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A concrete quantum quench experiment loaded a Bose-Einstein condensate of ${}^{87}\mathrm{Rb}$ chemical atoms from a magnetic field trap into an optical lattice: three orthogonal $838\mathrm{nm}$ laser standing waves, with approximately $125\mu \mathrm{m}$ beam waists. An $80\mathrm{ms}$ loading ramp preceded a $50\mu \mathrm{s}$ increase in lattice depth. This suppressed quantum tunnelling while retaining each well's vibrational ground state. The switch preserved atom-number populations rather than freezing every motional degree of freedom.^[ref]

After variable hold times, releasing the traps and taking photon absorption images after $16\mathrm{ms}$ of flight revealed disappearance and return of interference peaks. Background-subtracted peak counts supplied a coherent fraction. Varying final depth tested the predicted revival period $h/U$; varying initial depth tested the collapse's sensitivity to atom-number statistics. Peak broadening outside counting windows and spatial variation of $U$ limited later signals.^[ref]

Here $U$ is the on-site interaction energy. The approximate isolated-well Hamiltonian operator $H=U\hat{n}(\hat{n}-1)/2$, with number operator $\hat{n}$, gives different quantum phases to its number states, which later rephase. This is a many-level realization of the distinction between fixed populations and changing quantum phases.^[ref]

Gregor Jotzu and collaborators realized an artificial Haldane model in 2014. They loaded spin-polarized potassium-40 fermions into a laser-made honeycomb lattice. Piezoelectric mirrors shook the optical lattice along two axes; their relative phase controlled time-reversal symmetry breaking. Optical phase control adjusted the sublattice offset.^[ref]

A magnetic field gradient drove Bloch oscillations. Band mapping measured interband transfer to locate valley-gap closures. Opposite-force drift measurements distinguished cancelling from reinforcing Berry curvature contributions; reversing shaking handedness reversed the drift. Interband transfer and harmonic confinement affected this readout, including drift beyond ideal transition lines. This neutral-atom Floquet Hamiltonian realization did not measure electrical sheet conductivity. Magnetic fields were used for preparation and forcing.^[ref]

The five DiVincenzo criteria are: a scalable collection of well-characterized qubits; preparation in a known fiducial quantum state; relevant coherence times long compared with quantum gate times; a universal quantum gate set; and qubit-specific quantum measurement. The criterion is useful control of the whole computation, including errors, rather than merely observing two resolved levels.^[ref]

For trapped ions, the implementation is:

For trapped neutral atoms, the corresponding implementation is:

Thus neither platform is forbidden from meeting a criterion in principle. Weak neutral-ground-state interactions make entangling control harder, but claiming that trapped neutral atoms cannot implement a universal quantum gate set would be false. Gate-model quantum algorithms using Rydberg blockade had already been demonstrated in a programmable neutral-atom processor by 2022.^[ref]

Use the stationary quantum vortex fieldwith no dependence on the coordinate along the vortex line. The polar Laplacian contributes $R^{\prime \prime }+R^{\prime }/r-{\mathcal{N}}^{2}R/r^2$. ConsequentlyA regular vortex with nonzero integer winding number has $R\sim C{r}^{|\mathcal{N}|}$ near the origin. The angular phase gives the superfluid velocity $\hslash \mathcal{N}/(mr)$ outside the core. For $s\ne 0$, the exterior amplitude responds to the periodically varying potential; it is not generally a constant bulk value. The radial potential here is not a Cartesian translation-invariant optical lattice, so the Cartesian Bloch theorem is not an exact theorem for its full radial equation.

Recoil energy is the kinetic energy gained by an initially stationary atom absorbing a photon of momentum $p_R$. In a standing-wave optical lattice of spacing $d$, the photon wavenumber is $\pi /d$, giving $E_R={\hslash }^2{\pi }^2/(2m{d}^2)$.

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]