Electron-mediated entanglement of two distant macroscopic ferromagnets within a nonequilibrium spintronic device A. Suresh1R. D. Soares2P. Mondal1J. P. Santos Pires2 3J. M. Viana Parente Lopes2 3

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Electron-mediated entanglement of two distant macroscopic ferromagnets within a

nonequilibrium spintronic device

A. Suresh,1R. D. Soares,2P. Mondal,1J. P. Santos Pires,2, 3 J. M. Viana Parente Lopes,2, 3

Aires Ferreira,4A. E. Feiguin,5P. Plech´aˇc,6and B. K. Nikoli´c1, ∗

1Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA

2Departamento de F´ısica e Astronomia, Faculdade de Ciˆencias da Universidade do Porto,

Rua do Campo Alegre, s/n, 4169-007 Porto, Portugal

3Centro de F´ısica das Universidades do Minho e do Porto (CF-UM-UP) and Laborat´orio de F´ısica

para Materiais e Tecnologias Emergentes LaPMET, University of Porto, 4169-007 Porto, Portugal

4School of Physics, Engineering and Technology and York Centre for Quantum Technologies,

University of York, York YO105DD, United Kingdom

5Department of Physics, Northeastern University, Boston, MA 02115, USA

6Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA

Using the nascent concept of quantum spin-transfer torque [A. Zholud et al., Phys. Rev. Lett. 119,

257201 (2017); M. D. Petrovi´c et al., Phys. Rev. X 11, 021062 (2021)], we demonstrate that a current

pulse can be harnessed to entangle quantum localized spins of two spatially separated ferromagnets

(FMs) which are initially unentangled. The envisaged setup comprises a spin-polarizer (FMp) and a

spin-analyzer (FMa) FM layers separated by normal metal (NM) spacer. The injection of a current

pulse into the device leads to a time-dependent superposition of many-body states characterized by

a high degree of entanglement between the spin degrees of freedom of the two distant FM layers.

The non-equilibrium dynamics are due to the transfer of spin angular momentum from itinerant

electrons to the localized spins via a quantum spin-torque mechanism that remains active even for

collinear but antiparallel arrangements of the FMpand FMamagnetizations (a situation in which the

conventional spin-torque is absent). We quantify the mixed-state entanglement generated between

the FM layers by tracking the time-evolution of the full density matrix and analyzing the build-up

of the mutual logarithmic negativity over time. The eﬀect of decoherence and dissipation in the

FM layers due to coupling to bosonic baths at ﬁnite temperature, the use of multi-electron current

pulses and the dependence on the number of spins are also considered in an eﬀort to ascertain the

robustness of our predictions under realistic conditions. Finally, we propose a “current-pump/X-ray-

probe” scheme, utilizing ultrafast X-ray spectroscopy, that can witness nonequilibrium and transient

entanglement of the FM layers by extracting its time-dependent quantum Fisher information.

I. INTRODUCTION

Entanglement describes genuinely quantum and non-

local correlations between diﬀerent parts of a physical

system. Formally, it stems from a many-body wavefunc-

tion that is not expressible in a separable fashion, i.e.,

as the direct product of multiple single-particle states in

some basis. Initially explored in gedanken experiments

and tests of Bell-type inequalities involving spin-1/2 par-

ticles [1–4], quantum entanglement nowadays has risen

to the forefront of many applications, including quantum

cryptography and quantum computation [5–10].

The question of whether quantum entanglement can

survive beyond the microscopic domain into the realm

of macroscopic phenomena [11–13] has fascinated physi-

cists since the inception of quantum theory. Even though

the laws of quantum physics are believed to govern

the behavior of small quantum units and large objects

alike, the fast decoherence of massive quantum super-

positions makes deviations from a classical description

very challenging to detect on a macroscopic scale [14,15].

Notwithstanding these practical diﬃculties, continuous

∗bnikolic@udel.edu

experimental eﬀorts over the past two decades in quan-

tum state preparation and readout of mechanical sys-

tems have highlighted the possibility to entangle the in-

ternal degrees of freedom of larger and larger systems.

This includes putting the phonon modes of two dis-

tant macroscopic mechanical oscillators (each containing

≳1012 atoms) into a nonclassical state [16–20], even

at room temperature [21]. Recent experiments have also

achieved macroscopic entanglement of a mechanical os-

cillator (several millimeters long and ∼10 nm thick) with

a cloud made up of a billion cesium atoms (a collective

atomic spin oscillator) placed at a distance of ∼1 m us-

ing photons propagating between the two objects, as an

entanglement mediator [22]. Besides its fundamental in-

terest [23], these advances can pave the way to a new class

of quantum information technologies and quantum sen-

sors. The possibility to engineer entangled states of large

objects also opens up opportunities to improve the sen-

sitive of gravitational wave detectors, such as the Laser

Interferometer Gravitational-Wave Observatory (LIGO),

as illustrated by a recent proposal to exploit entangle-

ment between optical ﬁelds and atomic clouds to surpass

standard quantum limit [24].

These demonstrations of nonclassical bipartite states

generating entanglement between well-separated objects,

as well as earlier proposals that exploited radiation pres-

arXiv:2210.06634v2 [cond-mat.str-el] 21 Dec 2023

FIG. 1. Schematic view of 1D model of a FMp/NM/FMaspin

valve where a spin-unpolarized current pulse I(t)—carrying

charge Q=RI(t)dt =Neecomprised of one (Ne= 1) or

more (Ne>1) electrons—is injected into the polarizing FMp

layer. After traversing it to become spin-polarized, it im-

pinges onto the analyzing FMalayer where it transfers part

of its spin angular momentum onto the localized spins via

quantum STT [55,56]. Unlike in conventional Slonczewski-

Berger STT studies [52]—where localized spins within FMp

and FMalayers are modeled by classical vectors [51]—we re-

tain a fully quantum description based on spin operators.

Their ground state expectation values ⟨ˆ

Si⟩(t≤0) are de-

picted by red arrows and are arranged into a collinear but

antiparallel geometry in which conventional STT is identi-

cally zero [52]. This process dynamically generates a mixed

entangled quantum state of the FMp∪FMasubsystem. To

account for its dissipation and phase decoherence, we cou-

ple each spin to its own bosonic bath [57,58], where all such

baths are kept in equilibrium at temperature Tbb. Delayed X-

ray pulses, with incoming momentum and energy (ℏki,ℏωi),

are assumed to be shined during and after the current pulse

duration. X-rays, with momentum and energy (ℏkf,ℏωf),

scattered oﬀ the FMp∪FMasubsystem facilitate a witness-

ing scheme of nonequilibrium entanglement we propose based

on extraction [45] of time-dependent quantum Fisher infor-

mation from trRIXS response function [66–68].

sure eﬀects inside microcavities [25–27], have relied upon

the use of photons as mediators of quantum correlations

[28,29]. Recently, a greater interest has been placed on

trying to reproduce these eﬀects in a solid state setup. In

eﬀect, a small ∼10 [30] or moderate ∼103[31] number of

spins in solids have already been entangled at distances of

a few lattice constants, while predictions [32,33] of long-

range entanglement among spin-1/2 probes in strongly

correlated systems have also been recently realized in ex-

periments with superconducting ﬂux circuits [34]. More-

over, several recent theoretical works [35–38] prescribe a

way to entangle much larger spin ensembles (N≳1016)

residing within two distant spheres carved out of a fer-

rimagnetic insulator, using the cavity photon modes as

entanglement mediators. It is important to note that

quantum entanglement of a macroscopic number of de-

grees of freedom is ubiquitous in the ground or low-lying

excited states of strongly electron-correlated materials,

such as superconductors [11], quantum spin liquids [39],

antiferromagnets [12,40–43], and Hubbard model mate-

rials [44,45]. However, in practice, it is extremely chal-

lenging to isolate diﬀerent subsystems of such systems

and then probe their mutual entanglement.

At ﬁrst sight, it seems that none of the plethora

of nonequilibrium spin-dependent phenomena, involving

itinerant electrons and localized spins in typical spin-

tronic devices, like spin valves (SVs) and magnetic tunnel

junctions (MTJs), would be useful for investigating large-

scale entanglement of well-separated quantum units. An-

ticipating the deception of this expectation, we still recall

that SVs and MTJs are composed of two macroscopic

FM layers hosting a very large number of localized spins,

usually derived from d-orbitals of Fe, Ni, or Co. These

FM layers are separated by a few nanometers thick NM

spacer (such as Cu) in SVs (as illustrated schematically

by our 1D model in Fig. 1) or by an insulating barrier

(such as MgO) in the case of MTJs. At standard room-

temperature, the value of these localized spins within

FMs are typically S > 1, which falls outside [47] of the

“ultra-quantum” limit, where quantum corrections to the

S2(1 + 1/S) eigenvalue of the ˆ

ioperator are signiﬁ-

cant [48]. This fact has been used to intuitively (but not

rigorously [49]) justify the modeling of spin dynamics [50]

and injected electronic currents [51] in the presence of

magnetic ﬁelds, by means of the Landau-Lifshitz-Gilbert

(LLG) equation [50,51], which treats localized spins as

classical vectors of ﬁxed length. The extended LLG equa-

tion [51] includes a conventional (Slonczewski-Berger)

spin-transfer torque (STT) [52] term describing spin an-

gular momentum exchange at a semiclassical level. Such

a term may be phenomenological, as in classical micro-

magnetics codes [51], or it can be computed microscopi-

cally from some steady-state [53] or time-dependent [54]

single-particle quantum transport theory.

Defying conventional wisdom, recent experiments at

ultralow temperatures (T∼1 K) [55] have observed

current-driven magnetization dynamics in SVs that

started from collinear magnetizations in the two FM

layers. In this situation, the conventional STT is iden-

tically zero and the system’s dynamics cannot be un-

derstood within the LLG paradigm. This has motivated

the development of a quantum STT theory, where both

the ﬂowing electronic spins and localized spins must be

treated quantum-mechanically [56,59–62]. Even though

a ferromagnet in equilibrium remains in a separable (un-

entangled) quantum state under various externally im-

posed conditions [63], a single FM layer can be driven by

a spin-polarized current to experience a quantum STT

which induces a dynamical build-up of long-range entan-

glement [56,64]. The simplest signature of such entan-

glement is a shrinking in the expectation value of the

localized spin magnitude, i.e.,|⟨ˆ

Si⟩(t)|<Sℏ(ibeing the

site of the crystalline lattice). In some circumstances,

these expectation values can even be reduced to zero [56],

which further explains the failure of the classical LLG

equation [51,52] to describe the STT-driven magnetiza-

tion dynamics on these systems [49,65]. The quantum

nature of the problem then calls for a full-ﬂedged many-

body approach that captures the intrinsic quantum na-

ture of localized spins and thus goes beyond the common

paradigm of classical magnetization dynamics.

Here, we exploit the quantum STT mechanism as a

means to entangle two distant FM layers of a nonequi-

librium SV device that is achievable by modern tech-

niques of nanofabrication (depicted in Fig. 1). The quan-

tum STT is driven by a spin-unpolarized electronic cur-

rent pulse that is injected into the NM and plays the

role of an entanglement mediator. As the pulse trav-

els through the device, it ﬁrst becomes (partially) spin-

polarized by interacting with the polarizing FM layer

(FMp) and then, after traversing the spacer, it even-

tually exchanges spin angular momentum via quantum

STT with the analyzing FM layer (FMa). As required

by general theorems [28,29], the mediating pulse must

be intrinsically quantum mechanical, which assumes a

suﬃciently low working temperature [55] and requires

that decoherence mechanisms aﬀecting the charge-spin

dynamics are suppressed during its traveling time across

the entire device. Under these conditions, we predict that

both FM layers will become mutually entangled over

time, to a degree that can be easily controlled by pa-

rameters such as the magnitude and duration of injected

current pulse. Furthermore, as the device operates, the

FMp∪FMasubsystem also becomes entangled with the

mediating pulse, placing the former into a mixed and en-

tangled bipartite quantum state.

The paper is organized as follows. Section II overviews

the measures of mixed-state entanglement employed in

this study, while Sec. III introduces useful concepts and

notation, including the second-quantized Hamiltonian as

a microscopic model of our SV device and the many-body

algorithms employed in this work. The results for single-

electron current pulse as mediator of entanglement, in-

cluding eﬀects due to thermal ﬂuctuations and coupling

to bosonic baths, are reported in Secs. IV A and IV B;

while zero-temperature results for a many-electron pulse

as mediator of entanglement are analyzed in Sec. IV C.

In Sec. Vwe propose an experimental scheme for wit-

nessing macroscopic and nonequilibrium entanglement of

two FM layers based on application [45] of the state-of-

the-art time-resolved resonant inelastic X-ray scattering

(trRIXS) technique [66–68]. We conclude in Sec. VI.

II. MIXED ENTANGLED STATES AND

MEASURES OF THEIR ENTANGLEMENT

The quantum state of localized spins of the FMp∪FMa

subsystem will become a mixed entangled one in the

course of time evolution (Figs. 3-4). That is, it will be

described by a reduced density matrix which is not ex-

pressible as a convex combination of direct product states

ˆρFMp∪FMa̸=X

piˆρi

FMp⊗ˆρi

FMa,(1)

acting in the bipartite Hilbert space HFMp⊗HFMa. Here

ˆρFMp∪FMa= Treˆρ(t), where ˆρ(t) is density matrix of the

total system FMp∪FMa∪electrons and the partial trace

is performed over the degrees of freedom of the itiner-

ant electrons. Moreover, ˆρFMpand ˆρFMaare analogously

deﬁned density matrices of the individual layers FMp

and FMa. An equality in Eq. (1) would signify sepa-

rable (unentangled) mixed quantum state [69–74]. Al-

though enormous progress has been made in the last two

decades in understanding entanglement of pure quantum

many-body states [10,75,76], much less is understood

regarding the nature of quantum correlations in interact-

ing quantum many-body systems in mixed states. Thus,

how to detect and quantify mixed-state entanglement in

quantum devices containing many interacting particles is

a topic of great and emerging interest [70,71,73]. The

mixed states arise due to thermal ﬂuctuations [70], due to

decoherence by an external environment or because they

describe a subsystem of interest within a much larger and

globally entangled system described by a pure state.

To describe the entanglement of a quantum many-body

mixed state ˆρFMp∪FMaof a FMa∪FMpsubsystem, we cal-

culate the mutual logarithmic negativity (MLN) between

the FMpand FMalayers deﬁned by [74]

EN(FMp|FMa)≡EN(ˆρFMp∪FMa) = ln ||ˆρTFMp

FMp∪FMa||1

= ln ||ˆρTFMa

FMp∪FMa||1= ln X

n|λn|,(2)

where || ˆ

A||1= Tr|ˆ

A|= Trpˆ

A†ˆ

Ais the trace norm of

operator ˆ

A; and λnare the eigenvalues of ˆρTFMp

FMp∪FMaor

ˆρTFMa

FMp∪FMa. The matrix elements of the partial transpose

with respect to, e.g., FMpare given by [10]

ˆρTFMp

FMp∪FMaiα;jβ =ˆρFMp∪FMajα;iβ ,(3)

using matrix elements of ˆρFMp∪FMa

ˆρFMp∪FMaiα;jβ =FMp⟨i|FMa⟨α|ˆρFMp∪FMa|j⟩FMp|β⟩FMa.

(4)

Although the MLN can be zero for an entangled mixed

state, a nonzero MLN necessarily implies existence of en-

tanglement between the two parts. The MLN also oﬀers a

useful probe for distinguishing bipartite and multipartite

quantum correlations in a pure state of the total system—

for tripartite pure state |Ψ⟩FMp∪FMa∪eof a system com-

posed of all localized spins and injected electrons, as de-

noted by FMp∪FMa∪electrons, MLN of ˆρFMp∪FMain

Eq. (2) detects genuine quantum correlations between

FMaand FMp. In the SV device depicted in Fig. 1, they

build-up dynamically [Figs. 3(d) and Fig. 4] for t > 0

when the device is out of equilibrium, while not being

present prior (t≤0) to the injection of current pulse

when the SV remains in equilibrium.

In addition, we also use the mutual information (MI)

[Fig. 3(c)] between FMpand FMalayers

I(FMp|FMa) = SFMp+SFMa− SFMp∪FMa,(5)

obtained from the standard [75,76] von Neumann entan-

glement entropy [Figs. 3(a) and 3(b)]

Ssub(t) = −Tr [ˆρsub(t) ln ˆρsub(t)] ,(6)

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Electron-mediatedentanglementoftwodistantmacroscopicferromagnetswithinanonequilibriumspintronicdeviceA.Suresh,1R.D.Soares,2P.Mondal,1J.P.SantosPires,2,3J.M.VianaParenteLopes,2,3AiresFerreira,4A.E.Feiguin,5P.Plech´aˇc,6andB.K.Nikoli´c1,∗1DepartmentofPhysicsandAstronomy,UniversityofDelaware,Newark,DE1...

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Electron-mediated entanglement of two distant macroscopic ferromagnets within a nonequilibrium spintronic device A. Suresh1R. D. Soares2P. Mondal1J. P. Santos Pires2 3J. M. Viana Parente Lopes2 3.pdf

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Electron-mediated entanglement of two distant macroscopic ferromagnets within a nonequilibrium spintronic device A. Suresh1R. D. Soares2P. Mondal1J. P. Santos Pires2 3J. M. Viana Parente Lopes2 3

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