Pulse area theorem in a single mode waveguide and its application to photon echo and optical memory in Tm3Y3Al5O12 S.A. Moiseev1and M.M. Minnegaliev1 E.S. Moiseev1 K.I. Gerasimov1

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Pulse area theorem in a single mode waveguide and its application to photon echo and

optical memory in Tm3+:Y3Al5O12

S.A. Moiseev1∗and M.M. Minnegaliev1, E.S. Moiseev 1, K.I. Gerasimov1,

A.V. Pavlov1, T.A. Rupasov1, N.N. Skryabin2, A.A. Kalinkin2, S.P. Kulik2,3

1Kazan Quantum Center, Kazan National Research Technical University

n.a. A.N. Tupolev-KAI, 10 K. Marx St., 420111, Kazan, Russia

2Quantum Technologies Center and Faculty of Physics,

M.V. Lomonosov Moscow State University, Leninskie Gory, 119991, Moscow, Russia and

3Laboratory of quantum engineering of light, South Ural State University (SUSU), Lenin Avenue, 454080, Chelyabinsk, Russia

(Dated: April 11, 2023)

We derive the area theorem for light pulses interacting with inhomogeneously broadened ensem-

ble of two-level atoms in a single-mode optical waveguide and present its analytical solution for

Gaussian-type modes, which demonstrates the signiﬁcant diﬀerence from the formation of 2πpulses

by plane waves. We generalize this theorem to the description of photon echo and apply it to the

two-pulse (primary) echo and the revival of silenced echo (ROSE) protocol of photon echo quan-

tum memory. For the ﬁrst time, we implemented ROSE protocol in a single-mode laser-written

waveguide made of an optically thin crystal Tm3+:Y3Al5O12 . The experimental data obtained

are satisfactorily explained by the developed theory. Finally, we discuss the obtained experimental

results and possible applications of the derived pulse area approach.

I. INTRODUCTION

The coherent interaction of a light pulse with resonant

atomic ensembles plays a signiﬁcant role in modern op-

tics and quantum technologies [1–4]. These interactions

often poses nonlinear character, study of which is a dif-

ﬁcult theoretical task. The pulse area theorem [5] pro-

vides a simple but powerful tool for general analysis of

nonlinear coherent light-atoms dynamics in self-induced

transparency [5], optical solitons [6], superradiance [7],

photon echo in optically dense media [8–10], to name a

few. The approach was developed for propagating plane

light waves that interact with atoms in free space. Recent

progress in integrated quantum photonics [11–14] moti-

vates the study of the coherent interaction between light

pulses and resonant atomic ensembles in optical waveg-

uides, where the development of waveguide optical quan-

tum memory (QM) attracts growing attention [15–20].

The goal of an optical QM is to store quantum states

of light for subsequent on-demand retrieval at an arbi-

trary time [21–26]. QM is a vital component for numer-

ous quantum technologies, such as long-distance quan-

tum communications [4, 27, 28], quantum state prepara-

tion [29], and a synchronization unit for optical quantum

processing [24].

Great promises are associated with the photon-echo-

based optical QM in crystals doped with rare earth ions

(REI) [23] that have a long lifetime of quantum coherence

at optical and microwave transitions. Such optical QM

has advantages in its multiplexing capacity for storing a

large number of temporary light modes and demonstrates

high eﬃciency in REIs-doped crystals, for example, 58%

in the cavity assistant scheme [30] and 69% in the free

∗s.a.moiseev@kazanqc.org

space [31], which are comparable to the 76% eﬃciency of

single mode storage achieved with QM protocol based on

electromagnetically induced transparency in REI-doped

crystal [32].

Currently, there is growing interest in the implementa-

tion of photon echo QM in optical waveguides [16, 18, 33–

35] doped by REIs which seem as a convenient platform

for implementation of on-chip QM. Quantum storage in

REI-doped waveguides was demonstrated in experiments

on heralded single photon storage [36, 37], on-demand

qubit storage [19] and frequency-multiplexed storage [39].

All of the above experiments are based on the scheme of

reversible photon echo in an optically dense medium [40–

42], realized for inhomogeneous broadening in the form of

a periodic narrow atomic frequency combs that is called

AFC protocol [43]. There is a particular interest in the

revival of silenced echo (ROSE) protocol [44, 45] for im-

plementation of optically controlled on-demand QM with

low quantum noise background in atomic ensemble with

naturally inhomogeneous broadening and narrow homo-

geneous lines that is embedded in an optical waveguide.

Recently the ROSE protocol was implemented in the

151Eu3+ : Y2SiO5crystal with the type-II single mode

laser-written waveguide [35]. This type of the waveg-

uide supports propagation of light modes with only one

polarization. At the same time, light modes of arbitrary

polarization can propagate in a type-III waveguide. Such

type-III depressed cladding single-mode waveguides were

fabricated by femtosecond laser writing technique in the

crystal Tm3+:Y3Al5O12 [46]. Therefore, it is desirable to

implement the ROSE protocol in this waveguide, which

is the subject of experimental studies of this work.

The mentioned need for a highly eﬃcient implementa-

tion of photon echo QMs and its integration with waveg-

uide schemes makes it relevant to develop a general the-

oretical approach for describing the coherent interaction

between light and atoms in optical waveguides. The ap-

arXiv:2210.10835v2 [physics.optics] 8 Apr 2023

proach should properly simulate the application of in-

tense control laser ﬁelds, e.g., π−pulses, that induce non-

linear dynamics in resonant atoms.

Since the McCall-Hahn work [5], it was discovered that

the most general patterns of linear and nonlinear inter-

action between light pulses and coherent two-level media

can be described by the area theorem [1, 6, 7]. The area

theorem was also applied to the description of photon

echo in optically dense media [8, 9, 47, 48], in Fabry-

Perot resonator [10] as well as it was applied for studies

of photon echo CRIB protocol in free space [49] and for

description of cavity assistant ROSE protocol [45]. It is

worth noting that in recent experiments [45, 48], it has

been found that in media with a symmetrical form of in-

homogeneous broadening, the behavior of the pulse area

of the echo signal almost ﬂatly reproduces the behavior

of the amplitude of echo signals. It makes this approach

universal for describing linear and nonlinear behaviour in

various scenario of photon echo, as well as photon echo

based QM protocols in optically dense resonant media.

In this work, we derive the waveguide pulse area

(WPA) theorem for the resonant interaction between

light pulse and two-level atoms in a single-mode opti-

cal waveguide (see Fig. 1). We found the analytical form

of the WPA theorem, demonstrating signiﬁcant diﬀer-

ences in a formation of stable 2πpulses compared to the

well-known McCall-Hahn pulse area theorem [5]. Then

we generalize WPA theorem for the echo signal emission

and apply it for description of the two-pulse (primary)

photon echo and ROSE protocol of photon echo QM.

For the ﬁrst time, we present experimental results on

ROSE protocol in a laser-written waveguide of a Tm3+

crystal:Y3Al5O12. Then we discuss the obtained exper-

imental data using the WPA theorem, focusing on the

factors leading to a negative impact on the eﬃciency of

the QM protocol implementation and outlining the pos-

sible applications of the developed theoretical approach

to other problems.

II. WAVEGUIDE PULSE AREA THEOREM

A. Maxwell-Bloch equations in single mode

waveguide and pulse area theorem

We derive the equations of motion by starting with

the Hamiltonian ˆ

H0, which is composed of Hamiltonian

of non-interacting two-level atoms ˆ

Hawith inhomoge-

neously broadened resonant transition, Hamiltonian of

waveguide light modes ˆ

Hf, and an interaction Hamilto-

nian between the atoms and waveguide modes in dipole

and rotating wave approximations ˆ

Vfa:

H0=ˆ

Ha+ˆ

Hf+ˆ

Vfa.(1)

We consider the free atomic Hamiltonian to contain sev-

eral types of two-level atoms with diﬀerent dipole mo-

ments of resonant transition dm. It is a typical case with

rare-earth ions, where active ions may substitute for host

atoms at diﬀerent positions in a crystal, leading to sev-

eral (M) groups of atoms with diﬀerent dipole moments

[50]. The free atomic Hamiltonian is

Ha=

m=1

j=1

2(ω0+ ∆j)σj

3;m,(2)

where Nmis a number of atoms within m-th group, ω0is

the carrier frequency of the light ﬁeld coinciding with the

center of the line of the optical atomic transition (see Fig.

1), σj

1,m, σj

2,m, σj

3,m are standard set of Pauli matrices

related to the j-th two-level atom from m-th group, ∆j

is detuning of j-th atom from the carrier frequency.

The Hamiltonian of the waveguide light mode propa-

gating along the zdirection is [51–53]

Hf=~Zdza†(z)[ω0a(z)−ivg

∂

∂z a(z)],(3)

where vg=∂ω0

∂β is a group velocity (see Fig. 1), β=

pk2(ω0)−(2π/Λ)2,k(ω) = n(ω)ω/c,n(ω) = pε(ω),

ε(ω) is an electric permittivity of dielectric medium and

Λ being a critical waveguide wavelength [54], a(z) (a†(z))

is bosonic annihilation (creation) operator of light ﬁeld

mode at given coordinate zwith commutation relation

ˆa(z),ˆa†(z0)=δ(z−z0). The operators are expressed

with a help of their one dimensional Fourier image a(k):

ˆa(z) = 1

√2πZdkei(k−β)za(k),(4)

The electric ﬁeld of the waveguide mode [55], taking into

account their structure in an optical waveguide, has the

form:

E(r) = ieE0f(r⊥)ˆa(z)eiβzj+h.c., (5)

where E0f(r⊥) is a single photon electric ﬁeld ampli-

tude [2] at the point r⊥in the transverse plane of the

single mode waveguide with reference frame being cho-

sen as rj=rj

⊥+zjez.f(r⊥) is a membrane function

of the light mode [54, 56], eis a polarization vector of

the light ﬁeld, E0∼

=(~ω0

2ε0ε(ω0)S)1/2,~is reduced Planck’s

constant, Sis the cross-section of the light beam, ε0is

the electric permittivity of vacuum, respectively [2]. We

assume that permittivity of the dielectric medium is con-

stant and does not change over the bandwidth of interest.

The interaction Hamiltonian between the atoms and

the waveguide mode in the rotating wave approximation

takes form

Vfa =−~

m=1

j=1

Ω0,m(rj

⊥)σj

−,mˆa†(zj)e−iβzj+h.c.,

(6)

Light Pulse

Single

Mode

Waveguide

Atoms

Membrane

Function f(r )

Light Pulse

Spectrum

ω0

FIG. 1. Spatial scheme of interaction of light pulses with an ensemble of two-level atoms in a single-mode optical waveguide.

G(∆/∆in) is an inhomogeneous broadening with linewidth ∆in,cand vgare the speeds of light in free space and in the

waveguide; f(r⊥) is a membrane function of light ﬁeld in the the single mode waveguide.

where σj

±;m=1

2(σj

1;m±iσ2;m) are the isospin ﬂip opera-

tors for j-th atom of m-th group, Ω0,m(rj

⊥)=Ωmf(rj

⊥) is

a coupling constant between j-th atom of m-th group and

waveguide mode with Ωm=E0hdm·ei/~being a single

photon Rabi-frequency of j-th atom located at rj. The

membrane function f(r⊥) determines the dependence of

the interaction constant of an atom on its position r⊥in

the transverse plane of the waveguide. For sake of sim-

plicity but without losing generality, we assume f(r⊥) to

be a real-valued function.

We introduce slowly varied operators

ˆap(z, t) = ieiω0t−iβz ˆa(z, t),(7)

˜σj

−,m(t) = eiω0t−iβz ˆσj

−,m(t),(8)

˜σj

+,m(t) = e−iω0t+iβz ˆσj

+,m(t),(9)

with the same commutation relations as ˆa(z, t),ˆσj

−,m(t)

and ˆ

˜σj

+,m(t), respectively. The index pin ap(z, t) indi-

cates name of the studied pulse for further echo analysis,

i.e. p= (s, 1,2, e) correspond to signal, ﬁrst control, sec-

ond control and echo pulses, respectively.

The resulted Heisenberg equations for the slowly varied

operators are

∂

∂t +vg

∂

∂z ˆap(z, t) =

m=1

j=1

Ω0,m(rj

⊥)ˆ

˜σj

−,m(t)δ(z−zj),(10)

∂ˆ

˜σj

−,m

∂t =−i∆jˆ

˜σj

−,m −i

2Ω0,m(rj

⊥)ˆap(zj, t)σj

3,m,(11)

∂ˆ

˜σj

+,m

∂t =i∆jˆ

˜σj

+,m +i

2Ω0,m(rj

⊥)ˆa†

p(zj, t)σj

3,m,(12)

∂σj

3,m

∂t =−iΩ0,m(rj

⊥)hˆa†

p(zj, t)ˆ

˜σj

−,m −ˆap(zj, t)ˆ

˜σj

+,mi.

(13)

It is worth noting, that at relatively large number of

atoms with negligible small change of the population

(σj

3,m(t)∼

=σj

3,m(t0)), the system of Eqs. (10) - (13) is lin-

earized. Such conditions are often used in diﬀerent pho-

ton echo quantum memory schemes [23, 57, 58], where

their solutions can be found for the arbitrary quantum

states of lights.

The operator Eqs.(10)-(13) can be simpliﬁed in classi-

cal limit of light ﬁeld into c-number equations on aver-

age values of the operators by a proper splitting of two-

particle correlators. For an atomic ensemble with large

Nmand the waveguide mode being in coherent state we

replace the product of operator by a product of their

mutual average values [1, 59]

hˆa(†)

p(zj, t)hσj

3,m(t)i∼

=hˆa(†)

p(zj, t)ihσj

3,m(t)i,(14)

hˆa(†)

p(zj, t)σj

+(−),m(t)i∼

=hˆa(†)

p(zj, t)ihσj

+(−),m(t)i.(15)

For further convenience we switch to real-value equations

by separating the constant phase shift φ0and amplitude

from the averaged ﬁeld operator hˆa0(z, t)i=a0(z, t)eiφ0

and including the phase shift in the atomic operators

σj

+,m(t) = e−iφ0σj

0;+,m(t), σj

−,m(t) = eiφ0σj

0;−,m(t). Fi-

nally, by switching to the components of Bloch vector for

jmatom

m(t) = ih(σj

0;−,m(t)−σj

0;+,m(t))i,(16)

m(t) = h(σj

0;−,m(t) + σj

0;+,m(t))i,(17)

m(t) = hσj

3,m(t)i,(18)

we get the following system of equations

∂

∂t +γw

2+vg

∂

∂z ap(z, t) =

m=1

j=1

Ω0,m(rj

⊥)

4vj

m(t)δ(z−zj),(19)

∂uj

m(t)

∂t =−γ

2uj

m(t)−∆jvj

m(t),(20)

∂vj

m(t)

∂t =−γ

2vj

m(t)+∆juj

m(t)+

Ω0,m(rj

⊥)ap(zj, t)wj

m(t),(21)

∂wj

m(t)

∂t =−Ω0,m(rj

⊥)ap(zj, t)vj

m,(22)

where we have added phenomenologically the decay con-

stant γwdescribing non-resonant losses of the waveguide

modes and the decay constant of atomic phase relaxation

γ= 2/T2(see Appendix A, where the decay constants γw

and γare introduced together with the related Langevin

forces). Although Eqs. (19)-(22) do not describe all the

quantum properties of light and atoms, they are suﬃ-

cient for analysing the eﬃciency, coherence, and spectral

properties of optical QM for weak light ﬁelds.

Next, we derive a pulse area theorem for general de-

scription of the nonlinear properties of coherent inter-

action of light pulse with two-level medium in a single

mode optical waveguide. The presence of several dipole

moments makes it diﬃcult to deﬁne the total pulse area

of the atomic ensemble. Hence instead of pulse area, we

deﬁne envelope area of the ﬁeld that is agnostic to the

presence of several dipole moments

θp(z) = Ztδtf

dtap(z, t).(23)

We assume that pulse duration of the light pulses δtfis

signiﬁcantly shorter than the phase relaxation time of the

atomic transition (γδtf1). By integrating Eq. (19)

over a duration of the light pulse from its beginning t0

to its end similarly to [1, 5] and using formal solution

m(t) = vj

m(t) + iuj

m(t) of Eqs. (20)-(22) with initial

condition vj

m(t0)=0, wj

m(t0) = w0, we get equation for

the envelope area

∂

∂z +γw

2vgθp(z) =

Re{

m=1

j=1

Ω0,m(rj

⊥)

4vgZtδtf

dtrj

m(t)}δ(z−zj) =

Re{

m=1

j=1

Ω2

0,m(rj

⊥)

4vgZtδtf

dt Zt

dt0·

e−(γ/2+i∆j)(t−t0)ap(zj, t0)wj

m(t0)}δ(z−zj).(24)

Taking into account that ap(zj, t δtf) = 0, we

can extend the limits of integration over time from

Rtδtf

t0dt... to R∞

t0dt... By changing the order of integra-

tion further R∞

t0dt Rt

t0dt0... →R∞

t0dt0R∞

t0dt... and per-

forming integration over twe ﬁnd

∂

∂z +γw

2vgθp(z) = Re{

m=1

j=1

Ω2

0,m(rj

⊥)

4vg

γ/2 + i∆j·

Z∞

dt0ap(zj, t0)wj

m(t0)}δ(z−zj) =

m=1

j=1

2γΩ2

0,m(rj

⊥)

4vg(1

4γ2+ ∆2

j)Z∞

dt0αp(zj

m, t0)wj

m(t0)δ(z−zj).

(25)

The coupling constant between an atom and light

mode Ω0,m(r⊥) is contained in the equations for the

components of the Bloch vector (19), (21), (22) and

in the equation for the envelope area (25). For fur-

ther convinience the summation of an arbitrary function

Fm(rj

⊥, zj,∆j, t) over the atoms in continuous limit may

be approximated as an integration

j=1

Fm(rj

⊥, zj,∆j, t)δ(z−zj) =

ρmZd∆G(∆

∆in

)ZS

dxdyFm(r⊥, z, ∆, t),(26)

where Sis a cross-section of the waveguide, ρm=Nm

LS is

a density of m-th atomic group with L being length the

waveguide. For sake of simplicity we assume the identical

inhomogeneous broadening proﬁles G(∆

∆in ) for diﬀerent

atomic groups with linewidth ∆in. Substituting Eq. (26)

in Eq. (25) we get

∂

∂z +γw

2vgθp(z) =

m=1

πρm

4vgZS

dxdyΩ0,m(r⊥)Zd∆

2γG(∆

∆in )

π(1

4γ2+ ∆2)·

Ω0,m(r⊥)Z∞

dt0αp(z, t0)wm(r⊥, z, ∆, t0).(27)

For large enough inhomogeneous broadening (∆in γ)

single atom spectral response is assumed to be delta func-

tion γ

π(γ2+∆2)∼

=δ(∆). If there is no any initial coherence

m(t0) = 0, the solution for resonant atomic coherence

in Eqs. (21-22) is

Ω0,m(r⊥)Ztδtf

dt0αp(zj

m, t0)wm(r⊥, z, ∆=0, t0)

=vm(r⊥, z, ∆=0, t)|t

t0=wj

m(t0) sin Θm,p(r⊥, z),

(28)

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PulseareatheoreminasinglemodewaveguideanditsapplicationtophotonechoandopticalmemoryinTm3+:Y3Al5O12S.A.Moiseev1andM.M.Minnegaliev1,E.S.Moiseev1,K.I.Gerasimov1,A.V.Pavlov1,T.A.Rupasov1,N.N.Skryabin2,A.A.Kalinkin2,S.P.Kulik2;31KazanQuantumCenter,KazanNationalResearchTechnicalUniversityn.a.A.N.Tupolev-...

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Pulse area theorem in a single mode waveguide and its application to photon echo and optical memory in Tm3Y3Al5O12 S.A. Moiseev1and M.M. Minnegaliev1 E.S. Moiseev1 K.I. Gerasimov1.pdf

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Pulse area theorem in a single mode waveguide and its application to photon echo and optical memory in Tm3Y3Al5O12 S.A. Moiseev1and M.M. Minnegaliev1 E.S. Moiseev1 K.I. Gerasimov1

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