# Documentation:Momentum distribution and time of flight images

# Momentum distribution and time-of-flight images

## Theory

### Momentum distribution

The momentum distribution of bosons in an optical lattice at equilbrium can be expressed as:

Expressing the fourier transform of the wannier function as

and the interference term:

the momentum distribution reduces to the following form:

### Time of flight images (far-field approximation)

To measure it in experiments, the optical lattice is momentarily switched off and the bosons expand freely by assumption with momentum gained from the lattice momentum previously. Measurements are performed in our classical world, and therefore semiclassical treatment is already sufficient, i.e.

where is the time-of-flight taken by the bosons to move from the origin (experiment) to the detector probe at position . Here, we assume the simplest picture, ie. 1) no interaction, 2) no collision, and 3) uniform motion.

Of course, at equilibrium, i.e. "after the bosons have travelled for a long time", the time-of-flight image will capture a density of

which indirectly probes the original momentum distribution of the lattice bosons. This form is only correct in the far-field limit or in the limit of infinite time-of-flight (tof).

### Time of flight images (beyond far-field approximation)

To improve upon the accuracy, we shall correct it with semiclassical dynamics during the time of flight. For the bosons that originate from lattice site at to the detector probe at , their time-dependent wavefunction must be:

where the kinetic energy of the free bosons must be:

Therefore, the corrected time-of-flight image must be:

where the dependence of the initial site on the wannier enveloped function is neglected. Finally, we arrive at:

with time-of-flight phase .

### Symmetry leads to further simplification

Symmetries:

Therefore:

Define green function:

In reduced coordinates, , we redefine green function:

and therefore the momentum distribution/ time-of-flight image:

Here, we distinguish the momentum distribution from the interference

## Implementation in Python

### Example(s) of obtaining time-of-flight (tof)

For example, the time-of-flight phase for Rb-87 bosons (mass 86.99 a.m.u.) with a time-of-flight of 15.5 ms

a. expanded from an isotropic optical lattice of wavelength 765 nm:

>>> pyalps.dwa.tofPhase(time_of_flight=15.5, wavelength=765, mass=86.99) 0.006464602556863682

b. expanded from an optical lattice of wavelength 843 nm along x-direction and 765 nm along y- and z- directions:

>>> pyalps.dwa.tofPhase(time_of_flight=15.5, wavelength=[843,765,765], mass=86.99) [0.007850080468935228, 0.006464602556863682, 0.006464602556863682]

### Quantum Monte Carlo simulations

Both the momentum distribution and the time-of-flight images can be easily obtained from Quantum Monte Carlo simulations, implemented in the directed worm algorithm (dwa).

Read here to see the dwa documenation.

© 2013 - Matthias Troyer, and Ping Nang Ma