Supersolidity in a dipolar quantum gas

Ultracold quantum gases have nowadays become an invaluable tool in the study of quantum many-body problems. The high level of experimental control available on these systems and well established theoretical tools make ultracold quantum gases ideal platforms for quantum simulations of other systems currently inaccessible in experiments as well as for studies of fundamental properties of matter in the quantum degenerate regime. A key manifestation of quantum degeneracy in samples of ultracold bosonic neutral atoms is the formation of a Bose-Einstein condensate (BEC), a peculiar state of matter in which a macroscopic number of atoms occupy the same single-particle state. Bose-Einstein condensation occurs in extremely rarefied gases of bosonic atoms at temperatures around the nanoKelvin. At such temperatures, the equilibrium state of all known elements (except for helium) in ordinary conditions of density and pressure would be the solid phase. To obtain a BEC it is thus necessary to consider very dilute samples with a density of the order of 1014-1015 atoms/cm3, around eight orders of magnitude smaller then the density of ordinary matter. At such densities, the three-body recombination mechanisms responsible for the formation of molecules, that cluster to form solids, are suppressed. However, despite the extreme diluteness, two-body inter-atomic interactions play a prominent role in determining the physical properties of these systems. In the temperature and density regimes typical of BECs, the theoretical description of the system can be greatly simplified by noticing that the low-energy scattering properties of the real, generally involved, inter-atomic potential, can be perfectly reproduced by a simpler pseudo-potential, usually of the form of an isotropic contact repulsion, and described by a single parameter, the s-wave scattering length. Such parameter can even be tuned, in experiments, via the so-called Feschbach resonances. Despite its simplicity, this zero-range, isotropic interaction is responsible for an enormous variety of physical effects characterizing atomic BECs. This fact stimulated, over the last twenty years, the research of different possible types of interactions, that can eventually lead to the formation of new and exotic phases of matter. In this quest, the dipole-dipole interaction attracted great attention for different reasons. First, there are several experimental techniques to efficiently trap and cool atoms (or molecules) possessing a strong dipole moment. This led, for example, to the experimental realization of BECs of Chromium, Dysprosium and Erbium, which have, in the hyperfine state trapped for condensation, a magnetic dipole moment around ten times larger then the one typical of the particles in a BEC of alkali atoms. Moreover, since the dipole-dipole interaction is anisotropic and long-ranged, its low-energy scattering properties cannot be described by a simple short-range isotropic pseudo-potential. As a consequence, dipolar BECs show unique observable properties. The partially attractive nature of the dipole-dipole interaction can make a dipolar BEC unstable against collapse, similarly to the case of an ordinary (non-dipolar) BEC with negative scattering length. This happens, in particular, if a sample of magnetic atoms, polarized along a certain direction by some magnetic field, is not confined enough along such direction (for example via a harmonic potential). However, differently from ordinary BECs, where the collapse of the system is followed by a rapid loss of atoms and the destruction of the condensed phase, in the dipolar case such instability is followed by the formation of self-bound, (relatively) high density liquid-like droplets. If the geometry of the confinement potential allows it, the droplets spontaneously arrange into a regular, periodic configuration, in a sort of "droplet crystal". Moreover, by fine-tuning the interaction parameters, it is possible to achieve global phase coherence between these droplets. The spatially modulated, phase coherent system that forms is known as supersolid, and is a very peculiar system showing simultaneously the properties of a crystal and a superfluid. Ordinary mean-field theory, so successful in describing the vast phenomenology of ordinary BECs, fails in predicting the existence of the exotic phases of supersolids, quantum droplets and droplet crystals in a dipolar quantum gas. The state of the art description of dipolar BECs in such conditions is instead based on quantum fluctuations, taking into account the local density approximation of the first-order beyond-mean-field correction of the ground state energy of the system. This correction, known as the Lee-Huang-Yang correction, results in a repulsive energy term that balances the mean-field attraction at the relatively high densities that characterize the collapsing state. Using state-of-the-art simulation techniques, in this thesis I study the behavior of a dipolar Bose gas confined in a variety of trapping configurations, focusing on ground-state properties, elementary excitations, and the dynamical behavior under several kinds of external perturbations, focusing in particular on the supersolid phase. After reviewing the basic theory of dipolar Bose gases, setting the theoretical background, and describing the numerical techniques used, I first study the behavior of the dipolar Bose gas in an ideal situation, namely when the gas is confined in a harmonic trap along the polarization direction of the dipoles as well as one of the orthogonal directions. Along the unconfined direction, instead, I set periodic boundary conditions, in order to simulate the geometry of a ring. I study in particular the phase diagram of the system, focusing on how the ground state evolves from a superfluid, homogeneous along the ring, to the supersolid regime, and eventually to an array of independent droplets, by tuning a single interaction parameter, namely the s-wave scattering length. The superfluid phase is here characterized by the occurrence of a roton minimum in the energy-momentum dispersion relation. The energy of the roton, called roton gap, decreases when the s-wave scattering length of the system is decreased and the dipole-dipole interaction becomes the dominant interaction mechanism. When the roton minimum touches the zero-energy axis, the superfluid system is not stable anymore against mechanical collapse. The system thus tend to form denser clusters of atoms, regularly arranged in an equally-spaced array of droplets, whose relative distance is fixed by the inverse of the roton momentum. Such droplets are stabilized by quantum fluctuations, which enters in the energy functional of the system via the Lee-Huang-Yang correction. The density profiles of these droplets maintain a finite overlap if the scattering length is not too small. The phase characterized by overlapping, dense droplets of dipolar atoms is called supersolid. The main signatures of supersolid behavior, which in the thesis are shown to occur in this system, are 1. The occurrence of two Goldstone modes, associated with the two symmetries spontaneously broken in the supersolid, namely the symmetry for continuous translations, which is broken in favor of a discrete one, and the U(1) symmetry associated with Bose-Einstein condensation. 2. The manifestation of Non-Classical Rotational Inertia, due to the partially superfluid character of the system. Simply speaking, since the system behaves only partially as a superfluid, any rotational perturbation drags only the non superfluid part of the system. Hence, any measurement of the moment of inertia would give a value which is smaller then the one of a classical system with the same density distribution. Having studied the behavior of the dipolar Bose gas in a ring trap, I move on to explore possible manifestations of supersolid behavior in a fully trapped configuration, namely when the system is confined in an elongated (cigar-shaped) harmonic trap, with the long axis orthogonal to the polarization direction. Part of the results obtained in the three-dimensional harmonic trap have been compared with the first available experiments. The two key signatures of supersolid behavior, namely the occurrence of two Goldstone modes and Non-Classical Rotational Inertia, can be detected, in this case, by studying the low-energy collective oscillations of the system. First, a behavior equivalent to the one of the two Goldstone modes predicted in the ring trap, can be found in the axial compressional oscillations of the harmonically trapped system, which bifurcate at the superfluid-supersolid phase transition. When the system is driven through the supersolid-independent droplet transition, the lower-energy mode, associated with phase coherence, tends to disappear, while the higher energy mode, associated with lattice excitations, tends to assume a constant frequency. This behavior is specular to the one of the two Goldstone modes in the ideal system, and thus signal the presence of supersolidity in the trapped system. Important experimental confirmation of the predictions reported in the thesis have already been found. Instead, as shown in the thesis, a key manifestation non-classical inertia in a trapped dipolar supersolid can be found by studying the rotational oscillation mode known as scissors mode, whose frequency is directly related to the value of the moment of inertia (similar to the frequency of oscillation of a torsional pendulum for a classical system). Studying the behavior of the frequency of the scissors mode across the superfluid-supersolid-independent droplets phase transitions, I demonstrate the actual occurrence of non-classical inertia in a harmonically trapped dipolar supersolid. Another key manifestation of superfluidity in general many-body systems is given by the occurrence of quantized vortices, which I study in the case of the trapped dipolar Bose gas in a harmonic trap which is isotropic in the plan orthogonal to the polarization direction. I study in particular the size of the core of the vortex as function of the interaction parameters, showing that, in the superfluid phase, it increases as the superfluid-supersolid phase transition is approached. Then, in the supersolid phase, I show that quantized vortices settle in the interstices between the density peaks, and their size and even their shape are fixed respectively by the droplet distance and the shape of the lattice cell. I also study the critical frequency for the vortex nucleation under a rotating quadrupolar deformation of the trap, showing that it is related to the frequency of the lower-energy quadrupole mode, associated with the partial superfluid character of the system. In fact, in this configuration, the quadrupole mode splits into three modes, two of which can be associated to lattice excitations, and one to superfluid excitations. I find that the critical rotational frequency for vortex nucleation is related to the lower frequency quadrupole mode only, i.e. the one related to the superfluid character of the system. In ordinary BECs, when many vortices nucleates, they typically tend to arrange in a trinagular lattice. In a supersolid, however, vortices do not form on top of a uniform superfluid background, but rather on the background of the supersolid lattice, which is itself typically triangular. I thus show that the lattice formed by the vortices in the supersolid lattice is not triangular, but rather hexagonal, since the vortices settle in the interstices between the density peaks. Finally, I show that all these features can be observed in an expansion experiment. In the last part of the thesis, I study the behavior of the dipolar Bose gas confined by hard walls. In particular, I investigate the novel density distributions, with special focus on the effects of supersolidity. Differently from the case of harmonic trapping, in this case, the ground state density shows a strong depletion in the bulk region and an accumulation of atoms near the walls, well separated from the bulk, as a consequence of the competition between the attractive and the repulsive nature of the dipolar force. In a quasi two-dimensional geometry characterized by cylindrical box trapping, the consequence is that the superfluid accumulating along the walls forms spontaneously a ring shape, showing eventually also supersolidity. For sufficiently large values of the atom density, also the bulk region can exhibit supersolidity, the resulting geometry reflecting the symmetry of the confining potential even for large systems.