This is a blog article in progress, written by Jacob Biamonte. See also the quantum network theory series. To see discussions of the article as it was being written, visit the Azimuth Forum. For the final polished version, go to the Azimuth Blog.
NOTE: Here’s Part II
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People typically get married a few times in their lives, tops. In contrast to those shopping for say, a wedding ring, experimental physicists actually scavenge for defects when they’re diamond browsing.
Recently a few friends have taught me a thing or two about the quantum properties that result from these perfect defects in the structure of diamonds. It’s really neat.
I can’t help but be fascinated with this topic and I wanted to share my enthusiasm by writing this blog post. I think I can even get straight to the core concepts and explain this in an understandable fashion, so if you read on, you can tell me later if my conjecture was correct or not.
The main person who sparked my new found interest is Ville Bergholm. Together, we’ve been working with two NV groups: one’s in Stuttgart and another’s in Ulm. They’re in different German cities, but they collaborate on aspects of this topic, called “NV-centers” in diamond, which is parlance for
Before I explain what these tiny gizmos actually are, let me tell you about some of the exciting things they can be used for.
One thing to mention right away is that defects interact with other defects, but not much with the diamond itself. So these defects are isolated and this in turn lets controllable quantum effects take over. In fact, the experiments probing these effects typically operate at room temperatures. In contrast to this, many of the quantum systems studied to date have instead operated at ultra-low, cryogenic temperatures. These and other facts have opened up a lot of hope in the viability of NV-based technologies.
Along this line of thought, the technology that will likely happen first seems to be NV-based magnetic field sensors, at the very small scale with applications in biology. Today I want to explain to you some of the building blocks behind how this might work in practice.
NV-centers have complex physics and finding a good mathematical model that takes into account all the relevant degrees of freedom in a realistic system is complicated. However, we can study simplified models and these turn out to match experimental data and tell us a lot. Such approximations are known to work well, most of the time.
To understand the first decent approximation, all we need to do is solve an equation for the roots of a cubic polynomial! This is really the way it is done in practice, and you can read more about it here
So let’s get started with explaining an NV-center. I should warn you: I’m no expert here! I’m lucky enough to be able to write, ask for feedback and if there are questions, Florian Dolde who has just finished his PhD in Stuttgart and started at MIT as a Postdoctoral Scholar agreed to jump in and sort us out. Thanks Florian!
Oh and if you want other reasons why polynomials are so amazing, regular readers of Azimuth might recall that roots of polynomials have been discussed in the “beauty of roots series”.
Let’s motivate this topic — for a slightly more complete account of the tale, might I suggest
In our simplified version, first you start with a diamond, like this
As you know, diamond is known to be very strong, and to have a very regular crystal structure. In a perfect diamond, this structure is built from carbon bonds, like this
To get a diamond with the perfect defect, there seems to be two options. (1). The first is to look closely at a diamond found in nature: these are known to have just the right defects sometimes.
The problem with this approach is that diamonds in nature have many other defects. To address this concern, (2). In a second approach, you start with an ideally perfect diamond and blast it with a small particle accelerator.
After doing this, you still have to look at the diamond and find just the right defect for your needs, which can take months sometimes(!), but finding one opens the door to conduct interesting experiments. A good defect can be used for several years of experimental research.
So I guess if you like to break things (don’t laugh, it’s common to like to break things among scientists) then breaking the crystal structure of diamonds might be new and exciting to you. There are two choices to break the structure of diamonds. The choice is in which particle you’re going to smash the diamond with. Everyone seems to be using nitrogen. Specifically, they’re using either an $^{14}N$ or $^{15}N$ isotope. So pick your favourite isotope and get your particle accelerator ready. The difference is in the so called, nuclear spin. We’ll touch on this later on.
Ok, so after we pick our favourite stable nitrogen isotope, we propel them at the crystal. Some of them stick, smashing out of their way carbons. Whenever a nitrogen stops in a place where a carbon used to be, it results in unpaired electrons. You might want to think of something like this:
The nitrogen defect is supposed to be the blue ball. The missing carbon ball is what we’re concerned with today. Here, two unpaired electrons join together to make a so called spin-1 particle. We’re going to talk more about this soon.
To avoid daunting numerical calculations to solve quantum many-body problems, chemists, materials scientists and physicists alike have develop elegant ways and rules to approximate and to reason about the structure of matter. These tricks have many advantages, particularly, it avoids intractable calculations — the disadvantage is that these rules only work some of the time. In this particular case, the rules are perfectly valid. So the heuristic argument goes something like this.
In a negatively charged NV$^-$ center the nitrogen atom and the vacancy next to it have a total of 6 “unpaired” electrons, which do not directly take part in forming covalent bonds. Each of these electrons, in addition to its position, has a fixed amount of angular momentum associated with it called “spin-1/2”.
(Maybe explain what spin is better here?)
Due to the fermionic nature of electrons their combined wave function, consisting of both spatial and spin degrees of freedom, must be antisymmetric under any exchange of two particles. In general the interaction of the electrons with each other and the surrounding crystal field is rather involved, and solving the eigenstates of the system is a complicated problem. However, it turns out that to a very good approximation the electronic part of the ground state manifold of an NV$^-$ center behaves like a single electronic spin-1.
These electrons start to combine to form molecular orbitals. In the orbital with the lowest energy, we have a pair of electrons without what is called “spin”, and in the orbital with the second lowest energy, we again have a pair without spin. Finally, in the third orbital we have an electron pair that has spin. Spin can be thought of as a strange type of angular momentum. An electron can be measured to have spin angular momentum in two possible directions. Pairing electrons does not mean that you can get four possibilities—due to another rule called the exclusion principle a pair of electrons can only be found in 3 possible configurations.
Talk about units here. Set $\hbar=1$.
In this particular case were concerned with, the unpaired electrons form what is called an electron pair. As stated, the pair can be found in only one of three distinct configuration, and each of these configurations corresponds to a certain level of energy.
Physicists use a sort of short hand notation for vectors, called Dirac’s bracket notation. For the three configurations of observed spin angular momentum of an electron pair, we will use these kets:
for the three possible spin configurations of our electron pair. For each observed spin configuration, there is corresponding energy level. The amount of this energy depends on a few things. This is where things start to get interesting.
First, without doing anything to our pair, we only have two energy levels.
We don’t lose any information at all to say that the lower of these energy levels has zero energy and the higher energy level has energy $\Delta$. This common approach is exactly what we’ll do. We can then write down a matrix which describes this situation by putting the energy levels on the diagonal, and where the basis of the matrix corresponds to spin configurations. This matrix will have the following properties (which fully define it in our 3 configuration basis). We’re going to use a known matrix
So there we have it. The so called, zero-field effective Hamiltonian is given as
And this is the simplest way to view the static picture of an NV-center. You can just think of the system starting off in one of these energy states (say $|0\rangle$) and to make transitions requires adding energy to the system. (We recall from the network theory series that time dependent quantum evolution is given by the Schrödinger equation.
To make our model a bit better, we should include:
To do that, we are going to add additional terms to the systems Hamiltonian.
For those of you that skipped the last section, we’re dealing with a pair of electrons which can be found in three distinct “spin configurations”. Each of these configurations has a different quanta of energy associated to it. This quanta of energy depends mainly on two things that are external to the diamond itself. It depends on the magnitude of a static electromagnetic field, and the angle of the NV symmetry axis relative to this field.
To do this, we are going to be using the Pauli spin matrices for the case $J = 1$.
Spin matrices J=1
The Hamiltonian for a quantum system governs both how a quantum system changes in time, and also in the static picture, the energy eigenstates of a quantum system. In our case, for an NV-center we have
Technically speaking, we could have also included two more things. The first are $S_y$ terms, but we will assume without much loss of generality that we lined our system up in this physical direction, and so we just neglect them here. The second is the so called, strain term. Strain is very small in a good NV-center, but if we wanted to include it all we’d have to do is add the following term to our Hamiltonian. It accounts for a deformity of the lattice surrounding the NV-center.
This approximation is totally valid in our case, and generally whenever $E \ll \Delta$.
Determining these levels relates to solving a cubic equation. Studying the solutions to such polynomial equations lead to all sorts of amazing mathematics.
Experimentally one determines not the energy levels of a system, but how much energy it takes to make a system jump from one level to another.
We are given transition frequencies.
Here list the equations solving for the angle of the NV sym axis and the magnetic field
Florian says, “And in your explanation I would not explain about the 4 other electrons to heavily, just say they are bonded in lower engergy states and have therefore no spin left … that should be good enough”
So that’s about it for today. All we did was talk about where NV-Centers come from, and then sketch a bit of the intuition behind how a mathematical model for such systems can arise. Next time, we’re going to solve this model and then visualize this solution.