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arXiv:1504.03350 · Physical Review A

Quantum theory of light scattering in a one-dimensional channel: Interaction effect on photon statistics and entanglement entropy

Mikhail Pletyukhov, Vladimir Gritsev

Published Jun 29, 2015• 26 citations
arXivHTMLPDF

Abstract

We provide a complete and exact quantum description of coherent light scattering in a one-dimensional multimode transmission line coupled to a two-level emitter. Using a recently developed scattering approach, we discuss transmission properties, the power spectrum, the full counting statistics, and the entanglement entropy of transmitted and reflected states of light. Our approach takes into account spatial parameters of an incident coherent pulse as well as waiting and counting times of a detector. We describe the time evolution of the power spectrum and observe deviations from the Poissonian statistics for reflected and transmitted fields. In particular, the statistics of reflected photons can change from sub-Poissonian to super-Poissonian for increasing values of the detuning, while the statistics of transmitted photons is strictly super-Poissonian in all parametric regimes. We study the entanglement entropy of some spatial part of the scattered pulse and observe that it obeys the area laws and that it is bounded by the maximal entropy of the effective four-level system.

DOI
10.1103/physreva.91.063841

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We provide a complete and exact quantum description of coherent light scattering in a one-dimensional multimode transmission line coupled to a two-level emitter. Using a recently developed scattering approach, we discuss transmission properties, the power spectrum, the full counting statistics, and the entanglement entropy of transmitted and reflected states of light. Our approach takes into account spatial parameters of an incident coherent pulse as well as waiting and counting times of a detector. We describe the time evolution of the power spectrum and observe deviations from the Poissonian statistics for reflected and transmitted fields. In particular, the statistics of reflected photons can change from sub-Poissonian to super-Poissonian for increasing values of the detuning, while the statistics of transmitted photons is strictly super-Poissonian in all parametric regimes. We study the entanglement entropy of some spatial part of the scattered pulse and observe that it obeys the area laws and that it is bounded by the maximal entropy of the effective four-level system.
A complete characterization of quantum fluctuations in many-body systems is accessible through the full counting statistics. We present an exact computation of statistical properties of light in a basic model of light-matter interaction: a multimode photonic field coupled to a single two-level emitter. We mostly consider an initial coherent state in a given mode and demonstrate how the original Poissonian statistics gets modified because of quantum many-body scattering effects leading to non-Poissonian distributions. We argue that measuring this statistics in a simple quantum optical setup provides an insight into many-body correlation effects with photons.
We introduce and study a photonic analog of the Kondo model. The model is defined as a far-detuned regime of photonic scattering off a three-level emitter in a $\mathrm{\ensuremath{\Lambda}}$-type configuration coupled to a one-dimensional waveguide with linear dispersion. We study dynamics of this local emitter driven by a coherent field pulse as well as the first- and second-order correlation functions of a scattered light. Various polarization-dependent correlation effects and an entanglement between the emitter and transmitted photons are quantified by the purity of an emitter's state. We also show that statistical properties of the scattered light are very sensitive to a polarization of the incoming coherent pulse.
Strong coupling between a two-level system (TLS) and bosonic modes produces dramatic quantum optics effects. We consider a one-dimensional continuum of bosons coupled to a single localized TLS, a system which may be realized in a variety of plasmonic, photonic, or electronic contexts. We present the exact many-body scattering eigenstate obtained by imposing open boundary conditions. Multiphoton bound states appear in the scattering of two or more photons due to the coupling between the photons and the TLS. Such bound states are shown to have a large effect on scattering of both Fock- and coherent-state wave packets, especially in the intermediate coupling-strength regime. We compare the statistics of the transmitted light with a coherent state having the same mean photon number: as the interaction strength increases, the one-photon probability is suppressed rapidly, and the two- and three-photon probabilities are greatly enhanced due to the many-body bound states. This results in non-Poissonian light.
In a preceding paper we introduced a formalism to study the scattering of low-intensity fields from a system of multilevel emitters embedded in a three-dimensional (3D3\mathrm{D}3D) dielectric medium. Here we show how this photon-scattering relation can be used to analyze the scattering of single photons and weak coherent states from any generic multilevel quantum emitter coupled to a one-dimensional (1D1\mathrm{D}1D) waveguide. The reduction of the photon-scattering relation to 1D1\mathrm{D}1D waveguides provides a direct solution of the scattering problem involving low-intensity fields in the waveguide QED regime. To show how our formalism works, we consider examples of multilevel emitters and evaluate the transmitted and reflected field amplitude. Furthermore, we extend our study to include the dynamical response of the emitters for scattering of a weak coherent photon pulse. As our photon-scattering relation is based on the Heisenberg picture, it is quite useful for problems involving photodetection in the waveguide architecture. We show this by considering a specific problem of state generation by photodetection in a multilevel emitter, where our formalism exhibits its full potential. Since the considered emitters are generic, the 1D1\mathrm{D}1D results apply to a plethora of physical systems such as atoms, ions, quantum dots, superconducting qubits, and nitrogen-vacancy centers coupled to a 1D1\mathrm{D}1D waveguide or transmission line.
We present a full multimode treatment of the interaction of the quantized radiation field with a single two-level atom in a one-dimensional waveguide configuration. Starting with an incident pulse consisting of an arbitrary (finite) number of photons in a general initial state, we derive the equations of motion and a closed-form expression for the shape of the pulse after its interaction with the atom. We then specialize our results to the two-photon case where a number of analytical results can be derived, for both unidirectional and bidirectional systems. We study the effects of different pulse shapes, the manifestations of the entangled, so-called bound state of the two photons, single- and two-photon detection probabilities, and provide simple approximate results for the strong-coupling (or adiabatic, or long-pulse) regime. We also discuss the requirements for a true unidirectional setup, and the application of such a setup to photon sorting proposed by Witthaut, Lukin, and S\o{}rensen [Europhys. Lett. 97, 50007 (2012)].
Entanglement is usually associated with compound systems. We first show that a one-dimensional (1D) completed scattering of a particle on a static potential barrier represents an entanglement of two alternative one-particle sub-processes, transmission and reflection, macroscopically distinct at the final stage of scattering. The wave function for the whole ensemble of scattering particles can be uniquely presented as the sum of two isometrically evolved wave packets to describe the (to-be-)transmitted and (to-be-)reflected subensembles of particles at all stages of scattering. A noninvasive Larmor-clock timing procedure adapted to either subensemble shows that namely the dwell time gives the time spent, on the average, by a particle in the barrier region, and it denies the Hartman effect. As regards the group time, it cannot be measured and hence it cannot be accepted as a measure of the tunneling time. We argue that nonlocality of entangled states appears in quantum mechanics due to inconsistency of its superposition principle with the corpuscular properties of a particle. For example, this principle associates a 1D completed scattering with a single (one-way) process, while a particle, as an indivisible object, cannot take part in transmission and reflection, simultaneously.