Quantum tomography for time domain Balanced Homodyne Detection with high electronic noise

0.73 MB·English

by  Martina Esposito

Quantum tomography for time domain Balanced
Homodyne Detection with high electronic noise
Martina Esposito1, Daniele Fausti1,2,*, Stefano Olivares3, Fabio Benatti1,4, Roberto Floreanini4,
Fabio Novelli1, Federico Cilento2, Francesco Randi1, and Fulvio Parmigiani1,2
1Department of Physics, Università degli Studi d Trieste, 34127 Trieste, Italy
2Sincrotrone Trieste S.C.p.A., 34127 Basovizza, Italy
3Department of Physics, Università degli Studi di Milano, 20133 Milano, Italy
4Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, 34151 Trieste, Italy
*Correspondence to [[email protected]]
3
1
0
2 Herewereportonthedesign,constructionandcharacterizationofatimedomainBalancedHomodyneDetection

n apparatus operating in the regime of high electronic noise. The experimental setup has been commissioned by
a studying coherent ultrashort (∼80 fs) light pulses using Pattern-function Quantum Tomography. Furthermore,
J we proved that the electronic noise can be treated as a detector inefficiency showing the effectiveness of this

1 approach by measuring light coherent states with different energies.
1

] In Quantum Mechanics the measurement of the state Results
s
ofaphysicalsystemimpliestoestimatetheexpectation
c
i value of any observable of the system. For a generic In BHD the signal, i. e. the field under investigation, is
t
p operatortheexpectationvalueisobtainedstartingfrom mixed with a strong coherent reference field, the local
o the measurement of an ensemble of observables defined oscillator (LO), by a 50/50 beam splitter. The outputs
s. “quorum”1. In particular, for the electromagnetic field are collected by two photodiodes and the difference
c the quorum is a continuum set of the field quadra- photocurrent (homodyne photocurrent) is measured. It
i
s tures2,3. can be proven that, when the LO is significantly more
y Balanced Homodyne Detection (BHD) is an experimen- intense than the signal, the homodyne photocurrent is
h
tal method for measuring a discrete set of the field proportional to the signal field quadrature17. Given aˆ
p
[ quadratures for a single mode cw field4–7. Such mea- and aˆ† the field operators associated with the optical
surements can be analyzed by the Pattern-function mode of the signal, the quadrature operator is defined
1
Quantum Tomography (PFQT) technique in order to as:
1v characterize the quantum state of the optical mode8. xˆ = (aˆe−iΦ√+aˆ†eiΦ), (1)
Φ
7 Instead, for pulsed radiation fields a BHD operating 2
4 in the time domain must be used9–15. In this case the
whereΦistherelativephasebetweenthesignal andthe
2
optical pulses are equally prepared quantum states and
LO. The continuum set of quadratures with Φ∈[0,2π]
.
1 the set of measurements fed to PFQT comprises single
constitutes a quorum characterizing the signal mode.
0 pulse quadrature measurements. Although this is a
Noteworthy the concept of light mode, usually associ-
3
standard method in quantum information, its effective-
1 ated with a monochromatic field, needs to be general-
ness under high noise condition has to be proven.
: ized for pulsed light. A thorough theoretical treatment
v Here we report the characterization of a time domain
for BHD in pulsed regime is presented herewith in the
i
X BHDapparatusoperatinginaregimeofhighelectronic Methods section.
noise (EN). We treat the low shot-to-electronic-noise
r The opto-mechanical scheme of our experimental ap-
a ratio (≈0.8 dB) as a detector inefficiency16. In order
paratus is shown in Fig. 1. The laser source is a
to prove the applicability of this treatment to high EN
mode-locked Ti:Sapphire oscillator with 80 MHz repeti-
homodyne detection, the PFQT technique has been ex-
tion rate. A beam splitter divides the incoming beam
ploited. The expectation value of the number operator
intwopartswhichtheninterfereinasecondbeamsplit-
of pulsed coherent states with different energies has
ter (NPBS in Fig. 1). The outputs are detected and
been found consistent with the measured mean number
subtracted by a differential photodetector (Thorlabs
of photons per pulse. Furthermore, the analyzed states
PDB430A). The latter is made up of two Si/PIN pho-
areconfirmedtobeGaussianandminimumuncertainty
todiodes with nominal quantum efficiency of 0.77. We
states.
verified that the response of the photodiodes is linear
up to 0.6 mW LO power, while a non-linear response
sets in at higher powers. The detector subtraction ef-
ficiency is quantified by the common mode rejection
1
Figure 1: Scheme of the opto-mechanical setup.
ratio(CMRR),definedastheratiobetweenthedetector
output power when both photodiodes are illuminated
and the power when one of the two is screened. The
measured CMRR for our apparatus is larger than 36
dB.
The homodyne photocurrent is recorded by a digital
Figure 2: Detector characterization measurements. (a)
oscilloscope (Tektronix TDS3000B) with a bandwidth
Correlation coefficient between pulse n and n+m (LO
of 500 MHz and a sampling rate of 5 GSamples/s. The
power: 0.6 mW). Inset: parametric plot of 2 × 104
digitized output is numerically integrated over time in-
subsequentpulseswherethepulsen+1isplottedagainst
tervalscorrespondingtothedurationofthepulse. Each
the pulse n. (b) HD noise variance versus LO power;
integral is associated with a single quadrature measure-
shot noise contribution (dashed curve); electronic noise
ment. Wecheckedtheindependenceofeachquadrature
background (dashed-dotted curve).
measurement by performing a correlation test between
two subsequent pulses. The Correlation Coefficients
(CCs)14 between pulse n and n+m (m=0,1...9) are
with respect to the LO by means of a neutral density
shown in Fig. 2 (a). In the inset a parametric plot of
filter(F inFig. 1). ThephasedifferenceΦbetweenthe
1
2×104 subsequentpulsesisshown,wherethepulsen+1
two arms of the interferometer can be modulated in the
is plotted against the pulse n. The weak correlation
[0,2π] range using a piezoelectric translator in the LO
in this plot demonstrates that there is no significant
arm (PZT in Fig. 1). The homodyne photocurrent is
impact on the measured integral of the pulse n + 1
acquired by the digital oscilloscope and each difference
from that of the pulse n. Each CC in Fig. 2 (a) is
pulse is integrated. The obtained value N , correspond-
i
calculated using the data obtained from 200 difference
ing to a certain piezo position p , is proportional to a
i
pulses. The CC between pulse n and pulse n+1 is
quadrature measurement x :
Φ
0.30±0.08; this value is slightly higher with respect to
other reported BHDs, i. e. smaller than 0.112–15. We N =γx . (2)
i Φ
can conclude that there is a small correlation between
adjacent pulses, since the CC has approximately the The proportionality factor γ is obtained using the vac-
same value for m=2,...9. This could be attributed to uum state as a reference. This is allowed since the
noise at a frequency lower than the repetition rate of quadraturevarianceforthevacuumstateisσ2[xˆ ] =
Φ |0(cid:105)
the pulses. 1/2 for any phase value. Defining γ as:
Inshot noise regime thehomodynedetector(HD)noise
(cid:113)
varianceisexpectedtochangelinearlyintheLO power γ = 2(cid:104)N2(cid:105), (3)
withaconstantoffsetrepresentingtheelectronicnoise18. i0
We calculated the HD noise variance of 8×103 pulses the vacuum homodyne measurement has been used to
for different values of the LO power. The result is extract the N values. (cid:104)N2(cid:105) is the variance of 8×104
i0 i0
shown in Fig. 2 (b): the HD noise variance contains experimental data of the vacuum. The homodyne data
an electronic background and a linear contribution up have beencalibrated dividingN byγ. Notethatγ will
i
to 0.6 mW LO power; for higher powers the non-linear give a misleading estimation of the actual mean photon
effects due to the photodiodes start. We work at 0.6 number. Therefore, a correction taking into account
mWLO oscillatorpowerinordertohavethemaximum the EN effect is necessary, as discussed in the following.
shot-to-electronic noise ratio (S = 1.2) achievable in The calibrated measurements for three optical coherent
the linear regime. The measurements reported in Fig. states with different energies are reported in Fig. 3.
2 are obtained with the signal beam blocked, i.e. with The optical density OD of the filter F used for each
1
the signal in the vacuum state. measurement is given in the caption. The homodyne
When the signal beam is not blocked, it is attenuated traces are composed of M =8×104 experimental data:
2
Thesedatahavebeenusedtocharacterizethequantum
state of the signal field via PFQT8.
Discussions
The PFQT technique can be summarized as follows.
For a generic observable Oˆ of the optical mode it is
possible to find a pattern function R (x;Φ) for which
Oˆ
the statistical average over the experimental data,
M
1 (cid:88)
S = R (x ;Φ ), (6)
M M Oˆ k k
k=1
gives an estimation of the expectation value (cid:104)Oˆ(cid:105) with
an error
(cid:115)
(cid:80)M [R (x ;Φ )−S ]2
(cid:15) = k=1 Oˆ k k M . (7)
M M(M −1)
Thistechniqueisusedtoestimatetheexpectationvalue
of the number operator (aˆ†aˆ), which can be evaluated
in an independent way.
The pattern function of the number operator is:
1
R (x ;Φ)=(x )2− , (8)
aˆ†aˆ Φ Φ 2η
where the factor η is the quantum efficiency of the
homodyne detection apparatus. In order to take into
account the EN loss effect, we put η = η given in
eq
equation (4). Finally we estimate the mean photon
Figure 3: Homodyne traces of three optical coherent
number as:
states. Each trace is acquired using a different optical
densityOD ofthefilterF1. (a)OD =4.5; (b)OD =7; 1 (cid:88)M 1
(cid:104)nˆ(cid:105) = R (x ;Φ ) . (9)
(c)OD =8. Intheinsetof(a)weshowtheinterference PFQT M aˆ†aˆ k k η
eq
figure obtained by measuring the mean value of 4 inte- k=1
grated pulses versus the piezo position. The homodyne The error on this estimation is calculated using equa-
traces are measured in the piezo range corresponding tion (7). The results for the three different homodyne
to the central optical cycle of the interference figure. traces are reported in Table 1. The expectation values
OD (cid:104)nˆ(cid:105)
{(x ;Φ ),(x ;Φ ),...,(x ;Φ ),...,(x ;Φ )}. PFQT
1 1 2 2 k k M M
However, the effect of the EN, which adds a random 4.5 596±2
quantity to each field quadrature measurement, must 7 1.6±0.1
be accounted for. Appel at al. demonstrate16 that this 8 0.6±0.1
noise can be treated as an optical loss channel with
equivalent transmission efficiency,
Table 1: Expectation values of the number operator of
1 three optical coherent states for three different optical
η =1− , (4)
eq S densities ODs. The values are obtained via PFQT from
the homodyne measurements reported in Fig. 3 taking
whereS istheratiobetweenshotandelectronicnoiseat
into account the equivalent quantum efficiency η as
the chosen LO power. In particular, the vacuum state eq
in equation (4).
measurement is distorted by the EN and consequently
the rescaling factor γ must be changed. Following the
calculated using the PFQT are consistent, in the order
Appel’s treatment16 and considering equation (3), the
of magnitude, with the mean number of photons per
following expression is obtained,
pulse obtained from the nominal OD of the filter. In
γ(cid:48) =√η γ, (5) this way we verify that the treatment of low EN16 is
eq
valid also for high EN detection systems.
where γ(cid:48) is the conversion factor in case of absence of Now we focus on the optical state with the lowest en-
electronic noise. Thus, in order to take into account ergy [Fig. 3 (c)]. The analysis for the other states is
theEN,thehomodynedatahavebeennormalizedtoγ(cid:48). analogous.
3
Methods
FirstweverifytheGaussianityoftheexperimentaldata
through the evaluation of the kurtosis excess κ, which
is zero for Gaussian distributions19. Considering 103 BHDtheoreticaltreatmentinthepulsedregime.
experimental data in the first phase range of the ho- BHDinapulsedregimerequiresaformalgeneralization
modyne trace in Fig. 3 (c) and we obtain a kurtosis of its theoretical description with respect to the one-
excess κ=−0.008. Since all the phase ranges lead to mode regime; indeed, the LO and the signal at the
a κ≈0, this proves the Gaussianity of the homodyne beam splitter are not monochromatic. Classically, the
probability distribution function. electric field of a pulsed laser beam is in a mode-locked
Then we use the PFQT to evaluate the expectation superposition of amplitudes:
values of the the two quadratures associated with
M
txˆhe posit=ionpˆ)aannddmofotmheeinrtvuamriaonpceersa(tσo2r[sqˆ](;xˆσ(Φ2[=pˆ0])).=Thqˆe; E(t)= (cid:88) |αl|eiΦl(t) , Φl(t)=ωlt+ϕl , (12)
(Φ=π/2) l=−M
results, shown in Table 2, demonstrate that the ana-
lyzed state is a minimum uncertainty state. where the phases Φ (t) are mode-locked by the condi-
l
tion ϕ = lϕ . Each frequency ω contributes to the
l 0 l
(cid:104)qˆ(cid:105) σ2[qˆ] field with the amplitude αl =|αl|eiΦl and the number
0.36±0.01 0.53±0.05 of contributing frequencies depends on the shape of
the pulse. Quantized pulsed laser light is described
(cid:104)pˆ(cid:105) σ2[pˆ]
by associating to each monochromatic component a
0.00±0.01 0.53±0.05
coherent state |α (cid:105), that is an eigenstate of the annihi-
l
lation operator aˆ of photons in the mode of frequency
l
Table 2: Expectation values of the position and mo- ω , aˆ |α (cid:105)=α |α (cid:105), and to the entire pulse the tensor
l l l l l
mentum operators and their variances. The values are product
obtained via PFQT from the homodyne measurements (cid:79)M
|α¯(cid:105)= |α (cid:105), (13)
reported in Fig. 3 (c). l
l=−M
where α¯ is the vector whose components are the am-
Finally we evaluate the covariance matrix C, that fully
plitudes α . By means of the creation and annihilation
characterizes Gaussian states and directly depends on l
the field quadratures20. The elements of the covariance operators aˆ and aˆ†l each monochromatic coherent state
reads
matrix are defined as:
C = 1(cid:104){Rˆ ,Rˆ }(cid:105)−(cid:104)Rˆ (cid:105)(cid:104)Rˆ (cid:105), (10) |αl(cid:105)=D(αl)|0(cid:105) , D(αl)=eαlaˆ†l−α∗laˆl , (14)
kj 2 k j j k
where |0(cid:105) is the vacuum state and D(α) is the so-called
displacement operator.
whereRˆ =(qˆ,pˆ)isthefirst-moments vector oftheopti-
Sincethecreationandannihilationoperatorspertaining
calstateand{Aˆ,Bˆ}=AˆBˆ+BˆAˆistheanti-commutator
to different modes commute, the pulsed coherent state
between two operators20. From the PFQT we obtain
(13) can conveniently be recast as:
the following estimations:
(cid:18) (cid:19) |α¯(cid:105)=D(α¯)|0(cid:105) , D(α¯)=eAˆ†(α¯)−Aˆ(α¯) , (15)
0.53±0.5 −0.02±0.03
C = . (11)
−0.02±0.03 0.53±0.05
in terms of a multi-mode displacement operator D(α¯)
where
This is consistent with what is expected for a coherent
state. The other two homodyne measurements, Aˆ†(α¯)=(cid:88)α aˆ† , Aˆ(α¯)=(cid:88)α∗aˆ . (16)
l l l l
reported in Fig. 3 (a) and (b), lead to similar results. l l
Thereasonforlabelingthepulsedcoherentstateby|α¯(cid:105)
In conclusion, we reported the characterization of
can now be easily understood. The state of one photon
a time domain BHD apparatus operating in the high
of frequency ω is given by aˆ†|0(cid:105)=|1 (cid:105), while a generic
electronic noise regime. The apparatus has been char- l l l
non-monochromaticsuperpositionoffrequenciesω with
acterized via the PFQT of optical coherent states with (cid:80) l
amplitudesα correspondstothestate|1 (cid:105)= α |1 (cid:105)
differentmeannumberofphotonsperpulse. Wetreated l α¯ l l l
that results from
the electronic noise trough the approach used by Appel
at al.16 and we showed that such a treatment is valid Aˆ†(α¯)|0(cid:105)=(cid:88)α aˆ†|0(cid:105)=(cid:88)α |1 (cid:105)=|1 (cid:105). (17)
l l l l α¯
for high EN homodyne detectors. This result is of rele-
l l
vance in the perspective of using BHD and PFQT in
experiments performed in high noise environments such Therefore, Aˆ†(α¯) is the creator operator of a single-
astimedomainpump-probespectroscopy. Inparticular, photoninthe(notnormalized)superpositionstate|1 (cid:105),
α¯
ultrashortlightpulsescouldbeusedtopreparecoherent while Aˆ(α¯) destroys a photon in the same state; thus,
states into matter and BHD to have a full character- the quantum state of the pulsed laser is a coherent
ization of the light pulses which have interacted with state associated not with a single amplitude α , but
l
transient matter states. with the vector α¯ of all the amplitudes contributing
4
to the pulse: in other words, we have a Poissonian For this reason, in the main text of the letter we have
distribution not with respect to the number of photons limited the discussion to the simpler case of a single
inamonochromaticwave,buttothenumberofphotons mode BHD, with the proviso that whenever a quadra-
in the superposition |1 (cid:105). ture is used, it actually refers to its expression in the
α¯
The normalized operators pulsed regime. In particular, the measured data can be
used to reconstruct the expectation values of all signal
Aˆ(α¯) Aˆ(α¯)
Aˆ= , Aˆ† = , (18) observables that can be expressed as functions of the
|α¯| |α¯| operators Aˆ and Aˆ†. For instance, the mean value of
where |α¯|2 = (cid:104)1 |1 (cid:105), satisfy the canonical commuta- thephoto-currentoperatorsecondmomentwithrespect
α¯ α¯
to the LO pulsed coherent state |z¯(cid:105)(cid:104)z¯| is
tion relations
(cid:104)Aˆ,Aˆ†(cid:105)= |α¯1|2 (cid:88)αi∗αj[aˆi,aˆ†j]=1 . (19) (cid:104)z¯| IˆΦ2 |z¯(cid:105)=2|z¯|2XˆΦ2 + (cid:88)aˆ†laˆl (24)
ij l
By means of them we can now extend the state- and differs from the pulsed quadrature second moment
tomography techniques to the case in which the signal Xˆ2 bythenumberofphotonsinthepulsedsignal which
Φ
and the LO are pulsed. In the monochromatic case, is to be taken much smaller than the intensity |z¯|2 of
when the signal mode a interferes with the LO mode thepulsedLO.Asimilarsuppressionby|z¯|−2 occursfor
b on the beam splitter, the photo-current operator is the correction terms appearing in higher moments so
Iˆ=aˆ†ˆb+ˆb†aˆ17. that the distribution of the outcomes of the homodyne
Ifmorefrequenciesω arepresentbothintheLO andin photocurrent is equal to that of the corresponding field
l
thesignal,eachoneofthecorrespondingmodeoperators quadratures.
will be subjected to the beam splitting transformation Notice that, in the case of only one frequency mode,
and the detectors will ideally register photons of all the above setting reduces to the standard homodyne
involved frequencies. Then, the photo-current operator tomographic techniques. Therefore, in conclusion, we
becomes Iˆ=(cid:80) aˆ†ˆb +ˆb†aˆ . have generalized the treatment of a monochromatic
l l l l l
If the LO incoming field is in a pulsed coherent state BHD to the case of pulsed input light. It explicitly
|z¯(cid:105) = eBˆ†(z¯)−Bˆ(z¯)|0(cid:105), with generalized creation and takes into account the fact that the state of a pulsed
annihilation operators Bˆ†(z¯) = (cid:80) zˆb† and Bˆ(z¯) = laseristhetensorproductofthecoherentstatesofeach
l l l
(cid:80) z∗ˆb , the phase difference Φ between the LO and contributing frequency.
l l l
the signal is changed by the action of the piezoelectric
translator placed in the LO arm on all the LO modes: Acknowledgement
ˆb →ˆb eiΦ , ˆb† →ˆb†e−iΦ . (20)
l l l l
Useful discussions with Simone Cialdi and Marco Bar-
The photo-current operator which is measured by the bieri are acknowledged. This work has been supported
pulsed homodyne setup is thus given by byMIUR(FIRB"LiCHIS"-RBFR10YQ3H),University
Iˆ =(cid:88)(cid:16)aˆ†ˆb eiΦ + aˆ ˆb†e−iΦ(cid:17) . (21) of Trieste ("FRA 2009" and "FRA 2011") and Euro-
Φ l l l l peanUnion,SeventhFrameworkProgramme,underthe
l project GO FAST, grant agreement no. 280555.
Let ρˆ denote the quantum state (density matrix) of
s
the signal field and by |z¯(cid:105)(cid:104)z¯| the projector onto the
References
(coherent) state of the incoming pulse. Then, using
that ˆb |z¯(cid:105)=z |z¯(cid:105) and the expressions in (16), the ex-
l l (cid:16) (cid:17) 1. Fano, U. Description of states in quantum mechan-
pectation value IΦ =Tr ρs⊗|z¯(cid:105)(cid:104)z¯| IˆΦ of the photo- ics by density matrix and operator techniques. Re-
current is calculated as follows: views of Modern 29, 74 (1957).
IΦ = (cid:88)(cid:16)Tr(cid:104)ρˆsaˆ†l(cid:105)·(cid:104)z¯|ˆbl|z¯(cid:105) eiΦ+h.c.(cid:17) 2. Lvovsky & Raymer Continuous-variable optical
l quantum-state tomography. Reviews of Modern
= (cid:88)(cid:16)Tr(cid:104)ρˆ (aˆ†z eiΦ+aˆ z∗e−iΦ)(cid:105)(cid:17) Physics 81, 299 (2009).
s l l l l
l 3. Leonhardt, U. Measuring the Quantum State of
(cid:104) (cid:16) (cid:17)(cid:105)
= Tr ρˆ Aˆ†(z¯)eiΦ+Aˆ(z¯)e−iΦ . (22) Light (Cambridge Studies in Modern Optics, 1997).
s
Using (18) and comparing these results in the standard 4. Vogel, K.&Risken, H.Determinationofquasiprob-
treatmentofBHDforthesingle-modecase,onerealizes ability distributions in terms of probability distri-
that the pulsed BHD measures a quantity I propor- butionsfortherotatedquadraturephase.Phys.Rev.
Φ
tional to a quadrature which generalizes that in (1): A 40, 2847 (1989).
5. Smithey, D. T., Beck, M., Raymer, M. G. & Fari-
√ (cid:104) (cid:105) Aˆ†eiΦ+Aˆe−iΦ dani, A. Measurement of the Wigner distribution
I = 2|z¯|Tr ρˆ Xˆ , Xˆ = √ .
Φ s Φ Φ 2 andthedensitymatrixofalightmodeusingoptical
(23) homodyne tomography: Application to squeezed
5
states and the vacuum. Phys. Rev. A 70, 1244
(1993).
6. Welsch,Vogel&OpatrnyHomodyneDetectionand
Quantum State Reconstruction. Progress in Optics
39, 63 (1999).
7. Zavatta, Viciani & Bellini Non-classical field char-
acterization by high-frequency,time-domain quan-
tum homodyne tomography. Laser Phys. Lett. 3, 3
(2006).
8. D’Ariano, Paris & Sacchi Quantum tomography.
Advances in Imaging and Electron Physics 128,
205 (2003).
9. Hansenet al.Anultra-sensitivepulsedbalancedho-
modynedetector: Applicationtotime-domainquan-
tum measurements. Optics letter 26, 1714 (2001).
10. Zavatta, Bellini, Ramazza, Marin & Arecchi Time
domain analysis of quantum states of light: noise
characterizationandhomodynetomography.J.Opt.
Soc. Am. B 19, 1189 (2002).
11. Kumar et al. Versatile Wideband Balanced Detec-
tor for Quantum Optical Homodyne Tomography.
Optics Communications 285 (2012).
12. Haderka, Michálek, Urbášek & Ježek Fast time do-
mainbalancedhomodynedetectionoflight.Applied
Optics 48, 2884 (2009).
13. Cooper, Soller & Smith High-stability time domain
balanced homodyne detector for ultrafast optical
pulse applications. Conference Paper: Quantum
Electronics and Laser Science Conference (QELS)
San Jose, California (May, 2012).
14. Chi et al. A balanced homodyne detector for high
rate Gaussian-modulated coherent-state quantum
keydistribution.New Journal of Physics 13(2011).
15. Okubo, Hirano, Zhang & Hirano Pulse-resolved
measurement of quadrature phase amplitudes of
squeezed pulse trains at a repetition rate of 76
MHz. Opt. Lett. 33, 1458 (2008).
16. Appel, Hoffman, Figueroa & Lvovsky Electronic
noise in optical homodyne tomography. Physical
Review A 75, 035802 (2007).
17. Ferraro, A., Olivares, S. & Paris, M., Gaus-
sian states in quantum information (Bibiopolis,
Napoli,2005).
18. Bachor, H. A. & Ralph, T. C., A Guide to experi-
ments in Quantum Optics (WileY-VCH, 2004).
19. Buono et al. Quantum characterization of bipartite
Gaussian states. J. Opt. Soc. Am. B 27, Section 5
(2010).
20. Olivares, S. Quantum optics in the phase space.
The European Physics Journal Special Topics 203,
3 (2012).
6