Multiparticle Wannier states and Thouless pumping of interacting bosons

Yongguan Ke (柯勇贯), Xizhou Qin (秦锡洲), Yuri S. Kivshar, and Chaohong Lee (李朝红)

Phys. Rev. A 95, 063630 – Published 30 June 2017

Abstract

The study of topological effects in physics is a hot area of research, and only recently have researchers been able to address the important issues of topological properties of interacting quantum systems. But it is still a great challenge to describe multiparticle and interaction effects. Here, we introduce the multiparticle Wannier states for interacting systems with cotranslational symmetry, which provide an orthogonal basis for constructing effective Hamiltonians for the isolated bands. We reveal how the shift of multiparticle Wannier state relates to the Chern number of the multiparticle Bloch band and study the Thouless pumping of two interacting bosons in a one-dimensional superlattice. In addition to the Thouless pumping of bound states when two bosons move unidirectionally as a whole, we describe topologically resonant tunneling when two bosons move unidirectionally, one by the other, provided the neighboring-well potential bias matches the interaction energy. Our work creates a paradigm for multiparticle topological effects and provides a way to detect topological states in interacting multiparticle systems.

Received 23 February 2017

DOI:https://doi.org/10.1103/PhysRevA.95.063630

Physics Subject Headings (PhySH)

Cold atoms & matter waves Quantum transport Topological phases of matter

Atomic, Molecular & OpticalNonlinear DynamicsCondensed Matter, Materials & Applied PhysicsGeneral PhysicsInterdisciplinary PhysicsQuantum Information, Science & Technology

Authors & Affiliations

Yongguan Ke (柯勇贯)^1,2, Xizhou Qin (秦锡洲)¹, Yuri S. Kivshar³, and Chaohong Lee (李朝红)^1,2,*

¹TianQin Research Center & School of Physics and Astronomy, Sun Yat-Sen University (Zhuhai Campus), Zhuhai 519082, China
²State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-Sen University (Guangzhou Campus), Guangzhou 510275, China
³Nonlinear Physics Center, Research School of Physics and Engineering, Australian National University, Canberra, Australian Capital Territory 2601, Australia

^*lichaoh2@mail.sysu.edu.cn

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Issue

Vol. 95, Iss. 6 — June 2017

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Images

Figure 1
Thouless pumping of interacting particles. (a) Two interacting bosonic atoms in a double-well superlattice. The double-well potential bias $2 Δ$ and the hopping modulation $δ$ can be controlled by the phase difference between the short and long lattice, and the interaction strength $U$ can be tuned by the Feshbach resonance. (b) The parameters $Δ$ and $δ$ change with time in one pumping cycle. There are two distinct cases: (i) $| Δ_{0} | < | U / 2 |$ (blue ellipse) and (ii) $| Δ_{0} | > | U / 2 |$ (red ellipse). In the case (i), the bias $2 Δ$ is always smaller than the interaction strength during the pumping process. Then two bosons in the same site form a bound state. In case (ii), resonant tunneling happens when the bias equals the interaction strength, $| 2 Δ | = U$ . (c), (d) Thouless pumping of bound states for case (i). The two bosons initially prepared in either the lower or upper well will be unidirectionally transported a whole. (e) Topologically resonant tunneling for case (ii). The two bosons initially prepared in the lower well will be unidirectionally transported through the double-well barrier one by one when the bias matches the interaction.
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Figure 2
Multiparticle Bloch bands, multiparticle Bloch states, and maximally localized multiparticle Wannier states (MPWSs). (a) Multiparticle Bloch band of two noninteracting bosons. The parameters are chosen as $J = 1, δ = 0, Δ = 2, U = 0$ and the system size is $2 L = 98$ . Insets (bar graphs): the density distributions of multiparticle Bloch states from the three types of continuum bands. (b) Multiparticle Bloch band of two interacting bosons. Three isolated multiparticle Bloch bands (A, B, C) are separated from the continuum scattering-state bands. Insets (bar graphs): the density distributions of maximally localized MPWSs for the isolated bands (A, B, C). The corresponding schematic pictures for the maximally localized MPWSs are shown on the right-hand side. The parameters are the same as those in (a) except for $U = 10$ .
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Figure 3
Thouless pumping of bound states during one pumping cycle. The blue-solid and red-dashed lines denote the c.m. position shifts of the initial states ${| 2 〉}_{29}$ and ${| 2 〉}_{30}$ , respectively. Upper insets: The schematic diagram for the motion of a maximally localized multiparticle Wannier state occupying the isolated band B during one pumping cycle. Lower insets: The schematic diagram for the motion of a maximally localized multiparticle Wannier state occupying the isolated band C during one pumping cycle. The parameters are set as $J = 1, δ_{0} = 0.8, Δ_{0} = 2, U = 30, φ_{0} = 0, ω = 0.005$ and the system size is $2 L = 58$ .
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Figure 4
Topologically resonant tunnelings during one pumping cycle. The blue solid (black dashed) line denotes the c.m. position shifts of the initial state ${| 2 〉}_{29}$ (maximally localized Wannier state). Insets: The schematic diagram for the motion of a maximally localized multiparticle Wannier state occupying the corresponding isolated band A during one pumping cycle. The parameters are set as $J = 1, δ_{0} = 0.8, Δ_{0} = 20, U = 30, φ_{0} = 0, ω = 0.005$ and the system size is $2 L = 58$ .
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Figure 5
Multiparticle Bloch band. (a) The 3D view of Bloch band in the $κ - t$ Brillouin zone. (b) The $t - E$ view of Bloch band. The parameters are chosen as $J = 1, δ_{0} = 0.8, Δ_{0} = 2, φ_{0} = 0, U = 30$ and the system size is 58.
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Figure 6
Multiparticle Bloch band. (a) The 3D view of Bloch band in the $κ - t$ Brillouin zone. (b) The $t - E$ view of Bloch band. The parameters are chosen as $J = 1, δ_{0} = 0.8, Δ_{0} = 20, φ_{0} = 0, U = 30$ and the system size is 58.
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