Photon generation by time-dependent dielectric: A soluble model
Markus Cirone
Istituto Nazionale di Fisica della Materia and Istituto di Fisica dell’Universita `, via Archirafi 36, I-90123 Palermo, Italy
Kazimierz Rza ¸z
˙
ewski
Centrum Fizyki Teoretycznej, Polska Akademia Nauk and College of Science, Aleja Lotniko ´w 46, 02-668 Warszawa, Poland
Jan Mostowski
Instytut Fizyki, Polska Akademia Nauk and College of Science, Aleja Lotniko ´w 46, 02-668 Warszawa, Poland
Received 1 April 1996; revised manuscript received 1 August 1996
A soluble model of electromagnetic field interacting with time-dependent dielectric medium is given. The
main assumption is that the dielectric constant is switched rapidly from the initial to the final value. Generation
of electromagnetic field and photon statistics are found. S1050-29479607712-8
PACS numbers: 12.20.-m
I. INTRODUCTION
The notion of the photon is the most basic concept in
quantum electrodynamics. However, the quantum nature of
the electromagnetic field is well understood in the case of
electromagnetic field in empty space only. In the case of
fields interacting with external currents and other sources the
notion of photons is not so clear. Of particular interest has
become the case of electromagnetic field in cavities. If the
cavity consists of perfect conductors proper boundary condi-
tions for the field mimic the interaction of the field with
charges in the mirrors. The case of field quantization in the
presence of dielectrics without dispersion was studied in
some detail 1. Dispersion inevitably leads to damping,
which makes the field quantization and hence the notion of
photons rather complicated and not yet fully understood 2.
The even more complicated case of photons in a cavity with
oscillating perfect mirrors has attracted a lot of interest re-
cently. The corresponding classical problem of electromag-
netic field in a cavity with oscillating mirrors does not have
a solution in the closed form, but many approximate solu-
tions have been found and discussed.
Quantum effects in the electromagnetic field confined to a
cavity with moving mirrors has attracted a lot of interest 3.
This problem is also interesting from the point of view of
plasma physics 4. The quantization of the field in such a
situation with explicit time dependence does not follow in
any natural way from the standard quantization procedures.
Some purely quantum effects such as photon noise reduction
have been discussed 5.
Yet another aspect of quantum electrodynamics in rapidly
changing geometry was discussed recently in Ref. 6. The
fully inhibited spontaneous emission of the atom placed in a
node of the resonant cavity mode is assumed. Then, suddenly
one of the cavity mirrors is removed. Now the atom is free to
emit a photon. The paper predicts an instantaneous response
of the detector exposed by the removed mirror.
Quantization of the electromagnetic field in the case of
time-dependent boundaries is thus needed. Deeper under-
standing of the quantization, the idea of photons, will pro-
vide a tool for the study of these effects.
The present paper is devoted to a detailed study of a
simple system with an explicitly time-dependent environ-
ment. Instead of discussing a cavity with oscillating mirrors,
which is a rather difficult problem, we will discuss a much
simpler, yet nontrivial case of a cavity from which a dielec-
tric slab is rapidly removed. This system contains most of the
ingredients of much more complicated ones, especially with
time-dependent components. Our model also contains an ex-
ternal time-dependent element, the dielectric slab. Its time
dependence leads to a variety of phenomena — photon cre-
ation, squeezing of electromagnetic field, etc. On the other
hand, the model is simple enough to be solved exactly, as
opposed to much more complicated ones. Our discussion of
the model sheds some light on cavity QED processes with
explicit time dependence.
II. THE SYSTEM
The system under consideration consists of a cavity with
perfectly reflecting metallic mirrors. The mirrors are of the
shape of two parallel planes separated by a distance denoted
by 2 L +d . Because of the planar symmetry we will consider
electromagnetic waves propagating in the direction perpen-
dicular to the mirrors, in this way the problem is reduced to
one dimension. The dielectric medium is present between the
mirrors forming the cavity. It forms a slab parallel to the
cavity mirrors, has width d , and is characterized by dielectric
constant , which for simplicity is assumed to be indepen-
dent of the frequency. The geometry of the system is shown
in Fig. 1. We will consider a case when the electromagnetic
field is in the state of vacuum i.e., the lowest-energy eigen-
state for times t 0. At time t =0 the dielectric slab is re-
moved. This is done instantaneously, i.e., faster than any
other characteristic time scale in the problem. The question
we will ask and solve in the paper is what is the state of the
electromagnetic field for t 0.
PHYSICAL REVIEW A JANUARY 1997 VOLUME 55, NUMBER 1
55 1050-2947/97/551/625/$10.00 62 © 1997 The American Physical Society