Optical lattices and tunneling calibration

For a one-dimensional standing wave, the intensity modulation and its scalar optical potential have period half the laser wavelength. Write the potential with a positive depth multiplying sine squared, choosing an additive energy constant to place a minimum at zero. The photon recoil energy gives a natural unit for comparing depths across species and wavelengths. The same dimensionless depth can correspond to different energies in hertz because recoil depends on atomic mass and wavelength.

Near a deep potential minimum, expanding sine squared yields a harmonic oscillator. Its level spacing grows with the square root of dimensionless depth. This approximates the lowest on-site vibrational separation and eventually the band gap, but it does not give the width of the lowest band. Anharmonic corrections and exact Bloch-band calculations improve the estimate outside the deep-lattice limit. A Gaussian beam envelope also adds slow confinement that breaks the ideal infinite periodic symmetry.

Expanding the atomic field in localized lowest-band Wannier functions gives a hopping term and an on-site interaction. The hopping depends on overlap through the barrier and falls approximately exponentially with square root of lattice depth in a deep sinusoidal lattice. On-site interaction depends on the fourth-power Wannier overlap and the scattering coupling. Calibration errors in depth can therefore produce substantial errors in J and in the interaction-to-hopping ratio.

A single-band description requires temperature, relevant interaction energies per particle, and drive energies small compared with the separation to neglected bands. At higher filling, interactions can reshape local orbitals and create multiband corrections. A rapid lattice quench can also populate higher bands if its time profile is too fast on the local gap scale. A quench sudden compared with many-body tunneling can still be slow enough to preserve the lowest band, but those are separate inequalities that must be checked.

For one noninteracting atom in an unbiased isolated double well, the localized states form a two-level Hamiltonian with off-diagonal matrix element minus J. Starting in the left site gives a sinusoidal right-site probability. The energy splitting is twice J, so its probability oscillation frequency is twice J divided by h. Confusing that splitting with J itself gives a factor-of-two calibration error. The first complete transfer occurs after a quarter of h divided by J.

Bias, interactions, coupling to additional sites, and dephasing alter that simple trace. A fit from an interacting cloud cannot be interpreted as a single-particle J without checking the appropriate dynamics. Band-modulation spectroscopy, diffraction, and independent two-site transfer measurements can constrain different parts of a lattice model. An experimental quench should be reported with its initial state, ramp duration, populated bands, and measured observable; a static Hubbard parameter does not by itself determine all subsequent many-body evolution.