Thursday, September 13, 2012

Strong electronic correlations in diamond qubits

Yesterday I read a nice paper
Mechanism for optical initialization of spin in NV− center in diamond
by SangKook Choi, Mahish Jain, and Steven Louie

The NV- center in diamond has attracted renewed interest in the past few years because it can be used as a qubit and experimentalists have created entangled states associated with it.

Outstanding questions concern the nature, ordering, and quantum numbers of the low-lying states of a single NV- center. This is key to understanding the process whereby one creates a qubit with a coherence time of order of milliseconds following optical excitation. A previous post discussed this and a recent article by Doherty, Manson, Delaney and Hollenberg provides a helpful review.

The authors obtain definitive results by considering a 4 site Hubbard model with 6 electrons. The model parameters are evaluated from ab initio electronic structure calculations. The 4 sites correspond to orbitals localised on the 3 carbon atoms and one nitrogen atom next to the carbon vacancy.

They find that the Hubbard U is of order 3-4 times the hopping integral and that electronic correlations play a significant role, having a significant effect on the relative energies of the excited states. Previous studies have not treated these correlation effects adequately and so have obtained spurious results. Specifically, the singlet excited states cannot be described in terms of a single Slater determinant.

The effect of correlations is illustrated in the Figure below which shows the energy of the states as a function a structural relaxation [the N atom moves towards the vacancy while the neighbouring C atoms move away]. GEG denotes Ground state Equilibrium Geometry, and EEG denotes Excited state Equilibrium Geometry.
The left panel gives results for exact diagonalisation of the Hubbard model and the right panel for a perturbative treatment based in Density Functional Theory [GW-BSE].

A few minor comments.

1. The authors point out an outstanding question concerns the mechanism whereby the 1A1 excited singlet state decays non-radiatively in 1 nsec to the 1E ground singlet state.

2. Calculating the energy levels for defects in diamond has a long history which seems to have been forgotten. A seminal paper from 1957 was by [my hero] Charles Coulson and Mary Kearsley. They pointed out the utility of a molecular orbital description.

3. Because the nitrogen atomic orbitals are much lower in energy (about 2.6 eV) than the carbon orbitals it should be possible to "integrate them out" and reduce the 4 site 6 electron model to a 3 site 4 electron model where the "carbon" sites have some admixture of the nitrogen orbitals.

4. Using the C3 symmetry of the system it is possible to block diagonalise the Hubbard model into blocks with dimension no larger than 2. The notes below show the energy eigenvalues and quantum numbers for 2 holes on 3 sites with the sign of the hopping integral the same as for the NV- center. The triplet states involve a single Slater determinant. The singlet states all involve two Slater determinants. In the parameter regime relevant to the NV- center, the two determinants have comparable coefficients in all the wavefunctions.

I thank Taras Plakhotnik for bringing the paper to my attention and reviving my interest in this problem.


  1. Since the electronic structure is local, this would be amenable to the sort of analysis that we have done with methines.

    I'm not sure if you're right about reducing it to a 4,3 model, at least not in the sense that a self-consistent solution could be achieved.

    In Crystal Violet, one can engineer a 4,3 CASSCF solution, but it breaks symmetry. It does not predict degeneracy in the 1E excited state. I'm still thinking about why this is, but it seems important that the two E orbitals need to have different orbitals to correlate with.

    The problem may be that you can't do this while simultaneously insisting on orthogonality; in this case, you would need to have different signs of the different carbon orbital tails on the nitrogen, but this might be hard to do in a way that maintains the symmetry.

  2. Maybe its not a problem if you don't insist on real orbitals...