Fig. 2: Quench phase shift (QPS) Φq(tf) of a sensor qubit coupled to a Gaussian quantum environment, acquired during a Hahn-echo sequence, as a function of protocol time tf.
From: Intrinsic and induced quantum quenches for enhancing qubit-based quantum noise spectroscopy
We assume the environment has spectral function which behaves as a power law at low frequencies, i.e., \({{{{{{{\mathcal{J}}}}}}}}[\omega ]=(\alpha /\pi ){\omega }_{{{{{{{{\rm{c}}}}}}}}}{\left(\omega /{\omega }_{{{{{{{{\rm{c}}}}}}}}}\right)}^{s}{e}^{-{\left(\omega /{\omega }_{{{{{{{{\rm{c}}}}}}}}}\right)}^{2}}\). Curves correspond to different power laws: Ohmic (s = 1, dark blue curve), sub-Ohmic (s = 1/2, purple curve), and super-Ohmic (s = 3/2, dark green curve). We see that the QPS is extremely sensitive to the spectral function power law s. Light-colored lines depict the asymptotic long-time dependence of the QPS \({{{\Phi }}}_{{{{{{{{\rm{q}}}}}}}}}({t}_{f}) \sim {t}_{f}^{1-s}\) (see also Eq. (29)), which shows excellent agreement with the exact results in the long-time regime \({t}_{f}\gg {\omega }_{{{{{{{{\rm{c}}}}}}}}}^{-1}\), as expected. Note that for Gaussian, bosonic environments, the QPS is independent of temperature.
