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Scientific Reports

The X-like shaped spatiotemporal structure of the biphoton entangled state in a cold two-level atomic ensemble

Dasen Zhang, Zhi‐Ming Zhang

Published Feb 20, 2017• 1 citation
Publisher

Abstract

We study the spatiotemporal structure of the biphoton entangled state generated by the four-wave mixing (FWM) process in a cold two-level atomic ensemble. We analyze, for the first time, the X-like shaped structure of the biphoton entangled state and the geometry of the biphoton correlation for different lengths and densities of the cold atomic ensemble. The propagation equations of the photon pairs generated from FWM process are derived in a spatiotemporal framework. By means of the input-output relations of the propagation equations, the biphoton amplitude function is obtained in a spatiotemporal domain. In the given frequency range, the biphoton amplitude displays an X-like shaped geometry, nonfactorizable in the space-time domain. Such an X-like shaped spatiotemporal structure is caused by the phase matching and the FWM gain. The former leads to the X-like shaped envelope of the biphoton correlation, while the latter gives rise to the oscillations around the X-like shaped envelope.

DOI
10.1038/srep42373

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We study the spatiotemporal structure of the biphoton entangled state generated by the four-wave mixing (FWM) process in a cold two-level atomic ensemble. We analyze, for the first time, the X-like shaped structure of the biphoton entangled state and the geometry of the biphoton correlation for different lengths and densities of the cold atomic ensemble. The propagation equations of the photon pairs generated from FWM process are derived in a spatiotemporal framework. By means of the input-output relations of the propagation equations, the biphoton amplitude function is obtained in a spatiotemporal domain. In the given frequency range, the biphoton amplitude displays an X-like shaped geometry, nonfactorizable in the space-time domain. Such an X-like shaped spatiotemporal structure is caused by the phase matching and the FWM gain. The former leads to the X-like shaped envelope of the biphoton correlation, while the latter gives rise to the oscillations around the X-like shaped envelope.
We investigate the spatiotemporal structure of the biphoton entangled state produced by parametric down-conversion (PDC) at the output face of the nonlinear crystal. We analyze the geometry of biphoton correlation for different gain regimes (from ultralow to high), different crystal lengths, and different tuning angles of the crystal. While for collinear or quasicollinear phase matching a X-shaped geometry, nonfactorizable in space and time, dominates, in the highly noncollinear conditions we observe a remarkable transition to a factorizable geometry. We show that the geometry of spatiotemporal correlation is a consequence of the angle-frequency relationship imposed by phase matching and that the fully spatiotemporal analysis provides a key to control the spatiotemporal properties of the PDC entangled state and in particular to access a biphoton localization in time and space in the femtosecond and micrometer range, respectively.
We investigate the spatiotemporal structure of the biphoton correlation in type-II parametric down-conversion (PDC). As in type-I PDC [Phys. Rev. Lett. 102, 223601 (2009)], we find that the correlation is nonfactorizable in space and time. Differently from type I, the type-II correlation in the spontaneous regime displays an asymmetric V-shaped structure in any cross section including time and one transverse dimension. This asymmetry along the temporal coordinate originates from the signal-idler group velocity mismatch and tends to disappear as the parametric gain is raised. We observe a progressive transition toward a symmetric X-shaped geometry similar to that found in type I when stimulated PDC becomes dominant. We also give quantitative evaluations of the localization in space and in time of the correlation, analyze its behavior for different crystal tuning angles, and underline qualitative differences with respect to type-I PDC.
We investigate the nonlinear interaction between two photons in a single-input pulse at an atomic two-level nonlinearity. A one-dimensional model for the propagation of light to and from the atom is used to describe the precise spatiotemporal coherence of the two-photon state. It is shown that the interaction generates spatiotemporal entanglement in the output state similar to the entanglement observed in parametric down-conversion. A method of generating photon pairs from coherent pump light using this quantum-mechanical four-wave mixing process is proposed.
We have theoretically studied the space-time entangled biphoton state generated from a two-level atomic system. In the photon counting measurement, the two-photon coincidence counting rate is a damped oscillation. The oscillation period is determined by the effective Rabi frequency and the damping rate is determined by the linewidth of the inhomogeneous-broadened ground state and the dipole dephasing rate. In an optical-pathway-balanced configuration, the two-photon temporal correlation shows an antibunching effect which corresponds to the interference between two types of nonlinear four-wave mixing processes occurring in such a two-level system. The visibility of the normalized second-order quantum coherence function ${g}^{(2)}(\ensuremath{\tau})$ increases along with the increase of the effective Rabi frequency, but has an upper limit at 45%. We find agreement between the theory and the experiments [P. Kolchin et al., Phys. Rev. Lett. 97, 113602 (2006); S. Du, J.-M. Wen, M. H. Rubin, and G. Y. Yin, ibid 98, 053601 (2007)].
We investigate the spectral properties of the biphoton state from the cascade emissions of cold atomic ensembles, which are composed of a telecommunication photon (signal) followed by an infrared one (idler) via four-wave mixing. With adiabatic conditions for Gaussian driving pulses of width $\ensuremath{\tau}$, the spectrum of the biphoton state has the form of a Gaussian that conserves signal and idler photon energies within $\ensuremath{\hbar}/\ensuremath{\tau}$ modulated by a Lorentzian with a superradiant linewidth. Multiplexing the atomic ensembles with frequency-shifted cascade emissions, we may manipulate and shape the spectrum of the biphoton state. The entropy of entanglement is derived from Schmidt decomposition, which can be larger if we multiplex the atomic ensembles in a way that conserves signal and idler photon central energies. The eigenvalues in Schmidt bases are degenerate in pairs for symmetric spectral shaping in which the mode probability densities show interference patterns. We also demonstrate the excess entropy of entanglement that comes from continuous frequency space, which scales up the total entropy. The scheme of the multiplexed cascade-emitted biphoton state provides multimode structures that are useful in long-distance quantum communication and multimode quantum information processing.