Theory of Decoherence at Solid Surfaces
G. Doyen1 and D. Drakova2
1 Ludwig-Maximilians
2 Faculty
Universität, München, Germany
of Chemistry, University of Sofia, Bulgaria
Abstract
Decoherence processes at solid surfaces are observed at all time scales. The most common surface processes are
classified according to the presently common view on decoherence theory. Prominent examples of decoherent
surface processes are electronic relaxation and deexcitation, vibrational relaxation, diffusion, inelastic scattering, sticking, STM-induced chemical reactions and desorption, localization of adsorbates. Various mechanisms,
suggested at the present state of the art of decoherence theory, are investigated for their ability of providing
the understanding of decoherence at solid surfaces. In some cases environmental decoherence by coupling to
phonons and electron-hole pairs in the surface is a viable mechanism. Some new ideas are introduced, which
have not been discussed in the framework of decoherence theory so far.
1
Introduction
Decoherence has been used to imply a classical behaviour displayed by a quantum system in interaction with
its environment. Though it might not have been pointed out explicitly, many experimental observations on surfaces involve decoherence. We speak of decoherence, if the time development of a local system does not obey
Schrödinger’s equation as set up for the local system. Decoherence can mean missing interference patterns and/or
selection of preferred basis states (e.g. localization). The entanglement of the system with the environment is the
key feature of decoherence theory and it is utilized to explain the suppressed interference within the local system
and its description with a selected preferred basis state (so called pointer state).
Pointer states are the states, in which a particle in a local system is observed with the help of a large class of
measuring techniques. They are specified for a given local system in interaction with its environment, without any
influence from the measurement process. Pointer states are the result of this interaction. Decoherence theory is
summoned to define how they are generated. It is known, however, what their properties are. Though the pointer
states are transient states, they are robust, i.e., they should exist for the time the experimental measurement is
carried out. Some examples of pointer states used for several specific observations on adsorbate systems include:
• Localized wave packets, in which single atoms and molecules are observed in the STM (cf. Xe adsorbed on
Ni(110) [1]).
• In weak inelastic scattering of noble gas atomic beams from solid surfaces the pointer states are the local
resonances with the transient local deformation of the substrate surface due to the short-time residence of
the projectile close to it (so called deformation resonances).
• In elastic scattering of He beams the pointer states are just the diffraction states, eigenfunctions of the static
surface potential of the local system.
• In the current-induced desorption of CO from Cu(111) (low temperature STM experiment by Bartels et al.
[2]) the pointer state is a product state between the localized wave packet for the CO core movement and a
wave packet for the tunnelling electron in a negative ion resonance on the copper surface.
• In thermal desorption the pointer state is the deformation resonance.
Decoherence occurs at all time scales. A collection of typical examples of processes at solid surfaces in table
1 illustrates that decoherent surface processes can be found on any timescale. In order to treat decoherence at
surfaces, we have to define within a world model: (i) the system; (ii) the environment; (iii) the interaction between
1
time
scale [s]
10−18
10−18
−17
10
10−17
10−16
10−16
10
−15
10
−14
10
−14
−13
10
10−12
10−12
10−12
10−12
10−11
10−11
10
−11
10−10
10−10
10
−10
10
−7
10
−5
10
0
100
∞
phenomenon
X-ray diffraction
adsorbate core
state spectr.
LEED
adsorbate
valence st. spectr.
resonance ioniz.
+Auger neutr.
INS
resonance neutral.
surface state
linewidths
thermal
desorption
sticking at
374 K
2PPE
Penning
transitions
He diffraction
molecular
vibrations
vibrational
linewidths
frustrated
molecular
rotations
Kr scattering
sticking at
cold surfaces
positron
lifetime
adsorbate
localization
transition
state complex
surface
diffusion
quantum
diffusion
oscillatory
surface
reactions
Eigler
cascades
tunnelling
adsorbate
induced surface
reconstruction
quantum
reflection
typical
example
O/Rh(111)
coherence or
type of decoherence
coherent
coherent
decoherence
mechanism
fcc(111) Ag,Cu,Au
CO/Ru(0001)
classical
electron
capture
surface state
internal Auger
phonons
CO/Pd(111)
CO/Rh(111)
Cs/Cu(111)
He*/CO/Pd
classical
phonons
He+ /Ni,Cu,Ge
He/LiF
C2 H2 /Cu(100)
CO/Cu(001), CO/Cu(110)
CO/Pd(100)
dim. effect
electron emission
electron emission
coherent
classical
statistics
coherent
coherent
dim. effect
electron emission
decoherent
C2 H2 /Cu(100)
coherent
Kr/LiF
decoherent
He accomodation
classical
Xe/Ni(110)
classical
statistics
classical
STM-induced
CO desorption
Xe/Ni(100)
decoherent
H/W(110)
coherent
CO/Pt(110)
decoherent
CO cascades
on Cu(111)
coherent
H/Ni(110)
decoherent
none
coherent
dephasing
phonons,
gravitons
dim. effect,
phonons
phonons,
electon gas
gravitons
gravitons,
phonons
gravitons,
phonons
classical
experimental
group (leader)
year
Baraldi et al. [3]
2003
electron gas
Davisson/Germer
Küppers, Ertl [4]
Doyen et al. [5]
Ertl et al. [6]
1927
1978
1992
1987
electron gas
Hagstrum [7]
1966
2D electron gas
Echenique et al. [8]
Berndt et al. [9]
Menzel et al. [10]
2006
2000
2005
Engel [11]
Beutl et al. [12]
Petek [13]
Ertl et al.
[14, 15]
Boato et al. [16]
Ho et al. [17]
[18]
Bradshow et al. [19]
1978
1999
2000
1979
1982
1976
1998
1999
1978
Ho et al. [20]
1998
phonons,
gravitons
phonons
Palmer et al. [21]
1971
Goodman et al. [22]
1967
phonons,
electron gas
hidden
dimensions
hidden
dimensions
solid
surface
solid
surface
solid
surface
Lahtinen et al. [23]
1991
Eigler et al. [1]
1990
Ertl et al. [2]
1998
Zhu et al. [24]
2000
Gomer et al. [25]
1985
Ertl et al. [26]
1986
solid
surface
Eigler et al. [27]
2002
solid
surface
vibrations
solid
surface
Ertl et al. [28]
1984
solid
solid
coherent
classical
statistics
classical
statistics
classical
statistics
decoherent
O on gr. VIII metal surfaces
CO/NiAl(110)
He*/Pd,Cu,W
environment
phonons,
electrons,
gravitons
phonons
solid surf.
electron gas
solid
surface
solid
surface
electron gas
phonons,
electron gas
solid surface
phonons
electrons,
phonons
phonons
Table 1: Coherent and decoherent phenomena on solid surfaces and their time scales
system and environment. The decoherence mechanism and the timescale will result from a solution of the world
model.
It is well known that solids on their own exhibit classical features (e.g. localization of atoms, symmetry of crystals,
thermal and electric conduction, thermodynamic laws, etc.). In the past, however, the main theoretical interest has
been to uncover the quantum nature of solids which led to phonons, plasmons and other collective excitations.
How does the classical nature of a solid arise from decoherence of quantum atoms has not been an urgent issue
in solid state theory. A similar attitude among surface scientists, especially practitioners (experimentalists and ”ab
initio” theoretical engineers), is prevailing, concerning the movement of gas particles near sufaces. As a prominent
example take the appodictic (and unjustified) statement by Stampfl et al. [29]: "For atoms heavier than hydrogen,
the nuclear motion is indeed classical."
Concerning the theory of surface science this complete neglect of the decoherence concept has, however, led to
controversies sometimes resulting in absurd expectations like the one that at extremely cold surfaces all classical
behaviour will suddenly change to obey quantum predictions [30]. Many experimental results in surface science
have been accumulated over the last decades, which remained completely unexplained by the standard quantum
theory of surface science: unexpected peaks in UPS termed ”satellites”, localization of adsorbed particles, sticking
at cold surfaces, survival of excited atoms [31], missing diffraction peaks for heavy rare gas atoms [21], angular
distributions of scattered (metastable) gas particles [32], thresholds for STM induced reactions [2], etc., etc. It
2
appears now essential for a deeper understanding of surfaces science to investigate decoherence effects in order to
clarify their impact on the mentioned unresolved issues.
The organization of the paper is a follows. In section 2 the formal foundations of decoherence theory are summarized. The next section describes Quantum Nano Dynamics which is used to construct model worlds. Section 4
illustrates this approach to decoherence at surfaces by studying electronic and phononic decoherence for CO on
Cu(100). Afterwards the difficult problem of localization of adsorbed particles on surfaces is addressed.
2
2.1
Formal Theory of Decoherence
Reduced Density Matrices
In the framework of decoherence theory [33] it is supposed that any closed quantum world can be described by a
Hilbert space Hworld :
Hworld = Hsystem ⊗ Henviron
(1)
A full description of the quantum world is given by the pure density operator ρ =| ΨihΨ | in terms of a single
wavefunction | Ψi. This density operator evolves according to Schrödinger’s equation. Decoherence then arises as
a consequence of the assumption that operators, tied to the environmental degrees of freedom, are unobservable,
i.e., operators which correspond to observables take the form A ⊗ I where A acts on Hsystem and I is the identity
operator on Henviron . The expectation value of any such observable in a pure quantum world state will be given
by
hAi = tr(ρsystem A)
(2)
where the reduced density matrix ρsystem is the partial trace of ρ over Henviron .
A potential pointer state in Hworld would now be described by a a direct product of a system part | f i and an
environmental part | γi:
| Ψpp i =| f i⊗ | γi.
(3)
The system part of a pointer state | f i is assumed to be ”robust”, i.e., not to change significantly due to interaction
with the environment. There is no demolition of the system part of a pointer state due to scattering of environmental
particles (”environs”) off the system. Examples of environs are plasmons [34], solvent polarizations [35], electronhole pairs [36, 37, 38, 39], phonons. Their combined effects have to be considered as well.
Of course, not every state of the kind | Ψpp i will be a pointer state, an additional necessary (not sufficient)
requirement being that the environment parts of the pointer states {| γi i} will become mutually orthogonal. In the
following the system part of a pointer state will sometimes be referred to just as a pointer state, if no confusion is
to be expected.
According to Joos and Zeh [40] permanent scattering of environs from the system will lead to the required orthogonality. This environmental decoherence, i.e. permanent ”measurement” by the environment, is then assumed as
the mechanism of localization of atoms and molecules at surfaces. In these calculations simplifying assumptions,
such as von Neumann non-demolition scattering, are often introduced.
2.2
Generalized Ehrenfest Theorem
The formalism of the reduced density matrices ρsystem can directly be related to formal scattering theory. If in eq.
(2) the sums are explicitly displayed one obtains with | Ψi =| i+i⊗ | γi i:
X
(4)
hAi = tr(ρsystem A) =
hg+ | i+ihi+ | A | g+ihγl | γi ihγi | γl i
g,l
3
Choosing for A
A=
2π X
V | f ihf | V δ(Ef − Ei )
h̄
(5)
f
and taking into account the mutual orthogonality of the scattering states {| i+i}, the expectation value of A is
written as:
2π X
hAi =
(6)
| hf | V | i+i |2 | hγl | γi i |2 δ(Ef − Ei )
h̄
l,f
This is just the generalized Ehrenfest Theorem [41] which is the exact expression for scattering rates.
Now assume that the world state is a superposition of potential pointer states:
X
| Ψi =
ci | i+i⊗ | γi i
(7)
i
This yields for the expectation value of A:
hAi =
2π X
ci c∗j hj+ | V | f ihf | V | i+ihγl | γi ihγj | γl iδ(Ef − Ei )
h̄
(8)
l,i,j,f
Clearly, if the environmental states {| γl i} are mutually orthogonal, one obtains:
hAi =
2π X
| ci |2 | hf | V | i+i |2 δ(Ef − Ei ).
h̄
(9)
i,f
The result is a stochastic sum over initial pointer states, weighted by the modulus squared of the coefficients.
Summarizing, we see that in the presence of decoherence, scattering can only occur out of pointer states. In the
same way we find that in the presence of decoherence scattering out of arbitrary initial states can only occur into
pointer states.
Equation (8) is the basis of our decoherence investigations in sections 4 and 5.1. The matrix elements hf | V | ii,
hj | V | f i, hγl | γf i, etc. are evaluated within a world model defining a separation between the system and
the environment. This means that the mentioned matrix elements are evaluated on purely physical grounds within
the world model without assuming any decoherence mechanism or decoherence process a priori. If then, after
inserting the evaluated matrix elements, eq. (8) reduces (approximately) to eq. (9), we know that our model world
will decohere and our theory has revealed the decoherence mechanism at work.
2.3
Time Dependent Decoherence
Measurements on local systems are often performed in such a way that probing particles are scattered from the
local system. This might be different from the scattering discussed in the previous subsection, where scattering
processes within a stationary world were studied.
A different procedure is to calculate only the time dependence of the finite local system and to treat the interaction
with the environment as scattering of environs from the local system. The result of the environ scattering is then
projection on the final states as they occur in the generalized Ehrenfest theorem. This is a kind of collapse and the
time evolution of the system occurs along a piecewise deterministic path. Between two scattering events the system
develops unitary according to Schrödinger’s equation and at a scattering event the system changes instantaneously
to a particluar state. If the scattering events occur sufficiently rapidly the time development of the system can
deviate appreciably from the unitary predictions (concerning the system alone). The deviation is then interpreted
as decoherence. This approach will be used below in the case of adsorbate localization at solid surfaces.
4
3
Quantum Nano Dynamics (QND)
The theoretical approch consists of building simple, but non-trivial, models of a world composed of a local system
under investigation and an environment [42]:
Hworld = Hsystem + Henviron + Hsys−env
(10)
Hsystem is further decomposed into
Hsystem = Hsurf ace + Hvacuum + Vsurf −vac
(11)
The environment comprises elementary excitations in the bulk of the solid like electrons and electron-hole pairs,
phonons, plasmons, photons, gravitons, etc.:
Henviron = Helectron + Hphonon + Hplasmon + Hphoton + Hgraviton + ...
(12)
The local system polarizes the environment and these polarizations act back on the local system. The selfconsistent static part of this back reaction is included in Hsurf ace .
For the case that the environment can be completely bosonized the environmental hamiltonian can with this notation formally be written as:
X
X
J † J
(13)
Henviron =
ωQ
ξJQ bQQ bQQ
Q
JQ
where the ξJQ are possible polarization tensors, JQ are quantum numbers used to label the different environs. It
is not always possible to bosonize the electronic degrees of freedom. In the next section an example, where the
electronic degrees of freedom cannot be bozonized, is discussed.
4
Example of Decoherence by Electronic and Phononic Degrees of Freedom in
the Environment: CO scattering from Cu(100)
The surface part of the system hamiltonian is written in the form:
Hsurf ace = Hel + Hvib + Tcore + Wcore−core +
X hBQ i
Q
ωQ
(hBQ i − 2BQ )
(14)
Here the electronic part Hel involves the CO molecule and the electron states on the surface. BQ describes the
displacement of the local environs (phonons and plasmons) of frequency ωQ . The environmental hamiltonian eq.
(13) contains the plasmons and the phonons. The decoupled electrons and plasmons of the metal surface are not
explicitely included in the hamiltonians Hel and Henviron . The interaction between the system and the environment
contains in addition to the boson coupling a coupling between the metal electrons and those on the CO-molecule:
X
X JQ †
XX
J
s
Hsys−env =
(BQ − hBQ i)
(bQ + bQQ ) +
(| λ0 ihλ0 | VAk
(z) | λihλ | c†As cks + herm.conj.)
Q
JQ
λ0 ,λ A,k,s
(15)
s (z) = hAs | W
with VAk
el−el | ksi.
Evaluating the transition matrix elements needed in eq. (8) for scattering of environs (electrons and phonons) from
the system one obtains:
sys
hγf | ⊗hf | Hsys−env | i+i⊗ | γi i = hγf | γi ihf | Venv
| i+i
q
X X
1
d
=
− ihBQ
i hnQ i + 1) | dihd | i+i
hQf | Qd ihSf | Sd ihf | ΓGAs (
πρel
A,s,Q d
5
(16)
2
1
0
-1
distance (Å)
Figure 1: Potential energy curve (full curve) and the space part of the deformation resonance (dashed curve) for
CO/Cu(100).
sys
Venv
is the part of the system - environment interaction acting in the system alone. ΓGAs is a quantity derived
from the electronic gas particle Green’s function, which is responsible for electron-hole pair excitation in the
environment [39]. ρel is the density of metal electron states. hQf | Qd ihSf | Sd i is the overlap between the
perturbed (by the gas particle) and unperturbed gas particle projected environ states, hQf | Qd i being the environ
mode overlap and hSf | Sd i the overlap of the Slater determinants of the metal electrons. In the case of a single
deformation resonance | di and a single gas particle projected environ mode Q, we have:
hγf | γi i = hQf | Qd ihSf | Sd i
(17)
In eq. (8) hf | V | i+ihγl | γi i is replaced by eq. (16) and hγj | γl ihj+ | V | f i by its conjugate complex (here
now, of course, replacing j for i in P
eq. (16)). When doing so, we replace l by f . Then hj+ | di still appears in eq.
(8). But using hj+ | i+i = δij = d hj+ | dihd | i+i ≈ hj+ | dihd | i+i allows us to substitute | i+i for | j+i.
This means that eq. (8) reduces to
hAi =
2π X
sys
| ci |2 | hd | i+i |2 | hf | Venv
| di |2 | hQf | Qd ihSf | Sd i |2 δ(Ef − Ei ).
h̄
(18)
i,f
which is obviously of the form of eq. (9). Hence scattering into any final state occurs only out of the deformation resonance. Our solution of the world model has resulted into decoherence and in this case the deformation
resonance is therefore the pointer state.
The decoherence mechanism is the interaction between system and environment, which occurs only in the deformation resonance. While in the deformation resonance, phonons and electron-hole pairs are created and are
radiated into the interior of the environment. This entangles the deformation resonance with the environment and
information about the deformation resonance (the pointer state) is stored in the environment.
The potential energy curve and the space part of the deformation resonance are displayed in figure 1. The results
of the totally quantum mechanical solution of our model world exhibit in fig. 2 a scattering behaviour which could
qualitatively be explained by a classical hard cube model [43]. The decoherence theory rationalizes this through
6
intensity (arb. units)
270 meV
70 meV
135 meV
20
40
scattering angle
60
Figure 2: Angular distributions of CO scattered from Cu(100) at substrate temperature Ts = 0 K. The three curves
refer to three different values of the incoming energy Ekin (CO) normal to the surface: 70 meV (full curve), 135
meV (short dashes) and 270 meV (long dashes). The incoming angle is 45o with respect to the copper surface.
the fact that the gas particle decoheres transiently in the deformation resonance. The further evolution occurs as if
the gas particle would have been prepared in this state. This means that previous quantum effects like resonances
above the attractive adsorption well or diffraction have been ”forgotten”.
The sticking coefficient displays also a classical behaviour, increasing to unity for low incoming kinetic energy of
the CO molecule (cf. fig. 3). This behaviour can also be traced back to the fact that the deformation resonance is the
origin of any further gas particle evolution. For any kind of scattering the gas particle has to be trapped transiently
in the deformation resonance, for sticking as well. If the incoming energy of the gas particle is sufficiently low
and the surface cold, the gas particle will necessarily lose so much of its kinetic energy that escape is impossible.
Technical details of the calculation leading to this conclusion can be found in the literature [44].
5
Adsorbate Localization
The observed fact that an adsorbed particle localizes on a particular adsorption site is difficult to understand within
decoherence theory. Though there are examples of quantum motion of adsorbates on solid surfaces, for instance
quantum diffusion of hydrogen atoms on Ni, Cu and W surfaces [45]-[48] and tunnelling of CO molecules in the
low-temperature STM [27], a lot of surface phenomena display adsorbate localization. Adsorbate diffusion on solid
surfaces is very often of stochastic character, implying that the particle localizes before diffusing. An interesting
manifestation of this kind of behaviour is the theoretical explanation of the transfer of a Xe-atom between a tip
and a metal surface as observed by Eigler et al. [49]. The theoretical explanation [50] is based purely on a series
of consecutive stochastic processes.
The time evolution of a free wave packet, localized initially over 5 Å simulating the space extension of a Xe atom
on Ni(110) (as it is seen in the STM image of Eigler et al. [1]) and energy in the range 5-7 meV, according to the
7
1.0
total
sticking coefficient
0.8
0.6
v (1 <- 1)
0.4
v (0 <- 1)
0.2
0
50
100
150
incoming energy (meV)
200
250
Figure 3: Variations of the sticking coefficient of CO on Cu(100) as a function of the incoming kinetic energy of
the molecule (full curve) at Ts = 0 K. The two contributions to the sticking coefficient are due to: (i) vibrationally
elastic scattering (long-dashed curve), i.e. the scattered CO is in the first excited vibrational state v(1 ← 1) as the
incoming molecule and (ii) vibrational relaxation of the CO molecule v(0 ← 1) in the final state (short-dashed
curve).
time-dependent Schrödinger equation shows that within 2.4 × 10−10 seconds it delocalizes over a region larger
than 40 Å (fig. 4). In contrast to this result, in the low temperature STM experiments of Eigler et al. Xe is localized
for days on end and it is localized at the same place on the metal surface.
Possible decoherence mechanisms to be discussed are via interaction of the adparticle core movement with substrate phonon modes and electron-hole pairs. The classical diffusion of adparticles on solid surfaces is due to
energy exchange with the same surface excitations. The adsorbed particle exchanges energy with the surface
bosons to escape out of or to get trapped into the adsorption potential well. We compare the exparimental data on
Xe diffusion on Ni(100) at 30K with theoretical data to show that despite the fact that the coupling to the phonons
can provide the explanation of the experimental diffusion coefficient, it is too weak to explain the localization of
Xe by means of coupling to phonons and e-h-pairs. There must be a different decoherence 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.
5.1
Scattering theory of adsorbate diffusion
The diffusion of an adsorbate can be treated as in section 4 as a scattering process of the substrate phonons and
e-h-pairs from the adparticle core movement state in the deformation resonance, which is transiently created when
the substrate surface is deformed due to the interaction with the adsorbing particle. The emission or absorption
of phonons in the deformation resonance is accompanied by a change of the adsorbate core movement state,
which may result in localization of the adsorbate on a different adsorption site, i.e. decoherence in a different
localized adsorption state. The rate of adsorbate diffusion can be evaluated as the rate of phonon scattering in the
deformation resonance, taking into account the probability for the adparticle to be in the deformation resonance
and the probability for projection onto a different localized adsorption state out of the deformation resonance. This
picture is in agreement with the conclusion in section 2.2, namely that in the presence of decoherence, scattering
8
0.000 10-11 s
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
-20
-10
0
distance [a.u.]
10
20
0.3
0
10
20
0.05
0.1
s
-10
0.10
0.2
4.836 10
-20
0.15
0.4
-11
2.418 10-11 s
0.0
0.00
-0.1
-0.05
-0.2
2.418 10-10 s
-0.10
-0.3
-0.15
-0.4
-20
-10
0
10
20
-20
-10
0
10
20
Figure 4: Time evolution of a wave packet (energy range 5-7 meV, space extension of approximately 5 Å) simulating a free single xenon atom. Snapshots of the wave packet at times [s] indicated on the left and right handsides
of the panels.
can occur only out of pointer states, the deformation resonance being the pointer state in the case of adsorbate
surface diffusion.
Absolute rate theory predicts values of the order of 102 − 10−4 cm2 s−1 for the diffusivity Do in adsorbate surface
diffusion [51]. The value of the diffusivity for Xe diffusion on Ni(111), provided by Nabighian and Zhu [24], is of
the order of Do = 2 × 10−9±0.2 cm2 s−1 at 30K and low xenon coverage of Θ = 0.04, unusually small compared to
the values due to absolute rate theory. A commonly used relation between the frequency factor ν and the diffusivity
Do = νl2 /2 (l: the nearest neighbour distance in the substrate surface) yields in the case of Xe/Ni(111) the value
of ν = 7 × 106 s−1 , which means that the characteristic interaction time with the phonons of the substrate surface
is of the order of 10−7 s. The theoretical QND frequency factor, estimated with the help of eqn. (??) via the
energetically lowest deformation resonance, is of the order of ν = 6 × 106 s−1 , which compares favourably to the
experimental data by Nabighian and Zhu [24].
The experimentally measured rate of Xe diffusion is reproduced within the QND theory as due to coupling between
the Xe core movement state and the metal phonon modes. However, the unusually small frequency factors, both
from experiment and from the QND theory, mean that the diffusion rate at low temperature is too slow compared to
the rate of coherent time evolution of the Xe wave packet. In 10−7 s, needed for a single interaction event between
Xe and a surface phonon, the Xe wavepacket will be delocalized over many lattice constants. Even if it is localized
again at a later time via decoherence with the phonons, the probability that this will be on the same adsorption site
as before is negligible. Therefore we have to discard decoherence via permanent measurement with the phonons
as a viable localization mechanism.
The localization has to be due to a very short range interaction with the adsorbate. Low energy phonon modes
and tomonagons have wavelengths much too long to be able to localize wave packets of the dimensions of several
Angströms, typical for single adatoms seen in STM images.
9
5.2
Scattering theory of gravitational decoherence
5.2.1 Gravitational interaction in 11-dimensional space-time
The decoherence and localization mechanism we suggest is the interaction with the gravitons. The gravitational
interaction in 4D space is very weak compared to the electromagnetic interaction. However, as suggested by
string theory, increasing the number of dimensions increases the gravitational interaction at small distances and
makes it of short range because of the r−8 power law (cf. eq. (20) below). The higher hidden dimensions of
11-dimensional space-time have to be compactified, in order that the classical law of gravitaty will not be violated
at distances where it has been proved to be valid.
In 4-dimensional space-time the gravitational interaction between two point masses M1 and M2 at distance r is
according to Newton’s gravitational law:
(4)
Vgrav
(r) = −G
M1 M2
r
(19)
where G is Newton’s gravitational constant 6.67−11 [N m2 /kg 2 ]. In 11-dimensional space-time (10 space dimensions) the gravitational interaction is:
(11)
Vgrav
(r) = −G(11)
M1 M2
π 7 r8
(20)
G(11) is the gravitational constant in 11-dimensional space-time and has to be determined by comparison with
experiment. The gravitational law eq. (20) cannot be valid for large separations r as this would violate the
experimentally verified classical law eq. (19). Therefore the hidden dimensions (7 out of 11) are wrapped up 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 11-dimensional gravitational law at large distances yields:
G(11) = (2aπ)7 G
(21)
Assuming that at r = 1 bohr the gravitational interaction energy between two electrons equals 10−9 Hartree
implies that the gravitational interaction energy between a Xe atom and a Ni atom at 1 bohr is ≈ 10 Hartree. At
a distance of 6 bohr the gravitational interaction between Xe and Ni reduces to 10−5 Hartree. Using Heisenberg’s
uncertainty relation ∆E∆t = h̄, this corresponds to a time t ≈ 10−11 s, which is the order of the decoherence
time we need to explain a lot of experiments. To have interaction energy 10−9 Hartree at 1 bohr we need an
enhancement of G in eq. (21) of 1030 , i.e., a ≈ 10000 bohr. If this is correct a deviation from Newton’s classical
gravitational law will be found at distances smaller than 10−3 mm. The best experiment we know of up to now
verified that at 100 nm the deviation from Newton’s classical gravitational law are smaller than a factor of 109
[52]. With our rough estimates we predict at 100 nm a deviation by a factor 107 .
5.2.2 Warp resonances, gravitons
In analogy with the transient deformation resonances (a transient local deformation of the metal surface, in which
the adsorbate-metal surface interaction leads to creation and/or annihilation of phonons and e-h-pairs) we introduce the warp resonance. It is a transient resonance state in which the adsorbate and substrate atoms are so close
to each other, that the hidden compactified dimensions contribute to the gravitational interaction (11 space-time
dimensional interaction), which is mediated by gravitons. Gravitons, emitted in 10-dimensional space, transfer the
information on the state of the local system into the hidden dimensions of the environment. This is the essence
of a measurement by the environment, it is a decoherence process via interaction with the gravitons. In the warp
resonance the gravitational interaction increases according to eq. (20), therefore it is localized and can lead to
adsorbate localization via emission or absorption of gravitons.
10
5.2.3 Gravitational decoherence
In the framework of scattering theory the rate of localization of an adparticle via interaction with gravitons is
determined by the rate of graviton emission or absorption in the transient warp resonance times the probability that
the adparticle is in the warp resonance. The adparticle is localized on its equilibrium adsorption site both before
and after the scattering process.
The model world hamiltonian is now:
H = Hsystem + Hgrav + Vgrav
X
X
X
=
εloci c†loci cloci +
εwi c†wi cwi +
εk c†k ck
i
+
i
X
(Vloci ,k c†loci ck
k
+ herm. conj.) + Hgrav + Vgrav
(22)
i,k
The first three terms of the hamiltonian eq. (22) refer to the adsorbate in a non-perturbed localized state | loci i,
to basis states for the warp resonances | wi i and basis states (plane waves) for the movement of the delocalized
adsorbate | ki, respectively. (c†i and ci are creation and destruction operators for the adparticle in the basis state i.)
The states {| loci i} und {| ki} are not orthogonal to each other.
The hamiltonian for the gravitons (spin equal 2) can be written as [53]:
Hgrav =
X
p+ ,p
T
h̄ωp+ ,pT
D
X
bIJ
ξIJ bIJ†
p+ ,pT p+ ,pT
(23)
I,J=2
p+
where
is the momentum along a light cone direction, pT is the transverse momentum, ωp+ ,pT is the corresponding graviton frequency, D is the number of space dimensions in which the gravitons live (10 in our case) and ξIJ
is an arbitrary symmetric traceless polarization matrix (responsible for the spin two of the gravitons). bIJ†
and
p+ ,pT
IJ
bp+ ,pT satisfy the usual boson commutation relations. The summation in eq. (23) runs over indices I, J from 2
to D because the value 0 is reserved for the time variable and the value 1 is for the propagation direction of the
graviton. Therefore the graviton can have vibrational modes in (D − 1)-space dimensions.
In the final state | f i a graviton has been emitted into or absorbed from the environment, the adparticle is in a
local state which does not feel the gravitational interaction Vgrav with the metal atom, hence | f i is a solution to
Hsystem alone. The scattering potential is the gravitational potential in the warp resonance Vgrav .
Defining an external gravitational potential Φ provided by the Ni(110) surface by:
Z
hwi | Vgrav | wi i = MXe dD xρXe (x)Φ(x)
(24)
where x is a D-dimensional position vector, we obtain the usual multipole expansion:
µ
1 X µν ∂Egrav
Q
Vgrav (x) = MXe Φ(xi ) − p · Egrav (xi ) −
(xi ) + ...
2D µν
∂xµ
It is well known that the dipole contribution (second term) will be exactly zero. The third (quadrupole) term will be
orders of magnitude smaller than any non-vanishing dipole term and therefore cannot be of any use for localizing
the Xe atom.
It has to be emphasized that this situation is dramatically different from phonon scattering and from photon scattering. In the case of photon scattering the monopole contribution is exactly zero for an electrically neutral piece
of matter and the dipole term will be the first leading term and it is just this term which we think of and which we
discuss, if we are talking about light emission and absorption. In the case of phonon scattering there might be a
monopole term, but the dipole term is dominating and mainly responsible for phonon emission and absorption.
In the case of gravitational interaction we are left with the monopole term and we have to figure out, how this term
can lead to graviton emission and absorption.
11
Quantum Zeno effect
warp resonances
local core movement states
hwgrav1
Eloc1, |loc1,kground>
Ew2,|w2,kground>
Eloc2+hwgrav, |loc2,kgrav>
Ew1+ hwgrav1, |w1,kgrav1>
hwgrav1
hwgrav
Ew1,|w1>
Eloc2, |loc2,kground>
Figure 5: Two consecutive decoherence events of the Xe core movement state: the first one (depicted with full thin
lines) is associated with graviton absorption in the initial state and leads to an energy conserving collapse of the
Xe core movement state in the | loc1 , κground i state. The second decoherence event (depicted with thin dashed
lines) is associated with graviton emission and leads to collapse the Xe core movement state in the | loc2 , κground i
state.
5.2.4 Gravitational quantum Zeno process
The localization of an adsorbate via graviton emission or absorption in a warp resonance can be envisaged as a
gravitational quantum Zeno process. The most simple continuous decoherence process of the Xe core movement at
temperatures of a few Kelvin, that one can imagine, is displayed in fig. 5. It represents an illustration of a quantum
Zeno effect, where two consecutive decoherence events with the participation of gravitons are involved (one of
them associated with graviton absorption, the other one - with graviton emission) leading to a final adparticle
localized state, which is identical with the initial state. Two localized Xe core movement states | loc1 i and | loc2 i
are involved, which do not feel the gravitational interaction in the warp resonances. (We discuss the components
of the adsorbate core movement states parallel to the substrate surface alone.) The core movement states feel the
gravitational potential within the warp resonances, where further two states, | w1 i and | w2 i are involved in the
decoherence events. The warp resonance | w1 i is more contracted than | w2 i and its kinetic energy is higher.
However, exactly because of the stronger localization, the attractive gravitational interaction in | w1 i is stronger.
If we want to study the time development of decoherence, as it was described in section 2.3, the time evolution of
the system starts from the initial state | i1i =| loc2 , κground i with the graviton in its ground state κground . Absorption of a graviton with wave vector κgrav and energy h̄ωgrav (from an assumed background of gravitons: masses
and energy are abundant everywhere in the environment, but gravitons have never been detected experimentally) is
the first decoherence event, the Xe core movement being decohered in the state | loc2 , κgrav i =| loc2 i⊗ | κgrav i.
This state is energetically degenerate with the warp resonance | w2 , κground i. The scattering state which evolves
from | loc2 , κgrav i due to the gravitational interaction leads to mixing with the warp resonance | w2 , κground i with
a high coefficient Cw2 because of the degeneracy coupling: | i1+i =| loc2 , κgrav i + Cw2 | w2 , κground i. The
transiton from | i1+i onto | loc1 , κground i represents the collapse of the Xe core movement state onto a different
localized core movement state | f 1i =| loc1 , κground i, whereby the energy is conserved because both in the initial
state | i1+i and the final state of this collapse the gravitons are in their ground state | κground i. A second decoherence event can be envisaged, starting from the collapsed final state | loc1 , κground i of the former decoherence
event, which serves now as an initial state (depicted with thin dashed lines in fig. 5): | i2i =| loc1 , κground i. The
scattering state, which evolves from | i2i due to gravitational interaction within the warp resonance is a superposition: | i2+i =| loc1 , κground i + Cw1 | w1 , κgrav1 i of two degenerate states, | loc1 , κground i and the warp
resonance | w1 , κgrav1 i, the condition for degeneracy coupling being satisfied once again. Hence, the coefficient
Cw1 must have a high value. The decoherence is now associated with graviton emission h̄ωgrav1 and a collapse of
12
the Xe core movement on a localized final state | f 2i =| loc2 , κground i | κgrav1 i.
The rate of the first decoherence process (depicted by the full thin lines in fig. 5) equals according to the generalized
Ehrenfest theorem:
Rloc1 ,i1 ≈
2π
| Vgrav |2 | hloc1 | w2 i] |2 | Cw2 |2 δ(Eloc1 − Ew2 )
h̄
with the scattering potential V = Vgrav = − G
nance, which is missing in the final state.
11 M
Xe MN i
π7 r8
(25)
equal to the gravitational interaction in the warp reso-
The described decoherence mechanism is indeed a non-demolition measurement process in the sense discussed in
refs. [54]-[57]. The ”measurement” of the system by the environment leads to the localization of the Xe adparticle
in exactly the same state, from which it has started, a graviton has been first absorbed from the environment and
then emitted in the evironment, which is equivalent to reflection of a graviton. This non-demolition measurement
process is not just a mathematical construction, as it was originally suggested by Von Neumann [58]. Two physical
processes involving gravitons lead to exactly the same state of the local system from which its time-evolution has
started, the environment receives information about the state of the local system. This is the essence of a Von
Neumann non-demolition measurement in action.
5.2.5 Coherent time evolution in QND versus collapse
We use the idea of gravitational decoherence described in the last section to model the competition between the
coherent time-evolution and delocalization of an adsorbate, initially in a localized wave packet, and energy conserving collapses on localized states. The system consists of a Xe atom, described by a wave packet, interacting
with delocalized core movement states, subject to the time-evolution as defined by Schrödinger’s time dependent
equation. A gravitational collapse is allowed at random time intervals accompanied by the reflection of a graviton
in the environment. Whenever a collapse on the localized basis state occurs, we have a decoherence event leading to localization. The system has the option to decohere in the final localized state | loci via the gravitational
quantum Zeno process with rate provided by eq. (25), or to evolve further coherently.
As it is known from earlier studies (cf. for instance ref. [59]) the density operator ρsystem defined in a mixed
basis with matrix elements hi | ρsystem S −1 | ji = hi | ρ̃system | ji (S: overlap matrix in the local system) has
the property that its trace equals the total number of particles in the system. The matrix hi | ρ̃system | ji has the
meaning of a reduced density matrix in a non-orthogonal basis. Therefore this mixed representation of the density
operator provides a suitable estimate of the quasi-classical population of the basis states. Its time evolution can
be used to follow the variation with time of the population of different basis states. In the lowest panel of fig.
6 the quasi-classical populations | hloc | ρ̃system | ki |2 are plotted at time t0 = 0 as a function of the wave
vector k. This plot represents the distribution of | loci over the continuum of 2D plane waves {| ki} before the
time evolution of the wave packet starts. The localized Xe wave packet in position space appears as a smooth
spectral distribution in k-space. This will change significantly once the coherent time-evolution of the wave packet
is switched on.
The eigenfunctions of the QND hamiltonian Hsystem in eq. (22) are used to determine the time dependence
according to Schrödinger’s time dependent equation. The result is displayed in fig. 6 for a piecewise deterministic
evolution of the Xe wave packet. The plot represents the spectral distribution of the quasi-classical population of
the Xe core movement wave packet in the continuum of delocalized plane waves as a function of time. The time
evolution of the matrix elements of the reduced density matrix | hloc | ρ̃system | ki |2 for the first three timesteps
shows that the Xe wave packet develops plane wave components with different wave vectors which is equivalent to
Xe delocalization. A successful collapse after the third timestep of coherent evolution, as a result of gravitational
interaction with the substrate atom in the warp resonance, reduces the Xe core movement state to a localized wave
packet.
The time evolution of the matrix elements of the reduced density matrix show that the gravitational decoherence
mechanism can lead to successful Xe localization within time intervals of the order of 10−11 s. The localization
13
0.90
0.85
0.80
0.75
0.70
0.65
0.60
0.55
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
0
5
10
15
20
wave vector [a.u.]
Figure 6: Time evolution of the matrix elements | hloc | ρ̃system | ki |2 of the reduced density matrix: the coherent
time evolution and delocalization of the Xe core movement state in front of a nickel surface is interrupted by
collapse on local Xe core movement states at random time intervals, leading to the localization of the Xe atom
on a time scale of the order of 10−11 s. The different panels refer to nearly equally spaced time intervals from 0
(bottom) to 2 × 10−10 s (top).
of Xe via gravitational decoherence is fast enough compared to the timescale of the interaction with the phonons
of the order of 10−7 s, which was discussed in a previous section 5.1. This decoherence mechanism can ensure
fast localization of adsorbates at low temperatures, which is a necessary precondition for energy exchange with
phonons and for adsorbate surface diffusion. In this sense we suggest the gravitational decoherence as a viable
mechanism for adsorbate localization.
6
Conclusion
Entanglement of a local system with the environment, as suggested by the state of the art decoherence theory,
gives rise to the transient pointer states (deformation and warp resonances). Scattering events within these pointer
states involving the collision of environs and the differences in the dimensionality of the pointer states and the
final states of the scattering event can explain the transition to classical behaviour. Scattering between pointer
states within a stationary world leads in a natural way to decoherence. Adsorbate sticking with probability tending
to unity at low temperatures, as it is measured experimentally and is reproduced with classical models, is one
example, where a full quantum mechanical approach taking into account the weak interaction of the adsorbate
in the deformation resonance with phonons and electron-hole pairs, reproduces the classical result. Classical
reflection of some adsorbates from corrugated solid surfaces (e.g. CO from Cu(100), metastable He∗ from solid
surfaces), in contrast to diffraction observed with others (ground state He, Ne, Ar), is another example where
the classical behaviour results from a totally quantum mechanical treatment. The physics behind this behaviour is
weak interaction of the gas particle with the environs in the transient deformation resonance and loss of the memory
about the quantum diffraction states. With these examples we illustrate that decoherence, i.e. classical behaviour,
is a natural consequence of the quantum mechanical entanglement with the environmental polarizations.
A decoherence mechanism for adsorbate localization relying on permanent ”measurement” by phonons, electron-
14
hole excitations, etc. in the environment is too slow to compete with the coherent delocalization of adsorbate wave
packets as it is implied by the time dependent Schrödinger equation. Adsorbate localization times significantly
longer than 10−10 s would be in contradiction with many surface phenomena which require localization and occur
on a faster time scale (cf. the data in table 1). A new environmental decoherence mechanism is proposed via
coupling to gravitons and graviton scattering in the 11 dimensional space-time of string theory. The gravitational
interaction increases with the 8-th power upon decreasing the distance (for distances smaller than the warped hidden dimensions) and is very localized. The interaction with the environment, treated as scattering of gravitons
from the local system, interrupts the coherent time evolution of the adparticle wave packet. This is a kind of
instantaneous collapse on a particluar state which breaks the time evolution of the system along a piecewise deterministic path. If the scattering events are fast enough, the time development of the system can deviate appreciably
from the unitary predictions (concerning the system alone). The deviation is then interpreted as decoherence. The
characteristic time scale of gravitational decoherence proves to be of the order of 10−11 s. If adparticle localization
occurs on this time scale, then many classical surface phenomena needing longer characteristic times like surface
diffusion, classical reflection of heavy particles, STM induced desorption of adsorbates at low temperatures via
a transition state (cf. the time scales of different surface phenomena in table 1) can be explained as preceded by
particle localization.
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