DTIC ADA343354: Proceedings of the International Conference "Quantum Optics IV" Jaszowiec, Poland, June 17-24, 1997. Volume 93, Number 1

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POLISH ACADEMY OF SCIENCES
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POLISH PHYSICAL SOCIETY
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JANUARY 1998
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Vol. 93 - No. 1
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POLISH ACADEMY OF SCIENCES
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POLISH PHYSICAL SOCIETY
ACTA PHYSICA
POLONICA A
JANUARY 1998
Vol. 93 - No. 1
Proceedings of the International Conference
"Quantum Optics IV"
Jaszowiec, Poland, June 17-24, 1997
91s RECOGNIZED BY THE EUROPEAN
PHYSICAL SOCIETY
WARSAW
POLISH ACADEMY OF SCIENCES
INSTITUTE OF PHYSICS
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PROCEEDINGS OF THE INTERNATIONAL CONFERENCE
QUANTUM OPTICS IV
JASZOWIEC, POLAND, JUNE 17-24, 1997
Editors of the Proceedings
Jan Mostowski
Arkadiusz Orlowski
WARSAW
POLISH ACADEMY OF SCIENCES
INSTITUTE OF PHYSICS
The Conference Quantum Optics IV was held in Jaszowiec, Poland from
June 17 to June 24, 1997.
The Conference was organized by the Institute of Physics of the Polish
Academy of Sciences and the Center for Theoretical Physics of the Polish Academy
of Sciences.
PROGRAM COMMITTEE
I. Bialynicki-Birula (Poland)
N. Bigelow (USA)
J.H. Eberly (USA)
M. Fedorov (Russia)
J. Mostowski (Poland)
K. Rzqewski (Poland)
L. Sirko (Poland)
H. Walther (Germany)
ORGANIZING COMMITTEE
I. Bialynicki-Birula
J. Mostowski
K. Rzqiewski
L. Sirko
SPONSORS
We wish to thank the following institutions for their contribution to the
success of this Conference:
Committee for Scientific Research (Poland)
European Commission, Directorate General III
National Science Foundation (USA)
United States Air Force European Office of Aerospace Research and Development
Polish Physical Society
Committee of Physics of the Polish Academy of Sciences
PREFACE
This volume contains lectures presented at the Quantum Optics IV meeting
which took place at Jaszowiec (Poland) from June 17 to June 24, 1997. The lead-
ing themes of the meeting were: cold atoms, strong laser field-atom interactions
and quantum chaos. It turned out that the first topic got most attention. It is
worth noting that our meeting took place before the announcement of the Nobel
Prize in physics for 1997. This Nobel Prize was awarded to C. Cohen-Tannoudji,
W. Philips and S. Chu for their contribution to cooling and trapping of atoms.
Of particular interest were lectures on the "hot topic" of ultracold atoms forming
Bose-Einstein condensate. This effect is often considered as a part of condensed
matter physics but it became accessible to experiments due to progress in laser
cooling of dilute atomic gases. Thus we have witnessed yet another example of a
new subject encompassed by quantum optics. Some earlier examples include clas-
sical and quantum chaos. Although not directly related to quantum optics they
draw from experimental possibilities provided by modern laser techniques. Our
third theme: strong laser field-atom interactions is a more traditional subject of
quantum optics.
Some 140 physicists from 13 countries participated in the meeting. We had
25 invited lectures, of which 18 are included in this volume. In addition there were
120 posters presented at two poster sessions. The conference offered an overview
of the most important issues of quantum optics in its broad sense.
Organizers
Vol. 93 (1998) ACTA PHYSICA POLONICA A No. 1
Proceedings of the International Conference "Quantum Optics IV", Jaszowiec, Poland, 1997
QUANTUM IMPLICATIONS OF RAY SPLITTING
R. BLOMEL
Fakultit ffir Physik, Albert-Ludwigs-Universitiit
Hermann-Herder-Str. 3, 79104 Freiburg, Germany
Ray splitting is a universal phenomenon that occurs in all wave systems
with sharp interfaces. Quantum implications of ray splitting are: (i) the im-
portance of non-Newtonian orbits for the density of states in the semiclassical
limit, (ii) ray-splitting corrections to the average density of states and (iii)
the need to include non-Newtonian orbits in trace formulas for the oscillat-
ing part of the density of states. The signatures of non-Newtonian orbits in
the density of states have recently been identified experimentally (L. Sirko,
P.M. Koch, R. Bliimel, Phys. Rev. Lett. 78, 2940 (1997)).
PACS numbers: 05.45.+b
In 1948 Feynman introduced a particularly illuminating representation of
quantum mechanics [1]. According to Feynman the transition amplitude of a par-
ticle from point P to point Q in the phase space is given by a sum of complex phases
computed on the basis of all possible phase-space paths connecting P with Q. In
order to obtain the exact quantum transition amplitude all paths in S are equally
important. It is, however, possible to bring out particular subclasses of S by a
judicious choice of quantum problems. Let us focus on the class of potential prob-
lems. For such problems it is convenient to divide S into the two disjoint subsets of
Newtonian (AO) and non-Newtonian (A7") trajectories. The Newtonian trajectories
are the solutions of the classical canonical equations. For smooth potentials and
sufficiently small h very good approximations to the quantum transition ampli-
tudes can be obtained on the basis of A( alone. The contribution of the rest of the
paths is near zero because of destructive interference. In the case of non-smooth
potentials particular subclasses of V have to be kept besides the trajectories con-
tained in N for a good representation of transition amplitudes. In other words,
potentials with steps and other types of irregularities may be used as "projectors"
to bring out the effects of particular classes of non-Newtonian trajectories. In the
case of step potentials the importance of the non-Newtonian orbits survives the
h -* 0 limit. Thus Newtonian mechanics is not the only mechanics important
in the semiclassical limit of quantum mechanics. In the case of step potentials,
e.g., the underlying orbit structure in the semiclassical limit is obtained from a
nondeterministic, non-Newtonian mechanics [2-6].
(7)
8 R. Bhimel
In order to illuminate the new concept of a non-Newtonian mechanics con-
sider the following potential:
0, for x < 0,
V(x, w)= Vox/w, for 0 < X<w (1)
V0, I for x > w.
For w -- 0 we obtain a step potential. We are particularly interested in the
scattering of waves off the potential (1) for E > VO. In the asymptotic region to
the left of the potential the wave function is given by
O(x) = eikx + re- ik, x < 0, k = 2mE/h2. (2)
To the right of the potential we have
(x) = teik, x > w, r = V12m(E - Vo)/h 2. (3)
Since (1) is a piecewise linear potential, r can be computed with the help of Airy
functions. We obtain
ikKC1 + kqC2 - KqC3 - iq2C4 (4)
ikgC1 + kqC2 + nqC3 + iq2C4'
where
3
q (2mVo) 1/ (5)
h 2W\(cid:127)
The constants C are given by
C Ai(-fl)Bi(-a) - Ai(-e)Bi(-f),
1
C = Ai(-a)Bi'(-fl) - Ai'(-/)Bi(-a),
2
C = Ai(-fl)Bi'(-a) - Ai'(-a)Bi(-f6),
3
C = Ai'(-#)Bi'(-a) - Ai'(-a)Bi'(-/3), (6)
4
where
a = qwElVo, 8l = qw ( T--- _ 1) .(7)
We are mainly interested in the double limit w --* 0, h -+ 0. Investigation of Eq. (4)
shows that the two limits do not commute. Moreover, let w and h approach 0,
but keep their ratio v = h/w constant. Then it is easy to show that for h --+ 0 the
reflection amplitude is r 5 0 and depends on v. A finite r for h -+ 0 means that in
the classical limit there exist trajectories reflecting off the step although E > Vo.
These must clearly be non-Newtonian trajectories since Newtonian trajectories
transmit with probability 1 for E > V . The probability for a particle to go left
0
is p = IrJ2, the probability to go right is 1 - p. The decision about whether to
reflect (go left) or to transmit (go right) is left to chance, governed by p. Thus, the
non-Newtonian dynamics for a step potential is non-deterministic. If we represent
the incoming path of the particle by a ray, there are two possibilities for this ray
to leave the step: the particle may reflect (with probability p) or transmit (with
Quantum Implications of Ray Splitting 9
probability l-p). Thus, one incident ray creates two (or sometimes more) outgoing
rays. We call this situation "ray splitting".
Ray splitting is a universal phenomenon that occurs in all wave systems
with sharp interfaces in the limit of small wavelength. Examples are the splitting
of light rays at the interface between two transparent dielectrics, the splitting of
acoustic rays at the interface between two media of different density, the splitting
of rays associated with water surface waves at the interface between two different
depths, and finally the splitting of rays associated with de Broglie matter waves
in quantum mechanics at the position of a potential step.
The wave implications of ray splitting were recently studied in the context
of acoustic and quantum systems [2-7]. Major findings were the importance of
non-Newtonian orbits for the oscillating part of the density of states [2-6], the
necessity of correcting the Weyl formula [8] for the average density of states of
ray-splitting systems [7] and the need to modify existing trace formulas [8] to
include non-Newtonian periodic orbits [2]. Additional quantum implications are
the existence of new classes of scars in the quantum wave functions [3].
The signatures of periodic non-Newtonian orbits were recently identified ex-
perimentally [5, 6] in the context of microwave resonance spectroscopy. We used
thin dielectric- and metal-loaded cavities to generate ray-splitting of microwaves
a'. sharp air/teflon and air/metal interfaces. The Fourier transform of the measured
density of resonances shows peaks at the optical path lengths of non-Newtonian or-
bits. Since for thin microwave resonators the electromagnetic Helmholtz equation
and the quantum Schr6dinger equation are equivalent [9, 10], these experiments
are of direct relevance for quantum ray-splitting systems.
Having established the importance of non-Newtonian orbits experiments
should now aim at testing the ray-splitting correction of the Weyl formula. In order
to do this the experiments have to be improved in such a way that
; 500 levels can be measured without missing a single one. Not missing a sin-
gle level is a stringent experimental constraint which may be achieved with the
help of numerical support. It was recently demonstrated [4] that a few hundred
levels are indeed enough for a first qualitative test of the ray-splitting correction.
The theory of ray-splitting systems is also not complete yet. For example, the
ray-splitting correction derived analytically in Ref. [7] applies only to rectilinear
ray-splitting boundaries. What is missing is the computation of the correction for
curved ray-splitting boundaries.
Another promising route for theoretical research is the identification of pre-
bifurcation ghosts [11] in ray-splitting systems. As non-Newtonian orbits undergo
much the same type of bifurcations as Newtonian orbits, we expect the existence
of non-Newtonian ghosts in ray-splitting systems. Theoretical work on the identi-
fication of the signatures of non-Newtonian ghosts is currently in progress.
Fruitful discussion with Prof. Fritz Haake on the nature of ghost orbits
are gratefully acknowledged. The author is grateful for financial support by the
Deutsche Forschungsgemeinschaft.