Spectral Analysis of Loschmidt Echo in Nuclear Spin Baths

The revivals of the background spins coupled to a qubit constitute Loschmidt echo which plays a crucial role in the storage of quantum information within a solid-state matrix. Its characterization and quantum control are currently of wide interest. Here, we present the spectral analysis of Loschmidt echo in a nuclear spin bath. The model spin-Hamiltonian consists of both hyperfine and quadrupolar couplings together with nearest-neighbor two-body interactions. For this so-called central spin problem, we use the exact diagonalization for small spin systems, and for the larger ones we resort to the cluster correlation expansion technique. Predominantly through the hyperfine term, a feedback loop is formed which introduces nonlinearity and harmonics into the system. Our Fourier analysis of the Loschmidt echo sheds light on how the spectral profile is synthesized through various interactions. This provides an invaluable insight on the qubit decoherence and control of the nuclear spin reservoir.