Quantum proton entanglement on a nanocrystalline silicon surface

We investigate the quantum entangled state of two protons terminating on a silicon surface. The entangled states were detected using the surface vibrational dynamics of nanocrystalline silicon with inelastic neutron scattering spectroscopy. The protons are identical, therefore the harmonic oscillator parity constrains the spin degrees of freedom, forming strongly entangled states for all the energy levels of surface vibrations. Compared to the proton entanglement previously observed in hydrogen molecules, this entanglement is characterized by an enormous energy difference of 113 meV between the spin singlet ground state and the spin triplet excited state. We theoretically demonstrate the cascade transition of terahertz entangled photon pairs utilizing proton entanglement. A combination of proton qubits and a modern silicon technology can result in a natural unification of computing platforms, thereby achieving unprecedented levels of massive parallelism processing.

Figure 1

(a) Schematic of n-Si composed of 377 Si atoms (blue balls) and 196 H atoms, with a diameter of 2.4 nm. The n-Si surface consists of SiH (green balls) or $\mathrm{Si}{\mathrm{H}}_2$ terminations (red balls). The ratio of $\mathrm{Si}{\mathrm{H}}_2$ to SiH terminations is modeled as 1:1 to coincide with the Fourier-transform infrared spectrum measurements. (b) Schematic to analyze $\mathrm{Si}{\mathrm{H}}_2$ vibration dynamics. The position of each atom ($\mathrm{Si}:{\mathbf{r}}_{\mathbf{1}}$, $\mathrm{H}:{\mathbf{r}}_{\mathbf{2}}$, and $\mathrm{H}:{\mathbf{r}}_{\mathbf{3}}$) is expressed by introducing the displacement vectors ${\mathbf{u}}_{\mathbf{1}}$, ${\mathbf{u}}_{\mathbf{2}}$, and ${\mathbf{u}}_{\mathbf{3}}$ from the equilibrium positions ${\mathbf{R}}_{\mathbf{1}}$, ${\mathbf{R}}_{\mathbf{2}}$, and ${\mathbf{R}}_{\mathbf{3}}$ as ${\mathbf{r}}_{\mathbf{1}}={\mathbf{u}}_{\mathbf{1}}+{\mathbf{R}}_{\mathbf{1}}$, ${\mathbf{r}}_{\mathbf{2}}={\mathbf{u}}_{\mathbf{2}}+{\mathbf{R}}_{\mathbf{2}}$, and ${\mathbf{r}}_{\mathbf{3}}={\mathbf{u}}_{\mathbf{3}}+{\mathbf{R}}_{\mathbf{3}}$. The force constants of Si-Si and Si-H bonds and H-H interactions are expressed as ${\mathbf{k}}_{\mathbf{1}}$, ${\mathbf{k}}_{\mathbf{2}}$, and ${\mathbf{k}}_{\mathbf{3}}$, respectively. (c) Entanglement energy levels using scissor modes. The evenly spaced energy levels have wave functions (Ψ) composed of the product of the harmonic oscillator and proton spin. For $E=0$ and 226 meV, the scissor modes have spin singlet states (angular momentum $J=0$), and for $E=113$ meV the scissor mode has spin triplet states ($J=1$, $m=-1$, 0, +1). A cascade transition of 27-THz entangled photon pairs can be achieved utilizing the scissor modes. Here, $|R_{\mathrm{nn}-1}\rangle$ (blue arrow) and $|L_{\mathrm{nn}-1}\rangle$ (red arrow) denote the transition from the $n$ to $n-1$ level with right and left circulation polarization, respectively.

Figure 2

Two-dimensional $S$($Q$, $\hslash \omega$) plots of (a) 1SC (113 meV) and (b) 2SC (226 meV) modes. (c) Sliced $S$($Q$) spectrum of 1SC mode (red circles) and 2SC mode (blue circles) derived from the two-dimensional $S$($Q$, $\hslash \omega$) plots. The red and blue lines are theoretically fitted $S$($Q$) curves with the effects from proton entanglement for 1SC and 2SC modes. The dotted red (inset) and blue lines are theoretically calculated $S$($Q$) curves without the entanglement effect (distinguishable case) for comparison. (d) FT spectra of both sliced (red circles) and theoretically fitted (blue line) $S$($Q$) for the 1SC mode. Inset shows the first excited energy states of $\mathrm{Si}{\mathrm{H}}_2$ vibrations. Here the red or blue lines represent triplet or singlet states, and $E(n_{1\mathrm{X}}+n_{1\mathrm{Y}}+n_{1\mathrm{Z}}=1)=58\mathrm{meV}$, $E(n_{2\mathrm{X}}+n_{2\mathrm{Y}}=1)=82\mathrm{meV}$, $E(n_{3\mathrm{X}}+n_{3\mathrm{Y}}=1)=113\mathrm{meV}$, $E(n_{2\mathrm{Z}}=1)=265\mathrm{meV}$, and $E(n_{3\mathrm{Z}}=1)=269\mathrm{meV}$.

Figure 3

Two-dimensional $S$($Q$, $\hslash \omega$) plots of (a) 1DSC (97 meV) and (b) 2DSC (194 meV) modes. (c) Sliced $S$($Q$) spectrum of 1DSC mode (red circles) and 2DSC mode (blue circles) derived from the two-dimensional $S$($Q$, $\hslash \omega$) plots. The red and blue lines are theoretically fitted $S$($Q$) curves consisting of both the distinguishable case (dotted red or blue line) and other scattering components (broken red line: Si-O bending, broken blue line: Si-D stretching). (d) FT spectra of both sliced (red circles) and theoretically fitted (blue line) $S$($Q$) for the 1DSC mode. The peak at 2.5 Å disappeared owing to loss of the proton entanglement effect. Inset shows the first excited energy states of SiHD vibrations, where $E(n_{1\mathrm{X}}+n_{1\mathrm{Y}}=1)=52\mathrm{meV}$, $E(n_{1\mathrm{Z}}=1)=58\mathrm{meV}$, $E(n_{2\mathrm{X}}+n_{2\mathrm{Y}}=1)=72\mathrm{meV}$, $E(n_{3\mathrm{X}}+n_{3\mathrm{Y}}=1)=97\mathrm{meV}$, $E(n_{2\mathrm{Z}}=1)=192\mathrm{meV}$, and $E(n_{3\mathrm{Z}}=1)=267\mathrm{meV}$.