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Physical Review C

Entanglement and coherence of the wobbling mode in the particle–triaxial-rotor model

Q. B. Chen, S. Frauendorf

Published Dec 23, 2024• 11 citations
Publisher

Abstract

The entanglement and coherence of the wobbling mode are studied in the framework of the particle plus triaxial-rotor model for the one-quasiparticle nucleus $^{135}\mathrm{Pr}$ and the two-quasiparticles nucleus $^{130}\mathrm{Ba}$. The focus lies on the coupling between the total and the particle angular momenta. Using the Schmidt decomposing, it is quantified in terms of the von Neumann entropy of the respective subsystems, which measures their mutual entanglement. The entropy and the entanglement increase with spin $I$ and number of wobbling quanta $n$. The coherence of the wobbling mode is studied by means of the eigenstate decomposition of its reduced density matrix. To a good approximation, the probability distributions of the total angular momentum can be interpreted as the incoherent combination of the coherent contributions from the first two pairs of eigenvectors with the largest weight of the reduced density matrix. Decoherence measures are defined, which, in accordance, scatter between 0.1 to 0.2 at low spin and between 0.1 and 0.3 at high spin. Entanglement in the framework of the adiabatic approximation is further analyzed. In general, the coherent eigenstates of the effective collective Hamiltonian approximate the reduced density matrix with the limited accuracy of its pair of eigenstates with the largest weight. As the adiabatic approximation becomes more accurate with decreasing excitation energy, the probability distribution of the angle of the total angular momentum around a principal axis approaches the one of the full reduced density matrix. The $E2$ transition probabilities and spectroscopic quadrupole moments reflect this trend.

DOI
10.1103/physrevc.110.064322

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The entanglement and coherence of the wobbling mode are studied in the framework of the particle plus triaxial-rotor model for the one-quasiparticle nucleus 135Pr^{135}\mathrm{Pr}135Pr and the two-quasiparticles nucleus 130Ba^{130}\mathrm{Ba}130Ba. The focus lies on the coupling between the total and the particle angular momenta. Using the Schmidt decomposing, it is quantified in terms of the von Neumann entropy of the respective subsystems, which measures their mutual entanglement. The entropy and the entanglement increase with spin III and number of wobbling quanta nnn. The coherence of the wobbling mode is studied by means of the eigenstate decomposition of its reduced density matrix. To a good approximation, the probability distributions of the total angular momentum can be interpreted as the incoherent combination of the coherent contributions from the first two pairs of eigenvectors with the largest weight of the reduced density matrix. Decoherence measures are defined, which, in accordance, scatter between 0.1 to 0.2 at low spin and between 0.1 and 0.3 at high spin. Entanglement in the framework of the adiabatic approximation is further analyzed. In general, the coherent eigenstates of the effective collective Hamiltonian approximate the reduced density matrix with the limited accuracy of its pair of eigenstates with the largest weight. As the adiabatic approximation becomes more accurate with decreasing excitation energy, the probability distribution of the angle of the total angular momentum around a principal axis approaches the one of the full reduced density matrix. The E2E2E2 transition probabilities and spectroscopic quadrupole moments reflect this trend.
The entanglement and coherence of the wobbling mode are studied in the framework of the particle plus triaxial rotor model for the one-quasiparticle nucleus 135^{135}135Pr and the two-quasiparticles nucleus 130^{130}130Ba. The focus lies on the coupling between the total and the particle angular momenta. Using the Schmidt decomposing, it is quantified in terms of the von Neumann entropy of the respective sub-systems, which measures their mutual entanglement. The entropy and the entanglement increase with spin III and number of wobbling quanta nnn. The coherence of the wobbling mode is studied by means of the eigenstate decomposition of its reduced density matrix. To a good approximation, the probability distributions of the total angular momentum can be interpreted as the incoherent combination of the coherent contributions from the first two pairs of eigenvectors with the largest weight of the reduced density matrix. Decoherence measures are defined, which, in accordance, scatter between 0.1 to 0.2 at low spin and between 0.1 and 0.3 at high spin. Entanglement in the framework of the adiabatic approximation is further analyzed. In general, the coherent eigenstates of the effective collective Hamiltonian approximate the reduced density matrix with the limited accuracy of its pair of eigenstates with the largest weight. As the adiabatic approximation becomes more accurate with decreasing excitation energy, the probability distribution of the angle of the total angular momentum around a principal axis approaches the one of the full reduced density matrix. The E2E2E2 transition probabilities and spectroscopic quadrupole moments reflect this trend.
We investigate the entanglement in two-quasiparticle plus triaxial-rotor (PTR) model for the particle-hole configuration π(1h11/2)1⊗ν(1h11/2)−1π(1h_{11/2})^1 \otimes ν(1h_{11/2})^{-1}π(1h11/2​)1⊗ν(1h11/2​)−1, the particle-particle configuration π(1h11/2)1⊗ν(1h11/2)1π(1h_{11/2})^1 \otimes ν(1h_{11/2})^1π(1h11/2​)1⊗ν(1h11/2​)1, and two-proton particles configuration π(1h11/2)2π(1h_{11/2})^2π(1h11/2​)2 for different values of the triaxiality parameter. The entanglement between the angular momenta of the two quasiparticles and the total angular momentum is quantified by the three bipartite concurrences C\mathcal{C}C of one type of angular momentum with the other two angular momenta and the area F\mathcal{F}F of the triangle formed by the bipartite concurrences. Collective chiral and wobbling modes are identified for γ>15∘γ>15^\circγ>15∘ via spin coherent state (SCS) maps and spin squeezed state (SSS) plots. Their entanglement increases from moderate values at the band head to near-maximal values at I=20I=20I=20. The area F\mathcal{F}F of the chiral partners changes order as function of III which reflects the crossing of the partner bands as a signature of chirality. For the π(1h11/2)2π(1h_{11/2})^2π(1h11/2​)2 configuration, the antisymmetrization required by the Pauli exclusion principle causes strong entanglement between the two protons, which significantly amplifies the area F\mathcal{F}F. For γ<15∘γ<15^\circγ<15∘, the lowest bands become various uniformly rotating quasiparticle configurations, which have large values of F\mathcal{F}F for all values III.
Abstract We examine mode entanglement and correlation of two fermionic particles analytically and numerically. We study the one- and two-mode entropies and a global characteristic, the one-body entanglement entropy considering angular momentum coupled state with a single configuration. We show that with rearrangement of the single-particle orbitals the Slater decomposition can be obtained which can also be applied for configuration mixing subject to special restriction. With the help of the Slater decomposition, we derive analytical expressions for the entanglement measures, which becomes very simple for zero total angular momentum. The Slater decomposition allows us to define associated modes, and it turns out that they have identical one- and two-mode entropies. Furthermore, we show that specific single configurations describe maximally entangled states when the total angular momentum is zero. The numerical shell model study of two valence neutrons in the sd shell is revealed that the one-body entanglement entropy of the ground state is close to the maximal value, and the associated modes have the largest mutual information.
We examine the mode entanglement and correlation of two fermionic particles. We study the one- and two-mode entropy and a global characteristic, the one-body entanglement entropy. We consider not only angular momentum coupled states with single configuration but use the configuration interaction method. With the help of the Slater decomposition, we derive analytical expressions for the entanglement measures. We show that when the total angular momentum is zero specific single configurations describe maximally entangled states. It turns out that for a finite number of associated modes the one- and two-mode entropies have identical values. In the shell model framework, we numerically study two valence neutrons in the sdsdsd shell. The one-body entanglement entropy of the ground state is close to the maximal value and the associated modes have the largest mutual information.
The ground-state entanglement of a single particle of the N-harmonium system (i.e., a completely-integrable model of NNN particles where both the confinement and the two-particle interaction are harmonic) is shown to be analytically determined in terms of NNN and the relative interaction strength. For bosons, we compute the von Neumann entropy of the one-body reduced density matrix by using the corresponding natural occupation numbers. There exists a critical number NcN_cNc​ of particles so that below it, for positive values of the coupling constant, the entanglement grows when the number of particles is increasing; the opposite occurs for N>NcN > N_cN>Nc​. For fermions, we compute the one-body reduced density matrix for the closed-shell spinned case. In the strong coupling regime, the linear entropy of the system decreases when NNN is growing. For fixed NNN, the entanglement is found (a) to decrease (increase) for negatively (positively) increasing values of the coupling constant, and (b) to grow when the energy is increasing. Moreover, the spatial and spin contributions to the total entanglement are found to be of comparable size.
A quantum mechanical system of two coupled rotors (particles constrained to move on a circle) is studied from an open quantum systems point of view. One of the rotors is integrated out and the reduced density operator of the other rotor is studied. Its eigenvalues are worked out explicitly using the properties of Mathieu functions, and the von Neumann entropy, which is a standard measure of entanglement, is computed in terms of the Fourier coefficients defining the Mathieu functions. Furthermore, upon introducing a time-periodic delta kick and making one of the rotors much heavier than the other, the two-rotor system can be interpreted as a system-bath model, allowing us to introduce a series of approximations to derive a master equation of the Lindblad type describing the time-evolution of the reduced density operator.