Monthly Archives: June 2016

Schrodinger’s Cat and Macroscopic Quantum Mechanics

A persisting question that we inherited from the forefathers of the quantum formalism is why quantum mechanics, which works emphatically well on the micro-scale, seem at odds with our intuition at the macro-scale. Intended to demonstrate the absurdity of applying quantum mechanics on the macro-scale, the mirco/macro logical disconnect was famously captured by Schrodinger in his description of a cat being in a superposition of both alive and dead states. There have been many attempts in the theoretical literature to come to grips with this apparent contradiction, the most popular of which goes under the umbrella of decoherence, where interaction with the environment results in a loss of information.

Back in 1999, Arndt, Zellinger and co-workers observed a two-slit interference of C60 molecules (i.e. buckyballs), in what was the largest molecule to exhibit such interference phenomena at the time. The grating used had a period of about 100 nm in the experiment, while the approximate de Broglie wavelength of the C60 molecules was 2.5 picometers. This was a startling discovery for a couple reasons:

  1. The beam of C60 molecules used here was far from being perfectly monochromatic. In fact, there was a pretty significant spread of initial velocities, with the full width at half maximum (\Delta v/v) getting to be as broad as 60%.
  2. The C60 molecules were not in their ground state. The initial beam was prepared by sublimating the molecules in an oven which was heated to 900-1000K. It is estimated, therefore, that there were likely 3 to 4 photons exchanged with the background blackbody field during the beam’s passage through the instrument. Hence the C60 molecules can be said to have been strongly interacting with the environment.
  3. The molecule consists of approximately 360 protons, 360 neutrons and 360 electrons (about 720 amu), which means that treating the C60 molecule as a purely classical object would be perfectly adequate for most purposes.

In the present, the record set by the C60 molecule has since been smashed by the larger molecules with mass up to 10,000 amu. This is now within one order of magnitude of a small virus. If I was a betting man, I wouldn’t put money against viruses exhibiting interference effects as well.

This of course raises the question as to how far these experiments can go and to what extent they can be applied to the human scale. Unfortunately, we will probably have to wait for a while to be able to definitively have an answer to that question. However, these experiments are a tour-de-force and make us face some of our deepest discomforts concerning the quantum formalism.

The Mystery of URu2Si2 – Experimental Dump

Heavy fermion compounds are known to exhibit a wide range of ground states encompassing ferromagnetism, anti-ferromagnetism, superconductivity, insulating and a host of others. A number of these compounds also exhibit more than one of these phases simultaneously.

There is one of these heavy fermion materials that stands out among the rest, however, and that is URu2Si2. The reason for this is that there is an unidentified phase transition that occurs in this compound at ~17.5K. What I mean by “unidentified” is that the order parameter is unknown, the elementary excitations are not understood and there is a consensus emerging that we currently may not have the experimental capability to identify this phase unambiguously. This has led researchers to refer to this phase in URu2Si2 as “hidden order”. Our inability to understand this phase has now persisted for three decades and well over 600 papers have been written on this single material. For experimentalists and theorists that love a challenge, URu2Si2 presents a rather unique and interesting one.

Let me give a quick rundown of the experimental signatures of this phase. Firstly, to convince you that there actually is a thermodynamic phase transition that happens in URu2Si2, take a look at this plot of the specific heat as a function of temperature:

In the lower image, one can see two transitions, one into the hidden order phase at 17.5K and one into the superconducting phase at ~1.5K. One can see that there is a large entropy change at the phase transition into the hidden order phase, which makes it all the more remarkable that we don’t know what it going on! I should mention that the resistivity also shows an anomaly going into the hidden order phase both along the a- and c-axis (the unit cell is tetragonal).

Furthermore, the thermal expansion coefficient, \alpha = L^{-1}(\Delta L/\Delta T), has a peak for the in-plane coefficient and a smaller dip for the c-axis coefficient at the transition temperature. This implies that the volume of the unit cell gets larger through the transition, indicating that the hidden order phase exhibits a strong coupling to the lattice degrees of freedom.

For those familiar with the other uranium-based heavy fermion compounds, one of the most natural questions to ask is whether the hidden order phase is associated with the onset of some sort of magnetism. Indeed, x-ray resonance magnetic scattering and neutron scattering experiments were carried out in the late 80s and early 90s to investigate this possibility. The structure found corresponded to one where there was a ferromagnetic arrangement in the a-b plane with antiferromagnetic coupling along the out-of-plane c-axis. However, this was not the whole story. The magnetic moments were extremely weak (0.02\mu_B per Uranium atom) and the magnetic Bragg peaks found were not resolution-limited (correlation length ~400 Angstroms). This means that order was not of the true long-range variety!

Also, rather strangely, the integrated intensity of the magnetic Bragg peak was shown to be linear as a function of temperature, saturating at ~3K (shown below). All these results seemed to imply that the magnetism in the compound was of a rather unconventional kind.

The next logical question to ask was what the inelastic magnetic spectrum looked like. Below is an image exhibiting the dispersion of the magnetic modes. Two different modes can identified, one at the magnetic Bragg peak wavevectors (e.g. (1, 0, 0)) and one at “incommensurate” positions (e.g. 1 \pm 0.4, 0, 0). The “incommensurate” excitations exhibit approximately a ~4meV gap while the gap at (1, 0, 0) is about 2meV. These excitations show up with the hidden order and are thought to be closely associated with it. They have been shown to have longitudinal character.

The penultimate thing I will mention is that if one examines the optical conductivity of URu2Si2, a gap of ~5meV in the charge spectrum is also manifest. This is shown below:

And lastly, if one pressurizes a sample up to 0.5 GPa, the URu2Si2 becomes a  full-blown large-moment antiferromagnet with a magnetic moment of approximately 0.4\mu_B per Uranium atom. The transition temperature into the Neel state is about 18K.

So let me summarize the main observations concerning the hidden order phase:

  1. Weak short-range antiferromagnetism
  2. Strong coupling to the lattice
  3. Dispersive and gapped incommensurate and commensurate magnetic excitations
  4. Gapped charge excitations
  5. Lives nearby anti-ferromagnetism
  6. Can coexist with superconductivity

I should stress that I am no expert of heavy fermion compounds, which is why this is my first real post on them, so please feel free to point out any oversights I may have made!

More information can be found in these two excellent review articles:


Crises in Confidence

While pursuing a PhD in physics, it seems almost inevitable that at some point one will suffer a crisis in confidence. This is usually accompanied by asking oneself some of the following questions, especially if one is intending to go down the academic route:

  1. Am I good enough to be here?
  2. Should I leave with a Masters degree? Have I been in graduate school too long to leave with just a Masters degree?
  3. Should I start developing other skills to make myself a more marketable candidate to pursue other careers?
  4. Do I really like this enough to continue doing this?
  5. Have I made a huge mistake in going to graduate school? My friends who started working right away seem happier.
  6. Is the modern academic climate, where there is pressure to publish, where I want to be?

Obviously, I can’t answer all these questions. Everyone’s answers will be different. The reason I bring these questions up, though, is that they are on everyone’s mind, that is, unless you are going to be the next Feynman. Even the most successful of graduate students will likely go through periods where they are low on confidence.

The only thing that one can do is be honest and work to the best of one’s ability. Despite the immense pressure to publish, I think it is worth pursuing a project that will enable one to say after graduate school, “I accomplished A and I developed skills in B” and not “I published X papers”.

I also think it is worth talking to older graduate students and postdocs about how they combated their periods of low confidence — it may help you get through yours. Talking to one’s advisor about these issues can also help, but be wary that they are sometimes far removed from the graduate school experience.

I think that everyone can and should acknowledge that there is certainly a large element of luck involved in determining one’s scientific path. Sometimes you roll double-sixes and sometimes you roll a two-three combo.

Related: Inna has also written an excellent article about her experience in getting a PhD. You can read it here.

Non-sequitur: I was recently at another beam time run and as with most runs, got little sleep. As fatigue starts to kick in at 4-5 AM, I sometimes (for some bizarre reason) find myself listening to a song on repeat. “Ageless Beauty” happened to be the one this week, a cover of a song originally sung by Canadian group Stars:

Neither Energy Gap Nor Meissner Effect Imply Superflow

I have read several times in lecture notes, textbooks and online forums that the persistent current in a superconductor of annular geometry is a result of either:

  1. The opening of a superconducting gap at the Fermi surface
  2. The Meissner Effect

This is not correct, actually.

The energy gap at the Fermi surface is neither a sufficient nor necessary condition for the existence of persistent supercurrents in a superconducting ring. It is not sufficient because gaps can occur for all sorts of reasons — semiconductors, Mott insulators, charge density wave systems all exhibit energy gaps separating the occupied states from the unoccupied states. Yet these systems do not exhibit superconductivity.

Superconductivity does not require the existence of a gap either. It is possible to come up with models that exhibit superconductivity yet do not have a gap in the single-particle spectra (see de Gennes Chapter 8 or Rickayzen Chaper 8). Moreover, the cuprate and heavy fermion superconductors possess nodes in their single-particle spectra and still exhibit persistent currents.

Secondly, the Meissner effect is often conflated with superflow in a superconductor, but it is an equilibrium phenomenon, whereas persistent currents are a non-equilibrium phenomenon. Therefore, any conceptual attempts to make a conclusion about persistent currents in a superconducting ring from the Meissner effect is fraught with this inherent obstacle.

So, obviously, I must address the lurking $64k question: why does the current in a superconducting ring not decay within an observable time-frame?

Getting this answer right is much more difficult than pointing out the flaws in the other arguments! The answer has to do with a certain “topological protection” of the current-carrying state in a superconductor. However one chooses to understand the superconducting state (i.e. through broken gauge symmetry, the existence of a macroscopic wavefunction, off-diagonal long-range order, etc.), it is the existence of a particular type of condensate and the ability to adequately define the superfluid velocity that enables superflow:

\textbf{v}_s = \frac{\hbar}{2m} \nabla \phi

where \phi is the phase of the order parameter and the superfluid velocity obeys:

\oint \textbf{v}_s \cdot d\textbf{l} = n\hbar/2m

The details behind these ideas are further discussed in this set of lecture notes, though I have to admit that these notes are quite dense. I still have some pretty major difficulties understanding some of the main ideas in them.

I welcome comments concerning these concepts, especially ones challenging the ideas put forth here.

Science for Science’s Sake

A couple weeks ago, Sheila Patek was on the PBS News Hour and discussed eloquently how science and scientists work. She talked about how scientists are driven by curiosity to venture into the unknown, which may or may not lead to applications for humans. When you go somewhere no one has gone before, who knows if it’s going to be of any use? However, that doesn’t mean you shouldn’t go. Here’s the video: