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Highly controllable atomic, molecular, and optical systems have emerged as an increasingly powerful toolkit in advancing the frontiers of quantum simulation, metrology, computation, and fundamental physics. In this thesis, we present theoretical work on manipulation of contact and photon-mediated interactions in optical lattice clocks and cavity QED systems, as well as explorations of their possible applications on problems relevant for quantum simulation and metrology.We start from an overview of the relevant theoretical background for this thesis, including optical lattices, contact interactions, spin systems, photon-mediated interactions and measurements, as well as metrological concepts. We then present research in three closely related directions.Firstly, we discuss theory ideas for optimizing the performance of optical lattice clocks via Hamiltonian engineering. Based on the tunable delocalization of Wannier-Stark states in tilted lattices, we can fine-tune the relative strength of on-site p-wave and nearest-neighbor s-wave interactions, leading to a minimization of density shifts in a 1D optical lattice clock. We also discuss the tunability of nearest-neighbor interactions by lattice geometry. Considering the improved sensitivity of optical lattice clocks, we further analyze the manifestation of general relativistic effects in a quantum many-body optical lattice clock and discuss protocols for their near-term observation.Additionally, we discuss theory ideas for exploring emergent collective behaviors and dynamical phases in interacting arrays. We utilize sideband transitions in trapped bosonic gases to engineer Lipkin-Meshkov-Glick model and identify dynamical phase transitions between ferromagnetic and paramagnetic phases. We also provide a theoretical proposal for correlated hopping processes facilitated by multilevel atoms in cavity QED systems, which features intriguing phenomena such as chiral transport and correlation spreading behaviors. We then consider protocols for the control and amplification of atomic Bloch oscillations via cavity-mediated interactions. Moreover, we realize the Bardeen–Cooper–Schrieffer (BCS) model, an iconic model that describes the behavior of superfluids and superconductors, by photon-mediated spin exchange interactions using the Anderson pseudospin mapping, and for the first time observe a dynamical phase with persistent oscillations of the BCS order parameter.Finally, we discuss theory ideas for entanglement generation via photon-mediated interactions and measurements. We provide a theory proposal for implementing homogeneous one-axis twisting interactions in a lattice-based atom interferometer using partial delocalized Wannier-Stark states in tilted lattices, which suppresses inhomogeneities in atom-light couplings at a magic lattice depth. We also compare two common approaches experimentally used to generate spin squeezing in cavity QED systems, quantum nondemolition measurements and unitary one-axis twisting dynamics. We derived simple criteria to determine the best protocol based on the detector's quantum efficiency.

In conventional Bardeen–Cooper–Schrieffer superconductors 1 , electrons with opposite momenta bind into Cooper pairs due to an attractive interaction mediated by phonons in the material. Although superconductivity naturally emerges at thermal equilibrium, it can also emerge out of equilibrium when the system parameters are abruptly changed 2 – 8 . The resulting out-of-equilibrium phases are predicted to occur in real materials and ultracold fermionic atoms, but not all have yet been directly observed. Here we realize an alternative way to generate the proposed dynamical phases using cavity quantum electrodynamics (QED). Our system encodes the presence or absence of a Cooper pair in a long-lived electronic transition in 88 Sr atoms coupled to an optical cavity and represents interactions between electrons as photon-mediated interactions through the cavity 9 , 10 . To fully explore the phase diagram, we manipulate the ratio between the single-particle dispersion and the interactions after a quench and perform real-time tracking of the subsequent dynamics of the superconducting order parameter using nondestructive measurements. We observe regimes in which the order parameter decays to zero (phase I) 3 , 4 , assumes a non-equilibrium steady-state value (phase II) 2 , 3 or exhibits persistent oscillations (phase III) 2 , 3 . This opens up exciting prospects for quantum simulation, including the potential to engineer unconventional superconductors and to probe beyond mean-field effects like the spectral form factor 11 , 12 , and for increasing the coherence time for quantum sensing. The dynamical phases of out-of-equilibrium Bardeen–Cooper–Schrieffer superconductors have been simulated using cold atoms levitated inside an optical cavity.

We propose a scheme to control and enhance atomic Bloch oscillations via photon-mediated interactions in an optical lattice supported by a standing-wave cavity with incommensurate lattice and cavity wavelengths. Our scheme uses position-dependent atom-light couplings in an optical cavity to spatially prepare an array of atoms at targeted lattice sites starting from a thermal gas. On this initial state we take advantage of dispersive position-dependent atom-cavity couplings to perform non-destructive measurements of single-particle Bloch oscillations, and to generate long-range interactions self-tuned by atomic motion. The latter leads to the generation of dynamical phase transitions in the deep lattice regime and the amplification of Bloch oscillations in the shallow lattice regime. Our work introduces new possibilities accessible in state-of-the-art cavity QED experiments for the exploration of many-body dynamics in self-tunable potentials.

We propose a quantum enhanced interferometric protocol for gravimetry and force sensing using cold atoms in an optical lattice supported by a standing-wave cavity. By loading the atoms in partially delocalized Wannier-Stark states, it is possible to cancel the undesirable inhomogeneities arising from the mismatch between the lattice and cavity fields and to generate spin squeezed states via a uniform one-axis twisting model. The quantum enhanced sensitivity of the states is combined with the subsequent application of a compound pulse sequence that allows to separate atoms by several lattice sites. This, together with the capability to load small atomic clouds in the lattice at micrometric distances from a surface, make our setup ideal for sensing short-range forces. We show that for arrays of \\(10^4\\) atoms, our protocol can reduce the required averaging time by a factor of \\(10\\) compared to unentangled lattice-based interferometers after accounting for primary sources of decoherence.

We propose a new direction in quantum simulation that uses multilevel atoms in an optical cavity as a toolbox to engineer new types of bosonic models featuring correlated hopping processes in a synthetic ladder spanned by atomic ground states. The underlying mechanisms responsible for correlated hopping are collective cavity-mediated interactions that dress a manifold of excited levels in the far detuned limit. By weakly coupling the ground state levels to these dressed states using two laser drives with appropriate detunings, one can engineer correlated hopping processes while suppressing undesired single-particle and collective shifts of the ground state levels. We discuss the rich many-body dynamics that can be realized in the synthetic ladder including pair production processes, chiral transport and light-cone correlation spreading. The latter illustrates that an effective notion of locality can be engineered in a system with fully collective interactions.

In conventional Bardeen-Cooper-Schrieffer (BCS) superconductors, electrons with opposite momenta bind into Cooper pairs due to an attractive interaction mediated by phonons in the material. While superconductivity naturally emerges at thermal equilibrium, it can also emerge out of equilibrium when the system's parameters are abruptly changed. The resulting out-of-equilibrium phases are predicted to occur in real materials and ultracold fermionic atoms but have not yet all been directly observed. Here we realize an alternate way to generate the proposed dynamical phases using cavity quantum electrodynamics (cavity QED). Our system encodes the presence or absence of a Cooper pair in a long-lived electronic transition in \\(^{88}\\)Sr atoms coupled to an optical cavity and represents interactions between electrons as photon-mediated interactions through the cavity. To fully explore the phase diagram, we manipulate the ratio between the single-particle dispersion and the interactions after a quench and perform real-time tracking of subsequent dynamics of the superconducting order parameter using non-destructive measurements. We observe regimes where the order parameter decays to zero (phase I), assumes a non-equilibrium steady-state value (phase II), or exhibits persistent oscillations (phase III). This opens up exciting prospects for quantum simulation, including the potential to engineer unconventional superconductors and to probe beyond mean-field effects like the spectral form factor, and for increasing coherence time for quantum sensing.

We study the combined effects of measurements and unitary evolution on the preparation of spin squeezing in an ensemble of atoms interacting with a single electromagnetic field mode inside a cavity. We derive simple criteria that determine the conditions at which measurement based entanglement generation overperforms unitary protocols. We include all relevant sources of decoherence and study both their effect on the optimal spin squeezing and the overall size of the measurement noise, which limits the dynamical range of quantum-enhanced phase measurements. Our conclusions are relevant for state-of-the-art atomic clocks that aim to operate below the standard quantum limit.

We propose to simulate bosonic pair creation using large arrays of long-lived dipoles with multilevel internal structure coupled to an undriven optical cavity. Entanglement between the atoms, generated by the exchange of virtual photons through a common cavity mode, grows exponentially fast and is described by two-mode squeezing of effective bosonic quadratures. The mapping between an effective bosonic model and the natural spin description of the dipoles allows us to realize the analog of optical homodyne measurements via straightforward global rotations and population measurements of the electronic states, and we propose to exploit this for quantum-enhanced sensing of an optical phase (common and differential between two ensembles). We discuss a specific implementation based on Sr atoms and show that our sensing protocol is robust to sources of decoherence intrinsic to cavity platforms. Our proposal can open unique opportunities for next-generation optical atomic clocks.

Engineering a Hamiltonian system with tunable interactions provides opportunities to optimize performance for quantum sensing and explore emerging phenomena of many-body systems. An optical lattice clock based on partially delocalized Wannier-Stark states in a gravity-tilted shallow lattice supports superior quantum coherence and adjustable interactions via spin-orbit coupling, thus presenting a powerful spin model realization. The relative strength of the on-site and off-site interactions can be tuned to achieve a zero density shift at a `magic' lattice depth. This mechanism, together with a large number of atoms, enables the demonstration of the most stable atomic clock while minimizing a key systematic uncertainty related to atomic density. Interactions can also be maximized by driving off-site Wannier-Stark transitions, realizing a ferromagnetic to paramagnetic dynamical phase transition.

We theoretically propose and experimentally demonstrate the use of motional sidebands in a trapped ensemble of \\(^{87}\\)Rb atoms to engineer tunable long-range XXZ spin models. We benchmark our simulator by probing a ferromagnetic to paramagnetic dynamical phase transition in the Lipkin-Meshkov-Glick (LMG) model, a collective XXZ model plus additional transverse and longitudinal fields, via Rabi spectroscopy. We experimentally reconstruct the boundary between the dynamical phases, which is in good agreement with mean-field theoretical predictions. Our work introduces new possibilities in quantum simulation of anisotropic spin-spin interactions and quantum metrology enhanced by many-body entanglement.