Parametric excitation of a ferrimagnetic sphere resonator

TL;DR

The paper addresses multistability in finite quantum systems by studying a ferrimagnetic sphere resonator (FMSR) under parametric excitation, challenging the linear QM constraint on such behavior. It compares a nonlinear spontaneous disentanglement master equation (the rapid disentanglement model) with a Bosonization-based nonlinear spin model, testing both against FMSR measurements with parallel pumping. The results show that the disentanglement model qualitatively accounts for the observed instability and finite bistability region, while the Bosonization model predicts an unbounded bistability region not supported by the data, with the rapid disentanglement predictions aligning more closely with the experiments. This work assesses a nonlinear extension to quantum dynamics in a finite system and suggests spontaneous disentanglement as a viable mechanism for observed nonlinear spin phenomena, with potential implications for quantum measurement and magnonics; the framework is falsifiable and applicable to other finite-dimensional quantum systems. $|\,\omega_K\,| \ge \sqrt{3}\,\gamma_3$ and cubic steady-state relations for key observables underpin the analysis.$

Abstract

The response of a ferrimagnetic sphere resonator to an externally applied parametric excitation is experimentally studied. Measurement results are compared with predictions derived from a theoretical model, which is based on the hypothesis that disentanglement spontaneously occurs in quantum systems. According to this hypothesis, time evolution is governed by a modified master equation having an added nonlinear term that deterministically generates disentanglement. It is found that the disentanglement--based model is compatible with the experimental results. In particular, the model can qualitatively account for an experimentally observed instability in the system under study, which cannot be derived from any theoretical model that is based on a linear master equation.

Figures (6)

• Figure 1: Two spins. Time evolution of the magnetization $\left\langle \mathbf{S}\right\rangle =\left( k_{x},k_{y},k_{z}\right)$ for the case $L=2$. Initial states along the Bloch sphere equator are labelled by green $\times$ symbols, and the two locally--stable steady states by red $\times$ symbols. Assumed parameters' values are $\omega_{\mathrm{K}}=0$, $\Omega_{\mathrm{T}1}=0$, $\omega_{\mathrm{d}}=0$, $\gamma_{\mathrm{D}}/\omega_{\mathrm{A}}=100$, $\Omega_{\mathrm{L}1}/\omega_{\mathrm{A}}=3/2$ and $\omega_{\mathrm{A}} T_{1}=\omega_{\mathrm{A}} T_{2}=0.2$.
• Figure 2: FMSR. (a) Experimental setup. The FMSR is inductively coupled to two loop antennas (LA), which allow both driving and detection of FMSR magnetic resonance. The signal generators labeled as SG L and SG T drive the longitudinal and transverse loop antennas (LLA and TLA), respectively. A tunable phase shifter (PS) controls the relative phase $\phi_{\mathrm{T}}$ between longitudinal and transverse driving tones [see Eq. (\ref{['H DLS']})]. A circulator (C) and a splitter/combiner (SC) are used to direct the input and output microwave signals. The LLA (TLA) axis is parallel (perpendicular) to the applied static magnetic field $\mathbf{H}_{\mathrm{s}}$. All measurements are performed at room temperature. The transverse driving frequency $\omega_{\mathrm{T}}/\left( 2\pi\right)$ is set to $1.874 \operatorname{GHz}$, and the resonance frequency $\omega_{0}/\left( 2\pi\right)$ is tuned by adjusting the electromagnet current. (b) TLA reflectivity $R_{\mathrm{TLA}}$ as a function of transverse driving detuning frequency $\omega_{\mathrm{d}}/\left( 2\pi\right) =\left( \omega_{\mathrm{T}}-\omega_{0}\right) /\left( 2\pi\right)$ for 3 different values of the driving power applied to the LLA, which is denoted by $P_{\mathrm{L}}$. The transverse driving power, which is denoted by $P_{\mathrm{T}}$, is $10$ dBm. (c) TLA reflectivity $R_{\mathrm{TLA}}$ (in dB units) as a function of transverse driving detuning frequency $\omega_{\mathrm{d}}/\left( 2\pi\right)$ and $P_{\mathrm{T}}$ (for this measurement no longitudinal driving is applied). (d) Intermodulation peak intensity (in dBm units) as a function of transverse driving detuning frequency $\omega_{\mathrm{d}}/\left( 2\pi\right)$ and relative phase $\phi_{\mathrm{T}}$ (controlled by the PS). The longitudinal driving detuning frequency $\omega_{\mathrm{f}}/\left( 2\pi\right)$ is set to $5 \operatorname{kHz}$. The intermodulation peak at frequency $\left( \omega_{\mathrm{T}}+2\omega_{\mathrm{f}}\right) /\left( 2\pi\right)$ is measured using the RFSA. For this measurement $P_{\mathrm{T}}=10$ dBm and $P_{\mathrm{L}}=30$ dBm. (e) Mixing with low--frequency longitudinal driving. The measured RFSA intensity $I_{\mathrm{RFSA}}$ is plotted as a function of $\omega_{\mathrm{s}} \equiv\omega_{\mathrm{RFSA}} - \omega_{\mathrm{T}}$, where $\omega_{\mathrm{RFSA}}$ is the RFSA angular frequency. Longitudinal driving frequency is $\omega_{\mathrm{L}}/\left( 2\pi\right) =0.5\operatorname{MHz}$, and power is 0 dBm. (f) Theoretical calculation of $I_{\mathrm{RFSA}}$ based on Eq. (D11) of Ref. Buks_033807. FMSR measured parameters that are used for the calculation are $\omega_{\mathrm{L}}T_{1}=0.6$ and $T_{1}/T_{2}=2.1$.
• Figure 3: Peak points. (a) The peak point frequency shift $f_{\mathrm{dPP}}$ as a function of transverse driving power $P_{\mathrm{T}}$ for longitudinal driving power $P_{\mathrm{L}}$ values of $0 \operatorname{mW}$ (blue), $320 \operatorname{mW}$ (green) and $560 \operatorname{mW}$ (red). (b) The dependency on longitudinal driving power $P_{\mathrm{L}}$, for the case $P_{\mathrm{T}}=31.6\operatorname{mW}$. For both plots, solid lines represent predictions derived using Eq. (\ref{['dePP']}) of SM section \ref{['SM_RDM']}. The dimensionless parameter $D$ in Eq. (\ref{['dePP']}) is determined by measuring the transverse driving power and frequency detuning at the lower bistability onset point (see SM section \ref{['SM_RDM']}). A calibration yields the dimensionless parameters $W=P_{\mathrm{T}}/\left( 37 \operatorname{mW}\right)$ and $W_{\mathrm{A}}^{2}T_{2}^{2}=P_{\mathrm{L}}/\left( 980 \operatorname{mW}\right)$.
• Figure S1: Dimensionless polarization$\ z$ as a function of dimensionless detuning $\delta$ for the rapid disentanglement model. Calculation of steady state is based on Eq. (\ref{['z= DLS']}). Assumed parameters's values are $\alpha=0.8$ and $D=2\alpha$. The five curves, which are labelled by the numbers 1, 2, 3, 4, and 5, are calculated for five different values of the dimensionless driving amplitude $W$ respectively given by $W_{-}/2$, $W_{-}$, $\left( W_{-}+W_{+}\right) /2$, $W_{+}$ and $2W_{+}$, where $W_{\pm}$ is the value of $W$ corresponding to the bistability onset point having value of $z$ given by $z_{\pm}$. Peak points are labeled by green triangles [see Eq. (\ref{['dePP']})].
• Figure S2: Bosonization. The steady state value of magnon number expectation value $E$ is calculated as a function of the detuning angular frequency $\Omega_{\mathrm{d}}$ by solving the cubic polynomial equation given by Eq. (\ref{['E MM SS']}) of the main text. Assumed parameters' values are $\gamma _{1}/\gamma=0.5$, $\gamma_{3}/\gamma=0.1$ and $\omega_{\mathrm{K}}/\gamma=1$. The driving amplitudes $\Omega_{1}$ for the curves labelled by the integers 1, 2 and 3 are $0.5\Omega_{\mathrm{1c}}$, $\Omega_{\mathrm{1c}}$ and $10\Omega_{\mathrm{1c}}$, respectively, where $\Omega_{\mathrm{1c}}$ is the value of $\Omega_{\mathrm{1}}$ at the bistability onset point, which is labeled by a red cross symbol. Peak points are labeled by green triangles.