The researchers then added, and later subtracted, a photon from the beam before calculating its mean number of photons. They also did the opposite: They subtracted first and then added a photon to the beam. The researchers found that, under certain conditions, subtracting a photon changed the beam's quantum state, increasing the mean number of photons.
https://www.science.org/doi/abs/10.1126/science.1146204
In ordinary arithmetic, multiplication obeys
a commutative law. That is, for any two
numbers n and m, the product nm is always
equal to mn. In classical physics, measure-
ments of physical properties also obey a com-
mutative law. For example, if one first meas-
ures the position of a particle and then its
momentum, one obtains the same result by
first measuring the particle’s momentum and
then its position. However, quantum mechani-
cal quantities do not in general obey this com-
mutation relation (1). In fact, the breakdown
of the commutative law lies at the heart of
many fundamental quantum properties, such
as the Heisenberg uncertainty principle. In the
example of position and momentum, the lack
of commutativity is conventionally stated
by means of the relation ,
where and are the quantum mechanical
operators (2) associated with position and
momentum, respectively, and where h is
Planck’s constant.
In an intriguing and illustrative report on
page 1890 of this issue, Parigi et al. (3) present
the results of a laboratory demonstration of
what happens in the quantum mechanical
operations of photon creation and annihi-
lation, which lacks commutativity. These
authors add a single photon to a light beam,
which corresponds to the action of the stan-
dard quantum mechanical creation operator â†.
They can also subtract a single photon from
the light beam, which corresponds to the anni-
hilation operator â.
Parigi et al. measure the quantum mechan-
ical state of a thermal light field after perform-
ing these two operations on it, and they show
that the final state depends on the order in
which the operations are performed. This
result is a striking confirmation of the lack
of commutativity of quantum mechanical
operators. Moreover, the authors present the
strongly counterintuitive result that, under
certain conditions, the removal of a photon
from a light field can lead to an increase in the
mean number of photons in that light field, as
predicted earlier (4).
The basic idea of the
experiment of Parigi et al.
and some of their results are
shown in the figure. In the
top row, a laser beam passes
through a rotating ground
glass plate (th) to mimic the
random fluctuations of a ther-
mal source and is detected
by a quantum state analyzer
(QSA). The results of the mea-
surement are shown on the
right. Here, p(E) gives the pro-
bability distribution of the
electric field amplitude E.
Rows B through E illustrate
the consequences of acting
on the input state by various
quantum mechanical opera-
tions. Row B shows the result
of removing a single photon
from the field with a beam
splitter. Counterintuitively, the
mean number of photons in
the output field is increased
by this operation. Row C
illustrates the consequence of
adding a single photon to the
input state with an optical
parametric amplifier (a device
that splits one photon into two,
each with approximately half
the energy of the original
photon). Row D illustrates the consequence of
first adding a photon to the field and then sub-
tracting a photon, whereas row E illustrates the
situation in which a photon is first subtracted
and then a photon is added. One sees that the
fields created in these two situations are
markedly different.
Beyond the conceptual interest in the
The authors are at the Institute of Optics, University of
Rochester, Rochester, NY 14627, USA. E-mail: boyd@
optics.rochester.edu
Quantum arithmetic. Schematic experimental procedure of Parigi et al.
and some of their laboratory results. The order in which photons are added
and subtracted from a light field strongly influences the field’s properties.
CREDIT: ADAPTED BY P. HUEY/SCIENCE
Published by AAAS
niques they describe could pave the way
toward new possibilities in the fields of quan-
tum information science and quantum optics
(5–8). These results show how one can con-
vert a purely thermal light field, which pos-
sesses no nonclassical properties, into a light
field with strongly nonclassical features. This
work thus constitutes a step toward the devel-
opment of techniques for “quantum state
engineering,” that is, the creation of states
with specified quantum properties. States of
this sort are expected to play a key role in
quantum computing, quantum cryptography,
and control of quantum systems.
References and Notes
1. P. A. M. Dirac, Principles of Quantum Mechanics
(Clarendon, Oxford, UK, 1958).
2. In quantum mechanics, physical quantities such as posi-
tion and momentum are obtained by applying “opera-
tors” to a system’s wave function.
3. V. Parigi, A. Zavatta, M. Kim, M. Bellini, Science 317,
1890 (2007).
4. M. Ueda, N. Omoto, T. Ogawa, Phys. Rev. A 41, 3891 (1990).
5. L. Mandel, E. Wolf, Optical Coherence and Quantum
Optics (Cambridge Univ. Press, Cambridge, UK, 1995).
6. M. O. Scully, M. Suhail Zubairy, Quantum Optics
(Cambridge Univ. Press, Cambridge, UK, 1997).
7. R. Loudon, The Quantum Theory of Light (Oxford Univ.
Press, Oxford, UK, ed. 3, 2000).
8. C. C. Gerry, P. L. Knight, Introductory Quantum Optics
(Cambridge Univ. Press, Cambridge, U
https://servizi.ino.it/AllegatiPubblici/Stampa/Stampa13.pdf
Interview with Valentina Parigi (in French)
https://www.sorbonne-universite.fr/node/5486
What excites you about your current research?
Dr Valentina Parigi: I currently work with multimode quantum optical fields and in particular, my project consists of arranging them in complex network structures, which are used in quantum information protocols. I’m really excited by the idea that complex network theory, which has been developed for real-world social, biological and technological networks, has turned out to be apt at describing our quantum optical systems.
Presentation in English Valentina Parigi
http://www.2physics.com/2009/01/adding-and-subtracting-photons-for.html
the photon creation operator acts on a state with a well-defined number of photons (also called a Fock state) by increasing this number by one. Conversely, when the photon annihilation operator acts on the same state, it subtracts a quantum of excitation, thus reducing the number of photons in the state by exactly one.
However, the situation becomes completely different as soon as one starts dealing with general superpositions or mixtures of Fock states. If the magician were using a distribution of quantum rabbits, the operation of adding one animal to the hat by a “rabbit creation operator” and then, immediately after, subtracting another by a “rabbit annihilation operator”, would lead to a final probability distribution of rabbits in the hat completely different from the initial one. Furthermore, the reverse sequence of operations would lead to a third outcome, different from both, i.e. the two operations do not commute.
Finally, when we tried to subtract a photon from a coherent state (the most classical, wave-like, state of light) we found that nothing changed in the state. In other words, we performed the first experimental demonstration that coherent states are invariant under photon annihilation.
how a nonlinear crystal creates two photons from one photon
John G. Cramer:
To take a simple example, consider a half-silvered mirror. This is a piece of glass that has had just enough reflective material so that exactly half the light striking it at 45deg. incidence goes straight through the glass and the other half bounces from the reflective surface at a right angle. If a single photon of light encounters this half-silvered mirror, there is a 50% chance that it will be transmitted (pass through) and a 50% chance that it will be reflected. One would think that the photon must do one thing or the other, but quantum mechanics tells us that it can do both. This can be verified experimentally.
a photon that was transmitted comes back to the same path as a photon that was reflected, we can observe "quantum interference". The quantum mechanical wave describing the transmitted photon adds or subtracts from the wave for the reflected photon. If they subtract completely, the photon waves on the two paths cancel each other and no photons can be observed at the exit. If they add, the photons waves on the two paths help each other and can be observed. The two quantum states of the particle, "photon reflected" and "photon transmitted", are superimposed, with definite observable consequences. Quantum superposition is not limited to photons, which are somewhat special because they have no rest mass. It works equally well for massive particles and has been demonstrated using very slow neutrons, for example.
The mathematics of state superposition is special because quantum mechanics describes each state function as a complex variable that has a real part and an "imaginary" part (that behaves like the square root of -1). Each state has its own quantum mechanical "phase", the angle in the complex plane that its complex state function makes with the real number axis. The result of superimposing several states is determined by the quantum phases of the states. Some combinations of phases produce cancellation while others produce reinforcement.
This is a consequence of the fact that in quantum mechanics the "averaged" states are complex (real plus imaginary) space-time functions with complex weighting factors (not real functions with real weighting factors).
https://www.npl.washington.edu/av/altvw45.html
They show that the superposition of a group of these time-evolving systems can result in a net time displacement which is quite different (positive or negative) from the elapsed time experienced by the external observer of the system. Thus, the system inside the shell had been translated forward or backward in time with respect to the observer.
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