Symmetry-Induced Emergence of a Pseudo-Qutrit in the Dipolar Coupling of Two Qubits

Abstract

We investigate a system of two identical and distinguishable spins 1/2, with a direct magnetic dipole–dipole interaction, in an external magnetic field. Constraining the hyperfine tensor to exhibit axial symmetry generates the notable symmetry properties of the corresponding Hamiltonian model. In fact, we show that the reduction of the anisotropy induces the invariance of the Hamiltonian in the 3×3 subspace of the Hilbert space of the two spins in which S^2 invariably assumes its highest eigenvalue of 2. By means of appropriate mapping, it is then possible to choose initial density matrices of the two-spin system that evolve in such a way as to exactly simulate the time evolution of a pseudo-qutrit, in the sense that the the actual two-spin system nests the subdynamics of a qutrit regardless of the strength of the magnetic field. The occurrence of this dynamic similitude is investigated using two types of representation for the initial density matrix of the two spins. We show that the qutrit state emerges when the initial polarizations and probability vectors of the two spins are equal to each other. Further restrictions on the components of the probability vectors are reported and discussed.

Pseudo-Qutrit Formed by Two Interacting Identical Spins (s = 1/2) in a Variable External Magnetic Field

“…As was shown in [15], the square of the total spin commutes with the Hamiltonian (5); thus, first, we need to choose a base set with a determined total spin S = Ŝ1 + Ŝ2 , which is achieved with the following transformation matrix…”

“…The problem of two interacting neighboring nuclear spins in a crystal lattice is of interest. The most frequently used nuclei isotopes with nonzero spins I = 1/2 are 13 C with a magnetic moment µ C = +0.70238µ n , 1.108%, 15 N with µ N = −0.283049µ n , 0.365%. Naturally, we must not forget hydrogen with "the main" nuclear isotope 1 H with µ p = +2.79867µ n , where the nuclear magneton µ n = 5.05095 • 10 −24 Erg/G.…”

“…Naturally, we must not forget hydrogen with "the main" nuclear isotope 1 H with µ p = +2.79867µ n , where the nuclear magneton µ n = 5.05095 • 10 −24 Erg/G. Isotopes 13 C and 15 N are interesting because they are the main impurities in diamond crystals, and are popular in many research studies on quantum technologies. Silicon, another popular crystal for quantum technologies, has the same diamond lattice structure.…”

“…Recently, the problem of two interacting identical nuclear spins situated in a crystal lattice as nearest neighbors or in some other "frozen" state has been studied [14,15]. In [15], it was shown that the formation of a pseudo-qutrit state is possible for identical spin 1/2 particles.…”

“…Recently, the problem of two interacting identical nuclear spins situated in a crystal lattice as nearest neighbors or in some other "frozen" state has been studied [14,15]. In [15], it was shown that the formation of a pseudo-qutrit state is possible for identical spin 1/2 particles. Furthermore, the problem was solved for both standard quantum and probability representations of quantum states [16,17], where spin states were described by standard classical probability distributions [18][19][20] in the stationary case.…”