Quantum correlations of the photon fields in a waveguide quantum electrodynamics

TL;DR

The paper develops a time-dependent quantum framework for waveguide quantum electrodynamics by deriving the explicit positive-frequency field operator $E^+(x,t)$ under Markov approximation and using it to compute $G^{(1)}$ and $G^{(2)}$ for subspaces with one or two excitations. The approach yields analytic expressions for photon amplitudes and spectra, including a Hanbury Brown and Twiss–type interference term in $G^{(2)}$ arising from two indistinguishable pathways (spontaneous emission and scattered photons) whose visibility depends on detector geometry. In the plane-wave limit, the results reproduce known transmission/reflection spectra and reveal stimulated Rabi features at frequency $2\sqrt{\Lambda}$, with $\Lambda=\int d\omega\,g^2(\omega)$. The framework accommodates arbitrary single-photon pulse shapes via $f(\omega)$, enabling broad applicability to 1D open-waveguide setups and informing quantum photonic-network design.

Abstract

We present a time-dependent quantum calculations of the first order and second order photon correlation functions for the scattering of a single-photon pulse on a two-level atom (qubit) embedded in a one-dimensional open waveguide. Within Markov approximation we find the analytic expression for the quantum operator of positive frequency electric field. We restricted Hilbert space of initial states by the states with one and two excitations and show that the photon probability amplitudes are given by the off-diagonal matrix elements of the electric field operator between these states. For two-excitation initial state where the atom is excited and there exists a single photon in a waveguide we calculate the second order correlation function which describes the measurements by two detectors at two different space-time points. The second order correlation function exhibits the interference term showing that the measurements of two detectors are correlated. This interference is similar to that found in the Hanbury Brown and Twiss correlation experiment with two indistinguishable photons.

Figures (3)

• Figure 1: A schematic diagram for a two-photon interference. First pathway (solid arrows): the spontaneously emitted photon $\textit{sp}$ is measured by detector $D_1$ and then the scattering photon $\textit{sc}$ is measured by detector $D_2$. The other pathway (dashed arrows): the scattering photon $\textit{sc}$ first is measured by detector $D_1$ and then the spontaneous emited photon $\textit{sp}$ is measured by detector $D_2$.
• Figure 2: Frequency spectra of the normalized first-order correlation function (\ref{['g1_norm_posit']}) for transmitted field, $x >0$. Different lines correspond to different frequency shifts: blue dashed line, $2\sqrt{\Lambda} = \Gamma$; red solid line, $2\sqrt{\Lambda} = 2\Gamma$; Green dotted line, $2\sqrt{\Lambda} = 5\Gamma$; Other parameters: $\Gamma /\Omega = 0.02, \; \Delta/\Omega = 0.1$.
• Figure 3: The dependence of second-order correlation function on delay time $\Delta T$ for different positions of detectors (shown at the top of each plot). Blue dashed line corresponds to the sum of bare squared modulus of the correlation function $g^{(2)}_{mod}=(G^{(2)}_{path1}+G^{(2)}_{path2})2\pi d^4/\hbar^4\Delta\Gamma^3$; red line corresponds to the interference term $g^{(2)}_{int}=G^{(2)}_{int}2\pi d^4/\hbar^4\Delta\Gamma^3$; black solid line corresponds to full correlation function $g^{(2)}_{full}=G^{(2)}_{full}2\pi d^4/\hbar^4\Delta\Gamma^3$. Top row of graphs (a)-(d) is plotted for resonant frequency of incident field, $\omega_0/\Omega = 1$. Bottom row (e)-(h) is plotted for detuned incident field, $\omega_0/\Omega = 1.1$. Decay rate $\Gamma /\Omega = 0.02$.