Table Of ContentElectronic Noise in Optical Homodyne Tomography
Ju¨rgen Appel, Dallas Hoffman, Eden Figueroa, A. I. Lvovsky
Institute for Quantum Information Science, University of Calgary, Calgary, Alberta T2N 1N4, Canada∗
(Dated: February1, 2008)
The effect of the detector electronic noise in an optical homodyne tomography experiment is
shown to be equivalent to an optical loss if the detector is calibrated by measuring the quadrature
noiseofthevacuumstate. Anexplicitrelationbetweentheelectronicnoiselevelandtheequivalent
optical efficiency is obtained and confirmed in an experiment with a narrowband squeezed vacuum
source operating at an atomic rubidium wavelength.
PACSnumbers: 42.50.Dv,03.65.Wj,05.40.Ca
7
a. Introduction Optical homodyne tomography detector. The squeezed vacuum source is a narrowband,
0
0 (OHT) is a method of characterizing quantum states of tunablecontinuous-waveopticalparametricamplifierop-
2 light by measuring the quantum noise statistics of the erating at a 795-nm wavelength.
electromagnetic field quadratures at different phases. We begin our theory by writing the effects of both
n
a Theoretically proposed in [1] and first experimentally inefficiency types on the reconstructed Wigner function
J implemented in the early 1990s [2], OHT has become (WF).
5 a standard tool of quantum technology of light, in b. The beam splitter model for optical signal loss
1 particular in quantum information applications. Accordingtothebeamsplittermodelofabsorption,non-
OHT is prone to a variety of inefficiencies, of which unitary opticalefficiency η modifies the WF W(X,P) of
2 the main one — the optical losses (OLs) — is due to the signal state as follows [4]:
v
absorption in beam paths, non-unitary efficiency of the
6
1
1 photodiodes, imperfect balancing of the detector, and, W (X,P) = W(η−1/2X′,η−1/2P′) (1)
OL
1 partially, imperfect mode matching between the signal πη(1 η)ZZ
−
0 and the local oscillator modes [3]. Also, the presence of (X X′)2+(P P′)2
′ ′
1 non-mode-matched signal light appears as added noise exp − − dX dP .
× (cid:20)− 1 η (cid:21)
6 in the detector, raising the apparent noise floor [3]. All −
0
OLs can be modeled by an absorber in the signal beam This transformation can be understood by visualizing
/
h path. The effect of OL in quantum state reconstruction the absorber as a fictional beam splitter with the signal
p iswellunderstood,andtherearewaysofbothitsquanti- beam entering one port and the vacuum state entering
t- tative evaluation and compensation for its effects in the the other. Transmission of the signal field through the
n quantum state reconstruction procedure [4, 5]. beam splitter rescales the field amplitudes by a factor of
a Animportantadditionalinefficiencysourceistheelec- η−1/2. The vacuum state reflected from the beam split-
u
q tronic noise (EN) of the homodyne detector. Its effect ter into the signalmode adds noise,whichexpressesas a
: is the addition of a random value to each quadrature convolutionof the WF with a two-dimensionalGaussian
v
measurement, which blurs the marginaldistribution and function of width √1 η.
i −
X causes errors in quantum state reconstruction. Unlike c. Effectoftheelectronicnoise Intheabsenceofthe
r the optical inefficiency, the EN, to our knowledge, has electronicnoise,theelectricsignal(voltageorcurrent)V
a not yet been studied, and is usually neglected whenever generated by the homodyne detector is proportional to
an efficiency analysis is made [6–12]. the quadrature sample X: V = αX. The coefficient α
The purpose of this paper is to fill this gap and to an- hastobedeterminedwhenquantumstatereconstruction
alyze the role of the electronic noise in OHT. We show isthegoal. Thisisusuallydonebymeansofacalibration
its effect to be equivalent to that of optical losses. The procedure,whichconsistsofacquiringasetofquadrature
key to this equivalence is the detector calibration proce- noise samples V0i of the vacuum state, prepared by
{ }
dure, which involves measuring the quadrature noise of letting only the local oscillator into the detector. The
the vacuum state. Applying this procedure to the vac- vacuum marginal distribution,
uumstate contaminatedwiththe ENresultsinrescaling
of the field quadratures, analogous to that taking place pr0(X)=π−1/2exp(−X2), (2)
due to optical absorption.
is loss-independent, and has a mean square of X2 =
We confirm our experimental findings in a squeezing h i
1/2. One can thus use the experimental data to obtain
experiment, in which we vary the local oscillator power
toachievedifferentsignal-to-noiseratiosofthehomodyne
α= 2 V2 . (3)
q h 0ii
When the EN is present, the observedprobability dis-
∗http://www.iqis.org/ tribution of the electric signal is a convolution of the
2
scaledmarginaldistributionpr(X)ofthestatemeasured d. Equivalence of the efficiency loss mechanisms
and the electronic noise histogram p (V): Comparing the above expression with Eq. (1) for the
e
1 V′ WF of a state affected by opticallosses,we find that the
p˜r (V)= pr p (V V′)dV′ (4) two are equivalent if we set
EN αZ (cid:18) α (cid:19) e −
α2 α2
(the factor of α is added to retain the normalization of η = = . (12)
the marginal distribution). Under realistic experimental eq α′2 α2+T
conditions of a broadband electronic noise, its distribu-
In other words, we may treat the electronic noise as an
tion can be assumed Gaussian:
additional optical attenuator with a transmission equal
1 V2 to η . Accordingly,the knownnumericalprocedures for
p (V)= exp , (5) eq
e √πT (cid:18)− T (cid:19) correctingfor opticallosses canbe readilyappliedto the
electronic noise.
where T is the mean square noise magnitude.
This result may appear surprising. The field quadra-
Consider the detector calibration procedure in the
turenoisemeasuredinthepresenceoftheENisaconvo-
presence of the EN. According to Eqs. (2), (4), and (5),
lution of the WF’s marginal distribution with the noise
the vacuum state measurement yields a distribution
histogram. In the case of OL, on the other hand, the
1 V2 convolution is preceded by rescaling of the quadrature
p˜r (V )= exp 0 (6)
0,EN 0 √π√α2+T (cid:18)−α2+T(cid:19) variables. The similarity between the effects of the two
inefficiency emerges due to the detector calibration pro-
withameansquareofhV02i=(α2+T)/2. Ifnocorrection cedure. The vacuum state measurement is distorted by
for the EN is made, the quantity the EN but not the OL, hence there is additional rescal-
ing of the quadratures in the presence of the EN.
α′ =q2hV02i=pα2+T (7) expFroersesxtpheereimqueinvtaallenatppeffiliccaietinocnys,inittiesrmmsoroefctohnevseignnieanl-tttoo-
is interpretedaccording to Eq. (3) as the conversionfac-
noise ratio of the detector. By the latter we understand
tor between the electric signal and the field quadrature.
the ratio between the observedmean square noise of the
Inahomodynemeasurementofanunknownstate,one
vacuum state (in the presence of the EN) and the mean
acquires a set of samples from the homodyne detector,
and then applies the calibration factor α′ to rescale its square electronic noise, i.e. the quantity
histogram p˜rEN(V) and obtain the associated marginal α2+T
distribution. Using Eq. (4), we write S = . (13)
T
′ ′
pr (X) = αp˜r (αX) (8)
EN EN Re-expressing Eq. (12) in terms of S yields
α′ V′
′ ′ ′
= pr p (αX V )dV .
α Z (cid:18) α (cid:19) e − S 1
η = − . (14)
eq
Introducing a new integration variable X′ = V′/α′ and S
utilizingtheENdistribution(5),Eq.(8)canberewritten As evidenced by this equation, reasonablylow electronic
as noisedoesnotconstitute a significantefficiencyloss. For
α′2 α′ example, the original time-domain detector of Smithey
′
prEN(X) = √πTαZ pr(cid:18)αX (cid:19) (9) and co-workers [2, 6], featuring a 6 dB signal-to-noise
ratio (i.e. the rms quadrature noise being only twice as
(X X′)2
′ highasthermsEN)hasanequivalentquantumefficiency
exp − dX .
× (cid:20)− T/α′2 (cid:21) of 75%. A similar equivalent efficiency is featured by
a high-speed detector by Zavatta et al. [13]. A 14-dB
This rescaled, noise contaminated marginal distribution
time-domain detector of Hansen and co-workers[14] has
is then used to reconstruct the Wigner function.
an equivalent efficiency of 96%, and a typical frequency-
Because the marginal distribution is tomographically
domaindetectorwithasignal-to-noiseratioof20dBhas
linked to the WF,
an efficiency of 99%.
e. Experiment We verify our theoretical findings in
pr (X)= W(Xcosθ P sinθ,Xsinθ+P cosθ)dP,
θ Z − a squeezing experiment. The squeezed vacuum source is
(10) a continuous-wave optical parametric amplifier (OPA),
theWignerfunctionreconstructedfromanoisymeasure- operatingatthe 795-nmwavelengthofthe D1 transition
ment is a convolution in atomic rubidium. We have constructed this source
α′4 α′ α′ to perform experiments on interfacing quantum infor-
′ ′ ′ ′
W (X,P) = dX dP W X , P
EN πTα2 ZZ (cid:18)α α (cid:19) mation between light and atoms, in particular storage
[15,16]andRamanadiabatictransferring[17,18]oflight
(X X′)2+(P P′)2
exp − − . (11) inatomicvapor. Similarsourceshavebeenreportedvery
× (cid:20)− T/α′2 (cid:21) recently by two groups [19, 20]
3
FIG. 1: Quadrature noise spectrum of the squeezed vacuum
generated by the OPA, obtained by a simultaneous scan of
the local oscillator phase and the detection frequency. The
green line is theshot noise spectrum.
The optical cavity of the OPA is of bow-tie configura-
tion composed of four mirrors,among which two are flat
and two are concave with a curvature radius of 100 mm.
The cavity is singly resonant for the generated 795-nm
wavelength. As the output couplerwe use one ofthe flat
cavity mirrors, which had a reflectivity of 93%. Other FIG. 2: Experimental results. a) Observed shot noise as a
cavitymirrorshadanover99.9%reflectivity. Withthese function of the local oscillator power. Also displayed is a
linearfitandtheelectronicnoiselevel. b)Observedsqueezing
parameters,thecavityhada460-MHzthreefreespectral
as a function of the local oscillator power. The solid line is
range with the resonant line width of 6 MHz
calculated usingEq. (4).
The source employs a 5-mm long periodically poled
KTiOPO crystalas the nonlinear element. The crystal,
4
manufactured by Raicol, features a 3.175 µm poling pe-
analyzer.
riod and double antireflection coating on both sides. It
is placed at the beam waist between the curved mirrors. Figure1showstheelectronicspectrumofsqueezedvac-
The OPA is pumped by up to 240 mW of 397.5 nm uumfieldgeneratedbytheOPA.Upto3dBofsqueezing
light. The pump field is the second harmonic of a fre- is observed within the cavity linewidth.
quency stabilized titanium-sapphire master laser. The To verify the predictions of Eq. (14), we stabilize the
same laser is used to provide the local oscillator for the pump levelat 140 mW and set the spectrumanalyzerto
homodyne detector. The OPA cavity is locked at reso- run in the zero span mode at a detection frequency of 1
nancebymeansofanadditionaldiodelaserphaselocked MHz. The signal-to-noise ratio of the detector is varied
to the master oscillator at 3 free spectral ranges of the by changing the local oscillator intensity. At each in-
OPA cavity. The cavity lock is implemented using the tensity value, we record three spectrum analyzer traces:
Pound-Drever-Hall method, using a beam modulated at electronic noise (measured by blocking both photodi-
20MHz andpropagatinginthe directionoppositetothe odes of the homodyne detector), shot noise (measured
pumped mode. by blocking the squeezed vacuum signal) and the phase-
The circuit for homodyne detection uses two Hama- dependent squeezed vacuum quadrature. The highest
matsu S3883 photodiodes of 94% quantum efficiency in (hQ2+i) and lowest (hQ2−i) quadrature noise levels of the
a“back-to-back”configuration, akin to that employed in squeezingmeasurementarethendeterminedandnormal-
Refs. [13, 14], followedby a OPA847operationalpream- ized to the shot noise level. The uncertainty of Q2± is
h i
plifier. Thedetectorexhibitsexcellentlinearityandanup estimatedas0.2dB;itis muchsmallerforthe shot-noise
to 12-17 dB signal-to-noise ratio over a 100-MHz band- and electronic-noise data. The results of these measure-
width, with local oscillator powers up to 10 mW. At a ments are shown in Fig. 2.
1-MHz sideband relevant for this experiment, the high- To determine the quantum efficiency, we use the fact
est signal-to-noise ratio of the detector exceeds 17 dB, thatforthepuresqueezedstate,theproductofthemean
which, according to Eq. (14), corresponds to equivalent square quadrature uncertainties is the minimum allowed
opticallossesofonly2%. Thedetectoroutputisobserved by the uncertainty principle, i.e. Q2 Q2 =
h pure,+ih pure,−i
in the frequency domainusing anAgilentESA spectrum 1/4, where the normalization factor corresponds to the
4
Measuring Q2 and Q2 andsolvingEqs. (15),wecan
h +i h −i
find the value of η, i.e. how much loss a pure squeezed
state would have experienced to generate the state with
the quadrature noise levels observed[21]:
(2 Q2 1)(1 2 Q2 )
η = h +i− − h −i . (16)
2 Q2 +2 Q2 2
h +i h −i−
The result of applying this method to our squeezed vac-
uum data is displayed in Fig. 3 along with a fit with
Eq. (14) multiplied by an overallefficiency factor of 0.51
FIG.3: Equivalentquantumefficiencyηeq asafunctionofthe (emergingduetovariousopticallosses). Theexperimen-
detector’s signal-to-noise ratio S. This ratio is expressed in tal data matches well to the theoretical prediction.
decibels as 10log S. Both theexperimental results and the
10 f. Conclusion We have found that there is a strong
theoreticalpreediction(14)(multipliedbyanopticalefficiency
similarity between the effects of optical absorption and
factor of 0.51) are displayed.
electronic noise on the state reconstructed by means of
opticalhomodyne tomography. Therefore,the detector’s
shot noise level. After undergoing an optical loss, the Gaussianelectronicnoisecanbetreatedasanadditional
squeezedstateremainsGaussian,butlosesitspurity. Us- optical loss which is related to the detector’s signal-to-
ing the beam splitter model of absorption, we find that noise ratio by Eq. (14). The finding is verified in a
uponpropagatingthroughanattenuatoroftransmission squeezing experiment involving a narrowband nonclas-
η,thequadraturenoisereducesbyafactorofη,butgains sical light source suitable for experiments on interfacing
excess vacuum noise of intensity 1 η: quantum information between light and atoms.
−
Acknowledgements We thank Michael Raymer for
Q2 = ηQ2 +(1 η)/2; (15a)
h +i h pure,+i − discussions. A. L.’s research is supported by Quan-
Q2 = ηQ2 +(1 η)/2. (15b) tumWorks, NSERC, CFI, AIF, and CIAR.
h −i h pure,−i −
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