The flow of carriers following Poissonian statistics generates current fluctuations called shot noise. The electronic shot noise is utilized to determine an effective charge in electronic transport. Similarly, the study of spin shot noise is expected to provide insight into the nature of the elementary unit of angular momentum quanta $\hbar$, and to provide information on fundamental spin transport properties [1,2]. However, spin shot noise has not yet been observed experimentally, since it is difficult to detect physically meaningful noise by separating from other noises in the method proposed in the theoretical studies [1,2].

In our study [3], we propose an all-optical approach to detect spin shot noise. The key idea is to use an ultrafast pump laser pulse to impulsively drive the uniform magnetization of a ferromagnet far from equilibrium, as shown in Fig. 1(a). To model this experimental situation, we theoretically calculate the magnon population dynamics using the Lindblad equation and the Fokker-Planck equation, considering the function of the spin component (see Fig. 1(b)). We analyze the time derivative of the autocorrelation function, which mimics the fluctuation of the spin flow, i.e., the spin current. Furthermore, we define the Fano factor as the ratio between the nonequilibrium spin current flowing out of the spin system and its nonequilibrium fluctuation.

We show the calculated temperature dependence of the Fano factor under the magnetic field of 3 T in Fig. 1(c). At low temperatures ($k_{B}T \ll \hbar\omega_{0}$, $\omega_{0}$: the Larmor frequency), only the energy relaxation process predominates, resulting in the Fano factor becoming $\hbar$. Through an intuitive discussion grounded in the Poisson process, this outcome signifies the transfer of angular momentum from the spin system to the bath in units of $\hbar$. The conditions required for this experiment, ―low temperature ($<$1 K) and high magnetic field ($\sim$3 T)―, are experimentally feasible. It is worth noting that a similar rationale has been applied in determining the unit of charge in electronic transport, derived from nonequilibrium current noise (shot noise). As the temperature increases, the energy gain process also becomes significant. Consequently, the presence of two distinct transition processes diminishes the average spin flow, although it contributes additively to its fluctuation, leading to the increase of the Fano factor with increasing temperature.

Finally, we outline a feasible experimental protocol to observe the Fano factor. We propose using a thin film of ferromagnetic permalloy as the sample. Ultrafast dynamics of the $z''-''$component of the sample’s magnetization, $S_{z}(t)$, is measured following laser pulse irradiation (see Fig. 1(b)), employing time-resolved magneto-optical Kerr effect measurements. We can deduce the ensemble average of the spin flow from the time derivative of the average across trials of the magnetization dynamics measurements, i.e., $d\langle S_{z}(t)\rangle/dt$. Then, the correlation function $C(t^{\prime}, t)$ is obtained by calculating the covariance of $S_{z}$ at different times, that is, $\langle S_{z}(t^{\prime})S_{z}(t)\rangle$. From a cusp in $C(t^{\prime}, t)$ along the line $t^{\prime} = t$, we can obtain the Fano factor, combining the time derivative of the mean magnetization value.

In summary, we investigated the nonequilibrium fluctuation arising from the ferromagnetic magnetization under pulse irradiation. We calculated the Fano factor, which is defined as the ratio between the nonequilibrium spin current flowing out of the spin system and its nonequilibrium fluctuation and observed that the Fano factor measured at low temperature offers insight into the unit of angular momentum transferred per spin relaxation process in a bulk ferromagnet. Our proposal sets the stage for nonequilibrium spin-noise spectroscopy, offering an advanced technique to access information that is inaccessible by other experimental means.

References

• [1] A. Kamra and W. Belzig, Phys. Rev. Lett. 116, 146601 (2016).
• [2] M. Matsuo, Y. Ohnuma, T. Kato, and S. Maekawa, Phys. Rev. Lett. 120, 037201 (2018).
• [3] T. Sato, S. Watanabe, M. Matsuo, and T. Kato, Phys. Rev. Lett. 134, 106702 (2025).

Authors

• T. Sato, S. Watanabe^a, M. Matsuo^b, and  T. Kato
• ^aKeio University
• ^bUniversity of Chinese Academy of Sciences