Inter-transition interference in spectrum of Kerr parametric oscillators

TL;DR

The paper addresses the challenge of spectroscopic parameter extraction in Kerr parametric oscillators (KPOs) by incorporating inter-transition interference arising from near-degenerate energy levels and nonzero zero-frequency off-diagonal density-matrix elements. It extends prior reflection- spectroscopy theory with a modified method that couples inter-level transitions through $\rho^{\rm (F)}_{nm}[-\tilde{\omega}_{\rm in}]$ and includes off-diagonal coherence, yielding analytic expressions for two-transition interference and comprehensive conditions for when interference occurs. Numerical results for two-photon and four-photon KPOs demonstrate sharper, deeper spectral dips and pronounced differences from previous theory, particularly at small pump amplitudes where off-diagonal coherences are large. The framework is generalized to transmission measurements and provides tools for parameter estimation in a broader class of driven nonlinear quantum systems, with implications for KPO-based quantum information processing and spectroscopic diagnostics.

Abstract

We theoretically investigate reflection and transmission measurements of two-photon and four-photon Kerr parametric oscillators (KPOs), introducing interference effects between inter-level transitions. Due to the level degeneracy of a KPO, a probe field can be resonant with multiple inter-level transitions. We extend the previous theory of reflection measurements by incorporating the interaction between inter-level transitions and off-diagonal elements of the density matrix which had previously been neglected. We demonstrate that interference among these transitions substantially modifies the spectrum. We identify the conditions for the interference, as well as those under which the off-diagonal elements of the density matrix affect the spectrum. The theory is also generalized to transmission measurements, and is applicable to a broad class of systems beyond KPOs.

Figures (7)

• Figure 1: Schematic of (a) the reflection and (b) the transmission measurements. A KPO (blue circle) is attached to a transmission line (grey bar), where an incoming and an outgoing microwaves propagate. The dashed lines illustrate the coupling between the KPO and the transmission line. (c) Schematic energy diagram of a two-photon KPO showing the four relevant energy levels. The left panel corresponds to smaller pump amplitude, and the right panel corresponds to larger pump amplitude. The arrows indicate relevant transitions. The solid lines and the dashed lines represent energy levels with even and odd parity, respectively. In the left panel, the transition frequencies for $|\tilde{0}\rangle \rightarrow |\tilde{3}\rangle$ and $|\tilde{1}\rangle \rightarrow |\tilde{2}\rangle$ are different, and thus the interference effect is small. In contrast, the transition frequencies are approximately the same in the right panel, resulting in significant interference (indicated by an orange zigzag line).
• Figure 2: (a) Amplitude of the reflection coefficient as a function of $\omega_{\rm in}$ for a two-photon KPO in the stationary state. The purple solid (green dashed) curve corresponds to the modified (previous) theory. We used $p/2\pi=68$ MHz, $\Delta/2\pi=0$ MHz, $K/2\pi=17$ MHz, $\kappa_{\rm ex}/2\pi=1$ MHz, and $\kappa_{\rm int}/2\pi=0.45~$MHz. Panel (b) shows the transition frequencies between energy levels indicated by the vertical lines overlaid on the data presented in panel (a). The inset illustrates five relevant energy levels. Amplitude of the reflection coefficient as a function of $\omega_{\rm in}$ for the previous theory (c) and for the modified theory (d). The pump amplitude, $p$, ranges from $50~{\rm MHz}$ to $100~{\rm MHz}$ in steps of 10 MHz. The darker color of the curves denote larger pump amplitude. The other parameters are the same as panel (a). The factor $2p$ in the horizontal axis is introduced to cancel the shift of dips as $p$ is changed. (e) Amplitude of the reflection coefficient as a function of $\omega_{\rm in}$ for $p/2\pi=100$ MHz. The analytical result in equation (\ref{['Gamma_0312_7_24_25']}) overlaps with the numerical result.
• Figure 3: Amplitude of the reflection coefficient as a function of $\omega_{\rm in}$ and $p$ for a two-photon KPO in the stationary state. The parameters used are $K/2\pi=17$ MHz, $\Delta=1.5K$, $\kappa_{\rm ex}/2\pi=1$ MHz, and $\kappa_{\rm int}/2\pi=0.1~$MHz. Panels (a) and (b) correspond to the modified theory and the previous one, respectively. Panels (c) and (d) show the same data as panels (a) and (b), but over a shorter range of $p$. Panel (e) shows the frequencies of relevant transitions overlaid on the data presented in panel (d). Panel (f) presents the energy diagram of the system. Letter $\tilde{n}$ on the figure indicates the energy eigenstate $|\tilde{n}\rangle$.
• Figure 4: Amplitude of the density matrix element $|\rho^{\rm (F)}_{\tilde{0}\tilde{4}}[0]|(=|\rho^{\rm (F)}_{\tilde{4}\tilde{0}}[0]|)$ of the stationary state as a function of $p$. The parameters are the same as in figure \ref{['fourier_com_6_8_25']}. The solid line is shown as a guide to the eye.
• Figure 5: Energy diagram of a four-photon KPO as a function of $p$. The parameter set used is $K/2\pi=10$ MHz and $\Delta=1.5K$. Letter $\tilde{n}$ on the figure indicates the energy eigenstate $|\tilde{n}\rangle$.