Demultiplexing of photonic temporal modes by a linear system

Temporally and spatially overlapping but field-orthogonal photonic temporal modes (TMs) that intrinsically span a high-dimensional Hilbert space are recently suggested as a promising means of encoding information on photons. Presently, the realization of photonic TM technology, particularly to retrieve the information it carries, i.e., demultiplexing of photonic TMs, is mostly dependent on nonlinear medium and frequency conversion. Meanwhile, its miniaturization, simplification, and optimization remain the focus of research. In this paper, we propose a scheme of TM demultiplexing using linear systems consisting of resonators with linear couplings. Specifically, we examine a unidirectional array of identical resonators with short environment correlations. For both situations with and without tunable couplers, propagation formulas are derived to demonstrate photonic TM demultiplexing capabilities. The proposed scheme, being entirely feasible with current technologies, might find potential applications in quantum information processing.

Figure 1

General illustration of a linear system TM demultiplexer.

Figure 2

Examples of tensor $g$ with $T=10$ ns and ${\kappa }_0=1{\mathrm{ns}}^{-1}$. (a) Time-independent system. (b) Tunable coupling system in the form given by Eq. (27) with $T_c=5$ ns.

Figure 3

Illustration of a unidirectional array of $N$ resonators.

Figure 4

Numerical demonstration of the vanishing condition for a time-independent array of four resonators with ${\kappa }_0=1{\mathrm{ns}}^{-1}$. (a) The vanishing initial excitation in the network. (b) The integrations $S_{mn}\equiv {\int }_0^{T}dt{B}_m\left(t\right)B_n^{*}\left(t\right)$.

Figure 5

Rotating frame wave function of orthogonal TMs (a) for a time-independent array of resonators with ${\kappa }_0=1{\mathrm{ns}}^{-1}$, and the corresponding dynamics of the array upon their entry (b)–(e).

Figure 6

Rotating frame wave function of orthogonal TMs (a) for an array of resonators with tunable coupling under ${\kappa }_0=1{\mathrm{ns}}^{-1}$ and $T_c=20$ ns, and the corresponding dynamics of the array upon their entry (b)–(e).

Figure 7

Effect of time error on the average fidelities ${\mathcal{F}}_d$ of the demultiplexer, where $d$ is the dimension of the TM qudit. Tunable coupling in the previously given form is used with ${\kappa }_0=1{\mathrm{ns}}^{-1}$. (a) The time mismatch between the demultiplexer and an ideal sender. (b) Delay of operation by the tunable coupler of the first resonator.

Figure 8

Numerical results with frequency mismatch of the first resonator. Tunable coupling in the previously given form is used with ${\kappa }_0=1{\mathrm{ns}}^{-1}$. (a) Average fidelities ${\mathcal{F}}_d$ of the demultiplexer, where $d$ is the dimension of the TM qudit. (b) Fidelities ${\mathcal{F}}_n^{\prime }$ for the transfer of a single photon in a TM eigenstate associated with resonator $n$. (c) Four transfer matrix elements as functions of detuning.

Figure 9

Effect of photon loss on the average fidelities ${\mathcal{F}}_d$ of the demultiplexer, where $d$ is the dimension of the TM qudit. Tunable coupling in previously given form is used with ${\kappa }_0=1{\mathrm{ns}}^{-1}$. TM photons are assumed as designed with photon loss considered.

Figure 10

Flow chart for solving the coupling functions to the demultiplexing of arbitrary orthogonal TMs.

Figure 11

Demultiplexing of Schmidt modes with nonuniform coupling functions. Parameter $\lambda$ controls the pulse length of Schmidt modes. (a) First four Schmidt modes in dimensionless time representation. (b) The rotating frame wave function of each Schmidt mode photon upon arrival of the corresponding resonator. $B_{nm}$ denotes the output photon wave function of the $m\mathrm{th}$ resonator when the input photon is the $n\mathrm{th}$ Schmidt mode. (c) Coupling function of each resonator. (d) Population of each resonator in the absorption of its corresponding Schmidt single photon.