Generalized cluster decomposition principle illustrated in waveguide quantum electrodynamics

Abstract

We present the general structure of two-photon S matrix for a waveguide coupled to a local quantum system that supports multiple ground states. The presence of the multiple ground states results in a non-commutative aspect of the system with respect to the exchange of the orders of photons. Consequently, the two-photon S matrix significantly differs from the standard form as described by the cluster decomposition principle in the quantum field theory. PACS numbers:The scattering matrices (S matrices) are of essential importance for characterizing the interaction of quantum particles. On one hand, each element of a scattering matrix describes the probability amplitude of a particular scattering event. Thus every element of a scattering matrix is of direct experimental significance. On the other hand, from a theoretical point of view, the analytic structure of an S matrix is strongly constrained by symmetries and causalities, as well as by other general aspects such as the local nature of the interactions. Consequently, much of the literature on quantum field theory is devoted to the computation and elucidation of the structure of S matrices [1-4].Using the cluster decomposition principle [1,5,6], the standard form of two-particle S matrix listed in quantum field theory textbooks is S = S 0 + i T , where S 0 , the non-interacting part of the S matrix, is of the formand contains the product of two δ functions. The T matrix, which describes the interaction, is of the formand contains a single δ functions. Here, k 1,2 and p 1,2 are the momenta of the incident and outgoing particles, respectively. t k is the individual particle transmission amplitude and C p1p2k1k2 characterizes the strength of the interactions between two particles. Recently, this form is also shown to apply in waveguide quantum electrodynamics arXiv:1610.01727v1 [quant-ph]

Emergent causality and theN-photon scattering matrix in waveguide QED

“…Here, n n ¢ ¹ , e.g., n 2 ¢ = if n=1. The computation is detailed in appendix F. This structure has recently been found by Xu and Fan for a Λ atom (M = 2, M 1 ¢ = ) within the RWA and Markovian approximations [38]. At first sight (30) may look striking.…”

“…Here, we find the shape of the S-matrixfrom general principles, including the inelastic processes. We recover the nontrivial form computed by Xu and Fan for a particular case in [38] and find the generalization of the standard cluster decomposition to the waveguide-QED model given by equation (2), with several ground states.The paper has the following organization. Section 2 presents the nonrelativistic Hamiltonian that models the interaction between propagating photons and quantum impurities, the concept of wave packet, a review of the scattering theory needed, and two conditions necessary for the validity of our results.…”

“…Here, we find the shape of the S-matrixfrom general principles, including the inelastic processes. We recover the nontrivial form computed by Xu and Fan for a particular case in [38] and find the generalization of the standard cluster decomposition to the waveguide-QED model given by equation (2), with several ground states.…”

“…The final result is that S pk 0 mn ( ) cannot be written as a product of one-photon S-matrices. This has been recently pointed out in the particular example of a Λ atom coupled to a waveguide within the RWA and Markovian approximations with point-like coupling by Xu and Fan [38] using the input-output formalism [27].…”

Violation of Cluster Property in Quantum Antiferromagnet

“…It has been also pointed out that the violation of this property is deeply connected with confinement in QCD [18] [19], because we cannot separate the quark operator far from the anti-quark operator. With these interests there are active studies on the cluster property in quantum field theory [20] [21].…”