Classical particle in a Penning trap

[kelly0303]

Hello! I have a particle in a Penning trap (moving only along the axial direction) and I have a resonant circuit connected to one of the electrodes, measuring the current induced by the particle. Assume that the energy of the particle is much bigger than the thermal energy of the circuit ($k_B T$). If I were to measure the current induced by the particle in real time (not the FFT of it), would I see a sinusoidal signal with amplitude given by the amplitude of the particle and on top of this a jitter given by the amplitude of the thermal noise? So basically, would the trajectory of the particle be a perfect harmonic oscillation, and on top of that a random motion given by the thermal noise?

 

[Twigg]

Yep! The trajectory will be a solution of the equation $$\frac{d^2 z}{dt^2} + \gamma \frac{dz}{dt} + \omega_0^2 z = h(t)$$ where $\gamma$ is the damping factor (related to the resonant circuit's resistance), $\omega_0$ is the axial trap frequency, and $h(t)$ is the acceleration due to thermal noise. For the case where the particle's oscillatory motion is much larger than thermal motion, then the particle moves in a nearly harmonic (but damped) path but with a little thermal noise randomly jiggling it around.

The fluctuation-dissipation theorem states that the rate at which the particle's oscillatory motion damps out is related to magnitude of the thermal motion. The easy way to see this is to recall that $\gamma \propto R$ and $h(t) \propto \sqrt{R}$, so $h(t) \propto \sqrt{\gamma}$. (Edit: R is the resistance of the resonant circuit.)

 

[kelly0303]

Yep! The trajectory will be a solution of the equation $$\frac{d^2 z}{dt^2} + \gamma \frac{dz}{dt} + \omega_0^2 z = h(t)$$ where $\gamma$ is the damping factor (related to the resonant circuit's resistance), $\omega_0$ is the axial trap frequency, and $h(t)$ is the acceleration due to thermal noise. For the case where the particle's oscillatory motion is much larger than thermal motion, then the particle moves in a nearly harmonic (but damped) path but with a little thermal noise randomly jiggling it around.

The fluctuation-dissipation theorem states that the rate at which the particle's oscillatory motion damps out is related to magnitude of the thermal motion. The easy way to see this is to recall that $\gamma \propto R$ and $h(t) \propto \sqrt{R}$, so $h(t) \propto \sqrt{\gamma}$. (Edit: R is the resistance of the resonant circuit.)

Thank you! So if we had no external circuit (or if the circuit was not resonant with the ion motion), would we still have this jitter? In principle we still have image charges on the electrodes of the trap, which couples the ion with the motion of the electrons inside the trap walls, so naively I would assume that the jitter is still there. Also for the same temperature, the thermal energy is the same (with or without resonator) i.e. $k_bT$. However I am not sure how the coupling between the ion and the resonator comes into play. Is the amplitude of the jitter proportional to the Q value of the circuit (I thought that has to do with the damping, not with the thermal noise)?

 

[hutchphd]

Yep! The trajectory will be a solution of the equation d2zdt2+γdzdt+ω02z=h(t) where γ is the damping factor (related to the resonant circuit's resistance), ω0 is the axial trap frequency, and h(t) is the acceleration due to thermal noise.

You have confused me slightly. Is the damping from the sample or is it from the rest of the tank circuit or both? I am always fascinated by the fluctuation-dissipation theorem... it seems to be lurking everywhere.

Is the amplitude of the jitter proportional to the Q value of the circuit (I thought that has to do with the damping, not with the thermal noise)?

Please acquaint yourself thoroughly with the fluctuation-dissipation theorem; IMHO a very important and universal result. It says essentially that "thermal noise" and damping are different sides of the same coin.

.

 

[Twigg]

So if we had no external circuit (or if the circuit was not resonant with the ion motion), would we still have this jitter?

Nope!

In principle we still have image charges on the electrodes of the trap, which couples the ion with the motion of the electrons inside the trap walls, so naively I would assume that the jitter is still there.

So when I said thermal noise earlier, I meant thermal noise due to the Johnson-Nyquist noise of the tank circuit, which gets applied to the electrodes and accelerates the ion. If you disconnect the tank circuit, it doesn't add any noise to the ion. There might be other sources of thermal noise, but they small relative to the tank circuit's contribution.

Also, for others following this thread and confused what a Penning trap is, check out this article, sections 3.1 and 3.2.

However I am not sure how the coupling between the ion and the resonator comes into play. Is the amplitude of the jitter proportional to the Q value of the circuit (I thought that has to do with the damping, not with the thermal noise)?

It's both! (Although I think the amplitude of the jitter should be *inverse* proportional to the Q-factor.) Damping and thermal noise are intrinsically related, and this is the content of the fluctuation-dissipation theorem (FDT). Honestly, I wouldn't bother remembering the mathematical statement for FDT. I've wasted so many hours on that, only to end up back where I started. The juicy part of the theorem is that the noise power (noise amplitude squared) is proportional to the damping coefficient. This fact shows up everywhere: Brownian motion, electrical thermal noise, thermally-driven electromagnetic fields, even thermal jitter of mechanical resonators.

P.S., Technically, the "juicy part" of the FDT states that the autocorrelation of the thermal noise (as opposed to the noise power) is proportional to the damping coefficient. Again, the exact math isn't terribly useful since it's usually easier to derive the FDT for a given system from scratch than to use a generalized FDT. Just my two cents.

Specifically, for the Penning trap, here's how you get the FDT:

First you need to find an expression for $\gamma$ in terms of the resonator circuit resistance $R$. You know that the induced charge on the electrode is given by $q' = q \frac{z}{D}$ for some length $D$ that depends on the trap geometry. Thus, the current going through the resonator circuit is $\dot{q'} = q \frac{\dot{z}}{D}$, and the power dissipated is $$\begin{equation} \tag{1}P = I^2 R = q^2 \frac{\dot{z}^2}{D^2} R \end{equation}$$. Likewise, you know the energy dissipated in the ion's motion is $F_{damp} \cdot \dot{z}$, so $$\begin{equation} \tag{2} P = \gamma \dot{z}^2 \end{equation}$$ Combining equations (1) and (2) yields $$\begin{equation} \tag{3} \gamma = \frac{q^2}{D^2}R \end{equation}$$

Next, you need to evaluate the spectral density of the thermal noise. Let $\tilde{h}(\omega)$ be the Fourier transform of $h(t)$, the force on the ion due to thermal disturbances divided by mass. Then we know that $$\begin{equation} \tag{4} \tilde{h}(\omega) = \frac{q}{m} \tilde{E_t}(\omega) = \frac{q}{m} \frac{\tilde{V_t} (\omega)}{L} \end{equation}$$ where $E_t$ is the electric field at the ion's position due to thermal noise, $V_t$ is the voltage at the electrode due to thermal noise, and $L$ is another geometry-dependent length scale that converts electrode voltage to electric field at the trap center. We know that the voltage on the electrode $V_t$ will be given by Johnson-Nyquist noise: $$\begin{equation} \tag{5} |\tilde{V_t}(\omega)|^2 = 4 k_B T R \end{equation}$$. So, combining equations (4) and (5), $$\begin{equation} \tag{6} |\tilde{h} (\omega)|^2 = \left(\frac{q}{m}\right)^2 L^{-2} \times 4k_B T R \end{equation}$$

Combining equations (3) and (6) gives us the FDT: $$|\tilde{h} (\omega)|^2 = \left( \frac{D}{L} \right)^2 \frac{4 k_B T}{m^2} \gamma$$ As was promised, $|h|^2 \propto \gamma$.

Edit: I left out a factor of R in equation 6. Fixed that!

 

[kelly0303]

Nope!So when I said thermal noise earlier, I meant thermal noise due to the Johnson-Nyquist noise of the tank circuit, which gets applied to the electrodes and accelerates the ion. If you disconnect the tank circuit, it doesn't add any noise to the ion. There might be other sources of thermal noise, but they small relative to the tank circuit's contribution.

Also, for others following this thread and confused what a Penning trap is, check out this article, sections 3.1 and 3.2.It's both! (Although I think the amplitude of the jitter should be *inverse* proportional to the Q-factor.) Damping and thermal noise are intrinsically related, and this is the content of the fluctuation-dissipation theorem (FDT). Honestly, I wouldn't bother remembering the mathematical statement for FDT. I've wasted so many hours on that, only to end up back where I started. The juicy part of the theorem is that the noise power (noise amplitude squared) is proportional to the damping coefficient. This fact shows up everywhere: Brownian motion, electrical thermal noise, thermally-driven electromagnetic fields, even thermal jitter of mechanical resonators.

P.S., Technically, the "juicy part" of the FDT states that the autocorrelation of the thermal noise (as opposed to the noise power) is proportional to the damping coefficient. Again, the exact math isn't terribly useful since it's usually easier to derive the FDT for a given system from scratch than to use a generalized FDT. Just my two cents.

Specifically, for the Penning trap, here's how you get the FDT:

First you need to find an expression for $\gamma$ in terms of the resonator circuit resistance $R$. You know that the induced charge on the electrode is given by $q' = q \frac{z}{D}$ for some length $D$ that depends on the trap geometry. Thus, the current going through the resonator circuit is $\dot{q'} = q \frac{\dot{z}}{D}$, and the power dissipated is $$\begin{equation} \tag{1}P = I^2 R = q^2 \frac{\dot{z}^2}{D^2} R \end{equation}$$. Likewise, you know the energy dissipated in the ion's motion is $F_{damp} \cdot \dot{z}$, so $$\begin{equation} \tag{2} P = \gamma \dot{z}^2 \end{equation}$$ Combining equations (1) and (2) yields $$\begin{equation} \tag{3} \gamma = \frac{q^2}{D^2}R \end{equation}$$

Next, you need to evaluate the spectral density of the thermal noise. Let $\tilde{h}(\omega)$ be the Fourier transform of $h(t)$, the force on the ion due to thermal disturbances divided by mass. Then we know that $$\begin{equation} \tag{4} \tilde{h}(\omega) = \frac{q}{m} \tilde{E_t}(\omega) = \frac{q}{m} \frac{\tilde{V_t} (\omega)}{L} \end{equation}$$ where $E_t$ is the electric field at the ion's position due to thermal noise, $V_t$ is the voltage at the electrode due to thermal noise, and $L$ is another geometry-dependent length scale that converts electrode voltage to electric field at the trap center. We know that the voltage on the electrode $V_t$ will be given by Johnson-Nyquist noise: $$\begin{equation} \tag{5} |\tilde{V_t}(\omega)|^2 = 4 k_B T R \end{equation}$$. So, combining equations (4) and (5), $$\begin{equation} \tag{6} |\tilde{h} (\omega)|^2 = \left(\frac{q}{m}\right)^2 L^{-2} \times 4k_B T \end{equation}$$

Combining equations (3) and (6) gives us the FDT: $$|\tilde{h} (\omega)|^2 = \left( \frac{D}{L} \right)^2 \frac{4 k_B T}{m^2} \gamma$$ As was promised, $|h|^2 \propto \gamma$.

Thanks a lot! This makes things a lot more clear! Does the frequency dependence of $h(\omega)$ come from the $R$ in $\gamma$, which is the real part of the impedance of the circuit (which has a dependence on $\omega$)? So if I want to calculate $h(t)$ (this is what I need if I want to check the induced amplitude due to the thermal noise relative to the actual ion amplitude, right?), I need to FT the impedance of the external circuit?

 

[Twigg]

Does the frequency dependence of h(ω) come from the R in γ, which is the real part of the impedance of the circuit (which has a dependence on ω)?

In the derivation I did above, the dependence of $\tilde{h}(\omega)$ on R doesn't come from $\gamma$ (which is a property of the ion motion). Rather, it comes from the thermal voltage noise from the tank circuit, which is proportional to $R$, as in Equation (5) of my last post.
That expression (equation 5) is the standard formula for thermal voltage noise of a resistance $R$. See this chapter of Elementary Statistical Physics by C. Kittel for a derivation from statistical mechanics. The short version is that he uses stat. mech. to get the expected value of the energy of a single mode of a transmission line, and then he sets this equal to the power spectral density $|\tilde{V}(\omega)|^2/R$. There is a simpler derivation of equation (5) from the Drude model, but the electron's equation of motion in the Drude model is identical to the ion's equation of motion in the Penning trap, and I felt that the notation would get very confusing. I can still reproduce that Drude model derivation if you find Kittel's derivation unhelpful, just let me know!

So if I want to calculate h(t) (this is what I need if I want to check the induced amplitude due to the thermal noise relative to the actual ion amplitude, right?)

To answer the question in parentheses, almost! There's one more step to getting the motion induced by thermal noise.

Let me write the total motion of the ion as $$\begin{equation} \tag{1} z(t) = z_0 e^{-\gamma t} \cos(\omega_1 t + \phi) + z_t(t) \end{equation}$$ where $z_0$ is the amplitude of the oscillatory motion, $\omega_1 = \sqrt{\omega_0^2 - \frac{\gamma^2}{4}}$ is the damped ion resonance frequency, $\phi$ is the initial phase of the oscillation, and where $z_t(t)$ is the noise induced on the ion by thermal noise on the tank circuit.

If we stick this into the equation of motion, we get $$\begin{equation} \tag{2} \frac{d^2 z_t}{dt^2} + \gamma \frac{dz_t}{dt} + \omega_0^2 z_t = h(t) \end{equation}$$ The oscillation doesn't appear at all in equation (2) because it's a solution to the homogenous equation so it cancels.

Now, if we take the Fourier transform of equation (2) and solve for $\tilde{z_t}(\omega)$ (FT of the thermally-induced ion motion), that gives $$\begin{equation} \tag{3} \tilde{z_t}(\omega) = \frac{\tilde{h}(\omega)}{ \omega_0^2 - \omega^2 + i\gamma \omega} \end{equation}$$

Now, if we plug this result into the final result of my last thread, we get $$\begin{equation} \tag{4} |\tilde{z_t}(\omega)|^2 = \frac{\gamma}{|\omega_0^2 - \omega^2 + i\gamma \omega|^2} \left(\frac{D}{L} \right)^2 \frac{4 k_B T}{m^2} \end{equation}$$ And I think this is the quantity you are looking for, right? The amplitude of the ion's motion induced by the tank circuit's thermal noise?

I need to FT the impedance of the external circuit?

As shown above, it's more that you need the FT of the ion's "impedance". What made you think you needed to FT the circuit's impedance? I might be missing something.

Was that helpful?

 

[kelly0303]

In the derivation I did above, the dependence of $\tilde{h}(\omega)$ on R doesn't come from $\gamma$ (which is a property of the ion motion). Rather, it comes from the thermal voltage noise from the tank circuit, which is proportional to $R$, as in Equation (5) of my last post.
That expression (equation 5) is the standard formula for thermal voltage noise of a resistance $R$. See this chapter of Elementary Statistical Physics by C. Kittel for a derivation from statistical mechanics. The short version is that he uses stat. mech. to get the expected value of the energy of a single mode of a transmission line, and then he sets this equal to the power spectral density $|\tilde{V}(\omega)|^2/R$. There is a simpler derivation of equation (5) from the Drude model, but the electron's equation of motion in the Drude model is identical to the ion's equation of motion in the Penning trap, and I felt that the notation would get very confusing. I can still reproduce that Drude model derivation if you find Kittel's derivation unhelpful, just let me know!To answer the question in parentheses, almost! There's one more step to getting the motion induced by thermal noise.

Let me write the total motion of the ion as $$\begin{equation} \tag{1} z(t) = z_0 e^{-\gamma t} \cos(\omega_1 t + \phi) + z_t(t) \end{equation}$$ where $z_0$ is the amplitude of the oscillatory motion, $\omega_1 = \sqrt{\omega_0^2 - \frac{\gamma^2}{4}}$ is the damped ion resonance frequency, $\phi$ is the initial phase of the oscillation, and where $z_t(t)$ is the noise induced on the ion by thermal noise on the tank circuit.

If we stick this into the equation of motion, we get $$\begin{equation} \tag{2} \frac{d^2 z_t}{dt^2} + \gamma \frac{dz_t}{dt} + \omega_0^2 z_t = h(t) \end{equation}$$ The oscillation doesn't appear at all in equation (2) because it's a solution to the homogenous equation so it cancels.

Now, if we take the Fourier transform of equation (2) and solve for $\tilde{z_t}(\omega)$ (FT of the thermally-induced ion motion), that gives $$\begin{equation} \tag{3} \tilde{z_t}(\omega) = \frac{\tilde{h}(\omega)}{ \omega_0^2 - \omega^2 + i\gamma \omega} \end{equation}$$

Now, if we plug this result into the final result of my last thread, we get $$\begin{equation} \tag{4} |\tilde{z_t}(\omega)|^2 = \frac{\gamma}{|\omega_0^2 - \omega^2 + i\gamma \omega|^2} \left(\frac{D}{L} \right)^2 \frac{4 k_B T}{m^2} \end{equation}$$ And I think this is the quantity you are looking for, right? The amplitude of the ion's motion induced by the tank circuit's thermal noise?As shown above, it's more that you need the FT of the ion's "impedance". What made you think you needed to FT the circuit's impedance? I might be missing something.

Was that helpful?

Thank you for this! I might not be sure myself what I need (sorry for that!). So basically, I want to measure the ion oscillation in real time (in the regime where the amplitude is much bigger than the thermal noise). If I do that, what I would measure is equation (1) i.e. a damping sinusoidal with noise on top. What I want to know is the magnitude of this noise (so that would be $z_t(t)$, right?). What I meant (I think) is that equation (4) gives $z_t(\omega)$, so in order to get $z_t(t)$ I would need to FT that. Or I can solve directly equation (2), but for that, again I need $h(t)$, while these formulas give me $h(\omega)$. Am I asking the wrong question?

 

[Twigg]

Thank you for this! I might not be sure myself what I need (sorry for that!).

No worries! That's just how science goes

Your comment about needing the circuit impedance has been making me re-think the math I posted. In fact, I'm quite sure I was wrong. Specifically, I think I didn't account for the Johnson-Nyquist noise correctly, because I gave a formula for white noise $|\tilde{V_t}(\omega)|^2 = 4k_B T R$ when in reality the voltage noise you'd measure on the electrode should be limited to the bandwidth of the resonant circuit, as in Figure 7 of this article. I think the formula I put up is only accurate if the tank circuit is only a resistor and no resonant circuit. Also, I recommend you take a look at that figure, because it's exactly the situation you're interested in just in the frequency-axis instead of the time-axis. Note that the broad peak around 651400 Hz (0 Hz on the plot's x-axis) is $|\tilde{V_t}(\omega)^2|$ and the narrow peak about 75 Hz away is the ion's oscillatory motion. The SNR on that figure 7 is the SNR you'd get if you measured the amplitude of the ion's oscillating motion. Could that be what you're interested in?

That paper's equation 16 gives the correct result. In the notation we've been using, it's $$\gamma = \frac{q^2 \mathrm{Re}(Z(\omega))}{m D^2}$$ (Looks like I also lost a factor of $m$ in my equation 3 of post #5, whoops!) Sorry again that I messed that up.