Direct Wigner Tomography of a Superconducting Anharmonic Oscillator

The analysis of wave-packet dynamics may be greatly simplified when viewed in phase space. While harmonic oscillators are often used as a convenient platform to study wave packets, arbitrary state preparation in these systems is more challenging. Here, we demonstrate a direct measurement of the Wigner distribution of complex photon states in an anharmonic oscillator—a superconducting phase circuit, biased in the small anharmonicity regime. We apply our method on nondispersive wave packets to explicitly show phase locking in states prepared by a frequency chirp. This method requires a simple calibration, and is easily applicable in our system out to the fifth level.

Figure 1

Experimental techniques. (a) Measurement of the occupation probabilities using short pulses in bias $I_{\mathrm{meas}}$ that cause selective tunneling of states $n<k$ which are detected later with an on-chip SQUID. (b) Time sequence of the Wigner tomography measurement of Fock-type superpositions. (c) Tunneling measurements of optimized versus nonoptimized states at small anharmonicity as a function of measurement pulse amplitude. For the optimized states we used pulse sequences of length 20, 20, and 12 ns for the first, second, and third excited levels, respectively.

Figure 2

Tomographic pulse. Occupation probabilities for levels $n\le 12$, as a function of $|\alpha |$ in the experiment (a) and theory (b). (c), (d) Histograms along the dashed lines shown in (a) and (b).

Figure 3

Superposition of Fock-type states. (a) Wigner tomography of genetically optimized superposition of Fock-type states and (b) density matrices, extracted from the measurements shown in (a).

Figure 4

Dynamics of a phase-locked wave packet. (a) Wigner tomography during a chirp ($T=20\mathrm{ns}$, $\Omega /2\pi =66\mathrm{MHz}$, $f_{\mathrm{in}}-f_{01}=320\mathrm{MHz}$, $f_{\mathrm{fin}}-f_{01}=-50\mathrm{MHz}$) and (b) during drive-free evolution. (c), (d) Extracted density matrices for (a) and (b), respectively.

Figure 5

State purity. Extracted state purity $\wp$ from experiments (red circles) and simulation (solid line) during chirp and decay. The simulation includes the effect of the decay and dephasing and agrees with the measured decay time.