The description of an impurity atom in a Bose-Einstein condensate can be cast in the form of Fröhlich's polaron Hamiltonian, where the Bogoliubov excitations play the role of the phonons. An expression for the corresponding polaronic coupling strength is derived, relating the coupling strength to the scattering lengths, the trap size and the number of Bose condensed atoms. This allows to identify several approaches to reach the strong-coupling limit for the quantum gas polarons, whereas this limit was hitherto experimentally inaccessible in solids. We apply Feynman's path-integral method to calculate for all coupling strengths the polaronic shift in the free energy and the increase in the effective mass. The effect of temperature on these quantities is included in the description. We find similarities to the acoustic polaron results and indications of a transition between free polarons and self-trapped polarons. The prospects, based on the current theory, of investigating the polaron physics with ultracold gases are discussed for lithium atoms in a sodium condensate.

We demonstrate the existence of a collective excitation branch in the pair-breaking continuum of superfluid Fermi gases and BCS superconductors, as suggested by Littlewood and Varma in 1982. We analytically continue the RPA equation on the collective mode energy through its branch cut associated with the continuum, and obtain the full complex dispersion relation, including in the strong coupling regime. For ∆/µ > 1.210 (very close to unitarity in a superfluid Fermi gas), where ∆ is the order parameter and µ the chemical potential, the real part of the branch is wholly within the band gap [0, 2∆]. In the long wavelength limit, the branch varies quadratically with the wave number, with a complex effective mass that we compute analytically. This contradicts the result of Littlewood and Varma that prevailed so far.

We consider two large polaron systems that are described by a Fröhlich type of Hamiltonian, namely the Bose-Einstein condensate (BEC) polaron in the continuum and the acoustic polaron in a solid. We present ground-state energies of these two systems calculated with the Diagrammatic Monte Carlo (DiagMC) method and with a Feynman all-coupling approach. The DiagMC method evaluates up to very high order a diagrammatic series for the polaron's self-energy. The Feynman all-coupling approach is a variational method that has been used for a wide range of polaronic problems. For the acoustic and BEC polaron both methods provide remarkably similar non-renormalized ground-state energies that are obtained after introducing a finite momentum cutoff. For the renormalized groundstate energies of the BEC polaron, there are relatively large discrepancies between the DiagMC and the Feynman predictions. These differences can be attributed to the renormalization procedure for the contact interaction. k k k k k k k q k q k q pol 2 2 † † , † †Here, the ĉ k † (ĉ k ) are the creation (annihilation) operators of the charge carriers with band mass m and momentum k. The second term in the above Hamiltonian gives the energy of the phonons which carry the polarization. Thereby, the operator bˆk † (bˆk ) creates (annihilates) a phonon with wave vector k and energy ω k ( ). The last term in equation (1) denotes the interaction between the charge carrier and the phonons. A