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Abstract
In this paper, we directly image the orbital angular momentum (OAM) correlations, expressed in the Bessel-Gauss (BG) basis, present in quantum-entangled photon pairs produced by the process of spontaneous parametric downconversion (SPDC). We use a spatial light modulator that displays an appropriate phase mask, followed by coupling into a single-mode fiber, so as to project the signal, or heralding, photon onto a BG mode, and verify that the idler, or heralded, photon is projected non-locally as expected. In contrast with similar experiments relying on Laguerre-Gauss (LG) modes, our current experiment permits, firstly, full index control (allowing the experimenter to define both the azimuthal index ℓs and the continuous radial index krs, also referred to as scaling parameter). Importantly, while not resolving the radial index leaves the heralded single photon in a statistical mixture of all available radial modes, the ability to determine both indices allows us to herald single photons in a particular BG mode, described by a quantum-mechanically pure state. Our use of BG modes permits, secondly, the spatially-resolved detection of the heralded single photon, both in the near and far fields, with a time-gated intensified CCD camera, making it possible to experimentally determine both the radial and azimuthal indices of the heralded single photon. Thirdly, the fact that for BG modes the radial index (scaling parameter) is continuous makes it possible to use the scaling parameter as a continuous adjustment in quantum state engineering, for precise mode matching, or for the selection of photon-pair properties such as the spiral bandwidth, as we experimentally demonstrate. We believe that our work opens up interesting new possibilities in the field of quantum communications based on the spatial degree of freedom of photon pairs.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Quantum entanglement is one of the most intriguing features of quantum mechanics and forms the basis for the development of next-generation technologies, such as quantum communications [1–3], quantum computation [4–6], and quantum imaging [7–10]. While the early development of quantum communication protocols exploited polarization entanglement [11,12], the spatial degree of freedom expressed discretely through the orbital angular momentum (OAM), is highly promising since it permits straightforward scaling to higher dimensions, in contrast to polarization which is limited to a dimension of 2.
Spontaneous parametric down conversion (SPDC) represents the process of choice for the generation of photon pairs correlated or entangled in their OAM degree of freedom. Each of the three waves, pump ($p$), signal ($s$), and idler ($i$), involved in the SPDC process may exhibit OAM, as quantified by the topological charge $\ell _\mu$ with $\mu =p,s,i$. Under conditions of azimuthal symmetry, OAM is conserved [13–15] i.e. $\ell _p=\ell _s+\ell _i$, and the resulting two-photon quantum state presents entanglement. Thus, for the pump with a specific topological charge $\ell _p$, the state is a coherent superposition of $\vert \ell _s \rangle _s \vert \ell _p-\ell _s\rangle _i$, over a range of $\ell _s$ values. The Laguerre-Gauss (LG) mode family has been used as basis in a number of relevant experiments which explore OAM photon pair correlations and entanglement [16–23]. Spatial light modulators (SLM’s) have been successfully used in some of these works for displaying appropriate phase masks which can project the signal and idler photons to particular modes and thus probe the underlying correlations [23]. Nevertheless, it is experimentally challenging to obtain phase-only masks yielding LG modes with arbitrary values of both the azimuthal index $\ell$ (corresponding to the topological charge), and the radial index $p$ [18,24]. In most SPDC experiments to date using LG modes and SLM’s, a helical phase holographic mask $\exp (-i \ell \phi )$ is used to convert an LG mode field component with the opposite phase $\exp (i \ell \phi )$ to a an $\ell =0$ mode (Gaussian mode) which can couple well into a single mode fiber. In this manner one can selectively project the incoming field to a desired $\ell$ value. An important caveat, is that through this approach the resulting projected mode is left in an uncontrolled statistical mixture of modes with different $p$ values (and a fixed $\ell$ value). It therefore becomes impossible to address an arbitrary LG mode, as defined by particular values of both indices $\ell$ and $p$, with the resulting inability to exploit the full LG basis for quantum communication protocols.
Of course, we have the freedom to choose a different basis in which to perform our experiment. In this paper we show that the Bessel-Gauss (BG) mode family leads to some key advantages over LG modes. First, by switching to the BG mode family we are now able to define an appropriate SLM phase mask which allows us to determine both indices, the continuous radial scaling parameter $k_{r}$ as well as the azimuthal index $\ell$. Defining both indices allows us to herald single photons in a single, well-defined BG mode, described by a quantum-mechanically pure quantum state, to be possibly used in a wide variety of quantum communications protocols. Second, BG modes have a particularly simple angular spectrum morphology (in the form of a single intensity ring with its radius in transverse momentum coordinates equal to the scaling parameter $k_{rs}$). This implies that from a spatially-resolved, far-field measurement of the heralded photon it becomes possible to directly experimentally access the scaling parameter $k_{rs}$, complementing a near-field measurement from which the absolute value of the azimuthal index can be obtained. Third, the fact that the radial index (scaling parameter) is continuous makes it possible to use it as a continuous adjustment in quantum state engineering, for precise mode matching, or for the selection of photon-pair properties; indeed, we experimentally demonstrate in this paper how the scaling parameter can be used, in conjunction with the SPDC crystal length and pump focusing strength, to control the resulting spiral bandwidth.
In this paper we thus extend previous work from our group in which we directly imaged the OAM correlations in the two-photon state generated by type-I SPDC [25]. We project the signal (heralding) photon onto a specific BG mode (defined by user-selected radial and azimuthal indices), and spatially-resolve, using a time-gated intensified CCD camera [26–28], the idler (heralded) in both the near and far fields. Our work explores further the capabilities of spatially-resolved detection schemes at the single photon level, which we believe will be useful in future implementations of free-space quantum communications and quantum information technologies.
2. Theory
The process of type-I spontaneous parametric down conversion (SPDC) in a collinear configuration conserves orbital angular momentum (OAM) if the experimental arrangement allows for the collection of all the emitted wavevectors [25,29,30], i.e. $\hbar \ell _p = \hbar \ell _s + \hbar \ell _i$ is satisfied for the pump(p), signal(s) and idler(i) photons carrying OAM values of $\hbar \ell _j$, with $j = p, s, i$, where $\ell _j$ is the azimuthal index for each of the three fields. It is then natural to decompose the two-photon state in a a rotational symmetric basis. In this work we will exploit the Bessel-Gauss (BG) family of modes as basis, by writing the quantum state as [26]
A Bessel-Gauss beam can be defined as a conical superposition of Gaussian beams, characterized by a beam waist parameter $\omega _0$, with a cone opening half-angle $\arcsin (k_r/k)$, where $k$ is the wavenumber and $k_r$ is the transverse wavenumber. For a BG beam of order $\ell$ as experimentally observed in the near-field, the transverse amplitude as a function of the propagation distance $z$ and in terms of the polar coordinates $\{\rho ^\bot ,\phi \}$, can be written as follows [31,32]
In the far-field, the angular spectrum (AS) of the BG beams presents a characteristic ring shaped transverse intensity distribution, as described by
Note that Bessel Gauss beams (as opposed to Bessel beams) are defined by the width of the Gaussian envelope $w_0$, in addition to the azimuthal and radial indices. The angular spectrum (i.e. the intensity pattern in the far-field) is characterized by a single intensity ring with its radius in transverse momentum space equal to the scaling parameter $k_{rs}$, and its width inversely proportional to $w_0$, according to $w_0=4/\delta k$, where $\delta k$ is the ring width. Therefore, the width of the Gaussian envelope can be conveniently retrieved from an experimental measurement of the angular spectrum (obtained as the far-field intensity pattern).
In this work we use a spatially-resolved heralded single photon detection scheme to directly observe the idler-photon transverse distribution in the near-field and in the far-field, while projecting the entangled two-photon state in the Bessel-Gauss basis, as described by Eq. (4) and Eq. (5) respectively.
3. Experiment
The experimental setup is shown in Fig. 1. A diode laser (PL) at $405.2nm$ is used to pump a $\beta$ barium borate (BBO) crystal to produce collinear degenerate photon pairs via type-I spontaneous parametric down conversion (SPDC). The pump beam is spatially-filtered by coupling through an aspheric lens (AL$_1$) into a single mode fiber (SMF$_1$) and subsequent outcoupling, with a second aspheric lend (AL$_2$) back into free space. The resulting spatially-filtered beam pass a half wave plate (HWP) for fine adjustment of the linear polarization orientation and is then focused into the crystal, defining a spot size of radius $w_0$ (as controlled by the focal length of the lens used, labelled as L$_1$). In the experiments described below, we have used two source configurations: configuration A, involving a BBO crystal of length $L=2mm$ and a pump spot size of $w_0=300\mu m$, and configuration B, involving $L=1mm$ and $w_0=600\mu m$.
Fig. 1. Experimental setup for producing OAM-entangled photon pairs via collinear type-I SPDC, with projection of the heralding signal photon into a well-defined BG mode and a spatially-resolved detection of the heralded idler photon in the near-field (DP) and the far-field (DP$_2$ in the inset). The measurement in the near-field provides information about the azimuthal index $\ell$, while the spatially-resolved observation of the BG mode in the far-field represents a direct measurement of the scaling factor $k_r$ (see below). The inset shows the additional lens L$_{11}$ required to probe the far-field (DP$_2$) of the heralded idler photon.
The photon pairs are first spectrally filtered by the BPF element (composed of a long-pass filter which transmits wavelengths $\lambda > 500nm$, followed by a $810 \pm 5nm$ band-pass filter), and then split probabilistically into two separate arms by a beamsplitter (BS). The crystal’s output plane is imaged onto two identical image planes (IP$_1$ in the reflected arm, for the idler photon, and IP$_2$ in the transmitted arm, for the signal photon), using a bifurcated 4f telescope with 2.4$\times$ magnification, formed by a plano-convex lens L$_2$ (with focal length f$_2$ = 125$mm$) and a plano-convex lenses L$_3$ (with f$_3$ = 300$mm$).
Our experiment operates as follows (the relevant details are provided below). The signal photon in the transmitted arm is phase modulated (by a spatial light modulator, SLM) and coupled into a single mode fiber (SMF$_2$), which leads to detection in an avalanche photodiode (APD). The idler photon in the reflected arm is detected, with spatial resolution, by an intensified CCD (ICCD) camera which is time-gated by the electronic pulse produced for each signal-photon detection event at the APD. Note that prior to reaching ICCD, the idler photon is transmitted through an image-preserving delay line (OD) designed to compensate for the insertion delays of the ICCD camera and of the APD (see appendix A). In this manner, we observe the spatial structure of the idler photon in the near and far fields, as heralded by the detection of the phase-modulated signal photon.
The phase mask displayed on the SLM corresponds to the following transmission function [26]
where $\rho ^\bot$ is the radial coordinate, $\phi$ is the azimuthal angle, and $\mbox {sign}(.)$ represents the sign function. The effect of the phase $\exp (-i \ell _0 \phi )$ is to suppress an optical vortex phase component in the incoming field proportional to $\exp (i \ell \phi )$ with $\ell =\ell _0$, so that it may couple well into the single mode optical fiber (SMF$_2$). The result of this phase $\exp (-i \ell _0 \phi )$, acting on its own, is that a field component corresponding to a superposition of BG modes with different scaling factors $k_r$ but the same azimuthal index $\ell _0$ is permitted to couple into the fiber. By adding the factor $\mbox {sign}[ J_\ell (k_r \rho ^\bot ) ]$, we are able to effectively also determine a fixed value for the scaling factor $k_r$. The end result is that the field reflected from the SLM is projected onto a BG mode with user-selected values for both the azimuthal index $\ell _0$ and for the scaling factor $k_r$. Note that the slight ellipticity in the modes produced is linked to astigmatism in the overall optical system.In Fig. 2 we show examples of the phase distribution used on the SLM for all combinations of a selection of $\ell$ values ($\ell =1,5,8$) and a selection of $k_r$ values ($k_r=0,15,30 rad/mm$). As a test of these SLM transmission functions, we couple the beam from a diode laser at $808 nm$, in backpropagation through the single mode fiber SMF$_2$, so that a Gaussian mode reaches the SLM and the resulting intensity pattern is recorded on the crystal plane. These resulting intensity patterns are experimentally recorded, and shown in each of the insets, indicating that the system correctly projects onto the desired mode determined by $\ell$ and $k_r$.
Fig. 2. Phase masks displayed on the SLM used to convert a $k_{rs} = k_r$ and $\ell _s = \ell$ BG signal-photon into a $\ell _s = 0$ Gaussian mode. The inset shows the BG spatial modes defined by the scaling parameter $k_r$ and azimuthal index $\ell$ obtained, as a test, in backward propagation from the single-mode fiber SMF$_2$ to the crystal plane.
Returning to the detailed description of our setup, in the transmitted arm, a 4f telescope with 2$\times$ magnification – consisting of lenses L$_4$ (with focal length f$_4$ = 100$mm$) and L$_5$ (f$_5$ = 200$mm$) – images the single photon on plane IP$_2$ to a new plane IP$_3$, set to coincide with a spatial light modulator (SLM). Plane IP$_3$ (phase-modulated by the SLM) is then relayed onto plane IP$_4$ by three consecutive demagnifying 4f telescopes (with magnifications $0.24\times$, $0.4\times$, and $0.016\times$, respectively), formed by: i) lenses L$_6$ (with focal length f$_6= 250mm$) and L$_7$ (with f$_7= 60mm$), ii) lenses L$_8$ (f$_8= 250mm$) with L$_9$ (f$_9= 100mm$), and iii) lens L$_{10}$ (with f$_{10}= 500mm$) and aspheric lens AL$_3$ (with $f_{al3}= 8mm$). The single photon on plane IP$_4$ is then coupled into a single mode fiber (SMF$_2$) with its input plane set to coincide with this plane.The fiber SMF$_2$ leads to a Si avalanche photodiode (APD) which detects the signal photon.
In the reflected arm, the idler photon on plane IP$_1$ is transmitted through an image-preserving optical delay line (OD), which introduces a delay of around $115 ns$ (as determined by the insertion delay in both the ICCD and the APD), in order to reach the detection plane (DP) of an ICCD camera (Andor iStar 334T); see appendix A for details of the OD. Since plane DP corresponds to an image of plane IP$_1$, placing the detection array of the ICCD camera on DP leads to the detection of the idler photon in the near-field of the crystal. Alternatively, the idler photon’s transverse amplitude may be Fourier transformed by transmission through a lens $L_{11}$ (with focal length $f_{11}=250mm$) placed a distance $f_{11}$ from DP to define a new plane DP$_2$ a distance $f_{11}$ from L$_{11}$, as shown in the inset of Fig. 1. Placing the ICCD on plane DP$_2$ leads to the detection of the idler photon in the far-field [7]. An important feature of our experiment is the ability to observe the transverse intensity of the heralded idler photon either in the near-field or in the far-field, as selected by the experimenter.
Let us turn to a discussion of how the parameters defining a mode can be obtained experimentally. We note that the radius of the inner-most intensity ring in the near-field intensity pattern (for $\vert \ell \vert \ge 1$) is a monotonically increasing function of $\vert \ell \vert$ so that it becomes possible with adequate calibration to infer the value of $\vert \ell \vert$ in a particular experiment (note that in the case of $\ell =0$ can be straightforwardly identified since it is the only one which yields an intensity maximum at the origin). We also note that the radius of the single intensity ring appearing in the far-field directly yields the scaling parameter $k_r$. A Bessel-Gauss mode, as opposed to a Bessel mode, is determined by the width of the Gaussian apodization envelope $w_0$, in addition to the $\ell$ and $k_r$ parameters already discussed. If the $w_0$ value is required, it can be conveniently inferred from the single intensity ring width in the far-field (as discussed in the theory section above) [32].
Note that a direct determination of the topological charge $\ell$ of the heralded idler photon, including its sign, could be accomplished through far-field diffraction through a triangular aperture. We followed such a strategy for photons encoded in the LG basis, and of course also could be used in the present case for BG modes [25,32].
In a full experimental run, for each combination of signal-photon projection values $\ell _s$ and $k_{rs}$, we record for the idler photon, both, the near-field and far-field intensity patterns (obtained as the spatially-resolved coincidence counting rate). We have performed such an experimental run, for source configuration A (crystal length $L=2mm$ and pump spot size $w_0=300 rad/mm$), and for $\ell _p=0$, for all combinations of $\ell _s$ values within the range $-15 \le \ell _s \le 15$ with the following $k_{rs}$ values: $0,15,30 rad/mm$. For each combination of $k_{rs}$ and $\ell _s$ we have collected data over a period of $400$s. We present in Figs. 3 (for near-field measurements) and 4 (for far-field measurements) data for a subset of the full resulting measurement set (for $\ell _s$ values $-10,-5,0,5,10$). The full data, with other $\ell _s$ and $k_{rs}$ value combinations, is available upon request. In the case of the near-field measurements an inset is included for each measurement, showing the phase distribution programmed on the SLM for that specific measurement (the phase masks are identical for the far-field measurements). The measured near-field and far-field idler-photon intensity patterns, as a function of the user-selected values $\ell _s$ and $k_{rs}$, constitute the direct imaging of the signal-idler correlations which result from the OAM entangled two-photon state.
Fig. 3. Spatially-resolved coincidence count rate between the signal photon, projected onto particular $k_{rs}$ and $\ell _s$ values with the help of a spatial light modulator (SLM) followed by coupling into a single-mode fiber, and the idler photon detected in the near-field by a time-gated ICCD camera. Each plot is labelled by the $\ell _s$ and $k_{rs}$ (in $rad/mm$ units) values. The axes represent the camera pixels, with the colorbars expressed in coincidence events per 400 seconds. The insets shows the corresponding phase masks (central region shown) displayed on the SLM.
Fig. 4. Spatially-resolved coincidence count rate between the signal photon, projected onto particular $k_{rs}$ and $\ell _s$ values with the help of a spatial light modulator (SLM) followed by coupling into a single-mode fiber, and the idler photon detected in the far-field by a time-gated ICCD camera. Each plot is labelled by the $\ell _s$ and $k_{rs}$ (in $rad/mm$ units) values. The axes represent the camera pixels, with the colorbars expressed in coincidence events per 400 seconds. For $k_{rs} \neq 0$ cases, we show the measured $k_{ri}$ values (in $rad/mm$ units) near the bottom of each plot, corresponding well with the $k_{rs}$ projection values used.
From the far-field correlation imaging data (Fig. 4) it is clear that, as expected, each given $k_{rs}$ projection value (with $k_{rs}\neq 0$) defines a family of heralded (idler) modes, each with its angular spectrum in the form of a single intensity ring of radius $k_{ri}=k_{rs}$. Thus, the signal-idler correlations in the two-photon state imply that the resulting heralded (idler) photon scaling parameter value $k_{ri}$ should be identical to the corresponding heralding (signal) photon value $k_{rs}$. In each panel of Fig. 4, for $k_{rs}\neq 0$ values, we show near the bottom the experimentally obtained value of $k_{ri}$, clearly matching well the corresponding $k_{rs}$ value. Note that the ability to determine both indices $\ell _s$ and $k_{rs}$ for the heralding photon, implies that the heralded photon will be described by a single, well-defined BG mode corresponding to a quantum-mechanically pure state, with indices $\ell _i=-\ell _s$ and $k_{ri}=k_{rs}$.
Let us now turn our attention to the question of how to obtain the spiral spectrum from near-field correlation imaging data (Fig. 3). Indeed, we may directly obtain, for a given selection of scaling factor $k_{rs}$, the coefficients $|C_{(\ell _s,-\ell _s)}|^2$ from the total intensity summed over all pixels in the data corresponding to a particular $\ell _s$. In Fig. 5 we show for source configuration A ($L=2mm$ and $w_0=300\mu m$) the resulting spiral spectrum for the three values of $k_{rs}$ which we have considered ($k_r=0$, shaded in black, $k_r=15 rad/mm$, shaded in dark blue, and $k_r=30 rad/mm$, shaded in light blue). It may may be appreciated that the spiral spectrum becomes flatter (leading to a larger spiral bandwidth) as one increases the value of $k_{rs}$.
Fig. 5. Measured spiral spectrum $|C_{(l,-l)}|^2$ of the two-photon state (for source configuration A), obtained from the projection of the signal photon to BG modes with three different values of the scaling parameter $k_{rs}$ ($0,15,30 rad/mm$) and the azimuthal index $\ell _s$.
In Fig. 6 we show, for comparison, the effect on the spiral spectrum of switching from source configuration A ($L=2mm$ and $w_0=300 \mu m$ ), together with $k_{rs}=0$, to configuration B ($L=1mm$ and $w_0=600 \mu m$), together with $k_{rs}=30 rad/mm$. These two situations have been selected since they are highly contrasting, as quantified by the spiral bandwidth. While in the first case the spiral bandwidth (full width at half maximum) is around $\Delta \ell =16$, in the second case the bandwidth is much larger than the experimentally accessible $\ell$ range, so that $\Delta \ell \gg 31$. Note that because the radial index for BG modes (scaling parameter) is continuous, in contrast to LG modes for which it is discrete, it may be used to continuously adjust photon pair quantities such as the spiral bandwidth as indeed our experimental results in Figs. 4 and 5 indicate. This could certainly constitute a useful feature for quantum state engineering.
Fig. 6. Comparison of the measured spiral spectrum $|C_{(l,-l)}|^2$ in two situations: i) in black source configuration A with $k_{rs}=0$ and ii) in red source configuration B with $k_{rs}=30 rad/mm$.
Thus, as known from previous work [26,27], it is possible to engineer the photon-pair quantum state, along with the scaling parameter of the heralding photon, to select the resulting spiral bandwidth over a wide range of possible values.
4. Conclusions
We have carried out an experiment in which, for a type-I collinear spontaneous parametric downconversion (SPDC) photon-pair source, we directly image the resulting OAM signal-idler correlations when expressed in the Bessel-Gauss basis. In particular, we project the heralding signal photon onto a Bessel-Gauss (BG) mode with user-selected indices $k_{rs}$ and $\ell _s$, and detect the idler photon, with transverse-spatial resolution, in both the near and far fields. The signal-photon projection is accomplished with a spatial light modulator which displays an appropriate phase mask, followed by coupling into a single-mode fiber. The idler photon is detected with spatial resolution by an intensified CCD camera, which is time-gated by the electronic pulse produced by each signal-photon detection event. We show how to retrieve the spiral spectrum from our imaged OAM correlation data, and in addition show how the scaling parameter, along with crystal length and pump focusing strength, can be used to select the resulting spiral bandwidth. Switching to an experimental analysis based on BG, instead of Laguerre-Gauss, modes leads to three distinct advantages: i) the ability to set values for both azimuthal and radial indices, and therefore herald a single photon in a unique BG mode, described by a quantum-mechanically pure state, ii) the ability to experimentally determine both indices solely from the heralded arm: the radial index from a far-field spatially-resolved measurement, and the absolute value of the azimuthal index from a corresponding near-field measurement, and iii) allows us to use the scaling parameter (radial index) as a continuous adjustment for photon-pair properties, in particular for the resulting spiral bandwidth. We believe that these results open up interesting possibilities for future work on the exploitation of the spatial degree of freedom in photon-pair sources.
Appendix A: Image-preserving optical delay line
The image preserving optical delay line (OD in Fig. 1) exploits the same configuration as reported in an earlier work form our group [25]. The delay line, shown in Fig. 7, is formed by eight consecutive 1 $\times$ telescopes set in a forward and backward folded-path configuration, propagating the crystal’s image plane located at IP$_1$ up to the ICCD’s near-field detection plane defined at DP, introducing a delay of around $115ns$.
The idler photon at IP$_1$ propagates with the $p$-polarization transmitted by the polarizing beamsplitter (PBS), placed at the focal plane of the first 1$\times$ telescope, formed by two bi-convex $2"$ diameter, $500mm$ focal length lenses (LD$_1$ and LD$_2$). The next three consecutive 1$\times$ telescopes are formed by two bi-convex ($2"$ diameter and $1000mm$ focal length) lenses, LD$_3$ to LD$_8$. The folded path configuration is accomplished by placing a quarter wave plate (QWP) prior to the last mirror (M$_8$), flipping the polarization of the idler photon from $p$ to $s$. After passing again through the three $1000mm$ 1 $\times$ telescopes, the $s$-polarization photon is reflected at the PBS, defining a new optical path including an additional 1 $\times$ telescope, formed with a $2"$ diameter $500mm$ focal length lens (LD$_9$), relaying the propagated image in a near-field configuration to plane DP.
Funding
Consejo Nacional de Ciencia y Tecnología (Fronteras de la Ciencia grant 217559, grant 293471); Universidad Nacional Autónoma de México (UNAM) (grant IN104418); Air Force Office of Scientific Research (grant FA9550-18-1-0346).
Acknowledgments
We acknowledge support from CONAYCT, Mexico, PAPIIT (UNAM), and AFOSR.
Disclosures
The authors declare no conflicts of interest.
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