Multiphoton-scattering theory and generalized master equations

We develop a scattering theory to investigate the multiphoton transmission in a one-dimensional waveguide in the presence of quantum emitters. It is based on a path integral formalism, uses displacement transformations, and does not require the Markov approximation. We obtain the full time evolution of the global system, including the emitters and the photonic field. Our theory allows us to compute the transition amplitude between arbitrary initial and final states, as well as the $S$ matrix of the asymptotic in and out states. For the case of few incident photons in the waveguide, we also rederive a generalized master equation in the Markov limit. We compare the predictions of the developed scattering theory and that with the Markov approximation. We illustrate our methods with five examples of few-photon scattering: (i) by a two-level emitter, (ii) in the Jaynes-Cummings model; (iii) by an array of two-level emitters; (iv) by a two-level emitter in the half-end waveguide; and (v) by an array of atoms coupled to Rydberg levels. In the first two, we show the application of the scattering theory in the photon scattering by a single emitter, and examine the correctness of our theory with the well-known results. In the third example, we analyze the condition of the Markov approximation for the photon scattering in the array of emitters. In the fourth one, we show how a quantum emitter can generate entanglement of outgoing photons. Finally, we highlight the interplay between the phenomenon of electromagnetic-induced transparency and the Rydberg interaction, and show how this results in a rich variety of possibilities in the quantum statistics of the scattering photons.

Figure 1

Four examples of the problem studied here: (a) single two-level system coupled to a waveguide; (b) JC system coupled to a waveguide; (c) single two-level system coupled to a one-sided waveguide; and (d) an array of atoms coupled to a waveguide.

Figure 2

Emission and absorption of a single photon by a single emitter: (a) The spatial amplitude ${\left|\psi \left(x\right)\right|}^2$ of a right-propagating single-photon wave packet, produced by an emitter initially prepared in the excited state at $T=0$. The wave packet shown here is after an evolution time $T=10/\mathrm{\Gamma }$. (b) The probability ${\left|A\left(T\right)\right|}^2$ of the emitter in the excited state for the incident wave packet with the width $1/\gamma$ and localized at $x_0$. Here, the emission rate $\mathrm{\Gamma }=1$ into the waveguide is taken to be the unit.

Figure 3

(a) Single incident photon reflection spectra in a two-emitter system, where the distance between emitters is $d$. (b) Excitation probability of second emitter ${\left|A(2,T)\right|}^2$ as a function of time $T$ in a two-emitter system, when the second (first) emitter is initially prepared in the ground (excited) state. Here, ${\mathrm{\Gamma }}_f=0$ and $\mathrm{\Gamma }$ is taken as the unit. The Markovian results are also given by the solid (blue) curves.

Figure 4

The solid (blue) and the dashed (red) curves show the wave functions of the right- and left-moving photons after the evolution time $T$, where the first (second) emitter is initially prepared in the excited (ground) state. Here, ${\mathrm{\Gamma }}_f=0$ and $\mathrm{\Gamma }$ is taken as the unit, and the green dots denote the two emitters.

Figure 5

(a),(b) The exact and Markovian $T$-matrix elements for $E=1$. (c),(d) The second order correlation functions of outgoing photons: In (c) $\tau =0$, in (d) $E=0$, and Markovian results are shown by the solid (blue) curves. Here, ${\mathrm{\Gamma }}_f=0$ and $\mathrm{\Gamma }$ is taken as the unit.

Figure 6

(a) The phase shift of the single photon. (b) The von Neumann entropy of the reflective photons. Here, $d={10}^{-4}$, and $\mathrm{\Gamma }$ is taken as the unit.

Figure 7

The single-photon transmission spectra, where $\mathrm{\Gamma }=1, {\mathrm{\Delta }}_e=0, \mathrm{\Omega }=1$, and the atom number is 20. (a) ${\mathrm{\Gamma }}_f=0$ and $d={10}^{-4}$; (b) ${\mathrm{\Gamma }}_f=1$ and $d={10}^{-4}$; (c) ${\mathrm{\Gamma }}_f=0$ and $d={10}^{-2}$; (d) ${\mathrm{\Gamma }}_f=1$ and $d={10}^{-2}$.

Figure 8

Bunching and antibunching behaviors of transmitted photons, where $\mathrm{\Gamma }=1, {\mathrm{\Gamma }}_f=1, {\mathrm{\Delta }}_e=0, \mathrm{\Omega }=1, k_{0}d=\pi /2, U_0={10}^8$, and the atom number is 20: (a) the Feynman diagram for the $T$ matrix; (b) the second order correlation functions $g^{\left(2\right)}\left(x\right)$ for $C=C_3=C_6=1, E=0$, and $k=0$; (c) the second order correlation functions $ln{g}^{\left(2\right)}\left(0\right)$ for the uniform case $C=0.46$, where the relative momentum $k=0$, and the inset shows $g^{\left(2\right)}\left(x\right)$ for the frequency $E/2=0.2$ of each photon; (d) the schematic for the generation of the bunched and antibunched photons, where the incident photons have different frequencies. $s^{\prime }$ is the shifted energy level due to the Rydberg interaction.

Figure 9

(a) Probability of Rydberg states for the single incident photon wave packet, where $\mathrm{\Gamma }=1, {\mathrm{\Gamma }}_f=0, \mathrm{\Delta }=0, \mathrm{\Omega }=0.1, d={10}^{-4}, \gamma =0.01$, and the atom number is 20. (b) The Feynman diagram for the two excitation propagation.

Figure 10

Probabilities in the state $\left|s\right.〉$ for two counter- and copropagating polaritons with the dipolar case $C_3=1$, where $\mathrm{\Gamma }=1, {\mathrm{\Gamma }}_f=0, {\mathrm{\Delta }}_e=0, d={10}^{-4}$, and the atom number is 20. (a)–(c) show the probabilities at the instants $T=600$, 1200, and 1800 for the counterpropagation case: $\mathrm{\Omega }=0.1$ and $\gamma =0.01$; (d)–(f) show the probabilities at the instants $T=12$, 20, and 28 for the copropagation case: $\mathrm{\Omega }=1$ and $\gamma =0.1$.