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arXiv:1608.01280 · Physical Review A

Quantum-optical description of losses in ring resonators based on field-operator transformations

Paul M. Alsing, Edwin E. Hach, Christopher C. Tison, A. Matthew Smith

Published May 10, 2017• 15 citations
arXivHTMLPDF

Abstract

In this work we examine loss in ring resonator networks from an operator valued phasor addition approach which considers the multiple transmission and cross coupling paths of a quantum field traversing a ring resonator coupled to one or two external waveguide buses. We demonstrate the consistency of our approach by the preservation of the operator commutation relation of the out-coupled bus mode. We compare our results to those obtained from the conventional quantum Langevin approach which introduces noise operators in addition to the quantum Heisenberg equations in order to preserve commutation relations in the presence of loss. It is shown that the two expressions agree in the neighborhood of a cavity resonance where the Langevin approach is applicable, whereas the operator valued phasor addition expression we derive is more general, remaining valid far from resonances. In addition, we examine the effects of internal and coupling losses on the Hong-Ou-Mandel manifold discussed in Hach et al. [Phys. Rev. A 89, 043805 (2014)] that generalizes the destructive interference of two incident photons interfering on a 50:50 beam splitter (HOM effect) to the case of an add-drop double bus ring resonator.

DOI
10.1103/physreva.95.053828

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In this work we examine loss in ring resonator networks from an operator valued phasor addition approach which considers the multiple transmission and cross coupling paths of a quantum field traversing a ring resonator coupled to one or two external waveguide buses. We demonstrate the consistency of our approach by the preservation of the operator commutation relation of the out-coupled bus mode. We compare our results to those obtained from the conventional quantum Langevin approach which introduces noise operators in addition to the quantum Heisenberg equations in order to preserve commutation relations in the presence of loss. It is shown that the two expressions agree in the neighborhood of a cavity resonance where the Langevin approach is applicable, whereas the operator valued phasor addition expression we derive is more general, remaining valid far from resonances. In addition, we examine the effects of internal and coupling losses on the Hong-Ou-Mandel manifold discussed in Hach et al. [Phys. Rev. A 89, 043805 (2014)] that generalizes the destructive interference of two incident photons interfering on a 50:50 beam splitter (HOM effect) to the case of an add-drop double bus ring resonator.
We provide a general macroscopic phenomenological formula of quantum Langevin equations for two coupled phase-conjugated electromagnetic fields with linear loss (gain) and complex nonlinear coupling coefficient. The macroscopic phenomenological formula is obtained from the coupling matrix to preserve the field commutation relations and correlations, which does not require knowing the microscopic details of light-matter interaction and internal atomic structures. To validate this phenomenological formula, we take spontaneous four-wave mixing in a double-$\mathrm{\ensuremath{\Lambda}}$ four-level atomic system as an example to numerically confirm that our macroscopic phenomenological result is consistent with that obtained from the microscopic Heisenberg-Langevin theory. We find that a complex-valued nonlinear coupling coefficient can lead to noises even without linear gain or loss. Finally, we apply the quantum Langevin equations to study the effects of linear gain and loss, complex phase mismatching, as well as complex nonlinear coupling coefficient in entangled photon pair (biphoton) generation, particularly to their temporal quantum correlations.
Several years ago, Guo et al. [Phys. Rev. Lett. 103, 093902 (2009)] demonstrated a counterintuitive phenomenon wherein the transmission of classical light through a coupled pair of waveguides is enhanced as the level of loss in one waveguide surpasses a critical threshold. In this paper, we employ the Heisenberg-Langevin formalism to explore a quantum perspective of this phenomenon. In the case where light is incident in both waveguides, a generally nonzero interference term in the presence of loss appears. Our analysis reveals that loss-induced transparency can manifest at the single-photon level for both separable and entangled photonic states.
Here we present a fully quantum mechanical transfer function model for travelling wave whispering gallery mode resonators. Micro-resonators, such as ring and disk resonators, have been key to the development of high performance chip-scale photonic systems due to their compact footprint, sensitivity and low power operation. In this work we present the first understanding of these resonators to any arbitrary multi-photon state. This was achieved by developing a model that utilizes an efficient scheme for determining the quantum electrodynamic transfer functions relating the Bosonic input/output mode operators in the resonator. This approach has been applied to the understanding of both single photon and two-photon states. In this work we will present a key result on a resonant Hong-Ou-Mandel effect that is inherently realized for any resonator-waveguide coupling constants and can operate over a wide range of resonance conditions. Furthermore, the transfer function approach allows for the straightforward understanding of any resonator-waveguide network with arbitrary modes. This will directly enable the application of quantum resonators to the realization of robust, scalable and efficient Linear Optical Quantum Computing (LOQC) gates. Consequently, it is expected that resonators can be used for both Nonlinear Sign Shift and CNOT gates. And these gates can robustly controlled and efficiently tuned using standard electro-optic effects available in a variety of material systems, such as, Silicon.
The equations of motion for the resonant modes in a lossy cavity, which couples them through the nonresonant thermal loss mechanism described by Senitzky, are examined. The second-order perturbation treatment of dissipation is justified for long time intervals by reference to an intermittent similarity transformation. Self-consistent expressions for commutators such as 〈[Pn(t1), Pm(t2)]〉d, Pn being the operator-amplitude of the electric field in the nth mode, as averaged over the loss mechanism d with respect to its diagonal density matrix, are obtained from Laplace transform expressions for these operators. These averaged commutators describe correlations of fluctuations in different modes, in a way compatible with the ordinary commutation relations for lossless modes. The self-consistent anticommutator expressions, averaged over the loss mechanism, imply the Bose-Einstein formula for thermal energy in each mode at time t2 = t1. This is shown from direct examination of anticommutators and also from an integration of the correlation function. Commutators for the total electric and magnetic fields at two space-time points in the cavity have the proper causal properties. The dissipation terms in the field equations of motion cannot be derived from an equivalent Hamiltonian unless a modified analysis such as that of Kemeny is undertaken. Finally, the approximate equations of motion for the mode operator-amplitudes of a parametric amplifier are derived in the creation-annihilation operator framework and the spectral density of noise obtained at the signal frequency. The equilibrium noise present due to coupling with the idle modes is proportional to the number of photons in the idle frequency mode instead of that number plus one, in contrast to the conclusion of Wagner and Hellwarth. In this system the loss-averaged operators evolve from initial thermal equilibrium values to parametric equilibrium values according to nonlinear equations of motion containing time dependent commutators.
We investigate the effect of backscattering on the Hong-Ou-Mandel manifold (HOMM) that manifests in double-bus micro-ring resonators (MRRs). The HOMM represents higher-dimensional parameter solutions for the complete destructive interference of coincident detection in the HOM effect. To model the backscattering, we introduce a set of internal ‘beam splitters’ inside the ring that allow photons to ‘reflect’ into new counterpropagating modes inside the MRR. We find that the one-dimensional HOMM in MRRs investigated here is extremely robust against deterioration due to backscattering, even in a linear chain of identical MRRs. Further we find that a small amount of backscattering introduced into a chain of non-identical MRRs connected in parallel could be desirable, causing them to behave more like chain of identical MRRs.
We present two complementary strategies for modeling nonlinear quantum optics in realistic resonant integrated optical devices, where scattering loss is present. In the first strategy, we model scattering loss as an effective absorption; in the second, we employ a Hamiltonian treatment, with the loss modelled as a coupling to a 'phantom' channel. As an example, we use these two approaches to model spontaneous four-wave mixing in (i) a ring-channel system and (ii) an add-drop system. We present the rates of photon pairs, broken pairs, and lost pairs for both systems, as well as the full biphoton wavefunction (BWF) including the effects of scattering. We find that for a high-finesse resonator coupled to an arbitrary number of channels, the full BWF can be extracted from the BWF associated with photon pairs in an experimentally accessible output channel.