Abstract We consider the inelastic scattering of few photons by a ladder-type quantum system residing in a semi-infinite waveguide. Two scenarios, namely 2-photon scattering and 3-photon scattering, are investigated. The single photons are described by continuous temporal pulse functions, according to which the exact forms of the stationary output field states can be derived. Based on the exact analysis of the nonlinear dynamics, the spectral entanglement among the output photons mediated by the ladder-type quantum system is simulated. In the 2-photon scattering case, strong correlation between the output photons can be observed if each input photon is in resonance with a transition frequency of the system. In particular, there exists a weak two-photon process for the non-resonance case, where two input photons can couple to the transition between the first and third levels of the system if a two-photon resonance condition is satisfied. In the 3-photon scattering case, the presence of an ancillary photon could significantly influence the correlation pattern. There exist two nonlinear terms, one is for 2-photon inelastic scattering and the other for 3-photon inelastic scattering, in the 3-photon output state. As a result, both two-photon and three-photon processes can be observed in the pattern of spectral entanglement.
Abstract We consider the inelastic scattering of few photons by a ladder-type quantum system residing in a semi-infinite waveguide. Two scenarios, namely 2-photon scattering and 3-photon scattering, are investigated. The single photons are described by continuous temporal pulse functions, according to which the exact forms of the stationary output field states can be derived. Based on the exact analysis of the nonlinear dynamics, the spectral entanglement among the output photons mediated by the ladder-type quantum system is simulated. In the 2-photon scattering case, strong correlation between the output photons can be observed if each input photon is in resonance with a transition frequency of the system. In particular, there exists a weak two-photon process for the non-resonance case, where two input photons can couple to the transition between the first and third levels of the system if a two-photon resonance condition is satisfied. In the 3-photon scattering case, the presence of an ancillary photon could significantly influence the correlation pattern. There exist two nonlinear terms, one is for 2-photon inelastic scattering and the other for 3-photon inelastic scattering, in the 3-photon output state. As a result, both two-photon and three-photon processes can be observed in the pattern of spectral entanglement.
Quantum nonlinearity is an essential ingredient for many quantum technologies, but often the nonlinearity is too weak to be exploited at the few-photon level. However, few photons interacting strongly with single quantum emitters in a waveguide environment can impact a significant nonlinear response, opening up a wide range of photon-photon correlations. Using a waveguide-QED system containing a single atom (treated as a two-level system) chirally coupled to a waveguide, we theoretically investigate two-photon nonlinearities with delay-controlled temporal correlations. We use both matrix product states (MPS) and a frequency-dependent scattering theory approach to analyze the exact population dynamics, as well as the first-order and second-order photon correlation functions in transmission of the system, when pumped by a two-photon Fock-state pulse with a bimodal temporal pulse envelope. The two-photon Fock-state pulses are considered to be either two single photons localized to each peak of the pulse, or both photons delocalized (but correlated) between the two peaks. We consider the regimes of a short, moderate, and (relatively) long distance between the two pulse peaks, comparing the important differences in the temporal correlations with the two types of two-photon pulses. We demonstrate the strikingly different nonlinear features and quantum correlations that occur for uncorrelated and correlated two-photon pairs in experimentally accessible regimes.
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.
We present a detailed quantum theory and simulations of few-photon Fock state pulses interacting with a two-level system (TLS) in an open waveguide-QED system. For a rectangular pulse shape, we present an exact temporal scattering theory to derive analytical expressions for the TLS population, using one-photon and two-photon pulses, for both chiral and symmetric emitters. We also derive the stationary (long-time) and time-dependent spectra for one-photon excitation, and show how these differ at a fundamental level when considering TLS population effects. Numerically, we also present matrix product state (MPS) simulations, which allow us to compute more general photon correlation functions for arbitrary quantum pulses, and we use this approach to also show results for Gaussian quantum pulses and to confirm the accuracy of our analytical solutions. In the case of a chiral TLS, we show how a one-photon pulse, of any temporal shape, yields a transmitted long-time spectrum that is identical to the input pulse, despite significant TLS population effects. However, the dynamical spectra and spectral intensity show rich population effects, allowing for their detection in spectroscopic experiments. We also show the striking differences between one-photon and two-photon excitation, where the latter shows clear signatures of nonlinear saturation effects. In addition, we demonstrate how significant population TLS dynamics also occur for pulses that are relatively long compared to the radiative decay time (showing that a weak excitation approximation, which neglects finite TLS population effects, cannot be made) and investigate the population signatures, nonlinear features and dynamical behavior as a function of quantum pulse length.
Realising a platform for efficiently guiding and manipulating light at the singlephoton level lies at the heart of optical quantum information processing. In this thesis we aid in the pursuit of this goal by analysing few-photon transport properties in photonic devices while keeping in mind possible applications and experimental challenges. We especially focus on single and two-photon transport properties in one-dimensional waveguides coupled to (artificial) atoms modelled as two and three-level emitters. We begin by introducing key concepts relevant for this thesis and then explain how to model such waveguide geometries. We show how directional-dependent coupling between a waveguide and a localized quantum system can be modelled from a chain of harmonic oscillators by allowing the system to couple to more than one site. Using the input-output formalism and scattering matrix theory we determine a general form for the single and two-photon S-matrices for a waveguide coupled to a two-level system. We also explain how to engineer a quantum optical Fanoresonance waveguide geometry and calculate how this modifies the S-matrices, the few-photon scattering probabilities, and the scattered output spectra. We show that the nonlinear effects in such a system generates highly correlated directional scattering probabilities. We device a protocol that can generate maximal entanglement between two spectrally distinct solid-state emitters by the use of multi-photon scattering. The results also reveal a rich structure for multi-photon states interacting with non-identical emitters. As a final investigation, we quantify how spectral diffusion in quantum dots affects finite-width single and two-photon scattering. Single-photon scattering probabilities are described by a simple convolution but this is not the case for a two-photon state scattering off a two-level emitter. We show that the effect of spectral wandering depletes the nonlinear scattering properties, but that it affects co and counter-propagating output probabilities differently.
We study two-level systems (2LS) coupled at different points to a one-dimensional waveguide in which one end is open and the other is either open (infinite waveguide) or closed by a mirror (semi-infinite). Upon injection of two photons (corresponding to weak coherent driving), the resonance fluorescence and photon correlations are shaped by the effective qubit transition frequencies and decay rates, which are substantially modified by interference effects. In contrast to the well-known result in an infinite waveguide, photons reflected by a single 2LS coupled to a semi-infinite waveguide are initially bunched, a result that can be simply explained by stimulated emission. As the number of 2LS increases (up to 10 are considered here), rapid oscillations build up in the correlations that persist for a very long time. For instance, when the incoming photons are slightly detuned, the transmitted photons in the infinite waveguide are highly antibunched. On the other hand, upon resonant driving, incoherently reflected photons are mostly distributed within the photonic band gap and several sharp side peaks. These features can be explained by considering the poles of the single-particle Green function in the Markovian regime combined with the time delay. Our calculation is not restricted to the Markovian regime, and we obtain several fully non-Markovian results. We show that a single 2LS in a semi-infinite waveguide can not be decoupled by placing it at the node of the photonic field, in contrast to recent results in the Markovian regime. Our results illustrate the complexities that ensue when several qubits are strongly coupled to a bus (the waveguide) as might happen in quantum information processing.
The purpose of this paper is to study the interaction between a two-level system (qubit) and two continuous-mode photons. Two scenarios are investigated: Case 1, how a two-level system changes the pulse shapes of two input photons propagating in a single input channel; and Case 2, how a two-level system responds to two counterpropagating photons, one in each input channel. The steady-state output field states for both cases are derived analytically in both the time and frequency domains. Numerical simulations demonstrate rich and interesting two-photon scattering phenomena induced by a two-level system.