# Monthly Archives: June 2021

## 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.