How to understand the concept of a fluxon, the dual of the Cooper pair?

Question

I've been reading the paper Bifluxon: Fluxon-Parity-Protected Superconducting Qubit by Kalashnikov et al. which deals with a novel type of superconducting qubit. Without going into too much of the details, it is a qubit consisting of a superconducting island, Josephson junctions, and a highly inductive loop. It forms a qubit protected against certain forms of energy decay and decoherence, and they write that at the core of this protected design lies that the parity of fluxons in the loop is preserved via Aharonov-Casher interference.

Now, I am a little ashamed to admit that I can follow most of the paper, given that I ignore a very basic question: how do I understand the concept of fluxons themselves? What are they? I have read that they are the dual to a Cooper pair. I have read that they can enter loops, and I often encounter them in circuits containing Josephson junctions. If I follow Wikipedia, it is a quantum of electromagnetic flux, made of circulating supercurrents. Is there an intuitive way that I can think about them, similar to how one can Cooper pairs? That is, in superconducting circuits like the Cooper pair box I think of those as 2e-charged particles that can jump across a Josephson junction. Does a picture like that hold for fluxons? Do they require the presence of a loop, or can they also exist without? Are they related to a flux quantum, being a quantum of flux?

Maybe a final remark would be if an intuitive picture of a fluxon can also make it intuitive how they relate to the aforementioned Aharonov-Casher effect. What I know about it is that it relates to magnetic moments picking up a phase when enclosing a charge. So I imagine that the fluxon carries this magnetic moment?

Answer 1

Yes, it is a magnetic flux quantum, but technically you are only supposed to call it a fluxon when it is actually a flux soliton, meaning a soliton solution to the sine-Gordon equation in a long Josephson junction. In other contexts, the preferred term is single or superconducting flux quantum or SFQ. The reason why magnetic flux in superconducting electronics often exists in these quantized units is simply that this minimizes the total energy in the loop when the loop includes a Josephson junction, where the Josephson energy $(\Phi_0/2\pi) I_c (1 - 2\cos(\theta))$ where $\theta$ is the difference in the superconducting order parameter (characterized by a phase) across the junction. Since this function has periodic minima, you can see that the flux in the loop likes to sit at values that make the phase across the junction a multiple of $2\pi$ as long as the current is less than the critical current. When the phase across the junction rotates by $2\pi$, that is equivalent to one $\Phi_0$ of flux crossing the loop. The duality you mentioned has to do with a duality that exists between these circuits, in which the circuit can be converted to a functionally isomorphic one in which the roles of flux and charge are reversed. The dual of a Josephson junction is called a quantum phase slip junction; you can think of it as a JJ where the critical current is so low that you only occasionally get a single Cooper pair tunneling across the junction; this event corresponds to when a fluxon crosses the junction in the perpendicular direction in an ordinary JJ circuit. So whereas in the normal circuit you have individual flux quanta moving about, in the dual circuit you have individual Cooper pairs moving about. I’m not familiar with the specific qubit you mentioned, but from the name I’m guessing it uses two flux quanta. :)