## Saturday, October 12, 2019

### On the entanglement between superfluidity, superconductivity, and entanglement.

On the entanglement between superfluidity, superconductivity, and entanglement.
A lot of misconceptions and mistakes is being done simply because people do not know what they are talking about; meaning, they do not know the exact meaning of the words they use, so they use those words very loosely, without employing the specific meaning of them – because they don’t know that, because they don’t know the definition of that, because they did not have a good education on the foundations of science, and they are too confident, or lazy to do it on their own.

When I talk science, I know what I’m talking about, and I want everyone to know that as well, so we would use the same words assuming the same meaning for all of them.

That is why I start this post from some important definitions (that I borrowed from another post: On a Definition Of Science, but sine it is also mine, that’s not plagiarism).

An object is something that represents the focus of our attention. This is what we are talking about. An object can be physical (e.g. we can touch it; often, we also call it a “system”) or abstract (e.g. a symbol). A small physical object localized at a specific place in space is called a particle (usually, we say a “system” when we focus on many particles or on a large object). For each particle we can assign a set of parameters, and a specific set of values of those parameters we call “a state of a particle”. That state may evolve, i.e. change in time; when that happens, we call it "a process", or a “behavior”.

Now we can talk about the meaning of another term – correlation.

In general, correlation is interdependence between two objects – abstract (i.e. states) or physical (i.e. particles). I will illustrate what “interdependence” means in a couple of paragraphs.

I believe that this definition is better than saying that correlation is a relationship, and it goes well beyond statistics.

There are different types of correlations, for example:
1) A temporal correlation – when two states of the same object correlate.
2) A processual, or behavioral, or spatial, or inter-systemic correlation – when states of at least two different objects correlate (most often not at the same time).

We concentrate on the latter correlation.

For a behavioral correlation interdependence means that a change in a state of one object affects a change in a state of another one, and can be illustrated using a simple model.

Let's say, we observe a behavior (or an evolution) of a state of object #1. Then we repeat the observation; we observe the evolution of the same object keeping all but one circumstances to be exactly the same as in the previous observation, but this time we either remove, or add another object or change a state of one of the previously present objects. If that action does NOT affect the behavior of object #1, we say, that objects #1 and #2 do not correlate, there is no correlation between the states of those two object, correlation is absent. Otherwise, a correlation exists, it is present, evolutions of the objects correlate.

Just to make things absolutely clear, “otherwise” means, that the action of adding or removing an object or changing the state of an object that is not object #1 affects the evolution of object #1, as compared to the previous observation; and hence a correlation exists, it is present, evolutions of the objects correlate.

There are plenty of well-known examples of correlations.

In an outer space, in the absence of any stars a comet travels straight ahead without speeding up or slowing down. But if a star is present on its path, the path is getting bent, hence the star affects the comet, there is correlation between those two objects, and that specific correlation has specific name – interaction, and described by a specific physical quantity – a force; specifically, the force of gravity, that is described by the Newton’s law of gravity.

If you take two plastic cups and connect them with a string in a certain manner, stretch the string and start making sounds on one end, speaking into one cup, a second cup also starts vibrate and making a sound.
In physics there are important laws that describe correlations between different states of different objects; those laws have a name of conservation laws.

For example, if in outer space a grenade blows up and separates in exactly two identical halves; those two objects – halves – will fly away from each other. But if we catch one half of this grenade and measure its velocity, we will be able to predict the exact velocity of another half; and that prediction is governed by the law of consideration of a linear momentum.

Quantum mechanics also describes many correlations between quantum objects, quantum particles. Some of those correlations are very similar to correlations in the macroscopic world; for example, the law of conservation of linear momentum works for quantum particles as well.

However, the vast majority of physicists believe that quantum objects also exhibit a different type of a correlation - a correlation that is specific only for the quantum world, a correlation that does not exists for macroscopic objects, a correlation that has a name “entanglement”.

Many people have been writing about quantum entanglement, including yours truly.

The discussion about entanglement goes deep to the foundations of quantum mechanics. Albert Einstein didn't like it; he thought that entanglement as it was understood in his time, and how it is still understood, demonstrates that quantum mechanics is either not local theory, or includes “spooky” interactions that happen faster than the speed of light (and naturally, Einstein could not accept it).

And idea of entanglement can be described by a thought experiment proposed by Albert Einstein (and that has different variations).

In order to conduct this is experiment, first one needs to prepare two entangled particles, so those two particles would be a part of one quantum system. Then they fly away from each other; and when an observer affects a state of one particle, the second particle should immediately change it state in a specific and predictable way.

Some physicists argue that the term is wrong; that this not some kind of new quantum type correlation called “entanglement”, this is just another manifestation of some law of conservation, that works on the quantum level in the same way it works in the classical world.

The majority of physicists believe that entanglement is a uniquely quantum phenomenon and cannot be explained by application of laws of conservation.

I’ve been back-and-force on this, but eventually I settled on the idea of entanglement, but not because of to-be-fashionable experiments with a couple of entangled particles (even if they are produced in bulk) – those experiments are ideologically trivial, although technologically challenging and advanced.

There are much more interesting demonstrations of quantum entanglement, and those are superfluidity and superconductivity.

Let's start from superfluidity.

It was discovered in 1937 in liquid Helium below about 2 K (at the normal pressure). In its superfluid state, quote: “the liquid … behaves as if it consists of two components: a normal component, which behaves like a normal fluid, and a superfluid component with zero viscosity

The two-fluid model is clear and intuitive. At the absolute zero, T = 0 K, all Helium is superfluid, i.e. does not experience friction and can flow without experiencing resistance. Above a certain temperate called critical (at a given pressure), Helium behaves like a regular normal fluid with internal and external viscosity. Above absolute zero, but below a critical temperate Helium is “composed” of two components: one components is “normal” and behaves like a regular liquid as it was above the critical temperature, but the second component is superfluid, like it would be at zero temperature.

The mystery here is what happens to atoms below the critical temperature?

If you would look at the Helium atoms above or below the critical temperature, you would not see any difference. And yet, they behave in the very different way! Why?

There must be some difference on the atomic level, otherwise what would be the source/reason for a superfluid component to exist?

The answer is – quantum entanglement!

When temperature drops below the critical one, quantum correlations between many atoms make those atoms to be entangled and to form one highly/strongly correlated quantum state, also called a coherent state. Atoms in that coherent state form the superfluid component.

At T = 0 K, all atoms belong to this coherent state. Above zero, some atoms belong to the coherent state, and others don’t, and those that do not form the normal component of Helium. Above critical temperature coherent state does not exist.

The difference between atoms in a normal state and atoms in a superfluid coherent state is very significant (although, atoms constantly “jump” from one state into another and back, on average, the number of atoms in each state remains constant). That difference is described by how those atoms react to an attempt to be excited, for example by a simple hit from another atom.

If we hit (a fancy term is “collide”) an atom in a normal component (state, phase) we can transfer to that atom practical any amount of energy, from very little to very large. When we make this atom, that is a part of a normal component, collide with a different atom, for example with an atom in a wall of a vessel or a tube holding Helium, this collision results in a transfer of energy from an atom in a wall to the atom in the normal component, or back. This interaction is responsible for internal friction and for the friction between the normal component and the walls of a pipe, and when the fluid travels through a pipe this interaction is responsible for the existence of resistance/friction/viscosity.

However, if we try to hit/knock/collide an atom in the coherent state, that atom is entangled with all other atoms in that state, with all other atoms that compose a superfluid component. So, when we hit a single atom in that state, we hit all of them in that state. And in order to excite this whole system of correlated entangled atoms, a small energy is not enough anymore to excite it. The energy has to be above a certain threshold; if the energy is below that threshold, the atoms in the entangled stated do not “feel” it, “ignore” the interaction. And that is why this component of a fluid, that is composed of correlated atoms in a coherent entangled state, can travel through pipes without feeling any resistance. That makes this component superfluid.

When the system goes through the critical temperature in one direction or another, atoms do not disappear, or new atoms do not appear; but when the temperature drops below the critical one, some atoms become entangled, they become strongly correlated in a specific quantum way, and they form a new phase, a superfluid phase.

What is happening when the temperature rises above the critical?

Note, when we said that above critical temperature coherent state does not exist, that statement was not exactly accurate. According to the general theory of critical transitions, a critical transition does not happen instantly. Even above a critical temperature in some parts of a material and for a short periods of time coherent state may exist. But it is very short-living, or metastable, and the regions with the coherent state are small and do not overlap so they cannot cover the whole sample of a material. When temperature is getting closer to the critical value, those short-living short-range pockets of a new phase getting larger and live longer and below the critical temperature finally occupy the whole material space. So, in general, the traces of a new phase may be observed even above the critical temperature.

A superfluid transition is not related to existence or absence of new particles, but related to the absence or presence of strong quantum correlations between large numbers of particles. Those correlations may exist even above the critical temperature in a form of metastable, short-range and short-living pockets.

Superconductivity was discovered in 1911 but explained only in 1957.

To that time, a well-developed model could explain many electrical phenomena based on the idea that electrons in conductors could be treated as a carrying electric charge fluid.

Electric current could be seen as a river, as a stream of electric charges traveling in the same direction. Atoms or ions would present islands that could slow that stream down. If you would give this electric fluid a push, those islands would eventually make it stop - and this is a nature of electric resistance in conductors.

Turns out, however, that under certain circumstances those islands could also help fluid to travel.

We could imagine that when electron fluid flows around an island, sometimes some of those electrons may start traveling together in a common whirl. Of course, this is just a visual representation of a fact that some electrons can move in a correlated manner - not an actual motion of the electrons.

So, the islands (atoms, ions) could not just slow down the motion of the electrons, but also could make them moving in a correlated manner. And then those correlated electrons can correlate with other correlated electrons all across a conductor, forming one strongly correlated entangled state. When that happens, a macroscopic number of electrons get entangled in a “superfluid electric component”, that can travel with no resistance, and is called a “supercurrent”.

When electrons form one highly correlated fully entangled macroscopic state similar to a superfluid component in Helium, and when electrons in that state interact with atoms, in the event when that interaction is not strong enough, if it doesn't go over a certain threshold of energy, those entangled electrons just ignore that interaction and keep exist in the unchanged entangled state.

Those electrons that get correlated in “a whirl”, due their motion around the same island (atom, ion) has a name a Cooper pair. Below a critical temperature electrons in those pairs form an entangled superconductive state.

However, like Helium atoms do not disappear above critical temperature, Cooper pairs also should not completely disappear immediately above critical temperature. Although, the number of Cooper pairs should significantly decrease with even the slightest increase in the temperature, I bet, accurate experiments would demonstrate their existence even in the normal state.

And now it is a natural time to talk about high-temperature superconductivity.

It was discovered in 1986, and to this day there is no commonly accepted explanation of this phenomenon.

In high-temperature superconductors electrons cannot be treated any more as a fluid. A better model would treat electrons rather as a solid, or as a crystal where electrons spend most of the time at certain locations, and from time to time hop from one location to another one. However, since electrons form a superconductive state, it means that under certain circumstances, at least some of the electrons are getting entangled in a coherent strongly correlated state that allows them to travel through the material without feeling the existence of other electrons or atoms or ions.

We can try to understand the nature of this phenomenon building on another very important and common feature that all those highly entangled states have in common.

In a superfluid or in a superconductor, that highly correlated state with entangled particles (atoms of electrons), due to strong correlations represents a state with a very high order, or a very law chaos, as compared with the regular normal state or a component of a material. Hence, the entropy of this state is equal to zero. Entropy is a measure of order v. chaos; more chaos - higher entropy; and when a macroscopic state is strongly correlated and highly entangled, the chaos is so low that the entropy is zero.

In all superfluid or superconductive systems, below a certain temperature macroscopic parts of a system simply stop “generating” an entropy.
At the absolute zero temperature, the entropy of all systems is zero, but in certain systems under certain circumstances when temperature is rising above zero, some parts of this system still remain “feeling” themselves like they still at zero temperature, forming a highly ordered, strongly correlated entangled state. With the rise in temperature, the portion of the entangled part of a system gradually decreases to zero and becomes zero at a critical temperature (with some short-living short-range pockets above the critical temperature).

This idea may guide us toward better understanding of the nature of high-temperature superconductivity.

It is assumed that in materials exhibiting high-temperature superconductivity, it happens when an original order of electrons is destroyed or disturbed by doping.

In an original material (under ideal circumstances, at zero temperature) electrons from a “crystal”. The simplest model would look like this:
A plus or a minus in the picture represents that fact that electrons have a feature called a “spin”; we can imagine a spin as an arrow attached to an electron that can have only two directions, in or our (or up v. down, or left v. right – but always only TWO). And in an ideal case, two neighboring electrons would prefer having opposite spins (because that makes their “life” easier, or energy lower). In this “crystal”, electrons prefer stay at their places, because they “hate” – meaning, electrically repel – each other. Hence, jumping to a place already occupied by another electrons is restricted, almost forbidden. Hence, in this state the material a is not a good conductor, but an insulator. However, doping can change that.

Doping is procedure that can change the number of electrons, for example, take some electrons out form their locations. This is done by inserting some other atoms that can attract electrons and keep those electrons on those atoms. In that case, the structure of the material is not so ordered anymore, it has some random holes – empty spaces at the locations from where electrons were removed.

For a better visibility, in the picture those holes are noted by black circles (instead of just keeping those places empty).

Now we have to use the most powerful human ability – imagination. Other animals do not have it. Albert Einstein said that imagination is the true sign of intelligence. No single AI professional has any idea how to model it (e.g. The new stage of the race for AI domination AI).

Imagine that ALL black holes would move on step diagonally down and the electron moved from that place one step diagonally up (to take the place where the hole was). This action is shown by arrows in the next picture.
As the result, you get this:
Now, compare the two pictures below, the first one is an excerpt from the ordinal picture (before the jump), and the second on is an excerpt from the final picture (after the jump):
----------------------------------------------------------------------------------------------
These two pictures are identical!

They show two DIFFERENT but IDENTICAL states of the system. Of course, because our pictures have a finite size, we could see some differences at the edges, but for a very large system (or with periodic boundary conditions) the deference would be negligible.

This simple illustration makes us to make a conclusion that when all electrons would jump simultaneously in the same direction – that is diagonal – this actions does NOT change the state of the system. In a certain sense – this action does not make the state more chaotic, less ordered – as long as all electrons are entangled.

It is naturally to assume, that this is the state we are looking for, the state with zero entropy.

Based on this idea, we could assume that in high-temperature superconductors electrons jump:
(a) in the same direction;
(b) diagonally.

I do not follow specialized literature, but I know that I am not the first who suggested that in high-temperature superconductors electrons may travel in the same direction (even I wrote on this and published it on arXiv; although in a very primitive way – sorry for the visual picture used to support my model, I like visual pictures, as you may have noticed already). I also heard of a suggestion that electrons jump diagonally (honestly, do not remember where). But to my best knowledge, I am the first one who makes both statements as a model for high-temperature superconductivity.

Due to very specific geometry of the model, it could be tested using a simple mechanical motion of a sample of a high-temperature superconductor – the movement along diagonals or along the edges would affect supercurrent differently. However, for this experiment one would need a thin, ideally a single, sheet of a high-temperature superconductor. This still maybe a challenging technological obstacle.

Dr. Valentin Voroshilov

P.S. On this page (if you click on this link) you will find some piece on the foundation of quantum mechanics. Recently, it has become fashionable to run actual or thought experiments on quantum entanglement. However, ALL of them are based on ONE specific interpretation of quantum mechanics and completely ignore and do not mention the fact that so far there are several different interpretations of it. In my publications I attacked different aspects of different publications on the matter.

Appendix
Everyone who claims he/she knows what quantum mechanics is about must read the original EPR paper (so, ask the guy - have you read EPR? that is a litmus test for you should you even listen to the guy).
It has many layers, more than just the thought experiment they use to claim that quantum mechanics is not a complete theory (e.g. click on this link and scroll down to Appendix III).
The fact of the matter is that this experiment does show that quantum mechanics is different from classical mechanics. When this matter is accepted, one has a choice: (a) follow the strategy "shut up and calculate" and do not spend any time on trying to make the theory "complete" or (b) spend some time on trying to make the theory "complete".
In the latter case, one can be inventing different approaches - some are mentioned in the four pieces about a cat:

But the simplest (thank you Occame!) way to resolve all the mysteries of quantum mechanics would be to assume that - yes, faster than light interactions do exist! Tachyons are responsible for that "spooky action at distance". There is a whole world of particles that cannot travel slower than the speed of light! And that world interacts with our world, where particles cannot travel faster than the speed of light. Simple!
Imagine a sea of tachyons. Every known particle can have its counterpart in that sea - tachyo-electron, tachyo-proton, etc. Due to fluctuations, for a teeny-tiny instant of time, those tachyons may enter our world, become a so called virtual particle, and interact with our particles. But even more interesting process happens when our particles can disappear from our world and enter the world of tachyons, spend there a teeny-tiny instant of time and come back again - but at a different location, or with a different speed.
Of course, until tachyons are found, they are just a theory, a mathematical abstract. But so was the Higgs boson.
BTW: tachyons, or in general the world of faster than light particles, can explain such phenomenon as tunneling. A classical particle cannot escape a potential well - when it has not enough energy. But a quantum particle can "tunnel" through. Why? Because due to interactions with tachyons it may "accidentally" (a scientific name - fluctuations) gain energy enough to get "over the well".