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Journal of Russian Laser Research

Squeezing of Relative and Center-of-Orbit Coordinates of a Charged Particle by Step-Wise Variations of a Uniform Magnetic Field with an Arbitrary Linear Vector Potential

V. V. Dodonov, Matheus B. Horovits

Published Jul 1, 2018• 11 citations
Publisher
DOI
10.1007/s10946-018-9733-1

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For a charged particle in a homogeneous magnetic field, we construct stationary squeezed states which are eigenfunctions of the Hamiltonian and the non-Hermitian operator % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVy0df9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lqpe0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-xir-f0-yqaqVeLsFr0-vr% 0-vr0dc8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGybGbaK% aadaWgaaWcbaGaeuOPdyeabeaakiabg2da9iqadIfagaqcaiGacoga% caGGVbGaai4CaiabfA6agjabgUcaRiqadMfagaqcaiGacohacaGGPb% GaaiOBaiabfA6agbaa!4894! $$\hat X_\Phi = \hat X\cos \Phi + \hat Y\sin \Phi $$ , % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVy0df9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lqpe0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-xir-f0-yqaqVeLsFr0-vr% 0-vr0dc8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGybGbaK% aaaaa!3A82! $$\hat X$$ and Ŷ being the coordinates of the Larmor circle center and Φ is a complex parameter. In the family of the squeezed states, the quantum uncertainty in the Larmor circle position is minimal. The wave functions of the squeezed states in the coordinate representation are found and their properties are discussed. Besides, for arbitrary gauge of the vector potential we derive the symmetry operators of translations and rotations.

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We consider a quantum charged particle moving in the x y plane under the action of a time-dependent magnetic field described by means of the linear vector potential of the form A = B ( t ) − y ( 1 + β ) , x ( 1 − β ) / 2 . Such potentials with β ≠ 0 exist inside infinite solenoids with non-circular cross sections. The systems with different values of β are not equivalent for nonstationary magnetic fields or time-dependent parameters β ( t ) , due to different structures of induced electric fields. Using the approximation of the stepwise variations of parameters, we obtain explicit formulas describing the change of the mean energy and magnetic moment. The generation of squeezing with respect to the relative and guiding center coordinates is also studied. The change of magnetic moment can be twice bigger for the Landau gauge than for the circular gauge, and this change can happen without any change of the angular momentum. A strong amplification of the magnetic moment can happen even for rapidly decreasing magnetic fields.
In the case of a constant uniform magnetic field it can be assumed, without the loss of generality, that the vector potential (the gauge) is a linear function of position, i.e. it could be considered as a three-dimensional real matrix or, more generally in an n-dimensional space, as a tensor A of the rank two. The magnetic tensor H is obtained from A by antisymmetrization, i.e. H=A-A^T. It is shown that the transpose of A plays a special role, since it determines the operator of the orbit center of a charged particle moving in an external magnetic field H. Moreover, this movement can be considered as a combination of N<=n independent cyclotronic movements in orthogonal planes (cyclotron orbits) with quantized energies, whereas in other n-2N dimensions the particle is completely free with a continuous energy spectrum. The proposed approach enables introduction of the four-dimensional space-time and, after some generalizations, non-linear gauges.
For a charged particle in a homogeneous magnetic field, we construct stationary squeezed states which are eigenfunctions of the Hamiltonian and the non-Hermitian operator % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVy0df9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lqpe0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-xir-f0-yqaqVeLsFr0-vr% 0-vr0dc8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGybGbaK% aadaWgaaWcbaGaeuOPdyeabeaakiabg2da9iqadIfagaqcaiGacoga% caGGVbGaai4CaiabfA6agjabgUcaRiqadMfagaqcaiGacohacaGGPb% GaaiOBaiabfA6agbaa!4894! X^Φ=X^cos⁡Φ+Y^sin⁡Φ\hat X_\Phi = \hat X\cos \Phi + \hat Y\sin \Phi X^Φ​=X^cosΦ+Y^sinΦ , % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVy0df9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lqpe0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-xir-f0-yqaqVeLsFr0-vr% 0-vr0dc8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGybGbaK% aaaaa!3A82! X^\hat XX^ and Ŷ being the coordinates of the Larmor circle center and Φ is a complex parameter. In the family of the squeezed states, the quantum uncertainty in the Larmor circle position is minimal. The wave functions of the squeezed states in the coordinate representation are found and their properties are discussed. Besides, for arbitrary gauge of the vector potential we derive the symmetry operators of translations and rotations.