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Physical Review Research

Probing the evolution of quantum superposition states in a nuclear spin-5/2 system

Byoung Jin Suh, F. Borsa, S.-H. Baek

Published Aug 15, 2023• 0 citations
Publisher

Abstract

We report the effects of off-diagonal coupling on eigenstates and eigenvalues of a realistic quantum Hamiltonian, a $^{121}\mathrm{Sb}$ (spin $I=5/2$) nuclear spin Hamiltonian in an antimony single crystal. In our nuclear resonance experiment, off-diagonal matrix elements are generated and controlled by the angle $\ensuremath{\theta}$ between the directions of the electric field gradient and an external magnetic field. With a very small $\ensuremath{\theta}$, a two-state-like avoided level crossing is directly probed as a function of an external magnetic field. On the other hand, by increasing $\ensuremath{\theta}$ under a weak magnetic field, we find that the nuclear spin transitions undergo dramatic changes as $\ensuremath{\theta}$ approaches ${90}^{\ensuremath{\circ}}$. All our experimental observations are in excellent agreement with exact diagonalization calculations for the Hamiltonian, demonstrating how superposition states of a multistate quantum system evolve as a function of a well-defined control parameter.

DOI
10.1103/physrevresearch.5.033105

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We report the effects of off-diagonal coupling on eigenstates and eigenvalues of a realistic quantum Hamiltonian, a 121Sb^{121}\mathrm{Sb}121Sb (spin I=5/2I=5/2I=5/2) nuclear spin Hamiltonian in an antimony single crystal. In our nuclear resonance experiment, off-diagonal matrix elements are generated and controlled by the angle \ensuremathθ\ensuremath{\theta}\ensuremathθ between the directions of the electric field gradient and an external magnetic field. With a very small \ensuremathθ\ensuremath{\theta}\ensuremathθ, a two-state-like avoided level crossing is directly probed as a function of an external magnetic field. On the other hand, by increasing \ensuremathθ\ensuremath{\theta}\ensuremathθ under a weak magnetic field, we find that the nuclear spin transitions undergo dramatic changes as \ensuremathθ\ensuremath{\theta}\ensuremathθ approaches 90\ensuremath∘{90}^{\ensuremath{\circ}}90\ensuremath∘. All our experimental observations are in excellent agreement with exact diagonalization calculations for the Hamiltonian, demonstrating how superposition states of a multistate quantum system evolve as a function of a well-defined control parameter.
Preserving quantum coherence with the increase of a system's size and complexity is a major challenge. Molecules, with their diverse sizes and complexities and many degrees of freedom, are an excellent platform for studying the transition from quantum to classical behavior. While most quantum-control studies of molecules focus on vibrations and rotations, we focus here on creating a quantum superposition between two nuclear-spin isomers of the same molecule. We present a scheme that exploits an avoided crossing in the spectrum to create strong coupling between two uncoupled nuclear-spin-isomer states, hence creating an isomeric qubit. We model our scheme using a four-level Hamiltonian and explore the coherent dynamics in the different regimes and parameters of our system. Our four-level model and approach can be applied to other systems with a similar energy-level structure.
The measurement of the nuclear spin state in the Kane's solid-state quantum computer was investigated for a simple two-qubit system. To simplify the calculations, the Hamiltonian matrix was divided into five invariant subspaces based on the projection of the total spins along the external magnetic field. The calculations analyzed the relations between the eigenvalues and eigenstates of the Hamiltonian matrix and the interactions between the two electrons. The results show that measurement of the nuclear spin state by exchanging the nuclear and election spin wave functions is restricted. Only four of the sixteen possible states in a two-qubit system can be measured using this method.
As a nuclear spin model of scalable quantum register, the one-dimensional chain of the magnetic atoms with nuclear spins 1/2 substituting the basic atoms in the plate of nuclear spin free easy-axis 3D antiferromagnet is considered. It is formulated the generalized antiferromagnet Hamiltonian in spin-wave approximation (low temperatures) considering the inhomogeneous external magnetic field, which is directed along the easy axis normally to plane of the plate and has a constant gradient along the nuclear spin chain. Assuming a weak gradient, the asymptotic expression for coefficients of unitary transformations to the diagonal form of antiferromagnet Hamiltonian is found. With this result the expression for indirect interspin coupling, which is due to hyperfine nuclear electron coupling in atoms and the virtual spin wave propagation in antiferromagnet ground state, was evaluated. It is shown that the inhomogeneous magnetic field essentially modifies the characteristics of indirect interspin coupling. The indirect interaction essentially grows and even oscillates in relation to the interspin distance when the local field value in the middle point of two considered nuclear spin is close to the critical field for quantum phase transition of spin-flop type in bulk antiferromagnet or close to antiferromagnetic resonance. Thus, the external magnetic field, its gradient, microwave frequency and power can play the role of control parameters for qubit states. Finally, the one and two qubit states decoherence and longitudinal relaxation rate are caused by the interaction of nuclear spins with virtual spin waves in antiferromagnet ground state are calculated.
The nuclear spin is a prime candidate for quantum information applications due to its weak coupling to the environment and inherently long coherence times. However, this weak coupling also challenges the addressability of the nuclear spin. Here we demonstrate nuclear magnetic resonance (NMR) on a single on-surface atom using a local scanning probe. We employ an electron-nuclear double resonance measurement scheme and resolve nuclear spin transitions of a single 47Ti isotope with a nuclear spin of I = 5/2. The quadrupole interaction enables to resolve multiple NMR transitions, which are consistent with our eigenenergy calculations. Our experimental results indicate that the nuclear spin can be driven efficiently irrespective of its hybridization with the electron spin, which is required for direct control of the nuclear spin in the long-lifetime regime. This investigation of NMR on a single atom in a platform with atomic-scale control is a valuable development for other platforms deploying nuclear spins for characterization techniques or quantum information technology.
Quantum coherence in small single-domain ferromagnetic particles is studied using dynamical calculations of the reversal of a single quantum spin resulting from a rotation of an external magnetic field, for \ensuremath{\leqslant}S\ensuremath{\leqslant}75. The analysis of the time evolution of the spin, the temporal correlation, and the spectrum leads to the conclusion that, for small spin S\ensuremath{\leqslant}15, coherent tunneling back and forth between the easy directions occurs at some specific resonant fields only. Away from resonance, quantum coherence in the spin temporal evolution is absent. These results are in qualitative accordance with those obtained for a Heisenberg Hamiltonian with exact quantum calculations and are at variance with semiclassical approximations. However, as the spin becomes larger, the number of resonant fields increases, the resonances become very small, and coherence in time evolution is lost for any value of the magnetic field.
A central challenge in nuclear physics is understanding quantum many-body systems governed by the strong nuclear force. The inherent complexity of these systems, combined with the limitations of classical computational methods, underscores the need for new approaches to study nuclear structure and dynamics. Here, we demonstrate that a spin-based digital quantum simulator using nuclear magnetic resonance, where nuclear spins simulate interacting fermions, offers a powerful tool to address this challenge. As a first step, we experimentally simulate the Agassi model, which encapsulates the interplay between collective and single-particle behaviors in finite nuclei. By representing nucleons as both bosons (nucleon pairs) and fermions (individual unpaired nucleons), the Agassi model captures highly non-linear interactions and is particularly suited for studying nuclear phase transitions, such as those between spherical and deformed shapes. We experimentally measure the correlation function as an order parameter during the evolution of the many-body system, successfully detecting a quantum phase transition. Specifically, we observe a sharp transition between the symmetric phase and the broken symmetry phase. This work underscores the potential of quantum simulation as a transformative tool in nuclear physics, particularly for exploring complex quantum many-body systems with applications in nuclear structure and reaction dynamics.