# Noise

class Noise(*, power_densities, frequency_step, time_domain_sample_count, name=None)

The noise amplitude $$\beta(t)$$ associated to a Hamiltonian term. You must provide its one-sided power spectral density $$S(f)$$ (sampled at some frequencies $$\{f_k = k \Delta f\}$$). From it, the function generates a random discrete time series to represent $$\beta(t)$$ between 0 and $$\tau$$ for each simulation trajectory. The function builds a two-sided power spectral density $$\mathcal{S}(f)$$, defined as $$\mathcal{S}(0) = S(0)$$ and $$\mathcal{S}(f_k) = \mathcal{S}(-f_k) = S(|f_k|)/2$$ for $$k > 0$$; and an amplitude spectral density $$B(f_k) \equiv e^{i \phi(f_k)} \sqrt{\mathcal{S}(f_k)}$$, such that $$|B(f_k)|^2 = \mathcal{S}(f_k)$$. Any realization of $$B(f_k)$$ with phases $$\phi(f_k)$$ fulfilling $$\phi(f_k) = -\phi(-f_k)$$ is a Hermitian function, and as such, corresponds to the spectrum of a real time domain process $$\beta(t)$$. In particular, the function takes random phases $$\phi(f_k)$$ from a flat distribution in the $$(-\pi, \pi)$$ range and generates a realization of $$\beta(t)$$ from the inverse (discrete) Fourier transform of $$B(f)$$. Namely, $$\beta(t_n) = \sqrt{\Delta f} \sum_{k=0}^{2N-2} B(f_k) \exp(2\pi \, i \, n \, k / (2N-1) )$$ , with $$t_n = n \Delta t$$ for $$n \in (0, ..., 2N-1)$$ and $$\Delta t = ( (2N-1) \Delta f)^{-1}$$. The function then upsamples this time series at $$N_\mathrm{segments}^{(\beta)}$$ points between 0 and $$\tau$$ via Whittaker–Shannon interpolation.

Variables
• power_densities (List[float]) – The one-sided power spectral density of the noise, $$S(f)$$, sampled at frequencies $$(0, \Delta f, 2 \Delta f, \ldots)$$.

• frequency_step (float) – The difference between adjacent sample frequencies of the one-sided power spectral density, $$\Delta f$$.

• time_domain_sample_count (int) – The number of time domain samples, $$N_\mathrm{segments}^{(\beta)}$$, to take for the noise amplitude across the duration of the simulation $$\tau$$. The noise is treated as piecewise-constant with this number of segments.

• name (str, optional) – The name used to identify this noise in the output. If you don’t provide it, the function generates one.