Cloud parameter estimation for interacting BEC after time-of-flight

Technical Summary: Cloud Parameter Estimation for Interacting BEC after Time-of-Flight

Problem Statement
In ultracold atomic physics, absorption imaging following a time-of-flight (TOF) expansion is the standard method for extracting system parameters such as temperature ($T$), atom number ($N$), and condensed fraction from Bose-Einstein condensates (BECs). While analytical models exist for the free expansion of pure thermal clouds (modeled as ideal gases with Bose enhancement) and pure BECs (modeled via the Gross-Pitaevskii equation and Thomas-Fermi approximation), these models fail for partially condensed clouds. Specifically, the standard analytical approach assumes that the thermal and condensed components are decoupled. However, in reality, the high-density condensed atoms exert repulsive mean-field forces on the thermal atoms, altering the thermal cloud's expansion profile. Because an analytical solution for the free expansion of interacting partially condensed clouds is unavailable, researchers typically rely on non-interacting fits, introducing systematic uncertainties in the extracted parameters.

Methodology
The authors employ a numerical simulation based on a ballistic approximation to model the expansion of partially condensed clouds. The methodology proceeds as follows:

1. Mean-Field Potential: The simulation incorporates the repulsive interaction between the BEC and thermal atoms within the trap. The potential experienced by thermal atoms is modified to include a term $2U_0n_0$, where $U_0$ is the interaction energy and $n_0$ is the BEC density. The BEC itself is treated as unaffected by the thermal cloud due to the latter's low density.
2. Numerical Integration: Unlike the analytical solution for non-interacting clouds, the integral for the thermal density distribution $n_{th}(r, t)$ cannot be solved analytically when the effective potential $V_{eff}$ includes the BEC density term. The authors calculate the distribution numerically by summing over an initial grid (representing the in-trap cloud) and an expansion grid (representing the TOF profile). For each point in the expansion grid, the probability of an atom originating from a specific point in the initial grid with the requisite momentum is calculated.
3. Error Estimation Protocol: To quantify the errors introduced by ignoring interactions, the authors generate synthetic TOF images using the simulation with known input parameters ($N_{s}, T_{s}$). These synthetic images are then fitted using the standard analytical Bose-enhanced distribution (Eq. 7) described in Section 2.3. The difference between the input simulation values and the fitted values yields the relative errors in thermal atom number ($\Delta N_{th}$) and temperature ($\Delta T$).
4. Experimental Validation: The simulation is also used as a fitting function for experimental data from a $^{87}$Rb system. The results of fitting experimental images with the simulation are compared against fits using the standard Bose-enhanced function and the theoretical semi-ideal model.

Key Results
The study reveals that neglecting interactions leads to significant, non-negligible errors in parameter estimation, even at long times of flight:

• Profile Deformation: The repulsive force from the BEC creates a "dip" in the thermal cloud density at the center during short TOF. As the cloud expands, thermal atoms fill this hole, eventually leading to a distribution that is more peaked than the standard Bose-enhanced prediction at longer TOF.
• Parameter Errors:
  • Atom Number: Fitting with the non-interacting model generally overestimates the thermal atom number at short TOF (due to the central dip being interpreted as lower density) and can underestimate it at long TOF. Errors can reach up to ~64% at short TOF and converge to approximately -5% at long TOF.
  • Temperature: Temperature errors are generally within $\pm 10\%$. The sign of the error depends on the TOF duration and trap geometry. For example, in spherical traps at short TOF, the temperature is overestimated, while at long TOF, it is underestimated.
• Dependence on Geometry: The magnitude and sign of the errors depend heavily on trap geometry (cigar, pancake, or spherical) and the aspect ratio. Spherical traps are found to be optimal for temperature estimation (minimizing temperature error) but worst for atom number estimation. Conversely, asymmetric traps (cigar/pancake) allow for better atom number determination but introduce larger temperature errors.
• Experimental Agreement: When applied to experimental data, fitting with the simulation-derived distribution yields a condensed fraction that aligns significantly better with the semi-ideal model than the standard Bose-enhanced fit. The standard fit tends to underestimate temperature and slightly overestimate the BEC fraction.

Significance and Claims
The paper claims that the repulsive interactions between condensed and thermal atoms fundamentally reshape the thermal distribution during expansion, rendering standard analytical fits insufficient for high-precision measurements in partially condensed regimes.

• Systematic Error Quantification: The authors provide a systematic quantification of the errors inherent in current experimental practices, demonstrating that these errors persist even at long expansion times because the BEC alters the momentum distribution of the thermal atoms.
• Improved Fitting Model: The primary contribution is the demonstration that using the numerical simulation as a fitting function for experimental data yields more accurate parameters. This approach bridges the gap between the ideal gas assumption and the complex reality of interacting clouds.
• Trade-offs: The work highlights a trade-off in parameter estimation: the choice of trap geometry and TOF duration dictates whether one prioritizes accurate temperature or atom number determination when using simplified models.
• Limitations: The authors modestly note that their simulation assumes ballistic expansion (neglecting interactions during the TOF). They acknowledge that while this model improves agreement with the semi-ideal model over standard fits, small discrepancies may remain due to the omission of in-flight interactions, which would require a fundamentally different theoretical approach to model.

In conclusion, the paper establishes that for experiments involving partially condensed clouds, relying on non-interacting analytical fits introduces systematic biases. The proposed numerical fitting method offers a path to higher precision in extracting cloud parameters without requiring a full analytical solution to the interacting expansion problem.