Transverse force orbits

A natural idea to obtain circular orbits was to give the droplet an electric charge and to place it in a magnetic field. If an analogy to quantum mechanics exists we should then observe a form of Landau quantization of the orbits. However, it is difficult to control the electric charge of a droplet so that we turned to a variant. It relies on an analogy first used by Michael Berry [ref 1]. Its starting point is the similarity of relation B=∇×A in electromagnetism with 2Ω=∇×U in fluid mechanics. In these relations, the vorticity 2Ω is the equivalent of the magnetic field B, the fluid velocity U that of the vector potential A. Using this analogy M. Berry found a remarkable hydrodynamic analogue of the Aharonov-Bohm effect (1). Wishing to investigate circular orbits we use the same analogy but differently. A charge q moving at velocity V in a homogeneous external magnetic field B is submitted to F_B=q(V × B) . In a system rotating at an angular velocity Ω, a mass m moving with velocity V is submitted to a Coriolis force F_Ω=-m(V × 2Ω) where 2Ω is the vorticity of the solid-body rotation. In classical physics, both these forces lead to orbiting motions in the planes perpendicular to B or Ω respectively. In a magnetic field the orbit has a radius ρ_L=mV / qB. On a rotating surface a mobile of velocity V moves on a circle of radius R_C=V / 2Ω.

We thus devised a system in which the walker moves in a rotating system [P1, T2]. If it was a normal classical particle it should move on a circle of radius R_C decreasing continuously with increasing Ω. This is what is observed in the regime of low memory. In contrast, when the memory is long enough, so that the particle interacts with the wave originated along the whole orbit, the circular orbits present a spontaneous quantization. An additional force due to the cumulated waves then limits the possible orbits to a discrete set. The wave-sustained path-memory is thus demonstrated to generate a quantization of the orbit size and thus of the angular momentum.

This experiment was revisited by D. Harris and J. Bush [ref 2] and by A. Oza et al. [ref 3 & ref 5]. They showed that in the very high memory range there were regimes with intermittent transitions involving several of the quantized states so that in these regimes only the probabilities of being in one state are defined.

Central force orbits

In order to establish the analogy to quantum physics on a firm ground a basic step should be to observe a quantization of energy. Take for instance a quantum particle trapped in a harmonic potential well. It is well known that in quantum physics it can be in successive eigenstates of increasing energy and decreasing mean wavelength as sketched in the one-dimensional case in Fig. 1a. This is unfortunately an experiment that cannot be done with a walker because: the droplet velocity being approximately constant, so is the particle’s kinetic energy. As for the wave field, it is characterized by a fixed wavelength λ_F. However, it can be noted that in the quantum situation, the successive eigenstates could be obtained at a fixed energy and wave-length by tuning the width of the well (see Fig. 1b). Here we will use this idea to seek the possible emergence of eigenstates when the spatial confinement is varied.

In order to trap the walker in a potential well, the millimetre size drop is loaded with a small amount of ferrofluid and submitted to a magnetic field. Two coils in a Helmholtz configuration provide a uniform magnetic field so that the drop becomes a small dipole of moment m_B. A small cylindrical magnet placed at a distance d above the liquid bath is the source of an additional non-uniform magnetic field so that the magnetized drop is submitted to a centripetal central force. The potential well can be considered as harmonic. The width of this well can be tuned by changing the distance d of the magnet to the liquid surface. For a drop of mass m, the dynamics is then reduced to the motion of a bouncing particle propelled by a wave and submitted to an external force F_ext = -κr, where κ the spring constant depends on d the distance from magnet to the bath.

In the low memory limit, the droplet motion is given by the classical laws of mechanics and its orbit is circular [P18 & P20]. In the high memory limit the most generally observed trajectory, has a complex aspect with a looped structure. But a fine tuning of the potential width reveals the existence of simple stable trajectories, each existing in a narrow range of values of Λ. When these orbits are circular (film V.3) they have the same discrete radii as those observed in the Coriolis experiment. In these conditions other types of stable trajectories also show-up, having the shape of lemniscates (as in film V.4), eccentric loops or trefoils.

These discrete orbits observed in the experiment (P18, P21, T3) as well as in its simulation (P21 & T4) can be compared by measuring their main characteristics: their mean spatial extension R and their mean angular momentum L_z. A remarkable feature is that both these quantities are quantized. Figure shows them as plotted against the non-dimensional width of the potential well Λ. Both are related to the wavelength only: the control parameter Λ only plays a role in selecting amongst the possible R_n. For each level R_n, only a discrete set of possible angular momentum values L_z is reachable by the system. The same discrete values of L_z are observed for different R_n.

A two dimensional graph can be obtained by plotting against each other R and L_z. The various types of trajectories appear to be well defined in this 2D plot. From these results, we can define two quantization numbers n that defines the orbit size and m that defines the in a discrete set of values of the angular momentum. All the trajectories can be positioned at nodes (n,m) of a lattice with the generic rules: n is an integer, and m can only take the values m∈{- n,-n+2,...,n+2,n} (see Fig. V-5)(P18, T3 & T4)

What is the origin of these quantization effects that are also perfectly recovered in numerical simulations [T4] shown in the films IV.6 and IV.7 ?

In the high memory situations the drop keeps emitting waves in a bounded central region. The accumulation of these waves will necessarily be related to the global wave eigenmodes of this domain. This will have a feedback effect on the droplet motion. The trajectories that eventually emerge are those for which the trajectory shape and the global wave field have achieved a mutual adaptation. We examined the quantization of the mean radii of the various trajectories in this framework. We used a general approach that consists in decomposing (using Graf’s theorem) the entire wave field on the basis formed by the Bessel functions centred on the axis of the potential well. The total wave generates a virtual landscape of wave potential energy that traps the particle in specific orbits. This model [P18, T3, T4] gives an excellent interpretation of the possible radii of the circular orbits: they corresponds to the successive zeros of a J₀ function centered on the magnet axis.

The intermittency and the "wave function collapse"

We have thus shown that, as expected from the situation sketched in Figure V.2, there were specific values of the confinement that permitted stable orbital motions. These "states" are characterized by a non-quantum double quantization of the orbit size and the mean angular momentum. Naturally our system has no relation with the Planck constant but, as discussed in [P11], an analogy appears by considering that the Faraday wavelength plays here a role comparable to the de Broglie wavelength in quantum mechanics. We can now examine the motion of the walker when the width Λ of the potential well does not correspond to a pure state.

As shown in Film V.5 and Fig. V.6 the motion appears complex. However, direct observation reveals that regular motions still exist but only during limited time intervals (Fig. V.6 a). If the temporal evolution of the angular momentum L_z is recorded as a function of time, (Fig. V.6 b) the signal is not erratic, but composed of well-defined domains separated by abrupt transitions. The trajectory is thus formed of a succession of sequences of pure eigenstates with intermittent transitions between them. From long recordings probability distributions function of Lz can be obtained (Fig. V.6 c).

As demonstrated in [P19], this chaotic motion is characterized by an intermittency phenomenon inducing transitions between different periodic orbits. In this type of complex situation, the walker, at any given time, is in one of the possible discrete modes. This justifies the use of the term eigenstates for the states (n,m): they form a basis of decomposition on which the complex trajectories can be decomposed. In general the value of Λ defines the time spent on the average in each of the states. Only a fine-tuning of Λ permits the preparation of a "pure" state for which any time evolution is blocked.

Finally, we can note that we have in this macroscopic system the possibility of performing a continuous non-intrusive observation. It is interesting to consider a gedanken situation in which this observation would be impossible and we could only perform single intrusive measurements. Each of these destructive glimpses would lead to the observation of one of the eigenstates only but they would differ from each other. With many realizations, a probability of each result would emerge. A superposition of states would then be the best reachable description, the probabilities would seem intrinsic and the measurement would appear as a resulting from a "wave function collapse".