Isolated quantum-state networks in ultracold molecules

TL;DR

This work tackles the challenge of navigating the dense hyperfine structure of ultracold bialkali molecules to achieve fast, high-fidelity state transfers and robust, isolated state networks. It introduces a simple 3-level off-resonant heuristic to rapidly evaluate many transitions, then maps molecular transitions to a graph and applies graph-theoretic methods to identify optimal routes and closed networks. The authors demonstrate two concrete applications in ^87Rb^133Cs: a 4-state closed loop suitable for a synthetic dimension with high fidelity and a 3-state Λ system for an iSWAP-style quantum gate, including considerations of magnetic-field noise. The approach is general, scalable, and supported by open-source code (Diatomic-Py and diatomic-networks), offering a practical framework to design state-coupling schemes for quantum information processing and simulation with ultracold molecules.

Abstract

Precise control over rotational angular momentum is at the heart of recent advances in quantum chemistry, quantum simulation, and quantum computation with ultracold bialkali molecules. Each rotational state comprises a rich manifold of hyperfine states arising from combinations of rotation and nuclear spins; this often yields hundreds of transitions available between a given pair of rotational states, and the efficient navigation of this complex space is a current challenge for experiments. Here, we describe a general approach based on a simple heuristic and graph theory to quickly identify optimal sets of states in ultracold bialkali molecules. We explain how to find pathways through the many available transitions to prepare the molecule in a specific state with maximum speed for any desired fidelity. We then examine networks of states where multiple couplings are present at the same time. As example applications, we first identify a closed loop of four states in the RbCs molecule where there is minimal population leakage out of the loop during simultaneous microwave coupling; we then extend the optimisation procedure to account for decoherence induced by magnetic-field noise and obtain an optimal set of 3 states for quantum computation applications.

Figures (9)

• Figure 1: Rotational and hyperfine structure of $^{87}$Rb$^{133}$Cs, as an example bialkali molecule. Hyperfine Zeeman structure for (a) $N=1$ and (b) $N=0$ are shown. In (b) the $(N=0,M_F=5)_0$ and $(0,4)_1$ states are highlighted. In (a) the states are coloured by the strength of one-photon transitions from these states for unpolarised microwaves. Transitions from $(0,5)_0$ are red, and those from $(0,4)_1$ are blue. The lowest energy $N=1$ states with $M_F=3,4,5,6$ are labeled in (a) and their composition in the uncoupled basis given as a function of magnetic field in (c). The compositions of the initial $N=0$ states are given in (d). For all composition plots, The sign of the state coefficient is indicated by $+/-$, and hatchings represent the $M_N$ of the uncoupled component; $M_N=0$ is shown with dots, $M_N= \pm 1$ are shown with positive/negative gradient lines respectively. The largest uncoupled component at high field [and second largest for $(0,4)_1, (1,5)_0$] is labeled by $(N,M_N,m_\mathrm{Rb},m_\mathrm{Cs})$. The arrows between composition plots indicate allowed transitions with dotted, solid, and dashed lines indicating $\sigma^-,\pi$ and $\sigma^+$ transitions respectively. The magnitude of the transition dipoles associated with each of these transitions is given in (e), with red lines indicating transitions from $(0,5)_0$, blue lines from $(0,4)_1$ with the line style matching the arrows above.
• Figure 2: Energy level schematics. (a) The 3-level model that we use as a heuristic to quickly evaluate off-resonant couplings in the many-level molecule. We consider 2 states $\ket{0}$ and $\ket{1}$ coupled resonantly with Rabi frequency $\Omega$ and detuning $\delta$, in the presence of a third state $\ket{2}$ that has an allowed transition from $\ket{0}$ and that is detuned from $\ket{1}$ by $\Delta$. (b) Each rotational level of the molecule is comprised of a manifold of densely packed hyperfine states. We wish to resonantly couple single hyperfine states occupying neighboring rotational levels (black lines) with a driving field that induces Rabi oscillations with a frequency $\Omega$ (shown in red). Off-resonant couplings to other states (gray lines) can lead to deviation from the ideal two-level system.
• Figure 3: Analytic 3-level dynamics of populations $P_i$ for $\kappa=10$, $D=2.3$. (a) $P_1$ and $1-P_2$ against detuning fraction $f$ after evolution for time $\tau = \pi$. The inset highlights the AC-Stark shift of the transition caused by the applied driving field and the off-resonant state. (b) Populations against normalised time $\tau$, while $f=\delta_\text{AC}/\Delta$. The circular point in the inset of (a) and in (b) highlight evaluation at the same parameters.
• Figure 4: Phase space plot of equation $2\left< P_\mathrm{1} \right>^\text{comp}_t$, showing the average expected transfer to the intended state while compensating for the AC-Stark shift as a function of the normalised off-resonant detuning ratio $\kappa$, and the relative coupling strength of the states $D=d_{01}/d_{02}$. The high-fidelity contour lines when $\kappa>1$ follow $D\propto \kappa$ as is highlighted and is evident from Eq. \ref{['eqn:analytic-transfer-2-asmp-expanded']}.
• Figure 5: Deviation of the fidelity value predicted from numerical simulation to that predicted from the heuristic. The solid red line represents perfect equivalence.