We propose and demonstrate experimentally a single lens design for an astigmatic mode converter that transforms the transverse mode of paraxial optical beams. As an application, we implement a controlled-not gate based on a Michelson interferometer in which the photon polarization is the control bit and the first order transverse mode is the target. As a further application, we also build a transverse mode parity sorter which can be useful for quantum information processing as a measurement device for the transverse mode qubit.
©2010 Optical Society of America
Optical implementations of quantum computation and quantum information processing can be done by manipulation of different photonic degrees of freedom. Polarization and orbital angular momentum (OAM) are two examples widely used for these implementations . The polarization state of a single photon is a typical realization of a qubit and can be mapped on the Poincaré Sphere. Recently, optical beams carrying OAM (vortices) have been also considered as an additional resource for quantum information schemes [2, 3, 4]. The Laguerre-Gaussian (LG) modes are the simplest examples of optical vortices . They are solutions of the paraxial wave equation in cylindrical coordinates. In cartesian coordinates, the Hermite-Gaussian (HG) modes are the solutions of the paraxial equation. Experimentally, both LG and HG modes are easily created by computer-generated holograms [6, 7] and transformed into each other by cylindrical lenses . Exploring the correspondence between the mathematical structure used for polarization and first order paraxial beams, a Poincaré representation of transverse modes was proposed in refs.[8, 9]. The first order modes are then considered as an additional photonic qubit.
As in classical computation with wires and logic gates, the realization of quantum computation also needs the utilization of quantum gates to manipulate the quantum information or the qubits  and quantum channels to exchange quantum information. While the polarization state of the photon can be transformed by birrefringent wave plates, the transverse mode can be transformed by astigmatic mode converters based on cylindrical lenses . Recently, it has been demonstrated the possibility to manipulate the OAM carried by photons through the polarization with a q-plate optical device that generates single-photon entangled states between polarization and orbital angular momentum [11, 12, 13].
In this work we propose and demonstrate a simplified design for the astigmatic mode converter which can be used in one arm of a Michelson inerferometer to implement a controlled-not (CNOT) gate where the photon polarization is the control qubit and its first-order transverse mode is the target. The same mode conversion scheme is then used to produce a one-dimensional parity mode sorter. Parity is one important photonic degree of freedom that can be useful in quantum information protocols, so that both one- and two-dimensional parity sorters have been proposed [14, 15].
The implementation of a mode converter that transforms an LG mode into an HG one and vice versa was initially done by Beijersbergen et al in . In their experiment, the converter consisted of two cylindrical lenses mounted parallel to each other and placed at a suitable distance forming a telescope. The astigmatic mode converter works analogously to a birrefringent wave plate that rephases the polarization components according to its optical axis. In the mode converter, the distance between the cylindrical lenses provides appropriate control of the relative phase between the components of the incoming beam in the natural basis of the mode converter (HG modes parallel and perpendicular to the cylindrical lenses). The relative phase acquired by the HG components of the incoming beam has its origin in the evolution of their individual Gouy phases inside the mode converter, as detailed in ref.. The transformation implemented by the mode converter is designated by the phase difference it impinges in the orthogonal HG components, so that a ϕ mode converter is one adjusted to implement a phase difference equal to ϕ.
In Fig. 1 we sketch the single lens π/2 mode converter. In order to test it, we first prepared either an LG or an HG mode by sending a 632.8nm He-Ne laser beam through holographic masks. The diffracted beam passes through a HWP to adjust its polarization horizontally and is transmitted through a polarizing beam splitter (PBS), then a quarter wave plate (QWP) oriented at 45° turns the beam polarization to a circular one. A cylindrical lens oriented along the vertical direction is followed by a flat mirror placed at the focal plane, where the wavefront also becomes flat. The astigmatic wavefront gets adapted to the flat mirror and is reflected back to the input lens where the trasnformation is resumed. In this sense, the single-lens mode converter (SLMC) is a folded version of the usual design. It requires less astigmatic components and provides a much easier alignment. The distance between the cylindrical lens and the flat mirror defines the Gouy phase difference between the HG components in the natural basis of the mode converter. After the double pass through the cylindrical lens, the laser beam is finally transformed to vertical polarization by a second pass through the QWP and gets reflected by the PBS, where the transformed beam is separated from the input one.
Since the SLMC is a folded version of the usual design , the separation between the lens and the flat mirror is d = f/√2 for the π/2 converter and d = f for the π converter. In Fig.(1) we also show experimental images obtained with the π/2 SLMC. The first two images show, respectively, the HG to LG and LG to HG conversion of first order modes. The third and fourth images show the LG to HG conversion of LG modes with topological charge l = 2 and l = 3, respectively. Finally, the last two images show the LG to HG conversion of LG modes with higher order radial numbers produced by a Fabry-Perot cavity. All images demonstrate the efficient work of the SLMC.
In order to estimate the fidelity of the transformations, we used a confocal Fabry-Perot cavity as the measurement device of the transverse mode. First, we sent an input Laguerre-Gaussian (LG) beam through the Fabry-Perot cavity and measured the transmitted intensity. This allowed us to calibrate the measurement device for cavity losses. Then the input beam was transformed by the SLMC, and the Hermite-Gaussian (HG) mode so produced was also sent through the Fabry-Perot cavity. At this point the beam intensity was increased to compensate for losses in the SLMC and provide the same intensity entering the Fabry-Perot cavity.
For a perfect conversion of the LG into an HG mode, the intensity of the HG mode transmitted by the confocal cavity should be the same as the transmitted LG mode. The ratio between the transmitted intensities of the HG and LG modes gives us an estimate of the probability that a single photon passing through the mode converter is correctly transformed to an HG mode. The fidelity of the state ρ produced by the SLMC with respect to the ideal ∣HG〉 state is . Therefore, the square root of the intensity ratio gives us an estimate of the fidelity provided by the mode converter. Our experimental results give a ratio around 70%, corresponding to a fidelity of 0.84.
2. The CNOT gate
A controlled NOT (CNOT) gate has two input qubits, known as the control and the target. The action of the gate is described as follows, if the control is set to 0, then the gate leaves the target unchanged, if the control qubit is set to 1, then the gate flips the target qubit. The optical implementations of the CNOT gate reported so far involve either a Mach-Zender (MZ)  or a Sagnac  interferometer where the target qubit was flipped depending on the path followed by the photon in the interferometer. In a practical implementation, each design has its advantages. While a Sagnac interferometer offers a robust phase balance between the two paths, the MZ interferometer makes it easier to intervent in each path independently.
Here, we propose an implementation of an optical CNOT gate with the single-lens mode converter inserted in one arm of a Michelson interferometer. In our proposal, the linear polarization is the control qubit represented by ∣0〉1 = ∣H〉 and ∣1〉1 ≡ ∣V〉 for horizontal and vertical polarization, respectively, and the first order HG mode is the target qubit represented by ∣0〉2 = ∣h〉 and ∣1〉2 ≡ ∣ν〉 for (HG10) and (HG01), respectively.
The experiment is illustrated in Fig. 2. An HG mode is produced by diffraction of a linearly polarized laser on a holographic mask. Both the initial orientation of the HG mode and the linear polarization are controlled before the interferometer. An incoming horizontally polarized beam is transmitted through a PBS at the entrance of the Michelson interferometer, goes through the quarter wave plate QWP1, is converted by a double pass in the π SLMC oriented at 45°, and comes back making a second pass in QWP1, being finally reflected by the PBS with vertical polarization. The π SLMC at 45° flips the horizontal and vertical HG modes. Otherwise, an incoming vertically polarized beam, is initially reflected and finally transmitted by the PBS after a double pass through QWP2 without any transformation on its transverse mode. Finally a HWP at 45° is used to restore the initial polarization state.
The experimental input and output images of the CNOT gate are presented in Fig. 3 in correspondence with the schematic representation of the gate operation. The first four images show clearly that the transverse mode is rotated for horizontal polarization and remains unchanged for vertical polarization. The last two images correspond to an input beam polarized at 45° and prepared in either a horizontal or vertical HG mode. In this case, the output beam is a spin-orbit entangled mode of the type
These modes are polarization vortices with doughnut shape images.
After the CNOT gate we introduced a second PBS (not shown in Fig. 2) and registered the image in each polarization output. The results are displayed in the rightmost column of Fig. 3), thus confirming the expected operations performed by the gate. In the last two images, the orientation of the HG mode in each PBS output is in agreement with Eq.(1).
3. Transverse mode sorter
As a further application of the SLMC, we now describe a transverse mode sorter (TMS) that splits an incoming first order transverse mode in its HG components. It is sketched in Fig. 4. In order to test this device, we used the same He-Ne laser with holographic masks designed to produce first order LG and HG modes. The input mode with horizontal polarization passes through a PBS and a QWP that transforms its linear polarization into circular. Then, half of its intensity is transmitted through a regular 50%-50% beam splitter (BS) and makes a round trip through a π SLMC placed in one arm of a Michelson interferometer. On the other arm, a mirror is mounted on a piezoelectric transducer (PZT) that controls the relative phase between the two arms. Two outputs result from the interference between the two arms. One output exits the interferometer at counterpropagation with respect to the input beam and is separated from the input by a second pass through the QWP, followed by a reflection at the PBS with vertical polarization (port 1). The other output exits the interferometer through port 2.
When phase balanced, the interferometer delivers the HG components parallel and perpendicular to the SLMC at different ports. It works as a measurement device for the transverse mode qubit in the natural HG basis of the SLMC. In order to test this functioning, we sent different input beams and measured the corresponding outputs as shown in Fig. 5. First, we sent an input beam prepared in one of the first order HG modes belonging to the natural basis of the SLMC (image 1 in Fig. 5(a)). In this case, when the interferometer is phase balanced, only one output lights up while the other one remains dark (image 2). Both outputs light up when the interferometer is unbalanced (image 3). When the phase difference between the two arms of the TMS is π, the output ports are interchanged (image 4).
We then sent either a vertical HG mode or an LG mode through the TMS, and registered the images at the two outputs for different voltages at the PZT, but now with the SLMC oriented at θ = 45° (image 1 in Figs. 5(b) and 5(c)). When the interferometer is balanced, the two output ports exhibit the expected images of the HG components parallel and perpendicular to the SLMC (image 2). The unbalanced interferometer delivers in both outputs arbitrary combinations of the elements in the natural basis of the SLMC (image 3). As before, the output ports are interchanged for a π phase difference between the two arms of the TMS (image 4). These results clearly demonstrate that the TMS works properly as a measurement device for the photonic transverse mode qubit.
In conclusion, we have implemented a single lens design for an astigmatic mode converter used to transform first order paraxial beams. Our setup requires less astigmatic components and provides a much easier alignment of the mode converter. We have also demonstrated two applications of the single lens mode converter to quantum information devices. First, we built a CNOT gate using the photon polarization as the control qubit and the transverse spatial mode as the target. We also demonstrated the operation of a transverse mode sorter that works as a measurement device for the spatial qubit. Both devices were developed with a strategy that employs a single-lens mode converter in one arm of a Michelson interferometer.
The authors acknowledge financial support provided by Instituto Nacional de Ciência e Tecnologia de Informação Quântica (INCT-CNPq-Brazil), Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ-Brazil), and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES-Brazil).
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