Active spin lattice hyperpolarization Application to hexagonal boron nitride color centers F. T. Tabesh1M. Fani1J. S. Pedernales2M. B. Plenio2and M. Abdi1y 1Department of Physics Isfahan University of Technology Isfahan 84156-83111 Iran

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Active spin lattice hyperpolarization: Application to hexagonal boron nitride color centers

F. T. Tabesh,1, ∗M. Fani,1J. S. Pedernales,2M. B. Plenio,2and M. Abdi1, †

1Department of Physics, Isfahan University of Technology, Isfahan 84156-83111, Iran

2Institut f¨ur Theoretische Physik, Albert-Einstein-Allee 11, Universit¨at Ulm, 89069 Ulm, Germany

(Dated: October 10, 2022)

The active driving of the electron spin of a color center is known as a method for the hyperpolarization of the

surrounding nuclear spin bath and to initialize a system with large number of spins. Here, we investigate the

eﬃciency of this approach for various spin coupling schemes in a one-dimensional Heisenberg chain coupled to

a central spin. To extend our study to the realistic systems with a large number of interacting spins, we employ

an approximate method based on Holstein-Primakoﬀtransformation. The validity of the method for describing

spin polarization dynamics is benchmarked by the exact numerics for a small lattice, where the accuracy of the

bosonic Holstein-Primakoﬀapproximation approach is conﬁrmed. We, thus, extend our analysis to larger spin

systems where the exact numerics are out of reach. The results prove the eﬃciency of the active driving method

when the central spin interaction with the spin bath is long range and the inter-spin interactions in the bath spins

is large enough. The method is then applied to the realistic case of optically active negatively charged boron

vacancy centers (VB) in hexagonal boron nitride. Our results suggest that a high degree of hyperpolarization

in the boron and nitrogen nuclear spin lattices is achievable even starting from a fully thermal bath. As an

initialization, our work provides the ﬁrst step toward the realization of a two-dimensional quantum simulator

based on natural nuclear spins and it can prove useful for extending the coherence time of the VBcenters.

I. INTRODUCTION

Various properties of defect centers in solid-state materials

have been studied considerably for a long time [1], especially

Nitrogen-Vacancy (NV) centers in diamond [2] (and refer-

ences therein) and diﬀerent defects in silicon carbide [3,4].

In particular, it has been identiﬁed that electron spins of de-

fect centers in wide-band gap semiconductors, most notably

diamond, can be initialized optically and controlled by mi-

crowaves [2]. In addition, the controlled coupling of these

electron spins with proximal nuclear spins (of e.g. nitrogen

atoms for the NV center cases or 13C) have been achieved by

using microwave pulses [5], electrical [6], or optical detec-

tion [7].

On the other hand, extended systems such as nuclear spins

on the surface of diamond [8] or thin 13C layers in diamond [9]

have been recently proposed as potential quantum simulators.

However, a key challenge of these implementations is the ini-

tialization of such nuclear spin ensemble, i.e. generation of a

robust hyper-polarized state with nearly 100% spin polariza-

tion [8,9]. Therefore, various methods have been studied to

achieve this high level of spin polarization by employing color

centers in diamond [8,10–16] and in silicon carbide [17]. The

highly controllable color centers can be polarized eﬃciently at

room temperature via optical and microwave drives, then their

polarization is transferred to other interacting spin species.

In spite of the various valuable works on NV centers in

diamond, spin defects in non-carbon lattices have mostly

been overlooked, while there are tremendous unexplored ar-

eas outside the carbon realm. More recently, defect centers

in 2D materials such as hexagonal boron nitride (hBN) have

been identiﬁed experimentally [18] and characterized theoret-

ically [19,20]. In this case, due to the simultaneous presence

∗fatemeh.tabesh@gmail.com

†mehabdi@gmail.com

of diﬀerent nuclear spin species in a lattice structure, the ini-

tialization of spin ensemble is more complex. Primary exper-

imental attempts to initialize the spin ensemble in hBN based

on the anti-crossing levels have been recently reported [21].

In this paper, we adopt approaches developed in the ﬁeld of

color centers in diamond based on the microwave control of

color centers, to examine scalabale schemes for the hyper-

polarization of the nuclear spins in hBN.

Here, we study the hyperpolarization of nuclear spins

(Borons and Nitrogens) in a mono-layer of hBN lattice via

electromagnetic manipulation of the electron spin of VB. As

an immediate application, this can signiﬁcantly decrease the

pure dephasing contribution of the spin bath, and thus, en-

hancing coherence time of the defect spin state. A longer co-

herence time shall prove useful in every follow-up quantum

technological applications, see e.g. [22]. Unlike the hyper-

polarization of the nuclear spins in diamond, here one should

deal with the polarization of two sub-lattices with diﬀerent

nuclear species and diﬀerent spin values. We survey the hy-

perpolarization of the hBN lattice by optical pumping and mi-

crowave driving and ﬁnds its rapid and eﬃcient performance

well-beyond the low temperature and high magnetic ﬁeld lev-

els. In particular, we examine the direct polarization swap

between the VBdefect and the surrounding nuclei in such a

way that a microwave ﬁeld is applied to handle the electron

spin of the VBdefect. In this scheme, the population transfer

takes place when the Rabi frequency of the microwave driving

ﬁeld is resonant with the energy splitting of the nuclear spins.

Therefore, the ﬂip-ﬂop processes between the VBdefect and

the nuclear spins can result-in the polarization of the nuclear

spin lattice. Moreover, we investigate the optimal control over

the nuclei through adjustment of the magnetic ﬁeld orientation

as well as frequency and amplitude of the microwave drive

that excites the electron spin.

In order to corroborate our study with numerical analyses

on such large spin systems, we employ an approximate numer-

ical method that overcomes the typical limitations in computa-

arXiv:2210.03334v1 [quant-ph] 7 Oct 2022

tional resources for dealing with large scale spin systems. The

numerical method is based on using a bosonic approximation

through the Holstein-Primakoﬀ(HP) transformation [23]. We

justify validity of the approximation in the working regime of

our interest by benchmarking its results with the exact numer-

ics for small sized lattices. The study is then extended to the

larger lattices with faster and yet considerably less computa-

tional resources. The HP transformation has been used for

investigating hyperpolarization of oil molecules [13], sensing

phases of water via NV centers [24], and simulating on a dia-

mond surface [8].

Before addressing the hBN problem, we explore the abil-

ities and limitations of the employed numerical bosonic

method and examine the range of its validity by applying it

to the simple model of inhomogeneous central spin model.

In this toy model, the spins in a one-dimensional Heisenberg

chain inhomogeneously interact with a distinct spin, the ‘cen-

tral spin’. We argue that the bosonic approach is in good

agreement with exact numerical simulation if the interaction

between the central spin and the spin bath is long range. Also,

we ﬁnd that the interaction among bath spins plays an eﬀective

role to transfer polarization throughout the one-dimensional

lattice. Therefore, we introduce the Heisenberg model and

the Gaussian state method in Sec. II. In this example, we are

interested in the study of possibility of strong polarization by

driving the central spin. In addition, we compare the results of

the exact solution and bosonic approximation for a few nuclei.

In Sec. III, we introduce the Hamiltonian of hBN lattice with

a negatively charged VBdefect and the nuclear bath model.

We also study the hyperpolarization in this lattice. Finally, we

summarize and conclude in Sec. IV.

II. CENTRAL SPIN MODEL

In order to better understand the hyperpolarization process

through active manipulation of a defect spin and to examine

the area of validity of the employed approximate numerical

method, here we investigate the familiar problem of central

spin model.

A. Hamiltonian

We begin with the nearest neighbor Heisenberg model for a

chain of one-half spins [25–27] described by the Hamiltonian

(~=1)

HB=h

n=1

n+λ

N−1

n=1

(ˆ

nˆ

n+1+ˆ

S+

nˆ

S−

n+1+ˆ

S−

nˆ

S+

n+1) , (1)

where ˆ

n=1

2σz

ndenotes the spin operator in z-direction and

S±

nis raising/lowering operator on site n. Here, his the Lar-

mor frequency which is proportional to the background mag-

netic ﬁeld and λis the coupling strength. The ﬁrst term of

Hamiltonian (1) is the free energy of spins and the second

term describes the nearest neighbor one-dimensional interac-

tions among the spins. At zero temperature and in the limit

FIG. 1. (a) Sketch of the central spin model studied in this work (top

panel) and the hyperpolarization of the bath spin chain as a result

of active manipulation of the control spin (bottom panel). (b) The

geometry of VBdefect in hBN: A negatively charged boron vacancy

(gray) surrounded by nitrogen (blue) and boron (orange) atoms. The

optical polarizing and microwave drives are also indicated. (c) VB

defect simpliﬁed energy level diagram: The straight green lines show

the exciting laser transitions, while the curly green and dashed blue

lines denote the radiative and non-radiative decay to the ground state,

respectively. The red circle arrows represents the microwave drive.

The inset presents a closer look at the ground state manifold and its

manipulation via external magnetic ﬁeld and microwave drive.

of λh, the ground state of the system is unique and given

by |0i=|0i⊗N, where |0iand |1iare the eigenstates of σz

(Pauli matrix in the z-direction) respectively corresponding to

−1 and +1.

Now, we consider a central spin one-half particle as the

control quantum entity, whose free dynamics is described by

the Hamiltonian ˆ

Hcs =ω0ˆszwith Larmor frequency ω0. In

a possible realistic physical implementation, the control spin

is an optically active spin that a resonant driving ﬁeld is ap-

plied for its manipulation, see Fig. 1(a). The central (control)

spin interacts with the Heisenberg chain through the following

Hamiltonian

Hint =

n=1

Jnˆsxˆ

n, (2)

where Jnare the long-range coupling strengths, while ˆszand

ˆsxare the spin operators of the control spin. The whole system

dynamics is thus given by the Hamiltonian ˆ

H=ˆ

Hcs+ˆ

HB+ˆ

Hint.

For the sake of simplicity, we assume that λ, Jnh, ω0.

Hence, the rotating wave approximation (RWA) is safely sat-

isﬁed and the total Hamiltonian reads

H=ω0ˆsz+h

n=1

n+λ

N−1

n=1

(ˆ

nˆ

n+1+ˆ

S+

nˆ

S−

n+1+ˆ

S−

nˆ

S+

n+1)

n=1

Jn(ˆs+ˆ

S−

n+ˆs−ˆ

S+

n) . (3)

The second line of the above Hamiltonian implies the ﬂip-ﬂop

interactions between the control spin and Heisenberg chain

spins.

B. Hyperpolarization

The hyperpolarization is attainable by performing cycles of

polarization transfer that is made up of the following steps:

(i) The control spin is initialized in the |0istate in each cy-

cle realization. This can be done, e.g. by applying a laser

pulse. (ii) By bringing the control spin into resonance with

the ‘bath’ spins the Hartmann-Hahn condition is satisﬁed, and

thus, the polarization of the control spin is transferred to the

bath spins [28]. The frequency tuning of the control spin

for satisfying the resonance condition can be done in various

ways. One possibility is to apply a resonant driving ﬁeld with

a proper Rabi frequency, see Sec. III. The two above polariz-

ing steps are repeated as many as necessary times to achieve

the desired level of polarization in the Heisenberg chain spins.

To put it in the mathematical form, the total initial state is

given by ρ(0) =|0ih0|cs NN

n=1ρth

nwhere ρth

nis the initial un-

polarized thermal state of the nth spin in the chain. After the

(R+1)th iteration of the polarization cycle, the state of the

Heisenberg spins is found by

ˆρB(R+1)τ=Trcs nˆ

U(τ)|0ih0|cs ⊗ˆρB(Rτ)ˆ

U†(τ)o, (4)

where ˆ

U(t)≡exp{−iˆ

Ht}is the time evolution operator with

Hthe Hamiltonian in Eq. (3) and Trcs {} indicates the partial

trace over the control spin. Here, the period of all cycles is

taken identical and equal to τ. We must emphasize that the

polarization can be eﬀectively conveyed from the control spin

to the Heisenberg spins when ω0=h. Thus, after a perfect

hyperpolarization procedure the ﬁnal state of the bath spins

approaches to |0i=|0i⊗N.

In the following we shall perform exact and approximate

numerical analysis to study this procedure. In order to quan-

tify the level of lattice polarization, we deﬁne the average col-

lective expectation value of the operators ˆ

Szas

Sz=1

n=1

hˆ

, (5)

with hˆ

niand sndenoting the expectation value and amount

of the spin of the nth bath spin in the chain, respectively. In

the current case we have sn=1/2. Note that the total polar-

ization is normalized to unity and −1≤Sz≤+1. The exact

numerics are performed by the QuTiP package [29]. Nonethe-

less, due to the limited computational power the hyperpolar-

ization of large spin systems is studied by a method based

on the Holstein-Primakoﬀapproximation, which is discussed

next.

C. Gaussian states method

In order to model the behavior of large spin baths, we now

employ the Holstein-Primakoﬀtransformation (HPT) and the

corresponding approximation (HPA) to compute the time evo-

lution of Eq. (4) for large number of spins based on bosonic

states [13,23]. In this method, a highly polarized spin is

treated as a boson close to its ground state. It is worth men-

tioning that despite this fact, in the hyperpolarization prob-

lem one usually takes the initial state of the spin bath in a

fully thermal state. Nevertheless, since state of the bath spins

get closer to their respective ground state after each hyper-

polarization cycle the amount of total error committed in this

method is tractable. On the other hand, bosonic ﬁelds whose

dynamics is described by Hamiltonians quadratic in the cre-

ation and annihilation operators of such ﬁelds preserve their

Gaussian character in all future times, provided they are ini-

tially in a Gaussian state. Given the fact that Gaussian states

are completely characterized by their ﬁrst and second mo-

ments, the equations of motion for those moments is enough

for describing the system dynamics. Notably, those equations

of motion have a dimensionality that grows linearly with the

number of constituents. Therefore, their simulation has con-

siderably less complexity compared to the original dynamics.

The Hamiltonian (3) can be exactly mapped into bosons un-

der the HPT. However, the resulting bosonic Hamiltonian is

not quadratic in the creation and annihilation operators. This

makes it diﬃcult to study the system dynamics. One, there-

fore, takes its linear approximation, the lowest order of the

HPA, that works the best for spin states that are close to their

ground state. In this approximation, the spin operators of a

spin-sparticle are transformed as the following

ˆsz=ˆa†ˆa−s11 ,

ˆs+=ˆa†p2s−ˆa†ˆa≈ˆa†√2s, (6)

ˆs−=p2s−ˆa†ˆaˆa≈√2sˆa.

where ˆa†(ˆa) is the bosonic creation (annihilation) operator

with commutator [ˆa,ˆa†]=11. Similarly, we perform the trans-

formation for the Heisenberg chain operators by assigning the

bosonic operators ˆ

bnwho satisfy the commutation relation

[ˆ

bn,ˆ

b†

m]=11δnm. By applying the above transformations the

Hamiltonian in Eq. (3) reads

H(t)=ω0ˆa†ˆa+λ

2(ˆ

b†

1ˆ

b1+ˆ

b†

Nˆ

bN)+(h−λ)

n=1

b†

nˆ

+λ

N−1

n=1

b†

nˆ

bn+1+ˆ

bnˆ

b†

n+1+1

2hˆ

b†

nˆ

bnnn+1(t)+ˆ

b†

n+1ˆ

bn+1nn(t)i

n=1

Jn(ˆa†ˆ

bn+ˆaˆ

b†

n) . (7)

where to maintain the quadratic form of the Hamiltonian we

have employed the mean-ﬁeld approximation to deal with the

terms arising from the ˆ

Szˆ

Szinteractions. Here, ni=hˆ

b†

iˆ

bii

is the instant occupation number of the ith spin site. The

above Hamiltonian can be cast into the compact form of

H(t)=ˆ

R†V(t)ˆ

Rwhere we have introduced the bosonic op-

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Activespinlatticehyperpolarization:ApplicationtohexagonalboronnitridecolorcentersF.T.Tabesh,1,M.Fani,1J.S.Pedernales,2M.B.Plenio,2andM.Abdi1,y1DepartmentofPhysics,IsfahanUniversityofTechnology,Isfahan84156-83111,Iran2Institutf¨urTheoretischePhysik,Albert-Einstein-Allee11,Universit¨atUlm,89069Ulm,Ge...

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Active spin lattice hyperpolarization Application to hexagonal boron nitride color centers F. T. Tabesh1M. Fani1J. S. Pedernales2M. B. Plenio2and M. Abdi1y 1Department of Physics Isfahan University of Technology Isfahan 84156-83111 Iran.pdf

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Active spin lattice hyperpolarization Application to hexagonal boron nitride color centers F. T. Tabesh1M. Fani1J. S. Pedernales2M. B. Plenio2and M. Abdi1y 1Department of Physics Isfahan University of Technology Isfahan 84156-83111 Iran

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