# Real–time ghost imaging of Bell–nonlocal entanglement between a photon and a
quantum memory

Mateusz Mazelanik  Centre for Quantum Optical Technologies, Centre of New
Technologies, University of Warsaw, Banacha 2c, 02-097 Warsaw, Poland  Faculty
of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland  Adam
Leszczyński  Centre for Quantum Optical Technologies, Centre of New
Technologies, University of Warsaw, Banacha 2c, 02-097 Warsaw, Poland  Faculty
of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland  Michał
Lipka  Centre for Quantum Optical Technologies, Centre of New Technologies,
University of Warsaw, Banacha 2c, 02-097 Warsaw, Poland  Faculty of Physics,
University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland  Wojciech Wasilewski
Centre for Quantum Optical Technologies, Centre of New Technologies,
University of Warsaw, Banacha 2c, 02-097 Warsaw, Poland  Michał Parniak
Centre for Quantum Optical Technologies, Centre of New Technologies,
University of Warsaw, Banacha 2c, 02-097 Warsaw, Poland  Niels Bohr Institute,
University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen, Denmark
<EMAIL_ADDRESS>
###### Abstract

Certification of nonlocality of quantum mechanics is an important fundamental
test that typically requires prolonged data collection and is only revealed in
an in-depth analysis. These features are often particularly exposed in hybrid
systems, such as interfaces between light and atomic ensembles. Certification
of entanglement from images acquired with single-photon camera can mitigate
this issue by exploiting multiplexed photon generation. Here we demonstrate
this feature in a quantum memory (QM) operating in a real-time feedback mode.
Through spatially-multimode spin-wave storage the QM enables operation of the
real-time ghost imaging (GI) protocol. By properly preparing the spatial phase
of light emitted by the atoms we enable observation of Bell-type nonlocality
from a single image acquired in the far field as witnessed by the Bell
parameter of $S=2.227\pm 0.007>2$. Our results are an important step towards
fast and efficient utilization of multimode quantum memories both in protocols
and in fundamental tests.

Correlated photon pairs not only provide a basic tool for testing the quantum
theory, but also constitute a bedrock of modern quantum optics where the
applications as secure communication or quantum-enhanced imaging are gathering
a continuously rising interest. Starting from seminal works of Clauser and
Aspect [1, 2] polarization-entangled photons have been used for decades to
demonstrate the quantum spookiness, namely by demonstration of the Bell-type
correlations. Tremendous efforts have been put to close numerous loopholes [3,
4, 5, 6] and to utilize different degrees of freedom (DoF) such as time-bin
[6, 7], position [8] and momentum [9] or angular momentum of photons [10] to
trustfully rule out the local hidden variables theories by violation of the
Bell inequality [11]. With the development of experimental quantum optics, the
Bell test emerged as a versatile benchmarking tool providing a performance
measure for various quantum protocols [12, 13].

The Bell inequality [11] arose as a response to work by Einstein, Podolsky and
Rosen (EPR) [14] concerning completeness of quantum theory. The so-called EPR
paradox suggesting the failure of local realism has been initially
demonstrated with quadratures of light [15, 16] and lately with position and
momenta correlated photons [17, 18, 19, 20]. The quantum ghost imaging (GI)
[21, 22] \- a technique that uses correlated photon pairs to reconstruct an
image from photons that do not interact with the object being imaged - emerged
from the latter.

In the modern quantum GI protocol the object being imaged is placed in either
near or far field of an EPR photon source. The source generates correlated
photons in two distinct beams - a _signal_ and _idler_. The object is
illuminated by the _signal_ photons which are then detected using non-
spatially-resolving _bucket_ detector acting as a trigger source for second,
spatially-resolving detector - a camera placed in the _idler_ beam in the same
optical plane as the object. As the camera registers _idler_ photons only when
a _signal_ photon is transmitted through the object (a trigger signal is
generated) the _idler_ photon events build up the image. In such scenario the
_idler_ photons have to be additionally time-delayed to surpass the trigger
signal electronic propagation delay, which was up to date only achieved via an
image-preserving optical delay line [23], while in other cases researchers
relied on post-selection of correlated events. On the other hand, a spatially-
multimode QM [24] is a natural choice when one needs to store the idler photon
and release it upon the trigger signal, thus gaining more versatility,
potentially much larger storage times and simultaneously avoiding difficult
engineering of delay lines.

Here we employ a wavevector-multiplexed emissive QM [24] to demonstrate a
real-time GI of polarization and wavevector DoF Bell-type correlations. We
utilize our ability of EPR-type entanglement generation [25] to prepare
polarization Bell-states in many wave-vector modes. The tens of $\mu s$-long
storage time of our memory enables us to easily realize a heralded GI setup
without any additional optical delay line. The images of Bell-type
correlations are acquired using a state-of-the art single photon sensitive
camera giving almost 79% visibility and violation of the Bell inequality by 32
standard deviations. With that, we bring this extremally versatile approach to
Bell inequalities to a hybrid atom-photon system. This is not only of
fundamental interest, but the atomic medium brings about a set of possible
manipulations [26] and interactions [27].

The role of quantumness in GI has been a topic of debate for past two decades.
While it is currently known that the ghost images can be produced with
classical light [28, 29] the quantum GI can lead to improved contrast or
signal to noise ratio (SNR) [30, 31] and resolution [32] in the produced
image. Therefore, the current question is not the quantum nature of GI but
rather if with the quantum illumination a new features can be observed.
Finally, the recent demonstrations of imaging of Bell-type nonlocal behavior
[33] and quantum GI utilizing higher order correlations [34] suggest a
sparking interest in this topic, with our work bringing this issue of
fundamental interest into a hybrid atom-photon system.

Figure 1: Experimental setup for QM-assisted phase sensitive GI. (a) The
wavevector multiplexed quantum memory W-Mux QM generates time delayed EPR
_signal_ and _idler_ pairs. The Mach-Zehnder interferometers (MZI)
superimposes two halfs of the wavevetor modes onto each other generating
wavevector indexed superposition of polarization DoF Bell-states. The MZIs
provides also additional control of the Bell-state phase
$\varphi_{s}-\varphi_{i}$. The _signal_ photons are detected by _bucket_
detector placed after polaryzer at angle $\theta_{s}$. The _signal_ detection
events are registered by FPGA controlling the experimental sequence
(especially the readout pulse r) and providing a trigger for the I-CMOS
camera. (b) The experimental sequence. The readout is performed only if a
_signal_ photon is detected. (c) Relevant $\mathrm{{}^{87}Rb}$ energy levels
used in the QM protocol.

We use wavevector multiplexed multimode QM as a quantum light source for phase
sensitive GI [24]. Our experimental setup is shown in Fig. 1(a) and described
in more detail in the Supplementary Information. In the memory, the signal
photons are generated together with spin waves. After a programmable delay,
the spin wave is converted into idler photons. The resulting photon pair is
correlated in positions and anti-correlated in momenta (EPR state), while the
polarizations are uncorrelated and simply set by collection optics. Therefore,
we may write the resulting two-photon state as:

$|\psi\rangle=\int\psi(\mathbf{k}_{s},\mathbf{k}_{i})|\mathbf{k}_{s},H\rangle|\mathbf{k}_{i},H\rangle
d\mathbf{k}_{s}\mathrm{d}\mathbf{k}_{i},$ (1)

where $H$ stands for horizontal polarization and the biphoton wavefunction
$\psi_{\mathrm{}}$ in the realistic scenario can be approximated by [25, 24,
18]:

$\psi(\mathbf{k}_{s},\mathbf{k}_{i})=\mathcal{\frac{\sigma}{\pi\kappa}}e^{-\frac{(\mathbf{k}_{s}+\mathbf{k}_{i})^{2}}{4\kappa^{2}}-\frac{\sigma^{2}(\mathbf{k}_{s}-\mathbf{k}_{i})^{2}}{4}},$
(2)

where the Gaussian widths $\kappa$ and $\sigma^{-1}$ corresponds respectively
to strength of the momenta and position correlations. Furthermore, as we are
limited by the numerical aperture rather than spread of the phase matching
spectrum determining the set of available emission angles, we will assume
perfect correlation in position and take $\sigma\to 0$.

The _signal_ and _idler_ photons are collected by two identical far-field
imaging setups (represented as a single lenses on Fig. 1(a)). Next, the
photons pass through setups similar to MZI, where each of the beams (either
_signal_ or _idler_) is split in half and resulting components (upper-ⓤ and
lower-\raisebox{-.9pt}{d}⃝) are joined together on the polarizing beam
splitter (PBS), with a help of a half-wave plate. Additionally, as the MZIs
are placed in the far-filed of the ensemble, tilting one of the MZI mirrors
results in a wavevector-dependent phase between the two joined paths (see Fig.
1(a)). The resulting state can be written as:

$\displaystyle|\psi_{\mathrm{B}}\rangle=\mathcal{N}\iint_{[0,\delta\mathbf{k}]}e^{i\varphi_{s}(\mathbf{k}_{s})}\psi(\mathbf{k}_{s},\mathbf{k}_{i}-\delta\mathbf{k})|\mathbf{k}_{s},H\rangle|\mathbf{k}_{i},V\rangle+e^{i\varphi_{i}(\mathbf{k}_{i})}\psi(\mathbf{k}_{s}-\delta\mathbf{k},\mathbf{k}_{i})|\mathbf{k}_{s},V\rangle|\mathbf{k}_{i},H\rangle)\mathrm{d}\mathbf{k}_{s}\mathrm{d}\mathbf{k}_{i}\equiv$
$\displaystyle\iint_{[0,\delta\mathbf{k}]}\tilde{\psi}(\mathbf{k}_{s},\mathbf{k}_{i})|\mathbf{k}_{s}\rangle|\mathbf{k}_{i}\rangle(|H\rangle|V\rangle+e^{i\varphi_{s}(\mathbf{k}_{s})-i\varphi_{i}(\mathbf{k}_{i})}|V\rangle|H\rangle)\mathrm{d}\mathbf{k}_{s}\mathrm{d}\mathbf{k}_{i},$
(3)

where $\mathcal{N}$ is the normalization constant,
$\tilde{\psi}(\mathbf{k}_{s},\mathbf{k}_{i})$ is the renormalized
$\psi(\mathbf{k}_{s}-\delta\mathbf{k}/2,\mathbf{k}_{i}-\delta\mathbf{k}/2)$
and $\delta\mathbf{k}=\delta k\hat{\mathbf{y}}$ with $\delta
k=286\,\mathrm{mm^{-1}}$ is the wavevector shift applied to the upper (ⓤ on
Fig. 1(a)) part of beam to superimpose it on the lower part
(\raisebox{-.9pt}{d}⃝ on Fig. 1(a)). By $\varphi_{s(i)}(\mathbf{k}_{s(i)})$ we
denote the additional phase added before the superimposition at each side. The
$|\psi_{\mathrm{\mathrm{B}}}\rangle$ represents a wavevector-indexed
superposition of polarization DoF Bell-states and exhibits both EPR and Bell-
type correlations.

Figure 2: Phase profiles for Bell entanglement preparation and verification.
(a) Measured phase profiles in the _idler_ and _signal_ arms and the resulting
phase difference profile relevant for the generated state. (b) Comparison
between the classically simulated (left) and measured (right) correlation
function $\mathcal{C}^{0,0}(\mathbf{k}_{i})$. (c) The $k_{x}$–averaged fringes
from (b): line–simulation, points–measurement.

To exploit the latter we perform a polarization measurement over large set of
_signal_ and _idler_ wavevectors using a GI technique with feedback enabled by
the QM feature of our source. In contrast to “single-channel” [35] or “dual-
channel” [2] experiments we implement a hybrid scenario in which one party
(_signal_) measures using a “single channel” polarizer and a _bucket_ detector
when the second one (_idler_) uses a “dual-channel” polarizer (BD on Fig.
1(a)) and single-photon-sensitive camera (I-CMOS) [36] placed in the far-field
of the atomic ensemble. The _bucket_ detector [a multimode-fiber-coupled
avalanche photiodiode (APD)] is connected to the experimental sequence
controller (FPGA), which decides whether to perform the readout and gate the
camera or not (experimental sequence in Fig. 1(b)). This way the readout
process, and most importantly, the camera measurement is performed only if a
_signal_ photon is detected. This feedback is the central point of our
experimental implementation and provides a performance boost compared with
hitherto post-selection approaches.

The joint measurement of _signal_ -_idler_ pairs can be described by
wavevector-indexed set of POVMs:

$\Pi_{\pm}^{\theta_{s},\theta_{i}}(\mathbf{k}_{i})=\int|\mathbf{k},\theta_{s}\rangle\langle\mathbf{k},\theta_{s}|\otimes|\mathbf{k}_{i},\pm\theta_{i}\rangle\langle\mathbf{k}_{i},\pm\theta_{i}|\mathrm{d}\mathbf{k},$
(4)

where $|\theta\rangle=(|H\rangle+e^{i\theta}|V\rangle)/2$ represents the
measurement setting on the Bloch sphere’s equator and by $\pm$ we denote the
two possible outcomes on the “dual-channel” polarizer. Then, the outcome
probability distribution is calculated as:

$\displaystyle
p_{\pm}^{\theta_{s},\theta_{i}}(\mathbf{k}_{i})=\langle\psi_{\mathrm{B}}|\Pi_{\pm}^{\theta_{s},\theta_{i}}(\mathbf{k}_{i})|\psi_{\mathrm{B}}\rangle$
$\displaystyle=\int|\tilde{\psi}(\mathbf{k}_{s},\mathbf{k}_{i})|^{2}\cos^{2}\left(\frac{\phi(\mathbf{k}_{s},\mathbf{k}_{i})+\theta_{s}\mp\theta_{i}}{2}\right)\mathrm{d}\mathbf{k}_{s}$
(5)

where
$\phi(\mathbf{k}_{s},\mathbf{k}_{i})\coloneqq\varphi_{s}(\mathbf{k}_{s})-\varphi_{i}(\mathbf{k}_{i})$
is the phase difference profile and the integration goes over the _bucket_
detector area, which in our case is a circle bounded by the multimode fiber
numerical aperture (NA=0.2). From Eq. (5) we see that the resolution of the
phase-phase sensitive GI is limited by the strength of the wavevector
correlation described by Eq. (2), i.e the phase change in the _signal_ arm
$\varphi_{s}(\mathbf{k}_{s})$ should be slow when compared to anti-correlation
width $\kappa\approx 6\,\mathrm{mm^{-1}}$ (see Supplementary Information).

Figure 3: Ghost images for different measurement settings in Bell state
characterization. (a) Ghost images of _signal_ -_idler_ correlations for five
$(\theta_{s},\theta_{i})$ measurement settings. The $+$⃝ and $-$⃝ mark regions
corresponding to two channels ($\pm$) of the “dual channel” polarizer. (b)
Horizontal ($k_{x}$) averages of the correlation function
$\mathcal{C}^{\theta_{s},\theta_{i}}(k_{x},k_{y})$ for each measurement
setting from (a). The local visiblity is up to $80\%$.

In our setup we choose the $\varphi_{s}(\mathbf{k}_{s})$ to be linearly and
slowly varying mostly along the $y$ direction and similarly for
$\varphi_{i}(\mathbf{k}_{i})$ but with a higher slope. To measure the phase
profiles in the _signal_ and _idler_ arms we classically simulate the outcome
by seeding the memory with coherent state, using an additional laser beam
focused at the atomic cloud centre (see Supplementary Information and [26]).
For this measurement we replace the _bucket_ detector with a camera and
calcite beam displacer (see Supplementary Information for details), and
register intensity fringes in both arms. We collect frames for various
measurement settings $\\{\theta_{s},\theta_{i}\\}\in[0,2\pi]\times[0,2\pi]$
then for each frame we retrieve the phase using standard Fourier-transform
based procedure [37], and finally average the results. The retreived phase
profiles are presented in Fig. 2(a), where we also plot the combined profile
corresponding to the $\phi(\mathbf{k}_{s},\mathbf{k}_{i})$ phase difference
profile. Figure 2(b) presents the comparison between the classically simulated
(from retrieved phase) and measured correlation function
$\mathcal{C}^{0,0}(\mathbf{k}_{i})$ defined as:

$\mathcal{C}^{\theta_{s},\theta_{i}}(\mathbf{k}_{i})=\frac{n_{+}^{\theta_{s},\theta_{i}}(\mathbf{k}_{i})-n_{-}^{\theta_{s},\theta_{i}}(\mathbf{k}_{i})}{n_{+}^{\theta_{s},\theta_{i}}(\mathbf{k}_{i})+n_{-}^{\theta_{s},\theta_{i}}(\mathbf{k}_{i})},$
(6)

where by $n_{\pm}^{\theta_{s},\theta_{i}}(\mathbf{k}_{i})$ we denote the
number of registered idler photons with $\mathbf{k}_{i}$ wavevector in each
port of the “dual channel” polarizer. From Eq. (5) we expect the correlation
function to be proportional to $\cos(\phi(-\mathbf{k}_{s},\mathbf{k}_{i}))$,
and this is indeed what we see. The $\mathcal{C}^{0,0}(\mathbf{k}_{i})$ map is
build from $2\times 10^{7}$ experiment repetitions and contain $3\times
10^{5}$ counts corresponding to _signal_ -_idler_ coincidences. In Fig. 2(c)
we plot $k_{x}$-averaged fringes for both maps where we see good agreement
between the expectation and measurement. The only parameter fitted there is
the proportionality constant interpreted as the visibility $\mathcal{V}=78\%$.

To perform the Bell test in our hybrid scenario we acquire ghost images for
four different measurement settings
$(\theta_{s},\theta_{i})\in\\{(0,0),(\nicefrac{{\pi}}{{2}},0),(0,\nicefrac{{\pi}}{{2}}),(\nicefrac{{\pi}}{{2}},\nicefrac{{\pi}}{{2}})\\}\eqqcolon\mathcal{M}$
and for one additional (marginal) setting denoted as $\\{\infty,0\\}$ with the
the single channel polarizer removed. We use a variation of the CHSH
inequality [38, 39, 2]:

$|S|\leq 2,$ (7)

with the Bell parameter $S$ defined as follows (see Supplementary Information
for derivation):

$S=\mathcal{C}^{\theta_{s},\theta_{i}}(\mathbf{k}_{i})-\mathcal{C}^{\theta_{s},\theta_{i}^{\prime}}(\mathbf{k}_{i})+\mathcal{C}^{\theta_{s}^{\prime},\theta_{i}}(\mathbf{k}_{i})+\mathcal{C}^{\theta_{s}^{\prime},\theta_{i}^{\prime}}(\mathbf{k}_{i})\\\
-2\mathcal{C}^{\infty,\theta_{i}}(\mathbf{k}_{i}).$ (8)

Thanks to the wavevector dependent phase difference
$\phi(\mathbf{k}_{s},\mathbf{k}_{i})$ that in our case changes more than
$6\pi$ over observation region, the $|\psi_{\mathrm{B}}\rangle$ state (Eq.
(3)) maximally violates the inequality (7) for any two pairs of measurement
settings separated by $\pi/2$ :
$|\theta_{s}-\theta_{s}^{\prime}|=|\theta_{i}-\theta_{i}^{\prime}|=\pi/2$. It
stems from the fact that there is always a particular $\mathbf{k}_{i}$ and
thus $\phi(\mathbf{k}_{s},\mathbf{k}_{i})$ for which these settings are
optimal, i.e. lead to maximal violation of inequality (7). Therefore, for the
sake of simplicity we choose the angles to be $0$ or $\pi/2$.

In Fig. 3(a) we present acquired ghost images for the five particular
measurement settings, for both polarizer channels denoted as $+$ and $-$. From
these images we calculate the correlation functions as defined by Eq. (6). The
$k_{x}$-averaged results are presented in Fig. 3(b). From these, by fitting
the cosine function we obtain the following visibilities
{$77.9\%,78.6\%,77.2\%,78.9\%\\}\pm 0.5\%$ corresponding to the four
measurement settings from $\mathcal{M}.$ For the marginal setting (with the
“single channel” polarizer removed) we obtain
$\mathcal{C}^{\infty,\theta_{i}}(\mathbf{k}_{i})=(1\pm 3)\times 10^{-3}$ that
together with the visibilities give $S=2.213\pm 0.008$. This violates the CHSH
inequality (7) by 26 standard deviations (SDs), indicating the quantum nature
of the observed fringes. The less than 100% visibility is a result of some
imperfections present at different stages of our setup, which are discussed in
detail in the Supplementary Information.

To directly demonstrate the high visibility of the correlation function for
the four measurement bases we introduced the wavevector-dependent phase
difference profile $\phi(\mathbf{k}_{s},\mathbf{k}_{i})$ in the generated
state. Alternatively, as this phase is built from two independent phase
profiles in _signal_ and _idler_ arms we, could interpret it as to be
responsible for different measurement settings for different wavevectors
$\mathbf{k}_{i}$. This way, by treating the MZIs with the known phase profiles
$\varphi_{s(i)}(\mathbf{k}_{i})$ as a parts of the measurement devices we can
run the Bell test on the wavevector DoF entangled state given by Eq.(1). In
this way, we register many independent $(\varphi_{s},\varphi_{i})$ measurement
settings on a single ghost image. Consequently, it is convenient to redefine
the Bell $S$ parameter using the Freedman and Clauser formula [1] modified for
our hybrid scenario (see Supplementary Information for derivation):

$S=3\mathcal{C}(\phi)-\mathcal{C}(3\phi)-2\mathcal{C}^{\infty}(\phi),$ (9)

where $\mathcal{C}(\phi=\varphi_{s}-\varphi_{i})$ is defined using Eq. (6)
with $n_{\pm}^{\theta_{s},\theta_{i}}(\mathbf{k}_{i})$ averaged over the
wavevectors $\mathbf{k}_{i}$ corresponding to the same phase $\phi$:
$n_{\pm}^{\theta_{s},\theta_{i}}(\mathbf{k}_{i})\to
n_{\pm}^{\theta_{s},\theta_{i}}(\phi)=\langle
n_{\pm}^{\theta_{s},\theta_{i}}(\mathbf{k}_{i})\rangle_{\mathbf{k}_{i}:\phi(-\mathbf{k}_{i},\mathbf{k}_{i})=\phi}$.
The marginal correlation function $\mathcal{C}^{\infty}(\alpha)$ is defined in
the same manner and is experimentally found to be independent of
$\mathbf{k}_{i}$ and thus $\phi$ as can be seen in Fig. 3(b).

Figure 4 presents values of $\mathcal{C}(\phi)$ correlation function obtained
from single ghost image (the $(0,0)$ setting of Fig. 3(a)). The ensemble of
_signal_ and _idler_ phase samples used to calculate $\mathcal{C}(\phi)$ is
drawn in the inset (Fig. 4(i)) where the coloring, similarly to Fig. 2(b),
represents the value of the correlation function for each
$(\varphi_{s},\varphi_{i})$. We clearly see that the correlation varies with
the phase difference $\phi=\varphi_{s}-\varphi_{i}$ as expected. With the
definition of the Bell parameter given by Eq. (9) the inequality Eq.(7) is
maximally violated for four phases from the $[0,2\pi]$ range:
$\phi\in\\{\pi/4$, $3\pi/4,5\pi/4,7\pi/4$}. Those points and the corresponding
values of $\mathcal{C}(\phi$) are marked in the Fig. 4 using dashed lines.

Figure 4: Entanglement certification with a singule ghost image. Correlation
function $\mathcal{C}(\phi)$ from single ghost image ($(0,0)$ settings from
Fig.3(a)) along with fitted theoretical curve. Inset (i) shows the ensemble of
angular samples colored according to the correlation function value for each
point, similarly as in Fig. 2(b).

By fitting the visibility $\mathcal{V}=78.6\%\pm 0.3\%$ of the expected
correlation curve $\mathcal{V}\cos(\phi)$ to match the observed correlation
values $\mathcal{C}(\phi)$ we obtain $S=2.227\pm 0.007$ which violates the
inequality Eq.(7) by more than 32 SDs.

In summary, we have employed an emissive QM as a quantum light source for
phase-sensitive GI setup and demonstrated images of Bell-type correlations.
The QM generates entangled _signal_ and _idler_ photon pairs in many
wavevector modes and stores the _idler_ photon for controllable delay time.
The _idler_ photon release is controlled by real-time feedback allowing us to
realize the ghost imaging protocol without any additional image-preserving
delay line, as comapred with Ref. [33]. To the best of our knowledge, this is
the first demonstration of quantum-memory-assisted GI experiment, and
effectively a demonstration of atom-photon Bell-type entanglement in a single
image.

Let us finally discuss potential applications and direct advantages over
previosuly demonstrated ghost imaging experiments, based on spontaneous
parametric down-conversion (SPDC). In the Table 1 we compare our results with
other works that use delay line (DL) instead of quantum memory (QM) to acquire
the ghost images. In the SPDC experiments the _signal_ -_idler_ coincidence
rate (R) is limited by the maximum gating frequency of the image sensor and
varies from few to few hundreds coincidences per second (cps). In our setup we
achieve $\approx 6$ cps as we are limited by the atom cooling and trapping
period that occupies 89% of the sequence. The instantaneous rate is therefore
around 60 cps and can be doubled in non-phase sensitive approach by removing
the interferometers. This is comparable with most SPDC approaches although
with faster camera gating SPDC-based GI could potentially reach much higher
rates. The available number of modes (M) and achievable visibility (or
equivalently signal to noise ratio) is comparable in both cases. The parameter
that we want to focus on is the available time delay ($\tau$) needed to
properly gate the camera which in case of SPDC experiments never exceeds
$100\,$ns ($30\,$m of additional optical path). In our setup the delay can be
set programatically and can be as long as the wavevector-dependent memory
lifetime [40], that for $45\,\mu$s allows retrieval of 50% of the modes. Such
long delay can be useful in applications as quantum ghost LIDAR that can be
resistant to detection path turbulences [41] and quantum radar [42, 43] where
the quantum illumination leads to improvement of signal to noise ratio [44].
Another difference is that the _signal_ and _idler_ photons emitted from our
memory have few-MHz linewidths and thus they are easier to filter from the
broadband background noise and can be used for remote sensing of inherently
narrowband optomechanical and atomic systems, facilitating for example
magnetic field imaging. Moreover, presented system equipped with a more
advanced _signal_ photon detection and combined with wavefront shaping
techniques based on fast micro-mirror arrays or spatial light modulators could
lead to new adaptive quantum GI schemes [45]. As the phase modulation can be
realized by fast all-optical spin-wave modulation techniques [26, 46, 37] that
effectively reshape the _idler_ photon spatial structure, the phase reshaping
can be applied conditionally on _signal_ photon measurement outcome. We also
envisage that via feedback one may shape the idler light not only spatially,
but also in terms of its statistics (akin to heralding single or multi-photon
states) for protocols such as quantum image distillation [47]. Our technique
could also be applied in quantum information schemes such as quantum-secured
imaging [48], quantum secure ghost imaging [49].

Type |  | R
---
$(\frac{1}{\mathrm{s}})$
$\mathcal{V}$ |  | $\tau$
---
($\mathrm{ns}$)
M | PS | Ref.
QM | 6 | 0.78 | $>150$ | 370 | yes |  | This
---
work
DL | 35 | 0.9 | 73 | 500 | no |  [50]
DL | 528 | 0.71 | 84 | 113 | no |  [51]
DL | 4 | 0.86 | 67 |  | yes |  [33]
DL | $<6$ | 0.67 |  |  | yes |  [30]
Table 1: Comparison of two GI approaches: QM - Quantum Memory, DL - delay line
in means of coincidence rate (R), visibility ($\mathcal{V}$), number of modes
(M) and phase sensitivity (PS).

We thank K. Banaszek for the generous support and J. Kołodyński for insightful
discussions. This work was funded by MNiSW (DI2016 014846, DI2018 010848),
National Science Centre (2016/21/B/ST2/02559, 2017/25/N/ST2/01163,
2017/25/N/ST2/00713) and by the Foundation for Polish Science (MAB/2018/4
“Quantum Optical Technologies”). The "Quantum Optical Technologies ” project
is carried out within the International Research Agendas programme of the
Foundation for Polish Science co-financed by the European Union under the
European Regional Development Fund. M.P. was supported by the Foundation for
Polish Science via the START scholarship.

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## Supplementary Information

### Quantum Memory implementation

The quantum memory is based on a pencil shaped cold (T$\approx
50\,\mu\mathrm{K}$) cloud of ${}^{87}\mathrm{Rb}$ atoms prepared in a 3D
magnetooptical trap operating at around $400\,\mathrm{Hz}$ repetition rate.
The memory protocol utilizes two atomic ground states
$|g\rangle=|5^{2}S_{1/2},F\text{\,}=\text{\,}1\,\rangle$ and
$|h\rangle=|5^{2}S_{1/2},F\text{\,}=\text{\,}2\rangle$ to store _idler_
photons in a form of collective atomic excitations known as spin waves [see
also Fig. 1(c) of the main text]. The spin waves are generated together with
_signal_ photons in a write-out process for which a strong $150\,\mathrm{ns}$
long write (w) laser pulse red detuned by $30\,\mathrm{MHz}$ from
$|g\rangle\to|e_{w}\rangle=|5^{2}P_{3/2},F\text{\,}=\text{\,}2\rangle$
transition is employed. To read-out the _idler_ photon we use a strong read
(r) laser pulse ($150\,\mathrm{ns}$) resonant with
$|h\rangle\to|e_{r}\rangle=\text{\,}|5^{2}P_{1/2},F\text{\,}=\text{\,}2\rangle$
transition. The double $\Lambda$ scheme employed in the memory protocol allow
us to efficiently filter out any stray laser light present in either _signal_
or _idler_ beam (see [24] for details of the filtering system). Due to the
phase matching involved in the readout process the resulting photon pairs are
anti-correlated in momenta. As we choose the write and read beams to counter
propagate through the ensemble the anti-correlation reads:
$\frac{\mathbf{k}_{i}}{|\mathbf{k}_{i}|}=-\frac{\mathbf{k}_{s}}{|\mathbf{k}_{s}|}$,
where $\mathbf{k}_{s(i)}$, correspond to _signal_ and _idler_ photon
wavevector respectively. Moreover, as demonstrated previously [25] the
resulting photons are additionally correlated in positions, which is a feature
of the EPR state.

### CHSH inequality in a hybrid scenario

To perform the Bell test in our hybrid scenario, we use a variation of the
CHSH inequality adjusted for our correlation function defined (neglecting the
wavevector dependence) as follows:

$\mathcal{C}^{\theta_{s},\theta_{i}}=\frac{n_{+}^{\theta_{s},\theta_{i}}-n_{-}^{\theta_{s},\theta_{i}}}{n_{+}^{\theta_{s},\theta_{i}}+n_{-}^{\theta_{s},\theta_{i}}},$
(10)

where $n_{\pm}^{\theta_{s},\theta_{i}}$ denotes the number of registered
_signal_ -_idler_ coincidences at each port ($+$ or $-$) of the “dual channel”
polarizer. To derive the $S$ parameter for such correlation function we follow
the derivation of the single channel inequality [39] with assumptions adapted
to our hybrid scenario. For the convenience let us denote the _signal_ and
_idler_ polarizer orientations as $a\,(a^{\prime})$ and $b\,(b^{\prime})$.
Then using the non enhancement assumption [39] we can write the four
inequalities for _signal_ and _idler_ photon detection:

$\displaystyle 0\leq p_{s}^{a}(\lambda)\leq p_{s}^{\infty}(\lambda),$ (11)
$\displaystyle 0\leq p_{s}^{a^{\prime}}(\lambda)\leq p_{s}^{\infty}(\lambda),$
(12) $\displaystyle 0\leq p_{i,\pm}^{b}(\lambda)\leq
p_{i,+}^{b}(\lambda)+p_{i,-}^{b}(\lambda),$ (13) $\displaystyle 0\leq
p_{i,\pm}^{b^{\prime}}(\lambda)\leq
p_{i,+}^{b^{\prime}}(\lambda)+p_{i,-}^{b^{\prime}}(\lambda),$ (14)

where $\lambda$ is the hidden variable defining the quantum state of the
source and has associated probability density $\rho(\lambda)$. Additionally,
without loss of generality we can assume
$p_{i,+}^{b^{\prime}}(\lambda)+p_{i,-}^{b^{\prime}}(\lambda)\leq
p_{i,+}^{b}(\lambda)+p_{i,-}^{b}(\lambda)$ (which in the most sensible cases
is just equality) to arrive with the following expressions:

$-p_{s}^{\infty}(\lambda)p_{i,+}^{b}(\lambda)\leq
p_{s}^{a}(\lambda)p_{i,+}^{b}(\lambda)-p_{s}^{a}(\lambda)p_{i,+}^{b^{\prime}}(\lambda)+p_{s}^{a^{\prime}}(\lambda)p_{i,+}^{b}(\lambda)+p_{s}^{a^{\prime}}(\lambda)p_{i,+}^{b^{\prime}}(\lambda)\leq
p_{s}^{\infty}(\lambda)p_{i,-}^{b}(\lambda),$ (15)
$-p_{s}^{\infty}(\lambda)p_{i,-}^{b}(\lambda)\leq
p_{s}^{a}(\lambda)p_{i,-}^{b}(\lambda)-p_{s}^{a}(\lambda)p_{i,-}^{b^{\prime}}(\lambda)+p_{s}^{a^{\prime}}(\lambda)p_{i,-}^{b}(\lambda)+p_{s}^{a^{\prime}}(\lambda)p_{i,-}^{b^{\prime}}(\lambda)\leq
p_{s}^{\infty}(\lambda)p_{i,+}^{b}(\lambda).$ (16)

Then, using the standard expression for the LHVT coincidence probability
$p_{\pm}^{a,b}(\lambda)=-p_{s}^{a}(\lambda)p_{i,\pm}^{b}(\lambda)$ and
integrating inequalities (15) and (16) with the hidden parameter distribution
$\rho(\lambda)$ and finally adding them with opposite signs we obtain:

$-p_{+}^{\infty,b}-p_{-}^{\infty,b}\leq\mathcal{P}^{a,b}-\mathcal{P}^{a,b^{\prime}}+\mathcal{P}^{a^{\prime},b}+\mathcal{P}^{a^{\prime},b^{\prime}}-\mathcal{P}^{\infty,b}\leq
p_{+}^{\infty,b}+p_{-}^{\infty,b},$ (17)

where $\mathcal{P}^{a,b}=p_{+}^{a,b}-p_{-}^{a,b}$ is the probability
correlation function. As all of the probabilities in (17) are associated with
_signal_ -_idler_ coincidences the inequality can be tested without the total
number of emissions being known. However, to trustfully perform measurements
needed to test this inequality the source has to be very stable in time,
especially when it comes to the polarizer removal ($\infty$ setting).
Therefore, to perform the Bell test in our case we use the standard fair
sampling assumption and take the coincidences correlation function (10) as a
fair estimate of the probability correlation function
$\mathcal{P}^{\theta_{s},\theta_{i}}=\mathcal{C}^{\theta_{s},\theta_{i}}/2$
for any pair of measurements settings $(\theta_{s},\theta_{i})$ excluding the
special case with the _signal_ polarizer removed, for which
$\mathcal{P}^{\infty,\theta_{i}}=\mathcal{C}^{\infty,\theta_{i}}$. With these
assumptions $p_{+}^{\infty,b}+p_{-}^{\infty,b}=1$ and the inequality (17)
finally becomes:

$-2\leq\mathcal{C}^{\theta_{s},\theta_{i}}-\mathcal{C}^{\theta_{s},\theta_{i}^{\prime}}+\mathcal{C}^{\theta_{s}^{\prime},\theta_{i}}+\mathcal{C}^{\theta_{s}^{\prime},\theta_{i}^{\prime}}-2\mathcal{C}^{\infty,\theta_{i}}\leq
2.$ (18)

Further, when
$\mathcal{C}^{\theta_{s},\theta_{i}}=\mathcal{C}(\phi=|\theta_{s}-\theta_{i}|)$
and we choose $\theta_{s}$, $\theta_{s}^{\prime}$, $\theta_{i}$,
$\theta_{i}^{\prime}$ to satisfy
$|\theta_{s}-\theta_{i}|=|\theta_{s}^{\prime}-\theta_{i}|=|\theta_{s}^{\prime}-\theta_{i}^{\prime}|=\frac{1}{3}|\theta_{s}-\theta_{i}^{\prime}|=\phi$
the inequality (18) becomes:

$-2\leq 3\mathcal{C}(\phi)-\mathcal{C}(3\phi)-2\mathcal{C}^{\infty}(\phi)\leq
2,$ (19)

which is the correlation function equivalent of the inequality derived by
Freedman for probabilities [1].

### Visibility limitations

Here we summarize all factors contributing to reduced visibility in
correlation function fringes.

First of all, the ultimate visibility (assuming perfect optical components) is
limited by the purity of the generated photons pairs. Any stray noise present
in the _idler_ and indirectly _signal_ arm __ will appear as a nonzero floor
on the acquired ghost images. As the noise comes from uncorrelated photon
pairs it will appear also outside the region determined by the _signal_
observation region (limited by the bucket detector sensitivity area).
Therefore, by looking outside the correlation region we can get the average
amount of the noise present $\bar{n}_{\mathrm{bckg}}$ and then estimate the
ultimate visibility as:
$\mathcal{V}_{\mathrm{ult}}=(\bar{n}_{\mathrm{sig}}-\bar{n}_{\mathrm{bckg}})/(\bar{n}_{\mathrm{sig}}+\bar{n}_{\mathrm{bckg}})$,
where
$\bar{n}_{\mathrm{sig}}=\nicefrac{{1}}{{\mathcal{A}_{\circ}}}\int(n_{+}^{\infty,0}(\mathbf{k}_{i})+n_{+}^{\infty,0}(\mathbf{k}_{i}))\mathrm{d}\mathbf{k}_{i}$
is average number of registered coincidences in the circular area
$\mathcal{A}_{\circ}$. In our case the ultimate visibility amounts about
$89\%$. It is also noteworthy that this formula can be rewritten using Glauber
second order correlation function of _signal_ and _idler_ pairs, as
$g_{si}^{(2)}\simeq\bar{n}_{\mathrm{tot}}/\bar{n}_{\mathrm{bckg}}\approx
16.5$. The imperfections of the optical components, including mirrors,
polarizers and waveplates lower the achievable visibility by a factor of
$0.97^{2}$. Additionally, finite correlation strength characterized by
$\kappa$ limits the visibility for fast varying phase profiles in the _signal_
arm, however as this phase slope in our case is low this effect has minimal
contribution (less than $1\%$) to the total visibility. Finally, the remaining
drop (by a factor equivalent to about $97\%$) is most probably caused by the
residual mismatch of the wavevector shift $\delta\mathbf{k}$ between the two
MZIs.

The next sections of this Supplementary Information treat the details of
various effects impacting the visbility.

### Noise and photon pairs purity

The ultimate visibility that could be achieved in our setup when ideal optical
components are used is limited by purity of the generated photon pair state.
As the _signal_ -_idler_ pair generation process is purely probabilistic there
always is a chance of multiple-pair generation in one shot. The higher order
pairs similarly as spurious noise photons coming from the read laser leakage
and I-CMOS sensor dark counts will appear at the acquired images as a constant
noise floor, present even outside the correlation region determined by the
bucked detector area. Therefore, by looking outside the correlation circle we
can estimate the total amount of the noise present and hence calculate the
ultimate visibility by comparing the noise level with the correlated
coincidences.

In Fig. (5) we present the percentage noise and signal values taken from
measurement with $(0,0)$ analyzer settings. From these we estimate the
ultimate visibility to be
$\mathcal{V}_{\mathrm{ult}}=(\bar{n}_{\mathrm{sig}}-\bar{n}_{\mathrm{bckg}})/(\bar{n}_{\mathrm{sig}}+\bar{n}_{\mathrm{bckg}})\approx
88.6\%$.

Figure 5: The percentage contributions of the noise to the observed signal.
The bar represents the average value taken from the appropriate region in the
ghost image. The dark counts are taken from the region of the I-CMOS camera
frame that is not illuminated by any optical field (purple frame). The noise
counts are taken from regions (golden frame) outside the ghost-imaged circular
bucked detector area, these includes also the dark counts as they are present
uniformly over whole frame. The signal counts, that include also the noise
counts are taken from the circular regions inside the virtually-visible bucket
detector area (red circles). The wavevector variables are only for scale
purposes and they (in absolute sense) do not refer to the true wavevector
values.

The quality of the _signal_ -_idler_ pairs can be characterized by Glauber
second order correlation function defined as

$g_{si}^{2}=\frac{\langle n_{s}n_{i}\rangle}{\bar{n}_{s}\bar{n}_{i}},$ (20)

where $\langle n_{s}n_{i}\rangle$ is the average number of coincidences and
$\bar{n}_{s(i)}$ represents average number of _s_(_i_)-_signal_(_idler_)
photons. As we are working with low pair generation probability $p$ we can
approximate the above averages with proper probabilities. Let us we denote the
total transition-and-detection efficiencies as $\chi_{s}$, $\chi_{i}$ and
introduce the two spurious noise probabilities $\zeta_{s}$ ,$\zeta_{s}$ for
_s_ and _i_ arm respectively. Then, we can write the coincidence probability
by analyzing the possible events up to the $\mathcal{O}(p^{2})$ order:

$p_{si}=p\chi_{s}\chi_{i}+p\chi_{s}\zeta_{i}+p\chi_{i}\zeta_{s}+\zeta_{s}\zeta_{i}+p^{2}\chi_{s}\chi_{i}.$
(21)

The subsequent terms have the following interpretation: $p\chi_{s}\chi_{i}$
represent the probability of generating and detecting a genuine _s_ -_i_ pair;
$p\chi_{s}\zeta_{i}$, $p\chi_{i}\zeta_{s}$ are probabilities of coincidences
between the _signal_ (or _idler_) photon an the noise photon in the second
arm; $\zeta_{s}\zeta_{i}$ is the probability of noise-noise coincidence, and
$p^{2}\chi_{s}\chi_{i}$ is the _s_ -_i_ from two generated (and uncorrelated)
pairs. The single-photon detection probabilities read:
$p_{s}=p\chi_{s}+\zeta_{s}$ and $p_{i}=p\chi_{i}+\zeta_{i}$, which, after
dropping the $\zeta_{s}\zeta_{i}\ll p\chi_{s(i)}\zeta_{s(i)}$ term finally
give:

$g_{si}^{2}\simeq 1+\frac{1}{p+\zeta_{s}/\chi_{s}+\zeta_{i}/\chi_{i}},$ (22)

which for $\bar{n}_{s}\simeq p_{s}$ transform into:

$g_{si}^{2}\simeq 1+\frac{1}{\bar{n}_{s}/\chi_{w}+\zeta_{i}/\chi_{i}}.$ (23)

Additionally, as mentioned earlier the genuine coincidences appear on the
camera frame only in region limited by the bucket detector area, hence, by
looking outside this region we can estimate the denominator of (20) as
$\bar{n}_{s}\bar{n}_{i}\simeq p_{s}p_{i}=p_{si}-p\chi_{s}\chi_{i}$ and arrive
at $g_{si}^{(2)}\simeq\bar{n}_{\mathrm{tot}}/\bar{n}_{\mathrm{bckg}}$, which
in our case (Fig. 5) amounts to $g_{si}^{(2)}\approx 16.5$.

In Fig. 6 we present additional measurement of the $g_{si}^{2}$ function for
different _signal_ photon mean numbers. The solid curve is fit of (23) to the
data with $\chi_{s}=(7.5\pm 0.1)\%$ and $\zeta_{i}/\chi_{i}=(4\pm 2)\permil$.
The measurement for the GI experiment were performed around
$\bar{n}_{s}\approx 4\times 10^{-3}$, corresponding to $p\approx 0.05$.

Figure 6: Measured second order correlation function $g_{si}^{2}$ between
_signal_ and _idler_ photons as a function of mean _signal_ photon number
$\bar{n}_{s}$. The solid curve represents a simple model fitted to the data.
The ultimate visibility $\mathcal{V}_{\mathrm{ult}}$ corresponding to the
measured _$g_{si}^{(2)}$_ function values is present on the second vertical
axis.

### Correlation strength $\kappa$ and its impact on visibility

The generated biphoton wavefunction $\psi(\mathbf{k}_{s},\mathbf{k}_{i})$ is
characterized by near-field ($\sigma$) and far-field ($\kappa$) correlations
strengths. In our case, since we are working with wavevector correlations, the
$\kappa$ parameter is of the most importance. It limits, the achievable
resolution and in the case of the Bell-correlations demonstration it leads to
the drop of the visibility of observed quantum fringes. The visibility
reduction due to finite $\kappa$ in this case can be estimated by evaluating
the integral (see Eq. (5) of the main article)
$p_{\pm}(\mathbf{k}_{i})=\int|\tilde{\psi}(\mathbf{k}_{s}-\delta\mathbf{k},\mathbf{k}_{i})|^{2}\cos^{2}\left(\frac{\varphi(\mathbf{k}_{s})-\varphi(\mathbf{k}_{i})}{2}\right)\mathrm{d}\mathbf{k}_{s}$
for a linearly varying $\varphi(\mathbf{k}_{s})$, which without loss of
generality can by assumed to vary only in the $\hat{x}$ direction,
$\varphi(\mathbf{k}_{s})=\alpha k_{x}$. Additionally, as mentioned in the main
article the wavevector-space field of view is limited by technical constraints
rather than the spread $\sigma$ of the generated state and thus we take
$\sigma\to 0.$ In this case, the integral gives
$(1+e^{-\alpha^{2}\kappa^{2}/2}\cos(\varphi_{s}(-\mathbf{k}_{i})-\varphi(\mathbf{k}_{i}))/2$,
where we can identify the visibility to be
$\mathcal{V}_{\kappa}=e^{-\alpha^{2}\kappa^{2}/2}$. To estimate the
correlation strength $\kappa$ and thus the reduction of visibility we
performed additional measurement with the bucked detector replaced by camera.
The results in form of correlation maps in center of mass coordinates
$(\mathbf{k}_{s}+\mathbf{k}_{i})$ integrated over the second direction
$(\mathbf{k}_{s}-\mathbf{k}_{i})$ are present in Fig. (7). The maps are drawn
for two channels ($+$ and $-$) of the beam displacer (BD) with the noise
background $\bar{n}_{s}\bar{n}_{i}$ subtracted. The solid lines represents the
fitted Gaussian curves which give estimates of $\kappa$ in each direction.
Since, the estimates of $\kappa$ have similar values as the final estimate we
take the average over the four values and obtain $\kappa=(5.9\pm
0.7)\,\mathrm{mm^{-1}}$. This, together with our _signal_ phase slope
$\alpha\approx 12.4\,\mathrm{mrad\times mm}$ gives
$\mathcal{V}_{\kappa}\approx 99.7\%$ which is negligible when compared to
other imperfections.

Figure 7: Noise-subtracted _signal_ -_idler_ coincidences in the center of
mass coordinates measured with camera. The solid lines represent the cross
sections through the maximum of the fitted theoretical probability
distribution $|\psi(\mathbf{k}_{s},\mathbf{k}_{i})|^{2}$. The labels mark the
$2\kappa$ widths of the distributions.

### Visibility drop due to the MZIs $\delta\mathbf{k}$ mismatch

To derive the formula for the desired Bell-like sate
$|\psi_{\mathrm{B}}\rangle$ we assumed that the two MZIs have the same
(matched) wavevector shift $\delta\mathbf{k}$. In the experiential
realization, however, these two shifts will be always matched with finite
precision $\xi\mathbf{k}$. If this is the case, the generated state can be no
longed described by the compact form, where the superposition is only at the
polarization level as in Eq. (3) of the main text. To quantify the effect of
the residual mismatch we can to use the intermediate form and add the mismatch
to the one of the parts:

$|\psi_{\mathrm{B}}\rangle=\mathcal{N}\iint_{[0,\delta\mathbf{k}]^{2}}(e^{i\varphi_{s}(\mathbf{k}_{s})}\psi(\mathbf{k}_{s},\mathbf{k}_{i}-\delta\mathbf{k})|\mathbf{k}_{s},H\rangle|\mathbf{k}_{i},V\rangle+e^{i\varphi_{i}(\mathbf{k}_{i})}\psi(\mathbf{k}_{s}-\delta\mathbf{k-\xi\mathbf{k}},\mathbf{k}_{i})|\mathbf{k}_{s},V\rangle|\mathbf{k}_{i},H\rangle)\mathrm{d}\mathbf{k}_{s}\mathrm{d}\mathbf{k}_{i}.$
(24)

Then, after plugging (24) into Eq. (5) of the main text and evaluating the
integral in the $\delta\mathbf{k}\gg\kappa$ limit we obtain:

$p\propto(1+e^{-\xi\mathbf{k}^{2}/(8\kappa^{2})}\cos(\varphi_{s}(-\mathbf{k}_{i})-\varphi_{i}(\mathbf{k}_{i}))/2,$

where we identify the visibility as
$\mathcal{V}_{\xi\mathbf{k}}=e^{-\xi\mathbf{k}^{2}/(8\kappa^{2})}$ which we
estimate to be $97\%$ that correspond to $\xi\mathbf{k}\approx 0.5\kappa$.

### Classical interference measurements

To measure the phase profiles we replaced the _bucket_ detector with the
camera and beam displacer as in the _idler_ path and performed interference
measurements using classical light. The classical _idler_ light was generated
in the memory by seeding the write-out process using additional laser beam
that was injected to the memory simultaneously with the write lase pulse. The
seed beam was phase-coherent with the write laser and focused in the center of
the atomic cloud. To maintain the phase coherence the seed beam is derived
from the write laser by frequency shifting small fraction of the laser output
by $6.834\,\mathrm{GHz}$ using EOM and filtering cavity (see [26] for details
of the setup). This beam was used both to measure the phase profile in the
_signal_ arm and to generate strong atomic coherence in the memory. In the
read-out process this coherence was retrieved to light that has been used to
measure phase profile in the _idler_ arm. Each phase profile measurement
consisted of running the experiment with continuously changing the measurement
settings $\\{\theta_{s},\theta_{i}\\}\in[0,2\pi]\times[0,2\pi]$ and
registering the intensity fringes $I_{+\,(-)}$ in the $+$ and $-$ measurement
channels corresponding to two separate regions on the camera. The phase
profiles were then retrieved using Fourier-transform based procedure [37]
evaluated on the correlation function calculated as
$\mathcal{C}=\frac{I_{+}-I_{-}}{I_{+}+I_{-}}$. In this procedure we first take
a two-dimensional Fourier transform of the real-valued correlation frame, then
we select spatial-frequency region around one of two main (identical)
frequency components that are located symmetrically to the zero-frequency
point. We filter out that region by setting remaining (outside this region)
values to zero. Finally inverse two-dimensional Fourier transform gives us a
complex signal with phase equal to the interferometer phase profile shifted by
$\theta_{s\,(i)}$. We repeat the procedure for each correlation frame for both
_signal_ and _idler_ arm. For each calculated complex frame we set the global
phase $\theta_{s\,(i)}$ to 0 and finally we average the results. The final
phase profiles are retrieved from the complex average by simply evaluating the
$\arg(\cdot)$ function.


0 Research

# Toward a Bias-Aware Future for Mixed-Initiative Visual Analytics

Adam Coscia e-mail<EMAIL_ADDRESS>Duen Horng Chau e-mail:
<EMAIL_ADDRESS>Alex Endert e-mail<EMAIL_ADDRESS>Georgia Tech

###### Abstract

Mixed-initiative visual analytics systems incorporate well-established design
principles that improve users’ abilities to solve problems. As these systems
consider whether to take initiative towards achieving user goals, many current
systems address the potential for cognitive bias in human initiatives
statically, relying on fixed initiatives they can take instead of identifying,
communicating and addressing the bias as it occurs. We argue that mixed-
initiative design principles can and should incorporate cognitive bias
mitigation strategies directly through development of mitigation techniques
embedded in the system to address cognitive biases in situ. We identify domain
experts in machine learning adopting visual analytics techniques and systems
that incorporate existing mixed-initiative principles and examine their
potential to support bias mitigation strategies. This examination considers
the unique perspective these experts bring to visual analytics and is situated
in existing user-centered systems that make exemplary use of design principles
informed by cognitive theory. We then suggest informed opportunities for
domain experts to take initiative toward addressing cognitive biases in light
of their existing contributions to the field. Finally, we contribute open
questions and research directions for designers seeking to adopt visual
analytics techniques that incorporate bias-aware initiatives in future
systems.

Introduction

Mixed-initiative systems provide a collaborative approach to performing tasks
that balance human and machine effort. This design metaphor exists in visual
analytics (VA) systems in the form of machine-learning (ML) models and
algorithms that seek to combine human-in-the-loop goals with techniques for
automation while leveraging the power of visualization for creating insights
from information. Horvitz [9] expanded the literature of design principles
that seek to couple direct manipulation of interfaces with automated services
by addressing the uncertainty of ascertaining user goals under a cost and
benefits framework. These principles have been explored in increasingly
diverse visualization research [1, 5, 11] since then as designers develop
systems that explore the roles they have in augmenting the user experience
using VA techniques and systems. In particular, Ceneda et al. [2] recognize
the need for understanding when that role encompasses principles of guidance,
and establish a model for designing system initiatives that seek to close a
knowledge gap in data analysis.

As mixed-initiative VA research balances human and machine initiatives towards
this role of guidance, another role is being investigated: one that
concurrently considers the effects of cognitive biases on the analysis
process. Research teams across diverse domains of expertise consider the
impacts of cognitive bias regularly as they apply to adoption of VA within
their domains [3, 4, 13, 14, 18, 20]. However, little research has yet been
devoted to understanding this interplay of cognitive theory and design of
mixed-initiative VA systems as an interdisciplinary framework of future
research. Specifically, such frameworks have the potential to spawn lines of
research towards designing and developing tools that can help mitigate
cognitive biases and promote fairness – and lead toward a bias-aware future
for mixed-initiative VA systems. This paper identifies opportunities and
challenges that exist towards solving this challenge, and discusses them in
context of related work and open questions.

##  1 Existing Domain Applications of Visual Analytics Techniques and Systems

VA systems today see increased usage in a wider variety of contexts than ever
before, with researchers being encouraged to expand the scope of contributions
to the field into more intellectually diverse areas[10]. The prominence of
these systems in the ML community is underscored by their ability to create
insight into previously indecipherable streams of information that come from
the “black boxes” of code powering them[11]. In particular, the deep learning
community is beginning to leverage the power of visually representing
relationships and structures embedded in immensely complex neural networks to
great success[8]. Urban computing, with its emphasis on large, complex data
sets, and urban computing experts have increasingly turned to VA for data
exploration, pattern interpretation, and visual learning goals[21]. Expanding
research avenues that focus on interactive data analysis and exploration are
looking toward VA systems to extract human traits encoded in interaction
processes [20]. These teams investigating machine initiatives in support of
analytical reasoning show the impact of adopting VA across ever more diverse
domains. This diversity suggests that there is a great desire for VA systems
to explore the boundaries of conventional visual design principles, resist a
singular focus on domain-specific development practices, and test the limits
of machine-driven data analysis and exploration using VA techniques and
systems.

From the perspective of mixed-initiative literature, this desire for smarter
autonomous processes and initiatives should be equally mirrored by a focus on
understanding the human endeavors that also drive the analysis process.
However, examples in this direction lack the wider adoption seen above,
despite bringing much desired insight and contextualization to the domains
above. In other words, the marriage of cognitive theory with VA system design
principles remains limited in application, though teams are beginning to show
how domains might take advantage of VA in understanding this research area.
Nussbaumer et al. [13] develop a preliminary framework for addressing specific
cognitive biases in the field of criminal analysis aided by VA systems. Wall
et al. [17] consider the impact of cognitive bias on the analytic process in
existing VA systems, and establish a preliminary framework informed by the
fields of cognitive theory and decision-making. Addressing this
interdisciplinary approach, Padilla et al. [14] suggest a lack of unifying
framework limits cross-domain communication of findings informed by cognitive
theory, and develop a decision-making model to address this. While support
continues to grow for developing informed autonomous initiatives for VA
systems in ML, urban computing, criminal analysis, and other fields, such
systems also comprise a role of understanding and addressing the cognitive
behaviors of users during the analysis process.

##  2 Opportunities to Take Initiative

We present a collective vision consisting of multiple ideas for identifying
opportunities in designing mixed-initiative VA systems across various domains
with the goal of mitigating cognitive bias. Developers of mixed-initiative
systems design specific “initiatives” that such tools deliver to guide and
support users in their tasks. Our ideas highlight how such initiatives can be
grounded in what different communities know about the cognitive processes of
users and we discuss them in context of existing VA systems and literature.

Cognitive processes of people can be captured and interpreted systematically
to identify cognitive traits, tasks, biases, and more. Mixed-initiative
systems can employ automated initiatives that strive to address the
uncertainty in user’s goals and attention [9]. Much of the value added by
those principles is determined by the uncertain nature of how those
initiatives serve the personal goals of the user [20]. FairVis [1], a mixed-
initiative VA system for auditing fairness of ML models, addresses such
uncertainty outright using domain knowledge provided by the user to inform the
automated uncovering of model subgroups relevant to the user. Other models of
interpreting user goals have been developed that can capture interaction
provenance and interpret various aspects of user behavior or cognitive traits
[20]. We believe systems have information about user goals and intent
available in the analytic process itself and can thus help guide and support
users better if such information is acted on. When it comes to addressing bias
in analysis, such frameworks are critical. Current methods of bias mitigation
rely largely on strategies that reside outside of the systems themselves, such
as manual provenance tracking solutions and other structured analytic
techniques that help illuminate errors in the cognitive process of analysis
[7]. Instead, we envision visual analytic systems built to consider a wide
design space for how they can respond once biases are detected [19]. In other
words, designing systems that take initiative towards correcting these
analysis errors when detected.

Showing users a history of their process can help promote awareness and
reflection of personal bias. Inherent in the push for utilizing provenance
data is a belief that mixed-initiative systems can act on this information.
Supporting this idea, Ragan et al. [15] develop an organizational framework
that expands the literature of provenance types and purpose in the analytic
process of visualization and data analysis research. Padilla et al. [14]
further address the ability of cross-discipline research to inform the user
and machine, and contribute an integrative model that acts on decision-making
principles informed by cognitive theory used in visualization for
comprehension. The marriage of these critical components of mixed-initiative
VA systems provides a framework in which to realize the potential for
impacting the cognitive process of users by showing them what they’ve done
already. Feng et al. [6] consider this potential in developing HindSight, a
visual encoding technique for capturing and presenting user interaction
history while analyzing relatively simple visual charts. Their findings
indicate an ability to alter user behavior as a result of promoting awareness
of interaction history, a key aspect supporting the potential to integrate
cognitive bias mitigation strategies in mixed-initiative VA systems.

Systematic interpretation of analytic provenance can illuminate the direction
of guidance. Looking at user behaviors and the impact of visualizing
provenance data during the analytic process strongly suggests support for
adopting a design space that can engage with analytic provenance for
addressing cognitive bias in user interactions. Toward this end, we claim that
if systems can measure cognitive bias in the analysis process, then they
should compute what the user should do in response. We are informed by Ceneda
et al. [2] that guidance can be characterized and realized as a process for
solving problems by knowing where the user is, and where they wish to go, in
the analytic process. Madangopal et al. [12] integrate this understanding of
guidance with the expanded literature of analytic provenance in visualization
above through use-cases in national security, highlighting practical
applications for considering provenance-supported methods. These research
avenues reveal an interdisciplinary opportunity to develop bias mitigation
strategies using computational techniques that illuminate the direction of
guidance during the analysis process. Wall et al. [16] have captured anchoring
bias from user interactions using computational bias metrics integrated
directly into a VA system, building towards the goal of developing mixed-
initiative approaches for addressing these and other biases. Computational
models which operate over provenance data from the analytic process are
beginning to show promise in developing principles and techniques for guiding
user behavior, particularly in the context of cognitive bias mitigation.

##  3 A Bias-Aware Future

Research communities are exploring these opportunities for initiative to
expand the literature of cognitive theory as it applies to developing mixed-
initiative VA systems. Yet interdisciplinary research is inherently
challenging—differing goals, competing funding sources, distinct channels of
communication across scientific venues and other considerations can work
against the growth of expanding research directions. However, while
challenging, we posit that the next generation of responsive user interfaces
will come from such cross-domain approaches and work towards a bias-aware
future for mixed-initiative VA systems. We want to understand how such systems
can push, coax, and guide users towards a bias-aware workflow between machine
and human, balancing embedded bias mitigation strategies with increasing
productivity for users. Towards this goal, there are several open challenges
and questions to address.

How can bias-aware design reconcile mixed-initiative approaches with diverse
goals? Many current VA systems employ fixed heuristics or models to determine
which actions or initiatives should be taken when certain user actions are
recognized. For example, they may guide users towards improving the accuracy
of prediction models, clustering results, or other well-defined goals in ML
applications. Consider, however, the implications of systems designed to also
improve the analysis process by mitigating potential bias, or otherwise
guiding users towards goals other than task performance or accuracy. We
understand that perspectives on human bias in VA [18] reveal a number of ways
that bias manifests in practice, intentionally or otherwise, that certainly
impact these design considerations. Yet, at the end of the day, we still aim
to design VA systems in service of user goals without inhibiting performance.

This juxtaposition of diverse goals inspires challenges to overcome and
questions to be answered:

  * •

Are model accuracy and bias mitigation always in opposition, or can they
instead support one another? Perhaps traditional performance metrics need to
be revisited to include aspects of fairness or bias?

  * •

How should the treatment of such trade-offs between accuracy and
interpretation influence the implementation of bias-aware principles on either
end?

  * •

How can the growing desire for faster computation be reconciled with
additional layers of bias recognition and mitigation? Faster and more accurate
may not be better than bias-aware or fair.

  * •

At the end of the day, how can design find the balance along this spectrum
between productivity and proclivity for bias?

These and other questions involving multiple fields of research will continue
to pervade the conversation surrounding the future of mixed-initiative VA
research. As such, we hope this need for balance can be discussed across
disciplines and addressed with informed opportunities to take initiative.

How far should systems go to balance multiple goals and initiatives? When
designing systems with numerous diverse goals or initiatives, situations will
arise that require deciding between potentially competing goals. The
application of mixed-initiative principles introduces several patterns of
design that sacrifice opportunities to engage with cognition in the analytic
process in service of more immediate user goals [19]. To what degree then
should mixed-initiative VA systems strive to balance the influences of
cognitive biases with developed mixed-initiative user interface principles?

Consider once again the prior example of interfaces for predictive modeling,
clustering, or other well-defined goals:

  * •

Clustering metrics exist which can inform how systems should guide users
towards more accurate clusters; however, what if such systems also had metrics
that guide users towards reducing the potential for biased interactions?

  * •

Further, what if the guidance or initiative these systems recommended to users
differed between these two example goals?

  * •

Can existing frameworks account for the integration of metrics informed by
cognitive theory with existing interaction tracing techniques?

  * •

To what extent would these cognitive bias influences in mixed-initiative
applications correlate with interaction patterns and traces consistent with
existing VA literature?

  * •

How would such testing be implemented in practice, and to what extent be made
available to the user, if at all?

We believe these limitations in cross-domain communication can be a motivation
to designers in VA, ML, and other research areas, helping them consider how
informed bias-aware initiatives in mixed-initiative VA systems might take
shape in light of interdisciplinary research goals.

##  4 Conclusion

If mixed-initiative VA systems are going to continue to support the user, then
opportunities to take initiative should provide a re-contextualized way
forward from our evolving understanding of cognitive bias in the analytic
process. This paper identifies a number of open research directions for visual
analytics, where bias-awareness and fairness play a central role. It is our
hope that the discussion of these opportunities to create a bias-aware future
invigorate the community to spawn new lines of research towards addressing
these important challenges.

###### Acknowledgements.

This work was supported in part by the National Science Foundation grant
IIS-1813281.

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