# Tag Archives: Experiment

## Decoherence does not imply wavefunction “collapse”

While most physicists will agree with the statement in the title, some (especially those not working in quantum information) occasionally do get confused about this point. But there are a very beautiful set of experiments demonstrating the idea of “false decoherence”, which explicitly show that entangling a particle with the environment does not necessarily induce “wavefunction collapse”. One can simply “disentangle” a particle from its environment and interference phenomena (or coherence) can be recovered. Sometimes, these experiments fall under the heading of quantum eraser experiments, but false decoherence happens so often that most of the time it isn’t even noticed!

To my mind, the most elegant of these experiments was first performed by Zou, Wang and Mandel in 1991. However, the original article is a little opaque in describing the experiment, but, luckily, it is described very accessibly in an exceptional Physics Today article by Greenberger, Horne and Zeilinger that was written in 1993.

As a starting point, it is worth reviewing the Mach-Zehnder interferometer. An image of this interferometer is below (taken from here):

Quickly, laser light from the left is incident on a 50/50 beam splitter. A photon incident on the beam splitter thus has a probability amplitude of going through either the upper path or the lower path. Along each path, the photon hits a mirror and is then recombined at a second 50/50 beam splitter. The surprising thing about this experimental arrangement is that if the two path lengths are identical and the laser light is incident as shown in the image above, the photon will emerge at detector D1 with 100% probability. Now, if we add a phase shifter, $\phi$, to our interferometer (either by inserting a piece of glass or by varying the path length of, say, the upper path, the photon will have a non-zero probability of emerging at detector D2. As the path length is varied, the probability oscillates between D1 and D2, exhibiting interference. In a previous post, I referred to the beautiful interference pattern from the Mach-Zehnder interferometer with single photons, taken by Aspect, Grangier and Roger in 1986. Below is the interference pattern:

Now that the physics of the Mach-Zehnder interferometer is hopefully clear, let us move onto the variant of the experiment performed by Zou, Wang and Mandel. In their experiment, after the first beam splitter, they inserted a pair of non-linear crystals into each branch of the interferometer (see image below). These non-linear crystals serve to “split” the photon into two photons. For instance, a green incident photon may split into a red and yellow photon where $\omega_{green} = \omega_{red} +\omega_{yellow}$ and energy is conserved. (For those that like big words, this process is referred to as spontaneous parametric down-conversion). Now, what we can do is to form a Mach-Zehnder interferometer with the yellow photons. (Let’s ignore the red photons for now). The \$64k question is: will we observe interference like the in the original Mach-Zehnder interferometer? Think about this for a second before you continue reading. Why would or why wouldn’t you expect interference of the yellow photons? Do the non-linear crystals even make a difference?

It turns out that the yellow photons will not interfere. Why not? Because the red photons provide “which-path information”. If one were to put a detector at O in the figure or below the dichroic mirror D3, we would be able to detect the red photon, which would tell us which path the yellow photon took! So we know that if there is a red photon at O (D3), the yellow photon would have taken the upper (lower) path. We can no longer have single-particle interference when the object to be interfered is entangled with another object (or an environment) that can yield which-path information. Mathematically, interference can be observed when an object is in a superposition state:

$\left|\psi\right\rangle = 1/\sqrt{2} ( \left|U\right\rangle + e^{i\phi}\left|L\right\rangle )$

But single-particle interference of the yellow photon cannot be observed when it is entangled with a red photon that yields which-path information (though two-particle interference effects can be observed — we’ll save that topic for another day):

$\left|\psi\right\rangle = 1/\sqrt{2} ( \left|Y_U\right\rangle \left|R_U\right\rangle + e^{i\phi} \left|Y_L\right\rangle \left|R_L\right\rangle )$

In this experiment, it is helpful to think of the red photon as the “environment”. In this interpretation, this experiment provides us with a very beautiful example of decoherence, where entangling a photon with the “environment” (red photon), disallows the observation of an interference pattern. I should mention that a detector is not required after D3 or at O for the superposition of the yellow photons to be disallowed. As long as it is possible, in principle, to detect the red photon that would yield “which-path information”, interference will not be observed.

Now comes the most spectacular part of this experiment, which is what makes this Zou, Wang, Mandel experiment notable. In their experiment, Zou, Wang and Mandel overlap the two red beams (spatially and temporally), such that it becomes impossible to tell which path a red photon takes. To do this, the experimenters needed to ensure that the time it takes for a photon to go from BS1 (through a and d) to NL2 is identical to the time from BS1 (through b) to NL2. This guarantees that were one to measure the red photon after D3, it would not be possible to tell whether the red photon was generated in NL1 or NL2.

So the question then arises again: if we overlap the two red beams in this way, can we observe interference of the yellow photons at BS2 now that the “which-path information” has been erased? The answer is yes! Mathematically, what we do by overlapping two red beams is to make them indistinguishable:

$\left|\psi\right\rangle = 1/\sqrt{2} ( \left|Y_U\right\rangle \left|R_U\right\rangle + e^{i\phi} \left|Y_L\right\rangle \left|R_L\right\rangle )$

$\rightarrow 1/\sqrt{2} ( \left|Y_U\right\rangle + e^{i\phi} \left|Y_L\right\rangle ) \left|R\right\rangle$

Here, the yellow and red photons are effectively decoupled or disentangled, so that single-particle superposition is recovered! Note that by engineering the “environment” so that “which-path information” is destroyed, coherence returns! Also note that just by inserting an opaque object O in the path d, we can destroy the interference of the yellow beams, which aren’t even touched in the experiment!

Thinking about this experiment also gives us deeper insight into what happens in the two-slit experiment performed with buckyballs. In that experiment, the buckyballs are interacting strongly with the environment, but by the time the buckyballs reach the screen, the “environment” wavefunctions are identical and effectively factor out:

$\left|\psi\right\rangle = 1/\sqrt{2} ( \left|slit1\right\rangle \left|env1\right\rangle + e^{i\phi} \left|slit2\right\rangle \left|env2\right\rangle )$

$\rightarrow 1/\sqrt{2} ( \left|slit1\right\rangle + e^{i\phi} \left|slit2\right\rangle ) \left|env\right\rangle$

To my mind, the Zou-Wang-Mandel experiment is superlative because it extends our understanding of the two-slit experiment to a remarkable degree. It shows that decoherence does not imply “wavefunction collapse”, because it is possible to engineer “re-coherence”. Thus, one needs to distinguish reversible or “false” decoherence from irreversible or “true” decoherence.

## Mott Switches and Resistive RAMs

Over the past few years, there have been some interesting developments concerning narrow gap correlated insulators. In particular, it has been found that it is particularly easy to induce an insulator to metal transition (in the very least, one can say that the resistivity changes by a few orders of magnitude!) in materials such as VO2, GaTa4Se8 and NiS2-xSx with an electric field. There appears to be a threshold electric field above which the material turns into a metal. Here is a plot demonstrating this rather interesting phenomenon in Ca2RuO4, taken from this paper:

It can be seen that the transition is hysteretic, thereby indicating that the insulator-metal transition as a function of field is first-order. It turns out that in most of the materials in which this kind of behavior is observed, there usually exists an insulator-metal transition as a function of temperature and pressure as well. Therefore, in cases such as in (V1-xCrx)2O3, it is likely that the electric field induced insulator-metal transition is caused by Joule heating. However, there are several other cases where it seems like Joule heating is likely not the culprit causing the transition.

While Zener breakdown has been put forth as a possible mechanism causing this transition when Joule heating has been ruled out, back-of-the-envelope calculations suggest that the electric field required to cause a Zener-type breakdown would be several orders of magnitude larger than that observed in these correlated insulators.

On the experimental side, things get even more interesting when applying pulsed electric fields. While the insulator-metal transition observed is usually hysteretic, as shown in the plot above, in some of these correlated insulators, electrical pulses can maintain the metallic state. What I mean is that when certain pulse profiles are applied to the material, it gets stuck in a metastable metallic state. This means that even when the applied voltage is turned off, the material remains a metal! This is shown here for instance for a 30 microsecond / 120V 7-pulse train with each pulse applied every 200 microseconds to GaV4S8 (taken from this paper):

Electric field pulses applied to GaV4S8. A single pulse induces a insulator-metal transition, but reverts back to the insulating state after the pulse disappears. A pulse train induces a transition to a metastable metallic state.

Now, if your thought process is similar to mine, you would be wondering if applying another voltage pulse would switch the material back to an insulator. The answer is that with a specific pulse profile this is possible. In the same paper as the one above, the authors apply a series of 500 microsecond pulses (up to 20V) to the same sample, and they don’t see any change. However, the application of a 12V/2ms pulse does indeed reset the sample back to (almost) its original state. In the paper, the authors attribute the need for a longer pulse to Joule heating, enabling the sample to revert back to the insulating state. Here is the image showing the data for the metastable-metal/insulator transition (taken from the same paper):

So, at the moment, it seems like the mechanism causing this transition is not very well understood (at least I don’t understand it very well!). It is thought that there are filamentary channels between the contacts causing the insulator-metal transition. However, STM has revealed the existence of granular metallic islands in GaTa4S8. The STM results, of course, should be taken with a grain of salt since STM is surface sensitive and something different might be happening in the bulk. Anyway, some hypotheses have been put forth to figure out what is going on microscopically in these materials. Here is a recent theoretical paper putting forth a plausible explanation for some of the observed phenomena.

Before concluding, I would just like to point out that the relatively recent (and remarkable) results on the hidden metallic state in TaS2 (see here as well), which again is a Mott-like insulator in the low temperature state, is likely related to the phenomena in the other materials. The relationship between the “hidden state” in TaS2 and the switching in the other insulators discussed here seems to not have been recognized in the literature.

Anyway, I heartily recommend reading this review article to gain more insight into these topics for those who are interested.

## 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 :

## 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 is 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 was key to observing Bloch oscillations in semiconductor superlattices.

Let me use the two-well potential to give you a picture of how this localization might occur. Imagine symmetric potential wells, 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 form 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) in a semiconducting superlattice since it is an insulator! 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.

## 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!).