# Documentation:Momentum distribution and time of flight images

## Contents

# Bosons in an optical lattice trap

## Boson Hubbard model

### Hamiltonian

Bosons in an optical lattice trap can be effectively described by the single band boson Hubbard model

with hopping strength , onsite interaction strength , and chemical potential at finite temperature via Quantum Monte Carlo implemented in the directed worm algorithm. Here, () is the annihilation (creation) operator, and being the number operator at site i. Bosons in an optical lattice are confined, say in a 3D parabolic trapping potential, i.e.

.

due to the gaussian beam waists as well as other sources of trapping.

### Finite temperature

At finite temperature T, the physics is essentially captured by the partition function

and physical quantities such as the local density

for some configuration in the complete configuration space, with inverse temperature . Here, the units will be cleverly normalized later on.

### 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 (in experiment)

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).

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 .

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]

The following table summarizes the basic option to turn on time-of-flight imaging options.

Option | Default | Remark |
---|---|---|

tof_phase | 0. | 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

Please refer to the list of measurement observables provided by the DWA code.