Control and Tomography of a Three Level Superconducting Artificial Atom

A number of superconducting qubits, such as the transmon or the phase qubit, have an energy level structure with small anharmonicity. This allows for convenient access of higher excited states with similar frequencies. However, special care has to be taken to avoid unwanted higher-level populations when using short control pulses. Here we demonstrate the preparation of arbitrary three level superposition states using optimal control techniques in a transmon. Performing dispersive readout, we extract the populations of all three levels of the qutrit and study the coherence of its excited states. Finally we demonstrate full quantum state tomography of the prepared qutrit states and evaluate the fidelities of a set of states, finding on average 95%.

Figure 1

(a) Calculated in-phase transmission through the resonator for transmon states $j=0$, 1, 2. The dashed curve indicates the bare resonator response. The vertical (blue) arrow indicates the detuning ${\Delta }_{rm}$ of the measurement tone. (b) Pulsed I quadrature measurement responses for prepared states $|0⟩$, $|1⟩$, and $|2⟩$. The solid lines indicate least-squares fits to a Cavity-Bloch equation based model. (c) Measured dispersive shifts $s_j$ versus transmon transition frequency ${\omega }_{01}$. The solid lines are calculated within the linear dispersive approximation of Eq. (1).

Figure 2

(a) Measured $Q$ quadrature of the resonator transmission versus time and measurement detuning for a preparation of the transmon in state $|2⟩$. (b) Calculation based on Cavity-Bloch equations.

Figure 3

Reconstructed transmon populations in a Ramsey oscillations experiment on the 1–2 transition using a 5 MHz detuned drive field. For a given pulse delay $|0⟩$ [(blue) dots], $|1⟩$ [(red) diamonds] and $|2⟩$ [(green) squares] populations are extracted from a time-resolved averaged measurement trace. The lines are calculated using Bloch equations.

Figure 4

Reconstructed states $|{\Psi }_a⟩=1/\sqrt{2}(|1⟩-|2⟩)$ and $|{\Psi }_b⟩=1/\sqrt{3}(|0⟩+i|1⟩-|2⟩)$ represented by the expectation values of the Gell-Mann matrices ${\lambda }_i$. ${\lambda }_{1,4,6}={\sigma }_x^{\{0,1\},\{0,2\},\{1,2\}}$ and ${\lambda }_{2,5,7}={\sigma }_y^{\{0,1\},\{0,2\},\{1,2\}}$ [solid (red) bars] correspond to the Pauli matrices ${\sigma }_x$ and ${\sigma }_y$, i.e., the coherences, in the $\{|0⟩,|1⟩\}$, $\{|0⟩,|2⟩\}$ and $\{|1⟩,|2⟩\}$ subspace, respectively. The diagonal matrices [dotted (blue) bars] ${\lambda }_3={\sigma }_z^{\{0,1\}}$ and ${\lambda }_8=({\sigma }_z^{\{0,1\}}+2{\sigma }_z^{\{1,2\}})/\sqrt{3}$ relate to the population of the eigenstates and ${\lambda }_0$ denotes the identity. The measured values are depicted as (colored) bars while the expected outcomes for the perfectly prepared state are shown as (blue) outlines.