Matter-wave collimation to picokelvin energies with scattering length and potential shape control

TL;DR

This work demonstrates picokelvin-scale matter-wave collimation in a compact 39K Bose-Einstein condensate by tuning interactions with a Feshbach resonance during a matter-wave lens. The authors achieve 1D energies as low as $340 \pm 12$ pK and extrapolate 2D energies to $438 \pm 77$ pK, validating the approach with simulations and showing strong improvement over non-collimated cases. They propose an advanced delta-kick-based scheme to reach 3D energies below $16$ pK, requiring precise control of trapping frequencies and short, intense kick pulses, potentially enabling dense, high-coherence sources in standard lab setups without microgravity. The results open avenues for large-atom-number, ultra-cold ensembles in precision sensing and fundamental physics tests using compact dipole-trap platforms.

Abstract

The sensitivity of atom interferometers depends on their ability to realize long pulse separation times and prevent loss of contrast by limiting the expansion of the atomic ensemble within the interferometer beam through matter-wave collimation. Here we investigate the impact of atomic interactions on collimation by applying a lensing protocol to a $^{39}$K Bose-Einstein condensate at different scattering lengths. Tailoring interactions, we measure energies corresponding to $340 \pm 12$ pK in one direction. Our results are supported by an accurate simulation, which allows us to extrapolate a 2D ballistic expansion energy of $438 \pm 77$ pK. Based on our findings we propose an advanced scenario, which enables 3D expansion energies below $16$ pK by implementing an additional pulsed delta-kick. Our results pave the way to realize ensembles with more than $1\times10^5$ atoms and 3D energies in the two-digit pK range in typical dipole trap setups without the need for micro-gravity or long baseline environments.

Figures (4)

• Figure 1: Measured and simulated expansion rate energies in one and two dimensions. All measurements (blue circles and red squares) are performed within the camera frame and based on time-of-flight (TOF) series with a total length of 25ms. The error bars arise from the fit uncertainty of the expansion in the individual TOF series (c.f. data acquisition and analysis section). The dynamics of the ensemble are simulated (lines) simultaneously for all directions within the trap frame for 25ms TOF and are subsequently transformed into the camera frame. Uncertainty bands are obtained by a Monte Carlo method based on the detection angle and trap frequency errors matching the oscillations of the ensemble size. Panel a shows the results obtained in the strong interaction regime for a scattering length of 158a_0, using the Thomas-Fermi approximation in the theoretical description. Panel b shows the results obtained in the weak interaction regime at 10a_0 scattering length. Here we simulate the dynamics based on a variational approach (c.f. theoretical model section). For the data we choose a lower sampling rate, allowing to increase the number of points per TOF measurement to resolve the lower expansion energies, effectively. In both interaction regimes the measurements agree well with the simulation and the coupling of the dynamics in all dimensions allows to extrapolate the behavior in the entire horizontal plane as shown by the purple dashed lines. We find an overall improvement by 3.7 dB in the extrapolated 2D expansion rate energy, when reducing the scattering length. The purple star highlights the minimum 2D energy at 10a_0, as prominently featured in the inset.
• Figure 2: Ensemble dynamics in the collimated plane. Panel a shows the simulated 2D expansion energy E$^\text{2D}$ in the collimated plane for different scaling factors $\alpha_i$ and for the configuration used in the experiment (see lensing protocol section) for 250ms time-of-flight (TOF). For a common frequency reduction along both lensed directions with $\alpha_{x}=\alpha_{y}$ taking the values $\{1, 4/9, 1/4, 1/9\}$, the minimal energies are obtained for a scattering length $a$ being respectively $\{0, 0.1, 1.8, 13.5\}$ a_0 and identified by the squares for each case. The sequence becomes more robust against changes of the scattering length with larger frequency reduction, as the minima become more shallow. The purple star resembles the lowest 2D experimental expansion energy presented in Fig. 1b (obtained here for 250 ms TOF). While the curves are simulated for a fixed set of parameters, this point is obtained within a Monte-Carlo simulation including all experimental uncertainties. The error bar denotes 2-$\sigma$ deviation while the central point stands for the mean value (see data acquisition and analysis section). Qualitatively, the experimental configuration closely resembles the case of $\alpha_{x}=\alpha_{y} = 1/9$. The resulting expansion energies are globally shifted towards higher values, since $\alpha_{x}\approx 1/16$ and $\alpha_{y}\approx 1/9$. This causes the optimal release points to differ for each axis as marked by the triangle symbols in panel b, highlighting the importance of a symmetric choice of $\alpha$-values. Here, the measured ensemble width in $\mathbf{x'}$-direction is shown as blue circles for 10ms TOF at 10a_0 and the error bars represent the standard deviation of at least four measurements, The simulated size after 10 ms TOF is shown as solid blue line, while the corresponding oscillations of the ensemble widths in $\mathbf{x}$- and $\mathbf{y}$-direction within the trapping potential are shown with blue dashed and dotted lines, respectively.
• Figure 3: Generating a delta-kick collimated Bose-Einstein condensate in the regime of tens of pK. We take advantage of the holding process after trap relaxation to minimize the energy in the $\{x,y\}$-plane (a) (red line). Subsequently, a short free-fall time (pre-TOF) $\tau_z$ allows the ensemble to expand, followed by a delta-kick collimation (DKC) to collimate the third direction (b). We show the width evolution in all three directions after the optimal holding period without (non-solid lines) and with a DKC (solid lines). The inset shows the dynamics in the trapping potential, highlighting release (solid black line) and DKC (black dashed line) timings. This process leads to a reduced 3D expansion energy expressed as a function of the lensing time and the DKC duration after a pre-TOF of 10 ms (c) and 25 ms (d), leading to respectively 24.5pK and 15.7pK.
• Figure 4: Optical dipole trap setup. Time-averaged potentials are implemented with an acousto-optical modulator (AOM). A three-lens system (L1, L2, L3) with focal lengths $f_1=100mm$, $f_2=300mm$ and $f_3=150mm$, translates the change in AOM deflection angle into a parallel displacement, while simultaneously focusing the beam to a waist of 30.0 (45.0) µm in horizontal (vertical) direction. The lenses L4 and L5 ($f_{4,5}=150mm$) are used to re-collimate the beam after passing the experimental chamber and to re-focus it on the atoms. At each point optimal polarization is ensured by wave plates (WP1 - WP4) with additional orthogonal oriented polarizing beam splitters (PBS1 and PBS2) in front of the chamber for polarization cleaning. Dielectric mirrors (M1 - M4) are used to guide the beam through the setup. For perfect alignment and equal beam power the trap frame $\{x,y,z\}$, as given by the principal axes of the optical potential, resembles the symmetry axes of the vacuum chamber. The camera frame $\{x',y',z\}$ is obtained by a rotation around the $z$-axis, with the exact detection angle depending on the beam configuration and alignment. The figure is taken from Albers et al. Albers2022Commun and openly licensed via https://creativecommons.org/licenses/by/4.0/. Here the orientation of the detection arrow and the naming of the coordinate systems was altered to account for changes of the apparatus compared to the source material.