Ghost imaging with engineered quantum states by Hong-Ou-Mandel interference

Traditional ghost imaging experiments exploit position correlations between correlated states of light. These correlations occur directly in spontaneous parametric down-conversion (SPDC), and in such a scenario, the two-photon state used for ghost imaging is symmetric. Here we perform ghost imaging using an anti-symmetric state, engineering the two-photon state symmetry by means of Hong-Ou-Mandel interference. We use both symmetric and anti-symmetric states and show that the ghost imaging setup configuration results in object-image rotations depending on the state selected. Further, the object and imaging arms employ spatial light modulators for the all-digital control of the projections, being able to dynamically change the measuring technique and the spatial properties of the states under study. Finally, we provide a detailed theory that explains the reported observations.


Introduction
Ghost imaging as first observed by Pittman et. al [1], utilized entangled photon pairs to study EPR correlations in position and momentum. One photon of the pair interacts with an arbitrary object and is collected with a bucket detector with no spatial resolution. The other photon, in the imaging arm, does not interact with the object but rather is sent directly to a spatially-resolved device for detection, usually a 2D scanning detection system or a camera. Despite neither photon having information of the object, an image can be reconstructed when measuring in coincidences due to the spatial correlations attained prior interaction with the object, i.e., within the nonlinear crystal.
Although the first ghost imaging tests made use of entangled photon pairs, it is sufficient to rely on spatial correlations resulting, for example, from the spontaneous parametric down-conversion (SPDC) process [2]. Classical intensity correlations between thermal light have been also used to demonstrate ghost imaging [3,4,5], showing the analogy between the two scenarios [6,7]. Subsequently, ghost imaging has been studied from a computational perspective (a technique which only requires bucket detectors) [8,9] and using compressive sensing to reduce the number of required measurements [10]. Ghost imaging has also been observed in various degrees of freedom (DoF), such as the orbital angular momentum of light [11], correlations in the time domain [12], in momentumposition [13] and spectral [14] DoFs. 3D ghost images have been reconstructed using single-pixel detectors [15], and ghost imaging has even been studied in the presence of turbulence [16]. See Ref. [17] for a comprehensive review. Recently, the concept of ghost imaging was extended to entanglement swapped photons, demonstrating ghost imaging with initially independent photons [18]. In this case the role of state symmetry was crucial to the outcome of the object/image contrast.
Here we demonstrate a new form of ghost imaging where the object and image arms are placed after a Hong-Ou-Mandel (HOM) interference filter, allowing the biphoton imaging to be carried out using either symmetric or antisymmetric states. Furthermore, we employ spatial light modulators (SLMs) to dynamically control both the object and image, in particular, using digitallycontrolled holograms on the image arm SLM to reconstruct the object without a mechanical scanning system or a spatially-resolved camera. We show that our ghost imaging setup comprising the HOM filter results in the reconstruction of an image comprised of a 'double object', with each reconstructed object rotated in opposite directions. This is explained by the action of the symmetry selection step comprising a beamsplitter and Dove prisms.

Experiment
We start describing the experimental setup to easily identify the task of each optical element involved latter on in the Theory section. The experiment is divided conceptually into three sections. In the first, an entangled biphoton state is produced using a SPDC photon pair source, resulting in a state that is always symmetric. In the second, we pass the photon pair through a quantum state engineering system comprising Dove prisms (to control state phases) and a HOM interference filter to select out specific states based on their symmetry. Finally, in the third part we perform ghost imaging using the engineered two photon state, consisting in the object and mask projections and photon pair detection. We employ tools common in computational ghost imaging, namely digital projections for the image reconstruction, thus removing the need for cameras or scanning systems. This allows for an all-digital ghost imaging experiment.
The experimental setup is shown in Fig. 1 with the five parts comprising the creation (green), state engineering (pink), object (orange), mask (purple) and detection (red) steps. A mode-locked laser operating in the picosecond regime, centred at a wavelength of 355 nm with an average power of 350 mW, was used to pump a 3-mm-long β-barium borate (BBO) nonlinear crystal. Spatially-entangled photon pairs centered both at a wavelength of 710 nm (labelled as A and B) were generated in the nonlinear crystal by means of a spontaneous parametric down-conversion (SPDC) type-I process. A small difference in the angle of emission existed between photons A and B (of around 3 • ), making it easier to separate them with a D-shaped mirror. The photon in path A traversed two Dove prisms (DP 1 and DP 2 ). One of the Dove prisms was fixed to a rotation mount which is rotated by an angle θ about the optical axis to introduce a specific phase between spatial modes; the photon in path B was path length adjusted in order to achieve HOM interference.
The photons in A and B were then passed through a 50:50 beamsplitter (BS), the core element in the HOM filter. Only anti-symmetric input states result in a single photon in each arm, and so can be considered engineered when conditioned on coincidences. The symmetric states were tested by either removing the BS or working outside the HOM dip. Next, the photons were directed to the ghost imaging section of the setup: the object arm and image arm, the control of which were achieved using a single SLM encoded with amplitude and phase holograms (one half of the screen for the object and the other for the image). The plane of the crystal was relayed onto the SLM via paths A and B with a 5× magnification system (a 4f -system with f1 = 100 mm and f2 = 500 mm, not shown in Fig. 1), obtaining a ∼5 mm SPDC beam diameter at the SLM. The SLM plane was then relayed again with a 375× de-magnification system (a 4fsystem with f3 = 750 mm and f4 = 2 mm, also not shown) onto few-mode fibres (FMFs). FMFs (with ∼ 13 µm core diameters) were used in order to increase the collection area as opposed to using single-mode fibres (with ∼ 5 µm core diameters) and to reduce the noise which was observed when using multi-mode fibres (with ∼ 62.5 µm core diameters). In combination with few-mode fibres, the SLM allowed for joint projective measurements of particular spatial modes to be made. Interference band-pass filters (BPF) with bandwidths of 10 nm were used prior to the FMFs, which were in turn connected to avalanche photodiodes to detect the single photons, with coincidences registered via a coincidence counter.
To perform the ghost imaging measurements, the binary object, O, that we wished to reconstruct was encoded on the half of the SLM screen in path A, and the scan was performed by dynamically modifying the hologram encoded on the other SLM screen half in path B. The different procedures used to reconstruct the image, single pixel and random mask scans, are introduced in the Results section and explained in detail in the Supplementary Information.

Theory
Spatially-entangled photon pairs are generated in the nonlinear crystal (BBO). After propagating along the optical elements comprising the symmetry filter, the photons of each pair, A and B, are sent to the SLM screen, one half of which, in path A, is masked with a binary object O of our choosing, and the other half in path B is used to perform measurements. Based on said measurements on photon B, O can be reconstructed when detected in coincidence with photon A.
To study the effect of state symmetry on the reconstructed object, we first study the setup using the orbital angular momentum (OAM) basis of the photons [19]. Any set of spatial modes which form a basis can be used to express a mode of light with an arbitrary spatial profile, e.g. the Laguerre-Gaussian (LG), or Hermite-Gaussian (HG) modes. It is also evident that any arbitrary state can be written as the sum of a symmetric part and an anti-symmetric part. The effect that a state's symmetry has on, for example, coincidence events in an entanglement experiment has recently been studied [20], where it was shown how to control the spatial state symmetry by exploiting an HOM interferometric measurement [21], also known as an HOM filter. Such techniques work regardless of the spatial basis [22]. The HOM filter passes only anti-symmetric states when conditioned on coincidences and the symmetry of the input state is tuned by adjusting the relative phases using two Dove prisms rotated by an angle of θ relative to one another.
To begin, consider the state generated by SPDC at the crystal plane in the OAM basis and a the appropriate amplitude. The presence of the Dove prisms at a relative angle θ in path A has the effect When the relative angle is set to θ = π 4 , the only Ψ + ( Ψ − ) terms that survive are those with even (odd). With this state passed through the HOM filter, only the anti-symmetric modes (i.e. those with odd values) remain when conditioned on coincidences after the filter [20]. All symmetric states are removed, since they result in no coincidences.
One might ask whether such symmetry filtering holds when any DOF other than OAM is considered. Symmetry is an intrinsic property of a quantum state: a state which is (anti-)symmetric in one basis is (anti-)symmetric in all bases (see Supplementary Information). Hence, we can express a state in any basis we choose without affecting the symmetry. Thus, from here onward we elect to consider states expressed in the position basis, so Eq. (1) can be re-expressed as where the sum runs over all SLM pixels, a set we call S. We consider this discrete case since the SLM itself consists of discrete pixels. Here c(r) is the probability amplitude for photons A and B to be found in the crystal plane at the transverse position r = (x, y); they have the same position since they originate at the same point in the crystal. Photon A passes through two Dove prisms (which are initially set to have a relative angle of θ = 0), as depicted in Fig. 1. Later, when one of the Dove prisms in path A is rotated at an angle θ with respect to the other, R(2θ) will represent a rotation of the transverse position of photon A (for a setup without the Dove prisms, or with θ = 0, we have R(2θ) = I). The explicit θ dependence of R is suppressed for brevity. Note also that we assume paths A and B have the same path length unless stated otherwise. Therefore at the BS plane Eq. (3) becomes In the absence of a BS and hence an HOM filter, the SLM is placed at the crystal plane and so our 'no beamsplitter' state, |Ψ nbs , at the SLM plane is which shows a rotation of the transverse position of photons in path A. In such a case, it is predicted that the outcome will match that of a conventional ghost imaging experiment, save for the measured image being rotated by an angle of 2θ relative to the object. This is a corollary of the main study.

Ghost imaging with an HOM filter
In the presence of a 50:50 BS for HOM interference, and accounting for the number of mirror reflections in each path, the action of the filter is so that our 'beamsplitter' state, |Ψ bs , is We post-select on coincidences, allowing us to drop the latter two terms in Eq. (7), so with K the normalisation constant.
A comparison of all the imaging scenarios will be easier if all R dependence is moved to photon B. With this in mind, we exploit the mathematical fact that, given a bijective mapping σ : S 1 → S 2 from a finite set S 1 to a finite set S 2 with the same cardinality as S 1 , the summation of a function of mapped elements of S 1 , i.e. m∈S 1 f (σ(m)), is equal to summation of the same function of elements in the mapped set S 2 , n∈S 2 f (n). That is to say, since σ(S 1 ) = S 2 , we have m∈S 1 f (σ(m)) = m∈σ(S 1 ) f (m). This idea can be extended to the following identity for arbitrary functions f and g, where σ −1 is the inverse bijection If we assume that the pixels of the SLM screen are small enough that every pixel in the transverse plane after the rotation R can be associated, or 'matched', with a unique pixel in the original, un-rotated plane, then R is a bijection. In fact, R is a special bijection, a permutation, since the domain and range of R are the same set. Hence there exists an inverse permutation (rotation), R −1 , representing a rotation of the transverse plane by the same magnitude, but opposite direction, to R. Therefore, with R −1 as the permutation and applying Eq. (9) to the first term in Eq. (8), we obtain with S the set of discrete pixel positions. However, R −1 is a permutation, or rearranging, of the elements of S. Since we are summing over all pixel positions, we can replace R −1 (S) with S above, so Eq. (8) can be written as We therefore predict that ghost imaging with an HOM filter setup will produce a result consisting of a juxtaposition of the original object O rotated by an angle 2θ, and O rotated by −2θ.

Beamsplitter without an HOM filter
In order to bring into effect HOM filtering, it is experimentally necessary to make use of a BS and perfectly match the lengths of paths A and B. Photons A and B then have identical time stamps and are indistinguishable. All of this gives rise to the well-known 'HOM dip'.
However, we wish to study the effect of turning off the HOM filtering, but leaving the BS in place. This is achieved by slightly increasing the length of path B by way of the translation stage (the delay in Fig. 1) so that the difference in path length is larger than the coherence length of the SPDC detected photons. Photon B is ergo slightly delayed with respect to photon A and the photons are distinguishable. We indicate the presence of this time delay of photon B by means of a prime symbol, |r B → |r B . Effecting this change in photon B in Eq. (4) while applying the BS transformations in Eq. (6), and thereafter post-selecting on coincidences, gives Be that as it may, since the object masking SLM A is static and the time taken for each step of the measurement protocol carried out using SLM B is orders of magnitude larger than the time taken for photon B to travel the extra distance of the mismatched path B, experimentally, the time delay of photon B cannot be observed. Therefore, results obtained for the mismatched path length case (i.e. with a non-zero θ and BS present, but no HOM filtering) appear identical to the HOM filtering case, so |Ψ bs ≡ |Ψ bs .

Object reconstruction
Given either engineered state |Ψ nbs or |Ψ bs , the detection section of the experiment is carried out by masking SLM A with a binary object O, the information of which is contained in the function O(r): O(r) = 0 if the pixel at position r in SLM A is black in the object, and 1 if pixel r is white. Here, black means the SPDC photons are blocked (or deviated from the optical axis to be more precise) and white means the reflected photons are properly detected. The operator describing this masking process is |O A = N r O(r) |r A , with N the appropriate normalization. After masking SLM A with O and absorbing K into N , the state of photon B, in the absence of the beamsplitter, is In the case of HOM filtering, as well as the case of a non-zero θ-BS combination but mismatched path lengths, the state is If we set the weighting coefficients c to unity, we can visualize the outcome more clearly where the operator R = R(2θ) is the rotation in the transverse plane. Both of these formulae match the earlier predictions, namely: a single image rotated relative to the object in the case of Eq. (15), and a juxtaposed 'double' image with opposite rotations in the case of Eq. (16). The intensity of pixel |r B in the reconstructed object in each case is respectively

Results and discussion
First we confirm the SPDC spiral bandwidth and the HOM filtering (the first two sections of the experiment in Fig. 1), with the results given in Fig. 2. Here, the OAM spiral bandwidth of the SPDC photons is experimentally measured within the range = [ −15, 15], with the data in Fig. 2 (a) taken without a BS, and that of Fig. 2 (b) taken after introducing a BS and setting θ = π 4 , forming an HOM filter.
In what is to follow, we analyze the most important experimental results as predicted in the theory section. We first give the reconstructed object obtained in a standard ghost imaging setup, but instead use the SLM to dynamically encode the masks needed for each measurement. Next we show the effect of rotating one of the Dove prisms with respect to the other, and finally we implement the HOM filter before performing ghost imaging.

Rotated ghost imaging reconstruction
First, an experiment was run with the setup as depicted in Fig. 1, but without the HOM filter (the BS was removed). The SLM in path A was masked with a  Fig. 3 (a) and (b) while performing a digital raster scan using the SLM in path B (with a 48×48 resolution 'on pixel'). The results are shown in Fig. 3 (c) and (d) with a Dove prism angle of θ = 0 and in Fig. 3 (e) and (f) when θ = π 4 . The ghost images were reconstructed using the set of coincidence counts {c i } for every raster position in SLM B as

960×960 resolution object O as shown in
where c 1 is the coincidence count recorded for raster position P 1 , and n is a normalization constant (see Supplementary Information). The results confirm the accuracy of the digital scan approach. A different measurement scheme, a random mask scan [23] based on the compressed sensing concept [24], was also tested in the object's reconstruction in order to overcome the noise for low signal objects without needing to decrease the resolution [25], as shown in the examples of Fig. 3 (g) and (h). As before, SLM A is masked with a static 960×960 binary object O. However, instead of scanning over every pixel in SLM B individually and recording the corresponding coincidence count, the random mask scheme involves first generating a set of N random binary masks, with 50% of the pixels white and 50% of the pixels black, randomly so, for each mask. Then, SLM B is encoded with a random binary mask and the corresponding coincidence counts recorded. This process is repeated for every random mask. Finally, with the set of random binary masks {M i } and their corresponding coincidence counts {c i }, for a large enough N , the object is reconstructed by again taking a convex combination of images, with the images in this scheme the random masks themselves, i.e.
To test this measurement technique in a ghost imaging setup, the experiment was run with the objects given in Figs. 4 (a-d), using N = 4000 different random masks, recording the coincidences with an integration time of 1 second per mask, and setting the relative Dove prism angle to θ = π 4 for the results in Figs. 4 (e-h), and θ = − π 8 for those in Figs. 4 (m-p). From these results, as predicted, the reconstructed image is rotated by an angle of 2θ with respect to the original object. This confirms the effect of Dove prisms on ghost imaging and lends credence to the idea of performing such calculations in the chosen position basis.

Double ghost images
Next, to implement an HOM filter and investigate its effect on the reconstructed image, the relative Dove prism angle was set to a non-zero value and a beamsplitter (BS) inserted into the setup, which selects the state |Ψ bs . As per Eq. (16), the amplitude (related to the intensity) of pixel r in the reconstructed image is a combination of the intensity of pixel r in O rotated by R(2θ), and by R −1 (2θ) = R(−2θ). As stated, the reconstructed image will hence be a juxtaposition of O rotated by 2θ and O rotated by −2θ. This is confirmed experimentally in Figs. 4 (il) for a relative Dove prism angle of θ = π 4 , and in Figs. 4 (q-t) for θ = − π 8 . The experimental results in each row are for the object encoded in path A, in the first column. Note that the results in the last row of Fig. 4 are identical, with or without the beamsplitter and θ = π 4 , and match the intensity profile of the object, save for the rotation. In other words, we do not see the 'double' image in the reconstructed images. This is a result of the original object being invariant under a rotation by π. This image invariance under rotations could play a role in future applications where the study of the innate geometric symmetry of an object is important; or it may also find application in the field of quantum communication, wherein one could ascertain the centre of an SPDC beam source and align a system accordingly by using the counter-rotated reconstructed object. Note that the experimental results slightly differ from their simulations shown in the insets, due to the difference in reflection/transmission ratios depending on the input/output ports from the beamsplitter. What we estimated to be the reason for the dimmer anti-clockwise with respect to the clockwise rotation reconstructed image. In the other hand, the deliberate displacement of the object mask from the SPDC beam's center of coordinates, to properly identify the double rotation effect, might have added extra space between the images. Finally, Fig. 5 gives a summary of all possible scenarios considered with the setup in Fig. 1. In particular, the image in Fig. 5 (e) was recorded after the length of path B was increased by 100 µm in order to remove the HOM effect. That is to say, Fig. 5 (e) shows the results for the |Ψ bs state. It was anticipated that |Ψ bs ≡ |Ψ bs , which is confirmed experimentally given the fact that Fig. 5 (d) and (e) are qualitatively identical.
This image doubling can be understood as the beamsplitter 'splitting' the image in two, and then being recombined after changing the path conditions. When measured in coincidence, a rotated photon A is either transmitted by the beamsplitter and interacts with the object, in which case the unrotated photon B (whose phase is −2θ with respect to photon A) is measured by the detection scheme, or the unrotated photon B is reflected by the beamsplitter and interacts with the object, with the rotated photon A (with a 2θ phase relative to photon B) being measured. The |Ψ bs case could be distinguished from the |Ψ bs case by significantly increasing the path length in B but not changing the resulting image doubling reconstruction.
Moreover, such 'splitting' of the object into two rotated images is not restricted to any specific optical plane. This was tested by moving the BS to the Fourier plane of the crystal (and the SLM), with the results obtained in such cases being identical to those reported here for the image plane.

Conclusions
We have used an HOM filter to engineer particular quantum states and used them in ghost imaging experiments. The results are in agreement with the theory and confirm the image rotation and image 'doubling' as a consequence of the state preparation and filtering steps. Although such filtering is often understood in terms of the OAM basis, we translate it here to the position basis by virtue of the invariance of a quantum state's intrinsic symmetry under basis changes. In addition to an intriguing ghost imaging setup, we also employ all-digital control over the imaging arm for fast and convenient image reconstruction. Our work highlights important aspects of this form of ghost imaging and paves the way for further investigations and applications that employ imaging with specially engineered states.

The effect of a change of basis on a state's symmetry
A well-known result from high energy physics is that a change of basis does not change the nature of a state's symmetric character. Here we outline a simple proof of this.
Firstly, let H n := (V, ·, · ) be a complex Hilbert space of dimension n, and let {u 1 , u 2 , · · · , u n }, {v 1 , v 2 , · · · , v n } be two orthonormal bases of V . We define the linear operator 'change of basis' matrix M such that M u i = v i ∀ i. It is easy to see that M is then unitary, i.e. M x, M y = x, y ∀ x, y ∈ V . Let x = n i=1 α i u i , y = n j=1 β j u j be arbitrary, so The converse (i.e. a unitary matrix is a change of basis matrix) can be shown too: let U be a unitary matrix, let {|u i } be an orthonormal basis, and let |t i := U |u i for some set of vectors {|t i }. Then t i |t j = u i |U † U |u j = u i |u j = δ i,j , so {|t i } is an orthonormal basis too. Therefore, given a matrix U , U is unitary iff U represents a change of basis.
Next, let H A , H B be Hilbert spaces of dimension n and m and let {|i A }, {|j B } be respective orthonormal bases. Any arbitrary (anti-)symmetric state |Ψ ∈ H A ⊗ H B can be written as with ν = 1(−1) for the (anti-)symmetric case. Next, define the exchange operator P which switches the two particles in a state |x 1 , y 2 : P |x 1 , y 2 = |y 2 , x 1 . If the state |x 1 , y 2 is symmetric, then |x 1 , y 2 = |y 2 , x 1 ; if it is anti-symmetric, then |x 1 , y 2 = − |y 2 , x 1 . To apply P to |Ψ requires H A = H B , and hence n = m (this is clearly in line with the requisite indistinguishably of the two particles; it doesn't make much sense to talk about symmetric/anti-symmetric states if the constituent particles are distinguishable), so The eigenvalue of P , i.e., ν, tells us whether the state |Ψ is symmetric or antisymmetric. Next, use two n × n unitary matrices U 1 and U 2 to change the bases of H A and H B , respectively, to any other bases. It turns out that P commutes with the change of basis transformation, U 1 ⊗ U 2 , if U 1 = U 2 , so we have and, since U 1 maps a basis to another basis So, P |Ψ = ν |Ψ =⇒ P (U 1 ⊗ U 1 ) |Ψ = ν(U 1 ⊗ U 1 ) |Ψ , so the symmetry of a state |Ψ is maintained by an arbitrary change of basis.