Dynamic stimulated emission for deterministic addition and subtraction of propagating photons

TL;DR

The paper presents dynamic stimulated emission as a time-dependent coupling mechanism to deterministically add or subtract photons from a propagating optical mode, achievable with both a two-level system (TLS) and a driven three-level system (3LS). By solving the input-output problem in the $n$-excitation subspaces, employing linearisation and variational parameters, and applying time-reversal and parity transformations, it derives explicit $g(t)$-profiles enabling near-unit fidelities (${\cal F}>0.996$) for up to five excitations and even cascaded implementations with multiple emitters. It extends the framework to generate non-Gaussian superposition states, notably Schrödinger cat states from squeezed inputs and photon-added Gaussian states, with high fidelities and practical success probabilities, illustrating a path to integrated, efficient non-Gaussian light sources. These results suggest a versatile route to optical non-Gaussian state engineering using quantum emitters, potentially transforming single-photon sources into sources of photon-added Gaussian states and enabling scalable, mode-selective non-Gaussian optics. ${}$

Abstract

Photon subtraction and addition are essential non-Gaussian processes in quantum optics, where conventional methods using linear optics and number-resolving detection often suffer from low success probability. Here, we introduce the concept of \textit{dynamic stimulated emission}, whereby a quantum emitter undergoes stimulated emission with a time-dependent coupling. We show that, for both two- and three-level emitters, this process can be used to deterministically add or subtract a photon to a single propagating optical mode. We provide semi-analytic solutions to this problem for Fock states, enabling deterministic and unconditional single-photon subtraction and addition with fidelity ${\cal F}>0.996$. Our semi-analytic solutions are provided for both dynamically coupled two-level systems and for three-level systems whose dynamical coupling is controlled by a coherent laser drive. Moving beyond individual Fock states, we further showcase the ability to subtract and add single photons to photon-number superposition states. We show that Schrödinger cat states can be prepared from squeezed vacuum input via cascaded subtraction or cascaded addition. Finally, we show that our photon-addition process can be used to add a photon to any squeezed and displaced state with high success probability and fidelity ${\cal F}>0.99$, thereby potentially converting quantum emitters from single-photon sources to sources of single-photon-added Gaussian states without the need for inline squeezing. Our protocols provide a path towards integrating quantum emitters to construct efficient sources of single-mode non-Gaussian light beyond single photons.

Figures (8)

• Figure 1: (a) Schematic of dynamic stimulated emission for the addition of a single-photon to produce an $n$-photon Fock state in a one-dimensional waveguide QED system. When a $(n-1)$-photon Fock state pulse is incident on a two-level system (TLS) prepared in the excited state, a carefully chosen dynamic coupling strength $g(t)$ enables the TLS to emit a photon such that ouput is a $n$-photon Fock state. (b) Conversely, the schematic of the reverse process (subtraction). The TLS is excited by an $n$-photon Fock state. The dynamic coupling strengths, $g(t)$ are distinct for each addition and subtraction processes and they dependent on the excitation number and the input mode shape; explicit expressions are given by Eq. \ref{['n photon add g']} and Eq. \ref{['n photon sub g']}.
• Figure 2: Simulation results for the addition and subtraction processes, labeled by colours blue and red respectively. The schematic of both processes are depicted in (a), where $f_{\rm in}$ is the input temporal mode and $f_{\rm out}$ the output mode. The excitation in the TLS of dynamic stimulated emission follows the red arrows, the reverse process where a photon is absorbed by the TLS follows the blue arrows. Results in all sub-figures (b-f) are simulated with a Gaussian input temporal mode of width $\sigma g^2=2/\sqrt{\pi}$, where $g^2=|g_{{\rm sub},1}(t_0)|^2$ is the reference coupling rate of the system (see main text). (b) The lossless unconditional fidelities for the subtraction and addition processes for up to five excitations and (c) in the presence of loss for two- and four-excitations. The reduced addition fidelities in (c) are due to the TLS's idling time (see main text). (d) At final time, $T$, the excited state population for the subtraction process versus variational parameter $s_n$ for up to five excitations. (e) The input Gaussian temporal mode and the output temporal modes for the two- and four-excitation subtraction processes. (f) The ground (solid) and excited (dashed) state population dynamics for the two-excitation subtraction and addition processes.
• Figure 3: Simulation results for cascading up to five TLSs each with unique dynamic couplings. The subtraction and addition processes throughout this figure are labeled by colours blue and red respectively. (a) The schematic for cascading five TLSs for dynamic stimulated emission (red arrows) and absorption (blue arrows) with the corresponding ideal transitions, $\ket{5}\ket{g}^{\otimes 5}\rightarrow \ket{0}\ket{e}^{\otimes 5}$ and $\ket{0}\ket{e}^{\otimes 5}\rightarrow\ket{5}\ket{g}^{\otimes 5}$, respectively. (b) The unconditional fidelities from single- to five-excitation cascaded addition and subtraction processes; $n$ denotes the total number of excitations or equivalently the number of TLSs. The subtraction processes are all initialised with a Gaussian input temporal mode of width $\sigma g^2=2/\sqrt{\pi}$, where $g^2$ is a reference coupling rate (see main text). On the other hand, for the cascaded addition processes, the coupling strength of the first TLS is chosen such that it emits a photon with a Gaussian temporal mode of width $\sigma g^2=2/\sqrt{\pi}$. Sub-figures (c-e) show features of the five-excitation cascaded subtraction and addition. (c) The dynamics of the excited state populations for each cascaded TLS labeled by $j$. The temporal mode $f_j(-t)$ for subtraction (d) and addition (e) occupying the highest mean photon number before and after interacting with the $j$-th cascaded TLS respectively.
• Figure 4: $\Lambda$-type three-level system (3LS) and the effective two-level system (TLS). On the left-hand side, a full 3LS with the excited state $\ket{r}$ and degenerate ground states $\ket{e}$, $\ket{g}$; that is under a large detuned coherent drive with time-dependent Rabi frequency and phase. Adiabatically eliminating the excited state, $\ket{r}$, yields the right-hand side, an effective TLS with AC Stark shifts to the energy of the ground states.
• Figure 5: Simulation results with the full 3LS. The subtraction and addition processes throughout this figure are labeled by colours blue and red respectively. (a) The schematic of the 3LS dynamic stimulated emission (red arrows) and absorption (blue arrows) with input and output temporal modes labeled by $f_{\rm in}$ and $f_{\rm out}$ respectively. Results in all sub-figures are simulated with detuning $\Delta/\Gamma=5$ and with a Gaussian input temporal mode of width $\sigma\Gamma=25$. The lossless unconditional fidelities for the subtraction and addition processes for up to five excitations are presented in (b) and with loss for two- and four-excitations in (c). In (d), the time-dependent Rabi frequency (solid) and phase (inset, dashed) for the two-excitation subtraction and addition processes. (e) The input Gaussian temporal mode (black) and the output temporal modes for the two- and four-excitation subtraction processes. (f) The ground states, $\ket{g}$ (solid), $\ket{e}$ (dashed) and the excited state (dotted) population dynamics for the two-excitation subtraction and addition processes.