Analytical results for the superflow of spin-orbit-coupled Bose-Einstein condensates in optical lattices

Xiaobing Luo, Zhou Hu, Zhao-Yun Zeng, Yunrong Luo, Baiyuan Yang, Jinpeng Xiao, Lei Li, and Ai-Xi Chen

Phys. Rev. A 103, 063324 – Published 29 June 2021

Abstract

In this paper we show that for sufficiently strong atomic interactions, there exist analytical solutions of current-carrying nonlinear Bloch states at the Brillouin zone edge to the model of spin-orbit-coupled Bose-Einstein condensates (BECs) with symmetric spin interaction loaded into optical lattices. These simple but generic exact solutions provide an analytical demonstration of some intriguing properties which have neither an analog in the regular BEC lattice systems nor in the uniform spin-orbit-coupled BEC systems. It is an analytical example for understanding the superfluid and other related properties of the spin-orbit-coupled BEC lattice systems.

Received 22 April 2021
Accepted 15 June 2021

DOI:https://doi.org/10.1103/PhysRevA.103.063324

Physics Subject Headings (PhySH)

Bose-Einstein condensates Cold gases in optical lattices Spin-orbit coupling

Atomic, Molecular & Optical

Authors & Affiliations

Xiaobing Luo ^1,2,3,*, Zhou Hu¹, Zhao-Yun Zeng ², Yunrong Luo ⁴, Baiyuan Yang², Jinpeng Xiao ², Lei Li², and Ai-Xi Chen¹

¹Department of Physics, Zhejiang Sci-Tech University, Hangzhou 310018, China
²School of Mathematics and Physics, Jinggangshan University, Ji'an 343009, China
³State Key Laboratory of Low-Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084, China
⁴Department of Physics and Key Laboratory for Matter Microstructure and Function of Hunan Province, and Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Hunan Normal University, Changsha 410081, China

^*Corresponding author: xiaobingluo2013@aliyun.com

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Vol. 103, Iss. 6 — June 2021

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Images

Figure 1
The critical values of $g / V_{0}$ vs the SO coupling strength $k_{0}$ for different Rabi frequencies $Ω = 0.3, Ω = 0.5$ , and $Ω = 1$ . The critical lines are given by $\tilde{g} = 2 g sin θ / V_{0} = 1$ , above which the exact solution (7) exists for the stationary GP equation (6). Plotted quantities are in normalized units.
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Figure 2
Density profiles along the $x$ direction and the corresponding potential functions. (a, b) $V_{0} = 0.34$ , satisfying $\tilde{g} > 1$ , and (c) $V_{0} = 0.3511$ at the critical point $\tilde{g} = 1$ . Panel (a) is for the condensate in the exact Bloch state (7), and panel (b) is for the condensate in the exact Bloch state (11). Solutions (7) and (11) are twofold-degenerate Bloch states which coalesce into one single state at the critical point. All figures show that the density peaks situate at the valleys of lattice potential wells. The other parameters are $g = 0.5, Ω = 0.3, k_{0} = 0.4$ . Plotted quantities are in normalized units.
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Figure 3
Longitudinal and transverse spin polarization $〈 σ_{z} 〉$ and $〈 σ_{x} 〉$ as a function of SO coupling strength $k_{0}$ . The blue lines and red dashed lines denote spin polarizations of exact solutions (7) and (11), respectively, and shaded areas correspond to the regions where these exact solutions (7) and (11) no longer exist. The other parameters are $g = 0.5, V_{0} = 0.1, Ω = 0.3$ . Plotted quantities are in normalized units.
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Figure 4
(a) The atomic current density $J$ vs nonlinearity value $g$ . The other parameters are chosen as $V_{0} = 0.1, k_{0} = 0.4, Ω = 0.3$ . (b) The atomic current density $J$ vs lattice depth $V_{0}$ . The other parameters are chosen as $g = 0.5, k_{0} = 0.4, Ω = 0.3$ . (c) The atomic current density $J$ vs SO coupling strength $k_{0}$ . The other parameters are chosen as $g = 0.5, V_{0} = 0.1, Ω = 0.3$ . In all plots, the blue lines and red dashed lines describe the current densities of exact solutions (7) and (11), respectively, and shaded areas correspond to the regions where these exact solutions (7) and (11) no longer exist. Plotted quantities are in normalized units.
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Figure 5
Dynamical stability of the exact Bloch solution (7) with crystal momentum $k = 1$ . Maximum imaginary part of $w$ vs nonlinearity strength $g$ for different values of $k_{0}$ and $V_{0}$ : (a) $k_{0} = 0, V_{0} = 0.01$ ; (b) $k_{0} = 0, V_{0} = 0.1$ ; (c) $k_{0} = 0.9, V_{0} = 0.01$ ; (d) $k_{0} = 0.9, V_{0} = 0.1$ . Here $g_{11} = g_{22} = g_{12} = g$ and $Ω = 0.3$ . The largest imaginary value of $w, Max [Im (w)]$ , is used to measure the dynamical stability. $Max [Im (w)] = 0$ indicates dynamical stability, and $Max [Im (w)] \neq 0$ indicates dynamical instability. Plotted quantities are in normalized units.
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Figure 6
Landau stability of the exact Bloch solution (7) with crystal momentum $k = 1$ . (a) Plot of minimum (i.e., negative maximum) of $β$ vs nonlinearity strength $g$ with the same system parameters as those in 5. (b) Plot of minimum (i.e., negative maximum) of $β$ vs nonlinearity strength $g$ with the same system parameters as those in 5. The minimum value of $β, Min (β)$ , is used to measure the Landau stability. $Min (β) = 0$ indicates Landau stability, and $Min (β) < 0$ indicates Landau instability. Shaded areas represent the regions where the exact solution (7) no longer exists. Plotted quantities are in normalized units.
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