Localized to delocalized spatial quantum correlation evolution in structured bright twin beams

TL;DR

The paper addresses how spatial quantum correlations in bright twin beams evolve during propagation and how the spatial structure of the pump influences this evolution. It develops an analytical framework for the two-photon amplitude $\mathcal{F}$ and its propagated form $\tilde{\mathbb{F}}$ in a four-wave-mixing double-$\Lambda$ system and corroborates the theory with experiments in hot $^{85}$Rb, comparing Gaussian and Laguerre-Gaussian pumps. Key findings show that Gaussian pumps localize correlations in the near field with transitions to localized far-field anti-correlations, while Laguerre-Gaussian pumps drive delocalized correlations that form ring/doughnut patterns, with OAM conservation $l_c=2l_{pump}-l_{pr}$ observed and coherence areas evolving with propagation. These results demonstrate that pump shaping can tailor high-dimensional spatial quantum correlations for applications in quantum imaging, cryptography, and information processing.

Abstract

Quantum correlations in the spatial domain hold great promise for applications in quantum imaging, quantum cryptography and quantum information processing, owing to the infinite dimensionality of the associated Hilbert space. Here, we present a theoretical investigation, complemented by experimental measurements, of the propagation dynamics of the spatial quantum correlations in bright structured twin beams generated via a four-wave mixing process in a double-$Λ$ configuration in atomic vapor. We derive an analytical expression describing the evolution of the spatial quantum correlation distribution from the near field to the far field. To qualitatively support the theoretical predictions, we perform experiments measuring intensity-difference noise between different spatial subregions of the twin beams as they propagate from the near field to the far field. The presence of quantum correlations is manifested as squeezing in the intensity difference noise measurement. With a Gaussian pump, we observe localized correlations in the near field and localized anti-correlations in the far field. In contrast, with a structured Laguerre-Gaussian pump, there is a transition from localized correlations in the near field to delocalized correlations in the far field. The present results offer valuable insights into the fundamental behavior of spatial quantum correlations and open possibilities for potential applications in quantum information, quantum imaging and sensing.

Figures (8)

• Figure 1: Energy levels of $^{85}$Rb involved in four-wave mixing in double-$\Lambda$ configuration.
• Figure 2: Analytical results with the Gaussian pump: (top row) two-photon probability distribution obtained by fixing $x_{pr}$ = 0.25 mm and $x_c$ = 0.25 mm, (middle row) conditional probability distribution of the conjugate with the probe coordinates fixed at $x_{pr}$ = 0.25 mm and $y_{pr}$ = 0, (bottom row) line profile obtained from the conditional probability distribution (dashed line in the line profile plot correspond to $x_c$ = 0). The red dashed lines denote the beam width defined at 1/$e^2$ of the maximum value. All propagation distances are written in terms of the pump Rayleigh range ($z{_R}$)
• Figure 3: Analytical results with the Laguerre-Gaussian pump: (top row) two-photon probability distribution obtained by fixing $x_{pr}$ = 0.25 mm and $x_c$ = 0.25 mm, (middle row) conditional probability distribution of the conjugate with the probe coordinates fixed at $x_{pr}$ = 0.25 mm and $y_{pr}$ = 0, (bottom row) line profile obtained from the conditional probability distribution (dashed line in the line profile plot correspond to $x_c$ = 0). All propagation distances are written in terms of the pump Rayleigh range ($z{_R}$).
• Figure 4: The theoretically estimated coherence areas of the twin beams plotted against the propagation distance from the cell for the Gaussian and LG pump.
• Figure 5: Schematic layout of the experimental setup used to study spatial quantum correlation dynamics of bright twin beams. SPP, Spiral phase plate; HWP, Half-wave plate; QWP, Quarter-wave plate; PBS, Polarizing beamsplitter; SA, Spectrum analyzer, L1-L8, lenses; PD1, PD2, Photodetector; BD, Beam dump.