’09 STATISTICAL OPTICS
The Weird Math of
Photon Subtraction
A. Zavatta, V. Parigi, M.S. Kim and M. Bellini
Accurate single-photon-level ma- nipulation of light is needed for
the development of optical quantum
technologies that overcome the limits
of classical physics. Such technologies
may usher in a revolution in the way we
exchange and process information or
perform accurate measurements.
;e basic processes of adding or
subtracting single photons to or from a
radiation field are described by photon
creation and annihilation operators:
When applied to states with a well-defined number of photons (Fock states),
such operators increase or decrease the
photon number by one unit. However,
this intuitive behavior does not hold
when one is dealing with general superpositions or mixtures of Fock states. A
few years ago, we applied sequences of
creation and annihilation operators to
ordinary light pulses by making use of
beam-splitters1 and nonlinear crystals, 2
and verified fundamental quantum commutation rules by demonstrating that
the order of the operations makes a big
di;erence to the outcome. 3
During those experiments, we also
found that, under particular conditions,
subtracting a single photon changed the
quantum state of light to the extent that
its mean number of photons increased
instead of diminishing. Pushed by those
surprising findings, we have recently
decided to systematically analyze the action of photon annihilation upon some
paradigmatic states of light. 4
For Fock states, we have confirmed
the intuitive decrease of the photon
number by exactly one unit. However,
we achieved interesting results when
subtracting a single photon from a
thermal state, the most common form
of light (both the sun and ordinary light
bulbs emit chaotic thermal light). In this
case, taking one photon away exactly
doubled the mean number of photons in
Simpli;ed scheme for conditional single-photon subtraction from a light ;eld. BS is a low-re;ectivity beam-splitter. A click in the on/off photodetector heralds the success of the
photon annihilation operation on the initial ;eld state. Depending on the photon statistics of
the input ;eld, the output state may contain the same mean number of photons or a smaller
or larger number.
the pulse. Finally, subtracting a photon
from a coherent state (the most classical,
wave-like, state of light) did not change it
at all. ;is last result is an experimental
demonstration of the fact that coherent
states are invariant under photon annihilation. Since their introduction by Nobel
laureate Roy Glauber in the 1960s,
coherent states have been a cornerstone
in the quantum description of light.
However, their definition as eigenstates
of the annihilation operator had not been
verified so directly in an experiment.
Although counterintuitive, the strange
behavior of these quantum operations is
not unphysical and does not put energy
conservation at stake. Most of its weirdness simply derives from the misleading
implicit assumption that a deterministic
addition and subtraction of particles
can be represented by the creation and
annihilation operators, which work in a
probabilistic way. 5
Apart from providing some beautiful
demonstrations of the inner working of
quantum mechanics, the techniques used
in these experiments may be used to arbi-
trarily engineer light at the most accurate
levels by making the appropriate sequence of photon additions and subtractions. ;is capability will open the way
to tailor-made quantum light for future
technologies, such as the secure exchange
of information or the development of
novel protocols for quantum-enhanced
measurements and communications.
M. Bellini ( bellini@inoa.it) and A. Zavatta are with the
Istituto Nazionale di Ottica Applicata in Firenze, Italy.
V. Parigi is with the European Laboratory for Nonlinear
Spectroscopy in Firenze, Italy. M.S. Kim is with the
School of Mathematics and Physics, The Queen’s
University, Belfast, Northern Ireland, United Kingdom.
References
1. J. Wenger et al. “Non-Gaussian Statistics from Individual
Pulses of Squeezed Light,” Phys. Rev. Lett. 92, 153601
(2004).
2. A. Zavatta et al. “Quantum-to-classical transition with
single-photon-added coherent states of light,” Science
306, 660-2 (2004).
3. V. Parigi et al. “Probing quantum commutation rules by
addition and subtraction of single photons to/from a light
field,” Science 317, 1890-3 (2007).
4. A. Zavatta et al. “Subtracting photons from arbitrary light
fields: experimental test of coherent state invariance by
single-photon annihilation,” New J. Phys. 10, 123006
(2008).
5. M.S. Kim. “Recent developments in photon-level operations on travelling light fields,” J. Phys. B 41, 133001
(2008).
