Observation of anti-PT symmetry phase transition in the magnon-cavity-magnon coupled system

TL;DR

This paper experimentally demonstrates anti-PT symmetry phase transition in a magnon-cavity-magnon system using dissipative coupling via a microwave cavity, revealing level attraction and opening new research avenues.

Contribution

It introduces a novel experimental realization of anti-PT symmetry in a magnon system with dissipative coupling, utilizing a new magnon-readout method.

Findings

Demonstrated anti-PT symmetry phase transition in a magnon-cavity system.

Observed counterintuitive level attraction process.

Established a new platform for studying anti-PT symmetry properties.

Abstract

As the counterpart of PT symmetry, abundant phenomena and potential applications of anti-PT symmetry have been predicted or demonstrated theoretically. However, experimental realization of the coupling required in the anti-PT symmetry is difficult. Here, by coupling two YIG spheres to a microwave cavity, the large cavity dissipation rate makes the magnons coupled dissipatively with each other, thereby obeying a two-dimensional anti-PT Hamiltonian. In terms of the magnon-readout method, a new method adopted here, we demonstrate the validity of our method in constructing an anti-PT system and present the counterintuitive level attraction process. Our work provides a new platform to explore the anti-PT symmetry properties and paves the way to study multi-magnoncavity-polariton systems.

Tables

Table 1. Table 1: Parameters used in cavity-readout and magnon-readout methods. γ 1 subscript 𝛾 1 \gamma_{1} and γ 2 subscript 𝛾 2 \gamma_{2} are the dissipation rates of magnon 1 and magnon 2, respectively. g 13 subscript 𝑔 13 g_{13} or g 23 subscript 𝑔 23 g_{23} is the coupling strength between the cavity and the magnon 1 or magnon 2. | Ω | Ω \left|\Omega\right| is the effective detuning. κ int subscript 𝜅 int \kappa_{\rm int} is the intrinsic dissipation rate of the cavity (without additional ports). κ 1 subscript 𝜅 1 \kappa_{1} , κ 2 subscript 𝜅 2 \kappa_{2} and κ 3 subscript 𝜅 3 \kappa_{3} are the dissipation rates introduced by antenna 1, antenna 2 and antenna 3, respectively.

Probe Method	System Parameters (units: $2\pi \times \mathrm{MHz}$)
	${\gamma }_1$	${\gamma }_2$	$g_{13}$	$g_{23}$	$\left|\Omega \right|$	${\kappa }_{\mathrm{int}}$	${\kappa }_1$	${\kappa }_2$	${\kappa }_3$
Cavity-readout	$1.11$	$1.11$	$9.77$	$9.61$	$2.7$	tunable	0	0	$\approx {\kappa }_{\mathrm{int}}$
Magnon-readout	$2.22$	$2.22$	$6.65$	$6.41$	$2.7$	$1.5$	$0.45$	$0.92$	tunable

Equations

$$H_{\mathrm{eff}}=\left[\begin{array}{cc}\Omega -i(\gamma +\Gamma )& -i\Gamma \\ -i\Gamma & -\Omega -i(\gamma +\Gamma )\end{array}\right].$$

Full text

In the real world, quantum systems interact with surrounding environment and evolve from being closed originally into open ones de Vega and Alonso (2017); Xiang et al. (2013). Hamiltonians describing open systems are generally non-Hermitian. Due to the non-conserving nature, the eigen-energies are complex numbers and the corresponding dynamics are complicated Miri and Alù (2019). One special type of non-Hermitian systems, which respects parity-time ($\mathcal{PT}$) symmetry, has triggered unprecedented interest, and been widely explored theoretically and experimentally El-Ganainy et al. (2018); Feng et al. (2017); Özdemir et al. (2019); Wu et al. (2019). Another special type of non-Hermitian systems is the anti-$\mathcal{PT}$ symmetric system, which is the counterpart of the $\mathcal{PT}$ symmetric system and always preserve conjugated properties to those observed in $\mathcal{PT}$-symmetric ones Peng et al. (2016); Choi et al. (2018). Based on the conjugated properties, abundant phenomena and potential applications of anti-$\mathcal{PT}$ systems have been predicted or demonstrated theoretically. Examples include unidirectional light propagation Lau and Clerk (2018), flat full transmission bands Yang et al. (2017), enhanced sensor sensitivity Wiersig (2014), constructing topological superconductor Gong et al. (2018), and potential effects on quantum measurement back-action evading Bernier et al. (2018). Motivated by the intriguing phenomena and various potential applications, experimental realizations of anti-$\mathcal{PT}$ symmetric systems are highly desirable. However, because of the requirement of purely imaginary coupling constants between two bare states, there have been few experimental works about anti-$\mathcal{PT}$ symmetry Peng et al. (2016); Choi et al. (2018); Zhang et al. (2018); Li et al. (2019).

Recently, collective excitations of spin ensembles in ferromagnetic systems (also called as magnons) have drawn considerable attentions due to their very high spin density, low damping rate, and high-cooperativity with the microwave photons Lachance-Quirion et al. (2019); Goryachev et al. (2018). Especially the ferromagnetic mode in an yttrium iron garnet (YIG) sphere can strongly Goryachev et al. (2014); Zhang et al. (2014); Tabuchi et al. (2014); Zhang et al. (2015a) and even ultra-strongly Zhang et al. (2016a); Bourhill et al. (2016) couple to the microwave cavity photons, leading to cavity-magnon polaritons. Based on cavity-magnon polaritons, quantum memories have been realized Zhang et al. (2015b), remote coherent coupling between two magnons has been proposed Rameshti and Bauer (2018) and observed Lambert et al. (2016). At the same time, coupled cavity-magnon polaritons are attractive systems for exploring non-Hermitian physics Zhang and You (2019); Zhang et al. (2018, 2017), because of their easy reconfiguration, flexible tunability, and especially the strong compatibility with microwave Tabuchi et al. (2015); Lachance-Quirion et al. (2017), optics Kusminskiy et al. (2016); Hisatomi et al. (2016); Zhang et al. (2016b); Graf et al. (2018), as well as mechanical resonators Holanda et al. (2018); Zhang et al. (2016c).

Here, we propose a coupled magnon-cavity-magnon polariton system to experimentally demonstrate the anti-$\mathcal{PT}$ symmetry Yang et al. (2017). The pure imaginary coupling between two spatially separated and frequency detuned magnon modes is realized by engineering the dissipative reservoir of the cavity field. Different from previous cavity-magnon-polariton experiments, in which the signals are extracted from cavity Goryachev et al. (2014); Zhang et al. (2014); Tabuchi et al. (2014); Zhang et al. (2015a, 2016a); Bourhill et al. (2016), we need to extract the signals from the magnons. The experimental data are not only fitted well with the original experimental Hamiltonian calculated transmission spectrum but also the one predicted by the standard anti-$\mathcal{PT}$ Hamiltonian. By continuously tuning non-Hermitian control parameter, e.g. cavity decay rate, we present the spontaneous symmetry-breaking transition, which is accompanied by the energy level attraction. The results are compared with the data normally obtained from the cavity. This comparison demonstrates that the magnon-readout technique enables us to measure the magnon state separately in a multi-magnon-cavity coupled system and allow the exploration of many significant phenomena.

Our experimental setup is schematically shown in FIG. 1 (a). Two YIG spheres are placed inside a three-dimensional (3D) oxygen-free copper cavity with inner dimensions $40\times 20\times 8\ \rm mm^{3}$. The YIG spheres with 0.3 mm diameter are glued on one end of two glass capillaries, which are anchored at two mechanical stages. The YIG spheres are placed near the magnetic-field antinode of the cavity mode $\rm TE_{101}$ through two holes in the cavity wall. Two grounded loop readout antennae, antenna 1 and antenna 2, are coupled with the YIG sphere 1 and sphere 2, respectively. In this setup, we can change the position of YIG spheres relative to loop antennae by tuning the mechanical stages. In our experiment, we focus on the Kittle mode, which is a spatially uniform ferromagnetic mode. To avoid involving other magnetostatic modes, the antennae are carefully designed and assembled. The antenna 3 with a length tunable pin is used to control the dissipation rate of the cavity. When we probe the system from the cavity, the antenna 1 and antenna 2 are removed. The whole system is placed in a static magnetic bias field, which is created by a high-precision room temperature electromagnet. The bias magnetic field and the magnetic field of the $\rm TE_{101}$ cavity mode are nearly perpendicular at the site of two YIG spheres.

In our system, the two YIG spheres work at low excitation regime, thus the collective spin excitation of YIG spheres can be simply regarded as harmonic resonators. In dissipative regime, our system can be approximately described by the standard anti-$\mathcal{PT}$ Hamiltonian Yang et al. (2017) (Supplementary Materials A):

[TABLE]

Here $i\Gamma$ is the dissipative coupling rate, $\Omega=(\omega_{1}-\omega_{2})/2$ is the effective detuning in the rotating reference frame with frequency $(\omega_{1}+\omega_{2})/2$, where $\omega_{1}$ ($\omega_{2}$) is the resonant frequency of magnon 1 (2). For the Kittle mode, the frequency of a magnon linearly depends on the bias field $\vec{B}_{i}$, i.e., $\omega_{i}=\gamma_{0}\left|\vec{B}_{i}\right|+\omega_{m,0}\ (i=1,\ 2)$, where $\gamma_{0}=28\ \rm GHz/T$ is the gyromagnetic ratio and $\omega_{m,0}$ is determined by the anisotropy field. To obtain the effective Hamiltonian in Eq. (1), we further require that the dissipation rates of two magnons are nearly equal, i.e., $\gamma_{1}\approx\gamma_{2}=\gamma$, and the magnon 1 - cavity coupling rate $g_{13}$ approximately equals to the magnon 2 - cavity coupling rate $g_{23}$, i.e., $g_{13}\approx g_{23}=g$. In the regime of $\kappa\gg\gamma$ and $\kappa\gg\left|\omega_{3}-\omega_{1(2)}\right|$, with the cavity dissipation rate $\kappa$, the effective coupling rate is $\Gamma=g^{2}/\kappa$. We can conveniently obtain the eigenvalues of the Hamiltonian in Eq. (1), $\lambda_{\pm}=-i(\gamma+\Gamma)\pm\sqrt{\Omega^{2}-\Gamma^{2}}$. When $\left|\Omega\right|>\left|\Gamma\right|$, the eigenvalues are normally complex, and the system works in anti-$\mathcal{PT}$ symmetry broken phase regime. If $\left|\Omega\right|<\left|\Gamma\right|$, the eigenvalues are purely imaginary and the system works in anti-$\mathcal{PT}$ symmetry phase regime. The condition of $\left|\Omega\right|=\left|\Gamma\right|$ defines the EP.

We can probe the magnon-cavity-polariton system from either the magnon or the cavity. When we probe the system from the magnon, we carefully tune the mechanical stage and change the position of YIG spheres relative to the readout antennae to change the external dissipation rate $\gamma_{11}$ ($\gamma_{21}$) of magnon 1 (2), so that the readout antennae are critically coupled to the magnons, i.e., $\gamma_{i0}\approx\gamma_{i1}$ ($i=1,\ 2$), where $\gamma_{i0}$ is the intrinsic dissipation rate of magnon. In this situation, the total dissipation rate of magnon 1 (2) should be $\gamma_{i}=\gamma_{i0}+\gamma_{i1}\approx 2\gamma_{i0}$ ($i=1,\ 2$). In this setup, the dissipation rate of the cavity is controlled by solely changing the pin length of the antenna 3. When we probe the system from the cavity, the signal is injected into the cavity from antenna 3, and the reflected signal is measured from the same port. In this case, the overall dissipation rate of magnon 1 (2) equals to the intrinsic dissipation rates, i.e., $\gamma_{i}=\gamma_{i0}$ ($i=1,\ 2$). In this measurement setup, we require that antenna 3 is critically coupled to the cavity. To accomplish this requirement, we paste carbon tape at the electric-field antinode of the cavity mode to change the cavity intrinsic dissipation rate $\kappa_{\rm int}$ and change the pin length of antenna 3 to change the dissipation rate $\kappa_{3}$, such that the condition $\kappa_{\rm int}\approx\kappa_{3}$ can be satisfied. All system parameters used in both readout methods are presented in TABLE 1, which shows that the difference between $g_{13}$ and $g_{23}$, and the difference between $\gamma_{1}$ and $\gamma_{2}$ are both less than 5 percent of their average values. Therefore, we can safely neglect the difference between dissipation rates $\gamma_{1}$ and $\gamma_{2}$, and the difference between coupling rates $g_{13}$ and $g_{23}$.

Using the magnon-readout method, we read the reflection parameters $\rm S_{11}$ and $\rm S_{22}$ from antenna 1 and antenna 2, respectively. In this case, the magnon readout antennae coupled to the YIG spheres and to the cavity simultaneously. In other words, the applied probe microwave signal through antenna 1 (2) drives not only the magnon 1 (2) but also the cavity with a relative phase $\varphi_{13}$ ($\varphi_{23}$) simultaneously. The reflected signals from the magnon 1 (2) and the cavity also preserve the same relative phase $\varphi_{13}$ ($\varphi_{23}$). Based on the mechanism, we can solve the input and output field relation as $s_{\rm out}=-s_{\rm in}+\sqrt{\kappa_{k}}e^{i\varphi_{k3}}c+\sqrt{\gamma_{k1}}a$ ($k=1,\ 2$). Comparing with the magnon-readout method, the cavity-readout method is much simpler. The injected signal from the antenna 3 only drives the cavity, and the input-output field relation preserves the normal form, $s_{\rm out}=-s_{\rm in}+\sqrt{\kappa_{3}}c$. Using the magnon-cavity-magnon coupled original Hamiltonian and the input-output field relation, we can solve the whole spectra with the standard input-output theory in both readout methods, as shown in Supplementary Material B.

Based on the magnon-readout method, we can demonstrate that the approximation used in our system is valid and construct the anti-$\mathcal{PT}$ symmetry. We apply bias magnetic fields $B_{1}$ and $B_{2}$ to bias the magnon 1 and 2 at frequency at $\omega_{1}$ and $\omega_{2}$, respectively. In our experiment, the resonant frequencies $\omega_{1}$ and $\omega_{2}$ are set to satisfy the relationship $\omega_{1}-\omega_{2}=5.4\ \rm MHz$, thus the effective detuning in this configuration is $\left|\Omega\right|=2.7\ \rm MHz$, which is constant in all experiments. And then, we measure the reflection parameters $\rm S_{11}$ and $\rm S_{22}$ from antenna 1 and antenna 2, respectively. As shown in FIG. 2 (a), the measured $\rm S_{11}$ and $\rm S_{22}$ data are fitted well with the calculated spectra using the original experiment Hamiltonian, as shown in Supplementary Materials B. This result proves that the physical model used in solving the measurement spectra is sufficient. In the other side, the anti-$\mathcal{PT}$ Hamiltonian in Eq. (1) describes a system with dissipatively coupled detuned resonators. We can solve the corresponding reflection spectra with the standard anti-$\mathcal{PT}$ Hamiltonian in Eq. (1), as shown in FIG. 2 (b), in which the resonant dips are marked with triangles. In order to compare the experimental results with the spectra predicted by the standard anti-$\mathcal{PT}$ Hamiltonian, we draw the triangles at the same position in FIG. 2 (a). We conclude from this comparison that the resonance occurs at the right frequency and amplitude which are predicted by the standard anti-$\mathcal{PT}$ Hamiltonian. The measurement data demonstrate that the approximations are sufficient and indicate that we successfully construct the anti-$\mathcal{PT}$ symmetry in a magnon-cavity-magnon coupled system.

Based on the cavity-readout method, we can only probe the system through the antenna 3. As shown in FIG. 2 (c), although the measured data can be fitted well with the spectra given by the original experiment Hamiltonian, the results cannot prove that we successfully construct an anti-$\mathcal{PT}$ system. Because the cavity mode $c$ is eliminated in the large dissipation rate approximation, we cannot compare the measurement results with those obtained by the anti-$\mathcal{PT}$ Hamiltonian.

We now discuss the spontaneous phase transition of the anti-$\mathcal{PT}$ system. In our experiments, the coupling rates between magnons and the cavity are fixed values, which are around 6.5 MHz. In order to observe the anti-$\mathcal{PT}$ symmetry phase transition, we need to increase the effective coupling rate $\Gamma=g^{2}/\kappa$ by decreasing the overall dissipation rate of the cavity $\kappa$, where $\kappa=\kappa_{\rm int}+\kappa_{1}+\kappa_{2}+\kappa_{3}$ in the magnon-readout, $\kappa_{\rm int}$ is the intrinsic dissipation rate of the cavity (without additional ports), $\kappa_{1}$, $\kappa_{2}$ and $\kappa_{3}$ are the dissipation rates introduced by the antenna 1, antenna 2 and antenna 3, respectively. With different cavity dissipation rates, we obtain the corresponding transmission spectra $\rm S_{11}$ and $\rm S_{22}$, as shown in FIG. 2 (a). When the cavity dissipation rate $\kappa$ is large, the corresponding effective coupling rate is smaller than the effective magnon detuning (i.e., $\Gamma<\Omega$). The system works in the anti-$\mathcal{PT}$ symmetry broken phase, and the separation between two dips in the spectrum is larger than the full width at half maximum (FWHM). Using the definition of EP, we can obtain the corresponding cavity dissipation rate $\kappa_{0}=15.8\ \rm MHz$. Continuously decreasing the cavity dissipation across the EP results in two main counterintuitive phenomena: (i) decreasing the cavity loss, the measured spectra show mode attraction; (ii) increasing the effective coupling strength between the magnon modes, we observe the energy attraction instead of the mode splitting. These two counterintuitive phenomena are basically induced by the broken anti-$\mathcal{PT}$ symmetry phase transition. When our system works in anti-$\mathcal{PT}$ symmetry phase, the separation between two dips is smaller than the full width at half maximum (FWHM). In order to formulate the relationship between the dip separation and the FWHM, we can define the combined spectrum by $\rm\bar{S}=(S_{11}+S_{22})/2$, as shown in Supplementary Material C.

The anti-$\mathcal{PT}$ symmetry induced level attraction can be expressed even more clearly by examining the eigenvalues of different dissipation rate $\kappa$. Using the method elaborated in Supplementary Material D, we extract the eigenvalues and plot the real and imaginary parts as a function of $\kappa$ in FIG. 3 (a) and (b) respectively, which show excellent agreement with theoretical results. The experimental data in FIG. 3 (a) reveal that the exceptional point occurs at $\kappa_{0}=15.8\ \rm MHz$, which corresponds to a dissipative coupling rate $\Gamma=2.7\ \rm MHz$. According to Eq. (1), the two real parts of eigenvalues should be $\pm 2.7\ \rm MHz$ when the cavity dissipation rate $\kappa$ approaches infinity. As shown in FIG. 3 (a), the real parts of eigenvalues corresponding to $\kappa=105\ \rm MHz$, are approximate $\pm 2.7\ \rm MHz$, which are compatible with the theoretical results. When we decrease the value of $\kappa$, the difference between two real parts becomes smaller and is reduced to zero at the EP. The theory predicts that there should be two different imaginary parts in anti-$\mathcal{PT}$ symmetry regime, and a single value of imaginary part in symmetry broken regime. We have also observed this phenomenon in our experiment, as shown in FIG. 3 (b).

In conclusion, we have successfully constructed anti-$\mathcal{PT}$ symmetry in a magnon-cavity-magnon coupled system without any gain medium, and observed anti-$\mathcal{PT}$ symmetry from the magnon side. From the magnon-readout results, we have observed the broken anti-$\mathcal{PT}$ symmetry at the phase transition point (i.e., the EP), resulting in a counterintuitive energy attraction phenomenon instead of the energy repulsion widely reported in strongly coupled-resonator systems Goryachev et al. (2014); Zhang et al. (2014); Tabuchi et al. (2014); Zhang et al. (2015a). Encircling around such exceptional point in the future may allow us to observe various topological operations based on non-adiabatic transitions. The negative frequencies (negative-energy modes) in anti-$\mathcal{PT}$ symmetric Hamiltonian equivalent to harmonic oscillators with negative mass also have a close connection to evade quantum measurement backaction Bernier et al. (2018). Comparing with the cavity-readout results, we uncover the unique ability of magnon-readout method in exploring multi-magnon-cavity couped systems. Our experiment illustrates the power of the magnon-readout, and motivates further explorations on macroscopic quantum phenomena and the fundamental limit on the quantum sensing based on EPs Wiersig (2014).

This work was supported by the National Key R&D Program of China (Grant No. 2018YFA0306600), the CAS (Grants No. GJJSTD20170001 and No. QYZDY-SSW-SLH004), and Anhui Initiative in Quantum Information Technologies (Grant No. AHY050000).