Robustness of longitudinal transmon readout to ionization

TL;DR

Addressing the challenge of multi-photon induced non-QND effects in circuit QED readout, the paper assesses the longitudinal transmon readout under the full cosine potential. By nonperturbatively analyzing the Hamiltonian with branch analysis and Schrieffer-Wolff transformations, the authors show that the longitudinal coupling $g_z$ is detuning-independent and that increasing detuning raises the ionization threshold $n_{\rm crit}$ without reducing the dispersive shift. They demonstrate fast, high-fidelity QND readout with $|\alpha_f|^2$ on the order of tens of photons, achieving assignment errors below $10^{-4}$ in tens of nanoseconds for unit efficiency and still under $50$ ns for realistic efficiency; they further show robustness to circuit disorder, junction asymmetry $d$, and gate charge $n_g$, with $n_{\rm crit}$ remaining well above practical readout photon numbers. Classical modeling indicates the longitudinal readout behaves as a parametrically driven nonlinear oscillator and is more resistant to chaos than the dispersive readout, supporting experimental viability.

Abstract

Multi-photon processes deteriorate the quantum non-demolition (QND) character of the dispersive readout in circuit QED, causing readout to lag behind single and two-qubit gates, in both speed and fidelity. Alternative methods such as the longitudinal readout have been proposed, however, it is unknown to what extent multi-photon processes hinder this approach. Here we investigate the QND character of the longitudinal readout of the transmon qubit. We show that the deleterious effects that arise due to multi-photon transitions can be heavily suppressed with detuning, owing to the fact that the longitudinal interaction strength is independent of the transmon-resonator detuning. We consider the effect of circuit disorder, the selection rules that act on the transmon, as well as the description of longitudinal readout in the classical limit of the transmon to show qualitatively that longitudinal readout is robust. We show that fast, high-fidelity QND readout of transmon qubits is possible with longitudinal coupling.

Figures (7)

• Figure 1: Circuit diagram for longitudinal readout of the transmon qubit. A flux tunable transmon (green) is inductively coupled to a resonator (blue). The symmetric $\hat{\varphi}_t = (\hat{\varphi}_1 + \hat{\varphi}_2)/2$ and antisymmetric $\hat{\varphi}_r = (\hat{\varphi}_1 - \hat{\varphi}_2)/2$ modes of the circuit correspond to the transmon mode and the resonator mode, respectively. The transmon and resonator modes can be driven by a symmetric (in-phase) and antisymmetric (out-of-phase) combinations of the two drives. In the text, we assume $C_1 = C_2$. See \ref{['app_sec:circuit_quantization']} for details.
• Figure 2: (a) Transmon population as a function of the average resonator population. The ground and excited state branches, as well as the branches they swap with are highlighted. The transmon parameters are $E_C / 2 \pi = 0.215$ GHz, $E_J / E_C = 110$, with junction asymmetry $d = 0$ and gate charge $n_g = 0$. The resonator frequency is $\omega_r / 2 \pi = 8.8$ GHz. We keep 16 states in the transmon, and 100 states in the resonator. The qubit-resonator detuning is defined as $\Delta = \omega_q - \omega_r$. (b) The energy modulo $\omega_r$ spectrum as a function of the average resonator population. Parameters are the same as in (a). The wrapping of the energies is due to the spectrum being a modulo of the resonator frequency, indicating that when two lines cross, the two states are energetically separated by an integer number of resonator photons. Avoided crossings align with branch swappings occuring in (a). (c) Transmon population as a function of average resonator population with all parameters the same as in (a) but with $\omega_r / 2 \pi = 10.5$ GHz. The branch swappings seen in (a) are now removed due to the higher detuning between the resonator and the transmon. (d) Energy modulo $\omega_r$ spectrum for the parameters of panel (c). Importantly, the dispersive shift $\chi_z$ is the same in (a) and (c) as it is independent of detuning.
• Figure 3: (a) Critical photon numbers as a function of $E_J/E_C$ and transmon-resonator detunings $\Delta = \omega_q - \omega_r$. The diagonal white line indicates a strong resonance between the first excited and the fifth excited state of the transmon that spans across all values of $E_J/E_C$. The blue line indicates $E_J/E_C = 100$, corresponding to the fixed $E_J / E_C$ ratio used in panels (b) and (c). (b) Ionization critical photon numbers for various detunings $\Delta / 2\pi$ and Josephson junction asymmetries $d$. (c) Ionization critical photon numbers as a function of detuning $\Delta / 2\pi$ and gate charge $n_g$. For finite asymmetry and non-zero gate charge, the $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry is broken. Despite this, we observe that the critical photon numbers, especially at high detunings remain large. $E_J/E_C = 100$ in panels (b) and (c)
• Figure 4: Readout assignment error as a function of integration time $\tau$ for drive strengths corresponding to $\alpha_f = 2, 3$, and $4$. The parameters are $E_J / E_C = 50$, $d = 0$, $n_g = 0$, $\omega_r / 2 \pi = 9.3$ GHz, $\kappa / 2 \pi = 17$ MHz and qubit relaxation is ignored. The dispersive shift is $\chi_z / 2 \pi \approx -8.66$ MHz and the qubit frequency $\omega_q/2\pi \approx 4.07$ GHz, corresponding to $\Delta / 2 \pi = -5.3$ GHz. Solid lines are results obtained from heterodyne readout simulations with the measurement efficiency set to $\eta = 1$. Dashed lines are computed using the analytical expression for the signal-to-noise ratio with efficiency set to $\eta = 0.5$.
• Figure 5: (a) Poincare section for a transmon with no drive. The green line indicates the separatrix. (b) The average trajectory deviation with increased drive amplitude. The average trajectory deviation is calculated by averaging over many trajectory deviations calculated from \ref{['eqn:chaos_trajectory_deviation']}. (c), (e), and (g) show the Poincare section for a transmon coupled via the longitudinal interaction with drive strengths corresponding to $1$, $9$, and $49$ photons, respectively. The blue and red orbits indicate the Bohr-Sommerfeld orbits for the ground and first excited state of the transmon, respectively. (d), (f), and (h) show Poincare sections for a transmon coupled dispersively to a resonator again with $1$, $9$, and $49$ photons, respectively. The blue and red orbits indicate the Bohr-Sommerfeld orbits for the ground and first excited state of the transmon, respectively. In panel (h), the regular region is too small to support a Bohr-Sommerfeld orbits. For both longitudinal and dispersive readout cases, $E_J / E_C$ of the transmon was set to 110. For longitudinal readout the asymmetry was set to $d = 0$.