Observation of Scale Invariance in Two-Dimensional Matter-Wave Townes Solitons

Cheng-An Chen and Chen-Lung Hung

Phys. Rev. Lett. 127, 023604 – Published 9 July 2021

Abstract

We report near-deterministic generation of two-dimensional (2D) matter-wave Townes solitons and a precision test on scale invariance in attractive 2D Bose gases. We induce a shape-controlled modulational instability in an elongated 2D matter wave to create an array of isolated solitary waves of various sizes and peak densities. We confirm scale invariance by observing the collapse of solitary-wave density profiles onto a single curve in a dimensionless coordinate rescaled according to their peak densities and observe that the scale-invariant profiles measured at different coupling constants $g$ can further collapse onto the universal profile of Townes solitons. The reported scaling behavior is tested with a nearly 60-fold difference in soliton interaction energies and allows us to discuss the impact of a non-negligible magnetic dipole-dipole interaction (MDDI) on 2D scale invariance. We confirm that the effect of MDDI in our alkali cesium quasi-2D samples effectively conforms to the same scaling law governed by a contact interaction to well within our experiment uncertainty.

Received 5 March 2021
Accepted 21 June 2021

DOI:https://doi.org/10.1103/PhysRevLett.127.023604

Physics Subject Headings (PhySH)

Bose gases Cold atoms & matter waves

Solitons

Atomic, Molecular & OpticalNonlinear DynamicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Cheng-An Chen¹ and Chen-Lung Hung ^1,2,*

¹Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USA
²Purdue Quantum Science and Engineering Institute, Purdue University, West Lafayette, Indiana 47907, USA

^*[email protected]

Realization of a Townes Soliton in a Two-Component Planar Bose Gas

B. Bakkali-Hassani, C. Maury, Y.-Q. Zou, É. Le Cerf, R. Saint-Jalm, P. C. M. Castilho, S. Nascimbene, J. Dalibard, and J. Beugnon

Phys. Rev. Lett. 127, 023603 (2021)

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Vol. 127, Iss. 2 — 9 July 2021

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Images

Figure 1
Formation of 2D matter-wave soliton trains. (a) An elongated 2D Bose gas of peak density $n_{i} \approx 20 / μ m^{2}$ is held at an initial coupling constant $g_{i} \approx 0.129$ and quenched to a new coupling constant $g \approx - 0.0215$ , with simultaneous removal of the horizontal confinement in the $x - y$ plane. Arrays of solitary waves are observed in shot-to-shot images in (b), taken after a 50 ms wait time. A different sample in (c) is prepared at a much lower initial peak density $n_{i} \approx 6 / μ m^{2}$ and quenched to $g \approx - 0.0075$ , similarly generating solitary waves as observed in (d). Image size in (a),(b) is $19 \times 77 μ m^{2}$ . Image size in (c),(d) is $40 \times 160 μ m^{2}$ .
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Figure 2
Soliton formation statistics. (a) Probability $P_{tot}$ of finding $N_{s}$ solitons after the quench, evaluated using 68 samples as shown in Fig. 1. (b) Occurrence of solitons with peak density $n_{p}$ (bin size, $2 / μ m^{2}$ ). (c) Average peak density ${\bar{n}}_{p}$ versus position along the long ( $y$ ) axis (filled circles). Error bars represent standard deviation. Solid curve shows the density $n_{i}$ of the initial sample through the long axis. (d) Probability for observing a soliton at position $y$ in a quenched sample (bin size, $4 μ m$ ).
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Figure 3
Testing scale invariance. (a) Images at the top, from left to right, show solitons of low to high peak densities, selected from samples as shown in Fig. 1. Image size $19 \times 19 μ m^{2}$ . Their radial density profiles $n (r)$ (filled circles, inset) approximately collapse onto a single curve in the rescaled coordinate $\tilde{r} = \sqrt{n_{p}} r$ and $\tilde{n} = n / n_{p}$ . Error bars include statistical and systematic errors. Shaded band shows the standard deviation of 20 rescaled radial profiles around their mean $⟨ \tilde{n} ⟩$ (solid curve). (b) Shows soliton images and profiles observed in Fig. 1. Image size $60 \times 60 μ m^{2}$ .
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Figure 4
Universal soliton density profile. (a) Filled symbols show different scale-invariant mean profiles $⟨ \tilde{n} ⟩$ (inset), measured at interaction strengths $g \approx - 0.0075$ (triangle), $- 0.0170$ (circle), and $- 0.0215$ (square), respectively. Open circles display a scaled density profile reported in Ref. [17], for $g \approx - 0.034$ and with a fixed $n_{p} \approx 5 / μ m^{2}$ . These profiles collapse onto a single curve in the rescaled radial coordinate $R = \sqrt{| g |} \tilde{r}$ , and the magenta band marks their mean with standard error. Collapsed solid curves are the universal Townes profile (black) and the solutions of full GPE with the MDDI term Eq. (3), calculated using $g_{c} = - 0.009$ , $n_{p} = 1 / μ m^{2}$ (red), and $10 / μ m^{2}$ (blue), respectively, and rescaled using $g = g_{c} + 2 g_{DD}$ . For comparison, dashed curves show the same solutions rescaled using $g = g_{c}$ . (b) Universal atom number $N | g | = \int \tilde{n} d R$ using soliton profiles as in Fig. 3 and integrated up to $R = 4$ . Solid line and gray band indicate the mean and standard deviation.
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