This thesis describes three studies using ultracold gases of bosonic dysprosium atoms. We encode quantum states of interest in the large electronic ground state spin J=8 of each atom. Narrow-line optical transitions are used to manipulate these states by creating strong spin-dependent tensor light shifts with negligible heating. Our detection scheme uses arbitrary spin rotations followed by measurement of projection probabilities with single magnetic sublevel resolution. Firstly, we study a transverse-field Ising model of quantum spin-1/2’s with infinite-range interactions. This model shows a paramagnetic to ferromagnetic quantum phase transition in the thermodynamic limit. Our experiment relies on the formal equivalence between the states of a spin J=8 and exchange-symmetric states of 2J=16 qubits in the sector of maximum total spin. We probe the thermodynamic and dynamical properties, revealing finite-size quantum critical behaviour around the transition point. We also directly test the fundamental link between symmetry breaking and the appearance of a finite order parameter. This is enabled by access to the collective observable defining the underlying Z2 symmetry. Secondly, we partition the electronic spin to reveal entanglement in nonclassical states. By optically coupling the ground level to an excited state J' = J-1, we remove a pair of encoded qubits in a state defined by the light polarisation. We probe the concurrence of qubit pairs extracted from W and squeezed states to quantify their nonclassical character. We directly demonstrate entanglement between the 14- and 2-qubit subsystems via an increase in entropy upon partition. In a complementary set of experiments, we probe the decoherence of a state prepared in the excited level J' = J + 1 and interpret spontaneous emission as a loss of a qubit pair in a random state. This allows us to contrast the robustness of nonclassical pairwise correlations of the W state with the fragility of the coherence of a Schrödinger cat state. Thirdly, we simulate a quantum Hall system by exploiting the electronic ground state spin as a discrete synthetic dimension, and optically coupling it to atomic motion along a second real dimension. This creates an artificial magnetic field, with the two coupled dimensions forming a Hall ribbon with sharp edges along the synthetic axis. We demonstrate that the large number of magnetic sublevels leads to distinct bulk and edge behaviours, and a Hall response characteristic of a non-trivial topology. We perform a preliminary investigation of the entanglement spectrum associated with a partition at the centre of the synthetic dimension, and we explain how future studies could directly implement the corresponding entanglement Hamiltonian. We conclude with a roadmap for extending the techniques presented in this thesis in order to realise many-body topological states.
This thesis describes three studies using ultracold gases of bosonic dysprosium atoms. We encode quantum states of interest in the large electronic ground state spin J=8 of each atom. Narrow-line optical transitions are used to manipulate these states by creating strong spin-dependent tensor light shifts with negligible heating. Our detection scheme uses arbitrary spin rotations followed by measurement of projection probabilities with single magnetic sublevel resolution. Firstly, we study a transverse-field Ising model of quantum spin-1/2’s with infinite-range interactions. This model shows a paramagnetic to ferromagnetic quantum phase transition in the thermodynamic limit. Our experiment relies on the formal equivalence between the states of a spin J=8 and exchange-symmetric states of 2J=16 qubits in the sector of maximum total spin. We probe the thermodynamic and dynamical properties, revealing finite-size quantum critical behaviour around the transition point. We also directly test the fundamental link between symmetry breaking and the appearance of a finite order parameter. This is enabled by access to the collective observable defining the underlying Z2 symmetry. Secondly, we partition the electronic spin to reveal entanglement in nonclassical states. By optically coupling the ground level to an excited state J' = J-1, we remove a pair of encoded qubits in a state defined by the light polarisation. We probe the concurrence of qubit pairs extracted from W and squeezed states to quantify their nonclassical character. We directly demonstrate entanglement between the 14- and 2-qubit subsystems via an increase in entropy upon partition. In a complementary set of experiments, we probe the decoherence of a state prepared in the excited level J' = J + 1 and interpret spontaneous emission as a loss of a qubit pair in a random state. This allows us to contrast the robustness of nonclassical pairwise correlations of the W state with the fragility of the coherence of a Schrödinger cat state. Thirdly, we simulate a quantum Hall system by exploiting the electronic ground state spin as a discrete synthetic dimension, and optically coupling it to atomic motion along a second real dimension. This creates an artificial magnetic field, with the two coupled dimensions forming a Hall ribbon with sharp edges along the synthetic axis. We demonstrate that the large number of magnetic sublevels leads to distinct bulk and edge behaviours, and a Hall response characteristic of a non-trivial topology. We perform a preliminary investigation of the entanglement spectrum associated with a partition at the centre of the synthetic dimension, and we explain how future studies could directly implement the corresponding entanglement Hamiltonian. We conclude with a roadmap for extending the techniques presented in this thesis in order to realise many-body topological states.
This thesis manuscript presents experiments realized with ultracold atomic gases of dysprosium. This element has specific properties such as a large spin J=8 and tunable light-spin couplings originating from significant tensor light-shift contributions close to resonance. An overview of the experimental setup is given, from magneto-optical trapping to evaporative cooling. In particular we present the role of the tensor light-shift in the in-trap Doppler cooling process. Details on the use of spatial modulation to tune the shape of an optical dipole trap are also given. In the second part we describe the preparation of squeezed and ‘oversqueezed’ spin states as well as Schrödinger cat states using light-induced spin couplings. The magnetic sensitivity of these states is characterized using Ramsey spectroscopy. The Husimi function is reconstructed from population measurements along various directions, also allowing us to compute the purity of the prepared states. While squeezed states show an increased metrological gain, oversqueezed states exhibit a strong Ramsey oscillations contrast reduction. The Hellinger distance is used to extend the notion of metrological gain and the sensitivity measured in this case saturates the Cramér-Rao bound. Finally a system of 16 indistinguishable interacting spins 1/2 is realized. In the third part we present the measurement of pairwise entanglement from photon absorption measurements, which was not realized experimentally. Finally we present two projects where spatial degrees of freedom are added to the dynamics of internal levels. The first one consists in the realization of a system analogous to the Landau Hamiltonian of a charged particle in a magnetic field, with one spatial dimension and one synthetic dimension. The second project uses two spatial dimensions and exhibits an artificial magnetic field originating from positiondependent light-shifts.
Topological quantum many-body systems are characterized by a hidden order encoded in the entanglement between their constituents. While entanglement is often quantified using the entanglement entropy, its full description relies on the entanglement Hamiltonian, which is commonly used to identify complex phases arising in numerical simulations, but whose measurement remains an outstanding challenge. Here, we map entanglement to spectral properties by realizing a physical system whose single-particle dynamics is governed by the entanglement Hamiltonian of a quantum Hall system. We use a synthetic dimension, encoded in the electronic spin of dysprosium atoms, to implement spatially deformed dynamics, as suggested by the Bisognano-Wichmann prediction. The realized Hamiltonian, probed with bosonic atoms with negligible interactions, exhibits a chiral dispersion akin to a topological edge mode, revealing the fundamental link between entanglement and boundary physics. We numerically show that our protocol could be extended to interacting systems in fractional quantum Hall states. Spatial entanglement in many-body systems is fully characterized by the entanglement Hamiltonian, but its measurement has remained challenging. Here the authors realize this for a topological quantum Hall system using ultracold gases of dysprosium atoms.
This thesis reports on new experimental techniques for the study of strongly correlated states of ultracold atoms in optical lattices. We used a high numerical aperture imaging system to probe 87Rb atoms in a two-dimensional lattice with single-site resolution. Fluorescence imaging allows to detect single atoms with a large signal to noise ratio and to reconstruct the atom distribution on the lattice. We applied this new technique to a two-dimensional Mott insulator and directly observed number squeezing and the emerging shell structure. A comparison of the radial density and variance distributions to theory provides a precise in situ temperature and entropy measurement from single images. We find entropies around the critical value for quantum magnetism. In a second series of experiments, we demonstrated two-dimensional single-site spin control in the optical lattice. The differential light shift of a tightly focused laser beam shifts selected atoms into resonance with a microwave field driving a spin flip. In this way, we reach sub-diffraction limited spatial resolution well below the lattice spacing. Starting from a Mott insulator with unity filling we were able to create arbitrary spin patterns. We used this ability to prepare atom distributions to study one-dimensional single-particle tunneling dynamics in a lattice. By discriminating the dynamics of the ground state and of the first excited band, we find that our addressing scheme leaves most atoms in the vibrational ground state. Moreover, we studied coherent light scattering from the atoms in the optical lattice and found diffraction maxima in the far-field. We showed that an antiferromagnetic order leads to additional diffraction peaks which can be used to detect this order also when single-site resolution is not available. The new techniques described in this thesis open the path to a wide range of novel applications from quantum dynamics of spin impurities, entropy transport, implementation of novel cooling schemes, and engineering of quantum many-body phases to quantum information processing.
The three experiments described in this thesis investigate fundamental properties of ultracold atoms. Using laser cooling and evaporative cooling, a dilute gas of sodium atoms is cooled to ~100 nK. Under these circumstances a Bose-Einstein condensate (BEC) forms, where millions of atoms collapse into the lowest energy state of the system and share a macroscopic wavefunction. The experiments are done in an ultrahigh vacuum and the atoms are manipulated remotely using laser beams, magnetic fields, and RF-fields. Using ultracold atoms allows unprecedented control over the internal and external degrees of freedom of the system, making it very suitable for precision measurements. Ultracold atoms also serve as a model system for more complicated or remote fields such as solid-state physics and cosmology. In the first experiment the refractive index of an ultracold bosonic gas is studied. The atomic clouds are analyzed for temperature and number of particles using nearly non-destructive phase-contrast imaging. After each pulse of probe light a small fraction of the atoms is lost, while the cloud is simultaneously slightly heated. This makes it possible to study the scattering rate as a function of temperature using only a single sample. It is observed that the scattering rate increases below the critical temperature for BEC by more than a factor of 3 compared to the classical value. This enhancement is in good agreement with the predictions by theoretical work that takes two-body correlation and the Lorentz-Lorenz correction into account. The method is also used to perform calorimetry on the Bose gas and directly measure the heat capacity of the BEC. The second chapter deals with magnetism in ultracold gases. When cooling an ultracold gas containing different spin components below the critical temperature for BEC, stable magnetic domains form. The appearance of these domains and the dynamics of the domain walls are studied. The spin mixture is prepared by performing an RF-sweep on ground state atoms in the presence of a magnetic field. Spin dynamics are induced by applying magnetic gradients. Spin drag, which is the transport coefficient responsible for damping spin currents, was measured in a two-component system. Furthermore a new method for spin resolved phase contrast imaging is presented, which is used to study time evolution in situ non-destructively. The last experiment focuses on spontaneous symmetry breaking. When passing a continuous phase transition, spontaneous symmetry breaking causes the development of domains with an independently chosen order parameter. The domains are separated by boundary defects. In this experiment the spontaneous creation of solitons was observed, which are boundary defects in the phase of the condensate, as well as magnetic domain walls, which are the boundary defects in the magnetization. Through the Kibble-Zurek mechanism it is possible to determine the critical exponents that characterize the phase transition. The critical exponents are predicted to be universal, i.e. do not depend on the microphysics of the system, so the results from our ultracold atoms lab can help answer universal questions for example from cosmology and from the early universe.
Certain classes of quantum many-body systems, including those supporting phenomena like high-Tc superconductivity and spin liquids, are believed to be fundamentally intractable to classical modeling. Quantum simulations, in which synthetic materials are engineered by inducing well-controlled quantum systems like ultracold atoms to obey many-body Hamiltonians of interest, are a promising new approach to study this type of physics. In this work, I present several advances toward this ultimate goal of large-scale, highly controllable quantum simulations of many-body spin physics. We simulate long-range Ising and XY spin models in the presence of transverse and longitudinal magnetic fields using chains of up to 18 ultracold 171Yb+ ions held in a linear Paul trap, where two hyperfine levels in each ion encode spin-1/2 states. The tunable spin-spin interactions and effective magnetic fields are engineered using laser fields, and the individual spin states are directly imaged with state-dependent fluorescence. The results in this thesis address several of the ongoing challenges in the development of synthetic quantum matter platforms. One such challenge is establishing more flexible capabilities in the sorts of Hamiltonians we can model. By observing suppression of the ground state spin ordering, we have demonstrated our ability to continuously tune the interaction range in a power-law interaction pattern, and hence the amount of frustration present in the spin system. We have additionally begun developing tools to study particles of higher spin, which could eventually be used to create and study topological phases of matter. Another challenge is the necessity of identifying problems that the next generation of experiments, with flexible (but not arbitrary) controls and classically intractable (but not infinitely large) system sizes, can feasibly shed new light on. We have made measurements of how the range of interaction affects dynamics of spin correlations propagating through the chain, and the excellent agreement between our observations and numerical simulations indicate that at larger sizes, our experiment can meaningfully contribute to the open question of the fundamental speed limit on the transfer of information through such a spin chain. Finally, for classically intractable system sizes, it will be crucial to have multiple techniques at our disposal for validating our understanding of the exact microscopic model being implemented. We have developed and demonstrated an MRI-like spectroscopic technique for probing the energies of the many-body Hamiltonian, which serves as a new method for validating quantum simulations of the transverse Ising model. Our experiments can potentially be scaled up in the near future to study fully connected lattice spin models with several tens of spins, where classical computation begins to fail, and the results described in this thesis contribute to the effort to build experiments that can break new ground in the study of quantum many-body physics.
Magnetism is a subject that has fascinated mankind for countless generations. With the development of quantum mechanics around a century ago a fundamental understanding of many of the underlying causes of magnetism were obtained. However, the long range interaction combined with the complex systems magnetism appears in makes it very hard to investigate it. In our days this research area is more active than ever, mainly due to future demand of the electronic industry for components engineered down to the atomic level. Quantum effects gets more important at short scales, especially in lower dimensional materials like sheets and wires. The rapidly increasing amount of computational power available makes the numerical techniques a more important part of the research effort. Many of these are especially well suited to analyze lower dimensional quantum problems. Three of the most important techniques, (classical) Monte Carlo, Exact Diagonalization (ED) and matrix product states (MPS) based techniques, like density renormalization group (DMRG) and time evolving block decimation (TEBD) will be described in some detail and put to use later in this dissertation.With the rapid development of experimental techniques for ultracold gases in optical traps a new approach to investigate magnetic properties that are hard achieve or control in the solid state has emerged. We first study the ground-state phase diagram of a spin-1 condensate trapped in an optical trap when the magnetic dipole interaction between the atoms is taken into account along with confinement and spin precession. The boundaries between the regions of ferromagnetic and polar phases move as the dipole strength is varied and the ferromagnetic phases can be modulated. The magnetization of the ferromagnetic phase perpendicular to the field becomes modulated as a helix winding around the magnetic field direction, with a wavelength inversely proportional to the dipole strength. This modulation should be observable for current experimental parameters in 87Rb. Hence the much-sought supersolid state, with broken continuous translation invariance in one direction and broken global U(1) invariance, occurs generically as a metastable state in this system as a result of dipole interaction. The ferromagnetic state parallel to the applied magnetic field becomes striped in a finite system at strong dipolar coupling.The development of artificial gauge fields, that can mimic magnetic fields, in ultracold gases suggests that atomic realization of fractional quantum Hall physics will become experimentally practical in the near future. While it is known that bosons on lattices can support quantum Hall states, the universal edge excitations that provide the most likely experimental probe of the topological order have not been obtained. We find that the edge excitations of an interacting boson lattice model are surprisingly sensitive to interedge hybridization and edge-bulk mixing for some confining potentials. With properly chosen potentials and fluxes, the edge spectrum is surprisingly clear even for small systems with strong lattice effects such as bandwidth. Various fractional quantum Hall phases for bosons can be obtained, and the phases ν=1/2 and ν=2/3 have the edge spectra predicted by the chiral Luttinger liquid theory.Also, some of the traditional experimental techniques for detecting magnetic order and dynamics in solid state materials, like neutron scattering has had somewhat of a renaissance lately. In a recent experiment on CoNb2O6, Coldea et. al. found for the first time experimental evidence of the exceptional Lie algebra E8. The emergence of this symmetry was theoretically predicted long ago for the transverse quantum Ising chain in the presence of a weak longitudinal field. We consider an accurate microscopic model of CoNb2O6 incorporating additional couplings and calculate numerically the dynamical structure function using a recently developed matrix-product-state method. The excitation spectra show bound states characteristic of the weakly broken textrmE8 symmetry. We compare the observed bound state signatures in this model to those found in the transverse Ising chain in a longitudinal field and to experimental data.Finally, we investigate the ground state phase diagram of a related quantum spin chain, the S=2 XXZ chain with single-ion anisotropy. The interest in this system comes mainly from connecting the highly quantum mechanical spin-1 phase diagram with the classical S=∞ phase diagram. While most of these questions where believed to have been satisfactorily answered mainly with DMRG, some recent studies have questioned some of the conclusions. We use several of the recent advances within DMRG and perform a detailed analysis of the whole phase diagram. We extend the phase diagram by considering different types of single ion anisotropies which help us to answer two important questions: First we show that one can adiabatically move from the isotropic Heisenberg point to the so-called large-D phase with a continuous change of the Hamiltonian. Second, we can tune the model into a predicted intermediate phase which is equivalent to the topologically non-trivial spin-1 Haldane phase. Furthermore, we study the spin-3 XXZ chain to help explaining the development of the classical phase diagram.