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Quantum Rep., Volume 6, Issue 2 (June 2024) – 7 articles

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31 pages, 2408 KiB  
Article
A Dyson Brownian Motion Model for Weak Measurements in Chaotic Quantum Systems
by Federico Gerbino, Pierre Le Doussal, Guido Giachetti and Andrea De Luca
Quantum Rep. 2024, 6(2), 200-230; https://doi.org/10.3390/quantum6020016 - 16 May 2024
Viewed by 256
Abstract
We consider a toy model for the study of monitored dynamics in many-body quantum systems. We study the stochastic Schrödinger equation resulting from continuous monitoring with a rate Γ of a random Hermitian operator, drawn from the Gaussian unitary ensemble (GUE) at every [...] Read more.
We consider a toy model for the study of monitored dynamics in many-body quantum systems. We study the stochastic Schrödinger equation resulting from continuous monitoring with a rate Γ of a random Hermitian operator, drawn from the Gaussian unitary ensemble (GUE) at every time t. Due to invariance by unitary transformations, the dynamics of the eigenvalues {λα}α=1n of the density matrix decouples from that of the eigenvectors, and is exactly described by stochastic equations that we derive. We consider two regimes: in the presence of an extra dephasing term, which can be generated by imperfect quantum measurements, the density matrix has a stationary distribution, and we show that in the limit of large size n it matches with the inverse-Marchenko–Pastur distribution. In the case of perfect measurements, instead, purification eventually occurs and we focus on finite-time dynamics. In this case, remarkably, we find an exact solution for the joint probability distribution of λ’s at each time t and for each size n. Two relevant regimes emerge: at short times tΓ=O(1), the spectrum is in a Coulomb gas regime, with a well-defined continuous spectral distribution in the n limit. In that case, all moments of the density matrix become self-averaging and it is possible to exactly characterize the entanglement spectrum. In the limit of large times tΓ=O(n), one enters instead a regime in which the eigenvalues are exponentially separated log(λα/λβ)=O(Γt/n), but fluctuations O(Γt/n) play an essential role. We are still able to characterize the asymptotic behaviors of the entanglement entropy in this regime. Full article
(This article belongs to the Special Issue Exclusive Feature Papers of Quantum Reports in 2024–2025)
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16 pages, 1173 KiB  
Article
Fisher Information for a System Composed of a Combination of Similar Potential Models
by Clement Atachegbe Onate, Ituen B. Okon, Edwin Samson Eyube, Ekwevugbe Omugbe, Kizito O. Emeje, Michael C. Onyeaju, Olumide O. Ajani and Jacob A. Akinpelu
Quantum Rep. 2024, 6(2), 184-199; https://doi.org/10.3390/quantum6020015 - 13 May 2024
Viewed by 261
Abstract
The solutions to the radial Schrödinger equation for a pseudoharmonic potential and Kratzer potential have been studied separately in the past. Despite different reports on the Kratzer potential, the fundamental theoretical quantities such as Fisher information have not been reported. In this study, [...] Read more.
The solutions to the radial Schrödinger equation for a pseudoharmonic potential and Kratzer potential have been studied separately in the past. Despite different reports on the Kratzer potential, the fundamental theoretical quantities such as Fisher information have not been reported. In this study, we obtain the solution to the radial Schrödinger equation for the combination of the pseudoharmonic and Kratzer potentials in the presence of a constant-dependent potential, utilizing the concepts and formalism of the supersymmetric and shape invariance approach. The position expectation value and momentum expectation value are calculated employing the Hellmann–Feynman Theory. These expectation values are then used to calculate the Fisher information for both position and momentum spaces in both the absence and presence of the constant-dependent potential. The results obtained revealed that the presence of the constant-dependent potential leads to an increase in the energy eigenvalue, as well as in the position and momentum expectation values. Additionally, the constant-dependent potential increases the Fisher information for both position and momentum spaces. Furthermore, the product of the position expectation value and the momentum expectation value, along with the product of the Fisher information, satisfies both Fisher’s inequality and Cramer–Rao’s inequality. Full article
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12 pages, 4665 KiB  
Article
Spectral Analysis of Proton Eigenfunctions in Crystalline Environments
by Luca Gamberale and Giovanni Modanese
Quantum Rep. 2024, 6(2), 172-183; https://doi.org/10.3390/quantum6020014 - 6 May 2024
Viewed by 290
Abstract
The Schrödinger equation and Bloch theorem are applied to examine a system of protons confined within a periodic potential, accounting for deviations from ideal harmonic behavior due to real-world conditions like truncated and non-quadratic potentials, in both one-dimensional and three-dimensional scenarios. Numerical computation [...] Read more.
The Schrödinger equation and Bloch theorem are applied to examine a system of protons confined within a periodic potential, accounting for deviations from ideal harmonic behavior due to real-world conditions like truncated and non-quadratic potentials, in both one-dimensional and three-dimensional scenarios. Numerical computation of the energy spectrum of bound eigenfunctions in both cases reveals intriguing structures, including bound states with degeneracy matching the site number Nw, reminiscent of a finite harmonic oscillator spectrum. In contrast to electronic energy bands, the proton system displays a greater number of possible bound states due to the significant mass of protons. Extending previous research, this study rigorously determines the constraints on the energy gap and oscillation amplitude of the previously identified coherent states. The deviations in energy level spacing identified in the computed spectrum, leading to the minor splitting of electromagnetic modes, are analyzed and found not to hinder the onset of coherence. Finally, a more precise value of the energy gap is determined for the proton coherent states, ensuring their stability against thermal decoherence up to the melting temperature of the hosting metal. Full article
(This article belongs to the Special Issue Exclusive Feature Papers of Quantum Reports in 2024–2025)
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16 pages, 497 KiB  
Article
Measuring the Density Matrix of Quantum-Modeled Cognitive States
by Wendy Xiomara Chavarría-Garza, Osvaldo Aquines-Gutiérrez, Ayax Santos-Guevara, Humberto Martínez-Huerta, Jose Ruben Morones-Ibarra and Jonathan Rincon Saucedo
Quantum Rep. 2024, 6(2), 156-171; https://doi.org/10.3390/quantum6020013 - 27 Apr 2024
Viewed by 446
Abstract
Inspired by the principles of quantum mechanics, we constructed a model of students’ misconceptions about heat and temperature, conceptualized as a quantum system represented by a density matrix. Within this framework, the presence or absence of misconceptions is delineated as pure states, while [...] Read more.
Inspired by the principles of quantum mechanics, we constructed a model of students’ misconceptions about heat and temperature, conceptualized as a quantum system represented by a density matrix. Within this framework, the presence or absence of misconceptions is delineated as pure states, while the probability of mixed states is also considered, providing valuable insights into students’ cognition based on the mental models they employ when holding misconceptions. Using the analysis model previously employed by Lei Bao and Edward Redish, we represented these results in a density matrix. In our research, we utilized the Zeo and Zadnik Thermal Concept Evaluation among 282 students from a private university in Northeast Mexico. Our objective was to extract information from the analysis of multiple-choice questions designed to explore preconceptions, offering valuable educational insights beyond the typical Correct–Incorrect binary analysis of classical systems. Our findings reveal a probability of 0.72 for the appearance of misconceptions, 0.28 for their absence, and 0.43 for mixed states, while no significant disparities were observed based on gender or scholarship status, a notable difference was observed among programs (p < 0.05). These results are consistent with the previous literature, confirming a prevalence of misconceptions within the student population. Full article
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9 pages, 259 KiB  
Article
A Normalization Condition for the Probability Current in Some Remarkable Cases
by Antonio Feoli, Elmo Benedetto and Antonella Lucia Iannella
Quantum Rep. 2024, 6(2), 147-155; https://doi.org/10.3390/quantum6020012 - 23 Apr 2024
Viewed by 412
Abstract
Starting from the dynamics of a bouncing ball in classical and quantum regime, we have suggested in a previous paper to add an arbitrary function of time to the standard expression of the probability current in quantum mechanics. In this paper, we suggest [...] Read more.
Starting from the dynamics of a bouncing ball in classical and quantum regime, we have suggested in a previous paper to add an arbitrary function of time to the standard expression of the probability current in quantum mechanics. In this paper, we suggest a way to determine this function: imposing a suitable normalization condition. The application of our proposal to the case of the harmonic oscillator is discussed. Full article
5 pages, 199 KiB  
Editorial
The Many-Worlds Interpretation of Quantum Mechanics: Current Status and Relation to Other Interpretations
by Lev Vaidman
Quantum Rep. 2024, 6(2), 142-146; https://doi.org/10.3390/quantum6020011 - 18 Apr 2024
Viewed by 602
Abstract
This is a preface to a Special Issue of Quantum Reports devoted to the results of the workshop “The Many-Worlds Interpretation of Quantum Mechanics: Current Status and Relation to Other Interpretations” [...] Full article
(This article belongs to the Special Issue The Many-Worlds Interpretation of Quantum Mechanics)
8 pages, 320 KiB  
Communication
Continuum Limit of the Green Function in Scaled Affine φ44 Quantum Euclidean Covariant Relativistic Field Theory
by Riccardo Fantoni
Quantum Rep. 2024, 6(2), 134-141; https://doi.org/10.3390/quantum6020010 - 14 Apr 2024
Viewed by 886
Abstract
Through path integral Monte Carlo computer experiments, we prove that the affine quantization of the φ44-scaled Euclidean covariant relativistic scalar field theory is a valid quantum field theory with a well-defined continuum limit of the one- and two-point functions. Affine [...] Read more.
Through path integral Monte Carlo computer experiments, we prove that the affine quantization of the φ44-scaled Euclidean covariant relativistic scalar field theory is a valid quantum field theory with a well-defined continuum limit of the one- and two-point functions. Affine quantization leads to a completely satisfactory quantization of field theories in situations involving scaled behavior, leading to an unexpected term, 2/φ2, which arises only in the quantum aspects. Full article
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