Category Archives: Quantum Mechanics

Meissner effect as amplified atomic diamagnetism

As you can probably tell from my previous post, I have found the recent activism inspiring and genuinely hopeful of it translating into some form of justice and meaningful action. At the end of this post I share a few videos that I found particularly poignant.

It’s hard to imagine the history of condensed matter physics without both the discovery and theory of superconductivity. Superconductivity has played and continues to play an outsized role in our field, and it is quite easy to understand why this is the case. All one has to do is to imagine what our world would look like with room temperature superconductivity. Besides the potential technological implications, it has also garnered attention because of the wealth of stunning effects associated with it. A few examples include the Josephson effect, flux quantization, persistent superconducting currents, vortex lattices and the Meissner effect.

Now, these effects occur for various reasons, but there are a couple of them that can be viewed to some extent as a microscopic effect on a macroscopic scale. To show what I mean by that, I am going to focus on the Meissner effect and talk about how we can view it as an amplification of atomic diamagnetism. One could also extend the this microscopic to macroscopic amplification picture to the relationship between a Josephson junction in a superconducting ring and the Aharonov-Bohm effect, but I’ll leave that discussion to another day.

To understand what I mean by amplification, let’s first look at atomic diamagnetism. Here we can use a similar logic that led to the Bohr model of the atom. Two conditions are important here — (i) the de Broglie relation $\lambda = h/p$ and (ii) the Bohr quantization condition $n\lambda = 2\pi r$ which states that only integer wavelengths are allowed in a closed loop (such as an atomic orbit). See the image below for a simple picture (click the image for the source).

We can use the classical relation for the momentum $p=mv$ in addition to equations (i) and (ii) above to get $mvr = n\hbar$, which is what Bohr got in his atomic model. It’s worth noting here that when the atom is in its ground state (i.e. $n=0$), there is no “atomic current”, meaning that $j = ev = 0$. Without this current, however, it is not possible to have a diamagnetic response.

So how do we understand atomic diamagnetism? To do so, we need to incorporate the applied field into the deBroglie relation by using the canonical momentum. By making the “Peierls substitution”, we can write that $p = mv+eA$. Using the same logic as above, our quantization condition is now $mvr = n\hbar - eAr$. Now, however, something has changed; we do get a non-zero current in the ground state (i.e. $j = ev = -e^2A/m$ for $n=0$). Qualitatively, this current circulates to screen out the field that is trying to “mess up” the integer-number-of-wavelengths-around-the-loop condition. Note also that we have a response that is strictly quantum mechanical in nature; the current is responding to the vector potential. (I realize that the relation is not strictly gauge invariant, but it makes sense in the “Coulomb gauge”, i.e. when $\nabla\cdot A=0$ or when the vector potential is strictly transverse). In some sense, we already knew that our answer must look obviously quantum mechanical because of the Bohr-van Leeuwen theorem.

If we examine the equation for the electromagnetic response to a superconductor, i.e. the London equation, we obtain a similar equation $j = n_sev = -n_se^2A/m$, where $n_s$ is the superfluid density. The resemblance between the two equations is far from superficial. It is this London equation which allows us to understand the origin of the Meissner effect and the associated spectacular diamagnetism. Roughly speaking then, we can understand the Meissner effect as an amplification of an atomic effect that results in a non-zero ground state “screening” current.

I would also like to add that the Meissner effect is also visible in a multiply connected geometry (see below). This time, the magnetic field (for sufficiently small magnetic fields) is forbidden from going through the center of the ring.

What is particularly illuminating about this ring geometry is that you don’t have to have a magnetic field like in the image above. In fact, it is totally possible to have a superconducting ring under so-called Aharonov-Bohm conditions, where a solenoid passes through the center but the ring never sees the magnetic field. Instead, the superconducting ring “feels the vector potential”. In some sense, this latter experiment emphasizes the equation above where the current really responds (in a gauge-invariant way) to a vector potential and not just the magnetic field.

Understanding the Meissner effect in this way helps us divorce the Meissner effect from the at-first-sight similar effect of persistent currents in a superconducting ring. In the Meissner effect, as soon as the magnetic field is turned off, the current dies and goes back to zero. This is because through this entire process, the superconductor remains in its ground state. Only when the superconductor is excited to higher states (i.e. $n=1,2,3$…) does the current persist in a metastable fashion for a quasi-infinitely long time.

To me, understanding the Meissner effect in this way, which exposes the connection of the microscopic to the macroscopic, harks back to an old post I made about Frank Wilczek’s concept of upward inheritence. The Meissner effect somehow seems clearer through his lens.

Now as promised, here are the couple videos (if the videos don’t play, click on the panel to take you to the twitter website because these videos are worth watching!):

Whence we know the photon exists

In my previous post, I laid out the argument discussing why the photoelectric effect does not imply the existence of photons. In this post, I want to outline, not the first, but probably the conceptually simplest experiment that showed that photons do indeed exist. It was performed by Grangier, Roger and Aspect in 1986, and the paper can be found at this link (PDF!).

The idea can be described by considering the following simple experiment. Imagine I have light impinging on a 50/50 beamsplitter and detectors at both of the output ports, as pictured below. In this configuration, 50% of the light will be transmitted, labelled t below, and 50% of the light will be reflected, labeled r below.

Now, if a discrete and indivisible packet of light, i.e. a photon, is shone on the beam splitter, then it must either be reflected (and hit D1) or be transmitted (and hit D2). The detectors are forbidden from clicking in coincidence. However, there is one particularly tricky thing about this experiment. How do I ensure that I only fire a single photon at the beam splitter?

This is where Aspect, Roger and Grangier provide us with a rather ingenious solution. They used a two-photon cascade from a calcium atom to solve the issue. For the purpose of this post, one only needs to know that when a photon excites the calcium atom to an excited state, it emits two photons as it relaxes back down to the ground state. This is because it relaxes first to an intermediate state and then to the ground state. This process is so fast that the photons are essentially emitted simultaneously on experimental timescales.

Now, because the calcium atom relaxes in this way, the first photon can be used to trigger the detectors to turn them on, and the second photon can impinge on the beam splitter to determine whether there are coincidences among the detectors. A schematic of the experimental arrangement is shown below (image taken from here; careful, it’s a large PDF file!):

Famously, they were essentially able to extrapolate their results and show that the photons are perfectly anti-correlated, i.e. that when a photon reflects off of the beam splitter, there is no transmitted photon and vice versa. Alas the photon!

However, they did not stop there. To show that quantum mechanical superposition applies to single photons, they sent these single photons through a Mach-Zehnder interferometer (depicted schematically below, image taken from here).

They were able to show that single photons do indeed interfere. The fringes were observed with visibility of about 98%. A truly triumphant experiment that showed not only the existence of photons cleanly, but that their properties are non-classical and can be described by quantum mechanics!

The photoelectric effect does not imply photons

When I first learned quantum mechanics, I was told that we knew that the photon existed because of Einstein’s explanation of the photoelectric effect in 1905. As the frequency of light impinging on the cathode material was increased, electrons came out with higher kinetic energies. This led to Einstein’s famous formula:

$K.E. = \hbar\omega - W.F.$

where $K.E.$ is the kinetic energy of the outgoing electron, $\hbar\omega$ is the photon energy and $W.F.$ is the material-dependent work function.

Since the mid-1960s, however, we have known that the photoelectric effect does not definitively imply the existence of photons. From the photoelectric effect alone, it is actually ambiguous whether it is the electronic levels or the impinging radiation that should be quantized!

So, why do we still give the photon explanation to undergraduates? To be perfectly honest, I’m not sure whether we do this because of some sort of intellectual inertia or because many physicists don’t actually know that the photoelectric effect can be explained without invoking photons. It is worth noting that Willis E. Lamb, who played a large role in the development of quantum electrodynamics, implored other physicists to be more cautious when using the word photon (see for instance his 1995 article entitled Anti-Photon, which gives an interesting history of the photon nomenclature and his thoughts as to why we should be wary of its usage).

Let’s return to 1905, when Einstein came up with his explanation of the photoelectric effect. Just five years prior, Planck had heuristically explained the blackbody radiation spectrum and, in the process, evaded the ultraviolet catastrophe that plagued explanations based on the classical equipartition theorem. Planck’s distribution consequently provided the first evidence of “packets of light”, quantized in units of $\hbar$. At the time, Bohr had yet to come up with his atomic model that suggested that electron levels were quantized, which had to wait until 1913. Thus, from Einstein’s vantage point in 1905, he made the most reasonable assumption at the time — that it was the radiation that was quantized and not the electronic levels.

Today, however, we have the benefit of hindsight.

According to Lamb’s aforementioned Anti-Photon article, in 1926, G. Wentzel and G. Beck showed that one could use a semi-classical theory (i.e. electronic energy levels are quantized, but light is treated classically) to reproduce Einstein’s result. In the mid- to late 1960’s, Lamb and Scully extended the original treatment and made a point of emphasizing that one could get out the photoelectric effect without invoking photons. The main idea can be sketched if you’re familiar with the Fermi golden rule treatment to a harmonic electric field perturbation of the form $\textbf{E}(t) = \textbf{E}_0 e^{-i \omega t}$, where $\omega$ is the frequency of the incoming photon. In the dipole approximation, we can write the potential as $V(t) = -e\textbf{x}(t)\cdot\textbf{E}(t)$ and we get that the transition rate is:

$R_{i \rightarrow f} = \frac{1}{t} \frac{1}{\hbar^2}|\langle{f}|e\textbf{x}(t)\cdot\textbf{E}_0|i \rangle|^2 [\frac{\textrm{sin}((\omega_{fi}-\omega)t/2)}{(\omega_{fi}-\omega)/2}]^2$

Here, $\hbar\omega_{fi} = (E_f - E_i)$ is the difference in energies between the initial and final states. Now, there are a couple things to note about the above expression. Firstly, the term in brackets (containing the sinusoidal function) peaks up when $\omega_{fi} \approx \omega$. This means that when the incoming light is resonant between the ground state and a higher energy level, the transition rate sharply increases.

Let us now interpret this expression with regard to the photoelectric effect. In this case, there exists a continuum of final states which are of the form $\langle x|f\rangle \sim e^{i\textbf{k}\cdot\textbf{r}}$, and as long as $\hbar\omega > W.F.$, where $W.F.$ is the work function of the material, we recover $\hbar\omega = W.F. + K.E.$, where $K.E.$ represents the energy given to the electron in excess of the work function. Thus, we recover Einstein’s formula from above!

In addition to this, however, we also see from the above expression that the current on the photodetector is proportional to $\textbf{E}^2_0$, i.e. the intensity of light impinging on the cathode. Therefore, this semi-classical treatment improves upon Einstein’s treatment in the sense that the relation between the intensity and current also naturally falls out.

From this reasoning, we see that the photoelectric effect does not logically imply the existence of photons.

We do have many examples that non-classical light does exist and that quantum fluctuations of light play a significant role in experimental observations. Some examples are photon anti-bunching, spontaneous emission, the Lamb shift, etc. However, I do agree with Lamb and Scully that the notion of a photon is indeed a challenging one and that caution is needed!

A couple further interesting reads on this subject at a non-technical level can be found here: The Concept of the Photon in Physics Today by Scully and Sargent and The Concept of the Photon – Revisited in OPN Trends by Muthukrishnan, Scully and Zubairy (pdf!)

Jahn-Teller Distortion and Symmetry Breaking

The Jahn-Teller effect occurs in molecular systems, as well as solid state systems, where a molecular complex distorts, resulting in a lower symmetry. As a consequence, the energy of certain occupied molecular states is reduced. Let me first describe the phenomenon before giving you a little cartoon of the effect.

First, consider, just as an example, a manganese atom with valence $3d^4$, surrounded by an octahedral cage of oxygen atoms like so (image taken from this thesis):

The electrons are arranged such that the lower triplet of orbital states each contain a single “up-spin”, while the higher doublet of orbitals only contains a single “up-spin”, as shown on the image to the left. This scenario is ripe for a Jahn-Teller distortion, because the electronic energy can be lowered by splitting both the doublet and the triplet as shown on the image on the right.

There is a very simple, but quite elegant problem one can solve to describe this phenomenon at a cartoon level. This is the problem of a two-dimensional square well with adjustable walls. By solving the Schrodinger equation, it is known that the energy of the two-dimensional infinite well has solutions of the form:

$E_{i,j} = \frac{h}{8m}(i^2/a^2 + j^2/b^2)$                where $i,j$ are integers.

Here, $a$ and $b$ denote the lengths of the sides of the 2D well. Since it is only the quantity in the brackets that determine the energy levels, let me factor out a factor of $\gamma = a/b$ and write the energy dependence in the following way:

$E \sim i^2/\gamma + \gamma j^2$

Note that $\gamma$ is effectively an anisotropy parameter, giving a measure of the “squareness of the well”. Now, let’s consider filling up the levels with spinless electrons that obey the Pauli principle. These electrons will fill up in a “one-per-level” fashion in accordance with the fermionic statistics. We can therefore write the total energy of the $N-$fermion problem as so:

$E_{tot} \sim \alpha^2/ \gamma + \gamma \beta^2$

where $\alpha$ and $\beta$ parameterize the energy levels of the $N$ electrons.

Now, all of this has been pretty simple so far, and all that’s really been done is to re-write the 2D well problem in a different way. However, let’s just systematically look at what happens when we fill up the levels. At first, we fill up the $E_{1,1}$ level, where $\alpha^2 = \beta^2 = 1^2$. In this case, if we take the derivative of $E_{1,1}$ with respect to $\gamma$, we get that $\gamma_{min} = 1$ and the well is a square.

For two electrons, however, the well is no longer a square! The next electron will fill up the $E_{2,1}$ level and the total energy will therefore be:

$E_{tot} \sim 1/\gamma (1+4) + \gamma (1+1)$,

which gives a $\gamma_{min} = \sqrt{5/2}$!

Why did this breaking of square symmetry occur? In fact, this is very closely related to the Jahn-Teller effect. Since the level is two-fold degenerate (i.e. $E_{2,1} = E_{1,2}$), it is favorable for the 2D well to distort to lower its electronic energy.

Notice that when we add the third electron, we get that:

$E_{tot} \sim 1/\gamma (1+4+1) + \gamma (1+1+4)$

and $\gamma_{min} = 1$ again, and we return to the system with square symmetry! This is also quite similar to the Jahn-Teller problem, where, when all the states of the degenerate levels are filled up, there is no longer an energy to be gained from the symmetry-broken geometry.

This analogy is made more complete when looking at the following level scheme for different $d$-electron valence configurations, shown below (image taken from here).

The black configurations are Jahn-Teller active (i.e. prone to distortions of the oxygen octahedra), while the red are not.

In condensed matter physics, we usually think about spontaneous symmetry breaking in the context of the thermodynamic limit. What saves us here, though, is that the well will actually oscillate between the two rectangular configurations (i.e. horizontal vs. vertical), preserving the original symmetry! This is analogous to the case of the ammonia ($NH_3$) molecule I discussed in this post.

Fractional quasiparticles and reality

As a condensed matter physicist, one of the central themes that one must become accustomed to is the idea of a quasiparticle. These quasiparticles are not particles as nature made them per se, but only exist inside matter. (Yes, nature made matter too, and therefore quasiparticles as well, but come on — you know what I mean!)

Probably the first formulation of a quasiparticle was in Einstein’s theory of specific heat in a solid at low temperature. He postulated that the sound vibrations in a solid, much like photons from a blackbody, obeyed the Planck distribution, implying some sort of particulate nature to sound. This introduction was quite indirect, and the first really explicit formulation of quasiparticles was presented by Landau in his theory of He4. Here, he proposed that most physical observables could be described in terms of “phonons” and “rotons“, quantized sound vibrations at low and high momenta respectively.

In solid state physics, one of the most common quasiparticles is the hole; in the study of magnetism it is the magnon, in semiconductor physics, the exciton is ubiquitous and there are many other examples as well. So let me ask a seemingly benign question: are these quasiparticles real (i.e. are they real particles)?

In my experience in the condensed matter community, I suspect that most would answer in the affirmative, and if not, at least claim that the particles observed in condensed matter are just as real as any particle observed in particle physics.

Part of the reason I bring this issue up is because of concerns raised soon following the discovery of the fractional quantum Hall effect (FQHE). When the theory of the FQHE was formulated by Laughlin, it was thought that his formulation of quasiparticles of charge e/3 may have been a mere oddity in the mathematical description of the FQHE. Do these particles carrying e/3 current actually exist or is this just a convenient mathematical description?

In two papers that appeared almost concurrently, linked here and here, it was shown using quantum shot noise experiments that these e/3 particles did indeed exist. Briefly, quantum shot noise arises because of the discrete nature of particles and enables one to measure the charge of a current-carrying particle to a pretty good degree of accuracy. In comparing their results to the models of particles carrying charge e versus particles carrying charge e/3, the data shows no contest. Here is a plot below showing this result quite emphatically:

One may then pose the question: is there a true distinction between what really “exists out there” versus a theory that conveniently describes and predicts nature? Is the physicist’s job complete once the equations have been written down (i.e should he/she not care about questions like “are these fractional charges real”)?

These are tough questions to answer, and are largely personal, but I lean towards answering ‘yes’ to the former and ‘no’ to the latter. I would contend that the quantum shot noise experiments outlined above wouldn’t have even been conducted if the questions posed above were not serious considerations. While asking if something is real may not always be answerable, when it is, it usually results in a deepened understanding.

This discussion reminds me of an (8-year old!) YouTube video of David who, following oral surgery to remove a tooth, still feels the affects of anesthesia :