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Laser Physics Letters

Quasi-classical theory of cyclotron resonance with accounting for dissipation

С. В. Сазонов

Published Mar 27, 2024• 0 citations
Publisher

Abstract

Abstract Based on the quasi-classical version of the canonical Caldirola–Kanai quantization, non-relativistic cyclotron resonance in a dissipative medium is studied. The corresponding particle propagator in the coordinate representation is found. It is shown that the combined effect of dissipation and a constant magnetic field reduces to the suppression of the quantum properties of a charged particle. In turn, a time-varying electric field that causes cyclotron resonance does not exhibit similar properties and does not affect the uncertainties of the particle coordinates.

DOI
10.1088/1612-202x/ad3628

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Abstract Based on the quasi-classical version of the canonical Caldirola–Kanai quantization, non-relativistic cyclotron resonance in a dissipative medium is studied. The corresponding particle propagator in the coordinate representation is found. It is shown that the combined effect of dissipation and a constant magnetic field reduces to the suppression of the quantum properties of a charged particle. In turn, a time-varying electric field that causes cyclotron resonance does not exhibit similar properties and does not affect the uncertainties of the particle coordinates.
Abstract The quasi-classical dynamics of a charged particle in a weak magnetic field in the presence of dissipative losses caused by ionization of the medium is studied. The approximate approach proposed here is a generalization of the Caldirola—Kanai method for quantizing the translational motion of particles in dissipative media. It is shown that a weak curvature of a classical trajectory by the magnetic field is accompanied by an isotropic increase of uncertainty of the particle coordinates to the some maximum value at the moment the localized probability density wave packet stops. The limitation of the increase of coordinate uncertainty is due to irreversible ionization losses.
Abstract Based on the quasi-classical approximation, a general approach is proposed for constructing non-stationary quantum states of a charged particle in a magnetic field, when the dissipative forces of viscous friction and drag, proportional to the velocity and the square of the velocity, respectively, are also significant. The corresponding quasi-classical Green’s function is found, with the help of which the squeezed and coherent states of the particle are studied. It is shown that the dissipation and a magnetic field suppress the quantum properties of the particle. This is especially true for the transverse motion with respect to the magnetic field. Over time, the coherent and squeezed states transform into the same static state, which is characterized by a zero uncertainty of the transverse coordinates and an uncertainty of the longitudinal coordinate, which contains information about the initial velocity of the particle.
We deal with the motion of a charged particle in a constant magnetic field when immersed in a medium that exerts some type of friction. We analyse the problem classically and also quantum mechanically. In the latter case we use coherent state solutions of the problem that, for large energies, compared with the one associated with the cyclotron frequency, reduce to the classical limit. To study the dynamical behaviour we use time-dependent wave packets that can be constructed by superposition of the coherent states.
An approximate self-consistent approach is proposed that allows describing the quasi-classical translational dynamics of a non-relativistic particle in a dissipative medium with arbitrary dependence of the corresponding dissipative forces on velocity. It is shown that dissipation suppresses the quantum properties of the particle. This leads to the necessity of interpreting propagation in a dissipative medium as a continuous process of measuring the particle state. As examples, non-stationary coherent states of the particle are considered at three stages of its deceleration in the medium due to ionization losses. These stages correspond to high-energy losses, losses in the vicinity of the Bragg peak, and low-energy losses at the final stage of propagation.
The classical problem of planar cyclotron motion of a charged particle in a uniform magnetic field possesses symmetries which account for the ``accidental'' degeneracies of the analogous nonrelativistic Schrödinger equation, as found by Johnson and Lippman. The essentially quadratic nature of the Hamiltonian is not changed by considering the particle moving in a harmonic oscillator potential, a ``Zeeman effect'' for the harmonic oscillator. The transitions to the limiting cases of a weak magnetic field (pure harmonic oscillator) or a strong field (pure cyclotron motion) involve the contraction of the corresponding symmetry groups, yielding Larmor precession of the oscillator orbits in the first case, and the drift of the cyclotron orbit in the second. The constants of the motion generate the unitary unimodular group SU2 in all cases except for pure cyclotron motion, in which case one obtains the commutation rules of creation and annihilation operators. Only for certain ratios of magnetic field strength to the oscillator frequency does one obtain bounded closed orbits, and presumably only in these cases do degeneracies exist quantum-mechanically. A transition to a rotating coordinate system reduces the problem to that of a plane harmonic oscillator; however, the time dependencies of the transformation must be allowed for interpreting the constants thereby arising. Moreover, the velocity-dependent forces introduce gauge transformations which also affect the interpretation of the symmetries. There are two kinds of symmetries—inner symmetries involving the canonical coordinates and governing the shape of the orbits, and outer symmetries involving the mechanical coordinates and governing the location of the orbits.
Abstract A self-consistent quasi-classical generalization of the axiomatic quantization of the translational motion of a particle in a dissipative medium is proposed, which is modeled by a drag force F d , which depends in a power-law manner on the velocity v : F d ∼ v n , where n is an arbitrary real number. A detailed analysis of the quasi-classical coherent state is carried out. It is shown that under condition n < 3 the dissipative medium suppresses the quantum properties of particle motion, performing the function of a classical measuring device. Otherwise, there is no suppression of quantum properties.