Many-Body Chiral Edge Currents and Sliding Phases of Atomic Spin Waves in Momentum-Space Lattice

Yongqiang Li, Han Cai, Da-wei Wang, Lin Li, Jianmin Yuan, and Weibin Li

Phys. Rev. Lett. 124, 140401 – Published 9 April 2020

Abstract

Collective excitations (spin waves) of long-lived atomic hyperfine states can be synthesized into a Bose-Hubbard model in momentum space. We explore many-body ground states and dynamics of a two-leg momentum-space lattice formed by two coupled hyperfine states. Essential ingredients of this setting are a staggered artificial magnetic field engineered by lasers that couple the spin wave states and a state-dependent long-range interaction, which is induced by laser dressing a hyperfine state to a Rydberg state. The Rydberg dressed two-body interaction gives rise to a state-dependent blockade in momentum space and can amplify staggered flux-induced antichiral edge currents in the many-body ground state in the presence of magnetic flux. When the Rydberg dressing is applied to both hyperfine states, exotic sliding insulating and superfluid (supersolid) phases emerge. Because of the Rydberg dressed long-range interaction, spin waves slide along a leg of the momentum-space lattice without costing energy. Our study paves a route to the quantum simulation of topological phases and exotic dynamics with interacting spin waves of atomic hyperfine states in momentum-space lattice.

Received 6 September 2019
Revised 2 December 2019
Accepted 16 March 2020

DOI:https://doi.org/10.1103/PhysRevLett.124.140401

Physics Subject Headings (PhySH)

Cold atoms & matter waves Cold gases in optical lattices Phase diagrams Quasiparticles & collective excitations Synthetic gauge fields Van der Waals interaction

Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & Optical

Authors & Affiliations

Yongqiang Li^1,2, Han Cai³, Da-wei Wang ³, Lin Li⁴, Jianmin Yuan ^1,2, and Weibin Li ⁵

¹Department of Physics, National University of Defense Technology, Changsha 410073, People’s Republic of China
²Department of Physics, Graduate School of China Academy of Engineering Physics, Beijing 100193, People’s Republic of China
³Interdisciplinary Center for Quantum Information and State Key Laboratory of Modern Optical Instrumentation, Zhejiang Province Key Laboratory of Quantum Technology and Device and Department of Physics, Zhejiang University, Hangzhou 310027, China
⁴MOE Key Laboratory of Fundamental Physical Quantities Measurement, Hubei Key Laboratory of Gravitation and Quantum Physics, PGMF and School of Physics, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China
⁵School of Physics and Astronomy, and Centre for the Mathematics and Theoretical Physics of Quantum Non-equilibrium Systems, The University of Nottingham, Nottingham NG7 2RD, United Kingdom

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Vol. 124, Iss. 14 — 10 April 2020

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Figure 1
Momentum-space lattice. (a) Level scheme. Collective excitations in states $| a ⟩$ and $| b ⟩$ are coupled resonantly by a detuned (magenta, Rabi frequency $Ω_{0}$ ) and a resonant (blue, Rabi frequency $Ω_{r}$ ) standing wave laser, forming a momentum-space lattice. When weakly coupled to a Rydberg state (with Rabi frequency $Ω_{σ}$ and detuning $Δ_{σ}$ ), atoms in state $| σ ⟩ (σ = a, b)$ experience an effective interaction $V_{σ} (x)$ . (b) Two-leg zigzag lattice. The state $| a ⟩$ ( $| b ⟩$ ) sits on the $A$ leg ( $B$ leg) of the ladder. Hopping rate along the $A$ leg ( $B$ leg) is $h_{0} e^{i ϕ}$ ( $- h_{0} e^{i ϕ}$ ). The interleg hopping is determined by parameter $h_{r}$ . The interactions between sites in the $A$ leg ( $B$ leg) is determined by ${\tilde{V}}_{a} (k)$ [ ${\tilde{V}}_{b} (k)$ ]. (c) Momentum-dependent interaction ${\tilde{V}}_{σ} (k) / {\tilde{V}}_{σ} (0)$ (solid) and soft-core interaction $V_{σ} (x) / V_{σ} (0)$ (dashed). Using parameters $λ = 785 nm$ and $r_{c} = 4.5 μ m$ , the interaction is important when $k ≪ k_{c}$ . (d) Band structure of the noninteracting ladder for different hopping amplitudes and flux $ϕ = π / 4$ .
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Figure 2
Strongly correlated many-body ground states. (a),(b) Interaction effects on phase diagrams in momentum-space lattice in terms of hopping amplitude $h_{r}$ and $h_{o}$ for different flux. There are three quantum phases with different types of edge currents, including the ${CSF}_{m}$ and ${CSF}_{v}$ phases with chiral edge currents and the ${CSF}_{p}$ phase with antichiral edge current. Here, the filling $N_{tot} / L_{lat} = 0.125$ , with $N_{tot}$ being the total number of atoms in the lattice and $L_{lat}$ the lattice size [81], and the dashed lines denote the noninteracting system. Interaction effects on phase transitions for a (c) noninteracting and (d),(e) interacting systems. Note that, in the absence of two-body interactions, the currents for $ϕ = 5 π / 4$ can be obtained from (c) by using the chiral symmetry (i.e., swapping the state $| a ⟩$ and $| b ⟩$ ). The other parameters are (c) $h_{o} = 0.1$ and $V = 0$ , (d) $h_{o} = 0.1$ and $V = 1$ , (e) $h_{o} = 0.1$ and $V = 1$ . (f) Interplay between flux and interaction for a fixed hopping amplitude ratio. [Inset (g)] Interaction-induced ${CSF}_{v}$ - ${CSF}_{m}$ phase transition for a fixed flux $ϕ = 0.6 π$ . Periodic boundary condition is used in the calculation.
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Figure 3
Dynamics of excitations and currents. (a) Total excitation and (b) double occupation probability of the $B$ leg. Dotted and dashed curves correspond to noninteracting situations, and dot-dashed and solid lines correspond with two-body interactions. The flux is $ϕ = π / 4$ (solid and dashed) and $ϕ = 5 π / 4$ (dot-dashed and dotted). Time evolution of currents in the (c),(e) absence and (d),(f) presence of two-body interactions, with currents along the $A$ leg $J_{A A}$ (dotted), $B$ leg $J_{B B}$ (solid), and between the two legs $J_{A B}$ (dashed). Other parameters are (c),(e) $h_{o} = h_{r} = 0.02$ and $V = 0$ , and (d),(f) $h_{o} / V = h_{r} / V = 0.02$ . We have defined $τ \equiv h_{0} t$ .
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Figure 4
Emergent sliding phases in the double Rydberg dressing regime. (a) Representation of sliding insulator with random density distribution with degenerate energy. (b) Devil’s staircase structure of the system in the atomic limit, where $N_{tot}$ denotes the total number of particles. (c) Phase diagram with flux $ϕ = 0$ and particle number $N_{tot} = 32$ . The system supports the SI and SSF. (d) Phase diagram with flux $ϕ = π / 4$ and particle number $N_{tot} = 32$ . Both SI and SSF phases generate edge currents in the presence of flux. [Inset (e)] Averaged energy $E / N_{tot}$ as function of hopping $h_{r} = h_{o}$ in the strongly interacting regime. [Insets (f),(g)] Enlargement of the main figure, where normal superfluid phase (NSF) appears in the weakly interacting regime (CSFs appearing in the presence of flux).
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