Author: solvese

A while ago, in an episode of the popular Scandinavian talk show Skavlan, the founder of Norwegian Air Shuttle, Bjørn Kjos, met with famous mathematician and Fields medalist Cedric Villani. Kjos was talking about how he had been trying to solve the Schrödinger equation – seemingly impressing Villani quite a bit. “Wow, you are brave!” he exclaimed. However, I suppose he would have been even more impressed if Kjos had taken on the Dirac equation.

You may have learned that when material things start approaching the speed of light, the absolute speed limit of nature, our world starts looking strange. According to Einstein’s theory of relativity, things become shorter and time evolves more slowly.

Perhaps you have also heard that no-one has succeeded in formulating a theory comprising both quantum physics and relativity. Luckily, this is only partly true. More specifically, it is true when it comes to Einstein’s theory of general relativity, which deals with gravitation and is essential to our understanding of how the universe came about and evolves.

General theory’s predecessor, special relativity, however, is happily united by quantum physics. This union is entitled the Dirac equation, which the Brit Paul Dirac introduced in 1928 – only a couple of years after Erwin Schrödinger published his famous equation. Dirac and Schrödinger shared a Nobel price for that only a few years later.

The Dirac equation includes the Schrödinger equation as a special case – except that it also encompasses the strange notion of electron spin. This is, however, not the strangest thing that emerges from Dirac’s equation.

The equation is plagued with several issues which makes it hard to solve – also numerically. One of these issues is that it features solutions of negative energy – in addition to the expected ones. The meaning of these odd solutions remained a mystery for a while – even to Paul Dirac himself. In an interesting interview conducted by Friedrich Hund in 1982 – well worth seeing – he talks about how he spent a year or so pondering on how to deal with them. His solution, eventually, was to postulate a sea of occupied states – a sea from which particles could be promoted leaving behind a hole. Somewhat exotic, you may think.

These notions are, as it turns out, related to the existence of anti-matter. For every particle there is an odd twin with much of the same properties – except that most of them have opposite sign. Not a very intuitive conclusion; you cannot blame anyone – including Dirac – for being puzzled by this notion. I guess this is one of the many concepts emerging from quantum physics which merits the exclamation “who ordered that?!”. (The quote originates from Isodor Rabi, the father of the MR technique, when he heard about the discovery of the muon particle, one of the electron’s heavier siblings.)

Nonetheless, anti-matter exists; we have seen it. Moreover, we have seen that pairs of particle and anti-particle may be created – they may emerge out of nothing, so to speak. Needless to say, this complicates life for anyone who may want to describe quantum physics close to the speed of light theoretically or computationally. We cannot even say for sure how many particles we are dealing with!

Luckily for us, there is a window between the point where classical, classical in the meaning non-relativistic, breaks down and the point at which pair creation needs to be considered. For instance, with the strongest of lasers, electrons may be accelerated to a certain fraction of the speed of light – thus necessitating a relativistic treatment. Even in cases where these laser fields are not strong enough to start producing particle anti-particle pairs, the numerical description is still hard enough. Thus, we were thrilled to see that relativistic effects could, to a large extent, be treated by introducing the concept of relativistic mass into the Schrödinger equation. In effect, the electron becomes a bit heavier as it becomes faster. This, in turn, contributes to enhance the effect of atomic stabilization – the notion that an atom becomes less likely to be torn apart by a laser field as you increase the strength of the laser. In other words: Yet another non-intuitive feature of quantum physics – perhaps a topic for a future blog post.

So far there is only one animal was involved into quantum physics, that is the famous Schrödinger’s cat. Now some researchers claim that there is one more.

In February 2019, I came across the piece “Quantum theory: the weird world of teleportation, tardigrades and entanglement” [1]: Gröblacher is also interested in experiments involving living creatures … He is currently working on putting a sheet of nitride into a superposition of states … “A superposition state of these membranes would allow us to demonstrate that objects that are visible to the naked eye still behave quantum, and we can really test decoherence – the transition between classical and quantum mechanics,” he says. He hopes to extend the experiment by placing tiny living organisms called tardigrades onto the membrane of silicon nitride, putting them into superposition too.

Targidrates are known for being tough [2]. These micro-animals (their official title) can enter a ‘hibernation state’ of near complete dehydration and metabolism rate decreased by factor 1/1000, and, being in this state, survive an exposure to outer space (almost perfect vacuum), high-intensity radiation of all kinds (including gamma rays), and pressure up to 1200 atmosphere. Therefore, they might hopefully withstand the cryogenic environment required to achieve ground state cooling of the membrane. If there an animal able to survive superposition this must be a tardigrade.

After reading the piece, my immediate thought was: But would the tardy feel the difference between just the groundstate and superposition of the groundstate and the first excited state of the membrane? And in what measurable terms? Or is this difference is simply negligible on the background of the mere exposure to the cryogenic environment?

Another thought back then: The typical size of silicon nitride membranes Gröblacher is dealing with is 0.5 mm. This is also the typical size of tardigrades. I do not know the masses of the membranes and tardigrades but expect them to be comparable. It is not possible to maintain the extra-high quality factor of such membranes after placing on them a tardigrade – unless the membranes are on-purpose designed and curved, with ‘nests’ for tardys (a-la seats of Space Jockeys [3]).

Three years have passed and in 2022 a work with a sensational title, “Entanglement in a qubit-qubit-tardigrade system”, was published in New journal of Physics [4].

The authors claimed that hey set a tardigrade into entanglement with two qubits — and the former has survived it (“The animal is then observed to return to its active form after 420 hours at sub 10 mK temperatures and pressure of 6 × 10⁻⁶ mbar, setting a new record for the conditions that a complex form of life can survive“). But had the animal really been quantum entangled?

To prove it, one has to measure the quantum properties of the tardigrade, which the experiment does not do. Instead, a model was used (“The tomographic data shows entanglement in the qubit-qubit-tardigrade system, with the tardigrade modelled as an ensemble of harmonic oscillators or collection of electric dipoles“). Well, the quantum community is sceptical about such a ’proof’.

Also, there is a strong scepticism about that entanglement can be obtained by simply placing a tardy on top of a qubit. After that the qubit is no longer a resonator with well-tuned characteristics so one should not talk about ‘groundstate’ and ‘excited state’. From the point of view of quantum physics, a tardy is a system with macroscopically many degrees of freedom so it decoheres the qubit by working as an external environment. But doesn’t the environment become entangled with the object it is acting on? But is it measurable?

The problem is that the animal is not a quantum object in a sense that its state cannot be described as a superposition of a few basis states. So where is the entanglement? In such situation, the entanglement cannot be located and thus cannot be measured. We could also say that we are entangled with an electron in a oxygen atom of 0₂ floating somewhere inside out right lung – but it is not measurable and therefore does not exist (at least for quantum physics).

Anyway, the ‘entanglement’ claimed in the paper was not assessed in experiment and therefore the clam is much stronger than the experimental data can support. Schrödinger’s tardigrade is not yet here and Schrödinger’s cat has to wait for a companion.

[1] https://www.sciencefocus.com/science/quantum-theory-the-weird-world-of-teleportation-tardigrades-and-entanglement

[2] https://en.wikipedia.org/wiki/Tardigrade

[3] https://cdna.artstation.com/p/assets/images/images/004/247/248/large/mickael-goyec-orthographic-plansj.jpg?1481702424

[4] K. S. Lee at al., Entanglement in a qubit-qubit-tardigrade system, New Journal of Physics 24, 123024 (2022); https://iopscience.iop.org/article/10.1088/1367-2630/aca81f#:~:text=In%20particular%2C%20for%20any%20finite,electric%20field%2C%20see%20Appendix%20F%20.