We study the evolution of the energy of a harmonic oscillator when its frequency slowly varies with time and passes through a zero value. We consider both the classical and quantum descriptions of ...the system. We show that after a single frequency passage through a zero value, the famous adiabatic invariant ratio of energy to frequency (which does not hold for a zero frequency) is reestablished again, but with the proportionality coefficient dependent on the initial state. The dependence on the initial state disappears after averaging over the phases of initial states with the same energy (in particular, for the initial vacuum, the Fock and thermal quantum states). In this case, the mean proportionality coefficient is always greater than unity. The concrete value of the mean proportionality coefficient depends on the power index of the frequency dependence on a time near the zero point. In particular, the mean energy triplicates if the frequency tends to zero linearly. If the frequency attains zero more than once, the adiabatic proportionality coefficient strongly depends on the lengths of time intervals between zero points, so that the mean energy behavior becomes quasi-stochastic after many passages through a zero value. The original Born-Fock theorem does not work after the frequency passes through zero. However, its generalization is found: the initial Fock state becomes a wide superposition of many Fock states, whose weights do not depend on time in the new adiabatic regime. When the mean energy triplicates, the initial Nth Fock state becomes a superposition of, roughly speaking, 6N states, distributed nonuniformly. The initial vacuum and low-order Fock states become squeezed, as well as the initial thermal states with low values of the mean energy.
We study the evolution of the energy and magnetic moment of a quantum charged particle placed in a homogeneous magnetic field, when this field changes its sign adiabatically. We show that after a ...single magnetic field passage through zero value, the famous adiabatic invariant ratio of energy to frequency is reestablished again, but with a proportionality coefficient higher than in the initial state. The concrete value of this proportionality coefficient depends on the power index of the frequency dependence on time near zero point. In particular, the adiabatic ratio of the initial ground state (with zero radial and angular quantum numbers) triplicates if the frequency tends to zero linearly as a function of time. If the Larmor frequency attains zero more than once, the adiabatic proportionality coefficient strongly depends on the lengths of the time intervals between zero points, so that the mean energy behavior can be quasi-stochastic after many passages through zero value. The original Born-Fock adiabatic theorem does not work after the frequency passes through zero. However, its generalization is found: the initial Fock state becomes a wide superposition of many instantaneous Fock states, whose weights do not depend on time in the new adiabatic regime.
The problem of finding covariance matrices that remain constant in time for arbitrary multi-dimensional quadratic Hamiltonians (including those with time-dependent coefficients) is considered. ...General solutions are obtained.
A two-parameter family of quantum states preserving the mean value of the magnetic moment (proportional to the kinetic angular momentum) is found for a charged particle in a constant homogeneous ...magnetic field in the presence of an isotropic two-dimensional parabolic potential, which can be either attractive or repulsive (the case of the Penning trap). The evolution of such states in a specific time-dependent magnetic field of the Epstein–Eckart form is studied, with an emphasis on the limit cases of the sudden jump and adiabatic approximations. The behavior in the case of magnetic field inversion is shown to be qualitatively different from the case when the field does not change its sign. The case of time-dependent vector potentials with a constant magnetic field (arising due to deformations of the shape of a solenoid) is considered, as well.
We study the tunneling of slow quantum packets through a high Coulomb barrier. We show that the transmission coefficient can be quite different from the standard expression obtained in the plane wave ...(WKB) approximation (and larger by many orders of magnitude), even if the momentum dispersion is much smaller than the mean value of the momentum.
•Tunneling of quantum packets through high Coulomb barrier.•Transmission probability is much higher than for plain waves.•Transmission probability strongly depends on the packet shape.•Transmission probability can depend on the momentum variance only.•The concept of effective Planck constant does not work.
We study numerically the evolution of the cavity electromagnetic field mode which is in resonance with an oscillating boundary (dynamical Casimir effect), taking into account the interaction between ...the field and a two-level atom, that may or not be continuously monitored by a coupled atomic excitation detector. We analyze the behavior of the field statistics and the quadrature squeezing properties in different regimes, demonstrating that at the expense of decreasing the number of produced photons and the degree of squeezing, one can create qualitatively new types of cavity field states.
► We study the statistics of photons created in a cavity via dynamical Casimir effect. ► We take into account the interaction with a two-level atom placed inside the cavity. ► The field–atom dynamics is calculated numerically for the Rabi coupling. ► The interaction with a detector can totally change the statistics of created photons. ► The statistics can vary from weakly super-Poissonian to strong “hyper-Poissonian”.
New sum and product uncertainty relations, containing variances of up to five observables, but not containing explicitly their covariances, are derived. New inequalities for three observables, ...especially for the angular momentum and spin-1/2 operators, are also presented.
We consider a quantum spinless nonrelativistic charged particle moving in the xy plane under the action of a time-dependent magnetic field, described by means of the linear vector potential ...A=B(t)-y(1+α),x(1-α)/2, with two fixed values of the gauge parameter α: α=0 (the circular gauge) and α=1 (the Landau gauge). While the magnetic field is the same in all the cases, the systems with different values of the gauge parameter are not equivalent for nonstationary magnetic fields due to different structures of induced electric fields, whose lines of force are circles for α=0 and straight lines for α=1. We derive general formulas for the time-dependent mean values of the energy and magnetic moment, as well as for their variances, for an arbitrary function B(t). They are expressed in terms of solutions to the classical equation of motion ε¨+ωα2(t)ε=0, with ω1=2ω0. Explicit results are found in the cases of the sudden jump of magnetic field, the parametric resonance, the adiabatic evolution, and for several specific functions B(t), when solutions can be expressed in terms of elementary or hypergeometric functions. These examples show that the evolution of the mentioned mean values can be rather different for the two gauges, if the evolution is not adiabatic. It appears that the adiabatic approximation fails when the magnetic field goes to zero. Moreover, the sudden jump approximation can fail in this case as well. The case of a slowly varying field changing its sign seems especially interesting. In all the cases, fluctuations of the magnetic moment are very strong, frequently exceeding the square of the mean value.
We study the influence of the non-equidistancy of the frequency spectrum on the Dynamical Casimir effect in a rectangular cavity with a harmonically oscillating ideal wall. The transition from the ...linear growth of the mean photon number of photons created from vacuum in the cavity with equidistant spectrum to the exponential growth in a weakly non-equidistant case is shown explicitly.
► We consider the dynamical Casimir effect in different rectangular cavities. ► One-dimensional models predict the linear growth of the number of photons. ► Three-dimensional models predict the exponential growth. ► We show how one model is transformed continuously into the other. ► The key parameter is the degree of non-equidistancy of the frequency spectra.
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