# Supplementary document: Hiding images in quantum correlations

Chloé Vernière<EMAIL_ADDRESS>Sorbonne Université, CNRS,
Institut des NanoSciences de Paris, INSP, F-75005 Paris, France  Hugo Defienne
Sorbonne Université, CNRS, Institut des NanoSciences de Paris, INSP, F-75005
Paris, France

††preprint: APS/123-QED

##  I Details on the experimental setups

The laser is a 100mW-power continuous-wave vertically polarised diode laser at
$402$nm (Coherent). The polarisation can be tuned with a half-wave plate
placed in front of the laser. The EMCCD (iXon Life 897, Andor) has pixels of
size $16$x$16\mu$m2 and a quantum efficiency of $\sim 70\%$ at $810$nm. Its
temperature is set to -$60^{\circ}$C and is operated with a horizontal pixel
shift readout rate of $17$ MHz, a vertical pixel shift of $0.3$µs, a vertical
clock amplitude voltage of $4$ V above the factory setting, and an
amplification gain of $1000$. The exposure time is set to $3$ms. In most of
our results, $10^{6}$-$10^{7}$ frames were acquired.

The cat-shaped amplitude objects are imprinted in black on a plastic sheet.
Their size is $\sim 3\crossproduct 1.5$mm2. The grating used as an amplitude
object is a positive variable line grating test target (Thorlabs, R1L3S6P).

The non-linear crystal is a $0.5\times 5\times 5$mm type-I $\beta$-barium
borate (BBO) crystal (Newlight Photonics). It is slightly tilted around the
horizontal axis to ensure near-collinear phase matching of photons at the
output. For the phase object, the crystal is replaced by two identical type-I
BBO crystals with perpendicular axis to each other and optically contacted.
The photon pairs produced are thus entangled both in space and polarisation.
The crystals are slightly tilted around the vertical and horizontal axis to
ensure near-collinear phase-matching of the photons at the output.

Figure S1 shows the two different setups used in our experiments. Results
shown in Figure 1b-c and Figure 3a-b of the manuscript are acquired with the
setup presented in Figure S1a. Lenses $f_{1}=150$mm and $f_{2}=300$mm magnify
the laser. $f=100$mm projects the Fourier transform of the object on the
crystal. Lenses $f^{\prime}=35$mm, $f_{3}=150$mm and $f_{4}=50$mm do far-field
imaging of the crystal on the camera. The effective focal length is
$f^{\prime}_{eff}=\frac{f_{4}f^{\prime}}{f_{3}}=11.7$mm. All the other results
were acquired with the setup shwon in Figure S1b. In this configuration, there
is no magnification of the laser and $f=100$mm. The photon pairs produced by
the crystal are imaged onto an SLM by two telescopes with $f^{\prime}=35$mm -
$f_{1}=60$mm and $f_{2}=50$mm - $f_{3}=125$mm (magnification of $4.3$). They
are then de-magnified by two telescopes $f_{4}=200$mm - $f_{5}=75$mm and
$f_{6}=125$mm - $f_{7}=150$mm (magnification of 0.45) and Fourier imaged on
the camera with $f_{8}=50$mm. The effective focal length between the crystal
and the camera is $f^{\prime}_{eff}=\frac{f_{8}}{0.45.4.3}=26$mm. Unless
specified, all lenses are disposed in confocal configuration. The SLM is a
phase only modulator (Holoeye Pluto-2-NIR-015) with $1920\crossproduct 1080$
pixels and a $8\mu$m pixel pitch. It is here represented in transmission but
actually works in reflection. Both setups have a short-pass filter just before
the crystal to remove stray light emitted by the laser cavity (non
represented), a long-pass filter right after the crystal to filter out the
pump beam and a bandpass filter (810nm-10) in front of the camera.

Figure S1: Experimental setups. Two configurations of the setup used for the
amplitude objects data acquisition. See Figure S2.a for more details for the
phase object configuration inside the rectangular dashed box. BBO : non-linear
crystal in $\beta$-barium borate. LPF : long-pass filter. BPF : band-pass
filter. SLM : spatial light modulator. EMCCD : electron-multiplying charge-
coupled device.

##  II Theoretical demonstration of $G^{2}$

Following Ref. [1], we derive Equation $1$ in a 1D system and then extend the
results to a 2D system. Here we use the generic configuration introduced in
Figure $1$ of the manuscript, identifying only two lenses in the system,
namely $f$ between the object and the crystal, and $f^{\prime}$ between the
crystal and the camera. The crystal is thus illuminated by the Fourier
transform of an object described by $t(x)$, such that the illumination of the
crystal is

$\mathcal{FT}\left[t\right](x)=\int
dx^{\prime}\;t(x^{\prime})\exp(-2i\pi\frac{xx^{\prime}}{\lambda_{p}f})$

with $\lambda_{p}=405$nm the pump wavelength. Both signal and idler have the
same impulse-response functions $h(x,x^{\prime})$, that of a 2-f Fourier
system of effective focal length $f^{\prime}$, which reads

$h(x,x^{\prime})=\exp\left(-i\frac{2\pi}{\lambda
f^{\prime}}xx^{\prime}\right)$

with $\lambda=2\lambda_{p}=810$nm the wavelength of the SPDC light. The
correlation function of the two photons detected by the camera is then given
by

$\displaystyle G^{(2)}(x_{1},x_{2})$ $\displaystyle=\left|\int
dx\;\mathcal{FT}\left[t\right](x)h(x_{1},x)h(x_{2},x)\right|^{2}$
$\displaystyle=\left|\int
dx\;dx^{\prime}\;t(x^{\prime})\exp(-2i\pi\left(\frac{x^{\prime}}{\lambda_{p}f}+\frac{(x_{1}+x_{2})}{\lambda
f^{\prime}}\right)x)\right|^{2}$ $\displaystyle=\left|\int
dx^{\prime}\;t(x^{\prime})\delta\left(\frac{x^{\prime}}{\lambda_{p}f}+\frac{x_{1}+x_{2}}{\lambda
f^{\prime}}\right)\right|^{2}$
$\displaystyle=\left|t(-\frac{f}{f^{\prime}}x_{+})\right|^{2}.$ (B1)

We note
$x_{+}=\frac{\lambda_{p}}{\lambda}(x_{1}+x_{2})=\frac{x_{1}+x_{2}}{2}$. More
generally, considering an object in $2$D described by the vector
$\mathbf{r}=(x,y)$, we get

$G^{(2)}(\mathbf{r}_{1},\mathbf{r}_{2})=\left|t\left(-\frac{f}{f^{\prime}}\mathbf{r}_{+}\right)\right|^{2}.$

##  III $G^{2}$ measurement with an EMCCD camera

As detailed in Ref. [2], given a set of M frames acquired by the camera, we
compute (1) the intensity distribution $I(i,j)$ at pixel $(i,j)$ and (2) the
intensity correlations distribution $G^{(2)}(i,j,k,l)$ between pixels $(i,j)$
and $(k,l)$:

  1. 1.

The intensity distribution is computed by averaging the intensity over all the
frames : $I(i,j)=\frac{1}{M}\sum_{m=1}^{M}I_{m}(i,j)$, where $I_{m}(i,j)$ is
the intensity value measured at pixel $(i,j)$ of the $m^{th}$ frame.

  2. 2.

The second-order intensity correlation distribution between pixels $(i,j)$ and
$(k,l)$ is computed by multiplying the values from the same frame of the two
pixels and subtracting the product of the value of the two pixels from
consecutive frames. This is then averaged over all the frames and reads

$\small
G^{(2)}(i,j,k,l)=\frac{1}{M}\sum_{m=1}^{M}I_{m}(i,j)I_{m}(k,l)-\frac{1}{M-1}\sum_{m=1}^{M-1}I_{m}(i,j)I_{m+1}(k,l).$
(B2)

The second term enables to subtract the accidental coincidences, i.e. photons
from two different pairs or from noise. This gives a discrete version of the
continuous second-order intensity correlation function
$G^{(2)}(\mathbf{r_{1}},\mathbf{r_{2}})$, where $\mathbf{r_{1}}$ and
$\mathbf{r_{2}}$ are associated to pixel $(i,j)$ and $(k,l)$, respectively.

The sum coordinate projection $\Gamma^{+}(\mathbf{x^{+}},\mathbf{y^{+}})$, to
which we refer as the ’correlation image’, can be computed from
$G^{(2)}(\mathbf{r_{1}},\mathbf{r_{2}})$ by integrating along the minus-
coordinate. With continuous variables,

$\Gamma^{+}(\mathbf{r_{+}})=\int
G^{(2)}(\mathbf{r_{-}},\mathbf{r_{+}})d\mathbf{r_{-}}.$ (B3)

With discrete variables, we in practice compute :

$\Gamma^{+}(k,l)=\sum_{i=1}^{N_{x}}\sum_{j=1}^{N_{y}}G^{(2)}(k-i,l-j,i,j).$
(B4)

$N_{x}$ (resp. $N_{y}$) refers to the number of pixels along the horizontal
(resp. vertical) direction of the camera sensor.

##  IV Measurement with phase objects

Figure S2: Setup details for phase objects.a, For the phase experiment,
elements in the dashed box in Figure S1.b are replaced by the interferometric
system here represented. b, Conceptual scheme of the interferometer. A point-
like object is Fourier-imaged onto one crystal that generates photon pairs.
This photon pair flux interferes with photon pairs produced by another crystal
(used as a reference arm) through a beam splitter (BS). A lens $f^{\prime}$
images the point-like object onto the camera, and a Spatial Light Modulator
(SLM) positioned on the reference arm enables phase-shifting holography. c,
Values of $\Gamma^{+}(\mathbf{r_{+}}=0$) in function of the phase $\theta$
programmed on the SLM. $4.2\text{\times}{10}^{5}$ frames were taken for each
data point. HWP : half-wave plate. BD : beam displacer in calcite. BBO : non-
linear crystal in $\beta$-barium borate. PBS : polarized beam splitter. LPF :
long-pass filter. BS : beam splitter. SLM : spatial light modulator.

To retrieve the phase of an object, we modify the set-up as illustrated in
Figure S2.a. A half-wave plate at $22.5^{\circ}$ sets the pump polarisation to
be diagonally polarized and two beam displacers in calcite are inserted. They
separate the horizontal and vertical polarisation components of the pump, thus
effectively creating the two arms of an interferometer. In one of the arms, a
microscope slide is introduced to shift the phase. The two arms then recombine
and the beam is incident on the non-linear crystals. The two crystals have
their axes perpendicular to each other. The pairs are produced by type-I SPDC
and are thus cross-polarized at the output (vertical and horizontal). Only the
pairs with horizontal polarization carry the phase shift information. The
photon pairs then interfere before the camera using a polarizer rotated to
$45^{\circ}$. This configuration is a common-path interferometer i.e. pairs of
photons emitted by each crystal propagate along the same optical path and they
interfere only just before the camera.

To clarify its operation, our experiment can be conceptually represented by a
more general interferometric diagram in which there are two separate crystals
and their pairs are recombined by a beam-splitter, as illustrated in Figure
S2.b. The situation unfolds precisely as if we were employing the setup
illustrated in Figure 1 of the manuscript to image a point-like object and
using an external reference arm to measure its phase. By inserting or not the
microscope slide, the phase of such a virtual point-like object can be
changed.

To measure the phase of such a point-like object, we use a phase-shifting
holography method. To achieve this, it is necessary to precisely control the
phase of the reference arm, i.e., the phase of vertically polarized photon
pairs. For this purpose, a SLM is positioned along the path of the photons
between the crystals and the camera, as shown in Figure S1.b. The SLM enables
the modulation of the phase only for vertically polarized photons. At the
camera, the resulting second-order correlation function measured is

$\small
G^{(2)}(\mathbf{r_{-}},\mathbf{r_{+}})=\left|t(\mathbf{r_{+}})+e^{i2\theta}\right|^{2}\propto|\text{cos}(\theta_{t}(\mathbf{r_{+}})+2\theta-\theta_{ref})|^{2},$
(B5)

where $\theta_{t}$ is the phase of the object, $\theta$ is the phase displayed
by the SLM and $\theta_{ref}$ the reference phase. The presence of this phase
term is a consequence of the fact that the interferometer is not balanced. As
we will see bellow, it will be removed by subtraction. Figure S2.c shows
oscillations of $\Gamma^{+}$ at $\mathbf{r_{+}}=\mathbf{0}$ when scanning the
phase from $0$ to $2\pi$ using the SLM.

In practice, to retrieve the phase of the object, it is sufficient to measure
$G^{(2)}$ for only $4$ values of the SLM phase i.e.
$\theta\in[0,\pi/4,\pi/2,3\pi/4]$. This method is called phase-shifting
holography [3]. It has been used previously with photon pairs in Refs. [4, 5].
By applying phase-shifts $\theta\in[0,\pi/4,\pi/2,3\pi/4]$ and acquiring the
$G^{(2)}$, we retrieve the phase $\theta_{t}$ by combining the sum-projections
as follow :

$\theta_{t}-\theta_{ref}=\textrm{arg}(\Gamma^{+}_{\theta=0}-\Gamma^{+}_{\theta=\pi/2}+i(\Gamma^{+}_{\theta=\pi/4}-\Gamma^{+}_{\theta=3\pi/4})).$
(B6)

To eliminate the reference phase $\theta_{ref}$, it is necessary to calibrate
the system beforehand by taking a reference measurement without the object,
i.e., without the microscope slide. Figure S3.a-d show the four correlation
images for $\theta\in[0,\pi/4,\pi/2,3\pi/4]$ without the microscope slide in
the setup used to retrieve the reference phase $\theta_{ref}$ (Fig.S3.i).
Figure S3.e-h show the four correlation images with the microscope slide in
the setup used to retrieve $\theta_{t}-\theta_{ref}$ (Fig.S3.k). Finally,
subtracting the first phase measurement to the second one results to that of
the object $\theta_{t}$, shown in Figure S3.k and Figure $3$d of the
manuscript. $7.11\text{\times}{10}^{6}$ and $15.9\text{\times}{10}^{6}$ frames
where acquired for each measure of the holography procedure for respectively
the reference and the object measurements, which in total sums to respectively
$160$ and $72$ hours of acquisition.

Figure S3: Phase computation with four phase holographic measurement. a-d
Correlation images for $\theta\in[0,\pi/4,\pi/2,3\pi/4]$ without any object in
the setup. In inset, corresponding direct intensity image. i, Reconstructed
reference phase $\theta_{ref}$ of the correlation image . e-h, Correlation
images for $\theta\in[0,\pi/4,\pi/2,3\pi/4]$ with a phase object in the setup.
In inset, corresponding direct intensity image. j, Reconstructed phase
$\theta_{t}-\theta_{ref}$ of the correlation image with the object. k,
Reconstructed phase of the object $\theta_{t}$ by subtracting i to j. The gray
area corresponds to undefined phases.

##  V Theoretical demonstration of independance of $G^{(2)}$ with f and f’

In this section we show that the correlations do not depend on the distances
between the crystal and its two neighboring lenses $f$ and $f^{\prime}$. The
derivation is conducted in 1D then extended to the 2D case. We here follow the
generic formalism introduced in Figure $1$ of the manuscript. Let $f$ (resp.
$f^{\prime}$) be the distance between the object (resp. the camera) and the
lens $L$ (resp. $L^{\prime}$). Let $d$ (resp. $d^{\prime}$) be the distance
between the crystal and the lens $L$ (resp. $L^{\prime}$). The input on the
crystal then reads

$\mathcal{FT}\left[t\right](x^{\prime})=\int
dx\;t(x)\exp(i\frac{\pi}{\lambda_{p}}\left(\frac{x^{2}}{f}+\frac{x^{\prime
2}}{d}-\frac{(x^{\prime}f+xd)^{2}}{f^{2}d}\right))$

and the impulse-response function is

$h(x^{\prime},x)=\exp(i\frac{\pi}{\lambda}(\frac{x^{2}}{d^{\prime}}+\frac{{x^{\prime}}^{2}}{f^{\prime}}-\frac{(x^{\prime}d^{\prime}+xf^{\prime})^{2}}{d^{\prime}f^{\prime
2}})).$

Then :

$\displaystyle G^{(2)}(x_{1},x_{2})=\left|\int
dx^{\prime}\;\mathcal{FT}\left[t\right](x^{\prime})\;h(x_{1},x^{\prime})h(x_{2},x^{\prime})\right|^{2}$
$G^{(2)}(x_{1},x_{2})=|\iint
dx\;dx^{\prime}\;t(x)\;\exp(i\frac{\pi}{\lambda_{p}}(\frac{x^{2}}{f}+\frac{x^{\prime
2}}{d}-\frac{(x^{\prime}f+xd)^{2}}{f^{2}d}))\\\
\exp(i\frac{\pi}{\lambda}(\frac{x^{\prime
2}}{d^{\prime}}+\frac{{x_{1}}^{2}}{f^{\prime}}-\frac{(x_{1}d^{\prime}+x^{\prime}f^{\prime})^{2}}{d^{\prime}f^{\prime
2}}))\exp(i\frac{\pi}{\lambda}(\frac{x^{\prime
2}}{d^{\prime}}+\frac{{x_{2}}^{2}}{f^{\prime}}-\frac{(x_{2}d^{\prime}+x^{\prime}f^{\prime})^{2}}{d^{\prime}f^{\prime
2}}))|^{2}$ $\displaystyle
G^{(2)}(x_{1},x_{2})=\left|\exp(i\frac{\pi}{\lambda_{p}}(\frac{1}{2f^{\prime}}-\frac{d^{\prime}}{2f^{\prime
2}})(x_{1}^{2}+x_{2}^{2}))\int
dx\;t(x)\;\exp(i\frac{\pi}{\lambda_{p}}x^{2}(\frac{1}{f}-\frac{d}{f^{2}}))\;\int\;dx^{\prime}\exp(-i\frac{\pi}{\lambda_{p}}\frac{2x^{\prime}}{f}(x+\frac{f}{f^{\prime}}\frac{x_{1}+x_{2}}{2}))\right|^{2}$
(B7) $\displaystyle G^{(2)}(x_{1},x_{2})=\left|\int
dx\;t(x)\;\exp(i\frac{\pi}{\lambda_{p}}x^{2}(\frac{1}{f}-\frac{d}{f^{2}}))\delta(x+\frac{f}{f^{\prime}}\frac{x_{1}+x_{2}}{2})\right|^{2}$
(B8) $\displaystyle
G^{(2)}(x_{1},x_{2})=\left|t(-\frac{f}{f^{\prime}}\frac{x_{1}+x_{2}}{2})\exp(i\frac{\pi}{\lambda_{p}}(-\frac{f}{f^{\prime}}\frac{x_{1}+x_{2}}{2})^{2}(\frac{1}{f}-\frac{d}{{f^{2}}}))\right|^{2}$
(B9) $\displaystyle
G^{(2)}(x_{1},x_{2})=\left|t(-\frac{f}{f^{\prime}}x_{+})\right|^{2}.$ (B10)

The phase factors associated to $d$ and $d^{\prime}$ in B7 and B9 do not
modify the correlations measured on the camera. In B7, we use
$2\lambda_{p}=\lambda$.

##  VI Fitting process in Figure $5$ of the manuscript

Figure S4 shows the correlation images acquired to compute the signal-to-noise
ratio (SNR) and the correlation width for different values of the focal length
$f$. All were acquired with the same number of frames
($6.6\text{\times}{10}^{5}$). The inset in Figure S4.e shows a longer
acquisition ($4.9\text{\times}{10}^{6}$ frames) to be able to fit the width.

###  VI.1 Correlation width

To compute the width of the correlations, we fit the correlation spots
according to a double Gaussian model [6, 7]. The object $t(\mathbf{r})$ is in
this case the laser itself, which we model as a two dimensional Gaussian

$t(x,y)\propto\exp(-\frac{(x-x_{0})^{2}}{2\sigma_{x}^{2}})\exp(-\frac{(y-y_{0})^{2}}{2\sigma_{y}^{2}}),$

where $\sigma_{x}$ (resp. $\sigma_{y}$) is the width of the laser along the
horizontal (resp. vertical) axis. We subsequently fit the correlation spot
according to

$|t(\frac{f}{f^{\prime}}x_{+},\frac{f}{f^{\prime}}y_{+})|^{2}\propto\exp(-\frac{(x_{+}-\tilde{x}_{0})^{2}}{\tilde{\sigma}_{x}^{2}})\exp(-\frac{(y_{+}-\tilde{y}_{0})^{2}}{\tilde{\sigma}_{y}^{2}}),$

where $\tilde{\sigma}_{x}=\frac{f^{\prime}}{f}\sigma_{x}$ and
$\tilde{\sigma}_{y}=\frac{f^{\prime}}{f}\sigma_{y}$ are the theoretical values
for the width of the correlation spot. The values presented in Figure $5$ of
the manuscript correspond to the fitted value of $\tilde{\sigma}_{x}$ for the
data shown in Figure S4a-d,e (inset). Fitting the evolution of the correlation
width as $a/f$, we get $a_{exp}=$1.24\text{\times}{10}^{3}$$ pixel.mm which is
close within $5\%$ to the theoretical one $a_{th}=$1.30\text{\times}{10}^{3}$$
pixel.mm.

Figure S4: Width of the correlations. Correlations measurements with focal
length of the lens before the crystal of (a) $f=200$mm; (b) $f=150$mm; (c)
$f=100$mm; (d)$f=75$mm; (e) $f=50$mm. All were acquired with
$6.6\text{\times}{10}^{5}$ frames. e shows in inset a longer acquisition of
$4.9\text{\times}{10}^{6}$ frames for $f=50$mm to be able to fit the width of
the correlations.

###  VI.2 SNR

We define the SNR as the ratio between the mean value of the correlation spot
and the standard deviation of the noise around the correlation spot. This
standard deviation is proportional to that of the noise of the signal acquired
in the same experimental conditions, such that this computation of the SNR
actually corresponds to the ratio between the signal and its noise.
Experimentally, the correlation spot spreads proportionally to $1/f^{2}$, and
consequently, the signal scales as $f^{2}$. At a constant acquisition time,
and therefore with constant measurement noise, the SNR evolves as $f^{2}$ [8].

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