Temporal-mode measurement tomography of a quantum pulse gate

Encoding quantum information in the photon temporal mode (TM) offers a robust platform for high-dimensional quantum protocols. The main practical challenge, however, is to design a device that operates on single photons in specific TMs and all coherent superpositions. The quantum pulse gate (QPG) is a mode-selective sum-frequency generation designed for this task. Here, we perform a full modal characterization of a QPG using weak coherent states in well-defined TMs. We reconstruct a full set of measurement operators, which show an average fidelity of 0.85 to a theoretically ideal device when operating on a seven-dimensional space. Then we use these characterized measurement operators of the QPG to calibrate the device. Using the calibrated device and a tomographically complete set of measurements, we show that the QPG can perform high-dimensional TM state tomography with 0.99 fidelity.

Figure 1

Outline of QPG operation. The QPG is a beamsplitter operating on a TM defined by the index $\alpha$. For the measurement tomography, we send coherent states $|\beta \rangle$ to the QPG and at the converted (reflected) port we measure the number of converted photons using a bucket detector, noted as $n^{\alpha \beta }$.

Figure 2

Experimental setup. A femtosecond titanium:sapphire (Ti:Sa) oscillator with repetition rate of 80 MHz is used to pump an optical parametric oscillator (OPO). The pump of the QPG is obtained from a tap-off of the Ti:Sa laser. The input signal field is prepared by attenuating the OPO output to a mean photon number of 0.1 photon per pulse by using neutral density (ND) filters. For spectral shaping, we use SLMs in a folded $4f$ setup to shape the desired spectral amplitude and phase for the both fields. Then pump and input fields are combined on a dichroic mirror (DM) and coupled to an in-house built periodically poled lithium niobate (PPLN) waveguide, held at $207{}^{\circ }\mathrm{C}$. After the PPLN waveguide, the up-converted photons with a green color are selected by a $4f$ setup and coupled to a silicon avalanche photodiode (SiAPD), through a single-mode fiber (SMF).

Figure 3

Phase-matching function of the QPG. Right: The zero gradient of the phase-matching function $\mathrm{\Phi }({\omega }_{\mathrm{in}},{\omega }_{\mathrm{out}})$ is an indicator of group-velocity matching between input signal and pump field. The diagonal white lines are marking the orientation of the pump amplitude $\alpha ({\omega }_{\mathrm{out}}-{\omega }_{\mathrm{in}})$ and the bandwidth we use in this paper. The horizontal white lines are showing the bandwidth of the $4f$ setup used to filter the SFG signal. Left: Marginal distribution of the plot on the right side. Asymmetries are due to inhomogeneity of the effective refractive index along the waveguide.

Figure 4

The first eigenvectors of the $7\times 6$ measurement operators. For each plot, the $x$ axis corresponds to the frequency detuning (from the central frequency) and the $y$ axis to the amplitude and phase. Black and green lines are the measured amplitudes and phases, respectively; shaded areas and blue lines correspond to the theoretical MUB modes. Note that the phase is $2\pi$ periodic, which is also the interval of the $y$ axis. Please note that phases are only meaningful when a significant amplitude is present.

Figure 5

Two examples of state tomography with QPG in the Hermite-Gaussian basis in five (a) and seven (b) dimensions. State vectors corresponding to each density matrix are detailed in Appendix pp1. For each state the theoretical density matrix (left), the reconstructed density matrix without QPG calibration (middle), and the reconstructed density matrix with QPG calibration (right) are plotted.

Figure 6

Matrix of projections of MUB POVM elements for five (a) and seven (b) dimensions.

Figure 7

The impact of an imperfect QPG (parametrized in $\sigma$ with arbitrary units) on fidelity and purity of (a) measurement operators, (b) state tomography without calibration, and (c) state tomography with calibration of the POVMs. The y-axis in all cases indicates the purity or fidelity, scaling from zero to one.