Robust and stable delay interferometers with application to $d$-dimensional time-frequency quantum key distribution

We investigate experimentally a cascade of temperature-compensated unequal-path interferometers that can be used to measure frequency states in a high-dimensional quantum distribution system. In particular, we demonstrate that commercially-available interferometers have sufficient environmental isolation so that they maintain an interference visibility greater than 98.5\% at a wavelength of 1550 nm over extended periods with only moderate passive control of the interferometer temperature ($< \pm0.50 ^{\circ}$C). Specifically, we characterize two interferometers that have matched delays: one with a free-spectral range of 2.5 GHz, and the other with 1.25 GHz. We find that the relative path of these interferometers drifts less than 3~nm over a period of one hour during which the temperature fluctuates by $<\pm$0.10 $^{\circ}$C. When we purposely heat the interferometers over a temperature range of 20-50 $^{\circ}$C, we find that the temperature sensitivity is different for each interferometer, likely due to slight manufacturing errors during the temperature compensation procedure. Over this range, we measure a path-length shift of 26 $\pm$ 9 nm/$^{\circ}$C for the 2.5 GHz interferometer. For the 1.25 GHz interferometer, the path-length shift is nonlinear and is locally equal to zero at a temperature of 37.1 $^{\circ}$C and is 50 $\pm$ 17 nm/$^{\circ}$C at 22 $^{\circ}$C. With these devices, we realize a cascade of 1.25 GHz and 2.5 GHz interferometers to measure four-dimensional classical frequency states created by modulating a stable and continuous-wave laser. We observe a visibility $>99\%$ over an hour, which is mainly limited by our ability to precisely generate these states. Overall, our results indicate that these interferometers are well suited for realistic time-frequency quantum distribution protocols.