Bloch-Messiah decomposition and Magnus expansion for parametric down-conversion with monochromatic pump

Abstract

We discuss the Bloch-Messiah decomposition for the broadband squeezed light generated by type-I parametric down-conversion with monochromatic pump. Using an exact solution for this process, we evaluate the squeezing parameters and the corresponding squeezing eigenmodes. Next, we consider the Magnus expansion of the quantum-mechanical evolution operator for this process and obtain its first three approximation orders. Using these approximated solutions, we evaluate the corresponding approximations for the Bloch-Messiah decomposition. Our results allow us to conclude that the first-order approximation of the Magnus expansion is sufficient for description of the broadband squeezed light for squeezing values below 12.5 dB. For higher degrees of squeezing we show fast convergence of the Magnus series providing a good approximation for the exact solution already in the third order. We propose a quantitative criterion for this ultra-high-gain regime of parametric down-conversion when the higher-orders terms of the Magnus expansion, known in the literature as the operator-ordering effects, become necessary. arXiv:1806.09384v1 [quant-ph]

Bloch-Messiah reduction for twin beams of light

“…In the present work we limit our consideration to the first order of the Magnus expansion, which has been found a good approximation for not very high squeezing, below 12 dB [25,42]. An analytic treatment of higher orders of the Magnus expansion in PDC can be found in Refs.…”

“…In the case of degenerate PDC, where the signal and the idler photons are indistinguishable, the Bloch-Messiah reduction has proven to be a powerful tool for determining the squeezing eigenmodes for single-pass optical parametric amplifiers (OPA) [19,20] and multi-pass optical parametric oscillators [21][22][23][24]. Application of this formalism to a degenerate PDC with a monochromatic pump, undertaken recently by some of us [25], resulted in a successful identification of bichromatic squeezing eigenmodes. For nondegenerate high-gain PDC it has been found [26] that up to certain level of gain the Schmidt modes of photon pairs determine a modal decomposition for the signal and idler beams such that the corresponding modes are in a two-mode squeezed state.…”

Bloch-Messiah decomposition and Magnus expansion for parametric down-conversion with monochromatic pump: role of the gain

“…The role of operator ordering in PDC can be naturally studied in the framework of the Magnus expansion of a T -ordered exponential [10,11]. Neglecting the ordering corresponds to the first order of the Magnus series, which is enough for an approximate description of squeezed light for moderate squeezing, below 12 dB [6,12]. At higher squeezing higher orders of the series are necessary for a proper description of the field state.…”

“…At higher squeezing higher orders of the series are necessary for a proper description of the field state. Recently a general treatment of the Magnus expansion for PDC with a monochromatic pump has been undertaken [12], which allowed one to analyze the convergence of the Magnus series to the exact analytic solution existing in this particular case. In the present work we develop further the approach of Ref.…”

“…In the present work we develop further the approach of Ref. [12] and analyze explicitly the dependence of the squeezing spectrum and the squeezing angle on the parametric gain. We demonstrate the curious fact that the contribution of the higher orders of the Magnus expansion is important for intermediate phasemismatch angle only, being zero at the perfect phase-matching and negligible at a large phase mismatch.…”