Programming Quantum Measurements of Time inside a Complex Medium

TL;DR

The paper presents a scalable method to perform generalized, high-dimensional time-bin measurements by harnessing the spatial–temporal coupling in a multimode fiber. By measuring the fiber's multi-spectral transmission matrix and constructing a time-resolved transmission matrix, the authors define tau-modes that travel unscattered in time and can be coherently mixed to emulate large, unbalanced interferometers inside a single fiber. They demonstrate programmable time-bin measurements in dimensions up to $d=11$ using a $40\,\mathrm{m}$ MMF, achieving high-fidelity mutual unbiased basis measurements and validating the approach via quantum tomography. This fiber-based, common-path scheme reduces experimental overhead compared to cascaded Franson interferometers and holds promise for scalable quantum information tasks in long-distance networks and photonic quantum computing, albeit with ongoing refinements needed to mitigate drift and loss at higher dimensions.

Abstract

The temporal degree-of-freedom of light is incredibly powerful for modern quantum technologies, enabling large-scale quantum computing architectures and record key-rates in quantum key distribution. However, the generalized measurement of large and complex quantum superpositions of the time-of-arrival of a photon remains a unique experimental challenge. Conventional methods based on unbalanced Franson-type interferometers scale poorly with dimension, requiring multiple cascaded devices and active phase stabilization. In addition, these are limited by construction to a restricted set of phase-only superposition measurements. Here we show how the coupling of spatial and temporal information inside a single multi-mode fiber can be harnessed to program completely generalized measurements for high-dimensional superpositions of photonic time-bin. Using the multi-spectral transmission matrix of the fiber, we find special sets of spatial modes that experience distinct dispersive delays through the fiber. By exciting coherent superpositions of these spatial modes, we engineer the equivalent of large, unbalanced multi-mode interferometers inside the fiber and use them to perform high-quality measurements of arbitrary time-bin superpositions in up to dimension 11. The single fiber functions as a scalable, common-path interferometer for time-bin qudits that significantly eases the experimental overheads of standard approaches based on unbalanced Franson-type interferometers, serving as an essential tool for quantum technologies that harness the temporal properties of light.

Figures (9)

• Figure 1: $\tau$-modes of a multi-mode fiber (MMF): A random spatial mode $|r\rangle$ input into an MMF will generally scatter temporally and spectrally. The scattering behavior of an MMF is characterized by its multi-spectral transmission matrix $T(\omega)$, which can be Fourier-transformed to obtain its time-resolved transmission matrix $\tilde{T}(t)$. This matrix can be used to construct a special set of spatial $\tau$-modes ($|\tau_0\rangle$,$|\tau_1\rangle$,$|\tau_2\rangle$) that arrive at distinct times and are spectrally unscattered. A photon prepared in a coherent superposition of these spatial modes $|\tau_X\rangle=\frac{1}{\sqrt{3}}(|\tau_0\rangle+|\tau_1\rangle+|\tau_2\rangle)$ would arrive in a non-separable superposition of time and spatial modes $\frac{1}{\sqrt{3}}(|t_0\rangle|\tau_0\rangle+|t_1\rangle|\tau_1\rangle+|t_2\rangle|\tau_2\rangle)$.
• Figure 2: Programming generalized measurements of time-bin inside a multi-mode fiber (MMF).a) Schematic showing how a generalized, unbalanced Mach-Zehnder (Franson) interferometer can be used to measure an arbitrary 3-dimensional superposition of time-bins $|t_{in}\rangle$. The shapes (circle, triangle, square) are used as visual aids for labeling each time-bin. $|t_{in}\rangle$ traverses a color-coded, tunable phase-shifter $f_i$ and fixed delay in each arm, resulting in five peaks after the interferometer. The central peak contains a superposition of all three time-bins with desired phase-shifts introduced on each. In this manner, the Franson interferometer acts as a projective measurement of any phase-only time-bin superposition $|\psi\rangle = \frac{1}{\sqrt{3}}(f_0|t_0\rangle + f_1|t_1\rangle+ f_2|t_2\rangle)$. Abbreviations: Single-mode fiber, SMF; Superconducting nanowire single-photon detector, SNSPD. b) In our alternative approach, a digital micromirror device (DMD) is used to shape the spatial mode of the time-bin superposition $|t_{in}\rangle$ into an arbitrary superposition of MMF $\tau$-modes depicted in the inset (green). As each $\tau$-mode experiences a different delay in the MMF, the fiber effectively functions as a three-mode unbalanced interferometer as in a), with path replaced by spatial mode. A spatial light modulator (SLM) after the MMF is used to coherently transform the output $\tau$-mode superposition into a Gaussian mode that is coupled into an SMF. The resulting five peaks are the same as those produced in a), except that the MMF approach does not require interferometric stabilization and allows for arbitrary (not just phase-only) superpositions to be measured. In addition, the number of time-bins and their separations can be tuned simply by using an MMF with a different core diameter and length, respectively (see text for more details).
• Figure 3: Malus' law for a time-bin qubit. A sinusoidal intensity pattern is observed when we prepare a time-bin qubit $|{t}_{L}\rangle=\frac{1}{\sqrt{2}}(|t_0\rangle-i|t_1\rangle)$ and implement a tunable measurement $|t(\theta)\rangle\langle t(\theta)|$ on it. The phase $\theta$ is varied by changing the $\tau$-mode superposition hologram on the DMD (see Fig. \ref{['fig:concept_interferometer']}b). Sets of three interference peaks measured at the output when implementing time-bin measurements (i) $|t_L\rangle\langle t_L|$, (ii) $|t_+\rangle\langle t_+|$, and (iii) $|t_R\rangle\langle t_R|$ in our measurement device. The central peak corresponds to the probability of the measured time-bin state.
• Figure 4: Tomography of four-dimensional time-bin measurements. The left and right-most plots depict the phases and absolute values of 16 tomographically reconstructed measurement operators, $\hat{\mathcal{O}}$. These operators target the 16 possible MUB projectors for 4-dimensional time-bin superpositions $|{v}^\mu_{a}\rangle\langle{v}^\mu_{a}|$, which are shown in the middle for comparison. We observe an average fidelity of $96.24\%$ between the reconstructed measurement operators and their corresponding target projectors.
• Figure 5: Tomography of eleven-dimensional time-bin measurements. The top row shows the angle and absolute values of three reconstructed time-bin MUB measurement operators $\hat{\mathcal{O}}$ in dimension $d=11$ ($|v^{1}_1\rangle\langle v^{1}_1|$, $|v^{1}_2\rangle\langle v^{1}_2|$ and $|v^{1}_6\rangle\langle v^{1}_6|$). Their corresponding target projectors, shown in the bottom row, are seen to agree qualitatively with the reconstructed measurement operators. We calculate a fidelity of $84.2\%$, $86.3\%$, and $83.7\%$ to the target measurements with $a=1$,2, and 6 respectively.