Deterministic generation of entanglement in a quantum network by coherent absorption of a single photon

Advanced quantum information protocols rely on the operation of multinodal quantum networks where entanglement is distributed across the nodes. Existing protocols of entanglement generation are probabilistic, with the efficiency dropping exponentially with the size of the system. We formulate an approach for the deterministic generation of entangled states of a multinodal quantum network of arbitrary size by coupling a single photon standing wave with the nodes of the network. We show experimentally how this can be implemented in a simple binodal system. Since this approach relies on collective excitation of the network—not on local interaction with individual nodes—it allows generation of entanglement with unitary efficiency, independent of the size and the nature of the network.

Figure 1

Generation of nonclassical states of multinodal network by single-photon coupling. (a) Parallel scheme: photon is split on a series of beam splitters (BSs) and each “part” of the photon is independently coupled to corresponding nodes $B_1,...,B_M$ of the network. (b) Series scheme: photon is converted into a standing wave in the interferometer configuration and coherently coupled to the multinodal network.

Figure 2

Model of quantum light and multinodal network interaction. At initial time, system is described by amplitudes ${\widehat{a}}_R^{\left(\mathrm{in}\right)}$ and ${\widehat{a}}_L^{\left(\mathrm{in}\right)}$ of two coherent light modes propagating in opposite directions and carrying a single excitation and by input amplitudes ${\widehat{b}}_m^{\left(\mathrm{in}\right)}$ of the corresponding nodes in a vacuum state. At final time, system is characterized by two outgoing light modes, ${\widehat{a}}_R^{\left(\mathrm{out}\right)}$ and ${\widehat{a}}_L^{\left(\mathrm{out}\right)}$, and output amplitudes ${\widehat{b}}_m^{\left(\mathrm{out}\right)}$ of the nodes. The transformation is described by the scattering matrix $S$.

Figure 3

Coherent absorption of heralded single photons by a binodal system. (a) Schematic of the setup with a binodal absorber placed in the middle of the interferometer. Dark blue lines correspond to the optical fiber and red lines to the free space paths of the photons. (b) Measured probabilities of heralded signal photon detection by SPAD-1 (blue circles and fitting line) and SPAD-2 (red diamonds and fitting line), total transmission (purple triangles and fitting line), and total coupling (black line) probabilities as a function of phase delay $\phi$ between the two arms of the interferometer. The error bars represent the Poisson noise of randomly arriving photons. The highest photon absorption probability of 0.88 is marked by the white star.

Figure 4

Quantum information protocols with entangled multinodal networks. (a) Multiparty quantum key distribution. Two subsystems, $B$ and $C$, compose the network, where the $m\mathrm{th}$ party holds one node of each subsystem, $B_m$ and $C_m$ ($m=1,2,...,M$). The protocol starts with coherent distribution of a single photon in each subsystem, so that two excitations are present in the whole network. Next, each party performs measurement of conjugated nodes $B_m$ and $C_m$ by transferring their states to light modes, applying the phase shift ${\phi }_m$ (randomly chosen either 0 or $\pi$) to one of them, and mixing them on a beam splitter [for simplicity, measurement setups are shown only for the first (Alice) and last parties]. Quantum key distribution is based on correlation measurement between detectors’ counts belonging to different parties. (b) Efficient excitation of entangled states of a system composed of two weakly absorbing nodes. First, a single photon is deterministically coupled to a multinodal network of a higher dimension creating its entangled state. By measuring the state of nodes $B_3,B_4,...,B_M$, the wave function is reduced to the entangled state of two nodes $B_1$ and $B_2$. Ratio $P_2/P_1$ of probabilities of entanglement generation for the series scheme ($P_2$) compared to the parallel scheme ($P_1$) is shown in the graph as a function of the photon coupling probability $p_a$ in each node.

Figure 5

Optical response of a binodal system. (a) Model: two identical layers with $t_1=2/3$ and $r_1=-1/3$ spaced by variable distance $D$. Input monochromatic field with amplitude $E_0$ is partially transmitted $\left(t{E}_0\right)$ and partially reflected $\left(r{E}_0\right)$. (b) Intensity transmssion ${\left|t\right|}^2$, reflection ${\left|r\right|}^2$, and absorption ${\left|a\right|}^2=1-{\left|t\right|}^2-{\left|r\right|}^2$ coefficients of the structure as a function of the distance $D$. (c) Phases of transmission and reflection coefficients of the strutcure as a function of the distance $D$. Distance is expressed in units of the wavelength $\lambda$. Diamonds correspond to the regime with $r=-t=-1/2$, while circles correspond to $r=+t=-1/2$.