Spectrally-resolved quantum interferometry for absolute and high-precision determination of optical properties

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by  Florian Kaiser

Spectrally-resolved quantum interferometry
for absolute and high-precision determination of optical properties
Florian Kaiser1,∗ Panagiotis Vergyris1, Djeylan Aktas1,
Charles Babin1,2, Laurent Labonté1, and Sébastien Tanzilli1
1Université Côte d’Azur, CNRS, Laboratoire de Physique de la Matière Condensée, France
2École Normale Supérieure de Lyon, 46 Allée d’Italie, 69364 Lyon Cedex 07, France
Quantum optical metrology exploits superpositions of N-photon number states, referred to as
N00N-states,forphasesensingwithN timesimprovedsensitivitycomparedtoclassicalapproaches
involving N photons. Here, we introduce the concept of spectrally-resolved quantum white-light
interferometry based on energy-time entangled two-photon N00N-states as a tool for optical prop-
erty measurements. We apply this method to measure chromatic dispersion in an optical fibre and
demonstrate that employing such quantum states of light leads not only to better precision, but
7 alsotoabsoluteparameterdetermination. Notably,bycomparingthequantummethodtostate-of-
1 the-art classical realizations, we show a 2.6 times better precision, despite involving 60 times less
0 photonsonaverage. Thisunderlinesthattheimprovedresultsareessentiallyduetoconceptualad-
2 vantagesenabledbyquantumopticswhicharelikelytodefinenewstandardsinavarietyofapplied
research fields.
n
a
J
INTRODUCTION eters. Additionally, bothdatatreatmentandexperimen-
6
talproceduresaregreatlysimplified,makingthismethod

] Optical phase-sensing readytobeexploitedasanenablingtoolinalargevariety
h of fields.
p
Optical phase-sensing stands as one of the most ad-
-
t vancedtechniquesinclassicalmetrologyandhasrecently
n Standard white-light interferometry
a led to the direct detection of gravitational waves [1].
u Fromtheappliedside,phase-sensitivespectrally-resolved
q As shown in FIG. 1(a), the emission of a white-light
white-light interferometry (WLI) is a key enabling tool
[ source is usually directed to an interferometer in which
for improving optical technologies finding numerous ap-
the reference arm is free-space (with well known opti-
1 plications in physics, medicine, and biology. In the per-
v spectives of both research and development, WLI has cal properties) and the other arm comprises the sample
1 under test (SUT). Recombining both arms at the out-
been exploited for high-precision measurements such as
2 put beam-splitter leads to an interference pattern for
distance and displacement [2], strain, temperature and
6
which the intensity follows I ∝ 1 + cos(φ(λ)), with
1 pressure [3], surface profilometry [4], refractometry [5],
φ(λ)= 2π (L −n(λ)·L ). Here, λ represents the wave-
0 group delay dispersion [6], as well as chromatic disper- λ r s
1. sion [7–13]. length,LrandLsarethephysicallengthsofthereference
armandtheSUT,respectively,andn(λ)istherefractive
0 The optimal precision for measuring an unknown phase
index of the SUT. It is worth noting that interference is
7 φinanopticalinterferometerusingclassicallightisruled
√
1 onlyobservedwhentheinterferometerisbalancedwithin
by the standard quantum limit, δφ∼1/ N, where N is
: the coherence length of the white-light source, or the co-
v the average number of photons involved in the measure-
herence length imposed by the resolution of the spec-
i ment. The precision can be further improved by using
X
trometer, which is typically on the order of microns to
quantum light, e.g. so-called N00N-states which repre-
r millimetres [22, 23]. In this case, the phase term reads
a sent a coherent supersposition of having N photons in
(more details are given in the supplementary informa-
one optical mode with zero in an orthogonal mode, and
tion):
vice versa. This permits reaching the Heisenberg limit,
δφ ∼ 1/N [14–21]. However, due to experimental im- (cid:32)
1 d2n (∆λ)2
perfections, such as optical losses, noise and non-unity φ(λ +∆λ)≈2πL ·
0 s 2 dλ2 λ +∆λ
detector efficiencies, most of today’s real-world measure- 0
(cid:33)
ments operate far away from these fundamental limits. 1 d3n (∆λ)3
+ · +φ . (1)
Weintroduceanddemonstrateherespectrally-resolved 6 dλ3 λ +∆λ off
0
quantum white-light interferometry (Q-WLI) as a novel
approach for absolute and high-precision measurements Here,λ standsforthesymmetrycenterofthefringepat-
0
ofopticalproperties. Thepeculiaruseofenergy-timeen- tern,usuallyreferredtoasstationaryphasepoint[22,23],
tangled two-photon N00N-states permits achieving sig- ∆λ is the wavelength offset compared to λ , and φ is
0 off
nificant conceptual advantages compared to standard a constant offset phase. Fitting experimentally acquired
WLI, allowing absolute determination of optical param- data as a function of ∆λ allows inferring L , and/or the
s
2
sues. The quantum white-light source is composed of a
continuous-wave pump laser and a non-linear crystal in
which energy-time entangled photon pairs are generated
through spontaneous parametric downconversion [25].
This process obeys the conservation of the energy, i.e.
1 = 1 + 1 . Here, λ represents the wavelength of
λp λ1 λ2 p
the pump laser photons, and λ are the wavelengths
1,2
of the individual photons for each generated pair. An-
otherimplicationoftheconservationoftheenergyisthat
the degenerate wavelength of the emission spectrum is
λ =2λ .
0 p
The paired photons are sent to the interferometer which
is, intentionally, strongly unbalanced. This allows dis-
tinguishing contributions for which both photons take
opposite paths (delayed arrival times at the interferom-
eter’s outputs) or the same path (no arrival time differ-
ence)[26]. Atoneinterferometeroutput,asinglephoton
detector (SPD) is employed, and in the other a single-
photon sensitive spectrometer. Time tagging electronics
FIG. 1. Typical experimental setups for standard spectrally-
allowstoinferthephotonarrivaltimedifferences,andto
resolved WLI (a), and Q-WLI (b). BS, beam-splitter; SUT,
post-select only zero-time-delay events. This procedure
sample under test; SPD, single photon detector. &-symbol,
time-tagging and coincidence logic. leads to the formation of a two-photon NOON-state:
(cid:0)|2(cid:105)r|0(cid:105)s+eiφN00N|0(cid:105)r|2(cid:105)s(cid:1)
|ψ(cid:105)= √ . (2)
opticalmaterialparameters d2n and d3n. Assumingthat 2
dλ2 dλ3
Ls is precisely known, a typical fit to the data requires Here, the ket vectors, indexed by s and r, indicate the
three free parameters, i.e. λ0, dd2λn2 and dd3λn3, which are numberofphotonsinthereferenceandSUTarm,respec-
usually all interdependent in a non-trivial fashion. Con- tively, and φ = φ(λ )+φ(λ ). As detailed in the
N00N 1 2
sequently, uncertainties on one parameter induce errors supplementary information, after setting λ = λ +∆λ,
2 0
ontheothers. Infact,thehighnumberofrequiredfitting thespectraldependenceofthephasetermφ isgiven
N00N
parameters and the necessity to re-equilibrate the inter- by:
ferometer for every new SUT stand as the main issues of
d2n πL ·(∆λ)2
thistechnique,thereforelimitingitsprecisionandpoten- φ (λ +∆λ)≈ · s +φ , (3)
tial for absolute determination of optical properties, as N00N 0 dλ2 λ0 +∆λ off
2
well as its ease-of-use [12, 24].
However, more accurate optical measurements are ea- inwhichφoff = 4π(n(λ0λ)0Ls−Lr) isanoffsetterm. Inorder
toinferφ ,atwo-photoncoincidencemeasurementis
gerly demanded in almost all fields where optics is in- N00N
performed,takingadvantageofthefactthattherateRat
volved. A special focus is set on the material parameter
d2n, as it is directly related to the chromatic dispersion which pairs are projected onto the N00N-state is phase-
dλ2 dependent: R∝1+cos(φ ). Thetermφ hasbeen
coefficientD =−λ0·d2n,inwhichcrepresentsthespeed N00N off
c dλ2 object of research in the past [27–29], as it allows mea-
oflight[6–13]. MoreaccuratemeasurementsonD would
suring optical phase-shifts at constant wavelengths with
have tremendous repercussions for optimizing today’s
doubled sensitivity compared to the standard approach.
telecommunication networks, developing new-generation
Bytakingadvantageofenergy-timeentanglement,wead-
pulsed lasers and amplifiers, designing novel linear and
dress here, for the first time, the wavelength-dependent
nonlinear optical components and circuits, as well as for
terminequation3. Experimentally,thisismadepossible
assessing the properties of biological tissues.
by recording R as a function of ∆λ, i.e. the two-photon
coincidence rate is measured as a function of the paired-
photons’ wavelengths.
MATERIALS AND METHODS We now highlight a few pertinent purely quantum-
enabled features provided by equation 3. Compared to
Quantum white-light interferometry equation1,thedependenceonthethird-orderterm d3n is
dλ3
cancelledduetoenergy-timeentanglement[30]. Further-
FIG. 1(b) shows the experimental schematic for our more, λ does not have to be extracted from the data,
0
new concept of spectrally-resolved quantum WLI (Q- as it is exactly twice the wavelength of the continuous-
WLI), intended to overcome the above-mentioned is- wave pump laser, λ , and can therefore be known with
p
3
extremely high accuracy. This means that the quantum RESULTS AND DISCUSSION
strategy allows data fitting using exactly one free pa-
rameter, namely d2n which is an essential step towards Statistical analysis for comparing measurement
dλ2
absolute optical property determination with high preci- accuracy
sion. Additionally,wenotethattheN00N-state’sability
to interfere does not arise from the coherence length of Typical interference patterns for chromatic disper-
theindividualphotons,butfromthatofthephotonpair. sion measurements using both methods are shown in
For energy-time entanglement, this coherence length is FIG. 2(a,b). With the Q-WLI setup, twice as much
directly related to that of the continuous wave pump interference fringes are obtained for the same spectral
laser (typically from several to hundreds of meters) [25]. bandwidth which is a direct consequence of the doubled
This allows to operate the interferometer in largely un- phase sensitivity of the two-photon N00N-state. After
balanced conditions, such that no re-alignment is neces-
sary when changing the SUT. Finally, due to the use of
a two-photon N00N-state, a doubled sensitivity on d2n
dλ2
is achieved, allowing to perform measurements on much
shortersamplescomparedtostandardWLI(downtothe
mm to cm scale).
Detailed optical setup and data acquisition
AsaSUTweusea1mlongstandardsingle-modefibre
from Corning (SMF28e). For all measurements, the in-
terferometer is actively stabilized using a reference laser
and a piezoelectric transducer on one of the mirrors in
thereferencearm(moredetailsareprovidedinthemeth-
ods section). This ensures that φ remains constant.
off
For chromatic dispersion measurements using standard
WLI,asuperluminescentdiodeisused. Attheoutputof
theinterferometerwemeasureanaveragespectralinten-
sityof∼125pW/nmfrom1450−1650nm. Interferograms
arerecordedusingastandardspectrometerfromAnritsu
(modelMS9710B)with0.1sintegrationtimeand0.5nm
FIG. 2. Typical measurements when inferring chromatic dis-
resolution.
persion in a 1m long standard single-mode fibre using stan-
For the Q-WLI approach, the employed light source is dard WLI (a), and Q-WLI (b). Red dots, data points; Blue
made of a wavelength-stabilized 780.24nm laser, pump- lines, appropriate fits to the data from which D is extracted.
ingatype-0periodically-poledlithiumniobatewaveguide Error bars assume poissonian photon number statistics.
(PPLN/W). The quasi-phase matching in the PPLN/W
ischosensuchasto generateenergy-timeentangledpho- acquiring 2×100 measurements on the same SUT, we
ton pairs around the degenerate wavelength of λ = infertheprecisionofbothapproaches. Theresultsofthe
0
1560.48nm with a bandwidth of about 140nm. Here, statistical data analysis are shown in FIG. 3. For stan-
the spectral intensity is about 25fW/nm at the interfer- dard WLI, we obtain, on average, D = 17.047 ps at
nm·km
ometer output. To detect the paired photons, we use λ ≈ 1560.5nm with a standard deviation of σ =
0 classical
an InGaAs SPD (IDQ 220) at one interferometer out- 0.051 ps . This result is amongst the best reported
nm·km
put. Thesinglephotonspectrometerattheotheroutput to date in the literature [7–13]. For Q-WLI, we mea-
is made of a wavelength-tunable 0.5nm bandpass filter, sure, on average, D = 17.035 ps at λ = 1560.48nm
nm·km 0
followed by another InGaAs SPD (IDQ 230). To par- withasignificantlybetterstandarddeviationofσ =
N00N
tially compensate for the significantly reduced spectral 0.021 ps .
nm·km
intensityofthephotonpairsource,weuseanintegration Further statistical data analysis shows Kurtosis parame-
time of 8s. ters of 0.182 and 0.276 for standard and quantum WLI,
All measurements are repeated 100 times on the same respectively, proving that both data sets converge to-
SUT in order to infer the statistical accuracy of both wards a normal distribution. Comparing the data sets
WLI and Q-WLI approaches. with Levene’s null hypothesis test gives p < 5 · 10−4,
4
providingverystrongevidencethattheimprovedresults evaluatedandsubtractedfromthedatainordertoavoid
obtained with the quantum strategy are essentially im- systematic errors. For standard WLI, this implies that
possibletohaveoccurredbasedonrandomsamplingfrom the SUT has to be removed and that the length of the
the classical measurements. reference arm has to be reduced accordingly (typically
In our data, we observe a minimal offset of the central on the order of 1m). This procedure is technically chal-
values which is explained by either a slight wavelength lenging, time-consuming, and might lead to additional
offset of the spectrometer (< 0.2nm) used for the stan- systematic errors.
dard measurements, and/or that the interferometer was Atthispoint,Q-WLIshowsitsabilityforuser-friendly
unbalancedbyabout1.5µm. Theconsequenceisanerror operation. Interference is observed for interferometer
on the fitting parameter λ , which is then translated to path length differences up to the coherence length of the
0
an error in d2n [22, 23]. At this point, we highlight that pump laser, in our case ∼250m. Consequently, resid-
dλ2
for the Q-WLI approach, λ is essentially known with ual chromatic dispersion can be measured directly after
0
absolute accuracy, and only one free fitting parameter removing the SUT, without any realignment. FIG. 4
is required. Additionally, an unbalanced interferometer
doesnotinfluencethemeasurement. Itisthereforenatu-
raltoconsiderthatQ-WLIcanbeusedtodeterminethe
chromatic dispersion coefficient with absolute accuracy.
We also emphasize that the measurements performed
with Q-WLI involve ∼60 times less photons transmit-
ted through the SUT compared to standard WLI. This
clearly underlines that the improved results obtained
with the entanglement-enhanced strategy is due to as-
pects from both the fundamental and conceptual sides.
FIG.4. ExperimentalresultswhenusingQ-WLIforinferring
residual chromatic dispersion in our interferometer without
theSUT.Reddots,datapoints;Bluelines,appropriatefitto
the data.
shows the experimental results that we have obtained
when measuring chromatic dispersion in our bare inter-
ferometer, i.e. with the SUT. It turns out that in our
interferometer, residualchromaticdispersionamountsto
∼10% of the measured values on the 1m SUT. For all
the data discussed above, except for the raw data in
FIG. 2(a,b), we have therefore subtracted the residual
chromatic dispersion.
CONCLUSIONS
We have introduced and demonstrated the concept of
FIG. 3. Histogram of inferred chromatic dispersion coeffi-
spectrally-resolved Q-WLI exploiting energy-time entan-
cientsafter100repetitionsonthesameSUTusingbothstan-
gled two-photon N00N-states. Compared to standard
dard(blue)andentanglement-enhanced(red)measurements,
respectively. Fits to the data assume a normal distribtion. measurements, the N00N-state permits achieving a two
times higher phase sensitivity. More strikingly, the pe-
culiar use of such quantum states of light reduces the
number of free parameters for fitting experimental data
from three to one, representing a major advantage for
Device calibration using Q-WLI determining optical properties with high precision and
absolute accuracy. In addition, our setup does not re-
AnotheradvantageprovidedbyQ-WLIliesinstraight- quire a balanced interferometer for performing the mea-
forward device calibration. All the optical components surement which represents a significant time-saving ad-
in the interferometer actually show small residual chro- vantage compared to standard WLI, especially for de-
matic dispersion, and this undesired offset needs to be vice calibration and when considering measurements on
5
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ACKNOWLEDGEMENTS
The authors acknowledge financial support from the
Foundation Simone & Cino Del Duca, the Euro-
7
METHODS by dividing the data by the sum of both reference spec-
trograms.
Mach-Zehnder interferometer stabilization For the entanglement-enabled strategy, we normalize
the coincidence counts by taking advantage of the un-
desired contributions in which the paired photons take
Without active interferometer phase stabilization, we
oppositepathsinsidetheinterferometer. Thesecontribu-
observe 2π phase drifts every few seconds due tempera-
tions do not interfere at the interferometer output, such
ture drifts in the laboratory. This limits severely the in-
thattherelated(non-zerotimedelay)coincidencerateis
tegration times for both the classical and quantum mea-
directlyproportionaltothespectralintensityofthepho-
surements. Therefore, we employ an active phase sta-
tonpairgenerator. Normalizationisobtainedbydividing
bilization system. It is made of an actively wavelength-
theN00N-statecoincidencesbytwotimesthesumofthe
stabilized 1560.5nm reference laser sent in the counter-
non-N00N-state coincidences.
propagatingwaythroughtheinterferometer,andapiezo-
electric translation stage in the reference arm of the in-
terferometer [37]. The feedback loop has a bandwidth
Home-made single-photon spectrometer
of 100Hz which results in a long-term phase stability of
< 2π rad.
40 As a single-photon spectrometer, we use a wavelength
tunablemotorizedbandpassfilter(YenistaXTM-50)fol-
Spatial and polarization mode overlap lowed by a low noise single-photon avalanche photodi-
ode (id quantique id230) operated at 25% detection ef-
ficiency. The transmission loss of the filter is measured
In order to obtain high-visibility interference patterns
to be 4dB such that the total quantum efficiency of the
attheinterferometeroutput,thephoton(orphotonpair)
single-photon spectrometer is ∼10%.
contributions from both interferometer arms need to be
made indistinguishable in both the spatial and polariza-
tion modes. Spatial mode overlap is ensured by using
a fibre-optic beam-splitter at the interferometer output
and input [38]. Polarization mode overlap is obtained
usingfibre-opticpolarizationcontrollersinbothinterfer-
ometer arms. These components are not shown in the
main text figures in order to simplify the reading of the
manuscript.
Quality of the entangled photon pair source
We infer the entanglement quality of our photon pair
source in the following configuration. We fix the analy-
sis wavelength of the spectrometer at 1550nm and post-
selectthedesiredN00N-statebyacoincidencemeasure-
ment. Then, the path length difference of the MZI is
scannedandthetwo-photoncoincidencerateisrecorded.
We measure sinusoidal oscillations with a raw fringe vis-
ibilityof87.1±2.2%whichincreasesto95.5±2.6%after
thesubtractionofdetectors’darkcounts. Inotherwords,
we obtain a fidelity of 97.8% to the desired N00N-state.
We explain imperfections by unbalanced losses between
thetwoarmsoftheinterferometerandmulti-paircontri-
butions.
Normalization of intensity and coincidence
spectrograms
Fortheclassicalstrategy,normalizationisobtainedby
recording two reference spectrograms with either inter-
ferometer arm being blocked. Normalization is obtained
8
SUPPLEMENTARY INFORMATION and experiment (see FIG. 5) which leads to both an off-
set and a larger standard deviation, i.e. D = 17.070±
Derivation of the fitting function for standard WLI 0.054 ps .
nm·km
As outlined in the manuscript, the wavelength depen-
dent phase shift at the interferometer output is
2π
φ(λ)= (L −n(λ)·L ). (4)
λ r s
Weapproximatenown(λ)byathirdorderTaylorseries:
n(λ)=n(λ +∆λ)≈n(λ )+(cid:80)3 1 dkn ·(∆λ)k.
0 0 k=1 k! dλk
This leads to
(cid:32)
n(λ ) dn ∆λ
φ(λ +∆λ)≈2πL 0 + ·
0 s λ +∆λ dλ λ +∆λ
0 0
(cid:33)
1 d2n (∆λ)2 1 d3n (∆λ)3
+ · + · FIG.5. FittingstandardWLIdata(reddots)withafunction
2 dλ2 λ +∆λ 6 dλ3 λ +∆λ
0 0 taking into accound terms up to d2n (green line) and d2n
dλ2 dλ2
2πL (blue line). Only the later shows a perfect overlap with the
− r . (5)
experimental data.
λ +∆λ
0
Notethat,ingeneral,theinterferencefringesobtainedat
the interferometer output are usually too closely spaced
to be resolved by a commercial spectrometer because
of the strong phase-dependence on zero and first or- Derivation of the fitting function for Q-WLI
der derivatives. In order to cancel these terms, the in-
terferometer has to be precisely equilibrated to the so- For the Q-WLI, the phase term is given by the two-
called stationary phase point (SPP), which is found at photon phase φ = φ(λ )+φ(λ ), which can be cal-
N00N 1 2
L =(cid:0)n(λ )− dn ·λ (cid:1)L . Notethatthispointhastobe culated using equation 4. Respecting the conservation
r 0 dλ 0 s
foundindividuallyforeachnewsamplewithanaccuracy of the energy, i.e. 1 = 2 = 1 + 1 , and setting
on the order of a few micron. After finding the SPP, the λ =λ +∆λ leads tλop λ0 λ1 λ2
2 0
dominant term is d2n and the phase term simplifies to (cid:32)
dλ2 1d2n (∆λ)2
(cid:32)1 d2n (∆λ)2 φN00N(λ0+∆λ)≈2πLs· 2dλ2 · λ0 +∆λ
φ(λ +∆λ)≈2πL · 2
0 s 2 dλ2 λ +∆λ (cid:33)
0 1d3n (∆λ)4
1 d3n (∆λ)3 (cid:33) +6dλ3 · (cid:0)λ0 +∆λ(cid:1)2 +φoff,(7)
+ · +φ , (6) 2
6 dλ3 λ +∆λ off
0 where we consider the phase offset φ = 4π(n(λ0)Ls−Lr)
off λ0
in which φ =2πL dn is a constant phase offset. to be constant thanks to the active phase stabilization
off s dλ system. Letusstressthefourimportantfeaturesinequa-
AssumingthatL isknownprecisely, therequiredfitting
s
parameters are therefore d2n, d3n and λ . tion 7 compared to equation 6.
dλ2 dλ3 0 First, zero and first order derivatives are automatically
cancelled, and therefore no SPP has to be found.
Second, for ∆λ (cid:28) λ , a twofold enhanced sensitivity on
Data fitting up to d2n and d3n 0
dλ2 dλ3 the second order dispersion term is obtained, which is a
typical signature of N00N-states.
Data obtained with standard WLI require a fitting
Third, the third order dispersion term is strongly sup-
function taking into account terms up to d3n to obtain pressed by a factor of about 2∆λ, which means that it
the most precise and accurate results. dλ3 can be generally neglected, excλep0t for really exotic sam-
Fitting the data with the function described in equa- ples with extraordinarily high third order dispersion.
tion 6 leads to D =17.047nmp·skm at λ0 ≈1560.5nm and Fourth, λ0 =2λp is essentially known with arbitrary ac-
σclassical = 0.051nmp·skm after 100 measurements on the curacy such that is has not to be considered as a fitting
same standard single-mode fibre. parameter.
A fitting function taking into account only terms up Therefore,Q-WLIrequiresonlyonefreeparametertofit
to d2n is does not lead to a good overlap between data the data, i.e. d2n.
dλ2 dλ2