Probing excited-state dynamics of transmon ionization

TL;DR

This work investigates measurement-induced transmon ionization in dispersive readout by exploiting high-$E_J/E_C$ transmons that host up to 10 detectable energy levels. Through a combination of semiclassical driven-transmon dynamics and Floquet analysis, the authors identify a critical photon number at which ionization occurs, determine the post-ionization states (notably $ig|7ig>$), and demonstrate that the process behaves like a Landau-Zener transition controlled by pulse shaping. The experimental results closely match theoretical predictions, validating the driven-transmon and Floquet frameworks and outlining strategies to mitigate ionization for robust, high-fidelity QND readout. The study also highlights the role of Josephson harmonics and nonlinear Kerr effects in accurately predicting resonance conditions.

Abstract

The fidelity and quantum nondemolition character of the dispersive readout in circuit QED are limited by unwanted transitions to highly excited states at specific photon numbers in the readout resonator. This observation can be explained by multiphoton resonances between computational states and highly excited states in strongly driven nonlinear systems, analogous to multiphoton ionization in atoms and molecules. In this work, we utilize the multilevel nature of high-$E_J/E_C$ transmons to probe the excited-state dynamics induced by strong drives during readout. With up to 10 resolvable states, we quantify the critical photon number of ionization, the resulting state after ionization, and the fraction of the population transferred to highly excited states. Moreover, using pulse-shaping to control the photon number in the readout resonator in the high-power regime, we tune the adiabaticity of the transition and verify that transmon ionization is a Landau-Zener-type transition. Our experimental results agree well with the theoretical prediction from a semiclassical driven transmon model and may guide future exploration of strongly driven nonlinear oscillators.

Figures (10)

• Figure 1: Transmon ionization concepts. (a) The potentials and eigenstates of transmons. Here, we show examples of two transmons with $\omega_{01}/2\pi=\qty{5}{\GHz}$ and $E_J/E_C=100$ (left) or $E_J/E_C=270$ (right). The target state of ionization for a typical transmon is often close to (or even above) the top of its potential. High-$E_J/E_C$ transmons have a deeper potential and confine more energy levels, which makes the highly excited states accessible during the transmon ionization. The level diagram depicts a multiphoton resonance. In this example, the energy levels of states $\ket{1}$ and $\ket{7}$ are ac-Stark shifted by the drive to reach the resonance condition $\tilde{\omega}_7-\tilde{\omega}_1=n\omega_d$ at a certain drive power, with $\omega_d$ the drive frequency and $n$ the number of absorbed photons. Typically, $n > 1$. (b) Circuit diagrams. When the transmon is in one of its eigenstates, a readout pulse with frequency $\omega_d$ and amplitude $\varepsilon(t)$ creates a coherent state in the resonator. This coherent state can be effectively modeled as a classical drive applied directly to the transmon, which can induce transitions between transmon states. This driven transmon model is used for the numerical simulations in this work; see \ref{['eq:driven_harmonic_transmon_hamiltonian']}.
• Figure 2: Transmon ionization experiments. (a) Pulse sequence for the ionization experiments. The transmon is prepared in one of its eigenstates $\ket{j}$ and then evolves under a 2.2 stimulation drive on the resonator. After a 10 ring-down, a weak multitone readout pulse is applied to measure the transmon populations. At low stimulation power, an optional spectroscopy pulse can be used to probe the mean photon number $\bar{n}_r$. (b) The measured $\bar{n}_r(t)$ for an experiment with $\bar{n}_{r, \rm{max}} \sim 150$. (c) Populations of the transmon under different stimulation amplitudes when it is prepared in $\ket{1}$. The vertical dashed line marks the critical photon number at which the $\ket{1} \leftrightarrow \ket{7}$ transition can happen. (d) Populations of the transmon under different stimulation amplitudes when initially prepared in $\ket{7}$. The "deionization" shows the same critical photon number as the upward ionization.
• Figure 3: Comparisons between experiments and numerical simulations. (a-b) Transmon ionization associated with the $\ket{0} \leftrightarrow \ket{6}$ transition. Two different models, the conventional transmon model ($E_{J1}$, dashed lines) and the Josephson harmonics model ($E_{J8}$, solid lines), are used in simulations. We show the population of the qubit subspace $P_{\leq 1}$ and the populations of the higher excited states $P_{\geq 2}$. (c) Normalized Floquet quasienergies $\epsilon_j/\omega_d$ for each transmon branch when $\omega_d=\omega_{r, \ket{0}}$. The $\ket{0_f}$ branch has an avoided crossing with $\ket{6_f}$, which is also coupled to the higher-excited branch $\ket{19_f}$. (d-e) Similar to (a-b) but for the $\ket{1} \leftrightarrow \ket{7}$ transition. (f) Normalized Floquet quasienergies $\epsilon_j/\omega_d$ for each transmon branch when $\omega_d=\omega_{r, \ket{1}}$. The $\ket{1_f}$ branch has an avoided crossing with the $\ket{18_f}$ branch. The final state found in the dynamical simulations and the experiment is $\ket{7}$ instead of $\ket{18}$. This is due to a weak avoided crossing at lower photon number, at which most of the population in $\ket{18_f}$ is diabatically transferred to $\ket{7_f}$ during ramp-down. We take $n_g=0$ in (c) and (f).
• Figure 4: Landau-Zener transitions. (a) The measured photon numbers (red dots) of the steady-state sequence (top) and Landau-Zener sequence (bottom) for different steady-state times $t_s$. The data agree well with the numerical prediction (red lines), which uses parameters extracted from independent measurements. The transmon is prepared in $\ket{0}$ at the beginning of the sequence. The insets show the envelopes of the shaped stimulation pulses, each of which includes three segments: ramp-up, steady state, and ramp-down. In the Landau-Zener sequence, the amplitude during $t_s$ is intentionally increased above the amplitude used for the steady-state sequence to drive the resonator from $\bar{n}_{r, i}$ to $\bar{n}_{r, f}$. The pulse is followed by a 6 ring-down and an end-sequence measurement. We show three sequences with different Landau-Zener speeds identified by different symbols in the bottom panel. (b) The measured population of $\ket{0}$ from the end-sequence measurement for different steady-state durations $t_s$ and photon numbers $\bar{n}_{r, s}$. We observe a critical photon number for transmon ionization at around 1500 photons (see inset). Above 3000 photons, more resonances appear, while our pulse-shaping method fails to stabilize the photon number due to the higher-order nonlinearities of the resonator. (c) The measured populations of Landau-Zener experiments with different Landau-Zener speeds. The slope $d\bar{n}_r(t)/dt$ is controlled by changing the duration $t_s$ with fixed $\bar{n}_{r, i}=1300$, $\bar{n}_{r, f}=1700$ (circles), or by changing the difference $\bar{n}_{r, f}-\bar{n}_{r, i}$ with fixed $t_s=\qty{10}{\us}$. The horizontal gray dashed line shows the remaining ground-state population $P_0$ measured in the steady-state experiment, which has $\bar{n}_{r, f}-\bar{n}_{r, i} = 0$. The gray shaded area shows the range of theoretical predictions for 51 evenly spaced values of offset charge $n_g$. An adiabatic process results in more ionized population.
• Figure 5: Measured $\bar{n}_{r, \rm{max}}$ and the extrapolation results for initially prepared states $\ket{0}$, $\ket{1}$, $\ket{6}$, and $\ket{7}$.