A dilutely filled N -site optical lattice near zero temperature within a high-Q multimode cavity can be mapped to a spin ensemble with tailorable interactions at all length scales. The effective full site to site interaction matrix can be dynamically controlled by the application of up to N (N +1)/2 laser beams of suitable geometry, frequency and power, which allows for the implementation of quantum annealing dynamics relying on the all-to-all effective spin coupling controllable in real time. Via an adiabatic sweep starting from a superfluid initial state one can find the lowest energy stationary state of this system. As the cavity modes are lossy, errors can be amended and the ground state can still be reached even from a finite temperature state via ground state cavity cooling. The physical properties of the final atomic state can be directly and almost non-destructively read off from the cavity output fields. As example we simulate a quantum Hopfield associative memory scheme.

Mobile light scatterers in a high-Q optical cavity transversely illuminated by laser light close to a cavity resonance form ordered patterns, which maximize light scattering into the cavity and induce optical selftrapping. We show that a generalized form of such crystallization dynamics appears in multicolored pump fields with several cavity modes. Here the particles arrange in spatial patterns maximizing total light collection into the resonator. For changing input frequencies and strengths the particles dynamically adapt to the current illumination. Interestingly the system keeps some memory on past configurations, so that a later renewed application of the same pattern exhibits faster adaptation towards optimal collective scattering. In a noisy environment particles explore larger regions of configuration space spending most of the time close to optimum scattering configurations. This adaptive self-ordering dynamics should be implementable in a wide range of systems ranging from cold atoms in multimode cavities or nanofiber traps to molecules or mobile nano-particles within an optical resonator.

A cold dilute atomic gas in an optical resonator can be radiatively cooled by coherent scattering processes when the driving laser frequency is tuned close to but below the cavity resonance. When the atoms are sufficiently illuminated, their steady state undergoes a phase transition from a homogeneous distribution to a spatially organized Bragg grating. We characterize the dynamics of this self-ordering process in the semiclassical regime when distinct cavity modes with commensurate wavelengths are quasi-resonantly driven by laser fields via scattering by the atoms. The lasers are simultaneously applied and uniformly illuminate the atoms; their frequencies are chosen so that the atoms are cooled by the radiative processes, and their intensities are either suddenly switched or slowly ramped across the self-ordering transition. Numerical simulations for different ramp protocols predict that the system will exhibit long-lived metastable states, whose occurrence strongly depends on the initial temperature, ramp speed, and the number of atoms.

The N -queens problem is to find the position of N queens on an N by N chess board such that no queens attack each other. The excluded diagonals N -queens problem is a variation where queens cannot be placed on some predefined fields along diagonals. This variation is proven NP-complete and the parameter regime to generate hard instances that are intractable with current classical algorithms is known. We propose a special purpose quantum simulator that implements the excluded diagonals N -queens completion problem using atoms in an optical lattice and cavity-mediated long-range interactions. Our implementation has no overhead from the embedding allowing to directly probe for a possible quantum advantage in near term devices for optimization problems.

We dispersively couple a single trapped ion to an optical cavity to extract information about the cavity photon-number distribution in a nondestructive way. The photon-number-dependent AC-Stark shift experienced by the ion is measured via Ramsey spectroscopy. We use these measurements first to obtain the ion-cavity interaction strength. Next, we reconstruct the cavity photon-number distribution for coherent states and for a state with mixed thermal-coherent statistics, finding overlaps above 99% with the calibrated states.Cavity quantum electrodynamics (cavity QED) provides a conceptually simple and powerful platform for probing the quantized interaction between light and matter [1]. Early experiments opened a window into the dynamics of coherent atom-photon interactions, first through observations of collective Rabi oscillations and vacuum Rabi splittings [2][3][4][5] and later at the single-atom level [6][7][8][9][10][11]. More recently, building on measurements of the cavity field via the atomic phase [12,13], cavity photon statistics have been analyzed in experiments with Rydberg atoms or superconducting qubits in microwave resonators [14][15][16][17], culminating in the generation and stabilization of nonclassical cavity field states [18][19][20][21][22][23][24]. These experiments operate in a dispersive regime, in which information about the cavity field can be extracted via the qubits with minimal disturbance to the field [1].In parallel, it was pointed out that the Jaynes-Cummings Hamiltonian that describes cavity QED also describes the interaction of light and ions in a harmonic trapping potential [25]. This interaction underpins the generation of nonclassical states of motion [26][27][28][29] and the implementation of gates between trapped ions [30]. In analogy to the phase shifts experienced by qubits due to the cavity field, ions experience quantized AC-Stark shifts due to their coupling to the harmonic trap potential [31]. These shifts have been characterized using techniques similar to those introduced in Ref. [12]. Here, we have transferred the principle of dispersive measurement to an ion qubit coupled to a cavity. In contrast to experiments with flying Rydberg atoms, the ion is strongly confined; in contrast to both Rydberg and superconducting-qubit experiments, our cavity operates in the optical regime.We employ a single trapped 40 Ca + ion as a quantum sensor [32] to extract information about cavity photons without destroying them. Via Ramsey spectroscopy of the ion, we measure the phase shift and dephasing of the ion's state, both of which result from the interaction of the ion with the cavity field. The phase shift is induced by the mean number of cavity photons due to the AC-Stark effect, and the dephasing is caused by uncertainties in the cavity photon number. Reconstructing the cavity photon-number distribution from these measurements allows us to determine the mean and the width of the distribution and thus to distinguish between states with coherent photon statistics and mixed thermal-co...