Conference-ACSIN-10-Decoherence Mechanisms in Adsorbate Localization and Diffusion on Solid Surfaces

A long standing unsolved puzzle is the wide range of experimentally determined frequency factors for adsorbate diffusion spanning twelve orders of magnitude which are only poorly explained by modern quantum theories of surface diffusion (QTSD) based on ab initio potential energy surfaces (refs. [1]-[4]). This contribution investigates whether decoherence mechanisms, neglected in QTSD, can shed some light on these discrepancies. We suggest a quantum-mechanical theory treating the entanglement of adsorbates with their environment (phonons, electronhole pairs, plasmons, gravitons) [5]. Time dependent wave packet evolution and quantum Zeno effect (permenant “measurement” by specific environmental excitations) are investigated. [DOI: 10.1380/ejssnt.2010.6]


I. INTRODUCTION
Adsorbate core movement close to solid surfaces provides examples of both quantum and classical behaviour.In experiments on adsorbate sticking on cold surfaces [6][7][8], STM imaging of adsorbate localization [9], surface diffusion and desorption, induced by electron injection from the STM tip [10], a prominently classical behaviour is reported.Adsorbate quantum behaviour on the other hand is examplified by noble gas atom diffraction at solid surfaces [11], molecule tunnelling [12], molecule vibrational and rotational excitation in the STM [13,14], quantum diffusion of hydrogen on tungsten and copper single crystal surfaces [2,15], to mention only a few cases.The transition from quantum to classical behaviour is the essence of decoherence.In decoherence theory it is associated with entanglement of the local system with excitations due to its environment (phonons, electron-hole pairs, etc.) [16].
The experimental data raise questions: • The frequency factors for hydrogen diffusion vary over orders of magnitude depending upon the kind of measurement performed.The lowest values are due to single atom tracking in the low temperature STM experiments (T<65K) on 1 H diffusion on Cu(001) of the order of ν o ≈ 10 −3.5 s −1 [17].The value, reported using field-emission microscopy in the temperature-idependent tunnelling regime, is of the order of ν o ≈ 10 3.6 s −1 at T<100K for 1 H on Ni(001) at low coverage of 0.08 [18].The high-est value of the frequency factor for 1 H diffusion on Ni(001) of the order of ν o ≈ 10 4.5 s −1 (at 110K) is measured in the quantum diffusion regime with linear diffraction of a probe laser beam from a monolayer grating of adsorbed hydrogen [20].
• The STM results display the expected isotope effect on 1 H and 2 H diffusion on Cu(001) [2].Nearly no isotope effect is established in the FEM experiments by Lin and Gomer for H on Ni(100) [18].
The problems with the theoretical models of H quantum diffusion are: (i) Within the WKB approximation (ref.[2]) 1D hydrogen tunnelling is assumed from an initial state, a plane wave in a flat potential, through a potential barrier, disregarding the dynamics of the substrate atoms.Despite these deficits the values of the frequency factors and diffusivities are reproduced within the WKB approximation, if the theoretically determined barrier height is used, which differs from the experimentaly determined one [2].The isotope effect between 1 H and 2 H could be reproduced.(ii) 3D tunnelling, taking into account the lattice dynamics, leads to diffusivities which deviate by several orders of magnitude from the experimental values (ref.[4]).Large isotope effect on the diffusivity of 1 H and 2 H of the order of 5 × 10 5 is found [4], in contrast with the missing isotope effect, as shown by the experimental data [18].
We address the problem of adsorbate surface quantum diffusion.Surface diffusion may be envisaged as: starting from a localized adsorbate → delocalization → localization via environmental decoherence on a new adsorption site.Hence, diffusion is closely connected with the localization, i.e. with the decoherence mechanism.A competing channel for the localized adsorbate is the time evolution and delocalization as the time-dependent Schrödinger equation requires.
Adsorbate-phonon interaction is a possible candidate for environmental decoherence leading to localization of adsorbates.Regarding adsorbate surface diffusion as due to inelastic interaction with the substrate solid excitations, allows us to estimate the time scale of this environmental decoherence mechanism.Agreement between the theoretical data on the frequency factor of Xe diffusion on Ni(110) and the experimental value for Xe diffusion on Ni(111) [21] at 30-60K has indeed been established [5].The major consequence of adsorbate-phonon interaction, as it is provided by the theory, is that phonon scattering events are slow compared to the time scale of coherent time development of a wave packet used to describe a localized isolated Xe atom.Interaction with the substrate solid excitations is therefore not a fast enough decoherence mechanism to warrant the localization of a Xe atom on a solid surface in competition with the unitary time evolution of the wave packet.The conclusion we have to draw is that adsorbate-phonon interaction is not a viable deco-herence mechanism for Xe atom localization on Ni(110) in the low temperature STM.On the other hand Xe diffusion on the solid surface at low temperatures requires adsorbate localization prior to tunnelling or thermally induced hopping.This statement relies on the observation of Xe in the low temperature STM as a localized particle and never as a 2D Bloch wave.(The Xe-phonon interaction would be negligible, if Xe were delocalized over the whole surface, which would lead to discrepancy between the theoretical and the experimental diffusivity.)The conclusion is that there must be a different decoherhttp://www.sssj.org/ejssnt(J-Stage: http://www.jstage.jst.go.jp/browse/ejssnt/) ence process, a different "permanent measurement" process by the environment of the Xe adsorbate, occurring on a shorter timescale compared to the timescale of the diffusion process, which localizes the Xe atom so that the phonon mediated diffusion can be effective.The localization has to be due to a very local interaction with the adsorbate which will be able to localize wave packets of the dimensions of several Angströms, typical for single adatoms seen in STM images.The suggested decoherence mechanism is via entanglement with the gravitons, the messenger particles of the gravitational field.
In the theory of environmental decoherence and adsorbate localization two approaches can be followed: • Collapse model + rate equations.The results of this approach for Xe atom decoherence were described in a recent paper [5].
• Coherent time evolution in a world model.This is the focus of the present communication.
In the next section the theoretical formalism is presented, followed by typical examples and results and their discussion.

II. FORMALISM: COHERENT WORLD MODEL
In a coherent world model we follow the time evolution of an adsorbed particle localized on a substrate surface with 2D periodicity at its equilibrium position in the adsorption well.The localized wave packet used to represent the particle is not an eigenstate in a 2D periodic potential, hence it changes its shape with time as the timedependent Schrödinger equation requires.The entanglement with different elementary excitations in the environment will affect the evolution of the local adsorbate state, leading eventually to particle localization, hence to decoherence.The competition between the environmental decoherence and the coherent time evolution depends upon their relative rates.As we showed in a recent paper [5], the entanglement with the lattice phonons and tomonagons in the solid surface is not fast enough to decohere a single Xe atom as it is seen in the low temperature STM images.A faster decoherence mechanism was suggested due to entanglement with the gravitons within the spacetime deformation (warp resonance).The advantages are: it is fast and very local in contrast to the phonon modes and tomonagons at low temperature.
We use the philosophy and formalism of the Quantum Nano Dynamics (QND) theory, a field theoretical approach developed to treat correlated many particle systems [5,22,23].In the present paper we are concerned with the coherent time evolution of the core motion of a single adsorbed atom entangled with the continuum of delocalized plane waves and the gravitational field provided by the surface atoms.
In a many-particle picture the dynamics of the system is described with the help of both localized and delocalized basis states for all particles in the model: adsorbate, solid excitations, gravitons.The interactions are between many-body states, which are product states with components referring to the gas particle core movement parallel to the surface, its vibrational state and environmental excitations (cf.The meaning of the symbols is: n g0 , n e , n w : occupation number operators for the gas particle in the vibrationally ground and excited (parallel to the surface) core movement states and in the warp resonance; c + i , c j : creation and annihilation operators for the gas particle in the respective core movement states; V g0 loc , V e loc : interaction potential between a gas particle state and the warp resonance; V g0p , V ep : interaction potentials between the gas particle states and plane waves; E g0 , E e , E w , E p : energies of the respective gas particle states and plane waves.ω phon : energy of the phonon mode; b + , b: creation and annihilation operators for the phonon modes; ω o : excitation energy of the gas particle vibration parallel to the surface; σ + , σ − : operators transferring the system between the many particle states | g n g phon grav g and | e n e phon grav e ; ε grav , ε k : energy of local and continuum gravitons; a + grav , a grav : creation and annihilation operators for gravitons; V grav,w : interaction between the gas particle and the gravitons within the warp resonance (for the sake of a simpler notation the polarization degrees of freedom of the graviton are not explicitly displayed (cf.[5]); n p : occupation number operator for the plane waves.The sum over k in the last but one term in Eq. ( 1) includes summation over the local gravitons as well.
The coupling of the local gas particle states to the plane wave continuum leads to energy broadening and finite lifetime in the local gas particle states, hence to delocalization.We take into account the two lowest vibrational states of an adsorbate parallel to the solid surface | g and | e .They are generated by the interaction of two basis states of different spacial extension: | g 0 is diffuse (of the order of 10 bohr), | w is contracted (of the order of 1 bohr).Tunnelling out of the adsorption site into plane waves, hence delocalization and diffusion, can only occur via the diffuse core movement state | g .The coupling of the atom core movement to the graviton continuum occurs within the warped space which is represented with the more contracted basis function | w .

III. COHERENT TIME EVOLUTION OF THE LOCALIZED ADSORBATE ENTANGLED WITH THE PLANE WAVE AND GRAVITON CONTINUA
The time evolution of the adsorbate core movement state follows Schrödinger's time-dependent equation: with Ψ an eigenfunction of the total Hamiltonian Eq. ( 1).
We assume that at t 0 = 0 the adparticle is in the local state | g and a graviton of high energy is excited: | k .The eigenfunctions Ψ I (I=1,2,...) of the QND hamiltonian Eq. ( 1) are used to determine the time dependence according to the unitary time evolution: The expansion of the local adsorbate core movement state in the eigenfunctions {| I } at time t 0 = 0 is then: The time development of | g0 is obtained as: We need the representation of the density operator at t 1 in the input basis, for instance: The procedure is repeated for each next time step to provide the coherent time evolution of the density operator (in the input basis) which can be used to illustrate the delocalization and diffusion of a localized adsorbate on a solid surface in interaction with the continua of plane waves and gravitons.

IV. PARTICLE LOCALIZATION AND GRAVITATIONAL DECOHERENCE
The experimentally observed slow adsorbate surface diffusion in the quantum diffusion regime is associated with the decoherence and particle localization.Particle localization will hinder surface migration.Localization of a quantum particle on a solid surface with 2D periodicity is equivalent to classical behaviour.However, classical behaviour contradicts the expectation, based on quantum mechanics, for a fast decay of the localized particle wave packet [5].The appearance of quantum objects in interaction with their environment as classical particles implies decoherence.Decoherence can mean missing interference patterns and/or selection of preferred local states (pointer states), in which the particle is measured experimentally.Decoherence theory suggests the interaction with the environment as the physical reason for classical appearance of quantum objects [16].In a recent paper we proposed a collapse model for the decoherence of a quantum particle.The key features of the model refer to piecewise deterministic coherent time evolution of the local wave packet which is interrupted by non-demolition von Neuman-type measurements of the particle by gravitons.The collapse approach to decoherence via interaction with gravitons was successful to warrant adsorbate localization on a faster timescale, compared to diffusion, due to entanglement with solid excitations.Particle localization prior to diffusion is a necessary condition for the phonon-induced diffusion.
In the present article we revert, within a quantum mechanical model, to decoherence which is based on a completely coherent approach.The time evolution of the density matrix for an adsorbate entangled with gravitons in the spacetime curvature displays the salient features of decoherence.However, the classical gravitational interaction is very weak.At small distances superstring theory in its M-theory version as well as supergravity suggests a much stronger distance dependence of the gravitational interaction.According to these 11 dimensional space-time theories it is: G 11 is the gravitational constant in 11-dimensional spacetime and has to be determined by comparison with experiment, M 1 , M 2 : interacting masses, r: their separation.The gravitational law Eq.( 7) cannot be valid for large separations r as this would violate the experimentally verified classical law.Therefore the hidden dimensions (7 out of 11) are compactified to a small diameter 2a, so that at large distances the separation in the hidden dimensions never exceeds 2a.Equating the classical law and the 11dimensional gravitational law at large distances: yields: With this choice the classical law and Eq. ( 7) agree at separations significantly larger than 2a.Inserting Eq. ( 9) in Eq. ( 7) yields at r = 1 bohr: Parameters of the model.The parameter choice for the model is fixed.The conditions for the parameter choice are: • The gravitational interaction in 10D space should be chosen such that the classical gravitational law in 3D is not violated at the distances measured experimentally.
• The relationship between the DOS of gravitons and plane waves, which will be discussed in a separate contribution, should be satisfied.
• The time scale for diffusion of single adsorbates as provided by eleborate band structure calculations or by scattering theoretical approaches, should be reproduced in the limiting situation, neglecting the entanglement with gravitons.
• The splitting of the vibrational states of the adsorbate parallel to the surface should correspond to experimental data or should be chosen with the help of physically reasonable models, e.g.particle in a box, band structure calculation, etc.We used the value of 0.5 meV for the excitation energy of the adsorbate from the lowest parallel vibrational state.

V. ZENO EFFECT AND ANTI-ZENO EFFECT IN ADSORBATE SURFACE DIFFUSION, SURFACE DIFFUSION RATE ACCORDING TO STATIONARY STATE SCATTERING THEORY
The time dependence of a process in classical physics and chemistry, which is associated with the change of a measured quantity Q with time, is often expected to be exponential: with a time-independent rate constant γ.In particular, in the case of adsorbate surface diffusion, Q can be the number of particles on a unit surface area and the rate constant is just the diffusion coefficient.The exponential dependence of Q on time according to Eq. ( 11) may be approximately followed in a quantum mechanical process at intermediate time.However, the time dependence of a quantum mechanical process, e.g.particle localization or delocalization, at short time is with t 2 .This is so, because the localization of a particle scales with the probability P loc , which at small times varies with t 2 [24,25].
A simple derivation shows that this is indeed the case.
A localized adsorbed atom initially in the state | g at t o = 0 evolves with time under the action of the Hamiltonian H (assumed time-idependent) and the probability for its localization at time t equals: where k labels the decay channel.Therefore the initial probability for particle delocalization and diffusion is not proportional to t as it would follow from Eq. ( 11) at t → 0, but to t 2 as Eq.(12) shows.The first time domain, where the probability for delocalization is proportional to t, corresponds to particle diffusion with a constant time-independent diffusion rate.The time derivative of the probability for particle delocalization in the first linear time domain is equivalent to the diffusion rate, as it would have been evaluated within stationary state scattering theory using the generalized Ehrenfest theorem (GET).With a scattering state | g+ , which evolves from a localized initial state of the adsorbate | g , and decay channels with wave vectors k as final states, the delocalization rate, according to the GET, equals: for an energy-independent scattering potential V and constant density of states of the 2D plane waves ρ k .Examples of the behaviour in the different time domains will be shown in sections VI A and VI B. The deviation from the exponential time dependence of the delocalization rate, replaced by the t 2 dependence at http://www.sssj.org/ejssnt(J-Stage: http://www.jstage.jst.go.jp/browse/ejssnt/) e-Journal of Surface Science and Nanotechnology Volume 8 (2010) t → 0, leads to the quantum Zeno effect (QZE) [26].The essence of the QZE is that an unstable particle subject to continuous "measurement" will never decay, i.e. in our case the suppression of the unitary time evolution via the continuous observation prevents the delocalization of the adsorbate.The "permanent measurement" of the localized adsorbate by the gravitons is the decoherence mechanism leading to the suppression of its delocalization and slowing down of the diffusion.The probability for survival of the localized adsorbate at time t o after having been "measured" N times by the gravitons, amounts to: (14) which in the limit N → ∞ equals P loc = 1.This means that the adsorbate will remain frozen on its adsorption site for ever.The reader should observe that this does not happen for the time dependence Eq. ( 11), which would remain unchanged in a corresponding limit N → ∞.
Of course this is a limiting situation with an infinitely fast "measurement" of the adsorbate by its environment.In reality the entanglement with the environmental excitations occurs on a finite timescale, hence localization due to the QZE and environmental decoherence are well separated issues.However, as soon as in our model the timescale of interaction with environs is chosen very fast the QZE may predominate, distorting the physical picture of particle localization.
Another related issue is the quantum anti-Zeno effect (QAZE), which is an enhancement of the delocalization rate due to fast environmental measurement and localization, hence restoration of the quasi-classical exponential time-evolution [28,29].Depending on the frequency of the "measurement" by the environs a transition is possible between adsorbate delocalization which is suppressed or enhanced.Experimentally a transition from QZE to QAZE tunnelling of the Na atoms out of a potential trap has been observed with cold sodium atoms, trapped in far-detuned standing wave of light as a function of the frequency of the measurement [27].

A. Adsorbate diffusion: entanglement with plane waves
The interaction between the local basis core movement states for adsorbate motion parallel to the solid surface | g 0 and | w gives rise to the vibrational states: vibrational ground and first excited states.The entanglement of the vibrational states of the adsorbed particle parallel to the surface with the plane waves leads to their modification and energy broadening.An example is shown in Fig. 2 for a symmetric coupling between the ground and the excited vibrational states and the continuum of plane wave.In each of these states the particle will have a finite lifetime of the order of 0.08 seconds for this particular choice of the coupling to the plane waves.This is equivalent to the decay of approximately twelve particles per second.The particle oscillates (Rabi oscillations) between the two local states, the vibrational ground and the first excited state, with frequency of the order of 10 −11 seconds, corresponding to their energy difference of 5 • 10 8 peV (cf.Fig. 3 for a short time interval after t=0).The variation of the occupations of the two core movement states is plotted as a function of time with the dashed red and solid blue curves.The black curve represents the variation with time of the occupation of the delocalized plane waves.However, the entanglement with the plane waves is effective and can result into decay out of the adsorption site, hence into particle diffusion, only in the ground vibrational state.Each cycle of the Rabi oscillations brings the particle back in the vibrationally ground state, where it delocalized partially into the plane waves, the occupation of the plane waves being enhances at the cost of attenuation of the occupation of the local core movement state | g .The transfer of particle occupation from the local core movement states to the plane waves in this completely coherent process means particle diffusion.
At small times the delocalization increases with the square of time, as it is expected in a coherent time evolution at times t → 0, (Fig. 4).At later times within the lin-  no graviton entanglement graviton entanglement FIG.5: Time evolution of the occupation of delocalized plane waves: the adsorbate core movement state | g grav is entangled with the plane wave continuum and, via its warp resonance component, with the graviton continuum.The inset shows, on a longer time scale, a comparison of the time evolution of the plane wave occupation with and without the entanglement with the graviton continuum.
ergies corresponding to the ground and first vibrationally excited state of the adsorbate parallel to the substrate surface.With this procedure and with the present parameter choice the characteristic timescale for plane wave recurrence is long, of the order of 1.65 seconds (cf. the inset in Fig. 4, where the occupation of the plane waves with time is diplayed on a much longer timescale), and therefore it does not distort the coherent time evolution of the system at the smaller times of interest.

B. Adsorbate diffusion: entanglement with gravitons
The entanglement of the core movement of the adsorbate with gravitons within the warp resonance leads to slowing down of its delocalization in plane waves (cf. the inset in Fig. 5).For this choice of parameters the delocalization rate (inset, red curve) is nearly two times smaller than that for an adsorbed particle in interaction with plane waves only (inset, blue curve).The delocalization rate according to the GET, derived from the time derivative in the first linear time dependent region, is of the order of 7 particles per second, compared to 12 particles per second for the case when the entanglement with the plane wave continuum alone is taken into account.
The attenuation of the delocalization rate is due to the entanglement with the gravitons.The gravitational interaction is of short range and has a very strong distance dependence (cf.Eq. ( 7)).It is effective within the warp resonance alone and leads to redistribution of the particle occupation due to Rabi oscillations on a very short time scale of the order of 10 −11 seconds.These are Rabi oscillations in a more complicated system, the graviton continuum being also involved (cf. the upper panel of Fig. 6).
The gas particle core movement states (the ground | g grav and | g κ and the vibrationally excited states http://www.sssj.org/ejssnt(J-Stage: http://www.jstage.jst.go.jp/browse/ejssnt/) e-Journal of Surface Science and Nanotechnology 10 4 bohr and the graviton density of states it is easy to understand why the recurrence time of the gravitons is very short, of the order of 10 −11 seconds.Fast graviton recurrence from the hidden dimensions and entanglement with the local states leads to particle decoherence in the local states and stabilizes their occupation, however, each time at a lower occupation level because of delocalization in the plane waves during the previous time interval.The graviton decoherence is much faster than the coherent delocalization in plane waves.The overall result is slowing down and even flattenning of the variations with time of the occupation of the plane waves, hence in particle localization.The localization induced by entanglement to the gravitons hinders the surface diffusion.

C. Graviton assisted diffusion: limiting cases
We consider here the limiting cases that the system collapses (i.e., decoheres completely) in the system states | g grav and/or | w grav .We assume that the adsorbed particle (atom or molecule) is decohered in the state | g on each lattice site (cf.Fig. 8).Decoherence occurs, if the gas particle resides in the warp resonance | w where it is entangled with the gravitons.After emitting and/or absorbing a graviton, it is projected either onto the local state | g on the same site (localization) or on a local state on a neighbouring site (diffusion).In terms of decoherence theory these states represent pointer states, in which the particle resides as long as it is measured in experiment.
In the warp resonance the particle undergoes two consecutive "measurements" by the graviton, which implies that the warp resonance is a pointer state as well, in which the adparticle is transiently decohered.The question of whether or not the particle could in principle be experimentally observed in the warp resonance will not be addressed at the moment.Let us presently assume that it is just very difficult to observing a molecule in a transition state.
We can use the generalized Ehrenfest theorem to calculate transition rates between the pointer states.The relevant rates are illustrated in Fig. 8, the equation for the diffusion rate can be written as: ) One has to emphasize that the rate R tunnel might (but need not) be very different from the rate R loc tunnel .The latter is the usual rate for tunneling through a static barrier as it is very often estimated in the WKBapproximation.However, this estimate is unphysical because of the deficiencies of the WKB model: (i) WKB treats one-dimensional tunnelling and (ii) WKB assumes scattering states for the tunnelling particle, but experimentally the particle is obviously localized before diffusion.R tunnel , in contrast, describes particle tunnelling, starting from an elevated part of the potential energy surface.It could be much larger than the static tunneling rate.This has, of course, to be investigated by elaborate calculations.At present we shall examine diffusion rates assuming different values for R tunnel and R loc tunnel .The magnitude of R loc tunnel will depend on the importance of the graviton channel.If the particle is completely in the warp resonance, then the static tunnelling rate will tend to zero, as it is obvious from Eq. (15).
For R grav and R ref lec we might with some confidence assume the values: R grav = 10 10 sec −1 (16) In reality they may deviate by one or two orders of magnitude.The variation of the values of R grav may be even stronger, e.g. by factor 10 4 , because it depends on the product of the adparticle mass and the mass of the metal atom and on the static potential for adparticle core movement in the proximity of the solid surface.R grav for a Xe atom has been estimated to be of the order of 10 11 s −1 [5].As a first estimate R ref lec may be chosen as some fraction of the quantum diffusion rate of a particle in a harmonic potential well.Consider the following limiting cases: 1.The rate of adsorbate diffusion is dominated by the rate of gravitational decoherence in the warp resonance channel.
Then with the choice eqs.( 16) and ( 17) R dif f = R grav ≈ 10 10 sec −1 .This is the fastest possible diffusion rate.It is independent of the properties of the static barrier and is definitely dependent on the mass of the gas particle.In this case the diffusion is dominated by the penetration of the adparticle in the transient warp resonance.In the warp resonance the particle is in the repulsive branch of the potential energy surface, higher excited states (delocalized over a larger region on the surface) are virtually mixed in in the wave function and delocalization over neighbouring lattice sites is facilitated.The consequences for quantum diffusion are: • Since R grav increases only with the square of the adparticle mass, there will be negligible isotope effect compared to the exponential dependence of R loc tunnel on the square root of the mass in the WKB method and also much smaller than the value 10 5 due to the theory of Wahström et al. [4].In this case even inverse isotope effect may result.The strongly suppressed isotope effect in the diffusion of hydrogen isotopes on W(110), Ni(100) and Ni(111) [15,18] may be understood as due to the domination of the decoherence in the warp resonance R grav over the process of direct tunnelling (R loc tunnel ), which is the only diffusion channel in the theory of Wahnström et al.
• If the surface diffusion of hydrogen is dominated by the rate of decoherence in the warp resonance and by the gravitational interaction with the metal atoms, then, based on our result showing that hydrogen in its equilibrium adsorption site in a static potential is situated 0.2 bohr further away from the Cu(110) surface than on the Ni(110) surface [30], a more pronounced isotope effect is expected for hydrogen diffusion on Ni(110) than on Cu(110).This prediction has to proved by experiment and further calculations.
Obviously this limit is not applicable to hydrogen diffusion on the Cu(100) surface, where a very strong isotope effect is observed in the STM at temperature below 65 K [17].
In this case we have a diffusion process dominated by tunnelling through a static barrier with rate R tunnel .The large isotope effect from the WKBlike behaviour, if applicable to R tunnel , will survive, though the magnitude of R dif f might be orders of magnitude smaller than that calculated for static tunnelling R loc tunnel .This limit provides the understanding of hydrogen quantum diffusion on the Cu(100) surface, where a strong isotope effect is observed and very small values for the quantum diffusion rate, compared to a similar system H/Ni(100), have been measured [17].As we pointed out at the end of point 1, even the small elongation of the H-Cu bond compared to the H-Ni bond is sufficient to subdue the effect of the gravitational entanglement in the warp resonance and to promote the effect of the tunnelling channel, which is associated with a pronounced isotope effect.
3. Pure tunnelling through a static barrier is recovered when R grav = 0.In this case The adparticle is assumed to be localized before diffusion in this situation as well.This is in fact the assumption in most diffusion theories (cf.ref. [4]).However, the neglect of the diffusion channel via the warp resonance does not allow to reproduce the experimental observations.
Based on the discussion of the limiting cases, we may speculate on the very small tunnelling rate of a CO molecule out of the center of a CO-trimer on Cu(111) (measured in a single molecule tracking in the STM at T < 6 K) and the missing tunnelling of an isolated CO molecule [12].A CO molecule in the confined potential due to two CO molecules on neighbouring adsorption sites will experience an increase of the zero point energy.Hence its attempt frequency for tunnelling out of the potential well will increase compared to the attempt frequencies of the outmost CO molecules in the trimer.In contrast, there is no reason for an increase of the attempt frequency for tunnelling of a single CO molecule out of its potential well on the metal surface.A very low tunnelling attempt frequency of the order of 10 5.5±0.5 s −1 has been measured in quantum diffusion of CO out of the trimer, which in the context of our theory indicates a diffusion process dominated by tunnelling through the barrier and not by decoherence in the warp resonance.It is difficult to speculate on the isotope effect, estimated within the WKB-theory of 1D tunnelling by Eigler et al. [12], showing that the tunnelling mass is that of the carbon atom and not of the whole CO molecule.

VII. CONCLUSION
We have two points of view on the decoherence and localization of adsorbates which are (possibly) qualitatively equivalent: • The collapse due to coupling to gravitons is a quantum Zeno process.If collapse is allowed each time when the gravitons come back from the hidden dimensions and interact with the particle within the warp resonance leading to particle localization, the localization rate will be of the same order of magnitude as in the coherent approach.
• The coherent approach, based on the unitary time evolution of the local system entangled with the graviton environment, results in particle localization due to the recurrence of gravitons from the hidden dimensions.In the spirit of a collapse model this would be interpreted as "permanent measurement" by the gravitons with the rate of the recurrence.The recurrence time depends upon the graviton DOS, the size and the number of the hidden dimensions and the coupling to the gas atom.
This result shows how decoherence can be described within a purely quantum mechanical world model where phase information is temporarily stored in parts of the environment which in principle are not accessible to experiment.Decoherence leading to a slow down of quantum mechanical motion can be described within conventional quantum field theory.We do not need to modify Schrödinger's time dependent equation and introduce terms, as it is done for instance in stochastic approaches.

FIG. 1 :
FIG. 1: Experimental frequency factors for adsorbate surface diffusion on metal surfaces in the quantum tunnelling regime as a function of the activation barrier in the thermal diffusion regime.(References are provided below the figure.)The diffusion rate equals the diffusivity Do in the quantum diffusion regime; the frequency factor νo is calculated with the help of equation Do = 1 2d l 2 νo, where d = 1, 2 for diffusion in one-and two dimensions and l: the jump distance, assumed equal to an average separation between neighbouring adsorption sites of 3 bohr.

FIG. 6 :FIG. 7 :
FIG.6: Rabi oscillations of the gas particle between the basis state | g0 and the warp resonance | w , with the entanglement to the graviton continuum accounted for (lower panel).The oscillations of the graviton occupation are displayed in the upper panel for the same time interval.

TABLE I :
Localized and delocalized basis for different particles in the world model.
Table I).