1.2.4. Probing Quantum Commutation Rules by Addition and Subtraction of Single Photons to/from a Light Field [3]

As the final states depend on the order in which the two actions are performed, we directly observed the noncommutativity of the creation and annihilation operators

if the particles were photons in a single-mode radiation field, one would naturally use the bosonic creation and annihilation operators $\hat{a}^\dagger$ and $\hat{a}$ to perform the addition and the subtraction of single photons to and from the light field. Indeed, for the general case of a quantum light field described by a density matrix $\hat{\rho}$ , there is a broad consensus in calling the result of the application of the creation (annihilation) operator, $\hat{a}^\dagger\hat{\rho}\hat{a}(\hat{a}\hat{\rho}\hat{a}^\dagger)$, the “photon- added” (“-subtracted”) state, after proper normalization. This tendency derives from the fact that, when the photon creation operator a % † acts on a state with a well-defined number n of photons (also called a Fock or number state, and denoted by jn〉), it increases this number by one: $$\hat{a}^\dagger |n\rangle=\sqrt{n+1}|n+1\rangle$$ Conversely, when the photon annihilation operator $\hat{a}$ acts on the same state, it subtracts a quantum of excitation, thus reducing the number of photons in the state by exactly one $$\hat{a} |n\rangle=\sqrt{n}|n-1\rangle$$ When the initial number of particles is precisely known, the quantum and the classical cases give exactly the same results for arbitrary sequences of additions and subtractions. However, the situation changes completely for general superpositions or mixtures of Fock states.

It 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.

A single click simply conditions the generation of a photon- added or photon-subtracted thermal state, whereas a double click can either produce a first-subtracted- then-added thermal state or vice versa, depending on the combination of clicks.

The first interesting result appears when comparing the mean photon numbers of the states: For the photon-added state one finds, quite naturally, that the mean photon number is larger than in the original thermal light state, but unexpectedly the same result occurs for the photon- subtracted state. The operation of removing one photon from the field has increased (doubled) its final mean photon number. Such an increase also takes place for the sequence of operators that should intuitively bring the field back to the initial state.