Ballester, Tame and Kim (Phys. Rev. A 82, 012325, 2010) constructed the first fully quantized input-output (transfer-matrix) theory of surface-plasmon polariton (SPP) scattering at a metal–dielectric interface and used it to design single-SPP beam splitters, including a 50:50 splitter exhibiting Hong-Ou-Mandel interference. The construction left three items explicitly open: evanescent radiation excitations have no Hermitian photon-number operator and hence no physical observable; reciprocity is broken by the evanescent modes and was only checked numerically; and the treatment is single-mode (monochromatic), deferring wave-packet dynamics. In addition, its mode normalization assumes weakly dispersive, lossless media with real propagation constants — a stated working assumption of the 2010 construction rather than one of its announced open items. The problem is to complete this theory: build the quantum scattering formalism for SPPs at interfaces with rigorous observables for evanescent excitations, quantitative reciprocity conditions, and a multimode wave-packet input-output treatment of indistinguishability and interference in cascaded plasmonic circuits, reducing to the 2010 results in the appropriate limit.

Background

A surface-plasmon polariton (SPP) is a quantum of the electromagnetic surface wave that propagates along a metal–dielectric interface with its field coupled to a collective oscillation of the conduction electrons, confining light to subwavelength scales. Because integrated plasmonic circuits route single-quanta states through interfaces and junctions between waveguides with different materials, quantum-information applications require a scattering theory at the operator level: incoming SPP creation operators are mapped to outgoing SPP and radiation-mode operators by a transfer (input-output) matrix, generalizing the beamsplitter relations of quantum optics.

The quantized mode functions at a planar interface were introduced by Elson and Ritchie (Phys. Rev. B 4, 4129, 1971). Ballester, Tame and Kim (Phys. Rev. A 82, 012325, 2010) combined this mode quantization with electromagnetic field matching at a single interface between two regions with different metals and dielectrics, treating SPP modes together with transverse-magnetic (TM) and transverse-electric (TE) radiation modes, the latter including evanescent components — modes whose propagation constant along the interface exceeds the light line, so the field decays exponentially away from the interface rather than propagating. Normalizing each mode by the quantized Poynting vector so that energy flow equals ℏω\hbar\omega times a number operator, they built a unitary transfer matrix and showed that splitting ratios from near-total reflection to 50:50 are reachable with parasitic scattering into photon radiation suppressed to 5% or less, predicting a Hong-Ou-Mandel (HOM) effect — the disappearance of coincidence detections when one SPP enters each input port of a 50:50 splitter — as a direct test of the bosonic nature of SPPs.

Three idealizations of this theory were explicitly acknowledged as open. (1) Evanescent observables: extending the propagating-mode quantization straightforwardly to decaying evanescent radiation yields a photon-number operator that is not Hermitian, hence not a physical observable; the paper states that a more rigorous treatment 'along the lines of Ref. [41]' — the Carniglia–Mandel quantization of evanescent waves (Phys. Rev. D 3, 280, 1971), which restores Hermitian observables by pairing each evanescent wave with a partner mode in a second ('twin') Hilbert space, since joined by detector-mode approaches that build the field modes around the detecting apparatus (Inoue and Hori, Phys. Rev. A 63, 063805, 2001) — was out of scope. (2) Reciprocity: the asymmetric normalization of evanescent excitations breaks reciprocity (equality of transmission, reflection and loss for the two propagation directions), which was only assessed numerically. (3) Multimode dynamics: the theory is monochromatic; a 10-nm wave-packet numerical check showed the scattering coefficients are insensitive, but group-velocity differences between media broaden wave packets during propagation, and a full multimode analysis of indistinguishability and interference in cascaded or extended plasmonic circuits was deferred to a study 'in preparation' that never appeared. Beyond these deferred items, the construction rests on a stated working assumption about the media: the normalization and orthogonality relations assume weakly dispersive, passive media with real permittivities and real propagation constant kxk_x, so strongly dispersive, absorbing, or amplifying media — where kxk_x is complex and the SPP is no longer a normal mode of a closed system — fall outside the assumed regime. Rigorous quantization frameworks for such media do exist, namely the reservoir approach of Huttner and Barnett (Phys. Rev. A 46, 4306, 1992), in which each field mode is coupled to an auxiliary bath of oscillators that absorbs the dissipated energy, and macroscopic QED via Green functions — quantizing the field directly in terms of the lossy medium's classical Green function (its linear response kernel) — in dispersive absorbing dielectrics (Dung, Knöll and Welsch, Phys. Rev. A 57, 3931, 1998; Philbin, New J. Phys. 12, 123008, 2010), but these treat SPPs as resonances of the Green function coupled to a reservoir rather than as normal modes matched across an interface, and have not been merged with the interface scattering theory.

Problem Statement

Construct the complete quantum input-output theory of surface-plasmon polariton (SPP) scattering at metal–dielectric interfaces, completing the program initiated by Ballester, Tame and Kim (2010). The theory must jointly deliver:

  1. Rigorous evanescent-mode observables. Quantize the evanescent radiation excitations of the interface (e.g., along Carniglia–Mandel or detector-mode lines) so that they possess Hermitian number/intensity observables, and compute their contribution to measurable scattering quantities, replacing the non-Hermitian photon-number operator of the straightforward extension used in the 2010 transfer-matrix construction.

  2. Reciprocity. Quantify the reciprocity breaking induced by the evanescent excitations: state the conditions under which the input-output relation of the interface beamsplitter is reciprocal (equality of the transmission, reflection, and radiation-loss coefficients for the two propagation directions), and how symmetry can be engineered under practical material constraints.

  3. Multimode dynamics. Replace the monochromatic single-mode approximation by a full multimode wave-packet treatment, computing how group-velocity differences and dispersion across the media distort propagating SPP wave packets and how this distortion degrades indistinguishability and quantum-interference visibility (e.g., Hong-Ou-Mandel coincidence probabilities) in cascaded or extended plasmonic circuits.

The completion boundary: the three components jointly constitute the completion of the quantum theory of SPP scattering at interfaces with respect to the open items deferred by the 2010 paper; generalizing the 2010 construction's weakly dispersive, lossless media assumption to strongly dispersive, absorbing, or amplifying media with complex kxk_x is background context, not a required additional deliverable; the completed theory must reduce to the monochromatic results of the 2010 theory in the appropriate limit.

References

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