Ultracold atom-ion mixtures are gaining increasing interest due to their potential applications in ultracold and state-controlled chemistry, quantum computing, and many-body physics. Here, we studied the dynamics of a single ground-state cooled ion during few, to many, Langevin (spiraling) collisions with ultracold atoms. We measured the ion's energy distribution and observed a clear deviation from the Maxwell-Boltzmann distribution, characterized by an exponential tail, to a power-law distribution best described by a Tsallis function. Unlike previous experiments, the energy scale of atom-ion interactions is not determined by either the atomic cloud temperature or the ion's trap residual excess-micromotion energy. Instead, it is determined by the force the atom exerts on the ion during a collision which is then amplified by the trap dynamics. This effect is intrinsic to ion Paul traps and sets the lower bound of atom-ion steady-state interaction energy in these systems. Despite the fact that our system is eventually driven out of the ultracold regime, we are capable of studying quantum effects by limiting the interaction to the first collision when the ion is initialized in the ground state of the trap.

Experimental realizations of charged ions and neutral atoms in overlapping traps are gaining increasing interest due to their wide research application ranging from chemistry at the quantum level to quantum simulations of solid-state systems. Here, we describe a system in which we overlap a single ground-state cooled ion trapped in a linear Paul trap with a cloud of ultracold atoms such that both constituents are in the µK regime. Excess micromotion (EMM) currently limits atom-ion interaction energy to the mK energy scale and above. We demonstrate spectroscopy methods and compensation techniques which characterize and reduce the ion's parasitic EMM energy to the µK regime even for ion crystals of several ions. We give a substantial review on the non-equilibrium dynamics which governs atom-ion systems. The non-equilibrium dynamics is manifested by a powerlaw distribution of the ion's energy. We overview the coherent and non-coherent thermometry tools which we used to characterize the ion's energy distribution after single to many atom-ion collisions.

We study the steady-state motion of a single trapped ion oscillator driven to the nonlinear regime. Damping is achieved via Doppler laser cooling. The ion motion is found to be well described by the Duffing oscillator model with an additional nonlinear damping term. We demonstrate here the unique ability of tuning both the linear as well as the nonlinear damping coefficients by controlling the laser-cooling parameters. Our observations pave the way for the investigation of nonlinear dynamics on the quantum-to-classical interface as well as mechanical noise squeezing in laser-cooling dynamics.Nonlinear dynamics prevails in many dynamical systems in nature; it introduces a rich behavior that includes criticality, bifurcations, and chaos. Nonlinear dynamics on the microscopic scale is especially interesting because it can shed light on the quantum-to-classical transition [1] as well as provide a means to amplify or suppress thermal and quantum noise [2].All mechanical oscillators exhibit nonlinearity when driven far away from equilibrium. The simplest of such nonlinear oscillators is the Duffing oscillator, which includes a cubic term in the restoring force [3]. Duffing nonlinear dynamics has been studied with electrons in a Penning trap [4] and recently with nano-electromechanical beam resonators. The basins of attraction of a nanobeam oscillator were mapped [5]. Noise squeezing and stochastic resonances were observed close to the Duffing instability [6,7]. Noise squeezing was predicted to enable mass and force detection with precision below the standard thermal limit [8-10] and possibly below the standard quantum limit when operating close to the oscillator ground state [11].The mechanical motion of trapped ions is highly controllable and can be efficiently laser-cooled to the quantum ground state [12]. High-fidelity production of Fock, squeezed, and Schrödinger-cat states was demonstrated with a single trapped ion [13,14]. At the temperature range obtained with laser-cooling techniques, quadruple radiofrequency (rf) Paul traps are excellently approximated as harmonic. Nonlinearity in ion motion was observed when several ions were trapped due to their mutual Coulomb repulsion. Here, nonlinearity couples between the ion-crystal normal modes, even at the single quantum level [15,16]. Trap nonlinearities are important in the context of resonance ejection in high-resolution mass spectrometry [17] and were shown to introduce instabilities in the ion motion at certain trapping parameters [18]. Recently, amplification saturation of a single-ion "phonon laser," resulting from optical forces that are nonlinear in the ion velocity, was demonstrated [19].Here, we study the nonlinear mechanical response of a single laser-cooled 88 Sr + ion in a linear rf Paul trap. The nonlinearity originates from the higher than quadrupolar order terms in the trapping potential. We find that the ion's steady-state response is well described by the Duffing model with an additional nonlinear damping term [20]. Unlike other realizations of nonl...

The non-crystallographic phase problem arises in numerous scientific and technological fields. An important application is coherent diffractive imaging. Recent advances in X-ray free-electron lasers allow capturing of the diffraction pattern from a single nanoparticle before it disintegrates, in so-called ‘diffraction before destruction' experiments. Presently, the phase is reconstructed by iterative algorithms, imposing a non-convex computational challenge, or by Fourier holography, requiring a well-characterized reference field. Here we present a convex scheme for single-shot phase retrieval for two (or more) sufficiently separated objects, demonstrated in two dimensions. In our approach, the objects serve as unknown references to one another, reducing the phase problem to a solvable set of linear equations. We establish our method numerically and experimentally in the optical domain and demonstrate a proof-of-principle single-shot coherent diffractive imaging using X-ray free-electron lasers pulses. Our scheme alleviates several limitations of current methods, offering a new pathway towards direct reconstruction of complex objects.

We report on the measurement of the contribution of the magnetic-dipole hyperfine interaction to the tensor polarizaility of the electronic ground-state in 87 Rb. This contribution was isolated by measuring the differential shift of the clock transition frequency in 87 Rb atoms that were optically trapped in the focus of an intense CO2 laser beam. By comparing to previous tensor polarizability measurements in 87 Rb, the contribution of the interaction with the nuclear electric-quadrupole moment was isolated as well. Our measurement will enable better estimation of black-body shifts in Rb atomic clocks. The methods reported here are applicable for future spectroscopic studies of atoms and molecules under strong quasi-static fields.Atomic systems are a good experimental platform for the test of quantum many-body theories with high precision. Atomic structure calculations for heavy atoms, with a large number of electrons, require sophisticated approximations and numerical methods. The predictions of such calculations can be readily tested using spectroscopy experiments. One such prediction entails atomic polarizabilities.Due to their symmetry under parity, atoms do not have permanent electric-dipole moments. When placed in a static electric field, however, their electronic wavefunction is polarized and a dipole moment which is proportional to the applied field is induced. The atomic polarizability is the proportionality tensor between the induced dipole moment and the field. This tensor can be further reduced to scalar and rank-two tensor parts. The latter, also known as the tensor polarizability, is determined by the hyperfine interaction of the electron with different magnetic and electric moments of the nucleus. The ab-initio calculation of the different polarizabilites is a difficult task and requires the use of advanced quantum many-body methods [1]. The measurement of the tensor atomic polarizability requires precision spectroscopy under strong applied electric fields.The precision of spectroscopic experiments benefits from long interrogation times. Optical dipole traps with their long storage times were therefore considered as a promising platform to this end. However, it was soon realized that the perturbation of atomic levels by the trapping optical fields introduces systematic line shifts and broadenings, and therefore compromises the precision of spectroscopy. Here, we rather take advantage of the large Stark shifts of atoms that are trapped in the focus of an intense CO 2 laser field in order to measure the tensor differential polarizability of the clock transition in 87 Rb. The slowly varying field of the CO 2 laser, as compared with the atomic resonance frequencies in Rb, allows for the measurement of the static polarizability to a good approximation. The interaction of atoms with laser light has been previously used to investigate atomic structure [2][3][4][5] This tensor shift of the clock transition depends only on the contribution of the spin-dipolar hyperfine interaction to the polarizability a...