Multimode Time-Delay Interferometer for Free-Space Quantum Communication

Quantum communication schemes such as quantum key distribution (QKD) and superdense teleportation provide unique opportunities to communicate information securely. Increasingly, optical communication is being extended to free-space channels, but atmospheric turbulence in free-space channels requires optical receivers and measurement infrastructure to support many spatial modes. Here, we present a multimode Michelson-type time-delay interferometer using a field-widened design for the measurement of phase-encoded states in free-space communication schemes. The interferometer is constructed using glass beam paths to provide thermal stability, a field-widened angular tolerance, and a compact footprint. The performance of the interferometer is highlighted by measured visibilities of $99.02\pm 0.05\mathrm{\%}$ and $98.38\pm 0.01\mathrm{\%}$ for single- and multimode inputs, respectively. Additionally, high-quality multimode interference is demonstrated for arbitrary spatial-mode structures and for temperature changes of $\pm 1.0{}^{\circ }\mathrm{C}$. The interferometer has a measured optical-path-length drift of $130\mathrm{nm}/{}^{\circ }\mathrm{C}$ near room temperature. With this setup, we demonstrate the measurement of a two-peaked multimode single-photon state used in time-phase QKD with a visibility of $97.37\pm 0.01\mathrm{\%}$.

Figure 1

A simplified schematic of a Michelson-type interferometer. The beam splitter (BS) splits the input beam between two paths. The beams recombine at the beam splitter and exit at two separate output ports. This setup includes a small air gap at the end of the long arm ($d_2$).

Figure 2

The simulated change in the optical path difference as a function of deviations of the input about the chief ray of the multimode interferometer.

Figure 3

The simulated thermal performance of the multimode interferometer. (a) The thermal stability for small temperature drifts of the complete system (denoted by the red circle) and (c) the simulated optical-path-difference shift. (b) The interference-phase tuning via control over the temperature of the short arm $d_1$ (denoted by the red oval) and (d) the simulated optical-path-difference shift.

Figure 4

A schematic of the experimental setup used to characterize the interference quality and thermal performance of the interferometer (gray box): SM, single-mode; MM, multimode; TEC, thermoelectric cooler; ADC, analog-to-digital converter.

Figure 5

The temperature control of the interferometer bottom plate. (a) The settling time of the plate temperature after a step change of $2^{\circ }\mathrm{C}$. The target temperature is stable after approximately 3 min. (b) The long-term temperature stability at a fixed set point. The temperature is controlled with an accuracy of $\pm {0.0015}^{\circ }\mathrm{C}$.

Figure 6

The interference phase tuning with short-arm temperature control. The output from the interferometer is plotted as a function of the square of the voltage supplied to the heater element on the short arm, showing a tuning range over a full interference fringe.

Figure 7

Interference fringes for SM and MM (50-$\mu \mathrm{m}$-core-fiber) input beams as a function of the input voltage to scan the laser-diode current. Each data point (black) is the average of ten measurements and the error bars indicate the standard deviations of those ten measurements. The red lines show a fit to the data using the function given by Eq. (10). (a) An SM interference fringe. The visibility of the interference is ${\mathcal{V}}_{\mathrm{SM}}=99.02\pm 0.05\mathrm{\%}$. The high-order fringe structure in the constructive interference could be due to reflections from multiple surfaces within the interferometer. (b) An enlarged view of the SM-fringe minimum. (c) An MM interference fringe. The visibility of the interference is ${\mathcal{V}}_{\mathrm{MM}}=98.38\pm 0.01\mathrm{\%}$. (d) An enlarged view of the MM-fringe minimum.

Figure 8

Spatial-mode characterization from a $50\mu \mathrm{m}$ and a $105\mu \mathrm{m}$ core fiber. (a) A schematic showing the setup used to adjust the mode profile of the beam. The spatial mode is changed by inducing bends in the fiber controlled by a positioning stage and a micrometer. (b) Spatial-mode examples for different micrometer settings from a 50-$\mu \mathrm{m}$-core fiber. (c) Spatial-mode examples for different micrometer settings from a 105-$\mu \mathrm{m}$-core fiber. The inset label in each image gives the micrometer setting and the intensity profile is recorded using a beam profiler approximately 1.5 cm away from the end of the fiber (no collimation).

Figure 9

The multimode interference as a function of the input spatial mode. The correlation coefficient between the first frame and subsequent frames as a function of the micrometer setting for 50-$\mu \mathrm{m}$- (circles) and 105-$\mu \mathrm{m}$- (triangles) core fibers is shown against the left vertical axis. Also plotted is the visibility of the interference fringe over the same micrometer range for each fiber, shown in the right vertical axis. The error in the fringe visibility is $\le 5\times {10}^{-4}$.

Figure 10

An example measurement of the output power from the interferometer while the system reaches thermal equilibrium. This particular set of data shows a temperature change from $21.0{}^{\circ }\mathrm{C}$ to $21.5{}^{\circ }\mathrm{C}$.

Figure 11

The multimode interference as a function of the temperature. (a) Fitted functions to the measured interference fringe at each temperature setting. The temperature settings for the curves, moving from left to right, are 21.0, 21.5, 21.9, 22.0, 22.5, and $23.0{}^{\circ }\mathrm{C}$, respectively. The phase of the interference at $22.0{}^{\circ }\mathrm{C}$ is set at the data point marked with an “X.” The path-length shift is calculated using points where the fringe intersects the dotted line. (b) The experimentally measured (black) and theoretically predicted (red) path-length change as a function of the temperature. (c) The interference visibility as a function of the temperature.

Figure 12

The time-phase protocol with $d=2$. (a) Temporal and (b) phase-basis states. (c) The time-delay interferometer and its action on a single wave packet. (d) The interferometer setup for a $d=2$ phase-basis measurement. (e) The interference patterns for a measurement on the $|f_0⟩$ state.

Figure 13

The interference characterization and the spatial mode for phase-state measurement. (a) Output from the plus (minus) port of the interferometer shown in blue (black), with Eq. (10) fitted to each curve shown in red. The visibility between the plus and minus ports with the phase set at the maximum of the plus port is $98.7\mathrm{\%}$. (b) An example of the spatial mode under test, measured at the output of a 25-$\mu \mathrm{m}$-core fiber.

Figure 14

Phase-state $|f_0⟩$ preparation and measurement using a 25-$\mu \mathrm{m}$-core fiber. (a) Single-photon data (blue) with a sum of Gaussian functions fitted to the data (red) for the $|f_0⟩$-state preparation. Zeros above the data denote the phase of each bin, indicating a $|f_0⟩$ state. Time-bin boundaries are shown with black dotted lines. (b) Measurement of the $|f_0⟩$ state using the multimode interferometer, showing the plus $(+)$ output port collected at detector D0 (blue) and the minus $(-)$ output port collected at detector D1 (black). Time-bin boundaries are shown with dotted lines. The resulting visibility between the bright and dark fringes in the middle time bin is $97.37\pm 0.01\mathrm{\%}$.

Figure 15

A CAD schematic of the interferometer, with input and output beam-steering and coupling optics. An overhead CAD view (a) and a perspective view (b) of the interferometer layout. (c) A photograph of the assembled interferometer. The heater tape is shown on the short-arm glass and several temperature sensors are placed at various positions on the Kovar plate. (d) A detailed view of the beam splitter and glass-rod-alignment features and of the substrate cutout for thermal isolation of the short arm.