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 τR ≈ 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   . For example, optical excitation of NO2 in a molecular beam near the excitation wavelength λex = 593 nm induces a transition between a state of the ground electronic state (X2A1) and a state of the first excited electronic state (A2B2). However, the state is not stable   . The isolated molecule evolves in a radiationless and irreversible process from to a state in a time τ0 ≈ 3 μs, which is short compared to the radiative lifetime τR ≈ 40 μs of both and   , but long compared to the time domain of intramolecular dynamics in NO2 (e.g.   ). The two states and have the same radiative lifetime but differ in the degree of polarization of fluorescence light   . The transition is smooth. Zeeman coherence in the magnetic sublevels is conserved in the evolution of to . The experiments in Refs.   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 τR as coherence decay times. The lifetime τR is in agreement with results of radiative decay measurements revealing single-exponential decay (  and references given there).
The transition exhibits an unusual feature, named inversion effect, which was not seen on atoms and molecules before (   and references given there). The inversion effect is an inversion of the distribution of the occupation probabilities cm of the magnetic sublevels of versus the light intensity I or versus the transit time TL of the molecules through the light beam. Optical coherence (e.g. Dm = 0 for π excitation) reduces the decay rate of to by the coupling of to the ground state . The occupation probabilities are for and for with , where 2vm is the Rabi frequency. This gives cm ~ Zm for low values of ITL and for high values of ITL. The inversion effect shows that light-induced optical coherence between the states and works against the process driving the molecule from to . Obviously, the interaction causing the transition does not affect optical coherence and is most likely a non-electromagnetic interaction affecting primarily the nuclear dynamics in the molecule. In Ref.  , the transition is described by a decay process. A more consistent description uses a time-asymmetric evolution in the optically excited molecule  .
The time constant τR ≈ 40 μs measured by radiative decay  or with use of the Hanle effect  , agrees well with results of optical radio-frequency double resonance experiments (see Ref.  and references given there), and with results of “time of flight” experiments (see Ref.  and Sec. 2 below). The time constant τ0 was extracted from the width of the “broad” Hanle signal (Ref.  ), from the width of the “broad rf-resonance” (see Refs.   and Sec. 2), and from the width of the “n-resonance” (see Ref.  and Sec. 2). The time constant τ0 agrees also with the lifetime τin evaluated with use of the integrated absorption coefficient giving values for τin between 1 μs and 4 μs (Ref.  and references given there). In general, one expects τin = τR. The disagreement of τin with τR 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 τin was identified with the lifetime of the uncoupled A2B2 electronic state   . This theory is not in agreement with the occurrence of two time constants τ0 and τR simultaneously on a single isolated hfs-level of NO2. Moreover, radiative decay with the time constant τin was never detected by optical excitation near λex = 593 nm   . In this work, we identify τin with τ0 and give a different interpretation of the disagreement of τin with τR.
The present work aims to explain the transition and the time constant τ0 We have experimental evidence (see Ref.  and Sec. 2 of this work) that the transition 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.     ), and based on ADD-theory (see Refs.   ), 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.    ). 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 + ne)-dimensional space, where ne refers to the number of extra dimensions   . Axions are an essential ingredient of various compactification scenarios including string theory and other theories with nontrivial extra dimensions (e.g. Refs.     ). Our proposal is as follows. In , all nuclei of NO2 are completely 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 the dark matter field to stimulate the transition 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.   . 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 (Ts + TL) of the molecules from the centre of L1 to the centre of L2 with s = uTs and d = uTL. Here TL 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 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 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 = Po 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 λex = 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 and during TL and has a strong effect on the fluorescence intensity P. Depending on TL, this adjustment reduces P up to 50% of the P value for strongly focused L1 and L2 at constant laser power  . 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 , , and .
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.   ) 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 , , and 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 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.   ). 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 (see text). The Figure is taken from Ref.  .
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.  ) 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 or in . L1 depopulates some and populates the linked , which evolve fast into the (see Figure 2(b)). During the time of flight (Ts + TL) the rf-field induces Δm = ±1 transitions in the and , which are detected by L2 causing an increase (by about 1%) of the fluorescence intensity P. (The rf-transitions in contribute to the broad resonance structure.) The technique is well known (e.g.  ). Measurements of the width ΔB of the narrow resonances versus the time of flight (Ts + TL) give , where w1 is a constant comprising the Planck constant, the Bohr magnetron, the g-factor, and a numerical factor  . The width ΔB depends only on the apparative time constants and approaches zero for large values (Ts + TL). The ratio of the signal strengths of resonances in and in versus (Ts + TL) gives the lifetime τR = 40 ± 10 μs of 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 (Ts + TL) = 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 TL 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 TL = 5.7 μs, whereas in Figure 3 we used TL = 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”   . These resonances are connected with the inversion effect  . 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 (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 (TL)−1 → 0) and by the dependence of the signal strength of these resonances versus TL  . 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 versus 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”  . The signal shape is not Lorentzian but approximately Gaussian with the width (FWHM) Δn and the amplitude An. The result is with , with the transit time broadening
, and with being a contribution to the width Δν due to laser frequency jitter during the time of flight (Ts + TL). We have with τ0 = 2.5 μs. The width Δn is independent of the light intensity although the amplitude An shows strong saturation versus the light intensity. Measurements of An versus the time of flight (Ts + TL) yield with τ = 22 μs. The discrepancy between τ and τR ≈ 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 ≈ τR (for details see Ref.  ). We note that the recorded “n-resonance” signals have generally an oblique underground which is connected
Figure 5. The y-axis represents the quantity (see text) measured in the set-up S3 versus . This signal is named “n-resonance” in the text. The Figure is taken from Ref.  .
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.
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 with the radiative lifetime τR ≈ 40 μs, and the experiments in arrangement S2 (e.g. Figure 4) reveal that each hfs-level has a state with the decay time τ0 ≈ 3 μs. Moreover, a molecule evolves from into one state only and not into a bunch of states . The resonances in Figure 3 disagree with a molecule being in a superposition of several states , because rf-transitions between different states result in additional resonances and a broadening of the width. The assumption of a molecule being in a superposition of several states disagrees also with the results of Hanle experiments and Zeeman quantum beat experiments, which both reveal a coherence decay time of in agreement with the radiative lifetime τR   . Evidently, the transition transfers Zeeman coherence in the magnetic sublevels from a single state to a single state . Therefore, the transition 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  . We conclude that the level scheme (comprising the states , , and ) 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 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 , with the same dependence on the transit-time TL, and with the same time of flight (Ts + TL). 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 (Ts + TL) is , if we assume τ = τR taking into account the constraints in the detection geometry (see Sec. 2). A change of the population in (“hole burning”) does not affect An in a detectable way. This result shows that the “n-resonance” is predominantly due to induced emission from to as indicated in Figure 2(b). However, there is no absorption process from to . Light absorption from to contradicts the experimental results, in particular the inversion effect. What does it mean that and are both connected to by an electric-dipole transition, but a molecule being in is only excited into by an optical transition? We conclude that the states and but not the state are eigenstates of the hamiltonian Hmol of the unperturbed (no transition ) molecule. In the unperturbed molecule, there is no state and the state has the decay rate (τR)−1. This conclusion disproves the proposal in Ref.  . The perturbation causing the transition affects the molecule in and modifies into with the rate (τ0)−1 without changing the radiative decay rate significantly. The two “states” and represent a substructure of a single isolated hfs-level of the molecule.
The level width of represented e.g. by the width of the “broad rf-resonance” or by the width of the “n-resonance” corresponds to an energy spread of about 400 peV. This width is by a factor of about larger than the natural linewidth (2πτR)−1 ≈ 4 kHz of . A molecule evolves from into a single state . However, the level energy of seems to vary within the width of . We assign the near Gaussian shape of the “ν-resonance” to the distribution of the level energies of within the width of in the ensemble of excited molecules. The level energies of 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 . The level energies of seem to occupy an energy band having a width determined by the decay rate (τ0)−1 and representing an effective degeneracy of the excited hfs-level. A molecule is only in one of these levels.
An effective degeneracy of the excited hfs-levels explains the difference between the lifetime τR measured by radiative decay measurements and the lifetime τin measured by the integrated absorption coefficient   . According to Equation (22) of Ref.  , τin is given by , where the quantity Ain includes an integral over the whole of the absorption band concerned, and gl and gu are the degeneracy of the lower (l) and the upper (u) state involved, respectively. In Refs.   , τin was evaluated using , which gives . However, assuming an effective degeneracy of the excited hfs-level and gl = 1, we obtain with . This result eliminates the discrepancy between the radiative lifetime τR and the lifetime τin evaluated on the basis of the integrated absorption coefficient. Finally, we note that a disagreement between τin and τR is also known for the molecules SO2 and CS2   .
In , the molecule is in the vibrational ground state with no vibrational mode excited. The vibrational quantum numbers of and are not known. Spectroscopic studies (near λex = 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.  and a similar result in Ref.  ). These studies reveal also that the N-O bond length is significantly longer in than in  . However, the strongly collision disturbed fluorescence spectrum favours the study of only and gives no information on . The experiments in Refs.   show that and differ in the degree of polarization of the fluorescence light. In Ref.  , we conclude that has a symmetric configuration (equal N-O bond lengths) in agreement with the results in Ref.  , whereas has an asymmetric configuration (unequal N-O bond lengths). In Ref.  , we associate 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  .
What causes the transition ? 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 and works against the transition . Therefore, we conclude that the perturbation causing this transition affects the molecule in but not in , 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 C2v symmetry) are degenerate (intersect) and coupled e.g. by the antisymmetric stretch vibration mode (e.g.    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.
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” and 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   , 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.   ). 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 , , and with having the shortest and having the longest bond length. We propose the following. In , all nuclei of NO2 are completely confined in a single compactification space, and in , 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 and represent stable configurations of the nuclei, whereas is an unstable configuration of the nuclei. Here “stable” means that the dark matter field does not affect the configuration of the nuclei. In 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 with the rate (τ0)−1. A coherent superposition of and reduces this action of the axion field, because this field does not affect the molecule in . This explains the inversion effect. Moreover, molecule and axion field are a non-separable system with an effective degeneracy of about τR (τ0)−1 of the excited hfs-levels. This explains the difference between the lifetime τR measured by radiative decay measurements and the lifetime τin measured by the integrated absorption coefficient. The phenomenological description given here does not explain the dynamics of the transition . This is beyond the scope of the experimental work reported here and needs further clarification by theory.
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.