Level attraction from interference in two-tone driving

We consider the experimental setup of fig. 1, where a ferromagnetic sphere made of Yttrium-Iron-Garnet (YIG) is placed inside a microwave cavity. The second-quantised Hamiltonian of the system enclosed inside the cavity is that of the lowest-order magnon mode (quantised spin wave) coupled to a single cavity mode, described by Harder et al. (2021)

where $g/2\pi$ is the (complex) coherent coupling strength Flower et al. (2019); Gardin et al. (2023a) between the cavity mode (operators $c,c^{\dagger}$) and the magnon mode (operators $m,m^{\dagger}$). The frequency of the cavity mode $\omega_{c}/2\pi$ is fixed by the cavity’s geometry, while the ferromagnetic resonance of the magnon is tuned by a static magnetic field $\mathbf{H}_{0}=H_{0}\hat{\mathbf{z}}$ as $\omega_{m}=\gamma\absolutevalue{\mathbf{H}_{0}}$ where ${\gamma}{/2\pi}=28$ GHz/T is the gyromagnetic ratio.

The quantum Langevin equations (QLEs) allow to model the coupling of the cavity system described by eq. 1 to the environment, to model intrinsic dissipation or external drives. To model the intrinsic damping of the cavity mode $c$ and the magnon mode $m$, we assume that they each couple to a different bosonic bath with coupling constants $\kappa_{c}/2\pi$ and $\kappa_{m}/2\pi$. As per fig. 1, the cavity system described by eq. 1 is coupled to three ports $p_{i}$, labelled from 1 to 3, where the first two ports couple only to the cavity mode (with real-valued coupling constants $\kappa_{1}/2\pi$ and $\kappa_{2}/2\pi$ and phases $\phi_{1},\phi_{2}$ Bourcin et al. (2024)) and the third couples only to the magnon mode (real-valued coupling constant $\kappa_{3}/2\pi$ with phase $\phi_{3}$). Physically, the coupling of the magnon to port 3 is due to the Zeeman interaction with the magnetic field created by the loop of port 3. The QLE for the cavity and magnon modes then read (see appendix A for details)

where $\widetilde{\omega}_{c}=\omega_{c}-i\frac{\kappa_{c}+\kappa_{1}+\kappa_{2}}{2}$ and $\widetilde{\omega}_{m}=\omega_{m}-i\frac{\kappa_{m}+\kappa_{3}}{2}$. In the QLEs, $c^{\text{in}}$ and $m^{\text{in}}$ account for intrinsic damping and have zero mean $\expectationvalue{c^{\text{in}}}=\expectationvalue{m^{\text{in}}}=0$, while $p_{i}^{\text{in}}$ represent the inputs from the ports.

We now assume that $p_{1}^{\text{in}}$ and $p_{3}^{\text{in}}$ correspond to coherent drives at the same frequency, albeit with a phase and amplitude difference written $\expectationvalue{p_{3}^{\text{in}}}=\delta e^{i\phi}\expectationvalue{p_{1}^{\text{in}}}$. We also perform a semi-classical approximation and neglect quantum fluctuations, which amounts to only considering expectation values. Using the input-output formalism (see appendix B for details), the reflection at Port 1 when Port 3 is active is found to be

where $\widetilde{\Delta}_{c}=\omega-\widetilde{\omega}_{c}$ and $\widetilde{\Delta}_{m}=\omega-\widetilde{\omega}_{m}$. By comparing with the expressions of the standard S-parameters (detailed in appendix C), we note that the reflection $r_{1}$ can also be written $r_{1}(\omega)=S_{11}+\delta e^{i\phi}S_{13}$ which is expected given the linearity of the problem.

For $\delta\neq 0$, the reflection coefficient $r_{1}$ can be greater than one, because it is only normalised to the input power from Port 1, thus neglecting that of Port 3. Following the setup of fig. 1, and taking the power out of Port 1 as a reference, the VNA outputs a power $1+\delta^{2}$, so we can renormalise the reflection to $R_{1}(\omega)=\frac{1}{\sqrt{1+\delta^{2}}}r_{1}(\omega)$. Similarly, we can calculate the normalised transmission coefficient through Port 2 when both Port 1 and Port 3 are active, and we find (appendix D)

We plot $\absolutevalue{R_{1}}$ in fig. 2 and $\absolutevalue{T_{2}}$ in fig. 3 with the parameters $\omega_{c}/2\pi=10$ GHz, $\absolutevalue{g}/2\pi=20$ MHz, $\kappa_{c}/2\pi=\kappa_{m}/2\pi=1$ MHz, $\kappa_{1}/2\pi=\kappa_{2}/2\pi=10$ MHz, $\kappa_{3}/2\pi=5$ MHz, $\varphi=\mathrm{arg}\,g=-\frac{\pi}{2}$ and $\phi_{1}=\phi_{2}=\phi_{3}=0$. The reflection $R_{1}$ exhibits controllable level repulsion and attraction depending on the amplitude $\delta$ of the drives and the dephasing $\phi$, mirroring the experimental results of Boventer et al. (2019, 2020). On the other hand, we notice that the transmission $T_{2}$ only shows level repulsion, despite the system being driven in exactly the same way: between $R_{1}$ and $T_{2}$, only the measurement location changes, and thus the physics should be the same. In the next section, we will attribute the observed level attraction to interference.

An advantage of the input-output formalism is that it allows us to understand analytically the spectral features of fig. 2. Indeed, by writing $r_{1}(\omega)=N(\omega)/D(\omega)$ in terms of a nominator $N$ and a denominator $D$, the denominator can be factorised as

where

are the complex-valued solutions of the quadratic eq. 7.

On the other hand, the nominator $N$ can be written as

where we defined $\widetilde{\omega}_{c}^{\prime}=\widetilde{\omega}_{c}+i\kappa_{1}$ and

with $\arg{g}$ is the phase of the complex number $g$. Comparing eqs. 7 and 10, we see that the nominator can be factored similarly to the denominator as $N(\omega)=(\omega-\widetilde{\Omega}_{+}(\delta,\phi))(\omega-\widetilde{\Omega}_{-}(\delta,\phi))$, where the complex frequencies $\widetilde{\Omega}_{\pm}(\delta,\phi)$ are formally identical to eq. 9, after replacing $\omega_{c}\mapsto\omega_{c}^{\prime}$ and $\absolutevalue{g}\mapsto G$. Finally, we obtain

In general, the observed resonances and anti-resonances of eq. 12 are a combined effect of both the numerator $N$ and the denominator $D$. However, as discussed below, and numerically illustrated in fig. 2, the spectral features are well characterised by the poles (zeroes $\widetilde{\omega}_{\pm}$ of $D$) and zeroes (zeroes $\widetilde{\Omega}_{\pm}$ of $N$) of $r_{1}$. Notably, it is sufficient to solely examine the zeroes $\widetilde{\Omega}_{\pm}$ to understand the resonances, and the poles $\widetilde{\omega}_{\pm}$ to understand the anti-resonances, independently of each other.

For small dissipation rates $\kappa_{i}$, the imaginary part of $\widetilde{\omega}_{\pm}$ is small compared to its real part. Therefore, in eq. 12, when $\omega$ is close to $\omega_{\pm}=\Re{\widetilde{\omega}_{\pm}}$, the denominator $\absolutevalue{D}$ of $\absolutevalue{R_{1}}$ almost vanishes, leading to a resonance behaviour (maxima) of $\absolutevalue{R_{1}}$. Furthermore, $\omega_{\pm}$ corresponds to the spectrum of the Hamiltonian of eq. 1, and hence this resonance behaviour is expected to give information about the spectrum of the closed-system. As shown by the solid lines in figs. 2 and 3, the spectrum is that of coherent coupling, characterised by energy level repulsion with an angular frequency gap $2\absolutevalue{g}$. Hence, the denominator of $\absolutevalue{R_{1}}$, or equivalently its resonances, does inform on the underlying physics.

The situation is different for the nominator $N$ of $R_{1}$, which leads to anti-resonances. Indeed, while the expressions of $\widetilde{\omega}_{\pm}$ and $\widetilde{\Omega}_{\pm}$ are formally identical, the anti-resonance coupling strength $G/2\pi$ for $\widetilde{\Omega}_{\pm}$ is complex-valued (while it is real-valued, $\absolutevalue{g}/2\pi$, for $\widetilde{\omega}_{\pm}$) which can lead to level attraction. To see this more clearly, it is convenient to introduce the effective amplitude $\delta_{0}=\frac{\sqrt{\kappa_{1}\kappa_{3}}}{\absolutevalue{g}}\delta$ and the effective phase $\phi_{0}=\phi_{1}-\phi_{3}+\arg{g}-\frac{\pi}{2}$ of Port 3. We can then rewrite eq. 11 as

Therefore, when $\phi-\phi_{0}=0$ the anti-resonance coupling strength $G/2\pi$ is real-valued and increases as $\delta_{0}$ increases, leading to an increase in level repulsion (see the dashed lines in the first column of fig. 2). On the other hand, when $\phi-\phi_{0}=\pi$, $G/2\pi$ is real-valued when $\delta_{0}<1$ and diminishes when $\delta_{0}$ increases. Eventually, when $\delta_{0}>1$, the $G/2\pi$ becomes purely imaginary due to taking the square root of a negative number, leading to level attraction. In the limiting case where $\delta_{0}=1$, we have $G=0$ and we obtain two uncoupled anti-resonances as shown in fig. 2(g) (horizontal at the cavity mode frequency and diagonal at the magnon’s frequency $\omega_{m}/2\pi$). As can be seen from the dashed lines in fig. 2, these spectral features indeed correspond to anti-resonances (minima of $\absolutevalue{R_{1}}$) at the frequencies $\Omega_{\pm}$. Physically, these frequencies $\Omega_{\pm}$ are determined by the interference between the nominators of $S_{11}$ and $S_{13}$ since $r_{1}(\omega)=S_{11}+\delta e^{i\phi}S_{13}$. Hence, the input-output formalism shows that while the denominator of $\absolutevalue{R_{1}}$ informs on the physics, the nominator is interference-based. Furthermore, the denominators of both $R_{1}$ and $T_{2}$ are identical (see appendix D for the detailed expression), leading to similar resonant behaviour following the coherent coupling spectrum. However, their anti-resonance behaviours differ because their nominators differ (see appendix D).

Concerning the occurrence of level attraction, we note that it requires the total phase $\phi+\phi_{3}-\phi_{1}+\frac{\pi}{2}-\mathrm{arg}\,g$ equal to $\pi$. Physically, this is determined by the magnon-photon coupling phase $\mathrm{arg}\,g$, the coupling of the cavity and magnon modes to the probes ($\phi_{1}$ and $\phi_{3}$), and the chosen phase difference $\phi$ between Port 1 and Port 3. The strength of level attraction itself is determined by $|G|$, which depends on $|g|$ and the ratio $\frac{\sqrt{\kappa_{1}\kappa_{3}}}{g}$, i.e. the magnon-photon coupling strength and the coupling to the ports.

The input-output formalism provides interesting insights thanks to the resulting analytical expressions. However, it remains a toy-model of a two-tone driving experiment. Instead, a realistic modelling, taking into account the geometry of the cavity and of the ports, can be perfomed using COMSOL Multiphysics^®, a finite element modelling software. We use a two-post re-entrant cavity Goryachev et al. (2014); Kostylev et al. (2016), similar to that sketched in fig. 1, in which the loop antenna of Port 3 is inserted from the bottom of the cavity (see appendix H for more details). At the location of the YIG sphere, the cavity mode’s magnetic field is purely along the $\hat{\mathbf{x}}$ axis Bourhill et al. (2020). In our analysis, we assumed that Port 3 does not couple to the cavity mode, which requires us to orient the loop antenna so that it generates a magnetic field orthogonal to the cavity mode. Thus, noting $\hat{\mathbf{n}}$ the normal to the plane of the loop, we need $\hat{\mathbf{n}}$ to be in the $(x,z)$ plane. Furthermore, since only the components orthogonal to the static magnetic field $H_{0}\hat{\mathbf{z}}$ contribute to the coupling between Port 3 and the magnon, $\kappa_{3}/2\pi$ is maximum when $\hat{\mathbf{n}}=\hat{\mathbf{y}}$.

We first numerically calculated the S parameters for different values of $\mathbf{H}_{0}$ using COMSOL, as detailed in appendix H. From these two $S$ parameters, we can plot $R_{1}$ and $T_{2}$ after normalising $r_{1}(\omega)=S_{11}+\delta e^{i\phi}S_{13}$ and $t_{2}(\omega)=S_{21}+\delta e^{i\phi}S_{23}$, and we obtain the results of fig. 4, which successfully reproduce the input-output theory results of figs. 2 and 3. As a further verification, we also performed a fully two-tone experiment in COMSOL for $\delta=12$, and the results were identical to those of the second and fourth rows of fig. 4.