Quantum state tomography of orbital angular momentum photonic qubits via a projection-based technique

A qubit, i.e. a two-dimensional quantum system, evolves in a Hilbert space spanned by two basis vectors usually denoted $|0\rangle$ and $|1\rangle$ in analogy with classical information. For pure state, it can be represented by:

$$\begin{eqnarray}&&|\Psi \rangle =\alpha |0\rangle +\beta |1\rangle ,\end{eqnarray} \tag{ 1 }$$

with $|\alpha {{|}^{2}}+|\beta {{|}^{2}}=1$.

The tomography of such superposition states requires measurements performed in three different bases [23]. In addition to the logical basis $\{|0\rangle ,|1\rangle \}$, two additional mutually unbiased bases can be defined as the superpositions: $\{|\pm \rangle =(1/\sqrt{2})(|0\rangle \pm |1\rangle )\}$ and $\{|\pm {\rm i}\rangle =(1/\sqrt{2})(|0\rangle \pm {\rm i}|1\rangle )\}$. Measuring the qubit in the logical basis will yield the values of $|\alpha {{|}^{2}}$ and $|\beta {{|}^{2}}$ and measurements in the two superposition bases will provide the relative phase between the coefficients α and β. This sequence of measurement actually allows to perform the reconstruction of the density matrix of any mixed state, which in the general case can be expressed as :

$$\begin{eqnarray}\hat{\rho }=\frac{1}{2}\left( \hat{\mathbb{I}}+\mathop{\sum }\limits_{i=1}^{3}{{S}_{i}}{{{\hat{\sigma }}}_{i}} \right)=\frac{1}{2}\left( \begin{array}{ccccccccccccccc} 1+{{S}_{3}} & {{S}_{1}}-{\rm i}\;{{S}_{2}} \\ {{S}_{1}}+{\rm i}\;{{S}_{2}} & 1-{{S}_{3}} \\ \end{array} \right),\end{eqnarray} \tag{ 2 }$$

where the ${{\hat{\sigma }}_{i}}$ are the Pauli matrices and the ${{S}_{i}}={\rm Tr}\left( \hat{\rho }{{{\hat{\sigma }}}_{i}} \right)$ coefficients are usually called Stokes parameters when dealing with polarization states. The S_i coefficients indicate the relative weights of either basis state in the different bases. Indeed, they can written as ${{S}_{1}}={{p}_{|+\rangle }}-{{p}_{|-\rangle }}$, ${{S}_{2}}={{p}_{|+{\rm i}\rangle }}-{{p}_{|-{\rm i}\rangle }}$ and ${{S}_{3}}={{p}_{|0\rangle }}-{{p}_{|1\rangle }}$, where ${{p}_{|\Psi \rangle }}$ is the probability to measure the qubit in the state $|\Psi \rangle$. If the qubit is in a pure state as expressed in equation 1, then

$$\begin{eqnarray*}&&\hat{\rho }=|\Psi \rangle \langle \Psi |={{\left| \alpha \right|}^{2}}|0\rangle \langle 0|+{{\left| \beta \right|}^{2}}|1\rangle \langle 1|+\alpha {{\beta }^{*}}|0\rangle \langle 1|+\beta {{\alpha }^{*}}|1\rangle \langle 0|,\end{eqnarray*}$$

and the Stokes parameters can be easily related to the parameters α and β:

$$\begin{eqnarray*}&&{{S}_{1}}=2\operatorname{Re}(\alpha {{\beta }^{*}})\qquad {{S}_{2}}=-2{\rm Im}(\alpha {{\beta }^{*}})\qquad {{S}_{3}}=|\alpha {{|}^{2}}-|\beta {{|}^{2}}.\end{eqnarray*}$$

In this work, we focus on qubits encoded in the OAM degree of freedom. We now briefly review the essential features of the OAM of light, focusing on the Laguerre–Gaussian modes that offer a convenient basis to describe it.

The set of Laguerre–Gaussian modes is a complete orthonormal basis of solutions to the paraxial propagation equation. They are equivalently the eigensolutions of the 2D harmonic oscillator in cylindrical coordinates. Their transverse shape is propagation invariant, as can be seen from the following expression of their electric field amplitude:

$$\begin{eqnarray}\begin{array}{rcl} {\rm LG}_{p}^{l}(r,\theta ,z) & = & {{\mathcal{E}}_{0}}\frac{{{K}_{lp}}}{w(z)}{{\left( \frac{r\sqrt{2}}{w(z)} \right)}^{|l|}} \\ {} & {} & \times {{{\rm e}}^{{\rm i}l\theta }}{{{\rm e}}^{-{{(\frac{r}{w(z)})}^{2}}}}{\rm L}_{p}^{|l|}\left( \frac{2{{r}^{2}}}{w(z)} \right){{{\rm e}}^{-{\rm i}k\frac{{{r}^{2}}}{2R(z)}}}{{{\rm e}}^{{\rm i}(2p+|l|+1)\zeta (z)}}, \\ \end{array}\end{eqnarray} \tag{ 3 }$$

where ${{\mathcal{E}}_{0}}$ is the electric field amplitude, ${{K}_{lp}}=\sqrt{\frac{2}{\pi }\frac{p!}{(l+p)!}}$ is a normalization constant, and ${\rm L}_{n}^{\alpha }(x)$ are the generalized Laguerre polynomials. Here, we have chosen a wave traveling along the z axis and polar coordinates (r, θ) to parametrize the transverse plane. The parameters $w(z)={{w}_{0}}\sqrt{1+{{(z/{{z}_{R}})}^{2}}}$, ${{z}_{R}}=\pi w_{0}^{2}/\lambda$, $R(z)=z\left( 1+{{({{z}_{R}}/z)}^{2}} \right)$, and $\zeta (z)={\rm arctg}(z/{{z}_{R}})$ are the radius, Rayleigh length, radius of curvature and Gouy phase for a beam of waist w₀ at a wavelength λ.

In contrast to the standard ${\rm TE}{{{\rm M}}_{00}}={\rm LG}_{p=0}^{l=0}$ mode, the higher-order Laguerre–Gaussian modes have a rotating phase profile ${{{\rm e}}^{{\rm i}l\theta }}$ with a singularity at the origin. Due to this rotating phase, the local Poynting vector has a non-vanishing component along the orthoradial direction and the beam thus exhibits an OAM around this axis. The OAM carried by each photon in such a mode is equal to the index $l\in \mathbb{Z}$ (in ℏ units), which is also equal to the circulation of the phase around the axis divided by $2\pi$. As the LG modes are eigenfunctions of the propagation equation, their transverse shape, and hence the OAM of the photons, is preserved. This makes the OAM number l a relevant quantum number for information encoding. As required for the smoothness of the electric field amplitude, the phase singularity is associated with an intensity nulling in the middle as the $2l$ th power of the radial coordinate ${{(r/w(z))}^{2|l|}}$. This feature gives the LG modes their characteristic doughnut-shaped intensity profiles.

The other index, $p\in \mathbb{N}$, describes the radial shape of the beam. With each additional p unit, the amplitude has an additional sign change along the radius and the intensity an additional zero-value ring. This number has a less straightforward interpretation than the orbital index l and has been subject to various recent theoretical and experimental investigations [24–27].

One promise of the OAM degree of freedom for quantum information encoding is the large (potentially infinite) dimension of the available Hilbert space. We first restrict ourselves to a two-dimensional space spanned by the modes ${\rm LG}_{p=0}^{l=\pm 1}$. Extensions of the tomography scheme to higher-dimensional OAM spaces are discussed in section 6.

For a single photon living in this qubit space, we define the corresponding logical basis vectors as $|0\rangle \;=\;|l=+1\rangle =|{\rm R}\rangle$ and $|1\rangle =|l=-1\rangle =|{\rm L}\rangle$. R and L respectively refer to the right and left handedness of the helical wavefront. The OAM difference of two between the modes defining the logical basis ensures a very good distinguishability as will be explained later. As the index p does not play a significant role in the present study, we drop it in the following and we consider only modes with radial index p = 0.

Zoom In Zoom Out Reset image size

Equally-weighted superpositions $|\Psi \rangle =|{\rm R}\rangle +{{{\rm e}}^{{\rm i}\phi }}|{\rm L}\rangle$, which span the equatorial plane of the Bloch sphere as shown in figure 2, correspond to rotated Hermite–Gaussian (HG) modes of the type ${\rm TE}{{{\rm M}}_{01}}$. These modes consist of two bright spots, separated by a dark line. In the plane orthogonal to the propagation axis, the angle ${{\alpha }_{{\rm d}}}$ of the dark axis with respect to the horizontal axis is related to the relative phase ϕ by:

$$\begin{eqnarray}&&{{\alpha }_{{\rm d}}}=(\phi -\pi )/2.\end{eqnarray} \tag{ 4 }$$

Accordingly to their spatial shapes (see figure 2) and in analogy to the case of a polarization basis, we denote the specific following modes as vertical, anti-diagonal, diagonal and horizontal:

$$\begin{eqnarray}\begin{array}{rcl} |{\rm V}\rangle & = & (|{\rm R}\rangle -|{\rm L}\rangle )/\sqrt{2}, \\ |{\rm A}\rangle & = & (|{\rm R}\rangle -{\rm i}|{\rm L}\rangle )/\sqrt{2}, \\ |{\rm D}\rangle & = & (|{\rm R}\rangle +{\rm i}|{\rm L}\rangle )/\sqrt{2}, \\ |{\rm H}\rangle & = & (|{\rm R}\rangle +|{\rm L}\rangle )/\sqrt{2}. \\ \end{array}\end{eqnarray} \tag{ 5 }$$

The bases $\left\{ |{\rm R}\rangle ,|{\rm L}\rangle \right\}$, $\left\{ |{\rm H}\rangle ,|{\rm V}\rangle \right\}$ and $\left\{ |{\rm D}\rangle ,|{\rm A}\rangle \right\}$ will constitute the three mutually unbiased bases for the tomography measurement.

Zoom In Zoom Out Reset image size

The determination of the OAM state of a bright beam of light can be done by imaging and wavefront measurements (either using interferometry or a microlens array [28]). These methods require usually many photons, so that characterizing an OAM state at the single-photon level calls for other methods. The different techniques developed so far can be classified into two categories:

• Projective-based techniques. In these methods, the photons to be measured impinge on a device that performs a projection onto an OAM eigenstate before being measured. The mode projectors are typically made of a hologram that converts an input mode with non-zero l value into a ${\rm TE}{{{\rm M}}_{00}}$ mode followed by a spatial filter (pinhole or single-mode fiber). The holograms can be either fixed [8] or dynamically programmed with a spatial light modulator [9, 16], they can be either intensity [29] or phase holograms [8], they can diffract the light to all directions or be optimized for a single output direction. In any case, the projector only selects one mode and photons in other modes are lost.
• Mode-sorting techniques. Here, the propagation direction of the signal is changed depending on its OAM value. This feature overcomes the problem of losses inherent to mode projection. However, these methods are often more challenging than the previous ones. Among them, one can cite a Mach–Zehnder interferometer in the arms of which Dove prisms have been inserted [30]. Another method that has seen significant developments in the few past years relies on a log-polar coordinate interpolation realized with two phase-modulating elements. The radial and polar coordinates r, θ in one plane are mapped one by one onto the cartesian x and y coordinates in a subsequent plane. This approach was first implemented with two spatial light modulators [31] then with fixed refractive optics [32–34]. It is also possible to take benefit from the fact that any unitary manipulation of transverse modes (and hence mode sorting) can be achieved by multiple phase modulation steps separated by optical Fourier transforms [35].

As underlined before, quantum state tomography requires projection into different bases in order to access the coherence terms of state superpositions. In this context, building on the projective techniques in order to benefit from their simplicity and high distinction ratio, we developed an OAM tomography setup, as described in the next section.

The apparatus is schematized in figure 3. The incoming state is first split using a non-polarizing beamsplitter. Each of the subsequent paths includes a mode projector onto an OAM eigenstate. These mode projectors are based on the combination of a hologram and a single-mode fiber [8]. A blazed fork phase-hologram diffracts the light and performs OAM addition or subtraction depending on its orientation. Thus, on one path, the mode ${\rm L}{{{\rm G}}^{l=+1}}$ is converted into a mode ${\rm L}{{{\rm G}}^{l=0}}={\rm TE}{{{\rm M}}_{00}},$ which is then efficiently coupled to the single-mode fiber, while any other mode is converted into a Laguerre–Gaussian beam with a non-vanishing l value and hence not coupled to the subsequent fiber. There are two such paths, denoted R and L, that are arranged to project the incoming state onto the $|{\rm R}\rangle$ and $|{\rm L}\rangle$ states respectively. The diffraction efficiency of the holograms is 80% and the coupling efficiency to the single-mode fiber is also around 80%, leading to an overall transmission around 65% for the corresponding mode. The rejection of the other mode is optimized using the alignment described later, with a typical value around $25\;{\rm dB}$.

Zoom In Zoom Out Reset image size

As shown in figure 3, the two paths are then recombined via a fiber beamsplitter with two outputs labelled X and Y. The difference in propagation length along the two arms of the interferometer causes a phase shift denoted φ. If the input is in a Laguerre–Gaussian mode $|{\rm R}\rangle$ or $|{\rm L}\rangle ,$ then light will only be coupled into one of the interferometer arms, and this light will be equally distributed over X and Y regardless of φ. In contrast, if the input is in a superposition state, then there is a non-zero amplitude in both arms and these amplitudes will interfere. The probabilities to detect light at either outputs X or Y will vary sinusoidally with φ.

More specifically, let us take the example of a pure state $|\Psi \rangle =a|{\rm R}\rangle +b{{{\rm e}}^{{\rm i}\phi }}|{\rm L}\rangle$ (with $a,b\in \mathbb{R}$) entering the device. At the output, it will be transformed into the state:

$$\begin{eqnarray}&&(a+b{{{\rm e}}^{{\rm i}(\phi +\varphi )}})|X\rangle +(a-b{{{\rm e}}^{{\rm i}(\phi +\varphi )}})|Y\rangle ,\end{eqnarray} \tag{ 6 }$$

where we have assumed perfect transmission and mode rejection. The events detected at the output X for instance will thus be proportional to

$$\begin{eqnarray}&&P={{a}^{2}}+{{b}^{2}}+2ab{\rm cos} (\phi +\varphi ).\end{eqnarray} \tag{ 7 }$$

They correspond thus to the projection of the incoming state on the state $|{\rm R}\rangle +{{{\rm e}}^{{\rm i}\varphi }}|{\rm L}\rangle$. By choosing φ, any projection basis in the equatorial plane of the Bloch sphere can thus be chosen; this is the key feature of this setup.

In summary, the interferometer directs light in a given HG state towards output X and the orthogonal mode towards output Y. These two states, i.e. the basis in which this separation occurs, depend on the value of the interferometer phase φ. For $\varphi =0$, photons in the $|{\rm H}\rangle$ state will be directed towards output X while photons in state $|{\rm V}\rangle$ will be directed towards output Y. In the same way, for $\varphi =\pi /2$, the incoming photons are measured with respect to the basis $|{\rm A}\rangle$ and $|{\rm D}\rangle$. Finally, if one of the paths is blocked, then the device will act as a projector onto the OAM eigenstate $|{\rm R}\rangle$ or $|{\rm L}\rangle$. In this case, the detectors at X and Y receive the same signal. Therefore, when the device is properly calibrated (as will be explained in section 4), it can be regarded as a black box performing state projection and yielding photon counts in the outputs X and Y as summarized in table 1.

Table 1.  Detected projections in the output X and Y as a function of the configuration of the interferometer. The relative phase φ defines the projection basis.

	L path	R path	Full interferometer
	blocked	blocked	$\varphi =0$	$\varphi =\pi$/2	$\varphi =\pi$	$\varphi =3\pi$/2
Output X	$|{\rm R}\rangle$	$|{\rm L}\rangle$	$|{\rm H}\rangle$	$|{\rm A}\rangle$	$|{\rm V}\rangle$	$|{\rm D}\rangle$
Output Y	$|{\rm R}\rangle$	$|{\rm L}\rangle$	$|{\rm V}\rangle$	$|{\rm D}\rangle$	$|{\rm H}\rangle$	$|{\rm A}\rangle$

In order to change the projection basis, one mirror inside the interferometer is mounted on a piezoelectric transducer, allowing to vary φ. In the following, we explain how to access this phase using backward-propagating reference light.

3.2.1. Phase reference beam.

Thermal and mechanical drifts continuously change the interferometer phase $\varphi$ on a scale of a few degrees in a few seconds. To access this phase, an auxiliary phase-reference beam is injected backwards into the interferometer. While it propagates backwards, the reference beam crosses the two holograms. The ${\rm TE}{{{\rm M}}_{00}}$ modes emerging out of the fibers are thus converted into an ${\rm L}{{{\rm G}}^{l=-1}}$ mode in the R path, and into an ${\rm L}{{{\rm G}}^{l=+1}}$ mode in the L path. These modes are superimposed at the input beamsplitter with a phase difference that is precisely equal to the interferometer phase φ. As a result, we get the superposition $\left( {\rm L}{{{\rm G}}^{l=+1}}+{{{\rm e}}^{{\rm i}\varphi }}{\rm L}{{{\rm G}}^{l=-1}} \right)$. Similarly to equations (4) and (5), the equal weight superposition of ${\rm L}{{{\rm G}}^{l=+1}}$ and ${\rm L}{{{\rm G}}^{l=-1}}$ with a relative phase $\varphi$ results in a rotated HG ${\rm TE}{{{\rm M}}_{01}}$ mode, consisting of two bright spots with a dark line between them. The dark line makes an angle

$$\begin{eqnarray}&&{{\alpha }_{{\rm d}}}=(\varphi -\pi )/2\end{eqnarray} \tag{ 8 }$$

with the horizontal axis. Measuring this angle enables us to access the interferometer phase $\varphi$. We now detail how to analyze the images taken by the digital camera to efficiently extract this information.

3.2.2. Image analysis routine.

In a first step, all images are enhanced by applying a median filter (to reduce high-frequency noise and dead pixels) and a midtone-stretching filter (to increase the contrast in the middle intensity region and to reduce variations in the high- and low-intensity regions).

When φ varies, the dark line in the intensity pattern of the phase reference beam rotates around the beam axis, according to equation (8). First, the center around which the dark line rotates has to be determined. For this purpose, many images for different values of φ are required, covering roughly uniformly the whole range of $\varphi \in [0,2\pi ]$ (corresponding to the range of $[0,\pi ]$ for the angle ${{\alpha }_{{\rm d}}}$ of the dark line). Averaging them results in a ring-shaped image, as shown in table 2, to which a doughnut-like distribution is fitted. The fit provides two parameters: the position of the center and the radius of interest. As long as the alignment of the reference path is not changed, these values remain valid for all images. Consequently, this initial procedure has to be performed only once for a measurement series: either during the calibration, if a real-time analysis of the phase is desired, or using a part of the stored images, if post-processing is performed.

Table 2.  Illustration of the steps involved in determining the center of symmetry of the images. The image displayed in the first column is the average of a few hundred phase-reference pictures. After treating it with the median filtering and midtone stretching operations explained in the text, the second image is obtained. Their difference shows the importance of the image processing; without it, the background would be strong enough to shift the center out of its actual position. In the second image, the projections of the intensity onto x and y axes are shown. They are used to obtain the starting values for the fit: the first-order momenta give an estimated center position (indicated with a red dot), the second-order momenta a starting value for the width of the ring. The last column shows the fit output: the obtained center is depicted in red and the circle with the radius of interest in green.

The following analysis of the individual images is illustrated in figure 4. The center is used as the origin of polar coordinates, while the radius of interest defines the area that will be analyzed.

Zoom In Zoom Out Reset image size

• This circular area is first divided into N angular bins ('pie slices'), where N has to be divisible by 8 so that there will be angle bins corresponding to each of the four $|{\rm H}\rangle ,|{\rm D}\rangle ,|{\rm V}\rangle$ and $|{\rm A}\rangle$ modes² . In our analyses, a typical value of N was 120.
• For each angular bin k, the average intensity $I({{\alpha }_{k}})$ is calculated. The first half of these data (0–180°) is plotted in figure 4c as open red circles. We could now fit a sinusoidal function to $I({{\alpha }_{k}})$ to determine ${{\alpha }_{{\rm d}}}$ as its phase. However, since we have to process many images and since fits are computationally expensive, we choose the following straight-forward calculation instead.
• Since we are only interested in an axis and not a direction, the angle bins are folded: The intensities of each two opposing bins, i.e. at 180° from each other, is added. This leaves us with only half the number of bins.
• The dark line should be along the axis of least intensity, but also orthogonal to the axis with most intensity. To account for both conditions at once, we subtract from the intensity of each bin the intensity of the (unique) bin at 90° from it. These last two steps (folding and subtracting) also rectify slight image asymmetries that are always present (see explanations below and figure 5).
• To reduce the influence of remaining bin-to-bin noise, the data is smoothed: the processed intensity $\tilde{I}({{\alpha }_{k}})$ of bin k is calculated as the average over a 45° wide sector centered around that bin. The last bin $N/2-1$ is considered being next to bin 0, i.e. the angle is taken modulo 180°. The data $\tilde{I}$ is shown in figure 4c as blue dots.
• The angle of the bin with the smallest $\tilde{I}$ is the angle ${{\alpha }_{{\rm d}}}$ of the dark axis, providing the interferometer phase φ via equation (8).

Zoom In Zoom Out Reset image size

Since the alignment of the beams from the L and R paths is subjected to experimental imperfections, the image analysis routine has been tested against computer-generated images presenting various simulated defects. For this, we numerically generated superpositions of Laguerre–Gaussian beams with either positional or angular misalignment. Examples of such test images are given in figure 5.

The simulated beams have a more pronounced deformation than the experimental phase reference beam. Yet, the image analysis routine yields the correct result for each of them, i.e., the algorithm finds the bin corresponding to the angle ϕ that was used to calculate the respective mode superposition.

3.2.3. Timing and noise issues.

3.2.4. Polarization and wavelength of the reference beam.

We show now that it is highly desirable to use the same polarization and the same wavelength for the reference light as for the signal.

First, the polarization-maintaining fibers and fibered beamsplitter, but also dielectric mirrors are birefringent, so the interferometer phase will be different for signal and reference if they have different polarizations. Furthermore, the birefringence changes with the mechanical stress of the fibers and with the temperature drifts, thus this difference will not stay constant.

Second, if the two optical paths differ geometrically by $\Delta L$, two beams at different wavelengths will accumulate different interferometer phases, leading to a dephasing

$$\begin{eqnarray}&&\varphi =(2\pi /\lambda )\cdot n\cdot \Delta L.\end{eqnarray} \tag{ 9 }$$

The variation of the measured interferometer phase around a certain wavelength $\lambda =c/\nu$ can then be easily calculated as:

$$\begin{eqnarray}&&({\rm d}\varphi /{\rm d}\nu )\approx 12{}^\circ /({\rm cm}\;{\rm GHz})\times \Delta L,\end{eqnarray} \tag{ 10 }$$

where ν is the light frequency. So with a path difference $\Delta L$ on the order of a few centimeters, a difference by a few hundreds of MHz in the light providing the phase reference beam is already enough to change the inferred value of φ by several tens of degrees.

Finally, dispersion can also play a significant role since a part of the interferometer is fibered. In a single-mode fiber, the change of the effective refractive index is dominated by the dispersion of the material [39]. We can thus estimate the dispersion ${\rm d}n/{\rm d}\nu$ of our silica fibers to be on the order of ${{10}^{-3}}/100\;{\rm nm}$ [40]. In a perfectly symmetrical situation, the first-order contribution of dispersion vanishes. However, even if the optical path lengths are precisely equal, there might be a difference in their composition in terms of free-space and fibered lengths. Let us call this difference in fiber length $\Delta {{L}_{{\rm fib}}}$. With this, we find:

$$\begin{eqnarray}&&({\rm d}\varphi /{\rm d}\nu )\approx -0.1{}^\circ /({\rm cm}\;{\rm GHz})\times \Delta {{L}_{{\rm fib}}}.\end{eqnarray} \tag{ 11 }$$

This effect will be smaller than the previous one for typical configurations, but can still play a role if signal and reference are separated by several GHz.

If the frequencies of signal and reference are different, but stay constant, these two contributions lead, first, to a constant offset that could be determined, and second, to a different change of the phase when varying the path length. The latter difference is proportional to the relative wavelength difference and can thus in many cases be neglected for close wavelengths. As soon as the frequencies vary however, especially with respect to each other, the correlation between signal and reference interferometer phase will be lost.

In this section, we have presented a detailed description of our OAM detection setup, and discussed the relevant issues relative to the phase-reference light. We now turn to the detailed calibration procedure and provide the main figures of merit that have been measured thoroughly.

The goal of the alignment procedure is twofold: an optimal coupling of the respective LG mode into the fiber of the respective branch and a strong rejection of all the other modes, as demanded by the mode selection requirement. The alignment is performed using classical bright light fields aligned with the signal to be analyzed later, their path being defined by the same single-mode fiber. A spatial light modulator is inserted on the way and enables to send various spatial modes into the setup.

In a first step, the position of the holograms is set by monitoring the intensity distribution with a camera placed behind the hologram. This distribution strongly depends on the relative position of the incoming beam and the hologram center. Sending in a ${\rm L}{{{\rm G}}^{l=+1}}$ mode, we can thus optimize the observed intensity distribution to be close to a ${\rm TE}{{{\rm M}}_{00}}$ profile in the R path (table 3). In the same way, the position of the L path hologram is optimized by sending in a ${\rm L}{{{\rm G}}^{l=-1}}$ mode. Even a small deviation of the hologram by a few micrometers becomes clearly visible. Then, the coupling into the fibers is optimized. Using two mirrors behind each of the holograms, we are able to adapt the mode exiting the hologram by maximizing the coupling efficiency up to 80%.

Table 3.  Mode transformation and subsequent filtering. (a) Mode conversion performed by the holograms. After fine alignment of the hologram center, the impinging mode ${\rm L}{{{\rm G}}^{l=+1}}$ (respectively ${\rm L}{{{\rm G}}^{l=-1}}$) is converted into a ${\rm TE}{{{\rm M}}_{00}}$ mode in the far field of the R path (respectively L path), while other modes are converted to higher l-valued modes with doughnut shapes. The goal of this alignment is to maximize the subsequent coupling of the desired mode in the optical fiber and to minimize the coupling of all other modes. (b) Coupling efficiency in each path for various Laguerre–Gaussian modes at the input.

The next stage consists in sending a ${\rm TE}{{{\rm M}}_{00}}$ mode and using the same mirrors to now minimize its coupling. The rejection pattern of the setup is more pronounced than the acceptance and thus allows a better approach to the optimal point. Finally, a random search in the region around the found optimum allows to do some fine tuning. Here, all six degrees of freedom (two transversal positions of the hologram and four directions for the two mirrors) are slightly varied while switching between the coupled mode and the unwanted modes (such as ${\rm TE}{{{\rm M}}_{00}}$ ). This way, the ratio ${{\eta }_{{\rm others}}}/{{\eta }_{{\rm LG}}}$ is minimized (where ${{\eta }_{\Lambda }}$ is the coupling efficiency of a mode Λ into the single-mode fiber), while the coupling ${{\eta }_{{\rm LG}}}$ of the targeted mode is kept at or close to the maximum.

Each path is finally characterized by measuring the couplings for the different LG modes sent into the setup. In the R path for example, we record the transmission of the mode corresponding to that path (${\rm L}{{{\rm G}}^{l=+1}}$), the two neighbors in l number (${\rm L}{{{\rm G}}^{l=+2}}$ and ${\rm L}{{{\rm G}}^{l=0}}={\rm TE}{{{\rm M}}_{00}}$), and the mode corresponding to the other path (${\rm L}{{{\rm G}}^{l=-1}}$). An average rejection of 17 dB for the next neighbors (2% transmission) is typically obtained. For the suppression for the opposite mode (at $\Delta l=2$), a rejection of 23 dB was obtained in the worst case, and up to 37 dB in the best case. The typical value is 25 ± 3 db³ . An example of detailed coupling figure of merit is given in table 3.

An additional characterization is performed by sending classical beams carrying the four HG modes H, V, D and A, as shown in figure 6. Theoretically, all four modes should lead to the same power of coupled light into both fibers. However, the power balance is not strictly mode-independent. Most of this imbalance can be explained by the imperfect mode filtering given in table 3. In the R path for instance, 82% transmission for mode ${\rm L}{{{\rm G}}^{l=+1}}$ and $0.5\%$ for mode ${\rm L}{{{\rm G}}^{l=-1}}$ make up for a ±6% difference in transmission between different HG modes. These mode-selective losses decrease the count rate and can additionally lead to a reduction of the fringe visibility, leading in turn to a decrease in the observed fidelity (see section 4.4).

Zoom In Zoom Out Reset image size

Finally, the phase reference circuit is started and the interferometer phase slowly scanned. Sending again the modes H, V, D and A, the respective fringes are measured and correlated to the interferometer phase obtained from the reference. These fringes allow to check first for a good fringe visibility (typical values above 93% are obtained, mainly limited by the residual imbalance in the coupling of the HG modes and by the background noise) and second for the correct phase relation between signal and reference. Figure 6 illustrates this process. As seen before, the power detected at the output X should exhibit a sinusoidal dependence in $\varphi =2{{\alpha }_{{\rm d}}}+\pi$ (equation 4). The condition for minimum power, i.e. $\varphi =\theta +\pi \leftrightarrow 2{{\alpha }_{{\rm d}}}=\theta$ resulting from equation (6), allows to deduce the value of θ from the position of the resulting fringes. We checked that this was indeed the case and found good agreement within $\pm 3{}^\circ$.

The setup enables thus to accurately project an input state on to various target modes. We now discuss the overall detection efficiency of the setup.

The overall loss of the device comes from various contributions:

• Input-beamsplitter and filtering: $\sim 50\%$. The combination of the $50/50$ beamsplitter at the input and the subsequent mode selection causes a 50% fraction of the signal to go 'into the wrong path' whatever the input mode. This fraction is filtered out and lost. It does not account for the quality of the coupling into the mode selecting fibers.
• Hologram diffraction and other optics: $\sim 80\%$ transmission.
• Fiber coupling efficiency: $\sim 80\%$.
• Beamsplitter for back-propagating the phase reference beam: $\sim 25\%$ losses. Larger splitting ratios, such as 90:10 or even 99:1 would reduce these losses, which are thus not intrinsic to our scheme.

From all these parameters, one can extract the detection efficiency: $0.5\times 0.8\times 0.8\times 0.75\sim 24\%\pm 3\%$ (without including the quantum efficiency of the detectors).

In our implementation, the detection efficiency for the HG modes was slightly lower than for the LG modes. This was due not to the detection setup but to the imperfection in the mode preparation. Indeed, the probe modes were generated by a reflection on a phase-only spatial light modulator. Yet, this method is limited by the intrinsic mode overlap between an ideal (target) mode and the sharply phase-modulated beam with gaussian envelope produced by the SLM. This overlap is lower for HG modes (64%) than for LG modes (78%) [35]. The rest of the power goes to higher-order $p\gt 0$ modes that are not detected due to spatial filtering. In practice, this generates some additional mode-selective losses but cannot degrade the quality of the tomography (visibility and then fidelity), as they are the same for each mode in every pair of orthogonal modes.

Experimental imperfections prevent from measuring unit fidelities even for ideal qubit states at the input. In this section, we discuss the upper limits on the fidelity values that can be measured.

The fidelity for HG states is especially sensitive to a reduced visibility V, setting a limit of ${{F}_{{\rm max} }}=(1+V)/2$. The reduced visibility can originate from a contamination with background noise, an imperfect mode filtering or a mode-dependent fiber-coupling. An imbalance of $\pm \Delta \eta$ in the fiber coupling of orthogonal HG modes will decrease the visibility of the fringe by approximately ${{(\Delta \eta )}^{2}}/2$. A 99% fringe visibility, as achieved in some of the reported results without noise subtraction, leads to a maximum fidelity of $99.5\%$.

For LG modes at the entrance, no fringe should be present in the ideal case when scanning φ. Therefore, their fidelity is insensitive to the previous visibility reduction. However, a small relative leakage of mode L in path R (and vice versa) leads to a spurious fringe with a visibility $2\sqrt{\epsilon }$. This leakage translates into a maximum fidelity equal to $1-\epsilon$. With a leakage of less than −25 dB, the resulting error on the fidelity is limited to a fraction of a percent.

Finally, a shift in the interferometer calibration by an angle $\Delta \theta$ will also decrease the maximum fidelity by an amount ${{(\Delta \theta )}^{2}}$. Even for a $\Delta \theta$ on the order of $5{}^\circ$, the fidelity decreases thus by less than a percent. As can be seen from figure 6, this phase shift was controlled to better than $3{}^\circ$ in our implementation.

In summary, this section has shown the reliability of our setup for OAM qubit measurements. Before moving on to some example of measurements performed with (approximate) single photons, let us recall three key parameters:

• 1.  
  Cross-talk suppression (in LG basis): ∼25 dB.
• 2.  
  Precision in phase measurement: better than 3°.
• 3.  
  Detection efficiency (excluding detectors): $24\%\pm 3\%$.