Engineering quantum states from a spatially structured quantum eraser

In the Hong-Ou-Mandel (HOM) effect, quantum interference occurs when two indistinguishable photons, entering a beamsplitter (BS), take the same output path (photon bunching) [1]. Although the HOM effect is typically investigated by tuning the temporal matching between two identical photons, the indistinguishability required for the phenomenon makes it fundamentally sensitive to all degrees of freedom (DoFs), from polarisation [2, 3] to frequency [4, 5], and time-bins [6] as well as for collinear spatial modes [7], including multi-particle [8, 9] and high dimensional scenarios [10, 11, 12, 13]. Monitoring this effect has rendered it a versatile tool with innate sensitivity to measure a wide array of variations between the two inputs for quantum enhanced measurements [14, 3, 15]. With the advent of sensitive cameras enabling spatially resolved observation [16], measurements across the spatial DoF [17, 18] have been made possible, such as dip tracking for spatially-resolved height measurements [19] or insight into the spatial-temporal [20] effects from multimode spontaneous parametric down-conversion (SPDC).

The fundamental quantum nature of HOM interference also lends itself to tests of quantum mechanics [21, 22], among which a notable example is the quantum eraser [2, 23]. Such a paradigm allows one to restore quantum interference, even if two photons are made distinguishable before entering the BS by ‘marking’ a degree of freedom such as polarisation. This is achieved by projecting the outputs of both exit ports onto a basis that cannot yield information on which path the photon took through the BS and so making the photons effectively indistinguishable again. Moreover, the projection process can allow one to edit the state by introducing phases to vary bunching behaviour to anti-bunching and vice versa. One may accordingly imagine engineering a desired photon number in an augmented state, lending itself to be used as a structuring tool.

Structured light, in general, is a very powerful concept in modern optics, spanning from the classical regime to fundamental quanta [24]. For instance, classical implementations exhibit enhanced sensing [25, 26, 27, 28, 29], microscopy [30] and communication [31] capabilities, while quantum mechanically, it provides a test-bed for quantum mechanics [32, 33], secure high-dimensional communication [34] and increased resilience to noise [35, 36, 37]. As such, there is a strong interest in tailoring complex or new structures, such as designing across non-local degrees of freedom for quantum skyrmions [38] or harnessing non-homogeneity for complex entanglement structures [39, 40, 41]. It follows that, by designing a mode with structured photon coalescence, we can combine the advantages of both concepts, holding the possibility for direct impact in applications based on quantum interference or structured light as well as unlocking new prospects.

By altering a given DoF before the BS in HOM interference, we already find vanilla photon number engineering occurs by default, where bunching forms NOON states [42, 43, 44] and, similarly, anti-bunching engineers non-local entangled states [45]. Moreover, coincidence detection acts as a filter for particular Bell and high-dimensional spatial states [10], while altering the modal distinguishability between photons, facilitates space-time entanglement engineering [46, 47] or the population of spatial modes [7, 12]. Such state engineering can be extended to photon subtraction or additions with continuous variables [48]. Where, in these cases, interference was used to postselect or allocate photons to particular spatial modes, here we demonstrate a scheme to spatially structure the quantum erasing process itself, thus obtaining a photonic state with no classical counterpart, that is directly structured in a quantum property of the field, i.e., photonic coalescence.

To do this, we harness the innate dependency of the HOM effect on identical conditions to demonstrate how spatially tailoring a degree of freedom can allow one to spatially tailor quantum interference. We achieve this by exploiting geometric phase devices to generate non-uniform structure in the polarisation degree of freedom and, having done so, a spatially-varying structure in the distinguishability for the input photons. We therefore implement a quantum eraser [2] via polarisation projective measurements in order to study the caveats associated with these engineered states and elicit conditions where coalescent structures can be heralded or erased. We subsequently provide a framework for how one may engineer the bunching and anti-bunching distribution across the transverse mode profile by exploiting the simple generation of vector vortex (VV) modes. This approach and subsequent structures, however, may be generalised to achieve arbitrary freedom in engineering this fundamentally quantum property towards developing a unique and versatile tool for applications in fields such as quantum metrology, microscopy and communications.

In particular, without loss of generality, we consider two modes that present the same polarisation in some regions of the transverse profile, as depicted in Fig. 1(b). As such, we choose the radial ($q=0.5$) and $\pi$ [$q=0.5$ followed by a half-waveplate (HWP)] modes depicted there, which are two first order orthogonal VV modes [39], to be the input states of the BS. These may respectively be described by the azimuthally dependent polarisations,

where $e_{H}$ and $e_{V}$ are, respectively, the horizontal and vertical polarisation unit-vectors.

As in typical HOM experiments, the interference or coalescent behaviour of our modes are observed through the two-fold coincidences measured on the state

corresponding to the projection of the full BS-output state onto the subspace in which the two photons are separated in ports A and B, and may be calculated by the fourth-order correlation function [50], as detailed in the Supplementary Information (SI). With this, we find the spatially varying coincidence probabilities for arbitrary polarisation projections {$\alpha$,$\beta$} on the photons detected in each port (respectively corresponding to coordinates $\varphi_{1}$ and $\varphi_{2}$) when temporally tuned ($\tau=\tau^{\prime}$)

and temporally distinguishable ($\tau\neq\tau^{\prime}$)

Here $\mathbf{u}_{\alpha}$ and $\mathbf{u}_{\beta}$ are the unit-vectors directed along the detection polariser axes. For simplicity, we ignore here the radial distribution as the polarisation variation depends only on $\varphi$ (see SI for a more general treatment). Experimentally, we realise these polarisation state projections by respectively placing polarisers $P_{1}$ and $P_{2}$ in each output port of the BS.

As depicted in Figure 2, we adopted two strategies to observe the generated structure by means of these coincidences, both in and out of the temporal tuning. In one case, the photons from each output port were collected by coupling into multimode fibres connected to bucket detectors placed in each arm (flip mirror down), rendering no resolution of the spatial distribution. This corresponds to measuring the correlations given in Eqs. (4) and (5) after integrating over both azimuthal angles $\varphi_{1}$ and $\varphi_{2}$. In the second case (flip mirror up), we performed spatially-resolved measurements of one photon by replacing one bucket detector with a camera, which was then conditioned on the spatially-unresolved detection of another photon by the bucket detector in the other arm. This corresponds to measuring the correlations given in Eqs. (4) and (5) after integrating over one angle only, e.g. $\varphi_{2}$.

It is now possible to identify polarisation projection measurements that erase the which-path information and enable quantum interference between these structured modes, even for bucket detection. Intuitively, one may consider the VV spatial distributions depicted in Fig. 2 which shows they share the same polarisation state along the horizontal (H) and vertical (V) axes, while being completely orthogonal along the diagonal (D) and anti-diagonal (A) directions. It thus follows that a projection along H and V should destroy the which-path information for the latter pairing and recover HOM interference. To observe the related coalescence for such polarisations ($\alpha$ and $\beta$) we define the visibility, $\mathcal{V}_{\alpha,\beta}=(C^{\alpha,\beta}_{out}-C^{\alpha,\beta}_{in})/C^{\alpha,\beta}_{out}\in[-1,1]$ between the coincidences detected out ($C_{out}$) and in ($C_{in}$) the temporal indistinguishably criteria [as depicted in Fig. 1 (b)]. This allows us to easily recognise bunching as $\mathcal{V}$ becomes positive from anti-bunching where $\mathcal{V}$ becomes negative with extreme cases of perfect bunching (antibunching) being detected when $\mathcal{V}=1$ $(-1)$.

The graphs given in each panel of Figure 3 show the experimental outcomes of the bucket-only detection as the delay between the two input photons was varied by moving the delay line. Here, projections $(P_{1},P_{2})=$ (H,H) elicit a dip at $\Delta t=0$, coinciding with a erasure of the which-way information for the two photons and thereby restoring quantum interference. For $(P_{1},P_{2})=$ (H,V), we find a peak instead of a dip, corresponding to photon antibunching. [2, 11] Alternatively, cases $(P_{1},P_{2})=$ (H,A),(H,D),(A,D) and (A,A) result in a flat curve (no HOM dip nor peak).

On the other hand, for spatially-resolvable measurements (using the camera), we expect the following spatially structured visibilities, corresponding to a set of polarisation projections of $P_{1}\in\{H,A\}$ in the camera arm and $P_{2}\in\{H,V,D,A\}$ in the bucket detector arm as depicted by the arrows in Fig. 3:

obtained by integrating (4) and (5) over $\varphi_{2}$ due to bucket detection in arm 2. In this case, images conditioned on coincidence with the bucket detector were captured “out” (upper left inset of the curves) and “in” (upper right inset of the curve) the temporally matched condition. Some asymmetry towards the left in the distributions may be noted as the result of experimental misalignment in the delay line to the camera (see Methods) as well as the $q$-plate with respect to the photon distribution. This, however, does not significantly detract from the expected distributions. Each measured pair of coincidence distributions then allowed us to reconstruct the visibility map (lower row) for the projection settings.

The resulting outcomes relate well to what was observed in the case of no spatial resolution (bucket detection) as the coincidences measured by bucket detection are proportional to the integrated pixel distribution over the mode profile. We additionally compare these to the calculated profiles from Eq. 6 that are shown as upper-right insets of the visibility maps. Accordingly, $(P_{1},P_{2})=$ (H,H) shows uniform coalescence across the spatial distribution, while (H,V) exhibits a uniform anti-coalescence, within the expectations of the calculated distributions. The cases of HA, HD, AA and AD similarly yield a uniform structure in line with expectations of a zero visibility across the measured profile.

Crucially, projections AH and AV, reveal spatially varying features with two complementary lobes, showing coalescence and anti-coalescence across the structure as predicted by Eqs. 6. These structures are not detectable without spatial resolution, as shown by the flat curves in the corresponding graphs. As a result, we herald a mode of structured bosonic coalescence with some parts containing either zero or two photons as in the case of NOON states. At the same time, other regions of the mode, corresponding to anti-coalescence, will always be populated by one photon.

In conclusion, we have generated structured quantum states of light that have no classical counterpart, with a spatially tailored bosonic coalescence. We achieved this by exploiting a simple and robust scheme based on spatially tailored distinguishability between two photons in a quantum eraser setting. This additionally invites insight into the structured state where, in particular, the use of orthogonal modes for each interfered photon showed a global versus local HOM effect [6] such that detection without spatial resolution yielded no sign of interference, but spatially resolved measurements reveals its presence. Although these results were demonstrated for VV modes, our technique may be generalised to intelligently engineer arbitrary desired interference structures for varying purposes. Additionally, we point out that programmable approaches, for tailoring the input polarisation distributions, [51] would allow dynamic on-demand variations of these states. As distinguishabilty is the property one needs to tailor, one may extend our result to other degrees of freedom, such as temporally structured light [52]. By nature of the ability to spatially tailor the coalescent behaviour of photons, we believe these new states and engineering thereof could be beneficial for fields such as quantum microscopy, metrology and communications, including high-dimensional protocols.

that, acting on the state $\ket{H}$, produces the desired state $\ket{\pi}=\cos{(\varphi)}\ket{H}-\sin{(\varphi)}\ket{V}$ for $q=0.5$.

We acknowledge support from the Italian Ministry of Research (MUR) through the PRIN 2017 project “Interacting photons in polariton circuits” (INPhoPOL) and the PNRR project PE0000023-NQSTI. We also acknowledge support from NATO through SPS Project HADES - MYP G5839 and from the Italian Space Agency (ASI) through the ”High dimensional quantum information” (HDQI) project.