The supersolid phase of a dipolar Bose-Einstein condensate has an intriguing excitation spectrum displaying a band structure. Here, the dressing of an impurity in a one-dimensional dipolar supersolid with the excitations of the supersolid is studied. The ground-state energy of the supersolid polaron is calculated using a variational path integral approach, which obtained accurate results for other polaron systems within the Bogoliubov and Fr\"ohlich approximations. A divergence is observed at the superfluid-supersolid phase transition. The polaron radius is also computed, showing that, as a function of impurity-atom interactions, the polaron can become localized to a single droplet, behaving like a small solid-state polaron.
Motivated by recent experimental evidence for spin supersolid states in triangular-lattice compounds, we numerically investigate the dynamical properties of magnetic field-induced phases in the spin-1/2 easy-axis triangular antiferromagnetic Heisenberg model in the presence of magnetic impurities. In both weak- and strong-field spin supersolid states, the gapless Goldstone mode at the
K points remains robust against impurities, which is a direct manifestation of spin superfluidity. By contrast, at the same impurity density, impurities induce a splitting of the magnon bands in the conventional magnetic state, the so-called up-up-down state. In addition, the finite superfluid stiffness probed by the twisted phase in the spin supersolid states is consistent with the excitation spectrum. We argue that the excitation spectrum with impurities provides direct spectroscopic evidence for dissipationless spin dynamics in the spin supersolid states, which is experimentally accessible via inelastic neutron scattering.
Bose-Einstein condensation is a macroscopic quantum phenomenon where cooling down a gas of bosonic particles to ultracold temperatures results in the macroscopic occupation of the lowest energy state. This new state of matter has been used as a quantum simulator of many-body phenomena and theories due to its extreme tunability and controllability. An important example of such a many-body system is the polaron which, in its most general form, describes an impurity interacting with a bosonic bath. In the case of ultracold Bose gases, the impurity interacts with and is dressed by the Bogoliubov excitations of the gas. A supersolid is a state of matter that exhibits both superfluidity and crystalline order, which are seemingly irreconcilable properties. It has recently been observed in dipolar ultracold Bose gases, where it arises from the long-range anisotropic dipole-dipole interactions, and also other systems. The breaking of both translational symmetry and gauge symmetry results in multiple excitation modes in the supersolid. In this thesis, the charged Bose polaron and an impurity in a dipolar ultracold Bose gas exhibiting supersolidity (the supersolid Bose polaron) are studied. The ground-state properties of the charged Bose polaron are discussed using a variational path-integral approach. Absorption spectroscopy is proposed as a probe of the charged Bose polaron, and the optical response is studied for weak impurity-boson interactions. Next, the one-dimensional dipolar supersolid Bose polaron is investigated where multiple approaches are used for the derivation of the ground-state properties. The study of the polaron problem with a supersolid background is of particular interest because of the rich excitation spectrum of the supersolid. Interesting phenomena, such as possible localization of the polaron to a single droplet for strong interactions, are observed. The absorption spectrum of a charged impurity in the supersolid is also derived, which, because of the different excitation modes of the supersolid, shows multiple peaks from which information such as the superfluid fraction may be obtained.