Tunneling spectroscopy using a probe qubit

We describe a quantum tunneling spectroscopy technique that requires only low-bandwidth control. The method involves coupling a probe qubit to the system under study to create a localized probe state. The energy of the probe state is then scanned with respect to the unperturbed energy levels of the probed system. Incoherent tunneling transitions that flip the state of the probe qubit occur when the energy bias of the probe is close to an eigenenergy of the probed system. Monitoring these transitions allows the reconstruction of the probed system eigenspectrum. We demonstrate this method on an rf SQUID flux qubit.

Figure 1

Energy level diagram for the $Q+P$ qubit system described by the Hamiltonian in Eq. (4) at ${\epsilon }_Q=0$ and ${\epsilon }_{\text{comp}}=-2J$. The system separates into two distinct manifolds, ${\mathcal{M}}_{\downarrow }$ and ${\mathcal{M}}_{\uparrow }$, that differ in the orientation of the probe qubit $P$. Spinor notation indicates state of qubits $Q$ and $P$ on the left and right, respectively. Allowing for weak tunneling in qubit $P$ facilitates incoherent tunneling processes ${\Gamma }_{01}$ and ${\Gamma }_{02}$ between the manifolds. QTS proceeds by initializing the system in the lowest energy state of ${\mathcal{M}}_{\downarrow }$and finding values of ${\epsilon }_P$ for which transition rate $\Gamma$ peaks due to resonant processes ${\Gamma }_{01}$ and ${\Gamma }_{02}$.

Figure 2

(a) QTS tunneling rate data versus ${\epsilon }_p$ for ${\Phi }_Q^x=0.1$m${\Phi }_0$ and ${\Phi }_Q^{ccjj}=0.637$. Peaks in $\Gamma$ are readily associated with the processes ${\Gamma }_{01}$ and ${\Gamma }_{02}$ denoted in Fig. 1. The centers of the peaks are found by fitting to a sum of two Gaussian peaks. (b) Qubit $Q$ energy spectra obtained by QTS at control bias values ${\Phi }_Q^{ccjj}/{\Phi }_0=0.640$, (c) ${\Phi }_Q^{ccjj}/{\Phi }_0=0.637$. In both plots, the ordinate is the probe bias energy ${\epsilon }_P=2\left|I_P^p\right|{\Phi }_P^x$ and the abscissa is the flux bias ${\Phi }_Q^x\propto {\epsilon }_Q$ applied to qubit $Q$. The gray scale indicates the transition rate $\Gamma$ in the $({\Phi }_Q^x,{\epsilon }_P)$ plane. White circles denote the centers of peaks in $\Gamma$ found by the Gaussian fits. Avoided crossings between two localized states, explicitly labeled in (b) as $|\uparrow \rangle$${}_Q$ and $|\downarrow \rangle$${}_Q$, are visible. (d) The difference in probe energy $\delta {\epsilon }_P$ between the two peak centers as a function of ${\Phi }_q^x$ for the data sets in (a) and (b). Results have been fitted to the dispersion of the Hamiltonian in Eq. (5) using ${\Delta }_Q$ and $|I_Q^p|$ as free parameters.

Figure 3

$I_Q^p\left({\Phi }_Q^{ccjj}\right)$ vs ${\Delta }_Q\left({\Phi }_Q^{ccjj}\right)$ from the $Q$ spectra for a range of settings of the control bias ${\Phi }_Q^{ccjj}$. The data points from the fits in Fig. 2d are labeled with arrows. The data also have horizontal error bars approximately the size of the symbols. Results have been fitted to the rf SQUID model from Ref. 14 (solid line) with the rf SQUID capacitance $C$ as the only free parameter.