Dissipative stabilization of dark quantum dimers via squeezed vacuum R. Guti errez-J auregui1A. Asenjo-Garcia1and G. S. Agarwal2 1Department of Physics Columbia University New York New York 10027 USA

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Dissipative stabilization of dark quantum dimers via squeezed vacuum

R. Guti´errez-J´auregui,1, ∗A. Asenjo-Garcia,1and G. S. Agarwal2

1Department of Physics, Columbia University, New York, New York 10027, USA

2Institute for Quantum Science and Engineering,

and Departments of Biological and Agricultural Engineering,

and Physics and Astronomy Texas A&M University, College Station, TX 77843

Understanding the mechanism through which an open quantum system exchanges information

with an environment is central to the creation and stabilization of quantum states. This theme

has been explored recently, with attention mostly focused on system control or environment engi-

neering. Here, we bring these ideas together to describe the many-body dynamics of an extended

atomic array coupled to a squeezed vacuum. We show that ﬂuctuations can drive the array into a

pure dark state decoupled from the environment. The dark state is obtained for an even number of

atoms and consists of maximally entangled atomic pairs, or dimers, that mimic the behavior of the

squeezed ﬁeld. Each pair displays reduced ﬂuctuations in one polarization quadrature and ampliﬁed

in another. This dissipation-induced stabilization relies on an eﬃcient transfer of correlations be-

tween pairs of photons and atoms. It uncovers the mechanism through which squeezed light causes

an atomic array to self-organize and illustrates the increasing importance of spatial correlations in

modern quantum technologies where many-body eﬀects play a central role.

I. INTRODUCTION

The collective radiation of an atomic array is an iconic

example of many-body behavior in quantum open sys-

tems. It follows the loss of excitations from several atoms

to a common environment, and arises from vacuum ﬂuc-

tuations [1]. Recent interest in this process lies in the in-

sight it provides to stabilize quantum states by protecting

them from dissipation via destructive interference of indi-

vidual radiative paths. The paths depend on the spatial

arrangement of the array and on the spectral and spa-

tial properties of the environment. Their manipulation

builds upon a larger trend in quantum technologies: the

use of spatial correlations to generate, control, and probe

entangled states in extended many-body systems [2–8].

Current interest in quantum state stabilization via cor-

related radiation follows two experimental trends. On the

one hand, the ability to control atomic positions at the

single-particle level has lead to the creation of emitter

arrays whose patterns are tailored to achieve particu-

lar tasks in quantum simulation [5, 9], sensing [10–13],

or information processing [14, 15]. On the other hand,

ﬂuctuations of the environment have been engineered to

control the radiative response of single trapped ions and

superconducting circuits [17–23].

In this work we bring together ideas used to study

quantum systems extended in space with those of envi-

ronment engineering to describe the correlated decay of

an atomic array coupled to a squeezed vacuum. Squeezed

vacuum corresponds to an engineered environment com-

posed of correlated photonic pairs [24]. It displays a

phase-sensitive ampliﬁcation and deampliﬁcation of ﬂuc-

tuations that has been used to unveil the stochastic na-

ture of quantum optical processes, such as spontaneous

∗Email:r.gutierrez.jauregui@gmail.com

decay [19, 25] and resonance ﬂuorescence [20, 26]. We

show that—depending on the atomic positions and the

spatial proﬁle of the electromagnetic modes carrying the

squeezed ﬁeld—an atomic array can settle into highly

entangled pure states protected from the environment.

The states are built from atomic pairs that mimic the

underlying environment: displaying reduced ﬂuctuations

in one polarization quadrature and ampliﬁed in another.

We explore this phenomenon in one-dimensional arrays

of diﬀerent sizes and atomic positions to show how to ma-

nipulate the atom-atom correlations in the steady-state.

Depending on the system parameters, the stabilized state

is shown to be a pure dimerized state with pair-wise en-

tanglement, a melted dimer with all-to-all correlations,

or an uncorrelated mixed state.

The paper is organized as follows. We begin in Sec. II

by characterizing the broadband squeezed drive and de-

riving the atomic master equation using a cascaded-open-

quantum system perspective. Then, in Sec. III, we map

out changes in the steady state for diﬀerent array sep-

arations and centers. The array is shown to decouple

from the environment when atoms are placed, as pairs,

at points where the two-point correlations of the ﬁeld are

maximized. Numerical results are supported by analyti-

cal expressions obtained via an unraveling of the master

equation. The decoupled states are described in Sec. IV

where we introduce atom-atom interactions. The slow-

fast dynamics obtained from the interplay between coher-

ent interactions and collective dissipation are discussed in

Sec. V. Section VI is left for conclusion.

II. BACKGROUND

The experimental realization of an artiﬁcial atom ra-

diating into a squeezed vacuum by Siddiqi and collab-

orators [19, 20] demonstrated the ability to tailor the

environment and test the limits of conventional quantum

arXiv:2210.03141v2 [quant-ph] 21 Feb 2023

optics using superconducting circuits [27, 28]. It showed

that the atom undergoes a polarization-sensitive decay,

where it relaxes into a steady state following dramati-

cally diﬀerent timescales for each polarization quadra-

ture, as predicted by Gardiner [25]. Eventually, the atom

reaches a mixed steady state identical to that obtained

from the absorption and emission of uncorrelated thermal

photons.

Key to this observation was a source able to produce

correlated photons that covered all the spatial modes sur-

rounding the artiﬁcial atom. The atom was coupled to an

environment displaying a reduced dimensionality with re-

spect to free space to achieve this. At the time, attention

was focused on the temporal correlations of the squeezed

modes with their spatial structure used to determine the

coupling strength and local phase of the interaction. This

structure, however, plays a central role when the system

includes several atoms extended in space, with each one

probing a diﬀerent local environment.

A. Model: Atomic array and correlated travelling

photons

We consider a one-dimensional array of Nat atoms

separated a distance afrom their nearest neighbors, as

sketched in Fig. 1. Each atom is labelled by its position

znand is modelled as a two-state system with states |eni

and |gniseparated by a transition frequency ω0. The

atoms couple exclusively to the electromagnetic modes

of a waveguide, which acts as a source of dissipation and

mediates atom-atom interactions. This waveguide has a

length Land is driven by a broadband squeezed source

through two ports that control the input and output of

photons.

The electric ﬁeld inside the waveguide is naturally ex-

panded in terms of travelling modes. Its positive fre-

quency component reads

Es(z, t) = rLc

2πZdω Z∞

dkδ(ω−ck)e−iωt+sikzbs(k),

(1)

where the operator bs(k) annihilates a photon of wavevec-

tor kand frequency ωpropagating along the s=±di-

rection. In writing Eq. (1) we have assumed a linear

dispersion ω=c|k|, which results in a density of modes

g(ω) = L/2πc.

Correlations between diﬀerent waveguide modes are

grounded on the physical process used to produce and

transmit the squeezed light [29, 30]. We consider here a

parametric ampliﬁer source driving the waveguide. The

source runs on a photon-photon interaction mediated by

a nonlinear medium that is activated by an external

pump [31]. To be speciﬁc, we have in mind a Joseph-

son travelling wave ampliﬁer [32, 33] whose nonlinearity

works as an analogue to those of atomic gases [34, 35]

or optical ﬁbers [36] used in seminal experiments of

b+,in

azc

b-,in

ΔAθ

ΔA0ΔAπ

k0z

3π/2 2ππ/20

|en

|gn

ω0

Figure 1. An emitter array of lattice constant aand center zc

is coupled to a one-dimensional waveguide. The array is com-

posed of Nat qubits of resonance frequency ω0. The waveguide

is driven by broadband squeezed light through two ports. Su-

perposition of both drives causes a periodic ampliﬁcation and

deampliﬁcation in the ﬁeld quadratures ﬂuctuations, as shown

by the local variances ∆A2

0and ∆A2

πtaken from Eq. (6) with

ϕ= 0, Nph = 0.88, M2

ph =Nph(Nph + 1), and drawn as blue

and white solid lines.

squeezing. The ampliﬁer outputs correlated pairs of pho-

tons, aand b, whose frequencies and wavevectors fol-

low the phase matching conditions ωa+ωb= 2ωcand

ka+kb= 2kcwith ωcthe central frequency of the am-

pliﬁer. The input-output relation for each pair is

bs,out(kc+k) = ukbs,in(kc+k) + vkb†

s,in(kc−k) (2)

where the squeezing parameters ukand vkdepend on the

pump strength and nonlinear interaction. They satisfy

|uk|2− |vk|2= 1 [33].

The ampliﬁer outputs photons across a broad opera-

tional bandwidth, which are fed into the waveguide. We

set the central frequency of the ampliﬁer to match the

atomic resonance frequency (ωc=ω0) and consider a

bandwidth much broader than the atomic decay rate so

the squeezed drive covers all the modes relevant for the

atomic interaction. The squeezed ﬁeld inside the waveg-

uide appears as δ-correlated white noise characterized by

hE†

s(zn, tn)Es(zm, tm)i=Nphδ(τns −τms),(3a)

hEs(zn, tn)E†

s(zm, tm)i= (Nph + 1)δ(τns −τms),(3b)

hEs(zn, tn)Es(zm, tm)i=e−iωc(τns+τms)Mphδ(τns −τms),

(3c)

hE†

s(zn, tn)E†

s(zm, tm)i=e+iωc(τns+τms)M∗

phδ(τns −τms),

(3d)

and hEs(z, t)i= 0. Here, cτns =ctn−sznaccounts

for retardation and the parameters Nph and Mph follow

from ukand vkof Eq. (2) [37]. The parameters satisfy

Nph(Nph + 1) ≥ |Mph|2and reach the equality for states

of minimal uncertainty [26].

Equations (3) describe a ﬁeld with Nph photons per

mode that are correlated in pairs through Mph. The

correlations cause a phase-dependent ampliﬁcation and

deampliﬁcation of ﬂuctuations of the ﬁeld quadratures

Aθ(z, t) = 1

s=±

(Es(z, t)eiθ +E†

s(z, t)e−iθ).(4)

As the phase of each travelling ﬁeld Es(z) rotates while it

propagates along the waveguide, the maximally squeezed

quadrature of this ﬁeld rotates as well. The superposition

of left- and right-propagating ﬁelds then gives way to an

oscillating two-point correlation

hAθ(zn)Aθ(zm)i=1

2|Mph|cos(θ−ϕ) cos kc(zn+zm)

2(Nph +1

2),(5)

where the reference phase ϕis set by Mph =|Mph|eiϕ.

The local variances in ﬁeld quadratures

∆A2

θ(z) = hδAθ(z)δAθ(z)i,(6)

with δAθ=Aθ− hAθi, reﬂect an spatial dependence of

ﬂuctuations in this background ﬁeld, as shown in Fig. 1

for maximally and minimally squeezed quadratures.

B. Master equation for an atomic array submerged

in an squeezed environment

The waveguide modes carry the squeezed drive and

photons scattered in and out of the array. This com-

posite ﬁeld is, in general, non-classical and is described

by correlation functions of many orders. We follow the

theory of cascaded quantum systems [39, 40] to model

the evolution of the ﬁeld sources and derive the master

equation for an array radiating into a squeezed vacuum.

In the electric-dipole and rotating-wave approxima-

tions each atom probes the amplitude of the local elec-

tromagnetic ﬁeld via the interaction Hamiltonian

HSR =~X

n,s

√γs(Es(zn)σ(n)

++E†

s(zn)σ(n)

−),(7)

where σ(n)

+=|enihgn|and σ(n)

−=|gnihen|are raising and

lowering operators for the nth-atom; and γs=1

2γwith

γthe decay rate into the waveguide [38]. The total ﬁeld

is composed of free and scattered components

Es(z, t) = Efs(z, t)−iX

n,s

√γsσ(n)

−(t−st0

n)Θ(t−st0

n),(8)

obtained from the Heisenberg equation of motion using

Eqs. (1) and (7). Here ct0

n= (z−zn) describes a time

delay between emission and absorption of an excitation

and the step function Θ(x) ensures causality.

Equation (8) accounts for the evolution of a source as

its output ﬁeld reaches its target. The spatial separa-

tion between source and target is eﬀectively removed by

moving into an interaction picture where the sources are

retarded. For a small array—such that the only changes

on the source as the ﬁeld propagates from one end of the

array to the other are given from its free evolution—this

retardation produces a local phase only [40, 41]. The

Schr¨odinger picture operators (t= 0) of the ﬁeld under

this assumption are

E+(zn) = eik0(zn−z1)Ef+(z1)−i1

2√γ+X

m<n

eik0(zn−zm)σ(m)

−,

E−(zn) = eik0(zNat −zn)Ef−(zNat )−i1

2√γ−X

m>n

eik0(zm−zn)σ(m)

−,

(9)

where we have divided into right- and left-propagating

channels to deﬁne source and target consistently.

The master equation for the density matrix of the ar-

ray ρis derived by substituting Eq. (9) into Eq. (7) and

following the standard approach [42, 43]. The amplitude

and correlations of the free, squeezed ﬁeld are traced back

to its value at the edges of the array. In an interaction

picture with respect to the free term Pn~ω0σ(n)

+σ(n)

−, the

master equation reads

˙ρ=1

i~[Hscatt, ρ] + 1

2γ(Nph + 1)LJ+ρ+NphLJ†

+ρ+1

2|Mph|LJϕ,+ρ−1

2|Mph|LJϕ+π,+ρ

2γ(Nph + 1)LJ−ρ+NphLJ†

−ρ+1

2|Mph|LJϕ,−ρ−1

2|Mph|LJϕ+π,−ρ,(10)

where atom-atom interactions via the two counter-

propagating channels s=±sum to give

Hscatt =1

2~γ

Nat

n,m=1

sin k0|zn−zm|σ(n)

+σ(m)

−.(11)

Loss is accounted for by Lξ· ≡ ξ·ξ†−·1

2ξ†ξ−1

2ξ†ξ·with

collective jump operators

Js=X

e−isk0znσ(n)

−,(12a)

Jϕ,s =eiϕ/2Js+e−iϕ/2J†

s,(12b)

whose reference phase ϕfollows the from squeezed-light

source, deﬁned below Eq. (5). We take ϕ= 0 throughout.

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DissipativestabilizationofdarkquantumdimersviasqueezedvacuumR.Gutierrez-Jauregui,1,*A.Asenjo-Garcia,1andG.S.Agarwal21DepartmentofPhysics,ColumbiaUniversity,NewYork,NewYork10027,USA2InstituteforQuantumScienceandEngineering,andDepartmentsofBiologicalandAgriculturalEngineering,andPhysicsandAstronomyT...

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Dissipative stabilization of dark quantum dimers via squeezed vacuum R. Guti errez-J auregui1A. Asenjo-Garcia1and G. S. Agarwal2 1Department of Physics Columbia University New York New York 10027 USA

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