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Triphoton generation near atomic resonance via SSWM: Harmonic expansion for accurate optical response

Jianming Wen

Abstract

Quantum correlations of time-frequency-entangled photon pairs generated via parametric processes are critically influenced by both the linear and nonlinear optical responses of the medium. This sensitivity is especially significant in schemes utilizing atomic ensembles with well-defined energy level structures near resonance. However, conventional theoretical approaches often fall short in accurately calculating the optical responses--particularly when a single atomic transition is simultaneously driven by multiple light fields with (significantly) different intensities. To address this limitation, we generalize the harmonic expansion method originally introduced by Wen for biphoton generation near atomic resonance. As a case study, we apply this generalized approach to the reliable direct generation of time-energy-entangled W-state triphotons via spontaneous six-wave mixing in a five-level asymmetric-M atomic system. Our results demonstrate the method's superior accuracy and self-consistency, offering clear advantages over traditional calculation techniques.

Triphoton generation near atomic resonance via SSWM: Harmonic expansion for accurate optical response

Abstract

Quantum correlations of time-frequency-entangled photon pairs generated via parametric processes are critically influenced by both the linear and nonlinear optical responses of the medium. This sensitivity is especially significant in schemes utilizing atomic ensembles with well-defined energy level structures near resonance. However, conventional theoretical approaches often fall short in accurately calculating the optical responses--particularly when a single atomic transition is simultaneously driven by multiple light fields with (significantly) different intensities. To address this limitation, we generalize the harmonic expansion method originally introduced by Wen for biphoton generation near atomic resonance. As a case study, we apply this generalized approach to the reliable direct generation of time-energy-entangled W-state triphotons via spontaneous six-wave mixing in a five-level asymmetric-M atomic system. Our results demonstrate the method's superior accuracy and self-consistency, offering clear advantages over traditional calculation techniques.
Paper Structure (11 equations, 3 figures)

This paper contains 11 equations, 3 figures.

Figures (3)

  • Figure 1: Direct generation of continuous-mode, time-energy entangled W-triphotons via single-step SSWM in a laser-cooled five-level atomic system with an asymmetric-M structure. (a) Energy-level scheme: one weak pump field E$_p(\mathbf{k}_p,\omega_p)$ and two strong coupling fields E$_{c1}(\mathbf{k}_{c1},\omega_{c1})$ and E$_{c2}(\mathbf{k}_{c2},\omega_{c2})$ drive the system, leading to spontaneous W-triphoton simultaneous emission $\hat{E}_{s1}(\mathbf{k}_{s1},\omega_{s1})$, $\hat{E}_{s2}(\mathbf{k}_{s2},\omega_{s2})$, and $\hat{E}_{s3}(\mathbf{k}_{s3},\omega_{s3})$ through SSWM. (b) Spatial configuration of incident lasers and emitted triphotons.
  • Figure 2: Representative resonance profiles of the fifth-order nonlinear susceptibility $|\chi^{5}(\delta_1,\delta_2)|$ for the following parameters: $N=10^{17}$, $L=1.5$ cm, $\gamma_{31}=2\pi\times3$ MHz, $\gamma_{41}=\gamma_{31}$, $\gamma_{51}=0.2\gamma_{31}$, $\gamma_{21}=0.04\gamma_{31}$, $\Delta_{c2}=0$, and $\Delta_p=-2\pi\times300$ MHz. (a) $\Delta_{c1}=0$ MHz, $\Omega_{c1}=\Omega_{c2}=20\gamma_{31}$ and (b) $\Delta_{c1}=300$ MHz, $5\Omega_{c1}=\Omega_{c2}=50\gamma_{31}$.
  • Figure 3: Normalized photon coincidence detections $R_3$ and $R_2$ in three operating regimes. (a) Threefold coincidences in the damped Rabi-oscillation regime, using the same parameters as Fig. \ref{['fig2']} except $\Omega_{c1}=\Omega_{c2}=5\gamma_{31}$ and $\text{OD}=1.5$; (b) corresponding conditional twofold coincidences. (c) Three-photon coincidences in the group-delay regime, with $\Omega_{c1}=\Omega_{c2}=1.6\gamma_{31}$ and $\text{OD}=88$; (d) corresponding conditional two-photon coincidences. (e) Triple coincidences in the hybrid regime, using parameters from (c) except $\text{OD}=8$; (f) conditional two-photon coincidences in this regime. Insets: top-view contours.