Anomalous Quantum Correlations of Squeezed Light

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by  B. Kühn

Anomalous quantum correlations of squeezed light
B. Ku¨hn and W. Vogel
Arbeitsgruppe Theoretische Quantenoptik, Institut fu¨r Physik, Universita¨t Rostock, D-18051 Rostock, Germany
M. Mraz, S. K¨ohnke, and B. Hage
Arbeitsgruppe Experimentelle Quantenoptik, Institut fu¨r Physik, Universita¨t Rostock, D-18051 Rostock, Germany
(Dated: February 1, 2017)
Threedifferentnoisemomentsoffieldstrength,intensityandtheircorrelationsaresimultaneously
measured. For this purpose a homodyne cross correlation measurement [W. Vogel, Phys. Rev. A
51,4160(1995)]isimplementedbysuperimposingthesignalfieldandaweaklocaloscillatoronan
unbalanced beam splitter. The relevant information is obtained via the intensity noise correlation
of the output modes. Detection details like quantum efficiencies or uncorrelated dark noise are
meaningless for our technique. Yet unknown insight in the quantumness of a squeezed signal field
7
is retrieved from the anomalous moment, correlatingfield strength with intensity noise. A classical
1
inequality including this moment is violated for almost all signal phases. Precognition on quantum
0
theory is superfluous, as our analysis is solely based on classical physics.
2
PACS numbers: 03.65.Ta, 42.50.Dv, 42.50.Lc
n
a
J
I. INTRODUCTION nonclassicality of the signal field. The other technique
1
in [11] was called homodyne cross correlation measure-
3
ment (HCCM): signal and LO are interfered at a single
To distinguish nonclassical effects of light from classi-
] unbalanced beam splitter and the two output fields are
h cal ones and to conceive possible applications has been a
recorded with linear detectors. Unlike in BHD, a corre-
p centralquestionofquantumopticsforseveraldecades. It
lation measurement is performed. The detector currents
- isoffundamentalinterestiftheoutcomeofanopticalex-
t are multiplied and not subtracted, which yields second-
n perimentcanbeinterpretedintheframeworkofclassical
order intensity noise correlations. An experimental real-
a statistical electrodynamics, or if a quantum description
u ization of this method has been missing yet.
isnecessary. Apossiblewaytocertifynonclassicaleffects
q In this contribution we report the first experimental
isbasedonmoments,as,e.g.,quadraturesqueezing[1,2]
[
implementation of HCCM. Our signal field is prepared
orsub-Poissonstatistics[3], eachisbasedonasingleob-

1 servable quantity. in a phase-squeezed coherent state, generated via para-
v metric down-conversion. For the intensity regime we use
In contrast, anomalous moments composed of non-
5 for signal and LO standard linear photodiodes are suit-
6 commutingobservablesarehardtoaccessinexperiments. able. The contributions of different orders of the LO
9 An importantexample is the correlationof intensity and
field strength are extracted from the measured correla-
8 field strength noise, as it unifies the particle and wave
tion function. Our method certifies anomalous quantum
0 natureof quantumlight. Itsmeasurementwasoriginally
. correlations of squeezed light even for most of the anti-
1 proposed by a homodyne correlation technique with a
squeezed phase region.
0 weaklocaloscillator(LO)[4]. Anomalousmomentswere
7 detected in resonance fluorescence of a single trapped
1 atom [5]. In this setting, balanced homodyne detection
:
v (BHD)withaweakLOwasconditionedonthedetection II. HOMODYNE CROSS CORRELATION
i of a resonance fluorescence photon. Conditional homo- MEASUREMENT
X
dyne detection was also studied by simulations [6] and
ar experiments [7], which allows to observe large violations The basic setup of our measurement technique is il-
of a Schwarz inequality; see also Ref. [8]. However, this lustrated in Fig. 1. The investigated squeezed field
approach only applies to a Gaussian or weak source field was generated in a hemilithic, standing wave, nonlin-
anditrequiresthreedetectors. Higher-ordercorrelations ear cavity, used as optical parametric amplifier (OPA).
of multiple field modes are accessible by balanced or un- An 11mm long 7% magnesium oxide-doped lithium nio-
balanced homodyne correlation measurements [9, 10]. bate (7%MgO:LiNbO ) crystal served as χ(2)-nonlinear
3
In Ref. [11], two detection schemes have been theo- mediumwithnon-criticalphasematching. Astrongseed
reticallyanalyzed, whichusefour-porthomodyningwith beamwasinsertedintotheOPAtoproduceacoherently
comparableintensitiesofsignalandLO.Oneofthetech- displaced squeezed field with a signal power of 284µW.
niques, called homodyne intensity correlation measure- The OPA was pumped with 243mW at 532nm result-
ment,wasintroducedin[4]. Itwasrealizedonlyrecently ing in a gain of 2.3 with -2.7dB squeezing and 5.5dB
tocertifyquadraturesqueezinginresonancefluorescence anti-squeezing at 1064nm. The flip mirrors F1 and F2
light from a single quantum dot [12]. Negative values of allow us to send the signal field and the LO either to
the measured intensity noise correlation directly uncover the BHD or the HCCM device. For the HCCM the LO
2
power is of the magnitude of the signal power. Both the detectors k = 1,2. The intensity noise correlation
fields are combined on an unbalanced beam splitter and can be separated into three contributions with different
the two output beams are recorded with photodetectors powers of the LO field strength E ,
L
(PDs). InordertovalidateourHCCMtechnique,weim-
plemented an additional BHD setup. The latter is used ∆G(2,2)(φ)=∆G(2,2)+∆G(2,2)(φ)+∆G(2,2)(φ).(3)
0 1 2
to characterize the generated state with an established
Defining coefficients T by (T ,T ,T ) = (1,|R|/|T| −
technique. ThereisonlyonedifferencetoanormalBHD i 0 1 2
|T|/|R|,−1) and T = |T|2|R|2, the zeroth-order (in E )
device, an ND-filter is placed in the signal beam in front L
term is given by
of the 50:50 beam splitter to reduce the intensity of the
signal to 32µW. This avoids demolition of the PDs, as
∆G(2,2) =TT (cid:104)(∆I)2(cid:105) (4)
the LO power can be reduced to 1.03mW. Due to the 0 0
knowledge of the power reduction in the signal field, we
with the signal intensity I =E(−)E(+) and the intensity
are able to estimate the squeezing of the undamped sig- φ φ
noise ∆I =I−(cid:104)I(cid:105). It is independent of both phase and
nal. The visibility in the BHD setup is 97% and the
field strength of the LO. The first-order term,
quantumefficiency∼90%. IntheHCCMsetupthevisi-
bilityis96%andthequantumefficienciesofthePDsare
∆G(2,2)(φ)=TT E (cid:104)∆E ∆I(cid:105), (5)
∼ 94%. In both detection setups we used the technique 1 1 L φ
of continuous variation of the optical phase as presented
withthesignal(electric)fieldstrengthE =E(+)+E(−)
in [13]. This provides a uniformly distributed phase. φ φ φ
and the corresponding fluctuation ∆E =E −(cid:104)E (cid:105), in
φ φ φ
general is 2π-periodic in the phase and linear in the field
Pump
strength of the LO. Note that this anomalous moment is
Laser composed of two observables. A Fourier decomposition
OPA of the second-order term,
F1 ND BHD
Signal ∆G(2,2)(φ)=TT E2(cid:104)(∆E )2(cid:105), (6)
50:50 2 2 L φ
LO F2
which is quadratic in the LO field strength, is in general
PBS λ
2 composed of a π-periodic and a constant Fourier compo-
nent in the LO phase. The different dependences of the
14:86
terms (4)–(6) on the phase and field strength of the LO
HCCM allow us to separate them from ∆G(2,2)(φ), for details
see [11] and the discussion below.
Additionalcontributionsin(3)arisefromclassicalfluc-
tuations of the LO, which though very small in our case
are evaluated as follows. The dominant effect is a con-
FIG.1. (Coloronline)Experimentalsetupforthegeneration
stant offset, obtained from a correlation measurement
and detection of squeezed light. The squeezed field is gener-
with blocked signal. This yields a direct observation of
atedinanOPA.TheflipmirrorsF1andF2areusedtosend
the intensity fluctuation of the LO, including possibly
the squeezed field either to the BHD or the HCCM device.
occurring correlated dark noise in the two detectors. To
correct for LO and correlated dark noise, this offset is
The measurement outcome of the HCCM is the cor-
removed from the correlation C(φ) measured in the case
relation of electric current fluctuations (ac) of the two
with unblocked signal. A strong point of the technique
detectors. The ac-time-sequences, [c(1)···c(Nφ)] and
1 1 φ is that even if uncorrelated dark noise in both detectors
[c(1)···c(Nφ)] , measured for a particular LO-phase φ, were stronger than the quantum noise of the signal, it
2 2 φ
are same-time correlated, i.e., does not contaminate the measurement result. By con-
trast, uncorrelated dark noise is relevant in BHD.
C(φ)=c (φ)c (φ)= 1 (cid:88)Nφ c((cid:96))(φ)c((cid:96))(φ). (1) Note that the expressions (4)–(6) are also correct for
1 2 N 1 2 a lossy beam splitter, i.e., |T|2 + |R|2 < 1. The the-
φ
(cid:96)=1
ory of Ref. [11] can also be extended to an asymmetric
For the intensities present in our experiment, the detec- beam splitter; see, e.g., [14]. In this case, the intensity
tors respond linearly. Therefore, the quantity (1) is pro- reflection-transmission ratio of the beam splitter for the
portional to the intensity noise correlation ∆G(2,2)(φ)= LO (|R |2 : |T |2) and for the signal (|R |2 : |T |2)
L L S S
(cid:104)∆I1∆I2(cid:105)φ, i.e., can be different. This yields the more general coeffi-
cients (T ,T ,T )=((|R |/|R |)(|T |/|T |),|R |/|T |−
C(φ)=ζ ζ ∆G(2,2)(φ), (2) 0 1 2 S L S L S L
1 2 |T |/|R |,−1) and T = |T ||T ||R ||R |. We may as-
L S S L S L
where (cid:104)·(cid:105) is the classical expectation value and ζk is sume symmetric transmittance, i.e. |TS|2 =|TL|2.
the product of detector parameters such as detector effi- IftheLOisstrongcomparedwiththesignal, theterm
ciency, gain factor, and other positive scaling factors of ∆G(2,2)(φ) is dominant, and the correlation outcome is
2
3
proportional to the negative squeezing effect. Accord- of 4.58×105 data samples was used. For details on the
ingly, in this scenario the anomalous moment negligibly fit via regression analysis [15, 16] and the error calcula-
contributes to the total correlation and it is, therefore, tionseeSec.AoftheAppendix. Weobserveanexcellent
not accessible. Even if the LO intensity is comparable agreement of the experimental outcome with the theo-
to the signal intensity, the anomalous moment is only retical prediction. For the LO-blocked case we obtain
accessible if the beam splitter is unbalanced [11]. The C = 0.80153±0.00014 using 3×108 data samples.
block
maximum visibility is reached for a 14:86 intensity par- In addition, the extracted contributions C , C , and C
0 1 2
tition, which is approximately used in our experiment. are shown. One clearly observes the 2π-periodic anoma-
lous moment of intensity-field noise. Once a calibration
ofthesetupisperformed,i.e.,ζ andζ inEq.(2)andthe
1 2
III. SEPARATION OF MOMENTS LO strength are known, the moments can be quantified.
Let us study the separation of the contributions It is important to note that our method is quite sen-
sitive to drifts of the signal state, since one has to en-
Ck(φ)=ζ1ζ2∆G(k2,2)(φ) (7) sure that approximately the same signal state is present
in the LO-blocked and unblocked case. We incorporate
from the total correlation C(φ) by using a Fourier de-
a drift error of C as the difference of the result of
composition, block
twosubsequentmeasurements. Notethatdrifterrorscan
2
(cid:88) be further reduced by increasing the frequency of block-
C(φ)=a + [a cos(kφ)+b sin(kφ)], (8)
0 k k ing/unblocking the LO.
k=1
withrealparametersa andb ,asproposedin[11]. Since
k k
bothC (φ)andC (φ)containaphase-independentpart,
0 2
itisnecessarytoperforminadditionameasurementwith CΦ CΦ(cid:43)Π C
0
blocked LO, which yields the resulting correlation out-
come Cblock. The contributions Ck(φ) are obtained from C(cid:72)1 (cid:76) C2 (cid:72) (cid:76)
the latter and the Fourier coefficients as
2.0
C (φ)=C (9)
0 block
C1(φ)=a1cos(φ)+b1sin(φ) (10) 1.5
C (φ)=a cos(2φ)+b sin(2φ)+a −C . (11)
2 2 2 0 block
1.0
CΦ C C C
0 1 2 0.5
1.5
(cid:72) (cid:76)
0.0
1.0
(cid:45)0.5
0 0.5 1 1.5
0.5
(cid:142)
E
L
0.0
FIG. 3. (Color online) Measured correlation C(φ) (filled
markers) and C(φ+π) (unfilled markers) for φ = 3π/4 as a
(cid:45)0.5
functionoftherescaledLOfieldstrengthE˜ . Theerrorbars
L
of one standard deviation statistical uncertainty are within
(cid:45)1.0
the size of the markers. The quadratic fits are the thin solid
0 Π 2Π
anddashedcurves,composedofthecontributionsC ,C ,and
0 1
Φ C .
2
FIG. 2. (Color online) Measured correlation C(φ) (markers)
asafunctionofphase. Theerrorbarsofonestandarddevia-
tionstatisticaluncertaintyarewithinthesizeofthemarkers. Alternatively, the contributions Ck may be separated
ThefitaccordingtoEq.(8)isshownbythethinsolidcurve, bythedependenceontheLOfieldstrength,seeSec.Bof
composed of the contributions C , C , and C . the Appendix. In our experiment five different LO pow-
0 1 2
ersnamely0µW(blockedLO),117µW,166µW,216µW,
Figure 2 shows the measured correlation C(φ) for 120 and 275µW were probed for the phases φ = 3π/4 and
phases selected equidistantly in [0,2π] and the fit ac- φ+π. The result is shown in Fig. 3 together with the
cording to Eq. (8). For each phase the same number contributions C proportional to Ek.
k L
4
IV. CLASSICAL CORRELATIONS 0.01
Classical
Inaclassicalpictureaninequalitycanbederivedbased 0.00
on the extracted moments, which is always fulfilled. For
an arbitrary function f of Eφ(±), the expectation value Φ (cid:45)0.01(cid:76)(cid:68) Squeezing
(cid:104)|f|2(cid:105) is nonnegative. For our experimental outcome we L (cid:72)
(cid:64)
use a properly chosen function of the form f = h0∆I + et (cid:45)0.02
h ∆E and h ,h ∈C. Defining the matrix d
1 φ 0 1
(cid:18) (cid:104)(∆I)2(cid:105) (cid:104)∆E ∆I(cid:105)(cid:19) (cid:45)0.03
M(φ)= φ , (12)
(cid:104)∆E ∆I(cid:105) (cid:104)(∆E )2(cid:105)
φ φ
(cid:45)0.04
thedeterminantofM(φ)foraclassicallycorrelatedsignal 0 Π Π
2
field is non-negative for all phases φ ∈ [0,2π). This is
Φ
equivalent to the inequality
(cid:104)∆E ∆I(cid:105)2 ≤(cid:104)(∆I)2(cid:105)(cid:104)(∆E )2(cid:105). (13) FIG. 4. (Color online) The solid line shows det[L(φ)] as a
φ φ
functionofphaseasobtainedthroughseparationbydifferent
If the beam splitter transmittance and reflectance ratios phase periodicity. Due to the π-periodicity of the plot, we
are known, one can determine the matrix confine ourselves to the interval [0,π]. The thin dashed lines
correspond to an error of one standard deviation. The thick
(cid:18) (cid:19)
C (φ)/T C (φ)/T dashed line marks the border between the classical and non-
L(φ)= 0 0 1 1 (14)
C (φ)/T C (φ)/T classicalregions. Squeezingispresentwithinthelight-colored
1 1 2 2
interval. The marker at φ=3π/4 (anti-squeezed region) fol-
from the contributions C (φ) of C(φ). The determinant lowsfromtheseparationbytheLOfieldstrength, fromdata
k
of this matrix is related to the determinant of M(φ) as measured for φ and φ+π. The error corresponds to one
standard deviation.
det[L(φ)]=ζ2ζ2T2E2det[M(φ)]. (15)
1 2 L
Obviously, the sign of det[L(φ)] equals that of
Our method is especially beneficial, when the phase
det[M(φ)]. Thus,thenecessarycondition(13)foraclas-
intervalofsqueezingissmall,e.g.,forstrongsqueezingor
sicallycorrelatedsignalfieldcanbetesteddirectlybythe
phase diffused states. Then it is challenging to stabilize
matrixL(φ)throughdet[L(φ)]≥0. Notethatnoknowl-
the system onto the squeezed phase. In this regard, our
edge of the efficiencies and gain factors incorporated in
method may detect quantum effects under demanding
thedetectionprocessisrequired. Alsotheexactstrength
squeezing conditions of the input state. Note that the
of the (weak) LO is meaningless, cf. Eq. (15). The ratios
positivecorrelationoutcomeC forblockedLOshows
|R |2 :|T |2 and|R |2 :|T |2 havetobeknown,butnot block
L L S S that the necessary classicality condition (cid:104)(∆I)2(cid:105) ≥ 0 for
thereflectanceandtransmittanceitself,whichmakesthe
the variance of the signal intensity is valid.
test robust to beam splitter losses.
It is important that the whole previous analysis is
purely classical and does not require any bosonic com-
V. QUANTUM CORRELATIONS mutation relations [17–20]. This essential property has
the benefit, that the derived classicality condition based
on anomalous correlations applies without assumptions
Figure 4 shows the experimental result for det[L(φ)]
on the validity of quantum physics for the interpretation
as a function of the LO phase. The determinant is sig-
ofthemeasurementoutcome. Bycontrast,thesqueezing
nificantly negative in a wide range of phases φ, which is
condition(cid:104)(∆Eˆ )2(cid:105)<(cid:104)(∆Eˆ )2(cid:105) foraparticularphase
a clear violation of the classicality condition (13). Re- φ φ vac
φ, which is applied in balanced homodyne detection, in-
markably, the determinant is even negative for phases
trinsically utilizes non-vanishing commutators. Hence,
where no squeezing is present, e.g., for φ=3π/4 with 28
such quantumness tests require the postulate of the va-
standard deviations significance. Hence the anomalous
lidity of quantum physics. This consideration is closely
quantum correlations under study also exist in the anti-
related to the definition of nonclassicality in the sense of
squeezed phase region. For comparison, the determinant
TitulaerandGlauber[21],whichisbasedontheGlauber-
obtainedbyseparationthroughtheLOfieldstrengthde-
SudarshanP function[22,23]. Thatis,astateisnonclas-
pendence is shown for φ=3π/4. Since the LO intensity
sical if it violates a condition (cid:104): fˆ†fˆ:(cid:105) ≥ 0, wherein : · :
is not scanned continuously in our case, the drift of the
denotesnormal-ordering, classicalexpectationvaluesare
signal state yields a larger uncertainty than the separa-
replaced by the quantum mechanical ones, and classical
tion by phase. Nevertheless, this proof-of-principle ex-
periment certifies nonclassicality with a significance of field quantities f by the corresponding field operators fˆ.
4.7 standard deviations. With some technical effort, this It is eminent that our HCCM device accesses, based
technique could also be further improved. on quantum measurement theory [4, 11], three pairwise
5
non-commuting observables, : (∆Iˆ)2 :, : (∆Eˆ )2 :, and significance. Furthermore, a quantum correlation test
φ
: ∆Eˆ Iˆ :, within a single measurement scenario. The based on solely the measured moments shows the exis-
φ
anomalous correlations violating the classicality condi- tence of anomalous quantum correlations even outside
tion(13),cf. Fig.4,turnouttobeinexcellentagreement thesqueezedphaseregion. Asacentralbenefit, thedata
with the condition for anomalous quantum correlations, analysis of our technique is completely free of quantum
physical assumptions, such as non-vanishing commuta-
(cid:104):∆Eˆ ∆Iˆ:(cid:105)2 >(cid:104):(∆Iˆ)2 :(cid:105)(cid:104):(∆Eˆ )2 :(cid:105), (16) tion relations. Hence the technique visualizes directly
φ φ
violations of classical physics. The anomalous quantum
of the normal-ordered fluctuations of intensity and field correlations of squeezed light, which have been verified
strength. For the derivation of general criteria for quan- here for the first time, may pave the way for alternative
tum correlations of light, we refer to [24]. applicationsofsqueezedlightinquantumtechnology,be-
yond the phase interval of squeezing.
VI. CONCLUSIONS
ACKNOWLEDGMENTS
In conclusion, we have experimentally realized the ho-
modyne cross correlation measurement to observe up TheauthorsaregratefultoOskarSchlettweinforvalu-
to fourth-order moments of the field fluctuations of a able discussions. This work has been supported by
phase-squeezed coherent state. In particular, this allows the European Commission through the project QCUM-
us to determine the anomalous moment, which is com- bER (project reference: 665148) and by the Deutsche
posed of two non-commuting observables, namely inten- ForschungsgemeinschaftthroughSFB652(projectsB12,
sityandfieldstrengthnoise, whichisobservedwithhigh B13).
APPENDIX
Appendix A: Separation of moments by LO phase
In this section the separation of moments using the different dependence on the LO phase is studied. Consider
K points (φ ,C(φ )), j = 1,...,K for different phases φ , which are obtained independently from N data points
j j j φj
according to Eq. (1) of the main text. The standard error of the mean of C(φ ) is
j
(cid:118)
u =(cid:117)(cid:117)(cid:116)N(cid:88)φj [c(1(cid:96))(φj)c(2(cid:96))(φj)−C(φj)]2. (A1)
C(φj) N (N −1)
(cid:96)=1 φj φj
The statistical uncertainty of C is obtained analogously by
block
(cid:118)
u =(cid:117)(cid:117)(cid:116)N(cid:88)block [c1((cid:96),b)lockc(2(cid:96),b)lock−Cblock]2, (A2)
Cblock N (N −1)
block block
(cid:96)=1
where N is the number of data samples recorded for the LO-blocked scenario. This uncertainty is increased by
block
the error arising from a drift of the signal state. The measured values are fitted by
2
(cid:88)
C(φ)=a + [a cos(kφ)+b sin(kφ)] (A3)
0 k k
k=1
which corresponds to the minimization of the Euclidean norm (cid:107)Ua − C(cid:107) with respect to the vector of real fit
2
parameters a=(a ,a ,b ,a ,b )T. Here
0 1 1 2 2
 
1 cos(φ ) sin(φ ) cos(2φ ) sin(2φ )
1 1 1 1
1 cos(φ2) sin(φ2) cos(2φ2) sin(2φ2)
U =. . . . .  (A4)
. . . . . 
. . . . . 
1 cos(φ ) sin(φ ) cos(2φ ) sin(2φ )
K K K K
is a K×5 dimensional matrix and C =(C(φ ),··· ,C(φ ))T. The solution of the problem is given by
1 K
a=(UTU)−1UTC (A5)
6
(see e.g. [15, 16]) and the statistical uncertainty
(cid:118)
(cid:117) K
(cid:117)(cid:88)
(ua)i =(cid:116) [((UTU)−1UT)ij(uC)j]2 (A6)
j=1
withi=1,...,5followsfromerrorpropagation. FurthererroranalysisyieldsforthecomponentsC (φ)(cf.Eqs.(9)–
k
(11) of the main text) of C(φ) the uncertainties
u =u (A7)
C0(φ) Cblock
(cid:113)
u = u2 cos2(φ)+u2 sin2(φ) (A8)
C1(φ) a1 b1
(cid:113)
u = u2 cos2(2φ)+u2 sin2(2φ)+u2 +u2 . (A9)
C2(φ) a2 b2 a0 Cblock
The error of the quantities
C (φ)
B (φ)= k , (A10)
k T
k
which are required for the matrix L(φ) (cf. Eq. (14) of the main text), is given by
1 (cid:113)
u = [B (φ)]2u2 +u2 (A11)
Bk(φ) |Tk| k Tk Ck(φ)
with
(cid:113)
u = t2u2+r2u2 (A12)
T0 r t
(cid:18) (cid:19)
1
u = 1+ u (A13)
T1 m2 m
u =0. (A14)
T2
Here we define the ratios
|R | s
r = S = (A15)
|R | (cid:96)
L
|T |
t= S =1 (A16)
|T |
L
|R |
m= S =s·t (A17)
|T |
L
with the uncertainties
1(cid:113)
u = u2+r2u2 (A18)
r (cid:96) s (cid:96)
u =0 (A19)
t
(cid:113)
u = t2u2+s2u2, (A20)
m s t
as well as the ratios
|R |
s= S (A21)
|T |
S
|R |
(cid:96)= L , (A22)
|T |
L
together with their uncertainties u and u , which are determined by measuring the DC-levels of the two detectors
s (cid:96)
and incorporating a correction in case of different quantum efficiencies. Finally, one obtains for the statistical error
of det[L(φ)]
(cid:113)
u = [B (φ)]2u2 +[B (φ)]2u2 +4[B (φ)]2u2 . (A23)
det[L(φ)] 2 B0(φ) 0 B2(φ) 1 B1(φ)
The signed statistical significance is given by
det[L(φ)]
Σ= . (A24)
u
det[L(φ)]
7
Appendix B: Separation of moments by LO field strength
TheseparationofmomentsusingvariousLOfieldstrengthsE worksasfollows. ConsiderJ pairs(E˜ ,C(E˜ ;φ)),
L L,j L,j
j =1,...,J,measuredforagivenphaseφ,whereE˜ =E /E isarescaledLOfieldstrength. Onemaychoose,
L,j L,j L,ref
e.g., E =E with k =1,...,J. Since one has to measure anyway with blocked LO, one may include E˜ =0.
L,ref L,k L,1
Recalling Eq. (3) together with Eqs. (4)–(6) of the main text, one has to apply a quadratic fit of the form
C(E˜ ;φ)=C (E˜ ;φ)+C (E˜ ;φ)+C (E˜ ;φ), (B1)
L 0 L 1 L 2 L
where C (E˜ ;φ) is proportional to E˜k, i.e.,
k L L
C (E˜ ;φ)=D (φ)E˜k. (B2)
k L k L
The solution of the fitting problem (B1) is given by
D =(VTV)−1VTC (B3)
(see e.g. [15, 16]) with the Vandermonde matrix
1 E˜ E˜2 
L,1 L,1
V =1.. E˜L..,2 E˜L2..,2, (B4)
. . . 
1 E˜ E˜2
L,J L,J
D = (D (φ),D (φ),D (φ))T, and C = (C(E ;φ),··· ,C(E ;φ)). It is necessary to measure at least for three
0 1 2 L,1 L,J
(J = 3) different LO intensities including the blocked LO, however, more regression points ensure a more reliable
separation of moments. The statistical error can be determined by
(cid:118)
(cid:117) J
(cid:117)(cid:88)
uDk(φ) =(cid:116) [((VTV)−1VT)kj(uC)j]2, (B5)
j=1
with i = 0,...,2. Although this way works in principle if the measured data points sufficiently coincide with the
quadratic fit, it is very sensitive to systematic errors. That is why we apply a more robust analysis by using data
from a second phase φ+π to infer the D (φ). In particular, the relations
k
D (φ)=D (φ+π)=C (B6)
0 0 block
D (φ)=−D (φ+π) (B7)
1 1
D (φ)=D (φ+π), (B8)
2 2
are always fulfilled. Defining
C(E˜ ;φ)−C
y(E˜ ;φ)= L block, (B9)
L E˜
L
yields the two linear equations
y(E˜ ;φ)=D (φ)E˜ +D (φ) (B10)
L 2 L 1
y(E˜ ;φ+π)=D (φ)E˜ −D (φ). (B11)
L 2 L 1
ThroughacoupledlinearregressiononeobtainsD (φ)andD (φ). Inparticular,followingthemethodofleastsquares,
1 2
the function
J
E =(cid:88)[y (φ)−D (φ)E˜ −D (φ)]2 (B12)
j 2 L,j 1
j=1
J
+(cid:88)[y (φ+π)−D (φ)E˜ +D (φ)]2 (B13)
j 2 L,j 1
j=1
8
has to be minimized. Minimization with respect to D (φ) and D (φ) yields
1 2
J
1 (cid:88)
D (φ)= [y (φ)−y (φ+π)] (B14)
1 2J j j
j=1
(cid:80)J [y (φ)+y (φ+π)]E˜
D (φ)= j=1 j j L,j (B15)
2 2(cid:80)J E˜2
j=1 L,j
with the statistical uncertainties
(cid:118)
(cid:117) J
1 (cid:117)(cid:88)(cid:104) (cid:105)
u = (cid:116) u2 +u2 (B16)
D1(φ) 2J yj(φ) yj(φ+π)
j=1
(cid:118)
(cid:117) J
u = 1 (cid:117)(cid:116)(cid:88)E˜2 (cid:104)u2 +u2 (cid:105). (B17)
D2(φ) 2(cid:80)J E˜2 L,j yj(φ) yj(φ+π)
j=1 L,j j=1
Here, the uncertainty u is composed of the statistical error
yj(φ)
1 (cid:113)
u = u2 +u2 (B18)
yj(φ) E˜ C(EL,j;φ) Cblock
L,j
and an error due to a possible drift of the signal state, which is estimated by the deviation of y from the quadratic
j
fit. The error of the zeroth-order term is due to Eq. (B6) given by
u =u . (B19)
D0(φ) Cblock
Similar to Eq. (A11), for a proper application of our nonclassicality condition the quantities D have to be converted
k
to
D (φ)
D˜ (φ)= k k =0,1,2 (B20)
k T
k
with the statistical uncertainty
1 (cid:113)
u = [D˜ (φ)]2u2 +u2 (B21)
D˜k(φ) |Tk| k Tk Dk(φ)
and u determined according to Eqs. (A12)–(A14). A negative determinant of the matrix
Tk
(cid:18)D˜ (φ) D˜ (φ)(cid:19)
L˜(φ)= 0 1 (B22)
D˜ (φ) D˜ (φ)
1 2
indicates nonclassicality. Its error is obtained as
(cid:113)
u = [D˜ (φ)]2u2 +[D˜ (φ)]2u2 +4[D˜ (φ)]2u2 (B23)
det[L˜(φ)] 2 D˜0(φ) 0 D˜2(φ) 1 D˜1(φ)
and the signed statistical significance is given by
det[L˜(φ)]
Σ= . (B24)
u
det[L˜(φ)]
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