a plethora of phenomena
Quadratic light-matter interactions are nonlinear couplings such that quantum emitters interact with photonic or phononic modes exclusively via the exchange of excitation pairs.
这句话对“Quadratic light-matter interactions”做了一个解释，用到了长句以及一些副词来修饰，解释的非常清楚到位
Implementable with atomic and solid-state systems, these couplings lead to a plethora of phenomena that have been characterized in the context of cavity QED, where quantum emitters interact with localized bosonic modes.
We develop a general scattering theory under the Markov approximation and discuss paradigmatic examples for spontaneous emission and scattering of biphoton states.
Our analytical and semianalytical results unveil fundamental differences with respect to conventional waveguide QED systems, such as the spontaneous emission frequency-entangled photon pairs or the full transparency of the emitter to single-photon inputs.
This unlocks new opportunities in （重点句式）quantum information processing with propagating photons.
As a striking example（重点句式）, we show that a single quadratically coupled emitter can implement a two-photon logic gate with unit fidelity, circumventing a no-go theorem derived for conventional waveguide-QED interactions
The study of light-matter interactions is one of the main research pillars of quantum science.
介绍光与物质相互作用的重要性，如果是我写，肯定是：light-matter interaction is the central problem in quantum science，这里换了一个说法： main research pillars of
The implementation of systems where quantum emitters interact strongly with confined modes of the electromagnetic fields has allowed us to achieve an unprecedented level of control over quantum degrees of freedom
In the framework of cavity QED, the confinement of the electromagnetic fieldmakes it possible to observe a coherent exchange of excitations between localized photonic modes and single quantum emitters . Similarly, the coupling of quantum emitters to propagating fields can be strongly enhanced using waveguide structures, which confine photons to a one-dimensional environment
This setup, known as waveguide QED, has been realized in a variety of platforms such as atoms [6–9] or quantum dots [10,11] coupled to photonic waveguides, as well as superconducting qubits [12–20] coupled to microwave transmission lines.
Waveguide QED structures have a great potential to implement building blocks of quantum networks [21,22], since propagating photons are ideal to transport flying qubits over long distances while emitters can provide the strong quantum nonlinearity necessary for quantum information processing.
Therefore, there has been intense theoretical [23–30] and experimental [31–36] research in the control and characterization of few-photon correlations generated by single quantum emitters, as well as in the implementation of photonic devices working at the few-photon level
The rich quantum phenomenology arising in quadratic light-matter interactions motivates a fast-growing interest
In this work, we develop a quantum optics theory to describe a single quantum emitter interacting quadratically with the photons that propagate along a one-dimensional waveguide.
We study this problem with a two-photon scattering theory based on a Wigner-Weisskopf approach and the Born-Markov approximation.
We derive the general form of the scattering matrix, including semianalytical solutions for arbitrary photonic input states and full analytical solutions for Gaussian inputs.
Applying this theory, we unveil observable features of the emitter’s response that are fundamentally different with respect to conventional waveguide QED setups (see Fig. 1). These include (i) the spontaneous emission of correlated biphoton states, (ii) the strong interaction with spectrally narrow two-photon pulses, and (iii) full transparency to single-photon inputs.
Finally, we show that these effects can be exploited in quantum information applications, designing a deterministic controlled-phase gate that acts on pairs of propagating photons with perfect fidelity. This result seems to contradict a famous no-go theorem for photonic gates [40,79,80].
It is convenient to reparameterize the wave function’s element using the sum and difference of photon frequencies
In this limit one can formally solve Eq. (5) and replace the solution into Eq. (4) to obtain
For the sake of clarity, let us first write the scattering relations for a fixed propagation direction of the input field along the waveguide.
As demonstrated in this work, quadratic light-matter couplings are a compelling tool to process quantum information encoded in propagating photons.
This possibility should be thoroughly addressed considering each specific experimental setting, with a detailed microscopic model of the system and realistic noise sources.
The possibility of tailoring the spectral features of emitted and scattered biphoton states paves also the way to the generation of entanglement in the time-frequency domain, which is relevant for quantum computation  and sensing [86–88] applications.
对什么有意义，也可以用 be relevant for