# Finding the noise spectral density of a quantity made from different noisy components

I'm looking for the expression of the noise spectral density of the magnetic flux $\Phi$ generated by a noisy voltage signal $V$ applied to a resistor with Johnson-Nyquist noise $R$ which produces a current $I$. This current is then passed through a superconducting coil, generating the flux. In the end I believe the flux will be linearly proportional to the voltage: $$\Phi = \frac{\mu_0 N A \cos\theta }{H R}V = CV$$ where $\mu_0$ is the magnetic permittivity of free space, $N$ is the number of loops in the coil, $A$ is the relevant surface area for the flux, $\theta$ the angle between the magnetic field lines and the surface normal, $H$ is the length of the coil and $C$ is a constant that combines the quantities that should be (approximately) fixed.

So the idea is as follows. The voltage is set to some working point value $V$, but it has some small noise component (lets call the amplitudes $\delta V(t)$) that varies over time, specified by the noise spectral density $S_{VV}(f)$. Applied to the resistor, this produces a noisy current $I$. Moreover, the resistor produces thermal noise in the (idealized) form of some white noise spectrum, which also adds noise to the current produced. So my question is, knowing the noise spectral densities of the voltage and the thermal noise, what is the spectral density of the resulting current?

Circuit wise it's just a voltage source connected to a noisy resistor, connected to a coil.

To add some context, I'm working on an experiment where I control the dephasing time of a superconducting qubit which is related to the noise spectral density of the applied magnetic flux. That relationship I know, but now I'm trying to figure out how to engineer my magnetic flux in a specific way; given that I apply some voltage with a certain amount of noise to a resistor (with thermal noise) I'd like to predict the spectral density of the flux.

• I believe the Taylor expansion is a good approach. The first three terms give you a deterministic part and two noise parts, which as you can see just add. If the voltage and resistance noises are independent then I'm pretty sure the spectral densities also just add. Feb 28 '16 at 18:48
• Now, that said... resistance noise? Like, you have a resistor whose resistance is fluctuating? How much? How did you measure that? Is it low frequency noise coming from of temperature drift in the cryostat? Feb 28 '16 at 18:51
• I'm thinking about the Johnson-Nyquist noise from the resistor, which should add some noise to the generated current. The idea is we apply a voltage to the resistor and get a current, which goes through a superconducting coil to make a magnetic field and thus the flux. Important in that is that the resistor is actually outside the fridge, but I have to think a little bit about what happens to the noise as we have some attenuators at different cooling stages. As for the spectral densities just adding, I'm not 100% sure. I guess it makes sense intuitively, but I don't really see the math yet. Feb 28 '16 at 18:56
• In addition to that, I'm not entirely sure what to do in the sense of prefactors and such for the spectral densities using the Taylor expansion Feb 28 '16 at 18:58
• I think you may be confusing two different things. Johnson-Nyquist (JN) noise is a voltage (or current) noise coming from a resistor. That's not the same thing as the resistance itself fluctuating. If you want to understand the effect of the JN noise, please draw a circuit diagram (even if it's just one resistor and your bias coil) and ask how the JN noise adds to the other noises and I'll be happy to answer. By the way, this is one of my areas of expertise and this site could use more experimental questions, so please keep posting. Feb 28 '16 at 19:04

This circuit is linear so we can use superposition.

Denote the source voltage as $V_s$ and its spectral density as $S_{V_s}$. If this were the only noise source in the system then the flux noise would be $S_\Phi = C^2 S_{V_s}$. To see where the $C^2$ comes from, note that spectral density is equal to the Fourier transform of the autocorrelation function$^{[a,b]}$

$$S_V(\omega) = \int \, d\tau \, \langle V(0)V(\tau)\rangle \cos(\omega \tau) \, .$$

So then the flux noise would be

\begin{align} S_\Phi(\omega) =& \int \, d\tau \, \langle \Phi(0)\Phi(\tau)\rangle \cos(\omega \tau) \\ =& C^2 \int \, d\tau \, \langle V_s(0)V_s(\tau)\rangle \cos(\omega \tau) \\ =& C^2 S_{V_s}(\omega) \, . \end{align}

Now suppose we add the noisy resistor. The circuit model for a noisy resistor is a regular resistor in series with an ideal noisy voltage source $V_n$ which spits out noisy voltages that are Gaussian distributed and have a spectral density of open circuit voltage given as

$$S_{V_n}(f) = 4 k_b T R \frac{h f / k_b T}{\exp \left(h f / k_b T \right) - 1} \, .$$

If you're in the limit such that $h f \ll k_b T$, then this becomes the usual Johnson formula

$$S_{V_n}(f) = 4 k_b T R \, .$$

We can treat this resistor noisy voltage source and the other noisy voltage source as just two ideal voltages sources in series. Assuming these noises are independent, their resulting flux spectral densities just add.$^{[c]}$ Therefore, you just compute the flux spectral density coming from the voltage source noise, which is $S_{\Phi|V_s} = C^2 S_{V_s}$, and then you compute the spectral density coming from the resistor, which is $S_{\Phi|V_n} = C^2 S_{V_n}$, and add them.

Warning: It is really easy to get $C$ wrong in calculations like this. If your control wire is a coax line with 50$\,\Omega$ impedance and you're terminating it in a bias coil with much lower impedance, you have an impedance mismatch and there's probably a factor of 2 missing in your formula for $C$. If you post a question about that I'll answer it in detail.

Warning: Superconducting qubit generate their own flux noise (which I guess you know) so make sure your bias noises is below what you expect the intrinsic noise to be (usually around $1\,\mu\Phi_0 / \sqrt{\text{Hz}}$ at 1 Hz and with a roughly $1/f$ spectral density).

$[a]$: There are several conventions about spectral density which differ by factors of 2 etc. This convention is for single sided spectral density, i.e. what you see on a spectrum analyzer. However, I did not check this against my notes to make sure I got all the factors right, so use with caution if you see this note (will delete after I check).

$[b]$: The symbol $\langle \cdot \rangle$ means "ensemble average over many realizations of the noise".

$[c]$: You can prove this by using the formula for spectral density in terms of the autocorrelation function.

• That's a very nice answer, seems to make a lot of sense. It is kind of what I thought of on an intuitive level (after you helped me fix the actual question I was trying to ask) but this makes it better. As for C and the factor of 2, you make a good point. For now I'm still working out the concept on paper, once I move to the actual setup I will have to come back to this and a question along those lines might follow if I do not figure it out. In the end I suppose I will end up doing some spectroscopy to determine $C$ in any case, so the factor of two should show up then. Feb 28 '16 at 19:49