Engineering Optomechanically Induced Transparency by coupling a qubit to a spinning resonator Jessica Burns13 Owen Root23Hui Jing34 and Imran M. Mirza5y

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Engineering Optomechanically Induced Transparency by coupling

a qubit to a spinning resonator

Jessica Burns1,3,∗, Owen Root2,3,,∗Hui Jing3,4, and Imran M. Mirza5†

1Physics Program, University of Cincinnati, OH 45221, USA

2Physics Program, Nebraska Wesleyan University, Lincoln, NE 68504, USA

3Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education,

Department of Physics and Synergetic Innovation Center for Quantum Eﬀects and Applications,

Hunan Normal University, Changsha 410081, China

4Synergetic Innovation Academy for Quantum Science and Technology,

Zhengzhou University of Light Industry, Zhengzhou 450002, China

5Department of Physics, Miami University, Oxford, OH 45056, USA

(Dated: October 17, 2022)

We theoretically study the spectral properties of a pump-probe driven hybrid spinning optome-

chanical ring resonator optically coupled with a two-level quantum emitter (QE or qubit). Recently

we have shown [Optics Express, 27, 18, 25515–25530 (2019)] that in the absence of the emitter the

coupled cavity version of this setup is not only capable of nonreciprocal light propagation but can

also exhibit slow & fast light propagation. In this work, we investigate in what ways the presence

of a single QE coupled with the optical whispering gallery modes of the spinning optomechani-

cal resonator can alter the probe light nonreciprocity. Under the weak-excitation assumption and

mean-ﬁeld approximation, we ﬁnd that the interplay between the rotational/spinning Sagnac-eﬀect

and the qubit coupling can lead to the enhancement both in the optomechanically induced trans-

parency (OMIT) peak value and in the width of the transparency window due to the opening of

qubit-assisted back reﬂection channel. However, compared to the no-qubit case, we notice that such

an enhancement comes at the cost of degrading the group delay in probe light transmission by a

factor of 1/2 for clockwise rotary directions. The target applications of these results can be in the

areas of quantum circuitry and in non-reciprocal quantum communication protocols where QEs are

a key component.

I. INTRODUCTION

Bulk Faraday rotators, either based on magneto-

optical crystals (for instance Yttrium Aluminum Garnet-

YAG) [1] or alkali vapor cells (such as Rb) [2] present a

key example of nonreciprocal optical devices. Commer-

cially available nonreciprocal optical elements, for exam-

ple, the ones with 450shifting in the polarization plane

require Verdet constants V(λ)of almost 80 rad/T m for

a centimeter-long crystal when a magnetic ﬁeld of 1Tis

applied parallel to the propagation direction of electro-

magnetic radiation. However, when such an element is

brought into smaller scales for quantum photonics ap-

plications, due to magnetic ﬁeld strength limitations, it

turns out that even for a 100µm long crystal at 1Ta

Verdet constant of 8000 rad/T m is required to achieve

450shifting. Unfortunately, not all magneto-optical crys-

tals or Alkali vapors are capable of demonstrating such a

high value of V(λ)for a wide range of wavelength/λval-

ues. The matter is further worsened by the fact that even

if such a high value of V(λ)is attained for certain λvalues

it is achieved at the price of higher losses. These consider-

ations pose severe challenges to incorporating traditional

nonreciprocal elements in integrated quantum photonics.

To address these issues, in recent years, hybrid quan-

tum systems [3–8] have emerged as a potential solution.

∗These authors have contributed equally to this work.

†mirzaim@MiamiOH.edu

For instance, in 2013 Peng et al. studied non-reciprocity

in light transmission by breaking the PT -symmetry in

on-chip coupled microtoroid resonators [9]. Extending

this work to the hybrid domain, Zheng et al. consid-

ered two coupled cavity systems in which one of the cav-

ities was interacting with a single qubit that was utilized

to elevate the atom-ﬁeld nonlinearity through the gain

mechanism [10]. Around the same time, Miri et al. re-

ported a uniﬁed framework to establish optical isolation

and non-reciprocity in multimode optomechanical cavi-

ties [11]. Since then several studies have been conducted

to analyze the breaking of time-reversal symmetry of light

propagation in cavity quantum optomechanics (see, for

example, Refs. [12–14]).

One particularly important study in this context was

carried out by Lü et al. where they focused on a pump-

probe driven ﬁber coupled optomechanical ring resonator

which was capable of spinning [15]. Under steady-state

conditions, Lü et al. were able to theoretically show that

with the aid of rotational Sagnac eﬀect not only is non-

reciprocal probe light propagation possible to achieve but

additionally the dispersion properties in the optomechan-

ically induced transparency (OMIT) region allows one to

achieve slow light propagation. More recently, we and

others have further studied spinning ring resonator archi-

tectures and predicted irreversible refraction [16], better

control of non-reciprocity with slow & fast light propaga-

tion [17], nonreciprocal entanglement [18], breaking Anti-

PT -symmetry [19], photon blockade [20], and phonon

blockade [21] via the control of the rotational Sagnac ef-

arXiv:2210.07330v1 [quant-ph] 13 Oct 2022

FIG. 1: (Color online) Model for the hybrid spinning atom-optomechanical microresonator architecture

considered in this paper. In the right channel of the tapered ﬁber, ˆain and ˆaout respectively represent the probe

ﬁeld input and output operators. Whereas, in the left channel of the ﬁber (which is assumed to be driven by a

vacuum ﬁeld) the input and output operators are given by ˆ

bin and ˆ

bout, respectively. For further details about

the system see Sec. II(A).

fect.

Motivated by the aforementioned studies, in this paper

we have focused on the problem of qubit coupled spin-

ning optomechanical ring resonators. Here we asked the

question in what ways can the presence of a weakly cou-

pled two-level QE impact the probe light transmission

through such a hybrid architecture? In addition to be-

ing fundamentally an interesting problem, this study may

ﬁnd applications in non-reciprocal quantum circuitry [22]

and quantum networking/internet [23] where qubits play

a central role in the process of information storage and

manipulation. As some of the key ﬁndings of this work,

we notice (under a set of experimentally feasible param-

eters) that the presence of a weakly coupled qubit assists

in realizing 3% enhancement in the OMIT peak height,

and the transparency window broadens by 1MHz. These

diﬀerences appear due to the opening of a back reﬂection

channel for the probe light propagation which it is not

able to attain without the presence of a QE. Addition-

ally, we note that qubit results in a reduction in the group

delay of the probe ﬁeld compared to the no-qubit case.

These results indicated that even a weakly coupled qubit

can impact the probe light transmission considerably in

spinning optomechanical resonators.

The rest of the paper is organized as follows. In Sec-

tion II we present the theoretical model and calculate

the probe light transmission under steady-state condi-

tions. In Section III, we present results focusing on non-

reciprocal probe light propagation and controlling the

group velocity of light (slow light propagation). Finally,

in Section IV, we close with a summary of the main re-

sults and discuss possible future directions of this work.

II. THEORETICAL DESCRIPTION

A. Setup and Hamiltonian

As shown in Fig. 1, we consider a hybrid setup of a

spinning optomechanical resonator coupled with a sta-

tionary two-level QE (hereinafter also referred to as an

atom or a qubit) residing at the center of the resonator.

Under the rotating wave approximation, and in a frame

rotating at a frequency ωl, the Hamiltonian of the system

(with setting ~= 1) can be decomposed into three parts

H=ˆ

H0+ˆ

Hint +ˆ

Hdr,(1)

where ˆ

H0represents the free/non-interacting part of the

Hamiltonian which is given by

H0=X

ν=a,b

∆cνˆν†ˆν+ ∆eg ˆσ†ˆσ+ˆp2

2m+ˆp2

2mr2+1

2mω2

mˆx2.

(2)

The ﬁrst term on the right hand side of ˆ

H0describes the

Hamiltonian of two counter-propagating optical modes a

and bwith annihilation operators ˆaand ˆ

b, and frequencies

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EngineeringOptomechanicallyInducedTransparencybycouplingaqubittoaspinningresonatorJessicaBurns1;3;,OwenRoot2;3;,HuiJing3;4,andImranM.Mirza5y1PhysicsProgram,UniversityofCincinnati,OH45221,USA2PhysicsProgram,NebraskaWesleyanUniversity,Lincoln,NE68504,USA3KeyLaboratoryofLow-DimensionalQuantumStructur...

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Engineering Optomechanically Induced Transparency by coupling a qubit to a spinning resonator Jessica Burns13 Owen Root23Hui Jing34 and Imran M. Mirza5y

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