Experimental nonclassicality of single-photon-added thermal light states
Alessandro Zavatta,1, 2, ∗ Valentina Parigi,2,3 and Marco Bellini1, 3, †
1Istituto Nazionale di Ottica Applicata (CNR), L.go E. Fermi, 6, I-50125, Florence, Italy
2Department of Physics, University of Florence, I-50019 Sesto Fiorentino, Florence, Italy
3LENS, Via Nello Carrara 1, 50019 Sesto Fiorentino, Florence, Italy
(Dated: October 29, 2018)
We report the experimental realization and tomographic analysis of novel quantum light states
obtained by exciting a classical thermal field by a single photon. Such states, although completely
incoherent, possess a tunable degree of quantumness which is here exploited to put to a stringent
experimental test some of the criteria proposed for the proof and the measurement of state non-
classicality. The quantum character of the states is also given in quantum information terms by
evaluating the amount of entanglement that they can produce.
PACS numbers: 42.50.Dv, 03.65.Wj
INTRODUCTION
The definition and the measurement of the nonclassi-
cality of a quantum light state is a hot and widely dis-
cussed topic in the physics community; nonclassical light
is the starting point for generating even more nonclas-
sical states [1, 2] or producing the entanglement which
is essential to implement quantum information protocols
with continuous variables [3, 4]. A quantum state is said
to be nonclassical when it cannot be written as a mixture
of coherent states. In terms of the Glauber-Sudarshan
P representation [5, 6], the P function of a nonclassi-
cal state is highly singular or not positive, i.e. it cannot
be interpreted as a classical probability distribution. In
general however, since the P function can be badly be-
haved, it cannot be connected to any observable quan-
tity. In recent years, a nonclassicality criterion based on
the measurable quadrature distributions obtained from
homodyne detection has been proposed by Richter and
Vogel [7]. Moreover, a variety of nonclassical states has
recently been characterized by means of the negative-
ness of their Wigner function [8, 9, 10, 11], this however
being just a sufficient and not necessary condition for
nonclassicality [12]. It is still an open question which
is the universal way to experimentally characterize the
nonclassicality of a quantum state.
A conceptually simple way to generate a quantum light
state with a varying degree of nonclassicality consists in
adding a single photon to any completely classical one.
This is quite different from photon subtraction which, on
the other hand, produces a nonclassical state only when
starting from an already nonclassical one [13, 14].
In this Letter we report the generation and the analy-
sis of single-photon-added thermal states (SPATSs), i.e.,
completely classical states excited by a single photon,
∗Electronic address: azavatta@inoa.it
†Electronic address: bellini@inoa.it
first described by Agarwal and Tara in 1992 [15]. We
use the techniques of conditioned parametric amplifica-
tion recently demonstrated by our group [10, 11] to gen-
erate such states, and we employ ultrafast pulsed ho-
modyne detection and quantum tomography to investi-
gate their character. The peculiar nonclassical behavior
of SPATSs has recently triggered an interesting debate
[7, 16] and has been described in several theoretical pa-
pers [14, 15, 16, 17, 18]; their experimental generation
has already been proposed, although with more complex
schemes [14, 18, 19], but never realized. Thanks to their
adjustable degree of quantumness, these states are an
ideal benchmark to test the different experimental crite-
ria of nonclassicality recently proposed, and to investi-
gate the possibility of multi-photon entanglement gener-
ation. The nonclassicality of SPATSs is here analyzed by
reconstructing their negative-valued Wigner functions,
by using the quadrature-based Richter-Vogel (RV) crite-
rion, and finally comparing these with two other methods
based on quantum tomography. In particular, we show
that the so-called entanglement potential [20] is a sensi-
tive measurement of nonclassicality, and that it provides
quantitative data about the possible use of the states for
quantum information applications in terms of the entan-
glement that they would generate once sent to a 50-50
beam-splitter.
EXPERIMENTAL
The main source of our apparatus is a mode-locked
Ti:Sa laser which emits 1.5 ps pulses with a repetition
rate of 82 MHz. The pulse train is frequency-doubled to
393 nm by second harmonic generation in a LBO crystal.
The spatially-cleaned UV beam then serves as a pump for
a type-I BBO crystal which generates spontaneous para-
metric down-conversion (SPDC) at the same wavelength
of the laser source. Pairs of SPDC photons are emitted
in two distinct spatial channels called signal and idler.
Along the idler channel the photons are strongly filtered
http://arxiv.org/abs/0704.0179v1
mailto:azavatta@inoa.it
mailto:bellini@inoa.it
in the spectral and spatial domain by means of etalon
cavities and by a single-mode fiber which is directly con-
nected to a single-photon-counting module (further de-
tails are given in [9, 11]). The signal field is mixed with
a strong local oscillator (LO, an attenuated portion of
the main laser source) by means of a 50% beam-splitter
(BS). The BS outputs are detected by two photodiodes
connected to a wide-bandwidth amplifier which provides
the difference (homodyne) signal between the two pho-
tocurrents on a pulse-to-pulse basis [21]. Whenever a
single photon is detected in the idler channel, an homo-
dyne measurement is performed on the correlated spatio-
temporal mode of the signal channel by storing the corre-
sponding electrical signal (proportional to the quadrature
operator value) on a digital scope.
FIG. 1: (color online) Experimental setup. HR (HT) is a high
reflectivity (transmittivity) beam splitter; SPCM is a single-
photon-counting module; all other symbols are defined in the
text. The mode-cleaning fiber used to inject the thermal state
coming from the rotating ground glass disk (RD) into the
parametric crystal is not shown here for clarity.
When no field is injected in the SPDC crystal, con-
ditioned single-photon Fock states are generated from
spontaneous emission in the signal channel [8, 9]. We
have recently shown that, if the SPDC crystal is injected
with a coherent state, stimulated emission comes into
play and single-photon excitation of such a pure state is
obtained [10, 11]. However, a coherent state is still at the
border between the quantum and the classical regimes;
it is therefore extremely interesting to use a truly clas-
sical state, like the thermal one, as the input, and to
observe its degaussification [13]. In order to avoid the
technical problems connected to the handling of a true
high-temperature thermal source, we use pseudo-thermal
one, obtained by inserting a rotating ground glass disk
(RD) in a portion of the laser beam (see Fig.1). By cou-
pling a fraction (much smaller than the typical speckle
size) of the randomly scattered light into a single-mode
fiber, at the output we obtain a clean spatial mode with
random amplitude and phase yielding the photon distri-
bution typical of a thermal source [22] which is then used
to inject the parametric amplifier.
PROPERTIES OF SPATSS
In order to describe the state generated in our exper-
iment, we give a general treatment of photon addition
based on conditioned parametric amplification. By first-
order perturbation theory, the output of the parametric
amplifier when a pure state |ϕm〉 is injected along the
signal channel is given by
|ψm〉 = [1 + (gâ†sâ
i − g
∗âsâi)] |ϕm〉s |0〉i , (1)
where g accounts for the coupling and the amplitude of
the pump and â, â† are the usual noncommuting annihi-
lation and creation operators. For a generic signal input,
the output state of the parametric amplifier can be writ-
ten as
ρ̂out =
Pm |ψm〉 〈ψm| (2)
where the input mixed state is ρ̂s =
Pm |ϕm〉 〈ϕm| and
Pm is the probability for the state |ϕm〉. If we condition
the preparation of the signal state to single-photon de-
tection on the idler channel, we obtain the prepared state
ρ̂ = Tri(ρ̂out |1〉i 〈1|i) = |g|
2â†sρ̂sâs. (3)
When the input state ρ̂s is a thermal state with mean
photon number n̄, we obtain that the single-photon-
added thermal state is described by the following density
operator expressed in the Fock base:
n̄(n̄+ 1)
1 + n̄
n |n〉 〈n|. (4)
The lack of the vacuum term and the rescaling of higher
excited terms is evident in this expression. The P phase-
space representation can be easily calculated and is given
by (see also [15])
P (α) =
[(1 + n̄)|α|2 − n̄]e−|α|
2/n̄, (5)
while the corresponding Wigner function reads as
W (α) =
|2α|2(1 + n̄)− (1 + 2n̄)
(1 + 2n̄)3
e−2|α|
2/(1+2n̄) (6)
where α = x + iy. SPATSs have a well-behaved P func-
tion which is always negative around α = 0; this feature
is also present in the Wigner function and assures their
nonclassicality, however both P (0) andW (0) tend to zero
in the limit of n̄→ ∞.
DATA ANALYSIS AND DISCUSSION
After the acquisition of about 105 quadrature values
with random phases, we have performed the reconstruc-
tion of the diagonal density matrix elements using the
maximum likelihood estimation [23]. This method gives
the density matrix that most likely represents the mea-
sured homodyne data. Firstly, we build the likelihood
function contracted for a density matrix truncated to 25
diagonal elements (with the constraints of Hermiticity,
positivity and normalization), then the function is max-
imized by an iterative procedure [24, 25] and the errors
on the reconstructed density matrix elements are evalu-
ated using the Fisher information [25]. The results are
shown in Fig. 2, together with the corresponding recon-
structed [11] Wigner functions for two different temper-
FIG. 2: (color online) Experimentally reconstructed diagonal
density matrix elements (reconstruction errors of statistical
origin are of the order of 1%) and Wigner functions for ther-
mal states (left) and SPATSs (right): a) n̄ = 0.08; b) n̄ = 1.15.
Filled circles indicate the density matrix elements calculated
for thermal states and SPATSs with the expected efficiencies.
atures of the injected thermal state. Since in the low-
gain regime the count rate in the idler channel is given
by 〈n̂〉 = Tr(ρ̂outâ†i âi) = |g|2(1 + n̄), the mean photon
number values n̄ reported in Fig. 2 and in the following
are obtained from the ratio between the trigger count
rates when the thermal injection is present and when it
is blocked (see Ref. [11] and references therein).
The finite experimental efficiency in the preparation
and homodyne detection of SPATSs is fully accounted for
by a loss mechanism which can be modeled by the trans-
mission of the ideal state ρ̂ of Eq.(4) through a beam
splitter of trasmittivity η coupling vacuum into the de-
tection mode, such that the detected state ρ̂η is finally
found as:
ρ̂η = TrR{Uη(ρ̂ |0〉 〈0|)U †η} (7)
where Uη is the beam splitter operator acting on two in-
put modes containing the state ρ̂ and the vacuum, and
the states of the reflected mode (indicated by R) are
traced out. In the case of finite efficiency the expression
for the Wigner function thus results:
Wη(α) =
1 + 2η[n̄+ 2(1 + n̄)|α|2 − 2n̄η − 1]
(1 + 2n̄η)3
−2|α|2
1+2n̄η .
It should be noted that the value of experimental ef-
ficiency which best fits the data is the same (η = 0.62)
as that obtained for single-photon Fock states (i.e., with-
out injection), and implies that only a portion of vac-
uum due to losses enters the mode during the generation
of SPATS. Thanks to a very low rate of dark counts in
the trigger detector, the portion of the injected thermal
state which survives the conditional preparation proce-
dure and contributes to degradation of the SPATSs is in
fact completely negligible. However, since the nonclassi-
cal features of the state get weaker for large n̄, a limited
efficiency (η < 1) has the effect of progressively hiding
them among unwanted vacuum components.
Indeed, the measured negativity of the Wigner func-
tion at the origin (see Fig.3a and b) rapidly gets smaller
as the mean photon number of the input thermal state is
increased. With the current level of efficiency and recon-
struction accuracy we are able to prove the nonclassical-
ity of all the generated states (up to n̄ = 1.15), but one
may expect to experimentally detect negativity above the
reconstruction noise, and thus prove state nonclassical-
ity, up to about n̄ ≈ 1.5 (also see Fig.6a). It should
be noted that, even for a single-photon Fock state, the
Wigner function loses its negativity for efficiencies lower
than 50%, so that surpassing this experimental threshold
is an essential requisite in order to use this nonclassicality
criterion.
After having experimentally proved the nonclassical-
ity of the states for all the investigated values of n̄, it
is interesting to verify the nonclassical character of the
measured SPATSs also using different criteria.
The first one has been recently proposed by Richter
and Vogel [7] and is based on the characteristic func-
tion G(k, θ) = 〈eikx̂(θ)〉 of the quadratures (i.e., the
Fourier transform of the quadrature distribution), where
x̂(θ) = (âe−iθ + â†eiθ)/2 is the phase-dependent quadra-
ture operator. At the first-order, the criterion defines
a phase-independent state as nonclassical if there is a
value of k such that |G(k, θ)| ≡ |G(k)| > Ggr(k), where
-2 -1 0 1 2
0.0 0.4 0.8 1.2
-0.16
-0.12
-0.08
-0.04
 0.08
 0.34
 0.70
 1.15
Classical limit
FIG. 3: (color online) a) Sections of the experimentally re-
constructed Wigner functions for SPATSs with different n̄; b)
Experimental values for the minimum of the Wigner func-
tion W (0) as a function of n̄ for SPATSs (solid squares)
and for single-photon Fock states (empty circles) obtained
by blocking the injection; the values calculated from Eq.(8)
for η = 0.62 (solid curves) are in very good agreement with
experimental data and clearly show the appropriateness of
the model. Negativity of the Wigner function is a sufficient
condition for affirming the nonclassical character of the state.
Ggr(k) is the characteristic function for the vacuum mea-
sured when the signal beam is blocked before homodyne
detection. In other words, the evidence of structures nar-
rower than those associated to vacuum in the quadrature
distribution is a sufficient condition to define a nonclas-
sical state [12]. However, it has been shown that non-
classical states exist (as pointed out by Diósi [16] for a
vacuum-lacking thermal state [17], which is very similar
to SPATSs) which fail to fulfil such inequality; when this
happens, the first-order Richter-Vogel (RV) criterion has
to be extended to higher orders: the second-order RV
inequality reads as
2G2(k/2)Ggr(k/
2)−G(k) > Ggr(k). (9)
It is evident that, as higher orders are investigated,
the increasing sensitivity to experimental and statistical
noise may soon become unmanageable.
The measured |G(k)| and left hand side of Eq. (9) are
plotted in Fig. 4a) and b), together with the Ggr(k) char-
acteristic function, also obtained from the experimental
quadrature distribution of vacuum. While the detected
0 2 4 6 8
0 2 4 6 8
1.0a)
 Ggr(k)
 0.53
 0.70
 0.90
 1.15
4 6 8
 Ggr(k)
 0.08
 0.34
FIG. 4: (color online) Experimental characteristic functions
involved in the RV nonclassicality criterion for the detected
SPATSs: a) first order; b) second order (the inset shows a
magnified view of the region where the state with n̄ = 0.53 is
just slightly fulfilling the criterion).
SPATSs satisfy the nonclassical first-order RV criterion
only for the two lowest values of n̄, it is necessary to
extend the criterion to the second order to just barely
show nonclassicality at large values of k for n̄ = 0.53 (see
the inset of Fig.4b, where the shaded region indicates the
error area of the experimental Ggr(k)).
At higher temperatures, no sign of nonclassical be-
havior is experimentally evident with this approach, al-
though the Wigner function of the corresponding states
still clearly exhibits a measurable negativity (see Fig.3).
It should be noted that the second-order RV criterion for
the ideal state of Eq. (4) is expected to prove the nonclas-
sicality of SPATSs up to n̄ ≈ 0.6 [7]; however, when the
limited experimental efficiency and the statistical noise
is taken into account, it will start to fail even earlier.
The tomographic reconstruction of the state that was
earlier used for the nonclassicality test based on the neg-
ativity of the Wigner function, can also be exploited to
test alternative criteria: for example by reconstructing
the photon-number distribution ρn = 〈n| ρ̂meas |n〉 and
then looking for strong modulations in neighboring pho-
ton probabilities by the following relationship [26, 27]
B(n) ≡ (n+ 2)ρnρn+2 − (n+ 1)ρ2n+1 < 0, (10)
introduced by Klyshko in 1996, which is known to hold
for nonclassical states. In the ideal situation of unit ef-
ficiency SPATSs should always give B(0) < 0 due to
the absence of the vacuum term ρ0, in agreement with
Ref. [17]. The experimental results obtained for B(0)
by using the reconstructed density matrix ρ̂meas are pre-
sented in Fig.5a) together with those calculated for the
state described by ρ̂η (see Eq.(7)) with η = 0.62. The
agreement between the experimental data and the ex-
pected ones is again very satisfactory, showing that our
model state ρ̂η well represents the experimental one. Our
current efficiency should in principle allow us to find neg-
ative values of B(0) even for much larger values of n̄;
however, if one takes the current reconstruction errors
due to statistical noise into account, the maximum n̄ for
which the corresponding SPATS can be safely declared
nonclassical is of the order of 2. It should be noted that,
differently from the Wigner function approach, here the
nonclassicality can be proved even for experimental effi-
ciencies much lower than 50%, as far as the mean photon
number of the thermal state is not too high (see Fig.6b).
Finally, it is particularly interesting to measure the en-
tanglement potential (EP) of our states as recently pro-
posed by Asboth et al. [20]. This measurement is based
on the fact that, when a nonclassical state is mixed with
vacuum on a 50-50 beam splitter, some amount of entan-
glement (depending on the nonclassicality of the input
state) appears between the BS outputs. No entangle-
ment can be produced by a classical initial state. For
a given single-mode density operator ρ̂, one calculates
the entanglement of the bipartite state at the BS out-
puts ρ̂′ = UBS(ρ̂|0〉〈0|)U †BS by means of the logarithmic
negativity EN (ρ̂
′) based on the Peres separability cri-
terion and defined in [28], where UBS is the 50-50 BS
transformation. The computed entanglement potentials
for the reconstructed SPATS density matrices ρ̂meas are
shown in Fig. 5b) together with those expected at the
experimentally-evaluated efficiency (i.e., obtained from
ρ̂η with η = 0.62). The EP is definitely greater than zero
(by more than 13σ) for all the detected states, thus con-
firming that they are indeed nonclassical, in agreement
with the findings obtained by the measurement of B(0)
and W (0). As a comparison, the EP would be equal to
unity for a pure single-photon Fock state, while it would
reduce to 0.43 for a single-photon state mixed with vac-
uum ρ̂ = (1− η) |0〉 〈0|+ η |1〉 〈1| with η = 0.62.
To summarize, the three tomographic approaches to
test nonclassicality have all been able to experimentally
prove it for all the generated states (i.e., SPATSs with
0.0 0.4 0.8 1.2
0.0 0.4 0.8 1.2
Classical limit
Classical limit
FIG. 5: (color online) a) Experimental data (squares) and
calculated values (solid curve) of B(0) as a function of n̄;
negative values indicate nonclassicality of the state. b) The
same as above for the entanglement potential (EP) of the
SPATSs; here nonclassicality is demonstrated by EP values
greater than zero.
an average number of photons in the seed thermal state
up to n̄ = 1.15) for a global experimental efficiency of
η = 0.62. In order to gain a better view of the range of
values for n̄ and for the global experimental efficiency η
which allow to prove the nonclassical character of single-
photon-added thermal states under realistic experimen-
tal conditions, we have calculated the indicators W (0),
B(0), and EP from the model state described by ρ̂η. The
results are shown in Fig.6: the contour plots define the
regions of parameters where the detected state is classi-
cal (white areas), where it would result nonclassical if the
reconstruction errors coming from statistical noise could
be neglected (grey areas) and, finally, where it is defi-
nitely nonclassical even with the current level of noise
(black areas). From such plots it is evident that, as al-
ready noted, the Wigner function negativity only works
for sufficiently high efficiencies, while both B(0) and EP
are able to detect nonclassical behavior even for η < 50%.
In particular, the entanglement potential is clearly seen
to be the most powerful criterion, at least for these par-
ticular states, and to allow for an experimental proof of
a) b)
W(0) EPB(0)
FIG. 6: Calculated regions of nonclassical behavior of SPATSs as a function of n̄ and η according to: a) the negativity
of the Wigner function at the origin W (0); b) the Klyshko criterion B(0); c) the entanglement potential EP. White areas
indicate classical behavior; grey areas indicate where a potentially nonclassical character is not measurable due to experimental
reconstruction noise (estimated as the average error on the experimentally reconstructed parameters); black areas indicate
regions where the nonclassical character is measurable given the current statistical uncertainties.
nonclassicality for all combinations of n̄ and η, as long as
reconstruction errors can be neglected. Also considering
the current experimental parameters, EP should show
the quantum character of SPATSs even for n̄ > 3, thus
demonstrating its higher immunity to noise.
Although at a different degree, all three indicators are
however very sensitive to the presence of reconstruction
noise of statistical origin which may completely mask the
nonclassical character of the states, even for relatively low
values of n̄ or for low efficiencies. In order to unambigu-
ously prove the quantum character of higher-temperature
SPATSs in these circumstances the only possibility is to
reduce the “grey zone” by significantly increasing the
number of quadrature measurements.
CONCLUSIONS
In conclusion, we have generated a completely incoher-
ent light state possessing an adjustable degree of quan-
tumness which has been used to experimentally test and
compare different criteria of nonclassicality. Although
the direct analysis of quadrature distributions, done fol-
lowing the criterion proposed by Richter and Vogel, has
been able to show the nonclassical character of some of
the states with lower mean photon numbers, quantum to-
mography, with the reconstruction of the density matrix
and the Wigner function from the homodyne data, has
allowed us to unambiguously show the nonclassical char-
acter of all the generated states: three different criteria,
the negativity of the Wigner function, the Klyshko crite-
rion and the entanglement potential, have been used with
varying degree of effectiveness in revealing nonclassical-
ity. Besides being a useful tool for the measurement of
nonclassicality through the definition of the entanglement
potential, the combination of nonclassical field states -
such as those generated here - with a beam-splitter, can
be viewed as a simple entangling device generating multi-
photon states with varying degree of purity and entangle-
ment and allowing the future investigation of continuous-
variable mixed entangled states [29].
ACKNOWLEDGMENTS
The authors gratefully acknowledge Koji Usami for
giving the initial stimulus for this work and Milena
D’Angelo and Girish Agarwal for useful discussions and
comments. This work was partially supported by Ente
Cassa di Risparmio di Firenze and MIUR, under the
PRIN initiative and FIRB contract RBNE01KZ94.
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  We report the experimental realization and tomographic analysis of novel
quantum light states obtained by exciting a classical thermal field by a single
photon. Such states, although completely incoherent, possess a tunable degree
of quantumness which is here exploited to put to a stringent experimental test
some of the criteria proposed for the proof and the measurement of state
non-classicality. The quantum character of the states is also given in quantum
information terms by evaluating the amount of entanglement that they can
produce.

Experimental nonclassicality of single-photon-added thermal light states
Alessandro Zavatta,1, 2, ∗ Valentina Parigi,2,3 and Marco Bellini1, 3, †
1Istituto Nazionale di Ottica Applicata (CNR), L.go E. Fermi, 6, I-50125, Florence, Italy
2Department of Physics, University of Florence, I-50019 Sesto Fiorentino, Florence, Italy
3LENS, Via Nello Carrara 1, 50019 Sesto Fiorentino, Florence, Italy
(Dated: October 29, 2018)
We report the experimental realization and tomographic analysis of novel quantum light states
obtained by exciting a classical thermal field by a single photon. Such states, although completely
incoherent, possess a tunable degree of quantumness which is here exploited to put to a stringent
experimental test some of the criteria proposed for the proof and the measurement of state non-
classicality. The quantum character of the states is also given in quantum information terms by
evaluating the amount of entanglement that they can produce.
PACS numbers: 42.50.Dv, 03.65.Wj
INTRODUCTION
The definition and the measurement of the nonclassi-
cality of a quantum light state is a hot and widely dis-
cussed topic in the physics community; nonclassical light
is the starting point for generating even more nonclas-
sical states [1, 2] or producing the entanglement which
is essential to implement quantum information protocols
with continuous variables [3, 4]. A quantum state is said
to be nonclassical when it cannot be written as a mixture
of coherent states. In terms of the Glauber-Sudarshan
P representation [5, 6], the P function of a nonclassi-
cal state is highly singular or not positive, i.e. it cannot
be interpreted as a classical probability distribution. In
general however, since the P function can be badly be-
haved, it cannot be connected to any observable quan-
tity. In recent years, a nonclassicality criterion based on
the measurable quadrature distributions obtained from
homodyne detection has been proposed by Richter and
Vogel [7]. Moreover, a variety of nonclassical states has
recently been characterized by means of the negative-
ness of their Wigner function [8, 9, 10, 11], this however
being just a sufficient and not necessary condition for
nonclassicality [12]. It is still an open question which
is the universal way to experimentally characterize the
nonclassicality of a quantum state.
A conceptually simple way to generate a quantum light
state with a varying degree of nonclassicality consists in
adding a single photon to any completely classical one.
This is quite different from photon subtraction which, on
the other hand, produces a nonclassical state only when
starting from an already nonclassical one [13, 14].
In this Letter we report the generation and the analy-
sis of single-photon-added thermal states (SPATSs), i.e.,
completely classical states excited by a single photon,
∗Electronic address: azavatta@inoa.it
†Electronic address: bellini@inoa.it
first described by Agarwal and Tara in 1992 [15]. We
use the techniques of conditioned parametric amplifica-
tion recently demonstrated by our group [10, 11] to gen-
erate such states, and we employ ultrafast pulsed ho-
modyne detection and quantum tomography to investi-
gate their character. The peculiar nonclassical behavior
of SPATSs has recently triggered an interesting debate
[7, 16] and has been described in several theoretical pa-
pers [14, 15, 16, 17, 18]; their experimental generation
has already been proposed, although with more complex
schemes [14, 18, 19], but never realized. Thanks to their
adjustable degree of quantumness, these states are an
ideal benchmark to test the different experimental crite-
ria of nonclassicality recently proposed, and to investi-
gate the possibility of multi-photon entanglement gener-
ation. The nonclassicality of SPATSs is here analyzed by
reconstructing their negative-valued Wigner functions,
by using the quadrature-based Richter-Vogel (RV) crite-
rion, and finally comparing these with two other methods
based on quantum tomography. In particular, we show
that the so-called entanglement potential [20] is a sensi-
tive measurement of nonclassicality, and that it provides
quantitative data about the possible use of the states for
quantum information applications in terms of the entan-
glement that they would generate once sent to a 50-50
beam-splitter.
EXPERIMENTAL
The main source of our apparatus is a mode-locked
Ti:Sa laser which emits 1.5 ps pulses with a repetition
rate of 82 MHz. The pulse train is frequency-doubled to
393 nm by second harmonic generation in a LBO crystal.
The spatially-cleaned UV beam then serves as a pump for
a type-I BBO crystal which generates spontaneous para-
metric down-conversion (SPDC) at the same wavelength
of the laser source. Pairs of SPDC photons are emitted
in two distinct spatial channels called signal and idler.
Along the idler channel the photons are strongly filtered
http://arxiv.org/abs/0704.0179v1
mailto:azavatta@inoa.it
mailto:bellini@inoa.it
in the spectral and spatial domain by means of etalon
cavities and by a single-mode fiber which is directly con-
nected to a single-photon-counting module (further de-
tails are given in [9, 11]). The signal field is mixed with
a strong local oscillator (LO, an attenuated portion of
the main laser source) by means of a 50% beam-splitter
(BS). The BS outputs are detected by two photodiodes
connected to a wide-bandwidth amplifier which provides
the difference (homodyne) signal between the two pho-
tocurrents on a pulse-to-pulse basis [21]. Whenever a
single photon is detected in the idler channel, an homo-
dyne measurement is performed on the correlated spatio-
temporal mode of the signal channel by storing the corre-
sponding electrical signal (proportional to the quadrature
operator value) on a digital scope.
FIG. 1: (color online) Experimental setup. HR (HT) is a high
reflectivity (transmittivity) beam splitter; SPCM is a single-
photon-counting module; all other symbols are defined in the
text. The mode-cleaning fiber used to inject the thermal state
coming from the rotating ground glass disk (RD) into the
parametric crystal is not shown here for clarity.
When no field is injected in the SPDC crystal, con-
ditioned single-photon Fock states are generated from
spontaneous emission in the signal channel [8, 9]. We
have recently shown that, if the SPDC crystal is injected
with a coherent state, stimulated emission comes into
play and single-photon excitation of such a pure state is
obtained [10, 11]. However, a coherent state is still at the
border between the quantum and the classical regimes;
it is therefore extremely interesting to use a truly clas-
sical state, like the thermal one, as the input, and to
observe its degaussification [13]. In order to avoid the
technical problems connected to the handling of a true
high-temperature thermal source, we use pseudo-thermal
one, obtained by inserting a rotating ground glass disk
(RD) in a portion of the laser beam (see Fig.1). By cou-
pling a fraction (much smaller than the typical speckle
size) of the randomly scattered light into a single-mode
fiber, at the output we obtain a clean spatial mode with
random amplitude and phase yielding the photon distri-
bution typical of a thermal source [22] which is then used
to inject the parametric amplifier.
PROPERTIES OF SPATSS
In order to describe the state generated in our exper-
iment, we give a general treatment of photon addition
based on conditioned parametric amplification. By first-
order perturbation theory, the output of the parametric
amplifier when a pure state |ϕm〉 is injected along the
signal channel is given by
|ψm〉 = [1 + (gâ†sâ
i − g
∗âsâi)] |ϕm〉s |0〉i , (1)
where g accounts for the coupling and the amplitude of
the pump and â, â† are the usual noncommuting annihi-
lation and creation operators. For a generic signal input,
the output state of the parametric amplifier can be writ-
ten as
ρ̂out =
Pm |ψm〉 〈ψm| (2)
where the input mixed state is ρ̂s =
Pm |ϕm〉 〈ϕm| and
Pm is the probability for the state |ϕm〉. If we condition
the preparation of the signal state to single-photon de-
tection on the idler channel, we obtain the prepared state
ρ̂ = Tri(ρ̂out |1〉i 〈1|i) = |g|
2â†sρ̂sâs. (3)
When the input state ρ̂s is a thermal state with mean
photon number n̄, we obtain that the single-photon-
added thermal state is described by the following density
operator expressed in the Fock base:
n̄(n̄+ 1)
1 + n̄
n |n〉 〈n|. (4)
The lack of the vacuum term and the rescaling of higher
excited terms is evident in this expression. The P phase-
space representation can be easily calculated and is given
by (see also [15])
P (α) =
[(1 + n̄)|α|2 − n̄]e−|α|
2/n̄, (5)
while the corresponding Wigner function reads as
W (α) =
|2α|2(1 + n̄)− (1 + 2n̄)
(1 + 2n̄)3
e−2|α|
2/(1+2n̄) (6)
where α = x + iy. SPATSs have a well-behaved P func-
tion which is always negative around α = 0; this feature
is also present in the Wigner function and assures their
nonclassicality, however both P (0) andW (0) tend to zero
in the limit of n̄→ ∞.
DATA ANALYSIS AND DISCUSSION
After the acquisition of about 105 quadrature values
with random phases, we have performed the reconstruc-
tion of the diagonal density matrix elements using the
maximum likelihood estimation [23]. This method gives
the density matrix that most likely represents the mea-
sured homodyne data. Firstly, we build the likelihood
function contracted for a density matrix truncated to 25
diagonal elements (with the constraints of Hermiticity,
positivity and normalization), then the function is max-
imized by an iterative procedure [24, 25] and the errors
on the reconstructed density matrix elements are evalu-
ated using the Fisher information [25]. The results are
shown in Fig. 2, together with the corresponding recon-
structed [11] Wigner functions for two different temper-
FIG. 2: (color online) Experimentally reconstructed diagonal
density matrix elements (reconstruction errors of statistical
origin are of the order of 1%) and Wigner functions for ther-
mal states (left) and SPATSs (right): a) n̄ = 0.08; b) n̄ = 1.15.
Filled circles indicate the density matrix elements calculated
for thermal states and SPATSs with the expected efficiencies.
atures of the injected thermal state. Since in the low-
gain regime the count rate in the idler channel is given
by 〈n̂〉 = Tr(ρ̂outâ†i âi) = |g|2(1 + n̄), the mean photon
number values n̄ reported in Fig. 2 and in the following
are obtained from the ratio between the trigger count
rates when the thermal injection is present and when it
is blocked (see Ref. [11] and references therein).
The finite experimental efficiency in the preparation
and homodyne detection of SPATSs is fully accounted for
by a loss mechanism which can be modeled by the trans-
mission of the ideal state ρ̂ of Eq.(4) through a beam
splitter of trasmittivity η coupling vacuum into the de-
tection mode, such that the detected state ρ̂η is finally
found as:
ρ̂η = TrR{Uη(ρ̂ |0〉 〈0|)U †η} (7)
where Uη is the beam splitter operator acting on two in-
put modes containing the state ρ̂ and the vacuum, and
the states of the reflected mode (indicated by R) are
traced out. In the case of finite efficiency the expression
for the Wigner function thus results:
Wη(α) =
1 + 2η[n̄+ 2(1 + n̄)|α|2 − 2n̄η − 1]
(1 + 2n̄η)3
−2|α|2
1+2n̄η .
It should be noted that the value of experimental ef-
ficiency which best fits the data is the same (η = 0.62)
as that obtained for single-photon Fock states (i.e., with-
out injection), and implies that only a portion of vac-
uum due to losses enters the mode during the generation
of SPATS. Thanks to a very low rate of dark counts in
the trigger detector, the portion of the injected thermal
state which survives the conditional preparation proce-
dure and contributes to degradation of the SPATSs is in
fact completely negligible. However, since the nonclassi-
cal features of the state get weaker for large n̄, a limited
efficiency (η < 1) has the effect of progressively hiding
them among unwanted vacuum components.
Indeed, the measured negativity of the Wigner func-
tion at the origin (see Fig.3a and b) rapidly gets smaller
as the mean photon number of the input thermal state is
increased. With the current level of efficiency and recon-
struction accuracy we are able to prove the nonclassical-
ity of all the generated states (up to n̄ = 1.15), but one
may expect to experimentally detect negativity above the
reconstruction noise, and thus prove state nonclassical-
ity, up to about n̄ ≈ 1.5 (also see Fig.6a). It should
be noted that, even for a single-photon Fock state, the
Wigner function loses its negativity for efficiencies lower
than 50%, so that surpassing this experimental threshold
is an essential requisite in order to use this nonclassicality
criterion.
After having experimentally proved the nonclassical-
ity of the states for all the investigated values of n̄, it
is interesting to verify the nonclassical character of the
measured SPATSs also using different criteria.
The first one has been recently proposed by Richter
and Vogel [7] and is based on the characteristic func-
tion G(k, θ) = 〈eikx̂(θ)〉 of the quadratures (i.e., the
Fourier transform of the quadrature distribution), where
x̂(θ) = (âe−iθ + â†eiθ)/2 is the phase-dependent quadra-
ture operator. At the first-order, the criterion defines
a phase-independent state as nonclassical if there is a
value of k such that |G(k, θ)| ≡ |G(k)| > Ggr(k), where
-2 -1 0 1 2
0.0 0.4 0.8 1.2
-0.16
-0.12
-0.08
-0.04
 0.08
 0.34
 0.70
 1.15
Classical limit
FIG. 3: (color online) a) Sections of the experimentally re-
constructed Wigner functions for SPATSs with different n̄; b)
Experimental values for the minimum of the Wigner func-
tion W (0) as a function of n̄ for SPATSs (solid squares)
and for single-photon Fock states (empty circles) obtained
by blocking the injection; the values calculated from Eq.(8)
for η = 0.62 (solid curves) are in very good agreement with
experimental data and clearly show the appropriateness of
the model. Negativity of the Wigner function is a sufficient
condition for affirming the nonclassical character of the state.
Ggr(k) is the characteristic function for the vacuum mea-
sured when the signal beam is blocked before homodyne
detection. In other words, the evidence of structures nar-
rower than those associated to vacuum in the quadrature
distribution is a sufficient condition to define a nonclas-
sical state [12]. However, it has been shown that non-
classical states exist (as pointed out by Diósi [16] for a
vacuum-lacking thermal state [17], which is very similar
to SPATSs) which fail to fulfil such inequality; when this
happens, the first-order Richter-Vogel (RV) criterion has
to be extended to higher orders: the second-order RV
inequality reads as
2G2(k/2)Ggr(k/
2)−G(k) > Ggr(k). (9)
It is evident that, as higher orders are investigated,
the increasing sensitivity to experimental and statistical
noise may soon become unmanageable.
The measured |G(k)| and left hand side of Eq. (9) are
plotted in Fig. 4a) and b), together with the Ggr(k) char-
acteristic function, also obtained from the experimental
quadrature distribution of vacuum. While the detected
0 2 4 6 8
0 2 4 6 8
1.0a)
 Ggr(k)
 0.53
 0.70
 0.90
 1.15
4 6 8
 Ggr(k)
 0.08
 0.34
FIG. 4: (color online) Experimental characteristic functions
involved in the RV nonclassicality criterion for the detected
SPATSs: a) first order; b) second order (the inset shows a
magnified view of the region where the state with n̄ = 0.53 is
just slightly fulfilling the criterion).
SPATSs satisfy the nonclassical first-order RV criterion
only for the two lowest values of n̄, it is necessary to
extend the criterion to the second order to just barely
show nonclassicality at large values of k for n̄ = 0.53 (see
the inset of Fig.4b, where the shaded region indicates the
error area of the experimental Ggr(k)).
At higher temperatures, no sign of nonclassical be-
havior is experimentally evident with this approach, al-
though the Wigner function of the corresponding states
still clearly exhibits a measurable negativity (see Fig.3).
It should be noted that the second-order RV criterion for
the ideal state of Eq. (4) is expected to prove the nonclas-
sicality of SPATSs up to n̄ ≈ 0.6 [7]; however, when the
limited experimental efficiency and the statistical noise
is taken into account, it will start to fail even earlier.
The tomographic reconstruction of the state that was
earlier used for the nonclassicality test based on the neg-
ativity of the Wigner function, can also be exploited to
test alternative criteria: for example by reconstructing
the photon-number distribution ρn = 〈n| ρ̂meas |n〉 and
then looking for strong modulations in neighboring pho-
ton probabilities by the following relationship [26, 27]
B(n) ≡ (n+ 2)ρnρn+2 − (n+ 1)ρ2n+1 < 0, (10)
introduced by Klyshko in 1996, which is known to hold
for nonclassical states. In the ideal situation of unit ef-
ficiency SPATSs should always give B(0) < 0 due to
the absence of the vacuum term ρ0, in agreement with
Ref. [17]. The experimental results obtained for B(0)
by using the reconstructed density matrix ρ̂meas are pre-
sented in Fig.5a) together with those calculated for the
state described by ρ̂η (see Eq.(7)) with η = 0.62. The
agreement between the experimental data and the ex-
pected ones is again very satisfactory, showing that our
model state ρ̂η well represents the experimental one. Our
current efficiency should in principle allow us to find neg-
ative values of B(0) even for much larger values of n̄;
however, if one takes the current reconstruction errors
due to statistical noise into account, the maximum n̄ for
which the corresponding SPATS can be safely declared
nonclassical is of the order of 2. It should be noted that,
differently from the Wigner function approach, here the
nonclassicality can be proved even for experimental effi-
ciencies much lower than 50%, as far as the mean photon
number of the thermal state is not too high (see Fig.6b).
Finally, it is particularly interesting to measure the en-
tanglement potential (EP) of our states as recently pro-
posed by Asboth et al. [20]. This measurement is based
on the fact that, when a nonclassical state is mixed with
vacuum on a 50-50 beam splitter, some amount of entan-
glement (depending on the nonclassicality of the input
state) appears between the BS outputs. No entangle-
ment can be produced by a classical initial state. For
a given single-mode density operator ρ̂, one calculates
the entanglement of the bipartite state at the BS out-
puts ρ̂′ = UBS(ρ̂|0〉〈0|)U †BS by means of the logarithmic
negativity EN (ρ̂
′) based on the Peres separability cri-
terion and defined in [28], where UBS is the 50-50 BS
transformation. The computed entanglement potentials
for the reconstructed SPATS density matrices ρ̂meas are
shown in Fig. 5b) together with those expected at the
experimentally-evaluated efficiency (i.e., obtained from
ρ̂η with η = 0.62). The EP is definitely greater than zero
(by more than 13σ) for all the detected states, thus con-
firming that they are indeed nonclassical, in agreement
with the findings obtained by the measurement of B(0)
and W (0). As a comparison, the EP would be equal to
unity for a pure single-photon Fock state, while it would
reduce to 0.43 for a single-photon state mixed with vac-
uum ρ̂ = (1− η) |0〉 〈0|+ η |1〉 〈1| with η = 0.62.
To summarize, the three tomographic approaches to
test nonclassicality have all been able to experimentally
prove it for all the generated states (i.e., SPATSs with
0.0 0.4 0.8 1.2
0.0 0.4 0.8 1.2
Classical limit
Classical limit
FIG. 5: (color online) a) Experimental data (squares) and
calculated values (solid curve) of B(0) as a function of n̄;
negative values indicate nonclassicality of the state. b) The
same as above for the entanglement potential (EP) of the
SPATSs; here nonclassicality is demonstrated by EP values
greater than zero.
an average number of photons in the seed thermal state
up to n̄ = 1.15) for a global experimental efficiency of
η = 0.62. In order to gain a better view of the range of
values for n̄ and for the global experimental efficiency η
which allow to prove the nonclassical character of single-
photon-added thermal states under realistic experimen-
tal conditions, we have calculated the indicators W (0),
B(0), and EP from the model state described by ρ̂η. The
results are shown in Fig.6: the contour plots define the
regions of parameters where the detected state is classi-
cal (white areas), where it would result nonclassical if the
reconstruction errors coming from statistical noise could
be neglected (grey areas) and, finally, where it is defi-
nitely nonclassical even with the current level of noise
(black areas). From such plots it is evident that, as al-
ready noted, the Wigner function negativity only works
for sufficiently high efficiencies, while both B(0) and EP
are able to detect nonclassical behavior even for η < 50%.
In particular, the entanglement potential is clearly seen
to be the most powerful criterion, at least for these par-
ticular states, and to allow for an experimental proof of
a) b)
W(0) EPB(0)
FIG. 6: Calculated regions of nonclassical behavior of SPATSs as a function of n̄ and η according to: a) the negativity
of the Wigner function at the origin W (0); b) the Klyshko criterion B(0); c) the entanglement potential EP. White areas
indicate classical behavior; grey areas indicate where a potentially nonclassical character is not measurable due to experimental
reconstruction noise (estimated as the average error on the experimentally reconstructed parameters); black areas indicate
regions where the nonclassical character is measurable given the current statistical uncertainties.
nonclassicality for all combinations of n̄ and η, as long as
reconstruction errors can be neglected. Also considering
the current experimental parameters, EP should show
the quantum character of SPATSs even for n̄ > 3, thus
demonstrating its higher immunity to noise.
Although at a different degree, all three indicators are
however very sensitive to the presence of reconstruction
noise of statistical origin which may completely mask the
nonclassical character of the states, even for relatively low
values of n̄ or for low efficiencies. In order to unambigu-
ously prove the quantum character of higher-temperature
SPATSs in these circumstances the only possibility is to
reduce the “grey zone” by significantly increasing the
number of quadrature measurements.
CONCLUSIONS
In conclusion, we have generated a completely incoher-
ent light state possessing an adjustable degree of quan-
tumness which has been used to experimentally test and
compare different criteria of nonclassicality. Although
the direct analysis of quadrature distributions, done fol-
lowing the criterion proposed by Richter and Vogel, has
been able to show the nonclassical character of some of
the states with lower mean photon numbers, quantum to-
mography, with the reconstruction of the density matrix
and the Wigner function from the homodyne data, has
allowed us to unambiguously show the nonclassical char-
acter of all the generated states: three different criteria,
the negativity of the Wigner function, the Klyshko crite-
rion and the entanglement potential, have been used with
varying degree of effectiveness in revealing nonclassical-
ity. Besides being a useful tool for the measurement of
nonclassicality through the definition of the entanglement
potential, the combination of nonclassical field states -
such as those generated here - with a beam-splitter, can
be viewed as a simple entangling device generating multi-
photon states with varying degree of purity and entangle-
ment and allowing the future investigation of continuous-
variable mixed entangled states [29].
ACKNOWLEDGMENTS
The authors gratefully acknowledge Koji Usami for
giving the initial stimulus for this work and Milena
D’Angelo and Girish Agarwal for useful discussions and
comments. This work was partially supported by Ente
Cassa di Risparmio di Firenze and MIUR, under the
PRIN initiative and FIRB contract RBNE01KZ94.
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