Part III, Chapter 2

Degenerate Fermi Gases

Ultracold fermions obey the Pauli exclusion principle, forming Fermi seas with tunable interactions and exotic pairing.

2.1 Fermi Energy and Temperature

Identical fermions cannot occupy the same quantum state (Pauli exclusion principle). At T = 0, they fill energy levels from the bottom up to the Fermi energy E_F. The Fermi temperature T_F = E_F/k_B characterizes the onset of quantum degeneracy.

Derivation 1: Fermi Energy in a Harmonic Trap

Step 1. For a 3D isotropic harmonic trap with frequency ω, the single-particle levels are ε_n = (n_x + n_y + n_z + 3/2)ℏω. The number of states below energy ε is approximately:

$$N(\epsilon) \approx \frac{1}{6}\left(\frac{\epsilon}{\hbar\omega}\right)^3$$

Step 2. Setting N(ε_F) = N (for a single spin component):

$$E_F = \hbar\omega(6N)^{1/3}, \qquad T_F = \frac{\hbar\omega(6N)^{1/3}}{k_B}$$

For N = 10&sup5; &sup6;Li atoms in a trap with ω/2π = 1 kHz: T_F ≈ 1 μK. Experiments routinely reach T/T_F ≈ 0.05.

2.1.1 Signatures of Degeneracy

• Pauli pressure: Even at T = 0, fermions resist compression due to the exclusion principle. The cloud size scales as R_F = a_ho(48N)¹/&sup6;.
• Reduced compressibility: The equation of state deviates from the classical ideal gas as T approaches T_F.
• Modified thermodynamics: The specific heat becomes linear in T (instead of 3Nk_B/2), a hallmark of Fermi liquid behavior.

2.2 Feshbach Resonances

Feshbach resonances allow magnetic-field tuning of the interatomic interaction strength, including its sign. This is the key tool that enables exploration of the BCS-BEC crossover and strongly interacting Fermi gases.

Derivation 2: Scattering Length Near a Feshbach Resonance

Step 1. A Feshbach resonance occurs when a bound state in a closed channel becomes degenerate with the scattering threshold in the open channel. The magnetic field tunes the energy offset Δμ·B between channels.

Step 2. Near the resonance at B = B₀, the s-wave scattering length is described by:

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

where a_bg is the background scattering length and Δ is the resonance width. The scattering length diverges at B = B₀ (unitarity) and changes sign across the resonance: a < 0 (attractive) on one side, a > 0 (repulsive/bound state) on the other.

Key Feshbach Resonances

• &sup6;Li: Broad resonance at 832 G (Δ = 300 G) — the workhorse for BCS-BEC crossover studies
• &sup4;&sup0;K: Resonance at 202 G — first observation of Fermi condensates (JILA, 2004)
• &sup8;&sup7;Rb: Many resonances used for tuning BEC interactions

2.3 BCS-BEC Crossover

By tuning the interaction strength through a Feshbach resonance, a gas of fermions smoothly crosses over from a BCS superfluid of loosely bound Cooper pairs to a BEC of tightly bound diatomic molecules — the celebrated BCS-BEC crossover.

Derivation 3: BCS Gap Equation in the Crossover

Step 1. The BCS variational wave function is:

$$|\text{BCS}\rangle = \prod_\mathbf{k} (u_\mathbf{k} + v_\mathbf{k} \hat{c}_{\mathbf{k}\uparrow}^\dagger \hat{c}_{-\mathbf{k}\downarrow}^\dagger)|0\rangle$$

with |u_k|² + |v_k|² = 1. The pair amplitude v_krepresents the probability of a Cooper pair (k↑, -k↓).

Step 2. Minimizing the energy gives the gap equation and number equation:

$$\frac{1}{g} = \sum_\mathbf{k} \frac{1}{2E_\mathbf{k}}, \qquad N = \sum_\mathbf{k}\left(1 - \frac{\xi_\mathbf{k}}{E_\mathbf{k}}\right)$$

where E_k = √(ξ_k² + Δ²) is the quasiparticle energy, ξ_k = ℏ²k²/(2m) - μ, and Δ is the gap parameter.

Step 3. Three regimes emerge: (i) BCS limit (1/k_Fa → -∞): Δ << E_F, large Cooper pairs; (ii) Unitarity (1/k_Fa = 0): Δ ≈ E_F, strongly interacting; (iii) BEC limit (1/k_Fa → +∞): tightly bound dimers that Bose-condense. The crossover is smooth with no phase transition.

2.4 The Unitary Fermi Gas

At unitarity (|a| → ∞), the scattering length drops out of the problem. The gas is scale-invariant — its properties depend only on density and temperature. This universal regime connects neutron stars, quark-gluon plasma, and cold atoms.

Derivation 4: Universal Equation of State and the Bertsch Parameter

Step 1. At unitarity, the only energy scale is E_F. By dimensional analysis, the ground state energy per particle must be:

$$\frac{E}{N} = \xi \cdot \frac{3}{5}E_F$$

where ξ is the Bertsch parameter, a universal dimensionless constant.

Step 2. Quantum Monte Carlo calculations and experiments consistently find:

$$\xi = 0.370(5)$$

Step 3. The Bertsch parameter gives the effective mass, chemical potential, and pairing gap at unitarity. The superfluid transition temperature is T_c/T_F ≈ 0.167 — remarkably high compared to conventional superconductors (T_c/T_F ≈ 10&supmin;&sup4;).

2.5 Collective Excitations and Superfluidity

Collective modes of the trapped Fermi gas provide direct evidence of superfluidity and probe the equation of state across the BCS-BEC crossover.

Derivation 5: Breathing Mode Frequency

Step 1. For a gas in a harmonic trap with equation of state E ∝ V^-γ(polytropic index γ), the radial breathing mode frequency is:

$$\omega_B = \omega_\perp\sqrt{2 + 2\gamma}$$

Step 2. For a non-interacting Fermi gas: γ = 2/3, giving ω_B = 2ω_⊥. For a unitary Fermi gas, scale invariance requires γ = 2/3 as well (identical to ideal gas!):

$$\omega_B^{\text{unitary}} = 2\omega_\perp$$

Step 3. In the BEC limit, γ = 1 (molecular BEC with contact interactions):

$$\omega_B^{\text{BEC}} = \sqrt{5}\,\omega_\perp \approx 2.236\,\omega_\perp$$

Measuring the breathing mode frequency precisely tests the equation of state and confirms scale invariance at unitarity. Experiments agree with theory to better than 1%.

2.6 Experimental Evidence for Superfluidity

Several key experiments have established superfluidity in ultracold Fermi gases across the BCS-BEC crossover.

2.6.1 Vortex Lattices

Zwierlein et al. (MIT, 2005) rotated a Fermi gas near a Feshbach resonance and observed regular triangular lattices of quantized vortices — the definitive signature of superfluidity. Remarkably, vortices were observed across the entire BCS-BEC crossover, from the BEC side (tightly bound molecules) through unitarity to the BCS side (loosely bound Cooper pairs).

2.6.2 Condensation of Fermionic Pairs

Pair condensation is detected by rapidly sweeping the magnetic field from the BCS side to the BEC side, converting Cooper pairs into tightly bound molecules that can be imaged. The bimodal momentum distribution (narrow peak + broad background) confirms macroscopic occupation of the zero-momentum pair state — fermionic condensation.

2.6.3 RF Spectroscopy and the Pairing Gap

Radio-frequency spectroscopy drives atoms from one spin state to a third (unpaired) state. The threshold frequency for this transition measures the pairing gap Δ:

$$\hbar\omega_{\text{RF}} = \Delta + E_{\text{bind}}$$

Chin et al. (Innsbruck, 2004) and Schunck et al. (MIT, 2008) mapped the gap across the crossover, finding Δ ≈ 0.4 E_F at unitarity, consistent with theory.

2.6.4 Second Sound

In 2013, Sidorenkov et al. (Innsbruck) observed second sound — a temperature wave propagating at a speed distinct from first sound — in a unitary Fermi gas. Second sound is a hallmark of two-fluid hydrodynamics and superfluidity, previously observed only in liquid helium.

2.7 Fermi Polarons and Spin Imbalance

When one spin component has more atoms than the other (population imbalance), pairing is frustrated and exotic phases emerge.

2.7.1 The Fermi Polaron

A single impurity atom interacting with a Fermi sea forms a Fermi polaron — a quasiparticle dressed by particle-hole excitations of the sea. The polaron energy at unitarity:

$$E_P = -0.615\, E_F$$

The effective mass m* ≈ 1.20 m and the quasiparticle residue Z ≈ 0.78. These values have been precisely measured using RF spectroscopy by Schirotzek et al. (MIT, 2009) and Kohstall et al. (Innsbruck, 2012).

2.7.2 Phase Separation in Imbalanced Gases

At unitarity with population imbalance, the system phase separates: a superfluid core of equal spin populations surrounded by a normal shell of excess majority atoms. This was directly observed in-trap by Shin et al. (MIT, 2006) and Partridge et al. (Rice, 2006).

2.7.3 FFLO State

The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state — a superfluid with spatially modulated order parameter — is predicted for intermediate imbalance. Despite intense experimental effort, the FFLO state remains elusive in 3D cold atoms, though evidence has been found in 1D spin-imbalanced Fermi gases (Liao et al., Rice, 2010).

2.8 Fermions in Optical Lattices

Loading degenerate Fermi gases into optical lattices realizes the Fermi-Hubbard model, believed to contain the essential physics of high-temperature superconductivity:

$$\hat{H} = -J\sum_{\langle i,j\rangle,\sigma} \hat{c}_{i\sigma}^\dagger \hat{c}_{j\sigma} + U\sum_i \hat{n}_{i\uparrow}\hat{n}_{i\downarrow} - \mu\sum_{i,\sigma}\hat{n}_{i\sigma}$$

Experimental Achievements

• Band insulator: Filling factor n = 2 per site (one per spin). First observed by Köhl et al. (ETH Zurich, 2005).
• Mott insulator: U/J >> 1, one atom per site. Observed via the vanishing of compressibility (Jordens et al., 2008; Schneider et al., 2008).
• Antiferromagnetic correlations: Short-range Néel ordering observed in 2D and 3D lattices using quantum gas microscopes (Greif et al., 2013; Mazurenko et al., 2017).
• d-wave pairing: The holy grail — observing d-wave Cooper pairing in the doped Hubbard model remains an outstanding experimental challenge.

Quantum gas microscopes now image individual atoms on each lattice site with single-site resolution, enabling direct measurement of spin-spin correlations, string order parameters, and entanglement entropy — quantities inaccessible in solid-state experiments.

Applications

Understanding Neutron Stars

The interior of neutron stars contains strongly interacting fermionic nuclear matter at densities where 1/k_Fa ≈ 0 (unitarity). Cold atom experiments provide the most precise measurements of the equation of state at unitarity, directly constraining neutron star models.

High-Temperature Superconductivity

The BCS-BEC crossover in cold atoms provides insights into the pairing mechanism of unconventional superconductors. The pseudogap regime observed in cold Fermi gases has striking parallels to the cuprate superconductor phase diagram.

Quantum Simulation of Fermi-Hubbard Model

Fermions in optical lattices realize the Fermi-Hubbard model, believed to describe high-T_c superconductivity. Recent experiments have observed antiferromagnetic correlations and begun probing the d-wave pairing regime.

Topological Phases

Spin-orbit coupled Fermi gases and Fermi gases in optical lattices with artificial gauge fields can realize topological superfluids and insulators, providing clean platforms for studying Majorana fermions and other exotic quasiparticles.

2.9 Synthetic Spin-Orbit Coupling

Raman lasers can couple the internal (spin) and external (momentum) degrees of freedom of atoms, creating synthetic spin-orbit coupling (SOC). For fermions, this opens the door to topological phases and exotic pairing.

Two counter-propagating Raman beams drive transitions between spin states |↑〉 and |↓〉 with a momentum transfer 2ℏk_R. The effective single-particle Hamiltonian is equivalent to an equal Rashba-Dresselhaus spin-orbit coupling:

$$\hat{H}_{\text{SOC}} = \frac{(\hat{p} - \hbar k_R \hat{\sigma}_z)^2}{2m} + \frac{\Omega}{2}\hat{\sigma}_x + \frac{\delta}{2}\hat{\sigma}_z$$

First realized by Lin et al. (NIST, 2011) in bosons and extended to Fermi gases by Wang et al. (2012) and Cheuk et al. (MIT, 2012). With SOC, a Zeeman field, and s-wave pairing, the Fermi gas can enter a topological superfluid phase supporting Majorana zero modes — the foundation for topological quantum computing.

Key Equations Summary

Fermi Energy (harmonic trap):

$$E_F = \hbar\omega(6N)^{1/3}$$

Feshbach Resonance:

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

BCS Quasiparticle Energy:

$$E_\mathbf{k} = \sqrt{\xi_\mathbf{k}^2 + \Delta^2}$$

Universal Energy at Unitarity:

$$E/N = \xi \cdot \frac{3}{5}E_F, \qquad \xi = 0.370$$

Contact Parameter (Tan relations):

$$\mathcal{C} = \lim_{k\to\infty} k^4 n_\sigma(k)$$

Tan Relations (Universal at All Couplings)

Shina Tan discovered that a single parameter &Cscr; (the contact) governs many properties of strongly interacting Fermi gases:

• Momentum distribution: n(k) → &Cscr;/k&sup4; at large k
• Energy relation (adiabatic sweep): dE/d(-1/a) = ℏ²&Cscr;/(4πm)
• Pressure relation: P = 2E/(3V) + ℏ²&Cscr;/(12πmV)
• RF line shape: The high-frequency tail of the RF spectrum scales as ω^-3/2&Cscr;

The contact quantifies the probability of finding two atoms at short distance and is measurable in multiple independent ways, providing a consistency check of many-body theory.

2.9 Low-Dimensional Fermi Gases

Confining fermions to 2D (pancake) or 1D (cigar) geometries reveals physics qualitatively different from 3D.

2.9.1 Two-Dimensional Fermi Gases

In 2D, the bound state exists for any attractive interaction (unlike 3D where a threshold exists). The pairing gap and T_c are enhanced. The BKT (Berezinskii-Kosterlitz-Thouless) transition replaces the 3D superfluid transition, characterized by the proliferation of vortex-antivortex pairs.

Murthy et al. (Heidelberg, 2015) observed the BKT transition in a 2D Fermi gas using the algebraic decay of phase correlations measured through matter-wave interferometry.

2.9.2 One-Dimensional Fermi Gases

In 1D, the Yang-Gaudin model provides an exact solution for fermions with contact interactions. Unique 1D phenomena include:

• Tonks-Girardeau limit: Infinitely repulsive fermions map to non-interacting bosons (fermionization of bosons, bosonization of fermions)
• Spin-charge separation: Spin and density excitations propagate at different velocities
• Exact thermodynamics: The Yang-Yang thermodynamic Bethe ansatz provides exact finite-temperature results at all coupling strengths

Historical Context

1957 — Bardeen, Cooper, Schrieffer: BCS theory of superconductivity: fermion pairing by phonon-mediated attraction. Nobel Prize 1972.

1969 — Eagles; 1980 — Leggett: Predicted a smooth crossover from BCS pairing to BEC of bound pairs as interaction strength increases.

1999 — DeMarco & Jin: First observation of quantum degeneracy in a fermionic atomic gas (&sup4;&sup0;K at JILA), reaching T/T_F ≈ 0.5.

2003 — Regal, Greiner & Jin; Zwierlein et al.: Creation of molecular BEC from Fermi gases using Feshbach resonances.

2004 — Regal et al.; Zwierlein et al.: Observation of fermionic condensation and superfluidity across the BCS-BEC crossover.

2005 — Zwierlein et al.: Observation of quantized vortices in a Fermi gas, confirming superfluidity across the entire crossover.

Interactive Simulation

This simulation visualizes the Feshbach resonance, BCS-BEC crossover phase diagram, the Fermi-Dirac distribution at different temperatures, and the gap across the crossover.

Fermi Gases: Feshbach Resonance & BCS-BEC Crossover

Python

script.py124 lines

import numpy as np
import matplotlib.pyplot as plt

fig, axes = plt.subplots(2, 2, figsize=(12, 10))
fig.patch.set_facecolor("#0a0a0a")
for ax in axes.flat:
    ax.set_facecolor("#111111")

# Plot 1: Feshbach resonance - scattering length vs B field
ax1 = axes[0, 0]
B = np.linspace(-3, 3, 1000)
B0 = 0.0
Delta = 1.0
a_bg = 1.0
a = a_bg * (1 - Delta / (B - B0))
a = np.clip(a, -10, 10)

ax1.plot(B, a, color="#a78bfa", linewidth=2)
ax1.axhline(y=0, color="gray", linestyle="--", alpha=0.3)
ax1.axvline(x=B0, color="#c084fc", linestyle=":", alpha=0.5, label="B0 (resonance)")
ax1.fill_between(B[B < B0], -10, 10, alpha=0.05, color="#a78bfa")
ax1.set_xlabel("Magnetic field (B - B0) / Delta", color="white")
ax1.set_ylabel("Scattering length a / a_bg", color="white")
ax1.set_title("Feshbach Resonance", color="#c4b5fd", fontsize=14)
ax1.set_ylim(-10, 10)
ax1.legend(facecolor="#1a1a2e", edgecolor="#7c3aed", labelcolor="white")
ax1.tick_params(colors="white")
for spine in ax1.spines.values():
    spine.set_color("#7c3aed")

# Plot 2: Fermi-Dirac distribution at different T/TF
ax2 = axes[0, 1]
epsilon = np.linspace(0, 3, 500)  # in units of EF

for T_ratio, col, label in [(0.01, "#a78bfa", "T/TF = 0.01"),
                              (0.1, "#c084fc", "T/TF = 0.1"),
                              (0.3, "#f97316", "T/TF = 0.3"),
                              (1.0, "#ef4444", "T/TF = 1.0")]:
    if T_ratio < 0.02:
        f_FD = np.where(epsilon < 1, 1.0, 0.0)
    else:
        f_FD = 1 / (np.exp((epsilon - 1) / T_ratio) + 1)
    ax2.plot(epsilon, f_FD, color=col, linewidth=2, label=label)

ax2.axvline(x=1, color="gray", linestyle="--", alpha=0.3)
ax2.set_xlabel("Energy / EF", color="white")
ax2.set_ylabel("Occupation f(E)", color="white")
ax2.set_title("Fermi-Dirac Distribution", color="#c4b5fd", fontsize=14)
ax2.legend(facecolor="#1a1a2e", edgecolor="#7c3aed", labelcolor="white", fontsize=9)
ax2.tick_params(colors="white")
for spine in ax2.spines.values():
    spine.set_color("#7c3aed")

# Plot 3: BCS-BEC crossover phase diagram
ax3 = axes[1, 0]
inv_kfa = np.linspace(-2, 2, 300)

# Schematic Tc/TF across crossover
# BCS side: exponentially small
Tc_bcs = np.where(inv_kfa < -0.5,
                   0.6 * np.exp(np.pi / (2 * inv_kfa)),
                   np.nan)
# Crossover/BEC side
Tc_full = np.zeros_like(inv_kfa)
for i, x in enumerate(inv_kfa):
    if x < -1.5:
        Tc_full[i] = 0.6 * np.exp(np.pi / (2 * x))
    elif x < 0:
        Tc_full[i] = 0.167 * (1 + 0.5 * x)
    elif x < 0.5:
        Tc_full[i] = 0.167 + 0.05 * x
    else:
        Tc_full[i] = 0.218 * (1 - 0.3 / (1 + x))

Tc_full = np.clip(Tc_full, 0, 0.3)
ax3.plot(inv_kfa, Tc_full, color="#a78bfa", linewidth=2)
ax3.fill_between(inv_kfa, 0, Tc_full, alpha=0.15, color="#a78bfa", label="Superfluid")
ax3.fill_between(inv_kfa, Tc_full, 0.35, alpha=0.1, color="#f97316", label="Normal")

# Labels
ax3.text(-1.5, 0.05, "BCS", color="#c084fc", fontsize=14, fontweight="bold", ha="center")
ax3.text(0, 0.28, "Unitarity", color="white", fontsize=10, ha="center")
ax3.text(1.5, 0.05, "BEC", color="#f97316", fontsize=14, fontweight="bold", ha="center")

ax3.set_xlabel("1 / (kF * a)", color="white")
ax3.set_ylabel("Tc / TF", color="white")
ax3.set_title("BCS-BEC Crossover Phase Diagram", color="#c4b5fd", fontsize=14)
ax3.set_ylim(0, 0.35)
ax3.legend(facecolor="#1a1a2e", edgecolor="#7c3aed", labelcolor="white")
ax3.tick_params(colors="white")
for spine in ax3.spines.values():
    spine.set_color("#7c3aed")

# Plot 4: Pairing gap across crossover
ax4 = axes[1, 1]
inv_kfa_gap = np.linspace(-2, 2, 300)

# Schematic gap Delta/EF
gap = np.zeros_like(inv_kfa_gap)
for i, x in enumerate(inv_kfa_gap):
    if x < -1:
        gap[i] = 0.5 * np.exp(np.pi / (2 * x))
    elif x < 0:
        gap[i] = 0.5 * (1 + x)**2 + 0.5 * np.exp(np.pi / (2 * min(x, -0.01)))
    else:
        gap[i] = 0.69 + 0.5 * x

gap = np.clip(gap, 0, 3)

ax4.plot(inv_kfa_gap, gap, color="#a78bfa", linewidth=2, label="Delta / EF")
ax4.axhline(y=0.69, color="#c084fc", linestyle="--", alpha=0.5, label="Delta/EF at unitarity")
ax4.set_xlabel("1 / (kF * a)", color="white")
ax4.set_ylabel("Gap Delta / EF", color="white")
ax4.set_title("Pairing Gap Across Crossover", color="#c4b5fd", fontsize=14)
ax4.legend(facecolor="#1a1a2e", edgecolor="#7c3aed", labelcolor="white")
ax4.tick_params(colors="white")
for spine in ax4.spines.values():
    spine.set_color("#7c3aed")

plt.tight_layout()
plt.savefig("output.png", dpi=150, bbox_inches="tight", facecolor="#0a0a0a")
plt.show()
print("Plot saved: Fermi gases - Feshbach resonance, Fermi-Dirac, BCS-BEC crossover, pairing gap.")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Conceptual Questions

Q1: How can fermions form a superfluid?

Fermions pair into bosonic Cooper pairs and the pairs undergo BEC. In the BCS limit, pairs are loosely bound and overlap extensively. In the BEC limit, pairs are tightly bound molecules. At unitarity, the pairs are neither — they are strongly correlated with size comparable to the interparticle spacing.

Q2: What is special about unitarity?

At unitarity (1/a = 0), the scattering length diverges and drops out of all physical quantities. The system becomes scale-invariant — all properties depend only on density and temperature. This universality connects cold atoms to neutron stars and nuclear matter.

Q3: Why is the Fermi gas T_c/T_F so much higher than for metallic superconductors?

In metals, the attractive phonon-mediated interaction is weak (T_c/T_F ~ 10&supmin;&sup4;). At unitarity, the interaction is the strongest possible (limited only by unitarity of the S-matrix), giving T_c/T_F ≈ 0.17 — comparable to high-T_ccuprates and much higher than conventional superconductors.

Further Topics

Fermi Gas Frontiers

• Bose-Fermi mixtures: Mixtures of bosonic and fermionic atoms exhibit boson-mediated interactions between fermions, analogous to phonon-mediated Cooper pairing in BCS theory. These systems realize Bose-Fermi Hubbard models and polaronic physics.
• SU(N) Fermi gases: Alkaline-earth atoms (¹&sup7;³Yb, &sup8;&sup7;Sr) have nuclear spin I decoupled from electronic structure, giving SU(N=2I+1) symmetry with N up to 10. These large-spin systems have no analog in condensed matter and exhibit novel magnetic ordering and Kondo physics.
• Quantum gas microscopes: Single-site, single-atom resolved imaging of fermions in optical lattices enables direct measurement of density-density correlations, string order, and entanglement entropy — the most detailed probe of many-body quantum states ever achieved.
• Non-equilibrium dynamics: Quantum quenches across the BCS-BEC crossover reveal far-from-equilibrium pairing dynamics, prethermalization plateaus, and the emergence of Higgs modes (oscillations of the order parameter amplitude).