Proposal for automated transformations on single-photon multipath qudits

We propose a method for implementing automated state transformations on single-photon multipath qudits encoded in a one-dimensional transverse spatial domain. It relies on transferring the encoding from this domain to the orthogonal one by applying a spatial phase modulation with diffraction gratings, merging all the initial propagation paths by using a stable interferometric network, and filtering out the unwanted diffraction orders. The automation feature is attained by utilizing a programmable phase-only spatial light modulator (SLM) where properly designed diffraction gratings displayed on its screen will implement the desired transformations, including, among others, projections, permutations, and random operations. We discuss the losses in the process which is, in general, inherently nonunitary. Some examples of transformations are presented and, considering a realistic scenario, we analyze how they will be affected by the pixelated structure of the SLM screen. The method proposed here enables one to implement much more general transformations on multipath qudits than is possible with a SLM alone operating in the diagonal basis of which-path states. Therefore, it will extend the range of applicability for this encoding in high-dimensional quantum information and computing protocols as well as fundamental studies in quantum theory.

Figure 1

Example of a phase mask addressed at the SLM that we will consider in this work. Its transmission function is given by Eq. (12) for $D=8$, where the ${\mathrm{\Phi }}_l\left(y\right)$'s are diffraction gratings.

Figure 2

Sketch of the proposed setup for implementing automated state transformations on single-photon multipath qudit states. A source generates such states encoded in the $x$ direction [see Eq. (7)] with horizontal polarization. A programmable phase-only SLM addressed by a phase mask given by (12) modulates the $y$ component of the field profile, transforming the qudit state into (14). The first inset (I1) shows a representation of the photon path state at this point. The interferometric arrangement described in the text accomplishes the transfer of the encoding from $x$ to $y$ direction, generating the state (17). The cylindrical lens, which has the SLM in its focus, ensures that the diffraction orders will not separate too much and have approximately parallel propagation along the interferometer. It also ensures that they will be described at the posterior focal plane by the Fourier coefficients of the gratings displayed at the SLM screen. The transformations are defined by the phase gratings displayed at the SLM screen, as we show in Sec. 3c, and produce the unfiltered state (26). After a proper spatial filtering, the transformation (11) is accomplished, and the transformed multipath qudit state is now encoded in the $y$ direction, as illustrated in the second inset (I2).

Figure 3

The coefficients of the linear grating for the ideal (30), $N=6$, and $N=10$ pixelated (36) cases as functions of $\phi$, for the orders $j=0,\pm 1$. The phase in the plot is modulo $2\pi$. The values of $\phi$ with negative sign represent the descending sawtooth grating while the positive sign represents an ascending sawtooth grating. For each value of $\phi$, a different column of the matrix representing the transformation is defined. The interval $\{-4\pi ,4\pi \}$ shows a typical behavior for the orders $j=\pm 1$ and $j=0$. It is possible to see that this grating can be used at the regions of the SLM in order to make permutation operations, since $C_{\overline{m}}=1$ for $\phi =\overline{m}2\pi$, in the ideal case (35). The pixelation affects the diffraction efficiency for the modulus squared and can impinge differences of up to $\pi$ at the phase calculated in the ideal case. In the range and orders used in the examples of this section, the diffraction efficiency is affected by approximately 8% for the $N=6$ case. In the limit $N\to \infty$ the pixelated and ideal coefficients as functions of $\phi$ coincide.

Figure 4

The behavior of the modulus squared of the coefficients of the binary grating as well as their phase in terms of $\phi$ for the $j=0,\pm 1$ orders (40). These grating coefficients are periodic, with period $2\pi$; hence the interval chosen for $\phi$ in the plots is $\{0,2\pi \}$. At $\phi =(2k+1)\pi$, with $k\in {\mathbb{N}}_{+}$, the order $j=0$ is null and changes its phase by $\pi$; all the other orders reach their maximal modulus value. The coefficients of $j<0$ have the same squared modulus as the $j>0$ coefficients, but with a phase difference of $\pi$. This grating has $C_j=0$ for orders with even $j$. We show a qutrit projection possible with this grating for particular values of $\phi$, using this grating in three regions of the SLM, for a qutrit initial state [Eq. (41)].

Figure 5

The behavior of the modulus squared and phases of the coefficients of the ideal and pixelated ($N=6$ and $N=10$) triangular gratings as a function of $\phi$, for the orders $j=0,\pm 1$ [Eqs. (44) and (48)]. As $\phi$ goes further from zero, higher orders starts to have $|C_j{|}^2$ not negligible, making this grating a good candidate to work with higher dimensions for the Hilbert space of the final state. The coefficients of $j<0$ are identical to the $j>0$ coefficients and at $\phi =2\pi$, the order $j=0$ for the ideal case is null. However, the differences in the squared modulus due to pixelation can be severe and the plot of the modulus is made up to $\phi =10\pi$ to show this effect and its dependence on $\phi$: $C_0$ reaches unity for more values of $\phi$ than in the ideal case and $C_j$ is no longer equal to $C_{-j}$, differing by a phase factor. This odd behavior in the modulus happens for higher values of $\phi$ as $N$ increases and as the phase difference decreases; in the limit $N\to \infty$ the pixelated and ideal coefficients coincide, as expected. The phase plots are made modulo $2\pi$.