Simple source for large linear cluster photonic states

The experimental realization of many-body entangled states is one of the main goals of quantum technology as these states are a key resource for quantum computation and quantum sensing. However, increasing the number of photons in an entangled state has been proved to be a painstakingly hard task. This is a result of the nondeterministic emission of current photon sources and the distinguishability between photons from different sources. Moreover, the generation rate and the complexity of the optical setups hinder scalability. Here we present a scheme that is compact, requires a very modest number of components, and avoids the distinguishability issues by using only one single-photon source. States of any number of photons are generated with the same configuration, with no need for increasing the optical setup. The basic operation of this scheme is experimentally demonstrated, and its sensitivity to imperfections is considered.

Figure 1

A scheme for generating multiphoton cluster states. A single-photon source emits photons consecutively into an optical path, leading into a Sagnac-like loop. Inside the loop there is a half-wave plate (WP) at $22.5^{\circ }$, and its delay time is matched to the time between consecutive photon emissions. At the point of intersection of the optical paths, a photon-entangling operation entangles one photon from the source and another from the delay line. One photon continues into the delay loop, and the other one continues further towards a polarization controller (PC), a polarizing beam splitter (PBS), and detectors. The state is characterized by different detection sequences. See the main text for more details.

Figure 2

The entanglement length dependence on the number of modes. The length of the longest generated chain where entanglement can be measured between its two end points. The minimal length 2 is attained for more than $\sim 2.41$ modes. For chains of at least 10, 84, and 825 photons, no more than 1.2, 1.02, and 1.002 modes are allowed, respectively.

Figure 3

The optical setup used to demonstrate the scheme. A heralded single-photon source is used, and the entangling operation is performed with linear optics and postselection. A nonlinear ${\mathrm{BaB}}_2{\mathrm{O}}_4$ is pumped by a 390-nm double Ti:sapphire laser.

Figure 4

Nonlocal interference visibility between two entangled photons. A constructive (destructive) interference is observed as the delay is scanned for the $|{\varphi }^{+}\rangle \left(|{\varphi }^{-}\rangle \right)$ state. The fourfold signal is composed of the detection of the two heralding photons and the two entangled photons. Each data point was integrated over 16 min. The event rate was 1 Hz at an average pump power of 350 mW. Errors are calculated assuming Poisson distribution.

Figure 5

Nonlocal interference visibility between three entangled photons. A constructive (destructive) interference is observed as the delay is scanned for the $\frac{1}{\sqrt{2}}\left(\right|{\varphi }^{+}h\rangle +|{\varphi }^{-}v\rangle )[\frac{1}{\sqrt{2}}\left(\right|{\varphi }^{+}h\rangle -|{\varphi }^{-}v\rangle \left)\right]$ state. The sixfold signal is composed of the detection of the three heralding photons and the three entangled photons. Each data point was integrated over 34 h. The event rate was 1 per 35 min at an average pump power of 550 mW. Errors are calculated assuming Poisson distribution.

Figure 6

$\mathcal{C}$ as a function of the number of photons.

Figure 7

$\mathcal{F}$ as a function of the number of photons.

Figure 8

$\mathcal{C}$ as a function of the number of modes.

Figure 9

$\mathcal{F}$ as a function of the number of modes.

Figure 10

${\mathcal{C}}_{1,N}$ as a function of the number of qubits for different values of $p$ (depolarizing channel).

Figure 11

$\mathcal{F}$ as a function of the number of qubits for different values of $p$ (depolarizing channel).

Figure 12

$\mathcal{L}$ as a function of $p$ (depolarizing channel).

Figure 13

Imperfect PBS: ${\mathcal{C}}_{1,N}$ as a function of the number of qubits.

Figure 14

Imperfect PBS: $\mathcal{F}$ as a function of the number of qubits.

Figure 15

Lower bound on the probability for a successful generation of an $N$-qubit cluster state given a nonzero probability ($p=0.01$) for a two-photon emission by the single-photon source.

Figure 16

$\mathcal{L}$ as a function of $\epsilon$, $0.022\le \epsilon \le 0.098$.

Figure 17

$\mathcal{L}$ as a function of $\epsilon$, $0.11\le \epsilon \le 0.49$.