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Qudits, d-level quantum systems, have been shown to provide a better resource for quantum key distribution and other Quantum Information protocols. It is customary to generate photonic qudits using more than one degree of freedom of the same photon. In much the same way, multi-qubit states are generated using only a pair of photons and ingenious ways to manipulate more than one degree of freedom independently. In contrast to such costly implementations in terms of quantum resources, we present the controlled generation of two copies of two-qudit states using four photons and a single degree of freedom, transverse momentum. The degree of entanglement within each pair was inferred by exploiting the availability of two copies of the same state, without the need of a full tomographic reconstruction of the states, and both highly-entangled and separable states were generated. We show theoretically that the set of states obtainable using our setup is very diverse, ranging from maximally entangled states of qudits to separable states.
© 2016 Optical Society of America
1. Introduction
Much work has been done in expanding the dimension of discrete Hilbert spaces accessible to experiments using photons, either by using the path through a slit [1–7], using the orbital angular momenta [8–12], polarization states [13], exploiting more than one degree of freedom at the same time on few photons [14–16] or by creating many photons, each serving as a qubit [17–20].
The expansion of Hilbert space allows the preparation of single photon, two photons or even multi-photon systems in d-level quantum states called qudits (d > 2) [21]. Even though progress towards larger dimensions has been encouraging [22–24], the simultaneous use of more than two photons and a single degree of freedom with more than two levels, thus expanding the Hilbert space in two different ways at the same time, remains underexplored. Multipartite entanglement of four-photons in their orbital angular momentum degrees of freedom was only recently demonstrated [25, 26]. This expansion to multi-qudit states opens the possibility of implementing quantum key distribution protocols that are more secure than their qubit counterparts [27] as well as quantum secret sharing strategies [28–31], quantum imaging [32] and even entanglement distribution protocols, such as qudit teleportation [33] and entanglement swapping [34, 35]. In [36], the authors discuss the implementation of a Toffoli gate using 2 qubits and 1 qutrit instead the usual picture of 3 qubits, and they show that the gate implementation is more efficient in the first scheme.
Multiphoton interferometry is a useful tool for revealing nonclassical phenomena. Different applications in quantum information using multiphoton interferometry have been demonstrated [40]. The slit implementation used in the present work is also particularly interesting due to the fact that qudits with a prime number of levels can be generated with ease, and it is always possible to associate sets of mutually unbiased basis to such prime-dimensional spaces [41,42]. The slit states were also used to verify, experimentally [37], an entanglement criteria using modular variables in spatial interference patterns [38, 39].
Our experiment used a pulsed femtosecond laser to pump a high-efficiency periodically-poled KTiOPO4 crystal, grown for type-II spontaneous parametric down-conversion (SPDC). It has been shown that such crystals are efficient enough for multi-pair contributions to be significant [6, 43–45]. In addition to that, the photons were encoded into logical qudits, by passing them through a multiple-slit setup [2,3]. The resulting state is usually referred to as a path or slit state. Thus, our setup can give us both the higher number of particles and of system levels, as it was desired.
In this article we first discuss the experimental setup and how it is used to manipulate the state that comes out of the multiple slit, as well as detail the detection apparatus, following with a brief overview of the form taken by states generated with our experimental setup. Finally, our experimental results are presented, showing that we were able to generate both a state with two highly-entangled qudit pairs and also a state of four nearly-separable qudits, using the von Neumann entropy of the reduced single-particle state to quantify entanglement, since we assume the global state is pure.
2. Experimental setup
Our main objective was to be able to generate two copies of qudit states in a controlled manner. In order to do so we employed a 200 fs pulsed laser in the infrared (λ = 826 nm) with repetition rate of 76 MHz. The infrared beam with average power of 1.5 W is directed to the second harmonic generator and produces a 200 mW blue beam with wavelength λ = 413 nm. This beam was used to pump a periodically poled KTP crystal (PPKTP) grown for collinear, degenerate type II SPDC from 413 to 826 nm. Such crystal is known to be a highly efficient source for SPDC with no transverse walk-off between horizontal and vertical polarizations, due to the phenomenon of quasi-phase matching [46].
It has previously been shown that it is possible to prepare the quantum state of qubits encoded in the transverse degrees of freedom of position and momentum by using slits and manipulating the transverse amplitude of biphotons [2–6]. In our experiment, control of the state generated was achieved by introducing linear optic elements between the PPKTP and the multiple slits used to encode the transverse position-momentum degree of freedom of the photons into a qudit space. Therefore, the SPDC source, linear optical elements, and slits composed the state preparation part of our setup. After the slits a spherical lens was placed in the f -f configuration between the apertures and the detection plane, allowing us to map the far field of the slits and observe conditional interference in coincidence detections, as a function of the position of two separate detectors, moving independently.
In order to manipulate the biphoton amplitude we used two linear optical setups, indicated in Fig. 2. The first was a set of two lenses, one cylindrical and one spherical, sharing a focal plane coinciding with the center of the PPKTP crystal, in order to project a magnified image of the biphoton amplitude at the center of the crystal onto the slits plane, that were placed at the back focal plane of the second lens [5]. By propagating the amplitude of an electromagnetic field through the set of lenses using Fourier Optics one finds that in addition to projecting a magnified image of the object at the forward focal plane of the spherical lens, this type of telescope introduces a parabolic phase profile defined by
where k is the wavenumber of the down-converted photons, f2 is the focal length of the spherical lens and M is the magnification factor of the telescope, given by f2/f1.For the second setup we removed the cylindrical lens and left just the spherical lens, in an f -f configuration between the crystal center and the slits, thus projecting an optical Fourier transform of the biphoton amplitude at the center of the crystal onto the slits.
One important difference between qubit and qudit encoding is that the larger number of slits gives rise to more symmetries on the system. This fact means that even though we use the same symmetric illumination strategy of a previous work [6], our qudits are sensitive to a parabolic phase profile introduced by the type of telescope used in the setup indicated in Fig. 2(a).
The setup is completed with a detection apparatus, represented in Fig. 1 by the elements to the right of the slits, consisting of a lens in the f -f configuration between the slits and the detection plane, a polarized beamsplitter (PBS) and two 50/50 beamsplitters, one after each output of the PBS. At the end of each optical path, on the detection plane, is an avalanche photodiode (APD), that is used to detect photons and feed electrical pulses to the coincidence counting electronics, which uses a 5.6 ns temporal window to determine if pulses arriving from two (or more) different detectors constitute a (multiple) coincidence. In this way we are able to detect coincidences between photons of same polarization, always originated from different two-pair SPDC events, and also coincidences between photons with orthogonal polarizations, that can occur due to photons belonging to the same pair and also to photons from different pairs.
Fig. 1 Complete setup for our experiment, comprised of a four-qudit state source and a detection apparatus. The source contains a 10 mm long periodically-poled KTP crystal that is pumped by a pulsed beam with λ = 413 nm, 200 fs pulse width and 76 MHz repetition rate, a dichroic mirror (DM) that reflects the pump light and allows the down-converted photons to pass, a cylindrical lens (CL) of focal length fCL = 5 cm that can be taken out of the optical path by a translation stage, a spherical lens (SL) of focal length fSL = 20 cm and a slide with either triple metallic slits or quadruple slits made of photographic film. The detection apparatus is composed of a lens with a focal distance of 30 cm in an f -f configuration between the apertures and the detection plane, so that the far field of the slits can be imaged. A polarizing beamsplitter (PBS) splits the photons coming from the slits into two branches, one with horizontally polarized photons and another with vertically polarized ones. This allows us to detect coincidences due exclusively to photons from different pairs. Each branch has a 50/50 beamsplitter (BS) that sends photons to the two avalanche photodiode detectors of each branch, D1 and D3 on the vertically polarized branch and D2 and D4 on the horizontally polarized one. The electrical pulses are then sent to custom electronics that process detections and registers as coincidences all pulses that arrive within a 5.6 ns window. Thus, we are able to detect two-, three- and fourfold coincidences between all possible combinations of detectors.
Fig. 2 Optical setups used to control the qudit state generated after the slits. Figure (a) shows a telescope composed of a cylindrical lens with a focal distance 5 cm and a spherical lens with a focal distance 20 cm. Both share the same forward focal plane that coincides with the pump beam waist, positioned using the spherical lens SL1 that has a focal length of 30 cm, and the center of the PPKTP. The back focal plane of the spherical lens corresponds to the plane of the slits, so that a magnified astigmatic image of the crystal center is projected onto the apertures. Figure (b) shows the same spherical lens of Fig. (a) in an f -f configuration, that is obtained when the cylindrical lens is removed from the optical path. This configuration projects an Optical Fourier transform of the field at the center of the crystal onto the aperture plane.
The fact that we are able to detect signatures of two-pair events without fourfold coincidence detection is important, since the effective detection efficiency for fourfold coincidences is much lower than that of simple coincidences between two detectors. This signal, that we call same-branch coincidences (SBC), allows us to remove the contributions of photons originating from different pairs from the coincidence signal from detectors in different polarization branches (different branch coincidences, DBC).
3. Obtainable states
A collinear and frequency degenerate type-II SPDC source emits pairs of photons, also called biphotons, in a state described by [3, 47]
where the amplitude function is defined as Φ(xh, xv) = E(xh, xv)ξ(xh, xv) and |1π, xπ〉 indicates the single-photon Fock state for polarization π ∈ {h, v} with a transverse position degree of freedom xπ. E(xh + xv) is the complex field amplitude of the Gaussian pump beam, and is the Fourier transform of the phase-matching function for the type II PPKTP crystal used in our experiment, which has a length L and an effective index of refraction neff for biphotons of central frequency , qπ (π ∈ h, v) is the photon transversal momentum with polarization π, and ϕ is a phase-mismatch term that we will consider to be negligible. Both E and ξ are even functions and, more importantly, are each a function of only one of a set of two orthogonal variables, {x+, x−}, where x± = (xh ± xv)/2.Our qudit implementation uses multiple-slit apertures to encode the transverse amplitude of the biphoton into a discrete space with a number of levels defined by the number of slits. Such apertures are described by a transmission function T(x) = Σj Tj (x), where
x being the coordinate on the direction along which the slits are narrow. The width of each slit is 2a and a slit j is centered on x = dj. The separation between the centers of two adjacent slits is 2d, meaning |dj − dj−1| = 2d. In our case, the slits are indexed according to the computational basis for qudits of dimension D, so l ∈ [0, D − 1].The state of a biphoton after being transmitted through the apertures is
and can be simplified by assuming that slits are narrow enough to consider Φ′(xh, xv), the amplitude propagated from the crystal to the aperture plane, constant over a single slit region and then using the definition for T(x). This results in [2, 3] where l and m vary independently from 0 to D − 1. The so called slit states |l〉h and |m〉v are written in terms of Fock states for photons of polarization h and v, |1h, xh〉 and |1v, xv〉, with a spatial distribution over the x transverse coordinate given by where π ∈ {h, v} and j ∈ {0, D − 1}. A slit state |j〉π, with π ∈ {h, v} means a state of a photon that crossed slit j, with polarization π.Coefficients clm are determined by the amplitude of the two-photon field at the plane of the slits, clm = Φ′ (dl, dm). Therefore, each coefficient depends of the product E(dl, dm)ξ(dl, dm) which can also be written as , so that and . As in [5, 6], c+ and c− are modified individually by manipulating the propagation of the amplitude function using the optical setups indicated in Fig. 2. Setup of Fig. 2(a) makes E effectively constant over the plane of the apertures, while setup of Fig. 2(b) does the same for ξ.
The symmetry considerations presented above allow us to define two sets of states obtainable using our experimental setup: {|Ψ+〉} for constant and {|Ψ−〉} for constant . In the case of qutrits the generic element of each set can be written as a function of four angles −α and β for |Ψ+〉3, ϵ and θ for |Ψ−〉3. For ququarts six angles are needed −α, β, and γ for |Ψ+〉4, ϵ, θ, and ν for for |Ψ−〉4.
The angle parameters are set by the following relations that depend of Φ(xs, xi) at the position of the slits. For qutrits, the parameters are given by:
and with β given by Eq. (8). To calculate θ and ϵ, we have: andIn the case of ququarts, we have:
for |Ψ+〉4 and for |Ψ−〉4It is important to bear in mind that, in our experimental setup, the amplitudes Φ(xs, xi) for {|Ψ+〉} depend essentially on the optical Fourier transform of the pump field on the plane of the slits, Ẽp(kx+/f), while for {|Ψ−〉} they depend on ξ(x−/M), the Fourier transform of the phase matching function with M being the magnification of the telescope. The variables x+ and x− are defined as xs ± xi, respectively.
We present below the generic elements of {|Ψ+〉} and {|Ψ−〉} sets of states:
and for qutrits and and for ququarts.In the case of {|Ψ−〉} states, the phase ς is due to the parabolic phase profile of Eq. (1) with x2 replaced by in the argument. For the sake of simplicity, since dl and dm are multiples of the separation 2d between two slits, we indicate the phase by using:
Thus, Eq. (1) can be written as ζ(n) = eiς(n). The phases φ are due to the wavefront curvature of a Gaussian beam outside of its beam waist. In our case, this curvature is minimal, since the optical system is made so that the beam waist coincides with the forward focal plane of both lenses used to manipulate the state.
It is interesting to notice the rich structure we can obtain by increasing the number of slits. The last term in each state corresponds to a maximally entangled state of effective qubits that exist on a larger space and we notice that larger dimensions are more influenced by the parabolic phase profile than lower ones.
4. Results and discussion
We present our experimental results in the form of two-dimensional maps of coincidence detections as a function of the position of each of the two detectors used, which are indicated above each figure. Experimental data is accompanied by numerical simulations using state parameters that best approached the observed interference patterns. Data accumulation in each experimental map took approximately four hours, as is indicated in Figs. (3)–(6).
Fig. 3 Maps corresponding to a state of four qutrits, generated using a telescope with a cylindrical and a spherical lens between the PPKTP and a three-slit apperture, as shown on Fig. 2(a). The slits were 40 µm wide and had a 125 µm center-to-center spacing. Measurements were made with a 30 s integration time and a grid of 21 by 21 points separated by steps of 300 µm. The top row, maps (a) to (c) show data obtained with coincidence detection between photons with the same polarization. The bottom row (d) to (f) show data obtained with coincidence detection between photons with orthogonal polarizations. Maps (a), (b), (d), and (e) are obtained from experimental data, while maps (c) and (f) are simulations based on Eq. (19). Data in map (a) has been normalized to a maximum of 129 coincidence counts, while (d) has been normalized to a maximum of 448 coincidence counts.
Fig. 4 Maps corresponding to a state of four qutrits, generated using a spherical lens between the PPKTP and a three-slit apperture, as shown on Fig. 2(b). The slits were 40 µm wide and had a 125 µm center-to-center spacing. Measurements were made with a 30 s integration time and a grid of 21 by 21 points separated by steps of 300 µm. The top row, maps (a) to (c) show data obtained with coincidence detection between photons with the same polarization. The bottom row (d) to (f) show data obtained with coincidence detection between photons with orthogonal polarizations. Maps (a), (b), (d), and (e) are obtained from experimental data, while maps (c) and (f) are simulations based on Eq. (18). Data in map (a) has been normalized to a maximum of 11 coincidence counts, while (d) has been normalized to a maximum of 33 coincidence counts.
Fig. 5 Maps corresponding to a state of four ququarts, generated using a telescope with a cylindrical and a spherical lens between the PPKTP and a four-slit apperture, as shown on Fig. 2(a). The slits were 80 µm wide and had a 160 µm center-to-center spacing. Measurements were made with a 60 s integration time and a grid of 21 by 21 points separated by steps of 200 µm. The top row, maps (a) to (c) show data obtained with coincidence detection between photons with the same polarization. The bottom row (d) to (f) show data obtained with coincidence detection between photons with orthogonal polarizations. Maps (a), (b), (d), and (e) are obtained from experimental data, while maps (c) and (f) are simulations based on Eq. (21). Data in map (a) has been normalized to a maximum of 1204 coincidence counts, while (d) has been normalized to a maximum of 2932 coincidence counts.
Fig. 6 Maps corresponding to a state of four ququarts, generated using a spherical lens between the PPKTP and a four-slit apperture, as shown on Fig. 2(b). The slits were 80 µm wide and had a 160 µm center-to-center spacing. Measurements were made with a 25 s integration time and a grid of 21 by 21 points separated by steps of 200 µm. The top row, maps (a) to (c) show data obtained with coincidence detection between photons with the same polarization. The bottom row (d) to (f) show data obtained with coincidence detection between photons with orthogonal polarizations. Maps (a), (b), (d), and (e) are obtained from experimental data, while maps (c) and (f) are simulations based on Eq. (20). Data in map (a) has been normalized to a maximum of 299 coincidence counts, while (d) has been normalized to a maximum of 459 coincidence counts.
The first two figures of each row, indicated respectively as Fig. 3(a), 3(d), Fig. 4(a), 4(d), Fig. 5(a), 5(d), Fig. 6(a), and 6(d), are maps of coincidence detections, while the final figure of each row, indicated respectively as Fig. 3(c), 3(f), Fig. 4(c), 4(f), Fig. 5(c), 5(f), Fig. 6(c), and 6(f) is a numerical simulation. Coincidences between detectors on the same polarization branch, indicated as SBC on the numerical simulation, are shown on the first row, while coincidences between detectors on different polarization branches, indicated as DBC on the simulation, are shown on the second row.
The first map of each set, indicated as Fig. 3(a), Fig. 4(a), Fig. 5(a), and Fig. 6(a), allows us to observe graphically the presence or absence of single-photon transverse coherence or, in quantum information terms, the degree of purity of the single-photon state. The second map, indicated as Fig. 3(b), Fig. 4(b), Fig. 5(b), and Fig. 6(b), confirms our assessment about single-photon purity by showing the product of single counts as a function of the position of both detectors. The agreement between Fig. 3(a) and Fig. 3(b) shows that there is no correlation between photons with the same polarization, as expected, and validates our inferences about single-photon state purity. The same is seen for Fig. 4(a) and 4(b), Fig. 5(a) and 5(b), Fig. 6(a) and 6(b).
Due to the fact that our detection apparatus admits coincidences between a horizontally and a vertically polarized photon belonging to different pairs, maps indicated by Fig. 3(d), Fig. 4(d), Fig. 5(d), and Fig. 6(d) contain uncorrelated noise, which lowers the visibility of interference patterns that may be present. In order to compensate for this characteristic of our apparatus, we subtracted map Fig. 3(a) from map Fig. 3(d), that is, we removed the uncorrelated contribution to the total coincidence detections. The same procedure was done for Figs. (4)–(6). The result, shown in maps of Fig. 3(e), Fig. 4(e), Fig. 5(e) and Fig. 6(e), has a higher visibility.
The state corresponding to each set of coincidence-detection maps was estimated by calculating the expected characteristic parameters via the propagation of the biphoton amplitude function Φ(xh, xv) from the crystal to the slits using Fourier Optics, and then adjusting to find values that generated simulated maps that best fit the ones obtained experimentally. Once estimated parameters were obtained, the degree of entanglement within each pair was calculated using the von Neumann entropy of the reduced state of a single photon of the pair, normalized for the dimension of the single-particle Hilbert space, that is,
where λi are the eigenvalues of ρB = TrA[ρAB], the reduced density matrix obtained by tracing over one of the particles. The normalization, meaning that Sd ∈ [0, 1] was adopted to give a clearer notion of the degree of entanglement regardless of the dimension of the Hilbert space.Due to the duality between single-particle purity and entanglement that lies at the heart of using the von Neumann entropy as an entanglement quantifier for bipartite systems, it is possible to gain information about the degree of entanglement present in each pair by observing how coincidences between one photon from each pair behave. It is important to note that, while this gives us information about the degree of entanglement, no information is gained about the state itself. For instance, every maximally entangled state will present the same map, exhibiting no single-photon coherence and therefore no interference in the coincidence map with photons from different pairs.
This analysis relies on the underlying assumption that the global two-photon state is pure. That argument is valid if we consider negligible any coupling that would lead to entanglement between the transverse position (and momentum) and other degrees of freedom, which we ignore for the purpose of our experiment. Confirmation of the validity of our assumption comes from the high visibility of interference patterns observed in coincidence detections between photons with orthogonal polarizations after subtracting spurious contributions from photons belonging to different pairs, shown in maps indicated as Fig. 3(e), Fig. 4(e), Fig. 5(e) and Fig. 6(e).
Photons from different pairs are not correlated and if there is entanglement in each pair their contribution to the DBC lowers the visibility of conditional interference patterns. This happens because, as explained before, the more a single pair is entangled, the less transverse coherence a single photon will present, thus generating a broad diffraction pattern instead of interference fringes. On the other hand, if entanglement is absent, the visibility is unaffected by the portion of DBC due to uncorrelated photons, since in this case all photons are uncorrelated and have high transverse coherence.
Figure 3 shows maps corresponding to a state of four qutrits, generated using a telescope with a cylindrical and a spherical lens between the PPKTP and a three-slit aperture, as shown on Fig. 2(a). The top row, maps Fig. 3(a) to 3(c), shows data related to coincidence detections between photons with the same polarization. In this case, vertically polarized photons were detected. It can be seen that no clear structure is present and, more specifically, no conditional interference. The simulated map, Fig. 3(c), based on estimated state parameters shows a general agreement with experimental data, albeit more noise can be observed in maps 3(a) and 3(b). On experimental maps Fig. 3(d) and 3(e) we observe a high degree of conditional interference, for photons with orthogonal polarizations. That means that the interference pattern observed in coincidences when sweeping one detector changes as a function of the position of the second detector. For example, we see a complete π change on the interference pattern detected by sweeping over x1 when x2 varies from x2 = 0 mm to approximately x2 = ±1.5 mm, despite the attenuation generated by the diffraction envelope. It is also possible to notice that the diffraction envelope is broader than predicted by simulation, as shown in maps Fig. 3(c) and 3(f). For map Fig. 3(f) the broadening is asymmetrical and occurs only along the x1 + x2 = 0 diagonal. This was also observed with double slits in [6] and is related to the degree of entanglement between photons of the same pair, since no such enlargement is observed in separable states. The effects of entanglement on the diffraction pattern were analysed on a separate work [45]. Simulated maps were generated using estimated state parameters ε = −6.2°, θ = −46.6° and the telescope phases ς(n) of Eq. (22) in the way indicated for states of the type |Ψ−〉3 presented in Eq. (19). This set of estimated parameters corresponds to a state with S3 ≈ 0.98, that is, a highly entangled state.
Figure 4 exhibits maps corresponding to a state of four qutrits, generated using a spherical lens between the PPKTP and a three-slit apperture, as shown on Fig. 2(b). An interference pattern with no conditionality can be seen on the experimental coincidence maps for photons with the same polarization, Fig. 4(a) and 4(b), and for photons with orthogonal polarizations, Fig. 4(d) and 4(e). This is a clear sign of single-photon interference, indicating that each photon has a high degree of transverse coherence. The simulated maps, Fig. 4(c) and 4(f), based on estimated state parameters shows a very good agreement with experimental data. This is consistent with our assessment that the state is pure and separable, in which case each photon behaves independently of all the others and has a high degree of transverse coherence, leading to high-visibility single-photon interference patterns. In contrast to what was observed for the entangled state of qutrits, there is no broadening of the diffraction envelope in any direction with respect to the expected for a slit of width compatible to that used in the experiment. Simulated maps were generated using estimated state parameters α = 51.8°, β = 30.8° and phases φ1 = φ2 = 0°, for the parametrization indicated for states of the type |Ψ+〉3 presented in Eq. (18). This set of estimated parameters corresponds to a state with S3 ≈ 0.03, that is, an almost separable state.
Figure 5 shows maps corresponding to a state of four ququarts, generated using a telescope with a cylindrical and a spherical lens between the PPKTP and a four-slit aperture, as shown on Fig. 2(a). The top row, maps Fig. (a) to (c) shows data related to coincidence detections between photons with the same polarization. Vertically polarized photons were detected, as in the top row of Fig. 3. We can observe the same behaviour of Fig. 3, i.e., no conditional interference is present. The simulated map, Fig. 5(c), based on estimated state parameters shows a general agreement with experimental data, maps Fig. 5(a) and 5(b). On experimental maps for photons with orthogonal polarizations, Fig. 5(d) and 5(e), we observe a high degree of conditional interference. For example, we see a complete π change on the interference pattern detected by sweeping over x1 when x2 varies from x2 = 0 mm to approximately x2 = 0.75 mm, but the attenuation caused by the diffraction envelope is still present [6, 45]. It is also possible to notice that the diffraction envelope is broader than predicted by simulation as shown in maps Fig. 5(c) and 5(f). For map Fig. 5(f) the broadening is asymmetrical and occurs only along the x1 + x2 = 0 diagonal, as occurred in the experimental data related to qutrits (Fig. 3). Simulated maps were generated using estimated state parameters ε = −5.6°, θ = 68.9°, and ν = 88.8°, as well as the telescope phases ς(n) of Eq. (22) in the way indicated for states of the type |Ψ−〉4 presented in Eq. (21). This set of estimated parameters corresponds to a state with S4 ≈ 0.98, that is, a highly entangled state.
Figure 6 exhibits maps corresponding to a state of four ququarts, generated using a spherical lens between the PPKTP and a three-slit apperture, as shown on Fig. 2(b). The top row, maps Fig. 6(a) to 6(c) shows data related to coincidence detections between photons with the same polarization. As in the previous results, vertically polarized photons were detected to obtain the experimental maps Fig. 6(a) and 6(b). The maps presented here are similar to those of Fig. 4, an interference pattern with no conditionality can be seen on both maps, which is a clear sign of single-photon interference, indicating that each photon has a high degree of transverse coherence. The simulated map, Fig. 6(c), based on estimated state parameters shows a very good agreement with experimental data. On experimental maps Fig. 6(d) and 6(e) we observe the same high-visibility interference pattern, with no conditionality, for photons with orthogonal polarizations. This is a clear indication that the state is pure and separable, in which case each photon behaves independently of all the others and has a high degree of transverse coherence, leading to high-visibility single-photon interference patterns. In contrast to what was observed for the entangled state of ququarts, there is no broadening of the diffraction envelope in any direction with respect to the expected for a slit of width compatible to that used in the experiment. Simulated maps were generated using estimated state parameters α = 63.7°, β = 49.6°, γ = 37.8°, and phases φ1 = φ2 = φ3 = 0°, for the parametrization indicated for states of the type |Ψ+〉4 in Eq. (20). This set of estimated parameters corresponds to a state with S4 ≈ 0.04, that is, an almost separable state.
5. Conclusions and perspectives
In this work we realized an experimental generalization to higher dimensions of a previous scheme for generating two copies of qubit states belonging to a subset of the two-qubit Hilbert space defined by experimental constraints. Much in the same way, the qudit states we generated belong to subsets constrained by the gaussian profile of the pump beam and the symmetric illumination of the multiple slits.
Although material restrictions allowed us to generate only a few states, we have shown that both high and low entanglement can be obtained. The degree of entanglement and overall characteristics of the state were manipulated by altering the propagation of the biphoton amplitude function to the plane of the slits. A two-lens astigmatic telescope generated two photon-pairs in a highly-entangled state with positive correlation between qudits, while a single lens in the f -f configuration generated two pairs in an almost-separable state belonging to a different subset of the two-qudit Hilbert space.
We have shown that by observing the behavior of two-photon interference patterns generated by photons originated from a single pair and also by one photon from each of the two pairs copies in the same state we are able to confirm the degree of entanglement within each pair and ascertain the general characteristics of the state, without performing a complete characterization. Such process would require a full tomographic reconstruction, which would be unfeasible for four qutrits or four ququarts due to the large amount of different measurements required. Moreover, if the degree of entanglement is the only information of interest, observation of patterns generated by photons with the same polarization is sufficient.
Such detection of entanglement is based on the duality between the purity of the single-particle state and the amount of entanglement between the two components of the global system, which we assume to be pure. This assumption is backed by the fact that there is very little coupling between the transverse modes used to encode the qudits and other degrees of freedom of the photons generated by SPDC.
The present experimental setup, together with a Spatial Light Modulator (SLM), operating to produce amplitude and phase modulation in the down-converted photons, can be used to create and manipulate all possibilities for the {|Ψ−〉} and {|Ψ+〉} sets of states.
Acknowledgments
The authors wish to acknowledge the financial support of CNPq, CAPES and FAPEMIG Brazilian funding agencies. P.-L. de Assis was supported by CAPES Young Talents Fellowship Project number 88887.059630/2014-00.
References and links
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