The Azimuth Project Blog - Visualization of diamond defect field response (Rev #2)

Visualization of the quantum energies of a defect in diamond

Sensing the magnetic field

Initially the system has a degenerate energy space configuration. This is called the zero-field splitting. In this case, the Hamiltonian is simply

(1)$H_0 = \Delta S_z^2$

With a static external magnetic field applied at angle $\theta$ with respect to the NV-symmetry axis and with strength $\beta$, one then arrives at the Hamiltonian together with a so called Zeeman splitting.

(2)$H = H_0 + \beta (\cos \theta S_x + \sin \theta S_z)$

When probing the optical properties of the NV-center by driving the system with microwave pulses and detecting optical emissions, one can determine two resonant peaks.

We can consider some relationships of the eigenvalues of the Hamiltonian. The first is that

(3)$\text{Tr}(H) = 2 \Delta = \lambda_0 + \lambda_1 + \lambda_2$

We know that the first transition occurs at the energy gap between the first excited energy state $\lambda_1$, and the ground state $\lambda_0$. In other words we know that the magic frequency is

(4)$f_1 = \lambda_1 - \lambda_0$

The system is initially in the low-energy state corresponding to $\lambda_0$. So the transitions occur with respect to this state. The second transition is then

(5)$f_2 = \lambda_2 - \lambda_0$

Combining this information into one equation gives us a value for $\lambda_0$ provided we know the zero-field splitting $\Delta$ and the transition frequencies from the ground state into the first and also second excited states.

(6)$\lambda_0 = \frac{2 \Delta - (f_1+f_2)}{3}$

We then consider the characteristic equation of the Hamiltonian, written in variable $\lambda$

(7)$x^3 - 2 \Delta x^2 + (\Delta^2 - \beta^2)x + \Delta \beta \sin^2(\theta) = 0$

There exists eigenvalues (roots) $\lambda_0$, $\lambda_1$, $\lambda_2$ such that the above polynomial can be expressed as

$(x-\lambda_0)(x-\lambda_1)(x-\lambda_2) = x^3 - x^2 \text{Tr}(H) + \frac{x}{2}(\text{Tr}(H)^2 - \text{Tr}(H^2)) - \text{det}(H)$

The coefficients of indeterminate $x$ are of interest to us (as opposed to e.g. $x^2$ or $x^0$). By the way, the expression of these terms into traces as above, are a special case of the (Newton–Girard identies)[], that’s how old this sort of trick is. If you really wanted to have fun, you might notice that they look just list a dispersion and write them like this …

The point is that by equating coefficients from the equation

(8)$\frac{1}{2}(\text{Tr}(H)^2 - \text{Tr}(H^2)) = (\Delta^2 - \beta^2)$

we arrive at an expression for the magnitude of the magnetic field $\beta$ in terms of the measured transition frequencies $f_1$ and $f_2$ as

(9)$\beta^2 = \frac{1}{3} (f_1^2 + f_2^2 - f_1f_2 - \Delta^2)$

In other words, we can can use this NV to probe the external magnetic field! So it’s really a sensor, as we stated from the get go.

Visualization

We then shift to the so called, center of mass of the polynomial. This can be thought of as adding a term proportional to the identity to make the Hamiltonian traceless. This amounts to the assignment

(10)$\kappa_i := \lambda_i - \frac{2 \Delta}{3}$

for $i = 0, 1, 2$ and from this we arrive at the depressed cubic

(11)$\kappa^3 + a \kappa + b = 0$

with

(12)$a := - \left( \frac{\Delta^2}{3} + \beta^2 \right)$
(13)$b := \Delta\left( \frac{2}{27} \Delta^2 + \beta^2 \left( \sin^2(\theta) - \frac{2}{3}\right)\right)$

We’ve already solved for $\lambda_0$ in terms of the resonant frequencies $f_1$ and $f_2$. Changing this into an equation for $\kappa_0$ we arrive at

(14)$\kappa_0 = -(f_1 + f_2)$

which we know is a root of $\kappa^3 + a \kappa + b = 0$. In other words, there exists $q, r, p$ such that

(15)$(\kappa - \kappa_0)(q \kappa^2 + p \kappa + r) = \kappa^3 + a \kappa + b = 0$

The cubic has a neat feature, if the roots are real, they are known to be related to the triangle as follows

Let’s first understand that, and then equate the quantities we just determine with the length of the sides, and the orientation (that is the rotation) of the triangle.

• The applied static magnetic field is related to the length of the sides
• The angle of the triangle depends on both the applied field and also the angle of the NV symmetry axis

Conclusion

So there you have it. NV-centers in a nut shell, or at least in a rather long blog article. I certainly find this stuff interesting. As I mentioned at the start, the content here was inspired mainly from

There are a few more related references, including

For those that want to read another quick introduction, with more of an experimental focus, maybe check out

As I said, we’ve been working with Florian Dolde and others. So we have two of our own papers on NV-centers:

The sort of things I hear from experts in NV, about what will happen next, are mainly about scaling up the numbers of interacting NV-centers. Right now, the record is an impressive, but small handful of interacting particles (like 4 or 5), but with new the new methods being developed to create samples, people tell me this will likely double in the next two year. This would certainly open the door up to a lot more interesting applications. For example, say they are able to get 5 NV-centers to interact, this means we would then need to consider 59,049 energy levels. Modelling such a system using existing computing technologies will be challenging. Turning this on its head: such a system offers a computational resource, to at least, model itself.

Scrap yard

Among other things, quantum physics explains how structures, such as crystals, form. In principle, using quantum physics, we could understand many more things than we actually do about the nature of mater.

• To understand the world around us, all we have to do is solve the quantum equations describing what we see in nature.

The problem is that these equations seem impossible to solve using any computer not governed by quantum physics! The sad fact is that quantum computers which exist today as far to small to do anything practical.

One of the fun aspects of this is that we’re actually going to give a geometric interpretation to some things my friends Florian and Ya are actually measuring in their Stuttgart lab.

category: experiments