Intrinsic non-Markovian magnetisation dynamics

Abstract

Memory effects arise in many complex systems, from protein folding, to the spreading of epidemics and financial decisions. While so-called non-Markovian dynamics is common in larger systems with interacting components, observations in fundamental physical systems have been confined to specifically engineered cases. Here, we report the experimental observation of non-Markovian dynamics in an elemental material, crystalline cobalt. By driving this material with an intense terahertz electromagnetic field, we bring its magnetisation into a non-equilibrium state and follow its evolution. We measure the sample's low temperature magnetic response in the time domain which leads to an unexpectedly rich multi-peaked spectrum in the Fourier domain, that cannot be explained by established models. We use open quantum system theory, which predicts a non-Markovian memory kernel in the dynamical equations to capture the fundamental interaction between the spin system and the phonon bath. Simulations based on this theory produce a multi-peaked spectrum, which matches the measured one. Our non-Markovian approach is also able to reproduce the modification of the spectrum at higher temperatures. Our findings demonstrate that non-Markovian effects are observable at a much more fundamental level than previously thought, opening the door to their exploration and control in a broad range of condensed matter systems.

Figures (12)

• Figure 1: Sketch of Markovian vs non-Markovian dynamics.a. A pendulum (black) interacting with its environment (shaded) has a trajectory (grey spiral) in position-momentum space that monotonously moves towards its state of rest, consistent with Markovian dynamics. b. In contrast, the trajectory (grey spiral-like) of a single pendulum (black) that interacts (orange spring) with another pendulum, shows complex dynamics with non-Markovian features, such as time excursions away from its state of rest. This is highlighted by the later-time arrow (green) which is longer than the earlier-time arrow (purple), each showing the distance from the state of rest.
• Figure 2: Sum and difference spectra of terahertz-induced dynamics in a ferromagnetic cobalt film.a. Fast Fourier transform of summed (non-magnetic) time-resolved signal (blue symbols on solid line) following spin excitation with a strong terahertz pump field. Lorentzian spectral density $I(\nu)$ (orange solid line) centred at frequency $\nu_0$ = $4.2$ THz to match signal, and with width $\Gamma = 0.2$ THz, see main text. b. Fast Fourier transform of the difference (magnetic) response following the same pump field and at the same temperature (blue symbols on solid line), Fourier spectra of the simulated LLG equation (grey solid line), iLLG equation (black dashed line) and of the nM-LLG equation (orange solid line). nM-LLG simulation solves Eq. \ref{['eq:mLLG']} with memory kernel $K(t-t')$ specified by Lorentzian spectral density $I(\nu)$, with fixed parameters $\nu_0$, ${\cal A}$ and $\Gamma$, see main text. Units for amplitudes of the experimental and simulation spectra are chosen such that the peak amplitudes at $1.4$ THz coincide.
• Figure 3: Temperature dependence of THz-frequency spectrum.a. Fourier transform of experimental signal at three temperatures, $T_{\sf exp} = 20$ K (blue line, circles), $T_{\sf exp} = 220$ K (purple line, triangles) and $T_{\sf exp} = 300$ K (red line, squares). The three experimental spectra were measured with a single THz polarity, and contain magnetic and non-magnetic contributions. b. Fourier transform of dynamics obtained by solving the nM-LLG equation for three different unit-free temperatures $\tilde{T} \propto T_{\sf exp}$ chosen at the same ratios as in the experiment. Simulations of the nM-LLG equation \ref{['eq:mLLG']} include coloured thermal noise, and the demagnetisation field is reduced for higher temperatures, see Appendix. Scale of the $\tilde{T} = 20$ graph in b is chosen such that first peak amplitude matches the experimental one in a. Amplitudes at higher temperatures follow relatively from it. All remaining simulation parameters are identical to Fig. \ref{['fig:multipeak']}.
• Figure 4: Illustration of Markovian and non-Markovian spin dynamics.a. Markovian dynamics: the magnetisation (red arrow) is coupled only to the instantaneous state of the phonon bath (connected grid) at each frame in time $t'$ and $t$. Viewed on the Bloch sphere, the magnetisation uniformly spirals towards its rest state (cf. Fig. \ref{['fig:nonMarkovianHO']}a). b. Non-Markovian dynamics: the magnetisation at a later time $t$ is coupled (red spring) also to the spin state at an earlier time $t'$. Viewed on the Bloch sphere, the magnetisation dynamics exhibits a complex behavior (grey trajectory). The presence of higher frequency dynamics leads to spin excursions away from its state of rest (green arrow), a clear non-Markovian characteristic. In the Fourier domain, such complex motion presents in the form of multiple THz-frequency spectral peaks as seen in Fig. \ref{['fig:multipeak']} and \ref{['fig:temperaturedep']}.
• Figure 5: Experimental schematic: THz pump-Magneto-Optical Kerr Effect (MOKE) probe experiment on epitaxial Cobalt thin film. The THz pump and optical probe are collinear with an angle of incidence of 45 degrees. The directions of external bias magnetic field ($H_{\mathrm{bias}}\,\hat{\mathbf{x}}$), THz magnetic field ($H_{\mathrm{THz}}(t)\,\hat{\mathbf{y}}$), and the demagnetizing field ($-H_{\mathrm{d}}(t)\,\hat{\mathbf{z}}$) are shown.