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The classic charge qubit is the cooper pair box which is a capacitor in series with a Josephson junction. In my understanding, by changing the gate voltage at the capacitor, one can create a superposition of $n$ and $n+1$ cooper pairs on the 'island' in between the junction and capacitor.

A transmon looks far more like a classic LC circuit. It is often depicted as a Josephson junction in parallel with a very large capacitor and thus it is manipulated using microwave frequencies, not gate voltages. However, in all literature I can find it is called a special case of a charge qubit. I cannot seem to make sense of these two ideas. How are they equivalent?

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There are two things to consider:

  1. What does the potential look like?

  2. Is the wave function of the qubit narrow in the flux or charge basis?

Potential shape

The Hamiltonian of the transmon (a junction in parallel with a capacitor) is $$H_{\text{charge qubit}} = - E_J \cos(\phi) + \frac{(-2en)^2}{2C}$$ where $E_J\equiv I_c \Phi_0 / 2\pi$, $I_c$ is the junction critical current, $\phi$ is the phase across the junction and $n$ is the number of Cooper pairs which have tunneled through the junction. For easier comparison to other qubits it's really useful to note that using the Josephson relation $$V = \frac{\Phi_0}{2\pi}\dot{\phi}$$ and noting that the magnetic flux $\Phi$ is the time integral of the voltage we can write $$\Phi = \int dt V(t) = \Phi_0 \frac{\phi}{2\pi} \, .$$ Using this and the charge $Q = -2en$ the Hamiltonian becomes $$H_{\text{charge qubit}} = -E_J \cos(2\pi \Phi / \Phi_0) + \frac{Q^2}{2C} \, .$$ The $Q^2/2C$ term is the kinetic energy (notice the similarity to $p^2/2m$), and the part depending on $\Phi$ is the potential energy. Notice that, like with the charge qubit, this Hamiltonian's potential energy term is periodic. That is unlike the case with the e.g. flux qubit where the Hamiltonian is $$H_{\text{flux qubit}} = \frac{\Phi^2}{2L} - E_J \cos(2\pi \hat{\Phi}/\Phi_0) + \frac{Q^2}{2C} \, .$$

This is one of the main differences: the transmon Hamiltonian (like the charge qubit Hamiltonian) is periodic in the flux basis while the flux qubit Hamiltonian is not periodic in the flux basis. The physical reason for this difference is that the transmon (and charge) qubit do not have a dc path to ground. The junction and capacitor are in parallel with no e.g. inductor going to ground. The flux qubit has an inductance to ground; this inductance introduces the parabolic term in the Hamiltonian making the potential non-periodic. This is the sense in which a transmon is like a charge qubit.

Wave function widths

As you noted, the transmon is nearly a harmonic oscillator. The reason for this is that although the potential is periodic, the wave function is narrow enough that it mostly sits localized in a sincle well of the potential. We can check this self-consistently in an easy way: let's just compute the width of the wave function of a Harmonic oscillator which has the same parameters as a typical transmon. For a harmonic oscillator with Hamiltonian $$H = \frac{1}{2} \alpha u^2 + \frac{1}{2} \beta v^2 \qquad [u,v] = i \gamma $$ The mean square of $u$ in the ground state is $$\langle 0 | u^2 | 0 \rangle = (1/2) \gamma \sqrt{\beta / \alpha} \, . $$ The Harmonic oscillator Hamiltonian is $$H = \frac{\Phi^2}{2L} + \frac{Q^2}{2C} \qquad [\Phi, Q] = i\hbar \, .$$ Therefore, we have $\alpha = 1/L$, $\beta = 1 / C$, and $\gamma = \hbar$ and our mean square fluctuation of $\Phi$ is $$\langle 0 | \Phi^2 | 0 \rangle = (1/2) \hbar \sqrt{\frac{L}{C}} \, .$$ The inductance of an (unbiased) Josephson junction is $L_{J_0} = \Phi_0 / (2 \pi I_c)$. For the transmon this comes out to about $L=10\,\text{nH}$. With $C\approx 85\,\text{fF}$ this gives us $$\sqrt{\langle 0 | \Phi^2 | 0 \rangle} \approx 0.06 \Phi_0 \, .$$ As one period of the cosine potential is $\Phi_0$ wide (corresponding to a change in $\phi$ of $2\pi$), this means that the transmon wave function is pretty narrow in the flux basis. In this sense, the transmon is very unlike the charge qubit, which has a wide wave function in the flux basis.

So in the end, while the transmon and charge qbits share a certain theoretical similarity in the form of their Hamiltonians, for all practical purposes the transmon is actually more like a flux qubit with a large $C$ and biased so that the flux qubit only has one potential well.

Note that the width of the wave function in the flux basis decreases as we increase $C$. The whole reasons the transmon was invented was that narrowing the wave function by increasing $C$ leads to less sensitivity to charge noise.

However, in all literature I can find it is called a special case of a charge qubit.

That's largely historical. The folks who invented the transmon came from a charge qubit background, and the transmon was invented by trying to make the charge qubit less sensitive to charge noise.

In fact, I have an amusing story about this. The problem with the charge qubit was that its sensitivity to charge noise gave it a low $T_2$. Charge noise is difficult to reduce so people looked for a way to make qubits which would just be less sensitive to it. Professor Rob Schoelkopf's idea was to add a dc shunt to ground to short circuit low frequency charge fluctuations; by making this shunt with a bit of transmission line, the shunt would be a short at dc but still have high impedance at the qubit's oscillation frequency, thus preserving the circuit's function as a qubit. Thinking of this as TRANSmission line shunting a Josephson junction plasMON oscillation they dubbed it the "transmon". However, in the end, the best design was to use a capacitor instead of a transmission line.

So it should have been called the "capmon" :-)

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The transmon and Cooper pair box share the same design, but operate in opposite limits: Cooper pair box is operated under the condition $E_C\gg E_J$, so the charging energy dominates. While for transmon, it is $E_C\gg E_J$, and it is less sensitive to charge noise because of this parameter choice.

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  • $\begingroup$ I don't understand why we refer to $E_J$ and $E_C$ in these discussions. We never measure energy in the lab, and energy is not a useful design parameters. It's better to use frequency or impedance scales instead. $\endgroup$ – DanielSank May 22 '15 at 17:27
  • $\begingroup$ This is simply not sure. Josephson energy is something you can measure rather directly in the lab (through the critical current of the junction). And these two parameters show up in the Hamiltonian which control the properties of the system. For example $E_C$ is just inversely proportional to the capacitance. If this is not useful, I'm not sure what else is useful. You can of course use whatever other characteristics you like, but eventually these are the two energy scales that determine the physics. $\endgroup$ – Meng Cheng May 22 '15 at 17:32
  • $\begingroup$ Impedance and frequency scales are much more natural. I'm not going to argue with you, but just say that in seven years of superconducting qubit experiments I have never once measured an energy. $\endgroup$ – DanielSank May 22 '15 at 18:26
  • $\begingroup$ Fine. I don't understand how difficult it is to convert capacitance into charging energy, or critical current into Josephson energy, or vice versa. Just a simple conversion, put in things like $e$ and $h$. And frequency -- in my opinion it is just energy divided by $\hbar$. So you did measure energy although you did not realize that. $\endgroup$ – Meng Cheng May 22 '15 at 18:34
  • $\begingroup$ Instead of writing things as energy and expecting to do conversions with $\hbar$ etc., why not just do the conversion up front and work with parameters that actually have relevance to experiment? :-) Also, I really don't think it makes sense to say that measuring $I_c$ is the same as measuring energy. If we think that way then measuring distance is the same as measuring time because I can convert using the speed of light. $\endgroup$ – DanielSank May 22 '15 at 18:40

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