Entanglement Spectroscopy of a Driven Solid-State Qubit and Its Detector

We study the asymptotic dynamics of a driven quantum two-level system coupled via a quantum detector to the environment. We find multiphoton resonances which are due to the entanglement of the qubit and the detector. Different regimes are studied by employing a perturbative Floquet-Born-Markov approach for the $\mathrm{\text{qubit}}+\mathrm{\text{detector}}$ system, as well as nonperturbative real-time path integral schemes for the driven spin-boson system. We find analytical results for the resonances, including the red and the blue sidebands. They agree well with those of exact ab initio calculations.

Figure 1

Schematic picture of the models we use. In (a) the system is a TSS coupled to a harmonic oscillator with the latter coupled to an Ohmic environment with spectral density $J_{\mathrm{O}\mathrm{h}\mathrm{m}}(\omega )$. In (b), the TSS is coupled to an environment with peaked spectral density $J_{\mathrm{e}\mathrm{f}\mathrm{f}}(\omega )$.

Figure 2

Left: quasienergy spectrum ${\epsilon }_{\alpha ,k}$ of the driven $\mathrm{T}\mathrm{S}\mathrm{S}+\mathrm{H}\mathrm{O}$ system vs dc bias ${\epsilon }_0$ (in units of $\Delta$). The quasienergies are defined up to an integer multiple of $\hslash \Omega$; i.e., ${\epsilon }_{\alpha ,k}={\epsilon }_{\alpha }+k\hslash \Omega$. Inset: zoom of an anticrossing. Right: $P_{\infty }$ exhibits sidebands corresponding to quasienergy level anticrossings. Parameters are $\Omega =10\Delta$, $s=4\Delta$, $g=0.4\Delta$, ${\Omega }_p=4\Delta$, $\kappa =0.014$, and $k_{B}T=0.1\hslash \Delta$.

Figure 3

The first blue sideband $P_{\infty }$ vs ${\epsilon }_0$ (in units of $\Delta$) at $\nu =\Omega -{\Omega }_p$. The solid lines are the analytical prediction (13) for (a) $g=0.05\Delta$, (b) $g=0.2\Delta$, and (c) $g=0.4\Delta$. The triangles are the results of a Floquet-Bloch-Redfield simulation, cf. Eq. (5), with one (upward triangles) and two (downward triangles) HO levels taken into account. The circles in (b) are the results from a QUAPI simulation with six HO levels. We choose $s=2\Delta$, $\Omega =10\Delta$, ${\Omega }_p=4\Delta$, $\kappa =0.014$, and $k_{B}T=0.1\hslash \Delta$.

Figure 4

$P_{\infty }$ vs ${\epsilon }_0$ (in units of $\Delta$). The solid line is the NIBA prediction, while the circles are from a QUAPI simulation with six HO levels ($g=3\Delta$, $s=4\Delta$, $\Omega =10\Delta$, $\kappa =0.014$, $k_{B}T=0.5\hslash \Delta$, and ${\Omega }_p=4\Delta$). Inset (a): NIBA result for $k_{B}T=2\hslash \Delta$. The arrow indicates the first red sideband at $\nu =\Omega +{\Omega }_p$. Inset (b): $Q^{\prime }(t)$ vs $t$ shows damped oscillations.