Fig. 2: Relative error bound and quantum Fisher information.

Relative error bound \(\xi\) for estimation of bath temperature \(T\) (in units of \({\omega }_{0}\)) by a harmonic-oscillator dynamically controlled quantum thermometer (DCQT) under sinusoidal modulation for the following bath spectra: nearly flat bath spectrum (magenta solid curve), sub-Ohmic bath spectrum with \(s=0.1,{\omega }_{c}=100\) (black dashed curve) and the same spectra in the absence of control (turquoise dotted curve). For low temperatures in the absence of control, \(\xi \to \infty\), thus making it impossible to measure low temperatures as shown. In contrast, our dynamical control scheme reduces \(\xi\) to finite values at low temperatures, for nearly flat, as well as sub-Ohmic bath spectra, thus showing the advantage of DCQT. Inset: Quantum Fisher Information (\({\mathcal{H}}\)) as a function of temperature \(T\) (in units of \({\omega }_{0}\)) for DCQT under sinusoidal modulation and in the absence of control, for the bath spectra in the main figure (same curve colors). DCQT increases the quantum Fisher information significantly at lower temperatures, giving rise to a peak at \(T\approx {T}_{-1}\) (magenta dot), in addition to the peak at \(T\approx {T}_{0}\) (turquoise dot). Here, the modulation amplitude \(\mu =0.2,{\omega }_{0}=1,\Delta =0.9\) (see Eq. (6)), \(m=0,\pm 1,\pm 2,\pm 3\) and the number of measurements \({\mathcal{M}}=1\). The thermalization time \(\sim {\gamma }^{-1}\) is assumed to be long enough such that the secular approximation is valid.