Boundary effects of electromagnetic
vacuum fluctuations on charged particles
Department of Physics
National Dong Hwa University
Da-Shin Lee
Talk given at
National Tsing-Hua Univeristy
4 December 2008
Topics to be covered
Influence on electron coherence from quantum
electromagnetic fields in the presence of
conducting plates
Jen-Tsung Hsiang and Da-Shin Lee:
Phys. Rev. D 73, 065022 (2006)
Stochastic Lorentz forces on a point charge
moving near the conducting plate
Jen-Tsung Hsiang, Tai-Hung Wu and Da-Shin Lee:
Phys. Rev. D 77, 105021 (2008)
Effects of smeared quantum noise on the
stochastic motion of the charged particle near a
conducting plate
Jen-Tsung Hsiang, Tai-Hung Wu and Da-Shin Lee:
submitted to Phys. Rev. A
Coherence reduction of the electron due
to electromagnetic vacuum fluctuations
The interest in the decoherence phenomenon is motivated by
the study of the experimental realization of quantum
computers in which the central obstacle is to prevent the
degradation of quantum coherence arising from a
unavoidable coupling to the environment.
Coherence reduction of the electron due
to electromagnetic vacuum fluctuations
Influence of electron coherence
from the coupling to quantum
electromagnetic fields can be
studied with an interference
experiment through the effects of
phase shift and contrast change of
the interference pattern.
The Lagrangian for a nonrelativistic electron coupled to electromagnetic
fields is given by such a particle-field interaction ( the Coulomb gauge):
Imposition of the boundary condition on quantum fields will result in
modification of vacuum fluctuations that may further influence electron
interference.
The closed time path formalism
The initial density matrix for the electron and gauge fields is
assumed to be factorizable as:
The fields are assumed to be in thermal equilibrium with the
density matrix given by:
where
is the free field Hamiltonian.
Then, in the Schroedinger picture, the density matrix evolves
in time as:
We will take the limits:
The reduced density matrix of the electron by tracing out the fields becomes:
Here we have introduced an identity in terms of a complete set of eigenstates
Then, the matrix element of the time evolution operator can be expressed by the
path integral.
Reduced density matrix
The closed-time-path formalism
Suggested review article:
D. Boyanovsky, M. D'Attanasio, H.J. de Vega, R.
Holman, D.-S. Lee, and A. Singh : Proceedings of
International School of Astrophysics, D. Chalonge:
4th Course: String Gravity and Physics at the Planck
Energy Scale, Erice, Italy (1995) , hep-ph/9511361.
Decoherence functional & Phase shift
Consider the electron initially being
in a coherent superposition of two
localized states with the distinct
mean trajectories.
Phase shift
Decoherence
functional
Leading order effect comes from the contribution of the mean trajectory given by
the external potential where the width of the wavefunction is ignored ( discussed
later).
Gauge invariant decoherence functional
where the closed worldline
is for a moving electron along its path
in the forward time direction and then
along the path
in the backward
time direction.
By means of the 4-dimensional Stokes' theorem,
where the area element
of the electron
of the integral is bounded by a closed worldline
in Minkowski spacetime.
Decoherence is found sensitive to the field strength in the region in
Minkowski spacetime where the electron is excluded. The decoherence
effect is essentially driven by the non-static features of quantum fields.
Evaluation of decoherence functional
Unbounded case: worldlines of the electrons are given by
Lorentz invariance of the W functional allows us to chose the observe moving
with the velocity ,
in which the electrons are
seen to have transverse motion in the z direction only.
Dipole approximation by
considering small k modes
consistent with
nonrelativistic motion has
been applied to account for
E fields only.
Single plate:
The tangential component of E fields and the normal
component of B fields on the perfectly conducting plate
surface located at the z=0 plane vanish.
The image charge method:
Decoherence for a single plate (parallel)
Single plate: worldlines of the electrons are given by
Under the dipole approximation (small k),
Electron coherence is restored for small z as in the case with no influence from
electromagnetic fields due to the fact that E fields parallel to the plate surface
vanish on the boundary.
The boundary effect becomes irrelevant for large z.
Decoherence for a single plate (perpendicular)
Single plate: worldlines of the electrons are given by
Under the dipole approximation (small k),
Boundary induced effects of vacuum fluctuations suppress electron
coherence for small z. In particular, near the plate,
since large E
fields normal to the plate surface are induced.
Decoherence reduces to the result without the boundary for large z .
Decoherence for double plates (parallel)
Double plates: an additional plate is located at z=a plane
The double prime in the summation assigns an
extra normalization factor
to the n=0 mode.
Worldlines of the electrons:
The presence of the second parallel plate further suppresses vacuum
fluctuations of E fields in the direction parallel to the plate surface, thus
again restores electron coherence.
Decoherence for double plates (perpendicular)
Double plates: worldlines of the electrons are given by
In this case, an additional parallel plate seems to boost vacuum fluctuations
of E fields in the direction normal to the plate surface so as to further reduce
electron coherence significantly.
Thus, the presence of the conducting plate anisotropically modifies the
electromagnetic vacuum fluctuations that in turn influence electron
coherence.
Discussion on involved approximations
The finite conductivity effect: Now consider the boundary plate with finite
conductivity  .
path length
Anglin & Zurek, quant-ph/9611049
The Joule energy loss rate for bulk currents inside the conductor induced by
the motion of the surface charge with the same velocity of the electron can
be given by:
However, mean energy fluctuations of the electron owing to
electromagnetic vacuum fluctuations along the plate surface are given by:
Yu &Ford, PRD 70, 065009 (2004)
Thus, the finite conductivity effect can be ignored as long as the Joule
energy loss during the electron’s flight time is much smaller than its mean
energy fluctuations driven by vacuum fluctuations:
Discussion on involved approximations
The electrostatic attraction arising from the image charge on the
electron:
It can be neglected as the time scale for the electron with a trajectory
at a height z above the plate, which might fall into the boundary due
to this attraction force, is much larger than the electron’s flight time.
Thus,
The spreading of the quantum state:
The increase in the size of the localized quantum state during the
electron’s flight time can be estimated as:
The spreading effect can be ignored when
leading to
The backreaction from the fields on the mean trajectory of the
electron ( for example: radiation reaction ) will contribute to the
decoherence function of order
, and thus, is ignored.
Summary
Coherence reduction of the electron due to electromagnetic
vacuum fluctuations in the presence of the conducting plates
is studied with an interference experiment within the context
of the closed time path formalism where corrections beyond
involved approximations can be systematically incorporated.
Decoherence of the electron driven by non-static quantum
electromagnetic fields is found sensitive to the field strength
in the region in Minkowski spacetime bounded by a closed
worldline of the electron.
The plate boundary anisotropically modifies vacuum
fluctuations that in turn affect the electron coherence, and it
is found that electron coherence is restored as in the case with
no influence from electromagnetic fields when the path plane
is parallel to the plate surface, but reduced in the normal case.
Decoherence effect for localized states turns out too weak to
be detected.
Q&A
Stochastic Lorentz forces on a point
charge moving near the conducting plate
When a charged particle interacts with quantized electromagnetic fields,
a nonuniform motion of the charge will result in radiation that
backreacts on itself through electromagnetic self-forces as well as the
stochastic noise manifested from quantum field fluctuations will drive
the charge into a zig-zag motion.
We wish to explore further the anisotropic nature of vacuum
fluctuations under the boundary by the motion of the charged particle
near the conducting plate.
The Lagrangian for a nonrelativistic charged particle coupled to
electromagnetic fields is given by such a particle-field interaction ( the
Coulomb gauge):
The initial density matrix for the particle and fields is assumed to be
factorizable by ignoring the initial correlations:
The fields are assumed to be in thermal equilibrium with the density
matrix given by:
where
is the free field Hamiltonian.
Then, in the Schroedinger picture, the density matrix evolves in time as:
The reduced density matrix of the particle by tracing out the fields becomes:
Here we have introduced an identity in terms of a complete set of eigenstates
Then, the matrix element of the time evolution operator can be expressed by the
path integral.
Reduced density matrix
We also assume that the particle is initially in a localized quantum state,
which can be approximated by the position eigenstate:
The nonequilibrium partition function can be defined by taking the trace of
the reduced density matrix over the particle variable.
The limits
have be taken at this moment.
The stochastic Langevin equation is then obtained by extremizing the
stochastic effective action. We ignore intrinsic quantum fluctuations of the
particle by assuming that the resolution of the length scale measurement is
greater than its position uncertainty.
Remarks:
The influence of electromagnetic fields appears as the nonMarkovian
backreaction in terms of electromagnetic self forces , and stochastic
noise, driving the charge into a fluctuating motion.
This is the nonlinear Langevin equation on the charge's trajectory since the
dissipation kernel as well as noise correlation are the functional of the trajectory.
The noise-averaged result arises from classical effects.
Fluctuations on the particle’s trajectory driven by the noise entirely are of the
quantum origin as seen from an explicit
dependence on the noise term.
Fluctuation-Dissipation theorem
Fluctuation-Dissipation theorem plays a vital role in balancing between
these two effects to dynamically stabilize the nonequilibrium Brownian
motion in the presence of external fluctuation forces.
The tangential component of E fields and the normal component of B
fields on the perfectly conducting plate surface located at the z=0 plane
vanish.
The corresponding fluctuation-dissipation theorem
can be derived from the first principles calculation:
The F-D theorem at finite-T
The F-D theorem in vacuum
Gauge invariant expression
Retarded E and B fields are obtained by introducing the Lienard-Wiechert
potentials together with the Coulomb potential.
Stochastic E and B fields involve only the transverse components of the gauge
potentials because in the Coulomb gauge, the Coulomb potential is not a
dynamical variable, and hence it has no corresponding stochastic component.
Langevin equation under the dipole approximation
Dipole approximation will be applied for this nonrelativistic motion to
account for the backreaction solely from E fields. The charged particle
undergoes the harmonic motion with the small amplitude at
.
An additional component of the external potential is applied to counteract
the Coulomb attraction from its image charge.
The initial conditions are specified as
which can be achieved by applying an appropriate external potential to hold
the particle at the starting position with zero velocity. Then the applied
potential is suddenly switched off to the harmonic motion potential.
The noise-averaged equation ( classical effect )
Backreaction from the free-space contribution entails the
retarded Green's function nonvanishing for the lightlike
spacetime intervals. The charge follows a timelike trajectory
where radiation due to the charge’s nonuniform motion can
backreact on itself at the moment just when radiation is emitted.
It is given by
, electromagnetic self force + UV-divergence
absorbed by mass renormalization =the ADL equation.
Backreaction owing to the boundary has a memory effect where emitted radiation
backscatters off the boundary, and in turn alters the charge's motion at a later
time.
The kernel can be found from inverse Laplace transform:
where the Browish contour is to enclose all singularities counterclockwisely
on the complex s plane.
The branch-cut arises from discontinuity of the kernel.
Since
the cut lies within the region of
where imaginary part of the self-energy nonvanishing.
．
The pole equation:
The poles originally in the first Rienmann sheet move to the second sheet due to
the interaction with environment fields as long as the poles are in the cuts.
The pole on the first sheet located in the positive real s axis corresponds to the
runaway solution to be discarded.
Breit-Wigner shape
The resonance mode with the peak around the oscillation frequency is found to
have dominant contributions to the late time behavior:
High frequency modes relevant to
very early evolution are ignored.
Velocity fluctuations ( quantum effect )
It is of interest to study velocity fluctuations of this charged oscillator under
fluctuating electromagnetic fields to see how they are affected by the boundary
and asymptotically saturated as a result of the fluctuation-dissipation relation.
Velocity fluctuations
grow linearly in time at
early stages, and then
saturate to a constant at
late times.
Although they for two
different orientations of
the motion start off at
different rates, the same
saturated value is
reached asymptotically.
The function has a Breit-Wigner feature
on k space peaked at about
and its
width being approximately of order
at early times or
at late times.
The spectral density reveals the
oscillatory behavior on k space
over the change in k by
.
The integrand has the linear k dependence for large k, leading to quadratic UVdivergence with the weak time dependence in velocity fluctuations.
Growing regime:
Backreaction dissipation can be ignored. Velocity fluctuations thus mainly
result from the stochastic noise.
Velocity fluctuations are found to grow linearly with time. The growing rate is
related to the relaxation constant out of the dissipation kernel due to the F-D
relation.
Quadratic UV-divergence is found to vary slowly in time.
The effect of the stochastic noise on the oscillator is much weaker, leading to a
smaller growing rate on the parallel motion than the normal one since E field
fluctuations parallel to the plate vanish, but its normal components become
doubled, compared with that without the boundary.
The relaxation constant shares the similar feature as a result of the F-D relation.
The presence of the boundary apparently modifies the behavior of the charged
oscillator in an anisotropic way.
Saturation regime:
We investigate the behavior of velocity fluctuations at late times by
incorporating backreaction dissipation.
Backreaction from the contribution of the resonance is isotropic due to delicate
balancing effects between fluctuations and dissipation, and thus is solely
determined by the motion of the charge.
The high-k modes probe UV-divergence as well as the strong boundary
dependence for small z on backreaction.
As expected, the enhancement in velocity fluctuations arises in the normal
motion for small z resulting from large E fields induced in that direction.
Discussion on the saturated value of velocity fluctuations
The change in velocity fluctuations, as compared with a static charge interacting
with electromagnetic fields in its Minkowski vacuum state, arises from the
imposition of the conducting plate
as well as the motion of the
charge
.
The relative importance between two effects will be estimated by taking an
electron as an example.
Fluctuations induced by the boundary :
Velocity fluctuations owing to the electron's
motion are overwhelmingly dominant
Fluctuations induced by the motion of the charge
constrained by the
electron’s plasma
frequency as well as the
width of the wave
function
Summary
The influence of electromagnetic fields on a nonrelativistic
point charge moving near the conducting plate is studied by
deriving the nonlinear, nonMarkovian stochastic Langevin
equation from Feynman-Vernon influence functional within the
context of the closed time path formalism.
This stochastic approach incorporates not only backreaction
dissipation on a charge in the form of retarded Lorentz forces,
but also the stochastic noise manifested from electromagnetic
vacuum fluctuations.
Under the dipole approximation, noise-averaged result reduces
to the known ADL equation plus the corrections from the
boundary, resulting from classical effects. Fluctuations on the
trajectory driven by the noise are of quantum origins where the
dynamics obeys the F-D relation.
Velocity fluctuations of the charged oscillator are to grow
linearly with time in the early stage of the evolution at the rate,
smaller in the parallel motion than that of the normal case.
Same saturated value is obtained asymptotically for both
orientations of the motions due to delicate balancing effects
between F & D by taking the electron as an example.
Q&A