Time mirrors are very cool. They seem a bit magical, undoing the damage that time has done, at least under the right circumstances. For light and for water waves, we know how to make a mirror that behaves like it reverses time. But for quantum mechanical waves, some types of time mirrors have proven to be difficult. A recent theory paper suggests that, under the right circumstances, a mirror that undoes the past might be possible. And, it comes with a bonus movie.

Before we get to the lovely movie and a bit of inexplicable quantum theory, let me give you a taste of what a time mirror means in a classical world.

Bouncing light pulses

Imagine a very short pulse of light that passes through a bit of glass. A pulse is made up of many different colors of light that are all traveling together. If you were to shine the pulse in your eye, it would appear as a blinding flash of white. But, when the pulse hits the glass, the red colors travel just a bit faster than the blue colors. If your other eye was fast enough, it would see a dot that went from red to yellow to green to blue after the light had gone through the glass (Our eyes are neither fast, nor robust—you would just blind yourself, so don't do it).

A mirror that shines the light back through the glass does not undo the damage done to the light pulse. The red part of the pulse hits the mirror first, returns to the glass first, and thus gets an added boost over the blue light, so the colors spread out even further. However, a time mirror will store the pulse as it hits, then emit the pulse in the reverse order. So, the red bits arrive at this mirror first, but leave last.

Since the pulse is inverted, the red parts of the pulse are behind the blue, but still travel faster within the glass. By the time the pulse leaves the glass, it looks like it never entered in the first place. All the damage done by the glass has been undone.

Because quantum

Quantum mechanics also has waves—waves that represent the probability amplitude of finding a particle in a particular position, for instance. But in quantum mechanics, any measurement is likely to change the wave in a major way. If we don't measure the wave, how do we know how to reverse it in time?

It turns out that, for some quantum systems, you can do that without measurement. The essential idea is a bit subtle, and to get the gist, we need to understand how electrons behave in solid materials.

In all solid materials, the atoms are bound to each other by sharing electrons. Now, electrons don't have labels and, like small children, they don't know where they live. So, if an electron is initially shared between atoms A and B, what stops it from also moving on and sharing the love with atom C?

The freedom with which electrons move around is determined by the spatial arrangement of atoms, which sets an energy value on electron mobility. Electrons that possess an energy above that threshold value are free to move around, while electrons below it will be faithful to their home atoms. There is no requirement that the two thresholds be the same. Indeed, for metals, the threshold for confining electrons is higher than the threshold for which electrons are free, so there are always electrons that are free to move. For insulators, there is a large gap between the two thresholds, and no electrons have enough energy to move around freely.

The gap, called a band gap, between the two thresholds contain electron energy levels that simply cannot be occupied in the material.

This occurs because of the wave nature of electrons. An electron is also a wave, and that wave scatters off all the surrounding atoms. If an electron had an energy in the middle of the band gap, its wave will destructively interfere with itself, meaning that the probability for its existence goes to zero.

The result is that electrons never have an energy that falls in the band gap. This, it turns out, can also be used to create a quantum time mirror.

Electrons in a band, don't mind the gap

Let's Imagine injecting an electron with enough energy to be free to move. The probability amplitude of the electron position spreads outwards like water waves from a dropped stone: a ripple of probability expanding outwards. Then, suddenly, we modify the material's properties (with a magnet or a light pulse, for instance), and the electron wave finds itself in the forbidden gap region. What does it do?

The problem it faces is that it does not have enough energy to jump to an available free state and it has no way to give up energy to drop to a confined state. The wave cannot move either, because it now suffers from destructive interference.

The destructive interference forces the wave to periodically change from a forward-moving wave to a backward-moving wave. Once you know when the wave will change from moving forward to backward, you can create a time mirror.

At the appropriate moment, you undo your modification so that the electron wave finds itself back in the land of the free. If you time it right, the electron probability wave starts to move again, but in the opposite direction. So it returns from whence it came and condenses to a point. (Yes, this is similar to spin-echo techniques used in nuclear magnetic resonance spectroscopy. In case you were about to ask.)

Of course, this is a calculation for a perfect material, but all materials have imperfections. Edges, missing atoms, and many other imperfections cause all sorts of havoc. In this particular case, an imperfection acts as a place that can absorb or give up energy. Essentially, the imperfection can boost the electron out of the stationary wave pattern and destroy the time mirror. But this isn't much of a problem in this contest, since the calculation only applies to 2D materials like graphene. Obtaining large areas of 2D material without imperfections is perfectly possible. On the time scale of a few picoseconds, this might all work.

Even so, I'm not even sure what you might use it for. Still, it makes for a cool movie, and I'm hoping that it can be realised in the lab. Actually, I'm pretty confident the experiment can be performed: all of these tricks have been managed in gases of ultra cold atoms, so I'd be absolutely astonished if researchers can't replicate them in solid materials too.

Physical Review B, 2017, DOI: 10.1103/PhysRevB.95.165421