Synthetic Frequency Protocol in the Ramsey Spectroscopy of Clock Transitions

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by  V. I. Yudin

Synthetic Frequency Protocol in the Ramsey Spectroscopy of Clock Transitions
V. I. Yudin,1,2,3,∗ A. V. Taichenachev,1,2 M. Yu. Basalaev,1,2 and T. Zanon-Willette4
1Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090, Russia
2Institute of Laser Physics SB RAS, pr. Akademika Lavrent’eva 13/3, Novosibirsk, 630090, Russia
3Novosibirsk State Technical University, pr. Karla Marksa 20, Novosibirsk, 630073, Russia
4LERMA, Observatoire de Paris, PSL Research University, CNRS,
Sorbonne Universite´s, UPMC Univ. Paris 06, F-75005, Paris, France
(Dated: April27, 2016)
6
1 We develop an universal method to significantly suppress probe-induced shifts in any types of
0 atomicclocksusingtheRamseyspectroscopy. Ourapproachisbasedonadaptationofthesynthetic
2 frequency concept [V. I. Yudin,et al., Phys. Rev. Lett. 107, 030801 (2011)] (previously developed
for BBR shift suppression) to the Ramsey spectroscopy with the use of interrogations for different
r
p dark time intervals. Universality of the method consists in arbitrariness of the possible Ramsey
A schemes. However,mostextremalresultsareobtainedincombinationwithso-called hyper-Ramsey
spectroscopy[V.I.Yudin,etal.,Phys. Rev. A82,011804(R)(2010)]. Inthelattercase,theprobe-

6 inducedfrequencyshiftscanbesuppressedconsiderablybelowafractionallevelof10−18 practically
2 for any optical atomic clocks, where this shift previously was metrologically significant. The main
advantage of our method in comparison with other radical hyper-Ramsey approaches [R. Hobson,
] et al., Phys. Rev. A 93, 010501(R) (2016); T. Zanon-Willette, et al., Phys. Rev. A 93, 042506
h
(2016)] consists in much greater efficiency and resistibility in thepresence of decoherence.
p
-
m PACSnumbers: 32.70.Jz, 06.30.Ft,32.60.+i,42.62.Fi
o
t
a At the present time, huge progress occurs for high- laser modes.
s. precision optical atomic clocks based on both neutral
c atoms in optical lattices [1–8] and trapped ions [9–12]. Unconventionalsolutiontothisimportantproblemwas
si Exceptional accuracy and stability at the 10−17-10−18 proposed in the paper [28], in which so-called hyper-
y level are achieved. Potential possibilities to achieve the Ramseymethodhasbeendeveloped. Soonthisapproach
h level of 10−19 become clearer for nuclear clocks [13–16] wassuccessfully realizedin[29], wherethe hugesuppres-
p
and for highly charged ions [17–19]. Great fundamental sion(byfourordersofmagnitude)ofprobe-inducedshifts
[
(e.g., in tests of fundamental physical theories such as was experimentally demonstrated (see also [9]). How-

2 QED, QCD, unification theories, cosmology, dark mat- ever, a potential of this method was not going to be set-
v ter searches, etc.) and practical (navigation and infor- tled. In the experimental-theoretical paper [30] a ‘stun-
1 mation systems, gravity-geopotential surveying) impor- ning’ result was recently shown: certain simple modifi-
3
tance of the current and long-range researches is well- cation allows, in principle, totally(!) to exclude probe-
3
known and unquestionable. Current state, concomitant induced shifts. Other hyper-Ramsey modification, hav-
0
0 problems, and future prospects are well presented in the ing the same efficiency, was very soon proposed in the
. review [20]. theoreticalpaper[31]. Becausethesephenomenalresults
2
can have far-reaching consequences for development of
0
On the way to these remarkable achievements, differ- atomic clocks, it requires utterly thorough investigation
6
1 ent barriers arise, which require the development of new of the schemes [30, 31]. Besides, undoubted importance
: unconventional approaches. As an example, for some has a search of new variants to suppress probe-induced
v
of the promising clock systems, one of the key prob- shifts with the near extremal efficiency.
i
X lems is the frequency shift of the clock transition due
r to the excitation pulses themselves. For the case of In this paper, we develop an universal method to dra-
a magnetically induced spectroscopy [21, 22] these shifts matically suppress probe-induced shifts and their fluc-
(quadratic Zeeman and ac-Stark shifts) could ultimately tuations in any type of atomic clocks. Our approach
limit the achievable performance. Moreover, for ultra- is based on adaptation of so-called synthetic frequency
narrow transitions (e.g., electric octupole [23] and two- concept [32] to the Ramsey spectroscopy with the use of
photon transitions [24, 25]) the ac-Stark shift can be so interrogations for different durations of free evaluation
largeinsomecasestoruleouthighaccuracyclockperfor- intervals. We show that this protocol in combination
mance at all. A similar limitation exists for clocks based with the original hyper-Ramsey scheme [28] makes most
on direct frequency comb spectroscopy [26, 27] due to extremal results and is quite stable with respect to the
ac-Stark shifts induced by large numbers of off-resonant decoherence. Moreover, our method leads to the much
better and robust suppression of the shifts in compari-
son with protocols [30, 31], which are more sensitive to
the decoherence and therefore do not show phenomenal
∗Electronicaddress: [email protected] results already.
2
I. GENERAL THEORY δ¯ on the value T can be expressed as the following de-
T
creasing series in terms of powers of 1/T:
The essence of our approach consists in the following.
A A A
Previously in the paper [32] the so-called synthetic fre- δ¯ = 1 + 2 +...+ n +..., (4)
quency method, allowing to radically suppress thermal T T T2 Tn
(BBR) shift in atomic clock, was proposed. However,an
wherethecoefficientsA dependonthepulseparameters
ideology of this method can be easy extended on can- n
(durations, amplitudes, phases, and the value ∆ ).
celling of arbitrary systematic shift. Indeed, let us con- sh
Because the time T is precisely controlled in experi-
(0) (0)
sidertwoclockfrequencies ω andω (differentinthe
1 2 ments, thenwe canset agoalto eliminate the maincon-
general case). Assume that due to a certain physical
tribution∝A /T inEq.(4)usingthesyntheticfrequency
causewehavethestabilizedfrequenciesω andω ,which 1
1 2 protocol. To solve this task we will apply two different
areshiftedrelativetotheunperturbedfrequenciesatthe
dark intervals T and T (but with the same Ramsey
values ∆ and ∆ : 1 2
1 2 pulses), which will give us the corresponding stabilized
frequencies ω and ω . Using Eqs.(2)-(4) we easy find
ω =ω(0)+∆ ; ω =ω(0)+∆ . (1) T1 T2
1 1 1 2 2 2 the synthetic frequency ω(1) and its residual shift δ¯(1):
syn syn
Also assume that the ratio ε =∆ /∆ =const does not
12 1 2
ω −(T /T )ω
fluctuate, while the shifts ∆1,2 can be varied during ex- ω(1) = T1 2 1 T2 ,
periment (i.e., ∆ 6=const). In this case, we can con- syn 1−(T /T )
1,2 2 1
struct the following superposition: δ¯ −(T /T )δ¯
δ¯(1) =ω(1) −ω = T1 2 1 T2 , (5)
syn syn 0 1−(T /T )
ω −ε ω ω(0)−ε ω(0) 2 1
ω = 1 12 2 = 1 12 2 , (2)
syn 1−ε12 1−ε12 where the expression for δ¯(1) does not contain the term
syn
∝A . Fordeterminacy,belowwewillinvestigateindetail
which is insensitive to the perturbations ∆ and their 1
1,2
the particular case of T =T and T =T/2:
fluctuations. This frequency we will call as ‘synthetic 1 2
frequency’. A key advantage of this concept consists in
ω(1) =2ω −ω ,
the following: to construct the shift-free frequency ωsyn syn T T/2
we do not need to know the real values of shifts ∆1,2, δ¯(1) =ω(1) −ω =2δ¯ −δ¯ . (6)
syn syn 0 T T/2
because we need to know only their ratio ε , which can
12
be exactly calculated (or measured) for many cases.
Asitwillbeshownbelow,thevalueδ¯(1) canbelessthan
Letusshowhowtoincorporatethesyntheticfrequency syn
δ¯ [see Eq. (4)] by several orders of magnitude.
protocol in the Ramsey spectroscopy for significant sup- T
Moreover, we can go further to define other synthetic
pression of probe-induced shifts in atomic clocks. The
general idea can be understandable from the reasonings frequency ωs(y2n), for which both contributions A1/T and
related to the two-level system with unperturbed fre- A2/T2 will be simultaneously canceled. Here we need to
quency ω0. It is well-known that the standard Ramsey use three different time intervals (T1, T2, T3) with the
spectroscopy [33] uses two exciting pulses of resonance corresponding stabilized frequencies (ω , ω , ω ). In
T1 T2 T3
field with the frequency ω, which are separated by the particular, we will consider the case of T1=T, T2=T/2
free evolution interval T (dark time) [see in Fig. 1(a)]. and T3=T/3, for which the required superposition takes
Inthis case,the spectroscopicsignalhasa functionalde- the form:
pendence on the detuning δ=ω −ω , which consists of
0
set of narrowresonanceswith width of order ofπ/T (so- ω(2) =3ω −3ω +ω ,
syn T T/2 T/3
calledRamseyfringes). Thecentralfringecanbeusedas δ¯(2) =ω(2) −ω =3δ¯ −3δ¯ +δ¯ . (7)
reference point for stabilisation of frequency ω in atomic syn syn 0 T T/2 T/3
clock.
Consider an influence of probe-induced shift ∆sh, Asitwillbeshownbelow,thevalueδ¯s(y2n) canbelessthan
whicharisesonlyduringtheRamseypulses[seetwo-level δ¯(1) [see Eq. (6)] by several orders of magnitude.
syn
scheme in Fig. 1], while this shift is absent during the Using the same logic, the above procedure can be for-
dark time T. As a result, the stabilized frequency ω mally extended to the n-th order, when we will consider
T
also becomes differing from unperturbed frequency ω0: the synthetic frequency ωs(ynn) and corresponding residual
ω =ω +δ¯ , (3) shift δ¯s(ynn)=ωs(ynn) −ω0. In this case, the frequency ωs(ynn) is
T 0 T a special superposition of (n+1) different stabilized fre-
quencies (ω , ω ,..., ω , ω ) corresponding to the
wherethe index T denotesthe fixedtime ofthe free evo- T1 T2 Tn Tn+1
lution interval under frequency stabilization, and the re- dark time intervals (T , T ,..., T , T ). For δ¯(n) the
1 2 n n+1 syn
sultingshiftδ¯ 6=0existsduetothe ∆ 6=0. Onthe basis contributions(A /T,A /T2,..., A /Tn)[seeEq.(4)]are
T sh 1 2 n
ofgeneralprinciples,itcanbeshownthatthedependence simultaneously canceled.
3
In the general case, the laser frequency during the pulse
doesnothavetobethesameasthefrequencyduringthe
darktime,i.e.,ω 6=ω[34]. Aswewillsee,attimesitcan
p
be useful to approximately offset the induced shift, ∆ ,
sh
by stepping the laser frequency only during the pulses
by a fixed ∆ , i.e., ω = ω + ∆ [see Fig. 1(b)].
step p step
Thus, in the general case the detuning during the pulses
can be written as δ = δ −∆, where ∆ = ∆ −∆
p sh step
is the effective frequency shift during the pulse (instead
of ∆ ). This manipulation allows us to stabilise the
sh
frequency ω under controlled condition |∆/Ω |≪1 (in-
0
dependently of the value ∆ ), which makes it possible
sh
to radically suppress the probe-induced shifts with the
use of hyper-Ramsey spectroscopy [28, 29] even for the
large actual shifts ∆ : |∆ /Ω |>1. Indeed, if the ac-
sh sh 0
FIG. 1: Ramsey pulses with Rabi frequency Ω of different tual level shift ∆ is comparable to or larger than Ω ,
0 sh 0
duration (τ1 and τ2; panel (a)). During the pulses, we step we can always apply a frequency step ∆step (e.g., with
the laser frequency ω by ∆step (panel (b) and text). Hyper- anacousto-opticmodulator)duringexcitationtoachieve
Ramsey scheme with composite second pulse 3τ (panel (c) the condition |∆/Ω |≪1 for an effective shift ∆. ∆
0 step
and text). Also shown is a two-level atom with splitting ω ,
0 canbe evaluatedexperimentallyby variationofthe dark
detuningδ of thelaser with frequencyω duringdark timeT,
period T. If ∆ 6= ∆ , the observed transition fre-
and excitation related shift ∆ duringpulses. step sh
sh quency will be dependent on T [34]. With a control of
the shift to 1% under typical conditions we can achieve
II. COMPUTATIONAL ALGORITHM |∆/Ω0|<0.01 to 0.1.
Formulas (8) and (9) are sufficient for description of
Let us describe a computational algorithm, which al- the signal in Ramsey spectroscopy. For example, if at
lows us to calculate the signal in the Ramsey spec- t = 0 atoms are in the lower level |gi, then after the
troscopy. The action of a single light pulse (with fre- action of two pulses of duration τ and τ separated by
1 2
quency ωp, duration τ, and Rabi frequency Ω0) on two- dark period T (see Fig. 1a) the population n(e) of atoms
0 in the excited state |ei is determined by
level atoms with ground and excited states, |gi=
1
(cid:18) (cid:19)
1
and |ei= (separated by the unperturbed energy 2
0 n(e) = he|W(τ ,Ω ,δ−∆)V(Tδ)W(τ ,Ω ,δ−∆)|gi .
~ω ), is d(cid:18)escri(cid:19)bed by the matrix: 2 0 1 0
0 (cid:12) (1(cid:12)0)
W(τ,Ω ,δ )= (8) This fo(cid:12)(cid:12)rmucla describes Rambsey frcinges as a function(cid:12)(cid:12)of
0 p
variable detuning δ (but with fixed ∆). The presence
cos Ωτ + iδp sin Ωτ iΩ0 sin Ωτ of the additional shift ∆ in the course of the pulse ac-
c 2 Ω 2 Ω 2 ,
(cid:0) iΩΩ(cid:1)0 sin Ω2τ (cid:0) (cid:1) cos Ω2τ − iΩδ(cid:0)p sin(cid:1) Ω2τ ! tion leads to the shift of the central Ramsey fringe with
respect to the unperturbed frequency ω .
0
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
where Ω = Ω2+δ2 is the generalized Rabi frequency.
0 p
The detuninqg during pulse δp = ωp − ω0 − ∆sh con-
tains the excitation related shift ∆ (see Fig. 1, level
sh
scheme) due to the influence of other (far-off-resonant)
transitions. Within the frequency intervalcorresponding
III. SYNTHETIC FREQUENCY PROTOCOL
to the narrowclockresonancethe variationof∆sh onωp FOR HYPER-RAMSEY SPECTROSCOPY
isnegligible,i.e.,∆ canbeconsideredasaconstant(for
sh
fixed |Ω |).
0
During the dark period between the pulses, excitation Before the consideration of the synthetic frequency
related shifts (which produce the total actual shift ∆ ) protocol, note some general points. First of all, we as-
sh
are absent(e.g., the ac-Stark shift from the laser)or can sume that the position of the central fringe ω is de-
T
be turned off (like the Zeeman shift). If during the dark termined by stepping the phase of one of the pulses by
periodT the laserfrequency isω,thenthe freeevolution ±π/2 in the way [35] and equalizing these signals. This
is describedby the matrix V(Tδ) with detuning δ =ω− approach is of greater relevance for clocks, because it
directly generates an error signal with high sensitivity.
ω , where the matrix V(x) is determined as:
0
In respect to the signal (10), this method is formulated
b
eix/2 0 as following. Let us introduce the phase steps φ after
V(x)=b . (9)
0 e−ix/2 dark time T, which can expressedby the use of function
(cid:18) (cid:19)
b
4
n(e)(φ) and the error signal S(err):
R R
n(e)(φ)=
R
2
he|W(τ ,Ω ,δ−∆)V(φ)V(Tδ)W(τ ,Ω ,δ−∆)|gi ,
2 0 1 0
S(cid:12)(cid:12)(cid:12)R(errc) =n(Re)(π/2)−nb(Re)(−bπ/2).c ((cid:12)(cid:12)(cid:12)11)
Then the shift δ¯ of stabilized frequency ω is deter-
T T
mined as solution of the equation S(err) = 0 relative to
R
the unknown δ.
At first, let us consider the standard Ramsey spec-
troscopy, in which both exciting pulses have the equal
duration τ =τ =τ. In the case of Ω τ=π/2 and 2τ≪T,
1 2 0
the dominating contributions have the following linear
dependencies on the small value |∆/Ω |<1:
0
δ¯T ≈ T2 Ω∆; δ¯s(y1n) ≈ π8T 2Tτ Ω∆; δ¯s(y2n) ≈ π428T 2Tτ 2 Ω∆. HFIRG1..2:CTalhcueldateipoennsdaernecidesonoef tfhore sΩh0iτft=sπδ¯T/2, δ¯as(ny1n)d,faonrddδ¯iffs(y2en)refnort
0 0 (cid:18) (cid:19) 0 values4τ/T: (a) 4τ/T=0.25; (b) 4τ/T=0.1.
(12)
Here due to smallness of the ratio (2τ/T)≪1 (i.e., for dominating dependencies on ∆/Ω0:
short Ramsey pulses) we have the chain of inequalities:
|cδ¯os(ly2n)ca|≪n|sδ¯isg(y1nn)i|fi≪ca|δ¯nTtl|y. Tsuhpupsr,etshsethsyensthhiefttsicefvreenqufoerncsytapnrdoatrod- δ¯T ∝(cid:18)Ω∆0(cid:19)3; (14)
Ramsey spectroscopy. ∆ 5 ∆ 7 ∆ 2n+3
δ¯(1) ∝ ; δ¯(2) ∝ ;...; δ¯(n) ∝ .
However,mostextremalresultscanbeobtainedbythe syn Ω syn Ω syn Ω
(cid:18) 0(cid:19) (cid:18) 0(cid:19) (cid:18) 0(cid:19)
useofhyper-Ramseyscheme[28,36]imagedinFig.1(c).
Here the main peculiarity is the composite pulse (with Thus,forsyntheticfrequenciesahigher-order(morethan
total duration 3τ), which consists of sub-pulse 2τ with cubic) nonlinearities appear. Moreover, this character is
inverted phase (−Ω0) and sub-pulse τ with initial phase not changed under variations of Ω0, τ, and T, i.e., we
(Ω0). If for the error signal we apply additional phase absolutely do not need the rigorous condition Ω0τ=π/2.
±π/2-stepsdirectlyafterdarktime(asitwasin[28,29]), This circumstance is a key point to successfully realize
thenthismethodwewilldenoteasHR1[seeinFig.1(c)]. ourmethodinatomicclocks,becauseinrealexperiments
Inthiscase,wedefinethefunctionn(e) (φ)andtheerror the value of Ω0 can be controlled only at the level of 1-
HR1 10%.
signal S(err):
HR1 Forinstance,wepresentformulasunderΩ0τ=π/2and
(4τ/T)<1 [see also Appendix]:
n(e) (φ)=|he|W(τ,Ω ,δ−∆)
HR1 0 4 ∆ 3
W(2τ,−Ω0,δ−∆)V(φ)V(Tδ)W(τ,Ω0,δ−∆)|gi|2, δ¯T ≈ T Ω , (15)
c (cid:18) 0(cid:19)
ScH(eRrr1) =n(HeR)1(π/2)b−n(HebR)1(−πc/2). (13) δ¯(1) ≈ 48 4τ ∆ 5; δ¯(2) ≈ 865 4τ 2 ∆ 7.
syn πT T Ω syn π2T T Ω
(cid:18) 0(cid:19) (cid:18) (cid:19) (cid:18) 0(cid:19)
Then the position of the stabilized frequency ω (i.e.,
T
shift δ¯ ) is determined by the solution of equation These formulas demonstrate that the chain of inequal-
S(err) =T 0 relative to the unknown δ. As it was first ities, |δ¯s(y2n)|≪|δ¯s(y1n)|≪|δ¯T|, can be realized due to the
HR1 controlled smallness |∆/Ω |2≪1, first of all. Besides,
shown in [28], for HR1 the dominating contribution in 0
the shift δ¯ has cubic dependence on the small value the condition (4τ/T)≪1 (i.e., the use of short Ramsey
T pulses) leads to an additional suppression of the shifts.
|∆/Ω |≪1.
0 Thus,relatively smallinitialshift δ¯ and its fluctuations
In the Fig. 2 we demonstrate the calculations for HR1 T
canbedramaticallysuppressed(byseveralordersofmag-
without synthetic frequency [see δ¯ in Fig. 2], as well
T nitude) to the metrologically negligible values at all.
as with the use of synthetic frequency protocol [see δ¯s(y1n) In the resent paper [30], authors have proposed the
and δ¯(2) in Fig. 2]. A huge advantage of the synthetic use of ±π/2 phase steps after inverted (−Ω ) sub-pulse
syn 0
frequency protocol is obvious. Under |∆/Ω |2≪1 our 2τ [see in Fig. 1(c)] to form an error signal for hyper-
0
calculations show the following general character of the Ramsey approach (we will denote this method as HR2).
5
For its theoreticaldescriptionwe introduce the following
function n(e) (φ) and the error signal S(err):
HR2 HR2
n(e) (φ)=|he|W(τ,Ω ,δ−∆)V(φ)
HR2 0
W(2τ,−Ω ,δ−∆)V(Tδ)W(τ,Ω ,δ−∆)|gi|2,
0 0
c b
S(err) =n(e) (π/2)−n(e) (−π/2). (16)
cHR2 HR2 b HRc2
Thentheshiftδ¯ ofstabilizedfrequencyω isdetermined
T T
by the solution of equation S(err) = 0 relative to the
HR2
unknownδ. Inthis case,our theoreticalestimationsgive
uspracticallythesameexpressionsasEqs.(15),butwith
opposite sign for δ¯ and δ¯(2):
T syn
3 FIG.3: Thedependenciesofshiftsδ¯T forMHR[seeEq.(18)]
4 ∆
δ¯ ≈− , (17) under the decoherence (Γ6=0). Calculations are done for
T T (cid:18)Ω0(cid:19) 4τ/T=0.25, for different Rabi frequencies: Ω0τ=π/2 (black
48 4τ ∆ 5 865 4τ 2 ∆ 7 dashed lines); Ω0τ=0.9π/2 (red lines); Ω0τ=1.1π/2 (green
δ¯(1) ≈ , δ¯(2) ≈− , lines), and for different Γ: (a) Γ=0.01π/T; (b) Γ=0.1π/T.
syn πT T (cid:18)Ω0(cid:19) syn π2T (cid:18)T (cid:19) (cid:18)Ω0(cid:19) Upperpictures are obtained for φ=+π/2, and lower pictures
areobtainedforφ=−π/2[seeEq.(18)]. Onecanseethatup-
under Ω τ=π/2 and |∆/Ω |≪1.
0 0 per and lower graphs correspond each other by the inversion
relative to thecentral point (0,0).
IV. COMPARISON BETWEEN DIFFERENT
HYPER-RAMSEY APPROACHES IN THE
PRESENCE OF DECOHERENCE
However, besides HR2 the paper [30] also describes
the theory andsuccessfulexperimentaldemonstrationof
modified hyper-Ramsey (MHR) method, which is based
on combination of both protocols HR1 and HR2. In the
case of MHR, the error signal is determined as following
S(err) =n(e) (φ)−n(e) (φ). (18)
MHR HR1 HR2
Then the shift δ¯ of stabilized frequency ω is deter-
T T
(err)
minedbytheequationS =0relativetotheunknown
MHR
δ. This equation leads to an exceptional result: δ¯T = 0 FIG.4: Thedependenciesofshiftsδ¯T forGHR[seeEq.(19)]
for arbitrary φ, ∆, Ω , τ, and T. An analogous result
0 under the decoherence (Γ6=0). Calculations are done for
takes play also for alternative scheme [so-called general- 4τ/T=0.25, for different Rabi frequencies: Ω τ=π/2 (black
0
ized hyper-Ramsey (GHR)], presented in the theoretical dashed lines); Ω τ=0.9π/2 (red lines); Ω τ=1.1π/2 (green
0 0
paper [31]. Mathematical description of GHR [including lines),andfordifferentΓ: (a)Γ=0.01(π/T);(b)Γ=0.1(π/T).
the errorsignal S(err)]canbe expressedby the formulas: Upper pictures are obtained for φ=3π/4, and lower pictures
GHR are obtained for φ=π/4 [see Eq. (19)].
n(e) (φ)=|he|W(τ,Ω ,δ−∆)V(−φ)
GHR 0
W(2τ,Ω ,δ−∆)V(φ)V(Tδ)W(τ,Ω ,δ−∆)|gi|2; shown below, MHR and GHR methods are unstable rel-
0 0
c b
ative to the decoherence, which leads to an appearance
ScG(eHrrR) =n(GeH)R(φ)b−n(GbeH)R(−φc), (19) of the shift δ¯T6=0. Moreover, this residual shift is very
sensitivetovariationsofRabifrequencyΩ . Atthesame
where the shift δ¯ is determinedby the solutionofequa- 0
T time,ourapproachismuchmorestablerelativetothede-
tion SG(eHrrR) = 0 relative to the unknown δ. As it was coherenceand it can be significantly better and robustly
shown in [31], there is the same result: δ¯T = 0 for arbi- in real experiments than both MHR and GHR.
trary φ, ∆, Ω0, τ, and T. TodescribetheRamseyspectroscopyinthepresenceof
Atfirstglance,bothMHRandGHRaproaches[30,31] decoherence,we willuse the formalismofdensity matrix
are absolutely ideal for the frequency stabilization, be- ρˆ, which has the form
cause they allow us totally to eliminate probe-induced
shifts, i.e., the aboveconcept of synthetic frequency pro- ρˆ(t)= |jiρ (t)hk|; ρ ≡n(g); ρ ≡n(e), (20)
jk gg ee
tocol becomes not so important. However, as it will be
j,k=g,e
X
6
FIG. 5: The dependencies of shifts δ¯T, δ¯s(y1n), and δ¯s(y2n) for FIG. 6: The dependencies of shifts δ¯T, δ¯s(y1n), and δ¯s(y2n) for
HR1underthedecoherence(Γ6=0). Calculations aredonefor HR2underthedecoherence(Γ6=0). Calculationsaredonefor
4τ/T=0.25, for different Rabi frequencies: Ω τ=π/2 (black 4τ/T=0.25, for different Rabi frequencies: Ω τ=π/2 (black
0 0
dashed lines); Ω τ=0.9π/2 (red lines); Ω τ=1.1π/2 (green dashed lines); Ω τ=0.9π/2 (red lines); Ω τ=1.1π/2 (green
0 0 0 0
lines), and for different Γ: (a) Γ=0.01π/T; (b) Γ=0.1π/T. lines), and for different Γ: (a) Γ=0.01π/T; (b) Γ=0.1π/T.
in the basis ofstates |giand |ei. Inour case,the density for ∆=0. In contrast to both MHR and GHR, the syn-
matrix components ρ (t) satisfy the following differen- thetic frequency protocol,combined with originalhyper-
jk
tial equations: Ramsey scheme HR1, shows very good stability relative
[∂ +Γ−iδ˜(t)]ρ =iΩ(t)[n(g)−n(e)]/2; ρ =ρ∗ ; to both the decoherence and variations Ω0 [see δ¯s(y1n) and
t eg ge eg δ¯(2) in Figs. 5(a),(b)]. While the ‘simple’ hyper-Ramsey
∂tn(e) =i[Ω(t)ρge−ρegΩ∗(t)]/2; n(g)+n(e) =1. (21) casynnbe worsethanMHRandGHR[compareδ¯ inFig.5
T
with δ¯ in Figs. 3,4].
HerethetimedependenciesΩ(t)andδ˜(t)aredetermined ThuTs,inthepresenceofdecoherence,thesyntheticfre-
bythefollowing: Ω(t)=Ω0(or−Ω0)andδ˜(t)=δ−∆dur- quency protocol in combination with HR1 [28] can be
ingtheactionofRamseypulses,butΩ(t)=0andδ˜(t)=δ much better (by one-three orders of magnitude) than
during the dark time T. The main difference of above both MHR [30] and GHR [31]. Moreover, our calcula-
equations from the Schr¨odinger equation consists in the tions have shown an uselessness of synthetic frequency
presenceoftherelaxationconstantΓ>0(foroff-diagonal protocolforbothMHRandGHR.Itcanbeexplainedby
matrix elements ρ and ρ ), which describes the deco- thedifferenceofthegeneraldependenceδ¯ onparameter
eg ge T
herence. In particular, such simple model allows us to T in comparison with Eq. (4): for MHR and GHR this
estimate the influence of nonzero spectral width of the dependence contains also constant contribution A ∝ Γ.
0
probe field. To achieve this goal, we can assume the Besides, our calculations show [see Fig. 6] that the syn-
order-of-magnitude agreement between the value Γ and thetic frequency protocol for HR2 is much worse (under
spectralwidthofthe probefield. Similarestimationsare the decoherence) than for HR1. Probably it can be ex-
very important, because even best modern lasers, used plained in the following way: ±π/2 phase steps for HR1
in atomic clocks, have the spectral width at the level are directly conjugated with the dark interval T, while
of 0.1 Hz. Moreover, there are other possible causes of these±π/2stepsforHR2areisolatedfromtheintervalT
decoherence, which are connected with an action of en- by the sub-pulse 2τ (with inverted phase −Ω ) [see HR1
0
vironment, an influence of regimes of traps (or lattices), and HR2 in the Fig. 1(c)].
etc.
Fig. 3 shows that in the presence of decoherence the
MHR leads to the residual shift, which significantly de- A. Additional synthetic frequency protocol for
pends on variations of Rabi frequency Ω . Moreover, GHR
0
under conditionΩ τ6=π/2the methodMHR makes‘par-
0
asitic’shift δ¯ 6=0 evenif ∆=0. The next Fig.4 for GHR IfwewillmoreattentivelylookatFig.4andwillcom-
T
also demonstrates residual shifts and their strong sensi- pare upper and lower graphs,then we can see that these
tivitytothevariationofvalueΩ underthedecoherence. dependencies are practically transformed each to other
0
However,methodGHRdoesnotproduce‘parasitic’shift under reversalof sign. It allows us for GHR to use other
7
optical pumping (POP) clocks [40]. All these examples
demonstrate an universality and efficiency of synthetic
frequency protocol, which can be used in any type of
clocks based on Ramsey spectroscopy.
In addition, the above analysis, taking into account
the decoherence [41], leads us to the question about ex-
istence/absence of some hyper-Ramsey protocol, which
FIG. 7: The dependencies of shifts δ¯s(yGnHR) for synthetic fre- shows zero shift δ¯T=0 for arbitrary Γ, ∆, Ω0, τ, and T.
quency (22) under the decoherence (Γ6=0). Calculations are In the case of the existence, such protocol can be called
donefor4τ/T=0.25,fordifferentRabifrequencies: Ω τ=π/2 as ‘ultimate hyper-Ramsey’.
0
(black dashed lines); Ω τ=0.9π/2 (red lines); Ω τ=1.1π/2 To conclude, the synthetic frequency protocol in the
0 0
(green lines), and for different Γ: (a) Γ=0.01π/T; (b) Ramsey spectroscopy is a novel technique that offers a
Γ=0.1π/T. spectroscopic signal that is virtually free from probe-
induced frequency shifts and their fluctuations. Our
method has broad applications for any types of clocks,
type of synthetic frequency, which is formed as half-sum
especially those based on ultra-narrow transitions, two-
of two frequencies:
photon transitions, lattice clocks based on bosonic iso-
topes with controlled collision shifts [42, 43], CPT-
1
ω(GHR) = ω | +ω | , (22) Ramsey and POP-Ramsey clocks. Moreover, our ap-
syn 2 T φ=3π/4 T φ=π/4
proach opens a prospect for the high-precision opti-
(cid:16) (cid:17)
for φ=3π/4 and φ=π/4 [see Eq. (19)]. In this case, as cal clocks based on direct frequency comb spectroscopy.
seenfromFig.7,theresidualshiftδ¯(GHR) =ω(GHR)−ω High resolution matter-wave sensors [44] are also ex-
syn syn 0
becomes much less than initial shifts δ¯ in Fig. 4. pected to benefit from the suppression of phase shifts
T
in the interference patterns due to the excitation pulses.
The work was supported by the Russian Scientific
V. CONCLUSION Foundation (projectNo. 16-12-00052). M. Yu. Basalaev
was supported by the Russian Foundation for Basic Re-
search (Projects No. 16-32-60050 mol a dk and No. 16-
Inourfiguresweusedimensionlessquantities,because
32-00127mol a).
it allows us to use these calculations for different exper-
imental conditions. In particular, the parameter δ¯T/π
corresponds to the ratio of the shifts (δ¯ , δ¯(1), δ¯(2)) to
T syn syn Appendix
the typical width (FWHM) of Ramsey resonances π/T
(in the s−1 units). For example, if this width is equal
We recalculate, with the formalism developed in [36],
to 10 Hz, then to suppress shifts to the level <0.1 mHz
(i.e., to the fractionallevelless than 10−18 for an optical the approximate synthetic frequency shifts given by
range)we need fulfillment of (δ¯T/π)<10−5. The same is Eqs. (14) and (15) based on the Hyper-Ramsey function
related to the decoherence constant Γ: this value is also n(HeR)1 from Section III [see Eq. (13)]. The generalized
givenon a scale of π/T (see Γ=0.01π/T or Γ=0.1π/T in Hyper-Ramsey phase Φ(δp) is expressed with laser pa-
the captions). For example, the value Γ=0.01π/T corre- rameters from Section II as following:
sponds to the Γ/2π=0.1 Hz for 10 Hz width of Ramsey
resonances. δΩp tan2θ+δΩptanθ + δΩp tanθ+Ωδcc tanθc
In general, our calculations show that for Γ/2π>0.01- Φ(δ )=arctan 1−δp2Ω−2Ω20 tan2θtanθ 1−δΩpΩδcc tanθtanθc ,
0tsii.oo0nn01wofHitthzheHthpRer1osby[2en-8ti]nhpdertuioccdeudfrceseqhsuifmetnsucctyho ptmhrooertfeorcarocoltbiuionnstaclsoulmepvbpeirlneaos--f p 1− 1−δΩpδp2tΩa−n2Ω220θt+anδΩp2θtatnanθθ1δΩ−p tδΩapΩnδccθ+tanΩδccθttaannθθcc
 (23)
10−18-10−19 in comparison with methods MHR [30] and
GHR [31]. Moreover, our approach allows us to achieve including a reduced composite variable as
this level even for (Γ/2π)>1Hz. Apart from the com-
bination with Ramsey and hyper-Ramsey spectroscopy δc tanθ ≡2δp tan2θtanθ , (24)
for two-level systems, we have applied the synthetic fre- Ω c Ω tanθ−tan2θ
c
quency protocol to the Ramsey spectroscopy of the co-
herent population trapping (CPT) resonances (e.g., see where θ =Ωτ/2, Ω2 =δp2+Ω20 and δp =δ−∆.
[37–39]). Note that CPT clock is one of perspective The corrected high-order clock frequency shift δω(Ti)
variants of compact rf clocks with relatively high metro- using different free evolution times T (i = 1,2,3) under
i
logical characteristics. Our calculations also show sig- the condition 4τ ≪Ti takes the explicit form:
nificant suppression of the light shift for CPT-Ramsey
Φ(δ ,δ →0)
csalomckesawpiptrhoathche ucasenobfesyanlstoheatpicpflireedqufeonrcsyo-pcraoltleodcopl.uTlsehde δω(Ti)= Ti+ p∂Φ∂(δδp)|δ→0 . (25)
8
In particular, we consider the case of T = T, T = T/2 These results given by Eqs. (26a)-(26c) are in very good
1 2
andT =T/3,forwhichwecouldhavethethreepossible agreement with approximated synthetic frequency shifts
3
synthetic clock frequencies given by: presented in Eq. (15) and reported in figure Fig. 2 of
Section III when the condition |∆/Ω |2 ≪1 is achieved.
0
δ¯ =δω(T), (26a)
T
δ¯(1) =2δω(T)−δω(T/2), (26b)
syn
δ¯(2) =3δω(T)−3δω(T/2)+δω(T/3). (26c)
syn
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