Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Nanomaterials

Quadrature Squeezing and Geometric-Phase Oscillations in Nano-Optics

Jeong Ryeol Choi

Published Jul 17, 2020• 5 citations
Publisher

Abstract

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.

DOI
10.3390/nano10071391

Related papers

Properties of the Geometric Phase in Electromechanical Oscillations of Carbon-Nanotube-Based Nanowire Resonators

Jeong Ryeol Choi, Sanghyun Ju

The geometric phase is an extra phase evolution in the wave function of vibrations that is potentially applicable in a broad range of science and technology. The characteristics of the geometric phase in the squeezed state for a carbon-nanotube-based nanowire resonator have been investigated by means of the invariant operator method. The introduction of a linear invariant operator, which is useful for treating a complicated time-dependent Hamiltonian system, enabled us to derive the analytical formula of the geometric phase. By making use of this, we have analyzed the time behavior of the geometric phase based on relevant illustrations. The influence of squeezing parameters on the evolution of the geometric phase has been investigated. The geometric phase, in large, oscillates, and the envelope of such oscillation increases over time. The rate of the increase of the geometric phase is large when the parameters, such as the classical amplitude of the oscillation, the damping factor, and the amplitude of the driving force, are large. We have confirmed a very sharp increase of the geometric phase over time in the case that the angular frequency of the system reaches near the resonance angular frequency. Our development regarding the characteristics of the geometric phase is crucial for understanding the topological features in nanowire oscillations.

Nanoscale Research Letters•Feb 5, 2019•3 citations

Effects of light-wave nonstaticity on accompanying geometric-phase evolutions

Jeong Ryeol Choi

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.

arXiv:2107.03622•Optics Express•Sep 21, 2021•5 citations

Geometric phases for nonlinear coherent and squeezed states

Da-Bao Yang, Ying Chen, Fu-Lin Zhang +1 more

The geometric phases for standard coherent states which are widely used in quantum optics have attracted considerable attention. Nevertheless, few physicists consider the counterparts of nonlinear coherent states, which are useful in the description of the motion of a trapped ion. In this paper, the non-unitary and non-cyclic geometric phases for two nonlinear coherent and one squeezed states are formulated, respectively. Moreover, some of their common properties are discussed, such as gauge invariance, non-locality and nonlinear effects. The nonlinear functions have dramatic impacts on the evolution of the corresponding geometric phases. They speed the evolution up or down. So this property may have an application in controlling or measuring geometric phase. For the squeezed case, when the squeezed parameter r → ∞ , the limiting value of the geometric phase is also determined by a nonlinear function at a given time and angular velocity. In addition, the geometric phases for standard coherent and squeezed states are obtained under a particular condition. When the time evolution undergoes a period, their corresponding cyclic geometric phases are achieved as well. And the distinction between the geometric phases of the two coherent states may be regarded as a geometric criterion.

arXiv:1011.0501•Journal of Physics B Atomic Molecular and Optical Physics•Mar 16, 2011•12 citations

Nonadiabatic geometric phase for the cyclic evolution of a time-dependent Hamiltonian system

Jie Liu, Bambi Hu, Baowen Li

The geometric phases of the cyclic states of a generalized harmonic oscillator with nonadiabatic time-periodic parameters are discussed in the framework of squeezed state. It is shown that the cyclic and quasicyclic squeezed states correspond to the periodic and quasiperiodic solutions of an effective Hamiltonian defined on an extended phase space, respectively. The geometric phase of the cyclic squeezed state is found to be a phase-space area swept out by a periodic orbit. Furthermore, a class of cyclic states is expressed as a superposition of an infinite number of squeezed states. Their geometric phases are found to be independent of $\ensuremath{\Elzxh},$ and equal to $\ensuremath{-}(n+1/2)$ times the classical nonadiabatic Hannay angle.

arXiv:quant-ph/9806076•Physical Review A•Nov 1, 1998•10 citations

Pancharatnam phase for ordinary and generalized squeezed states

Istok P. Mendaš

The Pancharatnam phase for ordinary and generalized squeezed states of the harmonic oscillator is discussed. In particular, the general expressions for the total and dynamical phases are obtained yielding, from their difference, the geometric phase for a nonadiabatic and noncyclic evolution from an initial generalized squeezed state at time ${\mathrm{t}}^{|\mathrm{I}\mathrm{H}}$ to the final state at a time t\ensuremath{''}. The special case of a two-photon interaction process, such as parametric amplification, is subsequently discussed. In particular, the conditions leading to cyclic evolution are examined and the corresponding Aharonov-Anandan phase is found from the general expression for the Pancharatnam phase.

Physical Review A•Feb 1, 1997•9 citations

Amplitude Noise Reduction in a Nano-Mechanical Oscillator

Aziz Kolkıran

We study the quantum properties of a nano-mechanical oscillator via the squeezing of the oscillator amplitude. The static longitudinal compressive force 0 F close to a critical value at the Euler buckling instability leads to an anharmonic term in the Hamiltonian and thus the squeezing properties of the nano-mechanical oscillator are to be obtained from the Hamiltonian of the form H = a+a + β(a+ + a) / 4 . This Hamiltonian has no exact solution unlike the other known models of nonlinear interactions of the forms a+2 a2, (a2a)2 and a+4 + a4 - (a+2 a2 + a2 a+2 ) previously employed in quantum optics to study squeezing. Here we solve the Schrodinger equation numerically and show that in-phase quadrature gets squeezed for both vacuum and coherent states. The squeezing can be controlled by bringing F0 close to or far from the critical value Fc. We further study the effect of the transverse driving force on the squeezing in nano-mechanical oscillator.

Mathematical and Computational Applications•Apr 1, 2011•2 citations
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.
The geometric phase is an extra phase evolution in the wave function of vibrations that is potentially applicable in a broad range of science and technology. The characteristics of the geometric phase in the squeezed state for a carbon-nanotube-based nanowire resonator have been investigated by means of the invariant operator method. The introduction of a linear invariant operator, which is useful for treating a complicated time-dependent Hamiltonian system, enabled us to derive the analytical formula of the geometric phase. By making use of this, we have analyzed the time behavior of the geometric phase based on relevant illustrations. The influence of squeezing parameters on the evolution of the geometric phase has been investigated. The geometric phase, in large, oscillates, and the envelope of such oscillation increases over time. The rate of the increase of the geometric phase is large when the parameters, such as the classical amplitude of the oscillation, the damping factor, and the amplitude of the driving force, are large. We have confirmed a very sharp increase of the geometric phase over time in the case that the angular frequency of the system reaches near the resonance angular frequency. Our development regarding the characteristics of the geometric phase is crucial for understanding the topological features in nanowire oscillations.
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.
The geometric phases for standard coherent states which are widely used in quantum optics have attracted considerable attention. Nevertheless, few physicists consider the counterparts of nonlinear coherent states, which are useful in the description of the motion of a trapped ion. In this paper, the non-unitary and non-cyclic geometric phases for two nonlinear coherent and one squeezed states are formulated, respectively. Moreover, some of their common properties are discussed, such as gauge invariance, non-locality and nonlinear effects. The nonlinear functions have dramatic impacts on the evolution of the corresponding geometric phases. They speed the evolution up or down. So this property may have an application in controlling or measuring geometric phase. For the squeezed case, when the squeezed parameter r → ∞ , the limiting value of the geometric phase is also determined by a nonlinear function at a given time and angular velocity. In addition, the geometric phases for standard coherent and squeezed states are obtained under a particular condition. When the time evolution undergoes a period, their corresponding cyclic geometric phases are achieved as well. And the distinction between the geometric phases of the two coherent states may be regarded as a geometric criterion.
The geometric phases of the cyclic states of a generalized harmonic oscillator with nonadiabatic time-periodic parameters are discussed in the framework of squeezed state. It is shown that the cyclic and quasicyclic squeezed states correspond to the periodic and quasiperiodic solutions of an effective Hamiltonian defined on an extended phase space, respectively. The geometric phase of the cyclic squeezed state is found to be a phase-space area swept out by a periodic orbit. Furthermore, a class of cyclic states is expressed as a superposition of an infinite number of squeezed states. Their geometric phases are found to be independent of $\ensuremath{\Elzxh},$ and equal to $\ensuremath{-}(n+1/2)$ times the classical nonadiabatic Hannay angle.
The Pancharatnam phase for ordinary and generalized squeezed states of the harmonic oscillator is discussed. In particular, the general expressions for the total and dynamical phases are obtained yielding, from their difference, the geometric phase for a nonadiabatic and noncyclic evolution from an initial generalized squeezed state at time t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH to the final state at a time t\ensuremath{''}. The special case of a two-photon interaction process, such as parametric amplification, is subsequently discussed. In particular, the conditions leading to cyclic evolution are examined and the corresponding Aharonov-Anandan phase is found from the general expression for the Pancharatnam phase.
We study the quantum properties of a nano-mechanical oscillator via the squeezing of the oscillator amplitude. The static longitudinal compressive force 0 F close to a critical value at the Euler buckling instability leads to an anharmonic term in the Hamiltonian and thus the squeezing properties of the nano-mechanical oscillator are to be obtained from the Hamiltonian of the form H = a+a + β(a+ + a) / 4 . This Hamiltonian has no exact solution unlike the other known models of nonlinear interactions of the forms a+2 a2, (a2a)2 and a+4 + a4 - (a+2 a2 + a2 a+2 ) previously employed in quantum optics to study squeezing. Here we solve the Schrodinger equation numerically and show that in-phase quadrature gets squeezed for both vacuum and coherent states. The squeezing can be controlled by bringing F0 close to or far from the critical value Fc. We further study the effect of the transverse driving force on the squeezing in nano-mechanical oscillator.