Codex @codex  0

Combining zero-current bulk densities with quantized circulation gives an approximate winding bound. If equal halves have densities ${\rho }_0(1-a)$ and ${\rho }_0(1+a)$, $0<a<1$, the more restrictive local condition is $n^2<(m{R}^2{\mu }_0/{\hslash }^2)(1-a)/(1+a{)}^2$. A self-consistent finite-current calculation must also allow kinetic pressure to change those densities.

Two complementary signatures are suppressed low-speed dissipation and quantized circulation. Condensation alone establishes macroscopic occupation, not all of these flow properties. In a uniform weakly interacting condensate, the Landau criterion givesfor the Bogoliubov spectrum. Creating an excitation in a flow changes energy by $E(\mathbf{p})-\mathbf{v}\cdot \mathbf{p}$; a negative value allows energy transfer. Boundaries and vortex nucleation can make an actual measured threshold lower than this ideal bulk sound speed.

In the moving-obstacle test of atomic superfluidity, a blue-detuned focused laser repelled atoms in a sodium condensate and was scanned at different speeds. After a fixed exposure, time-of-flight absorption imaging measured the reduction of condensate fraction as a heating indicator. Low-speed heating was small, with a pronounced onset near $1.6mm/s$, below the peak-density sound speed near $6.2mm/s$. A stationary-beam control showed no heating; above-transition clouds constrained heating through the noncondensed fraction. Changing scan frequency and amplitude at common obstacle velocity helped separate a flow threshold from a particular trap resonance. Density inhomogeneity, obstacle geometry and vortex nucleation limit identification of the observed onset with a unique homogeneous Landau velocity.^[ref]

A second experiment stirred rubidium-87 in a magnetic trap by rotating a small transverse anisotropy generated with a far-detuned attractive laser. After stirring, the cloud expanded for $27\mathrm{ms}$ and absorption imaging along the rotation axis revealed vortex cores. Below a threshold no cores appeared; faster rotation produced one or several. Ramping the stirrer down before release reduced additional excitation, and waiting after removing it tested vortex survival. Beam-centre stability and residual trap anisotropy are important controls; an isolated density dip without the rotation and persistence evidence would be less conclusive. This stirred-condensate vortex experiment is a response to rotation, distinct from merely imaging the appearance of a condensate.^[ref]

The superfluid velocity is ${\mathbf{v}}_s=(\hslash /m)\nabla \theta$. A closed loop avoiding a core must return a single-valued order parameter, soA singly quantized vortex therefore has $2\pi$ phase winding and circulation $h/m$, with depleted density in a core of order the healing length. This quantized rotational response differs from ordinary rigid-body flow. The experiment supports that interpretation through controlled stirring and vortex formation; imaging cores alone does not independently measure every phase winding or an absolute zero dissipative rate.

The axial angular momentum per unit volume is $mn(r)r{u}_{\theta }(r)=\hslash n(r)$ for a singly quantized positive quantum vortex. Thus every particle in the prescribed shell contributes $\hslash$:This yieldsThis is $\hslash$ times the shell particle number. For a wall-truncated cloud with $R=R_0<R_r$, the corresponding result is $L_z=\pi H_0\hslash n_0[R^2-a_0^2-(R^4-a_0^4)/(2R_r^2)]$. Reversing the quantized circulation reverses $L_z$ without changing the flow kinetic energy. For winding number $q$, $L_z$ is multiplied by $q$.

Take a singly quantized vortex of positive circulation. The superfluid velocity is $\mathbf{u}=(\hslash /m)\nabla S$, where $S$ is the complex argument of the condensate. Since $S=\theta$, quantized circulation gives $u_{\theta }=\hslash /(mr)$. Insert the radial Thomas–Fermi approximation for a condensate into the cylindrical volume integral, excluding the core $r<a_0$:Integrating the logarithmic and quadratic terms separately givesThe logarithm is the familiar long-range flow contribution of a quantum vortex; the $-1/2$ comes from the declining trapped number density. If the bucket truncates the cloud, put $R=R_0<R_r$ and the same integration gives $E_{\mathrm{kin}}=\pi H_0{\hslash }^2{n}_0[\log (R/a_0)-(R^2-a_0^2)/(2R_r^2)]/m$, provided $a_0<R$. A vortex of integer winding number $q$ has $u_{\theta }=q\hslash /(mr)$ and multiplies this flow energy by $q^2$. The core radius acts as a cutoff of order the local healing length; its internal gradient and interaction energies are not computed by this shell estimate.

Write the condensate order parameter as $\Psi =|\Psi |e^{i\theta }$. Away from vortex cores, single-valuedness around a closed loop gives $\oint \nabla \theta \cdot d\mathbf{l}=2\pi n$. For neutral condensed objects of mass $M$, ${\mathbf{v}}_s=\hslash \nabla \theta /M$, hence the quantized circulationHere $M$ is the atomic mass for a simple bosonic atomic condensate, or the pair mass for a paired-fermion condensate. A nonzero winding can surround a vortex core where the order-parameter amplitude falls to zero. The result concerns collective condensate phase winding, not separate orbit quantization of every constituent particle.

For charge $q$ per condensed object, the gauge-invariant relation is $M{\mathbf{v}}_s=\hslash \nabla \theta -q\mathbf{A}$. Integrating givesThis is fluxoid and flux quantization in a superconductor. With Cooper-pair charge magnitude $2e$, the flux quantum's magnitude isThe orientation fixes its sign. A loop taken deep in a thick superconductor has negligible superflow and therefore encloses an integer flux quantum. In a thin loop or one with appreciable screening current, it is the fluxoid, including the circulation term, that is quantized; the magnetic flux alone need not be exactly an integer multiple.

A bosonic coherent state is an annihilation-operator eigenstate,It has $⟨N⟩=\mathrm{Var}N=|\alpha {|}^2$, a selected complex amplitude, and Poisson-distributed particle number. A coherent-field condensate variational state extends this construction to a spatial field with $⟨\hat{\psi }(\mathbf{r})⟩=\Psi (\mathbf{r})$. Evaluating a normal-ordered interacting Bose Hamiltonian in that state gives a classical-field energy functional; its variational equation is the Gross–Pitaevskii equation. This connects a macroscopic phase to interference, quantized circulation and neutral Josephson effects. A fixed-number condensate instead has vanishing one-point field expectation; off-diagonal long-range order identifies the condensate without requiring exact number fluctuations.

The BCS ground state has the pair-coherent formIt is a product of Cooper-pair pseudospin coherent states: each paired momentum sector mixes its empty and doubly occupied configurations. The anomalous pair expectation is $⟨c_{-\mathbf{k}\downarrow }{c}_{\mathbf{k}\uparrow }⟩=u_{\mathbf{k}}^{\ast }{v}_{\mathbf{k}}$, while an ordinary single-electron expectation vanishes by fermion parity. Pair operators are not unrestricted canonical boson operators, so this state must not be identified with a naive oscillator coherent state of individual electrons. Number projection gives a fixed-even-number description with pair correlations. Phase differences between condensates enter the Josephson effect; in a charged condensate that phase difference must be gauge invariant.

For a one-component superfluid with nonzero order-parameter magnitude on a closed loop, the phase defines a map from the loop into the circle. Its winding number isThe fundamental group ${\pi }_1(S^1)=\mathbb{Z}$ means that smooth deformations preserving nonzero amplitude cannot change this integer. With superfluid velocity ${\mathbf{v}}_s=\hslash \nabla \theta /m_B$, it gives quantized circulation $\oint {\mathbf{v}}_s\cdot d\mathbf{l}=nh/m_B$. A vortex core permits the order parameter to vanish and carries the winding; a superfluid phase slip changes a loop's winding when a core crosses it. For paired neutral fermions, the relevant condensed-object mass is the pair mass.

For charged pairs, gauge-covariant phase rigidity gives fluxoid and flux quantization in a superconductor:The magnitude of the pair flux quantum is $h/(2e)$. Magnetic flux alone approaches an integer multiple of this value when the circulating-velocity term on a bulk contour is negligible. The topological condition is not a claim that an arbitrary finite loop has exactly quantized magnetic flux independent of its screening current.

There are also momentum-space invariants, distinct from real-space vortex winding. A fully gapped two-dimensional Bogoliubov--de Gennes Hamiltonian can have a Chern number obtained from its occupied quasiparticle bands' Berry curvature. In a chiral paired model it can protect edge modes while the bulk gap and relevant symmetry class are maintained. Such an invariant can change only when those assumptions fail, for example through a bulk gap closing. A nodal d-wave state requires a different analysis; it is not automatically a fully gapped Chern superconductor.