The simplest time crystal: a particle in a box in the lowest energy state
Let us consider the simplest textbook example of quantum mechanics at work: a particle in a box.
|
| spoon
0
particle
| <-- • --> |
box, length L
Classically, the "crystal" would be a state where the particle has been stopped at a certain location. In quantum mechanics, the particle bounces back and forth, so that its de Broglie wavelength is double the length of the box.
Suppose that we insert a spoon in the box and try to slow down the particle. Why this cannot succeed?
________
/ \
jumping rope rotates
The wave function of the particle is like the jumping rope of children rotating around its ends fixed at the walls of the box.
Any disturbance to the wave moves the system to a higher energy state. When we try to insert the spoon into the box, we feel "pressure" resisting the insertion procedure. We must do work against the pressure.
If the particle is at the lowest energy level, then the time crystal of its movement is the slowest possible movement. The time crystal is stable simply because the particle cannot move any slower.
A time crystal involves "frame-dragging": any change in the motion requires additional energy
From general relativity we know frame-dragging around a rotating neutron star or a black hole. The lowest energy state of an approaching test mass is one where the test mass moves along with the rotating body. If we want to keep the approaching mass static relative to the global frame, we have to supply "kinetic energy" to it.
The lowest energy state of a time crystal drags the frame of electrons.
Let us compare this to an ordinary crystal. There the lowest energy state is static (except for zero-point oscillation). A time crystal probably behaves just like an ordinary crystal, if we take into account that the frame is dragged here and there.
If we push a single atom in an ordinary crystal, we have to do work. Furthermore, the momentum which we gave to the atom quickly disperses to the entire crystal.
An ordinary crystal may act as an easy-to-grasp analogue of a time crystal.
The time crystal of conducting electrons plus lattice vibrations in a superconductor
At a high temperature, conducting electrons behave much like a gas. A gas is not a time crystal since it is not ordered.
lattice of ions
+ +
o o o o o o <----
o o o o o o <---- vibration
o o o o o o <----
o o o o o o <----
^ ^
| |
e- e-
conducting electrons
We conjecture that in a superconductor many, or all, conducting electrons form a time crystal together with lattice vibrations. Electrons like to move in zones where a vibration has concentrated positive charge.
The electrons move some 1,000 km/s. It cannot be an ordinary crystal where electrons would be confined each to a small space.
The orbitals of a molecule are an analogue for the time crystal in the lattice. Electrons must keep on moving. Some electrons may have orbits which ship them around for the whole length of the molecule.
We conjecture that a superconductor is like a giant molecule where denser zones of the lattice ions are like nuclei in a molecule, except that these denser zones move at the speed of 5 km/s. Electrons move 200 times faster and will order and synchronize their own movement according to these zones.
How does the superconductor time crystal deal with impurities, or borders of crystals of the metal?
An ordinary crystal grows in a way where it tries to avoid impurity atoms or molecules. A time crystal probably behaves in the same way. It avoids impurities in the lattice.
How about borders of metal crystals? In a non-superconducting metal, borders of crystals cause scattering of conducting electrons. For the time crystal there are two options at a border:
1. The structure of the time crystal is so strong that it prevents scattering. Any vibration caused in the structure by crossing the border is a part of the time crystal itself. There cannot be any dissipation because it is the lowest energy state. The vibration cannot escape from the time crystal to the outside world: it remains as a part of the time crystal itself.
2. Some of the electrons are scattered at the border, but they remain as a part of the time crystal. There cannot be any dissipation of phonons or electromagnetic waves because it is the lowest energy state.
How does a thermal phonon behave in the time crystal of a superconductor?
Let us have a time crystal, attached to the center of the laboratory floor, in a thermodynamic equilibrium with its environment. The environment is a cavity filled with black body radiation.
Photons in surrounding space are turned into phonons as they enter the time crystal.
Since it is an equilibrium, the movements in the time crystal cannot systematically change, e.g., the angular momentum relative to the center of the laboratory. The time crystal should behave just like an ordinary crystal in a thermodynamic equilibrium.
An ordinary crystal can refract or reflect photons. It can absorb energy from photons, as long as the total flux of energy in and out of the crystal is zero.
Let us try to model how a time crystal reacts to a phonon which enters the crystal at a random location.
"diffraction grating" by the phonon
e- electron flow
|
v
+ + + + zones of positive charge
^
|
e- electron flow
The phonon concentrates positive charge in the lattice into zones marked with symbols +. It is like a diffraction grating for the approaching electrons.
The phonon contains a small amount of energy. The energy, or a part of it, is absorbed as vibrations in the time crystal.
The vibrations may be longitudinal in the electron flows, or they may be transverse, making the flows move sideways.
If the time crystal would not be in the lowest energy state, then there would be free kinetic energy in the electron flows, and the vibrations would steal kinetic energy: there would be resistance.
But what exactly is the reason why we cannot disturb much the movement in the electron flows? A hydrogen atom may appear completely neutral to the outside world, even though the electron is moving rapidly around the proton. Could it be that the electron flow plus associated lattice vibrations are not coupled to the thermal phonon at all?
If the negative charge in the electron flows is completely canceled by the positive charge concentration in the lattice vibrations, then the system might appear neutral.
Theorem. Any system of charges in its lowest energy state must have a static electromagnetic field if observed from a distance. Otherwise, the system would radiate.
The theorem suggests that the diffraction grating meets a continuous flow of electrons. It must not see individual electrons, because if it would, there would be an electromagnetic wave radiating from the system.
+ charge
e- --->
o o o o o o o o o o o o
long molecule
| |
====================== frame
An analogous setup: we have a long molecule where an electron orbits from end to end. The molecule is attached to a frame. We put a positive charge close to the molecule. There probably will be some dipole force between the molecule and the electron. The force has to be constant. If it were periodic, then the molecule would radiate electromagnetic waves.
We conclude that the electron flows appear to the diffraction grating as essentially continuous flows of charge. The grating disturbs the flows only a little.
Electron flows in a time crystal are like "continuous" flows of charge
Hypothesis. Electron flows in a time crystal look like continuous flows of charge to a disturbance like a thermal phonon. There is no scattering of individual electrons.
\ scattered wave
\
\
------------------ ^
------------------ |
+ + + + periodic potential
------------------ ^
------------------ | e- free electron
The scattering of a free electron looks like the diagram above. The periodic potential perturbs the electron wave and creates a weak scattered wave.
If there can be no scattering, the diagram above cannot be the right depiction of the process. What kind of a wave might describe a continuous flow of charge?
The perpetuum mobile problem if we assume a potential wall between different states of the current in a superconductor
Suppose that we have a superconducting wire in the lowest energy state. It is a time crystal.
e- ----> e- ---->
---------------- ================== -------------
ordinary superconductor
wire
Then we add an electron to the left end of the superconductor, or alternatively, remove an electron from the right end.
Let us assume that the added electron "joins" a flow of electrons in the time crystal.
There are problems, though: how does the electron know to join a flow which goes exactly from end to end? Also, the claim that an extra electron can "join" a flow without spoiling the properties of the time crystal, is very ad hoc. Why should it be true?
resistor
------- ### -------
| |
=============
e- ---->
superconductor
<-------
I = ε
Let us consider the following setup. There is a superconductor in a loop with an ordinary conductor whose resistance is extremely small. We put a small initial current
I = ε
into the loop.
Let us assume that the electron flow in the superconductor now is in some kind of a local lowest energy state with the current ε flowing. There is a potential wall which prevents the system from decaying to an I = 0 state.
Then we would have a perpetuum mobile which generates heat in the resistor for ever.
How to prevent the existence of a perpetuum mobile? Since there is now a small voltage over the superconductor, maybe that voltage creates the opposite current -ε within the superconductor, so that the total current is zero?
But that does not work. If there is a potential wall which prevents the decay ε -> 0, then there probably is a potential wall which prevents the transition 0 -> -ε.
Question. If we try to explain superconductivity by a (local) energy minimum argument, does that always lead to the existence of a perpetuum mobile?
Conclusions
Energy minimum arguments may explain the behavior of a time crystal in the lowest energy state. The time crystal in this case is a closed system.
But a superconductor in a loop with an ordinary conductor is a not a closed system if there is a current ε flowing in the loop. A local energy minimum argument for the current ε seems to lead to the existence of a perpetuum mobile. That is, conservation of energy is broken.
We have to investigate this more. In earlier blog posts we tried to explain superconductivity with Bloch's theorem for amorphous matter. Maybe that is the way. Above we imagined that a single electron moves along an electron flow to the other end of the semiconductor. Maybe the current is a collective movement of the time crystal? The hypothetical "condensate" in BCS theory moves the charge with a collective motion.
How would we explain the magnetic flux quantum? If the fraction part of the flux is canceled by a superconducting current, is that compatible with our earlier claim that the time crystal of electrons circulating around the loop prefer to have a flux an integer times Φ₀ through the loop?
Since the fraction part of the flux is canceled by a superconducting current, that means that the magnetic field of a superconducting current is not quantized!
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