Quantum mechanics allows the emergence of nonstatic quantum light waves in the Fock state even in a transparent medium of which electromagnetic parameters do not vary over time. Such wave packets become broad and narrow in turn periodically in the quadrature space. We investigate the effects of wave nonstaticity arisen in a static environment on the behavior of accompanying geometric phases in the Fock states. In this case, the geometric phases appear only when the measure of nonstaticity is not zero and their time behavior is deeply related to the measure of nonstaticity. While the dynamical phases undergo linear decrease over time, the geometric phases exhibit somewhat oscillatory behavior where the center of oscillation increases linearly. In particular, if the measure of nonstaticity is sufficiently high, the geometric phases abruptly change whenever the waves become narrow in the quadrature space. The understanding for the phase evolution of nonstatic light waves is necessary in their technological applications regarding wave modulations.
Quantum mechanics allows the emergence of nonstatic quantum light waves in the Fock state even in a transparent medium of which electromagnetic parameters do not vary over time. Such wave packets become broad and narrow in turn periodically in the quadrature space. We investigate the effects of wave nonstaticity arisen in a static environment on the behavior of accompanying geometric phases in the Fock states. In this case, the geometric phases appear only when the measure of nonstaticity is not zero and their time behavior is deeply related to the measure of nonstaticity. While the dynamical phases undergo linear decrease over time, the geometric phases exhibit somewhat oscillatory behavior where the center of oscillation increases linearly. In particular, if the measure of nonstaticity is sufficiently high, the geometric phases abruptly change whenever the waves become narrow in the quadrature space. The understanding for the phase evolution of nonstatic light waves is necessary in their technological applications regarding wave modulations.
If electromagnetic parameters of a medium vary in time, quantum light waves traveling in it become nonstatic. A recent report shows that such nonstatic waves can also appear even when the environment is static where the parameters of the medium do not vary. In this work, the properties of nonstatic waves in a static environment are investigated from their p-space analysis, focusing on the interrelation between the q- and p-space nonstatic waves. The probability densities in p-space (as well as in q-space) for both the nostatic Fock and Gaussian states evolve in a periodic manner, i.e., they constitute belly and node in turn successively as time goes by. If we neglect the displacement of waves, the q- and p-space wave phases are opposite each other. Since the intensity of the wave in each space is relatively large whenever the wave forms a belly, such a phase difference indicates that periodical intensity exchange between the q- and p-component waves takes place through their nonstatic evolutions. This is the novel reciprocal optical phenomenon arisen on account of the wave nonstaticity.
The characteristics of nonstatic quantum light waves in the coherent state in a static environment is investigated. It is shown that the shape of the wave varies periodically as a manifestation of its peculiar properties of nonstaticity like the case of the Fock-state analysis for a nonstatic wave. A belly occurs in the graphic of wave evolution whenever the wave is maximally displaced in the quadrature space, whereas a node takes place every time the wave passes the equilibrium point during its oscillation. In this way, a belly and a node appear in turn successively. Whereas this change of wave profile is accompanied by the periodic variation of electric and magnetic energies, the total energy is conserved. The fluctuations of quadratures also vary in a regular manner according to the wave transformation in time. While the resultant time-varying uncertainty product is always larger than (or, at least, equal to) its quantum-mechanically allowed minimal value ([Formula: see text]), it is smallest whenever the wave constitutes a belly or a node. The mechanism underlying the abnormal features of nonstatic light waves demonstrated here can be interpreted by the rotation of the squeezed-shape contour of the Wigner distribution function in phase space.
The phases are the main factor that affects the outcome of various optical phenomena, such as quantum superposition, wave interference, and light-matter interaction. As a light wave becomes nonstatic, an additional phase, the so-called geometric phase, takes place in its evolution. Then, due to this phase, the overall phase of the quantum wave function varies in a nonlinear way with time. Interestingly, the phase exhibits a step-like evolution if the measure of nonstaticity is extremely high. Such an abnormal phase variation is analyzed in detail for better understanding of wave nonstaticity in this work. As the wave becomes highly nonstatic, the phase factor of the electromagnetic wave evolves in a rectangular manner. However, the shape of the electromagnetic field is still a sinusoidal form on account of the compensational variation of the wave amplitude. The electromagnetic field in this case very much resembles that of a standing wave. The effects accompanying the step-phase evolution, such as modification of the probability distribution and alteration of the wave-interference profile, are analyzed and their implications are illustrated.
Nonstatic quantum light waves arise in time-varying media in general. However, from a recent report, it turned out that nonstatic waves can also appear in a static environment where the electromagnetic parameters of the medium do not vary in time. Such waves in Fock states exhibit a belly and a node in turn periodically in the graphic of their evolution. This is due to the wave expansion and collapse in quadrature space, which manifest a unique nonstaticity of the wave. The principle for wave expansion and collapse is elucidated from rigorous analyses for the basic nonstatic waves which are dissipative and amplifying ones. The outcome of wave nonstaticity can be interpreted in terms of the coefficient of the quadratic exponent in the exponential function appearing in the wave eigenfunction; if the imaginary part of the coefficient is positive, the wave expands, whereas the wave collapses when it is negative. Using this principle, we further analyze novel nonstatic properties of light waves which exhibit complicated time behaviors, i.e., for the case that the waves not only undergo the periodical change of nodes and bellies but their envelopes exhibit gradual dissipation/expansion as well.
The geometric phase, as well as the familiar dynamical phase, occurs in the evolution of a squeezed state in nano-optics as an extra phase. The outcome of the geometric phase in that state is somewhat intricate: its time behavior exhibits a combination of a linear increase and periodic oscillations. We focus in this work on the periodic oscillations of the geometric phase, which are novel and interesting. We confirm that such oscillations are due purely to the effects of squeezing in the quantum states, whereas the oscillation disappears when we remove the squeezing. As the degree of squeezing increases in q-quadrature, the amplitude of the geometric-phase oscillation becomes large. This implies that we can adjust the strength of such an oscillation by tuning the squeezing parameters. We also investigate geometric-phase oscillations for the case of a more general optical phenomenon where the squeezed state undergoes one-photon processes. It is shown that the geometric phase in this case exhibits additional intricate oscillations with small amplitudes, besides the principal oscillation. Such a sub-oscillation exhibits a beating-like behavior in time. The effects of geometric-phase oscillations are crucial in a wide range of wave interferences which are accompanied by rich physical phenomena such as Aharonov–Bohm oscillations, conductance fluctuations, antilocalizations, and nondissipative current flows.
A simple elegant expression of nonadiabatic light wave evolution is necessary in order to have a deeper insight for complicated optical phenomena in light science as well as in everyday life. Light wave propagation in linear media which have time-dependent electromagnetic parameters is investigated by utilizing a quadratic invariant of the system. The time behavior of the nonadiabatic geometric phase of the waves that yield a cyclic nonadiabatic evolution is analyzed in detail. Various quantum properties of light waves in this situation, such as variances of electric and magnetic fields, uncertainty product, coherent and squeezed states, and their classical limits, are developed. For better understanding of our research, we applied our analysis in a particular case. The variances of the fields D and B are illustrated and their time behaviors are addressed. Equivalent results for the corresponding classical systems are deduced from the study of the time evolution of the appropriate coherent and squeezed states.