Carrying loads on the head has been a common practice in many parts of the world. I remember ancient paintings of women from Southern Italy dressed in traditional costumes, carrying water with the typical “conca”, a hourglass-shaped copper or tin container of 10 or 20 liters, held on their heads with a simple length of cloth shaped into a ring, the “cercine”. A similar feat could be adopted to carry clothes to wash at a fountain, or large baskets of grapes at harvest time, which were all carefully held without spilling a single drop, or a grain. One can still watch scenes of this kind in many developing countries, in places or at times when there are no vehicles available, for carrying a burden that can easily approach half the person’s weight or more, along dirt roads ridden with holes, bumps and humps (see enclosed). Carrying weight on the head is an exercise that requires perfect balance, and most importantly an unusual resistance to dynamic perturbations along the path. This makes for an interesting metaphor of today’s physics news, in which a team of scientists in Germany tried to establish the maximum velocity an atom can maintain along a rugged path, without losing a single bit of its cargo of quantum information. Even more interestingly, this physical problem is linked to an old, outstanding question in mathematical physics.

In June 1696, the Swiss mathematician Johann Bernoulli published a seemingly innocent problem in the Acta Eruditorum, the first scientific journal of the German-speaking part of Europe (I note that this journal was still entirely written in Latin until 1782, while the Philosophical Transactions of the Royal Society was published in English already in 1667): Datis in plano verticalis duobus punctis A et B, assignare mobili M, viam AMB, per quam gravitate suam descendens et moveri incipiens a puncto A, brevissimo tempore perveniat ad alterum punctum B. (Johann’s Latin is correct, but it looks like is written by a first-year student). Translation: Given two points A and B in a vertical plane, find the fastest path from A to B that a moving mass M would follow, driven down by gravity. This is the famous “brachistochrone” problem, which kept busy some beautiful minds between the XVII and XVIII century. (Funny to note that Bernoulli posted the problem in Latin, but the denomination comes from the Ancient Greek for “fastest time”, brakystòs cronòs). 

Apparently, Bernoulli already had found the solution when he sent the letter to the journal, although he bragged his brother’s Jakob more accurate solution for his own. However, following a typical custom of the scientists of that time, he challenged other mathematicians in Europe, and allowed six months to present the solutions. No answer was given in time, and even Gottfried Leibniz requested an extension of the deadline by at least one full year. However, on the afternoon of January 29, 1697, Isaac Newton found the challenge in his mail, in a letter sent from Bernoulli himself. He solved the problem in just one night and dispatched the solution anonymously in the Philosophical Magazine. When Bernoulli read it, he famously declared that he recognized Newton behind the anonymous solver “tamquam ex ungue leonem”, that is: as the lion by its claw. However, Newton never published the details of the complete solution, which was only later reconstructed from a letter to his protégé David Gregory. Newton straightforwardly declared the solution to be a cycloid, and provided a graphical method to construct a cycloid including the initial and final point.

How come that the fastest path connecting two points in a plane under the action of gravity should be such a strange curve, as a cycloid? I presume that if such a question was asked to the average layman, the most probable answer would be just: “M will take the straight slope going from A to B”. The role of gravity (and the approximation of frictionless movement) is the key to this question. The body M must gain speed to devour time, ideally by choosing a trajectory leaving point A as close as possible to the vertical, where the projected gravity is maximum; but then, once it reaches a sufficient speed, it has to move forward to get to its endpoint B, and the more vertical it moves in the first part of the trajectory, the longer the total path. Therefore, the fastest path connecting A to B must be a compromise, between a trajectory exploiting as much of the vertical force as possible, while at the same time making the pathlength as short as possible. Obviously, the straight line connecting A to B is the shortest path; however, the initial acceleration is not enough in this case, unless A and B lie exactly on top of each other in the plane. A steeper curve, tangent to A is needed. A simple arc of circle could represent already a good approximation, whereas a segment of parabola does a quite bad job (see enclosed figure). The cycloid does the job.

In practice, the brachistochrone problem makes clear the intuitive notion that a minimum time is required for any physical process to occur, thereby ruling out any possible notion of “immediate action at a distance”. To change its state, a physical system requires a minimum time corresponding to some sort of transition frequency: if the transition time were to be zero, the frequency would be infinite, requiring an infinite amount of energy. The Russian physicist Leonid Mandelstam and his student Igor Tamm published a paper in January 1945 (J. Phys. USSR 9, p.249-254, completed just a few days before Mandelstam’s death) that translates this principle into the realm of quantum mechanics. They considered the transformation between two quantum states with an energy difference ∆E, proving that the fastest quantum time allowed (“quantum brachistochrone”) cannot be smaller than (ℏ/2∆E), a finding that gives physical meaning to Heisenberg’s uncertainty principle for the energy. 

However, the Mandelstam-Tamm limit is rigorous only for a perfect two-level system, but cannot be proved for two “distant” states, that is quantum states whose wave functions have zero overlap. In this case, the operator connecting the two quantum states is non-local, and the simple Rabi oscillation connecting the two states in this case would give a vanishing (Franck-Condon) integral. Far from being a purely academic question of quantum measure theory, such a notion is central to quantum computing. In fact, the problem of preserving the quantum information in a computing device is crucial to the possibility that qubits could be practically used to build a real computer. A few weeks ago, the group of Andrea Alberti in Bonn performed experiments of quantum teleportation of a massive object (a Caesium atom) between distant locations, two sites of an optical periodic lattice, created by counter-propagating laser beams. They were able to measure the path of the atom — actually, of the center of the wave packet defining the atom’s position in space — while travelling between the two sites, which were separated by a distance ∆x about 15 times the (initial) size of the Cs atom wave packet; in practical terms, they moved a single atom over a distance of about 0.5 microns. The “fidelity” of the transport process was measured by the square of the overlap integral between the initial state and the state at each time : the closer such a squared modulus is to 1, the closer is the quantum state to its initial information contents. (Notably, the process only looks at the initial and final states, saying nothing about intermediate states.) By changing the depth and distance of the optical trap, Alberti and coworkers determined the minimum distance at which maximum fidelity is preserved, and from this, the maximum velocity the quantum wave packet can reach along its path: the multi-level quantum brachistochrone. 

If you want to know more about their beautiful experiments, you can read the enclosed paper from Physical Review X 11, 011035 (2021). On the other hand, if you are just curious to know whether the solution of the quantum transport problem is anywhere close to the classical cycloid, I will anticipate that… it is not: the trajectory is a mix of waves wiggling and giggling about the geodetic that connects the initial and final sites. Keeping the information safe while moving as fast as possible is very much analogous to carrying a bowl containing water from one location to another, at the highest possible speed that avoids a spill, like the above examples of women carrying water bowls on their head. Alberti et al. also found that the velocity fluctuations along the path cancel out the effect of intermediate excited states, much alike to the movements that one can do to counteract the water sloshing in the container, while walking on a rugged path… Isn’t that amazing?

Good vibrations

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