# Finding the more fundamental law of classical mechanics

V. Tanrıverdi
<EMAIL_ADDRESS>
Address: Haliç University, 5. Levent Mahallesi,
15 Temmuz Şehitler Caddesi, No: 14/12
34060 Eyüpsultan/İSTANBUL TURKEY

###### Abstract

We have studied the relationship between the conservation of energy and
Newton’s second law or equations of motion. Differently from recent works, we
show that equations of motion can be obtained only in some special cases, and
we show that their results are distinct in general. This distinctiveness makes
it possible experimentally find the more fundamental law.

##  1 Introduction

The fundamental laws constitute the bases of scientific understanding. The
conservation of energy and Newton’s second law or equation of motion, which
can be obtained from Lagrange equations, are two of the fundamental laws of
classical mechanics. In general, these two laws are considered separate
fundamental laws. This can be seen as natural, but there are some recent works
considering relations between these, mostly trying to obtain equations of
motion from the conservation of energy.

The history related to these laws is a long one and some parts are related to
the ”vis viva controversy” including Newton’s second law and Leibniz’s usage
of mass times velocity square [1, 2, 3]. D’Alembert, Lagrange and Carnot were
aware that the conservation of energy can be seen as a consequence of Newton’s
second law in some special cases [4].

More recently, Vinokurov gives a formalism, the principle of detailed energy
conservation, to obtain equations of motion from a Lagrangian which is
obtained in terms of energy by using the conservation of energy in his work
[5]. Carlson writes energy in terms of generalized coordinates and momenta and
then gives a formalism to get equations of motion by using the conservation of
energy, and states that his formulation does not describe properly
electromagnetic field and relativistic cases [6]. Lindgren presents a
prescription, generalized energy method, to obtain equations of motion which
is formed by using the relation that the work done by a force is equal to
change in kinetic energy, which can be considered as the basis of the
conservation of energy, and claims that equations of motion can be obtained
except the cases including Coriolis force and gyro moments [7]. Zhou and Wang
claim that they obtain equations of motion in different cases by using the
conservation of energy [8]. Hanc and Taylor give a summary of Newtonian
physics including Lagrange equations and conservation of energy, and obtain
the equation of motion for a one-dimensional motion from the conservation of
energy while giving this summary [9]. On the other hand, Neuenschwander,
Taylor and Tujela state that variation of energy is not a suitable quantity to
obtain equations of motion [10].

In this work, we will follow a different path and study the relationship
between the conservation of energy and equations of motion without trying to
obtain the same or different results. Though some authors claim that equations
of motion can be obtained from the conservation of energy, we show by
derivation and examples that it is possible only in some cases. More
importantly, we show that it is possible to determine the more fundamental law
among the conservation of energy and Newton’s second law or equations of
motion experimentally by using their distinct results.

##  2 Theoretical considerations

One can get equations of motion from Lagrange’s equations which are given by
[11]

$\frac{d}{dt}\left(\frac{\partial\mathscr{L}}{\partial\dot{q}_{i}}\right)-\frac{\partial\mathscr{L}}{\partial
q_{i}}=0,$ (1)

where $\mathscr{L}=T-U$ is Lagrangian, $T$ is kinetic energy, $U$ is the
potential energy, $t$ is the time, $q_{i}$ and $\dot{q}_{i}$ are generalized
coordinates and velocities, respectively.

Now let us consider energy, $E(q_{i},\dot{q}_{i})=T+U$, for energy-conserved
systems. So its total time derivative is equal to zero. Then, one can write

$\sum_{i}\left[\frac{\partial
E}{\partial\dot{q}_{i}}\frac{d\dot{q}_{i}}{dt}+\frac{\partial E}{\partial
q_{i}}\frac{dq_{i}}{dt}\right]=0.$ (2)

This equation is one of the starting points of previous works that are trying
to relate the conservation of energy with equations of motion [5, 6]. It is
hard to compare this equation with Lagrange equations in these forms.
Forasmuch as we will consider scleronomic systems having velocity-independent
potential energy, $U=U(q_{i})$. For scleronomic systems with many particles,
kinetic energy can be written as
$T=\frac{1}{2}\sum_{jk}M_{jk}\dot{q}_{j}\dot{q}_{k}$ where
$M_{jk}=\sum_{i}m_{i}\frac{\partial\vec{r}_{i}}{\partial
q_{j}}\cdot\frac{\partial\vec{r}_{i}}{\partial q_{k}}$, $m_{i}$ and
$\vec{r}_{i}$ are the mass and position vector of $i$th particle, respectively
[11]. Then, Lagrange equations for $i$th particle can be found as

$\sum_{k}M_{ik}\ddot{q}_{k}+\frac{1}{2}\sum_{jk}\frac{\partial
M_{jk}}{\partial q_{i}}\dot{q}_{j}\dot{q}_{k}+\frac{\partial U}{\partial
q_{i}}=0,$ (3)

and from Eq. (2), one can obtain

$\sum_{i}\dot{q}_{i}\left[\sum_{k}M_{ik}\ddot{q}_{k}+\frac{1}{2}\sum_{jk}\frac{\partial
M_{jk}}{\partial q_{i}}\dot{q}_{j}\dot{q}_{k}+\frac{\partial U}{\partial
q_{i}}\right]=0.$ (4)

This equation, obtained from the total time derivative of energy, has a
summation and is distinct from equations of motion. Differently from equations
of motion, the summation of all terms can be equal to zero.

It is necessary to continue to study for a better understanding. From these
two equations, similar to D’Alembert, Lagrange and Carnot, one may conclude
that Lagrange equations are enough to show the conservation of energy for
considered systems since, in general, the generalized velocity $\dot{q}_{i}$
is different from zero. This is shown previously for simpler cases without
considering scleronomic systems [12, 13]. But this topic requires further
considerations. There is a question worth studying further: Does conservation
of energy is more fundamental than equations of motion or vice versa?

To find a way for answering this question, we will study different
possibilities. If the motion is one-dimensional, then the sum in Eq. (4) drops
since, in general, $\dot{q}_{i}\neq 0$ and one obtains the same equation as
Lagrange equations.

In some cases, there can be more than one dimension and conserved momenta can
be used to write the energy in a one-dimensional form. As an example, one can
consider central force motion with potential energy $U=-\frac{k}{r}$. By using
conserved angular momentum $L_{\theta}=mr^{2}\dot{\theta}$, its energy can be
written as
$E=\frac{1}{2}m\dot{r}^{2}+\frac{L_{\theta}^{2}}{2mr^{2}}-\frac{k}{r}$. Then,
by using Eq. (2) or Eq. (4) one can get

$\dot{r}\left(m\ddot{r}-\frac{L_{\theta}^{2}}{mr^{3}}+\frac{k}{r^{2}}\right)=0.$
(5)

Since $\dot{r}\neq 0$ in general, the result is the same as Lagrange equation
[11].

As another example, one can consider the heavy symmetric top whose kinetic and
potential energies can be written as
$T=\frac{I_{x}}{2}(\dot{\theta}^{2}+\dot{\phi}^{2}\sin^{2}\theta)+\frac{I_{z}}{2}(\dot{\psi}+\dot{\phi}\cos\theta)^{2}$
and $U=mgl\cos\theta$ where $m$ is the mass of the top, $\theta$, $\phi$ and
$\psi$ are Euler angles, $I_{z}$ is the moment of inertia along the symmetry
axis, $I_{x}$ is the moment of inertia along the other two perpendicular axes,
$l$ is the distance from fixed point to the center of mass and $g$ is the
gravitational acceleration. In this case, there are two conserved angular
momenta, i.e. $L_{\psi}=I_{x}a=I_{z}(\dot{\psi}+\dot{\phi}\cos\theta)$ and
$L_{\phi}=I_{x}b=I_{x}\dot{\phi}\sin^{2}\theta+I_{x}(\dot{\psi}+\dot{\phi}\cos\theta)\cos\theta$.
Then, by using these conserved angular momenta, energy can be written as
$E=\frac{1}{2}I_{x}\dot{\theta}^{2}+\frac{I_{x}(b-a\cos\theta)^{2}}{2\sin^{2}\theta}+\frac{I_{x}^{2}a^{2}}{2I_{z}}+mgl\cos\theta$
[11]. By using Eq. (2), one can obtain

$\dot{\theta}[I_{x}\ddot{\theta}+\frac{I_{x}a(b-a\cos\theta)}{\sin\theta}-\frac{I_{x}\cos\theta(b-a\cos\theta)^{2}}{\sin^{3}\theta}-mgl\sin\theta]=0$
(6)

After writing $a$ and $b$ in terms of angular velocities, one can get the same
result as Lagrange equations, i.e.
$I_{x}\ddot{\theta}=\sin\theta[I_{x}\dot{\phi}^{2}\cos\theta-
I_{z}\dot{\phi}(\dot{\psi}+\dot{\phi}\cos\theta)+mgl]$, since, in general,
$\dot{\theta}\neq 0$ [14].

Multi-dimensional cases, where the reduction is impossible, give more
instructive results. As an example, one can consider the double pendulum whose
kinetic and potential energies can be written as
$T=\frac{1}{2}(m_{1}+m_{2})l_{1}^{2}\dot{\phi}_{1}^{2}+\frac{1}{2}m_{2}l_{2}^{2}\dot{\phi}_{2}^{2}+m_{2}l_{1}l_{2}\dot{\phi}_{1}\dot{\phi}_{2}\cos(\Delta\phi)$
and $U=(m_{1}+m_{2})gl_{1}(1-\cos\phi_{1})+m_{2}gl_{2}(1-\cos\phi_{2})$,
respectively, where $m_{i}$ are masses of bobs, $l_{i}$ are lengths of
pendula, $\phi_{i}$ are deviations from the equilibrium position and
$\Delta\phi=\phi_{1}-\phi_{2}$ [14]. By using Eq. (2), one can obtain

$\displaystyle\dot{\phi}_{1}$
$\displaystyle\left[\frac{m_{1}+m_{2}}{m_{2}}\frac{l_{1}}{l_{2}}\ddot{\phi}_{1}+\ddot{\phi}_{2}\cos(\Delta\phi)+\frac{m_{1}+m_{2}}{m_{2}}\frac{g}{l_{2}}\sin\phi_{1}+\dot{\phi}_{2}^{2}\sin(\Delta\phi)\right]$
(7) $\displaystyle+$
$\displaystyle\dot{\phi}_{2}\left[\frac{l_{2}}{l_{1}}\ddot{\phi}_{2}+\ddot{\phi}_{1}\cos(\Delta\phi)+\frac{g}{l_{1}}\sin\phi_{2}-\dot{\phi}_{1}^{2}\sin(\Delta\phi)\right]=0$

From Lagrange equations, equations of motion can be obtained as [15]

$\displaystyle\frac{m_{1}+m_{2}}{m_{2}}\frac{l_{1}}{l_{2}}\ddot{\phi}_{1}+\ddot{\phi}_{2}\cos(\Delta\phi)+\frac{m_{1}+m_{2}}{m_{2}}\frac{g}{l_{2}}\sin\phi_{1}+\dot{\phi}_{2}^{2}\sin(\Delta\phi)=0$
(8)
$\displaystyle\frac{l_{2}}{l_{1}}\ddot{\phi}_{2}+\ddot{\phi}_{1}\cos(\Delta\phi)+\frac{g}{l_{1}}\sin\phi_{2}-\dot{\phi}_{1}^{2}\sin(\Delta\phi)=0.$
(9)

The result of Eq. (7) and Lagrange equations are not equal to each other.
However, it can be seen that the conservation of energy, i.e.
$\frac{dE}{dt}=0$, can be obtained from equations of motion by using the fact
that it is equal to $\dot{\phi}_{1}$ times Eq. (8) plus $\dot{\phi}_{2}$ times
Eq. (9) and Eq.s (8) and Eq. (9) are equal to zero. On the other hand, though
the conservation of energy can be obtained from equations of motion, the
equation that is obtained from the conservation of energy, i.e. Eq. (7), is
different and does not give equations of motion. There is a subtle point here
which we will mention at the conclusion.

As another example, one can consider the heavy asymmetric top whose kinetic
and potential energies can be written as
$T=\frac{I_{x}}{2}(\dot{\theta}\cos\psi+\dot{\phi}\sin\theta\sin\psi)^{2}+\frac{I_{y}}{2}(-\dot{\theta}\sin\psi+\dot{\phi}\sin\theta\cos\psi)^{2}+\frac{I_{z}}{2}(\dot{\psi}+\dot{\phi}\cos\theta)^{2}$
and $U=mgl\cos\theta$, respectively, where $I_{y}$ is the moment of inertia
around the y-axis and other definitions are the same as the heavy symmetric
top case. There is a cyclic coordinate, $\phi$, however, it is not enough to
obtain energy in a one-dimensional form. For this reason and considering
further complications, we will not use it to eliminate one degree of freedom.
By using Eq. (2), one can obtain

$\displaystyle\dot{\theta}$
$\displaystyle\left[\ddot{\theta}(I_{x}\cos^{2}\psi+I_{y}\sin^{2}\psi)+\ddot{\phi}\sin\theta\sin\psi\cos\psi(I_{x}-I_{y})\right.$
$\displaystyle\ \ \
-2\dot{\theta}\dot{\psi}\sin\psi\cos\psi(I_{x}-I_{y})+\dot{\phi}\dot{\psi}\sin\theta[(I_{x}-I_{y})(\cos^{2}\psi-\sin^{2}\psi)+I_{z}]$
$\displaystyle\ \ \
\left.-\dot{\phi}^{2}\sin\theta\cos\theta(I_{x}\sin^{2}\psi+I_{y}\cos^{2}\psi-
I_{z})-mgl\sin\theta\right]$ $\displaystyle+$
$\displaystyle\dot{\psi}\left[\ddot{\psi}I_{z}+I_{z}\ddot{\phi}\cos\theta-
I_{z}\dot{\theta}\dot{\phi}\sin\theta+(\dot{\theta}^{2}-\dot{\phi}^{2}\sin^{2}\theta)(I_{x}-I_{y})\sin\psi\cos\psi\right.$
$\displaystyle\ \ \
\left.-\dot{\theta}\dot{\phi}\sin\theta(I_{x}-I_{y})(\cos^{2}\psi-\sin^{2}\psi)\right]$
$\displaystyle+$
$\displaystyle\dot{\phi}\left[\ddot{\phi}[\sin^{2}\theta(I_{x}\sin^{2}\psi+I_{y}\cos^{2}\psi)+I_{z}\cos^{2}\theta]+\ddot{\theta}\sin\theta\sin\psi\cos\psi(I_{x}-I_{y})\right.$
$\displaystyle\ \ \
+\ddot{\psi}I_{z}\cos\theta+2\dot{\phi}\dot{\theta}\sin\theta\cos\theta(I_{x}\sin^{2}\psi+I_{y}\cos^{2}\psi-
I_{z})$ $\displaystyle\ \ \
+\dot{\theta}^{2}\cos\theta\sin\psi\cos\psi(I_{x}-I_{y})+2\dot{\phi}\dot{\psi}\sin^{2}\theta\sin\psi\cos\psi(I_{x}-I_{y})$
$\displaystyle\ \ \
+\left.\dot{\theta}\dot{\psi}\sin\theta[(I_{x}-I_{y})(\cos^{2}\psi-\sin^{2}\psi)-I_{z}]\right]=0.$

From Lagrange equations, one can find equations of motion as

$\displaystyle\ddot{\theta}$
$\displaystyle(I_{x}\cos^{2}\psi+I_{y}\sin^{2}\psi)+\ddot{\phi}\sin\theta\sin\psi\cos\psi(I_{x}-I_{y})$
$\displaystyle-$ $\displaystyle
2\dot{\theta}\dot{\psi}\sin\psi\cos\psi(I_{x}-I_{y})+\dot{\phi}\dot{\psi}\sin\theta[(I_{x}-I_{y})(\cos^{2}\psi-\sin^{2}\psi)+I_{z}]$
$\displaystyle-$
$\displaystyle\dot{\phi}^{2}\sin\theta\cos\theta(I_{x}\sin^{2}\psi+I_{y}\cos^{2}\psi-
I_{z})-mgl\sin\theta=0,$ $\displaystyle\ddot{\psi}$ $\displaystyle
I_{z}+I_{z}\ddot{\phi}\cos\theta-
I_{z}\dot{\theta}\dot{\phi}\sin\theta+(\dot{\theta}^{2}-\dot{\phi}^{2}\sin^{2}\theta)(I_{x}-I_{y})\sin\psi\cos\psi$
(12) $\displaystyle-$
$\displaystyle\dot{\theta}\dot{\phi}\sin\theta(I_{x}-I_{y})(\cos^{2}\psi-\sin^{2}\psi)=0,$
$\displaystyle\ddot{\phi}$
$\displaystyle[\sin^{2}\theta(I_{x}\sin^{2}\psi+I_{y}\cos^{2}\psi)+I_{z}\cos^{2}\theta]+\ddot{\theta}\sin\theta\sin\psi\cos\psi(I_{x}-I_{y})$
$\displaystyle+$
$\displaystyle\ddot{\psi}I_{z}\cos\theta+2\dot{\phi}\dot{\theta}\sin\theta\cos\theta(I_{x}\sin^{2}\psi+I_{y}\cos^{2}\psi-
I_{z})$ $\displaystyle+$
$\displaystyle\dot{\theta}^{2}\cos\theta\sin\psi\cos\psi(I_{x}-I_{y})+2\dot{\phi}\dot{\psi}\sin^{2}\theta\sin\psi\cos\psi(I_{x}-I_{y})$
$\displaystyle+$
$\displaystyle\dot{\theta}\dot{\psi}\sin\theta[(I_{x}-I_{y})(\cos^{2}\psi-\sin^{2}\psi)-I_{z}]=0.$

Similar to the double pendulum case, the results are distinct, and one can see
that $\frac{dE}{dt}=0$ is satisfied if one uses equations of motion in Eq.
(2). Similar to the double pundulum case, conservation of energy does not give
equations of motion, and the same subtle point exists.

##  3 Conclusion

We have studied the relation between the conservation of energy and equations
of motion for scleronomic cases with velocity-independent potentials. We have
seen that one can get the same equations from the conservation of energy and
equations of motion for one-dimensional cases and some multi-dimensional cases
where energy can be written in one-dimensional form by using conserved
momenta.

For multi-dimensional cases, whose energy cannot be written in one-dimensional
form, the resultant equations are distinct. This distinctiveness shows that
Lagrange equations and conservation of energy does not designate the same
results, in general.

This showed itself in the last two examples. These two cases are chaotic [16,
17, 18, 19], however, this does not necessarily mean that only chaotic motions
will result in distinct equations.

There are some recent studies concerning these two examples, and most of these
are mainly on the double pendulum, and only some of these consider the
conservation of energy [16, 17, 20]. However, none of these studies include
any results considering any possible difference between Eq. (7) and Eq.s (8)&
(9). Studies related to the heavy asymmetric top are fewer in number, and
there is not any work considering similar difference.

Nevertheless, the double pendulum, the heavy asymmetric top or a similar case
can answer the previously asked question: Does the conservation of energy more
fundamental than equations of motion or vice versa?

If we consider the success of the Lagrangian formalism and Newton’s second
law, we may regard it as more fundamental than the conservation of energy. On
the other hand, if we consider how wide the principle of the conservation of
energy is, we may regard it as more fundamental.

It is better to leave our deliberations aside. An experiment, for example
concerning the double pendulum, can obey equations of motion, i.e. Eq.s (8)
and (9), and conservation of energy is obtained from equations of motion. If
this happens, then the equations of motion or Newton’s second law is more
fundamental for classical mechanics since the conservation of energy is the
result of the equations of motion. Or, an experiment, again for example
concerning the double pendulum, can obey the equation obtained from the
conservation of energy, i.e. Eq. (7), while not obeying equations of motion,
i.e. Eq.s (8) and (9). If this case holds, then the conservation of energy is
more fundamental, and equations of motion or Newton’s second law is a mere
consequence of the conservation of energy for cases either energy is one-
dimensional or energy can be written in the one-dimensional form.

To find out the answer, we should ask the nature. In other words, we should
follow the basic scientific procedure, and experimentally test these results
to see which is more fundamental for classical mechanics. Probably, one result
will not be enough, but results of different systems can satisfy necessary
information for understanding. Such a procedure will increase the
understanding of not only mechanics but also other fields of physics and can
help develop better formulations.

## References

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# Quantum ghost imaging of a transparent polarisation sensitive phase pattern

Aditya Saxena  Department of Physical Sciences, Indian Institute of Science
Education and Research (IISER) Mohali, Sector-81, Mohali, 140306, India.
Manpreet Kaur  Department of Physical Sciences, Indian Institute of Science
Education and Research (IISER) Mohali, Sector-81, Mohali, 140306, India.
Vipin Devrari  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 exhibits a position and
polarisation dependent phase shift of transmitted light and it represents a
unitary transformation. A quantum ghost image of this pattern is produced with
hyper-entangled photons consisting of Einstein-Podolsky-Rosen (EPR) and
polarisation entanglement. In quantum ghost imaging, a single photon interacts
with the pattern and is detected by a stationary detector and a non-
interacting photon is imaged on a coincidence camera. EPR entanglement
manifests spatial correlations between an object plane and a ghost image
plane, whereas a polarisation dependent phase shift exhibited by the pattern
is detected with polarisation entanglement. In this quantum ghost imaging, the
which-position-polarisation information of a photon interacting with the
pattern is not present in the experiment. A quantum ghost image is constructed
by measuring correlations of the polarisation-momentum of an interacting
photon with polarisation-position of a non-interacting photon. The experiment
is performed with a coincidence single photon detection camera, where a non-
interacting photon travels a long optical path length of 17.83 $m$ from source
to camera and a pattern is positioned at an optical distance of 19.16 $m$ from
the camera.

##  1 Introduction

Quantum ghost imaging of light absorbing objects relies on Einstein-Podolsky-
Rosen (EPR) quantum entanglement of photons [1, 2, 3, 4, 5]. A single photon
after interaction with an absorptive object is detected by a stationary
detector and a non-interacting photon is detected in the image plane of an
imaging lens in a coincidence measurement with the interacting photon. EPR
entanglement of photons manifests position correlations between the object
plane and its ghost image plane. In quantum ghost imaging, a momentum state of
the interacting photon is measured in the object plane by placing a lens after
the object plane, which focus the photon on a stationary single photon
detector. Therefore, a photon detection does not reveal any information of
location on the object where a photon has interacted with it. In other words,
momentum measurement projects an interacting photon quantum state in a quantum
superposition of its position states in the object plane. In contrast, a non-
interacting photon is measured in a position basis in the image plane. Quantum
ghost imaging techniques of photon-absorptive objects has been extensively
investigated theoretically as well as experimentally [2, 6, 7, 4, 5, 8, 9, 10,
11, 12, 13, 14, 3, 15, 16]. Classical ghost imaging has been of considerable
interest with thermal light [17, 18, 19, 20]. Quantum imaging of spiral
objects has been realised with random light [21, 22]. Furthermore, ghost
imaging experiments are performed with X-rays [23, 24], electrons [25] and
correlated atoms [26, 27]. A transparent phase object is imaged with quantum
entangled photons using ghost diffraction [28, 29] and computational ghost
imaging technique [30, 31]. In a direct coincidence imaging experiment, a
polarisation sensitive meta-surface is imaged with polarisation entangled
photons [32].

In this paper, a quantum ghost imaging experiment of a transparent
polarisation sensitive phase pattern is presented, which manifests a
polarisation and position dependent phase shift of transmitted photons. To
image such type of pattern represented by a unitary transformation, quantum
entanglement in the polarisation degree of freedom is incorporated in addition
to EPR entanglement of photon pairs. Photons are EPR entangled and
polarisation entangled separately in the form of a hyper-entangled state.
Polarisation entanglement leads to detection of polarisation sensitive phase
shift exhibited by the pattern and EPR entanglement correlates position states
of an interacting photon in the object plane with position states of a non-
interacting photon in the image plane. However, experiment presented in this
paper is performed in quantum mechanical way, where the which-position-
polarisation information of interacting photon on the pattern is not available
even in principle. This experiment is a quantum ghost imaging version of a
direct coincidence imaging experiment of a transparent polarisation sensitive
phase pattern described in Ref. [33]. However, in this paper the quantum
entanglement under consideration involves EPR and polarisation degrees of
freedom in hyper-entanglement of photon pairs, where in Ref.[33] the hyper-
entangled state involves momentum and polarisation degrees of freedom. These
two experiments are fundamentally different. To produce a quantum ghost image,
EPR-polarisation hyper-entanglement becomes necessary as further described in
this paper. Therefore, experimental and theoretical analysis given in Ref.
[33] cannot be applied directly to the present experiment. Furthermore, the
experiments described in this paper and in Ref.[33] are very different from
the experiment described in Ref.[32] which involves only polarisation
entanglement of photon pairs and a direct imaging is performed in the close
vicinity of object. In contrast, experiment presented in this paper and in
Ref.[33] are realised for a large optical path length, where the object is
situated at a large distance from the imaging camera . Hyper-entanglement is
necessary to realise a ghost and direct imaging of the distant object by
jointly measuring momentum-polarisation of interacting photon and position-
polarisation of non-interacting photon.

As a consequence of EPR-polarisation hyper-entanglement the polarisation,
momentum and position quantum states of an individual photon are completely
undefined prior to a measurement. Hyper-entangled photons are produced by
type-II spontaneous parametric down conversion (SPDC) at the intersection of
emission cones of down converted photons. A nonlinear crystal is pumped by a
broad diameter laser beam to produce the required quantum entangled state of
photons. A polarisation sensitive phase pattern is produced by a spatial light
modulator (SLM). A corresponding quantum ghost image is captured on an
intensified-charge-coupled-device (ICCD) single photon sensitive camera. ICCD
camera detects a non-interacting photon only if an interacting photon is
detected with a particular measurement outcome by a stationary single photon
detector, which triggers the camera. A quantum ghost image gradually emerges
after accumulating the detections of non-interacting photons by repeating the
experiment for the same measurement outcomes. Each detection event of the non-
interacting photon on the camera corresponds to its position measurement.
However, momentum measurement outcome of interacting photon is kept same in
all measurements presented in this paper, which reveals no position-
information of the interacting photon on the pattern. Different quantum ghost
images are produced for different polarisation detection outcomes of photons
in a selected diagonal polarisation basis. By ignoring the interacting photon
measurement outcomes or by letting this photon escape without any measurement,
no ghost image is produced by single photon accumulation on camera. This is
regarded as one of the main characteristics of quantum ghost imaging method,
where an individual photon reveals no information of the pattern because of
quantum entanglement.

##  2 Concept and experiment

Figure 1:  _A schematic of a quantum ghost imaging experiment of a
polarisation sensitive phase pattern. Photon- $1$ interacts with the pattern
and detected by a single photon detector after passing through a polariser and
a convex lens. A non-interacting photon-$2$ is imaged on camera after passing
through an imaging convex lens and a polariser. Camera detects photon-$2$
leading to a joint measurement of its polarisation and position only if a
particular measurement outcome of polarisation and momentum of photon-$1$ is
produced._

Consider a schematic diagram of a ghost imaging configuration as shown in Fig.
1, where a hyper-entangled photons pair is produced by a source positioned
around origin. Photon-1 passes through a polarisation sensitive phase pattern
without absorption. After that, it passes through a polariser and a convex
lens $L_{o}$, which focus photon-$1$ on a single photon detector. If the
detector detects photon-1, then a particular joint measurement outcome
corresponding to a specific polarisation and momentum state of photon-$1$ in
an object plane containing the pattern is produced. Whereas photon-2 never
interacts with the pattern. It is detected by a single photon sensitive
coincidence camera, which measures its position in an image plane of an
imaging convex lens after a polarisation selection made by a polariser-$2$.
Coincidence camera detects photon-2 only if it receives a trigger signal from
a single photon detector _i.e._ a signal corresponding to a particular joint
measurement outcome of photon-1. An imaging lens is placed in path of
photon-$2$ such that an object plane corresponds to a plane containing the
pattern and an image plane corresponds to a plane of camera image sensor.
Since the pattern under consideration is transparent and it introduces a
polarisation dependent phase shift therefore, it cannot be imaged with
conventional ghost imaging method for absorptive objects. The current setup
shown in Fig. 1 differs from a conventional setup in three main aspects, (a)
photons are polarisation entangled and EPR entangled separately, which is a
necessary condition to produce a quantum ghost image of a transparent
polarisation sensitive phase pattern. Polarisation entanglement detects the
polarisation dependent phase shift and EPR entanglement correlates locations
of photons between the object and image planes. (b) The polarisation and
momentum state of photon-1 are measured together in the object plane. (c) The
polarisation and position states of photon-$2$ are measured together in the
image plane. A quantum ghost image is produced by correlating the joint
measurement outcomes of both photons.

In one-dimension, EPR quantum state in position basis can be written as
$|\alpha\rangle=\int^{\infty}_{-\infty}|x\rangle_{1}|x+x_{o}\rangle_{2}\mathrm{d}x$,
where subscripts $1$ and $2$ are particle labels and $x_{o}$ is position
difference between position states of particles. In momentum basis, the same
quantum state 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 particles have opposite momenta and $\hslash=h/2\pi$ is the reduced
Planck’s constant. Therefore, both position and momentum of each particle are
completely unknown. If $x_{o}=0$, then both particles have same position in
the integral. In addition, for the EPR quantum state $|\alpha\rangle$, both
particles are equally likely to be found at all spatial and momentum locations
_i.e._ spatial and momentum extensions of their position and momentum
wavefunctions are infinite. In this paper, a three dimensional time
independent scenario for photons is considered, where a finite source of
photons is positioned around origin as shown in Fig. 1. Let
$\mathbf{r}^{\prime}$ be an arbitrary point in the source where both photons
are produced together. An amplitude to find photon-$1$ at point $o_{a}$ in
region-$a$ and photon-2 at point $o_{b}$ in region-$b$ originating from a
point $\mathbf{r}^{\prime}$ and prior to any optical element can be written as
$\frac{e^{ip_{1}|\mathbf{r}_{a}-\mathbf{r}^{\prime}|/\hslash}}{|\mathbf{r}_{a}-\mathbf{r}^{\prime}|}\frac{e^{ip_{2}|\mathbf{r}_{b}-\mathbf{r}^{\prime}|/\hslash}}{|\mathbf{r}_{b}-\mathbf{r}^{\prime}|}$
[34, 35], where $p_{1}$ and $p_{2}$ are magnitudes of momentum of photon-$1$
and of photon-$2$, respectively. Consider $\mathbf{r}_{a}$ and
$\mathbf{r}_{b}$ are position vectors of points $o_{a}$ and $o_{b}$ from
origin, respectively. Therefore, $|\mathbf{r}_{a}-\mathbf{r}^{\prime}|$ and
$|\mathbf{r}_{b}-\mathbf{r}^{\prime}|$ are distances of $o_{a}$ and $o_{b}$
from $\mathbf{r}^{\prime}$, respectively. In addition, $|H\rangle_{a}$ and
$|V\rangle_{b}$ are the polarisation quantum states of photon-$1$ propagating
in region-$a$ and of photon-$2$ propagating in region-$b$, respectively.

For an extended source, a quantum state leading to a finite amplitude to find
a photon at $o_{a}$ with polarisation state $|H\rangle_{a}$ and a photon at
$o_{b}$ with polarisation state $|V\rangle_{b}$ is a linear quantum
superposition of amplitudes originating from all the points in the source,
which 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 constant and $\psi(x^{\prime},y^{\prime},z^{\prime})$ is
an amplitude of pair creation at a location
$r^{\prime}(x^{\prime},y^{\prime},z^{\prime})$. For a special case of an
infinitely extended EPR state the amplitude of pair production
$\psi(x^{\prime},y^{\prime},z^{\prime})$ is constant for any arbitrary
position vector $\mathbf{r}^{\prime}$. A symbol $\otimes$ denotes separation
of quantum states of different uncoupled degrees of freedom. Consider
magnitudes of photons momenta are equal, $p_{1}=p_{2}=p$. Therefore, due to
Bosonic nature of identical photons, a symmetric quantum state of photons is
written as
$|\Psi\rangle=\frac{1}{\sqrt{2}}(|\Psi\rangle_{12}+|\Psi\rangle_{21})$, where
$|\Psi\rangle_{21}$ is a quantum state given in Eq. 1 after the exchange of
photons. Therefore, after symmetrization [33], a total symmetric quantum state
can be written as

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

This integral represents a continuous variable entangled quantum state for a
finite extension of EPR source and
$\frac{1}{\sqrt{2}}(|H\rangle_{1}|V\rangle_{2}+|V\rangle_{1}|H\rangle_{2})$
represents a discrete variable symmetric polarisation entangled state. Total
quantum state in Eq. 2 is a hyper-entangled state, this combined quantum state
is a tensor product of quantum entangled states in different degrees of
freedom.

A pair creation amplitude is considered be a gaussian function such that
$\psi(x^{\prime},y^{\prime},z^{\prime})=ae^{-(x^{\prime 2}+y^{\prime
2})/\sigma^{2}}e^{-z^{\prime 2}/w^{2}}$, where $a$ is a constant, $\sigma$ and
$w$ are the widths of the gaussian. Consider two planes parallel to $xy$-plane
located on $z$-axis at distance $s_{1}$ (plane-$1$) and $s_{2}$ (plane-$2$)
from origin. An amplitude to find photon-$1$ on plane-$1$ and photon-$2$ on
plane-$2$ can be evaluated close to $z$-axis as follows. Consider distances of
planes are such that $\sigma^{2}p/hs_{1}\geq 1$ and $\sigma^{2}p/hs_{2}\geq
1$, where $s_{1}$ and $s_{2}$ are much greater than $\sigma$ and $w$. This
approximation is applicable to the experimental considerations presented in
this paper. However, in Ref.[33], a small source extension is considered
_i.e._ $\sigma^{2}p/hs_{1}\ll 1$ and $\sigma^{2}p/hs_{2}\ll 1$, which leads to
a momentum-polarisation hyper-entanglement of photons.

Therefore, Eq. 2 becomes

$|\Psi\rangle=\frac{aA_{o}}{s_{1}s_{2}}e^{i\frac{p(s_{1}+s_{2})}{\hslash}}\int^{\infty}_{-\infty}e^{-\frac{x^{\prime
2}}{\sigma^{2}}}e^{\frac{ip}{\hslash}\left(\frac{(x_{1}-x^{\prime})^{2}}{2s_{1}}+\frac{(x_{2}-x^{\prime})^{2}}{2s_{2}}\right)}\mathrm{d}x^{\prime}\int^{\infty}_{-\infty}e^{-\frac{y^{\prime
2}}{\sigma^{2}}}e^{\frac{ip}{\hslash}\left(\frac{(y_{1}-y^{\prime})^{2}}{2s_{1}}+\frac{(y_{2}-y^{\prime})^{2}}{2s_{2}}\right)}\mathrm{d}y^{\prime}\int^{\infty}_{-\infty}e^{-\frac{z^{\prime
2}}{w^{2}}}e^{-i\frac{2p}{\hslash}z^{\prime}}\mathrm{d}z^{\prime}\\\
\otimes\frac{1}{\sqrt{2}}(|H\rangle_{1}|V\rangle_{2}+|V\rangle_{1}|H\rangle_{2})$
(3)

After solving this integral, Eq. 3 can be succinctly written as
$|\Psi\rangle=\Phi(x_{1},y_{1};x_{2},y_{2})\otimes(|H\rangle_{1}|V\rangle_{2}+|V\rangle_{1}|H\rangle_{2})/\sqrt{2}$,
where $\Phi(x_{1},y_{1};x_{2},y_{2})$ is a two-photon position amplitude with
variables in the argument separated by a semicolon denoting a position of
photon-$1$ on plane-$1$ and of photon-$2$ on plane-$2$ such that

$\Phi(x_{1},y_{1};x_{2},y_{2})=c_{n}\frac{e^{i\frac{p}{\hslash}(s_{1}+s_{2})}e^{-i\phi}}{(4s^{2}_{1}s^{2}_{2}+(\frac{p}{\hslash})^{2}\sigma^{4}(s_{1}+s_{2})^{2})^{1/2}}\exp\left(\frac{-(\frac{p}{\hslash})^{2}(s_{1}s_{2}\sigma)^{2}\left((\frac{x_{1}}{s_{1}}+\frac{x_{2}}{s_{2}})^{2}+(\frac{y_{1}}{s_{1}}+\frac{y_{2}}{s_{2}})^{2}\right)}{4s^{2}_{1}s^{2}_{2}+(\frac{p}{\hslash})^{2}\sigma^{4}(s_{1}+s_{2})^{2}}-\left(\frac{p}{\hslash}\right)^{2}w^{2}\right)\\\
\exp\left(\frac{-\frac{ip}{2\hslash}\left((\frac{x_{1}}{s_{1}}+\frac{x_{2}}{s_{2}})^{2}+(\frac{y_{1}}{s_{1}}+\frac{y_{2}}{s_{2}})^{2}\right)(\frac{p}{\hslash})^{2}\sigma^{4}s_{1}s_{2}(s_{1}+s_{2})}{4s^{2}_{1}s^{2}_{2}+(\frac{p}{\hslash})^{2}\sigma^{4}(s_{1}+s_{2})^{2}}\right)\exp\left(\frac{ip}{2\hslash}\left(\frac{x^{2}_{1}+y^{2}_{1}}{s_{1}}+\frac{x^{2}_{2}+y^{2}_{2}}{s_{2}}\right)\right)$
(4)

provided $\tan(\phi)=-p(s_{1}+s_{2})\sigma^{2}/2\hslash s_{1}s_{2}$ with
$c_{n}$ as a constant. The resulting quantum state of photons is a product of
a continuous variable EPR entangled quantum state, where position states of
photons on two planes are quantum entangled with each other, and a discretely
entangled quantum state in the polarisation degree of freedom. In this
analysis to evaluate $\Phi(x_{1},y_{1};x_{2},y_{2})$, photons propagation
close to $z$-axis is considered. However, by considering propagation along all
directions in three-dimensional space for a large distance from the source, it
is shown in Ref.[33] that photons propagate in opposite direction along
$z$-axis. Therefore, equation of plane-$1$ is $z=-s_{1}$ where photon-$1$ is
found and of plane-$2$ is $z=s_{2}$ where photon-$2$ is found. In actual
experiment, anti-symmetric polarisation entangled state is produced as it
remains same in the diagonal polarisation basis. Photons are propagating in
the opposite direction along $z$-axis from the source, in non-overlapping
regions. This imposes a condition that $\Phi(x_{2},y_{2};x_{1},y_{1})=0$, when
photons are further exchanged [33]. Therefore, a hyper-entangled state
produced by source can be written as

$|\Psi\rangle=\Phi(x_{1},y_{1},z_{1}=-s_{1};x_{2},y_{2},z_{2}=s_{2})\otimes\frac{1}{\sqrt{2}}(|H\rangle_{1}|V\rangle_{2}-|V\rangle_{1}|H\rangle_{2})$
(5)

where $z$-position of photons is incorporated in
$\Phi(x_{1},y_{1},z_{1}=-s_{1};x_{2},y_{2},z_{2}=s_{2})$.

###  2.1 Quantum ghost interference

EPR entanglement part of Eq. 5 produces a quantum ghost interference [36, 37,
38], which can be captured on a coincidence camera if a polarisation sensitive
phase pattern is replaced with a double-slit positioned in a plane $z=-s_{1}$,
which is plane-$1$. Since a double-slit under consideration is polarisation
independent therefore, polarisation entanglement does not contribute to ghost
interference. Polariser-$1$ and polariser-$2$ can be removed to make no
polarisation measurement. A convex lens $L_{o}$ focus photon-$1$ on a single
photon detector. Detection of photon-$1$ at the detector projects quantum
state of photon-$1$ onto a momentum state in a plane $z=-s_{1}$. In other
words, this measurement projects photon-$1$ quantum state onto a quantum
superposition of position states in a plane of the double-slit. This
measurement does not reveal any which-path information at the double-slit.
Interference pattern is produced in coincidence measurements of photons, where
position of photon-$2$ is measured in a plane defined as an interference
plane, which is same as $z=s_{2}$ plane-$2$, located in path of photon-$2$. To
obtain a quantum ghost interference, an imaging lens is removed and a camera
detector surface is placed in the interference plane to detect position of
photon-$2$. In another configuration, an imaging lens is placed such that its
object plane corresponds to an interference plane and a corresponding image
plane is further imaged by a telescope onto the camera. The latter
configuration is preferable in experiments involving a long optical path.

Figure 2:  _A quantum ghost interference pattern for a double-slit of slit
separation $d=2$ mm. Double-slit is positioned such that (a) slit separation
is along $x$-axis and (b) slit separation is along $y$-axis._

Consider a double-slit of slit separation $d$ along $x$-axis at $y=0$. A
transmission function of the double-slit can be written as
$A_{T}(x_{1},y_{1})=(\delta(x_{1}-d/2)\delta(y_{1})+\delta(x_{1}+d/2)\delta(y_{1}))/\sqrt{2}$.
Therefore, an amplitude of coincidence detection of photon-$1$ by the detector
and photon-$2$ in the interference plane at a location $(x_{2},y_{2})$ can be
written as

$A_{12}(x_{2},y_{2})=\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}A_{T}(x_{1},y_{1})\Phi(x_{1},y_{1},-s_{1};x_{2},y_{2},s_{2})\mathrm{d}x_{1}\mathrm{d}y_{1}$
(6)

Here, integration represents a projection onto a quantum superposition of
position states of photon-$1$ in plane-$1$ containing a double-slit.
Therefore, the probability of coincidence detection
$P_{12}(x_{2},y_{2})=|A_{12}(x_{2},y_{2})|^{2}$ represents a quantum ghost
interference. Quantum ghost interference patterns shown in Fig. 2 are obtained
by solving Eq. 6, for a double-slit located at
$z=-s_{1}=-1.33\leavevmode\nobreak\ m$ and an interference plane located at
$z=s_{2}=1\leavevmode\nobreak\ m$. A slit separation of double-slit is
$d=2\leavevmode\nobreak\ mm$ and $\sigma=3\leavevmode\nobreak\ mm$. In Fig. 2
(a), single slits of the double-slit are positioned at
$x_{1}=1\leavevmode\nobreak\ mm$ and $x_{1}=-1\leavevmode\nobreak\ mm$ and in
(b) single slits are positioned at $y_{1}=1\leavevmode\nobreak\ mm$ and
$y_{1}=-1\leavevmode\nobreak\ mm$. A quantum ghost interference pattern
originates due to EPR-entanglement of
$\Phi(x_{1},y_{1},-s_{1};x_{2},y_{2},s_{2})$.

###  2.2 Quantum ghost imaging of a polarisation sensitive phase pattern:
Theory

To produce a quantum ghost image of a transparent polarisation sensitive phase
pattern an imaging lens of focal length $f$ is positioned such that the
pattern is located in its object plane $z=-s_{1}$ at a distance $u$ from
imaging lens, where $u>f$. Coincidence camera is positioned in an image plane
of imaging lens located at a distance $v$ from the lens. Imaging lens and
pattern are positioned on opposite locations _w.r.t._ source as shown in Fig.
1. Therefore, position of imaging lens from origin is $s_{2}=u-s_{1}$. Neither
of the photons reveal information of the pattern therefore, a quantum ghost
image is obtained by making correlated joint measurements on both photons.

Consider, a cartesian coordinate $(\xi,\eta)$-plane, coinciding with a plane
$z=s_{2}$ with origin on $z$-axis, where a thin imaging lens is positioned
such that $\xi$-axis is parallel to $x$-axis and $\eta$-axis is parallel to
$y$-axis. A polarisation sensitive phase pattern transforms the polarisation
quantum state of photon-$1$ as a unitary transformation _i.e._
$|H\rangle_{1}\mapsto e^{i\phi(x_{1},y_{1})}|H\rangle_{1}$ and
$|V\rangle_{1}\mapsto|V\rangle_{1}$ at a position $(x_{1},y_{1})$ on the
pattern. Where $\phi(x_{1},y_{1})$ is an arbitrary position dependent phase
shift introduced by the pattern. This transformation couples the polarisation
entanglement with a position of photon-$1$ on the pattern. A quantum image
state is defined as a quantum state that results in coincidence detection of
photon-$1$ at a position $(x_{1},y_{1})$ just after passing through the
pattern and photon-$2$ at a position ($x_{2},y_{2}$) in the image plane of
imaging lens but without polarisation measurement. A quantum image state can
be written as

$|\Psi\rangle_{I}=c_{I}\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}\Phi(x_{1},y_{1},-s_{1};\xi,\eta,s_{2})a_{p2}(\xi,\eta)e^{-i\frac{p}{\hslash}\frac{(\xi^{2}+\eta^{2})}{2f}}e^{i\frac{p}{\hslash}\left(\frac{(x_{2}-\xi)^{2}}{2v}+\frac{(y_{2}-\eta)^{2}}{2v}\right)}\mathrm{d}\xi\mathrm{d}\eta\\\
\frac{1}{\sqrt{2}}(e^{i\phi(x_{1},y_{1})}|H\rangle_{1}|V\rangle_{2}-|V\rangle_{1}|H\rangle_{2})$
(7)

which is succinctly written as
$|\Psi\rangle_{I}=\Phi_{I}(x_{1},y_{1};x_{2},y_{2})(e^{i\phi(x_{1},y_{1})}|H\rangle_{1}|V\rangle_{2}-|V\rangle_{1}|H\rangle_{2})/\sqrt{2}$,
where $c_{I}$ is a constant, $a_{p2}(\xi,\eta)$ is a transmission function of
an aperture placed close to the imaging lens,
$e^{-ip(\xi^{2}+\eta^{2})/2f\hslash}$ is the phase shift term introduced by
the imaging convex lens. After transmission through the lens, photon-$2$
propagates to the imaging plane where propagation from a position ($\xi,\eta$)
to a position ($x_{2},y_{2}$) is given by the term
$e^{ip((x_{2}-\xi)^{2}+(y_{2}-\eta)^{2})/2v\hslash}$. For a circular lens of
radius $\rho$ without any additional aperture, the transmission aperture
function $a_{p2}(\xi,\eta)=1$ for $\xi^{2}+\eta^{2}\leq\rho^{2}$ and zero
otherwise.

A quantum ghost image is obtained by correlating the joint measurements of
polarisation and momentum of photon-$1$ in the object plane with measurements
of polarisation and position of photon-$2$ in the image plane. Consider a
diagonal polarisation basis for photon-$j$ ($j=1$ or $2$) such that
$|d^{+}\rangle_{j}=(|H\rangle_{j}+|V\rangle_{j})/\sqrt{2}$ and
$|d^{-}\rangle_{j}=(|H\rangle_{j}-|V\rangle_{j})/\sqrt{2}$, where
$|d^{+}\rangle_{j}$, $|d^{-}\rangle_{j}$ represent linear polarisation states
of photon-$j$ with planes of polarisation at $\delta_{j}=+45^{o}$ and
$\delta_{j}=-45^{o}$ _w.r.t._ $x$-axis, respectively. If pass axes of
polariser-$1$ and polariser-$2$ are oriented at
$\delta_{1}=\delta_{2}=-45^{o}$ then photon-$1$ is measured in a polarisation
state $|d^{-}\rangle_{1}$ and photon-$2$ is measured in a polarisation state
$|d^{-}\rangle_{2}$ after their detection. Therefore, from Eq. 7 the
probability of coincidence detection corresponding to a polarisation
measurement outcome $(|d^{-}\rangle_{1},|d^{-}\rangle_{2})$ can be calculated
as

$P_{d^{-}_{1},d^{-}_{2}}(x_{2},y_{2})=|\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}{{}_{2}\langle
d^{-}|}{{}_{1}\langle
d^{-}|}\Psi\rangle_{I}a_{p1}(x_{1},y_{1})\mathrm{d}x_{1}\mathrm{d}y_{1}|^{2}$
(8)

which can be written as

$P_{d^{-}_{1},d^{-}_{2}}(x_{2},y_{2})=|\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}a_{p1}(x_{1},y_{1})\Phi_{I}(x_{1},y_{1};x_{2},y_{2})\left(\frac{1-e^{i\phi(x_{1},y_{1})}}{2\sqrt{2}}\right)\mathrm{d}x_{1}\mathrm{d}y_{1}|^{2}$
(9)

where $a_{p1}(x_{1},y_{1})$ is the aperture transmission function of an
aperture placed just after the pattern being imaged.

Similarly, if polarisers are aligned such that $\delta_{1}=+45^{o}$ and
$\delta_{2}=-45^{o}$ for $(|d^{+}\rangle_{1},|d^{-}\rangle_{2})$ measurement,
then the probability of coincidence detection of photons can be written as

$P_{d^{+}_{1},d^{-}_{2}}(x_{2},y_{2})=|\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}a_{p1}(x_{1},y_{1})\Phi_{I}(x_{1},y_{1};x_{2},y_{2})\left(\frac{1+e^{i\phi(x_{1},y_{1})}}{2\sqrt{2}}\right)\mathrm{d}x_{1}\mathrm{d}y_{1}|^{2}$
(10)

The coincidence detection probability represents a quantum ghost image of a
polarisation sensitive phase pattern. Quantum ghost images
$P_{d^{-}_{1},d^{-}_{2}}(x_{2},y_{2})$ and
$P_{d^{+}_{1},d^{-}_{2}}(x_{2},y_{2})$ are inverted _w.r.t_ each other.

###  2.3 Experiment

Figure 3:  _A schematic of experiment of quantum ghost imaging of a
polarisation sensitive phase pattern. This is an experimental configuration of
a schematic diagram shown in Fig. 1. The diagram is not to the scale and
photon-$1$ is incident on SLM close to a normal incidence._ Figure 4:  _(a) A
polarisation sensitive phase pattern. (b) A quantum ghost image of the pattern
for $\delta_{1}=-45^{o}$ and $\delta_{2}=-45^{o}$. (b) An inverted quantum
ghost image of the pattern for $\delta_{1}=+45^{o}$ and $\delta_{2}=-45^{o}$.
Distance of pattern from camera is 19.16 m._

A polarisation sensitive phase pattern under consideration is shown in Fig. 4
(a), which is produced by SLM. A lighter region on the pattern introduces a
phase shift $\phi(x_{1},y_{1})=\pi$ for horizontally polarised photon-$1$ and
a darker region corresponds to $\phi(x_{1},y_{1})=0$. After a transformation
by the pattern _i.e._ $|H\rangle_{1}\mapsto
e^{i\phi(x_{1},y_{1})}|H\rangle_{1}$ and $|V\rangle_{1}\mapsto|V\rangle_{1}$,
a polarisation entangled state of photons becomes
$|\Psi^{-}\rangle_{p}=(|H\rangle_{1}|V\rangle_{2}-|V\rangle_{1}|H\rangle_{2})/\sqrt{2}$
for $\phi(x_{1},y_{1})=0$ and
$|\Psi^{+}\rangle_{p}=-(|H\rangle_{1}|V\rangle_{2}+|V\rangle_{1}|H\rangle_{2})/\sqrt{2}$
for $\phi(x_{1},y_{1})=\pi$. A schematic of experiment performed is shown in
Fig. 3. Hyper-entangled photons are produced by type-II SPDC process in a
beta-barium-borate (BBO) nonlinear anisotropic crystal. A broad beam diameter
pump laser light of wavelength $405\leavevmode\nobreak\ nm$ is incident on the
crystal, which is aligned to produce non-collinear degenerate
$810\leavevmode\nobreak\ nm$ photons. Photons are hyper-entangled at the
intersection of conical emission profiles of down converted photons.
Intersection regions expand with distance from BBO crystal and width of each
region is about $7\leavevmode\nobreak\ mm$ in a transverse plane at a distance
$1.33\leavevmode\nobreak\ m$ from the crystal. EPR entanglement and
polarisation entanglement are produced at the intersection regions. A
compensation crystal is placed after the first BBO crystal to compensate for
longitudinal and transverse walk-offs. Compensation crystal is aligned
parallel to the first crystal such that optic axes of both crystals are
parallel to each other and both photons pass through the compensation crystal.
A half wave-plate is inserted between the two crystals to interchange
orthogonal linear polarisation states. Photon-$1$ of an entangled pair is
incident on SLM close to a normal incidence, which produces a polarisation
sensitive phase pattern. Difference between angle of reflection and angle of
incidence on SLM is about $8^{o}$. After passing through a convex lens
$L_{o}$, polariser-$1$ and an optical bandpass filter of center wavelength
$810\leavevmode\nobreak\ nm$, photon-$1$ is detected by a single photon
detector. Photon-$2$ never interacts with the pattern and it passes through an
imaging convex lens. Imaging lens is positioned such that it produces an image
of SLM surface onto the image plane as shown in Fig. 3. This image plane acts
as an object for a telescope, which produces its image on ICCD camera.
Photon-$2$ travels a longer optical distance as compared to photon-$1$ in
order to provide time to ICCD camera to open its direct gate electronic
shutter after receiving a trigger signal from single photon detector. ICCD
camera direct gate shutter opening delay time is $20\leavevmode\nobreak\ ns$
after receiving a trigger signal. Optical distance of ICCD camera from BBO
crystal is $17.83\leavevmode\nobreak\ m$ and an optical distance of SLM
surface from ICCD camera is $19.16\leavevmode\nobreak\ m$. Optical distance of
SLM surface from BBO crystal is $1.33\leavevmode\nobreak\ m$. Focal length of
imaging lens is $1.5\leavevmode\nobreak\ m$, this lens images SLM surface onto
the image plane as shown in Fig. 3 however, imaging lens distance from BBO
crystal is same as its focal length to collimate non-interacting photons on
the telescope, otherwise photon coincidence counts reduce considerably.

Telescope consists of three convex lens system, where lens $L_{1}$ of focal
length $40\leavevmode\nobreak\ cm$ and diameter $5\leavevmode\nobreak\ cm$
produces a real image of an image plane of the imaging lens. Lens $L_{2}$ of
focal length $10\leavevmode\nobreak\ cm$ produces a virtual image of the real
image produced by lens $L_{1}$. Lens $L_{3}$ of focal length
$f_{3}=1\leavevmode\nobreak\ m$ is configured in a $2f_{3}$ imaging
configuration such that ICCD camera detector surface is placed at a distance
$2\leavevmode\nobreak\ m$ from lens $L_{3}$ to image the virtual image
produced by lens $L_{2}$. Since a virtual image is displaced by displacing
lens $L_{2}$ therefore, a sharp image is produced on ICCD camera sensor
surface if virtual image is located at a distance $2f_{3}$ from lens $L_{3}$.
In this configuration, only lens $L_{2}$ is displaced to image any plane under
consideration. Polariser-$2$ is placed close to ICCD camera after the
telescope. Polariser-$1$ is placed at a distance $22\leavevmode\nobreak\ cm$
from SLM as shown in Fig. 3.

Polarisation entanglement part of the hyper-entangled state is characterised
by violating Clauser, Horne, Shimony, Holt (CHSH) inequality _i.e._ $0<|S|<2$
[39, 40, 33]. Experimentally measured value of parameter is $S=-2.57$. A
polarisation sensitive phase pattern is shown in Fig. 4 (a). A quantum ghost
image of the pattern is constructed by measuring coincidence photon counts
between single photon detector and different locations on ICCD camera. Prior
to any quantum ghost image measurement, a coincidence background ghost image
is captured on ICCD camera by imprinting a uniform polarisation sensitive
phase pattern on SLM that leads to minimum coincidence photon counts for the
same inclination of polarisers that is $\delta_{1}=\delta_{2}=-45^{o}$. This
background image is subtracted from all subsequent images to obtain a quantum
ghost image. A polarisation sensitive phase pattern is shown in Fig. 4(a),
where a lighter region transforms the polarisation entangled state to
$|\Psi^{+}\rangle_{p}$ and a darker region represents an identity
transformation resulting same polarisation entangled state
$|\Psi^{-}\rangle_{p}$ as produced by the source. A quantum ghost image of
this polarisation sensitive phase pattern is shown in Fig. 4 (b) for same
inclination of pass axes of polarisers _i.e._ $\delta_{1}=\delta_{2}=-45^{o}$.
The quantum ghost image of the same pattern gets inverted as shown in Fig. 4
(c), when pass axis of polariser-$1$ is inclined at $\delta_{1}=+45^{o}$ and
pass axis of polariser-$2$ remains at same inclination, $\delta_{2}=-45^{o}$.
A coincidence background image and a coincidence image with the pattern under
consideration are taken for $30\leavevmode\nobreak\ min$ time of exposure of
ICCD camera under the direct gate operation in accumulation mode. In
accumulation mode, frames captured during the exposure time are accumulated.
However, a frame can register photon-2 only if direct gate shutter is opened
by the trigger signal during its acquisition. Negative coincidence counts
regions in Fig. 4 (b) and (c) are due to experimental fluctuations because
each quantum ghost image is constructed by subtracting a background image from
an image with the pattern. The quantum ghost image shown in Fig. 4(c) is an
inverted image _w.r.t._ the quantum ghost image shown in Fig. 4(b).
Demagnification of each image is 0.87. Single photon-$2$ accumulation on ICCD
camera without making any correlation with photon-$1$ measurements results in
no image formation.

##  3 Conclusion

This paper presented an experiment on quantum ghost imaging of a transparent
polarisation sensitive phase pattern with EPR-polarisation hyper-entangled
photons. Since the pattern is polarisation sensitive therefore, polarisation
entanglement is incorporated to detect a polarisation sensitive phase
exhibited by the pattern. A quantum ghost image is a combined result of EPR
entanglement and polarisation entanglement of photons. Paper presented a
detailed analysis of hyper-entanglement of photons, quantum ghost interference
and quantum ghost imaging. The required hyper-entangled state is produced with
type-II SPDC process in a BBO crystal. The experiment is performed with a
gated ICCD single photon sensitive camera to capture coincidence ghost images.
Each quantum ghost image is background corrected. Two joint measurements on
each photon, which correspond to polarisation, momentum of photon-$1$ and
polarisation, position of photon-$2$, are correlated to construct a quantum
ghost image.

##  4 Methods

Hyper-entangled photon pairs are produced by Type-II degenerate SPDC process
in a BBO crystal at 810 $nm$. This source configuration is different from the
configuration used in a previously published quantum imaging experiment based
on hyper-entangled photon pairs, where a hyper-entangled state consists of
momentum entanglement and polarisation entanglement, which is produced at a
very large distance from the source as described in Ref.[33]. In the present
paper, hyper-entangled state consists of EPR-entanglement and polarisation
entanglement, which is produced at the intersection of down converted cones.
However, in the present source the beam size of the pump laser beam is
approximately expanded by twenty times as compared to the source described in
Ref.[33]. In this configuration, the integral given by Eq. 2 cannot be
approximated as given in the previous paper Ref.[33]. EPR entanglement
produces quantum ghost interference however, momentum entanglement cannot
produce quantum ghost interference. EPR-entanglement and polarisation
entanglement produce a quantum ghost image of the polarisation sensitive phase
pattern. Typical pump laser power passing through the BBO crystal is 60 $mW$
at 405 $nm$. A walk-off compensation crystal of half thickness is placed
parallel to the first crystal and a half wave plate is placed between the two
crystals to interchange orthogonal linear polarisation states. Diameter of
each intersection region of cones expands with distance from the crystal. The
required EPR-polarisation hyper-entangled state is produced at the
intersection regions. No image is observed for non-interacting photon
accumulation on ICCD camera without correlating it with the interacting
photon. In a coincidence measurement, the direct gate electronic shutter of
ICCD camera is opened by a trigger pulse generated by the single photon
detector after detection of an interacting photon. Direct gate shutter opening
allows a detection of photon by the ICCD camera. Direct gate shutter remains
open for the duration of trigger pulse which is $10\leavevmode\nobreak\ nsec$.
Single photon detector produces $2\times 10^{4}$ trigger pulses per second.
Therefore, the direct gate shutter opens $3.6\times 10^{7}$ times in 30 min
while camera is accumulating frames continuously. A narrow time window and low
photon pair generation rate allow a selective detection of an entangled pair
of photons _i.e._ it significantly reduces the overlapping of detection events
of a selected pair of photons with another pairs. Ghost images shown in Fig. 4
correspond to accumulation of photon counts in the coincidence window for
$30\leavevmode\nobreak\ min$ time of frame-accumulation on the ICCD camera. A
separate background ghost image of a uniform pattern is taken under same
conditions. Background image is subtracted from each image of a pattern under
consideration to obtain its background corrected ghost image shown in Fig. 4.
Image accumulation and background substraction are performed by the ICCD
camera.

##  5 Acknowledgments

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).

##  6 Author contributions

MS setup the experiment. MK, AS and MS performed the experiment. MK, AS and VD
took data. MS developed concept and theory. MS drawn diagrams and wrote the
manuscript.

##  7 Competing interests

The authors declare no competing interest.

##  8 Data availability statement

All data generated or analysed during this study are included in this
published article.

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