State tomography for magnetization dynamics

State tomography is an essential tool for analyzing physical states in quantum science. In magnets, elementary excitation called a magnon, or a spin wave, dominates magnetization dynamics and various magnetic properties. Here, we propose and demonstrate state tomography for magnetization dynamics, enabling us to obtain a density matrix and Wigner function of the magnetization dynamics. Using the technique, we found that parametrically excited magnons can form a mixed state composed of two coherent states. Magnetization state tomography will pave a way to explore a wide range of states of magnons, such as a squeezed state in condensed matter.

Figure 1

(a) A schematic illustration of Wigner functions and marginal probability distribution in quadrature space. (b) Correspondence between quadrature amplitude and precession amplitude. (c) A schematic illustration of the measurement setup used in the present study.

Figure 2

(a) A schematic illustration of parametrically excited magnetization dynamics under parallel parametric pumping. Magnetic moment precesses with the angular frequency of ${\omega }_0=2\pi f_0$ with the initial phase of 0 or $\pi$ with respect to the pumping microwave. (b) A schematic illustration of the 0- and $\pi$-phase states plotted as a function of time $t$. $m_x$ is the $x$ component of magnetization. (c) Power dependence of the AC inverse spin-Hall voltage. (d) Magnetic field dependence of the AC inverse spin-Hall voltage. The inset shows the voltage near $H=243.6$ Oe, where we perform the MST measurement.

Figure 3

(a) Reconstructed Wigner function for the experimentally obtained 0-phase state at an excitation power of 72.4 mW in the YIG/Pt dot. (b) Reconstructed Wiger function for the experimentally obtained $\pi$-phase state at 72.4 mW. (c)–(f) Contour plot of the appearance frequency distribution of the ISHE voltage. (c) and (d) show the distribution of the real part of the lock-in voltage ($V_{\mathrm{LI}}^{\mathrm{R}}$) for the 0- and $\pi$-phase states obtained at 72.4 mW. (e) and (f) show the imaginary part of the lock-in voltage ($V_{\mathrm{LI}}^{\mathrm{I}}$). (g) Reconstructed Wigner function for the experimentally obtained 0-$\pi$ mixed state at 112.2 mW. (h) Contour plot of the appearance frequency distribution obtained by measuring $V_{\mathrm{LI}}^{\mathrm{R}}$ at 112.2 mW, (i) Contour plot of occurrence distribution obtained by measuring $V_{\mathrm{LI}}^{\mathrm{I}}$ at 112.2 mW. The noisy behavior of the data in the region where $\theta \ge {300}^{\circ }$ is attributed to the temporary lack of magnetic field stability.

Figure 4

(a) The absolute value of the density matrix for the experimentally obtained 0 state at 72.4 mW. (b) The absolute value of the density matrix for the experimentally obtained 0-$\pi$ mixed state at 112.2 mW, which exhibits a characteristic chessboard pattern. (c) The number distribution for the 0 state. (d) The number distribution for the 0-$\pi$ mixed state.