# Anisotropic Spatial Entanglement

Satyajeet Patil<EMAIL_ADDRESS>Quantum Science and Technology
Laboratory, Physical Research Laboratory, Ahmedabad, India 380009.  Indian
Institute of Technology, Gandhinagar, India 382355.  Shashi Prabhakar  Quantum
Science and Technology Laboratory, Physical Research Laboratory, Ahmedabad,
India 380009.  Ayan Biswas  Quantum Science and Technology Laboratory,
Physical Research Laboratory, Ahmedabad, India 380009.  Indian Institute of
Technology, Gandhinagar, India 382355.  Ashok Kumar  Department of Physics,
Indian Institute of Space Science and Technology, Thiruvananthapuram, Kerala,
India 695547.  Ravindra P. Singh<EMAIL_ADDRESS>Quantum Science and
Technology Laboratory, Physical Research Laboratory, Ahmedabad, India 380009.

###### Abstract

The photon-pairs generated through spontaneous parametric down-conversion
(SPDC) possess strong correlations in their transverse position and momentum
degrees of freedom. For such photon-pairs, the transverse position correlation
length depends on the crystal thickness and the pump wavelength, whereas the
transverse momentum correlation length depends on the beam waist and spatial
coherence length of the pump. By controlling these parameters, it is possible
to engineer the spatial entanglement. Here, we utilize the circular asymmetry
of the pump by using an elliptical-Gaussian beam to change the degree of
entanglement in transverse directions, which we call anisotropic entanglement.
We show the interrelation between the degree of anisotropic entanglement and
the asymmetry in the beam width. In addition, we also show that for a highly
asymmetric pump beam, the entanglement along the $y$ direction completely
vanishes, whereas the entanglement in $x$ direction remains intact.

Position-momentum entanglement, Joint probability distribution, anisotropy
effect

##  I Introduction

The spontaneous parametric down conversion (SPDC) processes obey the laws of
energy and momentum conservation, and depending upon the phase matching
condition, the generated two photons, signal and idler, are strongly
correlated in continuous as well as discrete variables. Discrete variable
entanglement includes polarization [1] and orbital angular momentum [2], and
have been studied and utilized in numerous applications as quantum key
distribution [3], superdense coding [4], quantum teleportation [5], etc. The
transverse position and momentum of the SPDC photons are the continuous
spatial variables that have gained attention in recent years [6, 7, 8, 9], and
found applications in the field of quantum imaging [10, 11, 12, 13, 14, 15],
quantum metrology [16] and quantum communication [17, 18, 19].

The momentum correlation length of the SPDC photon pair depends upon the beam
waist, and the spatial coherence of the pump beam [20]. Few studies have
reported the effect of the degree of spatial coherence on the momentum
correlation length of the SPDC photons while keeping the same beam waist [20,
8]. SPDC photon pairs produced using a symmetric Gaussian pump beam with
constant spatial coherence have an equal degree of entanglement along
transverse directions, which we call isotropic spatial entanglement. On the
other hand, an elliptic Gaussian beam has an unequal beam waist along
orthogonal transverse directions. The inequality in the beam waist influences
the momentum correlation lengths and gives different momentum correlation
lengths in different directions; we refer to it as anisotropic spatial
entanglement.

The position correlation length is written as $\Delta(x_{i}|x_{s})$, which is
also known as the conditional uncertainty and tells about the uncertainty in
the measurement of position of idler photon when the position of signal photon
is known. Similarly, momentum correlation length is given as
$\Delta(p_{x_{i}}|p_{x_{s}})$. It has been proved that when the product of
position and momentum conditional uncertainties is less than 0.5 $\hbar$, the
position-momentum correlations of the system exhibit EPR-like correlations
[21, 22].

There are few methods to measure the conditional uncertainty experimentally,
like raster scanning of a slit in transverse directions [23, 8], inversion-
based interferometer [24], and imaging using electron-multiplying charge-
coupled devices (EMCCD) [25, 26]. The slit method confines one of the photons
from the SPDC photon pair using a slit and scanning for its conjugate paired
photon’s position. Due to the finite slit width, the resolution is limited.
For higher resolution, a slit of narrow width is required. In the inversion-
based interferometric method, only the paired photons provide constructive
interference, and the rest of the photons overlap destructively. With the
advancement in detection technology, a weak signal can be detected and
registered with quantum efficiency $>90\%$ using an electron-multiplying CCD
(EMCCD) camera. Here we use EMCCD based technique to find the spatial
entanglement.

This work studies and compares the spatial entanglement generated using
elliptical Gaussian beams (EGB) having different asymmetry. The asymmetric
pump modes were prepared using the combination of two cylindrical lenses. We
experimentally measured the conditional joint detection probability (JDP) of
position and momentum of SPDC photon pairs and evaluated the EPR-like
correlations for the spatial entanglement by varying the ellipticity of the
pump beam. This article presents a brief on the theoretical background in
section II followed by the experimental details and image analysis in section
III. The results are presented in section IV and finally we concluded in
section V.

##  II Theoretical Background

We use one dimensional Gaussian model to describe the spatial structure of the
two SPDC photons field along $x$ and $y$ directions, as given below in Eqn. 1
& 2 respectively.

$\displaystyle\Psi\left(x_{i},x_{s}\right)$ $\displaystyle=$ $\displaystyle
N\exp\left(-\frac{\left|x_{i}+x_{s}\right|^{2}}{4\sigma_{x+}^{2}}-\frac{\left|x_{i}-x_{s}\right|^{2}}{4\sigma_{x-}^{2}}\right),$
(1) $\displaystyle\Psi\left(y_{i},y_{s}\right)$ $\displaystyle=$
$\displaystyle
N\exp\left(-\frac{\left|y_{i}+y_{s}\right|^{2}}{4\sigma_{y+}^{2}}-\frac{\left|y_{i}-y_{s}\right|^{2}}{4\sigma_{y-}^{2}}\right),$
(2)

where, $N$ is the normalization constant, $x_{j}$, $y_{j}$ are the transverse
positions of photon, $j=\\{s,i\\}$. The indices $s$, $i$ represent for signal
and idler photons, respectively. $\sigma_{x+}$ is the standard deviation of
the first Gaussian term (Eqn. 1), giving momentum correlation length of photon
pair, and $\sigma_{x-}$ is the standard deviation of the second Gaussian term
(Eqn. 1), giving position correlation length. Similarly along the $y$-axis in
Eqn. 2.

The JDP of two photons $J(x_{i}|x_{s})=|\Psi(x_{i},x_{s})|^{2}$ versus their
spatial locations on the transverse plane gives Gaussian distribution, and the
standard deviation of this Gaussian distribution gives position and momentum
correlation length. Theoretically, the momentum correlation length along $x$
and $y$ are given as

$\displaystyle\sigma_{x+}\propto 1/\omega_{x},$ (3)
$\displaystyle\sigma_{y+}\propto 1/\omega_{y},$ (4)

where $\omega_{x}$ and $\omega_{y}$ are the beam waists along $x$ and $y$
directions, respectively. Position correlation length in both transverse
directions are same and independent of the pump beam parameters, except
wavelength. Theoretically, it is represented as,

$\sigma_{(x,y)-}=\sqrt{\frac{\alpha L\lambda_{p}}{2\pi}},$ (5)

where $L$ is the length of the non-linear crystal, $\alpha=0.455$ is the
adjustment constant [25], and $\lambda_{p}$ is the wavelength of the pump
beam. In our experiment, we used $5$ mm thick type-I BBO crystal and the
theoretically obtained value of the $\sigma_{x-}=\sigma_{y-}=12.11$ $\mu$m.

We define the product of position and momentum correlation widths along $x$
and $y$ directions as

$\displaystyle\Delta(x_{i}|x_{s})\Delta(p_{x_{i}}|p_{x_{s}})$ $\displaystyle=$
$\displaystyle\gamma_{x},\quad{\rm and}$ (6a)
$\displaystyle\Delta(y_{i}|y_{s})\Delta(p_{y_{i}}|p_{y_{s}})$ $\displaystyle=$
$\displaystyle\gamma_{y}.$ (6b)

For having spatial entanglement along a given axis, $\gamma$ $<$ 0.5 $\hbar$
along that axis. The number of modes $\mathcal{N}$ along that axis,
respectively, is calculated as [25]

$\mathcal{N}_{x}=\left(\frac{\omega_{x}}{\sigma_{x-}}\right)^{2},\quad{\rm
and}\quad\mathcal{N}_{y}=\left(\frac{\omega_{y}}{\sigma_{y-}}\right)^{2}.$ (7)

##  III Experiment

A schematic of the experimental setup is shown in Fig. 1. We used a 10 mW
continuous-wave diode laser (TOPTICA, TopMode-405-H_40154) of the wavelength
405 nm. The shape of the pump beam is modified to an EGB using combinations of
two cylindrical lenses. A type-I non-linear BBO crystal of thickness 5 mm is
kept in the pump beam path at the focal point of the second cylindrical lens
in such a way that it emits collinear SPDC output at 810 nm. Throughout the
experiment, the distance between crystal and EMCCD was fixed at 200 mm. To
distil the SPDC output from the pump beam, we used a long pass filter (Edmund
Optics, high performance long-pass Filter at 500 nm) after the BBO crystal and
one more band-pass filter at 810$\pm$5 nm (Edmund Optics, high performance
band-pass filter) is placed in front of the EMCCD.

The plano-convex lens of focal length 50 mm is kept between the crystal and
EMCCD at 100 mm from the crystal to satisfy near-field imaging configuration,
Fig. 1(a). Throughout the experiment, the distance between the non-linear
crystal and EMCCD was kept fixed at 200 mm. The far-field imaging is obtained
by replacing 50 mm focal length lens with the 100 mm focal length plano-convex
lens, as shown in Fig. 1(b). The far-field imaging maps the transverse
momentum of the photons onto position coordinates of the EMCCD.

Figure 1: Experimental setup to study the position-momentum entanglement
generated from an elliptical pump beam using second-order nonlinear crystal.
CL1 & CL2: cylindrical lenses; NLC: nonlinear crystal (BBO); BPF: band-pass
filter. (a) Near-field configuration, (b) Far-field configuration.

The conditional position and momentum uncertainties are estimated using the
joint probability distribution functions for photon pairs obtained in the near
and far-field regime, respectively. We fit the joint probability functions
with Gaussian functions, and the standard deviations of the fitted functions
(say, $\sigma_{x(emccd)}$ and $\sigma_{p(emccd)}$) give a measure of
conditional uncertainties. After considering the experimental parameters that
are involved, we finally obtain the actual conditional uncertainties as,

$\displaystyle\Delta(x_{i}|x_{s})$ $\displaystyle=$
$\displaystyle\frac{\sigma_{x(emccd)}}{M},$ (8a)
$\displaystyle\Delta(p_{x_{i}}|p_{x_{s}})$ $\displaystyle=$
$\displaystyle\frac{k\sigma_{p(emccd)}}{f},$ (8b)

where $\sigma_{x(emccd)}$ is the position correlation length at the EMCCD
plane measured using near-field imaging with magnification $M$ (in our
experiment, $M$=1), $\sigma_{p(emccd)}$ is the momentum correlation length
mapped in position coordinates of the EMCCD at the far-field, $k$ is the wave-
number of the SPDC photons, and $f$ is the focal length of the lens used for
far-field imaging of the crystal’s exit plane on to the EMCCD.

###  III.1 Pump beam shaping

The shape of the initial circular Gaussian beam is engineered using a
combination of two cylindrical lenses. The pump beam is shaped into an EGB
beam such that there is a mismatch in beam width along the transverse
directions. Based on the mismatch in beam width ($\omega_{x},\omega_{y}$), we
define the asymmetry factor,

$\beta=\frac{\omega_{y}}{\omega_{x}}.$ (9)

Table 1 illustrates the combinations of two cylindrical lenses used and the
corresponding $\beta$ values obtained for the EGB. Note that we orient both
the cylindrical lenses so that the shape of the beam modulates only along the
$y$-axis while the profile along the $x$-axis remains intact. As
$\beta\rightarrow 1$, asymmetry in the beam width along transverse directions
disappears.

Cylindrical lens combinations, focal lengths (f1- f2) | $\beta$
---|---
150 mm-50 mm | 0.193
100 mm-50 mm | 0.357
75 mm-50 mm | 0.552
None | 0.833
Table 1: Different combinations of two cylindrical lenses with focal lengths
f1 and f2, and the corresponding asymmetry factor $\beta$ for the pump beam.

###  III.2 EMCCD settings and data acquisition

To record the data, we used Andor’s iXon 633 EMCCD having a pixel size of 16
$\mu$m, operating in kinetic mode at -60∘C. In the acquisition setup, we keep
the following settings of the EMCCD identical for both position and momentum
measurements: gain $\times$1000, horizontal readout rate 1 MHz with pre-
amplifier gain $\times$1, vertical shift speed 3.3 $\mu$s, and vertical clock
voltage amplitude as normal. We chose 64$\times$64 sensor area, encompassing
the complete transverse profile of the near and far-field imaging of the SPDC
output. Different exposure times, 3.5 ms for position and 5 ms for momentum,
were chosen to maintain a mean number of photons per pixel ($\approx$0.004
photons/pixel).

First, we collected the data for dark noise of the EMCCD for different
exposure times, 3.5 ms and 5 ms. To measure average photons per pixel due to
dark noise, we acquired 0.25 million frames with the internal shutter closed
such that no light enters the EMCCD. Later, with internal shutter open, we
collected the 0.25 million frames of the SPDC photons at 3.5 ms and 5 ms
exposure times for the near and far-field imaging, respectively. From the BBO
crystal to the EMCCD, the entire setup was covered with black anodized
aluminium sheets to block noise due to stray photons. For pump beams with
different $\beta$ values, we followed the same technique as discussed above.
The EMCCD takes $\sim$3 hours to capture 0.25 million frames.

###  III.3 Image analysis

References [25, 27, 10, 11] report the method of finding spatial correlations
using EMCCD in detail. Here we explain the method that we use to find the
position and momentum correlations. First, we calculate the average number of
photons ($\overline{I}_{d}$) and standard deviation of the photons
$\delta_{d}$ per pixel from the frames taken at 3.5 ms to quantify the dark
noise of EMCCD. To get rid of the internal noise of the EMCCD,
$\overline{I}_{d}$ is subtracted from all the pixels of every frame of the
SPDC signal. For a pixel, if the subtraction falls below the ($\delta_{d}$)
then that pixel value is assigned to zero, otherwise we keep its original
value. This process is known as thresholding, and it improves the signal-to-
noise ratio of the results. These thresholded frames are then used for further
analysis.

Apart from the dark noise, there are other sources of noise such as stray
photons, pump photons leaked from the long-pass and band-pass filters, SPDC
photons with a lost partner, etc. To get rid of such noise, we utilize pixel-
to-pixel intensity correlation. To find the position correlation length along
the $x$ direction, the first thresholded frame is summed column-wise along the
$y$ direction. This summation converts the 64$\times$64 square matrix into a
1$\times$64 row matrix. This row matrix is then multiplied by its transpose,
which is 64$\times$1 column matrix. The multiplication of column and row
matrix gives a 64$\times$64 matrix in which the diagonal elements correspond
to the intensity correlation of pixels with themselves. The first sub-diagonal
element corresponds to the intensity correlation between the photons for which
the spatial separation is 1 pixel. Similarly, the second sub-diagonal element
shows the intensity correlation between the pixels for which the spatial
separation is two pixels and so on. The distribution of intensity correlation
with the pixel separation gives the JDP. This JDP of photon-pairs can have
contributions from the stray noise as discussed above. To subtract out such
unwanted contributions, we follow the technique described in Ref.[10].

As described in Ref [10], JDP is a function of the position of photon-pairs
and is given as,

$JDP(x)=\frac{1}{N_{f}}\sum_{l=1}^{N_{f}}I_{l}\left(x_{i}\right)I_{l}\left(x_{s}\right)-\frac{1}{N^{2}_{f}}\sum_{l\neq
l^{{}^{\prime}}}^{N_{f}}I_{l}\left(x_{i}\right)I_{l^{{}^{\prime}}}\left(x_{s}\right),$
(10)

where $N_{f}$ is the number of frames. $I_{l}$ is the number of photons in the
$l^{th}$ thresholded frame, $x_{i}$ and $x_{s}$ are the position coordinates
of $idler$ and $signal$ photons, respectively.

To calculate the JDP, the first and second frames are summed along the $y$
axis to get a row matrix. Then, the row matrix of the second frame is
transposed to a column matrix and multiplied by the first frame’s row matrix.
This creates the 64$\times$64 matrix in which the pixel-to-pixel intensity
correlation between two frames is present (first term of the Eqn. 10). Since
both the photons from the SPDC photon pairs are simultaneously generated
within the inverse bandwidth of the process, they hit the EMCCD together in a
given exposure time, and there is zero probability of hitting on the two
consecutive frames separately, which are acquired within a time interval
larger than the inverse bandwidth of the process. Therefore, the JDP between
two consecutive frames gives the JDP due to the noise and not due to the SPDC
photon pairs (second term of the Eqn. 10). When JDP obtained from successive
frames is subtracted from the JDP obtained from the same frames, it gives the
resultant JDP only due to the SPDC photon pairs. This procedure is repeated
over 0.25 million frames, and the resultant JDP is averaged out, which looks
like Fig. 2(a1-c1).

In order to obtain a quantitative estimation of position and momentum
correlations, the JDP for the spatial separation between photon pairs is
fitted to a sum of two Gaussian distributions. The first Gaussian distribution
corresponds to the noise curve, whereas the other corresponds to the signal
curve. For each Gaussian distribution, the standard deviation is calculated,
and the resultant standard deviation is obtained using the following formula,

$\mathcal{S}=\frac{\mathcal{S}_{s}\mathcal{S}_{n}}{\sqrt{\mathcal{S}_{s}^{2}+\mathcal{S}_{n}^{2}}},$
(11)

where $\mathcal{S}_{s}$ and $\mathcal{S}_{n}$ are the standard deviations of
the Gaussian distributions corresponding to signal and noise, respectively. A
multiplication by the pixel size (16 $\mu$m) to the resultant standard
deviation gives the position correlation between SPDC photon pairs at the
EMCCD plane. As explained in Sec. III, the position correlation length at the
crystal is calculated by dividing the position correlation length at the EMCCD
plane with magnification factor $M$ . The JDP for momentum correlations looks
like Fig. 2(a2-c2). To calculate the momentum correlation length, the width of
JDP distribution is multiplied by $f/k$, as discussed in section III.

##  IV Results and discussion

Obtained results from the experiment are shown in Fig. 2. The first row of
Fig. 2(a-c) shows the transverse profile of the pump beams having different
$\beta$ values. The color bar shows the normalized intensity distribution.

Figure 2: Intensity distribution of the pump beam and the obtained JDP for the
position and momentum in $x$ and $y$ directions. (a), (b), and (c) are for
different pump beams with $\beta$= 0.193, 0.351, 0.833, respectively. For
$\beta$ = 0.193, (a1, a3) represent the JDP of the position along $x$ and $y$,
(a2, a4) the momentum along $x$ and $y$; The same follows for $\beta$ = 0.351
and $\beta$= 0.833 in (b1-b4) and (c1-c4), respectively.

The position correlation length remains the same for different $\beta$ as it
is independent of pump beam parameter, Eqn.5, and this is observed
experimentally too. The width of the JDP for position remains same for the $x$
and $y$ directions as shown in Fig. 2(a1-c1) & (a3-c3).

For momentum correlation length along $x$ direction, width of the momentum JDP
for different $\beta$ values is nearly equal because the beam width along $x$
direction is almost equal, Fig. 2(a2-c2). Contrary to the $x$ direction, width
of the momentum JDP along the $y$ direction changes with the $\beta$ value,
Fig. 2(a4-c4). For the lower values of $\beta$, the JDP width is greater than
that for the higher values of $\beta$.

We also calculate the variation of $\gamma$ as a function of the asymmetry,
Fig. 3. It can be seen from the plot that as the asymmetry parameter goes away
from one, the value of $\gamma$ increases. For different asymmetry parameters,
we have shown the conditional uncertainty products and the number of modes in
$x$ and $y$ directions in Table 2.

Figure 3: The variation of $\gamma$ along $x$-axis (blue) and $y$-axis (red) against the asymmetry factor $\beta$. $\gamma_{x}$ and $\gamma_{y}$ are defined in Sec. II has the dimensions of $\hbar$. $\bm{\beta}$ | $\bm{\gamma_{x}(\hbar)}$ | $\bm{\gamma_{y}(\hbar)}$ | $\bm{\mathcal{N}_{x}}$ | $\bm{\mathcal{N}_{y}}$
---|---|---|---|---
0.193 | 0.038 | 0.133 |  $\sim$ 3900 |  $\sim$ 150
0.351 | 0.045 | 0.120 |  $\sim$ 3500 |  $\sim$ 400
0.552 | 0.080 | 0.041 |  $\sim$ 3600 |  $\sim$ 1100
0.833 | 0.041 | 0.040 |  $\sim$ 4000 |  $\sim$ 3900
Table 2: $\gamma_{x}$, $\gamma_{y}$, $\mathcal{N}_{x}$ and $\mathcal{N}_{y}$
along $x$\- and $y$-direction for the four different values of $\beta$.

Furthermore, in the context of spatial entanglement distribution in the
transverse plane, it would be interesting to know the asymmetry parameter for
which the spatial entanglement is completely lost in one direction, while
still existing in its orthogonal direction. In Fig. 4, we show the
entanglement death in $y$ direction for $\beta$ = 0.061 while keeping it alive
along x-direction. To generate the pump beam with such a low asymmetry value,
we used a cylindrical lens of focal length of 200 mm to focus the EGB on the
non-linear crystal. As shown in the Fig. 4(b), the width of momentum
correlation along $x$ direction remains same as that in Fig. 2(b-c), but the
width of momentum correlation along $y$ direction is completely spread over
entire JDP matrix, Fig.4(c). Experimentally obtained values of $\gamma_{x}$,
$\gamma_{y}$, $\mathcal{N}_{x}$, and $\mathcal{N}_{y}$ for the $\beta$=0.061
are given in the Table 3.

Figure 4:  (a) Intensity distribution of the pump beam with $\beta$=0.061, (b-c) are the momentum JDP for $x$ and $y$ directions, respectively. Color bar represents the normalized intensity (a) and the normalized coincidence counts (b,c). $\bm{\beta}$ | $\bm{\gamma_{x}(\hbar)}$ | $\bm{\gamma_{y}(\hbar)}$ | $\bm{\mathcal{N}_{x}}$ | $\bm{\mathcal{N}_{y}}$
---|---|---|---|---
0.061 | 0.037 | 0.713 |  $\sim$ 3700 |  $\sim$ 15
Table 3: $\gamma_{x}$, $\gamma_{y}$, $\mathcal{N}_{x}$ and $\mathcal{N}_{y}$
along $x$\- and $y$-direction to observe entanglement death along one axis and
high degree of entanglement in another axis.

##  V Conclusion

In conclusion, we used the combination of two cylindrical lenses to shape the
elliptical-Gaussian beam with distinct beam widths along transverse
directions. This asymmetry in the beam widths reflects in the momentum
correlation length of the SPDC photon pairs. Since the position correlation
length is independent of the pump beam parameters, the spatial entanglement is
mainly affected by the momentum correlation lengths. We experimentally show
that the spatial entanglement along the $y$ direction is controlled using
circular asymmetry of the pump beam while the spatial entanglement remains
intact along the $x$ direction. As the asymmetry in the pump beam increases,
the quality of entanglement and the number of modes in $y$-direction decreases
while the number of modes in $x$-direction approximately remains constant.

We also show that for a highly asymmetric pump beam, entanglement in the
$y$-direction completely vanishes, whereas the entanglement in the
$x$-direction is preserved. Such anisotropic entangled states may find
application in quantum communication and quantum imaging, where correlation
with different strengths is required.

## Acknowledgement

We gratefully acknowledge Prof. Daniele Faccio for explaining the method of
analyzing the images to obtain correlation lengths. S. Patil acknowledges
Abhinandan Bhattacharjee for helpful discussion, and also acknowledges
VIKRAM-100 cluster at Physical Research Laboratory, Ahmedabad. AK would like
to acknowledge the support from Science and Engineering Research Board, India,
through DST-SERB under Grant SRG/2019/001631.

## Disclosures

The authors declare no conflicts of interest related to this article.

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