Direct Temporal Mode Measurement for the Characterization of Temporally Multiplexed High Dimensional Quantum Entanglement in Continuous Variables

Field-orthogonal temporal mode analysis of optical fields has recently been developed for a new framework of quantum information science. However, so far, the exact profiles of the temporal modes are not known, which makes it difficult to achieve mode selection and demultiplexing. Here, we report a novel method that measures directly the exact form of the temporal modes. This, in turn, enables us to make mode-orthogonal homodyne detection with mode-matched local oscillators. We apply the method to a pulse-pumped, specially engineered fiber parametric amplifier and demonstrate temporally multiplexed multidimensional quantum entanglement of continuous variables in telecom wavelength. The temporal mode characterization technique can be generalized to other pulse-excited systems to find their eigenmodes for multiplexing in the temporal domain.

Figure 1

Conceptual diagram for the entangled states in various temporal modes ($A_k$, $B_k$) from a pulse-pumped parametric process.

Figure 2

Simulation results for the amplitudes (top) and converged phases (bottom) of the output spectral functions ${\psi }_k({\mathrm{\Omega }}_s)$ (${\mathrm{\Omega }}_s\equiv {\omega }_s-{\omega }_{s0}$ in the unit of pump bandwidth ${\sigma }_p$) for (a) $k=1$, (b) $k=2$, (c) $k=3$. The green curves are the input spectral functions, while the blue and red curves are intermediate outputs after the iteration steps indicated in the legends. The dashed curves are the final outputs.

Figure 3

Experimental setup. WS, wave shaper; FM, flip mirror; BS, beam splitter; $T$, terminator; HD, homodyne detection; FOPA, fiber optical parametric amplifier; OSA, optical spectrum analyzer; CWDM, coarse wavelength division multiplexer.

Figure 4

Measured spectral intensity $|{\psi }_k(\omega ){|}^2$ for $k=$ (a) 1, (b) 2, (c) 3, and those [(d)–(f)] for the corresponding idler field. The dotted lines are the initially injected seed [not shown for (c) due to crowdedness]. The pink curve in (a) is the output after two iterations. The blue lines are the output signal and the red lines are the feedback to the input [red is covered by blue in (a)].

Figure 5

(a) Power gain for the first five modes under different pump power. (b) Extracted mode numbers. The dashed red boxes on order numbers 4 and 5 indicate less reliable results due to unstable output.

Figure 6

Covariance matrix elements for (a) amplitude quadrature ($X$) and (b) phase quadrature ($Y$). The trivial diagonal elements of $C_{ii}=1$ are taken out, while the nonzero antidiagonal elements show strong pairwise correlation between the corresponding modes of signal and idler beams. Other near-zero, off-diagonal elements indicate total independence between different modes. See Supplemental Material [32] for exact numerical values.