Detection of sub-shot-noise spatial correlation in high-gain parametric down conversion.

Abstract

Using a 1 GW, 1 ps pump laser pulse in high-gain parametric down conversion allows us to detect sub-shot-noise spatial quantum correlation with up to 100 photoelectrons per mode by means of a high efficiency charge coupled device. The statistics is performed in single shot over independent spatial replica of the system. Evident quantum correlations were observed between symmetrical signal and idler spatial areas in the far field. In accordance with the predictions of numerical calculations, the observed transition from the quantum to the classical regime is interpreted as a consequence of the narrowing of the down-converted beams in the very high-gain regime.

Full text

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arXiv:quant-ph/0407211v1 27 Jul 2004
Detection of sub-shot-noise spatial correlation in high-gain parametric
down-conversion
O. Jedrkiewicz1, Y.-K Jiang2, E. Brambilla1, A. Gatti1, M. Bache1, L. A. Lugiato1, and P. Di Trapani1
1INFM, Dipartimento di Fisica e Matematica, Universita’ dell’Insubria, Via Valleggio 11, 22100 Como, Italy
2Department of Electronic Science and Applied Physics, Fuzhou University, 350002 Fuzhou, China
(Dated: February 1, 2008)
Using a 1GW-1ps pump laser pulse in high gain parametric down-conversion allows us to detect
sub-shot-noise spatial quantum correlation with up to one hundred photoelectrons per mode, by
means of a high efficiency CCD. The statistics is performed in single-shot over independent spatial
replica of the system. The paper highlights the evidence of quantum correlation between symmetrical
signal and idler spatial areas in the far field, in the high gain regime. In accordance with the
predictions of numerical calculations the observed transition from the quantum to the classical
regime is interpreted as a consequence of the narrowing of the down-converted beams in the very
high gain regime.
PACS numbers:
Spatial quantum optical fluctuations are studied be-
cause of new potential applications of quantum optical
procedures in parallel processing and multi-channel op-
eration. Examples are quantum holography [1], the quan-
tum teleportation of optical images [2], and the measure-
ments of small displacements beyond the Rayleigh limit
[3]. There is now a large literature on spatial effects in
the spontaneous regime of parametric down-conversion
(PDC) where photons are created one pair at a time [4].
The process of PDC is in fact particularly suitable for the
study of spatial correlations because of its large emission
bandwidth in the spatial frequency domain [5]. Never-
theless to date most spatial correlation measurements in
PDC have been performed in single photon counts regime
[6, 7] without evidencing any relevant quantum effects.
The quantum twin beam character of the PDC emission
has been evidenced in [8] by using low-to-medium pump-
power lasers (≤ 1 MW) and relying on statistical ensem-
bles from different temporal replicas of the system. With
increasing gain a transition from the quantum to the clas-
sical regime has been observed [9]. However, recent the-
oretical investigations predict multi-mode spatial quan-
tum correlations (sub-shot-noise photon-number correla-
tion between symmetrical portions of the signal and idler
angular distributions) also for arbitrarily high gains, pro-
vided that the detection area exceeds the typical size of
the mode (coherence area) [10, 11].
Here we report on the first quantum spatial mea-
surements of PDC radiation performed by using a low-
repetition rate (2 Hz) pulsed high-power laser (1GW-
1ps). This enables us to tune the PDC to the high-gain
regime while keeping a large pump beam size (∼ 1mm).
The huge number of transverse modes (roughly given by
the ratio between: (i) the area of the near-field gain pro-
file and (ii) the inverse of the angular bandwidth of the
PDC process) allows us to identify regions of the sig-
nal and idler beams where symmetrical signal-idler pixel
pairs correspond to independent spatial replica of the
quantum system. We concentrate on a portion of the
parametric fluorescence emitted close to the collinear di-
rection and within a narrow frequency bandwidth around
degeneracy. The generated pairs of signal and idler
phase-conjugate modes propagate at symmetrical angles
with respect to the pump direction in order to fulfil the
phase-matching constraints, and each pair of symmetri-
cal spots characterizing the far field represents a spatial
replica. Thanks to the very large number of these, the
statistical ensemble averaging necessary for the quantum
measurement can be solely done over the spatial replicas
for each, single, pump-laser pulse. A characterization of
the system over several shots was only made in our case
in order to verify that the selected spatial replicas are
indeed statistically identical, as required for the suitable
definition of the ensemble. The single-shot measurements
reveal sub-shot-noise spatial correlations for a PDC gain
G (intensity amplification factor) leading to the detec-
tion of up to ≃ 100 photoelectrons per mode. Finally, by
numerically solving the three-waves coupled equations in
the framework of a 3D+1 quantum model, we are able to
attribute the observed transition from quantum to classi-
cal regime to the near-field gain narrowing that occurs in
presence of a bell-shaped pump beam, at very-high gain.
The experimental setup is sketched in Fig. 1.
third harmonic (352 nm) of a 1ps, chirped-pulse ampli-
fied Nd:glass laser (TWINKLE, Light Conversion Ltd.)
is used to pump a type II 5x7x4mm3β-barium borate
(BBO) non-linear crystal, operated in the regime of para-
metric amplification of the vacuum-state fluctuations.
The input and output facets of the crystal are anti-
reflection coated at 352 nm and 704 nm, respectively.
The pump beam is spatially filtered and collimated to a
beam waist of approximately 1 mm (FWHM) at the crys-
tal input facet. The energy of the 352nm pump pulse can
be continuously tuned in the range 0.1-0.4 mJ by means
of suitable attenuating filters and by changing the en-
ergy of the 1055nm pump laser pulse, allowing to have
The

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a gain in the range 10 ≤ G ≤ 103.
fluorescence at the horizontally polarized signal and ver-
tically polarized idler modes is emitted over two cones,
whose apertures depend on the specific wavelengths (see,
e.g., [13, 14]). The BBO crystal (θ = 49.05◦, φ = 0) is
oriented in order to generate signal and idler radiation
cones tangent in the collinear direction at the degener-
ate wavelength ωs= ωi= ωp/2 (s, i and p referring to
signal, idler and pump respectively). The fluorescence
around the collinear direction is selected by means of a
5mm x 8mm aperture, placed 15 cm from the output
facet of the BBO. The aperture turned out to be nec-
essary in order to prevent beam clipping by the PBS,
otherwise giving rise to substantial scattered radiation.
The selected portion of the beam is transmitted through
a polarizing beam splitter (PBS), which separates the
signal and idler beams. The latter are finally sent onto
two separate regions of a deep-depletion back illuminated
charged coupled device (CCD) camera [15] (Roper Scien-
tific NTE/CCD-400EHRBG1, 16 bits dynamical range,
quantum efficiency η≈ 89% at 704 nm at -40◦C, dark cur-
rent and read out noise < 1e/pixel/s and < 3e/pixel/s
respectively), placed in the common focal plane of the
two lenses (f =10 cm) used to image the signal and idler
far fields. The detection array has 1340x400 pixels, with
a pixel size of 20µmx20µm. Prior to the experiment the
CCD was calibrated with a coherent source allowing the
retrieval of spatial shot-noise statistics in its full dynamic
range [16]. In our setup the correlation measurements are
performed without using any narrow-band interferential
filters (IFs), in contrast to the case of photon-counting
experiments (coincidence measurements), since IFs un-
avoidably introduce relevant transmission losses reducing
the visibility of sub-shot noise correlations. The pump-
frequency contribution is removed by using normal in-
cidence (M5) and at 45 deg (M4) high-reflectivity (HR)
mirrors coated for 352 nm placed before and after the
PBS, respectively, and a low-band pass color filter (90%
transmission around 704 nm) placed in front of the CCD.
A further HR@352nm mirror (M′
the two arms at a suitable angle in order to balance the
unequal transmission of the PBS in the two arms. All
the optical components (except the color filter) have an-
tireflection coatings at 704 nm.
tum efficiency of each detection line, which accounts for
both the transmission losses and the detector efficiency,
is ηtot≃75%.
Fig. 2a shows a typical far field image recorded in a
single shot, where a fairly broadband radiation (i.e. the
one transmitted by the rectangular aperture) is acquired
in the signal (left) and idler (right) branches. The se-
lection of the desired temporal and angular bandwidth
around degeneracy is made by temporarily inserting in
front of the CCD a 10nm wide IF around 704 nm, allow-
ing us to locate the collinear degeneracy point (see Fig.
2b). The data analysis is limited within two rectangu-
The parametric
4) is placed in one of
The estimated quan-
Teflon pinhole
(200 µm diameter)
(352nm, 1ps)
Filter
PBS
Rectangular
aperture
Pump pulses
CCD
f=10cm
f=10cm
Type II BBO
M5
M4
M4
M’4
M1
M2
M3
FIG. 1: Scheme of the experimental set-up used for the spatial
correlation measurements (see text).
lar boxes (black frames in Fig.2a) corresponding to an
angular bandwidth of 20 mrad x 8 mrad and to a tempo-
ral bandwidth smaller than 10 nm. The selected regions
contain 4000 pixels each. In this work we investigate
pixel pair correlation, and since the size of the CCD pixel
approximately corresponds to the physical size of the
replica, the ensemble is large enough to perform the de-
sired statistics. We have observed that much larger boxes
worsen the level of the correlation; this is attributed to
residual scattering owing to diffraction from the borders
of the aperture but also to the contribution of other fre-
quency components (far from degeneracy) characterized
by a lower angular symmetry between signal and idler
cones. A zoom of the selected areas is presented in Fig.2c,
where the rather spectacular symmetry of the intensity
distribution in the signal and idler branches shows the
twin-beam character of the phase-conjugate modes.
The aim of the experiment is to quantify the sym-
metrical pixel pair correlation.
suring the variance σ2
s−iof the PDC photoelectron-
number difference ns− ni of the signal/idler pixel pair
versus the mean total number of down-converted pho-
toelectrons (pe) of the pixel pair.
σ2
spatial averages performed over all the symmetrical pixel
pairs contained in the chosen regions. Each single shot
of the laser provides a different ensemble, characterized
by its pixel pair average pe number ?ns+ ni?, in turn
related to the parametric gain. In the experiment, en-
sembles corresponding to different gains are obtained by
varying the pump-pulse energy. We note that the read-
out noise of the detector, its dark current, and some un-
avoidable light scattered from the pump, signal and idler
fields contribute with a non-negligible background noise
to the process. This is taken into account by applying a
This is done by mea-
This variance is
s−i= ?(ns− ni)2? − ?ns− ni?2where the averages are

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(a)
(b)
(c)
FIG. 2: (a) Single-shot far field image recorded by the CCD
for a pump beam waist w0 ≃ 1mm and pump energy εp ≃
0.3mJ. The spatial areas for statistics are delimited by the
white boxes selected within the degenerate signal and idler
modes, spatially localized from the single shot image recorded
with the 10nm-broad IF (b). (c) Zoom of two symmetrical
areas of the signal and idler far fields.
standard correction procedure (see for example [18]), by
subtracting the background fluctuations σ2
fectively measured variance σ2
sity difference (signal+background)-(idler+background)
obtaining σ2
noise, having a standard deviation of 7 counts (±0.1 from
shot to shot, estimated by repeating the measurement
with the same pump-pulse energy) is measured in pres-
ence of pulse illumination over an area of the same size
of the acquisition area and suitably displaced from the
directly illuminated region. The validity of the data cor-
rection procedure is tested by sending in the setup (with
no crystal) through the PBS a coherent circularly polar-
ized pulsed beam (@704nm), and verifying for different
laser energies that the intensity difference fluctuations
from the two coherent portions of beams recorded on the
CCD lie at the shot noise level.
Fig. 3 shows the experimental results where each point
is associated with a different laser shot. The data are
normalized to the shot noise level (SNL), and their statis-
tical spread accounts for the background correction. Al-
bfrom the ef-
(s+b)−(i+b)of the total inten-
s−i= σ2
(s+b)−(i+b)− 2σ2
b. This background

0.4


-0.2






-10
x shift (pixels)

20



SNL


10 100
0.0
0.5
1.0
1.5
2.0
2.5
3.0
σ
2
s-i/<ns+ni>
<ns+ni> (pe)


FWHM
~2 pixels






-200 10
-0.6
-0.4
0.0
0.2
0.6
0.8
1.0
1.2
Correlation γ
γ@0.99
FIG. 3: Intensity difference variance σ2
SNL ?ns+ ni?. Each point (white circle) corresponds to a
single shot measurement where the spatial ensemble statis-
tics has been performed over a 100 x 40 pixels region. The
triangles (each one obtained by averaging the experimental
points corresponding to a certain gain) and their linear fit il-
lustrate the trend of the data in the region between ?ns+ ni?
=8 and 20. Inset: Typical correlation degree profile in the
regime where ?ns+ ni? ≃ 8 (see text).
s−inormalized to the
though the noise on the individual signal and idler beams
is found to be very high and much greater than the stan-
dard quantum limit (=?ns? and ?ni? respectively), we
observe an evident sub-shot noise pixel pair correlation
up to gains characterized by ?ns+ ni?≈ 15 − 18. Since
in that regime the observed transverse size of the co-
herence areas (i.e. of the modes) is about 2-4 pixels,
this approximately corresponds to 100 pe per mode. We
can have an idea of the transverse size of the mode by
looking at the standard two-dimensional cross-correlation
degree γ = (?nsni? − ?ns??ni?)/?σ2
angularly symmetrical signal and idler pixels contained
within the black boxes (see Fig. 2a). This can be plotted
for instance as a function of the horizontal and vertical
shifts of the recorded image on the CCD, keeping fixed
the position of the boxes. In general | γ |≤ 1 with γ = 1
for perfect correlation. A transverse section of the corre-
lation function obtained from a single-shot image char-
acterized by ?ns+ ni?≈ 8 is plotted in the inset of Fig.
3 as a function of the horizontal shift x (in pixels units).
As expected, virtually perfect correlation (in our case the
peak value is ≃ 0.99) is obtained for perfect determina-
tion (i.e. within one pixel) of the center of symmetry
between the signal and the idler regions.
In order to interpret the observed transition from quan-
tum to classical regime we present in Fig. 4 the results of
the numerical calculations. The full quantum model ac-
counts for the two transverse and the temporal degrees
of freedom with propagation along the crystal, for the
sσ2
i, between all the

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0246810
0.0
0.5
1.0
1.5
2.0
<ns+ni> ~ 10 pe
<ns+ni> ~ 40 pe
<ns+ni> ~ 210 pe
<ns+ni> ~ 1140 pe
<ns+ni> ~ 1830 pe
σ2
s-i/<ns+ni>
N (pixels)
Pixel/pixel correlation
FIG. 4: Numerical calculation of σ2
between symmetrical portions of signal and idler plotted as a
function of the detection area represented by N x N binned
pixels. Different curves correspond to different values of the
gain characterized by the mean number of down-converted pe
per pixel pair ?ns+ ni?.
s−i(normalized to SNL)
angular and chromatic material dispersion up to the sec-
ond order, and for the finite spatial and temporal widths
of the Gaussian pump pulse.
pulse parameters are those relative to the experiment.
The figure presents σ2
s−i, normalized to the SNL, vs the
size of the detection area for different gains. Each point
is the result of a statistics performed over one single
laser-shot. The case N=1 corresponds to the experiment.
The simulations (data not shown) outline that, in spite
of the fixed pump-beam diameter, the signal and idler
beam diameters at the crystal output strongly depend
on the gain and decrease when the latter increases. This
can be easily interpreted when considering that the sig-
nal and idler beam size maps not the pump-beam pro-
file but the actual parametric amplification gain profile
G(r) ∼ cosh2[σA(r)L] [19] (L being the crystal length, A
the pump field amplitude and σ a parameter proportional
to the setting characteristics), as long as filtering due to
the limited spatial bandwidth does not take place [20].
On narrowing the size of the PDC beams, the coherence
areas in the far field (i.e. the modes) increase their size,
as straightforward consequence of the convolution theo-
rem in Fourier analysis [13]. Since revealingquantum cor-
relations requires detection areas larger (or comparable)
to the mode size (as also discussed in [10]), it is necessary
when increasing the gain to have larger detectors in or-
der to obtain below-shot-noise variance as shown in Fig.
4. Note that Fig. 4 evidences the transition from quan-
tum to classical regime in case of single-pixel detection
(N=1) for a gain that is higher than in the experiment.
Instead, in the experiment, excess noise is observed for
?ns+ ni? >20, which we attribute first to the effect of
residual scattered light whose contribution grows linearly
The crystal and input-
with the radiation fluence and is thus expected to over-
come the shot noise at large pumping, and second to the
uncertainty in the determination of the symmetry center
of the signal and idler image portions. In fact simulations
have shown that an uncertainty as small as a few microns
(i.e. a fraction of the pixel size, unavoidable experimen-
tally), prevents to observe sub-shot-noise correlation as
soon as ?ns+ ni? exceeds some tens of pe, while still pre-
serving sub-shot-noise correlation for smaller gain values.
Finally, the maximum level of noise reduction observed
experimentally agrees with the theoretical limit (dotted
line in Fig. 3) determined by the total losses of the system
(∼ 1 − ηtot[10]), in accordance with the result of Fig. 4.
In conclusion, we have shown that twin beams of light
generated in parametric down-conversion exhibit sub-
shot noise spatial correlation by measuring an evident
quantum noise reduction on the signal/idler intensity dif-
ference. A transition to above shot-noise correlation is
observed as the gain increases. This quantum-to-classical
transition, in agreement with numerical simulations, is
explained as a narrowing of the signal/idler beams with
increased gain. This leads in turn to a larger mode size
and therefore also to the need of larger pixels to observe
below shot-noise correlation [10]. This will be the aim of
a future work. To our knowledge, this is the first exper-
imental investigation of quantum spatial correlations in
the high gain regime, where the huge number of trans-
verse spatial modes is detected in single shot by means
of a high-quantum-efficiency CCD.
This work has been supported by the European Union
(QUANTIM contract IST-2000-26019). M. B. acknowl-
edges support from the Carlsberg foundation.
[1] A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and
M. C. Teich, Opt. Express 9, 498 (2001).
[2] I. V. Sokolov, M. I. Kolobov, A. Gatti, and L. A. Lugiato,
Opt. Commun. 193, 175 (2001).
[3] N. Treps, U. Andersen, B. Buchler, P. K. Lam, A. Maitre,
H.-A Bachor, and C. Fabre, Phys. Rev. Lett. 88, 203601
(2002).
[4] L. A. Lugiato, A. Gatti, and E. Brambilla, J. Opt. B:
Quantum Semiclassical Opt. 4, S176 (2002), and refer-
ences therein.
[5] F. Devaux and E. Lantz, Eur. Phys. J. D 8, 117 (2000).
[6] B. M. Jost, A. V. Sergienko, A. F. Abouraddy, B. E. A.
Saleh, and M. C. Teich, Opt. Express 81, 3 (1998).
[7] S. S. R. Oemrawsingh, W. J. van Drunen, E. R. Eliel, and
J. P. Woerdman, J. Opt. Soc. Am. B 19, 2391 (2002).
[8] O. Aytur and P. Kumar, Phys. Rev. Lett. 65, 1551
(1990).
[9] M. L. Marable, S.-K. Choi, and P. Kumar, Opt. Express
2, 84 (1998).
[10] E. Brambilla, A. Gatti, M. Bache, and L. A. Lugiato,
Phys. Rev. A 69, 023802 (2004).
[11] A. Gatti, E. Brambilla, and L. A. Lugiato, Phys. Rev.
Lett. 83, 1763 (1999).

Page 5

5
[12] A. Yariv, Quantum electronics (John Wiley and Sons,
New York, 1989).
[13] A. Berzanskis, W. Chinaglia, L. A. Lugiato, K.-H. Feller,
and P. Di Trapani, Phys. Rev. A 60, 1626 (1999).
[14] M. H. Rubin, Phys. Rev. A 54, 5349 (1996).
[15] J.R. Janesick, Scientific Charge-Coupled Devices (SPIE
Press Bellingham, Washington, 2001), pp. 204-205; see
also http://www.roperscientific.de/theory.html.
[16] Y.-K. Jiang, O. Jedrkiewicz, S. Minardi, P. Di Trapani,
A. Mosset, E. Lantz, and F. Devaux, Eur. Phys. J. D 22,
521 (2003).
[17] K. Koch, E. Cheung, G. T. Moore, S. H. Chakmakjian,
and J. M. Liu, IEEE J. of quant. electron. 31, 769 (1995).
[18] A. Mosset, F. Devaux, G. Fanjoux, and E. Lantz, Eur.
Phys. J. D 28, 447 (2004).
[19] S. A. Akhmanov, V. A. Vysloukh, A. S. Chirkin, Op-
tics of Femtosecond Laser Pulses (American Institute of
Physics, New York, 1992), p.151.
[20] P. Di Trapani, G. Valiulis, W. Chinaglia and A. An-
dreoni, Phys. Rev. Lett. 80, 265 (1998).