Photon-induced molecular implementation of a quantized magnetic flux photoelectron

A fundamental magnetic flux quantum can be implemented into a free rotating molecule when the interacting molecular electron experiences the maximum possible intrinsic energy uncertainty of a gaussian, transform-limited half-cycle optical photon wavepacket. A magnetic flux resonance condition can be defined at this limit, with photoionization quenching, and the excited molecular electron is drawn into a Rydberg-like spherical surface where a 3D-diffraction pattern is oscillating at the minimum of a bound potential around a primary formed closed electronic loop. The induced rotational motion of the molecular ion core is initiated at the threshold of a robust inertial effect and the dissipated information entropy is the lowest allowed. The integrated quantum possibilities occur in the process as structural properties of a quantized magnetic flux implementation threshold.


Introduction
The quantized nature of the magnetic flux has been observed experimentally more than a half century ago. Magnetic flux trapped in hollow superconducting cylinders has provided in two independent experiments [1,2] the very first evidence for the existence of distinct values in the magnetic flux. These values occurred as multiples of a fundamental unit p F =  e 0 , the same for any observed superconducting loop. This physical effect was predicted earlier by London [3] who used initially a phenomenological model in the description of the electromagnetic properties of superconductors. The presence of a quantum structure has been then suggested [4] at a macroscopic scale and Ginzburg and Landau [5] have further considered the space variation of the corresponding electronic density. At the scale of a single molecule the photon-induced magnetic flux must be also quantized. As the induced magnetic moment of the interacting molecular electron is characterized by the Landég factor e , and this factor contains the one loop quantum mechanical correction [6] relative to the g D =−2 Dirac value [7], we may naturally ask whether a well defined internal structure is present into a photon-induced magnetic flux quantum. Since the fine structure constant α characterizes the strength of the coupling of an elementary electric charge with an electromagnetic field [8], the dimensionless Schwinger-Diract ratioˆa p = = + g g g 1 2 e e D is expected to play a fundamental key role in such a quantized magnetic flux implementation process.
The interaction of a photon wavepacket of well defined energy with a single molecule can provide under resonance conditions an energy and angular momentum transfer. With the well defined oscillation of the photon's electric field the interacting molecular electron accumulates the transferred energy whereas the overall effect of the oscillating induced magnetic field vector occurs only as an average effect, whose magnitude is reduced better and better down to a zero value. The very first experimental evidence for this effect was the issue of the photoelectric effect [9,10]. An excited bound electron can be ejected out of a metal surface and propagate in space as a free photoelectron of well defined kinetic energy if a characteristic threshold frequency is attained. The observed threshold then remains insensitive to the incident light intensity or the time interval of the underlying single-photon bound-electron interaction process. There is no any noticeable magnetic field effect in this excitation process. For linearly polarized light, the direction of distribution of the ejected photoelectrons peaks always in the direction of the oscillating electric field.
A peculiar limiting situation can be attained however if the electric field of the photon's wave packet is constrained to perform a unique, half-cycle oscillation. The corresponding induced magnetic field then exhibits a fixed polarity in space during the time duration D = D t t FWHM GTL HCW 0 of the photon-molecule interaction process; for an optical photon wave packet of a Gaussian, Transform-Limited, Half-Cycle Wavepacket (GTL-HCW), the interaction will consist of a molecular electron experiencing, both, the sharp and monotonous time increase of the electric field, and the fixed space polarity of an induced uniform magnetic field. Such a limiting situation is first characterized by the peculiar property of a GTL-HCW optical photon wave packet whose center frequency ω 0 is badly defined. This is because the intrinsic uncertainty Δω 0 is larger than the unrestrained value of ω 0 . The intrinsic energy uncertainty ΔE 0 =ÿΔω 0 can thus overlap coherently one or even more electronic threshold energies E eg 0 in a specific molecular system. When the interaction is performed in the presence of two indistinguishable entangled electronic states, the joint effect of the GTL-HCW electric and magnetic fields results into, both, the spatial separation of the excited molecular electron from the molecular ion core and, more importantly, the indistinguishability of the electric charge position along a photon-induced closed electronic loop. The formation of this loop is then performed in the presence of a magnetic flux amount transversing a well defined surface. Since at the scale of the interacting molecule the induced magnetic flux and the transferred angular momentum are allowed to occur only as quantized quantities, the photon-induced implementation of a single, Fundamental Magnetic Flux Quantum (FMFQ) must exhibit the property of a FMFQ ionization quenching; i.e., with the increasing peak power of a GTL-HCW absorbed photon the excited molecular electron carrying a quantized amount of magnetic flux can be drawn only into the lower and lower Rydberg molecular  0 . The quantized nature of the magnetic flux and angular momentum will then remove entirely the initial indeterminacy of the center frequency ω 0 . It is also worth noting that the imaginary part of the GTL-HCW electric field exhibits in the time envelope a single and monotonous time increase. The corresponding induced magnetic field of the GTL-HCW photon wave packet can thus exhibit in space only a uniform magnetic field of fixed space polarity. The joint effect of such a peculiar pair of electric and magnetic fields is therefore expected to result into a distant closed electronic loop, formed about a hydrogen-like molecular ion core, where the excited molecular electron experiences the presence of a TEES system. The underlying issue of the temporal Shannon entropies [12] contains relevant information into this peculiar limit; i.e., given that the two conjugate fixed length scales D l t 0 and w D l 0 (which ensure homogeneity) must satisfy the relation = . At the GTL-HCW limit the FMFQ-resonance condition then unveils an intrinsic property of the temporal and spectral Shannon entropies. We readily obtain, both, the minimum allowed entropic uncertainty in the time and frequency spaces, and the corresponding maximum possible difference . This maximized entropic uncertainty difference is the logarithm of the interaction constant of equation (1).
0 therefore stand in the interaction process as the fundamental intrinsic properties of the GTL-HCW optical photon wavepacket. Such a photon's behaviour is neither of wave nor of corpuscular nature. The interacting molecular electron experiences an electromagnetic cyclotropic regime 2 where the key interaction parameters are the transferred intrinsic uncertainties.
1. We have drawn in figure 1 the imaginary part of ( ) which defines the magnetic flux resonance condition when the value of ω 0 approaches the threshold frequency w eg 0 of a two entangled electronic states system. The photon-molecule interaction is then performed in the presence of two well defined interaction constants: . The interacting molecular electron experiences the monotonous time increase of the electric field: 0 . It is not allowed to provide any stable closed electronic loop. 1 Fundamental Physical Constants, The NIST Reference on Constants, Units, and Uncertainty. 2 By the term 'cyclotropic effect' we refer to the joint effect of the electro-magnetic field of the photon wavepacket on the excited molecular electron at the GTL-HCW limit.
Let us further remark that the valueF = 4 in figure 1 is associated with multiple oscillations of the electric field. That is, the overall induced magnetic field is averaged in practice down to a zero value. The predominance of the interacting electric field is therefore enhanced and the excited molecular electron ends as an ejected photoelectron [9,10]. In contrast, in the cyclotropic regime (see footnote 2) defined in figure 1 by the interaction constantsQ 0 andĈ 0 , the joint effect of the electric and magnetic fields does not allow to the excited FMFQphotoelectron to quit the molecular ion core. We may see in the following that the FMFQ-photoelectron is captured into a TEES bound potential at the fundamental state. Such a photoionization quenching can be first apprehended within the frame of a semi-classical descriptionwhere the spectral properties of the space-fixed interacting molecule are those of a rigid rotor ion core, while the excited molecular electron experiences the quantized nature of the transferred magnetic flux and angular momentum. The photon-induced magnetic flux is intimately related with the formation of a proteiform quasi-classical closed electronic loop. Given the large extension of this loop relative to the dimension of the positively charged molecular ion core, we will first consider a semi-classical approach where the distant molecular electron experiences the cyclotropic regime of an optical GTL-HCW photon wave packet. The quantum mechanical description of this process will be developed in section 4.
2. The photon is a zero rest mass particle. It is not localizable. Let us then assume that our interaction is centered at time t=0 and the interacting molecule is located at the position x=0. We introduce the dimensionless time variableˆ= D t t t 0 which can be then identified with the dimensionless propagation variableˆ= D x x x 0 . We set for instance: x=ct and Δx 0 =cΔt 0 . The position of the interacting molecule therefore contains an intrinsic uncertainty as large as Dx 0 . The electric field (linearly polarized along the z-axis) of a GTL-HCW photon wave packet is written: The index Φ 0 indicates the presence of the FMFQ-resonance condition. That is, the FWHM width Δt 0 has been prepared in terms of some specific electronic threshold transition frequency w eg 0 . In absence of external charges and currents the Maxwell equations, where the electric and magnetic fields stand as wave functions, first ind y 0 0 then casts into the following expression:  . We will make a systematic use of properly defined dimensionless quantitiesDall over the development of the present work. Since the value of (3) is small compared to unity we can assume into the semi-classical description the presence of a half-cycle wave plane behaviour. For the optical energies E eg 0 the interacting molecular electron experiences indeed (at the scale of a single molecule) a uniform magnetic field of fixed space polarity. Although the energy formula of Rydberg series is a result of a hydrogen-like atom structure, Rydberg states [13,14] are also present in molecules which can then attain diameters considerably larger than the typical diameter  -D 10 m 0 0 9 of a molecule. At the Rydberg limit, any isolated neutral molecule behaves like a hydrogen-like atom.
The joint effect of the GTL-HCW electric and magnetic fields on the excited molecular electron and the infinite mass hydrogen-like ion core involves the presence of quantized constraints. First, the induced magnetic field effect in presence of the quantized nature of, both, the angular momentum and the photon-induced magnetic flux, must satisfy the following two constraints: We might first ask whether the increasing peak power of an absorbed GTL-HCW photon wavepacket under the FMFQ resonance condition results into an ejected photoelectron or instead, it can only provide a photoninduced proteiform circular electronic loop trapped into a bound state. The synergetic effect of the GTL-HCW electric and magnetic fields provides for instance the following relations:  (5)) in terms of the interaction peak power, follows a fourth power law.It is then worth noting that the GTL-HCW time dependence, drawn in figure 1 with large dot points, exhibits equally well a singularly simple fourth power law. Namely: . The increasing peak power of a GTL-HCW photon wave packet under the FMFQ resonance condition therefore reduces drastically the effective value of Δx 0 . Instead of ionization, the excited molecular electron is drawn into the lower and lower Rydberg states. We will call this shrinking process: FMFQ syriknosis effect . The diameter D eg 0 of the proteiform circular loop is fixed exclusively by the TEES-threshold energy E eg 0 , so that;the conjugate quantities E eg 0 and D eg 0 counterbalance the presence of a quantized fundamental magnetic flux quantum: The constant pa 4 is the elementary electric charge [ ] e nu measured in natural (Lorentz-Heaviside) units [15]. The Chirelson's bound constant [16] = C 2 2 QC max is a manifestation of the photon-induced nonlocal behaviour of the excited molecular electron and its maximum quantum correlation with the molecular ion core. We may see indeed in the following that a nonlocal structure is present into the photon-induced TEES system. The subsequent rotational motion of the ion core and the induced dynamics of the excited molecular electron are both associated with a photon-induced diffraction pattern.
The TEES-structure involves the presence of the key interaction constantsQ 0 andĈ 0 . Let us remind that the threshold energy condition < D E E eg 0 0 defines a two entangled electronic state system which is intimately related with the cyclotropic effect of the GTL-HCW electric and magnetic fields. We must therefore characterize, beside the displacement from equilibrium of the molecular electron's charge, also, the simultaneous presence of a photon-induced fundamental magnetic flux quantum. Such a badly defined electric dipole can be conceptually represented by the expression: m D = eR ; is the Planck electric charge [17]. We can now readily associate the intrinsic indeterminacy in the space orientation of the induced electric dipole m D eg 0 with the presence of a delocalized excited molecular electron which, for reasons of symmetry, is not allowed to quit the ion core and propagate in space as a free photoelectron.

Photon-induced nonlocal TEES structure
In the presence of the FMFQ resonance condition a photon-induced nonlocal TEES structure is implemented into the free rotating molecule. The spectral domain DE N 0 of the TEES system shown in figure 2 contains the first N 0 rotational states of a rigid rotor [18] which then belong in an indistinguishable way to both entangled electronic states | ñ g and | ñ e . These states can thus evolve in time only as a coherent superposition of closed magnetic field lines. Given the quantized nature of the magnetic flux and the presence of a FMFQ resonance condition, these magnetic field lines can only operate as the harmonics  F l 0 of the fundamental magnetic flux quantum Φ 0 . Equation (6a) first shows that the principal l=1 closed electronic loop carries a fundamental magnetic flux quantum fixed exclusively by the TEES threshold energy E eg 0 . The residual closed magnetic field lines of the indistinguishable rotational states are then expected to operate in the TEES system as equiprobable magnetic flux vortices of opposed helicity condensing into a minimized magnetic flux fluctuation. We may further observe that the TEES structure is the issue of the transferred Shannon entropy in the time and frequency spaces and that of the implemented energy uncertainty. Let us remark for instance that the maximum possible energy uncertainty ΔE 0 and the corresponding threshold energy E eg 0 (which defines the FMFQ resonance condition) are closely related through the relation: where I 0 denotes the moment of inertia of the interacting molecule. We may see in the following that these physical quantities define rigorously well the photon-induced nonlocal structure of the TEES system pictured in figure 2.
1. Since the TEES structure is carried by a free rotating molecule, an induced fundamental angular frequency w rot coh must be present in the subsequent rotational motion. This motion is shared equally well by the two entangled electronic states and, also, by their N 0 indistinguishable rigid rotor rotational states. Any frequency in the TEES system must occur as a harmonic frequency of some fundamental frequency w rot coh . The indistinguishable spectral lines in absorption and emission [18] must then obey the following relations: Doing so, we obtain: 0 stands in the interaction process as the ending value of the entropic uncertainty transfer and, this is equally well the ending step of the energy uncertainty transfer. That is to say, the photoionization quenching observed in equations (6a)-(6b) must be necessarily associated with the presence of a bound potential formed from the transferred entropic and energy uncertainties into a TEES system. It is indeed of relevant importance to introduce under the above conditions a temporalspectral Shannon entropy variable: m a x , we introduce the following entropic plus time-energy uncertainty transfer function: where the variable stands as the characteristic power series of a multiplicative sequence:   (1). It is of crucial importance to notice the fact that this constant verifies the fundamental property: the position-momentum uncertainty relation of a quantum particle confined in a one dimensional box. ζ (2) is the Rieman zeta function [19]. We have set: is the electron's Schwinger-Dirac ratio [6,7,20]. The angular momentum state [ˆˆ] [ˆˆ] s s s s = x p x p min 1 0 0 0 0 therefore denotes the fundamental angular momentum state of a 1D confined FMFQ photoelectron whose magnetic moment is characterized by the fundamental constantĝ e . We may see in the following that the TEES generic function ( ) ( ) q s TEES 0 contains relevant information on the photon-induced rotational motion of the ion core and, also, in the position-momentum and time-energy uncertainties of the trapped FMFQ molecular photoelectron. The intrinsic uncertainties in the position-momentum and the timeenergy experienced by the excited molecular electron cannot operate separately. This is because these uncertainties are intimately related with the transferred entropic uncertainty in the time and frequency spaces.
2. Let us first illustrate a hidden intrinsic property of equations (8a)-(8b):    provides the minimum allowed position-momentum uncertainty of a 1D confined quantum particle. We observe the following remarkable property: is pure imaginary; this constant relates intrinsically the lowest allowed uncertainty state of the space orientation of the rotationally frozen molecular ion core to the lowest allowed photon-induced position-momentum uncertainty of the excited molecular electron along the 1D proteiform closed electronic loop. Given that the imaginary unit is the generator of rotations in the complex plane, the angles β and b¢ of figure 2 must rotate in a plane with opposed helicities so that their initial orientation β 0 is kept well defined in time and becomes a constant of motion (see for this in the quantum mechanical description of the process). The functionˆ(ˆ) contains relevant information. We may observe in figure 3 the presence of a photon-induced FMFQ potential where the interaction (1) and (7) emerge naturally as the fundamental key dimensionless constants of the FMFQ implementation process. The properties of the bound potential reflect the minimized energy, entropic uncertainty, and angular momentum state of the trapped FMFQ photoelectron. They are all defined together with the formation of the coherent superposition of the TEES rotational states. These properties are characterized by the following three expressions: and the interaction constantˆ-Q 0 1 matches the Schwinger's correction constant [6][7][8] α /Δ f 0 . More precisely: x p . This minimized state is consistent with the Aharonov-Bohm effect [21] which has been observed in the past, more generally, in superconducting and non-supeconducting systems as well. The underlying mechanism of this effect is the coupling of the electromagnetic potential with the complex phase of the charged particle's wavefunction. The phase of the complex quantum mechanical wavefunction is then allowed to change continuously from some value f 0 to the value f f + D l 0 0 as one goes around the loop and ends into the same point. The value l=1 is therefore associated with the lowest allowed phase difference along a stable closed electronic loop. We may see in the following that the magnetic flux modes is the real single-valued Lambert function [22]. All of the structural properties of the photon-induced TEES-system have emerged naturally from the transfer functions:

Quantum mechanical description of the photon-induced FMFQ-implementation
The GTL-HCW photon-molecule interaction is initiated in the presence of the constraints: w D D = t 4 ln 2 0 0 and w w D   0 0 0 . The badly defined center frequency w 0 of the photon wavepacket then occurs with the maximum possible intrinsic uncertainty Δω 0 and, the interaction process contains an average amount of information entropyá ñ H HCW GTL . We must then account for the fact that the interacting molecular electron experiences transition frequencies w eg with an intrinsic uncertainty as large as Δω 0 . Let us say, the molecular electron experiences the complex frequenciesw w w = + D i eg eg 0 instead of a well defined transition frequency ω eg . We will thus consider the presence of complex levelsw . Let us then assume in the time-dependent uniform electric field the presence of complex frequency componentsw w w = + D i 0 . Namely: ,

1.
We need to precise the expectation value | | m áY Yñ p g of the photon-molecule interaction at the GTL-HCW limit from the time-dependent Schrödinger equation [23]: |Yñ g is the time-dependent vector state accounting for the presence of the intrinsic uncertainty Δω 0 in the interaction process. The expectation value will then remain a real quantity provided that the property . Doing so, we obtain: The quantity between parentheses must be identified with a polarizability tensor [24] accounting for the ω contributions, each contribution containing an intrinsic uncertainty Δω 0 . We can say equivalently; for a well defined electronic threshold transition w eg 0 experiencing the uncertainty w D 0 of theωfrequency components . Given that both contributions ω and −ω are equally probable in the TEES system, the polarizability must remain unchanged if we permute the indices g and e, change the sign of all frequencies and polarizations, and reverse the time. Such a symmetrized polarizability tensor for a fixed space orientation β 0 of the molecular axis, is the following: can be then determined from the following complex expression: The global factor (˜) b C N 00 0 is a coherent superposition of N 0 rigid rotor autocorrelation functions. It provides the following diffraction pattern: * performed over a unit circle. Since the LFF angles β and b¢ cannot be defined separately in the photon-induced TEES system, we have introduced the variables: The coherent excitation of the N 0 ? 1 rotational states is therefore associated with an initial space fixed orientation β 0 , and the subsequent ion core rotational motioñ ( ) b w = t t rot coh at the angular frequency w rot coh . Any frequency in the TEES system must be in fact a harmonic frequency of some fundamental angular frequency w rot coh . Provided that a maximum quantum correlation must be present between the induced dynamics of the ion core and the dynamics of the excited molecular electron, we can expect equally well the presence of a diffraction pattern into the induced dynamics of the excited molecular electron. We may see at the end of this work that this pattern is defined equally well by the (commonly shared) variableb. We may finally observe in figure 2 that; the initial orientation β 0 can be associated with a fixed space orientation only if the angles βand b¢of the two entangled electronic states are drawn to rotate in a plane with opposed helicities.
2. The induced electric dipole observed at the time instant t ä [ ] t 0, coh 0 , where the GTL-HCW electromagnetic field is still operating in the formation of the TEES-system, can be characterized by the following, frequency averaged, GTL-HCW photon-molecule dipole interaction:  are not allowed to occur. This holds equally well for the trivial case l=0.
The initial orientation β 0 of the rotationally-frozen ion core, the TEES threshold and the polarization pof the photon wavepacket, first define a global factor [ ( )] . Only this factor contains the polarization dependence. The integrantˆ(ˆˆ) ¢ S t t , l MFT is associated instead with the interacting molecular electron exclusively. It is independent of the polarization state p. This peculiar aspect can be apprehended by the limiting situation of an induced magnetic field that cannot be averaged down to a zero value. Either for linear or circular polarization, the joint effect of the GTL-HCW electric and magnetic fields into the dynamics of an interacting molecular electron is the formation of a quasi-classical closed electronic loop drawn away from the molecular ion core.
The integrantˆ(ˆˆ) ¢ S t t , l MFT of equation (20b) defines the following Magnetic Flux Transfer (MFT) function: . We observe the following integrated issue: When the oscillatory regime of the interacting electric field collapses into the GTL-HCW domain f w t D = eg coh 0 0 0 (delimited in figure 4(a) by the two dashed vertical lines) with the advent of the GTL-HCW limit, and the induced magnetic field cannot be averaged any longer down to a zero value, the cyclotropic effect (see footnote 2) of the FMFQ resonance condition becomes prominent; i.e., the enhancement of the FMFQ-dipp shown in figure 4(a) is due exclusively to the synergetic effect of the GTL-HCW electric and magnetic fields. This is performed in the presence of a maximum quantum correlation established between the rotating ion core and the confined FMFQ photoelectron. This fact can be easily verified if we assume that the excited molecular electron experiences an infinite-like number W N 0 of closed magnetic field lines. Their coherent superposition results into the following 3D diffraction pattern:  3 . The underlying photon-induced coherent superposition in the TEES system is then responsible for the subsequent time recurrences of the photon-induced diffraction pattern.Similarly to the two-slit diffraction pattern distributed over a plane surface, the overlapping rigid rotor rotational states of the two entangled electronic states cancel each other for some space orientations whereas they reinforce in other ones, causing in this way the emergence of a diffraction pattern oscillating over a Rydberg spherical surface around a distant proteiform circular electronic loop centered at the rotating molecular ion core.
4.Given that a physical process demands the dissipation of at least 1Sh of information entropy, let us further introduce theShannon-threshold amplitude:   We can thus write the following integral action balance at the minimum of the FMFQ potential:  . This is the lowest allowed threshold because: . This width unveils the peculiar aspect of the GTL-HCW photon-molecule interaction; i.e., the photon-induced fundamental state is defined by the intrinsic uncertainties of the interaction process exclusively: and equations (12a)-(12c) and (13a)-(13b) have defined in the process the fundamental properties of the FMFQ photoelectron's potential. We can now provide further evidence for the presence of the photon-induced TEES nonlocal structure from equations (21a) and (24b). These equations exhibit the following integral relation: can be readily developed in terms of the Bernoulli numbers [19,27]. That is, . Note that the Bernoulli numbers can be also expressed in terms of the Riemann zeta functions ζ (s) [19]; these transcendental numbers arise naturally in the higher order terms of the electron's gyromagnetic ratio. They constitute a somehow general feature of the higher order calculations in perturbative quantum field theory. It is then remarkable that the MFA-amplitude can be defined in terms of the fourth power term of the TEES generic function; i.e., 6. Our very last request will concern the simultaneous presence of the magnetic flux quanta ±2Φ 0 , · ·  F 3 0 They must be necessarily associated with the presence of the large angular momenta ±2ÿ, · ·   3 whose , we find: . The appropriate unit of the FMFQ electric dipole is the eμ . It is important to notice that there is no any preferred well defined space orientation for the FMFQ electric dipole. This is because the electric charge is first delocalized around a closed electronic loop and then diffracts over the corresponding Rydberg spherical surface. In contrast to the case of semiconductors where the close proximity of the paired electron-hole can only provide short lifetimes at the nanosecond scale, the lifetimes of the outermost FMFQ molecular excitons are expected to be much more longer. It may be then worth to provide some comments of general interest: Given that the molecular excitons can interact with photons and provide polaritons, we may first ask whether two or more FMFQ dressed excitons can combine to form FMFQ multiexciton systems. This can be realized for instance when a weak dipole-dipole attractive potential is present. When a molecule absorbs a GTL-HCW photon wave packet, it converts the energy uncertainty ΔE 0 into the form of a FMFQ molecular exciton carrying, both, a quantized amount of energy and a quantized amount of magnetic flux. A FMFQ molecular exciton can then jump within a given molecular alignment from a donor molecule to an adjacent acceptor molecule by means of a near-field dipole-dipole interaction. We can determine in this way how a quantized amount of energy carrying a quantized amount of magnetic flux can flow at the molecular scale. The FMFQ molecular exciton can be used in quantum devices as the constituent element of the transport of a quantized amount of magnetic flux of a preselected amount of Rydberg energy. On the other hand, since the FMFQ molecular exciton is an integer-spin quantum particle the presence of a repulsive interaction could result into the presence of a Bose-Einstein condensed state. Our minimum uncertainty coherent state induced into a single molecular constituent(written in terms of the appropriate physical units of the interaction process) obeys the following fundamental equation: is the free-space FMFQ coupling constant. We may see in the following that this constant contains relevant information on the entropic uncertainty threshold of the integrated magnetic flux transfer and, also, provides a physical interpretation to the requisite presence of an imaginary part into the complex propagation variable˜( A minimum uncertainty state which refers to a coherent state of a quantized electromagnetic field has been first studied by Schrö dinger [28]. A complete quantum description of optical coherence phenomena related to electromagnetic fields has been then developed almost four decades later [29]. Coherent field states and photon wave functions have been further investigated in the past [30,31]. It has been shown that the photon wave functions, the quantized fields wave packets and their underlying optical coherence are intimately related [32]. The photon wave function description occurred as a equivalent method for describing states and dynamics of quantum electromagnetic fields. The imaginary part of the photon wave function has been found in particular to play the role of a magnetic induction field [32]. It is then important to remark that the overall effect of the imaginary part of the complex propagation variablex of the GTL-HCW photon wave function [ (˜)] Using the appropriate units of our process we can write immediately: Let us now remind that the Φ 0 -implementation process occurred in the presence of an entropic uncertainty threshold.

Discussion and concluding remarks
A quantized magnetic flux resonance can be induced into a free-rotating molecule if the magnetic field vector of a photon wavepacket exhibits a fixed space polarity during the photon-molecule interaction process. This is the case for instance for the absorption of a GTL-HCW optical photon wavepacket. The key aspects of this process then emerge naturally from the presence of a maximum possible energy uncertainty ΔE 0 , overlapping coherently two or more distinct electronic states, and the transfer of the corresponding entropic uncertainty defined in the time and frequency spaces. The synergetic effect of the electric and magnetic fields will first result into the formation of a TEES system and the subsequent photon-induced implementation of a fundamental magnetic flux quantum F 0 . A requisite condition for the space confinement of such a molecular photoelectron is the FMFQ resonance condition. The FWHM time width Dt 0 of the photon wave packet must be fixed in terms of a preselected TEES threshold energy < D E E eg 0 0 . If the time-energy uncertainty relation is expressed in terms of the appropriate units of fs and eV we can readily observe the prominence of the temporal and spectral Shannon entropies into the photon-induced implementation of a fundamental magnetic flux quantum. The initial indeterminacy of the center frequency w 0 of the GTL-HCW photon wavepacket is entirely removed in presence of the FMFQ resonance condition. This is because the selected TEES threshold frequency w eg 0 becomes the operating value of w 0 . The induced rotational motion of the molecular ion core is then defined from the coherent superposition of a large number of rotational states shared in an indistinguishable way by the two entangled electronic states. We have shown in this work that a photon-induced magnetic flux internal structure is present into the TEES system. The structural properties occur as integrated quantum possibilities and reveal the fundamental physical laws relying on the dissipated information entropy, the entropic uncertainty transfer, and the photon-induced position-momentum uncertainty of a confined molecular photoelectron carrying a fundamental magnetic flux quantum. The space confinement of the molecular electron into the frame of a rotating molecular ion core has been found to follow the dynamics of a diffraction pattern oscillating over a Rydberg spherical surface around a primary formed quasi-classical circular electronic loop.
It is then instructive to provide an analogy with the case of a macroscopic electronic loop formed incoherently in a Penning trap [33,34]. When a charged particle experiences the presence of a static and uniform magnetic field and the quadrupole electric field of a Penning trap, it is drawn into a state of 3D space confinement. The trapped particle then follows a composite motion consisting of three distinct harmonic oscillations [35,36]; a slow magnetron motion of frequency wevolving in time over a plane transverse to the uniform magnetic field, a fast reduced-cyclotron motion of frequency w + superposed in the same plane, and an axial harmonic motion of frequency w 0 oscillating along the axis of the applied magnetic field. Although the axial motion depends explicitly on the size and shape of the electrode structure, the dimension of the Penning trap cannot affect the intrinsic properties of the 3D confinement process [35]. This is also the case for the quantized Hall effect [37] where, within the experimental uncertainty, the observed results remain independent of the specific material or the improved and growing techniques of the experimental devices [38]. The quantum Hall effect is not affected by structure irregularities present in a semiconductor like impurities or interface effects. It depends exclusively on fundamental constants. Experiments performed with single charged particles have provided in the past the value of the correspondingg factor as a simple measurement of a frequency ratio. In particular, for the fundamental electric charge e, we have the free-electron resonance condition: w w = g e L c where, ω L is the spin precession (Larmor) frequency, and the free-cyclotron (Lorentz-force) frequency w c matches the Penning trap invariance relation [35]: w w w w = + + + c 2 2 0 2 2 . Although the effective value of g e has been found to differ from the expected Dirac value g D =−2 [7] only by a small fraction, this small discrepancy was the precursor of the singular success [39,40]  . The position-momentum and time-energy uncertainties of a molecular electron carrying a magnetic flux quantum cannot operate separately. Let us remind to this regard the fact that, although the magnetic flux confined in macroscopic superconducting loop/cylinder systems has provided experimental evidence [1,2,41] for the quantized values of the magnetic flux, more generally, this phenomenon is a consequence of the Aharonov-Bohm effect [21] which can be observed equally well in non-superconducting systems.
We have shown in the present work that an excited molecular electron experiencing the impulse electromagnetic field of a GTL-HCW photon wave packet can be also confined into the rotating frame of a molecular ion core. The unique and monotonous time increase of the electric field of a GTL-HCW interacting photon induces a uniform magnetic field of fixed space polarity and the joint effect of these fields can result only into the formation of a proteiform closed electronic loop responsible for a FMFQ photoionization quenching. A very first analogy of such aMFF-3D confinement ofan electric charge with the incoherent 3D macroscopic confinement of a charge in a Penning trap provides valuable information. The axial harmonic component in the Penning trap can be first associated with the (moment of inertia depended) induced angular frequency of the ion core, and the slow magnetron motion with the prior formation of a closed electronic loop carrying a fundamental magnetic flux quantum Φ 0 . Due to the implemented TEES coherence into a single free-rotating molecule, the analogy with the fast reduced-cyclotron motion is much less intuitive; i.e., the excited molecular electron cannot be observed during the impulse GTL-HCW-photon-molecule interaction as a point-like particle. The interaction can be realized only in presence of a photon-induced TEES system because the intrinsic uncertainty ΔE 0 in the energy of the absorbed photon overlaps coherently the threshold electronic transition energy E eg 0 of a two electronic state entangled system. It is impossible to precisely determine the position and energy of the excited molecular electron along the photon-induced circular loop. The electric charge of the interacting molecular electron is equally well present over the circular path of a closed loop because of the nonlocality property of the quantum entanglement. Instead of the spiral fast cyclotron motion of a point-like charged particle in a Penning trap, where the point to point well defined electric and magnetic fields are operating, we have the simultaneous formation of an infinite-like number of equiprobable pairs of magnetic flux loops of opposed helicity superposed to the circular loop of the well defined fundamental magnetic flux quantum. Their presence at the minimum of a bound potential is due exclusively to the minimized effect of the | |  l 2 harmonics of the FMFQ resonance. Such a paired configuration of harmonic frequencies can only condense into the minimum of a bound potential, which is then responsible for the photoionization quenching of the well defined magnetic flux quantum F 0 . The FMFQ implementation process is performed with the lowest allowed amount of dissipated information entropy at the threshold of a bound potential following the principle of least action.
Although the present contribution may be of fundamental interest in Physics, it may as well target a wider audience in the field of Chemistry, Quantum Information and Quantum Computing. Given that the photon's wave-particle duality is found to be challenged by the transfer of the fundamental quantum intrinsic uncertainties into a free rotating molecule, the magnetic flux resonance condition unveils new aspects in the light-matter interaction.