Photon statistics in chiral waveguide QED: I Mean field and perturbative expansions

Abstract

Waveguide Quantum Electrodynamics (WQED) offers a suitable stage for controlling the interaction of light with atoms, allowing for collective phenomena such as super- and subradiance. In a chiral waveguide setup, the quantum state evolves through all the Hilbert space, rendering an exact theoretical treatment exponentially hard and unobtained to date for more than $\sim 20$ atoms. In this work, we use a computationally efficient higher order mean-field approximation to model the radiation dynamics in a chirally coupled array of atoms, showing good agreement with recent experimental results. Further, based on a perturbative approximation of the full dynamics, we develop an analytical solution that captures photon statistics for a moderate atom number, $N$, and a homogeneous atom-waveguide coupling, $β$. Finally, we show that capturing the onset of second-order coherence from a fully inverted state requires a fourth-order mean-field approximation, as lower-order treatments fail to account for the necessary four-body correlations. These results illustrate the complex behavior of a symmetry-lacking system, and the methods discussed here provide systematic analytical solutions to which semi-classical methods such as the cumulant expansion or the truncated Wigner approximation can be benchmarked.

Figures (6)

• Figure 1: (a) Schematic of the setup. $N$ two-level atoms are coupled to a unidirectional waveguide. The coupling constant to the right waveguided mode is $\beta$ and to freespace modes is $1 - \beta$. This realizes a cascaded quantum system. We are interested in correlators of the output mode $\hat{a}_{\text{out}}$, such as the output power $P(t) = \langle \hat{a}_{\text{out}}^\dagger(t) \hat{a}_{\text{out}}(t) \rangle$ and Glauber's second-order quantum correlation function $G^{(2)}(t_1,t_2) = \langle \hat{a}_{\text{out}}^\dagger(t_1) \hat{a}_{\text{out}}^\dagger(t_2) \hat{a}_{\text{out}}(t_2)\hat{a}_{\text{out}}(t_1) \rangle$. In Ref. bach2024emergence, this model is implemented using the D2 transition of nanofiber-coupled cold cesium atoms with excited state lifetime 30.5 ns (b) We show experimental data from Ref. bach2024emergence$g^{(2)}(4.5 \,\text{ns},t) = G^{(2)}(4.5\, \text{ns},t) / (P(4.5\, \text{ns})P(t))$ (black data points) for the ensemble decaying from near maximal inversion alongside the theoretical prediction by a second order cumulant expansion or mean field (MF2) method as laid out in this work. (c) we show experimental data for $g^{(2)}(t,t)$ as well as predictions by a MF2 method and the truncated Wigner approximation (TWA) used in Refs. TWA2024cascadedbach2024emergence. The shaded lightblue area indicates the statistical uncertainty of the TWA method. (d) We show experimental data of $P(t)$ alongside the mean field prediction. We also show the theoretically predicted coherent component of $P(t)$, as explained in the text. The inset shows the excited population fraction during and after the inversion pulse as predicted by MF2. Details of the experimental parameters are given in Sec. \ref{['sec:simulation_experiment']}.
• Figure 2: Comparison of the radiation and excitation dynamics for the MC, MCSA, MF2, and MF3 approaches with $N = 18$ atoms and $\beta = 0.5$ starting from the ground state. The total power emitted into the waveguide, $P(t)$, after the pulse and the small coherent component, $P_{coh}(t)$, are shown. $P_{coh}(t)$ is multiplied by $10$ for clarity. The inset shows the excitation during and after the pulse.
• Figure 3: Comparison of the emission power into the waveguide, $P(t)$, and the second-order correlator, $g^{(2)}(0,t)$, for the analytical solutions truncated at $k_{max} = 6,7,8$, MCSA, MF2, and MF3 approaches with $N = 900$ atoms and $\beta = 0.011$. (a) $P(t)$ starting from the fully inverted state. (b) $P(t)$ starting from a lower Dicke state $\ket{\psi}_{N-1} \propto \sum_n \hat{\sigma}_{n} \ket{ee\dots e}$. (c) the ratio of the power from (b) to the power from (a) gives $g^{(2)}(0,t)$. The insets show the excitation fraction decaying with time.
• Figure 4: Two-time second order correlation $g^{(2)}(t_1,t_2)$ using the MF2 method. (a) near maximal inversion using a pulse area $A = 1.1 \pi$. (b) at maximal inversion using a pulse area $A = 1.07 \pi$.
• Figure 5: Comparison of the dynamics of $g^{(2)}(t,t)$ starting from the totally inverted state via MF, analytical approximations of truncation orders $7$ and $8$, and the TWA method from Refs. TWA2024cascadedbach2024emergence. The cumulants in MF2 were done both at the 'standard' atomic level (denoted as 'MF2'), and at a 'field' level (denoted as 'MF2 field'). The gray shaded area represents the statistical uncertainty of the TWA data. Parameters are $\beta = 0.01$ and $N=300$ (a) or $N=150$ (b)