1. Introduction and Motivation
The Aharonov–Bohm (AB) effect [
1], which is an interference phenomenon, continues to exhibit quantum mechanical features that have not yet been fully understood, e.g., the anomalous AB interference in a quantum Hall Fabry–Pérot interferometer as observed in Ref. [
2], the time-dependent AB effect and the absence of its apparent experimental evidence (as also mentioned in a recent study Ref. [
3]), the topological nature of the AB effect (see, e.g., a recent study that challenges its conventional concept in Ref. [
4]). On the other hand, the Kuramoto model (KM) [
5] has been applied to many branches of science (see review papers, e.g., Refs. [
6,
7,
8]) but scarcely to quantum mechanics. For quantum applications of the KM, see the examples given in Refs. [
9,
10,
11,
12]. Describing the AB effect from the KM’s perspective is, therefore, of interest and not considered elsewhere.
Synchronizations in quantum systems have recently been studied based on either having a classical analog (see, e.g., Refs. [
13,
14]) or none at all, e.g., Ref. [
15]. Furthermore, synchronizations of interfering particles, such as between photons (see, e.g., Ref. [
16]) and between electrons (see, e.g., Ref. [
17]) have also been considered. Since the synchronization condition is a main characteristic of the KM, it is, therefore, of particular interest to use the model to study the synchronization conditions of the bound state AB effect and their relationship with the interference occurring within the quantum system. Alongside the synchronization conditions, we are also interested in determining the extent of the magnetic field region near the interference line of the AB effect.
In this letter, beginning with the general case of the AB effect [
1], we consider its simplest case, the bound state, by constraining the path taken by the electron to that of a circle with constant radius, e.g., Refs. [
18,
19,
20]. Then, with its known interference phase difference and wave functions (see, e.g., Ref. [
1]), we propose a version of the KM and study the phase dynamics of the system.
2. A Kuramoto Model for the System
Begin with a KM model of the form [
5]
where
is the phase of the phase oscillator
i (with
) as a function of time
,
its angular frequency,
K a uniform coupling strength coefficient and
N the total number of phase oscillators.
Consider now a two-electron system, with each electron initially in the angular position
moving along a circular path of constant radius
one in the upper semicircle and the other in the lower semicircle; see
Figure 1. The two paths, namely, Path 1 and Path 2, form a circular ring except at the endpoint
.
In the center of the ring, there is an AB effect source, a magnetically impenetrable solenoid of radius oriented perpendicular to the semicircular paths and generating a magnetic field of flux . The resulting magnetic field lines emerge from the top, flow down the sides of the solenoid beyond the trajectories, and return through the bottom. Since this magnetic field does not intersect the paths of the electrons, the paths (of radius ) of the electrons are in a region where the magnetic field is excluded. Consequently, there is a phase shift between the electron trajectories at their mutual endpoint, specifically at .
In the general case where the “detecting” particles (i.e., the charged particles moving along the semicircular paths that will sense the AB effect) have charge
, the phase difference between Path 1 and Path 2 when they interfere at
, is [
1,
18]
where
represents the action of the classical Path
i,
ℏ the reduced Planck’s constant, and
c the speed of light in vacuum. In Equation (
2), it is observed that the interference depends on the phase difference by a factor of
. This specifically entails that, when the parameter
(called the flux parameter) is shifted by an integer value, the interference phenomenon remains completely “unaffected”. Thus,
only takes specific discrete values of
, where
(known as the quantum of flux or London’s unit) and
is the charge used by the “source” of the magnetic flux in the solenoid, e.g., Refs. [
1,
18]. In particular, with
, the AB effect is detectable only for
, where
represents the fractional part of the number. In our case, where
, the AB effect is detectable whenever
. We can find a detailed theoretical description in Refs. [
1,
18] and some supporting experiments in the works [
21,
22].
Hence, with
, i.e., for Paths 1 and 2 in
Figure 1, Equation (
1) gives
and
This study aims to determine , , and K. To do so, we make the following assumptions:
The probability for an electron to take either of the two paths is the same;
The two electrons simultaneously start at , one then proceeds towards the end of the semicircular track through Path 1 and the other through Path 2;
The electrons travel the same amount of angular displacement (differing only in the sign of ), at the time interval thus enabling them to interfere as .
Now, in quantum mechanics, the wave function
for the (bound state) Aharonov–Bohm effect can be expressed in terms of
and
as [
1]
where
denotes the wave function for the free particle moving along path
i. Furthermore, since this system also undergoes a scattering process [
1], then
can further be expressed in terms of the incident wave function
and the scattered wave function
as [
1]
where, for
,
,
and
with
being the magnitude of the wave vector of the incident electron.
Let
and
where
and
are a form of phase frequency (i.e., having the dimensions of phase divided by time) for the incident and the scattered waves, respectively. Then, upon inspecting Equations (
2)–(
10), we propose that
and
can be obtained as having the same expression but different in the sign of
due to Path 1 (ccw) and Path 2 (cw), namely,
for
(let
), and
for
(let
). Therefore,
which follows
We now want to establish the connection between
and
. We first observe that, unlike
, the second term on the rhs of Equation (
1) (containing
K) is typically the fluctuating part (i.e., phase-oscillating) of the model. Similarly, the second term on the rhs of Equation (
6) (containing
cf. Equation (
10)) is a varying term, which depends, for example, on the type of intercept boundary causing the scattering (in contrast to
). Also, we are aiming to obtain the synchronization conditions (which is related to
) of the interference (which is related to
). Together with the fact that the phase difference experienced by the incident electrons can be observed at the interference, we consequently link
and
. In other words, looking at the rhs of Equation (
6), we link
to the argument of
and
to that of
. This entails saying that
In particular, from Equations (
8) and (
10), it follows that
or
from whence we deduce that
Notice that we have excluded the terms arising from the prefactor on the right-hand side of Equation (
8), which would contribute to an imaginary phase
z when represented as
. In other words, we have not included the real term
in modeling
K.
With Equations (
1), (
11) and (
12), Equations (
3) and (
4) now read, respectively, (for
),
and (for
)
Note, that in the case of Path 2 in
Figure 1,
is the angular velocity in the clockwise (cw) direction. At this point, using Equations (
16) and (
17), we obtain for
and for
which confirms the results of Ref. [
1] (see Equation (
2)) as
. Therefore, (for
),
and (for
)
on
Table 1 we list the corresponding values of
(for
) for different values of
. We plot these numbers on a diagram in
Figure 2.
Note here that at the interference point (in polar coordinates ), the phase of each of the electrons is independent of the radius of the circular path. However, it is essential to remember that the circle’s radius must meet the constraint to ensure the paths remain within the magnetically excluded region. Therefore, we must further analyze our KM to determine the “critical” radius . This radius is likely related to the radius through which the magnetic field lines pass, i.e., no longer magnetically excluded.
Now, looking at Equations (
2) and (
16)–(
19), we find (for
)
and (for
)
with which we plot the values of
versus
for
on a diagram in
Figure 3 for
and in
Figure 4 for
We need to determine the range of values for R in which the AB effect can be detected. This depends on the magnetic field and how much R is within the excluded region. A more precise question would be the following: with a given flux parameter , how can we determine the maximum or critical R to ensure that it remains in the exclusive region?
Looking at the graph in
Figure 3, one can see that the function
is concave (i.e., the extreme is a maximum) until a certain value
, when it becomes a horizontal line. After this, i.e., when
, increasing
would make the function convex, that is, the extreme is a minimum. On the other hand, a similar feature can be observed in the graph in
Figure 4 but slanting upward (from the left) in a pattern instead of just symmetric along the horizontal.
We propose that for , together with a given value of k, entails a unique radius R, at which the bound state AB effect cannot be detected, that is, the magnetic field lines pass through this radius, and thus are not magnetically excluded. Beyond , the bound state AB effect resumes as there are no magnetic field lines in the region once again. Although for , for example, , the function is not as symmetrical as that for ; however, the critical radius, , can still indicate the “flipping” behavior, which is only attributable to the presence of the magnetic field lines.
We now carefully examine such values of
R that are “candidates” for
in the diagram in
Figure 5. We, therefore, determine that in our model,
for
. Pretty much in the same manner, we can find for
that
using the diagram in
Figure 6.
2.1. The Critical Coupling Strength
In order to find the synchronization condition using the KM, at first we have to introduce the order parameter
, which measures the degree of synchronization, e.g., Refs. [
5,
23]:
where
is the average phase of the oscillators. Equating the imaginary part of both sides of Equation (
24), we have
With Equation (
25), we may rewrite Equation (
1) as
Now, Equation (
24) represents a mean field with which the phase oscillators “interact”. If the function
is small, then Equation (
26) implies
, which means that the oscillator does not interact much with the mean field. On the other hand, if
K is large, then
is entrained by the mean field. This entrainment in the long run produces order (i.e., synchronization) starting from a certain form of disorder (i.e., incoherence) on the phase oscillators. The threshold between these two states (synchronization/incoherence) occurs at a critical coupling strength coefficient, denoted by
.
Looking at Equation (
26) and the fact that
, one can distinguish two regimes in terms of
K for
:
In our case,
, which is the threshold for the jump from the state of incoherence to that of bound state AB effect synchronization, has the form
This further implies that
and so the threshold from incoherency to synchronization based on this KM of the bound state AB effect occurs when
This means, for example, if we use an electron as the moving charged particle (the detector), and the source of the magnetic flux is a particle with charge
, then synchronization can occur for
, as specified in Equation (
29). The condition in Equation (
29) confirms that our version of the KM agrees with the physics of the bound state AB effect in that we have obtained the synchronization condition that coincides with the allowable values of
for the AB effect to occur in the case when the detecting particle is the electron.
2.2. A More General Case
We now write a more general form of the KM, i.e., when
is not necessarily equal to
(
and
):
and
yielding more general conditions for synchronization in terms of the critical coupling strength coefficient
:
and
In the most general case of
N electrons, our version of the KM has the form
3. Discussion and Outlook
We have derived a version of the KM (see Equations (
22) and (
23)), which is capable of describing the phase of the bound state AB effect, starting from the expression of the overall wave function for the scattering problem of the quantum system, e.g., Section 4 of Ref. [
1].
In this model, to observe the AB effect with an electron as the detecting charged particle, for a non-zero angular velocity and for , the radius R of the circular path through which the electron can orbit around the solenoid, which has a constant magnetic flux , should be such that .
Notice that, when at , becomes constant w.r.t. . We interpret this as the radius through which the magnetic field lines pass, making it magnetically included. When and , the function “flips” w.r.t. that when , indicating its behavior beyond the magnetic field region. In this model, the magnetic field region is concentrated at , which means that the field lines in that area are uniformly vertical, perpendicular to the trajectories of the electrons. This suggests that for , our KM yields the characteristics we desire in the quantum system. For instance, it ensures that the solenoid is sufficiently long and its radius is small enough to create uniform magnetic field lines that pass along its sides at . The “flipping” behavior only confirms that indeed is the “interface” between the two regions (i.e., less than and beyond the magnetic field) when . In other words, utilizing our KM, we can now understand the magnetically excluded and “included” regions for .
In the case of
, the graph of the function
is not as symmetric as that of
, due to the fact that
; however, the radius
still indicates the magnetic field region, see
Figure 4 and
Figure 6.
One can approximately express the incident wave vector of the electron as
, where
is the de Broglie wavelength of the incident electron given by (see, e.g., Ref. [
24])
, with
being the mass of the electron and
its initial angular velocity. Then, for
, we can have an expression for
in terms of
:
where
.
For any “allowable” values of
(see, e.g.,
Table 1), we notice that as
, the function
versus
converges into one single plot; see and compare when
and
in both
Figure 3 and
Figure 4. Beyond the magnetic field lines (i.e.,
), it can be seen that for
the function
at
is symmetric w.r.t.
; any
R larger than this would render behavior that is no longer symmetrical to that as
. Further analysis of these graphs, or as an extension of the current study, would provide valuable insights into the behavior of the detecting electron well beyond this region, specifically
.
In addition, the condition for the system to undergo a synchronization is
, implying that the magnetic flux has the form
with
. Since the synchronization coincides with the condition (see Equation (
29)) when
yields a detectable AB effect we, therefore, say that in our KM, a synchronization means the detection of the AB effect, while incoherence does not involve the AB effect. We have also generalized our model into
N electrons (or, in general,
N charged particles); see Equation (
34).
This study also shows that the KM can be derived from a quantum mechanical system with central symmetry (similar to that of the bound state Aharonov–Bohm effect with constant radius and magnetic flux), by expressing its wave function in terms of the incident wave function and the scattered wave function; see
Appendix A for the outline of a slightly different alternative derivation of the KM.