Units and choice of input parameters

In this document we describe the choice of units implemented in SpiDy.jl (which follows the conventions utilised in NJP 24 033020 (2022)) and how to appropriately choose the parameters that the library takes as input.

Equations of motion

SpiDy.jl is designed to implement the equations of spins in presence of a bath of harmonic oscillators as derived in NJP 24 033020 (2022), which for a single spin $\mathbf{S}$ in presence of an external field $\mathbf{B}_\mathrm{ext}$ read

\[\frac{\mathrm{d}\mathbf{S}}{\mathrm{d}t} = \gamma_e\mathbf{S}\times\left(\mathbf{B}_\mathrm{ext} + \mathbf{b}(t) + \mathbf{V}(t)\right),\]

where $\gamma_e$ is the electron gyromagnetic ratio, $b(t)$ is the environment induced thermal stochastic field, and

\[\mathbf{V}(t) = \gamma_e\int_{-\infty}^{t}\mathrm{d}t' \, \mathbf{K}(t-t')\mathbf{S}(t'),\]

where $\mathbf{K}(\tau)$ is a memory kernel accounting for the non-Markovian evolution of the spin (see NJP 24 033020 (2022) for more details). SpiDy focuses on the case of an environment with a Lorentzian spectral density, in which case these equations of motion can be rewritten as

\[\begin{align} \frac{\mathrm{d}\mathbf{S}}{\mathrm{d}t} &= \gamma_e\mathbf{S}\times\left(\mathbf{B}_\mathrm{ext} + \mathbf{b} + \mathbf{V}\right), \\ \frac{\mathrm{d}\mathbf{V}}{\mathrm{d}t} &= \mathbf{W}, \\ \frac{\mathrm{d}\mathbf{W}}{\mathrm{d}t} &= \gamma_e A \mathbf{S} - \omega_0^2\mathbf{V} - \Gamma\mathbf{W}, \end{align}\]

where $A$, $\omega_0$, and $\Gamma$ parametrise the Lorentzian spectral density as

\[J(\omega) = \frac{A\Gamma}{\pi} \frac{\omega}{(\omega_0^2 - \omega^2)^2 + \omega^2\Gamma^2},\]

and the thermal stochastic $\mathbf{b}$ field is given by

\[\mathbf{b}(t) = \int_{-\infty}^{+\infty}\mathrm{d}t' F(t-t') \xi(t'),\]

with $\xi$ being white noise and

\[F(\tau) = \frac{1}{2\pi}\int_{-\infty}^{+\infty}\mathrm{d}\omega e^{-i\omega\tau} \sqrt{P(\omega)}.\]

Here, $P(\omega)$ is the power spectral density of the environment, and it is given in terms of the Lorentzian spectral density $J(\omega)$ and the environment thermal noise $N(\omega)$ by $P(\omega) = \hbar\pi J(\omega) N(\omega)$. The noise $N(\omega)$ can be classical, "quantum", or "quantum" with no zero point fluctuations. For example, for the "quantum" case we have

\[N_\mathrm{qu}(\omega) = \coth\left(\frac{\hbar\omega}{2k_\mathrm{B}T}\right).\]

Note that in these equations above, all quantities have standard units.

Units rescaling

SpiDy.jl implements these equations in a unit-free way, following the conventions of NJP 24 033020 (2022). In summary, consider a spin of length $\hbar S_0$ is in presence of a reference magnetic field of magnitude $B_0$. We then define the Larmor frequency

\[\omega_\mathrm{L} = |\gamma_e| B_0,\]

with $\gamma_e$ the electron gyromagnetic ratio.

SpiDy.jl then takes as input parameters for the simulation the following unit free parameters:

• $S_0$: the spin length (in units of $\hbar$).
• $\bar{B}_\mathrm{ext}$: external magnetic field.
• $\bar{t}_\mathrm{end}$: final time of the evolution.
• $\mathrm{d}\bar{t}$: time differential.
• $\bar{\omega}_0$: peak frequency of the Lorentzian spectral density.
• $\bar{\Gamma}$: width of the Lorentzian spectral density.
• $\bar{\alpha}$: amplitude of the Lorentzian spectral density.
• $\bar{T}$: the environment temperature.

These unit-free quantities (denoted hereon by a bar on top) are related to the unitful quantities in the previous section by the following conversions:

\[\begin{align} \mathbf{B}_\mathrm{ext} &= B_0 \, \bar{B}_\mathrm{ext}, \\ t_\mathrm{end} &= \omega_\mathrm{L}^{-1} \, \bar{t}_\mathrm{end}, \\ \mathrm{d}t &= \omega_\mathrm{L}^{-1} \, \mathrm{d}\bar{t}, \\ \omega_0 &= \omega_\mathrm{L} \, \bar{\omega}_0, \\ \Gamma &= \omega_\mathrm{L} \, \bar{\Gamma}, \\ \omega_0 &= \omega_\mathrm{L} \, \bar{\omega}_0, \\ A &= \frac{B_0^2\omega_\mathrm{L}}{\hbar S_0} \, \bar{\alpha}, \\ T &= \frac{\hbar\omega_\mathrm{L}}{k_\mathrm{B}} \, \bar{T}. \end{align}\]

Given these choices of rescaling and adimensionalisation and plugging them in the equations of motion of the previous section, one finally gets the equations being solved by SpiDy, that is

\[\begin{align} \frac{\mathrm{d}\bar{\mathbf{S}}}{\mathrm{d}\bar{t}} &= \bar{\mathbf{S}}\times\left(\bar{\mathbf{B}}_\mathrm{ext} + \frac{1}{\sqrt{S_0}}\bar{\mathbf{b}} + \bar{\mathbf{V}}\right), \\ \frac{\mathrm{d}\bar{\mathbf{V}}}{\mathrm{d}\bar{t}} &= \bar{\mathbf{W}}, \\ \frac{\mathrm{d}\bar{\mathbf{W}}}{\mathrm{d}\bar{t}} &= \bar{\alpha}\bar{\mathbf{S}} - \bar{\omega}_0^2\bar{\mathbf{V}} - \bar{\Gamma}\bar{\mathbf{W}}, \end{align}\]

with environment Lorentzian spectral density

\[\bar{J}(\bar{\omega}) = \frac{\bar{\alpha}\bar{\Gamma}}{\pi} \frac{\bar{\omega}}{(\bar{\omega}_0^2 - \bar{\omega}^2)^2 + \bar{\omega}^2\bar{\Gamma}^2},\]

and thermal stochastic noise $\bar{\mathbf{b}}$ with power spectral density

\[\bar{P}(\bar{\omega}) = \pi \bar{J}(\bar{\omega})\bar{N}(\bar{\omega}).\]

For the "quantum" noise, the noise term is given by

\[\bar{N}_\mathrm{qu}(\bar{\omega}) = \coth\left(\frac{\bar{\omega}}{2\bar{T}}\right),\]

while for "classical" noise we have

\[\bar{N}_\mathrm{cl}(\bar{\omega}) = \frac{2\bar{T}}{\bar{\omega}}.\]

Finally, note that with these definitions above, the unit-free Gilbert damping is given by (see NJP 24 033020 (2022))

\[\eta = \frac{\bar{\alpha}\bar{\Gamma}}{\bar{\omega}_0^4}.\]

Extracting Lorentzian units from experimental data

To be written. See also EPL 139 36002 (2022).