Measurement of the Excitation Spectrum of a Dipolar Gas in the Macrodroplet Regime

J. J. A. Houwman, D. Baillie, P. B. Blakie, G. Natale, F. Ferlaino, and M. J. Mark

Phys. Rev. Lett. 132, 103401 – Published 7 March 2024

Abstract

The excitation spectrum of a cigar-shaped strongly dipolar quantum gas at the crossover from a Bose-Einstein condensate to a trapped macrodroplet is predicted to exhibit peculiar features—a strong upward shift of low momentum excitation energies together with a strong multiband response for high momenta. By performing Bragg spectroscopy over a wide range of momenta, we observe both key elements and also confirm the predicted stiffening of excitation modes when approaching the macrodroplet regime. Our measurements are in good agreement with numerical calculations taking into account finite size effects.

Received 28 July 2023
Revised 27 November 2023
Accepted 8 January 2024

DOI:https://doi.org/10.1103/PhysRevLett.132.103401

Physics Subject Headings (PhySH)

Cold atoms & matter waves

Bose gases Bose-Einstein condensates

Bragg spectroscopy

Atomic, Molecular & Optical

Authors & Affiliations

J. J. A. Houwman ¹, D. Baillie ², P. B. Blakie ², G. Natale ^1,3, F. Ferlaino ^1,3, and M. J. Mark ^1,3

¹Universität Innsbruck, Institut für Experimentalphysik, 6020 Innsbruck, Austria
²Department of Physics, Centre for Quantum Science, and The Dodd-Walls Centre for Photonic and Quantum Technologies, University of Otago, Dunedin 9016, New Zealand
³Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria

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Vol. 132, Iss. 10 — 8 March 2024

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Images

Figure 1
Spectra of excitations of an infinitely elongated dipolar quantum gas in the roton (a) and antiroton regime (b), corresponding to dipoles oriented perpendicular or parallel to the long axis of the system, respectively. The calculations are performed for ${}^{166}{Er}$ in an infinite tube with transversal trapping frequency $ω_{ρ} = 2 π \times 198 Hz$ , a uniform density of $n = 2.3 \times 10^{3} μ m^{- 1}$ , where $ε_{d d} = 1.36$ for (a) and $ε_{d d} = 1.15$ for (b). The color map shows the strength of the zero-temperature dynamical structure factor $S_{0}$ . The red dashed line is the variational energy [see Eq. (2)] and the black dashed line is the free particle energy $ε_{k_{y}}$ . For visibility, a typical experimental Fourier broadening of 47 Hz is applied to the DSF.
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Figure 2
(a) Illustration of the experimental setup: A BEC of ${}^{166}{Er}$ in an elongated trap with the dipole moment aligned along the weak axis $y$ . Bragg spectroscopy is performed by intersecting two beams at the BEC which impart momentum $ℏ k_{L}$ and energy $ℏ ω_{L}$ . (b)–(d) Measured response of the system ${\bar{k}}_{y}$ as a function of the excitation frequency $ω_{L} / ω_{ρ}$ for three different values of the imparted momentum $k_{L} = [0.59, 1.0, 1.76] / l_{ρ}$ , respectively. The black line is the sum of three Gaussians (dashed lines) fit to the data. The green lines are the theoretically calculated values of $S_{0} (k_{y}, ω)$ , rescaled in amplitude and accounting for the experimental Fourier and Doppler broadening.
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Figure 3
Excitation spectrum of an elongated ${}^{166}{Er}$ cloud polarized along the weakly confined axis with $ε_{d d} = 1.13$ , $N = 25 (5.0) \times 10^{4}$ and trapping frequencies $ω_{x, y, z} / 2 π = (170, 30, 230) Hz$ . (a) Measured response of the system ${\bar{k}}_{y}$ as a function of $ω_{L} / ω_{ρ}$ and $k_{L}$ . The datapoints denote the experimentally detected resonance positions from fits as shown in Figs. 2 with the error bars indicating one standard error from the fit. The black lines represent the maxima of $S_{0} (k_{y}, ω)$ from our numerical calculation. (b) Detected resonance positions on top of the calculated $S_{0} (k_{y}, ω)$ as indicated by the color map, including Fourier and Doppler broadening. (c) Calculated $S_{0} (k_{y}, ω)$ for our finite system with minimal broadening. (a)–(c) Each momentum column is normalized to the maximum value.
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Figure 4
Stiffening and shapes of excitation modes. (a) Frequency of the first excitation branch versus $ε_{d d}$ at fixed excitation momentum $k_{L} = 1.30 / l_{ρ}$ . The error bars denote one standard error from the fit. The shaded area corresponds to the theoretical predictions within a range of atom number $N = [15, 35] \times 10^{4}$ . $(b_{2})$ , $(c_{2})$ Shapes of the numerically calculated excitation modes at $ε_{d d} = 1.13$ and energies $ε_{j} / ℏ = 0.51 ω_{ρ}$ (b) and $ε_{j^{'}} / ℏ = 1.87 ω_{ρ}$ (c) represented via plots of $| δ n_{j} |^{2} = | (u_{j}^{*} + v_{j}^{*}) ψ_{0} |^{2}$ . The dashed lines mark the 1/e size of the condensate. $(b_{1})$ , $(c_{1})$ Profiles of the excitation modes after integration along $y$ . The dashed lines mark the integrated shape of the condensate.
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