Time Ordering Dynamics in Ionization-Excitation of Helium
to HeII (2p) Magnetic Sublevels Using Extreme Ultraviolet
Spectropolarimetry
R. Bruch1, H. Merabet,1* J. H. McGuire2, A. L. Godunov2, and Kh. Kh. Shakov2
1
2
Department of Physics, University of Nevada Reno, Reno, NV 89557 USA
Department of Physics, Tulane University, New Orleans, LA 70118-5698 USA.
Abstract. Time ordering of interactions in dynamic quantum multi-electron systems provides a constraint that connects the time
evolution of different electrons in ion-atom collisions. When spatial correlation between electrons is combined with time
ordering, the time evolution operators for different electrons become correlated. We report in this study evidence for quantum
time correlation between electrons by comparing full second-Born calculations, which include both spatial and temporal electron
correlation, to recent magnetic sublevel cross section measurements corresponding to simultaneous ionization-excitation of He
(1s2) 1So to He+ (2p) 2Po levels by proton impact. The experimental substates cross sections σ0 and σ1 for ML =0, ±1 are
determined by combining corresponding total extreme ultraviolet (EUV) cross sections with our recent polarization fraction
measurements. Such results have important applications for the understanding of quantum phenomena like time entanglement.
from the time-dependent Schrödinger wave equation,
which determines (11) the equation for the time
evolution operator, U, namely, ∂U/∂t = - i ħ V(t) U .
Here V(t) is the interaction (or sum of interactions) of a
system of atoms or molecules with light or matter.
The formal solution for the evolution operator may be
expressed (10),
INTRODUCTION
Most complex processes cannot in nature be fully
described in terms of independent transitions of
electrons.
The electronic mixing in time can
redistribute energy and facilitate transitions that would
otherwise be forbidden. Such mixing may be either
caused by external interactions, such as strong electromagnetic fields (1-4) or by internal interactions, such as
the Coulomb interactions between electrons (5) or spinorbit, orbit-orbit , and spin-spin interactions (6). Both
the external and internal interactions can be used to
shape and dynamically control nanostructures (7).
Time dependent external interactions, such as the fields
of moving charged particles (5) or oscillating laser
fields (3,4) can cause time dependent changes in the
internal interactions. The problem of time correlation
in multi-electron systems has recently been addressed
theoretically with inclusion of needed energy nonconserving terms (8,9). However, the effect of time
correlation between electrons has not been previously
confirmed by experiment. We report such evidence
here.
U (t 2 , t1 ) =
t2
∫ dt
t1
n'
t2
t2
t1
t1
... ∫ dt ' ' ∫ dt ' T V (t n ' )...V (t ' ' )V (t ' ).
(1)
Here T is the Dyson time ordering operator (10), which
imposes the causality-like constraint that V(tl’)V(tk’) =
0 if any tl’ < tk’. Thus the time ordering operator T
provides a time connection between pairs of
interactions due to the constraint that the interactions
occur in the order of increasing time.
In this study, the time ordering operator T has been
decomposed (9) into two terms, namely, T = Tunc + Tcor.
The uncorrelated term, Tunc, is the time independent
part of T, which does not connect the various
interactions V(t) in time and thus eliminates both
sequencing and time correlation in the time evolution
of the system (8). The correlated term, Tcor , regulates
sequencing and gives time correlation of the
interactions V(t).
In dynamic systems (9) time
correlation is provided by enforcement of time ordering
on the sequence of interactions, V(tn’) ... V(t’’) V(t’),
which cause the quantum system to change. For a twoelectron transition occurring via V1(t) and V2(t)
THEORY
Interconnection of different electrons in time
requires both internal electron correlations, which
interconnects the electrons, as well as time ordering,
which imposes a causal-like sequencing of interactions
in quantum time propagation (10). Electron correlation
is relatively well understood (5). Time ordering comes
*
(−ih) n
n!
n =0
∞
∑
Corresponding author: H. Merabet, E-mail: [email protected]
CP680, Application of Accelerators in Research and Industry: 17th Int'l. Conference, edited by J. L. Duggan and I. L. Morgan
© 2003 American Institute of Physics 0-7354-0149-7/03/$20.00
152
respectively, Godunov and McGuire (8) have shown
that in second order,
ionization of atomic helium with concurrent excitation
of a second electron into various magnetic sublevels of
the 2p excited state now provides evidence for
correlation in the time evolution of different electrons.
This means, for example, that one sequence of
interactions may differ from another, as illustrated by
the equation for Tcor given above.
Tunc V2 (t ' ' )V1 (t ' ) = 1 2 (V2 (t ' ' )V1 (t ' ) + V2 (t ' )V1 (t ' ' ) ), (2)
and
Tcor V2 (t ' ' )V1 (t ' ) = 1 2 sign(t ' '−t ' )[V2 (t ' ' ),V1 (t ' )] ,
(3)
Here sign(t’’ – t’) = (t’’ – t’)/|t’’ – t’| is a unit vector in
the direction of increasing time, and [V2(t’’),V1(t’)] =
V2(t’’)V1(t’) – V1(t’)V2(t’’) explicitly requires a time
connection between interactions at different times, t’
and t’’. This represents non-local quantum time
entanglement between electrons.
The Fourier
transform of Tcor gives the principal value part of the
energy propagator (8). This quantum term violates
conservation of energy for short times. This effect is
not present in the propagation of classical waves (12).
EXPERIMENTAL METHOD
The experimental setup used in this work has been
described in more detail elsewhere (16-19).
Therefore, we give here only a brief overview. This
apparatus is comprised of a target chamber housing the
MLM polarimeter, gas cell and a Faraday cup, and a
1.5 meter grazing incidence monochromator. A PC
controlled data acquisition system has been used to
operate the apparatus and to record the data. The
polarimeter utilizes a molybdenum-silicon (Mo/Si)
multilayer mirror. The MLM and EUV filter are fixed
in a box that was shielded with aluminum foils in
order to prevent stray light and particles from entering
the polarimeter. Two different wedges were used with
this setup to provide an angle of 50E for measuring
Lyman-α (HeII (2p) 2Po) emission and an angle of 40E
used to measure HeI (1snp) 1Po emissions. Since
sufficiently strong magnetic fields can lead to a
depolarization of the observed radiation by the Hanle
effect, the gas cell used in this study was mounted
inside a cylindrical magnetic shielding. With this
shielding at the interaction region of the gas target a
magnetic field smaller than 0.05 Gauss has been
achieved. The corresponding total cross sections σ
measurements have been conducted using a 1.5 m high
resolution grazing incidence monochromator.
Projectile
V1(t’)
V2(t”)
1s
1s
2p
Excitation
8
l’
2
1
He (1s ) S Atom
Ionization
FIGURE 1: Time correlation occurs when V2(t’’) is
connected to V1(t’). With quantum time ordering V2(t’’)V1(t’)
may differ from V1(t’)V2(t’’). Then V2(t’’) is connected to
V1(t’). Spatial electron correlation occurs when two electrons
interact with one another (denoted by the arrows).
In the independent electron approximation (5),
where spatial correlation between electrons is removed,
the quantum commutator [V2(t’’),V1(t’)] is zero. Then
each electron evolves independently in time, i.e. the
evolution operator, U(t2,t1), reduces to a product of
single electron evolution operators. When spatial
correlation between electrons is included, e.g. via twobody
Coulomb
electron-electron
interactions,
[V2(t’’),V1(t’)] is non-zero (5), and the time evolution
of different electrons may become entangled. Time
correlation between electrons corresponds to cross
correlation (13) between the time propagation of
amplitudes for different electrons, and describes how
electrons communicate about time.
These HeII results have been put on an absolute
scale by renormalizing our proton data (20) to full
second-Born cross section values for high velocities
impact (v=6.1 a.u.). In addition, the obtained cross
section data have been corrected for alignment effects
(21). The statistical uncertainties of the measured line
intensities in this study were between 0.5% and 3%
over most of the range of impact energies. When
instrumental uncertainties related to energy resolution
of the Van de Graaff accelerator, target pressure
stability, polarization and charge normalization are
combined, the total uncertainty for magnetic scattering
angle-integrated substate cross sections was found to be
about 13% to 15% for HeI (1s2p) 1Po and HeII (2p) 2Po
states.
Time ordering is generally thought (5,14) to be
needed to understand the difference observed (15) in
double ionization of atoms by protons and anti-protons
at high velocities. Nonetheless, there has been no
previous direct experimental evidence for time
correlation between different electrons. Our data for
The degree of linear polarization of the emitted
light is related to ionization plus excitation magnetic
substate cross sections for population from the helium
ground state to the He+(2p) ml = 0 and ml = 1 magnetic
sublevels, namely (22),
153
(
) 37σ(σ +−11σσ ) .
P 2P0 =
0
0
1
Uncertainty Principle, ∆E ∆t ≥ ħ. In quantum
mechanics all paths from A to B are possible, although
those outside of an envelope of trajectories, whose
width is proportional to ħ, are statistically improbable.
Within this envelope of uncertainty, time and energy
may not be simultaneously localized. The non-locality
in energy means that there are quantum fluctuations in
energy, not present in Maxewell’s equations, which for
a short time violate conservation of energy. These offenergy-shell terms correspond to quantum time correla-
(4)
1
The total cross section is given by σ = σ0 + 2 σ1,
where σ-1 = σ+1 for our cylindrical collision symmetry.
RESULTS AND DISCUSSION
In figure 2 we present both data and theory for the
polarization of light emitted from the 2p level of
excited helium following excitation of one electron and
ionization of another by an incident proton. It is
evident that a first-order theory does not describe either
the magnitude or the energy dependence of the
polarized light observed. A second-order calculation in
the
on-energy-shell
(or
energy-conserving)
approximation with Tcor = 0 provides some
improvement, but still does not agree with the data.
Only the second-order calculation including time
correlation between electrons is in good agreement
with observation, except at the lowest energy shown,
where it is expected that higher order terms in V(t) are
significant. The method described here applies to
incident photons as well as electrons, protons and ions
as projectiles.
0.25
p+He excita tion-ionization
0.20
polarization
0.15
0.10
0.05
Born 1
Born 2 unc
Born 2
Exp
0.00
-0.05
2
Figure 3 shows experimental magnetic sublevel
angle-integrated cross sections σ0 and σ1 for proton
impact along with the results of our theoretical
compilations (20). The behavior of the σ0 and σ1 cross
sections is qualitatively similar to the total cross section
for ionization-excitation. The second Born cross
sections and the experimental data do not converge to
the first Born result until the collision velocity is above
10 a.u. The effect of off-shell time correlation terms is
to increase the second Born cross sections somewhat.
We note that such a slowly varying effect is not typical
for double excitation (9), but is typical of double
ionization (5), except for the sign. We also note that
the effect of the off-shell terms is greater in σ1 than in
σ0.
This means that the independent time
approximation (8) is more sensible for the ML= ±1
mangnetic substate population σ1 than for σ0.
3
4
5
6
collision velocity (a .u.)
FIGURE: 2. Calculations with and without time correlation
of electrons compared to experimental data. Here polarized
light is emitted from helium following 1s – 2p excitation of
one electron accompanied by ionization of the second
electron. The polarization fraction is plotted as a function of
the velocity of the incident proton. The first-order calculation
(Born 1) has no time correlation, nor does the second-order
calculation without time correlation (Born 2 unc). The full
second-order calculation (Born 2) includes time correlation
by including Tcor from Equations. (2) and (3).
-tions arising from the constraint of time ordering
(9,10). Time correlation between electrons occurs only
when electrons interact with one another. Then the
effects of time ordering of external interactions for
different electrons become coupled, and the electrons
become interconnected in time. This correlation
insures that the electrons cooperate in seeking out the
most efficient way to get from A to B, subject to both
the constraint of time ordering and the freedom of
quantum uncertainty.
Time correlation between
electrons produces the observable difference between
the full and approximate quantum calculations shown
in figure 2.
We interpret our results as follows. The time
correlation discussed in this paper is a quantum
phenomenon not present in classical descriptions of
Newtonian or Maxwellian approaches. In Newtonian
particle mechanics there is only one true trajectory for a
particle, which is determined by Fermat’s Principle
(23) that nature seeks the most (or possibly the least)
efficient way from A to B. The one true path is
determined by minimizing the action. Action (24) is an
integral of the energy of the system over time.
Historically quantum mechanics was obtained (11)
from classical mechanics by constraining the action to
be an integer multiple of ħ. This leads to the
154
σ(ML= 0)
2
cm )
100
calculations of dynamically correlated systems of
particles, an independent time (or on-shell or wide
band) approximation without time correlation is widely
used (25-27) to save computational time and effort.
When this relatively easy independent time
approximation is valid, calculations of relatively
complex systems become feasible.
Magnetic Sublevel Cross Section (10
-20
10
In summary, the excellent agreement of our full
second Born calculations with our magnetic sublevel
cross section ratios has revealed the importance of
temporal electron correlation in the complex
simultaneous ionization-excitation process. We have
therefore presented evidence that correlation between
the time evolution between electrons can be significant.
This time correlation between electrons arises from
energy-non-conserving quantum fluctuations occurring
for short times in intermediate states of the system.
Such time correlation between electrons may play a
useful role in controlling quantum transmission of
information via sequencing in complex multi-electron
systems important for nanotechnology.
1
100
σ(ML= 1)
10
1
2
3
4
5
6
7
8
9
Proton Velocity (a.u.)
FIGURE 3: Magnetic sublevel scattering angle-integrated
cross sections for ionization-excitation- to HeII (2p) 2Po states
with ML = 0 ( squares) and ML = ±1 (diamonds) by proton
impact (20). Notations for theory are the same as FIGURE 1.
ACKNOWLEDGEMENT
In many cases the effects of time correlation
between electrons are small (8). This is fortunate in
that the off-shell, time-entangled terms require more
than one hundred times more computer time to evaluate
than the simpler on-shell, energy-conserving terms. In
We thank Alan Goodman and H. Schmidt-Böcking
for useful discussion. This work was supported in part
by the Division of Chemical Sciences, Office of
Sciences, U.S. Department of Energy, the Nevada
Business and Science Foundation.
reversed propagation, sequencing is in the direction of
decreasing time.).
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155