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Quantum mechanical evolution of electron’s Landau state in the dissipative magnetic field

Cheng Da, Hong-Yi Fan

Published Jun 1, 2024• 1 citation
Publisher

Abstract

Physicists have long studied Landau states and energy levels of electrons in a stable magnetic field. However, when the magnetic field dissipates, the question of how Landau states evolve remains unanswered. This study investigates the magnetic dissipative effect from a quantum mechanics perspective. We establish a quantum master equation that describes a dissipative magnetic field. Subsequently, we solve the master equation using an entangled state representation. Consequently, we found that the Landau state evolves into a binomial quantum state (mixed state) when the magnetic field dissipated. This is in sharp contrast to the case where the Landau state becomes a negative binomial state when an additional harmonic oscillator potential is applied. We hope that this finding will be helpful for the evolution of quantized plasma physics.

DOI
10.1063/5.0216852

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Physicists have long studied Landau states and energy levels of electrons in a stable magnetic field. However, when the magnetic field dissipates, the question of how Landau states evolve remains unanswered. This study investigates the magnetic dissipative effect from a quantum mechanics perspective. We establish a quantum master equation that describes a dissipative magnetic field. Subsequently, we solve the master equation using an entangled state representation. Consequently, we found that the Landau state evolves into a binomial quantum state (mixed state) when the magnetic field dissipated. This is in sharp contrast to the case where the Landau state becomes a negative binomial state when an additional harmonic oscillator potential is applied. We hope that this finding will be helpful for the evolution of quantized plasma physics.
This study aims to explore the magnetic diffusion effect of Landau states from the perspective of quantum mechanics while the magnetic diffusion often occurs in plasmas physics. By introducing the entangled state representation, we examine how the Landau state diffuses in the presence of magnetic diffusion. We also establish the master equation of the density operator corresponding to magnetic diffusion and solve it, and we find that the Landau state evolves into a Laguerre-polynomial-weighted chaotic state.
The entanglement of many localized spins (LSs) within solid magnetic materials is a topic of great basic and applied interest, particularly after becoming amenable to experimental scrutiny where recent neutron scattering experiments have witnessed macroscopic entanglement in the ground state (GS) of antiferromagnets persisting even at elevated temperatures. On the other hand, spintronics and magnonics studies assume that LSs of antiferromagnets are in unentangled N\'eel GS, as well as that they evolve, when pushed out of equilibrium by current or external fields, according to the Landau-Lifshitz-Gilbert (LLG) equation viewing LSs as classical vectors of fixed length. The prerequisite for applicability of the LLG equation is zero entanglement in the underlying many-body quantum state of LSs. In this study, we initialize quantum Heisenberg ferro- or antiferromagnetic chains hosting S=1/2,S=1S=1/2, S=1S=1/2,S=1, or S=5/2S=5/2S=5/2 LSs into an unentangled pure state and then evolve them by quantum master equations (QMEs) of Lindblad or non-Markovian type, derived by coupling LSs weakly to a bosonic bath (due to phonons in real materials) or by using additional ``reaction coordinate'' in the latter case. The time evolution is initiated by applying an external magnetic field, and entanglement of the ensuing mixed quantum states is monitored by computing its negativity. We find that non-Markovian dynamics never brings entanglement to zero, in the presence of which the vector of spin expectation value changes its length to render the LLG equation inapplicable. Conversely, Lindbladian (i.e., Markovian) dynamics ensures that entanglement goes to 0, thereby enabling quantum-to-classical transition in all cases---S=1/2,S=1S=1/2, S=1S=1/2,S=1, and S=5/2S=5/2S=5/2 ferromagnets or S=5/2S=5/2S=5/2 antiferromagnets---except for S=1/2S=1/2S=1/2 and S=1S=1S=1 antiferromagnets. Finally, we investigate the stability of an entangled antiferromagnetic GS upon suddenly coupling it to bosonic baths.
We show that passing through the amplitude dissipative channel the initial pure number state density operator is evolved into the density operator of binomial distribution (a mixed state), and the binomial distribution parameter is just equal to e −2 kt , where k is the dissipative parameter of the channel. We solve the corresponding master equation to obtain the operator-sum representation of the density operator by virtue of the entangled state representation, which seems to be a convenient approach.
Abstract We revisit the quantum dynamics of a charged particle in a time-dependent magnetic field, a fundamental problem exhibiting rich non-adiabatic behaviour, from the complementary perspective of the Madelung fluid formulation. We first analyse the system within standard quantum mechanics using perturbation theory around the Landau levels, and then address the same problem through the Madelung perspective. We show that the hydrodynamic formulation not only yields an intuitive derivation of the exact solution, it also provides a clear physical interpretation of non-adiabatic quantum evolution in terms of mechanical energy transfers. In this picture, the sloshing oscillations of the wave function arise from deviations from the force balance between the magnetic Lorentz force and the gradient of the Bohm potential within the Landau levels. More broadly, our study illustrates how the Madelung approach reveals unexpected analogies between quantum dynamics and phenomena familiar from geophysical fluid dynamics.
This paper solves the newly constructed nonlinear master equation dρ/d t = κ[2 f ( N ) a ρ (1/ f ( N − 1)) a † − a † a ρ−ρ a † a ], where f ( N ) is an operator-valued function of N = a † a , for describing amplitude damping channel, and derives the infinite operator sum representation of quasi-Kraus operators for the density operator. It also shows that in this nonlinear process the initial pure number state density operator will evolve into the binomial field (a mixed state) when