Bloch-Messiah decomposition and Magnus expansion in parametric down-conversion of light

Introduction. Squeezed states of light, generated in parametric down-conversion (PDC) and other nonlinear optical processes, represent an interesting object of fundamental research and an important resource for quantum... [ view full abstract ]

Introduction. Squeezed states of light, generated in parametric down-conversion (PDC) and other nonlinear optical processes, represent an interesting object of fundamental research and an important resource for quantum metrology and continuous-variable quantum information processing [1]. Spectrally broadband PDC opens up the possibility for generating multiple modes of squeezed light at once. Efficient implementation and detection of squeezing requires precise definition of the squeezing eigenmodes [2], which is based on the Bloch-Messiah decomposition of the field transformation in the nonlinear crystal [3].

Methods. We apply the formalism of the Bloch-Messiah decomposition to broadband collinear type-I PDC with undepleted monochromatic plane-wave pump, for which an exact analytical solution is available. The analytical solution allows us to define the squeezing eigenmodes and find the squeezing spectrum. Alternatively, we apply the Magnus expansion [4] to the field evolution operator and investigate the influence of operator ordering on the degree of squeezing and the squeezing eigenmodes.

Results. We show that for a monochromatic pump the squeezing eigenmodes have the modal functions varying as cosine and sine of time with a phase depending on the crystal dispersion. For squeezing below 12.5 dB the first order of Magnus expansion gives a good approximation for the field evolution, in agreement with a recent numerical study [4]. The Magnus expansion is convergent up to 27 dB of squeezing at degeneracy. However, for a proper description of the degree and the angle of squeezing above 12.5 dB higher orders of Magnus expansion are necessary.

[1] U. L. Andersen, T. Gehring, C. Marquardt, and G. Leuchs, Phys. Scr. 91, 053001 (2016).

[2] R. S. Bennink and R. W. Boyd, Phys Rev. A 66, 053815 (2002).

[3] S. L. Braunstein, Phys. Rev. A 71, 055801 (2005).

[4] A. Christ, B. Brecht, W. Mauerer, and C. Silberhorn, New J. Phys. 15, 053038 (2013).