Probing topological edge states in a molecular synthetic dimension

Abstract

Engineering synthetic dimensions, where the physics of additional spatial dimensions is simulated within the internal states of a quantum system, allows the realisation of phenomena not otherwise accessible in experiments. Ultracold ground-state polar molecules are an ideal platform to encode synthetic dimensions, offering access to large Hilbert spaces of long-lived internal states associated with the rotational and hyperfine degrees of freedom, that can be coupled together with microwave fields to simulate tunnelling. Here, to benchmark the advantages of ultracold molecules, we encode a 1D synthetic lattice in the rotational states of ultracold RbCs molecules and use it to investigate the well-known Su-Schrieffer-Heeger (SSH) model, a minimal model displaying topological properties. To probe the system, we perform spectroscopy using an auxiliary rotational state and study the time dynamics after deterministic state preparation. We demonstrate long coherence times, typically ~500 times the lattice tunnelling period, even for a synthetic lattice using 8 rotational states. Observations of dynamics at long times with full site-resolved readout of the synthetic dimension allow us to test the effects of chiral and non-chiral perturbations on the topologically protected edge states. Our work lays the foundation for further quantum simulations using the rich internal structure of molecules, including dipolar string phases in interacting samples of molecules, and adiabatic state preparation of many-body Hamiltonians.

Figures (6)

• Figure 1: Encoding the SSH model in the rotational structure of RbCs molecules. (a) An illustration of a lattice with the SSH model showing the alternating tunnelling rates $J_1$ and $J_2$. (b) The SSH Hamiltonian is encoded onto a synthetic lattice corresponding to the $\ket1$ to $\ket{N}$ rotational states (where $N$ is the rotational quantum number). The sites are coupled using microwave fields with Rabi frequencies $\Omega_{NN'}$. The ground state $\ket{0}$ is used as an auxiliary probe state. (c) The target Hamiltonian is implemented stroboscopically by interleaving resonant microwave pulses which are applied to the sample of molecules confined in an optical trap. (d) The energies of the eigenstates of the 8-level SSH Hamiltonian as a function of $J_2/J_1$. The SSH model exhibits a topological phase transition in the thermodynamic limit at $J_2/J_1 = 1$, from (i) a trivial to (ii) a topological regime. This is characterised by the appearance of the two edge eigenstates.
• Figure 2: Spectroscopy of the dressed eigenstates of the SSH Hamiltonian. We observe loss of molecules from $N=0$ when the probe field is tuned close to resonance to an eigenstate. Results are shown for (a) 4-site and (b) 8-site SSH chains. For each, we show (i) a theoretical prediction and (ii) experimental measurements of the normalised number of molecules remaining in $N=0$ as a function of probe detuning and the SSH tunnelling ratio $J_2/J_1$. In panel (i), the red dashed lines indicate the eigenenergies calculated by exact diagonalisation of the applied Hamiltonian. Panel (iii) shows the experimental measurements at specific ratios of $J_2/J_1$ (indicated inset) corresponding to the system in the trivial phase, at the transition boundary, and in the non-trivial topological phase. The solid lines indicate the theoretical predictions at the calibrated Rabi frequencies, with the shaded regions denoting the 1-$\sigma$ error surface. Error bars on all plots show 1$\sigma$ confidence intervals.
• Figure 3: The time evolution of the site populations under the 4-site SSH Hamiltonian after initialising in (a) an edge eigenstate and (b) a superposition of the edge eigenstates. (a) We initialise the molecule in the state $\ket{\Phi_{e+}} = \frac{1}{\sqrt{2}} (\ket{1}$ + $\ket{4})$ and evolve under the SSH Hamiltonian with $J_1= 2.30(2)$ kHz and $J_2 = 13.9(2)$ kHz. We see that the population remains equally split between $\ket1$ and $\ket4$, indicating that $\ket{\Phi_{e+}}$ is a good approximation of an eigenstate. (b) Keeping $J_1$ and $J_2$ the same, we initialise in site $\ket{1}$, which is a linear superposition of the two edge eigenstates $\frac{1}{\sqrt{2}} (\ket{\Phi_{e+}} + \ket{\Phi_{e-}}$), and again evolve under the SSH Hamiltonian. The population now oscillates between $\ket1$ and $\ket4$ with a frequency determined by $E_{\Phi_{e+}} - E_{\Phi_{e-}}$. From a fit to the data, we obtain an oscillation frequency of 379.0(7) Hz, in agreement with the predicted value of 371(13) Hz. The solid lines in both (a) and (b) are the theoretical predictions for $J_1 = 2.31$ kHz and $J_2 = 13.7$ kHz. Error bars on all plots show 1$\sigma$ confidence intervals.
• Figure 4: Dependence of the energy splitting between the edge states on the length of the SSH chain. (a) We study dynamics in SSH chains of varying length, with $J_1 = 470(15)$ Hz and $J_2 = 2.5(1)$ kHz, and the system initialised in site $\ket1$. In each case, the evolution of the population in site $\ket1$ with time is plotted and the energy splitting of the edge states is extracted by fitting the observed oscillations to a damped sinusoid. (b) The fitted edge-state energy splitting as a function of the number of sites in the SSH chain. As expected, we observe that the energy splitting reduces exponentially as the length of the SSH chain increases, with the blue line showing the expected behaviour, and the shaded area the 1-$\sigma$ error bounds from the uncertainties in Rabi frequencies. Error bars on all plots show 1$\sigma$ confidence intervals.
• Figure 5: Topological protection of the edge eigenstates in the 4-site SSH Hamiltonian. We implement perturbed versions of the 4-site SSH Hamiltonian, and show that the edge eigenstates are protected against chiral perturbations, but not against non-chiral ones. We again initialise the molecules in the SSH eigenstate $\ket{\Phi_\text{e+}}$ (upper panels) and site $\ket{1}$ (lower panels), and plot the population dynamics for all four sites. The unperturbed tunnelling rates $J_1$ and $J_2$ are the same as in Fig. \ref{['fig:SSH_time']}. (a) A chiral perturbation, with imbalanced Rabi frequencies. We see that the state $\ket{\Phi_\text{e+}}$ remains a good approximation of the eigenstate, as evidenced by the absence of population dynamics in the upper panels. Comparing the frequency of the oscillations in the lower panels with that from Fig. \ref{['fig:SSH_time']} shows that the eigenergies are altered by the perturbation. (b) A non-chiral perturbation, with $\Delta_\text{34} \neq 0$, which raises the energy of site $\ket{4}$. The significant oscillations when the system is initialised in $\ket{\Phi_\text{e+}}$ show that $\ket{\Phi_\text{e+}}$ is no longer a good representation of the eigenstate. The solid lines in both (a) and (b) show the theoretical predictions for the evolution of the populations at the experimental parameters. Error bars on all plots show 1$\sigma$ confidence intervals.