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We study the problem of a single impurity of mass $M$ immersed in a Fermi seaof particles of mass $m$. The impurity and the fermions interact through as-wave narrow Feshbach resonance, so that the Feshbach length $R_*$ naturallyappears in the system. We use simple variational ansatz, limited to at most onepair of particle-hole excitations of the Fermi sea and we determine for thepolaronic and dimeronic branches the phase diagram between absolute groundstate, local minimum, thermodynamically unstable regions (with negativeeffective mass), and regions of complex energies (with negative imaginarypart). We also determine the closed channel population which is experimentallyaccessible. Finally we identify a non-trivial weakly attractive limit whereanalytical results can be obtained, in particular for the crossing pointbetween the polaronic and dimeronic energy branches.

An impurity in a Fermi sea on a narrow Feshbach resonance: A variational study of the polaronic and dimeronic branches