We develop a new computational tool and framework for characterizing the scattering of photons by energy-nonconserving Hamiltonians into unidirectional (chiral) waveguides, for example, with coherent pulsed excitation. The temporal waveguide modes are a natural basis for characterizing scattering in quantum optics, and afford a powerful technique based on a coarse discretization of time. This overcomes limitations imposed by singularities in the waveguide-system coupling. Moreover, the integrated discretized equations can be faithfully converted to a continuous-time result by taking the appropriate limit. This approach provides a complete solution to the scattered photon field in the waveguide, and can also be used to track system-waveguide entanglement during evolution. We further develop a direct connection between quantum measurement theory and evolution of the scattered field, demonstrating the correspondence between quantum trajectories and the scattered photon state. Our method is most applicable when the number of photons scattered is known to be small, i.e. for a single-photon or photon-pair source. We illustrate two examples: analytical solutions for short laser pulses scattering off a two-level system and numerically exact solutions for short laser pulses scattering off a spontaneous parametric downconversion (SPDC) or spontaneous four-wave mixing (SFWM) source. Finally, we note that our technique can easily be extended to systems with multiple ground states and generalized scattering problems with both finite photon number input and coherent state drive, potentially enhancing the understanding of, e.g., light-matter entanglement and photon phase gates.
We develop a new computational tool and framework for characterizing the scattering of photons by energy-nonconserving Hamiltonians into unidirectional (chiral) waveguides, for example, with coherent pulsed excitation. The temporal waveguide modes are a natural basis for characterizing scattering in quantum optics, and afford a powerful technique based on a coarse discretization of time. This overcomes limitations imposed by singularities in the waveguide-system coupling. Moreover, the integrated discretized equations can be faithfully converted to a continuous-time result by taking the appropriate limit. This approach provides a complete solution to the scattered photon field in the waveguide, and can also be used to track system-waveguide entanglement during evolution. We further develop a direct connection between quantum measurement theory and evolution of the scattered field, demonstrating the correspondence between quantum trajectories and the scattered photon state. Our method is most applicable when the number of photons scattered is known to be small, i.e. for a single-photon or photon-pair source. We illustrate two examples: analytical solutions for short laser pulses scattering off a two-level system and numerically exact solutions for short laser pulses scattering off a spontaneous parametric downconversion (SPDC) or spontaneous four-wave mixing (SFWM) source. Finally, we note that our technique can easily be extended to systems with multiple ground states and generalized scattering problems with both finite photon number input and coherent state drive, potentially enhancing the understanding of, e.g., light-matter entanglement and photon phase gates.
We present a quantum dynamical study of pulsed few-photon scattering from a single artificial atom, consisting of a two-level system (TLS) or qubit, in a waveguide QED system, directly comparing and contrasting two different quantum theoretical simulation methods: (i) an input-output scattering approach that uses frequency-dependent scattering matrices, and (ii) a matrix product states (MPS) approach, which uses quantum noise operators in time bins and a tensor network technique to solve the time-dependent waveguide function for the entire system. Beginning with pulsed excitation using one-photon and two-photon Fock state pulses, we first show how to compute time-dependent observables with the scattering matrix approach, in terms of frequency integrals that encode the pulse spectrum, including how to extract the population dynamics of the excited quantum emitter, as well as the linear and nonlinear contributions. We present solutions for both symmetric and chiral TLS coupling. We then show how to compute the qubit and field observables in a more direct way using MPS, and obtain the characteristic bird-like shape for the two-photon correlation function at two times, which has been observed in recent experiments. We compare and contrast both of these methods, for one and two-photon excitation pulses, and show excellent agreement. We also present a study of the linear and nonlinear contributions, which can easily be calculated using scattering theory, and show the important role of pulse duration. Finally, we demonstrate the clear advantages of MPS by easily going to higher N-photon excitations, and show selected example population dynamics of up to eight-photon Fock-state pulses, manifesting in clear nonlinear population oscillations during the pulse interaction, similar to classical Rabi oscillations, but with quantum input fields that have a vanishing electric field expectation value.
In this work, we discuss connections between different theoretical physics communities and their works, all related to systems that act as sources of particles such as photons, phonons, or electrons. Our interest is to understand how a low-dimensional quantum system driven by coherent fields, e.g., a two-level system, a Jaynes-Cummings system, or a photon pair source driven by a laser pulse, emits photons into a waveguide. Of particular relevance to solid-state sources is that we provide a way to include dissipation into the formalism for temporal-mode quantum optics. We will discuss the connections between temporal-mode quantum optics, scattering matrices, quantum stochastic calculus, continuous matrix product states and operators, and very traditional quantum optical concepts such as the Mandel photon counting formula and the Lindblad form of the quantum-optical master equation. We close with an example of how our formalism relates to single-photon sources with dephasing.
Sorting quantum fields into different modes according to their Fock-space quantum numbers is a highly desirable quantum operation. In this Letter, we show that a pair of two-level emitters, chirally coupled to a waveguide, may scatter single- and two-photon components of an input pulse into orthogonal temporal modes with a fidelity ≳0.9997. We develop a general theory to characterize and optimize this process and reveal that such a high fidelity is enabled by an interesting two-photon scattering dynamics: while the first emitter gives rise to a complex multimode field, the second emitter recombines the field amplitudes, and the net two-photon scattering induces a self-time reversal of the input pulse mode. The presented scheme can be employed to construct logic elements for propagating photons, such as a deterministic nonlinear-sign gate with a fidelity ≳0.9995.
Non-Gaussian quantum states of light are of paramount importance to quantum computing. Nevertheless, their deterministic generation is challenging problem due to the difficulty to control nonlinearities in physical systems. In this work, we characterize the light stemming from one of the most fundamental quantum optics configurations: the unidirectional scattering of multimode and multiphoton light by a two-level system. We provide an analytic and explicit description of the output light solely in terms of the corresponding input temporal modes which allows a straightforward physical interpretation and is computationally more effective compared to numerical methods. Then, we focus on the specific case of the scattering of two photons in a single mode. By numerically decomposing the output state in terms of its principal modes, we find that it is possible to map single-mode two-photon inputs to be into two-mode entangled output states, i.e., two-photon NOON states, to very good approximation. The latter states, in turn, are known to have more Wigner negativity compared to the associated input, which ultimately suggests a potential application of our considered setup in the deterministic generation of non-Gaussian states.
Abstract The future of quantum information processing hinges on chip-scale nanophotonics, specifically cavity QED and waveguide QED. One of the foremost processes underpinning quantum photonic technologies is the phenomenon of Rabi oscillations, which manifests when a qubit is irradiated by an intense laser source. Departing from the conventional semiclassical framework, we explore the more general, quantum theoretical case where the optical excitation takes the form of a multiphoton Fock state, and the qubit couples to a continuum of radiation modes. By employing the real-space formalism, we analytically explore the scattering dynamics of the photonic Fock state as it interfaces with a two-level emitter. The resulting amplitude for atomic excitation features a linear superposition of various independent scattering events that are triggered by the potential of sequential, time-delayed photon absorptions and emissions. The lowest-order excitation event, initiated by the stochastic scattering of one of the several photons, appropriately characterizes the dynamics in a weak-field environment. This event is complemented by a multitude of higher-order scattering events ensuing from repeated atom–photon interactions. The temporal evolution of the qubit excitation in our configuration closely mirrors the semiclassical predictions, particularly in the strong-pumping limit where Rabi oscillations unfold. Notably, this compatibility with the semiclassical paradigm transcends beyond the strong-excitation regime and applies both to the weak-driving and large-detuning limits. In a nutshell, our analysis extends the existing results on quantum Rabi oscillations pertinent to single-mode cavity QED, to the multimode, waveguide-QED configurations wherein flying photons are the information carriers. Additionally, we delve into the dynamics of pulsed wave packets, shedding light on the potential to substantially enhance excitation efficiency, even in scenarios involving just a few photons. Beyond their theoretical merit, these findings should hold practical relevance for future Fock-state-based quantum computing and emerging waveguide-integrated photonic technologies such as those involving superconducting circuitry.