SUPPLEMENTARY INFORMATION · 2011-04-06 · spin zero-ﬁeld splitting for the NV centre orbital...

SUPPLEMENTARY INFORMATIONdoi: 10.1038/nnano.2011.22

nature nanotechnology | www.nature.com/naturenanotechnology 1

Supplementary Information forAtomic-scale magnetometry of distant nuclear spin clusters via nitrogen-vacancy spin in diamond

Nan Zhao, Jian-Liang Hu, Sai-Wah Ho, Jones T. K. Wan & R. B. Liu

I. CALCULATION OF NV CENTRE SPIN COHERENCE INA 13C NUCLEAR SPIN BATH

A. Microscopic model

The coupled system of an NV centre and a 13C nuclear spinbath is described by the following Hamiltonian

H = HGSNV + Hbath + Hint, (3)

where

HGSNV = ∆S 2

z − γeB · S, (4a)

Hbath = −

i

γ(i)nucB · Ii + HN-N, (4b)

Hint = S ·

i

Ai · Ii ≡ S · b, (4c)

where S is the spin-1 of the NV centre electron, I0 is the spin-1of the 14N nucleus, Ii for i > 0 is the spin- 1

2 of the 13C nucleusat Ri, γe and γ(i)

nuc are the gyromagnetic ratios of the electronspin and the ith nuclear spin (γ(0)

nuc = γN for the 14N nucleus,and γ(i)

nuc = γC with i > 0 for 13C nuclei), ∆ is the electronspin zero-field splitting for the NV centre orbital ground state(see Fig. S1), B is the external magnetic field, and Ai is thehyperfine coupling tensor (a 3 × 3 matrix) for the ith nuclearspin. The bath Hamiltonian Hbath includes the nuclear spininteractions as

HN-N =i> j

µ0

4πγ(i)

nucγ( j)nuc

R3i j

Ii · I j −3Ii · Ri j

Ri j · I j

R2i j

+ ∆N

Iz0

2, (5)

where Ri j = Ri − R j, and ∆N = 5.1 MHz is the 14N nuclearspin quadrupole splitting [1]. The hyperfine interaction ten-sor A0 of the 14N nuclear spin has the diagonal form of A0 =

diag[A⊥0 , A⊥0 , A

0] with A⊥0 = 2.1 MHz and A0 = 2.3 MHz [1].

The hyperfine interaction tensor Ai for i > 0 of 13C nuclearspins are obtained from first-principles calculations for 13Cspins in a 512-atom supercell containing the NV centre in themiddle. For nuclear spins outside the supercell, i.e., far awayfrom the centre, as the electron wave function is localized, theFermi contact part vanishes and the detailed electron spin den-sity distribution becomes less important. Thus, the hyperfineinteraction is taken to be the dipolar form with the electronassumed a point spin located at the vacancy site.

We first diagonalize the centre spin Hamiltonian HGSNV, with

the eigenenergies and eigenstates denoted by ωα and |α forα = +1, 0, and −1. In the eigenstate basis, the electron spin

operator S is written as

S =α,β

Sαβ |α β| , (6a)

Sαβ = α| S |β . (6b)

Since the centre spin splitting (∼GHz) is usually much greaterthan the typical hyperfine interaction strength (<MHz, exceptfor a few closely located 13C spins and the 14N spin), the directelectron spin flip by the hyperfine interaction is suppressedby the large energy mismatch. With the hyperfine-inducedelectron spin flip neglected, the Hamiltonian is written in apure dephasing form in the basis of the eigenstates |α of HGS

NVas

H =α

|α α| ⊗ (ωα + Hα) , (7)

with the bath spin Hamiltonian Hα conditioned on the centrespin state |α

Hα = Hbath +

i

A(α)i · Ii, (8)

where A(α)i ≡ α |S|α · Ai is the effective field for the ith nu-

clear spin due to the hyperfine interaction, called a Knightfield.

Virtual flips of the electron spin can still be caused by thehyperfine interaction, which leads to a hyperfine-mediated in-teraction between nuclear spins in addition to the dipolar in-teractions in Eq. (5) [2–4]. The second-order perturbation the-ory leads to the hyperfine-mediated interaction between two

3E

MW3A2

1A1

Figure S1: NV centre electronic and spin energy levels. The spincoherence of the orbital ground state (with the 3A2 symmetry) is con-sidered in this paper. The spin state can be initialized and read outby optical excitations (green arrows) and fluorescence (red wavy ar-row) via the orbital excited states (with the 3E symmetry) and themetastable state (with the 1A1 symmetry). The spin transition (blackwavy arrow) is manipulated by the microwave (MW) pulses in thedynamical decoupling schemes.

© 2011 Macmillan Publishers Limited. All rights reserved.

2 nature nanotechnology | www.nature.com/naturenanotechnology

SUPPLEMENTARY INFORMATION doi: 10.1038/nnano.2011.22

2

nuclear spins Ii and I j as

Hi, j =α

|αα| ⊗βα

Sαβ · Ai · Ii

Sβα · Aj · I j

ωα − ωβ, (9)

for |ωα−ωβ| |Sαβ ·Ai| and |Sβα ·Aj|. The hyperfine mediatedinteraction is conditioned on the electron spin state |α. Whenboth Ii and I j are located near the NV centre, the hyperfine-mediated interaction could have significant influence on thesystem dynamics [5]. However, in this paper, we focus onthe nuclear spin clusters far from the NV centre. In this case,the effect of the hyperfine-mediated interaction is negligibleas compared with the intrinsic dipolar interaction in Eq. (5).Furthermore, in natural-abundance diamond, very few nuclearspins can have strong hyperfine interaction with the electronspin. The electron spin decoherence is caused by a large num-ber of weakly coupled nuclear spins. As shown in Fig. S2,the hyperfine-mediated interaction has negligible effect on theelectron spin decoherence.

The eigenstates |α of HGSNV are in general not the eigenstates

of the spin operator S z (i.e. mS is not a good quantum num-ber), unless the magnetic field is applied along the NV axis.The transverse components of the magnetic fields mixes theeigenstates of S z, and induces field-dependent Knight fieldson the nuclear spins. Consider a single nuclear spin in theapplied magnetic field B. The Hamiltonian reads

H(i)α = −γ(i)

nucB · Ii + A(α)i (B) · Ii. (10)

Notice that the Knight field A(α)i (B) depends on the applied

magnetic field B, and in the weak field limit, it is expanded to

Figure S2: Comparison of the NV centre spin coherence calculatedwith (black lines) and without (red symbols) the hyperfine-mediatednuclear spin interaction in Eq. (9), under a (a) zero, (b) strong, and (c)medium magnetic field and five-pulse Uhrig dynamical decoupling(UDD5) control. The effect of the hyperfine-mediated interaction isnegligible. The magnetic field is applied along the NV axis. Changeof the magnetic field direction does not affect the results. The insetof (c) shows a close-up of the oscillations due to the ESEEM effectunder a magnetic field B = 0.03 T for clarity.

the first order term of the field as

H(i)α = A(α)

i (0) · Ii − γ(i)nucB · g(α,i)

nuc · Ii. (11)

The effect of the magnetic field on the Knight field is de-scribed by an effective g-tensor defined as

g(α,i)nuc ≡ ∂

∂B

B −A(α)

i

γ(i)nuc

= 1 − 1

γ(i)nuc

∂Sαα∂B

· Ai. (12)

In the weak field limit, by taking the transverse fields Bx,y asperturbations, the eigenstates of HGS

NV is written as

|±1 = |±1z −γeB±

∆ ∓ γeBz, (13a)

|0 = |0z +γeB−

∆ − γeBz|+1z +

γeB+∆ + γeBz

|−1z , (13b)

where B± =Bx ± iBy

/√

2, and the states |mz are the unper-turbed states (eigenstates of S z, i.e. S z |mz = m |mz). Withthese perturbed eigenstates, the effective g-tensor is expressedexplicitly as

g(α,i)nuc = 1 +

(3 |α| − 2)∆

γe

γ(i)nuc

Axx

i Axyi Axz

iAyx

i Ayyi Ayz

i0 0 0

. (14)

Eq.(14) reproduces the hyperfine interaction induced mod-ulation of g-factor presented in Refs. [2, 3, 5], which wasobtained with the second-order perturbation method. Ourmethod of direct diagonalization of the Hamiltonian HGS

NV andworking in the energy eigenstate basis (instead of the S z eigenstates) goes beyond the perturbation regime and is not limitedto the weak transverse field regime.

B. Cluster-correlation expansion method

The coherence L (t) between two electron spin states, e.g.,|0 and |+1, under dynamical decoupling control is [6]

L (t) = Tr· · · e−iH+1τ2 e−iH0τ1ρBeiH+1τ1 eiH0τ2 · · ·

, (15)

where ρB is the density matrix of the spin bath, and τi is theevolution time between two pulses which flip the electron spinbetween |0 and |+1. In general, the calculation of the deco-herence needs the solution of the many-body dynamics of thebath, which is a hard problem. The cluster correlation expan-sion (CCE) is proved to be an efficient method to calculate thedecoherence [6, 7]. With this method, the decoherence func-tion L(t) is expressed as the product of cluster correlations

L (t) =

M

LM (t) , (16)

with the unfactorizable correlation LM (t) defined by

LM (t) =LM (t)

M⊂M LM (t), (17)




3

Figure S 3: Convergence test of calculation of the centre spin de-coherence (under UDD5 control) under a magnetic field B = 0.2 Tapplied along the NV axis, for three controlling parameters, (a) thebath size Rc, (b) the dipolar interaction cut-off distance dc, and (c)the CCE truncation order. In testing one parameter, the other twoparameters are fixed at values for which the calculation is converged,namely, Rc = 40 Å, dc = 8 Å, and Mc = 4 (CCE-4).

where LM (t) is calculated the same as in Eq. (15) but with onlyinteraction within the cluster M included in the bath Hamilto-nian. In realistic calculation, the expansion has to be truncatedat a certain size of clusters.

The numerical calculation has been tested for convergencewith the truncation cluster size Mc, the bath size Rc (the max-imum distance from the NV centre of nuclear spins includedin the bath), and the cut-off distance dc of the dipolar inter-action between nuclear spins (the maximum distance betweentwo spins in a cluster included in the calculation). For the de-coherence of an NV centre spin in the 13C bath, as shown inFig. S3, calculation of the electron spin decoherence is wellconverged for dc = 8 Å, Rc = 40 Å (for a bath containingabout 500 13C spins), and Mc = 4. For the single moleculedetection, all 13C spins within the distance of the moleculesfrom the NV centre are included in the calculation.

C. Centre spin states

For an external magnetic field applied along the NV axis,the spin eigenstates of the NV centre orbital ground state isdenoted by |0gz and | ± 1gz. In this subsection, the subscript“g” is used to denote the orbital ground state, which is omittedin other parts of the paper without confusion. A magnetic fieldtilted away from the NV axis will induce state mixing, leading

z

+1

x

B

S0

(a)

S-1

S+1

z (b)

Figure S4: (a) Schematic of the electron spin moment Sα = α|S|αfor α = +1, 0 and −1 (green, blue and red arrows in turn) under amagnetic field (black arrow) tilted away from the NV axis (assumedto be in the [111] direction, and defined as the z-axis) with a polarangle θ. The x-axis is chosen in the [110] direction. (b) The S z = 0components Cg/e [defined in Eq.(20)] under a magnetic field B forthe ground state (left, with parameter ∆ = 2.87 GHz) and the excitedstate (right, with parameters ∆ = 1.43 GHz, and = 70 MHz [8]), asfunctions of the field strength and the polar angle. In the edge regionseparated by the black line, the component is greater than 90%.

to eigenstates

|αg =

m=0,±1

cαm(B)|mgz. (18)

Such states are in general not eigenstates of the spin operator,i.e., the spin has no certain magnetic moment along any di-rection. But if the field is not too strong, or if the tilt angle isnot too large, the magnetic moment along the NV axis is closeto 0 or ±1 for the eigenstates (see Fig. S4), with only smallstate mixing. Thus, we can still label the electron spin statesby α = 0 or ±1.

According to Eqs. (6b) and (8), changing the external mag-netic field will modify the interaction between the electronspin and bath spins, and hence the electron spin decoherence.Therefore, the magnetic field provides a new degree of free-dom to control the decoherence. More importantly, a tiltedfield breaks the symmetry of the hyperfine interaction, whichis important to unambiguously identify a 13C dimer (other-wise dimers at symmetric positions have the same oscillationsignatures and therefore cannot be distinguished).

It is important to keep the magnetic field strength and tiltangle within the range where one eigenstate has large overlapwith S z = 0 states, |0gz and |0ez for both the ground and theexcited states, since the optical initialization and detection ofthe NV centre spin rely on the selective transitions from and tothese states. At room temperature, the electron spin of an NVcentre in the excited state is described by the Hamiltonian [8]

HESNV = ∆

S 2z +

S 2

x − S 2y

− γeB · S, (19)

where ∆ = 1.43 GHz is the spin zero-field splitting of the ex-cited states, and is the strain-induced transverse anisotropy.Diagonalizing the ground and excited state spin Hamiltoni-ans HGS

NV and HESNV, we obtain the S z = 0 components of each

eigenstates |αg and |αe for a given magnetic field B. Wequantify the S z = 0 component by the following overlap

Cg/e(B) ≡ maxα

z0g/e|αg/e2 . (20)




4

Previous experimental investigations [9] have shown that tilt-ing the magnetic field away from the NV axis reduces thephotoluminescence of an NV centre. Our calculations of theS z = 0 component agree with the experimental results. Withthe suitable magnetic field strength and tilting angle, the pho-toluminescence of an NV centre is still well observable asshown in Ref. [9]. In this paper, we always choose the pa-rameters within the range where the the overlap Cg/e 90%as indicated as a red region separated by the black line inFig. S4(b).

D. Basic physical processes in the bath

In this subsection, we summarize the basic physical pro-cesses in the nuclear spin bath which cause the NV centre spindecoherence. Two kinds of processes are important in differ-ent magnetic field regimes, namely, the single nuclear spinrotation and the nuclear spin pairwise flip-flop. In this subsec-tion, we will focus on the dynamics of the 13C nuclear spins,which is dominant for the coherence behaviors of the NV cen-tre electron spin. The 14N nuclear spin is usually fixed by thelarge quadrupole splitting [2], and does not contribute signif-icantly to the electron spin decoherence, except for a rapidelectron spin echo envelope modulation (ESEEM) effect in amedium magnetic field.

1. Single nuclear spin rotation

In CCE, the simplest cluster contains a single nuclear spin.For cluster I j, the jth nuclear spin feels an external mag-netic field B and an effective Knight field A(α)

j induced by thehyperfine interaction with the electron spin. The latter de-pends on the NV centre spin state |α [see Eq. (8)]. The nu-clear spin starting from the | ↑ state will rotate about the fieldh(α)

j = γCB + A(α)j , conditioned on the electron spin state |α.

When the electron spin is flipped between, e.g., |0 and | + 1,the two fields h(0)

j and h(1)j are exchanged. Taking the Hahn

echo for example (a single π-pulse is applied at time τ), thenuclear spin precession conditioned on the electron spin stateis

| ↑h(0)

j τ

−−−→ |ϕ(τ)h(1)

j τ

−−−→ |ϕ(2τ), (21a)

| ↑h(1)

j τ

−−−→ |φ(τ)h(0)

j τ

−−−→ |φ(2τ), (21b)

represented by the bifurcated trajectories on the Bloch sphere[Fig. S5(a)]. The distance δ j between the two trajectories onthe Bloch sphere, defined by δ2

j = 1− |ϕ(2τ)|φ(2τ)|2, quanti-fies the electron spin decoherence due to the jth nuclear spin.

In the weak field regime (B 100 Gauss), where the typicalhyperfine interaction exceeds the nuclear spin Zeeman energy,the nuclear spin rotation is mostly driven by the hyperfine in-teraction. The single spin rotations of several hundreds of 13Cnuclear spins in the bath contribute to a large fluctuation ofthe Overhauser field felt by the electron spin, which causes

hj

(0)

hj

(1)

xy

xy

z

hjk

(0)

hjk

(1)

(a) (b)

j j,k

Figure S5: Bloch sphere representation of the decoherence processinduced by (a) a single nuclear spin cluster I j, and (b) a nuclear spinpair I j, Ik. The external field is applied along the NV axis. The blueand red arrows represent the fields (or the pseudo-fields) on the sin-gle nuclear spin (or the pseudo-spin for the nuclear spin pair), corre-sponding to the electron spin states |0 and |1, respectively. The bathspin (or pseudo-spin) vector rotates about the two fields (or pseudo-fields), along bifurcated trajectories. Upon an electron spin flippingpulse, the spin (or pseudo-spin) precession directions are exchanged,and the bifurcated trajectories may intercross at a later time.

rapid electron spin decoherence in a microsecond timescale[see inset of Fig. S6(a)]. During such a short period of time,a weakly coupled nuclear spin (i.e., a spin beyond the firstfew atom shells of the NV centre) can hardly perform a fullcycle of rotation before the electron spin coherence vanishesand therefore can hardly imprint any distinguishable oscilla-tion features on the electron spin signal.

If the weak magnetic field is applied along the NV axis, theelectron spin coherence under the Hahn echo control presentsperiodic collapses and revivals [Fig. S6(a)], which was ob-served in previous experiments [2]. The periodic recoveryof the electron coherence can be understood as follows. TheKnight field of a nuclear spin vanishes when the electron spinis in the |0z state, and the effective field h(0)

j = B is the samefor all nuclear spins. When the evolution time τ is a mul-tiple of the precession period 2π/(γCB), the evolution underh(0)

j = B would bring a nuclear spin back to its initial state.Thus, as shown in Fig. S5(a), with an electron spin flippingpulse applied as τ = 2nπ/(γCB) for integer n, the bifurcatedtrajectories of a nuclear spin will intercross at the echo time2τ, regardless of the position of the nucleus. So a perfect re-covery of coherence will occur (except for the decoherencecaused by many-body interaction in the bath, which will betaken into account in the higher order CCE). This collapseand revival effect can be also predicted, with the similar anal-ysis, for the Carr-Purcell-Meiboom-Gill (CPMG) dynamicaldecoupling scheme, which consists of equally spaced pulses.

For a weak magnetic field applied with a tilt angle fromthe NV axis, the Knight fields A(α)

i will be different for differ-ent nuclear spins. There is no common precession frequencyshared by all the nuclear spins. In this case, the revival be-comes less perfect, or even disappears. The revival effect isalso eliminated if the pulse separations are not commensurate,




5

Figure S6: NV centre spin decoherence in various magnetic fields.(a) Hahn echo in a weak field (B = 50 Gauss). Single-nucleus ro-tation induces a rapid decay with a microsecond timescale (inset).The coherence recovers when the pulse delay time is a multiple ofthe nuclear Larmor precession period. (b) Electron spin coherenceunder UDD5 control in a medium magnetic field (B =400 Gauss.The high-frequency oscillations are due to precession of single nu-clear spins. The envelope decays due to many-body interaction in thebath, with a slow modulation due to a coherent pair. (c) Electron spincoherence under UDD5 in a strong magnetic field (B = 2000 Gauss).The single-nucleus rotations (and hence rapid oscillations of the sig-nal) are greatly suppressed due to the large Zeeman energy. Themillisecond-timescale oscillation is due to a coherent pair in the bath.(d) Contribution to the electron spin decoherence by a typical inco-herent pair. (e) Contribution by the coherent pair (a dimer located at1.3 nm away from the NV centre), which matches the millisecond-timescale oscillation in (b) and (c).

e.g. in the UDD case.To preserve the electron spin coherence without the re-

vivals, a larger magnetic field can be applied. Increasing themagnetic field to medium strength, e.g., 400 Gauss as shownin Fig. S6(b), the rotations of all weakly coupled single nu-clear spins are strongly suppressed by the large Zeeman en-ergy as compared with the hyperfine coupling. Only the 14Natom of the NV centre and the 13C atoms within a few atomshells of the NV centre can induce the ESEEM on the electronspin decoherence.

2. Nuclear spin pairwise flip-flops

Under a strong magnetic field, the nuclear spin pairwiseflip-flops are the dominant mechanism of the electron spin de-coherence since the single nuclear spin dynamics is largelysuppressed. The Hilbert space of a two-spin cluster, e.g.,I j, Ik

, is spanned by the four basis states |↑↑, |↑↓, |↓↑ and

|↓↓. For a nuclear spin Zeeman energy much greater thantheir dipolar interaction strength, the two polarized states |↑↑and |↓↓ are energetically separated, and the non-secular spinflipping is prohibited due to the large energy cost. The |↑↓and |↓↑ states form a two-dimensional invariant subspace.The nuclear spin pair in this subspace can be mapped to a

pseudo-spin, i.e., |↑↓ → |⇑ and |↓↑ → |⇓. Similar tothe single nuclear spin case, the coherent evolution of thepseudo-spin can be understood using the the picture depictedin Fig. S5(b).

The nuclear spin flip-flop caused by the dipolar interactiongives rise to the transition between the pseudo-spin states |⇑and |⇓. The pseudo-spin transition rate is calculated throughthe nuclear spin dipolar interaction as

Xjk =µ0

4πγ2

C

R3jk

12

1 − 3 cos2 θ jk

, (22)

where θ jk is the angle between R jk and the external magneticfield. In general, the two nuclear spins I j and Ik feel differentKnight fields for a given electron spin state |α. The differencebetween the Knight fields (projected to the direction of theexternal magnetic field) induces an energy cost of the flip-flop

Z(α)jk = A(α)

j,z − A(α)k,z . (23)

Thus the flip-flop is mapped to the precession of a pseudo-spinσ about a pseudo-field h(α)

jk =Xjk, 0, Z

(α)jk

conditioned on the

electrons spin state |α. The Hamiltonian of the pseudo-spinreads

H(α)ps =

12

h(α)jk · σ = 1

2

Xjkσx + Z(α)

jk σz

. (24)

A typical evolution path of the pseudo-spin is shown inFig. S5(b), where h(0)

jk =Xjk, 0, 0

and h(1)

jk =Xjk, 0, Z

(1)jk

in

the case of magnetic field aligned with the NV axis. Using thispseudo-spin model, the electron spin decoherence contributedby the flip-flop is

Lpair (t) =⇑

eiH(0)ps τ1 eiH(1)

ps τ2 · · · e−iH(0)ps τ2 e−iH(1)

ps τ1

⇑. (25)

Here we have applied a sequence of electron spin flippingpulses at τ1, τ1 + τ2, . . . for dynamical decoupling control.

For the Hahn echo, the decoherence contributed by a singlepair Lpair(t = 2τ), averaged over different pseudo-spin initialstates | ⇑ and | ⇓, can be explicitly worked out as [3, 4]

Lpair(t) = 1 −Z2

jk sin2Xjkτ/2

sin2

X2

jk + Z2jkτ/2

X2jk + Z2

jk

. (26)

In most cases, for the inter-nucleus distance of severalAngstroms, the flip-flop transition rate Xjk ( 100 Hz) is muchless than the energy cost Z(α)

jk (∼ kHz) due to the Knight fielddifference. In this case, the pair contributes only a nominalfraction to the electron spin decoherence, since in Eq. (26) theoscillation amplitude, during the millisecond time scale weare interested in, is the order of X2

jkτ2 1. A large num-

ber of such pairs (hereon called incoherent pairs) give rise toa smooth decoherence background, with a (sub)millisecondtimescale. In the opposite limit, for the bonded pair locatedfar away (e.g. several nanometers) from the NV centre, theKnight field difference is much weaker than the dipolar in-teraction, i.e. Zjk Xjk. Obviously, according to Eq. (26),




such kind of remote dimers also give small contributions ofthe order Z2

jk/X2jk to the Hahn echo signal. Between these two

limiting cases, a pair which has dipolar interaction as strong asthe Knight field difference, i.e. Xjk ∼ Z(α)

jk ∼ kHz, can inducestrong oscillations. For example, a dimer located about 1.5 nmfrom the NV centre satisfies this condition. For the electronspin coherence under higher order dynamical decoupling con-trols, the analytical expression is complicated, but the essen-tial physics is similar to the Hahn echo case. To demonstratethe qualitatively different behaviors of the pairs under higherorder dynamical decoupling controls, we show in Fig. S6(d)the contribution of an incoherent pair to the electron spin co-herence under UDD5, in comparison with the strong oscilla-tions in Fig. S6(e) provided by a dimer (coherent pair) nearthe NV centre under the same conditions.

II. DETECTABLE RANGE

A coherent pair cannot be located too far away from thecentre. Otherwise, e.g., if the pair is outside the bath of the 500spins, as discussed in the previous Section, the contributionby the pair to the centre spin decoherence would be too small.The (rough) maximum distance from the centre for a pair tohave strong modulation on the electron spin decoherence pro-file is defined as the detectable range. In a natural abundancesample, this range is about 1.5 nm. Note that both the hyper-fine interaction and the nuclear spin interaction are essentiallydipolar and have the same scaling with distance R as R−3. Sothe Hamiltonian in Eq. (3) scales linearly with the 13C abun-dance η (without considering the rare cases that there are 13Catoms located within a few atom shells of the centre). As aresult, the centre spin decoherence time scales with the abun-dance as η−1. By this scaling relation, the detectable range ofa coherent pair depends on the 13C abundance as η−1/3.

III. APPEARANCE PROBABILITY OF A COHERENTPAIR

For an abundance η of 13C, the probability of having two13C nuclei in a C-C bond (a dimer) is pdimer = η2. The proba-bility of having the first dimer to appear in the atom shell (ofthickness ∆R) at distance R from the NV centre is

∆P(R) =1 − (1 − pdimer)NC-C(R+∆R)−NC-C(R)

(1 − pdimer)NC-C(R)

≈e−η2NC-C(R) − e−η

2NC-C(R+∆R)

= exp−16η2 4πR3

3a30

− exp−16η2 4π(R + ∆R)3

3a30

,(27)

where NC-C(R) is the number of C-C bonds within the distanceR from the centre, and a3

0 is the volume of a unit cell. Theprobability of having the first dimer within the distance R fromthe centre is

P(R) = 1 − exp−16η2 4πR3

3a30

. (28)

(a) (b)

Figure S7: Details in identifying the hidden dimer. (a) For an elec-tron spin coherence L(t) with strong oscillation features (gray linewith shadow), the position and depth of the first dip are used as anevaluator of matching between the oscillations in L(t) and those inLpair(t) (discrete line). (b) A case of electron spin coherence whichso occurs as to have no strong oscillations (gray line with shadow).A pair having strong oscillation under the same conditions (red line)should be rejected, while a pair contributing only small decoherenceunder the same conditions (black line) should be kept for the nextstep of screening.

As shown in Fig. 1c of the main text, in diamond of naturalabundance, there is ∼ 50% probability to have at least onedimer appearing within 1.5 nm (the detectable range) fromthe NV centre.

IV. DOUBLE-BLINDED NUMERICAL EXPERIMENTSFOR IDENTIFYING A HIDDEN DIMER

For various magnetic field strengths and directions, the con-tribution to the centre spin decoherence Lpair (t) of each dimeraround the NV centre is calculated using Eq. (25) as a func-tion of time. The data for totally over 20,000 dimers within arange of ∼ 2.5 nm are stored as the fingerprint library.

For a randomly generated configuration of 13C nuclei ofnatural abundance, the electron decoherence L (t) is calculatedfor various magnetic fields and under various dynamical de-coupling control. The nuclear spin configuration is knownonly to one team of the authors (Team 1).

When obtaining from Team 1 the numerically simulateddata of L (t), which has strong modulation, Team 2 comparesthe oscillations L (t) to the oscillations of each dimer under thesame magnetic fields and dynamical decoupling conditions.

We define quantitative evaluators of matching between theoscillations in Lpair (t) and those in L (t). Under multi-pulsedynamical decoupling, the decoherence caused by incoherentpairs is largely suppressed in the initial stage of evolution andthe electron spin coherence presents a plateau if without con-tributions from coherent pairs. Thus both the position anddepth of the first dip are good parameters to characterize anoscillation. Thus we can choose the position and depth dif-ference of the first dip as the evaluators of matching betweenthe simulated curve and the dimer curve. To allow for someimprecision in experimental measurement and modeling, wedefine the evaluators with a certain degree of tolerance, ∆tand ∆L [see Fig. S7(a)]. A dimer having matching evaluatorsgreater than the tolerance is filtered out. In the double-blinded




Figure S8: (a) The Fourier transform F(ωt) of the pulse modulationfunction for a 100-pulse CPMG sequence. For the N-pulse CPMG,sharp peaks appear at ωt = (2k + 1)Nπ. The first three peaks withk = 1, 2, and 3 are shown, and the first peak is magnified in the inset.(b) The signal of a single 1H2

16O molecule located at 10 nm abovethe NV centre under zero field. The red line is the approximate resultobtained by Eq.(33), and the black line with symbols is the result ofthe exact numerical calculation.

experiments, we have chosen ∆t = 200 µsec and ∆L = 0.1.Similarly, the positions of subsequent dips and peaks are alsoused in the fingerprint matching procedure.

Besides, it is also possible that under certain magneticfields, the coherence signal does not exhibit clear oscillationfeatures, as shown in Fig. S7(b). The signal in this case is alsouseful in identifying coherent pairs. It can be used to rule outpairs which give incompatible oscillation features under thesame magnetic field [e.g., the red line in Fig. S7(b)].

V. SINGLE MOLECULE DETECTION AT ZERO FIELD

A. Noise spectrum of detected molecules

In Fig. 4c of the main text, we demonstrate the singlemolecule detection using many-pulse dynamical decouplingunder a zero magnetic field.

For the detected molecules at a relative large distance fromthe NV centre (e.g. 10 nm), the eigenstates |n and eigenen-ergies ωn of the coupled spins are determined by the intrinsicinteractions HM of the molecule,

HM|n = ωn|n. (29)

The hyperfine interaction perturbs the molecules’ states, andinduces the transition between the eigenstates |m and |n withstrength Wmn =

m|2bz|n2 [see Eq. (4c) for the definition of

b].As seen by the NV centre electron spin, the hyperfine in-

teraction with one molecule induces a noise on the transition

| − 1 ↔ | + 1, which in the interaction picture is

bM(t) ≡ 2eiHMtbze−iHMt. (30)

The noise correlation function due to NM molecules is

C(t1 − t2) = NM

bM(t1)bM(t2)

+

bM(t2)bM(t1)

2, (31)

which has a noise spectrum

S (ω) =

C(τ)eiωτdt = 2πNM

m,n

PnWmnδ(ω − ωmn)

≡α∈mn

S αδ(ω − ωα), (32)

where ωmn = ωm − ωn is the molecule transition frequencyfrom state |n to state |m, and Pn is the level population prob-ability of the initial state |n.

With this noise spectrum S (ω) due to the molecules, theelectron spin coherence under the influence of the moleculesis well approximated by [10]

L+−(t) = exp−1

2

t

0dt1

t

0dt2C(t1 − t2) f (t1) f (t2)

= exp−

∞

0

dω2π

S (ω)ω2 F(ωt)

, (33)

where the modulation function f (t) ≡ (−1)q for t ∈tq, tq+1

,

corresponding to a dynamical decoupling sequence with theN pulses applied at t1, t2, . . . , tN between t0 = 0 and tN+1 = t,and the dimensionless noise filter function

F(ωt) ≡ω2

t

0f (t)eiωtdt

2

=

N

q=0

(−1)qe−iωtq+1 − e−iωtq

2

. (34)

In particular, for the N-pulse CPMG sequence [10]

F(ωt) =

16 sin4 ωt

4N cos−2 ωt2N cos2 ωt

2 for odd N

16 sin4 ωt4N cos−2 ωt

2N sin2 ωt2 for even N

, (35)

which, similar to the optical grating effect, has sharp peaks lo-cated at ωt = (2k+1)Nπ with integer k, with height ∼ N2. Thefunction F(ωt) of the 100-pulse CPMG sequence is shown inFig. S8(a). The noise spectrum S (ω) consisting of δ-functionsat discrete molecule transition frequencies and the filter func-tion F(ωt) having the sharp peak structures, the electron spincoherence shows sharp dips [Fig. S8(b) and Fig. 4c in themain text] at the particular times tmn = (2k+1)Nπ/ωmn, corre-sponding to each nuclear spin transition frequency ωmn in themolecule.

The depths of the dips have a simple scaling relation withthe number of molecules NM , the distance from the NV centreD, and the number of dynamical decoupling pulses N,

ln [L+−(tmn)] ∝ −NMN2D−6, (36)




8

since the dipolar hyperfine interaction strength scales with thedistance by D−3, the filter function peaks has height ∼ N2, andthe noise strength is proportional to the number of molecules.One can increase the sensitivity (the minimum number ofmolecules which can induce visible dips in the NV centre spincoherence) and the detection range D by increasing the num-ber of dynamical decoupling pulses.

Because the gyromagnetic ratio of 1H is about 4 timesgreater than that of 13C, the coupled 1H nuclear spins in1H2

16O and 12C1H4 molecules have much greater transitionfrequencies than the 13C nuclear spin clusters (e.g., 51.8 kHzfor the strongest transition in a 1H2

16O molecule versus3.1 kHz for the strongest transition in a 13C dimer, under azero magnetic field). According to Eq. (33), the greater transi-tion frequencies correspond to the coherence dips appearing atearlier times. As a result, the coupled 1H nuclear spins in themolecules impose the fingerprint dip structures on the initialstage of the total electron spin coherence. This suggests thatthe detection scheme is not limited to the simple moleculesdemonstrated here. Instead, it is also useful in detecting otherorganic molecules, which are rich in 1H nuclear spins. Also,other kinds of nuclei can be detected via their coupling to 1Hnuclei.

B. Background signal under zero field

The detection of single molecules is assumed to be carriedout under a zero magnetic field. In this case, the nuclear spintransitions are completely determined by the intrinsic chem-ical structure of the molecules to be detected, regardless oftheir orientations. Such a feature is important for identifyinga kind of molecules through their fingerprint transitions.

In the absence of a magnetic field, the background decoher-ence due to the dynamics of 13C nuclear spins has differentbehavior from the finite field case.

First, the decoherence induced by single nuclear spin rota-tions is completely removed by spin echo. The Knight fieldfelt by the ith nuclear spin is zero for the electron spin in the|0 state. For the electron spin in the |+1 and | −1 states, theKnight fields are anti-parallel (i.e. A(±1)

i = ±z · Ai). Thus, theconditional Hamiltonians H(i)

α = A(α)i · Ii for different electron

spin states |α commute with each other, and the contributionof single 13C nuclear spins to the decoherence vanishes at theecho time.

Second, the absence of the magnetic field removes the Zee-man energy cost of the non-secular nuclear spin flip. The pic-ture depicted in Fig. S5(b) is not valid in this case. Taking thetwo spin cluster

Ii, I j

for example. All the four states |↑↑,

|↑↓, |↓↑ and |↓↓ are involved in the nuclear spin evolution.The additional non-secular flipping driven by the anisotropicdipolar interactions induces a much shorter coherence timethan in the case of finite magnetic field. Because of this short-ened total coherence time, the 13C dimers, although exists inthe natural abundance diamond sample, cannot give the char-acteristic oscillations before the total coherence vanishes. Asa result, the 13C bath under zero field provides a smooth back-

ground decoherence profile, which indeed benefits the singlemolecule detection.

C. Electron spin manipulation under zero field

Under zero magnetic field, the |±1 states are degenerate,and cannot be distinguished in the frequency domain. In thiscase, instead of the coherence between |0 and |+1 (or |−1)states, the spin coherence between |±1 states may be used todetect coherent nuclear spin clusters. The spin manipulationof |±1 states is described below.

The Hamiltonian of the NV centre driven by a linear po-larized microwave with frequency ω in the absence of a staticmagnetic field reads

H(t) = ∆S 2z − γeBac · S cos(ωt). (37)

Without loss of generality, we define the x-axis as the direc-tion of the component of Bac perpendicular to the z-axis (theNV direction). With the resonance condition ω = ∆, and therotating wave approximation, the time-independent Hamilto-nian in the rotating frame is

Hrot =ΩR

2|B 0| + h.c., (38)

where ΩR = −γeBac,x is the Rabi frequency, and the brightstate is defined as |B = (|+1 + |−1) /

√2. Thus, the super-

position of |±1 state of the electron spin is prepared after aπ rotation from the initial state |0, by a pulse with durationtp = π/ΩR.

An arbitrary superposition state |ψ = α |+1+β |−1 can bewritten in the basis of the bright and dark states as

|ψ = α + β√

2|B + α − β

√2|D , (39)

where the dark state |D = (|+1 − |−1) /√

2 is orthogonalto the bright state |B. After a 2π rotation in the |0 , |Bsubspace (by a pulse with duration tp = 2π/ΩR), the brightstate will acquire a phase factor −1, and the state |ψ becomes

|ψ →ψ = −α + β√

2|B + α − β

√2|D

= − (α |−1 + β |+1) . (40)

Thus, the spin flipping between the |±1 states is realized.Therefore the dynamical decoupling can be implemented. Fi-nally, the spin state can be detected by applying a π rotationto convert the bright state back to the |0 state. The coherenceis measured by the probability of finding the NV centre in the|B state.

VI. FIRST-PRINCIPLES CALCULATION OF THEHYPERFINE COUPLING TENSORS

To determine the hyperfine parameters, especially for nu-clear spins not too far away from the centre, we have per-formed first-principles electronic structure calculation to ob-tain the atomic configuration and the ground-state electronic




9

Figure S9: (a) Perspective view of the calculated spin density and(b) spin density viewed along the [111] axis. Only the atoms that areclose to the NV centre are shown. Light blue and dark blue corre-spond to spin density of +0.015 and −0.001 e/Bohr3, respectively.

Table SI: The calculated principal values of the hyperfine tensor (Aαα)of each atom (in MHz) and their corresponding distance from thevacancy (Rvac) in Angstroms. The Fermi-contact (Aiso) term is alsocalculated for reference. The dipolar term is given by bαα = Aαα −Aiso.

Atom Aiso A11 A22 A33 Rvac14N -3.671 -3.977 -3.862 -3.173 1.6953C 291.33 237.75 238.74 397.49 1.6436C -9.485 -12.013 -10.814 -5.629 2.5023C -16.296 -17.680 -17.574 -13.634 2.5253C 0.579 -0.587 0.364 1.961 2.5266C 7.592 6.286 6.819 9.670 2.939

structure. The first-principles calculation is based on the den-sity functional theory, within the generalized gradient approx-imation for the exchange-correlation functionals [11]. Thecalculation begins with atomic relaxation of an unrelaxed 512-atom supercell (4 × 4 × 4 unicells) with the periodic bound-ary condition, which contains a single NV centre defect and2 unpaired electrons. The N atom and vacancy are locatedat (0, 0, 0) and (1/4, 1/4, 1/4)a0, respectively. Projected aug-mented wave (PAW) [12] method is used for the pseudoizationof C and N atoms. A plane wave cutoff of 35 Ry is used to ex-pand the Kohn-Sham orbitals, and a cutoff of 500 Ry is usedto expand the charges density. PBE parameterization is usedfor the GGA calculation. All the calculations are performedusing Quantum-ESPRESSO package ver. 4.1.1 (available at

http://www.pwscf.org).The cell parameters and atomic degrees of freedom are re-

laxed until a convergence of pressure < 0.1 GPa and force< 10−4 eV/Bohr is achieved. The size of the relaxed supercellis ∼ 14.30 × 14.30 × 14.30 Å. To further verify the relaxedstructure, we have repeated the structural relaxation calcula-tion, but using the local density approximation (LDA) withPerdew-Zunger (PZ) parameterization [13]. The cell param-eter obtained is 14.15 Å, which is about ∼ 1% lower thanthat obtained by using GGA. Our cell parameter is in goodagreement with that in Ref. [14] (∼ 14.2 Å). This again ver-ifies the accuracy of our calculation. The relaxed structure isshown in Fig. S9. As can be seen, the N atom and the near-est C atoms relax away from the carbon vacancy (V), and theN atom relaxes further than the nearest C atoms do, with theNV distance and the nearest CV distance given by 1.69 Å and1.643 Å, respectively.

The hyperfine tensor of each atom Ai is given by

Aαβi = δαβAiso,i + bαβi , (41)

where Aiso,i is the isotropic contribution (Fermi contact) andbαβi is the dipolar contribution. These quantities are deter-mined by

Aiso,i =8π3µ0

4πγeγ

(i)nucρs(Ri), (42a)

bαβi =µ0

4πγeγ

(i)nuc

ρs(r)

r3

3rαrβ − δαβr2

r2 d3r, (42b)

where the spin density ρs(r) ≡ ρ↑(r) − ρ↓(r), is determinedby first-principles calculation, and r in the integral is the dis-placement from the ith atom at Ri. Upon diagonalization ofAi, we obtain the principle axes of each atom.

The PAW method allows one to capture accurately theelectronic structure within the atomic core region, which isessential for the calculation of the hyperfine tensors [15,16]. We use CP-PAW (available at https://orion.pt.tu-clausthal.de/paw/) code to calculate the hyperfine parameters,in which the PAW method is implemented. To calculate thehyperfine tensors, we use planewave cutoffs of 35 Ry and150 Ry to expand the wavefunction and charge densities, re-spectively. The calculated hyperfine parameters of each atomare tabulated in Table I, with respect to the CV distance. Asthe NV-defect possesses the C3v symmetry, the numbers of C-atom having the same hyperfine parameters are either 3 or 6.The calculated hyperfine parameters and atomic structure areconsistent with the results of Gali et al. [14].

[1] He, X.-F., Manson, N. B. & Fisk P. T. H. Paramagnetic res-onance of photoexcited NV defects in diamond. II. Hyperfineinteraction with the 14N nucleus. Phys. Rev. B 47, 8816 (1993).

[2] Childress, L. et al. Coherent Dynamics of Coupled Electron andNuclear Spin Qubits in Diamond. Science 314, 281 (2006).

[3] Maze, J. R., Taylor, J. M. & Lukin, M. D. Electron spin deco-

herence of single nitrogen-vacancy defects in diamond. Phys.Rev. B 78, 94303 (2008).

[4] Yao, W., Liu, R. B. & Sham, L. J. Theory of electron spin de-coherence by interacting nuclear spins in a quantum dot. Phys.Rev. B 74, 195301 (2006).

[5] Gurudev Dutt, M. V. et al. Quantum register based on individ-




ual electronic and nuclear spin qubits in diamond. Science 316,1312 (2007).

[6] Yang, W. & Liu, R.B. Quantum many-body theory of qubit de-coherence in a finite-size spin bath. Phys. Rev. B 78, 085315(2008).

[7] Yang, W. & Liu, R.B. Quantum many-body theory of qubitdecoherence in a finite-size spin bath. II. Ensemble dynamics.Phys. Rev. B 79, 115320 (2009).

[8] Fuchs, G. D. et al. Excited-state spectroscopy using single spinmanipulation in diamond. Phys. Rev. Lett. 101, 117601 (2008).

[9] Epstein, R. J., Mendoza, F. M., Kato, Y. K. & Awschalom, D.D. Anisotropic interactions of a single spin and dark-spin spec-troscopy in diamond. Nature Physics 1, 94 (2005).

[10] Cywinski, L., Lutchyn R., Nave C. & Das Sarma, S. How toenhance dephasing time in superconducting qubits. Phys. Rev.B 77, 174509 (2008).

[11] Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized GradientApproximation Made Simple. Phys. Rev. Lett. 77, 3865 (1996).

[12] Blochl, P. E. Projector augmented-wave method. Phys. Rev. B50, 17953 (1994).

[13] Perdew, J. P. & Zunger, A. Self-interaction correction todensity-functional approximations for many-electron systems.Phys. Rev. B 23, 5048 (1981).

[14] Gali, A., Fyta, M. & Kaxiras, E. Ab initio supercell calcula-tions on nitrogen-vacancy center in diamond: Electronic struc-ture and hyperfine tensors. Phys. Rev. B 77, 155206 (2008).

[15] Van de Walle, C. G. & Blochl, P. E. First-principles calculationsof hyperfine parameters. Phys. Rev. B 47, 4244 (1993).

[16] Blochl, P. E. First-principles calculations of defects in oxygen-deficient silica exposed to hydrogen. Phys. Rev. B 62, 6158(2000).


SUPPLEMENTARY INFORMATION · 2011-04-06 · spin zero-ﬁeld splitting for the NV centre orbital...

Documents

Transcript of SUPPLEMENTARY INFORMATION · 2011-04-06 · spin zero-ﬁeld splitting for the NV centre orbital...