Adding and Subtracting Photons for Fundamental Tests of Physics and for Optical Quantum Technologies
(from left to right) Alessandro Zavatta, Marco Bellini, and Valentina Parigi
Author: Marco Bellini
Affiliation: Istituto Nazionale di Ottica Applicata – CNR
European Laboratory for Non-linear Spectroscopy (LENS), Florence, Italy.
>>Link to the Group Homepage.
[This is an invited article based on recent work of the author and his colleagues -- 2Physics.com]
Imagine a magician’s hat containing some rabbits whose precise number is unknown, but whose number probability distribution is well defined, so that, for a large ensemble of identical hats with the same probability distribution, one may define an average number of rabbits.
Now, if the magician puts one more rabbit in each hat, the mean rabbit number will, quite naturally, increase by one, while it will decrease by one if he takes one away (unless, of course, the hat was initially empty, in which case he would not be able to extract anything). Moreover, whatever the initial distribution, if the magician performs the two actions in a sequence, by first adding one rabbit and then taking one away, he/she will end up with exactly the same distribution for the number of rabbits remaining in the hat.
What would happen if, instead of normal rabbits, the magician used a microscopic hat containing quantum rabbits?
According to quantum physics, an electromagnetic field is composed of photons, which are so small that even a laser pointer with a typical power of 1mW emits a few millions of billions of them each second. Pure single photons are the ideal means to carry and encode information in emerging quantum technologies, but generating and manipulating them is still a very challenging task.
If the rabbits were identical quantum particles, one could assimilate them to photons in a radiation field (the hat), and would naturally use the so-called creation and annihilation operators to perform the addition and the subtraction of quantum rabbits to/from the hat. Indeed, as undergraduate physics students know, 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.
This is the manifestation of one of the most profound laws in quantum physics. Indeed, the non-commutativity of particular quantum operations leads to many of the counterintuitive and fascinating aspects of quantum mechanics, including the famous Heisenberg uncertainty principle.
In 2007 our team (A. Zavatta, V. Parigi, and M. Bellini) at the Istituto Nazionale di Ottica Applicata – CNR (Florence, Italy), in collaboration with M. S. Kim from the Queen’s University (Belfast, UK), succeeded in performing the first direct tests of this fundamental principle of quantum physics in a laboratory . We chose to use photons (which are much easier to manipulate than rabbits) and applied sequences of the creation and annihilation operators to an ordinary light pulse by making use of beam-splitters  and non-linear crystals . As non-commutativity predicts, we found that the order of the operations makes a big difference to the outcome.
Figure 2: Setups to conditionally subtract (a) and add (b) a single photon from/to a light field. BS is a low-reflectivity beam-splitter; PDC is a nonlinear crystal where parametric down-conversion takes place; the two white boxes denote on/off photodetectors that herald the success of the corresponding quantum operation on the initial field state.
During those experiments we also found that the quantum operations behave so unusually that, under particular conditions, subtracting a photon changed the quantum state of the light pulse to the extent that its mean number of photons increased instead of diminishing. Taking a quantum rabbit away from the hat could actually increase the mean number of the remaining ones!
In one of our recent works  we decided to verify this behavior in a systematic way for some paradigmatic states of light. By applying photon annihilation to a Fock state with a well-defined number of photons we confirmed the intuitive decrease of the photon number by exactly one unit. Surprises appeared when we subtracted a single photon from a thermal state, the most common state of light (both the sun and ordinary light bulbs emit chaotic thermal light). We found that the mean number of photons in the pulse after subtraction was the double of the initial one.
Figure 3: Experimental density matrices and Wigner functions for a thermal state (left panel) and for the same state after a single-photon subtraction (right panel). The photon-subtracted state has a broader Wigner and photon number distribution than the original one.
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. Since their introduction by Nobel laureate Roy Glauber in the 60’s, coherent states have been a cornerstone in the quantum description of light, but their definition as eigenstates of the annihilation operator had never been verified so directly in an experiment.
Figure 4: Experimental density matrices and Wigner functions for a coherent state (left panel) and for the same state after a single-photon subtraction (right panel). Photon annihilation does not modify a coherent state.
Although counterintuitive, the strange behavior of 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, on the contrary, work in a probabilistic way (i.e., the probability of extracting a particle from the hat scales with the number of particles already there) .
Apart from providing some beautiful demonstrations of the inner working of quantum mechanics, the techniques used in these experiments could in principle be used to arbitrarily engineer light at the most accurate levels by the appropriate sequence of photon additions and subtractions. This capability will open the way to “tailor-made” quantum light for future technologies, like the secure exchange of information through quantum cryptography or the development of novel protocols for quantum-enhanced measurements and communications.
For further info, please contact: Dr. Marco Bellini, Email: firstname.lastname@example.org
 “Probing Quantum Commutation Rules by Addition and Subtraction of Single Photons to/from a Light Field”, V. Parigi, A. Zavatta, M.S. Kim, and M. Bellini, Science, 317, 1890-1893 (2007). Abstract.
 “Non-Gaussian Statistics from Individual Pulses of Squeezed Light”, J. Wenger, R. Tualle-Brouri, and P. Grangier, Phys. Rev. Lett. 92, 153601 (2004). Abstract.
 “Quantum-to-classical transition with single-photon-added coherent states of light”, A. Zavatta, S. Viciani and M. Bellini, Science, 306, 660-662 (2004). Abstract.
 “Subtracting photons from arbitrary light fields: experimental test of coherent state invariance by single-photon annihilation”, A. Zavatta, V. Parigi, M. S. Kim, and M. Bellini, New Journal of Physics, 10, 123006 (2008). Abstract.
 “Recent developments in photon-level operations on travelling light fields”, M. S. Kim, J. Phys. B 41, 133001 (2008). Abstract.