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Light Scattering from an Atomic Array Trapped Near a One-Dimensional Nanoscale Waveguide: a Microscopic Approach

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Light Scattering from an Atomic Array Trapped Near a One-Dimensional Nanoscale Waveguide: a Microscopic Approach PHYSICAL REVIEW A 97, 023827 (2018) Light scattering from an atomic array trapped near a one-dimensional nanoscale waveguide: A microscopic approach V. A. Pivovarov,1 A. S. Sheremet,2,3 L. V. Gerasimov,4 V. M. Porozova,5 N. V. Corzo,2 J. Laurat,2 and D. V. Kupriyanov5,* 1Physics Department, St.-Petersburg Academic University, Khlopina 8, 194021 St.-Petersburg, Russia 2Laboratoire Kastler Brossel, Sorbonne Université, CNRS, PSL Research University, 4 place Jussieu, 75005 Paris, France 3Russian Quantum Center, Novaya 100, 143025 Skolkovo, Moscow Region, Russia 4Faculty of Physics, M.V. Lomonosov Moscow State University, Leninskiye Gory 1-2, 119991 Moscow, Russia 5Department of Theoretical Physics, St-Petersburg State Polytechnic University, 195251 St.-Petersburg, Russia (Received 17 November 2017; published 16 February 2018) The coupling of atomic arrays and one-dimensional subwavelength waveguides gives rise to interesting photon transport properties, such as recent experimental demonstrations of large Bragg reflection, and paves the way for a variety of potential applications in the field of quantum nonlinear optics. Here, we present a theoretical analysis for the process of single-photon scattering in this configuration using a full microscopic approach. Based on this formalism, we analyze the spectral dependencies for different scattering channels from either ordered or disordered arrays. The developed approach is entirely applicable for a single-photon scattering from a quasi-one-dimensional array of multilevel atoms with degenerate ground-state energy structure. Our approach provides an important framework for including not only Rayleigh but also Raman channels in the microscopic description of the cooperative scattering process. DOI: 10.1103/PhysRevA.97.023827 I. INTRODUCTION with the evanescent field of the waveguide mode has important differences from the light scattering from atoms in free space. Efficient control of light-matter interaction at the single- The effects of Zeeman degeneracy of an atomic transition photon level is a central and challenging task for quantum should be taken into consideration for a relevant description optics and quantum information science [1–7]. At the fun- of the Raman channels, which at the present time has been damental level, this interaction manifests itself via the basic primarily studied in free space [37,38]. The problem becomes quantum electrodynamics processes of spontaneous emission, more complicated as the strong-field confinement provided absorption, and scattering of a single photon by a single atom. by the nanofiber imposes an inherent link between the local In free space, the efficiency of this interaction is limited by the polarization and propagation direction of light. In such a small atomic scattering cross section in comparison to the usual strongly nonparaxial regime, the spin and orbital angular large beam shining area. However, it can be greatly enhanced momentum of the guided light obey joint dynamics and cannot by placing a single atom in the vicinity of a subwavelength be independently considered. In particular, it leads to direction- waveguide due to the Purcell effect [8] or by considering large sensitive emission and absorption of the guided light by the atomic ensembles [9]. The combination of these approaches atoms [30,39–42]. has motivated recent experimental efforts towards the develop- All these experimental results and theoretical works demon- ment of novel platforms that integrate an atomic chain coupled strate that such a hybrid system is a versatile platform for with a dielectric nanoscale waveguide [10–16]. studying various cooperative effects emerging from both the One example of such platforms is the so-called subwave- collective atomic structuring and the waveguide-mediated length nanofiber [15–20]. Due to a large evanescent field, the long-range coupling between atoms. The internal correlations, guided light can be efficiently used for trapping atoms close to existing in a many-particle microscopic quantum entangled the surface [21–23] and interacting with them [24–27]. Recent state, can play an essential role in the cooperative scattering experiments include, for instance, the demonstration of optical process. In this context, an ab initio theoretical insight on the memories in this setting [28,29]. The trapping technique can problem of light-atoms interaction would provide a natural also be adjusted to arrange atoms in an optical lattice commen- extension of the convenient but sometimes insufficient self- surate or nearly commensurate with the resonant wavelength. consistent description, based on the density matrix approach Such capability enables one to investigate Bragg reflection and the Maxwell-Bloch formalism. [30,31] and long-range interactions [32]. To clarify the last statement, let us be more concrete and These recent experiments and supporting theoretical studies point out that any self-consistent description of a many-body [33–36] have shown that light scattering from atoms interacting problem conventionally assumes the physical equivalence in description of proximal particles and underestimates the role of internal correlations. But for dense atomic systems, the mutual correlations are an attribute of their exact microscopic dynam- *[email protected] ics, which can be manifestable when cooperative processes 2469-9926/2018/97(2)/023827(15) 023827-1 ©2018 American Physical Society V. A . P I VOVA ROV et al. PHYSICAL REVIEW A 97, 023827 (2018) of light spontaneous emission or scattering, generally associ- The latter requires infinitely long “interaction” time, τ → ated with the Dicke-type super- and subradiance phenomena, +∞.TheT matrix, considered as a function of arbitrary become sensitive to the spatial distribution of atomic dipoles complex energy argument E, is expressed as follows: and to the excitation conditions; see [7,43,44]. Such a situation 1 takes place when the cooperative phenomena are considered Tˆ (E) = Vˆ + Vˆ V,ˆ (2.3) for atoms confined and interacted with nanoscale structures. E − Hˆ For the subwavelength waveguide systems, this has motivated researchers to extend the Maxwell-Bloch formalism toward where a more rigorous Heisenberg-Langevin approach based on ˆ = ˆ + ˆ the effective Hamiltonian concept; see [33–36]. But for the H H0 V. (2.4) particular case of a single-photon scattering, the problem can Hˆ , Vˆ , and Hˆ are the unperturbed Hamiltonian, the interacting be described in the more natural formalism of the quantum 0 part, and the total system Hamiltonian, respectively. scattering theory. The present paper develops such a systematic For a particular case of a single-photon scattering on an microscopic theory of light scattering from an atomic array ensemble of N atomic dipoles, the interaction part is given by trapped near a nanofiber, taking into account the entire interac- tion dynamics together with the complete angular momentum N structure of the guided light and the degenerate energy structure (a) Vˆ =− dˆ · Eˆ (ra) + Hˆself . (2.5) of the atoms. a=1 The paper is organized as follows. In Sec. II, we first provide a general description of our microscopic approach of The first term accumulates partial interactions for each of an light scattering in a waveguide configuration and emphasize ath atomic dipole d(a) with an electric field Eˆ (r) at the point the differences with a similar process in free space. The of the dipole’s position. The second term is the dipole’s self- derivation details concerning the modification of the electric- energy part, which is mainly important for renormalization of field Green’s function near the nanofiber are given in Appendix the self-action divergencies [7,46,47]. The field operator can A. In Sec. III, we then present numerical simulations for light be expressed by a standard expansion in the basis of plane scattering from an array consisting of five atoms. Ordered waves as and disordered configurations are considered. The parameters + − of the waveguide mode used in the simulation are given in Eˆ (r) ≡ Eˆ ( )(r) + Eˆ ( )(r) Appendix B. Finally, Sec. IV concludes the paper. 1/2 2πhω¯ † = j i[e a eik·r − e∗a e−ik·r], (2.6) V j j j j II. THE SCATTERING PROBLEM REVISED j FOR A WAVEGUIDE CONFIGURATION where j ≡ k,α is the complex mode index with the mode wave In this section, we review the basic points of the quantum vector k, frequency ωj = ck, and α = 1,2 numerating two scattering problem. Normally introduced in terms of the orthogonal transverse polarization vectors ej ≡ ekα for each k. † transformation of plane waves, it is extended here to the Here, aj and aj are the annihilation and creation operators for specific configuration where the input and output states are the jth field’s mode in free space and the quantization scheme fundamental modes of the waveguide. includes the periodic boundary conditions in the quantization volume V = L3. A. Mathematical framework According to the standard assumptions of the time- dependent scattering theory in free space, the infinite “in- The mathematical framework of the quantum scattering teraction” time τ is physically associated with the duration problem is based on the concept of the asymptotic evolutionary with which the photon’s wave packet overlaps and passes the operator Sˆ that transforms the system states from infinite past | | scattering object. As far as the wave packet tends to approach ψ in to infinite future ψ out as a result of the interaction pro- the monochromatic state, its duration should approach infinity cess [45]. In the interaction representation, the corresponding and, as a consequence, this justifies the presence of the δ asymptotic transformation is given by function and energy conservation in Eq. (2.2). However, the i H τ − i Hτ i H τ transfer τ →+∞ should be done only in the final step |ψ = e 2¯h 0 e h¯ e 2¯h 0 |ψ ≡ Sˆ|ψ , (2.1) out in in of the derivation procedure together with constructing the where τ →+∞. The operator Sˆ can be represented as a matrix main physical characteristics of the process such as transition in a decoupled basis of two interacting subsystems, which we probability per unit time, scattering tensor, and cross section.
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