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A physicist who studied superfluidity and macroscopic quantum mechanics at Illinois.^[ref]

A Bose-Einstein condensate is a state with macroscopic occupation of a one-particle mode. A phase-selected mean-field description can use $⟨\psi ⟩\ne 0$, while a fixed-number finite-volume state need not have a nonzero field expectation. Bose-Einstein condensation and superfluidity are related but distinct notions; long-wavelength phase fluctuations can invalidate a uniform condensate saddle in low dimensions.

Bose-Einstein condensation concerns occupation of a one-particle state. Matter-wave coherence concerns field correlations between positions or times. Superfluidity concerns a flow response, including suppression of dissipation under appropriate conditions. A density peak diagnoses neither a relative quantum phase between two separately prepared clouds nor a critical flow speed. These observables require different experiments.

For example, in a single-mode fixed-number Fock state, $⟨\hat{a}⟩=0$ but $⟨{\hat{a}}^{†}\hat{a}⟩=N$. Writing $\hat{\psi }(\mathbf{r})=\varphi (\mathbf{r})\hat{a}$ for that occupied mode givesThus vanishing mean field does not imply vanishing first-order matter-wave coherence. For two independent fixed-number modes, $⟨{\hat{a}}^{†}\hat{b}⟩=0$: there is no ensemble mean interference term establishing a fixed relative quantum phase. Interpreting fringes in an individual detection record requires considering correlations and the measurement; assigning a pre-existing definite relative quantum phase is unnecessary.

Wolfgang Ketterle's lecture describes a separate coherence test: two condensates expanded into one another and produced fringes. Selectively removing one condensate made the fringes disappear on alternating preparation cycles, testing against diffraction from the confinement. This compared two-cloud and one-cloud signals; it did not measure dissipationless flow.^[ref]

A physicist who researched superfluidity and superconductivity in thin systems.^[ref]

In a dilute gas, Bose-Einstein condensation makes a single one-particle state hold a substantial fraction of the bosons. This is a population effect: cooling reduces the number that the excited energy eigenstates can accommodate at thermal equilibrium. The surplus accumulates in a Bose-Einstein condensate. For an ideal gas the occupied mode is the trap ground state; interactions can deform that mode, as described by the Gross–Pitaevskii equation. Dilute does not mean classical: a small number density can still give overlapping thermal de Broglie wavelengths at sufficiently low temperature.

The appeal is that changing the trap changes a calculable Quantum Hamiltonian, while absorption imaging makes its population distribution visible. The Bose condensation in a three-dimensional harmonic trap below turns density of states into a prediction for a critical temperature and condensate fraction. These are separate questions from measuring matter-wave coherence or superfluidity.

Eric A. Cornell and Carl E. Wieman's joint lecture emphasizes the competition between useful elastic collisions, which restore thermal equilibrium during evaporative cooling, and collisions that remove or heat the trapped sample. A long trap lifetime alone is insufficient: reequilibration must keep up with evaporation.^[ref]

David Morris Lee, Douglas Dean Osheroff and Robert Coleman Richardson compressed coexisting liquid and solid helium-3 in a cell precooled by a dilution refrigerator. Helium-4 pressure drove coupled bellows and a piston; a capacitance diaphragm gauge read pressure and platinum nuclear magnetic resonance supplied thermometry. Uniform compression revealed a slope break A and a small pressure drop B; decompression and varied solid fraction checked reproducibility. Their April 1972 paper, Evidence for a New Phase of Solid He3, interpreted a new solid phase, without establishing liquid superfluidity. ^[ref]

The October follow-up replaced the thermometer assembly with a sample nuclear magnetic resonance coil. An applied magnetic field gradient selected different sample layers, separating strong solid signals from the liquid background. At A the liquid resonance shifted; at B its shift disappeared and its magnetic susceptibility fell. Removing the gradient resolved the frequency shift directly. Independence from solid fraction and distinct relaxation times supported the liquid assignment. Thus the reinterpretation rested on phase-selective evidence, rather than relabeling the first paper's title. These magnetic observations identify changed liquid phases; they do not directly measure frictionless flow. ^[ref]

Douglas Dean Osheroff recalls the subsequent p-wave interpretation. ^[ref]

The inference has a useful limit. A pressure anomaly in two coexisting phases cannot, by itself, identify which phase changed: both contribute to the common phase coexistence curve. Thermometer lag, a temperature gradient or slow equilibration can also separate indicated temperature from the sample state. Correlating another response with the anomaly provides information that repeating the same pressure trace cannot supply. Conversely, a shifted magnetic resonance alone does not measure a frictionless flow rate.

For a uniform three-dimensional ideal gas of spinless bosons with energy $p^2/(2m)$ and ground energy set to zero, the excited number density is $n_{\mathrm{ex}}={\lambda }_T^{-3}{\mathrm{Li}}_{3/2}(z)$, with quantum gas fugacity $z=e^{\mu /(k_{B}T)}\le 1$. Its finite maximum $\zeta (3/2)/{\lambda }_T^3$ gives the displayed critical temperature. Below it, particle conservation gives condensate fraction $N_0/N=1-(T/T_c{)}^{3/2}$. The excited-state internal energy is $U=(3/2)V{k}_{B}T{\lambda }_T^{-3}{\mathrm{Li}}_{5/2}(z)$, with $z\to 1$ below the transition. The exponent differs from that of a harmonic trap. Condensation is not by itself a demonstration of superfluidity; the ideal quadratic dispersion has vanishing Landau critical speed.

The Bose-gas phase-only action gives a thermal infrared integral $\int d^{d}q/q^2$ and a zero-temperature equal-time integral $\int d^{d}q/|q|$. Thus an infinite uniform short-range interacting gas has no true finite-temperature condensate in $d\le 2$, and the one-dimensional ground state also lacks true condensate long-range order. This does not exclude local density coherence, a finite-size condensate description or two-dimensional algebraic order and superfluidity.^[ref]

The Landau criterion asks whether creating an excitation can lower a moving superfluid's energy. For a weakly coupled, heavy obstacle moving through a uniform liquid initially at zero temperature, let $ϵ(p)>0$ be the excitation energy in the fluid rest frame, with momentum magnitude $p$. Transferring momentum $\mathbf{p}$ from an obstacle moving at velocity $\mathbf{v}$ supplies, to leading order in its negligible recoil, energy $\mathbf{v}\cdot \mathbf{p}$. Emission requiresFor an isotropic dispersion relation, the easiest direction is parallel to $\mathbf{v}$, giving the necessary kinematic conditionAn infimum includes cases in which the limiting slope occurs only as $p\to 0$. A finite-mass obstacle of mass $M$ instead loses $\mathbf{v}\cdot \mathbf{p}-p^2/(2M)$, so the corresponding emission inequality includes the extra positive recoil term $p^2/(2M)$ on the left. This is an energy and momentum balance, not a calculation of an emission rate.

The Landau criterion excludes spontaneous emission into the specified spectrum below its threshold. Above it, kinematic permission alone does not guarantee strong coupling or rapid decay. It is not a universal observed flow threshold: walls, channel geometry and quantum vortex nucleation can enable other decay routes, and finite temperature permits scattering from existing excitations. Using only bulk phonons and rotons leaves those processes outside the model.

Landau's 1941 energy and momentum argument used a roton approximation different from the finite-momentum minimum illustrated below.^[ref]

Landau received his Nobel Prize in Physics in 1962 while hospitalized in Moscow after a car accident.^[ref]

For a deliberately simple Bose-Einstein condensation comparison, stipulate an ideal free-particle excitation law $ϵ=p^2/(2m)$. Then $ϵ/p=p/(2m)$ has infimum zero. Macroscopic one-particle occupation therefore supplies no positive emission threshold by this argument alone. By contrast a linear phonon law $ϵ=cp$ yields $v_L=c$. The dispersion relation and the collective response contain information that the occupation criterion alone does not supply; Bose-Einstein condensation and thermodynamic superfluidity must be tested separately.

To check the picture algebraically, on the stipulated roton branch,Thus tangency occurs at $q_L=\sqrt{4.3}$, slightly above the roton energy minimum. At an energy minimum $e^{\prime }=0$, but a tangent through the origin needs $e^{\prime }=e/q>0$. Minimizing energy and minimizing the energy-to-momentum ratio are different operations. This geometric construction makes the criterion useful even when the full dispersion relation is supplied numerically rather than by a simple formula.

Awarded in equal thirds to David M. Lee, Douglas D. Osheroff and Robert C. Richardson for discovering superfluidity in helium-3.^[ref]

Sound propagation. In the 1997 experiment, a focused laser beam abruptly modified the trapping potential of a dilute condensate, producing a local excitation. A rapid sequence of nondestructive phase-contrast imaging measurements followed the resulting density motion. Measuring propagation speed at different densities tested the sound-like collective excitation spectrum, rather than individual free-particle flight.^[ref] For a weakly interacting uniform condensate, the Bogoliubov spectrum isDensity dependence and trapped-geometry averaging matter when comparing a measured pulse speed with this local uniform formula. A strictly ideal gas would have no nonzero interaction-supported sound speed.

Moving-obstacle dissipation. In the 1999 experiment, a blue-detuned focused beam repelled atoms and acted as an obstacle swept through a sodium condensate. Heating was inferred from the condensate fraction after stirring and re-equilibration. The heating rose markedly near $1.6\mathrm{mm}{\mathrm{s}}^{-1}$, below the quoted peak sound speed $6.2\mathrm{mm}{\mathrm{s}}^{-1}$.^[ref] Landau's criterion $v_L={\inf }_pϵ(p)/p$ motivates a dissipation threshold, but density variation, flow acceleration around the obstacle and vortex nucleation make the observed threshold different from simply substituting the peak sound speed. Together these experiments probe a collective excitation spectrum and the onset of dissipative flow; condensation alone is not a complete demonstration of superfluidity.

For a uniform time-independent saddle, extremizing the action gives $(-\mu +g|{\psi }_0{|}^2){\psi }_0=0$. Its grand-potential density is $f(\rho )=-\mu \rho +g{\rho }^2/2$, with $\rho =|{\psi }_0{|}^2\ge 0$. Since $g>0$, the minimum at positive density for $\mu >0$ isFor $\mu \le 0$ the uniform mean-field minimum is the vacuum, rather than a positive-density condensate. At fixed nonzero density in the weakly interacting condensed regime, the saddle relation sets $\mu \simeq g{\rho }_0$. The mean-field one-body density matrix factorizes as $⟨a^{†}(\mathbf{r})a({\mathbf{r}}^{\prime })⟩\simeq {\rho }_0$, giving macroscopic occupation of the uniform mode. Depletion corrections are omitted at this stage.

The action has the global particle-number symmetry $\psi \mapsto e^{i\alpha }\psi$, $\bar{\psi }\mapsto e^{-i\alpha }\bar{\psi }$. Choosing one saddle phase breaks this $U(1)$ symmetry; the degenerate mean-field minima form the circle $U(1)\simeq S^1$. There is no energetic preference for a uniform value of ${\varphi }_0$, but spatially varying phase costs gradient energy. Consequently a neutral condensate has a gapless Goldstone mode and a nonzero phase stiffness. The mode becomes sound once density dynamics is included. With $\hslash =1$ the superfluid velocity is $\nabla \varphi /m$, so a single-valued field allows circulation in units $2\pi /m$ around a closed loop that avoids zeros of the amplitude. This phase coherence and rigidity are the physical content of the broken-symmetry description.

The claim is a mean-field saddle statement, not an assertion that every $d$ and temperature in the question supports true long-range order. In a finite number-conserving system $⟨a⟩=0$; the symmetry-breaking order parameter is defined with the thermodynamic limit taken before an infinitesimal source is removed. In a homogeneous infinite short-range gas at nonzero temperature, infrared phase fluctuations prohibit true condensation in $d\le 2$. In one dimension they also destroy true condensate long-range order at zero temperature. A two-dimensional gas can nevertheless have superfluidity and algebraically decaying correlations below a Berezinskii–Kosterlitz–Thouless transition. A finite trap can have substantial coherence without contradicting these thermodynamic statements.^[ref]

Bose-Einstein condensation is macroscopic occupation of a single lowest-energy one-particle state. For a homogeneous three-dimensional ideal gas of conserved massive bosons, set that energy to zero. The excited-state population isHere $g_{3/2}(z)={\sum }_{r\ge 1}{z}^r/r^{3/2}$. As $\mu \to 0^{-}$, the excited states can hold only $V\zeta (3/2)/{\lambda }_T^3$ particles. Additional particles occupy the ground state. ThusThe isolated ground state must be counted separately from the continuum density of states; otherwise one misses the condensate. A dilute atomic gas is a physical realization of the quantum-degeneracy regime, but trapping geometry and interactions modify the homogeneous ideal-gas formulas.^[ref]

In the homogeneous ideal gas, one and two dimensions do not have finite-temperature condensation in the thermodynamic limit because the low-energy population integral diverges. Finite size or a trap changes that conclusion's assumptions. Condensation and superfluidity are distinct: the ideal quadratic gas has zero Landau critical velocity, while interactions can support a sound mode and superfluid flow. Neither zero viscosity nor phase-rigid flow follows from the occupation formula alone.

A superfluid exhibits superfluidity, with phase stiffness and a phase-gradient current. In an appropriate ordered or quasi-ordered regime it supports low-speed flow without viscous dissipation, and its long-wavelength Goldstone boson has a sound-like dispersion. True thermodynamic phase selection and Bose-Einstein condensation need separate infrared conditions.

Superfluidity is a quantum-fluid response with phase stiffness and flow without viscous dissipation in an appropriate low-speed regime. A superfluid supports a phase-gradient current; defects and excitations can limit that flow. Bose-Einstein condensation concerns macroscopic one-particle occupation and is a distinct criterion: the Gaussian phase stiffness theory can have divergent absolute-phase fluctuations while retaining a cost for gradients.

Why does cooling a liquid make it flow more freely through a tiny gap? Helium-4 supplies an experimental entrance to this question: the helium slit-flow experiment compares the motion of liquid levels on opposite sides of the transition into superfluidity. A flow measurement probes resistance to motion; a measurement of Bose-Einstein condensation probes occupation of a quantum state. The two-fluid model of superfluidity and the Landau criterion connect these questions to entropy transport and to the energy required to create an excitation.

In 2006, Zoran Hadzibabic, Peter Krüger, Marc Cheneau, Baptiste Battelier and Jean Dalibard split a cooled rubidium-87 cloud into two independent planar gases using an optical lattice. After release, the expanding sheets overlapped; absorption imaging recorded their matter-wave interference. Smooth fringe waviness revealed thermal phase fluctuations, while sharp dislocations indicated isolated vortices. The initial evaporative-cooling endpoint controlled the preparation; central fringe contrast served as a degeneracy proxy because direct thermometry in the lattice was difficult.^[ref]

Loss of extended coherence coincided with increased vortex signatures. The optical trap and finite cloud produced a crossover; tightly bound pairs were unresolved, and short-wavelength modes could contaminate high-temperature counts. These were observations of the vortex-unbinding mechanism, not a direct measurement of phase stiffness or a sharp homogeneous universal stiffness jump. The authors left the sample's superfluidity open.^[ref]