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  • Open Access

Waveguide QED with Quadratic Light-Matter Interactions

Uesli Alushi, Tomás Ramos, Juan José García-Ripoll, Roberto Di Candia, and Simone Felicetti
PRX Quantum 4, 030326 – Published 25 August 2023
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  • INTRODUCTION
  • SCATTERING THEORY FOR TWO-PHOTON…
  • PARADIGMATIC CASES
  • IDEAL CONTROLLED-PHASE GATE FOR…
  • DISCUSSION ON PHYSICAL IMPLEMENTATIONS
  • CONCLUSIONS AND OUTLOOK
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    Quadratic light-matter interactions are nonlinear couplings such that quantum emitters interact with photonic or phononic modes exclusively via the exchange of excitation pairs. Implementable with atomic and solid-state systems, these couplings lead to a plethora of phenomena that have been characterized in the context of cavity QED, where quantum emitters interact with localized bosonic modes. Here, we explore quadratic interactions in a waveguide QED setting, where quantum emitters interact with propagating fields confined in a one-dimensional environment. We develop a general scattering theory under the Markov approximation and discuss paradigmatic examples for spontaneous emission and scattering of biphoton states. Our analytical and semianalytical results unveil fundamental differences with respect to conventional waveguide QED systems, such as the spontaneous emission frequency-entangled photon pairs or the full transparency of the emitter to single-photon inputs. This unlocks new opportunities in quantum information processing with propagating photons. As a striking example, we show that a single quadratically coupled emitter can implement a two-photon logic gate with unit fidelity, circumventing a no-go theorem derived for conventional waveguide-QED interactions.

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    • Received 21 March 2023
    • Accepted 19 July 2023

    DOI:https://doi.org/10.1103/PRXQuantum.4.030326

    Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

    Published by the American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Quantum engineeringQuantum gatesScattering theorySuperconducting quantum optics
    Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & OpticalQuantum Information, Science & Technology

    Authors & Affiliations

    Uesli Alushi1, Tomás Ramos2, Juan José García-Ripoll2, Roberto Di Candia1,3,*, and Simone Felicetti4

    • 1Department of Information and Communications Engineering, Aalto University, Espoo 02150, Finland
    • 2Institute of Fundamental Physics IFF-CSIC, Calle Serrano 113b, Madrid 28006, Spain
    • 3Dipartimento di Fisica, Università degli Studi di Pavia, Via Agostino Bassi 6, Pavia I-27100, Italy
    • 4Institute for Complex Systems, National Research Council (ISC-CNR) and Physics Department, Sapienza University, P.le A. Moro 2, Rome 00185, Italy

    • *rob.dicandia@gmail.com

    Popular Summary

    We explore an innovative way to manipulate light and matter at the quantum level. We study quadratic light-matter interactions, a type of nonlinear coupling between quantum emitters and photonic or phononic modes. This kind of quantum interaction has been extensively studied only in cavity QED, where photons or phonons are localized. We take things a step further by applying this concept to a waveguide-QED setting, uncovering fundamental differences from conventional systems.

    In quadratic waveguide QED, a single artificial atom can scatter a pair of photons while being completely transparent to single photons. Moreover, these interactions offer a wealth of possibilities for quantum information processing with propagating photons. We demonstrate a remarkable feat: a single quadratically coupled emitter can implement a two-photon logic gate with unit fidelity, something that was previously thought impossible with conventional waveguide-QED interactions.

    This research offers an exciting avenue for fundamental quantum science and provides an effective alternative approach to overcome the intrinsic limitations of current quantum technologies.

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    Vol. 4, Iss. 3 — August - October 2023

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    • Figure 1
      Figure 1

      Photon scattering on a quadratically coupled two-level emitter. The nonlinearity of the light-matter interaction is such that the emitter is completely transparent to one-photon pulses, while it strongly interacts with the multiphoton components of the input state. When a two-photon state is sent as input there are three allowed output channels: reflection, splitting, and transmission. The frequency distribution of the input photons plays a key role in determining each channel probability. The frequency of reflected and split photons are strongly anticorrelated, while the transmitted photons are positively correlated.

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    • Figure 2
      Figure 2

      Spontaneous decay: the plots show the frequency distribution (FD) of the emitted field as a function of the two output frequencies ω, ω′. We compare the output FDs for two different coupling functions gΔμμ′=u(Δ)γμμ′/2π, with an isotropic spontaneous emission rate γμμ′=Γ/4, where Γ=0.004ω0. In the first line, we consider the Gaussian function of Eq. (20), while in the second line we assume the Lorentzian coupling of Eq. (21). The parameters β of the coupling functions are chosen to have the same FWHM in both cases. Notice how the spectrum of the spontaneously emitted photons switches from frequency correlation to anticorrelation as the ratio β/Γ increases. An analysis of the entanglement properties of spontaneously emitted photon pairs is provided in Sec. 3c.

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    • Figure 3
      Figure 3

      The two-photon-scattering Gaussian case. We compare the transmitted and reflected (or split) frequency distributions (FDs) for two different values of the total emission rate, Γ=0.004ω0 for the first and Γ=0.012ω0 for the second line. The FD for the reflection and splitting processes are identical. The FDs are shown as a function of the two frequency variables ω, ω′. The input field has been chosen as the product of two independent Gaussian FDs in ω¯ and Δ, centered at ω0 (corresponding to the two-photon resonance condition), variance α2=(0.02ω0)2 and FWHM=2α2ln2=0.047ω0. We consider a Gaussian distributed coupling [Eq. (20)] with β=α and we assume an isotropic spontaneous emission rate, i.e., γμμ′=Γ/4. Focusing on the scattered field (rightmost plots), we see that the width in the sum-frequency variable ω¯ (main diagonal) is proportional to Γ, while the width in the difference-frequency variable Δ (antidiagonal) is given by the input field variance α2 (which has been set equal to that of the coupling function). Interestingly, we see that, when both photons are transmitted, their frequency distribution is correlated, while it is strongly anticorrelated for the reflection and splitting processes.

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    • Figure 4
      Figure 4

      Plot of the reflection probability, i.e., P−−, as a function of the ratio β/α. To obtain the plot, we considered a Gaussian input [Eq. (28)] centered in ω0 with variance α2 and a Gaussian coupling with variance β2 [Eq. (20)]. In addition, we assumed an isotropic spontaneous emission rate. We compare the reflection probability for different values of Γ and we show that the maximum value of P−− is reached when α=β and Γ≫α.

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    • Figure 5
      Figure 5

      Entanglement generation. (a) Setup of a measurement scheme to postselect spontaneously emitted photons traveling in opposite directions. (b) Entanglement entropy of the postselected state of Eq. (32), as a function of the detuning of the filters with respect to the two-photon resonance condition. (c) Entanglement entropy as a function of the ratio between the width β of the coupling function [see Eq. (21)] and the total emission rate Γ. (d) For β/Γ=1/8 and δ/Γ=10, the state is well approximated by a frequency correlated Bell state |ψ−⟩≡12(|ωa,ωa⟩−|ωb,ωb⟩). (e) For β/Γ=10 and δ/Γ=10, the state is well approximated by a frequency anticorrelated Bell state |ϕ+⟩≡12(|ωa,ωb⟩+|ωb,ωa⟩).

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    • Figure 6
      Figure 6

      Sketch of the circuit scheme that implements a controlled-phase gate using two-level quantum emitters. The scheme has been proposed in Ref. [40] to show intrinsic limitations to the achievable fidelity. Here, we show that in the case of quadratic interactions, there are no fundamental limits and the fidelity can in principle be arbitrarily close to 1. The origin of this advantage is that, for quadratic couplings, only the input state |1c,1s⟩ interacts with the emitters. For all other input states, only single photons impinge on the emitters, which are fully transparent in the one-photon subspace.

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    • Figure 7
      Figure 7

      Worst-case gate logarithmic infidelity as a function of Γ, expressed in units of the input FWHM, in the case of a chiral waveguide with coupling gΔ++=g++u(Δ). The input amplitude probability is chosen as Cω¯Δ++=f(ω¯)u(Δ)∗. We compare two different relevant inputs: f(ω¯) real normal distributed (blue) and f(ω¯) real Lorentzian distributed (orange). Both the input frequency distributions f(ω¯) are centered in ω0 and have the same FWHM. We see how in both cases the worst-case gate fidelity approaches unity. In particular, we observe a steeper slope for the normal distributed f(ω¯) with respect to the Lorentzian one.

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    • Figure 8
      Figure 8

      Sketch of proposed implementations of quadratic coupling with propagating modes, generalizing established methods for the cavity QED setting. (a) A quadratically coupled emitter embedded in an array of coupled resonators, that supports a discrete number of propagating modes. (b) A quadratic atom-cavity system linearly coupled to external waveguides. In the bad-cavity limit, the cavity mode can be adiabatically eliminated, obtaining an effective quadratic interaction between qubit and propagating modes.

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    • Figure 9
      Figure 9

      Two photon scattering, Lorentzian case. We compare the transmitted and reflected (or split) FDs for two different values of the total emission rate, Γ=0.004ω0 for the first and Γ=0.012ω0 for the second line. The FD for the reflection and splitting processes are identical. The FDs are shown in function of the two frequency variables ω, ω′. The input field has been chosen as the product of two independent Gaussian FDs in ω¯ and Δ, centered at ω0 (corresponding to the two-photon resonance condition), variance α2=(0.02ω0)2 and FWHM=2α2ln2=0.047ω0. We consider a Lorentzian distributed coupling as in [Eq. (21)] with β=2α2ln2, in order to have the same FWHM for the input field and for the coupling. We assume an isotropic spontaneous emission rate, i.e. γμμ′=Γ/4.

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