SDW_UUD_Cs2CoBr4_suppl.pdf

# Spin density wave versus fractional magnetization plateau in a triangular
antiferromagnet

L. Facheris<EMAIL_ADDRESS>Laboratory for Solid State Physics, ETH
Zürich, 8093 Zürich, Switzerland  K. Yu. Povarov  Laboratory for Solid State
Physics, ETH Zürich, 8093 Zürich, Switzerland  S. D. Nabi  Laboratory for
Solid State Physics, ETH Zürich, 8093 Zürich, Switzerland  D. G. Mazzone
Laboratory for Neutron Scattering and Imaging, Paul Scherrer Institute,
CH-5232 Villigen, Switzerland  J. Lass  Laboratory for Neutron Scattering and
Imaging, Paul Scherrer Institute, CH-5232 Villigen, Switzerland  Department of
Physics, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark  B.
Roessli  Laboratory for Neutron Scattering and Imaging, Paul Scherrer
Institute, CH-5232 Villigen, Switzerland  E. Ressouche  Université Grenoble
Alpes, CEA, IRIG, MEM, MDN, 38000 Grenoble, France  Z. Yan  Laboratory for
Solid State Physics, ETH Zürich, 8093 Zürich, Switzerland  S. Gvasaliya
Laboratory for Solid State Physics, ETH Zürich, 8093 Zürich, Switzerland  A.
Zheludev<EMAIL_ADDRESS>http://www.neutron.ethz.ch/ Laboratory for Solid
State Physics, ETH Zürich, 8093 Zürich, Switzerland

###### Abstract

We report an excellent realization of the highly non-classical incommensurate
spin-density wave (SDW) state in the quantum frustrated antiferromagnetic
insulator Cs2CoBr4. In contrast to the well-known Ising spin chain case, here
the SDW is stabilized by virtue of competing planar in-chain anisotropies and
frustrated interchain exchange. Adjacent to the SDW phase is a broad $m=1/3$
magnetization plateau that can be seen as a commensurate locking of the SDW
state into the up-up-down spin structure. This represents the first example of
long-sought SDW-UUD transition in triangular-type quantum magnets.

Of the various magnetically ordered phases of insulators, the spin-density
wave (SDW) is perhaps the least classical one [1, 2]. While easily envisioned
in metals where the spin carriers are itinerant [3], it cannot exist in
classical models with localized spins of given magnitude at each site.
Nevertheless, there are several purely quantum-mechanical routes to realizing
SDW states in insulators. To a greater or lesser extent they are all based on
the Tomonaga–Luttinger Spin Liquid (TLSL) properties of the $S=1/2$ quantum
spin chain with antiferromagnetic (AF) exchange interactions $J$. In applied
magnetic fields, a single chain develops incommensurate spin correlations in
the longitudinal channel [4,
*BogolyubovIzerginKorepin_NucPhysB_1986_TLLexponents,
*HikiharaFuruski_PRB_2004_XXZtll, 7]. In most cases though, the transverse
commensurate correlations dominate, eventually resulting in transverse AF or
helical long-range order in coupled chains. To create a SDW, one needs to
somehow boost the longitudinal correlations in each chain or to ensure that
they are favored by inter-chain interactions. The first approach, realized in
materials like BaCo2V2O8 [8, 9] and SrCo2V2O8 [10], is to simply endue the
chains with Ising-type anisotropy. Another route, realized quite recently [11,
12, *FanYu_ChinPhysB_2020_IsingCoupledChains], is to impose Ising anisotropy
on _interchain_ interactions $J^{\prime}$. This is arguably the case of YbAlO3
[14, 15]. The third route to a SDW state exploits _frustrated_ zig-zag
interchain bonds $J^{\prime}$ in the so-called “distorted triangular lattice”
geometry. Commensurate transverse TLSL correlations in each chain become
completely decoupled at the Mean Field (MF) level. Incommensurate longitudinal
ones are not, and are thus the ones to order in 3D. In this model theory
predicts a SDW phase in a very wide range of $J^{\prime}/J$ ratios [16, 17,
18]. A very special feature of this mechanism is the SDW “locking” to a
commensurate wavevector producing a $m=1/3$ up-up-down (UUD) magnetization
plateau state. The latter persists even if $J=J^{\prime}$, where no chains can
be identified, and even in the fully isotropic case [19]. It thus establishes
an important link between 1D TLSL and 2D triangular lattice physics. To date,
this connection remains poorly understood experimentally, for lack of a
suitable model system.

In the present Letter we demonstrate the existence of an incommensurate SDW
state and its locking into a UUD phase in the triangular-lattice magnet
Cs2CoBr4 [20]. Both phases are well-pronounced and occupy nearly a quarter of
the phase diagram each. The mechanism behind this phenomenon is likely to be a
blend of the three “routes to SDW” described above.

Figure 1: Diffraction results for Cs2CoBr4, $\mathbf{H}\parallel\bm{b}$. (a)
AF-stripe, SDW, and UUD phases shown in pseudospin representation. Color
denotes the relative depth of on-site modulation. Exchange couplings $J$ and
$J^{\prime}$ are also indicated. (b) Points: incommensurate Bragg peak
position of the SDW phase versus field. The blue line shows the “pseudospin”
relative magnetization [20]. (c) Magnetic order parameter (modulation
amplitude) $M_{\textbf{Q}}$ vs field for AF (diamonds), SDW (circles), and UUD
(squares) phases. The corresponding wave vectors of actual measurement are
also indicated. The blue line is the magnetization curve $M(H)$ [20]. All
measurements are performed at $T\lesssim 0.1$ K. Figure 2: Overview of
magnetic excitations in Cs2CoBr4 at $T\lesssim 0.1$ K in (a) zero field and
(b) $1.8$ T. Color indicates the neutron scattering intensity proportional to
the dynamic structure factor $\mathcal{S}(\mathbf{q},\omega)$. The data are
integrated along $\mathbf{c}^{\ast}$ direction within the limits indicated.
Background subtraction was performed as described in [21]. Gray areas mask
regions where the incoherent scattering dominates over the signal.

Our target material Cs2CoBr4 crystallizes in the orthorhombic $P_{nma}$
structure, same as that of Cs2CuCl4 [22, 23], Cs2CuBr4 [24], and Cs2CoCl4 [25,
26, *Breunig_PRB_2015_CsCoClPhD, 28]. The magnetic $3d$ ions (four per unit
cell) are arranged in triangular-patterned layers in the $bc$ plane. The two
Cu-based compounds mentioned above are straightforward $S=1/2$ Heisenberg
$J-J^{\prime}$ model magnets [16]. The physics of Cs2CoCl4 and our material is
more complicated. The Co2+ magnetic ions sit in a low-symmetry distorted
tetrahedral environment, thus their orbital momentum is quenched. Their
magnetism is described in the language of $S=3/2$ spins dominated by crystal-
field effects. The latter reduce the magnetism to _pseudospin_ $\tilde{S}=1/2$
degrees of freedom at low temperatures. In the pseudospin representation the
interactions are strongly modified, yielding a nearly XY-type coupling. The
effect is strongest for intra-chain $J$ interactions along the $\bm{b}$ axis.
These anisotropic interactions are mostly a crystal field effect, in contrast
to more symmetric situations where anisotropic interactions arise from direct
spin-orbit entanglement [29, 30]. In Cs2CoCl4 the zig-zag inter-chain
interactions $J^{\prime}$ are negligible, making the material a nearly-ideal
XY chain [31, 28]. A distinctive feature of Cs2CoBr4 is that $J^{\prime}$ is
much stronger and comparable to $J$ [20]. This creates a novel type of
frustration, in addition to the already-present geometric frustration of
$J^{\prime}$. The crystal structure dictates mutually perpendicular directions
of the planar anisotropy in neighboring chains. The $\bm{b}$ direction is
shared by both planes. In this way two easy-plane anisotropies conspire to
effectively produce an easy-$\bm{b}$-axis anisotropy for the pseudospins.

Below $T_{N}=1.3$ K, as a function of magnetic field applied along the
$\bm{b}$-direction, Cs2CoBr4 goes through a sequence of five magnetic phases
[20]. The 1st and the 3rd phases in increasing fields are magnetization
plateaux, with pseudospin magnetization $\tilde{m}\simeq 0$ and $1/3$,
correspondingly. In this study we use neutron diffraction to unambiguously
identify these states as antiferromagnetic stripe phase (AF) with propagation
vector $\mathbf{Q}=(0,~{}1/2,~{}1/2)$, and UUD phase with
$\mathbf{Q}=(0,~{}1/3,~{}0)$. The intermediate 2nd magnetic state turns out to
be a longitudinal incommensurate SDW with propagation vector
$\mathbf{Q}=(0,~{}\xi,~{}0)$. The experiment was performed on the CEA-CRG D23
lifting-counter diffractometer at ILL (Grenoble, France). The $24.90(4)$ mg
single crystal of Cs2CoBr4 was mounted on the cold finger of a dilution
refrigerator $T\lesssim 0.1$ K in a vertical $6$ T cryomagnet, with
$\mathbf{H}\parallel\bm{b}$ and $ac$ being in the horizontal scattering plane.
This setting allowed us to cover $|h|\leq 6$, $-1\leq k\leq 0$ and $|l|\leq 7$
r.l.u. portion of the $(h,k,l)$ reciprocal space with $\lambda=2.36$ Å
neutrons (PG002), in which the Bragg peaks of the types described above were
collected in magnetic fields of $0$, $2.2$, and $3.2$ T correspondingly. The
symmetry-based group theory analysis and model refinement (using SARA$h$ [32]
and FullProf Suite [33, 21]) suggest the collinear configurations shown in
Fig. 1(a) to be the optimal solutions with $R$-factors $7.1$, $14.8$, and
$11$% for AF, SDW, and UUD correspondingly. The field dependencies of magnetic
order parameters $M_{\mathbf{Q}}$ associated with each phase (_modulation
amplitudes_ at the given propagation vectors $\mathbf{Q}$) are shown in Fig.
1(c). For AF and UUD phases, the Bragg peak intensity was measured by counting
at fixed $\mathbf{Q}$ versus field. For the SDW phase, the peak positions and
intensities were extracted from broad $k$-scans at 8 field values [21]. The AF
order parameter disappears above the discontinuous AF-SDW transition at $1.5$
T in agreement with thermodynamics [20]. The apparent residual intensity at
$(0,-1/2,3/2)$ in Fig. 1(c) is due to a simple background model [21]. In
contrast, the SDW-UUD transition seems to be a quintessential incommensurate-
commensurate locking, with rather insignificant jump in the spin modulation
depth. The residual intensity of $(0,-1/3,1)$ below $2.8$ T is due to the poor
resolution along the vertical $k$-direction. The $M_{\mathbf{Q}}$ extracted
from the broad $k$-scan at $2.8$ T inside the UUD phase agrees well with the
more precise UUD dataset.

The field dependence of the SDW propagation vector $(0,\xi,0)$ was extracted
from the same $k$-scans mentioned above. The result is plotted in Fig. 1(b).
Remarkably, the propagation vector closely follows the pseudospin relative
magnetization [20]:

$|\xi|=1/2-\tilde{m}/2.$ (1)

Such behavior is typical of longitudinal incommensurate correlations specific
to $S=1/2$ chains. In the TLSL framework this incommensurability corresponds
to a nesting vector that spans the Fermi sea of _fractionalized spinon
quasiparticles_ [1]: $\xi=2k_{F}$. This picture may provide the basic idea for
understanding the physics of Cs2CoBr4, but the analogy is difficult to extend
beyond Eq. 1. The actual field dependence of magnetization shown in Fig. 1(b)
is entirely different from that of a XXZ chain in longitudinal field [1, 8,
9].

Eq. 1 also holds for coupled-chain models, the Heisenberg $J-J^{\prime}$ model
in particular [16, 17, 18]. Similar types of field dependencies were
previously also observed in the Ising chains [8, 9, 10], or Ising-coupled
Heisenberg chains [14, 15]. A very narrow “elliptical spiral” phase with
linear incommensuration-field dependence was also reported for the
structurally similar Heisenberg $J-J^{\prime}$ magnet Cs2CuCl4 [22]. In all
these cases, however, the UUD phase at $\xi=1/3$ is either absent altogether
(Ising chains), or is very narrow compared to the SDW state (YbAlO3). In the
latter case, theory does not predict any plateau for the corresponding Ising-
coupled Heisenberg chain model [11, 12,
*FanYu_ChinPhysB_2020_IsingCoupledChains]. It has been proposed that the
commensurate-incommensurate locking in YbAlO3 might be the consequence of e.g.
additional small interactions [15]. The coexistence of a magnetization-scaled
SDW and well-defined plateau is unique for Cs2CoBr4.

To get more insight into the mechanism of the AF-SDW-UUD sequence of
transitions, we have measured the magnetic excitation spectra at zero field
(AF) and at $1.8$ T (SDW). The experiment was performed on the new CAMEA
spectrometer at PSI (Switzerland) [34, *LassJakobsen_SoftX_2020_Mjolnir]. The
$m=1.16$ g crystal of Cs2CoBr4 was mounted on the cold finger of a dilution
refrigerator with $bc$ in the scattering plane. A $1.8$ T horizontal magnet
was used, with the direction of the field set along $\bm{b}$. The unique
combination of the multiplexing capabilities of CAMEA, “continuous angle” data
acquisition mode, and the open geometry of this horizontal magnet allowed us
to obtain a detailed neutron scattering intensity dataset vs
$(k,l,\hbar\omega)$ in both AF and SDW states. Two measurement series were
performed at each field, with $E_{\mathrm{i}}=5.1$ and $E_{\mathrm{i}}=3.6$
meV, for higher coverage and higher resolution correspondingly ($\sim 0.16$
meV FWHM). The projections from the cumulative datasets for AF ($\mu_{0}H=0$)
and SDW ($\mu_{0}H=1.8$ T) phases are shown in Fig. 2. In zero field the
spectrum is gapped ($\Delta\simeq 0.35$ meV) and mostly dispersive along the
$\bm{b}$ direction, with the bandwidth approaching $0.7$ meV. In addition to
the pronounced magnon-like excitation at low energy, a continuum with a sharp
upper boundary is clearly visible. At higher energies,
$E_{\mathrm{CF}}=2.1(1)$ meV, we observe a non-dispersive level (see [21] for
extra data), that can be understood as
$\left|1/2\right\rangle\rightarrow\left|3/2\right\rangle$ transition of cobalt
$S=3/2$ at energy $2D$. This gives $D=12.2(6)$ K, in good agreement with the
susceptibility-based estimate of $14(1)$ K reported earlier [20].

The strength of $J^{\prime}$ interactions is key to understand the physics of
Cs2CoBr4. As Fig. 3 shows, the bandwidth along the $\mathbf{c}$ axis is only
about $0.1$ meV. This however, is not a sign of an insignificant $J^{\prime}$
, but rather of geometric frustration in the zig-zag inter-chain coupling and
of its predominant Ising nature. For a crude estimate we can rely on a simple
spin wave theory (SWT) calculation (using the SpinW package [36, 21]). The
starting point is the Hamiltonian

${\hat{\mathcal{H}}}=\sum_{i,j}\sum_{\alpha=x,y,z}J^{\alpha\alpha}_{e}\hat{\tilde{S}}_{i,2j}^{\alpha}\hat{\tilde{S}}_{i+1,2j}^{\alpha}+J^{\alpha\alpha}_{o}\hat{\tilde{S}}_{i,2j+1}^{\alpha}\hat{\tilde{S}}_{i+1,2j+1}^{\alpha}+J^{\prime\alpha\alpha}\hat{\tilde{S}}_{i,2j}^{\alpha}[\hat{\tilde{S}}_{i,2j+1}^{\alpha}+\hat{\tilde{S}}_{i,2j-1}^{\alpha}],$
(2)

with $j$ and $i$ enumerating the chains and sites within, and the diagonal
exchange tensors being
$J^{\alpha\alpha}_{e,o}=J(1,1+\delta,1-\Delta_{\mathrm{XY}})$ and
$J(1-\Delta_{\mathrm{XY}},1+\delta,1)$ for even and odd chains;
$J^{\prime\alpha\alpha}=J^{\prime}(1-\delta^{\prime},1,1-\delta^{\prime})$ for
zig-zag bonds. Parameters $\Delta_{\mathrm{XY}}=0.75$ and
$\delta^{\prime}=0.5$ are fixed from the basic pseudospin representation
arguments [26, 20]. This is just a minor extension of the model presented
earlier [20], which corresponds to $\delta=0$. As discussed in the Supplement,
a good description of the main sharp spectral features is obtained with
$J\simeq 0.8$ meV, $J^{\prime}\simeq 0.35$ meV, and tiny additional Ising-type
anisotropy $\delta=0.1$ present on the main $J$-bond. Since the semiclassical
linear spin wave theory is ill-defined for systems involving strong quantum
fluctuations like Cs2CoBr4, these results are a good approximation only of the
low-energy spectrum [21, 37, 38]. For this reason, we adopted an independent
approach to estimate the ratio of $J^{\prime}/J$: By using the Ising spin
chain as the starting point, and then the Random Phase Approximation, the
bandwidth along and transverse to the chain direction can be obtained [21, 39,
40]. This approach also points to a significant $J^{\prime}/J\gtrsim 0.4$
exchange ratio (see [21]). We conclude that in Cs2CoBr4 the inter-chain
coupling is almost half as strong as the in-chain one. This is consistent with
the observed spectrum being entirely different from that in weakly coupled
Ising spin chains with a pronounced “Zeeman ladder” of spinonic bound states
[41], such as in BaCo2V2O8 [42] .

Figure 3: (a) Dispersion along the $\mathbf{c}^{\ast}$ direction measured in
Cs2CoBr4 at $T\lesssim 0.1$ K in zero field. The particular $k$ value and
integration range are indicated in the plot. (b) Intensity vs. energy transfer
cuts at different momenta, integrated in $0.05$ r.l.u. along $l$ (see orange
stripes in (a)). Dark gray areas mask the incoherent scattering, while light
gray bands hide points excluded from the fit. Figure 4: (a)-(b) Low-energy
dynamics in the SDW phase of Cs2CoBr4 measured at $T\lesssim 0.1$ K and
$\mu_{0}H=1.8$ T. The panels show overviews of the “continuum” part at even
and odd $l$ with (c)-(d) zoom into the lowest accessible energies
correspondingly. Magenta crosses mark the positions of the magnetic Bragg
peaks, orange dashed lines show the $k=1/2$ zone-centers. The scattering in
grayed areas is dominated by the incoherent line. All data are background-
substracted. (e) Intensity-momentum cut at two $l$-values. The energy
integration range is shown. Dashed lines are double-gaussian fit. The $l=3$
curve is shifted upwards for visibility. (f) $0kl$-plane at $\hbar\omega=0$
meV. Arrows indicate the positions of SDW Bragg peaks. Orange dashed lines
mark the effective “triangular” Brillouin zones. Here the background is not
subtracted.

We turn to the dynamics of the SDW phase. The corresponding data set from
CAMEA is actually quite unique, given the horizontal-field scattering
geometry, the low energy scales and the wide reciprocal-space coverage. The
data collected at $\mu_{0}H=1.8$ T are visualized in Fig. 2(b) and Fig. 4.
Compared to zero field, the spectrum measured in the SDW phase is much broader
and clearly gapless. The latter agrees with the conclusions drawn from
thermodynamic measurements [20]. As could be expected, the excitation energy
goes to zero at the incommensurate $(0,\xi,0)$ positions of the SDW where the
corresponding Bragg peaks are located. However, there is one crucial
difference with the single-chain TLSL spectrum of longitudinal excitations. In
the latter, the spectrum is symmetric with respect to the $k=1/2$ point. In
Cs2CoBr4 it is not. The sign of $\xi$ in Eq. 1 describing the Bragg peak and
soft mode location corresponds to the parity of $l$ for odd $k\sim 1/2$, but
is reversed for $k\sim 3/2$. This “staggered” pattern of soft modes is once
again dictated by the quasi-triangular, rather than chain-like, nature of the
underlying ionic lattice. Fig. 4(f) shows this clearly. The low-energy
intensity is condensed around the side edges of the hexagonal zones. The low-
energy excitations with linear dispersion emanate from the corresponding Bragg
peaks. Their shape at higher energy transfer appears symmetric around $k=1/2$,
as Fig. 4(e) shows. The high energy part of the spectrum seems to be rather
diffuse. We cannot identify any sharp modes above $0.4$ meV. This may be due
to the splitting and smearing of the zero-field spectrum by non-commuting
magnetic fields, although the scenario with spinon complex continua,
characteristic of longitudinal fluctuations in the TLSL phase [1], cannot be
ruled out either. Moreover, the non-dispersive level has now shifted to
$E^{\prime}_{\mathrm{CF}}\simeq 1.7(1)$ meV, in agreement with the
expectations for easy-plane Co2+ ions in transverse field with the anisotropy
constant $D$ given above. To conclude, spectral properties of the SDW state
are heavily affected by the 2D $J^{\prime}$ exchange, but only at quite low
energies. The high energy part seems to be a result of the interplay of
spin-$1/2$ chain-like physics and Co2+ crystal-field effects. Similar
arguments apply to the high-field D and E phases [20]. As said, non-commuting
fields admix $|\pm 3/2\rangle$ states to the ground state, making the
presudospin-$1/2$ model inadequate at finite field. Their gapless nature [20]
hints to complex structure, among which are incommensurate planar states or
spin-density waves [2].

In conclusion, for the first time we have observed the generation of
incommensurate longitudinal SDW phase and its locking into UUD magnetization
plateau state, driven by quasi-2D correlations in a $J-J^{\prime}$ distorted
triangular lattice magnet. While there may exist some easy-axis anisotropy in
both the intra- and interchain exchange, the frustration of zig-zag bonds
appears to be the primary mechanism defining the phase diagram of magnetized
Cs2CoBr4. This material appears to be an ideal platform for exploring the
interplay of anisotropy and frustration. This exotic frustrated physics,
interpolating between 1D and 2D worlds, calls for intense future experiments
and in-depth theoretical analysis.

###### Acknowledgements.

We acknowledge the beam time allocation at PSI and ILL (TASP id: 20200145,
CAMEA id: 20211018, D23 id: 5-41-1086). J. Lass was supported by the Danish
National Committee for Research Infrastructure through DanScatt. We thank Dr.
Denis Golosov (Bar Ilan University) for insightful discussions.

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in 1,…,0 See pages \x, of SDW_UUD_Cs2CoBr4_suppl.pdf


# Quantum imaging of a polarisation sensitive phase pattern with hyper-
entangled photons

Manpreet Kaur  Department of Physical Sciences, Indian Institute of Science
Education and Research (IISER) Mohali, Sector-81, Mohali, 140306, India.
Mandip Singh

###### Abstract

A transparent polarisation sensitive phase pattern makes a polarisation
dependent transformation of quantum state of photons without absorbing them.
Such an invisible pattern can be imaged with quantum entangled photons by
making joint quantum measurements on photons. This paper shows a long path
experiment to quantum image a transparent polarisation sensitive phase pattern
with hyper-entangled photon pairs involving momentum and polarisation degrees
of freedom. In the imaging configuration, a single photon interacts with the
pattern while the other photon, which has never interacted with the pattern,
is measured jointly in a chosen polarisation basis and in a quantum
superposition basis of its position which is equivalent to measuring its
momentum. Individual photons of each hyper-entangled pair cannot provide a
complete image information. The image is constructed by measuring the
polarisation state and position of the interacting photon corresponding to a
measurement outcome of the non-interacting photon. This paper presents a
detailed concept, theory and free space long path experiments on quantum
imaging of polarisation sensitive phase patterns.

## Introduction

A transparent polarisation sensitive phase pattern introduces a position-
dependent change in the phase of transmitted light depending on the
polarisation of incident light. A classical image of such a pattern can be
obtained by measuring the position-wise polarisation shift of the transmitted
classical electromagnetic field in the image plane by a method known as
polarisation contrast imaging [1, 6, 7, 2, 3, 4, 5]. However, an image formed
by a single photon exposure of the pattern in each execution of the experiment
is defined as a quantum image. In this way, one can gain the quantum
mechanical advantage over the classical imaging. One such advantage is the
quantum secure transfer of images. Furthermore, by incorporating quantum
entangled photons, one can utilise properties of quantum entanglement and
quantum measurements to construct an image of the pattern. Where a single
photon of an entangled pair of photons interacts with the pattern. However, in
contrast to the classical polarisation contrast imaging, complete information
of the pattern is shared non-locally by both photons as a consequence of
quantum entanglement, even if they are separated by a large distance when a
photon interacted with the pattern. Here, one cannot obtain complete pattern
information just by measuring a single photon. Therefore, a joint measurement
becomes necessary and a quantum image of the pattern can be constructed by
correlating the measurement outcomes. Because of quantum entanglement, quantum
images can be transferred to another location securely and directly.

This paper presents the first free space long path experiment of quantum
imaging of a transparent polarisation sensitive phase pattern with hyper-
entangled photons. Where a hyper-entangled state is a product of quantum
entangled states involving different degrees of freedom [8, 9]. In the
experiment, the hyper-entangled state consists of a product of polarisation
entanglement [10] and momentum entanglement of two photons [11, 12, 13, 14,
15, 16]. Where the momentum entanglement corresponds to Einstein-Podolsky-
Rosen type of quantum entangled state originating from a finite region. Each
photon of a hyper-entangled bi-photon state carries information of
polarisation and momentum of the other photon but an individual photon has no
well-defined position, momentum and polarisation.

A non-birefringent transparent phase object can be classically imaged with
well-known phase-contrast imaging methods where the phase information is
converted to intensity [17, 18, 19]. In the context of the development of
imaging experiments, a phase-contrast imaging method is applied for a non-
destructive detection of a Bose-Einstein condensate [20, 21]. Two-photon
coincidence quantum imaging of absorptive objects has been studied
theoretically and experimentally with spatially correlated photons from a
perspective of foundations of quantum mechanics [22, 23, 24, 25, 26, 27, 28,
29, 30, 31]. Quantum imaging of an absorptive object has been realised
experimentally without detecting a photon interacting with the object [32,
33]. Bell’s inequality violation experiments are performed with images [34,
35]. Two-atom ghost imaging of an absorptive pattern for atoms has been
experimentally realised with metastable helium atoms [36, 37]. A polarisation-
sensitive metasurface is imaged in the near field with polarisation entangled
photons [38].

In this paper, experiments involve a multi-dimensional entanglement in the
form of a hyper-entangled state, to quantum image a transparent polarisation
sensitive phase pattern in free space, where the pattern is positioned at a
distance 16.91 m from a coincidence imaging camera. However, this particular
experiment is not aiming to close loopholes. In the experiment, polarisation
as well as momentum degrees of freedom of the hyper entangled photons are
utilised for quantum imaging.

##  1 Hyper-entangled photons

Figure 1:  _A schematic diagram of a quantum imaging experiment of a
transparent polarisation sensitive phase pattern. A single photon-2 of a
hyper-entangled photon pair is passed through the pattern. Polarisation state
of each photon is measured by passing through polarisers. Photon- $1$ is
measured in a particular quantum superposition of its position after its
polarisation selection by passing through a lens $L_{o}$ followed by a
detection by a single photon detector. An event signal produced by the single
photon detector is sent to a single photon sensitive camera to detect a
position of photon-2 corresponding to a particular measurement outcome of
photon-$1$. An image is gradually formed by accumulating photon-$2$ detections
on the camera by repeating the same experiment. _

Consider a hyper-entangled photons pair emitted by a source in opposite
directions as shown in a schematic diagram Fig. 1. Where photon-$2$ is passed
through a transparent polarisation sensitive phase pattern. Since photons are
polarisation entangled and momentum entangled separately therefore, a
polarisation sensitive phase pattern induces a position-dependent variation of
the quantum entangled state of photons. Polarisation measurement outcomes of
photon-$1$ and photon-2 are determined by a measurement setting (orientation
of pass axis) of polariser-$1$ and polariser-$2$, respectively. After passing
through polariser-$1$, photon-$1$ is focused by a lens $L_{o}$ on a single
photon detector after passing through a narrow aperture. A lens $L_{o}$
permits detection in a quantum superposition basis of the position of
photon-1. A photon detection produces a pulse named an event signal, which is
sent to a single photon detection camera to allow registration of photon-$2$
corresponding to a selected measurement outcome. A quantum image is gradually
formed by accumulating the photon registrations on camera for each repetition
of the experiment with the same measurement setting. In this way, this is a
coincidence imaging where the individual photon has no well-defined momentum
and polarisation however, each photon is carrying momentum and polarisation
information of the other photon by quantum entanglement. Therefore,
measurement outcomes of both photons are required to construct a quantum
image.

Consider Einstein Rosen Podolsky state (EPR) [39] of two particles in a
position space,
$|\alpha\rangle=\int^{\infty}_{-\infty}|x\rangle_{1}|x+x_{o}\rangle_{2}\mathrm{d}x$,
where subscripts $1$ and $2$ are the labels of particles. In this state, both
particles are equally likely to exist at all points in the position space with
a constant separation $x_{o}$. In this way, they are spatially entangled. The
EPR state in a momentum space can be written as
$|\alpha\rangle=\int^{\infty}_{-\infty}e^{i\frac{px_{o}}{\hslash}}|p\rangle_{1}|-p\rangle_{2}\mathrm{d}p$,
where both particles have opposite momenta and are equally likely to exist at
all points in the momentum space such that they are momentum entangled. Where
$\hslash=h/2\pi$ is the reduced Planck’s constant. In this way, both the
position and momentum of each particle are completely unknown. Consider a
source of a finite extension producing a pair of photons such that both
photons are originating from the same position in the source. This is a case,
if $x_{o}=0$ in the one-dimensional EPR state and in this paper, it is
extended to three dimensions by including the polarisation of photons.

Quantum state of photons originating from a position vector
$\mathbf{r}^{\prime}$, in time independent case, leads to a finite amplitude
to find a photon at a point $o_{a}$ in region-$a$ and another photon at a
point $o_{b}$ in region-$b$ as shown in Fig. 1. Consider points $o_{a}$ and
$o_{b}$ prior to any optical element from the source. A joint amplitude for
photons to be at these points in three-dimensions is
$\frac{e^{ip_{1}|\mathbf{r}_{a}-\mathbf{r}^{\prime}|/\hslash}}{|\mathbf{r}_{a}-\mathbf{r}^{\prime}|}\frac{e^{ip_{2}|\mathbf{r}_{b}-\mathbf{r}^{\prime}|/\hslash}}{|\mathbf{r}_{b}-\mathbf{r}^{\prime}|}$
[40, 41] where $p_{1}$ and $p_{2}$ are magnitudes of momentum of photon-$1$
and of photon-$2$, respectively. Position vectors of points $o_{a}$ and
$o_{b}$ from origin are $\mathbf{r}_{a}$ and $\mathbf{r}_{b}$, respectively.
Distances of $o_{a}$ and $o_{b}$ from $\mathbf{r}^{\prime}$ are
$|\mathbf{r}_{a}-\mathbf{r}^{\prime}|$ and
$|\mathbf{r}_{b}-\mathbf{r}^{\prime}|$, respectively. Here, both photons are
produced from a same location within the source. Consider, polarisation state
of photon-$1$ propagating in region-$a$ is horizontal $|H\rangle_{a}$ and of
photon-$2$ propagating in region-$b$ is vertical $|V\rangle_{b}$. A total
amplitude to find photon-$1$ at $o_{a}$ and photon-2 at $o_{b}$ is a linear
quantum superposition of amplitudes originating from all points within the
source. Therefore, a combined two-photon quantum state can be written as

$|\Psi\rangle_{12}=A_{o}\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}\psi(x^{\prime},y^{\prime},z^{\prime})\frac{e^{ip_{1}|\mathbf{r}_{a}-\mathbf{r}^{\prime}|/\hslash}}{|\mathbf{r}_{a}-\mathbf{r}^{\prime}|}\frac{e^{ip_{2}|\mathbf{r}_{b}-\mathbf{r}^{\prime}|/\hslash}}{|\mathbf{r}_{b}-\mathbf{r}^{\prime}|}\mathrm{d}x^{\prime}\mathrm{d}y^{\prime}\mathrm{d}z^{\prime}\otimes|H\rangle_{a}|V\rangle_{b}$
(1)

where $A_{o}$ is a normalisation constant and
$\psi(x^{\prime},y^{\prime},z^{\prime})$ is an amplitude of pair creation at a
location $r^{\prime}(x^{\prime},y^{\prime},z^{\prime})$ that is considered to
be $\frac{e^{-\frac{x^{\prime
2}}{2\sigma^{2}_{x}}}}{(2\pi)^{1/2}\sigma_{x}}\frac{e^{-\frac{y^{\prime
2}}{2\sigma^{2}_{y}}}}{(2\pi)^{1/2}\sigma_{y}}\frac{e^{-\frac{z^{\prime
2}}{2\sigma^{2}_{z}}}}{(2\pi)^{1/2}\sigma_{z}}$.

For a source extension much smaller than distances of points $o_{a}$ and
$o_{b}$ from origin, Eq. 1 is written as

$|\Psi\rangle_{12}=\Phi_{12}(r_{a};r_{b})\otimes|H\rangle_{a}|V\rangle_{b}$
(2)

where $\Phi_{12}(r_{a};r_{b})$ is the amplitude to find photon-$1$ at $o_{a}$
and photon-$2$ at $o_{b}$, which is calculated by solving the integral in Eq.
1,

$\Phi_{12}(r_{a};r_{b})=A_{o}\frac{e^{i(p_{1}r_{a}+p_{2}r_{b})/\hslash}}{r_{a}r_{b}}e^{-((p_{1x}+p_{2x})\sigma_{x})^{2}/2\hslash^{2}}e^{-((p_{1y}+p_{2y})\sigma_{y})^{2}/2\hslash^{2}}e^{-((p_{1z}+p_{2z})\sigma_{z})^{2}/2\hslash^{2}}$
(3)

where $p_{1x}=p_{1}\sin\theta_{a}\cos\phi_{a}$,
$p_{1y}=p_{1}\sin\theta_{a}\sin\phi_{a}$ and $p_{1z}=p_{1}\cos\theta_{a}$ are
the $x$, $y$ and $z$ components of momentum of photon-$1$ in the spherical
polar coordinate system. Similarly $p_{2x}=p_{2}\sin\theta_{b}\cos\phi_{b}$,
$p_{2y}=p_{2}\sin\theta_{b}\sin\phi_{b}$ and $p_{2z}=p_{2}\cos\theta_{b}$ are
the $x$, $y$ and $z$ components of momentum of photon-$2$.

Photons are momentum entangled due to inseparability of
$\Phi_{12}(r_{a};r_{b})$. Assume that photons have same energy therefore,
magnitudes of their momenta are equal, $p_{1}=p_{2}$. Since photons are
identical particles and there exist different amplitudes that cannot be
distinguished if a photon is detected at $o_{a}$ and another photon is
detected at $o_{b}$. An amplitude to find photon-$1$ at $o_{a}$ and photon-2
at $o_{b}$ is indistinguishable from finding photon-1 at $o_{b}$ and photon-2
at $o_{a}$ after exchange of photons if their polarisation is not measured.
Therefore, due to the Bosonic nature of identical photons, their symmetric
quantum state is written as
$|\Psi\rangle=\frac{1}{\sqrt{2}}(|\Psi\rangle_{12}+|\Psi\rangle_{21})$, where
$|\Psi\rangle_{21}$ is a quantum state Eq. 1 after the exchange of photons.

After the exchange of photons, Eq. 2 becomes
$|\Psi\rangle_{21}=\Phi_{21}(r_{a};r_{b})\otimes|V\rangle_{a}|H\rangle_{b}$.
However, from Eq. 3, $\Phi_{21}(r_{a};r_{b})=\Phi_{12}(r_{a};r_{b})$.
Therefore, a combined quantum state becomes

$|\Psi\rangle=\Phi_{12}(r_{a};r_{b})\otimes\frac{1}{\sqrt{2}}\left(|H\rangle_{a}|V\rangle_{b}+|V\rangle_{a}|H\rangle_{b}\right)$
(4)

This is a hyper-entangled state since photons are momentum entangled due to
inseparability of $\Phi_{12}(r_{a};r_{b})$ and polarisation entangled due to
inseparability of
$\frac{1}{\sqrt{2}}(|H\rangle_{a}|V\rangle_{b}+|V\rangle_{a}|H\rangle_{b})$,
where each quantum entangled state is symmetric. However, a combined quantum
state is a product of each quantum entangled state. From here onwards, a
photon in region-$a$ is labeled as photon-1 and a photon in region-$b$ is
labeled as photon-2.

##  2 Imaging with hyper-entangled photons

In a realistic experiment, a source emits entangled photons in particular
directions. To incorporate this directional dependence, quantum state is
multiplied by an additional directional function such that the highest
probability of finding photons is around a point $o_{a}$ in region-$a$ and
$o_{b}$ in region-$b$. Let $\psi_{a}(x_{1},y_{1},z_{1})$ and
$\psi_{b}(x_{2},y_{2},z_{2})$ are the envelops of wavefunctions for a photon
around $o_{a}$ and a photon around $o_{b}$. From these considerations, a total
symmetric quantum state of photons can be written as

$|\Psi\rangle_{s}=\frac{1}{\sqrt{2}}[\psi_{a}(x_{1},y_{1},z_{1})\psi_{b}(x_{2},y_{2},z_{2})\pm\\\
\psi_{a}(x_{2},y_{2},z_{2})\psi_{b}(x_{1},y_{1},z_{1})]a_{s}\Phi_{12}(r_{a};r_{b})\otimes\frac{1}{\sqrt{2}}\left(|H\rangle_{1}|V\rangle_{2}+e^{i\phi}|V\rangle_{1}|H\rangle_{2}\right)$
(5)

Where the phase $\phi$ is not arbitrary, it is either zero or $\pi$. The total
quantum state is symmetrised with a plus sign if $\phi=0$. For $\phi=\pi$, the
polarisation entangled state is antisymmetric after the exchange of photons
therefore, the quantum state in the external degrees of freedom has to be
antisymmetric. Here, $a_{s}$ is a normalisation constant. In the experiment, a
phase difference $\phi$ between $|H\rangle$ and $|V\rangle$ quantum states of
a photon is introduced either by placing a half-wave plate in the path of a
photon or by tilting the nonlinear crystal source of entangled photons. In
this paper, an antisymmetric polarisation entanglement is produced by the
source. Since photons are propagating in different regions $a$ and $b$ such
that $\psi_{a}(x_{1},y_{1},z_{1})$ and $\psi_{b}(x_{2},y_{2},z_{2})$ are non-
overlapping therefore, a second term
$\psi_{a}(x_{2},y_{2},z_{2})\psi_{b}(x_{1},y_{1},z_{1})a_{s}\Phi_{21}(r_{a};r_{b})$,
in Eq. 5 is negligible.

To form an image with a hyper-entangled state, consider photon-$2$ in
region-$b$ is passed through a transparent polarisation sensitive phase
pattern oriented perpendicular to $z$-axis as shown in Fig. 1. The pattern
introduces a position-dependent phase difference between the horizontal and
the vertical polarisation of a photon passing through it such that
$|H\rangle_{2}\rightarrow e^{i\phi(x_{2},y_{2})}|H\rangle_{2}$ and
$|V\rangle_{2}\rightarrow|V\rangle_{2}$. In the experiment, a phase difference
$\phi(x_{2},y_{2})$ is either zero or $\pi$. Consider photon-2 of a hyper-
entangled state, with an antisymmetric polarisation entangled state
$|\Psi^{-}\rangle_{p}=\frac{1}{\sqrt{2}}\left(|H\rangle_{1}|V\rangle_{2}-|V\rangle_{1}|H\rangle_{2}\right)$,
is incident on the pattern. A polarisation dependent phase is imprinted on
photon-2 in region-$b$. Assume that the pattern is located at $z=d_{2}$ in
region-$b$. An arbitrary location of photon-2 on the pattern just after the
phase imprint is represented by coordinates ($x_{2}$, $y_{2}$) in a $x$-$y$
plane. An arbitrary location of photon-1 in a plane oriented perpendicular to
$z$-axis at $z=-d_{1}$ in region-$a$ is represented by coordinates ($x_{1}$,
$y_{1}$). Therefore, after the phase imprint and normalisation, the quantum
state of photons is written as

$|\Psi\rangle_{I}=a_{s}\psi_{a}(x_{1},y_{1},-d_{1})\psi_{b}(x_{2},y_{2},d_{2})\Phi_{12}(r_{a};r_{b})\otimes\frac{1}{\sqrt{2}}\left(|H\rangle_{1}|V\rangle_{2}-e^{i\phi(x_{2},y_{2})}|V\rangle_{1}|H\rangle_{2}\right)$
(6)

A transparent polarisation sensitive phase pattern is imprinted in the phase
of polarisation entangled state and a spread of a photon on the pattern is due
to the momentum entangled part of the hyper-entangled state. Photons are also
spatially correlated due to their spatial entanglement in position space. The
position dependent polarisation entangled state of photons is
$|\Psi^{-}\rangle_{p}$ for $\phi(x_{2},y_{2})=0$ and
$|\Psi^{+}\rangle_{p}=\frac{1}{\sqrt{2}}\left(|H\rangle_{1}|V\rangle_{2}+|V\rangle_{1}|H\rangle_{2}\right)$
for $\phi(x_{2},y_{2})=\pi$. The location and polarisation information of each
photon is carried by the other photon because of their quantum entanglement.

After the imprint of a pattern, the position coordinates ($x_{2}$, $y_{2}$) at
$z=d_{2}$ of photon-$2$ are measured by imaging with a lens on a single photon
sensitive camera after its polarisation selection by polariser-$2$ as shown in
Fig. 1. The camera registers photon-$2$ only if photon-$1$ is detected by a
single photon detector after passing through polariser-$1$. Consider a lens
$L_{o}$ is not placed but a narrow aperture single photon detector is placed
at a location $(x_{o1},y_{o1},z=-d_{1})$ and orientation of pass axis of
polariser-$1$ is vertical and of polariser-$2$ is horizontal. This measurement
setting corresponds to a coincidence detection when photon-1 is detected at
$(x_{o1},y_{o1},z=-d_{1})$ in the vertical polarisation and photon-$2$ is
detected in the horizontal polarisation state on the camera. The same
experiment is repeated for a chosen measurement setting many times, a location
of photon-$2$ on camera varies each time and gradually an image is formed on
the camera. A corresponding probability of coincidence detection is written as
$P_{V_{1},H_{2}}(x_{2},y_{2})=\frac{1}{2}|a_{s}\psi_{a}(x_{o1},y_{o1},-d_{1})\psi_{b}(x_{2},y_{2},d_{2})\Phi_{12}(r_{a};r_{b})(-e^{i\phi(x_{2},y_{2})})|^{2}$.
Where subscripts of $P_{V_{1},H_{2}}$ denote orientation of pass-axes of
polariser-$1$ and polariser-$2$. It is clear that a phase information is lost
for this polarisation setting. Similarly, a polarisation sensitive phase
information cannot be recovered for other combination of orientations of
polarisers along horizontal and vertical directions.

Let the orientation of pass axes of polarisers is now changed to perform
polarisation measurements in the diagonal basis that is
$|d^{+}\rangle_{j}=\frac{|H\rangle_{j}+|V\rangle_{j}}{\sqrt{2}}$ and
$|d^{-}\rangle_{j}=\frac{|H\rangle_{j}-|V\rangle_{j}}{\sqrt{2}}$, where a
photon label $j$ is $1$ or $2$. If both polarisers are oriented to pass
quantum states $|d^{-}\rangle_{j}$ and photon-$1$ is detected at a location
$(x_{o1},y_{o1},z=-d_{1})$, then the corresponding probability of coincidence
detection is

$P_{d^{-}_{1},d^{-}_{2}}(x_{2},y_{2})=\frac{1}{4}|a_{s}\psi_{a}(x_{o1},y_{o1},-d_{1})\psi_{b}(x_{2},y_{2},d_{2})\Phi_{12}(r_{a};r_{b})|^{2}(1-\cos\phi(x_{2},y_{2}))$
(7)

which contains a phase information of the pattern. Since $\phi(x_{2},y_{2})$
is either $\pi$ or zero therefore, $P_{d^{-}_{1},d^{-}_{2}}(x_{2},y_{2})$ is a
two level image. Where an image of the transparent polarisation sensitive
phase pattern is obtained by measuring coincidence detection probability. If
on the other hand, polariser-$1$ is aligned to pass $|d^{+}\rangle$ and
polariser-$2$ is aligned to pass $|d^{-}\rangle$ then the corresponding
probability of coincidence detection is written as

$P_{d^{+}_{1},d^{-}_{2}}(x_{2},y_{2})=\frac{1}{4}|a_{s}\psi_{a}(x_{o1},y_{o1},-d_{1})\psi_{b}(x_{2},y_{2},d_{2})\Phi_{12}(r_{a};r_{b})|^{2}(1+\cos\phi(x_{2},y_{2}))$
(8)

This image is exactly inverted as compared to Eq. 7, where the maximum and the
minimum levels of the image are interchanged.

To make a measurement on photon-$1$ in a quantum superposition basis of
position $(x_{1},y_{1})$, a convex lens $L_{o}$ is placed at $z=-d_{1}$ and
photon-$1$ is detected close to its focal point by a narrow aperture single
photon detector placed on $z$-axis. The lens transforms a particular quantum
superposition of position (or an incident momentum state) in a plane at
$z=-d_{1}$ to a single point in its focal plane. Therefore, probability of
coincidence detection of photons for two different orientation
$(d^{+}_{1},d^{-}_{2})$ and $(d^{-}_{1},d^{-}_{2})$ of polarisers is
succinctly written as

$P^{\prime}_{d^{\pm}_{1},d^{-}_{2}}(x_{2},y_{2})=\frac{1}{4}|\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}a_{s}\psi_{a}(x_{1},y_{1},-d_{1})\psi_{b}(x_{2},y_{2},d_{2})\Phi_{12}(r_{a};r_{b})\mathrm{d}x_{1}\mathrm{d}y_{1}|^{2}(1\pm\cos\phi(x_{2},y_{2}))$
(9)

It is evident that opposite level images are formed when orientation of
polarisers is changed from $(d^{-}_{1},d^{-}_{2})$ to $(d^{+}_{1},d^{-}_{2})$.
However, no image is formed if only photon-2 is detected without considering
the measurement outcome of photon-$1$ or if photon-$1$ is not measured at all.

##  3 Experiment and discussion

A schematic diagram of the experimental setup is shown in Fig. 2. Hyper-
entangled photons are produced by type-II spontaneous parametric down
conversion (SPDC) [42, 43, 44, 45, 46, 47] in a beta-barium-borate (BBO)
nonlinear crystal. A linearly polarised pump laser beam of wavelength 405 nm
and narrow beam diameter is passed through a BBO crystal. Hyper-entangled
photons are produced at wavelength 810 nm in a first crystal, where a second
crystal is placed to compensate transverse and longitudinal walk-offs. A
polarisation sensitive phase pattern is produced by a reflection type spatial
light modulator (SLM). After the collimation, photon-$2$ is reflected from the
spatial light modulator and a position-dependent phase is imprinted such that
$|H\rangle_{2}\rightarrow e^{i\phi(x_{2},y_{2})}|H\rangle_{2}$ and
$|V\rangle_{2}\rightarrow|V\rangle_{2}$. Photon-$2$ is imaged on an
intensified-charge-coupled-device (ICCD) camera, after passing through
polariser-$2$, with a three lens ($L_{1}$, $L_{2}$ and $L_{3}$) telescopic
configuration. The telescope is aligned such that the object plane coincides
with the reflecting surface of the spatial light modulator. In this way, a
location of photon-2 immediately after the phase imprint is measured by
imaging it on the camera. The focal length of lens $L_{1}$ of diameter 5 cm is
40 cm. The focal length of lenses $L_{2}$ and $L_{3}$ are 10 cm and 100 cm,
respectively. Photon-$1$ is passed through polariser-$1$ to perform a
polarisation measurement and a lens $L_{o}$ to perform a measurement in the
quantum superposition basis of position after detection by a narrow area
single photon detector of very low dark counts. These are joint measurements.
Pass-axis of polariser-$1$ is subtending an angle $\delta_{1}$ and of
polariser-$2$ is subtending an angle $\delta_{2}$ with the horizontal axis.
These angles can be varied independently according to a measurement setting.
An electrical signal produced by a single photon detector corresponds to a
measurement outcome for a chosen measurement setting for photon-$1$. This
signal is connected to a direct gate terminal of the ICCD camera. The direct
gate terminal activates an electronic shutter of the camera that allows
detection signal of photon-$2$ produced by the imaging sensor of camera to
reach the charge-coupled-device (CCD) sensor of the camera after
amplification. This process efficiently detects a coincidence event of photons
with a 20 ns insertion delay. The distance of the imaging sensor surface of
the ICCD camera from the spatial light modulator is 16.91 m. The spatial light
modulator is placed at a distance 89 cm from the first BBO crystal. A first
lens $L_{1}$ of the telescope is placed at a distance 15.24 m from the spatial
light modulator surface. This lens produces an inverted real image of the
spatial light modulator surface, which acts as a real image for a second lens
$L_{2}$. A second lens is placed such that it produces a magnified virtual
image in a virtual image plane as shown in Fig. 2. A third lens $L_{3}$ of
focal length $f$ = 100 cm is positioned such that the photon detection plane
of the ICCD camera and a virtual image plane are positioned at $+2f$ and
$-2f$, respectively from lens $L_{3}$. In this $2f$-$2f$ imaging
configuration, an image can be fine-tuned on the ICCD camera by displacing the
second lens $L_{2}$ only and keeping other lenses stationary. The image
demagnification of the telescope is 0.52 and its spatial resolution at the
spatial light modulator surface is about $\sim$ 0.3 mm. In the experiment, a
structure size of the pattern is 1 mm, which is imaged by the ICCD camera from
a distance 16.91 m.

Figure 2:  _Main schematic diagram of the experiment. Hyper-entangled photons
are produced by a BBO crystal. A polarisation sensitive phase pattern is
produced by an SLM. The pattern is imaged in free space by ICCD camera located
at 16.91 m from the pattern by a lens combination in a telescopic
configuration. ICCD camera is activated by a measurement outcome event signal
of photon-1 to register photon-2 to form a quantum image._ Figure 3:  _(a) A
transparent polarisation sensitive phase pattern, a darker region represents
$\phi(x_{2},y_{2})=0$ and a lighter region represents $\phi(x_{2},y_{2})=\pi$.
(b) An image captured by ICCD camera, when measured polarisation states of
photon-1 and photon-2 are $|d^{-}\rangle_{1}$ and $|d^{-}\rangle_{2}$. (c) An
image with inverted levels, when measured polarisation state of photon-1 is
$|d^{+}\rangle_{1}$ and of photon-2 is $|d^{-}\rangle_{2}$. Each image is
captured for ten minutes time of exposure._ Figure 4:  _(a) A transparent
polarisation sensitive phase pattern with $1\leavevmode\nobreak\ mm\times
1\leavevmode\nobreak\ mm$ size of each square, a darker region represents
$\phi(x_{2},y_{2})=0$ and a lighter region represents $\phi(x_{2},y_{2})=\pi$.
(b) An image of the pattern captured by ICCD camera, when measured
polarisation states of photon-1 and photon-2 are $|d^{-}\rangle_{1}$ and
$|d^{-}\rangle_{2}$. (c) When measured polarisation state of photon-1 is
$|V\rangle_{1}$ and of photon-2 is $|d^{-}\rangle_{2}$. In this case, only the
edges of squares are absent due to the diffraction limit of telescope. (d) No
image is formed when measured polarisation state of photon-1 is
$|H\rangle_{1}$ and of photon-2 is $|d^{-}\rangle_{2}$._

Consider, photon-$j$ is detected after passing through a polariser-$j$ with
its pass axis inclined at an angle $\delta_{j}$ _w.r.t._ horizontal axis. Its
measured state of polarisation is, $|H\rangle_{j}$ for $\delta_{j}=0$,
$|V\rangle_{j}$ for $\delta_{j}=90^{o}$, $|d^{+}\rangle_{j}$ for
$\delta_{j}=45^{o}$ and $|d^{-}\rangle_{j}$ for $\delta_{j}=-45^{o}$.
Polarisation state of photon-$2$ is measured once it is detected by ICCD
camera at any location after passing through polariser-$2$. If a single photon
detector detects photon-$1$ after passing through a polariser-$1$ but without
a lens $L_{o}$, then it corresponds to a position measurement of photon-1
$(x_{01},y_{01})$ in a plane at $z=-d_{1}$ along with an outcome of measured
state of polarisation of photon-$1$. On the other hand, if a single photon
detector detects a photon-$1$ after passing through a polariser and a lens
$L_{o}$, then it corresponds to a measurement in the quantum superposition
basis of position along with an outcome of a measured state of polarisation of
photon-$1$. For this measurement setting, a coincidence photon detection
probability is given by Eq. 9.

A transparent polarisation sensitive phase pattern shown in Fig. 3(a) is
displayed on the spatial light modulator, where a position-dependent grey
level of the pattern determines a phase shift, which is either zero or $\pi$.
Lighter grey regions represent a $\pi$ phase shift and since a source produces
an antisymmetric polarisation entangled state therefore, polarisation
entanglement of photons, after passing photon-$2$ through this region,
transformed to
$|\Psi^{+}\rangle_{p}=\frac{1}{\sqrt{2}}(|H\rangle_{1}|V\rangle_{2}+|V\rangle_{1}|H\rangle_{2})$.
Darker grey regions represents a zero phase shift and polarisation entangled
state of photons remains same as produced by the source that is
$|\Psi^{-}\rangle_{p}=\frac{1}{\sqrt{2}}(|H\rangle_{1}|V\rangle_{2}-|V\rangle_{1}|H\rangle_{2})$.
Therefore, the information of a two-photon quantum image is contained by the
momentum and the polarisation entanglement parts of $|\Psi\rangle_{I}$. In the
first experiment, to produced a quantum image, both polarisers are aligned at
$\delta_{1}=-45^{o}$ and $\delta_{2}=-45^{o}$ to detect a same state of
polarisation of photons in the diagonal basis that is $|d^{-}\rangle_{1}$ of
photon-$1$ and $|d^{-}\rangle_{2}$ of photon-$2$. Photon-1 is also measured in
the quantum superposition basis of position. The experiment is repeated for
this setting and an image, in the form of gated photon counts of the pattern,
is accumulated for 10 minutes exposure of the ICCD camera. A background image
is captured by displaying a uniform grey level corresponding to $\phi=0$ on
the spatial light modulator with the same experimental setting. A coincidence
photon image is constructed by subtracting a corresponding background image as
shown in Fig. 3(b). The ICCD camera also captures accidental counts in the
coincidence window. These accidental counts are originated from the scattered
photons and camera noise. These counts accumulate with the actual coincidence
photon counts. A typical ratio of a coincidence to accidental counts is
0.35-0.45. This is why a background correction is necessary. The noise of
accidental photon counts in the background image, which is defined as a ratio
of standard deviation to an average of counts for twenty-five hundred pixels
is typically 0.12. All images shown in this paper are background corrected.
Each image on the ICCD camera is reduced by 0.52 image demagnification. In a
similar experiment, polariser-$1$ is aligned at $\delta_{1}=+45^{o}$ to detect
photon-$1$ in $|d^{+}\rangle_{1}$ and polariser-2 is kept at
$\delta_{2}=-45^{o}$ to detect photon in $|d^{-}\rangle_{2}$ . The coincidence
image photon counts accumulated by the ICCD camera are shown in Fig. 3(c),
which corresponds to an image with inverted levels as compared to an image
shown in Fig. 3(b). This result is in agreement with Eq. 9 that shows an
inversion of levels of images in the diagonal basis. These images are two-
level images corresponding to the two-level phase shift of the pattern.

Figure 5:  _(a) Edges of a pattern shown in Fig. 3(a) appear when measured
polarisation state of photon-1 is $|V\rangle_{1}$ and of photon-2 is
$|d^{-}\rangle_{2}$. (b) No image is formed when measured polarisation state
of photon-1 is $|H\rangle_{1}$ and of photon-2 is $|d^{-}\rangle_{2}$. (c) A
single photon image of low depth captured by ICCD camera. Edges of the pattern
appear as low coincidence photon counts darker regions due to the resolution
limit of the telescope._

In another experiment, a different transparent polarisation sensitive phase
pattern as shown in Fig. 4(a) is displayed on the spatial light modulator.
Where the gray levels corresponding to the position-dependent phase shifts are
the same as in the previous pattern. A background-corrected image of this
pattern shown in Fig. 4(b) is accumulated with the ICCD camera for 10 minutes
time of exposure in the diagonal basis, for $\delta_{1}=\delta_{2}=-45^{o}$
and with a lens $L_{o}$. This image is then compared with another background-
corrected image captured with a different setting of polariser-$1$,
$\delta_{1}=+90^{o}$ as shown in Fig. 4(c) keeping all other settings the
same. According to the theoretical analysis shown in the previous section, the
coincidence probability corresponding to a measurement of $|H\rangle$ or
$|V\rangle$ state of photon-$1$ does not contain the phase information of the
pattern therefore, no image should be formed. A measurement of photon-$1$
polarisation state in $|V\rangle_{1}$ collapses photon-$2$ polarisation state
onto $|H\rangle_{2}$ and this polarisation state is transformed by the spatial
light modulator such that $|H\rangle_{2}\rightarrow
e^{i\phi(x_{2},y_{2})}|H\rangle_{2}$. This is a pure phase pattern and it
diffracts photons. However, a telescope is located at a far distance 15.24 m
from the pattern and its spatial resolution at this distance is about $\sim$
0.3 mm, where the size of each square in the pattern is 1 mm$\times$1 mm. Due
to the finite lens diameter, the telescope is not capturing all the diffracted
photons carrying information of sharp edges and it leads to the fundamental
diffraction limit of resolution. Thus, the edges or phase boundaries in the
pattern lead to low two-photon coincidence counts. Therefore, the edges appear
as low-level regions in the image. In this case, if all diffracted photons are
also captured then no image of the pattern will result. This effect is exactly
matching for all the sharp edges by a comparison of an image shown in Fig.
4(c) with a corresponding image shown in Fig. 4(b). Another image is captured
for setting, $\delta_{1}=0$ to detect photon-$1$ in $|H\rangle_{1}$ keeping
all other settings the same as for the image shown in Fig. 4(b). Since this
setting collapses the polarisation state of photon-$2$ to $|V\rangle_{2}$ and
this quantum state remains unchanged by the spatial light modulator therefore,
no pattern is formed on ICCD camera as shown in Fig. 4(d). A similar
experiment is repeated for $\delta_{1}=90^{o}$ to detect $|V\rangle_{1}$ and
$\delta_{2}=-45^{o}$ to detect $|d^{-}\rangle_{2}$ for a pattern shown in Fig.
3(a). A corresponding background-corrected image is shown in Fig. 5(a), where
the sharp edges appear due to the diffraction limit of the telescope. As
explained, for $\delta_{1}=0$ and $\delta_{2}=-45^{o}$, no pattern is formed
as shown in Fig. 5(b). A single photon background-corrected image of the same
pattern is shown in Fig. 5(c). In this case, photon-1 is not measured and the
ICCD camera shutter is kept open for continuous exposure of photon-2 for ten
minutes. This image has low depth and only the sharp edges appear because of
the diffraction limit of the telescope.

##  4 Conclusion

This paper presents quantum imaging experiments of transparent polarisation
sensitive phase patterns with hyper-entangled photons. The paper begins with
the main concept of the experiment and then it is shown, how a hyper-entangled
state arises from the indistinguishability of identical photons. Furthermore,
a detailed theoretical analysis of quantum imaging is presented. In the
experiment, hyper-entangled photons are produced by a type-II SPDC source and
each polarisation sensitive phase pattern is imaged from a distance 16.91 m
with the ICCD camera. In this type of coincidence imaging, polarisation
entanglement and momentum entanglement of photons are involved. Images of
patterns are constructed after background correction by accumulating
coincidence photons on the ICCD camera for different measurement settings for
both photons. In a diagonal basis, image levels are inverted if a measured
outcome of polarisation state of photon-$1$ is changed from
$|d^{-}\rangle_{1}$ to $|d^{+}\rangle_{1}$. Edges of the pattern are not
captured because of the diffraction limit of the telescope. Eventually, low-
level regions corresponding to edges appear in a two-photon captured image.
Which is experimentally shown in detail. If photon-$1$ is measured in a
quantum state $|H\rangle_{1}$ then photon-$2$ collapses to a quantum state
$|V\rangle_{2}$, which is unaffected by the spatial light modulator and it
leads to no pattern in the image. A single photon accumulation of photon-$2$
on the camera by continuous exposure without considering photon-$1$ produces a
shallow image due to the diffraction limit of the telescope, which contains
partial information of the pattern.

## Methods

###  4.1 Experimental setup

Hyper-entangled photons are produced by a type-II SPDC process in a first BBO
crystal, which is a negative uniaxial crystal. A focused pump laser beam of
extraordinary polarisation is incident on the crystal. Crystal is tuned such
that the emission cones of ordinary and extraordinary polarised photons are
intersecting. Hyper-entanglement is produced in the intersection regions. Down
converted photons at wavelength 810 nm are passed through a half-wave plate to
interchange their linear polarisation. A second crystal of half thickness is
placed parallel to the first crystal with a same orientation of its optic axis
as the first crystal. This crystal compensates for transverse and longitudinal
walk-offs of down converted photons. The source is aligned to produce an
antisymmetric polarisation entangled state $|\Psi^{-}\rangle_{p}$ and a
momentum entangled state. In a position basis, the momentum entangled photons
are spatially entangled. Momentum entanglement is a result of the linear
momentum conservation of photons. Photons are polarisation entangled and
momentum entangled separately. Photon-$1$ and photon-$2$ of an entangled pair
are passed through two polarisers aligned with their pass axes subtending
angles $\delta_{1}$ and $\delta_{2}$ with the horizontal direction. The
photons transmitted by the polarisers are passed through bandpass filters of
peak transmission at 810 nm. To check polarisation entanglement the photons
are then detected by single photon detectors where single photon counts of
each photon detector and coincidence photon counts are measured by a
coincidence event counting module. However, to image the pattern the
experimental setup is shown in Fig. 2.

###  4.2 Violation of CHSH inequality

Einstein’s locality condition strictly forbids the faster than light
influence. Consider two polarisation entangled photons separated by a large
distance. Any measurement performed on one photon should not affect the
measurement outcome of the other photon immidiately if the locality condition
is true. Consider two different observables $A_{1}$ and $A^{\prime}_{1}$ of
photon-$1$ where any one of them is measured for a photon pair. Similarly, any
one of the two different observables $B_{2}$ and $B^{\prime}_{2}$ of
photon-$2$ is measured for a photon pair. A measured observable outcome
corresponds to a polarisation measurement of a photon. A measurement outcome
corresponds to one of the two orthogonal polarisations with measured value
$+1$ for one polarisation and $-1$ for the other. In the local realistic
model, the values of all such observables are well-defined prior to any
measurement. Consider a correlation function where each observable can have a
value +1 or -1, therefore

$C_{l}=(A_{1}+A^{\prime}_{1})B^{\prime}_{2}+(A_{1}-A^{\prime}_{1})B_{2}=\pm 2$
(10)

its value cannot exceed the bound $|\langle C_{l}\rangle|\leq 2$. However,
quantum mechanical observables cannot always have a well-defined value and
incompatible observables cannot be measured simultaneously. An observable
value is obtained by the measurement as a possible outcome that depends on the
quantum state and observable being measured. Quantum mechanics violates local
realistic prediction expressed in quantum observables such that

$\displaystyle
S=\langle(\hat{A}_{1}+\hat{A}^{\prime}_{1})\hat{B}^{\prime}_{2}+(\hat{A}_{1}-\hat{A}^{\prime}_{1})\hat{B}_{2}\rangle=\langle\hat{A}_{1}\hat{B}_{2}\rangle-\langle\hat{A}^{\prime}_{1}\hat{B}_{2}\rangle+\langle\hat{A}_{1}\hat{B}^{\prime}_{2}\rangle+\langle\hat{A}^{\prime}_{1}\hat{B}^{\prime}_{2}\rangle$
(11)

Therefore, the following Clauser, Horne, Shimony and Holt (CHSH) inequality is
violated by quantum mechanics [48]

$0\leq|S|\leq 2$ (12)

Expectation value of each term of product of observables in Eq. 11 can be
evaluated from coincidence photon counts $C(\delta_{1},\delta_{2})$ measured
for a corresponding orientation of pass axes of polarisers and from the
coincidence photon counts $C(\delta^{\perp}_{1},\delta^{\perp}_{2})$ where
$\delta^{\perp}_{j}=\delta_{j}+90^{o}$ [10]. The CHSH inequality is violated
for the polarisation entanglement. A plot of coincidence photon counts
measurements is shown in Fig. 6 for different orientations of polarisers. The
CHSH correlation parameter $S$, Eq. 11, is expressed in the form of angle of
pass axes of polarisers as

$S=E(\delta_{1},\delta_{2})-E(\delta^{\prime}_{1},\delta_{2})+E(\delta_{1},\delta^{\prime}_{2})+E(\delta^{\prime}_{1},\delta^{\prime}_{2})$
(13)

which is evaluated from the coincidence photon measurements. Where
$E(\delta_{1},\delta_{2})$ represents an expectation value of the joint
observable measurements performed on both photons, which is written as

$E(\delta_{1},\delta_{2})=\frac{C(\delta_{1},\delta_{2})+C(\delta^{\perp}_{1},\delta^{\perp}_{2})-C(\delta_{1},\delta^{\perp}_{2})-C(\delta^{\perp}_{1},\delta_{2})}{C(\delta_{1},\delta_{2})+C(\delta^{\perp}_{1},\delta^{\perp}_{2})+C(\delta_{1},\delta^{\perp}_{2})+C(\delta^{\perp}_{1},\delta_{2})}$
(14)

A measured value of parameter $S=-2.66$ where $0\leq|S|\leq 2$ for a local
realistic model, thus CHSH inequality is violated. The CHSH inequality is also
violated by taking coincidence photon measurements of observables with the
ICCD camera gated by the single photon detector signal.

Figure 6:  _Coincidence photon counts measurements for $|\Psi^{-}\rangle_{p}$
at different orientations of pass axes of polarisers. Solid line represents a
fit of $\sin^{2}(\delta_{1}-\delta_{2})$ corresponding to the coincidence
detection probability._

###  4.3 Spatial correlations

In the regions of the intersection of cones, the down converted photons are
diverging and are spatially correlated in the transverse planes as a
consequence of their spatial entanglement. These correlations are a result of
momentum entanglement of photons which is represented by
$\Phi_{12}(r_{a};r_{b})$ in Eq. 4. Spatial entanglement correlates locations
of photons. It is due to the spread of photons wavefunction the whole pattern
is exposed by a single photon. Once a location of a photon is determined on
the pattern its information is shared with the other photon due to their
spatial correlations thus each photon carries the spatial location information
of the other photon. Position correlations are measured in two planes oriented
perpendicular to the mean direction of propagation of each photon. A single-
slit of width 0.8 mm is placed at a distance $d_{1}=1.247$ m, immediately in
front of a convex lens $L_{o}$, where the lens is placed in front of a narrow
area single photon detector such that the detector is almost at its focal
point. The length of the slit is very large as compared to its width.
Polariser-$1$ and polariser-$2$ are removed from paths of photons and the
remaining setup is the same as shown in Fig. 2.

Figure 7:  _(a) Vertical position measurements of photons. Each vertical
position of photon- $1$ is measured by passing it through a horizontally
aligned single narrow slit placed in front of a lens $L_{o}$. A corresponding
position of photon-$2$ is measured by the ICCD camera in the plane of the
spatial light modulator. Each data circle represents mean vertical positions
of photons corresponding to a particular position of a single slit. A straight
line is a linear fit which shows that vertical positions of photons are equal
and opposite at equal distances from the BBO crystal. (b) Horizontal position
measurements of photons when narrow single slit is aligned vertically and
displaced horizontally. A straight line is a linear fit which shows that
horizontal positions of photons are equal and opposite at equal distances from
the BBO crystal._

If the single-slit is oriented horizontally and photon-$1$ is passed through
it and detected then a vertical position of photon-1 is measured with a
precision equals to the slit width because photon-$1$ is detected only if it
is transmitted by the slit. However, this measurement does not measure
horizontal position because of large length of the slit in the horizontal
direction. Similarly, a corresponding mean position of photon-$2$ is measured
on the ICCD camera, which corresponds to a measured mean position of
photon-$2$ on the spatial light modulator placed at $d_{2}=0.89$ m reduced by
0.52 due to the image demagnification of the telescope. Mean vertical
positions of photon-$1$ and photon-$2$ are measured after repeating the
experiment for different vertical displacements of the single-slit.
Correlation between the mean vertical positions of photon-$1$ and of
photon-$2$ is shown in Fig. 7(a). Similar measurements are performed when a
single-slit is oriented vertically and displaced horizontally. This setting
performs a horizontal position measurement of photon-$1$ with a resolution
equals to the slit width. Mean horizontal position of photon-$1$ and of
photon-$2$ are measured by repeating the experiment for each horizontal
displacement of the single-slit. A corresponding correlation of mean positions
of photons is shown in Fig. 7(b). The telescope demagnification is 0.52 and
distances $d_{1}$ and $d_{2}$ are different in the position correlation plots.
For $d_{1}=d_{2}$ and after taking into account the demagnification, the
measured mean position of photon-$1$ is equal and opposite to the measured
mean position of photon-$2$ in the transverse planes.

###  4.4 Classical and quantum imaging

Most of the imaging methods in everyday applications are based on classical
imaging. These methods utilise properties of classical fields such as
intensity, polarisation, phase, frequency and coherence. In the case of
optical imaging, classical fields are electromagnetic wave fields. If more
than one electromagnetic wave fields are involved then a classical image can
be constructed by correlating their classical observable properties. This type
of imaging technique, which is known as classical coincidence imaging, has
many applications in biomedical imaging and astronomy. A classical ghost
imaging is also a coincidence imaging technique, where only one
electromagnetic field interacts with the object and its intensity is measured
position-wise while the other electromagnetic wave is detected by a bucket
detector. Most of these imaging methods can image absorptive objects only. A
pure phase object is transparent and cannot be imaged by such methods.
However, a method of phase-contrast imaging, which transforms a phase
variation to intensity variation across the object is used to image a pure
phase object. A phase-contrast microscope can see transparent micro-organisms
and transparent objects. Such a method can also image a Bose-Einstein
condensate non-destructively. In addition, a transparent polarisation
sensitive phase object can be imaged with a polarisation-contrast microscope
[1, 6, 7], which measures the polarisation shift caused by the object.
However, quantum imaging methods are based on the quantum mechanical
properties of photons originating from their polarisation quantum states,
their quantum entanglement and quantum coherence. Quantum entanglement
enhanced resolution microscope can image microscopic structures beyond the
diffraction limit [49]. Quantum image construction also depends on how
measurements are performed on each photon since incompatible observables can
not be measured simultaneously. In experiments presented in this paper, two
joint measurements of compatible observables are performed on each photon
since they are hyper entangled in two different degrees of freedom. The first
measurement is polarisation measurement in a chosen polarisation basis and the
second measurement corresponds to a measurement of the momentum of the non-
interacting photon and position for the interacted photon. The individual
photon carries no complete image information and this method can be useful to
send images of the polarisation sensitive phase pattern securely and directly
over a large distance. The experiment shown in this paper is the first
demonstration of quantum imaging with hyper-entangled photons over a long
path.

### Acknowledgement

Mandip Singh acknowledges research funding by the Department of Science and
Technology, Quantum Enabled Science and Technology grant for project No. Q.101
of theme title “Quantum Information Technologies with Photonic Devices",
DST/ICPS/QuST/Theme-1/2019 (General).

### Author contributions statement

MS conceptualised the idea and setup the experiment, both authors performed
the experiment, MK took data, MS wrote the manuscript and supervised the
project.

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