Wednesday, March 11, 2015

A brilliant insight about quantum decoherence in electronic circuits

Yesterday, Matthew Woolley gave an interesting Quantum science seminar at UQ about some of his recent work on Photon assisted tunnelling with non-classical light.

I just want to focus on one point that was deeply imbedded in the talk. It is a idea that is profound and central to the physics of quantum electronic circuits. The idea is so old now its profoundness and brilliance may be lost on a new generation.
The idea and result is easiest for me to explain in terms of the figure below which describes a superconducting (Josephson junction) qubit connected to an electrical circuit. It is taken from this review.

One can quantise the electromagnetic field and consider a spin-boson model to describe decoherence and dissipation of the qubit. This is associated with a spectral density that is proportional to frequency with a dimensionless pre factor alpha, which for this circuit is given by
where R_V is the electrical resistance of the circuit, R_K is the quantum of resistance, and the C's are capacitances.
Similar physics is at play in normal tunnel junctions (see for example this important paper, highlighted by Matthew in his talk).

Why do I find this profound?
First, this is a very simple formula that depends only on macroscopic parameters of the electrical circuit. One does not have to know anything about the microscopic details of  all the different electronic degrees of freedom in the circuit or how they individually couple to the qubit. I find this surprising.
Second, the underlying physics is the fluctuation-dissipation theorem. The quantum noise in the electronic circuit is related to fluctuations in the current. By Kubo and the fluctuation-dissipation relation tell us the fluctuations in the current are essentially the conductivity [the inverse of the resistivity].

Who was the first to have this insight and calculate this?
I feel it was Caldeira and Leggett, but I can't find the actual equation with the circuit resistance in their 1983 paper.
Or did someone else do this earlier?

Because of the above, whenever the spectral density depends linearly on the frequency, Leggett (and now everyone) calls it ohmic dissipation.

I first learnt this through the thesis work of my student Joel Gilmore, and described in this review. There we considered a more chemical problem, two excited electronic states of a molecule that are in a polar dielectric solvent. The coupling to the environment is completely specified in terms of the frequency dependent dielectric constant of the solvent (and some geometric factors).

Update: Caldeira answers the question in a comment below.

1 comment:

  1. Amir Caldeira kindly sent an email saying.

    "If I understood you correctly, the issue you raised was indeed addressed (for the first time, as far as I know) in our Annals of Physics paper. In equations (3.8) and (3.9) we write the spectral function in terms of the damping constant of a general system which is related to the circuit parameters in (1.4) and (2.9) and further on in (A.27) and (A.28). Thanks for your comments."