Coherence and Decay of Higher Energy Levels of a Superconducting Transmon Qubit

We present measurements of coherence and successive decay dynamics of higher energy levels of a superconducting transmon qubit. By applying consecutive $\pi$ pulses for each sequential transition frequency, we excite the qubit from the ground state up to its fourth excited level and characterize the decay and coherence of each state. We find the decay to proceed mainly sequentially, with relaxation times in excess of $20\mu \mathrm{s}$ for all transitions. We also provide a direct measurement of the charge dispersion of these levels by analyzing beating patterns in Ramsey fringes. The results demonstrate the feasibility of using higher levels in transmon qubits for encoding quantum information.

Figure 1

(a) Schematic of the physical transmon qubit (not to scale) housed in the 3D cavity. The cross represents the Josephson junction, situated between the two junction electrodes forming the capacitor. (b) Simulated energy spectrum of the transmon with parameters $E_J/E_C=58$, where $U[\delta ]$ is the Josephson potential. (c) Rabi oscillations between state $|0⟩$ and $|1⟩$ under a Rabi drive tone of varying duration at $f_{01}$. The white dashed line indicates the position of the first $\pi$ pulse at ${\pi }_{01}=40\mathrm{ns}$. (d) Rabi oscillations between $|1⟩$ and $|2⟩$ with ${\pi }_{12}=29\mathrm{ns}$, obtained by adding a Rabi drive tone at $f_{12}$ after initializing state $|1⟩$ by applying a ${\pi }_{01}$ pulse. (e) Rabi oscillations on each successive qubit transition up to state $|4⟩$ using the depopulation readout method. The corresponding excitation pulse sequence and respective depopulation sequence are shown for each Rabi drive. The solid lines are best-fit curves allowing the extraction of the Rabi frequencies ${\mathrm{\Omega }}_{ij}$ as ${\mathrm{\Omega }}_{01}=8.45\mathrm{MHz}$, ${\mathrm{\Omega }}_{12}=10.3\mathrm{MHz}$, ${\mathrm{\Omega }}_{23}=13.0\mathrm{MHz}$, and ${\mathrm{\Omega }}_{34}=15.6\mathrm{MHz}$, respectively.

Figure 2

(a)–(c) Population decay traces of the qubit states up to $|3⟩$, obtained by varying the time delay $\mathrm{\Delta }t$ before the depopulation sequence. The solid lines are state occupations from the multilevel decay model taking into account all decay channels. (d) The sequential decay rates ${\mathrm{\Gamma }}_{i,i-1}$ (green dots) for increasing energy state $i$, showing the roughly linear dependance (solid line). (e),(f) Zoom of decay curves in panels (b) and (c), respectively, showing the model with all transition rates allowed (solid line) compared to the model with only neighboring transitions allowed (dashed line). The extracted decay times are listed in Table 1.

Figure 3

(a) Ramsey oscillations experiment on each subsequent energy transition up to state $|4⟩$. (b) The power spectral density (PSD), obtained from the discrete Fourier transforms of the corresponding Ramsey fringes. The Ramsey pulse sequence (top) corresponds to the fourth-row panel, representing a Ramsey sequence on state $|4⟩$, with the black $\pi /2$ pulses representing the ${\pi }_{34}/2$ pulses performed at frequency $f_{34}$ to bring the transmon into the superposition state $(|3⟩+|4⟩)/\sqrt{2}$ before allowing the free evolution time $\mathrm{\Delta }t$.

Figure 4

(a) Calculated change in transition frequency ${\epsilon }_{ij}(n_g)$ as a function of the effective offset charge $n_g$, expressing the charge dispersion of each transition. (b) Zoom of the lowest two transitions 0-1 and 1-2.