Master equation approach to magnon relaxation and dephasing

Magnons are the collective excitation of ordered magnets and they can carry information for data storage and processing. Because of the low dissipation rate and long spin diffusion length of magnons in magnetic insulators, the information coded in magnons can, in principle, weaken Joule heating problems and thus it is a desirable candidate for durable information processing Chumak et al. (2015). As bosonic particles, magnons share considerable similarities with photons and the quantum states of magnons such as single-magnon state Yuan and Duine (2020); Lachance-Quirion et al. (2017), squeezed states Zhao et al. (2004); Kamra and Belzig (2016); Li et al. (2019); Kamra et al. (2020); Yuan et al. (2021), Schrödinger cat states Sharma et al. (2021); Sun et al. (2021) and quantum many-body states Yuan et al. (2022a) have already been proposed in magnonic systems for information processing. The emerging field, dubbed “quantum magnonics” Yuan et al. (2022a), has started to attract significant interest due to its potential to push the traditional studies of magnonics to the quantum limit, and for its convenient docking with mature solid-state quantum platforms, such as cavity photons, mechanical oscillators, superconducting qubits and single-spin qubits.

To achieve desirable information processing using hybrid magnonic systems, a crucial objective is to maintain the coherence of magnons for a longer time. The decoherence in a magnetic system may come from the coupling of magnons with conduction electrons, lattice vibrations or phonons, impurities and magnon-magnon scattering process. The standard approach to characterize the dissipation of magnons in the past century is to introduce a phenomenological parameter called the Gilbert damping in the classical magnetization dynamics Landau (1965); Gilbert (2004). The linewidth of the magnetic resonance under external microwave driving is proportional to the Gilbert damping. By analyzing the frequency and temperature dependence of the linewidth, one may extract the various contributions to the damping Arias and Mills (1999); McMichael et al. (2003); Nembach et al. (2013); Khodadadi et al. (2020). In the framework of quantum optics, one usually introduces a decay rate of the photon mode, together with a stochastic field acting on the photon in the Heisenberg-Langevin equation, according to the fluctuation-dissipation theorem Walls and Milburn (1995). However, different from photons, magnons in the magnets are naturally coupled with each other through either anisotropy or exchange interactions White (2007). The coupling strength can easily reach ultra-strong and even deep-strong coupling regimes Yuan et al. (2022a). For example, in a two-sublattice antiferromagnet (AFM), the coupling of sublattice magnons is on the order of exchange interaction, which is as strong as the on-site frequency of the magnons. One has to consistently include the interactions among magnons to get a reasonable prediction of both the dynamic and equilibrium properties of the magnonic systems, especially the decoherence time of magnons. It has been known that the strong interactions inside a hybrid quantum optical system can cause additional dissipation channels and modify the dynamic equations Carmichael and Walls (1973); Rivas et al. (2010); Ban (2017); Betzholz et al. (2020). For magnonic systems, how the strong interaction among magnons influences the relaxation processes channels, especially its influence on the robustness of the quantum properties of magnons, has not been examined so far.

On the other hand, up till now, most of the studies on the magnon decoherence only account for relaxation process, while how pure dephasing without relaxation process almost gains no attention. If one intends to use quantum states of magnons for quantum information processing, the pure dephasing channel should be treated equal footing as the relaxation channel and be addressed properly, resembling the situation in more conventional qubit systems Yang and Liu (2008); Boissonneault et al. (2009); Nazarov and Blanter (2009); Nielsen and Chuang (2000).

In this article, we study the relaxation and dephasing of magnons in both ferromagnets and antiferromagnets based on the Lindblad formalism. We derive a generalized master equation for the dynamics of magnons under the Born-Markov approximation. In particular, we identify the interaction induced collective relaxation channels of magnons. This channel could not only enhance the entanglement of magnons, but also makes the quantum correlations of magnons more stable against thermal fluctuations, which is very different from the local dissipation that will destroy the quantumness of the systems. Second, we show that the antiferromagnetic magnons are subject to a decoherence-free evolution in ideal conditions when they are subject to a common bath. The comparison with the relaxation process within classical equation of motion is discussed in detail. Further, we show how our theoretical formalism can recover the dephaisng channels of magnons through four-magnon interaction and magnon-phonon coupling.

Our work is organized as follows: In Sec. II, we present the generalized formalism to consider the magnon relaxation caused by their interaction with the environment. In Sec. III, we apply the general formalism to study the stability of the single-mode squeezed state in an anisotropic ferromagnet. Then we introduce the two-mode squeezed magnons Yuan et al. (2020); Kamra et al. (2019) in a two-sublattice AFM and their relaxation channel subject to both two separate and one common bath and show how decoherence-free evolution of magnons can be realized in Sec. IV. The classical-quantum analogue of the dissipation channels is also discussed in detail. In Sec. V, we apply the generalized master equation to address the dephasing of magnons caused by magnon-magnon interaction and magnon-phonon scattering. In Sec. VI, we come to the discussions and conclusions. Unless stated otherwise, the reduced Planck constant $\hbar$ is set to one.

Let us consider a magnonic system described by the Hamiltonian $\hat{\mathcal{H}}_{\mathrm{FM}}$, which is expressed in terms of the magnon creation ($\hat{a}^{\dagger}$) and annihilation ($\hat{a}$) operators in general. To study the decoherence of the magnon state against thermal fluctuation, we model the environment as a collection of harmonic oscillators, i.e., $\hat{\mathcal{H}}_{\mathrm{R}}=\sum_{i}\omega_{i}\hat{c}_{i}^{\dagger}\hat{c}_{i}$, with commutation relations $[\hat{c}_{i},\hat{c}_{j}^{\dagger}]=\delta_{ij}$. The interaction of magnons with the bath is through the coherent transfer between magnons and bath, $\hat{\mathcal{H}}_{\mathrm{int}}=\sum_{i}g_{i}(\hat{a}\hat{c}_{i}^{\dagger}+\hat{a}^{\dagger}\hat{c}_{i})$. Note that the bath modes in the environment may include phonons, electron-hole pairs, or impurities. Here we do not specify them and only assume a phenomenological coupling coefficient $g_{i}$ for simplicity. A more detailed consideration of the various baths will not influence the essential physics presented below. It will, however, determine the temperature and frequency dependence of the coupling coefficients McMichael and Kunz (2002); Safonov and Bertram (2003); Rückriegel et al. (2014). With these assumptions, the total Hamiltonian of the system reads

Since the total system is Hermitian, its dynamic evolution should follow the master equation $\partial\hat{\rho}_{T}(t)/\partial t=i[\hat{\rho}_{T}(t),\hat{\mathcal{H}}_{\mathrm{T}}]$, where $\hat{\rho}_{T}$ is the density matrix of the total system. To eliminate the non-interacting terms in the total Hamiltonian $\hat{\mathcal{H}}_{\mathrm{T}}$, we transfer to the interaction picture and trace out the degree of freedoms for the bath to derive the evolution of the ferromagnetic magnons as

where $\tilde{\rho}(t)\equiv tr_{R}\tilde{\rho}_{\mathrm{T}}(t)$. Here all the operators in the interaction picture are labeled with an overhead tilde. Under the Born-Markov approximation, the master equation (2) can be reduced to Lindblad (1976); Gorini et al. (1976); Carmochael (1991); Manzano (2020)

where $\tilde{\rho}_{R}(0)$ is the initial density matrix of the bath and is assumed to be independent of time due to its large degrees of freedoms. Now the state of magnons $\tilde{\rho}(t)$ does not explicitly depend on its history $\tilde{\rho}(t^{\prime})$ ($t^{\prime}<t$). Following the standard notations to rewrite the interaction in the form $\hat{\mathcal{H}}_{\mathrm{int}}=\sum_{i}\hat{s}_{i}\otimes\hat{\Gamma}_{i}$ Manzano (2020), where $\hat{s}_{i}$ and $\hat{\Gamma}_{i}$ are respectively the operators acting in the spaces of magnonic system and bath, the master equation (3) can be finally written as

This equation is very general and has been widely used to study the dynamic behavior of open quantum system in quantum optics Lindblad (1976); Gorini et al. (1976); Carmochael (1991); Manzano (2020). Below we shall study the relaxation of both the single-mode and two-mode squeezed magnons and pure dephasing of magnons based on this master equation Yuan et al. (2022b).

Let us consider a biaxial ferromagnet described by the Hamiltonian

where $\hat{\mathbf{S}}_{i}$ is the spin operator on the $i$th site with spin number $S$, $J>0$ is the ferromagnetic exchange interaction between neighboring spins, $K_{z}>0$ and $K_{x}>0$ are respectively the easy-axis and hard-axis anisotropies, and $H$ is an external field along the $z-$axis. The classical ground state of the system is $\langle\mathbf{S}_{i}\rangle=Se_{z}$. To consider the magnon excitation above this ground state, we perform Holstein-Primakoff (HP) transformation of the spin operators Holstein and Primakoff (1940), i.e., $\hat{S}_{iz}=S-\hat{a}^{\dagger}\hat{a}$, $\hat{S}^{+}_{i}=\sqrt{2S-\hat{a}^{\dagger}\hat{a}}\hat{a}$, $\hat{S}^{-}_{i}=\hat{a}^{\dagger}\sqrt{2S-\hat{a}^{\dagger}\hat{a}}$, and substitute them into (5) to derive the effective Hamiltonian of magnon excitations up to the quadratic order as

where $\omega_{a}=2K_{z}S+K_{x}S+H$, $g_{aa}=K_{x}S/2$. Here we considered uniform precession mode of spins such that magnon operator $\hat{a}$ is spatial independent and the exchange coefficient $J$ does not appear in the Hamiltonian. Nevertheless, the formalism presented below is straightforward to be generalized to study the evolution of the magnon modes with non-zero wavevectors.

The ground state of the magnonic system described by Hamiltonian (6) is a single-mode squeezed state $|GS\rangle=\mathrm{exp}(r/2(\hat{a}\hat{a}-\hat{a}^{\dagger}\hat{a}^{\dagger}))|0\rangle$ with squeezing parameter $r=\mathrm{arctanh}(2g_{aa}/\omega_{a})/2>0$ and magnon occupation $\langle\hat{a}^{\dagger}\hat{a}\rangle=\sinh^{2}r$. The uncertainties of the total spin components $\hat{S}_{x}=\sum_{i}\hat{S}_{i}^{x}$ and $\hat{S}_{y}=\sum_{i}\hat{S}_{i}^{y}$ in this ground state are respectively, $\Delta S_{x}\equiv\sqrt{\langle\hat{S}_{x}^{2}\rangle-\langle\hat{S}_{x}\rangle^{2}}=e^{-r}\sqrt{NS/2}$ and $\Delta S_{y}\equiv\sqrt{\langle S\hat{}_{y}^{2}\rangle-\langle\hat{S}_{y}\rangle^{2}}=e^{r}\sqrt{NS/2}$. It is clear that the noise of $\hat{S}_{x}$ is squeezed below the symmetric limit at the sacrifice of uncertainty increase in the $\hat{S}_{y}$ component. The Heisenberg uncertainty relation is still maintained.

Now we are ready to apply the general formalism to study the quantum decoherence of squeezed magnons. The magnon and bath operators in this special case respectively read

where $\omega_{r}=\sqrt{\omega_{a}^{2}-4g_{aa}^{2}}$ is the resonance frequency of the magnetic system. By substituting all these variables into the master equation (4), simplifying the integrals using bath correlations at equilibrium conditions, and finally transferring it back to the original Schrödinger picture, we have

where $\mathcal{L}_{AB}[\hat{\rho}]=2\hat{A}\hat{\rho}\hat{B}^{\dagger}-\hat{\rho}\hat{A}^{\dagger}\hat{B}-\hat{A}^{\dagger}\hat{B}\hat{\rho}$ is the generalized Lindblad operator, and $n_{\mathrm{th}}=(e^{\omega_{r}/k_{B}T}-1)^{-1}$ is the Bose-Einstein distribution function with $T$ being the bath temperature, $\kappa$ is the effective relaxation rate defined as $\kappa\equiv g(\omega_{r})D(\omega_{r})$, where $D(\omega_{r})$ is the total density states of bath modes at the resonance frequency $\omega_{r}$.

The three distinguishable relaxation coefficients $\eta_{\omega\omega},\eta_{\omega g},\eta_{gg}$ are respectively defined as

Here $\eta_{\omega\omega}$ is the leading-order relaxation rate, and $\eta_{\omega g}$ and $\eta_{gg}$ can respectively be viewed as the first and second order corrections of the dissipation channels caused by the squeezing interactions. Their dependencies on the hard-axis anisotropy of the system $g_{aa}$ (or equivalently $K_{x}$) are shown in Fig. 1(a). As the anisotropy parameter $g_{aa}$ increases, both $\eta_{\omega g}$ and $\eta_{gg}$ becomes comparable with the leading-order terms $\eta_{\omega\omega}$ and thus they can not be neglected any longer. Note that we can derive the dynamic equation (8) alternatively, by first transferring to the diagonal basis of Hamiltonian (6) through a Bogliubov transformation, deriving the master equation in this basis and then transfer back to the original basis. Equation (8) is a powerful result that allows us to study the quantum dynamics of squeezed magnons starting from any initial state, as well as the equilibrium properties of the squeezed magnons.

An intuitive picture to understand these corrections of dissipation channels is illustrated in Fig. 1(b). Without squeezing ($g_{aa}=0,\eta_{\omega\omega}=1,\eta_{\omega g}=\eta_{gg}=0$), magnons can only directly relax to the ground state at zero temperature. The resulting master equation is reduced to the form

With squeezing, the magnons may first be excited to higher energy levels ($g_{aa}\hat{a}^{\dagger}\hat{a}^{\dagger}$) and then relax, which gives the first-order correction ($\eta_{\omega g}$). Further, the magnons on energy level $|n\rangle$ may be excited to $|n+2\rangle$ and fall back to $|n\rangle$ and then relax, this correspond to the second-order correction ($\eta_{gg}$). These higher-order processes involve the simultaneous excitation/annihilation of two or more magnons, hence we refer to them as collective relaxation channels.

To study how the collective relaxation channel influences the quantumness of the system, we simulate the quantum master equation (8) and show the uncertainty of the $S_{x}$ component as a function of anisotropy in Fig. 1(c). Here the uncertainty is defined as $\Delta_{x}=\sqrt{2/NS}\Delta S_{x}$. As the anisotropy parameter $g_{aa}$ increases, the generalized master equation recovers a steady state with much stronger squeezing ($\Delta_{x,G}$, red solid line) than the result by only including the local dissipations ($\Delta_{x,L}$, red dashed lines). By zooming in the steady state, we found it is exactly the ground state of the system $|GS\rangle$, and thus they share the same magnon occupation and squeezing of $\hat{S}_{x}$ ($\Delta_{x,th}$, black dashed lines). As a comparison, the system cannot relax to ground state if only the local dissipation is considered ($\Delta_{x,L}$, red dashed lines). Therefore, strictly speaking, the generalized master equation is essential to recover the correct equilibrium properties of the interacting magnon system.

Figure 1(d) shows the influence of temperature on the magnon squeezing. Firstly, as the temperature increases, the uncertainty $\Delta_{x,G}$ gradually increases above the symmetric limit. The critical value of the temperature is around 2 K for $n_{\mathrm{th}}=0.46$ and $\omega_{r}/2\pi=48.5$ GHz. This behavior clearly shows that magnon squeezing is a low temperature quantum phenomenon. It is different from the elliptical trajectory of spin precession under the influence of anisotropy, which always exists in anisotropic systems regardless of temperature Yuan and Duine (2021). Secondly, when only the local dissipation channel is taken into account, the squeezing will die at $n_{\mathrm{th}}=0.27$ smaller than the value of 0.46 when both local and collective dissipation channels are taken into account. This comparison suggests that the collective relaxation channel can increase the thermal stability of the quantumness of magnons.

Now we study the relaxation of two-mode squeezed magnons. Two-sublattice AFM hosts two types of magnons excited on each sublattice and it provides a natural platform to study the quantum correlations of magnons. The ground-state properties of the system guarantees that the generation of magnons in one sublattice is always accompanied by the generation of another magnon in the other sublattice, i.e., the interaction of the sublattice magnons is parametric-type Yuan et al. (2022a). Therefore, they naturally form a two-mode squeezed state Yuan and Duine (2021); Yuan et al. (2020); Kamra et al. (2019, 2020). Furthermore, the sublattice magnons are coupled through exchange interaction, which is comparable with the on-site frequencies of magnons and thus reaches the deep-strong coupling regime. We will see how such a strong interaction influences the relaxation channel of the system.

We consider a two-sublattice AFM described by the Hamiltonian

where $J>0$ is the antiferromagnetic exchange coupling between neighboring spins, $K_{z}>0$ is the easy-axis anisotropy. Given the external field applied along the $z-$axis ($\mathbf{H}=He_{z}$), the classical ground state of the systems when the field $H$ is below the spin-flop transition is a Néel state, i.e., $\langle\hat{\mathbf{S}}_{2l}\rangle=Se_{z},\langle\hat{\mathbf{S}}_{2l+1}\rangle=-Se_{z}$. Then we can follow the standard HP transformation to quantize the magnon excitation above this ground state as

where $\omega_{a,b}=H_{\mathrm{ex}}+H_{\mathrm{an}}\pm H$, $H_{\mathrm{ex}}=2ZJS^{2},H_{\mathrm{an}}=2K_{z}S$ with $Z$ being the coordination number, $g_{\mathrm{ab}}=H_{\mathrm{ex}}$ is the coupling of magnon modes excited on the two sublattices. Again, we focus on the uniform precession mode. The ground-state entanglement of magnons taking the contribution of finite wavevectors into account can be found in Refs. Hartmann et al. (2021); Azimi Mousolou et al. (2021); Wuhrer et al. (2022)

To describe the interaction of magnons with the environment, we have to clarify whether the two types of magnons are subject to two separate bath or one common bath. Both cases contain local and collective dissipation channels, but they will result in very different relaxation process as we shall discuss below.