Two-dimensional Thouless pumping in time-space crystalline structures

Abstract: Dynamics of particle in a resonantly driven quantum well can be interpreted as that of a particle in a crystal-like structure, with the time playing the role of the coordinate. By introducing an adiabatically varied phase in the driving protocol, we demonstrate a realization of the Thouless pumping in such a time crystalline structure. Next, we extend the analysis beyond a single quantum well by considering a driven one-dimensional optical lattice, thereby engineering a 2D time-space crystalline structure. Such a setup allows us to explore adiabatic pumping in the spatial and the temporal dimensions separately, as well as to simulate simultaneous time-space pumping.

• The paper demonstrates two-dimensional Thouless pumping by integrating temporal and spatial adiabatic modulation in optical lattice systems.
• It employs externally driven optical lattices with controlled phase modulations to achieve quantized particle transport and simulate 4D quantum Hall effects.
• The work implies potential applications in synthetic quantum systems by using time-space crystalline dynamics to explore higher-dimensional topological phenomena.

Two-dimensional Thouless Pumping in Time-Space Crystalline Structures

Introduction

The study of time-space crystalline structures, which combine periodicity in both spatial and temporal dimensions, has recently expanded the possibilities of simulating condensed-matter phenomena using quantum systems. Time crystals, a subset of these structures, have attracted significant interest due to their periodic repetition enforced by external time-dependent signals. The research presented in "Two-dimensional Thouless pumping in time-space crystalline structures" (2206.14804) explores the adaptation of conventional space crystal phenomena, particularly Thouless pumping, to these synthetic dimensions.

Thouless pumping, a process wherein adiabatic changes in lattice parameters lead to quantized particle transport, is a well-studied aspect of conventional crystalline structures. This paper extends Thouless pumping to encompass both time and space dimensions, demonstrating adiabatic particle transport in driven optical lattices that act as 2D time-space crystals.

Model

The primary focus is on an optical lattice resonantly driven by external signals, which mimics a two-dimensional time-space crystal. The traditional Hamiltonian in space is reformulated with time as an analogous spatial coordinate. The Hamiltonian is composed of an unperturbed spatial component and time-dependent perturbations:

$$\hat{H} = \hat{h}(\hat{p}_{x}, x) + \xi_{\rm S}(x,t) + \xi_{\rm L}(x,t|\varphi_{t}).$$

Here, the spatial Hamiltonian, $\hat{h}$, is affected by optical lattice potentials, while $\xi_{\rm S}$ and $\xi_{\rm L}$ introduce oscillatory factors enabling dynamic control over the lattice periodicity in time. The challenge centers around maintaining adiabatic conditions to achieve effective coupling between energy bands while preserving the crystalline structure.

Simulations

Temporal Thouless Pumping

The study first analyzes temporal adiabatic pumping in a system with minimal spatial complexity (one spatial cell). Here, detectable changes in quasienergy levels reveal adiabatic transport within the temporal dimension.

Figure 1: Temporal progression of Wannier functions illustrating the transition between temporal cells during adiabatic phase modulation.

The quasienergy spectrum shows how adiabatic manipulation of the temporal phase $\varphi_t$ results in particle transport across temporal lattice cells, confirming the presence of a time crystal structure within the driven lattice.

Spatial Thouless Pumping

The paper next extends its analysis to spatial adiabatic pumping within a time-space crystal encompassing multiple spatial cells. This manipulation highlights the potential for particle transfer across spatial sites, reinforcing the multidimensional aspect of the study.

Figure 2: Spatial displacement of Wannier functions as the adiabatic phase modulates, revealing transitions between spatial lattice sites.

The spectrum's alignment corroborates the expected behavior as Wannier functions traverse spatial cells, analogously to their temporal counterparts.

2D Time-Space Pumping

Simultaneous temporal and spatial pumping is the culmination of the research, showcasing the integrated nature of these dimensions within the lattice. This dual approach facilitates multidimensional exploration of topological effects, such as the 4D quantum Hall effect.

Figure 3: Combined time-space displacement of Wannier functions, indicating simultaneous transition across both dimensions as phases evolve.

The temporal and spatial aspects blend seamlessly, demonstrating particle transitions across the synthetic dimensions facilitated by synchronized modulation of $\varphi_t$ and $\varphi_x$.

Conclusion

The exploration of two-dimensional Thouless pumping within time-space crystalline structures broadens the horizon for artificial dimensional constructs in quantum systems. These dynamics highlight the versatility of external signal-driven systems in emulating complex solid-state phenomena, opening avenues for probing higher-dimensional quantum mechanics, such as simulating the 4D quantum Hall effect, using relatively simple spatial setups. As future work, leveraging these lattices could deepen understanding and applications in synthetic quantum systems, showcasing the transformative potential of time-space crystals.