optimized quantum sensing with a single electron spin using real-time adaptive measurements

7/24/2019 Optimized quantum sensing with a single electron spin using real-time adaptive measurements

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Optimized quantum sensing with a single electron spin

using real-time adaptive measurements

C. Bonato1,

, M.S. Blok1,

, H. T. Dinani2,3

, D. W. Berry2,

M. L. Markham4, D. J. Twithen

4, !. Han"on

1,#

1

QuTech and Kavli Institute of Nanoscience, Delft University of Technology,PO Box !"#, $#!! %& Delft, The Netherlands

$De'art(ent of Physics and &strono(y, )ac*uarie University,

+ydney, Ne +outh -ales $1!., &ustralia/0enter for ngineered Quantu( +yste(s, )ac*uarie University,

+ydney, Ne +outh -ales $1!., &ustralia"le(ent +ix 2td, Kings 3ide Par4, &scot, Ber4shire +2 5BP, UK,

These authors contri6uted e*ually to this or4

7 corres'onding author8 r9hanson:tudelft9nl

Quantum sensors based on single solid-state spins promise a unique combination ofsensitivity and spatial resolution

1-20. he !ey challenge in sensing is to achieve minimum

estimation uncertainty within a given time and with a high dynamic range. "daptive

strategies have been proposed to achieve optimal performance but their implementation

in solid-state systems has been hindered by the demanding e#perimental requirements.

$ere we realize adaptive d.c. sensing by combining single-shot readout of an electron spin

in diamond with fast feedbac!. %y adapting the spin readout basis in real time based on

previous outcomes we demonstrate a sensitivity in &amsey interferometry surpassing the

standard measurement limit. 'urthermore( we find by simulations and e#periments that

adaptive protocols offer a distinctive advantage over the best-!nown non-adaptive

protocols when overhead and limited estimation time are ta!en into account. )sing an

optimized adaptive protocol we achieve a magnetic field sensitivity of *.1 + 1., n $z-12

over a wide range of 1., m. hese results open up a new class of e#periments for solid-

state sensors in which real-time !nowledge of the measurement history is e#ploited to

obtain optimal performance.

$%ant%m "en"or" ha&e the 'otential to ahie&e

%n'ree(ente( "en"iti&ity )y e*'loitin+ ontrol o&er

in(i&i(%al %ant%m "y"tem"1,2

. -" a 'rominent

e*am'le, "en"or" )a"e( on "in+le eletron "'in"

a""oiate( with itro+en/0aany 0 enter" in

(iamon( a'italie on the "'in" %ant%m

oherene an( the hi+h "'atial re"ol%tion re"%ltin+

5rom the atomi/like eletroni wa&e 5%ntion3,4

.

6ioneerin+ e*'eriment" ha&e alrea(y (emon"trate(

"in+le/"'in "en"in+ o5 ma+neti 5iel("7/8

, eletri

5iel("9, tem'erat%re

:,1;an( "train

11. 0 "en"or" may

there5ore ha&e a re&ol%tionary im'at on )iolo+y12/

17, nanotehnolo+y

1


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/'!(, where '!(i" the "horte"t "en"in+ time%"e(. - maAor o'en %e"tion i" whether a(a'ti&e

'rotool", in whih the rea(o%t )a"i" i" o'timie( in

real time )a"e( on 're&io%" o%tome", an

o%t'er5orm non/a(a'ti&e 'rotool". While "alin+

)eatin+ the "tan(ar( mea"%rement limit ha" )eenre'orte( with non/a(a'ti&e 'rotool"

22,23, 5ee()ak

tehni%e" ha&e only reently )een (emon"trate(

5or "oli(/"tate %ant%m "y"tem"24/24MH, an( minim%m 'ro)a)ility 5or= 0 -12&4 -24 MH. Thi" in(eterminay in thee"timation ori+inate" 5rom the 5at that, 5or thi"

"en"in+ time, the a%ire( 'ha"e "'an" the ran+e Q/4?, 4?R, wra''in+ m%lti'le time" aro%n( the Q/?, ?R


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inter&al. The "en"in+ time i" then (erea"e(

to 2'!( . Ni&en the %rrent 7 5or o%tome1 )lak %r&e, the 5ilter 5%ntion" that wo%l( )ea''lie( to 7 a5ter the Baye"ian %'(ate 5or(etetion o%tome" 0 an( 1 are re're"ente(,re"'eti&ely, )y the li+ht re( an( )l%e area". or. = ;/2, ma*im%m (i"tin+%i"ha)ility i" en"%re(Eo%tome 0 wo%l( "elet the 'eak" aro%n(= ;,&24+16&>4 MH, while o%tome1wo%l("elet the 'eak" aro%n(= ;16&>4+,&24MH.The "ame 'roe"" i" then re'eate(, (erea"in+ the

"en"in+ time to '!(. The remainin+ %nertainty,orre"'on(in+ to the wi(th o5 the re"%ltin+ 'eak in

7, i" "et )y the lon+e"t "en"in+ time '!( &i+%re 3) "how" the 'ro)a)ility (en"ity yiel(e( )y

e*'erimental r%n" o5 the limite(/a(a'ti&e 'rotool

with (i55erent n%m)er" o5 "en"in+ time" N K1,3,7,8,:. Here, = 2 ?@A, an( eah e"timation"e%ene i" re'eate( 1;1 time", with 9 = 4: = >.or inrea"in+ , the wi(th o5 the (i"tri)%tion)eome" more narrowly 'eake( aro%n( the

e*'ete( &al%e o5 2 MH, while the win+" o5

(i"tri)%tion are "tron+ly "%''re""e(.

To &eri5y that the 'rotool work" o&er a lar+e

(ynami ran+e, we mea"%re the %nertainty a" a

5%ntion o5 (et%nin+. To ao%nt 5or the 'erio(inat%re o5 'ha"e we %"e the Hole&o &ariane

BC= | D)EFGHIJKLM |); 1 a" a mea"%re o5 the%nertainty. We e"timate NOP)y takin+ the meano5 the 'ro)a)ility (en"ity Pre"%ltin+ 5rom a"in+le r%n o5 the 'rotool. - 5i*e( initial 'ha"e

. = 0 in o%r e*'eriment" re"%lt" in a "'ei5i(e'en(ene o5 the &ariane on the ma+neti 5iel(.

or e*am'le, 5or N K 2, only 5o%r mea"%rement

o%tome" are 'o""i)le ;;, ;1, 1;, 11,

orre"'on(in+ to = ;, /27, /12.7, U12.7 MH,re"'eti&ely. The"e "'ei5i (et%nin+" an )e

mea"%re( with the hi+he"t a%ray "ine they

orre"'on( to mea"%rement" o5 an ei+en"tate o5o%r %ant%m "en"or at the en( o5 the !am"ey

e*'eriment, while 5or other 5re%enie" lar+er

"tati"tial 5l%t%ation" will )e 5o%n( (%e to "'in

'roAetion noi"e. i+%re 3 "how" BC a" a 5%ntiono5 (et%nin+ 5or the 'arameter" %K7, n the"e +ra'h, the SMS limit orre"'on(" to a

on"tant BC any "alin+ )eha&ior with a ne+ati&e"lo'e th%" im'ro&e" )eyon( the SMS.

We o)"er&e that, 5or the "ettin+ %K7,


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o'timie( a(a'ti&e 'rotool re%ire" "i+ni5iantly

le"" mea"%rement time. -t any 5i*e( mea"%rement

time, the a(a'ti&e 'rotool e"timate" the ma+neti

5iel( more a%rately, allowin+ a hi+her re'etition

rate 5or the e"timation "e%ene". Thi" i"

a(&anta+eo%" in the reali"ti "it%ation that thema+neti 5iel( to )e e"timate( i" not "tatiE in thi"

a"e, the e"timation time i" re%ire( to )e "horter

than the time"ale o5 the 5l%t%ation". F%r (ata

"how" that at an e"timation re'etition rate o5 2; H,

the non/a(a'ti&e 'rotool an e"timate a ma+neti

5iel( with an "en"iti&ity > K 84: V 37 nT H/1@2

,

while the o'timie(/a(a'ti&e 'rotool yiel(" >K 48

V 2 nT H/1@2

.

While the reor( "en"iti&ity re'orte( here i"

ena)le( )y "in+le/"hot "'in rea(o%t at low

tem'erat%re, a(a'ti&e tehni%e" an 'ro&e&al%a)le al"o in e*'eriment" at room tem'erat%re

23

where "'in/(e'en(ent l%mine"ene inten"ity

%n(er o55/re"onant e*itation i" ty'ially %"e( to

mea"%re the eletroni "'in. By a&era+in+ the "i+nal

o&er m%lti'le re'etition" an ar)itrarily hi+h rea(o%t

5i(elity an )e ahie&e(


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7V = 1|= 1 ; 7V = 0|where = 2('!(. D%e to it" 'erio(iity, it i" on&enient to e*'re"" 7V|in a o%rier"erie", re"%ltin+ in the 5ollowin+ %'(ate r%leE

fgY =1 + ;1hY:\; :Z

2 fgYZ

+ ]I_*`_ :\+ :Z ; 1 ihYEjkYfg)lmLYZ + hYEjkYfgj)lmLYZ nThe Baye"ian %'(ate i" 'er5orme( %"in+ the e*'erimental &al%e"


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&eferences

Q1R Nio&annetti, 0., Lloy(, S. Y Maone, L. -(&ane" in %ant%m metrolo+y. Nature Photonics3, 222/

22: 2;11.

Q2R Hi++in", B. L., et al. ntan+lement/5ree Hei"en)er+/limite( 'ha"e e"timation. Nature 430, 3:3

2;;8.

Q3R De+en, C. L. Sannin+ ma+neti 5iel( miro"o'e with a (iamon( "in+le/"'in "en"or. &''lied

Physics 2etters52, 243111 2;;9.

Q4R Taylor, J. M. et al. Hi+h/"en"iti&ity (iamon( ma+netometer with nano"ale re"ol%tion. Nature

Physics4, 91;/91< 2;;9.

Q7R Mae, J. !. et al. ano"ale ma+neti "en"in+ with an in(i&i(%al eletroni "'in in (iamon(.

Nature433,


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Q27R Blok, M. S. et al. Mani'%latin+ a %)it thro%+h the )akation o5 "e%ential 'artial

mea"%rement" an( real/time 5ee()ak. Nature Physics10, 19: 2;14.

Q2


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'igure 1. #periment concept and apparatus. a The a(a'ti&e 5re%eny e"timation 'rotool

on"i"t" o5 a "e%ene o5 initialiation, "en"in+, mea"%rement o'eration". -5ter eah mea"%rement

r%n, the o%tome ? i" %"e( to %'(ate the e"timate o5 the 5re%eny, whih i" then %"e( too'timie the "en"in+ 'arameter" 5or the 5ollowin+ r%n. *'erimentally, the 5re%eny e"timation an(

a(a'ti&e al%lation o5 the 'ha"e are 'er5orme( in real/time )y a miro'roe""or. ) The e*'eriment

i" 'er5orme( %"in+ the "tate" |0M= |RO= 0M |1M= |RO= ;1Mo5 the eletroni "'in o5 a 0 entrein (iamon(. The eletroni "'in i" rea(o%t )y re"onant o'tial e*itation an( 'hoton o%ntin+

28E

(etetion o5 l%mine"ene 'hoton" orre"'on(" to (etetion o5 the |0M"tate. We 'lot the 'ro)a)ilityo5 (etetin+ a 'hoton a5ter initialiin+ either in |0Mor |1M. The rea(o%t 5i(elitie" 5or the "tate" |0Mo%tome 0 an( |1Mo%tome 1 are :\= 0&66-0&02, :Z= 0&6-0&02, re"'eti&ely. ahmea"%rement r%n on"i"t" o5 a !am"ey e*'eriment, in whih the 'ha"e a%m%late( o&er time )y a

"'in "%'er'o"ition (%rin+ 5ree e&ol%tion i" mea"%re(. The mea"%rement )a"i" rotation i" ontrolle(

)y the 'ha"e o5 the 5inal ?@2 '%l"e. rom the mea"%re( 'ha"e, we an e*trat the 5re%eny ,orre"'on(in+ to an ener+y "hi5t )etween the le&el" |0Man( |1M+i&en )y an e*ternal 5iel( ma+neti5iel(, tem'erat%re, "trainZ. Here, to te"t the 'er5ormane o5 (i55erent 'rotool", we "et a" anarti5iial (et%nin+, "et )y the miro'roe""or )y a((in+ = 2 to the 'ha"e S%''lementaryi+%re S8.


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'igure 2. $igh dynamic-range adaptive magnetometryLimite(/a(a'ti&e 'rotool, in the a"e o5 one

!am"ey e*'eriment 'er "en"in+ time %K1, n eah "te', the %rrent 5re%eny 'ro)a)ility

(i"tri)%tion 7i" 'lotte( "oli( )lak line, to+ether with on(itional 'ro)a)ilitie" 7V|5or themea"%rement o%tome" V = 0 re( "ha(e( area an( V = 1 )l%e "ha(e( area. -5ter eahmea"%rement, 7 i" %'(ate( aor(in+ to Baye" r%le. The (etetion 'ha"e o5 the !am"eye*'eriment i" "et to the an+le whih attain" the )e"t (i"tin+%i"ha)ility )etween 'eak" in the %rrent

5re%eny 'ro)a)ility (i"tri)%tion 7. Xltimately, the 'rotool on&er+e" to a "in+le 'eak in the'ro)a)ility (i"tri)%tion, whih (eli&er" the 5re%eny e"timate.


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'igure 6. 'requency dependence of uncertainty.a/) re%eny e"timate e*am'le, 5or %K7,


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'igure 4. caling of sensitivity as a function of total time. a The three 'rotool" are om'are( )y

'lottin+ Q)= BC a" a 5%ntion o5 the total "en"in+ time T not inl%(in+ "'in initialiation an(rea(o%t. or %K7,


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Supplementary Information

Optimized quantum sensing with a single electron spin

using real-time adaptive measurements

C. Bonato, M.S. Blok, H. T. Dinani, D. W. Berry, M. L. Markham, D. J. Twitchen, R. Hanson

I. Comparison of sensing protocols: numerical simulations

In the followin, the !erformances of "ifferent sinle#$%&it fre$%ency estimation !rotocols will &e

com!are" thro%h n%merical sim%lations. We will "escri&e an" analyse three main sensin

alorithms, "efine" %sin a !se%"o#co"e in S%!!lementary Ta&les ', (, )* the limite"#a"a!ti+e, non#

a"a!ti+e an" o!timie"#a"a!ti+e !rotocols.In or"er to achie+e hih "ynamic rane, each estimation se$%ence consists of - "ifferent sensin

times, m%lti!les of the shortest sensin time = 20ns* = 2 = 1. . /.0fter each Ramsey, the electron s!in is meas%re"* the res%lt of each "etection is in"icate" in the

!se%"o#co"e &y the Ramsey (, ) f%nction. The in!%t !arameters of this f%nction are the sensin

time 1 an" the !hase 2 of the secon" 34( !%lse. In the sim%lation co"e, this f%nction enerates a

ran"om +al%e = 0, 1/, %sin the !ython li&rary numpy.random, chosen accor"in to the!ro&a&ility "istri&%tion 5!6, !'7'#!68, where*

= 0|= + !"#$ #%&'(2) + * (Eq. S-E1), are, res!ecti+ely the rea"o%t fi"elities for ms76 an" ms7'. In the followin sim%lations we %sethe +al%es* 7 6, 6.9:, 6.;; or '.66 -= 0/, 7 6.


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?or each !rotocol it is cr%cial to fin" the o!timal +al%es for an" U, i+en the e@!erimental rea"o%tfi"elities ?6an" ?'. The rele+ant fi%re of merit is the sensiti+ity

V, "efine" as

V

= ?@/.

Sim%lations are !erforme" &y r%nnin the !rotocol for )': "ifferent +al%es of the fre$%ency , o+er)' re!etitions for each +al%e. The "etection !hase *of the Ramsey is initially set to ero.

Limited-adaptive vs non-adaptive protocols.The limited-adaptive protocol5AS#'8 is "escri&e" &y the

!se%"o#co"e in Bo@#'. The controlle" !hase

*JKLMis %!"ate" e+ery time the sensin time is chane".

In this section, the !erformance of the limite"#a"a!ti+e !rotocol will &e com!are" to the non-

adaptive protocol"escri&e" &y Sai" et al. 5AS#(8 an" "emonstrate" e@!erimentally &y Wal"herr et al.

Supplementary a!le ":

Limited-adaptive protocol

for n = 1 to N:

tn= 2N-nchoose WXYZ[\ n= ! " #(n-1)

for m = 1 to n:

$ = %am&ey (*= *JKLM, ' = tn'min)aye&ianupdate (re& =$, *= *JKLM, ' = tn'min)

Supplementary Table 2:

Nonadapti!e protocol

for n = 1 to N:

tn= 2N-n

n= ! " #(n-1)

for m = 1 to n:Wn,m " (m#)$%&n$ = %am&ey (*= *n,m, ' = tn'min)aye&ianupdate (re& = $, *= *n,m, ' = tn'min)

Supplementary #igure S". Simulation& comparin* t+e limited-adaptive and non-adaptive protocol& for U= , for different value& of , it+ /$= ]$&. n t+e top ro, perfect readout fidelity (= 1), on t+e/ottom ro, = 0.^].0+e &+aded area& corre&pond to uncertaintie& (one &tandard deviation,calculated /y a /oot&trap tec+nique). Note t+at t+e &en&itivity i& not furt+er improved after t+e limit of/$i& reac+ed (total &en&in* time 0 1$&).


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5AS#8 an" -%sran et al. 5AS#:8. The !se%"o#co"e for the non#a"a!ti+e !rotocol is re!orte" in Bo@#(.

In this case, the !hase of the Ramsey e@!eriment is not %!"ate" in real#time &ase" on the estimation

of manetic fiel" i+en &y the !re+io%s meas%rement o%tcomes, &%t its +al%e is swe!t &etween 6 an"

3 accor"in to !re"efine" +al%es. If, for a i+en sensin time, T Ramsey e@!eriments are!erforme", the Ramsey !hase is increase" at each ste! &y

)_T.

0 com!arison of the sensiti+ity as a f%nction of sensin time /for "ifferent +al%es of fi@in U = ]/is shown in S%!!lementary ?i%re S'. The "ata#!oints corres!on" to estimation se$%ences withincreasin = 2. .10/. The total sensin time /, for each estimation se$%ence, is calc%late" as*/ = (U2I 1 + 2I I 1 `a. bc`d

In to! row of S%!!lementary ?i%re S', the sensiti+ities for the a"a!ti+e an" non#a"a!ti+e

!rotocols are com!are" in the case of !erfect rea"o%t fi"elities. In this case, the a"a!ti+e !rotocol

follows a Heisen&er#like scalin alrea"y for = 0, e+en tho%h the minim%m sensiti+ity can only &ereache" for = 1. n the other han", the non#a"a!ti+e !rotocol re$%ires at least = 2 to reachHeisen&er#like scalin. n the &ottom row, we com!are the sensiti+ities for re"%ce" rea"o%t fi"elity

= 0.^]/. Here, the a"a!ti+e !rotocol reaches HL#scalin for e 2, while the non#a"a!ti+e!rotocol can only et close to it with = ].It is im!ortant to stress that, in &oth cases, there is a &i im!ro+ement when nis a f%nction of f 0/ com!are" to the case where nis in"e!en"ent of = 0/. In other wor"s, it is &eneficialto re!eat more often Ramsey e@!eriments with shorter sensin time. The reason is two#fol"* on one

en" they contri&%te less to the total sensin time, on the other en" they are relate" to larer

fre$%encies which, if estimate" wron, wo%l" i+e a larer error.

Com!arison &etween !rotocols is easier when !lottin only the minim%m sensiti+ity +s ?. This is

shown in S%!!lementary ?i%re S( for !erfect rea"o%t fi"elity = 1. We fin" that for E(, thelimite"#a"a!ti+e !rotocol o%t!erforms the non#a"a!ti+e !rotocol. This is e@!ecte" since in this

reion only the limite"#a"a!ti+e !rotocol e@hi&its Heisen&er#like scalin. Howe+er, once the non#

a"a!ti+e !rotocol achie+es Heisen&er#like scalin e 2/ it reaches a lower sensiti+ity.

Supplementary #igure S$. Simulation re&ult& for t+e /e&t ac+ieved &en&itivity, comparin* t+e limited-

adaptive and non-adaptive protocol& a& a function of . ere, e a&&ume perfect readout fidelity (=1) and 024 = $&. n t+e top 5-a5i&, t+e total num/er of %am&ey e5periment& in t+e e&timation &equencefor = 10i& reported. Error /ar&, corre&pondin* to one &tandard deviation, are calculated /y /oot&trap.


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n the scale at the to! of S%!!lementary ?i%re S(, the n%m&er of Ramsey r%ns corres!on"in to an

estimation se$%ence with = 10"ifferent sensin times is re!orte". By increasin , the n%m&erof Ramsey e@!eriments increases as*

g= U + I 1

2 `a. bc`]

?or !erfect rea"o%t fi"elity, the limite"#a"a!ti+e !rotocol reaches HL#scalin for = 0* therefore itonly re$%ires R- 7 :6 Ramsey r%ns in the estimation se$%ence. n the other han", the non#a"a!ti+ere$%ires = 2, i.e. R- 7 '6 Ramsey r%ns. Fach Ramsey com!rises an initialiation4meas%rement"%ration, la&elle" as Go+erhea", not incl%"e" in the !lots where we only take the sensin time into

acco%nt/. In !ractice, it is howe+er necessary to minimie the total time of the se$%ence incl%"in

o+erhea"/, so that !rotocols that achie+e Heisen&er#like#scalin with smaller an" therefore less"etections R-/ are to &e !referre" as "isc%sse" in the main te@t ?i. /.

0 strikin res%lt is the fact that, once the non#a"a!ti+e !rotocol reaches Heisen&er#like#scalin, it

achie+es a &etter sensiti+ity than the limite"#a"a!ti+e one. Since non#a"a!ti+e !rotocols are a

!artic%lar case of the most eneral class of a"a!ti+e !rotocols, this in"icates that the limite"#

a"a!ti+e !rotocol is not o!timal an" that !rotocols with &etter !erformance m%st e@ist.

Optimized adaptive protocol.In or"er to im!ro+e the !erformance of the limite"#a"a!ti+e !rotocol,

we consi"er two mo"ifications*

". in the first one, the controlle" !hase is estimate" not only when chanin sensin time, &%t

&efore each Ramsey meas%rement 6full-adaptive7 protocol/. The im!ro+ement achie+e"

with this mo"ification can &e o&ser+e" in S%!!lementary ?i%re S), where we com!are

sim%lations for controlle" !hase %!"ate" only when chanin sensin time an" &efore each

Ramsey. In the left !lot, we com!are the sensiti+ity, for increasin n%m&er of meas%rements = 2. .10/ in the case U = h, = 0/. Both !rotocols scale &etter than the stan"ar"$%ant%m limit only for the first few "ata#!oints %ntil f d/. Howe+er, the a&sol%tesensiti+ity of the f%ll#a"a!ti+e !rotocol is a factor two &etter. In the central !lot, the same

c%r+es are "is!laye" for U = h, = ]/. ?or these !arameters, Heisen&er#like scalin ismaintaine" %ntil the coherence time limit is reache". 0ain, the f%ll#a"a!ti+e !rotocol is

&etter than the limite"#a"a!ti+e for all . In the riht !lot, we show the minim%m achie+e"sensiti+ity for &oth !rotocols, as a f%nction of . In all cases the f%ll#a"a!ti+e !rotocolo%t!erforms the !rotocol which %!"ates the o!timal !hase only when chanin the sensin

time.

Supplementary #igure S%.8daptive protocol&:&imulation re&ult& comparin* &en&itivitie& o/tained +en

updatin* t+e controlled p+a&e only +en c+an*in* &en&in* time (/lue) and updatin* it /efore eac+

%am&ey (red). 9e a&&ume perfect readout fidelity and /$= ]$&. S+aded area& repre&ent error /ar&corre&pondin* to one &tandard deviation (/oot&trap met+od).


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Sim%lations in S%!!lementary ?i%re S) are !erforme" ass%min !erfect rea"o%t fi"elity,U = h, /$= ]=s. 0""itional sim%lations not shown/ for "ifferent +al%es of U, or im!erfectrea"o%t fi"elity, confirm the im!ro+ements aine" &y the f%ll#a"a!ti+e stratey.

$. the secon" mo"ification was s%este" &y 0. J. Hayes an" D. W. Berry 5AS#)8. They !ro!ose"

a !rotocol where the "etection !hase of the Ramsey e@!eriment is *, = *,JKLM+ *,JL.0!hase increment *,JL, "e!en"ent only on the last meas%rement o%tcome, is a""e" to thecontrolle" !hase *,JKLM. S%ch !hase increment is o&taine" &y n%merically o!timiin the final+ariance in fre$%ency estimation for the s!ecific e@!erimental !arameters, thro%h a Gswarm

o!timiation !roce"%re 5AS#),AS#>, AS#98 an" ta&%late" in the f%nctions %6, %'.

In this stratey the !hase increments can &e mo"elle" &y a &inary "ecision tree. The sie an""irection of the ste!s "e!en"s on the last "etection res%lt an" the c%rrent +al%es of , -.The !article swarm o!timiation AS/ alorithm is then %se" to "etermine the ste!s in s%ch

a way that the final !hase +ariance is minimie". In the AS alorithm 5AS#98, the !ro&lem

s!ace the set of !hase increments in this case/ is searche" &y a swarm of !articles. Fach

!article, la&elle" &y i has a +elocity jkl/ an" a !osition mkl/ which are %!"ate"accor"in to its c%rrent &est !osition mk/ an" the &est !osition of the entire swarm mnkl/as*

jkl + 1= opjkl+ qnrnsmnkl I mklt + qMrM(mMkl I mklu

mkl + 1= mkl+ jkl `a. bc`v

Here rnan" rM are %niform ran"om n%m&ers in the inter+al 56 '8, o, qnan" qMare constants,wis the "imension of the s!ace an" lis the n%m&er of the iteration. In o%r sim%lations we%se" o = 0.^2x, qM = qn= 2.0]with 10!articles an" d00iterations.

The o!timie" a"a!ti+e !rotocol, "escri&e" in Bo@#), com&ines !hase increments with %!"ate of the

controlle" !hase &efore each Ramsey. 0 com!arison &etween the minim%m sensiti+ity achie+e" &y

the limite"#a"a!ti+e, non#a"a!ti+e an" o!timie"#a"a!ti+e !rotocols is re!orte" in S%!!lementary

?i%re S. We fi@ U = ]an" ass%me /$= xv=s.The o!timie" a"a!ti+e !rotocol a!!ears to !erform always at least as oo" as the &est &etween the

limite"#a"a!ti+e an" the non#a"a!ti+e !rotocols. ?or lower +al%es of , the non#a"a!ti+e !rotocolfails to reach HL#scalin, while &oth a"a!ti+e ones "o. ?or hiher +al%es of , &oth the non#a"a!ti+e

Supplementary Table ':

ptimied adapti!e protocol

y= (init)for n = 1 to N:

tn= 2N-n

n= ! " #(n-1)

for m = 1 to n:

c+oo&e *,JKLMif (y=) t+en*,JL= u m, n;el&e:

*,JL= u1 m, n;y= %am&ey (*= *,JKLM" *,JL, ' = tn'min)aye&ianupdate (y, *= *,JKLM" *,JL, ' = tn'min)


17/22

an" the o!timie" a"a!ti+e reach the minim%m sensiti+ity. -ote that, for a rea"o%t fi"elity =0.zz, while the o!timie" a"a!ti+e !rotocol reaches HL#scalin for U = ], = 2/ the non#a"a!ti+eone nee"s at least = d.

&pplication to room-temperature sensing. 0t room#tem!erat%re, the s!in#selecti+e o!tical

transitions within the ero#!honon line are not s!ectrally#resol+a&le an" therefore sinle#shot s!in

initialiation an" rea"o%t &y resonant o!tical e@citation is not !ossi&le. Instea", the rea"o%t e@!loits

the small "ifference in l%minescence intensity &y off#resonant o!tical e@citation in the reen/ when

the electron s!in is in -= 0, com!are" to -= 1.In the followin, we will %se n%m&ers from -%sran et al. 5AS#:8, that re!ort {= 0.0h1!hotons !ershot when the electron is in -= 0, {= 0.021!hotons !er shot when the electron is in -= 1.Since one shot is not eno%h to !erfectly "istin%ish &etween the two states, room#tem!erat%re

e@!eriments are re!eate" R times. The res%ltin sinal le+el is then, res!ecti+ely, {gan" {g. Thecontrast for a sinle re!etition scales as 5AS#;8*

1= }1 + 2{+ {{I {g `a. bc`^

Supplementary #igure S'. Simulation& comparin* t+e /e&t ac+ieved p+a&e &en&itivitie& for

different protocol& (left &ide, perfect readout fidelity = 1 < ri*+t &ide, = 0.zz). 9ea&&ume U = ], /$= xv$&. 0+e plot on t+e ri*+t report& &imulation re&ult& for t+e t+reeprotocol& di&cu&&ed in t+e main te5t, for e5perimental parameter& and &ettin* (a& in main

te5t).

Supplementary #igure S(.Effective readout fidelity a& a function of readout

repetition&, for mea&urement& at room-temperature (no &in*le-&+ot readout).


18/22

Supplementary #igure S).Simulation& comparin* t+e minimum ma*netic field &en&itivity (in n0 -1>2

)

for t+e optimied adaptive and t+e non-adaptive protocol& at room-temperature and lo-temperature

(/$= xv$&, = 10, U = ]).

This contrast is relate" to the fi"elity with which the two states can &e "istin%ishe" an", sincel%minescence "etection is shot#noise limite", the error scales at the stan"ar" $%ant%m limit asg_. -%sran et al. achie+e a fi"elity of 6., for consistency with !re+io%s res%lts, we ass%me

asymmetric rea"o%t fi"elity = 0.xxh, = + 1 I /, &ase" on the contrast achie+e" witha i+en n%m&er of rea"o%t re!etitions. The asymmetry of the rea"o%t fi"elity can &e controlle" at

will &y choosin the threshol" in !hoto#co%nts "istin%ishin -= 0from -= 1.Sim%lation res%lts show that, at room tem!erat%re, the %se of :6666 re!etitions can achie+e a

sensiti+ity of V1T H#'4(, either %sin the a"a!ti+e or non#a"a!ti+e !rotocols. Howe+er, %sin )>66re!etitions !er ste! with a lower effecti+e rea"o%t fi"elity/, a &etter sensiti+ity V0.dT H#'4(can&e reache". Moreo+er, for = 2, the !erformance of the o!timie"#a"a!ti+e !rotocol with )>66rea"o%t re!etitions !er ste! s%r!asses &oth the !erformance of the non#a"a!ti+e for the same

con"itions an" the !erformance of the !rotocols with :6666 re!etitions !er ste!. ?or e da"a!ti+ean" non#a"a!ti+e reach the same sensiti+ity* howe+er, as "isc%sse" a&o+e an" in the main te@t, a

smaller +al%e of allows a hiher re!etition rate of the estimation se$%ence.This s%ests the !ossi&ility that a"a!ti+e sensin, which reaches Heisen&er#limite" scalin for are"%ce" n%m&er of meas%rements e+en in sit%ation of lower fi"elity, may &e a"+antaeo%s for

room#tem!erat%re sensin, com!are" to non#a"a!ti+e !rotocols.

Sim%lations confirm the s%!erior !erformance of the !rotocol in the case where sinle#shot rea"o%t

is a+aila&le, ena&lin sensiti+ities on the or"er of a few nT H#'4(

, as "emonstrate" e@!erimentally &y

the "ata re!orte" in the main te@t.


19/22

II. Supplementary e*perimental details

a/ &dditional information a!out the e*perimental setup.

Sample. We %se an isoto!ically#!%rifie" "iamon" sam!le rown &y Flement Si@ Lt" ')

C

concentration 6.6'/ with a E'66K crystal orientation an" st%"y nat%rally occ%rrin - centres

locate" :':=m &elow the s%rface. 0 soli" immersion lens SIL/ is "eterministically etche" &yfoc%se"#ion#&eam ?FI Strata DB ():, )6 k alli%m ions/ aro%n" a s!ecific - centre to increase

collection efficiency. 0fter millin the lenses, the sam!le is cleane" for )6 min%tes in a &oilin

mi@t%re of e$%al !arts of !erchloric, s%lf%ric an" nitric aci". 0 sinle#layer al%mini%m#o@i"e

antireflection coatin, "esine" for &est !erformance at a wa+elenth of >)9 nm, is "e!osite" on

to! of the sam!le 5AS#;8. 0 (66 nm thick ol" microwa+e stri! line is "efine" alon the lenses +ia

electron &eam lithora!hy.

ptic&.The sam!le is mo%nte" on a ste!!er4scanner !ieo stack 0ttoc%&e/ in a Janis ST#:66 flow

cryostat with o!tical access an" ke!t at a tem!erat%re of T 7 ; N. 0t low tem!erat%res, -

centres e@hi&it narrow ero !honon lines OAL/ aro%n" >)9 nm which we a""ress resonantly with

two lasers -ew ?oc%s elocity t%na&le e@ternal ca+ity "io"e laser an" Sirah Matisse DS "ye laser

with DCM "ye/. Fach laser is locke" to a wa+emeter to ens%re lon#term fre$%ency sta&ility. Ina""ition, yellow e@citation at :9: nm from a fre$%ency#"o%&le" "io"e#laser To!tica/ is %se" for

chare#state control see Section IIc/. 0ll lasers are !%lse" &y aco%sto#o!tic mo"%lators Crystal

Technoloies/.

We "etect l%minescence e@cl%si+ely in the - !honon si"e&an" >:6 # 9:6 nm/, se!arate" from

the e@citation liht &y "ichroic filters Semrock LAD6'#>))R/. Ahoton "etection is !erforme" &y

a+alanche !hoto"io"es Aerkin Flmer SACM/.

Spin control.Microwa+e MW/ sinals to "ri+e the - centre electron s!in are enerate" &y a

Roh"e Schwar SMB'660 so%rce, with IP mo"%lation. To create the two 34( !%lses for each

Ramsey e@!eriment, the MW so%rce o%t!%t fre$%ency is mo"%late" sinle#si"e&an"

mo"%lation/ &y two !%lses with rectan%lar en+elo!e an" )6 MH carrier fre$%ency. The first

!%lse is enerate" "irectly &y an 0W 0r&itrary Wa+eform enerator, Tektroni@ 0W:6'/,while the secon" &y a fiel"#!roramma&le loic array ?A0/. The ?A0 recei+es the +al%e for the

!hase *chosen &y the control micro!rocessor 0Dwin ol"/ an" enerates the mo"%lation !%lsewith the !ro!er IP !arameters, with timin triere" &y the 0W. The +al%e for the !hase * iss!ecifie" as an ;#&it inteer (:> le+els/, lea"in to a resol%tion of '. "erees 6.6(: ra"ians/.

D%e to the hy!erfine co%!lin to the host'

- s!in, the electron s!in transition is s!lit into three

fre$%ency#se!arate" lines. 0s "isc%sse" in Section II&, these three lines are a""resse" se!arately

&y thee Ramsey e@!eriments, realie" &y "ri+in the electron s!in at the three fre$%encies. This

is achie+e" &y an a""itional fre$%ency mo"%lation, im!ose" to the +ector so%rce &y the 0W.

?ontrol microproce&&or. The micro!rocessor 0"win ol"/ manaes the se$%ence of control

!%lses o!tical an" microwa+e/ an" co%nts the l%minescence !hoton "%rin s!in rea"o%t.

Moreo+er, it !erforms the Bayesian %!"ate of the !ro&a&ility "ensity "istri&%tion an"chooses the !ro!er controlle" !hase an" !hase increments.The micro!rocessor co"e for Bayesian %!"ate minimies the n%m&er of coefficients 3F$. S#F(/to &e tracke" an" store", to a+oi" e@cee"in the memory &o%n"s of the micro!rocessor, an" to

o!timie s!ee". We only %se the coefficients which are known to &e non#ero an" contri&%te to

the choice of the controlle" !hase 2, nelectin the rest. Since the !ro&a&ility "istri&%tion is real

3$ = I3/, we can f%rther re"%ce the com!%tational re$%irements &y only storin thecoefficients for ~ f 0. Consi"erin all this, the n%m&er of coefficients that nee"s to &e!rocesse", at each ste! , is on the or"er of T = U + I1.


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In the case 7:, ?7(/, the time s!ent &y the micro!rocessor in the Bayesian %!"ate after each

Ramsey e@!eriment increases linearly from ;6=s for 7(/ to '

"$ (;.; 6.:9) '..;

Supplementary a!le '. emporal !udget of the estimation protocol. 0otal time, mea&ured /y t+e

internal microproce&&or cloc@, &pent /y t+e optimied-adaptive protocol in different ta&@& it+in t+e

+ole e&timation &equence. 0+e computational time (i.e. t+e time &pent /y t+e proce&&or in

performin* t+e aye&ian update), i& &imilar to t+at &pent on &pin initialiation. !iven t+at initialiation

and aye&ian update can /e performed &imultaneou&ly, t+e computational time repre&ent& no

additional over+ead.


21/22

We note that this metho" re$%ires the electron transition eneries an" therefore the static

manetic fiel" to &e known within the &an"wi"th of the !%lses Q '6 kH/. This is not a !ro&lem

for o%r im!lementation, where the effect of an e@ternal fiel" is im!lemente" as an artificial

"et%nin &y a"%stin the !hase of the final 34( !%lse. When estimatin a real manetic fiel"!ossi&le sol%tions wo%l" &e to initialie the nitroen s!in, a"%st the fre$%ency estimation

!rotocol to allow for sensin of m%lti!le fre$%encies with fi@e" offset or a"%st the interaction

times s%ch that the !hase ac$%ire" "%rin free e+ol%tion is in"e!en"ent of the state of the

nitroen s!in 2) = B/c/

,1 charge state and optical resonance pre- and post-selection.D%e to en+ironmental chare

noise, the o!tical transitions of the - centre shift in fre$%ency on a rane larer than the

linewi"th. Moreo+er, resonant e@citation can res%lt in ioniation of the -#chare" state into the

ne%tral -6state.

Before each estimation se$%ence, we check that the centre is in the -# state, with o!tical

transitions resonant with the e@citation lasers. We t%rn &oth the initialiation an" rea"o%t laserson transitions F an" Fy, res!ecti+ely/ for ':6 =s an" co%nt l%minescence !hotons. nly if the

Supplementary #igure S. -lectron spin dri!in. a) %a/i o&cillation& of t+e electron &pin

conditional on t+e &tate of t+e nitro*en &pin (1A

N , B=1). 9e tune t+e frequency of t+e microave

pul&e& in re&onance it+ one of t+e t+ree - = 0 to -= I1 tran&ition&, corre&pondin* to t+enitro*en &pin /ein* eit+er in -= I1, 0or +1(top, middle /ottom) and vary t+e len*t+ of t+epul&e. #rom a &inu&oidal fit (*rey line) e find %a/i frequencie& of (1AA, 1A and 1A2 2@)re&pectivelyb) Ener*y level &pectrum for t+e electron

-= 0to

-= I1tran&ition. 9e initialie

t+e electron &pin in m&= and t+en vary t+e frequency of a microave pul&e it+ fi5ed len*t+.

0+e pul&e detunin* i& it+ re&pect to a reference frequency of 2.CA33A !. 0+e &pectrum

&+o& t+ree line& oin* to t+e +yperfine interaction it+ t+e1A

N &pin it+ B= 2D 5 (2.1C .) . c)%a/i o&cillation of t+e electron &pin unconditional on t+e &tate of t+e nitro*en &pin.

9e apply t+ree &equential microave pul&e& eac+ on re&onance it+ one of t+e +yperfine line&.

#rom t+e &inu&oidal fit (*rey line) e find a %a/i frequency of (1A2 3) @.


22/22

Supplementary #igure S.i&to*ram& of t+e percenta*e of reFected

run& in c+ar*e and optical re&onance po&t-&election.

l%minescence !hoto#co%nts are larer than a i+en threshol" 6 co%nts/, the estimation

se$%ence is starte" c+ar*e and optical re&onance pre-&election/. We take the a&sence of

l%minescence !hoto#co%nts as an in"ication that the centre is ionie" into the -6 state* the

correct chare state is restore" &y resonant o!tical e@citation of the -6transition at :9: nm.

0n estimation se$%ence can consists of a lare series of Ramsey e@!eriments, with s!in

initialiation an" rea"o%t. Ioniation of the "efect or lare fre$%ency shifts of the o!ticaltransitions "%rin the se$%ence res%lts in incorrect s!in rea"o%t an" errors in the manetic fiel"

estimation. Therefore, we !erform a new check of the chare an" o!tical resonance con"itions

at the en" of the estimation se$%ence an" consi"er it as a +ali" estimation only if more than '6

l%minescence !hoto#co%nts are "etecte" c+ar*e and optical re&onance po&t-&election/.

In the historams in S%!!lementary ?i%re S;, we re!ort an e@am!le of the n%m&er of reecte"

r%ns for (:( re!etitions of the estimation se$%ence. While the a+erae n%m&er of reections in

the !ost#selection !rocess is aro%n" :6, we ha+e a consistent fraction of e+ents 9:4((:/ with

no reections, an" other r%ns with ;6 fail%re rate. This lare s!rea" is "%e to the fact that the

"ata was taken in lon a%tomate" meas%rement session "%rin nihts, with infre$%ent

o!timiations of the e@!erimental !arameters like s!atial o+erla! of laser &eams on the -

centre/. We &elie+e that the !ercentae of reecte" r%ns can &e "rastically re"%ce" &y o!timiinthe e@!erimental settins an" !roce"%res.

Supplementary 2eferences

5AS#'8 Ca!!ellaro, A. S!in &ath narrowin with a"a!ti+e !arameter estimation. G+y&. %ev. 8 (,

6)6)6' R/ (6''/.

5AS#(8 Sai", R. S., Berry, D. W. Twamley, J. -anoscale manetometry %sin a sinle#s!in system in

"iamon". G+y&. %ev. %, '(:'6 (6''/.

5AS#)8 Hayes, 0. J. ?. Berry, D. W. Swarm o!timiation for a"a!ti+e !hase meas%rements with low

+isi&ility. G+y&. %ev. 8/, 6');); (6'/.5AS#8 Wal"herr, . et al. Hih#"ynamic#rane manetometry with a sinle n%clear s!in in "iamon".

Nature Nanotec+.3'6: (6''/.

5AS#:8 -%sran, -. M et al. Hih#"ynamic#rane manetometry with a sinle electronic s!in in

"iamon". Nature Nanotec+. , '6< (6''/.

5AS#>8 Hentschel, 0. San"ers, B. C. Machine Learnin for Arecise P%ant%m Meas%rements. G+y&.

%ev. Hett."0', 6>)>6) (6'6/.

5AS#98 Bratton, D. Nenne"y, J. Definin a stan"ar" for !article swarm o!timiation. Groceedin*& of

t+e 2I BEEE Sarm Bntelli*ence Sympo&ium, !!. '(6#'(9 (669/

5AS#;8 Taylor, J. M. et al. Hih#sensiti+ity "iamon" manetometer with nanoscale resol%tion. Nature

G+y&ic&', ;'6#;'> (66;/.

5AS#

optimized quantum sensing with a single electron spin using real-time adaptive measurements

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