• image reconstruction
• imaging
• output coupling
• ricostruzione immagini

Motivations
The equation of state of a quantum gas completely characterizes the thermodynamics of the system, and it is mostly influenced by the fundamental properties of its constituents like particle statistics or their interactions. In bosonic systems, the interplay between all these elements gives rise to the phenomenon of Bose – Einstein condensation, which since its discovery in the early 1900s has been the subject of intensive research, both theoretically and experimentally.
In the last thirty years, the introduction of ultracold atomic gases made a significant upgrade in the experimental investigation of such fundamental physics. Their high versatility and the extreme control that can be achieved on the system’s parameters mark them as a privileged tool for studying the thermodynamic properties of many-body quantum systems, like the equation of state itself.
In the present literature there are many examples of equation of state measurement in ultracold gases, for both bosonic and fermionic systems and in different geometries. In the case of a 3D gas of interacting particles, the underlying thermodynamics is well understood, but there still lacks a characterization of the role of interactions in the regime of intermediate temperatures between zero and the critical threshold.
The purpose of my master thesis is to obtain a complete measure of the equation of state of a 3D homogeneous interacting Bose gas across the condensation threshold, highlighting the contribution of particle interactions below Tc ; in particular, we aim to observe the non-monotonic shift in the chemical potential predicted by the mean-field theory, of which there is no direct experimental observation up to now. The work is based on the assumption that the inhomogeneous profile of a trapped sample can be described through the bulk quantities of the same homogeneous system within the Local Density Approximation, and on a method for measuring the pressure and density profiles along the trap axis proposed by T.-L. Ho and Q. Zhou [Nature Physics 6, (2010)].

Methods
In order to obtain information on a large density range, from the rarefied thermal wings to the high density condensed region, one has to acquire the full spatial profile of the sample. In presence of a condensed fraction, the cloud density varies in a range spanning several orders of magnitude: when trying to measure such a highly dense sample with standard acquisition techniques, what one actually obtains is a saturated image with a severe loss of signal in the central area.
During my thesis I worked on an innovative data acquisition technique, based on a series of partial extractions of atoms from the sample through an output coupling mechanism, their sequential imaging, and the successive reconstruction of the original spatial profile. This technique aims to overcome the limitations of current imaging methods, allowing to measure the full spatial distribution of highly dense atomic samples.

Results
I worked in the Ultracold Gases Laboratory at the BEC Center, University of Trento. I first planned the experimental procedure and performed the necessary calibrations to include the new imaging scheme; then I did the actual measurements on Bose–Einstein condensates of sodium atoms produced in the laboratory.
I developed the image reconstruction algorithm, and wrote a user-interface Python software to process the acquired experimental data.
For testing the algorithm and checking its robustness against noise, I processed a number of synthetic realistic data series and checked that the original profiles were faithfully reconstructed. When applied to real data, it provided density profiles whose spatial distribution spanned the expected range of values: their validity was cross-checked by comparing some relevant parameters (like temperature or number of atoms) with those measured with a different technique on samples produced under the same conditions.
I also worked on the data elaboration and processing, and developed specific algorithms for calculating the thermodynamical quantities of interest from the in situ profiles acquired with the above discussed methods.
The performed analysis led us to identify the limitations of the implementation we chose for the new technique. We found the optimal settings for having a reliable acquisition, and within these constraints we obtained some preliminary results towards a complete characterization of the equation of state: all the measured pressure profiles clearly show the onset of condensation, with the characteristic bimodal shape; we measured a positive chemical potential in the trap center, in qualitative agreement with the mean-field prediction. In conclusion, all the necessary tools are availlable and ready for a quantitative determination of the equation of state.