Atomic physics, optics, many-body systems
Quantum statistics, condensates, superfluidity, lattices, dark states and electromagnetically induced transparency.
Exchange symmetry
Identical bosons give $+$, fermions give $-$.
Many-body ↗Pauli exclusion
Antisymmetry vanishes when two fermions share all quantum numbers, giving atomic shell structure and stability of matter.
Many-body ↗Bose-Einstein occupation
Mean occupation in equilibrium for bosons, valid for $\varepsilon>\mu$; it diverges as $\varepsilon\to\mu$, enabling condensation.
Many-body ↗Fermi-Dirac occupation
Mean occupation in equilibrium for fermions, always between 0 and 1.
Many-body ↗Zero-temperature Fermi sea
At $T=0$ ideal fermions fill all states up to the Fermi energy, producing degeneracy pressure.
Many-body ↗BEC phase-space criterion
Ideal homogeneous 3D Bose gas condenses when the phase-space density reaches this value, with $\lambda_{th}=h/\sqrt{2\pi mk_BT}$.
Many-body ↗Gross-Pitaevskii equation
Mean-field equation for a dilute weakly interacting condensate, with $g=4\pi\hbar^2a/m$.
Many-body ↗Bogoliubov dispersion
Excitations are phonons $E\propto k$ at low $k$ and free particles at high $k$, for a weakly interacting uniform BEC.
Many-body ↗Landau critical velocity
Below $v_c$ the energetic criterion forbids creating excitations, so flow is dissipationless: superfluidity.
Many-body ↗Quantized circulation
A single-valued order parameter has phase winding $2\pi\ell$, so circulation is quantized, giving quantized vortices.
Many-body ↗Bloch's theorem
In a periodic potential, eigenstates are plane waves times lattice-periodic functions, leading to energy bands.
Many-body ↗Bose-Hubbard Mott transition
At integer filling, when $U/J$ exceeds a critical value, the system changes from superfluid to Mott insulator.
Many-body ↗Lambda-system dark state
At two-photon resonance, a superposition of the two lower states decouples from the excited state and is not excited by light.
Many-body ↗Electromagnetically induced transparency
A strong control field creates a narrow transparency window by destructive interference of excitation pathways, with steep dispersion and slow light.
Many-body ↗Dark-state polariton
Mixing light and spin-wave excitation; reducing the control field to zero converts light into stored atomic coherence, which can later be retrieved.
Many-body ↗