Quantum thermometry using the ac Stark shift within the Rabi model

A quantum two-level system coupled to a harmonic oscillator represents a ubiquitous physical system. New experiments in circuit QED and nanoelectromechanical systems (NEMS) achieve unprecedented coupling strength at large detuning between qubit and oscillator, thus requiring a theoretical treatment beyond the Jaynes-Cummings model. Here we present a new method for describing the qubit dynamics in this regime, based on an oscillator correlation function expansion of a non-Markovian master equation in the polaron frame. Our technique yields a new numerical method as well as a succinct approximate expression for the qubit dynamics. These expressions are valid in the experimentally interesting regime of strong coupling at low temperature. We obtain a new expression for the ac Stark shift and show that this enables practical and precise qubit thermometry of an oscillator.

Figure 1

Comparison of the single term approximation (red, dashed) and a numerically exact approach (blue, solid) for different coupling strengths. Uncoupled Rabi oscillations are also shown as a reference (green, dotted). Left: the population ${\rho }_{00}\left(t\right)$ in the time-domain. Right: the same data in the frequency domain. The full numerical solution was Fourier transformed using Matlab's FFT algorithm. Other parameters are $\omega =1$ GHz, $\epsilon =\Delta =100$ MHz, and $T=10$ mK.

Figure 2

Main panel: comparison of dynamics calculated from truncating Eq. (16) at $N_{\mathrm{MAX}}=\pm 10$ (red, dashed) and a numerically exact approach (blue, solid). Lower left: Fourier transform of the dynamics. Lower right: the numerical weight of the $n\mathrm{th}$ term in the series expansion of Eq. (16), showing there are still only two dominant frequencies at $n=0$ and $n=-1$. Parameters: $\omega =0.5$ GHz, $g=0.1$ GHz, $\epsilon =0$, $\Delta =0.5$ GHz, $T=1\mathrm{mK}$.

Figure 3

Demonstration of qubit thermometry: $T_{\mathrm{in}}$ is the temperature supplied to the numerical simulation of the system and $T_{\mathrm{out}}$ is the temperature that would be predicted by fitting oscillations with frequency Eq. (19) to it. The inner blue line is the data and outer red lines show the effect of a 10 kHz error in the frequency measurement; the gray dashed line serves as a guide to the eye. The lower inset shows the variation of the qubit frequency $\Omega$ with temperature. The upper inset shows the dependence of the absolute error in the prediction against the signal length (see text). Other parameters: $\omega =1$ GHz, $g=0.01$ GHz, $\epsilon =0$, $\Delta =100$ MHz.