We employ a cascaded system approach to numerically simulate the interaction of photon pulses with a two-level scatterer in a chiral waveguide QED setup and show that the scattering of a two-photon pulse may lead to the predominant population of only two output wave packet modes in an entangled state, $|2,0\ensuremath{\rangle}--|0,2\ensuremath{\rangle}$. In a complementary wave-packet basis, this is a product state of two orthogonal single-photon wave packets. Operating this process in reverse allows a perfect merging of distinguishable single-photon wave packets into a single-mode pulse carrying two identical photons.
We employ a cascaded system approach to numerically simulate the interaction of photon pulses with a two-level scatterer in a chiral waveguide QED setup and show that the scattering of a two-photon pulse may lead to the predominant population of only two output wave packet modes in an entangled state, ∣2,0\ensuremath⟩−−∣0,2\ensuremath⟩. In a complementary wave-packet basis, this is a product state of two orthogonal single-photon wave packets. Operating this process in reverse allows a perfect merging of distinguishable single-photon wave packets into a single-mode pulse carrying two identical photons.
We employ a cascaded system approach to numerically simulate the interaction of photon pulses with a two-level scatterer in a chiral waveguide QED setup. It is possible to expand any pure state of two photons as a superposition of orthogonal two-photon pulses. We show that the scattering of a two-photon pulse of suitable duration may lead to an entangled output state between predominately two equally populated two-photon states. In a complementary wave packet basis, this state is a product state of two orthogonal single photon wave packets. The time reversal of the above scattering process allows for a perfect combination of distinguishable single-photon wave packet into a single-mode pulse carrying two identical photons.
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.
The study of waveguide-QED systems, where a continuum of quantum field modes is coupled to qubits or two-level systems, has improved our ability to manipulate quantum light-matter interactions on chip. In the typical theoretical approaches to waveguide QED, there are a few necessary approximations, e.g., considering the system in the weak excitation regime, or treating the waveguide as a bath. However, these inherent approximations can break down with short pulse excitation. Here, we investigate the few-photon quantum nonlinear response of chiral qubits, when excited with one and two-photon Fock states. Our theory uses a numerically exact approach, based on Matrix Product States, avoiding the limitations of the usual waveguide-QED approximations. Using a chiral-emitter waveguide system, we show explicitly the breakdown of the weak excitation approximation, and study the single and two-photon nonlinear responses. We demonstrate the impact on the qubit population, and discuss how the phase change can be examined from the photon quantum correlation functions, seeing a radical departure from scattering theory solutions.
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.
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.