Softening of Roton and Phonon Modes in a Bose-Einstein Condensate with Spin-Orbit Coupling

Si-Cong Ji, Long Zhang, Xiao-Tian Xu, Zhan Wu, Youjin Deng, Shuai Chen, and Jian-Wei Pan

Phys. Rev. Lett. 114, 105301 – Published 9 March 2015

Abstract

Roton-type excitations usually emerge from strong correlations or long-range interactions, as in superfluid helium or dipolar ultracold atoms. However, in a weakly short-range interacting quantum gas, the recently synthesized spin-orbit (SO) coupling can lead to various unconventional phases of superfluidity and give rise to an excitation spectrum of roton-maxon character. Using Bragg spectroscopy, we study a SO-coupled Bose-Einstein condensate of ${}^{87}{Rb}$ atoms and show that the excitation spectrum in a “magnetized” phase clearly possesses a two-branch and roton-maxon structure. As Raman coupling strength $Ω$ is decreased, a roton-mode softening is observed, as a precursor of the phase transition to a stripe phase that spontaneously breaks spatially translational symmetry. The measured roton gaps agree well with theoretical calculations. Furthermore, we determine sound velocities both in the magnetized and in the nonmagnetized phases, and a phonon-mode softening is observed around the phase transition in between. The validity of the $f$ -sum rule is examined.

Received 7 August 2014

DOI:https://doi.org/10.1103/PhysRevLett.114.105301

Authors & Affiliations

Si-Cong Ji, Long Zhang, Xiao-Tian Xu, Zhan Wu, Youjin Deng, Shuai Chen^*, and Jian-Wei Pan^†

Shanghai Branch, Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Shanghai 201315, China and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China

^*shuai@ustc.edu.cn
^†pan@ustc.edu.cn

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Vol. 114, Iss. 10 — 13 March 2015

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Images

Figure 1
Experimental setup and Bragg spectroscopy. (a) In the $x − y$ plane, there are a pair of Raman lasers with relative angle 105° and two Bragg beams separated by an angle $3 ° \leq θ \leq 180 °$ , which can produce a momentum transfer in the $x$ direction with $0.07 k_{r} \leq | q_{x} | \leq 2.60 k_{r}$ . A bias magnetic field in the $z$ direction generates the Zeeman splitting. [(b)–(d)] Spin-resolved TOF images (b1)–(d1), schematic diagrams of excitations (b2)–(d2), and excitation efficiency $P (q_{x}, ω)$ in (b3)–(d3) for $Ω = 2 E_{r}$ , with $θ = 86 °$ , $| q_{x} | = 1.77 k_{r}$ . The ellipses mark atoms kicked out by the Bragg pulse. The excitation frequencies $ω$ are, respectively, $- 7.38 kHz$ (b1), 0.23 kHz (c1), and 9.60 kHz (d1). The color bar in the second row indicates the spin proportion of each dressed state. The curves in (b3)–(d3) correspond to the Gaussian fits.
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Figure 2
Excitation spectrum and roton mode softening. (a) Excitation spectrum for $Ω = 2 E_{r}$ . The two branches of spectra are clearly seen and agree well with theoretical calculations (blue and red solid curves) based on a modified Bogoliubov theory. (b) Zoom in of the low-energy part (dashed box) of the excitation spectrum in (a), which clearly shows a roton-maxon structure. (c) Softening of roton mode. The measured roton gap $Δ$ (circles) becomes smaller as $Ω$ decreases and vanishes for small $Ω$ .
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Figure 3
Softening of phonon mode. Sound velocities $c_{1}$ (solid circles) and $c_{2}$ (open circles) are both plotted for $Ω < 4.3 E_{r}$ . For $Ω \geq 4.3 E_{r}$ , the sound velocity (diamonds) is taken as the average of the measurements in the negative and positive $x$ directions to minimize possible Doppler effect. Theoretical calculations for a spin- $1 / 2$ system (dot-dashed curve) predict a vanishing minimum at the phase transition point $Ω = 4 E_{r}$ . A more practical consideration based on a spin-1 system with the effectively suppressed state $| m_{F} = 1 ⟩$ (solid curve) shows that the sound velocity cannot touch zero, and the minimum is shifted to about $4.3 E_{r}$ for the set quadratic Zeeman shift $ε = 4.53 E_{r}$ . The insets show the low-energy excitation spectrum for $Ω = 4.3 E_{r}$ (left), where the spectrum becomes parabolalike, and $Ω = 5.5 E_{r}$ (right), where the linear mode recovers.
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Figure 4
Sum rules for $Ω = 2 E_{r}$ . (a) Static structure factor $S (q_{x})$ (red circles) and the contribution $S_{-} (q_{x})$ (blue circles) from the lower branch of the excitation spectrum. The theoretical calculations based on local density approximations are shown as red and blue solid curves. The relative contribution $S_{-} (q_{x}) / S (q_{x})$ is shown in the inset. (b) Energy-weighted moment $M_{1} (q_{x})$ . The good agreement between the experimental data (blue circles) demonstrates the validity of the $f$ -sum rule.
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