Commensurate-incommensurate supersolid ground state of a spin-orbit-coupled Bose-Einstein condensate in one-dimensional optical lattices

Shuai Li, Huan Wang, Fuli Li, Xiaoling Cui, and Bo Liu

Phys. Rev. A 102, 033328 – Published 18 September 2020

Abstract

Recent experimental advances to create tunable synthetic spin-orbit coupling (SOC) in ultracold gases provide new possibilities to access fruitful spin-orbit coupled quantum many-body physics. In this paper, we demonstrate that the combined effect of two-dimensional (2D) SOC and one-dimensional (1D) optical lattice in interacting bosons can provide an alternative scheme in ultracold gases to achieve the crossover of the commensurate-incommensurate supersolid ground state “with respect to the background optical lattice.” Interestingly, it is shown that the anharmonicity arising from the lattice potential leads to the “pin” effect and make the ground-state break the continuous translational symmetry along the direction perpendicular to the 1D lattice, whereas the competition between the lattice period and the SOC length results in the crossover of commensurate-incommensurate ground state along the direction of 1D lattice with discrete translational symmetry breaking. Such a combined effect of SOC and optical lattice, thus, induces a new 2D periodic pattern in the ground state, accompanying with nonzero 2D superfluid density characterizing the supersolid nature in 2D of the ground state. Furthermore, a skyrmion-anti-skyrmion lattice is found associated with the appearance of such supersolid ground state, indicating its topological nontrivial properties. Experimental signature of our proposed supersolid ground state is also predicted by means of the time-of-flight measurement.

Received 29 September 2019
Revised 16 July 2020
Accepted 10 August 2020

DOI:https://doi.org/10.1103/PhysRevA.102.033328

Physics Subject Headings (PhySH)

Cold atoms & matter waves

Supersolids

Atomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Shuai Li^1,2, Huan Wang^1,2, Fuli Li^1,2, Xiaoling Cui^3,4, and Bo Liu^1,2,*

¹Department of Applied Physics, School of Physics, Xi'an Jiaotong University, Xi'an 710049, Shaanxi, China
²Shaanxi Province Key Laboratory of Quantum Information and Quantum Optoelectronic Devices, Xi'an Jiaotong University, Xi'an 710049, Shaanxi, China
³Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing, 100190, China
⁴Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China

^*liubophy@gmail.com

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Vol. 102, Iss. 3 — September 2020

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Images

Figure 1
(a) Zero-temperature phase diagram of the model Hamiltonian in Eq. (1) as a function of the SOC strength $κ^{'}$ and lattice depth $V_{0} / E_{r} . E_{r}$ is the recoil energy. For certain SOC, there is a threshold of the lattice depth, beyond that the supersolid ground state appears. There are two regions of the supersolid ground state. SS-I stands for the region where the period of the supersolid ground-state wave function along the horizontal direction is fixed by $2 a_{L}$ . Whereas in SS-II, the supersolid ground-state wave function has a continuously changed period along the horizontal direction. The inset shows that there is a jump in the superfluid density (see details in the Appendix) when the transition from the superfluid to the supersolid ground state occurs. Panels (b) and (d) show the density profile of the supersolid ground state for the spin-up component. In (b), it is shown that a spatially periodic rectangle density distribution is formed. The periodicity of such a density profile is determined by the period of lattice potential and SOC for the horizontal and vertical directions, respectively. In (d), the corresponding momentum density distribution with respect to (b) is shown. To demonstrate the periodic structure, in (d), we just show the oscillatory portion of the density distribution by eliminating the effect of the background average density profile. Panels (c) and (e) show the supersolid ground-state wave function (real part) for the spin-up component. There is an additional $π$ -phase modulation in the ground-state wave function along the direction of the 1D lattice because of the existence of finite-momentum BECs and the period of ground-state wave function along the horizontal direction is $2 a_{L}$ . Therefore, such a ground state breaks the discrete translational symmetry of the background lattice. In (e), the corresponding wave function in momentum space with respect to (c) is shown. In (b)–(e), we choose $V_{0} / E_{r} = 2.98, κ^{'} = 10.8$ , whereas in the inset of (a), $V_{0} / E_{r} = 0.17$ . Other parameters are chosen as $c_{0}^{'} = 10, c_{2}^{'} = - 0.8 c_{0}^{'}$ , and $k_{L}^{'} = 3.5 π . a_{y}$ is the spatial periodicity of the density distribution along the $y$ direction.
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Figure 2
The period of the supersolid ground-state wave function along the horizontal direction as a function of the strength of SOC with certain lattice depth. Here, $k_{x}^{peak}$ labels the positive condensate momentum along the horizontal direction. It is shown that such period behaves as a continuous function of the strength of SOC, leading to a smooth crossover between the commensurate and the incommensurate supersolid ground states. The insets show the commensurate supersolid ground-state wave function (real part) in the momentum space with the horizontal periods $2 a_{L}$ and $3 a_{L}$ , respectively. The incommensurate supersolid ground state is shown, for example, by the red mark with horizontal period $6 / \sqrt{5} a_{L}$ . The other parameters are chosen as $c_{0}^{'} = 10, c_{2}^{'} = - 0.8 c_{0}^{'}, k_{L}^{'} = 3.5 π$ , and $V_{0} / E_{r} = 2.65$ .
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Figure 3
The expansion image of the supersolid ground state for the spin-up component. It is shown that the location of peaks coincide with the results as shown in Figs. 1 and 1. The appearance of larger periodicity with respect to the lattice period in the $x$ direction is indicative of the broken discrete translational symmetry of the background lattice. The other parameters are the same as in Fig. 1.
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Figure 4
The skyrmion-anti-skyrmion lattice spin texture of the proposed supersolid ground state. (a) Illustration of the skyrmion-anti-skyrmion lattice configuration formed by the vector $S$ defined in the main text. It is shown that the $x$ and $y$ components of $S$ form a vortex-anti-vortex lattice structure on the $x y$ plane. (b) and (c) show the skyrmion and antiskyrmion, respectively. Here, the arrows show ( $S_{x}, S_{y}$ ) and the colors index the $z$ component of $S$ . Other parameters are chosen as $c_{0}^{'} = 20, c_{2}^{'} = - 0.8 c_{0}^{'}, k_{L}^{'} = 8 π, κ^{'} = 30, V_{0}^{'} = 7$ .
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Figure 5
The expectation value of the energy $E$ contributed from the anharmonic trap $V_{Atrap, θ_{0}} (r)$ for different superfluid stripe phases pointing along the direction located at angle $θ$ with respect to the vertical direction. Here, we choose the long axis of the anharmonic trap located at angle $θ_{0} = π / 6$ . It is shown that the stripe phase pointing along the long axis of the trapping potential mostly minimizes the energy cost compared to all the other direction-pointing stripe phases. The inset shows the stripe phase pointing along $θ = π / 6$ where the dashed line indicates the direction determined by the polar angle of the condensate momentum. The polar axis, here, is set as the positive $y$ axis. Other parameters are chosen as $γ = 15, κ^{'} = 14, c_{0}^{'} = 2$ , and $c_{2}^{'} = - 0.8 c_{0}^{'}$ .
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Figure 6
Superfluid density $n_{x}$ as the function of strength of SOC. There is a jump in the superfluid density when the transition from the superfluid to the supersolid occurs. The other parameters are chosen as $c_{0}^{'} = 10, c_{2}^{'} = - 0.8 c_{0}^{'}, k_{L}^{'} = 3.5 π$ , and $V_{0} / E_{r} = 0.17$ .
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