Experimental Characterization of Bosonic Creation and Annihilation Operators

The photon creation and annihilation operators are cornerstones of the quantum description of the electromagnetic field. They signify the isomorphism of the optical Hilbert space to that of the harmonic oscillator and the bosonic nature of photons. We perform complete experimental characterization (quantum process tomography) of these operators. By measuring their effect on coherent states by means of homodyne tomography, we obtain their process tensor in the Fock basis, which explicitly shows the “raising” and “lowering” properties of these operators with respect to photon number states. This is the first experimental demonstration of complete tomography of nondeterministic quantum processes.

Figure 1

Experimental setups for photon annihilation (a) and creation (b). The processes are heralded by “clicks” in single-photon detectors.

Figure 2

Event count rate for the photon annihilation (a) and creation (b) operators as a function of the input coherent state amplitudes in the modes to which these operators are applied. The solid lines show the expected theoretical dependencies (5), with the vertical scale fit to the data.

Figure 3

Model of experimental imperfections for the photon annihilation (a) and creation (b) setups. The top and bottom schemes in (b) are equivalent to each other. Notation: BS, beam splitter; A, attenuator; the quantities in parentheses denote the transmission of the corresponding optical element. $\widehat{D}$ denotes the operator of phase-space displacement by $\Delta X=\sqrt{2T_1}(1-T_2){\alpha }_{\mathrm{in}}$. The amplitude of the input coherent probe state is ${\alpha }_{\mathrm{in}}$ while $\alpha$ is the effective amplitude of the coherent state in the mode acted upon by the operators.

Figure 4

The “diagonal” values of the process tensor ${\epsilon }_{kk}^{mm}$ of the photon annihilation (a) and creation (b) reconstructed from the experimental data, with correction for experimental imperfections. Each color corresponds to the photon number distribution in the output state for the Fock state $|m⟩$ at the input. Insets: Worst-case fidelities of the reconstructed processes acting within subspaces ${\mathcal{H}}_{\widehat{a},{\widehat{a}}^{†}}^{\prime }(n)$ of the optical Hilbert space spanned by number states $|1⟩,\dots ,|n⟩$ and $|0⟩,\dots ,|n-1⟩$, for $\widehat{a}$ and ${\widehat{a}}^{†}$, respectively.