Photon statistics in waveguide QED: II Exact solutions in a thermodynamic limit

Abstract

Waveguide quantum electrodynamics (WQED) offers a powerful framework for controlling light-matter interactions and realizing collective phenomena such as super- and subradiance. In general waveguide settings, the quantum dynamics spans the full Hilbert space, rendering exact theoretical treatments exponentially difficult and currently out of reach, and only a few models have exact, analytical solutions. Motivated by recent experiments, we treat the thermodynamic limit of the number of atoms, $N \rightarrow \infty$, while the homogeneous atom-waveguide coupling $β\rightarrow 0$ keeping the optical depth $4Nβ$ fixed. In this limit, a second order mean field method is exact, giving analytical solutions for the statistics of the photons emitted in the waveguide both for chiral and symmetric configurations starting from full inversion. As $N \rightarrow \infty$, the emission in freespace approaches that of an independent ensemble. However, until a special time, $\approx 1.59 \times$ the lifetime of a single-atom, we show an exponentially enhanced superradiance in the waveguide as the optical depth increases. After the special time, the emission into the waveguide exhibits subradiance. We also show that the initial shot-to-shot fluctuations in the rate of emission into the waveguide diminish in a chiral system and vanish in a symmetric system as $N$ approaches $\infty$. Additionally, the equal-time second-order correlation becomes trivial, showing that finite-size effects are essential to observe the emergence of second-order coherence. Finally, going beyond the thermodynamic limit requires higher order mean field methods. Our results illustrate finite- and infinite-body collective effects in symmetric and symmetry-lacking systems.

Figures (9)

• Figure 1: Schematic of the setup. $N$ two-level atoms are coupled to a waveguide. The coupling constant to the right waveguided mode is $\beta^+$, to the left waveguided mode is $\beta^-$, and to freespace modes is $1 - \beta^+ - \beta^-$. This paper investigates two extreme cases: 1) a unidirectional waveguide ($\beta^- = 0$, $\beta^+ = \beta$) and 2) a bidirectional and permutationally symmetric case where $\beta^+ = \beta^- = \beta/2$ and the atoms are placed in the mirror configuration with $\pi$ or $2 \pi$ phase between neighboring atoms.
• Figure 2: Comparison of the dynamics of the power radiated into a chiral waveguide starting from the inverted state for the MF2 approximation (orange), and the analytical solutions of truncation order ($k_{max}$) of 8 (red) and 7 (dark red) for $N \in \{ 900, 3 \times 10^4, 1 \times 10^5\}$ in solid, dashed, and dotted lines respectively. The inset shows the excitation for MF2.
• Figure 3: Comparison of the dynamics of the second-order correlation $g^{(2)}(0,t)$ starting from the inverted state for the MF2 approximation for a chiral system(orange), and a permutationally symmetric system (blue). $N \beta \equiv \mathcal{B} = 10$ and for $N \in \{ 900, 1 \times 10^4, 3 \times 10^4, \infty\}$. The infinite $N$ curves (in solid) come from the analytical formulas, while the other finite $N$ curves were numerically integrated.
• Figure 4: MF2 prediction of the power emitted in both directions of a symmetric waveguide starting from the fully inverted state. $\mathcal{B} = N \beta$ is fixed at $20$, while $N$ is increased until the thermodynamic limit is reached at $N \sim 2^{20}$. Legend shows $\log_2(N)$. Inset shows the enhancement factor of the total energy radiated in the waveguide due to collective effects.
• Figure 5: Normalized decay rate into the waveguide, $\Gamma(\mathcal{B},t)$, from an ensemble of a given optical depth $\mathcal{B}$ for a chiral system (solid lines) and a symmetric system (dashed lines). The $\mathcal{B} = 80, \, 100$ curves for a symmetric system grow out of figure bound and are not shown. Inset shows an oscillatory behavior in the radiation at late time for a chiral system, a signature of going through subradiant manifolds of the Hilbert space oscillationPhysRevLett.128.203601.