Excited State Properties Revisited : An Effect of Extra Compactified Dimensions

Experiments on NO2 reveal a substructure underlying the optically excited isolated hyperfine structure (hfs) levels of the molecule. This substructure is seen in a change of the symmetry of the excited molecule and is represented by the two “states” b and c of a hfs-level. Optical excitation induces a transition from the ground state a of the molecule to the excited state b . However, the molecule evolves from b to c in a time τ0 ≈ 3 μs. Both b and c have the radiative lifetime τR ≈ 40 μs, but b and c differ in the degree of polarization of the fluorescence light. Zeeman coherence in the magnetic sublevels is conserved in the transition b c → , and optical coherence of a and b is able to affect (inversion effect) the transition b c → . This substructure, which is not caused by collisions with baryonic matter or by intramolecular dynamics in the molecule, contradicts our knowledge on an isolated hfs-level. We describe the experimental results using the assumption of extra dimensions with a compactification space of the size of the molecule, in which dark matter affects the nuclei by gravity. In a , all nuclei of NO2 are confined in a single compactification space, and in c , the two O nuclei of NO2 are in two different compactification spaces. Whereas a and c represent stable configurations of the nuclei, b represents an unstable configuration because the vibrational motion in b shifts one of the two O nuclei periodically off the common compactification space, enabling dark matter interaction to stimulate the transition b c → with the rate (τ0). We revisit experimental results, which were not understood before, and we give a consistent description of these results based on the above assumption. How to cite this paper: Weber, H. G. (2017) NO2 Excited State Properties Revisited: An Effect of Extra Compactified Dimensions. Journal of Modern Physics, 8, 1749-1761. https://doi.org/10.4236/jmp.2017.811103 Received: September 2, 2017 Accepted: October 7, 2017 Published: October 10, 2017 Copyright © 2017 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/

with the rate (τ 0 ) −1 .We revisit experimental results, which were not understood before, and we give a consistent description of these results based on the above assumption.

Introduction
Various experiments on NO 2 reveal two characteristic time constants associated with the optically excited hyperfine structure (hfs) levels of the molecule, the radiative decay time τ R ≈ 40 μs and the time constant τ 0 ≈ 3 μs, which is no radiative decay time, not caused by collisions with baryonic matter, and not caused by intramolecular dynamics in the molecule [1] [2].For example, optical excitation of NO 2 in a molecular beam near the excitation wavelength λ ex = 593 nm induces a transition between a state a of the ground electronic state (X 2 A 1 ) and a state b of the first excited electronic state (A 2 B 2 ).However, the state b is not stable [1]  The time constant τ R ≈ 40 μs measured by radiative decay [5] or with use of the Hanle effect [1], agrees well with results of optical radio-frequency double resonance experiments (see Ref. [7] and references given there), and with results of "time of flight" experiments (see Ref. [8] and Sec. 2 below).The time constant τ 0 was extracted from the width of the "broad" Hanle signal (Ref.[1]), from the width of the "broad rf-resonance" (see Refs. [6] [9] and Sec. 2), and from the width of the "ν-resonance" (see Ref. [8] and Sec. 2).The time constant τ 0 agrees also with the lifetime τ in evaluated with use of the integrated absorption coefficient giving values for τ in between 1 μs and 4 μs (Ref.[10] and references given there).In general, one expects τ in = τ R .The disagreement of τ in with τ R by more than a factor 10 was assigned to a coupling of levels of the first excited electronic state A 2 B 2 with high lying vibrational levels of the ground electronic state and τ in was identified with the lifetime of the uncoupled A 2 B 2 electronic state [11] [12].
This theory is not in agreement with the occurrence of two time constants τ 0 and τ R simultaneously on a single isolated hfs-level of NO 2 .Moreover, radiative decay with the time constant τ in was never detected by optical excitation near λ ex = 593 nm [5] [10].In this work, we identify τ in with τ 0 and give a different interpretation of the disagreement of τ in with τ R .
The present work aims to explain the transition b c → and the time constant τ 0 We have experimental evidence (see Ref. [13] and Sec. 2 of this work) that the transition b c → with the time constant τ 0 and with the associated inversion effect is a property of the isolated hfs-levels of NO 2 and is not due to collisions with baryonic matter or due to an intrinsic (intramolecular) process in the molecule.We give a phenomenological description of the experimental results based on the following assumption: The molecule interacts by gravity with a background field, presumably the axion dark matter field (e.g.Refs.[14] [15] [16] [17]), and based on ADD-theory (see Refs. [18] [19]), gravity is strong in a compactification space of the size of the molecule.Most investigations of axion effects on atoms and molecules focus on non-gravitational interactions (e.g.Refs. [14] [15] [16]).However, gravitational interaction may become strong, if one assumes, as in ADD-theory, that the three forces of the standard model act in three dimensions, but gravity acts in a higher (3 + n e )-dimensional space, where n e refers to the number of extra dimensions [18]  with the rate (τ 0 ) −1 .In Sec. 2, we revisit experimental re-sults, which were not understood before, and in Sec. 3 we discuss these results.
Finally, in Sec. 4 we present our conclusion based on these results.

Experimental Results Revisited
We discuss experimental results, which were described in detail in Refs.[8] [9].We used the experimental apparatus depicted in Figure 1 in three different arrangements designated S 1 , S 2 , and S 3 in the following.In S 1 , the angle α is fixed at α = 0 and the molecules interact with a static magnetic field B parallel to the z-axis and with a radio-frequency (rf) field having constant frequency and linear polarization perpendicular to the z-axis.The measured quantity is There is a broad and unresolved resonance structure underlying the narrow resonances in Figure 3.This resonance structure can be resolved if T L is increased.We used arrangement S 2 , which is the same as S 1 but using L 1 only (L 2 is off).Figure 4 depicts a result obtained in S 2 with T L = 5.7 μs, whereas in Figure 3 we used T L = 1.6 μs.There are five Lorentzian shaped resonances associated with the same g-factors as in Figure 3.These resonances (an increase of P up to 5%) were named "broad rf-resonances" [6] [9].These resonances are connected with the inversion effect [6].The strength of these resonances depends strongly H. G. Weber to the resonance at A 3 in Figure 3. Taking these constraints into account, all resonances have the same properties in particular the same time constant τ 0 ≈ 3 μs as detailed investigations showed.
Figure 5 depicts the result of an α-scanning experiment using the set-up depicted in Figure 1 in the arrangement S 3 , i.e. using L 1 and L 2 , but no rf-field and no static magnetic field B (the earth magnetic field is compensated).The measured quantity is ( ) δν να = with P = P 0 if α is far off-resonance.The signal is a change (up to 20%) of the fluorescence intensity P with a minimum at α = 0 or δν = 0.It was named "ν-resonance" [8].The signal shape is not Lorentzian but approximately Gaussian with the width (FWHM) Δν and the amplitude A ν .The result is  )( ) with τ = 22 μs.The discrepancy between τ and τ R ≈ 40 μs was attributed to the change of the detection geometry when the gap width s was varied but the photodetector was fixed.The analysis of this geometrical constraint showed that τ should be increased by at least 30% yielding τ ≈ τ R (for details see Ref. [8]).We note that the recorded "ν-resonance" signals have generally an oblique underground which is connected with the angle tuning of the light beam L 2 .This underground is also present without L 1 .The underground (a straight line) is subtracted in the result shown in Figure 5.

Discussion
Figure 3 and Figure 4 show that the g-factors enable an assignment of measured data to the three excited hfs-levels depicted in Figure 2(a).The experiments in arrangement S 1 (e.g. Figure 3) reveal that each hfs-level has a state c with the radiative lifetime τ R ≈ 40 μs, and the experiments in arrangement S 2 (e.g. Figure 4) reveal that each hfs-level has a state b with the decay time τ 0 , with the same dependence on the transit-time T L , and with the same time of flight (T s + T L ).Therefore, the "ν-resonance" has the same properties as the corresponding signal in a single transition between an hfs-level in the ground state and an hfs-level in the excited state.The amplitude of the "ν-resonance" versus the time of flight (T s + T L ) is ( )( ) , if we assume τ = τ R taking into account the constraints in the detection geometry (see Sec. 2).A change of the population in a ("hole burning") does not affect A ν in a detectable way.This result shows that the "ν-resonance" is predominantly due to induced emission from c to a as indicated in Figure 2 , where the quantity A in includes an integral over the whole of the absorption band concerned, and g l and g u are the degeneracy of the lower (l) and the upper (u) state involved, respectively.In Refs.[10] [11], τ in was evaluated using 1 l u g g = = , state of the usual molecular hamiltonian.We propose the following interpretation of the experimental results.The molecule interacts by gravity with a background dark matter field, presumably the axion dark matter field, and, based on ADD-theory [18] [19], gravity is strong in a compactification space of the size of the molecule.The first assumption implies identifying the decay rate (τ 0 ) −1 with the oscillation frequency of the axion field (e.g.[14] [15]).This gives mc 2  .This is beyond the scope of the experimental work reported here and needs further clarification by theory.
by the two "states" b and c of a hfs-level.Optical excitation induces a transition from the ground state a of the molecule to the excited state b .However, the molecule evolves from b to c in a time τ 0 ≈ 3 μs.Both b and c have the radiative lifetime τ R ≈ 40 μs, but b and c differ in the degree of polarization of the fluorescence light.Zeeman coherence in the magnetic sublevels is conserved in the transition b c → , and optical coherence of a and b is able to affect (inversion effect) the transition b c → .This substructure, which is not caused by collisions with baryonic matter or by intramolecular dynamics in the molecule, contradicts our knowledge on an isolated hfs-level.We describe the experimental results using the assumption of extra dimensions with a compactification space of the size of the molecule, in which dark matter affects the nuclei by gravity.In a , all nuclei of NO 2 are confined in a single compactification space, and in c , the two O nuclei of NO 2 are in two different compactification spaces.Whereas a and c represent stable configurations of the nuclei, b represents an unstable configuration because the vibrational motion in b shifts one of the two O nuclei periodically off the common compactification space, enabling dark matter interaction to stimulate the transition b c → [2].The isolated molecule evolves in a radiationless and irreversible process from b to a state c in a time τ 0 ≈ 3 μs, which is short compared to the radiative lifetime τ R ≈ 40 μs of both b and c[1] [2], but long compared to the time domain of intramolecular dynamics in NO 2 (e.g.[3] [4]).The two states b and c have the same radiative lifetime but differ in the degree of polarization of fluorescence light [1] [2].The transition b c → is smooth.Zeeman coherence in the magnetic sublevels is conserved in the evolution of b to c .The experiments in Refs.[1] [2] were using magnetic field induced depolarization of the fluorescence light (zero-magnetic field level-crossing or Hanle effect measurement) as well as optical radio-frequency double resonance.These experiments give τ 0 and τ R as coherence decay times.The lifetime τ R is in agreement with results of radiative decay measurements revealing single-exponential decay ([5] and references given there).The transition b c → exhibits an unusual feature, named inversion effect, which was not seen on atoms and molecules before ([1] [2] and references given there).The inversion effect is an inversion of the distribution of the occupation probabilities c m of the magnetic sublevels , c m of c versus the light intensity I or versus the transit time T L of the molecules through the light beam.Optical coherence (e.g.∆m = 0 for π excitation) reduces the decay rate of , b m to , c m by the coupling of , b m to the ground state , m is the Rabi frequency.This gives c m ~ Z m for low values of IT L and of IT L .The inversion effect shows that light-induced optical coherence between the states a and b works against the process driving the molecule from b to c .Obviously, the interaction causing the transition b c → does not affect optical coherence and is most likely a non-electromagnetic interaction affecting primarily the nuclear dynamics in the molecule.In Ref. [2], the transition b c → is described by a decay process.A more consistent description uses a time-asymmetric evolu-H.G. Weber tion in the optically excited molecule [6].
figurations of the nuclei, b represents an unstable configuration because the vibrational motion in b shifts one of the two O nuclei periodically off the common compactification space, enabling the dark matter field to stimulate the transition b c →with the rate (τ 0 ) −1 .In Sec. 2, we revisit experimental re-

Figure 1
Figure 1 depicts schematically the experimental arrangement.NO 2 molecules are propagating freely in an effusive molecular beam along the z-axis.A 50% beam splitter splits the light beam of a single mode cw laser (spectral width < 10 MHz) into two beams L 1 and L 2 , which both cross the molecular beam.L 1 and L 2 have the same linear polarization parallel to the z-axis (π-excitation).The angle α can be varied continuously around α = 0.For α = 0 both light beams are parallel to each other (at right angle to the z-axis) and are separated by the gap width s.The gap width s as well as the aperture width d are both adjustable in the experiments.Here, d determines the diameters of L 1 and L 2 .Both s and d define the time of flight (T s + T L ) of the molecules from the centre of L 1 to the centre of L 2 with s = uT s and d = uT L .Here T L is the transit-time of the molecules through L 1 or L 2 .If ν is the light frequency as seen by the molecules in L 1 , the molecules in L 2 see ν + δν with ( ) u c δν να = for small α.Here u = 610 ± 25 ms −1 is the average velocity of the molecules along the z-axis and c the velocity of light.The measured quantity is ( ) o oA P P P = − either versus the angle α or (with α = 0) versus a magnetic field B, which is parallel to the z-axis.Here P is the fluorescence intensity as seen by a photomultiplier (perpendicular to the z-axis) and P = P o if α or B is off-resonance at a well-defined value.In the experiments, the laser light is tuned (with α = 0) to a molecular transition near λ ex = 593 nm.Then the beam divergence of L 1 and L 2 is adjusted to a maximum parallel light beam (flat wave surface at the intersection with the molecular beam).This adjustment seems to provide a maximum of optical coherence between a and b during T L and has a strong effect on the fluorescence intensity P. Depending on T L , this adjustment reduces P up to 50% of the P value for strongly focused L 1 and L 2 at constant laser power[8].This

Figure 2 .
Figure 2. (a) Laser induced transitions (λ −1 = 16850.29cm −1 ) between hfs-levels (F, g-factor) of the ground state fs-level (N = 0, J = 1/2) and the excited state fs-level (N = 1, J = 3/2).(b) Each optical transition between a single hfs-level in the ground state and a single hfs-level in the excited state comprises the states a , b , and c .

Figure 3 Figure 3 .
Figure 3. Magnetic resonance spectrum using the set-up S 1 with A 1 , A 2 , A 3 , und C 1 indicating expected magnetic resonances corresponding to the g-factors in Figure 2(a).The y-axis represents the quantity ( ) 0 0 A P P P = − (see text).The Figure is taken from Ref.

Figure 4 .
Figure 4. Magnetic resonance spectrum using the set-up S 2 with A 1 , A 2 , A 3 , und C 1 indicating expected magnetic resonances corresponding to the g-factors in Figure 2(a).The y-axis represents the quantity ( ) 0 0 A P P P = − (see text).
the width Δν due to laser frequency jitter during the time of flight (T s + T L ).We have 2.5 μs.The width Δν is independent of the light intensity although the amplitude A ν shows strong saturation versus the light intensity.Measurements of A ν versus the time of flight (T s + T L ) yield

Figure 5 .
Figure 5.The y-axis represents the quantity

≈ 3
Figure 2(b), applies to all transitions in Figure 2(a) between an hfs-level in the ground state and an hfs-level in the excited state.Moreover, the perturbation causing the transition b c → is neither an intrinsic process in the molecule nor caused by baryonic matter collisions.The "ν-resonance" in Figure5, is a superposition of at least three signals, of which each one is associated with the excitation of one of the three hfs-levels in Figure2(a).Each signal is centred at δν = 0 with the same width (b).However, there is no absorption process from a to c .Light absorption from a to c contradicts the experimental results, in particular the inversion effect.What does it mean that b and c are both connected to a by an electric-dipole transition, but a molecule being in a is only excited into b by an optical transition?We conclude that the states a and b but not the state c are eigenstates of the hamiltonian H mol of the unperturbed (no transition b c → ) molecule.In the unperturbed molecule, there is no state c and the state b has the decay rate (τ R ) −1 .This conclusion disproves the proposal in Ref. [13].The perturbation causing the transition b c → affects the molecule in b and modifies b into c with the rate (τ 0 ) −1 without changing the radiative decay rate significantly.The two "states" b and c represent a substructure of a single isolated hfs-level of the molecule.The level width of b represented e.g. by the width of the "broad rf-resonance" ν-resonance" corresponds to an energy spread of about 400 peV.This width is by a factor of about natural linewidth (2πτ R ) −1 ≈ 4 kHz of c .A mo- lecule evolves from b into a single state c .However, the level energy of c seems to vary within the width of b .We assign the near Gaussian shape of the "ν-resonance" to the distribution of the level energies of c within the width of b in the ensemble of excited molecules.The level energies of c are no eigenvalues the usual molecular hamiltonian of an isolated molecule.A complete description of the molecule requires to take account of the perturbation causing the transition b c → .The level energies of c seem to occupy an energy band having a width determined by the decay rate (τ 0 ) hfs-level.Amolecule is only in one of these c levels.hfs-levels explains the difference between the lifetime τ R measured by radiative decay measurements and the lifetime τ in measured by the integrated absorption coefficient[10] [11].According to Equation(22) of Ref.[27], τ in is given by ≈ 200 peV for the mass m of the axion.In applying the second assumption, we note that the N-O bond lengths differ in a , b , and c with a having the shortest and c having the longest bond length.We propose the following.In a , all nuclei of NO 2 are completely confined in a single compactification space, and in c , the two O nuclei are in two different compactification spaces.At the shorter bond length N-O, the N and O nuclei are confined in one compactification space, and at the longer bond length N-O, the O nucleus is isolated in a separate compactification space.We do not exclude a tunneling motion between the two configurations of an O nucleus.The experiments show that a and c represent stable configurations of the nuclei, whereas b is an unstable configuration of the nuclei.Here "stable" means that the dark matter field does not affect the configuration of the nuclei.In b the configuration of the nuclei is unstable, because presumably the vibrational motion shifts one of the two O nuclei periodically off the common compactification space.This enables the axion field to stimulate the transition b c → with the rate (τ 0 ) −1 .A coherent superposition of b and a reduces this action of the axion field, because this field does not affect the molecule in a .This explains the inversion effect.Moreover, molecule and axion field are a non-separable system with an effective degeneracy of about τ R (τ 0 ) −1 of the excited hfs-levels.This explains the difference between the lifetime τ R measured by radiative decay measurements and the lifetime τ in measured by the integrated absorption coefficient.The phenomenological description given here does not explain the dynamics of the transition b c →