Photon entanglement signatures in difference-frequency-generation

Abstract

In response to quantum optical fields, pairs of molecules generate coherent nonlinear spectroscopy signals. Homodyne signals are given by sums over terms each being a product of Liouville space pathways of the pair of molecules times the corresponding optical field correlation function. For classical fields all field correlation functions may be factorized and become identical products of field amplitudes. The signal is then given by the absolute square of a susceptibility which in turn is a sum over pathways of a single molecule. The molecular pathways of different molecules in the pair are uncorrelated in this case (each path of a given molecule can be accompanied by any path of the other). However, entangled photons create an entanglement between the molecular pathways.We use the superoperator nonequlibrium Green's functions formalism to demonstrate the signatures of this pathway-entanglement in the difference frequency generation signal. Comparison is made with an analogous incoherent two-photon fluorescence signal.

Fig. 1

DFG: (A) wave-vector configuration of the optical fields corresponding to the phase matching k₃ = k₁ − k₂. (B) molecular level scheme. (C) Liouville space pathways for the pair of molecules contributing to the signal molecule a (C1,C2) and b (C1*,C2*)

Fig. 2

Optical field SNGF which contribute to the DFG process. Interactions with molecule a occur at times t₄, t₂ (red arrows), and with molecule b at times t₃, t₁(blue arrows). Hilbert Space expressions for the signal are obtained by proceeding clockwise along the loop, starting at the bottom left.

Fig. 3

Nonlinear spectroscopy with entangled photons. A non-linear parametric down conversion χ ⁽²⁾ crystal PDC is used to obtain entangled photon pairs from the classical pump beam by parametric down conversion. BS are balanced 50 : 50 beam splitters. φ is a phase shift in one of the interferometer arms. The sample is a collection of N three-level molecules. a′₁, a′₂ are annihilation operators for the incoming non-entangled (canonical) modes and a₁, a₂ represent the entangled modes.

Fig. 4

Panels (A–C) are 2D spectra of coherent DFG signals. (A) generated by classical fields, (B) generated by maximally entangled photons (PDC/MZI) in the low pump intensity limit. (C) generated by fields in a coherent state of low intensity. (D) the incoherent TPEF signal with classical k₁, k₂ modes.

Fig. 5

(A) 1D section of the 2D DFG spectra along of Fig. 4 the line (d) in panels (A, dotted curve), (B, solid thick curve), (C, solid thin curve). (B) same as panel (A) but for a different section (the line (e)) in Fig. 4.

Fig. 6

(A) The incoherent TPF pathway contributing at ω₃ ≈ ω₁ − ω₂ resonance. Mode k₃ is spontaneously generated by classical modes k₁, k₂. (B) CTPL diagram for conventional incoherent two-photon emitted fluorescence (TPEF) signal, when mode k₃, k₂ are spontaneously generated by the classical mode k₁; (C) the loop diagram for two-photon induced fluorescence (TPIF) with classical k₁, k₂ modes; maximum of the signal corresponds to ω₃ ≈ ω₁ + ω₂.