Heisenberg treatment of multiphoton pulses in waveguide QED with time-delayed feedback

The dynamics of waveguide-QED systems involving coherent time-delayed feedback give rise to a hierarchy of multitime correlations within the Heisenberg picture due to the induced non-Markovianity. We propose to perform a projection onto a complete set of states in the Hilbert space to decompose the multitime correlations into single-time matrix elements. To illustrate the procedure, we consider the paradigmatic example of a two-level system that couples to a semi-infinite waveguide and interacts with quantum light pulses. Our approach complements the range of available methods as it allows calculating the dynamics under the inclusion of additional dissipation channels in a numerically exact and efficient manner for multiphoton pulses of arbitrary shape where memory requirements are known in advance.

Figure 1

TLS subjected to coherent time-delayed feedback and the resulting hierarchy of matrix elements. (a) Illustration of the setup which consists of a TLS with transition energy $\hslash {\omega }_0$ between ground-state $|g\rangle$ and excited state $|e\rangle$ coupling with decay rate $\mathrm{\Gamma }$ to a semi-infinite waveguide. The closed end of the waveguide at distance $d$ from the TLS provides feedback at the delay-time $\tau$ and the TLS can be excited through the waveguide via a quantum pulse of temporal shape $f\left(t\right)$. (b) Scheme of the contributing matrix elements for the dynamics without feedback for an excitation with up to three photons. The abbreviation ${\left[A\right]}_{m,n}^{i,j}, A\in \{{\sigma }_{+}{\sigma }_{-},{\sigma }_{-}\}$, encodes the matrix element $\langle i,m|A\left(t\right)|j,n\rangle$. The $n$-photon dynamics can be obtained recursively from the case of $n-1$ photons. (c) Scheme of the contributing matrix elements for the dynamics with feedback. When decomposing the expectation value of the TLS population, the matrix elements in the colored boxes have to be calculated and saved for the duration of one feedback interval $\tau$. The elements ${\left[A\right]}_{m,n}^{i,j}$ can be calculated recursively where, however, additionally the matrix elements outside the colored boxes couple to the dynamics. The underlined times indicate the scaling of the algorithm for the corresponding number of photons since they have to be integrated over and determine the required computational resources.

Figure 2

Benchmark for the dynamics of the TLS population in the initial state $|g,n\rangle$ with a rectangular pulse of duration $\mathrm{\Gamma }{t}_D=2$ (yellow, shaded area) containing $n$ photons, $n\in \{1–3\}$. (a) Benchmark for the decomposition of the TLS population without feedback where the expectation value is either directly calculated (solid lines) or decomposed via the insertion of a unity operator (dashed lines). (b) Benchmark of the feedback dynamics for feedback at $\mathrm{\Gamma }\tau =2$ with feedback phase ${\omega }_0\tau =2\pi m, m\in \mathbb{N}$, calculated using the matrix product state framework (solid lines) or the Heisenberg method (dashed lines).

Figure 3

Excitation of the atom-photon bound state via multiphoton pulses. (a) Expectation value of the TLS population as a function of time for a TLS with feedback at $\mathrm{\Gamma }\tau =2$ excited via a pulse containing two [red (medium gray) lines] or three photons [green (dark gray) lines] of shape $f\left(t\right)$ [yellow (light gray) lines]. From left to right: Rectangular pulse with $\mathrm{\Gamma }{t}_D=2$, Gaussian pulse with $\mathrm{\Gamma }\sigma =1$, exponentially decaying pulse with ${\mathrm{\Gamma }}_{\mathrm{pulse}}/\mathrm{\Gamma }=1$. (b) Steady-state population of the TLS in the case of a two-photon pulse as a function of the width of the pulse and the feedback delay time $\tau$. From left to right: Rectangular pulse with duration $t_D$, Gaussian pulse with width $\sigma$, exponentially decaying pulse with decay rate ${\mathrm{\Gamma }}_{\mathrm{pulse}}$.

Figure 4

Expectation value of the TLS population as a function of time under the influence of pure dephasing at different rates $\gamma$ for feedback at $\mathrm{\Gamma }\tau =1.2$ for a system in the initial state $|\psi \rangle$. (a) Initially excited emitter decaying in vacuum ($|\psi \rangle =|e,0\rangle$) with feedback phase ${\omega }_0\tau =2\pi m, m\in \mathbb{N}$. (b) Emitter initially in the ground state excited via a rectangular two-photon pulse of duration $\mathrm{\Gamma }{t}_D=1$ ($|\psi \rangle =|g,2\rangle$) with feedback phase ${\omega }_0\tau =2\pi m$. (c) Emitter initially in the ground state excited via a rectangular single-photon pulse of duration $\mathrm{\Gamma }{t}_D=1$ ($|\psi \rangle =|g,1\rangle$) with feedback phase ${\omega }_0\tau =2\pi m$. (d) Initially excited emitter decaying in vacuum ($|\psi \rangle =|e,0\rangle$) with feedback phase ${\omega }_0\tau =(2m+1)\pi$.