Theoretical Analysis of Photonic Resonances in Spectroscopic Measurements of a Kerr Nonlinear Resonator

TL;DR

PR in Kerr parametric oscillators occurs at detunings where the direct drive coupling vanishes, challenging conventional resonance. The authors combine degenerate perturbation theory and GKSL time-domain simulations with experiments to reveal that higher-order processes induce Rabi oscillations between the ground state and degenerate excited states, with frequencies scaling as $p^{n/2}/\\chi^{(n/2)-1}$. Decoherence then damps these oscillations, producing PR as a two-stage phenomenon observed near degeneracy points in experiments and reproduced in simulations. This work clarifies the microscopic origin of PR in KPOs and informs the use of KPOs for quantum information processing by linking spectroscopic signatures to coherent multi-photon dynamics.

Abstract

The Kerr parametric oscillator (KPO) has recently attracted considerable attention from the perspective of its applications to quantum information processing, and understanding its properties is an important challenge. Spectroscopic measurements serve as an effective means of elucidating detailed information about the system, such as the energy-level structure and the transition matrix elements of the KPO. Conventional spectroscopy requires the drive frequency to match an energy spacing with a nonzero transition matrix element. In recent years, a phenomenon called photonic resonance (PR) has been theoretically predicted in KPO spectroscopy. Specifically, resonance occurs under the condition that the detuning is set to $n/2$ times the Kerr nonlinearity, where $n$ is a natural number. However, under this condition the transition matrix element vanishes, and thus the mechanism by which photonic resonance (PR) arises has remained unclear. In this work, we aim to elucidate the physical origin of PR observed in KPO spectroscopy. We first performed theoretical calculations and experiments of spectroscopic measurements, confirming that PR can indeed be observed and that the theoretical and experimental results are in qualitative agreement. We then carried out an analytical study under the assumption of an ideal noiseless environment. Our analysis revealed that, although the transition matrix element of the external field expressed in the system's energy eigenbasis is zero, higher-order perturbative effects induce Rabi oscillations between the ground and excited states. Furthermore, numerical simulations in a time domain including the effect of decoherence demonstrated that coherent oscillations decay, leading to the appearance of PR.

Figures (16)

• Figure 1: Analytical plots of the steady-state photon number $\log_{10}\langle a^{\dagger}a\rangle$ of the KPO. The parameters are set to dissipation rate $\kappa/2\pi = 10^{-6}\,\mathrm{MHz}$ and nonlinearity $\chi/2\pi = -18.729\,\mathrm{MHz}$.
• Figure 2: Comparison between analytical and experimental results of the output power of the KPO. For the numerical calculations, we adopt experimental parameters: external dissipation rate $\kappa_e/2\pi=0.47$ MHz, internal dissipation rate $\kappa_i/2\pi=0.26$ MHz, total dissipation rate $\kappa/2\pi = (\kappa_e +\kappa_i)/2\pi = 0.73\,\mathrm{MHz}$, and nonlinearity $\chi/2\pi = -18.729\,\mathrm{MHz}$.
• Figure 3: Schematic circuit diagram of the KPO. The parametric pump is injected from the pump line, which is inductively coupled to the SQUID in the KPO. When the internal photon number increases due to PR and related effects, it is detected as an output signal. The output signal is amplified by amplifiers located both inside and outside the refrigerator (not shown) and then measured using a spectrum analyzer.
• Figure 4: Numerical calculation of the eigenstates $|E_+\rangle$ and $|E_-\rangle$ of the Hamiltonian in Eq. \ref{['h_kpo']} for the $|0\rangle$--$|n\rangle$ degeneracy. The fidelities $|\langle \phi_+ | E_+ \rangle|^2$ and $|\langle \phi_- | E_- \rangle|^2$ are plotted as a function of the parametric drive amplitude $p/2\pi$ [MHz]. The nonlinearity is set to $\chi/2\pi = 18$ MHz.
• Figure 5: Numerical calculation of the energy levels $E_+$ and $E_-$ of the Hamiltonian in Eq. \ref{['h_kpo']} for the $|0\rangle$--$|n\rangle$ degeneracy. The energy difference $(E_+ - E_-)/2\pi$ [MHz] is plotted as a function of the parametric drive amplitude $p/2\pi$ [MHz]. The nonlinearity is set to $\chi/2\pi = 18$ MHz.