Particle behavior in Penning trap

[Malamala]

Hello! In most of the modern mass measurements in a penning trap, they cool down the degrees of freedom of the ion (the 3 eigenmotions) using resistive cooling, in which they couple an external circuit to some of the electrodes of the trap and the ion is cooled down to the temperature of the external circuit by charged images. Usually this temperature is a few degrees kelvin, so the motion is still classical (the quantum number is of the order $10^6$).

I am not sure how the ion moves, once it reaches the temperature of the external circuit. It is still in the harmonic potential of the trap itself, but it is also in a thermal bath i.e. its energy is Boltzmann distributed. In the most general case (assume that we are looking only at the z-axis motion), the equation of motion would be $z(t)=z_0(t)cos(\omega t + \phi(t))$, where $z_0$, $\omega$ and $\phi$ are the amplitude, frequency and phase of the motion.

I assume that the frequency is constant (this is what they actually measure in practice to extract the mass), which would mean that $\phi$ should also be constant. Otherwise, by doing a Taylor expansion we would have a term linear in $t$ that would add some random noise to $\omega$ and we can't separate the 2 time dependencies in practice. So this would mean that in the end the equation of motion is given by $z(t)=z_0(t)cos(\omega t + \phi_0)$.

In this case the amplitude would be given by the energy of the external circuit at a given time t, which is Boltzmann distributed. Is this right? It seems a bit weird that despite the fact that it is in thermal equilibrium with electronic noise the ion in the trap still has a nice oscillatory motion. Can someone help me understand better how the ion actually behaves in these circumstances?

 

[Twigg]

In the absence of resistive cooling, the equation of motion for a single trapped ion (no collisions) is simply $$ \ddot{z} + \omega_z^2 z = 0$$ where $\omega_z$ is the axial trapping frequency and z is the displacement along the trap axis (z=0 is the trap center).
Note that, hypothetically, if you added a time-dependent axial electric field $\vec{E} = E_z(t)\hat{z}$ to the trapping potential (but still no resistive cooling), the resulting equation of motion would be $$\ddot{z} + \omega_z^2 z = \frac{Ze}{m} E_z(t)$$ where Z is the number of charges on the ion (singly ionized means Z=1, doubly ionized means Z=2, etc).

Now to resistive cooling. When the resistor is connected, the equation of motion becomes $$\ddot{z} - \gamma \dot{z} + \omega_z^2 z = 0$$ where $\gamma$ aka the cooling rate is a function of the resistance applied and the trap geometry. Note that the trajectories that solve this equation of motion take the form $z(t) = z(0) e^{-\gamma t} \cos(\omega_1 t + \phi)$ where $\omega_1 = \sqrt{\omega_z^2 - \frac{\gamma^2}{4}}$ and where $\phi$ is a constant.
Because there is thermal noise (see Johnson-Nyquist noise) on the resistor, those voltage fluctuations will reach the trapping electrodes and apply a random force on the ions. In other words, the full equation of motion is now $$\ddot{z} - \gamma \dot{z} + \omega_z^2 z = h(t)$$ where $h(t)$ is a completely random function of time (has autocorrelation equal to a delta function).
Consider two limiting cases: (1) when the ion's kinetic energy greatly exceeds the limiting temperature $\frac{1}{2}m\dot{z}^2 \gg k_B T$ and (2) when the kinetic energy is the same order as the limiting temperature $\frac{1}{2}m\dot{z}^2 \approx k_B T$.
In case (1), the ion trajectory barely notices the presence of the miniscule random force, and the trajectory is approximately the same as the homogenous case: $z(t) \approx z(0) e^{-\gamma t} \cos(\omega_1 t + \phi)$. However, as time goes on the effect of the random force will cause the phase of oscillation to accumulate random shifts back and forth, so what you will see is an unsteady phase $\phi(t)$. That's what your expression $z(t) = z_0(t) cos(\omega t + \phi(t))$ means. The phase will drift due to the force of thermal noise. The effect on the phase is usually more easily noticeable than the effect on the amplitude.
In case (2), the motion of the ion won't appear like a decaying sinusoid at all. It'll just do a jig of random motions. The power spectrum of these motions depend on the combined transfer function of your trapping electrode geometry and external circuit.

 

[Malamala]

In the absence of resistive cooling, the equation of motion for a single trapped ion (no collisions) is simply $$ \ddot{z} + \omega_z^2 z = 0$$ where $\omega_z$ is the axial trapping frequency and z is the displacement along the trap axis (z=0 is the trap center).
Note that, hypothetically, if you added a time-dependent axial electric field $\vec{E} = E_z(t)\hat{z}$ to the trapping potential (but still no resistive cooling), the resulting equation of motion would be $$\ddot{z} + \omega_z^2 z = \frac{Ze}{m} E_z(t)$$ where Z is the number of charges on the ion (singly ionized means Z=1, doubly ionized means Z=2, etc).

Now to resistive cooling. When the resistor is connected, the equation of motion becomes $$\ddot{z} - \gamma \dot{z} + \omega_z^2 z = 0$$ where $\gamma$ aka the cooling rate is a function of the resistance applied and the trap geometry. Note that the trajectories that solve this equation of motion take the form $z(t) = z(0) e^{-\gamma t} \cos(\omega_1 t + \phi)$ where $\omega_1 = \sqrt{\omega_z^2 - \frac{\gamma^2}{4}}$ and where $\phi$ is a constant.
Because there is thermal noise (see Johnson-Nyquist noise) on the resistor, those voltage fluctuations will reach the trapping electrodes and apply a random force on the ions. In other words, the full equation of motion is now $$\ddot{z} - \gamma \dot{z} + \omega_z^2 z = h(t)$$ where $h(t)$ is a completely random function of time (has autocorrelation equal to a delta function).
Consider two limiting cases: (1) when the ion's kinetic energy greatly exceeds the limiting temperature $\frac{1}{2}m\dot{z}^2 \gg k_B T$ and (2) when the kinetic energy is the same order as the limiting temperature $\frac{1}{2}m\dot{z}^2 \approx k_B T$.
In case (1), the ion trajectory barely notices the presence of the miniscule random force, and the trajectory is approximately the same as the homogenous case: $z(t) \approx z(0) e^{-\gamma t} \cos(\omega_1 t + \phi)$. However, as time goes on the effect of the random force will cause the phase of oscillation to accumulate random shifts back and forth, so what you will see is an unsteady phase $\phi(t)$. That's what your expression $z(t) = z_0(t) cos(\omega t + \phi(t))$ means. The phase will drift due to the force of thermal noise. The effect on the phase is usually more easily noticeable than the effect on the amplitude.
In case (2), the motion of the ion won't appear like a decaying sinusoid at all. It'll just do a jig of random motions. The power spectrum of these motions depend on the combined transfer function of your trapping electrode geometry and external circuit.

Hello! I am actually interested in the $\frac{1}{2}m\dot{z}^2 \gg k_B T$ case. So if the phase is a random function of time, doesn't it mean that the frequency is also a random function of time? I am a bit confused as in the mass measurements in a Penning trap, they measure the cyclotron frequency (which amounts to measuring the 3 eigenfrequency and adding them in quadrature) and from there, knowing the charge of the ion and the magnetic field they extract the mass. How come they are not affected by this random phase which affects the frequency?

 

[Twigg]

You'd disconnect the resistor from the endcaps for the axial frequency measurement. If you left the resistor connected, you would have the phase drift and other problems. For example, the axial frequency would be offset ($\omega_1 \neq \omega_0$ for a damped harmonic oscillator) and you'd see a systematic error of $\approx \frac{\gamma^2}{4\omega_z}$ in the cyclotron frequency.

 

[Malamala]

You'd disconnect the resistor from the endcaps for the axial frequency measurement. If you left the resistor connected, you would have the phase drift and other problems. For example, the axial frequency would be offset ($\omega_1 \neq \omega_0$ for a damped harmonic oscillator) and you'd see a systematic error of $\approx \frac{\gamma^2}{4\omega_z}$ in the cyclotron frequency.

But as far as I understand they use the signal from the circuit to measure the frequency. What exactly do you mean by removing the resistor?

 

[Twigg]

Right, sorry. I'm used to doing trap frequency measurements by imaging the ion cloud's axial position on a phosphor plate and I forgot that precision mass measurements use the pickup signal on the endcaps. Is there a particular paper or experiment you're interested in? It's easy to get off track talking about a non-specific system.

I would point out that there's no rule saying that you have to use the same circuit for cooling as you do for measurement. You can have a cooling circuit and a measurement circuit, selected by mosfets for example. For cooling, you may want a large resistor to dissipate power faster, but you might switch to a lower impedance circuit to do the axial frequency measurement since damping always make frequency measurements harder.

If you can refer us to a particular experiment of interest, I might be able to estimate how much of an effect this phase drift makes.

 

[Malamala]

Right, sorry. I'm used to doing trap frequency measurements by imaging the ion cloud's axial position on a phosphor plate and I forgot that precision mass measurements use the pickup signal on the endcaps. Is there a particular paper or experiment you're interested in? It's easy to get off track talking about a non-specific system.

I would point out that there's no rule saying that you have to use the same circuit for cooling as you do for measurement. You can have a cooling circuit and a measurement circuit, selected by mosfets for example. For cooling, you may want a large resistor to dissipate power faster, but you might switch to a lower impedance circuit to do the axial frequency measurement since damping always make frequency measurements harder.

If you can refer us to a particular experiment of interest, I might be able to estimate how much of an effect this phase drift makes.

The main one I am looking at is ALPHATRAP (the trap that has a uniform field in it), but I looked at several others from the same group and they use similar techniques. As far as I understand they use an external circuit for cooling and once the particle reaches thermal equilibrium with the circuit, they use the same circuit to read the signal. They use a dip that appears in the noise signal at the resonant frequency (the axial one in this case) to measure the frequency. I need to look more into the math of how that appears as it is not clear to me yet, but the main point is that it seems that they are able to extract precisely the axial frequency of the ion, despite the ion being in thermal equilibrium with the circuit. Also the width of this dip is given by a time constant (Chapter 3.2 equation 16) which appears to be fixed for a given ion and circuit, which means that no matter how long we do the measurement, the width would be constant, so the effect of this jitter due to the thermal noise would never affect the resolution of the measured frequency. But I am probably miss understanding something.

 

[Twigg]

Ah, ok now this is a lot clearer, thanks for sharing the paper!

For this noise dip measurement, you're really interested in the case where $\frac{1}{2} m\dot{z}^2 \approx k_B T$ and the motion of the ion is not sinusoidal at all. The ion's motion is Brownian motion or something like it. They're not plotting out the ion's position versus time, instead they're just observing a spectral feature, so it doesn't matter that the ion's trajectory is completely random. Does that make sense?

Edit: Just wanted to add, when the ion(s) reaches thermal equilibrium, that's the same as saying $\frac{1}{2} m\dot{z}^2 \approx k_B T$. In other words, once the ion cools down to equilibrium it no longer oscillates and instead moves randomly like Brownian motion. I think that may be part of the confusion

 

[Malamala]

Ah, ok now this is a lot clearer, thanks for sharing the paper!

For this noise dip measurement, you're really interested in the case where $\frac{1}{2} m\dot{z}^2 \approx k_B T$ and the motion of the ion is not sinusoidal at all. The ion's motion is Brownian motion or something like it. They're not plotting out the ion's position versus time, instead they're just observing a spectral feature, so it doesn't matter that the ion's trajectory is completely random. Does that make sense?

Edit: Just wanted to add, when the ion(s) reaches thermal equilibrium, that's the same as saying $\frac{1}{2} m\dot{z}^2 \approx k_B T$. In other words, once the ion cools down to equilibrium it no longer oscillates and instead moves randomly like Brownian motion. I think that may be part of the confusion

Ah, I said that we are in the $\frac{1}{2} m\dot{z}^2 \gg k_B T$ regime, I apologize for that. Of course we are at $\frac{1}{2} m\dot{z}^2 \approx k_B T$, as we reached thermal equilibrium, sorry for the confusion. So what you are saying is that the ion has a random motion along the z axis, but once we do a Fourier transform of that motion (of the measured voltage) we are able to see clearly a peak? But I still have some questions. If the axial frequency would be at the resonator frequency, I imagine that it gets amplified more than the other frequencies of the ion, but why is that so narrow, and not as wide as the electronic noise amplification? And why is it downwards (it has to do with the ion charge being positive)? Also in figure 7 the axial frequency is detuned from the resonator one and we still see a clear peak. How can that one get amplified so much when it is far from the resonator circuit? Thank you!

 

[Twigg]

Check out Fig. 8 in the paper you linked, noting both the circuit diagram and the plot. Keep in mind that the ion only moves in response to voltages applied to the endcaps. When you look at the Fourier spectrum of the voltage on the endcap, what you see is in fact the thermal noise of the tank circuit after being filtered by both the tank circuit and the ions themselves. The broad peak in the spectrum in Fig. 8 has a broad lineshape given by the bandwidth of the tank circuit, and the dip has a narrow lineshape given by the frequency response of the ions (namely, the cooling rate $\gamma$ is the HWHM of the dip).

Also in figure 7 the axial frequency is detuned from the resonator one and we still see a clear peak. How can that one get amplified so much when it is far from the resonator circuit?

In Fig 7, this is a hot ion that is oscillating with an overall motion $z(t) = z_0 e^{-\gamma y} \cos(\omega_1 t + \phi)$. The reason that you can still see the ion's spectrum despite being detuned off the resonator is simply that the oscillation has a large amplitude compared to the thermal background since "hot" by definition means $\frac{1}{2} m\dot{z}^2 \gg k_b T$.

If the axial frequency would be at the resonator frequency, I imagine that it gets amplified more than the other frequencies of the ion, but why is that so narrow, and not as wide as the electronic noise amplification?

This just means that $\gamma \ll R_t C_t$. In other words, the tank circuit has a lower Q than the ion's motion.

And why is it downwards (it has to do with the ion charge being positive)?

Understanding this part is really the key to understanding the measurement, and has nothing to do with the ion being positive or negatively charged. Let's just assume the ion is positively charged for convenience. Think about what happens when you apply a +1V shift to one endcap (but not the other). The ion will move in the opposite direction to get away from the higher potential. The image charge will tend to be stronger on the other endcap (the one without the +1V shift), so the voltage induced by the ion's motion will counteract the applied voltage. (This isn't a unique property of trapped ions, it's also true for plasmas, conduction electrons, and the bound charges in dielectrics. As a rule of thumb, free charges will move in a way that counteracts the applied field.)

We know that the ion's motion is being driven by thermal noise from the tank circuit. Since the ion moves in a way that induces a counteracting voltage, the effect of the ion will be a drop in the thermal noise spectrum.

Recall the ion's equation of motion: $$\ddot{z} - \gamma \dot{z} + \omega_z^2 z = h(t)$$ where h(t) is a function that describes the acceleration due to thermal voltages versus time. Taking a Fourier transform of both sides and solving for the FT of z, you get $$z(\omega) = \frac{h(\omega)}{(\omega_z ^2 - \omega^2) - i\gamma \omega}$$
In other words, the ion's motion spectrum is tightly confined around $\omega_z$ with HWHM $\gamma$. Since we know that the ion's motion will detract from the noise spectrum at the end of the day, we know to expect a narrow dip (with a HWHM of $\gamma$) at $\omega = \omega_z$.

 

[Malamala]

Check out Fig. 8 in the paper you linked, noting both the circuit diagram and the plot. Keep in mind that the ion only moves in response to voltages applied to the endcaps. When you look at the Fourier spectrum of the voltage on the endcap, what you see is in fact the thermal noise of the tank circuit after being filtered by both the tank circuit and the ions themselves. The broad peak in the spectrum in Fig. 8 has a broad lineshape given by the bandwidth of the tank circuit, and the dip has a narrow lineshape given by the frequency response of the ions (namely, the cooling rate $\gamma$ is the HWHM of the dip).
In Fig 7, this is a hot ion that is oscillating with an overall motion $z(t) = z_0 e^{-\gamma y} \cos(\omega_1 t + \phi)$. The reason that you can still see the ion's spectrum despite being detuned off the resonator is simply that the oscillation has a large amplitude compared to the thermal background since "hot" by definition means $\frac{1}{2} m\dot{z}^2 \gg k_b T$.
This just means that $\gamma \ll R_t C_t$. In other words, the tank circuit has a lower Q than the ion's motion.Understanding this part is really the key to understanding the measurement, and has nothing to do with the ion being positive or negatively charged. Let's just assume the ion is positively charged for convenience. Think about what happens when you apply a +1V shift to one endcap (but not the other). The ion will move in the opposite direction to get away from the higher potential. The image charge will tend to be stronger on the other endcap (the one without the +1V shift), so the voltage induced by the ion's motion will counteract the applied voltage. (This isn't a unique property of trapped ions, it's also true for plasmas, conduction electrons, and the bound charges in dielectrics. As a rule of thumb, free charges will move in a way that counteracts the applied field.)

We know that the ion's motion is being driven by thermal noise from the tank circuit. Since the ion moves in a way that induces a counteracting voltage, the effect of the ion will be a drop in the thermal noise spectrum.

Recall the ion's equation of motion: $$\ddot{z} - \gamma \dot{z} + \omega_z^2 z = h(t)$$ where h(t) is a function that describes the acceleration due to thermal voltages versus time. Taking a Fourier transform of both sides and solving for the FT of z, you get $$z(\omega) = \frac{h(\omega)}{(\omega_z ^2 - \omega^2) - i\gamma \omega}$$
In other words, the ion's motion spectrum is tightly confined around $\omega_z$ with HWHM $\gamma$. Since we know that the ion's motion will detract from the noise spectrum at the end of the day, we know to expect a narrow dip (with a HWHM of $\gamma$) at $\omega = \omega_z$.

This is such a good explanation! Thank you so much! So basically as a function of time the ion has a random motion once it reaches thermal equilibrium, but going to the frequency space, the motion is quite localized around $\omega_z$, which is what we want to measure in practice. And given that this random motion is in response to the random motion of the charges in the external circuit, the Fourier transform of the spectrum appears subtracted from the Fourier spectrum of the circuit alone. So is there a way to measure the axial frequency in a non-destructive way, without having an external circuit i.e. such that the motion of the ion is actually harmonic as a function if time?

EDIT: I am actually more curios if there is a way to extract the phase and/or the amplitude of the ion in a non-destructive way, after the ion reaches thermal equilibrium and the external circuit is disconnected (such that the motion is harmonic).

 

[Twigg]

Yes, there is a way to measure the axial frequency without an external circuit (probably several, but I can only think of the one off the top of my head), and that's using time-of-flight (TOF) in the axial direction or imaging from the side using an MCP (micro-channel plate detector) with a phosphor plate. I couldn't tell you how feasible these are in a Penning trap. I'm more used to Paul traps.

The main issue is that both of these methods require complicated electrode geometries that will make your ion deviate from the ideal motions. Basically, both methods require you to leave a path for the ions to fly out of the trap without being obstructed. Axial TOF requires ring-shaped endcaps so the ions can fly out along the z axis, and imaging from the side requires you to have a path for the electrons to fly out perpendicular to the z axis. These gaps will result in systematic errors in the observed cyclotron frequency.

In both of these methods, you'd apply a pulse to one endcap to drive the ion(s) into an axial oscillation. Then you'd wait for a certain amount of time, and kick all the ions onto the detector for TOF measurement or imaging. In the case of TOF measurement, the delay of the ion on the TOF trace corresponds to its position before being kicked. In the imaging case, you'd actually see the position of the ion(s) as fluoresence off the phosphor plate that's imaged on a camera.

The issue with imaging is the spatial resolution is on the order of a few microns, since that's the distance between adjacent pores on the MCP plate. There's also the responsivity of the MCP to take into account. I'm not savvy enough anymore to remember whether you can expect to see a single ion this way. The fluoresence is probably too small to be imaged unless you have a crazy high gain MCP.

The TOF is probably the way I'd go if I had to choose. The geometry is less funky and don't need a phosphor plate, making single ions a more realistic possibility (note that's just "more realistic", not "realistic" I've never worked on single ion experiments so I'm not betting on it)

I would say however, that these methods are more destructive than the endcap measurement. In fact, I would've called the endcap measurement "non-destructive" because you don't have to smash the ions into the MCP.

Edit: another fun systematic arises when you consider the fact that in all real ion traps, the trapping frequencies vary with position. So the larger the amplitude of your motions, the more systematics you get on the cyclotron frequency. It's just another reason why the noise dip measurement is really clever: you use very small thermal motions and get a very narrow spectral line to measure (broader line = harder to measure the central frequency accurately).

 

[Malamala]

Yes, there is a way to measure the axial frequency without an external circuit (probably several, but I can only think of the one off the top of my head), and that's using time-of-flight (TOF) in the axial direction or imaging from the side using an MCP (micro-channel plate detector) with a phosphor plate. I couldn't tell you how feasible these are in a Penning trap. I'm more used to Paul traps.

The main issue is that both of these methods require complicated electrode geometries that will make your ion deviate from the ideal motions. Basically, both methods require you to leave a path for the ions to fly out of the trap without being obstructed. Axial TOF requires ring-shaped endcaps so the ions can fly out along the z axis, and imaging from the side requires you to have a path for the electrons to fly out perpendicular to the z axis. These gaps will result in systematic errors in the observed cyclotron frequency.

In both of these methods, you'd apply a pulse to one endcap to drive the ion(s) into an axial oscillation. Then you'd wait for a certain amount of time, and kick all the ions onto the detector for TOF measurement or imaging. In the case of TOF measurement, the delay of the ion on the TOF trace corresponds to its position before being kicked. In the imaging case, you'd actually see the position of the ion(s) as fluoresence off the phosphor plate that's imaged on a camera.

The issue with imaging is the spatial resolution is on the order of a few microns, since that's the distance between adjacent pores on the MCP plate. There's also the responsivity of the MCP to take into account. I'm not savvy enough anymore to remember whether you can expect to see a single ion this way. The fluoresence is probably too small to be imaged unless you have a crazy high gain MCP.

The TOF is probably the way I'd go if I had to choose. The geometry is less funky and don't need a phosphor plate, making single ions a more realistic possibility (note that's just "more realistic", not "realistic" I've never worked on single ion experiments so I'm not betting on it)

I would say however, that these methods are more destructive than the endcap measurement. In fact, I would've called the endcap measurement "non-destructive" because you don't have to smash the ions into the MCP.

Edit: another fun systematic arises when you consider the fact that in all real ion traps, the trapping frequencies vary with position. So the larger the amplitude of your motions, the more systematics you get on the cyclotron frequency. It's just another reason why the noise dip measurement is really clever: you use very small thermal motions and get a very narrow spectral line to measure (broader line = harder to measure the central frequency accurately).

Thank you for this! I would have to look more into that. Is there any way to measure the amplitude and/or phase of the ion at a given moment? I am thinking of an experiment in which I just want the ion to oscillate as much as possible and measure the transition between 2 levels (something similar to a Stark interference experiment), but in order to extract any physics I would need to know the initial conditions of the ion. Ideally I would cool it down as much as possible and then disconnect the external circuit (I can even call the ion further with laser cooling if needed), but I haven't found anything in literature that can give me the phase and amplitude at $t_0$ so I know what to expect after the ion oscillates for a time $t$.

 

[Twigg]

Ah I see what you mean. The methods I described can measure amplitude and phase if you take multiple measurements, but you don't preserve the ion. For that, you'd actually want to use the endcap signal for a suitably low value of $\gamma$. Essentially, you'd be reproducing the situation in Fig. 7 of that paper you linked (although you wouldn't need a tank circuit anymore). You could leave the cooling circuit connected just long enough to take an amplitude and phase measurement, then you'd turn it off using a MOSFET like I mentioned earlier. You'd have to do a little math to find out how much the amplitude and phase will change when you abruptly turn off the cooling circuit, but that change will be deterministic.

However, if you're cooling the ions that aggressively, especially with laser cooling, then the amplitude and phase of your oscillation will probably be determined by the timing of the pulse you apply. The ion will initially be "at rest", because the pulse you apply will cause it to slosh with a much higher kinetic energy than its thermal energy. Does that make sense? Essentially, you control the phase and amplitude with the timing and peak voltage of the driving pulse respectively.