Quantum and Classical Chirps in an Anharmonic Oscillator

We measure the state dynamics of a tunable anharmonic quantum system, the Josephson phase circuit, under the excitation of a frequency-chirped drive. At small anharmonicity, the state evolves like a wave packet—a characteristic response in classical oscillators; in this regime, we report exponentially enhanced lifetimes of highly excited states, held by the drive. At large anharmonicity, we observe sharp steps, corresponding to the excitation of discrete energy levels. The continuous transition between the two regimes is mapped by measuring the threshold of these two effects.

Figure 1

Operation and measurement of the Josephson phase circuit. (a) Schematics of the circuit and the potential energy at different operating biases. The potential shape and anharmonicity ${\beta }_r$ are set by the current source $I_b$, and the state inside the well is controlled by the microwave drive $I_{\mu \mathrm{w}}$. (b) State measurement. A short pulse $I_{\mathrm{meas}}$ is applied in the flux bias to selectively tunnel excited levels $n>k$. The average phase $\delta$ is then measured with an on-chip SQUID to detect tunneling events. To determine the occupation probabilities of all the $N$ levels, this process is repeated with a series of different $I_{\mathrm{meas}}$ amplitudes [18]. (c) Time sequence of the chirp experiment. The drive amplitude $\Omega$ is expressed in units of the Rabi frequency, measured on the first transition.

Figure 2

State dynamics during the chirp. (a) Dressed energies of the lowest levels in the rotating frame as a function of the drive frequency detuning $\Delta$ from the first transition $f_{01}$. As the chirp progresses (decreasing $\Delta$), for a sufficiently small chirp rate the state remains on the adiabatic branch (solid black line). (b) Measured occupation probability (color scale) as a function of time and level number in the ladder-climbing regime (${\beta }_r=0.023$, $\alpha /2\pi =2\mathrm{MHz}/\mathrm{ns}$, $\Omega /2\pi =27\mathrm{MHz}$) and (c) autoresonance regime (${\beta }_r=0.002$, $\alpha /2\pi =10\mathrm{MHz}/\mathrm{ns}$, $\Omega /2\pi =190\mathrm{MHz}$). The detuning scale in (a) and the time scale in (b) are bound by the start and the end of the chirp. Insets: Simulated Wigner distribution at different times along the chirp.

Figure 3

Decay of a wave packet. (a) Time sequence of the decay measurement after the chirp. (b) Measured occupation probability (color scale) as a function of level number and time after the chirp shown in Fig. 2c, with ${\Omega }_{\mathrm{hold}}/2\pi =190\mathrm{MHz}$ and (c) ${\Omega }_{\mathrm{hold}}=0$. The insets in (b) and (c) show the simulated Wigner plot at different times along the decay. (d) Measured locking probability (color scale) as a function of time and amplitude of the drive after the chirp, with contours corresponding to $P_{\mathrm{locked}}(t_{\mathrm{hold}},{\Omega }_{\mathrm{hold}})=0.5$, obtained from data, theory, and simulation. (e) Escape probability (color scale) as a function of measurement amplitude $I_{\mathrm{meas}}$ and drive amplitude $\Omega$ after a chirp, with $\alpha /2\pi =10\mathrm{MHz}/\mathrm{ns}$ and ${\beta }_r=0.0046$. To measure the locking probability, an intermediate $I_{\mathrm{meas}}$ is used (dashed line) at the end of the chirp.

Figure 4

Transition from autoresonance to ladder climbing. (a) Measured locking probability (color scale) as a function of the dimensionless chirp parameters $\Omega /\sqrt{\alpha }$ and $\beta /\sqrt{\alpha }$. The red and black lines are the theoretical thresholds for autoresonance (${\Omega }_{\mathrm{th}}^{\mathrm{ar}}=0.82{\alpha }^{3/4}{\beta }^{-1/2}$) and ladder climbing (${\Omega }_{\mathrm{th}}^{\mathrm{lc}}=0.8{\alpha }^{1/2}$), respectively [13]. The blue line ($\Omega =\beta$) marks the separation between the quantum and classical regimes [12]. (b) A simulation of the experiment shown in (a) with the same parameters, including the effects of decay and measurement at different ${\beta }_r$ [18].