Molecules Simulate Extra Dimensions to Explore New Physics

Adarsh P. Raghuram and colleagues at Durham University have encoded a one-dimensional synthetic lattice within the rotational states of ultracold rubidium-caesium molecules, enabling investigation of the Su-Schrieffer-Heeger model and its topological properties. This collaboration between Durham University and Rice University achieves remarkably long coherence times, up to 500 times the lattice tunnelling period, and full site-resolved readout of the synthetic dimension, allowing detailed testing of perturbations on topologically protected edge states. These findings represent a key step towards advanced quantum simulations utilising molecular internal states, potentially enabling studies of complex systems such as dipolar string phases and adiabatic state preparation of many-body Hamiltonians

Topological edge states and long coherence in a molecular synthetic lattice

A 1D synthetic lattice is encoded in the rotational states of ultracold RbCs molecules to investigate the Su-Schrieffer-Heeger (SSH) model, a minimal model displaying topological properties. Spectroscopy utilising an auxiliary rotational state and study of the time dynamics after deterministic state preparation are performed to probe the system. Long coherence times, typically ∼500 times the lattice tunnelling period, are demonstrated, even for a synthetic lattice using 8 rotational states.

Dynamics at long times with full site-resolved readout of the synthetic dimension allow testing the effects of chiral and non-chiral perturbations on the topologically protected edge states. This lays the foundation for further quantum simulations utilising the rich internal structure of molecules, including dipolar string phases in interacting samples and adiabatic state preparation of many-body Hamiltonians. Ultracold polar molecules combine rich internal structures with access to tunable, long-range and anisotropic dipolar interactions.

These properties have led to many proposed applications in quantum computing and quantum simulation. Most proposals utilise the rotational states of the molecules to encode qubits, qudits, or pseudospins for the simulation of quantum spin models. An alternative approach, however, is to exploit the rotational states of polar molecules to realise synthetic dimensions, describing additional degrees of freedom encoded in the discrete internal states of a system used to simulate lattice sites in an additional spatial dimension.

Coherent coupling between the internal states is then interpreted as tunnelling in the synthetic lattice. This approach offers several advantages: it allows the simulation of higher-dimensional systems, engineering site-specific complex tunnelling amplitudes is straightforward, changing the geometry of the synthetic dimension simply requires changing the coupling fields, and resolving single sites in the synthetic dimension is readily achieved using state-selective imaging techniques. These advantages have been used to study phenomena including synthetic gauge fields, Anderson localisation, and Thouless pumping.

Synthetic dimensions have been implemented in experiments with photonic and cold atomic systems exploiting atomic hyperfine states, Rydberg states, quantised momentum states and atomic trap states. However, the rotational states of polar molecules offer a unique set of advantages as they are connected via strong electric-dipole transitions in the microwave domain and benefit from dipolar interactions between molecules, while being immune to radiative decay losses. This opens the possibility of preparing exotic many-body ground states using molecular synthetic dimensions.

The simplest synthetic dimension is a 1D tight-binding model, which can be encoded in a chain of microwave-coupled molecular rotational states. Such a chain can implement the well-known SSH model originally developed to understand the electrical conductivity of trans-polyacetylene. The model is characterised by a 1D chain comprising an even number of sites coupled with alternating strengths J1, J2. Despite its simplicity, the SSH model exhibits non-trivial topological features and has served as an excellent benchmark of various experimental platforms.

A 1D chain synthetic dimension within the rotational states of ultracold Rb133Cs molecules is realised and SSH models with up to 8 sites are studied. Tuning the tunnelling rates in the synthetic lattice probes the phase transition between trivial and non-trivial topological phases. The energies of the SSH eigenstates are experimentally measured using a combination of spectroscopic and interferometric methods. Demonstrating topological protection of the SSH edge states against chiral perturbations, the winding number is extracted from the dynamics of the system, showing that it changes across the topological phase transition.

Results show excellent agreement with theoretical predictions from exact diagonalisation over many tunnelling periods, highlighting the ability to engineer precisely controlled tunnelling rates and the long coherence times available in this system. The rotational states of RbCs molecules encode the 1D synthetic chain, with the first rotationally excited state (N = 1) serving as the first site in the 1D chain. The rotational states are labelled by the angular momentum quantum number N, utilising spin-stretched states where the rotational angular momentum projection and both nuclear spin projections take their maximum values.

To drive multiple rotational transitions in a scalable way, a stroboscopic approach is employed, rapidly alternating between driving each transition in turn using a single microwave source. The target Hamiltonian is realised as the time average of many pulses, each of which is 40 to 200 times shorter than the characteristic tunnelling times. Effective tunnelling rates, J1 and J2, in the synthetic dimension are controlled by controlling the amplitudes of the microwave fields.

This is an example of Floquet engineering or equivalently Trotterisation. The Trotter infidelity between the target and applied Hamiltonians is negligible, of the order 10−4 for the pulse durations used. Probing the synthetic chain with high spectroscopic resolution requires long interrogation times and long-lived coherence between rotational states. The molecules are confined in an optical trap and the total probe time is approximately π/. Spectroscopy is performed using the auxiliary N = 0 probe level.

The probe field is also pulsed stroboscopically, alternating with the SSH driving fields. Loss of molecules from N = 0 is observed when the probe field is tuned close to resonance to an eigenstate, for both 4-site and 8-site SSH chains. Normalised number of molecules remaining in N = 0, as measured experimentally, aligns with theoretical predictions as a function of probe detuning and the SSH tunnelling ratio J2/J1. The topological phase transition occurs at J2/J1 = 1. When J2/J1 > 1, the system enters a non-trivial topological phase, characterised by the emergence of two edge eigenstates localised on the ends of the chain.

Increasingly sophisticated tools are being built to simulate quantum systems, and encoding physics within the internal states of molecules offers a promising route. Scaling to more complex systems, particularly those with interacting molecules, remains a significant challenge. This demonstration of a synthetic dimension using ultracold molecules is significant, establishing a viable platform for quantum simulation and exceeding the coherence times of many existing methods by a factor of 500 within the simulated system.

This improved coherence allows for detailed observation of subtle quantum effects, paving the way for investigations into more intricate physical models and potentially novel materials. This demonstration of a synthetic dimension built within ultracold rubidium-caesium molecules establishes a new technique for exploring quantum systems, moving beyond simply mimicking spatial dimensions to actively engineering them within molecular properties. Achieving coherence 500 times longer than typical lattice tunnelling periods unlocks detailed observation of subtle quantum behaviours, particularly within topologically protected states. The ability to fully resolve individual sites within this synthetic dimension allows precise testing of how external influences affect these states, opening avenues to investigate more complex quantum phenomena.

Researchers successfully encoded a one-dimensional synthetic lattice within the rotational states of ultracold rubidium-caesium molecules and used it to investigate the Su-Schrieffer-Heeger model. This demonstrates a method for simulating complex physical phenomena by engineering additional dimensions within the internal properties of molecules, rather than relying on physical space. The experiment achieved coherence times approximately 500 times longer than the lattice tunnelling period, enabling detailed observation of quantum dynamics and topologically protected edge states. The authors suggest this platform could be extended to explore interacting molecular systems and dipolar string phases.