J-coupling NMR spectroscopy with nitrogen vacancy centers at high fields

PHYSICAL REVIEW RESEARCH 6, 043017 (2024)
J-coupling NMR spec oscopy wi h ni ogen acancy cen e s a high ields
Pol Alsina-Bolí a ,1,2,*A. Bi e i-U iba en,1,2,*C. Munue a-Ja aloy,1,2and J. Casano a1,2,3,†
1Depa men o Physical Chemis y, Uni e si y o he Basque Coun y UPV/EHU, Apa ado 644, 48080 Bilbao, Spain
2EHU Quan um Cen e , Uni e si y o he Basque Coun y UPV/EHU, Leioa, Spain
3IKERBASQUE, Basque Founda ion o Science, Plaza Euskadi 5, 48009 Bilbao, Spain
(Recei ed 5 Decembe 2023; accep ed 16 Sep embe 2024; published 4 Oc obe 2024)
A diamond-based senso u ilizing ni ogen- acancy (NV) cen e ensembles pe mi s he analysis o mic on-
sized samples h ough nuclea magne ic esonance (NMR) echniques a oom empe a u e. Cu en e o s a e
di ec ed owa ds ex ending he ope a ing ange o NV cen e s in o high magne ic ields, d i en by he po en ial
o la ge nuclea spin pola iza ion o he a ge sample and he p esence o enhanced chemical shi s. Especially
in e es ing is he access o J-couplings as hey ca y in o ma ion o chemical connec i i y inside molecules. In
his wo k, we p esen a p o ocol o access J-couplings in bo h homonuclea and he e onuclea cases wi h NV
cen e s a high magne ic ields. Ou p o ocol leads o a clea spec um exclusi ely con aining J-coupling ea u es
wi h high esolu ion. This esolu ion is limi ed p ima ily by he decohe ence o he a ge sample, which is
mi iga ed by he noise- il e ing capaci ies o ou me hod.
DOI: 10.1103/PhysRe Resea ch.6.043017
I. INTRODUCTION
Nuclea magne ic esonance (NMR) spec oscopy has es-
ablished i sel as a undamen al ool o explo ing ma e ials
since i s incep ion in he 1950s [1–3]. This echnique is essen-
ial o analyzing molecula s uc u e and unc ion. I inds
ex ensi e applica ions in he examina ion o mac oscopic
biological issues and unc ional s udies in medical imag-
ing [4–6]. Howe e , he inhe en low sensi i i y o s anda d
NMR spec oscopy es ic s i s applica ion o bulky sam-
ples, ypically equi ing mic oli e olumes wi h con en ional
mac oscopic coils o nanoli e olumes wi h mic ocoils [7].
In his scena io, newly de eloped solid-s a e quan um sen-
so s open up new a enues o spec oscopic measu emen s
wi h unp eceden ed sensi i i y and spa ial esolu ion, enabling
s udies a he cellula [8] and single-molecula le el [9–11]. In
pa icula , quan um senso s consis ing on NV ensembles ha e
demons a ed hei u ili y in conduc ing NMR spec oscopy
on mic oscopic samples a oom empe a u e, eaching sample
olumes o picoli e s [7,12–14]. The pu sui o ex ending he
ope a ing ange o NV-based senso s in o he high magne ic
ield egime is d i en by he ad an ages o e ed by ele a ed
ields. Namely, a la ge he mal nuclea spin pola iza ion and
enhanced ene gy shi s ha simpli y da a ex ac ion.
The abili y o access J-couplings—also known as scala
o indi ec couplings—is o pa icula in e es in he ac ual
*These au ho s con ibu ed equally o his wo k.
†Con ac au ho : jcasano [email p o ec ed]
Published by he Ame ican Physical Socie y unde he e ms o he
C ea i e Commons A ibu ion 4.0 In e na ional license. Fu he
dis ibu ion o his wo k mus main ain a ibu ion o he au ho (s)
and he published a icle’s i le, jou nal ci a ion, and DOI.
spec oscopic amewo k. J-couplings a e indi ec in amolec-
ula dipole-dipole in e ac ions media ed h ough chemical
bonds. They a e esponsible o complex spli ings in NMR
spec al lines, and he examina ion o J-coupling pa e ns
allows scien is s o ob ain in o ma ion ega ding nuclea con-
nec i i y wi hin a molecule. Hence, accessing J-couplings a
he mic oscale egime would open new a enues in mass-
limi ed scena ios o applica ions in, e.g., combina o ial
chemis y, as well as in cell and memb ane biology [15].
Ne e heless, J-couplings a e no i ial o de ec since (i)
hey a e independen o he ex e nally applied ield. Then,
a high magne ic ields o he ield-dependen e ms domina e
he dynamics, and (ii) long cohe ence imes a e equi ed o
esol e he spec al peaks co esponding o J-couplings owing
o hei weak alue ( ypically, on he o de o a ew He z).
In his wo k, we p esen J-coupling induced nuclea sig-
nal wi h ex ended cohe ence ime (J-INSECT). This p o ocol
enables he de ec ion o homonuclea and he e onuclea J-
couplings wi h NV ensembles ( hus, accessing mic oscale
samples) in he egime o high magne ic ields. J-INSECT
in eg a es he e odyne echniques [16–19] and exe s noise
il e ing o e he sample e ec i ely enhancing i s cohe ence.
This esul s in na owe peaks in he spec a, he eby in-
c easing equency esolu ion and hus enabling o esol e
quan i ies on he o de o a ew he z (such as J-couplings).
Addi ionally, ou me hod p o ides he capabili y o a ge
speci ic ypes o couplings. In complex molecules, whe e
nume ous peaks may a ise due o ex ensi e couplings be-
ween nuclei, his allows o he selec i e a ge ing o only he
couplings o in e es , simpli ying he spec a. We in es iga e
he pe o mance o J-INSECT in egimes including bo h a
educed and a la ge numbe o measu emen s, and analyze
he impac o dis inc noise sou ces (including he p esence
o always-on J-couplings) demons a ing he easibili y o he
me hod in common expe imen al condi ions.
2643-1564/2024/6(4)/043017(9) 043017-1 Published by he Ame ican Physical Socie y
POL ALSINA-BOLÍVAR e al. PHYSICAL REVIEW RESEARCH 6, 043017 (2024)
II. RESULTS AND DISCUSSION
A. Molecula a ge Hamil onian
To exempli y he me hod we conside an iso opic liquid
sample [20] con aining a a ge molecula ensemble wi h wo
ypes o a oms, i.e., each molecule comp ises NHhyd ogens
(H om now on) and a numbe NAo nuclei o he “A”
species. The esul ing Hamil onian is
H/¯h=ωH
NH

i=1
Sz
i+
NH

i=1
δH
iSz
i+
NH

i<j
JH
i,j
Si·
Sj
+ωA
NA

i
Iz
i+
NA

i=1
δA
iIz
i+
NA

i<j
JA
i,j
Ii·
Ij
+
NH

i
NA

j
JH−A
i,j
Si·
Ij
+
NH

i
H
i( )Sx
icos (ω1 )+
NA

i
A
i( )Ix
icos (ω2 ),
(1)
whe e we deno e he spin ope a o s o H nuclei as Sx,y,z
k, and
hose o he Aspecies as Ix,y,z
k. On he i s line we ha e La -
mo e ms o he NHH (wi h ωH=|γH|Bex ), hei chemical
shi s, and he homonuclea J-couplings o magni ude JH
i,j.The
e ms on he second line a e he equi alen in e ac ions o
he Aspecies in he a ge molecule. The hi d line comp ises
he e onuclea J-couplings be ween H and Anuclei (JH−A
i,j),
while he las one con ains he RF con ols o e H and A
nuclea spins. Due o signi ican di e ences in gy omagne ic
a ios among dis inc spin species, c oss alk e ec s a e sa ely
neglec ed using he o a ing-wa e app oxima ion (RWA), es-
pecially in high- ield scena ios. By se ing he RF ields such
ha ω1≡ωHand ω2≡ωA, in a o a ing ame wi h espec
o (w. . ) La mo e ms, Eq. (1) simpli ies o
H/¯h=Hshi +
NH

i<j
neq
JH
i,jSz
iSz
j+
NA

i<j
JA
i,jIz
iIz
j
+
NH

i
NA

j
JH−A
i,jSz
iIz
j+Hc,(2)
whe e Hshi =NH
i=1δH
iSz
i+NA
i=1δA
iIz
iand Hc=
NH
iH
iSx
i+NA
iA
iIx
i. Fo he sake o simplici y in he
p esen a ion o he me hod, Eq. (2) assumes he elimina ion
o he ans e sal spin ope a o s (Ix,y
i,Ix,y
j,Sx,y
i, and Sx,y
j).
This applies in a scena io such ha |ωH−ωi|JH−A
i,j,
|δH
i−δH
j| JH
i,j, and |δA
i−δA
j|JA
i,j(no e he alidi y
o hese condi ions is s onge a high magne ic ields). In
addi ion, he label “neq” in he i s summa ion indica es ha
he homonuclea J-couplings among magne ically equi alen
nuclei (i.e., hose wi h iden ical chemical shi s) a e no
included in Eq. (2) since hey do no ha e an impac in
he dynamics as con i med ia nume ical simula ions. As
a summa y, Eq. (2) ea u es only ZZ- ype in e ac ions and
se es as a model o explain he basics o ou p o ocol. We
ema k ha ou nume ical simula ions conside de ia ions
w. . . Eq. (2) due o sho comings in he applica ion o p io
app oxima ions and he conside a ion o all J-couplings
h oughou he en i e p o ocol. This encompasses ins ances
du ing he applica ion o ini e-wid h RF pulses, whe e
de ia ions in he Rabi equency a e also aken in o accoun .
B. P o ocol
J-INSECT is illus a ed in Fig. 1. This s a s wi h an ini ial
π
2pulse o e he H nuclei ha ans e hei he mal pola iza-
ions o he x-axis and con inues by al e na ing encoding and
de ec ion s ages.
Du ing ee e olu ions (o leng h τ/2) a he encoding
s ages, Eq. (2) simpli ies o
H/¯h=Hshi +
NH

i<j
neq
JH
i,jSz
iSz
j+
NH

i
NA

j
JH−A
i,jSz
iIz
j,(3)
as NA
i<jJA
i,jIz
iIz
jcommu es wi h e e y o he e m in Eq. (3)
and does no impac he dynamics o H nuclei— he emi e s in
J-INSECT—(check he eadou s age). πpulses a e deli e ed
in he middle o he encoding s age wi h a wo old objec i e.
On he one hand, hey cancel Hshi mi iga ing he b oadening
caused by spa ial inhomogenei ies in he ex e nal magne ic
ield. In o he wo ds, he e ocusing πpulses leng hen he
nuclea cohe ence ime om T∗
2 o T2. This enables a longe
scanning du a ion o he nuclea sample, which p o es espe-
cially ad an ageous o es ima ing J-couplings due o hei
ela i ely weak alues. On he o he hand, πpulses o e H
(and A) enable he selec i e encoding o JH
i,j(and JH−A
i,j)in
he sys em dynamics. This becomes e iden upon calcula ing
he e ec i e p opaga o a he end o he encoding s age a e
deli e ing πpulses o e H nuclei, esul ing in
U1( )=exp ⎡
⎢
⎢
⎣
−i
NH

i<j
neq
JH
i,jSz
iSz
j⎤
⎥
⎥
⎦
.(4)
Al e na i ely, in he scena io whe e synch onized πpulses
a e applied o H and A, esembling a double nucleus-nucleus
esonance, one ge s
U2( )=exp ⎡
⎢
⎢
⎣
−i ⎛
⎜
⎜
⎝
NH

i<j
neq
JH
i,jSz
iSz
j+
NH

i
NA

j
JH−A
i,jSz
iIz
j⎞
⎟
⎟
⎠
⎤
⎥
⎥
⎦
.
(5)
In addi ion, he in luence o addi ional nuclea species (such
as B) is nega ed wi hin he dynamics due o he e ocusing
in oduced by J-INSECT. Tha is, he una oidable p esence o
ex a nuclea spin iso opes in he a ge sample would no al e
he sys em dynamics unless a πpulse ha synch onously lips
he Bspecies is in oduced on pu pose (see la e ).
Then, he ini ial π
2pulse along he y-axis akes he he mal
s a e o hyd ogen nuclei o he x-axis (see Fig. 1) while ZZ
in e ac ions ha go e n he encoding s age—i.e., hose in
Eqs. (4)o (5)—encode J-couplings in he dynamics o Sx
i
and Sy
io H nuclei. Impo an ly, owing o he ini ializa ion
o he he mal pola iza ion along he x-axis, i can be
demons a ed ha ollowing he encoding s age, Sx
iexhibi s
043017-2
J-COUPLING NMR SPECTROSCOPY WITH NITROGEN … PHYSICAL REVIEW RESEARCH 6, 043017 (2024)
FIG. 1. Scheme o J-INSECT wi h RF and MW con ols. Each channel (black bold ho izon al line) deno es he adia ion pa e n o e each
sys em cons i uen . The uppe RF channel co esponds o he pulses o e nonhyd ogen nuclea species. In his espec : o de ec he e onuclea
J-couplings, simul aneous πpulses o e H and ano he iso ope a e deli e ed. The cen al channel pic u es he RF ha mus be applied o
H nuclei, i.e., an ini ial (π/2)ypulse, ollowed by an al e na ion o (π)xpulses and wo ull Rabi oscilla ions along Y o de ec ion, wi h
winding-unwinding mechanism o obus ness as o iginally p oposed in [17] and u he de eloped in [21]. The esul an magne iza ion o
hyd ogen nuclei, 
M∝NH
i[Sx
i,Sy
i,Sz
i] (see Appendix B), a di e en ins an s is ep esen ed wi h ed a ows in he sphe es: dashed a ows
indica e he magne iza ion a he beginning o each p o ocol s ep (pulses o ee e olu ions), while he solid a ow indica es he magne iza ion
a he end o he s ep. The e olu ion o he magne iza ion ou side he xy-plane (induced by RF d i ings) is depic ed wi h blue a ows indica ing
he magne iza ion’s ajec o y. Du ing he de ec ion s age, he Y¯
Ypulse induces a magne ic ield (Bz) on he NV ensemble along he NV axis
ha p ecess a a con ollable speed depending on he RF Rabi equency. The ampli ude o Bza he n h de ec ion pe iod is p opo ional o he
ampli ude o he ed solid a ow (i.e., o he magne iza ion a he end o he n h encoding pe iod). The Y¯
Ypulses b ing he magne iza ion back
o he s a e a he s a o he de ec ion s age as hey induce wo 2π o a ions. Bzis cap u ed by he NV ensemble ia a MW con ol sequence
(e.g., he XY4 in he lowe channel), while g een pulses indica e he s anda d ini ializa ion and eadou o he NVs.
signi ican ly g ea e ampli ude han Sy
i(app oxima ely 105
imes la ge ), as de ailed in Appendix A1. In his manne ,
J-INSECT is uned o a ge Sx
iin he de ec ion s ages.
A a gene ic n h de ec ion s age, owing o a o a ion o he
H nuclei along he y-axis (see Fig. 1), he magne ic ield Bn
z
induced by he sample o e he NV ensemble is along he NV
axis and eads [22] (see Appendix B)
Bn
z( )=B0(nτ)sin(H ),(6)
whe e His he Rabi equency o he Yand ¯
Ypulses
(see Fig. 1) while, impo an ly,
B0(nτ)∝2
NHSx
i =nτ.(7)
In conclusion, he subsequen magne ic ield Bn
z( ) has an
ampli ude p opo ional o Sx
io H nuclei, and oscilla es
wi h a con ollable equency H. This is, i espec i e o
he in ol ed La mo equencies, wha enables o ope a e a
high magne ic ields. No e ha , a 2 T hyd ogen La mo s
each ≈(2π)×85 MHz, posing a echnically challenging
ask o hei acking. On he con a y, Hcan be uned, o
ins ance, o ens o kHz [23,24], such ha he oscilla ions o
Bn
z( ) a e easily ollowed by he NV ensemble by applying,
e.g., a s anda d XY4 sequence. Reco ding he esul an phase
acquisi ion o he NV ensemble a e e e y de ec ion pe iod
leads o a di ec econs uc ion o he e olu ion o Sx
io
H nuclei, hus o he J-couplings encoded in Sx
i.In his
ega d, i is wo h no ing ha , unlike s anda d he e odyne
measu emen schemes, J-INSECT does no impose es ic-
ions on he sepa a ion be ween consecu i e measu emen s.
This means hey can be placed a bi a ily in ime, p o ided
ha he elapsed ime be ween measu emen s is eco ded, as
his is c ucial o ex ac ing chemical in o ma ion. Fo sim-
plici y in he p o ocol, hey a e placed equidis an , sepa a ed
by τ. No ice ha since hyd ogens a e le in he same s a e as
be o e he measu emen (due o he o a ion being a mul iple
o 2π, he nex encoding pe iod can be applied di ec ly, wi h-
ou he need o epola ize he sample). Finally, i is impo an
o no e ha he NV cen e s a e ac i e only du ing he de ec ion
pe iods; his implies a high spec al esolu ion cons ained no
by he senso , bu by he T2(>T∗
2) o H nuclei.
C. Nume ical esul s
We conside a picoli e sized liquid sample on op o a
diamond hos ing an NV ensemble a mic ome e dep h [12]
and a s a ic magne ic ield o Bex =2 T aligned wi h he NV
axis.
To exempli y J-INSECT we add ess a sample consis -
ing on luo ome hanol molecules (CH2FOH) ha con ains
i e spin-1/2 nuclei o h ee di e en species: H, ca bon,
and luo ine (13Cand F om now on, no e oxygen has no
spin) in a he mal s a e, see he scheme o he molecule in
043017-3
POL ALSINA-BOLÍVAR e al. PHYSICAL REVIEW RESEARCH 6, 043017 (2024)
FIG. 2. Simula ed spec a o he luo ome hanol molecule. Ob ained om 1000 epe i ions o he p o ocol wi h 600 encoding +de ec ion
s ages. Due o he dephasing ime T2=0.6 s, he FWHM is app oxima ely 1 Hz. Wi hin each igu e, he e is an illus a i e depic ion o
he molecule, highligh ing he a ge ed J-couplings. Colo ed nuclei signi y he species upon which πpulses a e applied. Addi ionally, each
igu e includes a boxed scheme de ailing he RF pulse ain applied on each case. (a) Ta ge ing only H and 13C, i e peaks a ise, which encode
J,J1and J2. (b) Ta ge ing he H, he 13Cand he F, en peaks a ise, encoding: J,J1,J2,JF
1,andJF
2.
inse o Fig. 2. Ou p o ocol enables accessing he e onu-
clea J-couplings among H and he 13Cnucleus, as well
as wi h he F. Mo eo e , i gi es access o he homonu-
clea coupling be ween nonequi alen H ( he wo H nuclei
in Fig. 2labeled as Haa e magne ically equi alen , whe eas
Hbis magne ically unequi alen wi h espec o he o me ).
We choose alues alling wi hin he ypical ange ound in
he li e a u e [2], namely: JHaHb≡J=(2π)×8Hz,JHaC≡
J1=(2π)×130 Hz, JHbC≡J2=(2π)×6 Hz, while in e -
ac ions wi h he F nuclei a e JHaF≡JF
1=(2π)×80 Hz and
JHbF≡JF
2=(2π)×4 Hz, whe eas JCF =(2π)×160. Fu -
he mo e, o a magne ic ield Bex =2 T, he chemical
shi s a e δHa=(2π)×512 Hz, δHb=(2π)×236Hz, δC=
(2π)×85Hz,and δF=(2π)×450 Hz.
Ou nume ical simula ions s a wi h Hamil onian
(1) in an in e ac ion pic u e w. . he La mo e ms
ωHNH
i=1Sz
i+ωANA
iIz
i. In his manne , he only
app oxima ion made is simpli ying he he e onuclea coupling
e m NH
iNA
jJH−A
i,j
Si·
Ijin Eq. (1) o only ZZ in e ac ions
[i.e., o NH
iNA
jJH−A
i,jSz
iIz
j,likeinEq.(2)]. This can
be done since |ωH−ωi|∼(2π)×10 MHz JH−A
i,j∼
(2π)×100 Hz, hus, he o a ing wa e app oxima ion sa ely
applies. The eade can ind he ull simula ed Hamil onian in
Appendix B.
To main ain he Hamil onian close o Eq. (2) he condi ion
τ>1/(|δHa−δHa|) mus hold, such ha he homonuclea
e m o H spins app oaches o a ZZ in e ac ion. The p e ious
condi ion imposes a lowe bound on τ, limi ing he amoun
o measu emen s ha can be ca ied ou o a gi en expe -
imen al ime. In ou case, we se τ=1.2/(|δHa−δHa|)=
4.3 ms (no e la e we explo e he J-INSECT pe o mance
wi h a la ge numbe o measu emen s, i.e., depa ing om
his condi ion). In addi ion, he ampli ude o he RF a ge ing
HisH=(2π)×50 kHz while, o obus ness pu poses,
πpulses o e H a e implemen ed ia a CORPSE a chi ec u e
[25] which educes he impac o de unings du ing pulse
execu ion. We obse e no disce nible impac on he sys em
dynamics when s anda d op-ha pulses a e applied o 13Cand
F, hus we exclusi ely deploy CORPSE pulses o e Hs. This
esul s in CORPSE
π=43.3µs, C
π=39.7 µs and F
π=10.6µs.
Ou simula ions i s conside 600 measu emen s ( ha is,
600 encoding and de ec ion s ages in a single expe imen al
un) hence he du a ion o he J-INSECT pe expe imen al
un is T o al =600 (τ+ CORPSE
π+2/H)≈2.66 s, whe e he
e m 2/Hco esponds o he du a ion o wo ull RF Rabi
oscilla ions du ing he eadou s age.
Rega ding e o and impe ec ions, ou nume ical simula-
ions comp ise he ollowing: (1) always on chemical shi s
and J-couplings; (2) RF c oss- alk e ec s du ing RF i adi-
a ion; (3) e o s on he ampli ude o he RF d i ings ( hese
a e simula ed h ough an O ns ein-Uhlenbeck p ocess [26–28]
wi h a 1% ampli ude shi and a noise co ela ion ime o 1
ms); and (4) a nuclea dephasing ime T2=0.6s[12], inco -
po a ed ia an exponen ial decay in he ield emi ed by he
sample.
We conside wo cases. When he J-INSECT a ge s H and
13Cnuclei (case 1) we ge he spec um in Fig. 2(a). No e ha
in his case, F does no ac i ely pa icipa e since we do no
apply πpulses o e i , al hough i is included in ou nume ical
model. Case 1 exhibi s i e esonances a di e en equencies
iencoding he e onuclea J-couplings be ween H and 13Cand
homonuclea J-coupling be ween H nuclei. The assignmen
o J-couplings based on he ob ained ican be accomplished
ei he by analy ically sol ing he e olu ion in Eq. (5)(see
Appendix A2) o by employing s anda d spin mul iplici y
echniques [29]. Con e sely, simul aneous a ge ing o he
F nuclei wi h an addi ional πpulse (case 2) esul s in he
eme gence o en esonances, as illus a ed in Fig. 2(b).No e
in his second scena io he he e onuclea coupling be ween
he F and H nuclei is also encoded in B0(nτ). The esul ing
es ima ions o he J-couplings in cases 1 and 2 a e collec ed
in Table I.
In summa y, J-INSECT e ec i ely discloses J-couplings
wi h la ge spec al esolu ion esul ing in a ull wid h a
hal maximum (FWHM) o , a mos , 1 Hz, co esponding
o he chosen alue o T2=0.6 s. Rega ding sensi i i ies,
he use o ele a ed magne ic ields allows achie ing a
signal- o-noise a io (SNR) o 30 wi h app oxima ely 18 000
epe i ions o he p o ocol. (The es ima ion ca ied ou o
TABLE I. J-coupling alues ob ained om he spec a shown in
Fig. 2(a) ( i s ow) and Fig. 2(b) (second ow). Values and hei
unce ain y a e gi en in He z.
JJ1J2JF
1JF
1
Case 1 8.1±0.5 130.1±0.86.1±0.9– –
Case 2 8.1±0.5 130.1±0.96±180.0±0.94±1
043017-4
J-COUPLING NMR SPECTROSCOPY WITH NITROGEN … PHYSICAL REVIEW RESEARCH 6, 043017 (2024)
case 1 wi h pho on s a is ics o NV expe imen s, assuming a
7% o luminescence con as . Addi ional de ails conce ning
case 1 and he de elopmen o case 2, which yields sim-
ila esul s, a e a ailable in Appendix C.) This implies an
expe imen al ime o ∼13 h. Hence, J-INSECT acili a es
a high- ideli y e ie al o J-couplings wi hin a easonable
signal-collec ing ime, wi h he po en ial o u he educe
he expe imen al du a ion by inco po a ing complemen a y
me hods, such as p io hype pola iza ion o he sample e-
po ed in Re s. [13,14] o ad anced eadou echniques such
as epe i i e eadou [30,31]. In e ms o sensi i i y, p e ious
wo ks using quan um he e odyne measu emen s a mode a e
magne ic ields (88 mT) [12] epo sensi i i ies in he ange
o ∼25–75pT ·Hz−1/2. Since ou wo k ope a es a high mag-
ne ic ields, which enhance pola iza ion by a ac o o ∼30,
we an icipa e a sensi i i y o app oxima ely ∼1–5pT ·Hz−1/2.
Addi ionally, ou app oach ex ends he cohe ence ime o he
sample om T∗
2 o T2, which will u he educe he sensi i i y.
An in e es ing inal aside o conside is enhancing he a e
o measu emen s, as a s a egy o ex a educe expe imen-
al ime. To exempli y he p ocedu e we inspec he simple
case 1. Enhancing he measu emen a e implies educing
τin he encoding s ages, hence depa ing om he condi-
ion τ>1/(|δHa−δHa|). In his scena io he spec um is no
longe gi en by he usual mul iplici y ules [i.e., he homonu-
clea e ms, H-H, he in e ac ion Hamil onian Eq. (2)a eno
longe ZZ in e ac ions], hus we es ima e he alues o he
J-couplings by pe o ming Bayesian in e ence. In pa icula ,
we choose o ex end he numbe o measu emen s on each
un (i.e., he numbe o de ec ion s ages) om 600 o 6553
and educe he o e all expe imen al ime ou imes ( om
∼13 h o ∼3h30
). In his case, he in e ence gi es
J=8.2±0.5, J1=130.1±0.6,and J2=5.7±0.4
(see Appendix D).
III. CONCLUSION
We in oduce J-INSECT, a p o ocol ha enables he de ec-
ion o homonuclea and he e onuclea J-couplings wi h NV
ensembles a high magne ic ields. J-INSECT inco po a es
he e odyne echniques and exe s noise il e ing, enhancing
he sample’s cohe ence hus enabling highly p ecise es-
ima ions o J-couplings limi ed by T2>T∗
2. We explo e
J-INSECT’s pe o mance in wo scena ios, conside ing bo h
educed and enhanced da a ha es ing. We assess he in-
luence o di e en noise sou ces, including he pe sis en
p esence o always-on J-couplings, showcasing he me hod’s
easibili y in expe imen al condi ions.
The comple e da ase suppo ing he indings o his wo k
is a ailable upon eques om he co esponding au ho . On
he one hand, seconda y da a eused and/o analyzed om
o he s udies is speci ied along he ex . On he o he hand,
o iginal da a gene a ed in his s udy ( ia nume ical simula-
ions) a e also a ailable upon eques .
The sou ce code and associa ed ma e ials used in his e-
sea ch a e a ailable upon eques .
ACKNOWLEDGMENTS
P.A.B. and A.B.U. acknowledge he inancial suppo o
he IKUR STRATEGY (IKUR-IKA-23/22) and (IKUR-IKA-
23/04), espec i ely. C.M.J. acknowledges he p edoc o al
MICINN G an No. PRE2019-088519. J.C. acknowledges
he Ramón y Cajal (RYC2018-025197-I) esea ch ellowship.
The au ho s acknowledge he Quench p ojec ha ecei ed
unding om he Eu opean Union’s Ho izon Eu ope–The
EU Resea ch and Inno a ion P og amme unde G an Ag ee-
men No. 101135742, he inancial suppo om he Spanish
Go e nmen ia he Nanoscale NMR and complex sys ems
(PID2021-126694NB-C21) p ojec , he ELKARTEK p ojec
Disposi i os en Tecnologías Cuán icas (KK-2022/00062), and
he Basque Go e nmen G an No. IT1470-22.
APPENDIX A: SPIN DYNAMICS OF THE SAMPLE
In his sec ion, we p o ide u he de ails ega ding he
nuclea spin dynamics in he sample.
1. Ampli udes o he σx
iand σy
icomponen s
We conside an e olu ion go e ned by he nex p opaga o
[no e his is simila o hose in Eqs. (4) and (5)in hemain
ex ]
U =i,je−iφi,j
2σz
iσz
j,(A1)
o e an ini ial s a e such ha
ρx
1... ρx
kρz
k+1... ρz
N,(A2)
whe e ρx,z
ja e he mal s a es pola ized along xand zdi ec-
ions. This is, ρz
j=1
2I+Bn
4σzwhile ρx
j=eiπ
4σy
jρz
je−iπ
4σy
j=
1
2I−BH
4σx, wi h BH=¯hγhBex
KbTand Bn=¯hγnBex
KbTbeing small
quan i ies a oom empe a u e (on he o de o 10−5 o a
magne ic ield o 2 T). In he p e ious exp essions, Bex =2T
is he ex e nal magne ic ield, ¯h=1.054×10−34 J·sis he
Planck cons an di ided by 2π,γH=(2π)×42.57 MHz/Tis
he gy omagne ic a io o he H and γn he co esponding one
o o he nuclei, KB=1.38×10−23 J/K he Bol zmann con-
s an , and he empe a u e T=300 K. He e we di e en ia e
among he mal s a es co esponding o he H nuclei, i.e., he
nuclea spin which is o a ed o he Xaxis a he s a o he
p o ocol (deno ed by ρx
j), and he o he gene ic nuclea spin,
which is ep esen ed wi h ρz
j.
On he one hand, one can demons a e ha
σx
1=T U ρx
1... ρx
kρz
1... ρz
kU†
σx
1
=T =1e|ρx
1|g... ρx
kρz
k+1... ρz
Nj>1e−iφ1,jσz
j
+g|ρx
1|e... ρx
kρz
k+1...ρz
Nj>1eiφ1,jσz
j,(A3)
whe e T =1[...] ep esen s he ace o e all spins excep ing
he i s one. Using e|ρx
1|g=g|ρx
1|e=−
BH
4one eaches
σx
1=−
BH
4k
j=1cos φ1,jN
j=k+1cos φ1,j−iBn
2sin φ1,j
−BH
4k
j=1cos φ1,jN
j=k+1cos φ1,j+iBn
2sin φ1,j.
(A4)
043017-5

POL ALSINA-BOLÍVAR e al. PHYSICAL REVIEW RESEARCH 6, 043017 (2024)
On he o he hand, one can demons a e ha σy
1leads o
σy
1=−
iBH
4k
j=1cos φ1,jN
j=k+1cos φ1,j−iBn
2sin φ1,j
+iBH
4k
j=1cos φ1,jN
j=k+1cos φ1,j+iBn
2sin φ1,j.
(A5)
No e he abo e exp essions hold o any H, hus o any
quan i y σx
iand σy
i ha in ol es a H spin in he a ge
sample.
No ice how he leading o de o each expec a ion alue
di e s by i e o de s o magni ude: σx
1∝O(BH) and σy
1∝
O(BHBn). Thus, σx
iis ∼105 imes g ea e han σy
i.
2. Fluo ome hanol esonances
We compu e he esonance posi ions o he speci ic case
o he luo ome hanol molecule. Fo he sake o simplici y we
conside a pola ized H, and he o he nuclei in he maximally
mixed s a e, ha is, ρx
i=1
2(I+σx
i) o H and ρm
i=1
2I o
o he nuclei. No e hese conside a ions only al e he ampli-
ude o he signal, no he esonance posi ion. Howe e , he
esul s in he main ex conside he mal s a es o all nuclei
in he luo ome hanol molecule. Thus, ollowing Eq. (A3), we
ha e
Sx
i=σx
i
2=1
2N
j=1cos φi,j,
whe e φ1,ico esponds o he phase accumula ed due o he
in e ac ion be ween he i h and j h spins du ing a ime , ha
is, φi,j( ) =Ji,j
2 .
When H nuclei and 13Ca e a ge ed in he luo ome hanol
molecule (co esponding o case 1 in he main ex ), he e-
sul ing signal is
3

i=1
Sx
i( ) =1
6cos J
2 2 cos J1
2 
+cos J
2 cos J2
2 ,
which leads o he esonance peaks in Fig. 2(a) o he main
ex , 1=±J2/2, 2=±(J2−2J)/2, 3=±(J2+2J)/2,
4=±(J1−J)/2, and 5=±(J1+J).
In he case whe e he F is also a ge ed wi h πpulses
(co esponding o case 2 in he main ex ), one inds
3

i=1
Sx
i( ) =1
6cos J
2 2 cos J1
2 cos JF
1
2 
+cos J
2 cos J2
2 cos JF
2
2 ,
wi h he co esponding esonances depic ed in Fig. 2(b) o he
main ex . These a e 1=±(J2−JF
2)/2, 2=±(+2J−J2−
JF
2)/2, 3=±(J2+JF
2)/2, 4=±(2J+JF
2−J2)/2, 5=
±(2J+J2−JF
2)/2, 6=±(2J+J1+JF
1)/2, 7=±(J1−
JF
1−J)/2, 8=±(J+J1−JF
1)/2, 9=±(J1+JF
1−J)/2,
and 10 =±(J+J1+JF
1)/2.
Al e na i ely, he esonan equencies can be ob ained ia
s anda d NMR mul iplici y ules [29]. Tha is, when a nuclei A
is coupled wi h nxspin-1
2nuclei o an iso ope X, heAsignal
spli s in o nx+1 lines. I Ais also coupled o an elemen di -
e en o X(le s sayY) his induces addi ional ny+1 spli ings
in each p e ious peak.
In ou case, since he b oadening is mi iga ed, all J-
coupling peaks end o be cen e ed a 0 Hz. Thus, when H
and 13Ca e a ge ed, he signal o Ha( ecall he Haand Hba e
magne ically inequi alen hyd ogen g oups in he molecule,
e e o Fig. 2o he main ex ) is spli in o 2×2=4 peaks
due o couplings wi h Hband he 13C, while he signal o Hb
is spli in o 3×2=6 peaks, h ee due o he pai o equi alen
Ha, and wo due o he 13C. Then, we a e le wi h 6 +4=10
peaks a ound 0 Hz. I J coupling in e ac ions wi h F a e
also exci ed, he signal o Hais spli in o 2×2×2=8( wo
pe e e y o he elemen ), while he signal o Hbspli s in o
3×2×2=12 peaks, leading o a o al o 20 peaks, see, e.g.,
Re . [6] o mo e in o ma ion.
APPENDIX B: FROM THE SAMPLE TO THE NV CENTERS
The oscilla ion o hyd ogens in he de ec ion s age o J-
INSECT impac s on he NV ensemble (which is conside ed
a a mic ome e dep h om diamond su ace) as a classical
magne ic ield. In his espec , see, e.g., Re s. [12,22]). In he
high ield egime, he magne ic signal induced by he sample
in he o hogonal di ec ions (i.e., xand y) wi h espec o he
ex e nal magne ic ield (o ien ed wi h he NV axis, i.e., z)
can be neglec ed as i apidly oscilla e a a speed γNBzwi h
Bzbeing se e al Tesla. The esul an ield emana ed om he
sample du ing an n h de ec ion pe iod (n∈N) can be well
app oxima ed by [12,22]
Bz( )=Bn
0sin(H ),(B1)
whe e H=γHBRF is he ampli ude o he RF d i ing. On he
o he hand, he ampli ude o he magne ic ield is
Bn
0=(2π)2(¯hγH)2μ0ρHBzF3
16πKBTMx,(B2)
whe e Mxis he xcomponen o he no malized magne iza-
ion, de ined as 
M=2
NHi
Si(nτ). No e ha he quan i y
i
Si(nτ) is compu ed ia nume ically e ol ing he nex
molecula Hamil onian
H/¯h=δHASz
1+Sz
2+δHBSz
3+δCIz+δFPz
+JHaHa
S1·
S2+JHaHb(
S1·
S3+
S2·
S3)
+JHaCSz
1+Sz
2Iz+JHbCSz
3Iz
+JHaFSz
1+Sz
2Pz+JHbFSz
3Pz+JCFIzPz.(B3)
In addi ion, ¯h=1.054×10−34 J·s is he Planck’s cons an
di ided by 2π,γH=(2π)×42.57 MHz/T is he gy omag-
ne ic a io o he H, μ0=4π×10−7H/m he magne ic
pe meabili y o ee space, ρH≈6.6×1028 m−3 he H den-
si y o he sample, Bz=2 T he ex e nal magne ic ield,
KB=1.38×10−23 J/K he Bol zmann cons an , T=300 K
he empe a u e and F3=4.1 cha ac e izes he geome y o
he sample [12].
043017-6
J-COUPLING NMR SPECTROSCOPY WITH NITROGEN … PHYSICAL REVIEW RESEARCH 6, 043017 (2024)
FIG. 3. Simula ing an expe imen . Panel (a) co esponds o case 1, while (b) holds o case 2. Inse s: In blue a e he signals wi hou
conside ing luminescence issues, while ed squa es a e he ou come ha esul s om a e aging 18000 epe i ions o he espec i e p o ocol
wi h 2.5×108NV cen e s assuming a con as o 7%. Main igu es: In blue a e he spec a ha co espond o he signals wi hou luminescence
issues, while he ed squa es a e he FT o a e aged signal in he inse ( hus, ca ying eadou noise). Bo h spec a a e no malized by a common
ac o , ensu ing ha he maximum peak in case 1 has ampli ude 1. In case 1, e e encing he ampli ude o he smalles peak, he co esponding
SNR is jus abo e 30. In case 2 unde he same e e ence c i e ion, i is sligh ly below 15.
The Hamil onian o a gene ic NV in he ensemble unde
he in luence o Bz( )is
HNV =σzγe
2Bz( )+Hc,(B4)
whe e Hcaccoun s o he MW d i ing on he NV ensemble.
In pa icula , applying a XY4 con ol sequence du ing he
de ec ion s age o e he NV ensemble wi h RF =21
Hand he
spacing be ween he MW pulses TMW = RF/4, yields
σyNV ≈2γe RF
πBn
0,(B5)
p o ided ha 2γe RF
πBn
01.
APPENDIX C: READOUT CONSIDERATIONS
To e alua e he impac o he measu emen a e o he
p o ocol unde he high ex e nal magne ic- ield pa adigm, we
make an es ima ion o he equi ed signal a e aging ime
leading o a signal o noise a ion (SNR) o 30.
In Re . [30] he au ho s eco ded he mean numbe o
pho on-coun pe measu emen when a single NV is in |0,
ha is, n0=0.016, and a pho on coun di e ence be ween he
g ound and exci ed spin s a es o n0−n1=0.005, yielding a
30% inc ease o luminescence con as be ween bo h s a es.
In his case, since we a e wo king wi h NV ensembles, we
conside a luo escence con as o 7%, which was achie ed
by lowe ing he pho on emission n0. Employing he la e pho-
on s a is ics, and aking as a basis he signal wi h no eadou
e o [in pa icula , he in e se Fou ie ans o m o he spec a
in Fig. 2(a), no e ha his signal ca ies he es o e o s
desc ibed in he main ex ], we ep oduce an expe imen al
ou come.
In pa icula , we assume a con igu a ion simila o ha in
Re . [12]. This is an NV cen e densi y o 0.8×1023 m3,a
beam diame e o 20 µm, and a 10 µm hick laye o NV
cen e s sensi i e o he he mal spin signal. This leads o a
senso con aining ≈2.5×108ac i e NV cen e s. The p o ocol
is epea ed 18 000 imes. Tha is, we a e age he emi ed
signal ac oss 18 000 epe i ions o he p o ocol, which leads
o a du a ion o 18 000 ×2.66s ≈13h20.
Speci ically, in case 1, he noise in he esul ing a e aged
signal has a s anda d de ia ion o app oxima ely 6 ×10−5
while he signal’s ampli ude is σz
NV∼1×10−3,seeinse o
Fig. 3(a). This leads o a SNR o abou 30 in he spec um
by aking he he heigh o he smalles peak. Fo case 2,
he noise s anda d de ia ion is simila . Howe e , he SNR is
app oxima ely 15. This is because e en hough bo h signals
( om case 1 and case 2) ha e simila ampli udes, he spec um
o case 2 exhibi s mo e peaks which esul s in peaks o smalle
ampli ude as i can be seen in Fig. 3 o cla i ica ion.
In conclusion, he he mal pola iza ion induced by he high
ex e nal magne ic ield is enough o balance ou he ela i ely
043017-7
POL ALSINA-BOLÍVAR e al. PHYSICAL REVIEW RESEARCH 6, 043017 (2024)
FIG. 4. (a) Top, scheme o he RF sequence employed o he high measu emen - a e p o ocol [wi h (π)xand (π)−xpulses al oge he wi h
CORPSE s uc u es o e Hs du ing encoding o enhanced obus ness] and he co esponding spec um (bo om). This shows he esul ob ained
wi h 6553 measu emen s (solid blue) while, o compa ison, we include he spec um ob ained wi h 600 measu emen s (dashed ed), which
co esponds o he same spec um shown in Fig. (a) o he main ex . (b) Pos e io dis ibu ions o each scanned pa ame e ia Bayesian
in e ence. I: Pos e io dis ibu ion co esponding o J; II: o J1; and III: o J2. The exac alue o each pa ame e is ma ked by a e ical do ed
line in I, II, and III. The unce ain ies a e he s anda d de ia ion σθ,gi enbyσ2
θ=(θ−θ)2=(θ−θ)2P(θ|D) dθ,whe eθs ands o
J,J1,andJ2.
low measu emen a e, yielding a de ec able signal ( hus o
clean J-coupling spec a) unde easonable expe imen al ime
in bo h cases.
APPENDIX D: BEYOND τ>1.2/(|δHa−δHa|)
To inc ease he numbe o measu emen s on each expe i-
men al un, we educe τin encoding s ages om τ=4.3ms
o τ=75 µs and pe o m 6553 measu emen s ins ead o 600.
We conside impe ec pulses (wi h 1% e o on he ampli-
ude d i ing) as well as always-on J-couplings and eadou
e o s (see p e ious sec ion). This inc ease in he numbe o
measu emen s leads o a educ ion on he o al expe imen al
epe i ions o be pe o med. In his case he p o ocol includes
an ex a (π) pulse du ing he encoding pe iod o u he educe
he impac o impe ec d i ings (see he scheme o he pulse
sequence in Fig. 4).
On he o he hand, he educ ion on τleads o an in i-
ca e spec um wi hou clea esonance peaks, see Fig. 4(a).
Thus, o in e he alues o J-couplings we pe o m Bayesian
analysis. As an inpu o he Bayesian in e ence we employ
he p e iously desc ibed noisy signal wi h impe ec pulses
and eadou e o s. Rega ding he pa ame e se , as he p io
knowledge we assume a uni o m dis ibu ion. To sum up, ia
Bayes in e ence, simplici y o he spec a can be aded o
sho e expe imen al imes.
As a igu e o me i , we simula e he esul ing sig-
nal pe o ming ou imes less a e ages ( hus a ou - imes
sho e expe imen ). This esul s in o a eadou e o wice
g ea e . The pos e io dis ibu ions a e depic ed in Fig 4(b),
om which he ollowing alues and unce ain ies a ise ( e-
spec i ely): J=8.2±0.5Hz,J1=130.1±0.6 Hz, and
J2=5.7±0.4Hz.
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Be lin, 1978).
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in oduc ion o he basics o NMR spec oscopy and se-
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[4] K. V. R. Cha y and G. Go il, NMR in Biological Sys ems. F om
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