Two interesting arXiv papers this week:

Adiabatic quantum computation along quasienergies

A potentially new model of Quantum Computation, which is a discretized variant of Adiabatic Quantum Computation (AQC). Is it equivalent to the standard model? Is it useful? No-one knows.

This paper also got me thinking:

Electronic structure of superposition states in flux qubits.

How do you measure the *cattiness* of a flux qubit? Cattiness being defined as the ability of a system to exhibit quantum properties as it approaches a classical limit in terms of mass, size, or some other measure. The name comes from the question of whether or not it is possible to put an entire ‘Schrodinger’s cat’ into a macroscopic superposition of states.

I have been wondering about this problem with regards to flux qubits for a while. You might think it is possible just to ‘count’ the number of electrons involved in the Josephson tunneling, giving around 1^10 particles. But wait, the electrons all form a macroscopic state – do you count the condensate as a single particle instead? This paper argues that the actual cat state is somewhere between these two extremes. This is good news, because although the upper bound would have been cooler in terms of Macroscopic Quantum Coherence, the superconducting flux qubit might still be the ‘cattiest thing in town’.

I’m also wondering about the cattiness of nanomechanical resonators coupled to optical or microwave cavities. This system can be put in a superposition of two mechanical states relating to the position and motion of the atoms in the bar. For example, the ground state can be thought of as the fundamental harmonic of the bar (think of it like a guitar string), with an antinode in the centre, wheras the first excited state has a node in the centre and two antinodes at 1/4 and 3/4 of the way along the bar. But here we find a similar problem to that of the flux qubit: Does the number of atoms in the bar matter?

For fun let’s calculate the number of atoms in a Niobium nanomechanical resonator:

Let’s say the mechanical bar is 20nm x 20nm x 1um.

The volume of the bar is therefore 4e-22m^3

The density of Nb is 8.57g/cm^3

The mass of the bar is therefore 3.428e-17kg

The atomic mass of Niobium is 92.906amu = 1.54e-25kg.

**The number of atoms in the bar is ~2.2e8**

To check that value:

The atomic radius of a Nb atom: 142.9pm = 0.1429nm

In 20nm there are 139.958 atoms,

and in 1um there are 6997.9 atoms.

**Therefore in the bar there are 1.37e8 atoms**

which is roughly the same as by the previous method.

So does that mean the ‘cattiness of the bar’ has an upper bound of 2e8? This would make it more catty than the flux qubit. Or do you have to assign more (or less) than one ‘quantum degree of freedom’ per atom? It’s not as simple as tunneling electrons, where the quantum state is determined by the direction of current flow around the loop. If anyone has any thoughts on this they would be appreciated. Just what exactly are the quantum degrees of freedom here?

The bar is obviously constrained by its end points, albeit not ideally. The displacement of the bar may therefore probably behave more classically near the ends, or the wavefunction may extend into the structural supporting region. This may affect the actual number of atoms in the superposition. What fraction of the length of the bar is behaving quantum mechanically?

Note that the mass of both the electron condensate in the case of the flux qubit AND that of the nanomechanical bar are both much lower than Penrose’s quantum mass limit of about 1e-8kg – so we can’t test that hypothesis in the lab yet. Note this relates to a post I wrote a while ago about electrons in a lump of superconductor – there are enough electrons in a bulk sample for the mass to be greater than the Penrose limit, but they aren’t doing any useful quantum computation, you can’t put them into a well defined superposition of states for example. We need to ENGINEER and CONTROL these cat states…

Anyhow, after that complicated Physics we are definitely in need of some cake:

We had this type of cake yesterday (amongst others) to celebrate a colleague passing his PhD viva ðŸ™‚