QCJC: Wallraff 2004

Welcome to the new season of QCJC! It’s still more or less the same as before, but instead of exclusively broadcasting to myself, I think there will be a few friends joining soon :)

Today’s paper will be Wallraff’s 2004 paper – it’s also Professor Schoelkopf’s most widely cited paper, and the first paper to show that superconducting circuits could be coupled to photon resonators for readout and control. It is very much an experimental paper, as the researchers were showing that this kind of coupling was not only true in theory, in the way that they can describe a Hamiltonian with mixing terms, but that they were able to observe avoided crossings through a special design of the architecture. As the authors state, “to our knowledge, our experiments constitute the first experimental observation of strong coupling cavity QED…”, which is really quite incredibly.

One of the key aspects of this experiment is achieving the “strong coupling regime” of coupling between the photon resonator and the superconducting qubit. By using a capacitor, they tune the coupling strength $g$ of the system such that it is greater than than the photon decay rate and the qubit decay rate. This means that the photon and the qubit are able to interact with one another before either the photon dissipates or the qubit decoheres. Having a strong coupling strength means that the Jaynes-Cummings Hamiltonian, which is as follows:

$H_{JC} = H_r + H_a + \hbar g (a^\dagger \sigma^- + a \sigma^+)$

where $a^\dagger$ is the creation operator for photons in the resonator, $a$ is the destruction operator for photons, $\sigma^-$ is the annihilation operators for cavity excitations, and $\sigma^+$ is the creation operator for excitations in the qubit. As an aside, the superconducting qubit that they use is called a Cooper Box here, but I believe that with the addition of the coupling capacitors, this is what is commonly refered to as a transmon today.

Therefore, as you can see from $H_{JC}$, there is an interaction between the resonator and the qubit. A photon can disappear in the resonator and result in an excitation in the qubit, or vice versa. This is significant because it leads to certain kinds of control in the qubit, where microwave signals can be used to manipulate the photon number in the resonator without destroying the fragile state of the superconducting qubit itself.

One significant aspect is that the primary source of dissipation is through the resonator losing photons, which is limited by the quality factor $Q$. In the paper, the initial choice is $Q = 10,000$, resulting in a photon lifetime of $0.1 \mu$s. This helps explain some of the work in my lab as being focused on resonator design and making sure that there is an incredibly high quality factor for them. The Cooper Box itself is created using two Josephson Junctions, and is shown to be fabricated on a silicon chip using optical lithography.

In addition, there is explanation for why the cryostat needs to be kept at such a low temperature. I had been puzzled for some time about this, as it seems that the critical temperature for aluminum is $T_c = 1.2K$ – and this is for the material we currently use in the lab, while the original paper uses niobium which has an even higher critical temperature at $T_c = 9.26K!$ Therefore, why does the system need to be cooled down to $20mK$? It seems that this drastic cooling is to prevent thermal fluctuations from occurring in the resonator cavity. The temperature needs to be less than the operating temperature of the resonator so that it remains in its ground state. The paper computes that $\hbar \omega_r / k_B = 300mK$ for their resonator, so therefore they need to achieve a temperature lower than that. Thus, the mean photon number inside the cavity is $n = 0.6$.

To execute their experiment, the group carried out spectroscopy on the cavity resonator in order to determine known properties of the superconducting qubit. By examining the phase and the transmission coefficients of the superconducting cavity using the resonator as a mediator, they were able to find that there was a faithful measurement conducted.

Afterwards, the group also use the resonator as a manner to control the qubit, first by tuning the qubit so that it is in resonance with the resonator. This is done by changing the magnetic flux bias that passes through the qubit, as shown in Figure 4. They are able to observe an anti-crossing between the resonator and the qubit, which agrees with theory!

So this is the paper that launched a thousand ships… and may a thousand more set sail!

Source: Wallraff, A. et al. Strong Couping of a Single Photon to a Superconducting Qubit using Circuit Quantum Electrodynamics Nature 431 162-167 (2004)