vacuum#
- pynlo.utility.vacuum(v_grid, rng=None)[source]#
Generate a root-power spectrum due to quantum vacuum fluctuations.
- Parameters:
- v_gridarray_like of float
The frequency grid.
- rngnp.random.Generator, optional
A NumPy random number generator. The default initializes a new Generator on each function call.
- Returns:
- a_vndarray of complex
The randomly-generated vacuum state root-power spectrum.
Notes
The combined noise of a coherent state’s amplitude and phase quadratures is equal to that of the vacuum. A coherent state \(|\alpha\rangle\) is defined by the displacement \(\alpha = x_1 + i \, x_2\), where \(x_1\) and \(x_2\) are the “amplitude” and “phase” (real and imaginary) quadrature amplitudes. In the number state basis \(|n\rangle\), a coherent state takes the form of a Poissonian distribution:
\[|\alpha\rangle = e^{-\frac{|\alpha|^2}{2}} \sum_{n=0}^\infty \frac{\alpha^n}{\sqrt{n!}} |n\rangle\]The probability \(P[\alpha]\) of measuring displacement \(\alpha\) from a coherent state with average displacement \(\beta\), a simultaneous measurement of \(x_1\) and \(x_2\), is as follows:
\[P[\alpha] = \frac{1}{\pi} |\langle \alpha | \beta\rangle|^2 = \frac{1}{\pi} e^{-|\alpha - \beta|^2}\]This probability distribution is Gaussian, and its noise can be completely described by calculating the variance of each quadrature component. Scaled to the number of photons (\(N=\alpha^2\)), the combined noise from both quadratures gives a total variance of one photon per measurement:
\[\sigma_{x_1}^2 = \sigma_{x_2}^2 = \frac{1}{2}\]\[\sigma_\alpha^2 = \sigma_{x_1}^2 + \sigma_{x_2}^2 = 1\]The width of the probability distribution is independent of the coherent state’s average displacement, which can be zero. This means that the root-photon noise can be generated independent of the state by sampling a normal distribution centered about zero mean. Also, since the Fourier transform of Gaussian noise is also Gaussian noise, the root-photon noise can be equivalently generated in either the time or frequency domains. Normalizing to the number of photons per measurement interval, the root photon noise for both quadratures becomes
1/(2 * dt)**0.5for the time domain and1/(2 * dv)**0.5for the frequency domain. The final root-power noise is found by multiplying the frequency domain root-photon noise by the square root of the photon energy associated with each bin’s frequency.