Boundary effects of electromagnetic vacuum fluctuations on charged particles Department of Physics N

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Boundary effects of electromagnetic vacuum
fluctuations on charged particlesDepartment of
Physics National Dong Hwa University Da-Shin
Lee Talk given atNational Tsing-Hua
Univeristy4 December 2008
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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

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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.
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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.
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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
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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.
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Reduced density matrix
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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.

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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).
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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 integral is
bounded by a closed worldline of the electron
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.
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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.
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Single plate
The tangential component of E fields and the
normal component of B fields on the perfectly
conducting plate surface located at the z0 plane
vanish.
The image charge method
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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.
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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 .
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Decoherence for double plates (parallel)
Double plates an additional plate is located at
za plane
The double prime in the summation assigns an
extra normalization factor to the n0
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.
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Decoherence for double plates (perpendicular)
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.
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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 electrons flight time is much smaller than
its mean energy fluctuations driven by vacuum
fluctuations
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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 electrons flight time.
• Thus,

• The spreading of the quantum state
• The increase in the size of the localized
  quantum state during the
• electrons 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.

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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.

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Q A
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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.

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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

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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.
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Reduced density matrix
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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.
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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.
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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. Fluctuation
s on the particles trajectory driven by the
noise entirely are of the quantum origin as seen
from an explicit dependence on the noise
term.
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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 z0 plane
vanish.
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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
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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.
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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.
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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 charges
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.
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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.
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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.
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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-dissipati
on 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.
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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 UV-divergence with
the weak time dependence in velocity
fluctuations.
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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.
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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.
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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
constrained by the electrons plasma frequency as
well as the width of the wave function
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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.

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Q A