Natural and magnetically induced entanglement of hyperfine-structure states in atomic hydrogen

The spectrum of atomic hydrogen has long been viewed as a Rosetta stone that bears the key to decode the writings of quantum mechanics in a vast variety of physical, chemical, and biological systems. Here, we show that, in addition to its role as a basic model of quantum mechanics, the hydrogen atom provides a fundamental building block of quantum information. Through its electron- and nuclear-spin degrees of freedom, the hydrogen atom is shown to lend a physically meaningful frame and a suitable Hilbert space for bipartite entanglement, the two-qubit concurrence and quantum coherence of which can be expressed in terms of the fundamental physical constants—the Planck and Boltzmann constants, electron and proton masses, the fine-structure constant, as well as the Bohr radius and the Bohr magneton. The intrinsic, natural entanglement that the hyperfine-structure (HFS) states of the H atom store at low temperatures rapidly decreases with a growth in temperature, vanishing above a ${\tau }_c\approx 5.35\mu \mathrm{eV}$ threshold. An external magnetic field, however, can overcome this thermal loss of HFS entanglement. As one of the central findings of this paper, we show that an external magnetic field can induce and sustain an HFS entanglement, against all the odds of thermal effects, at temperatures well above the ${\tau }_c$ threshold, thus enabling magnetic-field-assisted entanglement engineering in low-temperature gases and solids.

Figure 1

(a) Energy-level diagram of the hyperfine structure of a ground-state H atom driven by a magnetic field $B$. (b) The concurrence of the HFS states of the H atom as a function of the temperature $T$ for different values of the normalized magnetic field $\xi$.

Figure 2

The concurrence of the HFS states of the H atom as a function of the normalized magnetic field $\xi$ below (a) and above (b) the critical temperature $T_c$.

Figure 3

The $l_1$-norm coherence of the HFS states of the H atom as a function of the temperature $T$ (a) and the normalized magnetic field $\xi$ (b).

Figure 4

The concurrence $C$ quantifying the two-qubit entanglement in a 1D Heisenberg model as a function of the temperature $T$ (a) and the normalized magnetic field $\xi$ (b).