Coherent Dynamics of Strongly Interacting Electronic Spin Defects in Hexagonal Boron Nitride Ruotian Gong1Guanghui He1Xingyu Gao2Peng Ju2Zhongyuan Liu1Bingtian Ye34

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Coherent Dynamics of Strongly Interacting Electronic Spin Defects in Hexagonal

Boron Nitride

Ruotian Gong,1Guanghui He,1Xingyu Gao,2Peng Ju,2Zhongyuan Liu,1Bingtian Ye,3,4

Erik A. Henriksen,1,5Tongcang Li,2,6Chong Zu1,5,†

1Department of Physics, Washington University, St. Louis, MO 63130, USA

2Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USA

3Department of Physics, Harvard University, Cambridge, MA 02138, USA

4Department of Physics, University of California, Berkeley, CA 94720, USA

5Institute of Materials Science and Engineering, Washington University, St. Louis, MO 63130, USA

6Elmore Family School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907, USA

†To whom correspondence should be addressed; E-mail: zu@wustl.edu

(Dated: July 14, 2023)

Optically active spin defects in van der Waals materials are promising platforms for modern

quantum technologies. Here we investigate the coherent dynamics of strongly interacting ensembles

of negatively charged boron-vacancy (V−

B) centers in hexagonal boron nitride (hBN) with vary-

ing defect density. By employing advanced dynamical decoupling sequences to selectively isolate

diﬀerent dephasing sources, we observe more than 5-fold improvement in the measured coherence

times across all hBN samples. Crucially, we identify that the many-body interaction within the V−

ensemble plays a substantial role in the coherent dynamics, which is then used to directly estimate

the concentration of V−

B. We ﬁnd that at high ion implantation dosage, only a small portion of the

created boron vacancy defects are in the desired negatively charged state. Finally, we investigate

the spin response of V−

Bto the local charged defects induced electric ﬁeld signals, and estimate its

ground state transverse electric ﬁeld susceptibility. Our results provide new insights on the spin and

charge properties of V−

B, which are important for future use of defects in hBN as quantum sensors

and simulators.

Introduction— Solid-state point defects with optically

addressable electronic spin states have become some of

the most fertile playgrounds for new quantum technolo-

gies [1–18]. Signiﬁcant recent progress has been made in

creation and control of such spin-active quantum emit-

ters in atomic-thin van der Waals materials. The two-

dimensional (2D) nature of the host materials can enable

seamless integration with heterogeneous, optoelectronic,

and nanophotoic devices, providing a pathway to investi-

gating light-matter interactions at the nanoscale [19–22].

From a wide range of contestant spin defects in 2D

materials, the negatively charged boron vacancy center,

V−

B, in hexagonal boron nitride (hBN) has particularly

attracted substantial research interest in the past few

years [23–31]. Importantly, it has been demonstrated

that the spin degree of freedom of V−

Bcan be optically

initialized and readout, as well as coherently manipulated

at room temperature. Compared to conventional spin

qubits in three-dimensional materials, such as nitrogen-

vacancy (NV) center in diamond, V−

Bfeatures several

unique advantages in quantum sensing and simulation.

From the perspective of quantum sensing, the

atomically-thin structure of hBN can allow the V−

Bsen-

sor to be positioned in close proximity with the target

materials, facilitating the imaging of inter-facial phenom-

ena with unprecedented spatial resolution and sensitivity

[25, 32–34]. Moreover, since hBN has been widely em-

ployed as the encapsulation and gating dielectric material

in 2D heterostructure devices, introducing the embedded

V−

Bsensors does not require any additional complexity in

the fabrication process [35–39]. On the quantum simula-

tion front, the ability to prepare and control strongly in-

teracting, two-dimensional spin ensembles opens the door

to exploring a number of intriguing many-body quan-

tum phenomena [40–42]. For instance, dipolar interac-

tion in 2D is particularly prominent from the perspective

of localization and thermalization, allowing one to exper-

imentally investigate the eﬀect of many-body resonances

[43–50].

V−

Bin hBN, like solid-state spin defects in general,

suﬀers from decoherence. To this end, research eﬀort

has been devoted to characterizing the coherence time of

V−

B. However, the measured spin echo timescale, TEcho

in several studies varies from tens of nanoseconds to a

few microseconds [24, 51–53]. This immediately begs

the question that where does such discrepancy originate

from, and what are the diﬀerent decoherence mechanisms

in dense ensemble of V−

In this letter, we present three main results. First,

we introduce a robust diﬀerential measurement scheme

to reliably characterize the spin coherent dynamics of

V−

Bensemble (Fig. 1 and Fig. 2). We observe spin-echo

TEcho

2≈70 ns across three hBN samples with distinct

V−

Bdensities (created via ion implantation with dosages

spanning two orders of magnitude), consistent with the

expectation that the spin-echo coherence time is domi-

nated by the Ising coupling to the nearby nuclear spin

and dark electronic spin bath [51, 54]. By applying a

arXiv:2210.11485v3 [quant-ph] 12 Jul 2023

0 500 1000 1500 2000 2500 3000

Time (ns)

0.2

0.4

0.6

0.8

Normalized Contrast C(t)

Laser Power = 5 mW

Laser Power = 10 mW

0 1000 2000 3000

Time (ns)

0.96

0.98

SB / SR

Polarization

Detection

Laser

Microwave

2∓y

III

πx

20 μs

10 μs

PL Signal

SB, SD

Bath Spin

V−

(d)

(e)

(a)

XY8

DROID

π|x

π/2|x

π/2|y

π|y

Dgs

|ms=±1⟩

|ms= +1⟩

|ms=−1⟩

∝2B

(b) (c)

FIG. 1. Spin dynamic of V−

Bensemble (a) Schematic of V−

Bspin ensemble (red spins) inside hBN crystal lattice (Nitrogen–

blue; Boron–white); ˆzis deﬁned along the c-axis (perpendicular to the lattice plane). ˆxand ˆylie in the lattice plane, with ˆx

oriented along one of the three V−

BNitrogen bonds. Here we only include two layers for the purpose of demonstration, but all

our samples have a thickness ∼100 nm. Two types of decoherence sources are presented here for V−

Bspin ensemble: the Ising

coupling (grey wavy lines) to the bath spins (grey), and the dipolar interaction within V−

Bthemselves (red wavy lines). (b)

Energy level diagram of the defect spin ground-state. In the absence of any external perturbation, the |ms=±1⟩states are

degenerate and separated by Dgs ≈3.48 GHz from the |ms= 0⟩state. Under an external magnetic ﬁeld Balong the c-axis of

hBN, the degeneracy between |ms=±1⟩states are lifted via the Zeeman eﬀect, with a splitting ∝2B. We choose |ms= 0⟩

and |ms=−1⟩states as our two-level system. (c) Experimental pulse sequences for XY-8 (top) and DROID (bottom). The

rotations along the positive ˆxand ˆyaxes are plotted above the line, while the rotations along the negative axes are plotted

below the line. (d) Diﬀerential measurement sequence for spin echo. I: 20 µs wait time to reach charge state equilibration.

II: 10 µs laser pulse to initialize the V−

Bspin to |ms= 0⟩, with the reference signal, SR(t), collected at the end of the laser

pulse. III: microwave wave pulses for spin echo measurement; for the bright signal, a ﬁnal π

2pulse along the −ˆyaxis is applied;

while for the dark signal, a ﬁnal π

2pulse along the +ˆyaxis is applied to rotate the spin to an orthogonal state. IV: laser

pulse to detect the spin state. (e) Spin echo measurement on sample S3 at two diﬀerent laser powers. Without diﬀerential

measurement, the measured signal, SB/SRexhibits a laser power dependence which comes from charge relaxation dynamics

(inset). Using diﬀerential measurement, the measured contrast, C(t), is independent of the laser power. Error bars represent

1 s.d. accounting statistical uncertainties.

more advanced dynamical decoupling sequence, XY-8, to

better isolate V−

Bfrom the bath spin environment [55–57],

we observe substantial extensions in the measured coher-

ent timescales, TXY8

2. Interestingly, the extracted TXY8

decreases with increasing V−

Bdensity, indicating that the

dipolar interaction within the V−

Bensemble is critical for

understanding the coherent dynamics. To further cor-

roborate this, we utilize the DROID pulse sequence to

decouple the V−

B−V−

Bdipolar interaction [58, 59], and

achieve an additional ∼2-fold improvement in the mea-

sured coherence time, TD

2. Second, by comparing the ex-

perimentally measured TXY8

2and TD

2to numerical simu-

lations, we directly esimtate the spin density of V−

Bacross

three hBN samples. We ﬁnd that the ratio of negatively

charged V−

Bto total created boron vacancy defects (VB)

decreases signiﬁcantly with increasing ion implantation

dosage (Fig. 3). Third, based on the extracted V−

Bden-

sity, we introduce a microscopic model of local charges

surrounding a spin defect to account for the observed en-

ergy splitting between |ms=±1⟩states at zero magnetic

ﬁeld [60, 61], and estimate the transverse electric ﬁeld

susceptibility of V−

Bto be around d⊥≈40 Hz/(V ·cm−1)

(Fig. 4).

Experimental system— To investigate the coherent

spin dynamics of V−

Bensemble at various defect densities,

we prepare three hBN samples with diﬀerent implanta-

tion dosages. Speciﬁcally, we irradiate hBN ﬂakes (thick-

ness ∼100 nm) using 3 keV He+ion beams with dose

densities, 0.30 ±0.03 nm−2(sample S1), 1.1±0.1 nm−2

(sample S2), and 10 ±1 nm−2(sample S3), respectively,

Implantation Dosage (ion/nm2)

Time (ns)

5000

10000

15000 T1

Spin Locking

10-1 100101

200

400

600 DROID

XY8

Spin Echo

0 500 1000 1500

Time (ns)

0.2

0.4

0.6

0.8

Normalized Contrast C(t)

Spin Locking

DROID

XY8

Spin Echo

(a)

(b)

FIG. 2. Spin coherent and relaxation dynamics. (a) The

spin coherent and relaxation timescales measured on sample

S3 with the highest ion implantation dosage. Dashed lines are

data ﬁtting with single exponential decays. (b) The extracted

coherence timescales T2and relaxation timescales T1for the

three hBN samples.

to create V−

Bdefects [27, 52]. Here error bars on the im-

plantation dosages account for the current ﬂuctuations

during the implantation process. We remark that, given

an ion implantation dosage, the total created VBcon-

centration can be estimated via SRIM simulation (see

Methods) [62], yet the actual density of the negatively-

charged V−

Bhas remained unknown.

The V−

Bcenter has a spin triplet ground state (|ms=

0,±1⟩), which can be initialized and read out via optical

excitation and coherently manipulated using microwave

ﬁelds [23, 30]. In the absence of any external pertur-

bations, the |ms=±1⟩states are degenerate and sep-

arated from |ms= 0⟩by Dgs ≈3.48 GHz (Fig. 1b).

In the experiment, we apply an external magnetic ﬁeld

B≈250 G along the c-axis of the hBN lattice to sep-

arate the |ms=±1⟩states via the Zeeman eﬀect and

isolate an eﬀective two-level system |ms= 0,−1⟩. A mi-

crowave ﬁeld is used to coherently manipulate the spin

ensemble with a Rabi frequency Ω ≈83 MHz (π-pulse

length τπ= 6 ns). We note that such a strong Rabi drive

is crucial for the high ﬁdelity control of V−

B, as the spin

transition is largely broadened by the hyperﬁne interac-

tion to the nearby nuclear spin bath (see Methods).

Robust measurement scheme— To reliably probe the

spin dynamics of a dense ensemble of V−

B, we introduce

a robust diﬀerential measurement scheme illustrated in

Figure 1d [63, 64]. Speciﬁcally, after letting the spin

system reach charge state equilibration for 20 µs with-

out any laser illumination (I), we apply a 10 µs laser

pulse (532 nm) to initialize the spin state of V−

B(II),

followed by the measurement pulse sequences (III). Tak-

ing spin echo coherent measurement as an example, we

ﬁrst apply a π

2-pulse along the ˆyaxis to prepare the sys-

tem in a superposition state ⊗i|0⟩i+|−1⟩i

√2, and then let

it evolve for time t. A refocusing π-pulse along the ˆx

axis at time t/2 is used to decouple the spin ensemble

from static magnetic noise. A ﬁnal π

2-pulse along the

−ˆydirection rotates the spin back to the ˆzaxis for ﬂuo-

rescence detection (IV), and the measured photon count

is designated as the bright signal, SB(t). By repeating

the same sequence but with a ﬁnal π

2-pulse along the

positive +ˆyaxis before readout, we measure the ﬂuores-

cence of an orthogonal spin state to be the dark signal,

SD(t). The diﬀerence between the two measurements,

C(t) = [SB(t)−SD(t)]/SR(t), can faithfully represent the

measured spin coherent dynamics of V−

B, where SR(t) is

a reference signal we measure at the end of the initializa-

tion laser pulse (II).

Figure 1e shows the measured spin echo dynamics of

the highest dosage hBN sample S3. We ﬁnd that the mea-

sured ﬂuorescence contrast, SB(t)/SR(t) [SD(t)/SR(t)],

changes dramatically with diﬀerent laser powers (inset),

originating from the charge state relaxation dynamics af-

ter the laser pumping. This is particularly prominent at

high laser power, where the optical ionization of the de-

fect charge state is enhanced. This eﬀect can lead to an

artifact in the extracted spin echo timescales, which may

explain the previous discrepancy in the measured TEcho

However, the obtained ﬂuorescence contrast from diﬀer-

ential measurement, C(t), is consistent across diﬀerent

laser powers, enabling an accurate extraction of the spin

coherent timescales.

A few remarks are in order. First, this diﬀerential mea-

surement scheme has been widely employed in the stud-

ies of the dense ensemble of NV centers in diamond to

counter the ionization process [9, 40, 64–66]. Secondly,

previous theoretical studies predict that the ionization

of V−

Brequires signiﬁcantly higher energy (∼4.46 eV)

than the ionization of NV centers (∼2.7 eV) [65, 67, 68].

This may explain why our experimental observation that

the two-photon ionization process for V−

Bonly becomes

evident under strong laser power (∼10 mW); while the

ionization of NV centers happens at ∼10 −20 µW laser

[64, 65]. Third, we note that unlike neutral NV0cen-

ters which emit ﬂuorescence starting at 575 nm, neutral

boron-vacancy V0

Bhas not been directly observed from

photo-luminescence signals. Therefore the proposed ion-

ization process only oﬀers a potential explanation of the

experiment.

Coherent dynamics— Equipped with the robust diﬀer-

ential measurement scheme, we now turn to the investi-

gation of coherent dynamics of V−

Bensemble at various

defect densities. The decoherence mechanism of V−

Bcon-

sists of two major contributions: (1) the Ising coupling to

the bath spins in the environment; (2) the dipolar inter-

action between V−

Bensemble themselves (Figure 1a and

Methods) [58]. To isolate the eﬀect of each component,

we measure the coherent dynamics of V−

Busing three dif-

ferent dynamical decoupling pulse sequences.

We start with the spin echo pulse sequence, which

is commonly used to characterize the coherent proper-

ties of a quantum system. Spin echo can decouple the

static components of the Ising coupling between V−

Band

the spin bath. By ﬁtting the measured dynamics to

a single exponential decay, ∼e−(t/T Echo

2), we extract

TEcho

2≈70 ns across all three hBN samples (Figure 2b).

This observation indicates that the spin echo decoherence

of V−

Bis predominantly limited by the spin ﬂuctuation

within the environmental spin bath, which does not de-

pend on the V−

Bconcentration. Indeed, a previous study

has shown that the Ising coupling to the local nuclear

spin bath (nitrogen-14, boron-10, and boron-11), as well

as the dark electronic spins, can accurately account for

the measured spin echo timescales [51].

Next, we apply a more advanced dynamical decoupling

pulse sequence, XY-8, to better decouple the V−

Bensem-

ble from the environment. Instead of a single refocusing

π-pulse, XY-8 employs a series of π-pulses with alternat-

ing phases (Fig. 1c). We ﬁx the time intervals between

pulses, τ0= 4 ns, suﬃciently smaller than the corre-

lation timescale of the local spin bath (estimated from

the spin echo timescale) [40, 44]. As a result, XY-8 is

expected to further suppress the ﬂuctuations within the

local spin noise and improve the measured spin coher-

ent timescales. This is indeed borne out by our data.

As shown in Figure 2, the extracted coherence times,

TXY8

2, are signiﬁcantly extended in all three samples. In

contrast to the previous spin echo measurement where

TEcho

2does not depend on V−

Bdensity, here we observe

that TXY8

2= [250 ±35] ns of sample S1 is longer than

sample S3, TXY8

2= [167 ±10] ns. This suggests that V−

−V−

Binteraction plays a key role in the measured XY-8

coherent timescales. Indeed, in XY-8 measurement, since

the refocusing π-pulses ﬂip all V−

Bspins together, there

is no suppression of the dipolar interaction between V−

(see Methods).

To this end, we introduce DROID pulse sequence to

further decouple the dipolar interaction within V−

Bthem-

selves (Fig. 1c) [58]. By applying a series of π/2 rotations

along diﬀerent spin axes to change the frames of inter-

action (also known as toggling frames), DROID modi-

ﬁes the dipolar Hamiltonian to an isotropic Heisenberg

10-1 100101

Ion Dosage

(ion/nm2)

(%)

(a) (b)

100 200 300 400 500

VB Density (ppm)

102

103

Coherence Time (ns)

DROID

XY8

0 300

Counts (a.u.)

FIG. 3. Characterizing V−

Bdensity (a) Comparison be-

tween the experimentally measured and numerically simu-

lated coherent timescales, T2, for DROID and XY-8 pulse

sequences. The solid lines show the timescales extracted from

simulations with error bars plotted as semi-transparent col-

ored areas. To determine V−

Bdensities for the three hBN

samples, we minimize the relative squared residuals of TXY8

and TD

2between simulations and experiments. Inset: ﬂuo-

rescence counts versus extracted densities after contrast ad-

justment (see Methods). (b) The measured V−

Bcharge state

ratio η=ρV−

B/ρVBfor three hBN samples with diﬀerent ion

implantation dosages.

interaction, where the initial state, ⊗i|0⟩i+|−1⟩i

√2, consti-

tutes an eigenstate of the Heisenberg interaction, and

consequently does not dephase (see Methods). As shown

in Figure 2, the measured coherent timescales, TD

2, in-

deed exhibit an approximate two-fold increase compared

to TXY8

2across all three samples, agreeing with the can-

cellation of dipolar-induced decoherence. Interestingly,

we also observe that the spin relaxation time, T1, and

spin-locking time, Tρ

1, both decrease with increasing ion

implantation dosages (Figure 2b). In principle, the dipo-

lar interaction between V−

Bwill not lead to a decrease of

T1due to the conservation of total spin polarization dur-

ing the ﬂip-ﬂop process (see Supplementary Note 2.2).

This T1related trend may be attributed to the presence

of lattice damage during the implantation process or lo-

cal charge state ﬂuctuations [64]. We note that the spin

relaxation process will introduce an additional decay to

the coherent dynamics. However, the measured T1and

Tρ

1are much longer than T2across all three samples at

room temperature (Figure 2). Nevertheless, we ﬁx the

duration between the polarization (II) and the read-out

(IV) laser pulses to account for the eﬀect of T1relaxation

on the T2measurement (see Methods).

Extracting V−

Bdensity— The diﬀerence between TXY8

and TD

2originates from the V−

B−V−

Bdipolar interaction,

which can be used to estimate the density of V−

Bdirectly.

In particular, by randomly positioning 12 electronic spins

at diﬀerent sampling concentrations, we construct the

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CoherentDynamicsofStronglyInteractingElectronicSpinDefectsinHexagonalBoronNitrideRuotianGong,1GuanghuiHe,1XingyuGao,2PengJu,2ZhongyuanLiu,1BingtianYe,3,4ErikA.Henriksen,1,5TongcangLi,2,6ChongZu1,5,†1DepartmentofPhysics,WashingtonUniversity,St.Louis,MO63130,USA2DepartmentofPhysicsandAstronomy,PurdueU...

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Coherent Dynamics of Strongly Interacting Electronic Spin Defects in Hexagonal Boron Nitride Ruotian Gong1Guanghui He1Xingyu Gao2Peng Ju2Zhongyuan Liu1Bingtian Ye34.pdf

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Coherent Dynamics of Strongly Interacting Electronic Spin Defects in Hexagonal Boron Nitride Ruotian Gong1Guanghui He1Xingyu Gao2Peng Ju2Zhongyuan Liu1Bingtian Ye34

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