NO₂ Excited State Properties Revisited: An Effect of Extra Compactified Dimensions

Hans-Georg Weber^1,2

¹Heinrich-Hertz-Institut, Einsteinufer 37, Berlin, Germany.
²Sudetenstr. 16, Amorbach, Germany.

DOI: 10.4236/jmp.2017.811103                

Abstract

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” and of a hfs-level. Optical excitation induces a transition from the ground state of the molecule to the excited state . However, the molecule evolves from to in a time τ₀ ≈ 3 μs. Both and have the radiative lifetime τ_R ≈ 40 μs, but and differ in the degree of polarization of the fluorescence light. Zeeman coherence in the magnetic sublevels is conserved in the transition →, and optical coherence of and is able to affect (inversion effect) the transition →. 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 , all nuclei of NO₂ are confined in a single compactification space, and in , the two O nuclei of NO₂ are in two different compactification spaces. Whereas and represent stable configurations of the nuclei,represents an unstable configuration because the vibrational motion in shifts one of the two O nuclei periodically off the common compactification space, enabling dark matter interaction to stimulate the transition → with the rate (τ₀)⁻¹. We revisit experimental results, which were not understood before, and we give a consistent description of these results based on the above assumption.

Keywords

Extra Dimensions, Compactification Space, Dark Matter, Molecular Physics

1. Introduction

Various experiments on NO₂ 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 τ₀ ≈ 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 NO₂ in a molecular beam near the excitation wavelength λ_ex = 593 nm induces a transition between a state $|a\rangle$ of the ground electronic state (X²A₁) and a state $|b\rangle$ of the first excited electronic state (A²B₂). However, the state $|b\rangle$ is not stable [1] [2] . The isolated molecule evolves in a radiationless and irreversible process from $|b\rangle$ to a state $|c\rangle$ in a time τ₀ ≈ 3 μs, which is short compared to the radiative lifetime τ_R ≈ 40 μs of both $|b\rangle$ and $|c\rangle$ [1] [2] , but long compared to the time domain of intramolecular dynamics in NO₂ (e.g. [3] [4] ). The two states $|b\rangle$ and $|c\rangle$ have the same radiative lifetime but differ in the degree of polarization of fluorescence light [1] [2] . The transition $|b\rangle \to |c\rangle$ is smooth. Zeeman coherence in the magnetic sublevels is conserved in the evolution of $|b\rangle$ to $|c\rangle$ . 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 τ₀ 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 $|b\rangle \to |c\rangle$ 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 $|c,m\rangle$ of $|c\rangle$ versus the light intensity I or versus the transit time T_L of the molecules through the light beam. Optical coherence (e.g. Dm = 0 for π excitation) reduces the decay rate of $|b,m\rangle$ to $|c,m\rangle$ by the coupling of $|b,m\rangle$ to the ground state $|a,m\rangle$ . The occupation probabilities are $b_m~Z_m{\left(1+Z_m\right)}^{-1}$ for $|b,m\rangle$ and $c_m~Z_m{\left(1+Z_m\right)}^{-2}$ for $|c,m\rangle$ with $Z_m={\tau }_0{T}_L{\left(2v_m\right)}^2$ , where 2v_m is the Rabi frequency. This gives c_m ~ Z_m for low values of IT_L and $c_m~{\left(Z_m\right)}^{-1}$ for high values of IT_L. The inversion effect shows that light-induced optical coherence between the states $|a\rangle$ and $|b\rangle$ works against the process driving the molecule from $|b\rangle$ to $|c\rangle$ . Obviously, the interaction causing the transition $|b\rangle \to |c\rangle$ 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 $|b\rangle \to |c\rangle$ is described by a decay process. A more consistent description uses a time-asymmetric evolution in the optically excited molecule [6] .

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 τ₀ 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 “n-resonance” (see Ref. [8] and Sec. 2). The time constant τ₀ 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 A²B₂ with high lying vibrational levels of the ground electronic state and τ_in was identified with the lifetime of the uncoupled A²B₂ electronic state [11] [12] . This theory is not in agreement with the occurrence of two time constants τ₀ and τ_R simultaneously on a single isolated hfs-level of NO₂. 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 τ₀ and give a different interpretation of the disagreement of τ_in with τ_R.

The present work aims to explain the transition $|b\rangle \to |c\rangle$ and the time constant τ₀ We have experimental evidence (see Ref. [13] and Sec. 2 of this work) that the transition $|b\rangle \to |c\rangle$ with the time constant τ₀ and with the associated inversion effect is a property of the isolated hfs-levels of NO₂ 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] [19] . Axions are an essential ingredient of various compactification scenarios including string theory and other theories with nontrivial extra dimensions (e.g. Refs. [20] [21] [22] [23] ). Our proposal is as follows. In $|a\rangle$ , all nuclei of NO₂ are completely confined in a single compactification space, and in $|c\rangle$ , the two O nuclei of NO₂ are in two different compactification spaces. Whereas $|a\rangle$ and $|c\rangle$ represent stable configurations of the nuclei, $|b\rangle$ represents an unstable configuration because the vibrational motion in $|b\rangle$ shifts one of the two O nuclei periodically off the common compactification space, enabling the dark matter field to stimulate the transition $|b\rangle \to |c\rangle$ with the rate (τ₀)⁻¹. In Sec. 2, we revisit experimental results, 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.

2. Experimental Results Revisited

We discuss experimental results, which were described in detail in Refs. [8] [9] . Figure 1 depicts schematically the experimental arrangement. NO₂ 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 L₁ and L₂, which both cross the molecular beam. L₁ and L₂ have the same linear polarization parallel to the z-axis (π-excitation). The angle a 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 L₁ and L₂.

Both s and d define the time of flight (T_s + T_L) of the molecules from the centre of L₁ to the centre of L₂ with s = uT_s and d = uT_L. Here T_L is the transit-time of the molecules through L₁ or L₂. If n is the light frequency as seen by the molecules in L₁, the molecules in L₂ see n + dn with $\delta \nu =\nu \alpha \left(u/c\right)$ for small a. 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 $A=\left(P-P_o\right)/P_o$ 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 L₁ and L₂ 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 $|a\rangle$ and $|b\rangle$ 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 L₁ and L₂ at constant laser power [8] . This

Figure 1. Schematic diagram of the experimental apparatus enabling experiments in the arrangements S₁, S₂, and S₃ (see text).

Figure 2. (a) Laser induced transitions (l⁻¹ = 16850.29 cm⁻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 $|a\rangle$ , $|b\rangle$ , and $|c\rangle$ .

property of the optical excitation process is an essential ingredient to the experiments on NO₂ discussed here and was never reported for another molecule to the knowledge of the author.

The residual Doppler width in the molecular beam is about 50 MHz. The laser light induces a transition between a well-defined fine structure (fs) level in the ground state and a well-defined fs-level in the excited state. We investigated up to 20 such absorption lines in NO₂. Figure 2(a) depicts the transition between the ground state fs-level (N = 0, J = 1/2) and the excited state fs-level (N = 1, J = 3/2). The level structure is represented by the angular momentum coupling schema N + S = J and J + I = F, where N, S, I, and F are the rotational, electron spin, nuclear spin and total angular momentum quantum numbers of the predominant isotopic form ¹⁴N¹⁶O₂. The ¹⁴N nucleus has I = 1 and the ¹⁶O nucleus has I = 0. The hyperfine structure (hfs) splitting in the ground and excited state is larger than 5 MHz (see discussion in Refs. [8] [9] ) and the excitation width of the laser light is less than 0.2 MHz (see below). Consequently, the laser light induces in a molecule a transition between a single hfs-level in the ground state and a single hfs-level in the excited state. Figure 2(a) depicts possible transitions (∆F = +1 and ∆F = 0). We will show that each such transition between an hfs-level in the ground state and an hfs-level in the excited state comprises the states $|a\rangle$ , $|b\rangle$ , and $|c\rangle$ as depicted in Figure 2(b) and described above.

We used the experimental apparatus depicted in Figure 1 in three different arrangements designated S₁, S₂, and S₃ in the following. In S₁, 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 $A=\left(P-P_o\right)/P_o$ versus B. Figure 3 depicts the measured resonance spectrum with A₁, A₂, A₃, and C₁ indicating magnetic field values corresponding to the g-factors given in Figure 2(a), which were known before e.g. by optical-rf double resonance experiments (see discussion in Refs. [8] [9] ). The narrow resonances in Figure 3 appear on top of a broad resonance structure, which we discuss in the next paragraph. The

Figure 3. Magnetic resonance spectrum using the set-up S₁ with A₁, A₂, A₃, und C₁ indicating expected magnetic resonances corresponding to the g-factors in Figure 2(a). The y-axis represents the quantity $A=\left(P-P_0\right)/P_0$ (see text). The Figure is taken from Ref. [8] .

narrow resonances (but not the broad resonance structure) disappear if we use only L₁ or only L₂. The perturbation (due to optical-rf double resonance, see discussion in Ref. [8] ) appearing on top of the resonance at A₃ exists also if we use either L₁ or L₂ solely. The narrow resonances in Figure 3 are assigned to molecules being either in $|a\rangle$ or in $|c\rangle$ . L₁ depopulates some $|a,m\rangle$ and populates the linked $|b,m\rangle$ , which evolve fast into the $|c,m\rangle$ (see Figure 2(b)). During the time of flight (T_s + T_L) the rf-field induces Δm = ±1 transitions in the $|a,m\rangle$ and $|c,m\rangle$ , which are detected by L₂ causing an increase (by about 1%) of the fluorescence intensity P. (The rf-transitions in $|b\rangle$ contribute to the broad resonance structure.) The technique is well known (e.g. [24] ). Measurements of the width ΔB of the narrow resonances versus the time of flight (T_s + T_L) give $\Delta B=w_1{\left(T_s+T_L\right)}^{-1}$ , where w₁ is a constant comprising the Planck constant, the Bohr magnetron, the g-factor, and a numerical factor [8] . The width ΔB depends only on the apparative time constants and approaches zero for large values (T_s + T_L). The ratio of the signal strengths of resonances in $|c\rangle$ and in $|a\rangle$ versus (T_s + T_L) gives the lifetime τ_R = 40 ± 10 μs of $|c\rangle$ independent of constraints in the detection geometry (see below). These measurements show that the resonances are not affected by collisions during the time of flight measured up to (T_s + T_L) = 35 μs. This verifies that the molecules travel collision-free in the molecular beam.

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 S₂, which is the same as S₁ but using L₁ only (L₂ is off). Figure 4 depicts a result obtained in S₂ 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

Figure 4. Magnetic resonance spectrum using the set-up S₂ with A₁, A₂, A₃, und C₁ indicating expected magnetic resonances corresponding to the g-factors in Figure 2(a). The y-axis represents the quantity $A=\left(P-P_0\right)/P_0$ (see text).

on the adjustment of the divergence of L₁ as described above. These resonances enable measurements of the time constant τ₀ by the width of each resonance (after extrapolation of (T_L)⁻¹ → 0) and by the dependence of the signal strength of these resonances versus T_L [9] . The measurements of the “broad rf-resonances” are hampered by a compromise between nonlinearity of the Zeeman splitting and attainable resolution in the resonance spectrum. The width of the resonance at A₃ is already strongly affected by the nonlinear Zeeman splitting. Moreover, this resonance is also perturbed by an optical-rf double resonance signal similar to the resonance at A₃ in Figure 3. Taking these constraints into account, all resonances have the same properties in particular the same time constant τ₀ ≈ 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 S₃, i.e. using L₁ and L₂, but no rf-field and no static magnetic field B (the earth magnetic field is compensated). The measured quantity is $A=\left(P-P_0\right)/P_0$ versus $\delta \nu =\nu \alpha \left(u/c\right)$ with P = P₀ 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 δn = 0. It was named “n-resonance” [8] . The signal shape is not Lorentzian but approximately Gaussian with the width (FWHM) Δn and the amplitude A_n. The result is $\Delta \nu =\Delta {\nu }_0+\Delta {\nu }_t+\Delta {\nu }_j$ with $\Delta {\nu }_0=\left(0.13\pm 0.05\right)\text{MHz}$ , with the transit time broadening

$\Delta {\nu }_t=\left(0.87\pm 0.05\right){\left(T_L\right)}^{-1}$ , and with $\Delta {\nu }_j=\left(7.6\pm 1.0\right)\times {10}^9\left(T_s+T_L\right){\left(\text{Hz}\right)}^2$ being a contribution to the width Δν due to laser frequency jitter during the time of flight (T_s + T_L). We have $\Delta {\nu }_0={\left(\text{π}{\tau }_0\right)}^{-\text{1}}=0.\text{13}\text{MHz}$ with τ₀ = 2.5 μs. The width Δn is independent of the light intensity although the amplitude A_n shows strong saturation versus the light intensity. Measurements of A_n versus the time of flight (T_s + T_L) yield $A_{\nu }~\exp \left[-\left(T_s+T_L\right){\left(\tau \right)}^{-1}\right]$ 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 “n-resonance” signals have generally an oblique underground which is connected

Figure 5. The y-axis represents the quantity $A=\left(P-P_0\right)/P_0$ (see text) measured in the set-up S₃ versus $\delta \nu =\nu \alpha \left(u/c\right)$ . This signal is named “n-resonance” in the text. The Figure is taken from Ref. [8] .

with the angle tuning of the light beam L₂. This underground is also present without L₁. The underground (a straight line) is subtracted in the result shown in Figure 5.

3. Discussion

Figure 3 and Figure 4 show that the g-factors enable an assignment of the measured data to the three excited hfs-levels depicted in Figure 2(a). The experiments in arrangement S₁ (e.g. Figure 3) reveal that each hfs-level has a state $|c\rangle$ with the radiative lifetime τ_R ≈ 40 μs, and the experiments in arrangement S₂ (e.g. Figure 4) reveal that each hfs-level has a state $|b\rangle$ with the decay time τ₀ ≈ 3 μs. Moreover, a molecule evolves from $|b\rangle$ into one state $|c\rangle$ only and not into a bunch of states $|c\rangle$ . The resonances in Figure 3 disagree with a molecule being in a superposition of several states $|c\rangle$ , because rf-transitions between different states $|c\rangle$ result in additional resonances and a broadening of the width. The assumption of a molecule being in a superposition of several states $|c\rangle$ disagrees also with the results of Hanle experiments and Zeeman quantum beat experiments, which both reveal a coherence decay time of $|c\rangle$ in agreement with the radiative lifetime τ_R [1] [25] . Evidently, the transition $|b\rangle \to |c\rangle$ transfers Zeeman coherence in the magnetic sublevels from a single state $|b\rangle$ to a single state $|c\rangle$ . Therefore, the transition $|b\rangle \to |c\rangle$ is no intrinsic (intramolecular) process in the molecule, because there is no “sink” in the molecule to provide a recurrence time longer than the radiative lifetime [26] . We conclude that the level scheme (comprising the states $|a\rangle$ , $|b\rangle$ , and $|c\rangle$ ) depicted in 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 $|b\rangle \to |c\rangle$ is neither an intrinsic process in the molecule nor caused by baryonic matter collisions.

The “n-resonance” in Figure 5, 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 Figure 2(a). Each signal is centred at δn = 0 with the same width $\Delta {\nu }_0={\left(\text{π}{\tau }_0\right)}^{-1}=0.13\text{MHz}$ , with the same dependence on the transit-time T_L, and with the same time of flight (T_s + T_L). Therefore, the “n-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 “n-resonance” versus the time of flight (T_s + T_L) is $A_{\nu }~\exp \left[-\left(T_s+T_L\right){\left({\tau }_R\right)}^{-1}\right]$ , if we assume τ = τ_R taking into account the constraints in the detection geometry (see Sec. 2). A change of the population in $|a\rangle$ (“hole burning”) does not affect A_n in a detectable way. This result shows that the “n-resonance” is predominantly due to induced emission from $|c\rangle$ to $|a\rangle$ as indicated in Figure 2(b). However, there is no absorption process from $|a\rangle$ to $|c\rangle$ . Light absorption from $|a\rangle$ to $|c\rangle$ contradicts the experimental results, in particular the inversion effect. What does it mean that $|b\rangle$ and $|c\rangle$ are both connected to $|a\rangle$ by an electric-dipole transition, but a molecule being in $|a\rangle$ is only excited into $|b\rangle$ by an optical transition? We conclude that the states $|a\rangle$ and $|b\rangle$ but not the state $|c\rangle$ are eigenstates of the hamiltonian H_mol of the unperturbed (no transition $|b\rangle \to |c\rangle$ ) molecule. In the unperturbed molecule, there is no state $|c\rangle$ and the state $|b\rangle$ has the decay rate (τ_R)⁻¹. This conclusion disproves the proposal in Ref. [13] . The perturbation causing the transition $|b\rangle \to |c\rangle$ affects the molecule in $|b\rangle$ and modifies $|b\rangle$ into $|c\rangle$ with the rate (τ₀)⁻¹ without changing the radiative decay rate significantly. The two “states” $|b\rangle$ and $|c\rangle$ represent a substructure of a single isolated hfs-level of the molecule.

The level width of $|b\rangle$ represented e.g. by the width of the “broad rf-resonance” or by the width $\Delta {\nu }_0={\left(\text{π}{\tau }_0\right)}^{-1}\approx 100\text{MHz}$ of the “n-resonance” corresponds to an energy spread of about 400 peV. This width is by a factor of about ${\tau }_R{\left({\tau }_0\right)}^{-1}\ge 10$ larger than the natural linewidth (2πτ_R)⁻¹ ≈ 4 kHz of $|c\rangle$ . A molecule evolves from $|b\rangle$ into a single state $|c\rangle$ . However, the level energy of $|c\rangle$ seems to vary within the width of $|b\rangle$ . We assign the near Gaussian shape of the “ν-resonance” to the distribution of the level energies of $|c\rangle$ within the width of $|b\rangle$ in the ensemble of excited molecules. The level energies of $|c\rangle$ 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 $|b\rangle \to |c\rangle$ . The level energies of $|c\rangle$ seem to occupy an energy band having a width determined by the decay rate (τ₀)⁻¹ and representing an effective degeneracy $g_u\approx {\tau }_R{\left({\tau }_0\right)}^{-1}$ of the excited hfs-level. A molecule is only in one of these $|c\rangle$ levels.

An effective degeneracy $g_u\approx {\tau }_R{\left({\tau }_0\right)}^{-1}$ of the excited 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 ${\tau }_{in}=\left(g_u/g_l\right)A_{in}$ , 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 $g_l=g_u=1$ , which gives ${\tau }_{in}=A_{in}\approx 3\mu \text{s}\approx {\tau }_0$ . However, assuming an effective degeneracy $g_u\approx {\tau }_R{\left({\tau }_0\right)}^{-1}$ of the excited hfs-level and g_l = 1, we obtain ${\tau }_{in}=\left[{\tau }_R{\left({\tau }_0\right)}^{-1}\right]A_{in}={\tau }_R$ with $A_{in}={\tau }_0$ . This result eliminates the discrepancy between the radiative lifetime τ_R and the lifetime τ_in evaluated on the basis of the integrated absorption coefficient. Finally, we note that a disagreement between τ_in and τ_R is also known for the molecules SO₂ and CS₂ [11] [12] .

In $|a\rangle$ , the molecule is in the vibrational ground state with no vibrational mode excited. The vibrational quantum numbers of $|b\rangle$ and $|c\rangle$ are not known. Spectroscopic studies (near λ_ex = 593 nm) on a static NO₂ gas sample (50 mTorr) suggest totally symmetric vibrational symmetry (no asymmetric stretch mode excited) of the excited state establishing the A²B₂ electronic symmetry for this state (see Ref. [28] and a similar result in Ref. [29] ). These studies reveal also that the N-O bond length is significantly longer in $|b\rangle$ than in $|a\rangle$ [28] . However, the strongly collision disturbed fluorescence spectrum favours the study of $|b\rangle$ only and gives no information on $|c\rangle$ . The experiments in Refs. [1] [2] show that $|b\rangle$ and $|c\rangle$ differ in the degree of polarization of the fluorescence light. In Ref. [2] , we conclude that $|b\rangle$ has a symmetric configuration (equal N-O bond lengths) in agreement with the results in Ref. [28] , whereas $|c\rangle$ has an asymmetric configuration (unequal N-O bond lengths). In Ref. [2] , we associate $|c\rangle$ with a state having its energy minimum at linearity of the O-N-O angle and a strong asymmetry in the N-O bond length [30] .

What causes the transition $|b\rangle \to |c\rangle$ ? As already noted, we are able to exclude collisions with baryonic matter and an intrinsic (intramolecular) process in the molecule. The inversion effect shows that optical coherence between $|a\rangle$ and $|b\rangle$ works against the transition $|b\rangle \to |c\rangle$ . Therefore, we conclude that the perturbation causing this transition affects the molecule in $|b\rangle$ but not in $|a\rangle$ , and it does not affect optical coherence. As optical coherence is fast perturbed by electromagnetic interaction, we assume a non-electromagnetic interaction affecting primarily the nuclear dynamics in the molecule. However, a change e.g. in the N-O bond length affects easily also electronic dynamics in the excited molecule, because the potential energy surfaces of several electronic states (e.g. X²A₁, A²B₂, and B²B₁ in C_2v symmetry) are degenerate (intersect) and coupled e.g. by the antisymmetric stretch vibration mode (e.g. [31] [32] [33] and references given there). Therefore, also small perturbations of the nuclear configuration affect the symmetry of the electronic state and are thus enhanced to an optically detectable signal.

4. Conclusion

Experiments on NO₂ reveal a substructure underlying the isolated hyperfine structure (hfs) levels of the collision-free, optically excited molecule. This substructure is seen in a change of the symmetry of the excited molecule and is represented by the two “states” $|b\rangle$ and $|c\rangle$ underlying a single hfs-level. This finding contradicts our expectation on a molecule being excited into a stationary 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 (τ₀)⁻¹ with the oscillation frequency of the axion field (e.g. [14] [15] ). This gives mc² ≈ 200 peV for the mass m of the axion. In applying the second assumption, we note that the N-O bond lengths differ in $|a\rangle$ , $|b\rangle$ , and $|c\rangle$ with $|a\rangle$ having the shortest and $|c\rangle$ having the longest bond length. We propose the following. In $|a\rangle$ , all nuclei of NO₂ are completely confined in a single compactification space, and in $|c\rangle$ , 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 $|a\rangle$ and $|c\rangle$ represent stable configurations of the nuclei, whereas $|b\rangle$ is an unstable configuration of the nuclei. Here “stable” means that the dark matter field does not affect the configuration of the nuclei. In $|b\rangle$ 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 $|b\rangle \to |c\rangle$ with the rate (τ₀)⁻¹. A coherent superposition of $|b\rangle$ and $|a\rangle$ reduces this action of the axion field, because this field does not affect the molecule in $|a\rangle$ . This explains the inversion effect. Moreover, molecule and axion field are a non-separable system with an effective degeneracy of about τ_R (τ₀)⁻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 $|b\rangle \to |c\rangle$ . This is beyond the scope of the experimental work reported here and needs further clarification by theory.

Acknowledgements

This work is dedicated to my friend and colleague Dr. Franciszek Bylicki, Torun, Poland, who died in 2016. He contributed much to the experimental work reported here.

Conflicts of Interest

The authors declare no conflicts of interest.

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