A Quantum Charged Particle under Sudden Jumps of the Magnetic Field and Shape of Non-Circular Solenoids

Abstract

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.

Energy and Magnetic Moment of a Quantum Charged Particle in Time-Dependent Magnetic and Electric Fields of Circular and Plane Solenoids

“…Practically, the values µ = 0.1 and µ = 10 can be quite sufficient. This result is important, because it justifies the reasonableness of the "sudden jump" approach in numerous applications, in particular, in our papers [30,63,85]. However, such justifications are not universal: they work well if only the "transition time" is well defined, as in the cases of exponential-like decay.…”

“…However, this approximation is quite good in the non-relativistic case, because the spatial inhomogeneity scale of the electromagnetic field is proportional to the light velocity c, whereas the cyclotron radius of a charged particle (defining the admissible inhomogeneity scale of the magnetic field) is proportional to the particle velocity v c. For more details one can consult Refs. [19,85].…”

Magnetization dynamics of a harmonically confined quantum charged particle in time dependent magnetic fields inside a circular solenoid

“…However, they are not equivalent for time-dependent fields, due to different geometries of the induced electric field E = −∂A/(c∂t) [69,70]. Recently, such a nonequivalence was shown explicitly in different examples of sudden jumps of the magnetic field (without a confining potential) in papers [71][72][73]. It is a challenge to find exact or approximate solutions for smooth variations of arbitrary linear vector potentials.…”

Efficient algebraic solution for a time-dependent quantum harmonic oscillator

“…The main property of squeezed states is that they provide variances of certain quadratures smaller than the value associated to coherent states [26][27][28], enhancing the sensitivity of several systems [29,30]. Some remarkable examples are in telecommunications [31][32][33], spin-squeezed states [34][35][36] and some variations of the Landau problem with a time-dependent magnetic field [37,38]. Moreover, when squeezed states of light are employed, an astonishing improvement in the detection rate at LIGO [39,40] and sensitivity enhancement in the shot-noise limit at the Advanced Virgo gravitational wave detector [41] are observed.…”