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

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

$$\begin{eqnarray}&&{\left|{\boldsymbol{r}}\right\rangle }_{A}\to \displaystyle \frac{1}{\sqrt{2}}\left[{\left|{\boldsymbol{r}}\right\rangle }_{A}+{\left|{\boldsymbol{r}}\right\rangle }_{B}\right];\ {\left|{\boldsymbol{r}}\right\rangle }_{B}\to \displaystyle \frac{1}{\sqrt{2}}\left[{\left|{\boldsymbol{r}}\right\rangle }_{B}-{\left|{\boldsymbol{r}}\right\rangle }_{A}\right],\end{eqnarray} \tag{ 6 }$$

so that our 'BS' state, $\left|{{\rm{\Psi }}}_{{\rm{bs}}}\right\rangle$, is

$$\begin{eqnarray}\begin{array}{rcl}\left|{{\rm{\Psi }}}_{{\rm{bs}}}\right\rangle & = & \displaystyle \frac{1}{2}\displaystyle \sum _{{\boldsymbol{r}}}c({\boldsymbol{r}})\left[{\left|R{\boldsymbol{r}}\right\rangle }_{A}+{\left|R{\boldsymbol{r}}\right\rangle }_{B}\right]\left[{\left|{\boldsymbol{r}}\right\rangle }_{B}-{\left|{\boldsymbol{r}}\right\rangle }_{A}\right]\\ & = & \displaystyle \frac{1}{2}\displaystyle \sum _{{\boldsymbol{r}}}c({\boldsymbol{r}})\left[{\left|R{\boldsymbol{r}}\right\rangle }_{A}{\left|{\boldsymbol{r}}\right\rangle }_{B}-{\left|{\boldsymbol{r}}\right\rangle }_{A}{\left|R{\boldsymbol{r}}\right\rangle }_{B}+{\left|R{\boldsymbol{r}},{\boldsymbol{r}}\right\rangle }_{B}-{\left|R{\boldsymbol{r}},{\boldsymbol{r}}\right\rangle }_{A}\right].\end{array}\end{eqnarray} \tag{ 7 }$$

We post-select on coincidences, allowing us to drop the latter two terms in equation (7), so

$$\begin{eqnarray}&&\left|{{\rm{\Psi }}}_{{\rm{bs}}}\right\rangle ={ \mathcal K }\displaystyle \sum _{{\boldsymbol{r}}}c({\boldsymbol{r}})\left[{\left|R{\boldsymbol{r}}\right\rangle }_{A}{\left|{\boldsymbol{r}}\right\rangle }_{B}-{\left|{\boldsymbol{r}}\right\rangle }_{A}{\left|R{\boldsymbol{r}}\right\rangle }_{B}\right],\end{eqnarray} \tag{ 8 }$$

with ${ \mathcal K }$ the normalisation constant.

A comparison of all the imaging scenarios will be easier if all R dependence is moved to photon B. In the supplementary material we demonstrate how the rotational dependence can be shifted from photon A to photon B, substituting R by R⁻¹, so equation (8) can be written as

$$\begin{eqnarray}&&\left|{{\rm{\Psi }}}_{{\rm{bs}}}\right\rangle ={ \mathcal K }\displaystyle \sum _{{\boldsymbol{r}}}c({\boldsymbol{r}}){\left|{\boldsymbol{r}}\right\rangle }_{A}\left[{\left|{R}^{-1}{\boldsymbol{r}}\right\rangle }_{B}-{\left|R{\boldsymbol{r}}\right\rangle }_{B}\right].\end{eqnarray} \tag{ 9 }$$

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.

To affect 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 figure 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, ${\left|{\boldsymbol{r}}\right\rangle }_{B}\to {\left|{\boldsymbol{r}}^{\prime} \right\rangle }_{B}$. Effecting this change in photon B in equation (4) while applying the BS transformations in equation (6), and thereafter post-selecting on coincidences, gives

$$\begin{eqnarray}&&\left|{{\rm{\Psi }}}_{{\rm{bs}}}^{{\prime} }\right\rangle ={ \mathcal K }\displaystyle \sum _{{\boldsymbol{r}}}c({\boldsymbol{r}})\left[{\left|R{\boldsymbol{r}}\right\rangle }_{A}{\left|{\boldsymbol{r}}^{\prime} \right\rangle }_{B}-{\left|{\boldsymbol{r}}^{\prime} \right\rangle }_{A}{\left|R{\boldsymbol{r}}\right\rangle }_{B}\right].\end{eqnarray} \tag{ 10 }$$

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 $\left|{{\rm{\Psi }}}_{{\rm{bs}}}^{{\prime} }\right\rangle \equiv \left|{{\rm{\Psi }}}_{{\rm{bs}}}\right\rangle$.

Given either engineered state $\left|{{\rm{\Psi }}}_{{\rm{nbs}}}\right\rangle$ or $\left|{{\rm{\Psi }}}_{{\rm{bs}}}\right\rangle$, 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 $O({\boldsymbol{r}})$: $O({\boldsymbol{r}})=0$ if the pixel at position ${\boldsymbol{r}}$ in SLM_A is black in the object, and 1 if pixel ${\boldsymbol{r}}$ 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 ${\left|O\right\rangle }_{A}={ \mathcal N }{\sum }_{{\boldsymbol{r}}}O({\boldsymbol{r}}){\left|{\boldsymbol{r}}\right\rangle }_{A}$, with ${ \mathcal N }$ the appropriate normalization. After masking SLM_A with O and absorbing ${ \mathcal K }$ into ${ \mathcal N }$, the state of photon B, in the absence of the BS, is

$$\begin{eqnarray}&&\left\langle O\right|{{\rm{\Psi }}}_{{\rm{nbs}}}\rangle ={{ \mathcal N }}^{* }\displaystyle \sum _{{\boldsymbol{r}}}c({\boldsymbol{r}})O(R{\boldsymbol{r}}){\left|{\boldsymbol{r}}\right\rangle }_{B}.\end{eqnarray} \tag{ 11 }$$

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

$$\begin{eqnarray}&&\left\langle O\right|{{\rm{\Psi }}}_{{\rm{bs}}}\rangle ={{ \mathcal N }}^{* }\displaystyle \sum _{{\boldsymbol{r}}}c({\boldsymbol{r}})\left[O(R{\boldsymbol{r}})-O({R}^{-1}{\boldsymbol{r}})\right]{\left|{\boldsymbol{r}}\right\rangle }_{B}.\end{eqnarray} \tag{ 12 }$$

If we set the weighting coefficients c to unity, we can visualize the outcome more clearly

$$\begin{eqnarray}&&\left\langle O\right|{{\rm{\Psi }}}_{{\rm{nbs}}}\rangle \propto \displaystyle \sum _{{\boldsymbol{r}}}O(R{\boldsymbol{r}}){\left|{\boldsymbol{r}}\right\rangle }_{B},\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&\left\langle O\right|{{\rm{\Psi }}}_{{\rm{bs}}}\rangle \propto \displaystyle \sum _{{\boldsymbol{r}}}\left[O(R{\boldsymbol{r}})-O({R}^{-1}{\boldsymbol{r}})\right]{\left|{\boldsymbol{r}}\right\rangle }_{B},\end{eqnarray} \tag{ 14 }$$

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 equation (13), and a juxtaposed 'double' image with opposite rotations in the case of equation (14). The intensity of pixel ${\left|{\boldsymbol{r}}\right\rangle }_{B}$ in the reconstructed object in each case is respectively

$$\begin{eqnarray}&&{\left|{\left\langle {\boldsymbol{r}}\right|}_{B}\left\langle O\right|{{\rm{\Psi }}}_{{\rm{nbs}}}\rangle \right|}^{2}\propto {\left|O(R{\boldsymbol{r}})\right|}^{2},\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&{\left|{\left\langle {\boldsymbol{r}}\right|}_{B}\left\langle O\right|{{\rm{\Psi }}}_{{\rm{bs}}}\rangle \right|}^{2}\propto {\left|O(R{\boldsymbol{r}})-O({R}^{-1}{\boldsymbol{r}})\right|}^{2}.\end{eqnarray} \tag{ 16 }$$

Equations (15) and (16) are key to understanding the object reconstructions shown in the following section.

First, an experiment was run with the setup as depicted in figure 1, but without the HOM filter (the BS was removed). The SLM in path A was masked with a 960 × 960 resolution object O, as shown in figures 3(a), (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 figures 3(c), (d) with a Dove prism angle of θ = 0 and in figures 3(e), (f) when $\theta =\tfrac{\pi }{4}$. The ghost images were reconstructed using the set of coincidence counts { c_i } for every raster position in SLM_B as

$$\begin{eqnarray}&&\mathrm{Image}=\displaystyle \frac{{c}_{1}}{n}{P}_{1}+\displaystyle \frac{{c}_{2}}{n}{P}_{2}\,+\,\cdots ,\end{eqnarray} \tag{ 17 }$$

where c_i is the coincidence count recorded for raster position P_i, and n is a normalization constant (see supplementary material). The results confirm the accuracy of the digital scan approach. However, the resolution that can be used in such a single 'on pixel' reconstruction technique is limited by the strength of the signal arriving at the SLM. The integration time for each raster position increases as the pixel size decreases, in order to overcome the noise.

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A different measurement scheme, a random mask scan [24] based on the compressed sensing concept [25], was also tested for the object reconstruction in order to overcome the noise in low signal cases without the need to decrease the resolution [26], as shown in the examples of figures 3(g), (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 one of these random binary masks 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 being the weighted random masks themselves, i.e.

$$\begin{eqnarray}&&\mathrm{Image}\approx \displaystyle \frac{({c}_{1}-\bar{{\boldsymbol{c}}})}{n}{M}_{1}+\displaystyle \frac{({c}_{2}-\bar{{\boldsymbol{c}}})}{n}{M}_{2}\,+\,\cdots ,\end{eqnarray} \tag{ 18 }$$

where c_i is the coincidence count recorded for each random mask M_i, n is a normalization constant, and $\bar{{\boldsymbol{c}}}$ is the average of all coincidence counts measured [26]. This is done since the 'on' pixel would ordinarily correspond to a value of 1 and the 'off' pixel to −1, giving an average outcome of 0. But in our case, the 'off' pixel corresponds to 0, thus the non-zero average values must be subtracted to remove the noise. An animated example of the random mask reconstruction of figure 3(g) can be observed in the attached animation file (Lambda reconstruction), where the object is given in the leftmost, the real random mask used for each scan (iteration) is given in the middle, and the reconstructed image appears in the rightmost. The reconstructed image becomes clearer as the number of iterations, shown at the top, increase.

It is worth mentioning that this scheme can be generalised to cases with arbitrary proportions of black: white pixels. As we decrease the proportion of white pixels, we decrease the average of the measured coincidences which needs to be subtracted, i.e. the measurements are less noisy when not performing the average subtraction, with the extreme case being only 1 pixel as in equation (17). However, the maximum attainable resolution decreases, for a given signal arriving at the SLM, when decreasing the proportion of white pixels.

To test this measurement technique in a ghost imaging setup, the experiment was run with the objects given in figures 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 $\theta =\tfrac{\pi }{4}$ for the results in figures 4(e)–(h), and $\theta =-\tfrac{\pi }{8}$ for those in figures 4(m)–(p).

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From these results, the reconstructed image is rotated by an angle of 2θ with respect to the original object, as predicted in equation (15). 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.

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 BS inserted into the setup, which selects the state $\left|{{\rm{\Psi }}}_{{\rm{bs}}}\right\rangle$. As per equation (16), the intensity of pixel ${\boldsymbol{r}}$ in the reconstructed image is a combination of the intensity of pixel ${\boldsymbol{r}}$ in O, rotated by both 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 figures 4(i)–(l) for a relative Dove prism angle of $\theta =\tfrac{\pi }{4}$, and in figures 4(q)–(t) for $\theta =-\tfrac{\pi }{8}$.

The experimental results in each row are for the objects given in the first column. Note that the results in the last row of figure 4 are identical, with or without the BS and $\theta =\tfrac{\pi }{4}$, 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 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 of the BS. We expect this to be the reason of the anti-clockwise-rotated portion of the reconstructed image to be dimmer compared with the clockwise-rotated portion; each half of the SPDC state, one in arm A and the other in arm B, traverses different ports of the BS. On the other hand, we deliberately displaced the object from the SPDC beam center of coordinates adding extra space between the reconstructed images, to properly identify the double rotation effect.

Finally, figure 5 gives a summary of all possible scenarios considered with the setup in figure 1. In particular, the image in figure 5(e) was recorded after the length of path B was increased by 100 μm in order to remove the HOM effect but keeping the BS in. That is to say, figure 5(e) shows the results for the $\left|{{\rm{\Psi }}}_{{\rm{bs}}}^{{\prime} }\right\rangle$ state. It was anticipated that $\left|{{\rm{\Psi }}}_{{\rm{bs}}}\right\rangle \equiv \left|{{\rm{\Psi }}}_{{\rm{bs}}}^{{\prime} }\right\rangle$, which is confirmed experimentally given the fact that figures 5(d) and (e) are qualitatively identical.

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This image doubling can be understood as the BS '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 BS 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 BS and interacts with the object, with the rotated photon A (with a 2θ phase relative to photon B) being measured.

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 a case identical to those reported here for the image plane.