Differential thermal states: true states of heralded photons

Introduction. Non-Gaussian states of light play an important role in the modern quantum optics, being a necessary resource for many protocols of quantum information processing. We introduce a family of single-mode optical... [ view full abstract ]

Introduction. Non-Gaussian states of light play an important role in the modern quantum optics, being a necessary resource for many protocols of quantum information processing. We introduce a family of single-mode optical states characterized by negative non-singular P-functions. We show that these states correspond to the states of light, conditionally prepared by the technique of "photon heralding".

Methods. We consider the process of parametric down-conversion with undepleted classical pump wave, passing through a nonlinear crystal and producing signal wave A and idler wave B, which we consider single-mode with the annihilation operators a and b respectively. The joint state of two modes at the crystal output is a two-mode squeezed vacuum state

   sech(r) ∑_n tanhⁿ(r) |n>_A|n>_B , 

where r is the degree of squeezing determined by the pump amplitude, the nonlinear susceptibility of the crystal and its length, and |n> is the n-photon Fock state. We consider detection of the mode B by a detector not resolving the number of photons and characterized by a quantum efficiency η. Besides, we consider linear losses in the mode A, modeled  by a beam-splitter with intensity transmittance μ.

Results. We show that the conditional state of the mode A after a click of detector, monitoring the mode B, is a difference of two thermal states. A thermal state of one mode has the density operator ρ_th(q) = (1-q)∑_nqⁿ|n>

   ρ^(-)(q,p,γ) = N(ρ_th(q)/(1-q) - γρ_th(qp)/(1-qp)) = N ∑_n(qⁿ-γ(qp)ⁿ)|n>

where 0≤γ≤1 and N is a normalization factor. The parameters (q,p,γ) are functions of the physical parameters (r,η,μ).  It is easy to see that this density operator is always positive. The Glauber-Sudarshan P-function of this state is a square-integrable function for almost all values of parameters, and is negative for γ>p, which corresponds to a non-classical state. The P-function corresponding to parameters q=exp(-1), p=exp(-0.01), γ=1 is shown in Fig. 1a. The corresponding distribution of photons is shown in Fig. 1b.

Discussion. The obtained state describes the true state of a photon, generated by the technique of "photon heralding" [1], widely used in the modern quantum optics and quantum technologies.

[1] M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, Rev. Sci. Instr. 82, 071101 (2011).