Physics of dilute neutral-atom gases in the nK-μK temperature regime. Laser-cooling (Hansch-Schawlow 1975 proposal, Chu-Cohen-Tannoudji-Phillips 1997 Nobel) down to the Doppler limit T_D = ℏΓ/(2 k_B); sub-Doppler (Sisyphus,…
ultracold-atoms
BEC order parameter Ψ(r,t): macroscopic occupation of the ground state
Bogoliubov / Penrose-Onsager criterion for Bose-Einstein condensation: a one-body density matrix ρ_1(r, r') = ⟨Ψ̂†(r')Ψ̂(r)⟩ has a single…
Doppler cooling limit: T_D = ℏΓ/(2 k_B) (Hansch-Schawlow 1975)
Hansch-Schawlow 1975 / Wineland-Itano 1979 proposal; Chu 1985 experimental demonstration of optical molasses: six red-detuned…
Forced evaporative cooling: truncate the Boltzmann tail to cool the remainder
Hess 1986 proposal; Masuhara-Doyle-Luiten 1988 demonstration in hydrogen; Davis-Petrich-Ketterle 1995 in sodium. After laser cooling has…
BEC critical temperature: k_B T_c = (2πℏ²/m) · (n/ζ(3/2))^(2/3)
Einstein 1925: for a non-interacting ideal Bose gas of N indistinguishable bosons with mass m in a uniform volume V, the thermal de Broglie…
GP healing length: ξ = 1/√(8π n a_s) (kinetic ≡ interaction)
Healing length ξ is the lengthscale at which the kinetic energy ℏ²/(2mξ²) of a condensate density modulation equals the mean-field…
Feshbach resonance: a(B) = a_bg(1 − Δ/(B − B_0)) — tunable interactions
Feshbach 1958 (nuclear scattering); Tiesinga-Verhaar-Stoof 1993 (ultracold atoms); Inouye et al 1998 (observation in ²³Na condensate). …
Thomas–Fermi approximation for trapped BEC: kinetic term dropped, μ = V(r) + g·n(r)
Thomas–Fermi approximation for a weakly-interacting trapped Bose–Einstein condensate. Setup: a macroscopic condensate wavefunction Ψ(r)…
Polylog / Riemann-zeta Bose–Einstein integrals: g_ν(z) = Li_ν(z); g_ν(1) = ζ(ν)
Bose–Einstein thermodynamic integrals as polylogarithms. Setup: ideal Bose gas in 3D with chemical potential μ < 0 and fugacity z =…
Bose-Hubbard in optical lattice: H = -J·Σ b†b + (U/2)·Σ n(n-1) in tensor-product site basis
Bose-Hubbard model on an optical lattice. Setup: neutral atoms loaded into a standing-wave potential V(x) = V₀·sin²(k·x) forming an array…
Thomas–Fermi radius for trapped BEC: R_TF = sqrt(2μ/(m·ω²))
Thomas–Fermi radius of a trapped BEC. From framework F7, the TF density profile n(r) = (μ − (1/2)m·ω²·r²)/g vanishes at r = R_TF where the…
BEC condensate fraction: N₀/N = 1 − (T/Tc)^(3/2); k_B·Tc = 2π·ℏ²·n^(2/3)/(m·ζ(3/2)^(2/3))
BEC condensate fraction and critical temperature. Using framework F8 (polylog thermodynamic integrals), the thermal-cloud density above Tc…
Bose-Hubbard MI-SF critical ratio (strong-coupling): (U/J)_c = 2·√(2·n·(n+1)) at integer filling n
Mott-insulator to superfluid critical hopping ratio in the strong-coupling perturbation limit. Setup: Bose-Hubbard Hamiltonian from…
Cohen-Tannoudji sub-Doppler cooling
Cohen-Tannoudji 1989 / Dalibard-Cohen-Tannoudji 1989: Sisyphus + polarization-gradient + VSCPT cooling mechanisms reach below the Doppler…
BEC (Anderson 1995)
Anderson-Cornell-Wieman + Ketterle 1995 (Nobel 2001) BEC in 87Rb + Na; modern atom-chip + species (Sr/Yb/Cr/Er/Dy).
Optical lattice (Jaksch 1998)
Jaksch-Bruder-Cirac-Gardiner-Zoller 1998 cold-atoms in optical-lattice maps to Bose-Hubbard; basis of analog-quantum-simulation.
Feshbach tunable
Feshbach-resonance enables tunable-interaction; modern degenerate-Fermi-gas BCS-BEC crossover Greiner 2003 + Zwierlein 2005.
Rydberg arrays (Saffman 2010)
Saffman-Walker-Molmer 2010 Rydberg-blockade quantum-gates; modern QuEra + Atom-Computing 256+ qubit demonstrations.
Dipolar droplets (Kadau 2016)
Kadau-Pfau 2016 magnetic-dipolar quantum droplets in Dy; modern dipolar-supersolid demonstrations 2019.
BEC (Cornell-Wieman-Ketterle 1995)
E Cornell-C Wieman 1995 + Ketterle 1995 (Nobel 2001) BEC-Rb-Na; modern modern foundational text + 2024 BEC chip-atom + microgravity ISS.
Doppler cooling (Hänsch-Schawlow 1975)
T Hänsch-A Schawlow 1975 + Wineland Itano 1979; modern modern foundational + sub-Doppler Sisyphus + 2024 magic-wavelength clock.
MOT trap (Phillips-Chu Nobel 1997)
S Chu 1985 + W Phillips Cs-MOT + Cohen-Tannoudji Sisyphus (Nobel 1997); modern modern foundational + 2024 dual-species-tweezer arrays.
Hubbard lattice (Jaksch-Zoller 1998)
D Jaksch-P Zoller 1998 ultracold-Hubbard; modern modern foundational + 2024 quantum-gas microscope + Mott-superfluid imaging.
Feshbach (Inouye 1998)
S Inouye-Stamper-Kurn 1998 Feshbach-resonance Na; modern modern foundational + Cs-Cesium 2003 BCS-BEC crossover + 2024 dual-species.
Synthetic gauge (Lin 2009)
Y-J Lin-Spielman 2009 NIST synthetic-gauge-field; modern modern foundational + 2024 Hofstadter-Harper + topological-Hall ultracold.