Letter Optics Letters 1 Quantum electrodynamics of chiral and antichiral waveguide arrays

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Letter Optics Letters 1

Quantum electrodynamics of chiral and antichiral

waveguide arrays

JEREMY G. HOSKINS1, MANAS RACHH2,AND JOHN C. SCHOTLAND3*

1Department of Statistics, University of Chicago, Chicago, Illinois 60637, USA

2Center for Computational Mathematics, Flatiron Institute, New York, New York 10010, USA

3Department of Mathematics and Department of Physics, Yale University, New Haven, Connecticut 06511, USA

*Corresponding author: john.schotland@yale.edu

Compiled January 6, 2023

We consider the quantum electrodynamics of single

photons in arrays of one-way waveguides, each con-

taining many atoms. We investigate both chiral and

antichiral arrays, in which the group velocities of the

waveguides are the same or alternate in sign, respec-

tively. We ﬁnd that in the continuum limit, the one-

photon amplitude obeys a Dirac equation. In the chi-

ral case, the Dirac equation is hyperbolic, while in the

antichiral case it is elliptic. This distinction has impli-

cations for the nature of photon transport in waveguide

arrays. Our results are illustrated by numerical simula-

http://dx.doi.org/10.1364/ao.XX.XXXXXX

There has been considerable recent interest in the quantum

electrodynamics of light-matter interactions in waveguide sys-

tems [

–

]. The reduced dimensionality of such systems gives

rise to new physical phenomena, especially modiﬁcations of

spontaneous and stimulated emission. Moreover, the enhanced

coupling between atoms and photons can lead to the generation

of strong correlations between photons. Similarly, long-ranged

interactions between atoms bring about a variety of novel many-

body effects. We also note that architectures based on coupled

waveguide arrays have emerged as promising candidates for

quantum circuits and other quantum technologies.

Waveguides, in which the propagation of light is predomi-

nantly in one direction, allow for the intriguing possibility of

serving as one-way carriers of quantum information [

–

]. The

attendant violation of reciprocity in such waveguides is due to

enhanced spin-orbit coupling of light that is conﬁned at sub-

wavelength scales [

–

]. A model of single photons in a one-

way waveguide containing a collection of two-level atoms was

introduced in [

]. It was found that the absence of backscatter-

ing prohibits the existence of band structure in periodic systems.

Moreover, single-photon transport is unaffected by position dis-

order, but not by disorder in atomic transition frequencies, where

Anderson localization obtains. Related results for three-level

atoms and electromagnetically induced transparency have also

been described [14].

In this Letter, we consider the quantum electrodynamics of a

single photon in an array of one-way waveguides. The waveg-

uides, each of which contains many two-level atoms, are ar-

ranged in a one-dimesional lattice, as illustrated in Fig. 1. Two

physical settings are distinguished. For a chiral array, the fre-

quencies alternate in value between two interpenetrating sub-

lattices. In an antichiral array, the group velocities alternate in

sign between the sublattices. In both cases, we will show that

in the continuum limit, the one-photon amplitude of a single-

excitation state obeys a two-dimensional Dirac equation. In a

chiral array, the coordinate along the waveguide is timelike and

the transverse coordinate is spacelike. In contrast, in an antichi-

ral array, both coordinates are spacelike. We will see that the

distinction between the timelike and spacelike character of the

Dirac equations has physical consequences which we illustrate

with numerical simulations.

We begin by considering a one-dimensional array of chiral

waveguides. Each waveguide is assumed to contain many two-

level atoms. We employ a real-space quantization procedure that

treats the atoms and the optical ﬁeld on the same footing [

The Hamiltonian of the system is of the form

H=HA+HF+HI

Here the atomic Hamiltonian HAis given by

HA=ω0Zdx ∑

ρn(x)σ†(x)σ(x).(1)

Here we work in units where

¯h=

ω0

is the atomic transition

frequency of the atoms, and

ρn(x) = ∑jδ(x−xjn )

is the number

density of atoms in the

th waveguide, where

xjn

is the position

of atom

in waveguide

. In addition

σ(xjn) = |

jnih

jn|

the lowering operator for an atom at position

in waveguide

, where

jni

and

jni

denote the corresponding ground and

excited states of the atom. It follows that

obeys the anticom-

mutation relations

{σ(xin ),σ†(xjm )}=δijδnm (2)

and the commutation relations

[σ(xin ),σ(xjm )] = 0 , (3)

with all other commutators and anticommutators vanishing. The

Hamiltonian of the optical ﬁeld HFis of the form

HF=Zdx ∑

nφ†

n(x)(Ωn+ivn∂x)φn(x)(4)

+J0(φ†

n(x)φn+1(x) + φ†

n+1(x)φn(x)).(5)

arXiv:2210.04082v2 [physics.optics] 5 Jan 2023

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摘要：

LetterOpticsLetters1QuantumelectrodynamicsofchiralandantichiralwaveguidearraysJEREMYG.HOSKINS1,MANASRACHH2,ANDJOHNC.SCHOTLAND3*1DepartmentofStatistics,UniversityofChicago,Chicago,Illinois60637,USA2CenterforComputationalMathematics,FlatironInstitute,NewYork,NewYork10010,USA3DepartmentofMathematicsand...

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Letter Optics Letters 1 Quantum electrodynamics of chiral and antichiral waveguide arrays

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