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| Photon Shot Noise | ||||
| Cramér Rao Decaying Sine Wave | ||||
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Photon Shot NoiseDC light incident on a photodiode creates a photocurrent that creates a voltage drop across an attached resistor.Photon shot noise = \(\sqrt{2 e V R} \), where \(e = 1.60217\times 10^{-19}\) C is the electron charge, \(V\) is voltage measured, and \(R\) is the feedback resistor. These values also auto-seed the CRLB calculator below. Voltage \(V\) = Volts Resistance \(R_f\) = Ohms ------------------------------------------------------------ PSN = V/√ Hz ------------------------------------------------------------ The power \(P\) of the light incident on the photodiode is obtained from the current \(I \) divided by the efficiency of the photodiode \(\eta\). \(P = I/\eta = \frac{V }{R \eta} \) Photodiode efficiency \(\eta \) = A/W = Amperes/Watt ------------------------------------------------------------ \(P\) = W ------------------------------------------------------------ |
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Cramér Rao Decaying Sine WaveCramér Rao Wikipedia entryThe best-case frequency error from fitting a monoexponetially decaying sine wave, \(y(t) = A \exp(-t/T_2) \sin(2\pi t + \phi) \), can be found using the Cramér Rao Lower Bound of the variance of unbiased estimators (minimum variance unbiased MVU estimator). The lower bound of the standard deviation of frequency is given by $$\sigma_f = \frac{ \sqrt{12 C}}{2\pi (A/\rho) T^{3/2}} \text{Hz , }$$ where \(\rho\) is the spectral noise density (V/√ Hz) of the measurement, \(A\) is the amplitude (V) of the sine wave, and \(T\) is the measurement time (seconds). The dimensionless constant \(C\) takes into account the exponential decay. Note \(A/\rho = \text{SNR}\) is the signal-to-noise ratio. For a non-decaying sine wave, \(C = 1\). Courtesy of Eq. 3 in V.G. Lucivero, et al., we have in the limit of a fast sampling time (or large number of samples per measurement), $$C = \frac{2 T^3 e^{2\alpha}(e^{2\alpha} - 1)}{3T_2(T_2^2e^{2\alpha}( e^{2\alpha} - 2) + T_2^2 -4 T^2 e^{2\alpha} ) }, \text{with } \alpha = T/T_2 . $$ Amplitude \(A\) = Volts Spectral Noise Density \(\rho\) = V/√ Hz Detection Time \(T\) = Seconds Relaxation Time \(T_2\) = Seconds ------------------------------------------------------------ \(C\) = ------------------------------------------------------------ \(\sigma_f\) = Hz ------------------------------------------------------------ Often this frequency measurement is repeated with a repetition time of \(T_r\), giving a bandwidth of \(\text{BW}=1/2 T_r\). Thus, assuming white noise, an estimate of the frequency spectral density is \(\sigma_f /\sqrt{\text{BW}}\). For \(T_r = T\), this estimate becomes $$ \text{Spectral density} = \frac{ \sqrt{24 C} }{2\pi (A/\rho) T} \frac{ \text{Hz}}{\sqrt{\text{Hz}}}. $$ ------------------------------------------------------------ \(\sigma_f\)/√ BW = Hz/√ Hz ------------------------------------------------------------ Assume the frequency measurement is due to observing the Larmor precession of a quantum mechanical spin, with field coupling \(\gamma \mathbf{B}\cdot\mathbf{S}\). Here the coupling strength \(\gamma\) is the gyromagnetic ratio in Hz/Tesla (see table below), (Gyromagnetic ratio Wikipedia entry.) \(\mathbf{B}\) is the field strength in Tesla, and \(\mathbf{S}\) is the spin state. The estimate of a magnetometer spectral density for a system using this spin species is then $$ \text{Mag. Spectral density} = \frac{ \sqrt{24 C} }{2\pi \gamma (A/\rho) T} \frac{ \text{T}}{\sqrt{\text{Hz}}}. $$ Gyromagnetic Ratio \(\gamma\) = Hz/T ------------------------------------------------------------ \(\sigma_B/√ BW \) = T/√Hz, in a bandwidth of BW = Hz ------------------------------------------------------------
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Interestingly, the energy resolution lower bound estimate per measurement is ------------------------------------------------------------ \(\sigma_E\) = eV ------------------------------------------------------------ |