F}a)$ need not share the same many-body properties.

Field stability matters, but the divergent derivative of $a(B)$ must not be confused with a divergent physical response. At fixed $k_{\rm F}$ the crossover coordinate is

$$x(B)=\frac{B-B_0}{k_{\rm F}a_{\rm bg}(B-B_0-\Delta)}.$$

Its slope at the pole is finite: $x'(B_0)=-1/(k_{\rm F}a_{\rm bg}\Delta)$. Field uncertainty, density noise, and effective range must be propagated into the observable being measured. The scattering-length parameter alone is a poor error coordinate at its pole.

BEC–BCS Crossover Overview

Continuously connected paired states

For the equilibrium paired branch of a balanced, equal-mass, two-component gas with short-range s-wave interactions, the zero-temperature molecular-BEC and BCS superfluids are continuously connected. The crossover changes pair size, chemical potential, excitation spectrum, and equation of state without requiring a separate symmetry-breaking transition between its endpoints.

Regime	Two-body and pair structure	Many-body description
BEC side, $a>0$ and $a$ much larger than range	shallow dimers exist; pair size can be smaller than interparticle spacing	interacting bosonic molecules of mass $2m$
unitarity, $a^{-1}=0$	no scattering-length scale; in the ground state, pair size and spacing are comparable	strongly correlated universal Fermi gas
BCS side, $a<0$	no shallow two-body dimer; overlapping Cooper pairs are many-body objects	paired state organized around a Fermi surface

“BEC side” does not mean that unpaired fermions have become elementary bosons. It means that opposite-spin fermions bind into composite dimers whose low-energy center-of-mass motion can Bose condense.

Molecular side

For a large positive scattering length and negligible effective range, the shallow dimer binding energy is

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

for two equal-mass atoms. This universal formula requires $a$ to be much larger than the interaction range. The dimer density of a fully paired balanced gas is $n/2$, and its mass is $2m$.

The molecules interact with one another and can be lost in inelastic collisions. Treating them as an ideal Bose gas ignores dimer–dimer scattering, residual unpaired atoms, finite binding energy, and conversion dynamics.

Weak BCS side

For $k_{\mathrm F}a\to0^-$, the mean-field zero-temperature gap has the asymptotic form

$$\frac{\Delta_{\mathrm{BCS}}}{E_{\mathrm F}} \simeq \frac{8}{e^2} \exp\!\left[ \frac{\pi}{ 2k_{\mathrm F}a } \right].$$

Because $a<0$, the exponent is negative and the pairs are exponentially fragile in weak coupling. Their size is much larger than the interparticle spacing. Medium-induced corrections change the prefactor, and this asymptotic expression should not be extrapolated through unitarity.

The full gap equation, number equation, and quasiparticle transformation are derived in BCS mean-field theory.

Unitary gas

At unitarity,

$$a^{-1} = 0, \qquad k_{\mathrm F}\lvert r_{\mathrm e}\rvert \ll 1,$$

the scattering length drops out of the zero-range equilibrium problem. For a homogeneous balanced gas at zero temperature, dimensional analysis gives

$$\mu = \xi_{\mathrm B}E_{\mathrm F}$$

and

$$\frac EN = \frac35 \xi_{\mathrm B}E_{\mathrm F},$$

where the Bertsch parameter is measured and calculated to be approximately

$$\xi_{\mathrm B} \simeq 0.37.$$

The number is not determined by dimensional analysis; it is a universal many-body constant. Scale invariance also implies

$$P = \frac23 \frac EV$$

for the homogeneous zero-range gas. This pressure–energy relation remains valid at finite temperature and spin imbalance; those change the universal equation-of-state function, not the relation $P=2E/(3V)$ (Tan). In contrast, the specific constant $\xi_{\rm B}$ above refers to the balanced zero-temperature ground state. Finite scattering length or effective range introduces extra scales. In a trap, apply the relation locally where LDA is valid; a relation involving total trap energy also contains the trapping potential.

A resonance is not a phase transition

Sweeping $B$ through $B_0$ changes the sign of $a^{-1}$ continuously. In a balanced low-temperature paired gas, it connects molecular and Cooper pairing. The resonance field itself is not the superfluid transition.

At fixed interaction, raising temperature can produce:

1. a low-temperature superfluid;
2. a normal state with strong pair correlations or a pairing pseudogap over some regime;
3. a higher-temperature nondegenerate gas.

The locations and even the usefulness of these crossovers depend on the observable and model. A spectroscopic pairing scale need not equal the superfluid critical temperature.

Pairing, Condensation, and Superfluidity

Pair formation

Pairing means that opposite-spin atoms develop strong two-body or many-body correlations. Possible signatures include:

• a shifted or broadened radio-frequency spectrum;
• a pair-breaking threshold;
• a contact or high-momentum tail revealing short-range unlike-spin correlations;
• correlated opposite-momentum atoms;
• molecule production after a projection ramp.

Each probe has a forward model. Radio-frequency spectra can contain Hartree shifts, final-state interactions, trap averaging, and pulse broadening. Molecule conversion depends on ramp speed and overlap. Pairing evidence alone does not prove phase coherence or superfluid flow. A contact is also present in normal scattering states and does not by itself establish long-lived bound pairs or a pairing pseudogap.

Pair condensation

Fermionic pair condensation is associated with a macroscopic eigenvalue of an appropriate two-body density matrix, not of the fermionic one-body density matrix. Experimentally, rapid magnetic-field sweeps can project fragile pairs onto tightly bound molecules whose center-of-mass momentum distribution is easier to image. A low-momentum molecular component then supports pair condensation under the conversion model.

The ramp must be:

• fast compared with many-body motion that would rearrange pair center-of-mass momentum;
• slow enough to convert pairs efficiently through the two-body avoided crossing;
• characterized for unpaired-atom conversion and loss.

These requirements can compete. A “rapid projection” is not an instantaneous, model-free measurement.

The number-conserving order criterion and distinction from anomalous averages live in Off-Diagonal Long-Range Order.

Superfluid response

Superfluidity concerns response, not merely pair population. Strong evidence includes:

• quantized vortex lattices under rotation;
• second sound and two-fluid hydrodynamics;
• a critical velocity with controlled obstacle and heating analysis;
• persistent flow or phase stiffness;
• equation-of-state and collective-mode behavior across a thermodynamic transition.

Vortex lattices are particularly direct because circulation quantization and macroscopic rotational response are difficult to reproduce with a normal paired gas. Even then, nucleation thresholds, lifetime, imaging selection, and the rotation protocol should be reported.

Observables and Inference

What each probe measures

Observable	Primary inference	Important caveat
in situ density	equation of state, pressure, compressibility	trap and imaging calibration
momentum distribution	Fermi surface or low-momentum pairs	interactions during expansion
density fluctuations	Pauli suppression and correlations	optical resolution and binning
radio-frequency spectrum	pairing and excitation energies	final-state interactions and pulse response
pair projection	pair center-of-mass distribution	sweep dynamics and conversion efficiency
collective mode	hydrodynamics and equation of state	collision regime and anharmonicity
first or second sound	compressibility and two-fluid response	inhomogeneity and damping
vortex lattice	quantized superfluid circulation	nucleation and survival bias
site-resolved lattice image	occupations and correlations	parity projection and detection fidelity

The strongest analyses predict several observables from one parameter set. For example, a measured equation of state fixes pressure and compressibility; those predict sound speed and hydrodynamic profiles. Treating each data set with an independent effective temperature, interaction, and density hides inconsistency.

Equation of state from a trap

Under local-density conditions,

$$\mu_\sigma(\mathbf r) = \mu_{\sigma,0} - V_\sigma(\mathbf r).$$

A single trapped profile samples many local chemical potentials. With a calibrated potential and imaging response, hydrostatic balance can convert the density profile into pressure and compressibility. At unitarity these data can be organized into universal functions of reduced temperature and polarization.

LDA fails near small systems, sharp interfaces, rapidly varying lattice potentials, and length scales comparable with correlation or pair sizes.

Expansion

For a weakly interacting gas released suddenly, long time of flight can approximately map momentum to position:

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

Near unitarity, collisions and interaction pressure continue during expansion. Aspect-ratio inversion can indicate hydrodynamic behavior, but it is not unique to superfluidity: a strongly collisional normal gas can also expand hydrodynamically. Turning interactions off rapidly can improve momentum mapping, but that ramp itself must be modeled.

Thermometry

Deep degeneracy makes thermometry difficult because most atoms are insensitive to temperature. Common methods include:

• fits to weakly interacting momentum or density wings;
• equation-of-state thermometry against a calibrated universal curve;
• fluctuation thermometry through fluctuation–dissipation relations;
• impurities or minority components used as embedded thermometers;
• adiabatic ramps to a regime with a better calibrated thermometer;
• lattice spin or density correlations compared with controlled theory.

State the measured quantity and calibration range. A Gaussian width is not a universal thermometer for a degenerate or strongly interacting Fermi gas.

Preparation generally combines a spin mixture, conservative confinement, and evaporative or sympathetic cooling before the interaction ramp. Pauli blocking can slow late-stage rethermalization. A ramp may be adiabatic for two-body binding yet sudden for trap motion; choose its timescale for the degree of freedom that should follow it.

Quantum Simulation Uses

Continuum universal matter

The unitary gas is a clean realization of strongly interacting fermions without a material lattice. It supports quantitative studies of:

• universal thermodynamics and transport;
• pairing above and below the superfluid transition;
• spin imbalance and polarons;
• shear and spin transport;
• scale invariance and its anomalies;
• expansion and two-fluid hydrodynamics.

Universality is a hypothesis with corrections, not a label bestowed by setting $B=B_0$. Verify small effective range, adequate system size, three-dimensional kinematics, and controlled loss.

Optical lattices

Projecting two-component fermions into a sufficiently isolated lowest band can produce the Fermi–Hubbard model

$$\hat H = - t \sum_{\langle ij\rangle,\sigma} \left( \hat c_{i\sigma}^\dagger \hat c_{j\sigma} + \mathrm{h.c.} \right) + U \sum_i \hat n_{i\uparrow} \hat n_{i\downarrow} + \sum_{i,\sigma} V_i \hat n_{i\sigma}.$$

The lattice depth controls tunneling $t$ and Wannier functions; the scattering length contributes to on-site interaction $U$; confinement produces $V_i$. At large repulsive $U/t$ near half filling, the low-energy spin-exchange scale is

$$J \simeq \frac{4t^2}{U}.$$

Reaching $k_{\mathrm B}T\lesssim J$ is much harder than reaching $T<T_{\mathrm F}$ in the original continuum gas. Lattice loading can redistribute entropy, and off-resonant photon scattering or technical noise can heat the sample.

Hubbard Model develops the Hamiltonian, controlled limits, and many-body phases. Optical lattices explains bands, loading, and experimental calibration.

Quantum gas microscopy

Fermionic microscopes can resolve site occupations and correlations, allowing direct measurements of:

• density and spin correlation functions;
• antiferromagnetic domains;
• doublons and holes;
• transport of particles and correlations;
• full counting statistics over finite regions.

Detection can be parity projected, state selective, or lossy depending on the protocol. The reconstructed observable must match the actual detection channel.

Exercises

1. The thermally active shell

For a uniform ideal three-dimensional gas in the grand canonical ensemble, show at low temperature that $\operatorname{Var}N/N\simeq(3/2)T/T_{\rm F}$. Use the total density of states $D(E_{\rm F})=3N/(2E_{\rm F})$ and $f(1-f)=-k_{\rm B}T\,df/d\epsilon$. Does this formula describe the total number variance of an exactly number-fixed sample?

Solution

Independent mode occupations give $\operatorname{Var}N=\int D(\epsilon)f(1-f)d\epsilon$. The factor $f(1-f)$ is concentrated near the chemical potential. Replacing the smooth density of states there by $D(E_{\rm F})$ and integrating the derivative yields $\operatorname{Var}N\simeq k_{\rm B}TD(E_{\rm F})$. Dividing by $N$ gives the result. In a number-fixed ensemble, $\operatorname{Var}N=0$ exactly: mode correlations enforce the constraint. Fluctuations in a small observation region require its own spatial correlation and imaging model.

2. Trap Fermi temperature

A balanced two-component gas has total atom number $N=2.0\times10^5$ in a trap with $\bar\omega/(2\pi)=150\ \mathrm{Hz}$. Calculate the ideal trap Fermi temperature. If $T=120\ \mathrm{nK}$, what is $T/T_{\mathrm F}^{\mathrm{trap}}$?

Solution

Each component has $N_\sigma=N/2$, so

$$E_{\mathrm F}^{\mathrm{trap}} = \hbar\bar\omega \left( 6N_\sigma \right)^{1/3} = \hbar\bar\omega \left( 3N \right)^{1/3}.$$

Therefore

$$\begin{aligned} \frac{ E_{\mathrm F}^{\mathrm{trap}} }{h} &= \left( 150\ \mathrm{Hz} \right) \left( 6.0\times10^5 \right)^{1/3} \\ &\simeq 12.7\ \mathrm{kHz}, \end{aligned}$$

and

$$T_{\mathrm F}^{\mathrm{trap}} \simeq 607\ \mathrm{nK}.$$

Thus

$$\frac{T}{ T_{\mathrm F}^{\mathrm{trap}} } \simeq \frac{120}{607} \simeq 0.20.$$

This is a global ideal-trap convention, not the local reduced temperature at every point in the cloud.

3. Why finite temperature preserves the unitary pressure relation

For a homogeneous zero-range unitary gas, write its energy at fixed entropy and component numbers as

$$E=N E_{\rm F}\,F\!\left(\frac{S}{Nk_{\rm B}},\mathcal P\right), \qquad E_{\rm F}\propto(N/V)^{2/3}.$$

Derive $P=2E/(3V)$. Why does polarization not invalidate the argument?

Solution

At fixed $S,N_\uparrow,N_\downarrow$, both arguments of $F$ are fixed, while $E\propto V^{-2/3}$. Therefore $P=-(\partial E/\partial V)_{S,N_\uparrow,N_\downarrow}=2E/(3V)$. Entropy per particle and polarization change $F$, but introduce no new length in this scale transformation. A finite scattering length or range would add an argument that changes as the volume changes, giving extra terms. The zero-temperature balanced value is $F=3\xi_{\rm B}/5$.

4. Two Pauli suppressions

Explain separately why two identical spin-up fermions have no low-energy $s$-wave collision channel and why an up–down collision can still be suppressed in a deeply degenerate balanced gas.

Solution

For two identical spin-up fermions, the internal spin state is symmetric. The full two-fermion state must be antisymmetric, so the spatial state must be odd under exchange. An $s$ wave has even parity and is forbidden; the leading allowed low-energy channel is $p$ wave.

An up and a down fermion are distinguishable internal components, so an $s$-wave collision is allowed. In a degenerate many-body state, however, the collision rate contains

$$f_1f_2 \left( 1-f_3 \right) \left( 1-f_4 \right).$$

If candidate final states 3 and 4 are already occupied, the factors $1-f_3$ and $1-f_4$ suppress the event. The first mechanism is a two-particle exchange selection rule; the second is final-state blocking by the many-body occupation distribution.

5. Pole, zero, and crossover coordinate

A model resonance has

$$a_{\mathrm{bg}} = -1000a_0, \qquad B_0 = 200\ \mathrm G, \qquad \Delta = 10\ \mathrm G.$$

Find the pole and zero crossing. At $B=205\ \mathrm G$, calculate $a$ and $1/(k_{\mathrm F}a)$ for $k_{\mathrm F}=6.67\times10^6\ \mathrm{m^{-1}}$. Classify the side of the crossover.

Solution

The pole is at $B=B_0=200\ \mathrm G$. The zero is at

$$B = B_0+\Delta = 210\ \mathrm G.$$

At $205\ \mathrm G$,

$$\begin{aligned} a &= -1000a_0 \left[ 1- \frac{10}{5} \right] \\ &= +1000a_0 \simeq 5.29\times10^{-8}\ \mathrm m. \end{aligned}$$

Hence

$$k_{\mathrm F}a \simeq 0.353$$

and

$$\frac{1}{k_{\mathrm F}a} \simeq 2.83.$$

The scattering length is positive, so this point lies on the molecular-BEC side. Whether the universal shallow-dimer formula is accurate also requires $a$ to exceed the microscopic range.

6. Universal shallow dimer

For lithium-6 atoms with $m=9.99\times10^{-27}\ \mathrm{kg}$ and $a=2000a_0$, estimate the universal dimer binding energy in frequency and temperature units.

Solution

The scattering length is

$$a \simeq 1.058\times10^{-7}\ \mathrm m.$$

For equal masses,

$$E_b = \frac{\hbar^2}{ma^2} \simeq 9.94\times10^{-29}\ \mathrm J.$$

Therefore

$$\frac{E_b}{h} \simeq 1.50\times10^5\ \mathrm{Hz} = 150\ \mathrm{kHz},$$

and

$$\frac{E_b}{k_{\mathrm B}} \simeq 7.20\ \mu\mathrm K.$$

These values use the zero-range universal formula. Effective-range and closed-channel corrections matter if $a$ is not sufficiently larger than the interaction range.

7. Distinguish three experimental observations

Classify each observation as primary evidence for degeneracy, pairing, pair condensation, or superfluid response. State one caveat for each.

1. Suppressed local density fluctuations relative to a Poisson gas.
2. A radio-frequency pair-breaking threshold.
3. A stable triangular vortex lattice after rotation.

Solution

1. Suppressed fluctuations are primary evidence for Fermi degeneracy and correlations. Imaging resolution, binning, and interaction corrections must be included.
2. A pair-breaking threshold is primary evidence for pairing or an excitation gap. Final-state interactions, Hartree shifts, trap averaging, and pulse broadening must be modeled. It does not by itself prove superfluidity.
3. A stable vortex lattice is primary evidence for quantized superfluid rotational response. The stirring, nucleation threshold, survival during ramps, and imaging selection must still be characterized.

The observations become much stronger together because they constrain different logical levels.

8. Exchange scale in a Fermi–Hubbard simulator

A lattice simulator has $t/h=500\ \mathrm{Hz}$ and $U/t=8$. Estimate $J/h$ from $J=4t^2/U$. If $k_{\mathrm B}T=0.30t$, calculate $k_{\mathrm B}T/J$. Why is $T<T_{\mathrm F}$ before lattice loading not enough to claim an antiferromagnetic low-temperature state?

Solution

Since $U=8t$,

$$J = \frac{4t^2}{8t} = \frac t2.$$

Thus

$$\frac Jh = 250\ \mathrm{Hz}.$$

The temperature ratio is

$$\frac{k_{\mathrm B}T}{J} = \frac{0.30t}{0.50t} = 0.60.$$

The relevant magnetic scale in the strongly repulsive lattice is $J$, not the pre-lattice continuum Fermi energy alone. Loading changes the spectrum and can redistribute entropy or add heat. A credible antiferromagnetic claim therefore uses spin correlations, structure factors, or staggered magnetization together with calibrated $t$, $U$, entropy, and detection fidelity.

References

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