## Wannier-Stark Ladder, Wavefunction Localization and Bloch Oscillations

Most people who study solid state physics are told at some point that in a totally pure sample where there is no scattering, one should observe an AC response to a DC electric field, with oscillations at the Bloch frequency ($\omega_B$). These are the so-called Bloch oscillations, which were predicted by C. Zener in this paper.

However, the actual observation of Bloch oscillations are not as simple as the textbooks would make it seem. There is an excellent Physics Today article by E. Mendez and G. Bastard that outline some of the challenges associated with observing Bloch oscillations (which was written while this paper was being published!). Since the textbook treatments often use semi-classical equations of motion to demonstrate the existence of Bloch oscillations in a periodic potential, they implicitly assume transport of an electron wave-packet. To generate this wave-packet is non-trivial in a solid.

In fact, if one undertakes a full quantum mechanical treatment of electrons in a periodic potential under the influence of an electric field, one arrives at the Wannier-Stark ladder, which shows that an electric field can localize electrons! It is this ladder and the corresponding localization which is key to observing Bloch oscillations.

Let me use the two-well potential to give you a picture of how this localization might occur. Imagine a symmetric potential well, where the lowest energy eigenstates look like so (where S and A label the symmetric and anti-symmetric states):

Now, imagine that I start to make the wells a little asymmetric. What happens in this case? Well, it turns out that that the electrons start to localize in the following way (for the formerly symmetric and anti-symmetric states):

G. Wannier was able to solve the Schrodinger equation with an applied electric field in a periodic potential in full and showed that the eigenstates of the problem are actually of a form of a Stark ladder. This means that the eigenstates are of identical functional form from quantum well to quantum well (unlike in the double-well shown above) and the energies of the eigenstates are spaced apart by $\Delta E=\hbar \omega_B$! The potential is shown schematically below. It is also shown that as the potential wells slant more and more (i.e. with larger electric fields), the wavefunctions become more localized (the image is taken from here (pdf!)):

A nice numerical solution from the same document shows the wavefunctions for a periodic potential well profile with a strong electric field, exhibiting a strong wavefunction localization. Notice that the wavefunctions are of identical form from well to well.

What can be seen in this solution is that the stationary states are split by $\hbar \omega_B$, but much like the quantum harmonic oscillator (where the levels are split by $\hbar \omega$), nothing is actually oscillating until one has a wavepacket (or a linear superposition of eigenstates). Therefore, the Bloch oscillations cannot be observed in the ground state (which includes the the applied electric field) — one must first generate a wavepacket in the solid.

In the landmark paper that finally announced the existence of Bloch oscillations, Waschke et. al. generated a wavepacket in a GaAs-GaAlAs superlattice using a laser pulse. The pulse was incident on a sample with an applied electric field along the superlattice direction, and they were able to observe radiation emitted from the sample due to the Bloch oscillations. I should mention that superlattices must be used to observe the Wannier-Stark ladder and Bloch oscillations because $\omega_B$, which scales with the width of the quantum well, needs to be fast enough that the electrons don’t scatter from impurities and phonons. Here is the famous plot from the aforementioned paper showing that the frequency of the emitted radiation from the Bloch oscillations can be tuned using an electric field:

This is a pretty remarkable experiment, one of those which took 60 years from its first proposal to finally be observed.

## Consistency in the Hierarchy

When writing on this blog, I try to share nuggets here and there of phenomena, experiments, sociological observations and other peoples’ opinions I find illuminating. Unfortunately, this format can leave readers wanting when it comes to some sort of coherent message. Precisely because of this, I would like to revisit a few blog posts I’ve written in the past and highlight the common vein running through them.

Condensed matter physicists of the last couple generations have grown up ingrained with the idea that “More is Different”, a concept first coherently put forth by P. W. Anderson and carried further by others. Most discussions of these ideas tend to concentrate on the notion that there is a hierarchy of disciplines where each discipline is not logically dependent on the one beneath it. For instance, in solid state physics, we do not need to start out at the level of quarks and build up from there to obtain many properties of matter. More profoundly, one can observe phenomena which distinctly arise in the context of condensed matter physics, such as superconductivity, the quantum Hall effect and ferromagnetism that one wouldn’t necessarily predict by just studying particle physics.

While I have no objection to these claims (and actually agree with them quite strongly), it seems to me that one rather (almost trivial) fact is infrequently mentioned when these concepts are discussed. That is the role of consistency.

While it is true that one does not necessarily require the lower level theory to describe the theories at the higher level, these theories do need to be consistent with each other. This is why, after the publication of BCS theory, there were a slew of theoretical papers that tried to come to terms with various aspects of the theory (such as the approximation of particle number non-conservation and features associated with gauge invariance (pdf!)).

This requirement of consistency is what makes concepts like the Bohr-van Leeuwen theorem and Gibbs paradox so important. They bridge two levels of the “More is Different” hierarchy, exposing inconsistencies between the higher level theory (classical mechanics) and the lower level (the micro realm).

In the case of the Bohr-van Leeuwen theorem, it shows that classical mechanics, when applied to the microscopic scale, is not consistent with the observation of ferromagnetism. In the Gibbs paradox case, classical mechanics, when not taking into consideration particle indistinguishability (a quantum mechanical concept), is inconsistent with the idea the entropy must remain the same when dividing a gas tank into two equal partitions.

Today, we have the issue that ideas from the micro realm (quantum mechanics) appear to be inconsistent with our ideas on the macroscopic scale. This is why matter interference experiments are still carried out in the present time. It is imperative to know why it is possible for a C60 molecule (or a 10,000 amu molecule) to be described with a single wavefunction in a Schrodinger-like scheme, whereas this seems implausible for, say, a cat. There does again appear to be some inconsistency here, though there are some (but no consensus) frameworks, like decoherence, to get around this. I also can’t help but mention that non-locality, à la Bell, also seems totally at odds with one’s intuition on the macro-scale.

What I want to stress is that the inconsistency theorems (or paradoxes) contained seeds of some of the most important theoretical advances in physics. This is itself not a radical concept, but it often gets neglected when a generation grows up with a deep-rooted “More is Different” scientific outlook. We sometimes forget to look for concepts that bridge disparate levels of the hierarchy and subsequently look for inconsistencies between them.

## The Struggle

Haruki Murakami, the world-renowned Japanese novelist, has garnered a large following because one can easily relate to his protagonists. I have been reading his novels for around ten years now, and recently picked up his unique memoir What I Talk About When I Talk About Running. It is a quirky book, at once about his marathon and ultra-marathon running endeavors, his writing struggles, and how the two are interwoven.

To me, the most inspirational part of this book lies in how through mundaneness and mediocrity springs a rather unique exceptionalism. Murakami is an outstanding writer, but his talents have a limit, and he is honest about this. Most of the book is about struggling, with running and with writing. When I reflect on the book, the image I have in my mind is of a  truck wheel, bearing huge weight, going around and around, yet somehow trudging forward.

Here is a passage from the book I particularly enjoyed, which is applicable in many contexts:

…writers who aren’t blessed with much talent — those who barely make the grade — need to build up their strength at their own expense. They have to train themselves to improve their focus, to increase their endurance. To a certain extent, they’re forced to make these qualities stand in for talent. And while they’re getting by on these, they may actually discover real, hidden talent within them. They’re sweating, digging out a hole at their feet with a shovel, when they run across a deep, secret water vein. It’s a lucky thing, but what made this good fortune possible was all the training they did that gave them the strength to keep on digging. I imagine that late-blooming writers have all gone through a similar process.

Naturally, there are people in the world (only a handful, for sure) blessed with enormous talent that, from beginning to end, doesn’t fade, and whose works are always of the highest quality. These fortunate few have a water vein that never dries up, no matter how much they tap into it. For literature, this is something to be thankful for. It’s hard to imagine the history of literature without such figures as Shakespeare, Balzac and Dickens. But the giants are, in the end, giants — exceptional, legendary figures. The remaining majority of writers who can’t reach such heights (including me, of course) have to supplement what’s missing from their store of talent through whatever means they can. Otherwise it’s impossible for them to keep on writing novels of any value. The methods and directions a writer takes in order to supplement himself becomes part of that writer’s individuality, what makes him special.

Most of what I know about writing I’ve learned through running everyday. These are practical, physical lessons. How much can I push myself? How much rest is appropriate — and how much is too much? How far can I take something and still keep it decent and consistent? When does it become narrow-minded and inflexible? How much should I be aware of the world outside, and how much should I  focus on my inner world? To what extent should I be confident in my abilities, and when should I start doubting myself? I know that if I hadn’t become a long-distance runner when I became a novelist, my work would have been vastly different. How different? Hard to say, but something would have definitely been different.

The book ends with what Murakami hopes his tombstone will read:

Haruki Murakami

1949-20**

Writer (and Runner)

At Least He Never Walked

## Kapitza-Dirac Effect

We are all familiar with the fact that light can diffract from two (or multiple) slits in a Young-type experiment. After the advent of quantum mechanics and de Broglie’s wave description of matter, it was shown by Davisson and Germer that electrons could be diffracted by a crystal. In 1927, P. Kapitza and P. Dirac proposed that it should in principle be possible for electrons to be diffracted by standing waves of light, in effect using light as a diffraction grating.

In this scheme, the electrons would interact with light through the ponderomotive potential. If you’re not familiar with the ponderomotive potential, you wouldn’t be the only one — this is something I was totally ignorant of until reading about the Kapitza-Dirac effect. In 1995, Anton Zeilinger and co-workers were able to demonstrate the Kapitza-Dirac effect with atoms, obtaining a beautiful diffraction pattern in the process which you can take a look at in this paper. It probably took so long for this effect to be observed because it required the use of high-powered lasers.

Later, in 2001, this experiment was pushed a little further and an electron-beam was used to demonstrate the effect (as opposed to atoms), as Dirac and Kapitza originally proposed. Indeed, again a diffraction pattern was observed. The article is linked here and I reproduce the main result below:

(Top) The interference pattern observed in the presence of a standing light wave. (Bottom) The profile of the electron beam in the absence of the light wave.

Even though this experiment is conceptually quite simple, these basic quantum phenomena still manage to elicit awe (at least from me!).

## Diversity in and of Physics

When someone refers to a physicist from the early twentieth century, what kind of person do you imagine? Most people will think of an Einstein-like figure, but most likely, one will think of a white male from western Europe or the US.

Today, however, things have changed considerably; physics, both as a discipline and in the people that represent it, has become more diverse. This correlation is probably not an accident. In my mind, the increased diversity is an excellent development, but as with everything, it can be further improved. There are a couple excellent podcasts I listened to recently that have championed diversity in different contexts.

The first podcast was an episode of Reply All entitled Raising the Bar (which you should really start listening to at 11:52 after the rather cringe-worthy Yes-Yes-No segment!). The episode focuses on the lack of diversity in many companies in Silicon Valley. In doing so, they interview an African-American man named Leslie Miley who was a security guard at Apple and went on to work as a software developer and manager at Twitter, Apple, and Google among other companies (i.e. he possessed a completely unorthodox background by Silicon Valley standards). He makes an interesting statement about companies in general (while referring specifically to Twitter) saying:

If you don’t have people of diverse backgrounds building your product, you’re going to get a very narrowly focused product.

He also goes onto say that including people from different backgrounds is not just appropriate from a moral standpoint, but also that:

Diverse teams have better outcomes.

There is plenty of research to support this viewpoint. In particular, Scott Page from the Santa Fe institute and University of Michigan – Ann Arbor is interviewed in the episode and suggests that when teams of people are selected and asked to perform a task, teams of “good people” from diverse backgrounds generally outperform many “excellent people”/experts from similar backgrounds (i.e. the same Ivy League schools, socio-economic status, age etc.).

There is a caveat that is presented in this episode, however. They suggest that it may take longer for a diverse team to gel and to communicate and understand each other. But again, the outcomes in the long-term are generally better.

There is an excellent episode of Hidden Brain that also covers similar topics, but focuses on building a better workplace. The host of the podcast, Shankar Vendantam, interviews the (then) head of human resources at Google, Laszlo Bock, to gain some insight into how Google has been able to build their talent pool. Of specific interest to physicists was how much Google borrows from places like Bell Labs to build a creative workplace environment. Again, Bock stresses the importance of diversity among the employees at Google in order for the company to be successful.

In physics departments across the country, I think it is necessary to take a similar approach. Departments should strive to be diverse and hire people from different backgrounds, schools, genders, and countries. Not only that, graduate students with unorthodox backgrounds should also be welcomed. This again, is not just important for the health of the department, but for the health of the discipline in general.

I strongly suspect that Michael Faraday was one of the greatest experimental physicists in the past few hundred years not in spite of his lack of mathematical acuity, but probably because of it. His mathematical ability famously did not extend much beyond basic algebra and not even as far as trigonometry.

## Halloween Humor…

Click the image for similar cartoons…

Thanks to Judy Nguyen who shared these with me!

## Bohr-van Leeuwen Theorem and Micro/Macro Disconnect

A couple weeks ago, I wrote a post about the Gibbs paradox and how it represented a case where, if particle indistinguishability was not taken into account, led to some bizarre consequences on the macroscopic scale. In particular, it suggested that entropy should increase when partitioning a monatomic gas into two volumes. This paradox therefore contained within it the seeds of quantum mechanics (through particle indistinguishability), unbeknownst to Gibbs and his contemporaries.

Another historic case where a logical disconnect between the micro- and macroscale arose was in the context of the Bohr-van Leeuwen theorem. Colloquially, the theorem says that magnetism of any form (ferro-, dia-, paramagnetism, etc.) cannot exist within the realm of classical mechanics in equilibrium. It is quite easy to prove actually, so I’ll quickly sketch the main ideas. Firstly, the Hamiltonian with any electromagnetic field can be written in the form:

$H = \sum_i \frac{1}{2m_i}(\textbf{p}_i - e\textbf{A}_i)^2 + U_i(\textbf{r}_i)$

Now, because the classical partition function is of the form:

$Z \propto \int_{-\infty}^\infty d^3\textbf{r}_1...d^3\textbf{r}_N\int_{-\infty}^\infty d^3\textbf{p}_1...d^3\textbf{p}_N e^{-\beta\sum_i \frac{1}{2m_i}(\textbf{p}_i - e\textbf{A}_i)^2 + U_i(\textbf{r}_i)}$

we can just make the substitution:

$\textbf{p}'_i = \textbf{p}_i - e\textbf{A}_i$

without having to change the limits of the integral. Therefore, with this substitution, the partition function ends up looking like one without the presence of the vector potential (i.e. the partition function is independent of the vector potential and therefore cannot exhibit any magnetism!).

This theorem suggests, like in the Gibbs paradox case, that there is a logical inconsistency when one tries to apply macroscale physics (classical mechanics) to the microscale and attempts to build up from there (by applying statistical mechanics). The impressive thing about this kind of reasoning is that it requires little experimental input but nonetheless exhibits far-reaching consequences regarding a prevailing paradigm (in this case, classical mechanics).

Since the quantum mechanical revolution, it seems like we have the opposite problem, however. Quantum mechanics resolves both the Gibbs paradox and the Bohr-van Leeuwen theorem, but presents us with issues when we try to apply the microscale ideas to the macroscale!

What I mean is that while quantum mechanics is the rule of law on the microscale, we arrive at problems like the Schrodinger cat when we try to apply such reasoning on the macroscale. Furthermore, Bell’s theorem seems to disappear when we look at the world on the macroscale. One wonders whether such ideas, similar to the Gibbs paradox and the Bohr-van Leeuwen theorem, are subtle precursors suggesting where the limits of quantum mechanics may actually lie.