In this work, we experimentally demonstrate chiral transport in a synthetic rhombic chain in the momentum space of ultracold 87Rb atoms. Adopting the state-of-the-art momentum-lattice engineering technique, we are able to switch and tune the synthetic flux threaded through each plaquette on demand and probe the dynamics of the condensate along the synthetic lattice. We observe local breathing modes and biased oscillations in a binary staggered-flux configuration, both characteristic of the flat-band localization under the staggered flux. We then engineer a state-dependent chiral transport by introducing Floquet modulations to the flux. While the micromotion of the Floquet dynamics involves the occupation of multiple sublattice sites and is complicated in general, the observed chiral transport is consistent with the quantized winding of the Floquet Bloch bands of a coarse-grained effective Hamiltonian. While the topological robustness and fast time scale of the quantized transport can be useful for quantum device design, our setup further paves the way for the exploration of the rich dynamics under the impact of the flat-band localization, disorder, Floquet driving, and long-range interactions typical of the momentum lattice. To the best of our knowledge, the observed topological transport in a system with flat-band localization is the first of its kind and absent in all existing studies.
In our experiment, the flat bands are induced by implementing synthetic fluxes in a rhombic chain. The manipulation of synthetic flux in ultracold atomic systems is a powerful tool for manipulating band structures and dynamics, as well as for simulating and studying topological phases. For instance, synthetic flux plays key roles in generating flat bands in the AB cage and the Creutz lattice, where particles exhibit highly localized states due to interference. Synthetic flux is also crucial for the simulation of the Hofstadter model, which features fractal spectra and topological bands. The interplay of different forms of synthetic flux, lattice geometry and interaction can also give rise to dynamic phases and transitions. In open systems, the combination of synthetic flux and local loss leads to directional dynamics signaling important non-Hermitian effects.
Here we consider a binary-flux-ladder (BFL) configuration with recurring synthetic fluxes ϕ and ϕ, as illustrated in Fig. 1a. Each unit cell contains 6 sublattice sites, respectively labeled {A, B, C, A, B, C} in the j-th unit cell. Under the tight-binding approximation, the system is described by the effective Hamiltonian
where J is the nearest-neighbor hopping rate, , and (, and ) are the creation (annihilation) operators for atoms on the sublattice sites of the j-th unit cell, respectively. Here {A, B, C, A} encircle the n-th plaquette, consistent with Fig. 1a. Since the synthetic flux of the n-th plaquette ϕ is given by , we henceforth adopt the gauge convention: θ = ϕ(t), and θ = 0 otherwise.
The band dispersions of Hamiltonian (1) are E = 0, , and . Here the flux-dependent factor , k (- π≤k < π) is the Bloch wave number. For convenience, we have set the lattice constant to unity. Notably, when ϕ = π, for arbitrary ϕ. All the bands are then flat regardless of ϕ [see Fig. 1b-d]. When ϕ ≠ ± π, the binary-flux lattice has six flat bands with the central band being two-fold degenerate. As shown in Fig. 1b, we label these flat-bands as E, E, E respectively. Through changing the values of ϕ, we can adjust the splitting between E and E as shown in Fig. 1d. This gives rise to intricate localized states and opens up room for engineering the transport dynamics, as we show below.
Experimentally, we implement the rhombic chain in Fig. 1a along a Raman-coupled momentum lattice in a Rb Bose-Einstein condensate (BEC), which contains ~ 2 × 10 atoms. As illustrated in Fig. 1a, sublattice sites {A, B, C} are encoded in the momentum states of atomic hyperfine levels (see Methods): A and B correspond to different momentum states of , and C is encoded in those of , respectively. Hopping between adjacent momentum-lattice sites is implemented by two-photon Raman or Bragg processes, with a fixed hopping amplitude J/h = 0.95(5) kHz. The flux ϕ is meticulously controlled by locking the relative phases of the coupling lasers. The spatial (and temporal later) inhomogeneity of the flux pattern requires a much-improved stability in phase locking, compared to previous studies with a homogeneous flux. More details about implementing the Raman lattice are illustrated in the "Methods" section and Supplementary materials.
An outstanding dynamic signature of the flat-band configuration is the localized breathing mode under the AB-caging mechanism. The conventional AB caging occurs for ϕ = ϕ = π but persists under the BFL configurations with only ϕ = π. We experimentally confirm this by focusing on the case with ϕ = 0 and ϕ = π, which has the largest gap between the low- and high-lying bands E. The corresponding localized eigenstates of the two flat bands are . As such, an initial state can be expressed as an equal-weight superposition of and the subsequent breathing mode should acquire a frequency proportional to the energy gap between the bands. Note that similar phenomena have been observed in photonic systems.
We prepare the initial state , by coherently splitting the initial BEC on site B onto sites A and A, and detect the density dynamics as shown in Fig. 1e. While the ensuing dynamics is localized and centered around the initial site, the breathing-mode character is clearly visible in Fig. 1f. The observed breathing mode features a frequency ω = 2π × 4.9(2) kHz, consistent with the numerically calculated energy gap ω = 2π × 4.9J in Fig. 1d. Due to the decoherence of the Raman-coupling processes, the overall dynamics is damped, with a fitted decay lifetime of 0.40(5)ms, as illustrated in Fig. 1f. The limited lifetime of decoherence is mainly caused by the inhomogeneous interactions of trap potential and the phase noise of Raman-Bragg lasers.
For a more general choice of ϕ and local initial state, the dynamics of the condensate center of mass becomes oscillatory but is biased in direction depending on the initial state. For instance, Fig. 2a, b display the density dynamics under {ϕ = 0, ϕ = π} with different initial-site excitations. While the dynamics is largely localized within the initial unit cell under the flat-band condition (enforced by ϕ = π), atoms initialized on-site A (A) move toward the right (left) at the beginning of the time evolution, and the overall oscillation is accordingly biased in direction. The biased oscillation can be interpreted as an interference effect between the upper and lower legs of the rhombic plaquette, which is fully destructive (constructive) under the π (0) flux. As a result, the initial condensate prepared on-site A (A) tends to delocalize on the 0-flux plaquette, moving to the right (left) at short times.
Further, by varying ϕ, we can adjust the oscillated dynamics of the local oscillation, as shown in Fig. 2c. This becomes clearer in Fig. 2d, where we plot the evolution of second moment of the position operator, defined through , where n is the position of initial injection state, , and are the expectation values of particle number operators, respectively. We further extract the two frequency components Fig. 2d, with ω = 2π × 1.4(1) kHz, ω = 2π × 2.6(1) kHz for the {ϕ = 0, ϕ = π} case and ω = 2π × 1.7(1) kHz, ω = 2π × 2.3(1) kHz for the {ϕ = 2π/3, , ϕ = π} case, respectively. These are consistent with the theoretical prediction of , explaining the ϕ dependence of the biased oscillation.
The local dynamics observed above can be harnessed for quantized chiral transport through the Floquet engineering. We show that a topological chiral transport can be achieved by constructing a helical Floquet channel which leads to a perfect spin-momentum locking through the winding of the Floquet Bloch bands. The helical Floquet channel is the Floquet counterpart of the helical edge states in 2D topological insulators with time-reversal symmetry. These states feature directional quantum transport with intrinsic spin-dependent propagation, making them ideal candidates for quantum device design.
For this purpose, we map the central two sites A and A in each unit cell to pseudospins, with the spinor field operator ψ = (ψ, ψ) as shown in Fig. 3a. Then, we apply the Floquet protocol by exchanging the neighboring flux to achieve the helical Floquet channel in our BFL configuration. Under this mapping, the chirality of the transport-the propagation directions of atoms are locked with the spin degrees of freedom, is linked to the breaking of the time-reversal symmetry defined by the combination of spin reversal and complex conjugation by preparing different initial states.
Specifically, assuming the wave packet is initialized on a local sublattice site A, a perfect population transfer between A and A can be achieved under appropriate flux parameters and at a discrete set of times. Analytically, we find that, following an evolution time t, the wavefunction on A and A are respectively given as and , where ε and ε are the eigen energies of real-space Hamiltonian. It follows that a perfect population transfer occurs at T, under the conditions εT = (μ + ν)π and εT = μπ, where μ is a positive integer and ν is a positive odd integer. In terms of the flux parameters, the conditions for the perfect population transfer are given by {ϕ = ϕ, ϕ = π}, with
In Fig. 3b, we show the discrete values of ϕ that satisfy the conditions above. On the other hand, we also have
which gives the time of the perfect population transfer, as shown in Fig. 3c.
Note that the above conditions are highly non-trivial, as the full dynamics generically involve the occupation of all the sublattice sites including B and C. Qualitatively, the perfect population transfer derives from the interference between different flat-band components of the initial state.
Based on the flux-dependent local dynamics, we propose to implement a spin-dependent chiral transport as follows. First, we initialize the condensate on-site A to the left of the plaquette with flux ϕ. We then let the condensate evolve along the momentum lattice, switching the synthetic fluxes ϕ and ϕ within each unit cell at integer multiples of T. Taking ν = 1 and μ = 2 for instance, for a wavefunction initialized on-site A under the synthetic-flux configuration {ϕ = ϕ ≈ 0.442π, ϕ = π}, it propagates to the right and becomes fully localized on-site A at t = T ≈ 0.425 ms when the coupling J = h × 1.5 kHz. We then swap ϕ and ϕ so that the wavefunction continues to propagate toward the right, becoming fully localized on the central site of the next plaquette (site A) at t = 2T. By repeating this procedure, a persistent chiral current is realized, which is quantized at discrete time steps. By contrast, the transport is not quantized when, for instance, {ϕ = 0.2π, ϕ = π}, since the oscillatory dynamics become rather complicated, as illustrated in Fig. 3d.
For the experimental confirmation of the chiral dynamics, we excite both spin-up and spin-down components at t = 0, and observe the right- and left-going chiral transport in Fig. 3e, f, respectively. The quantized chiral transport can be understood by neglecting the complicated micromotion, and focusing on coarse-grained Floquet dynamics at discrete time steps t = 2mT (m = 0, 1, 2. . . ). The subsequent dynamics involve only the occupation of the central two sites A and A in each unit cell so that we can map them to pseudospins, with the spinor field operator ψ = (ψ, ψ). The stroboscopic Floquet dynamics in the two-dimensional spinor subspace can be described by the Floquet operator , where , , and . Here T = 2T, and H and H respectively describe the spin-flip process under the switching of flux and the transport process during each half period. Note that coarse-grained effective Hamiltonians are widely used to deal with the stroboscopic dynamics of periodically driven systems.
In the quasi-momentum space, we have , and it follows that the Floquet Bloch Hamiltonian reads
The Hamiltonian has the intrinsic feature of spin-momentum locking in the real space, which underlies the spin-dependent chiral transport. Indeed, this unique feature can be revealed by the Floquet band structure, shown in Fig. 3g, where the quasienergy spectrum is gapless, featuring decoupled linear dispersions for the two spin species. We note that these helical Floquet channels exist at discrete time steps when the interim occupations of sites B and C are neglected. This is different from previous studies where simpler engineered SSH lattice models were used.
In Fig. 3h, we show the measured chiral transport using the quantity . While different spin species flow in different directions, quantized chiral transport is achieved at discrete times mT/2 (m = 0, 1, 2, . . . ), consistent with the description of the stroboscopic Floquet dynamics. The quantized transport is topologically protected by the Floquet winding numbers
where are the irreducible blocks of U. The Floquet winding numbers reflect the state-dependent winding of quasi-energy bands as the quasi-momentum k traverses the Brillouin zone. The winding numbers can also be extracted from the measured in Fig. 3h. They are given by () for the rightward (leftward) transport, consistent the theoretical predictions ν = 1 (ν = - 1).