Polar molecules in optical lattices will be a new tool for exploring the quantum physics of strongly correlated many-body systems, bringing deep insight to many important areas of physics that are currently poorly understood.

#### Dipole-Dipole Interactions

The rich potential of polar molecules in lattices stems from the dipole-dipole interaction between molecules. Further richness originates from the many internal degrees of freedom in molecules. The rotational excited states are long lived and are easily coupled to the ground state with convenient microwave fields. The intrinsic narrow linewidth of such microwave transitions provides a sensitive spectroscopic probe of dipolar interactions. Moreover, dipole-dipole interactions can be realised by directly coupling the two rotational states with a microwave field, removing the need for an applied DC electric field. Here, the strong dipolar interaction in the rotating frame of the microwave transition can be used to model interesting many-body Hamiltonians, with the rotational state acting as a pseudo-spin. Finally, the detuning and power of the microwave field can be used to modify molecular collisions.

#### Dipolar Physics in Lattices

Optical lattices are a major theme of research in the field of ultracold atomic gases. Ever since the first observations of the transition from a superfluid to a Mott-insulator, many groups have studied the rich physics and strong connections to condensed-matter systems. Most experiments have been limited to systems with on-site “contact” interactions only. The prospect of introducing long-range interactions using molecules is very exciting, and the theory has been discussed extensively. Close collaboration is required between experiment and theory, both to understand the molecular structure and interactions and to gain insight into the many-body nature of the system. The control of the number of molecular states and the versatility of optical trapping geometries will help us to explore aspects of these challenges independently, yielding a better understanding of their contributions to the many-body physics.

Some of the exciting physics that motivates our investigations is as follows:

**Quantum Magnetism:**It should be possible to use multicomponent dipolar molecules in an optical lattice to observe quantum magnetism.**Exotic Quantum Phases:**The presence of long-range interactions is predicted to give rise to exotic spin-correlated phases for bosons, such as spin ice and quantum spin glasses, and supersolid phases, characterised by concomitant crystalline long-range and superfluid order.**Quantum Simulation:**The combination of microwave excitation with dipole-dipole interactions and spin-rotation couplings provide a complete toolbox for effective two-spin interactions with designable range, spatial anisotropy and coupling strengths significantly

larger than relevant decoherence rates. This allows spin-lattice Hamiltonians to be engineered as desired.

#### Experimental Methodology

The significant challenges and enormous rewards of dipolar molecules in lattices merit a multi-pronged approach. We will use our existing experiments on RbCs and CsYb, together with our KCs apparatus currently under development, to investigate different aspects of lattice physics. CaF may also contribute, if the phase-space density can be increased to a competitive level. In addition, we will develop a powerful new tool, the molecular microscope, to probe the spatial correlations in the lattice directly.

**RbCs:** We will use our RbCs apparatus to explore spectroscopic methods for (i) learning how to handle the molecular complexity and (ii) probing quantum magnetism applications in the lattice. Key to these studies will be the development of quantum control of the hyperfine state using microwaves and the implementation of optical lattice potentials in 1D, 2D and 3D. We are currently beginning to measure this structure using microwave transitions. We will develop methods to control the hyperfine state coherently, transfer population between states, and prepare quantum superpositions for use in Ramsey interferometry schemes. We will add “tunable” optical lattice potentials to our apparatus. Here, the relative polarisability of the two species can be tuned by changing the lattice wavelength, to aid in loading the atoms, and hence molecules. In close collaboration with theory, we will learn how to use sensitive microwave spectroscopy techniques to measure dipole-dipole interactions and gain insight into the complex many-body dynamics in the lattice. We will relate our results to those obtained from the tweezer arrays and to the literature on quantum magnetism in order to advance this important field.

**CsYb:** We will use our Cs-Yb apparatus to (i) refine the lattice loading methods and (ii) develop techniques to produce ^{2}Σ molecules in the lattice, with the aim of exploring quantum simulation of spin-lattice models. CsYb molecules are unstable against reactive collisions and so need to be produced in the protected environment of a 3D optical lattice. We will modify our apparatus to accommodate optical lattices in a glass cell with in-vacuum electrodes.

**KCs:** We expect our KCs experiment to come online around the mid-point of the Programme. Our focus will be on the physics accessible with fermionic molecules, as we originally proposed. Bringing fermionic molecules into QSUM will add considerably to our Programme. Our methods for this project are largely the same as for RbCs. However, we will focus on a simple lattice geometry where the molecules are confined in a lattice of 2D pancake traps, as shown below. Such a system will enable a wide array of investigations. For example, within a single 2D layer the application of external electric and microwave fields can be used to realise stable topological superfluid phases. When multiple layers are considered, the dipole–dipole interactions between molecules in different layers lead to new regimes of superfluidity ranging from a high-T_{C} BCS-like fermionic superfluid to quasi-condensation of interlayer bound states.

#### Many-Body Theory Methodology

To meet the above challenges, we need to develop our theoretical methods. Accurately treating long-range interactions and the resulting quantum correlations will be tackled by Tensor Network Theory (TNT). TNT allows numerical simulation of quantum systems with short-range interactions with unprecedented precision providing quasi-exact numerical results in one spatial dimension. It can handle coherent and open system dynamics as well as ground-state calculations. Efficient representations of thermal states enable studies at finite temperatures. More complex TNT algorithms for two spatial dimensions are being developed with early results suggesting they should perform as well as their 1D counterparts.

Studies of full 3D lattices will be conducted using Dynamical Mean Field Theory (DMFT) which has already successfully been applied to molecular lattices. An intermediate step towards this goal is an algorithm combining DMFT and TNT methods on classical computers. These methods will be ideally suited for investigating 3D molecular lattices.