Phase locking a clock oscillator to a coherent atomic ensemble

1.5 MB·English

by  R. Kohlhaas

APS/123-QED
Phase locking a clock oscillator to a coherent atomic ensemble
R. Kohlhaas1,2, A. Bertoldi3,∗ E. Cantin3,4, A. Aspect1, A. Landragin2, and P. Bouyer1,3
1Laboratoire Charles Fabry, Institut d’Optique, CNRS,
Universit´e Paris-Sud, avenue Augustine Fresnel, F-91127 Palaiseau, France
2LNE-SYRTE, Observatoire de Paris, CNRS and UPMC,
61 avenue de l’Observatoire, F-75014 Paris, France
3Laboratoire Photonique, Num´erique et Nanosciences - LP2N Universit´e
Bordeaux - IOGS - CNRS: UMR 5298, rue Mitterrand, F-33400 Talence, France
4Quantel, 4 rue Louis de Broglie, Building D, F-22300 Lannion, France
(Dated: May 5, 2015)
5 The sensitivity of an atomic interferometer increases when the phase evolution of its quantum
1 superposition state is measured over a longer interrogation interval. In practice, a limit is set by
0
the measurement process, which returns not the phase, but its projection in terms of population
2
differenceontwoenergeticlevels. Thephaseintervaloverwhichtherelationcanbeinvertedisthus

y limited to the interval [−π/2,π/2]; going beyond it introduces an ambiguity in the read out, hence
a a sensitivity loss. Here, we extend the unambiguous interval to probe the phase evolution of an
M atomicensembleusingcoherencepreservingmeasurementsandphasecorrections,anddemonstrate
the phase lock of the clock oscillator to an atomic superposition state. We propose a protocol

4 based on the phase lock to improve atomic clocks limited by local oscillator noise, and foresee the
application to other atomic interferometers such as inertial sensors.

]
h
PACSnumbers: 95.55.Sh,42.62.Eh,67.85.-d,37.25.+k
p
-
m
From the first observations of Huygens on the coordi- to local oscillator noise.
o nated motion of coupled nonlinear oscillators [1], phase In an atomic clock, the frequency of a LO is repeat-
t
a synchronization has evolved to an indispensable tool for edlyreferencedtoanatomictransitionfrequencybycom-
. time and frequency metrology and starts to be inves- paring their respective phase evolutions in an interroga-
s
c tigated for quantum systems [2–4]. Phase lock loops tion time T and applying a feedback correction. During
si (PLLs) [5], where a local oscillator (LO) is phase locked theinterrogation, theatomsareinasuperpositionstate,
y toareferencesignal,arewidelyusedforthegenerationof and the projection of the relative phase between the LO
h atomictimescales[6,7], thesynchronizationintelecom- and the atomic ensemble is measured as a population
p munication[8]orinradionavigation[9]. Inusualatomic imbalance of the two clock levels. The readout is thus
[
frequency standards, however, only the frequency of the a sinusoidal function of the phase drift, and the latter

3 local oscillator is locked on the atomic resonance. This can only be unambiguously determined if it stays within
v featurederivesfromthequantumnatureofthereference the[−π/2,π/2]interval,hereaftercalledinversionregion.
9 system, i.e., the quantum superposition of two internal Hence, for a given LO noise, the interrogation time of
0
statesofanensembleofatoms,moleculesorions,whichis the atomic transition must be kept short enough such
7
destroyedbythedetectionattheendoftheinterrogation thatphasedriftsbeyondtheinversionregionareavoided.
3
0 process. A similar limitation exists for any measurement Currently, LO noise limits the interrogation time in ion
. ofaquantumsystemasinmagnetometersorinertialsen- [11] and optical lattice clocks [12–15] and is expected to
1
sors [10]. Locking the phase of the local oscillator onto become a limit for microwave clocks with the recently
0
the phase of the quantum superposition of the two lev- discovered spin-self rephasing effect [16]. The standard
5
1 els of the quantum system would improve the long term approach to tackle this issue consists in improving the
: stability, as the phase is the integral of the frequency, quality of local oscillators [17–21]. As an alternative, it
v
and reduce the constraints on the stability of the LO hasbeenrecentlyproposedtotrackandstabilizetheLO
i
X or of the measured signal. More in general, phase lock- phase evolution using several atomic ensembles probed
r ing a classical system to a quantum system would give a withincreasinginterrogationtime[22,23],orbyenhanc-
a direct link in metrology to the fundamental oscillations ing the Ramsey interrogation interval by stabilizing the
of quantum particles and could lead to enhanced sensi- LO either via cascaded frequency corrections [24] or by
tivities and new applications in precision measurements. coherence preserving measurements on the same atomic
Here,wedemonstratethedirectphaselockofaLOtoan ensemble and feedback [25, 26].
atomic ensemble, based on repeated coherence preserv- In this letter we show for the first time how the phase
ingmeasurementsoftheatomicensemble. Wealsostudy lock of a classical oscillator to an atomic superposition
howthistechnologycouldimproveatomicclockssubject statecanbeexploitedtokeeptherelativephasebetween
theLOandtheatomicsystemintheinversionregion. In
addition, we demonstrate a protocol based on this phase
lock to operate an atomic clock beyond the limit set by
∗Electronicaddress: [email protected] theLOdecoherence,whichnowadaysrepresentsthebot-
2
tleneck of the best available frequency standards. We 1.) 2.) 3.)
z z z
begin with a minimally destructive measurement of the
LO phase drift when it is within the inversion region;
the measurement readout is then used to correct the LO
phase so as to reduce its drift. The cycle is repeated y y y
x x π x
using the residual atomic coherence. The result is a se- 2 φ
ries of successive, phase-related measurements of the rel- Initialdstate Superposition Dephasing
ative phase evolution, and the feedback keeps the phase state
in the inversion region, leading to an effectively longer 4.) 5.) 6.)
z z z
interrogation time [27]. We demonstrate this approach
with a trapped ensemble of neutral atoms probed on a
microwave transition. π
π 2 y π y y
x 2 x 2 x
ggφφ
Phased Backdrotation Feedback
measurement
FIG. 2: Bloch’s sphere representation of the phase
lock between the LO and the atomic superposition
state. The phase lock between the classical oscillator and
the atomic spin is obtained using repeated, time correlated
Ramseyinterrogationsandfeedback. Thesequencebeginsby
preparingtheatomicCSSinthe|↓(cid:105)stateviaopticalpumping
(step1.). Themeasurementoftherelativephasebetweenthe
FIG. 1: Experimental scheme. The evolution of the LO CSS and the LO starts when a π/2 microwave pulse around
phaseϕLOiscomparedtothephaseϕatofanatomicensemble the y axis brings the CSS into a balanced superposition of
in a superposition state using coherence preserving measure- the |↓(cid:105) and |↑(cid:105) states, depicted as a vector on the equato-
mentsinaRamseyspectroscopysequence. Therelativephase rial plane of the Bloch sphere (step 2.). The x axis is chosen
isobtainedfromthereadoutofthepopulationdifference,and to represent the phase of the local oscillator, and ϕ the rela-
is used to implement the phase lock between the two oscilla- tivephasebetweentheLOandtheatomicsuperpositionthat
tors by applying a feedback correction phase ϕFB on the LO evolvesbecauseoftheLOnoise(step3.). Afteraninterroga-
outputusingaphaseactuator. Thelightshiftinducedbythe tiontimeT,theprojectionofϕismappedontoapopulation
optical trap and by the probe has been engineered to have a difference by a projection π/2 pulse around the x axis and
homogeneous measurement of the atomic ensemble [28]. read out with the coherence preserving detection (step 4.).
TheCSSisrotatedbacktotheequatorialplanebyareintro-
ductionπ/2pulsearoundthexaxis(step5.),andfeedbackis
The experimental scheme shown in Fig. 1 has been
applied on the phase of the LO (step 6.). The PLL between
described in [28]. A cloud of cold 87Rb atoms is trapped
the LO and the atomic ensemble consists in the repetition of
in an optical potential (see Appendix A), prepared with
thestepsfrom3.) to6.),potentiallytilltheatomicensemble
a π/2 pulse of a resonant microwave field in a bal-
shows a residual coherence.
anced superposition state of two hyperfine levels |↓(cid:105) ≡
|F =1,m =0(cid:105)and|↑(cid:105)≡|F =2,m =0(cid:105)oftheelec-
F F
tronic ground state, and probed using a nondestructive
detection. Typically, 5 × 105 atoms at a temperature site π/2 microwave pulse (reintroduction pulse), which
of 10 µK are used in the measurements reported here. brings the atomic state back to the previous coherent
The phase ϕ of the superposition state oscillates at a superposition. Moreover, after each measurement and
at
frequency of 6.835 GHz corresponding to the energy dif- reintroduction pulse, the phase read out can be used
ference between the |↓(cid:105) and |↑(cid:105) atomic states, which is to correct the LO phase. The evolution and manipu-
the fundamental reference if atoms are protected from lation of the atomic ensemble can be illustrated using
perturbations. A microwave LO has a frequency close the Bloch sphere representation (Fig. 2): the collective
to the atomic frequency difference, so that the relative state of N two-level atoms in the same pure single par-
at
phase ϕ = ϕ −ϕ between the two oscillators drifts ticle state (also called coherent spin state (CSS)) forms
LO at
slowly because of the LO noise. ϕ can be measured us- a pseudo-spin with length J = N /2, where J denotes
at z
ing the Ramsey spectroscopy method (see Fig. 2): a the population difference and ϕ = arcsin(J /J ) is the
y x
second π/2 microwave pulse (projection pulse) maps it phase difference between the phase of the LO and that
onto a population difference, which we read out with a of the superposition state. A resonant microwave pulse
weakopticalprobeperturbingtheatomicquantumstate determines the rotation of J around an axis in the equa-
only negligibly and preserving the ensemble coherence torial plane of the Bloch sphere, and the axis direction
[28–30]. Unlike for destructive measurements, the inter- is set by the phase of the microwave signal. The rep-
rogation of ϕ can continue in a correlated way, once the etition of the manipulation, measurement and feedback
action of the projection pulse is inverted using an oppo- cycleimplementsthephaselockoftheLOontheatomic
3
superposition state, as shown in steps 2.-6. of Fig. 2. limit set by the inversion region between J and the rel-
z
We first show that we can reconstruct the time evolu- ative phase. We apply on the local oscillator two types
tionoftherelativephasebetweentheLOandtheCSSby of signals, first a frequency offset, and second periodic
monitoring the population difference and without apply- phase jumps, and use the output of the coherence pre-
ing feedback. For this purpose, we frequency offset the servingmeasurementstoactivelyminimizeϕ. Thephase
LO by 100 Hz from the nominal resonance, and periodi- lock is obtained by controlling the phase of the local os-
cally measure the projection of the relative phase sin(ϕ) cillatorbymeansofadigitalphaseshifter(seeAppendix
withonlyasmallreductionoftheatomicensemblecoher- D). The feedback is performed after the atomic spin is
ence (Fig. 3). Every 1 ms ϕ is mapped to a population rotatedback totheequatorialplaneof theBlochsphere.
difference via a projection π/2 microwave pulse around Whenthedisturbanceappliedonthelocaloscillatorcon-
the x-axis, a weak measurement of J is performed, and sists of a frequency offset, there is a linear phase drift
z
thecollectivespinisrotatedbacktotheequatorialplane between the LO and the atomic phase (Fig 4, red). The
oftheBlochsphereviaareintroductionπ/2pulsearound phase evolution in open loop is reconstructed from the
the x-axis. The π/2 microwave pulses are derived from data of Fig. 3 by taking into account the damping on
anamplifiedversionoftheLOat6.835GHzleadingtoa thesinusoidalsignalduetothedecoherencesources, and
pulse length of τ =47 µs, and the rotation axis is con- knowingthataconstantfrequencyoffsetisappliedonthe
π/2
trolled with a quadrature phase shifter. The coherence LO.Weremarkthatsuddensigninversionsoftheapplied
preserving, dispersive measurement relies on frequency frequency offset when ϕ = ±π/2 would produce exactly
modulationspectroscopy(seeAppendixB).Fig. 3shows thesameevolutionofthepopulationdifference,illustrat-
in a single experimental run how the spin state evolves ing the need to keep ϕ in the inversion region. In closed
aroundtheequatoroftheBlochsphere,withtherelative loop,thephasedriftduetothe100Hzfrequencyoffseton
phaseϕmappedonthenormalizedpopulationdifference the LO is periodically reset to zero, with a precision set
J /J. The signal-to-noise-ratio (SNR) of the weak mea- by the π/32 step size of the digital phase shifter and the
z
surementsis20forafullstatecoherenceandeachreadout uncertainty of the coherence preserving measurements.
of the relative phase drift reduces the state coherence by Asaconsequence,theJz/J signalshowsasaw-tooth-like
2%. The destructivity from the probe is the main deco- evolution (Fig. 4, blue signal). Without phase lock, the
herence source till 10 ms, then the inhomogeneous light phase drift leaves the inversion region after 2.5 ms and
shift of the dipole trap on the clock states becomes the rotates several times around the Bloch sphere, whereas
dominant decoherence source. withphaselockitstaysintheinversionregionforallthe
22msintervalshownintheimage. Whenthefeedbackis
activethetotalphasedriftresultsasthephasemeasured
1
at the end of the Ramsey interferometer, added to the
correction phase shifts on the LO via the feedback con-
0.5 troller. We next apply periodic phase jumps of π/3 back
and forth on the LO using a second phase shifter. The
signal obtained in open loop is shown at the top of Fig.
J / Jz 0 5. When the feedback controller is active, the jumps de-
tected on the relative phase are corrected to zero, with a
-0.5 precisionsetbytheresolutionofthephaseshifterandthe
uncertainty of the weak measurements (Fig. 5, bottom).
ThesolidlinesinFig. 4and5aredrawnfromtheknown
-1 timing for the applied phase signal and the feedback on
0 5 10 15 20
Time (ms) the phase. For a combination of a phase drift and phase
jumps, the relative phase can leave the inversion region
FIG. 3: Coherence preserving measurement of the while the noise action cannot be predicted from previ-
relative phase between the LO and the atomic super- ous measurements. This problem affects atomic clocks
position. Real-timemeasurementofthenormalizedpopula- and highlights the requirement of feedback on the LO
tiondifferencetowhichtherelativephaseϕbetweenthelocal
phase to keep track of the relative phase drifts. Without
oscillator and the atomic superposition state is periodically
feedback, the Ramsey interrogation time should be kept
mappedviamicrowaverotationpulses. Thephaseprecession
sufficiently short to avoid ambiguities for the measured
is induced by setting the LO frequency 100 Hz off the nomi-
phase shift.
nalatomictransitionfrequency. Theexperimentalpointsare
fittedwithasinusoidalevolution,dampedbecauseofthede-
We propose now a protocol to efficiently use thephase
coherence induced by the J measurement and the residual
z locktoimproveanatomicclock. Inaconventionalatomic
differential light shift on the clock transition.
clock, the phase drift ϕ is destructively read out after
a single interrogation and feedback is performed on the
We next introduce feedback and demonstrate that we LO by the addition of a feedback correction frequency
canphaselocktheLOontheatomicsuperpositionstate, ω = −ϕ/T considering unity gain. Our protocol of
FB
and increase the Ramsey interrogation time beyond the using the PLL between the LO and the atomic superpo-
4
phasedriftduringT . Thefinalphasemeasurementϕ ,
tot f
together with the phase corrections ϕ(i), gives a precise
4π FB
estimate of the phase drift during T . The total phase
tot
drift can be computed as ϕ = ϕ −(cid:80)N ϕ(i), where
d) 3π tot f i=1 FB
drift (ra tthhee fseuemdbiascokvceornatlrlotlhleercaonrdrescttoiornedpihnasteheshmifitcsroacpopnltierdollbeyr
ase 2π during the sequence. The feedback on the frequency is
h
P set accordingly to be ω = −ϕ /T . After the final
FB tot tot
π phasereadout,thephaseshifterisresettoitsinitialposi-
tiontoavoidanyimpactoftheintermediatephaseshifts
0 on the long term clock stability. The measurement SNR
0 5 10 15 20 of the total phase drift depends only on the final phase
Time (ms)
measurement, since it can make use of all the residual
ensemble coherence, whereas the SNR of the intermedi-
FIG. 4: Phase lock between the LO and the atomic
ate measurements has to be only sufficient to keep the
superpositionstate. TheevolutionoftheLO-atomrelative
phaseisreconstructedfromtheJ signalinFig. 3(red,solid atomic state in the inversion region. To further improve
z
line) and when feedback is applied on the phase of the LO the protocol, the clock operation could be optimized by
after each measurement (blue, solid line). In the open loop adapting the destructivity of the intermediate measure-
case, the points are reported at the values given by the fit in ments to the varying coherence of the atomic sample, or
thepreviousfigure. Intheclosedloopcase,therelativephase implementing an adaptive measurement protocol for the
is always smaller than π/2, and does not enter in the region last measurement [31].
representedingreywhereitcannotbeunivocallydetermined
For a proof-of-concept demonstration, we run an
from the measurement.
atomic clock that exploits the PLL between the LO and
theatomicsuperpositionstate,andthephasereconstruc-
tion protocol. The LO signal was deteriorated so as to
d)
a π/4 have an increased phase drift over the interrogation in-
drift (r 0 terval(seeAppendixC).Asabenchmark,wefirstrunan
ase -π/4 atomic clock adopting a standard Ramsey interrogation
Ph sequence,andwithoutanyfeedbackonthephase. Inthe
clock operation,the dead time for the preparation of the
d) new ensemble is TD =1.9 s and the interrogation time is
drift (ra π/04 sceothetoreTnc=el1ifemtism,em.uTchheshpohratseermtheaansutrheemmeneatsiusrpeedrfaotrommeidc
e
as -π/4 with the coherence preserving detection adopted for the
h
P phaselock. Thetwo-sampleAllanfrequencystandardde-
0 5 10 15 20 viationwascalculated from thesumoftheLOnoiseand
Time (ms) the correction signal applied by the feedback controller.
The clock instability reaches a τ−1/2 scaling after a few
FIG. 5: Correction of phase jumps between the LO
clock cycles (see Fig. 7, red) at a level consistent with
andtheatomicensemble. EvolutionoftheLO-CSSphase
an initial SNR = 20 and considering the coherence de-
whenperiodicphasejumpsofπ/3areappliedbackandforth
cay in the optical trap. The instability is far higher than
ontheLO:thepointsareobtainedfromthemeasuredpopula-
tiondifference,whereasthesolidlinerepresentstheLOphase with state-of-the art atomic clocks since the experimen-
measuredafterthecorrectionphaseactuator. Thecontroller, tal setup was not explicitly designed for the operation of
implemented using coherence preserving measurements and an atomic clock, and the Allan frequency standard devi-
feedback on a phase actuator, maintains the relative phase ation is 1.5×10−9 at 1 s. We then operate an atomic
close to zero. Above the system is operated in open loop, clockmakinguseofaPLLsequencewithN=9successive
belowinclosedloop. Inthelattercase,thephasecorrections interrogations, and again with T=1 ms, thus increasing
are applied 150 µs after each measurement.
thetotalinterrogationtimeto9ms(seeFig. 6). Forsim-
plicity, theintermediateandthefinalphasereadoutsare
set to have the same measurement strength. The Allan
sitionstateinanatomicclockisbasedonthereconstruc- frequencydeviationshowsaτ−1/2 scalingasexpectedfor
tion of the phase drift experienced by the LO over the atomic clocks, which demonstrates that the phase recon-
extendedinterrogationtimeT =N×T(Nisthenumber struction protocol is working properly. In the opposite
tot
ofphasecoherentinterrogations)bycombiningthephase case,frequencyoffsetswouldbecorrectedonlyatashort
shifts applied by feedback and the final phase readout time with the phase actuator, and the stability of the
(Fig. 6). TheknownphasecorrectionsappliedtotheLO clock would then diverge because of the non-zero dead
phaseservethedualpurposeofkeepingtherelativephase time interval. The comparison of the instability shows
intheinversionregionandgivingacoarseestimateofthe thattheclockadoptingthephaselockandreconstruction
5
T
C
10-9
TD σ(τ)
v
e
n d10-10
a
All
time
T T T T 10-11
1 2 3 N 10

σ / σ (τ)19 2468

0
1 10 100
Averaging time τ[s]
FIG.7: AtomicclockimplementingaPLL.(Top)Allan
FIG.6: ProtocolforatomicclockwithPLL.Operation frequency standard deviation σ for a normal Ramsey clock
1
oftheclock: ateachcycleofdurationTC,therelativephaseis with interrogation time T=1 ms (red) and σ9 for a clock im-
repeatedlymeasuredinacoherencepreservingwayduringthe plementing the phase lock between the LO and the atomic
phaselockinterval(above: shadedgreyareas). Eachinterro- superposition state for 9 successive, correlated interrogations
gation, represented in the inset by a light red peak between on the same atomic ensemble, for a total interrogation time
themanipulationπ/2pulses,isfollowedbyaphasecorrection of 9×T=9 ms (blue). The dead time is in both cases T =
D
ϕ(i) ontheLO,representedintheinsetbyalightbluepeakin 1.9s. Theredandbluelinesarefitstothedata,withaslope
FB
therightafterthereintroductionofthespinontheequatorial set to τ−1/2. The continuous black line lies a factor 9 below
plane of the Bloch sphere. The final phase readout ϕ (dark the red curve, and represents the best achievable level of the
f
redpeakintheinset),whoseSNRissetbytheresidualcoher- phase lock sequence for the same number of interrogations.
ence,togetherwiththepreviouslyappliedphaseshiftsonthe Thedashedblacklineliesafactor3belowtheredcurve,and
LO, provides the total phase drift ϕ experienced during the is the optimum level for 9 consecutive uncorrelated Ramsey
extendedinterrogationintervalT =N×T.Theinterrogation measurementswithdurationTeachandthesametotalcycle
tot
sequence ends with the application of a frequency correction time. (Bottom) The grey triangles represent the ratio of the
on the LO (dark blue peak in the inset), then a new atomic Allandeviationforthetwoclocks; thesolidanddashedlines
ensembleispreparedinthedeadtimeintervalT forthenext correspond to the factors 9 and 3 from the top plot.
D
cycle.
suppressed for 87Rb as reported in [16, 33], whereas in
method is at a lower level by a factor (4.76±0.25) with an optical lattice it is strongly reduced with the choice
respect to the clock implementing the standard Ramsey of light at the magic wavelength [34]. In the original
interrogation, as shown in Fig. 7 (blue). In the optimal proposal to lock the local oscillator phase on the atomic
case, theinstabilitywoulddecreasebyafactor9(Fig. 7, phase[25],frequencyfeedbackonthelocaloscillatorafter
black line) because of the correspondingly longer inter- each weak J measurement is performed. This scheme
z
rogation time. Experimentally, we obtain a lower value leads to a longer effective interrogation time, but to a
because of several detrimental effects, where the main SNR given by the weak measurements, which is lower
contributions are a reduced SNR due to the cumulated than that of a measurement at the quantum projection
destructivity from the probe and the decay from the op- noise and beyond. Our protocol can overcome this limit,
tical dipole trap, and the finite phase shifter accuracy, and reach projection limited readout while keeping an
equal to 38 mrad. The result can be further compared extended interrogation time, thanks to the feedback on
to the case that the 9 phase measurements would have the LO phase.
beenuncorrelated,forexamplebyrepreparingtheatomic Inthephaselocksequence,severaleffectsmustbecon-
state and starting a new Ramsey cycle after each mea- sidered to maximize the SNR of the last measurement
surement [32]. We would expect here a factor 3 (Fig. 7, while maintaining a high accuracy on the total phase
blackdashedline),whichisclearlyexceededbythephase drift over the increased interrogation time: the rotations
lock sequence. operated on the Bloch sphere must be fast, the measure-
The phase lock can be performed as long as the co- ment induced decoherence limited, and the phase shifter
herence of the state is maintained. For integration times used for the correction accurate. The decoherence re-
longer than the coherence lifetime of the trapped ensem- lated to the repeated interrogations of the relative phase
ble, the atomic phase is lost and our locking scheme be- canbestronglyreducedbytheuseofanopticalcavityto
comes again a frequency lock, like when the quantum enhance the probe interaction with the atomic ensemble
superposition is destroyed by the detection. In our ex- [35–37]. An optimized clock configuration would con-
periment, the coherence lifetime is limited to 20 ms by sist in a two atomic ensembles using the same LO: the
the dephasing in the optical dipole trap. Nevertheless, firstensembleprovidestheinformationtoimplementthe
trapped induced dephasing of the atomic state can be phase feedback algorithm on the LO; the resulting cor-
6
rectedphasefortheLOstaysintheinversionregionfora superposition,andwereproduceditonaclassicalreplica
much longer period, and this prestabilized LO is used to byintroducingfeedbacktoimplementaPLL.Phaselock-
interrogatethemasterensemblewiththestandardRam- ing a classical oscillator to an atomic superposition state
sey sequence. This scheme avoids the requirement of a canbeakeytechnologytoimprovethesensitivityofsys-
trade-off between the number of intermediate measure- temswheretheatomicphaseevolutionisusedtoimprove
ments and the SNR of the final measurement by sepa- thequalityoftheclassicalcounterpart,suchasinatomic
rating the two problems. It also removes the systemat- interferometers or frequency combs [44], whenever the
ics imposed by the intermediate coherent manipulations main decoherence source is determined by the classical
andmeasurementsoftheatomicstate,whichnowaffects subsystem. This development may open new directions
only the first ensemble. The solution promises the same andpossibilitiesinseveraltechnologicalfieldsandaswell
benefits foreseen for the phase reconstruction schemes in basic science.
proposed in [22, 23], but using only a single additional
ensemble.
InanatomicclockthephaselockbetweentheLOand Acknowledgments
theatomicsuperpositionstatecanreducetheDickeffect
[38],i.e. thealiasingoftheclockoscillator,thankstothe We acknowledge funding from DGA, CNES, EMRP
longer interrogation time. However, the most important (JRP-EXL01 QESOCAS), the European Union (EU)
advantage of the scheme is the reduction of the decoher- (iSENSE), ANR (MINIATOM), LAPHIA (APLL-
ence related to the local oscillator, which translates to a CLOCK, within ANR-10-IDEX-03-02) and ESF Euro-
lower white noise frequency for a fixed detection noise. QUAM.TheEMRPisjointlyfundedbytheEMRPpar-
The noise reduction can be exploited to lower the LO ticipating countries within EURAMET and the Euro-
stabilityrequirementstothebenefitofotherparameters, pean Union. LCFIO and SYRTE are members of the
like portability of the experimental setup, for mobile or Institut Francilien de Recherche sur les Atomes Froids
spatial applications, or viceversa, to remove the limita- (IFRAF). We thank T. Vanderbruggen for his contribu-
tion set by the LO to reach ultimate performances. tion during the initial phase of the experiment, as well
For example, the interrogation interval in the best as M. Prevedelli, G. Santarelli and T. Udem for useful
available optical clocks [12–15] is limited to (cid:39) 1 s by discussions. P.B. acknowledges support from a Chair of
the quality of the local oscillator; this limitation is not ExcellenceofR´egionAquitaine,E.C.fromQuantel. Any
fundamental, andourmethodcouldincreasetheinterro- mention of commercial products does not constitute an
gationintervalandhencethesensitivityinawayatbest endorsement.
proportional to the number of coherence preserving in-
terrogations. LOs with higher coherence will reduce the
number of intermediate operations required to obtain a Appendix A: Atomic sample preparation
given clock interrogation interval. The latter will then
be limited by other effects: practically, a first limit is set 87Rb atoms laser cooled with a magneto-optical trap
by the vacuum quality, which can reduce the ensemble aretransferredtoanopticaldipoletrapat1560nm,that
coherence via background collisions, but extremely long uses a 4 mirror optical resonator to enhance the laser in-
trappinglifetimeshavebeenalreadyreachedfor trapped tensity [45]. The atoms are trapped at the crossing of
atoms [39]. Ultimately, with the combination of exist- two cavity arms with a waist of 100 µm. The ensem-
ing techniques and the method presented in this paper, ble is evaporatively cooled by decreasing the intensity of
the1–10mHzexcitedstateslifetimeexpectedforalkaline thedipoletraptillatemperatureof10µKisreachedfor
earth-like atoms could be reached, which motivates the 5×105 atomsinacloudwith1/e2 radiusof50µm. Inthe
search for transitions with a lower linewidth [40–42]. last operation before starting the Ramsey interrogation
Phase locking the LO to the atomic state preserves theatomsareopticallypumpedinthe|↓(cid:105)state. These-
classical correlations in time against the decoherence quencetopreparetheensembleintheinitialstate,which
by the local oscillator. The technique and the related corresponds to the dead time T in the atomic clock se-
D
enhancement factor could thus be combined with spin quence, lasts 1.9 s.
squeezing, which improves the clock sensitivity by intro-
ducing quantum correlations between the particles to go
below the standard quantum limit [37, 43]. More gener- Appendix B: Non-destructive dispersive probe
ally, increasing the interrogation time using minimally
destructive measurements and feedback on the phase The measurement of J is based on the dispersion
z
could be applied to other atomic interferometers, such caused by the trapped atoms on a far off-resonance opti-
as atomic inertial sensors, where for example the phase calprobe. Theprobebeamhasawaistof47µmmatched
of the interrogation lasers could be locked to the phase tothesizeoftheatomiccloud. Itisphasemodulatedata
evolution of matter waves. frequencyof3.853GHzandfrequencyreferencedat3.377
In conclusion, using a coherence preserving detection GHz on the red of the the F=1→F’=2 transition; these
we tracked the phase evolution of an atomic collective conditions produce a symmetric mixing of the |↓(cid:105) and
7
|↑(cid:105) states because of probe induced spontaneous emis- 73C-C+,Minicircuits). Theelectronicintegrationofthis
sion,thusavoidingaverticaloffsetwhentheBlochsphere signal during the 60 ns interrogation pulse produces a
contracts. In this way, each sideband mainly probes the voltage proportional to the average atomic population
populationofoneofthetwolevels,withthesamemagni- differenceduringtheprobing. Suchavoltageisdigitized
tude and opposite sign for the couplings. We cancel the by a 14 bit resolution analog-to-digital converter embed-
probe induced light shift and the related decoherence by ded in the microcontroller unit (MCU) used to control
preciselycompensatingtheeffectofthecarrierwiththat the feedback loop (ADuC841, Analog Devices). In the
of the sidebands; the cancellation is obtained by setting experiments implementing the phase lock between the
a modulation depth of 14.8% for the phase modulation. LO and the atomic superposition the MCU controls the
Thedifferentiallight-shiftontheD2linefromthedipole phaseactuator,a6bitsstepphaseshifterwitharangeof
trap at 1560 nm was compensated with light blue de- 2π (RFPSHT0204N6, RF-Lambda). The total delay for
tuned from the 52P → 42D transitions at 1529 the feedback is approximately 150 µs depending on the
3/2 5/2,3/2
nm. Whenthetotalpoweroftheprobeissetto480µW, calculation time of the MCU. When running the clock
itcausesthedecayoftheatomiccoherencewithalifetime based on the PLL technique, the MCU acts as well on
of 2.85 µs; in the experiment here reported the interro- a frequency actuator, which is the frequency modulation
gation pulses, obtained using an amplitude electro-optic input on the 165 MHz synthesizer.
modulator,havebeensettolast60ns. Eachpulsedeter- The feedback controller we propose for the clock ex-
minesthena2%destructivityoftheensemblecoherence, ploiting the phase lock between the LO and the atomic
and a SNR of 20 for the J measurement on the initial superposition state consists of a cycle divided in three
z
sampleof5×105 atoms. Thepopulationimbalanceread- main steps: in the first one, successive correlated inter-
outs have been normalized to the signal when all atoms rogations with probe interval T are realized on the same
were repumped to the state F=2. coherentatomicensemble,andfeedbackontheLOphase
is applied. The control law is
ϕ(i) = ϕ(i−1)+ϕ(i)
Appendix C: Frequency chain LO LO FB
= ϕ(i−1)+g ϕ(i)
LO ϕ
The 6.835 GHz frequency used to coherently manip-
ulate the atomic spin is generated by a frequency chain where ϕ(i) is the estimated phase difference between the
based on a Spectra Dynamics DLR-100 system as a fre- LO and the atoms at the i-th cycle and ϕ(i) is the phase
LO
quency reference. The DLR-100 relies on an ultra-low of the LO only. The values of ϕ(i) =g ϕ(i) are saved in
FB ϕ
noise 100 MHz quartz, locked at low frequency to the the feedback controller and typically a gain g = −1 is
ϕ
10th harmonic of a frequency doubled 5 MHz quartz to chosen.
further improve the phase noise. The 100 MHz signal is The second step consists in the final phase readout:
multiplied to 7 GHz and then mixed with a tunable syn- at the end of the N-th interrogation interval of duration
thesizer at 165 MHz to obtain the signal resonant with T, a destructive measurement ϕ of the phase is per-
f
the transition between the |↓(cid:105) and |↑(cid:105) state. The noise formed, with the highest possible precision. The saved
addedbythemicrowavepulsestothemeasurementofJz phase shifts on the LO and ϕf are used to reconstruct
isnegligibleforourweakprobe,anditscontrolwellbelow thefullphasedriftbetweentheLOandtheatomsinthe
the atomic quantum projection noise level has already total interrogation time, equal to T =N×T.
tot
been shown in [46]. Frequency noise is added to the LO In the third step, one then performs feedback on the
signal using a frequency modulation port on the synthe- frequency as in a conventional atomic clock
sizer, with a conversion factor set to 200 Hz/V . The
rms
noise signal for the demonstration of the clock using the ω(n) = ω(n−1)+ω(n)
LO LO FB
PLLsequenceisgeneratedwithasignalgenerator,which = ω(n−1)+g ϕ(n)/T
produceswhitefrequencynoisewithaspectraldensityof LO ω tot tot
2.7×10−2 Hz2/Hz; this signal is low pass filtered at 1.85 where
kHz before being added to the LO. The result is a rms
phase drift of 430 mrad over 10 ms for the LO. ϕ(tnot) = ϕ(fn)− (cid:88) ϕF(iB)
i∈n−thcycle
= ϕ(n)− (cid:88) g ϕ(i)
f ϕ
Appendix D: Feedback controller
i∈n−thcycle
and n is the clock cycle. In addition, the phase shift set
The atomic populations on the |↓(cid:105) and |↑(cid:105) states
by the feedback controller on the LO is reset to zero
determine a differential phase shift of the probe side-
bands. As a consequence, the probe beam is modu- ϕ(n) =ϕ(0) =0.
lated in amplitude, and the modulation signal is de- LO LO
tected by a photodiode (1591NF, New Focus), amplified The cycle then repeats. The important feature of the
(two HMC716LP3E, Hittite) and demodulated (ZX05- feedbackcontrolleristhatthefeedbackactionsontheLO
8
oscillator phase during the interrogation time are saved. the precise final measurement at the end. As already re-
They are then used with the output of the final precise marked, in the experimental demonstration of the clock
measurement to determine the total phase drift. There operation, we adopted the same probe for the intermedi-
is no drawback from the uncertainty of the weak mea- ate and the final measurements.
surements, since any feedback errors are detected with
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