Temporally-long C-band heralded single photons generated from hot atoms

TL;DR

This work develops a theoretical framework and experimental demonstration for temporally long C-band heralded single photons generated from hot atoms via diamond-type SFWM. By solving optical Bloch equations and introducing the atomic velocity group $\omega_{D0}$ and the atom-frame detuning $\Delta_{\rm atom}$, the authors connect Doppler-broadened dynamics to the biphoton temporal width $\tau_b$. They achieve a record $\tau_b$ of $28.3$ ns (linewidth $11$ MHz) for C-band HSPs in a hot $^{87}$Rb vapor, validating the model and showing that hot-atom sources can reach regimes previously observed only with cold atoms. The results advance the understanding of SFWM biphoton generation in diamond-type schemes and have practical implications for quantum repeater architectures requiring narrow, tunable, fiber-compatible photons.

Abstract

C-band photons are recognized for having the lowest loss coefficient in optical fibers, making them highly favorable for optical fiber-based communication. In this study, we systematically investigated the temporal width of C-band heralded single photons and developed a theoretical model for biphoton generation via the spontaneous four-wave mixing (SFWM) process using a diamond-type transition scheme, which has not been previously reported. Our experimental data on temporal width closely aligns with the predictions of this model. Additionally, we introduced a new concept: the atomic velocity group relating to the two-photon resonance condition and the one-photon detuning in this atomic frame. These two parameters are crucial for understanding the behavior of the biphoton source. The concept indicates that the hot-atom source behaves similarly to the cold-atom source. Guided by our theoretical model, we observed 1529-nm (C-band) heralded single photons with a temporal width of 28.3$\pm$0.6 ns, corresponding to a linewidth of 11.0$\pm$0.2 MHz. For comparison, the ultimate linewidth limit is 6.1 MHz, determined by the natural linewidth of the atoms. Among all atom-based sources of 1300 to 1550 nm heralded single photons utilizing either cold or hot atoms, the temporal width achieved in this work represents the first instance of a width exceeding 10 ns, making it (or its linewidth) the longest (or narrowest) record to date. This work significantly enhances our understanding of diamond-type or cascade-type SFWM biphoton generation and marks an important milestone in achieving greater temporal width in atom-based sources of C-band heralded single photons.

Figures (14)

• Figure 1: (a) The diamond-type transition scheme or cascade-type scheme, where $|2\rangle$ and $|3\rangle$ are the same state. In the SFWM process, the coupling and pump laser fields, along with the signal and idler single photons form the four-photon resonance, where $\Delta_c$, $\Delta_p$, $\Delta_s$, and $\Delta_i$ (= $\Delta_c+\Delta_p-\Delta_s$) represent their one-photon detunings, and $\Omega_c$, $\Omega_p$, $\Omega_s$, and $\Omega_i$ are the Rabi frequencies. The actual energy levels of the atoms in the experiment are specified in Sec. \ref{['sec:setup']}. (b) The transition scheme viewed in the atom frame, where the Doppler shift of the atoms causes the coupling and pump fields to satisfy the two-photon resonance. $\Delta_{\rm atom}$ represents the one-photon detuning of the coupling-pump transition. The signal (or idler) photon decays with a frequency near the resonance frequency of the transition $|4\rangle$$\rightarrow$$|3\rangle$ (or $|3\rangle$$\rightarrow$$|1\rangle$).
• Figure 2: Theoretical predictions (red lines) of the representative two-photon correlation functions, or biphoton wave packets (a) before and (b) after passing through the etalons, whose characteristics are illustrated in Appendix \ref{['sec:Etalon']}. Since the signal photon appears before the idler photon, we calculated the generation probability of the idler photon right after the atomic vapor cell upon the appearance of the signal photon. Each inset shows the spectrum that gives the wave packet in the main plot. In the calculation, $(\alpha, \Omega_c, \Omega_p, \gamma)$ = $(420, 17\Gamma, 78\Gamma, 0.4\Gamma)$ and $(\omega_{D0}, \Delta_{\rm atom})$ = $(0, 380\Gamma)$ or 2$\pi$$\times$(0, 2.28 GHz). The spectral FWHMs are (a) 12.0 and (b) 11.4 MHz. The blue line in the main plot of (b) represents the best fit, with an FWHM of 25.9 ns.
• Figure 3: Theoretical predictions of the temporal FWHM as a function of $\Delta_{\rm{atom}}$ for different values of $\omega_{D0}$. Blue, green, and red lines correspond to $\omega_{D0}/2\pi$ = 0, 0.24, and 0.48 GHz, respectively. For all cases, we set $\alpha$ = 400 and $\gamma$ = 0.4$\Gamma$. We set $\Omega_c\times\Omega_p$ = 1300$\Gamma^2$ for the central solid and nearby dashed lines and increased (decreased) the value by 20% for the lower (upper) solid lines. We also set $\Omega_p/\Omega_c$ = 4.6 for the solid lines and increased (decreased) the value by 20% for the upper (lower) dashed lines. The transition dipole matrix element, Clebsch-Gordan coefficients, and ratio of pump to coupling powers determine $\Omega_p/\Omega_c$ = 4.6.
• Figure 4: Effect of the optical depth $\alpha$ on the temporal FWHM. Blue, green, and red lines represent $\omega_{D0}/2\pi$ = 0, 0.24, and 0.48 GHz, respectively. For all lines, $\gamma$ = 0.4$\Gamma$, $\Omega_c \times \Omega_p$ = 1300$\Gamma^2$, and $\Omega_p/\Omega_c$ = 4.6. We set $\alpha$ = 450, 400, and 350 for the upper, central, and lower lines of the same color.
• Figure 5: Effect of the decoherence rate $\gamma$ on the temporal FWHM. Blue and red lines represent $\omega_{D0}/2\pi$ = 0 and 0.48 GHz, respectively. For all lines, $\alpha$ = 400, $\Omega_c$$\times$$\Omega_p$ = 1300$\Gamma^2$, and $\Omega_p/\Omega_c$ = 4.6. We set $\gamma$ = 0.2, 0.4, and 0.6 for the upper, central, and lower lines of the same color, respectively.