Full Schmidt characterization of spatiotemporally entangled states produced from spontaneous parametric down-conversion

Abstract

The full Schmidt decomposition of spatiotemporally entangled states generated from spontaneous parametric down-conversion (SPDC) has not been carried out until now due to the immense computational complexity arising from the large dimensionalities of the states. In this Letter, we utilize the rotational symmetry of the states to reduce the complexity by at least four orders of magnitude and carry out the decomposition to reveal the precise forms of the spatiotemporal Schmidt modes and the Schmidt spectrum spanning over 10^4 modes. We show that the Schmidt modes have a phase profile with a transverse spatial vortex structure that endows them with orbital angular momentum at all frequencies. In the high-gain regime, these Schmidt modes broaden and the Schmidt spectrum narrows with increasing pump strength. Our work can spur novel applications at the intersection of quantum imaging and spectroscopy that utilize entangled states produced from SPDC.

Figures (3)

• Figure 1: Low-gain SPDC: (a) Spatiotemporal intensity profile $I(q_{sx},\omega_{s})=G^{(1)}(q_{sx},\omega_{s};q_{sx},\omega_{s})=\sum_{l,m}\lambda_{lm}|u_{lm}(q_{sx},\omega_{s})|^2$, and (b) Spatiotemporal Schmidt spectrum $\left\{\lambda_{lm}\right\}$.
• Figure 2: Low-gain SPDC: (a) Magnitude profiles $|u_{lm}(q_{sx},\omega_{s})|$ of the Schmidt modes for $|l|=0,1,2$ and $m=0,1,2,3$. (b) and (c) depict the variation of the Schmidt number $K$ with pump beam-waist $w_{p}$ and crystal length $L$, respectively.
• Figure 3: High-gain SPDC: (a) depicts the broadening of the Schmidt modes $|u_{00}(q_{sx},\omega_{s})|$ and $|u_{11}(q_{sx},\omega_{s})|$ with increasing gain for $g=1,4,$ and $8$ . (b) depicts the total integrated intensity of the signal field (left y-axis in violet) and Schmidt number $K$ (right y-axis in red) with increasing $g$.