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IEEE photonics journal

Robust Geometric Phase of Bloch Sphere Deformation in Quasiphase Matched Structures

Feiyan Zhao, Haojun Zhang, Xiaoxi Xu, Huancan Liang, Hexiang He, Yangui Zhou, Jiantao Lü, Yongyao Li

Published May 24, 2022• 8 citations
Publisher

Abstract

We consider the adiabatic geometric phase (AGP) in the sum frequency generation process under the undepleted pump approximation. The Bloch sphere is stretched or compressed by adjusting the modulation wave vector, Δ, which is one of the control parameters of quasiphase matching (QPM). In this paper, two kinds of parametric rotation schemes on the sphere are studied: elliptical rotation and elliptical wedge rotation. We find that the AGP generated in the elliptical rotation is related to the length of the Δ axis of the ellipse, but there is a certain point where the AGP is independent, while the AGP in the elliptical wedge rotation shows great stability and does not depend on the shape of the ellipsoid. The results further demonstrate the superior flexibility of the wedge scheme, which helps to suppress the uncertainty in the generation of AGPs.

DOI
10.1109/jphot.2022.3177687

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We consider the adiabatic geometric phase (AGP) in the sum frequency generation process under the undepleted pump approximation. The Bloch sphere is stretched or compressed by adjusting the modulation wave vector, Δ, which is one of the control parameters of quasiphase matching (QPM). In this paper, two kinds of parametric rotation schemes on the sphere are studied: elliptical rotation and elliptical wedge rotation. We find that the AGP generated in the elliptical rotation is related to the length of the Δ axis of the ellipse, but there is a certain point where the AGP is independent, while the AGP in the elliptical wedge rotation shows great stability and does not depend on the shape of the ellipsoid. The results further demonstrate the superior flexibility of the wedge scheme, which helps to suppress the uncertainty in the generation of AGPs.
The application of the adiabatic geometric phase (AGP) to nonlinear frequency conversion may help to develop new types of all-optical devices, which leads to all-optical modulation of the phase front of one wave by the intensity of other waves. In this paper, we develop the canonical Hamilton equation and a corresponding geometric representation for two schemes of four-wave mixing (FWM) processes (ω1 + ω2 = ω3 + ω4 and ω1 + ω2 + ω3 = ω4), which can precisely describe and calculate the AGP controlled by the quasi-phase matching technique. The AGPs of the idler (ω1) and signal (ω4) waves for these two schemes of FWM are studied systematically when the two pump waves (ω2 and ω3) are in either the undepleted or in the depleted pump cases, respectively. The analysis reveals that the proposed methods for calculating the AGP are universal in both cases. We expect that the analysis of AGP in FWM processes can be applied to all-optically shaping or encoding of ultrafast light pulse.
nonlinear crystal rotate along a closed path, geometric phase will be generated in the signal and idler waves that participate in the nonlinear frequency conversion. In this paper, we study two rotation schemes, full-wedge rotation and half-wedge rotation, of the QPM parameters in the process of fully nonlinear three-wave mixing. These two schemes can effectively suppress the uncertainty in creating the geometric phase in the nonlinear frequency conversion process when the intensity of the pump is depleted. The finding of this paper provides an avenue toward constant control of the geometric phase in nonlinear optics applications and quantum information processing.
Copyright © 2014 Ryuzi Yano and Hiroyuki Shinojima. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Simulations of the Bloch vector manipulation of two operations (one that utilizes the geometric phase (geometric operation) and the other that does not utilize the geometric phase (simple operation) in the adiabatic regime enabled us to study the tolerance of the geometric phase utilized operation against the laser noise. We simulated the Bloch vector manipulation by the laser field with the amplitude, the frequency, and the constant phase are proportionally, integrally, and derivatively controlled. The manipulation of the Bloch vector is a /2 rotation around 2-axis. There are few advantage by using the operation of the geometric phase over the simple operation. It has an advantage over only when the geometric phase operation is utilized such that the magnitude of the frequency of the noise is large, and that the noise correlation time is small.
Adiabatic perturbation is shown to be singular from the exact solution of a spin-1/2 particle in a uniformly rotating magnetic field. Due to a non-adiabatic effect, its quantum trajectory on a Bloch sphere is a cycloid traced by a circle rolling along an adiabatic path. As the magnetic field rotates more and more slowly, the time-energy uncertainty, proportional to the length of the quantum trajectory, calculated by the exact solution is entirely different from the one obtained by the adiabatic path traced by the instantaneous eigenstate. However, the non-adiabatic Aharonov-Anandan geometric phase, measured by the area enclosed by the exact path, approaches smoothly the adiabatic Berry phase, proportional to the area enclosed by the adiabatic path. The singular limit of the time-energy uncertainty and the regular limit of the geometric phase are associated with the arc length and arc area of the cycloid on a Bloch sphere, respectively. Prolate and curtate cycloids are also traced by different initial states outside and inside of the rolling circle, respectively. The axis trajectory of the rolling circle, parallel to the adiabatic path, is shown to be an example of transitionless driving. The non-adiabatic resonance is visualized by the number of cycloid arcs.
As is well known, when an SU (2) operation acts on a two-level system, its Bloch vector rotates without change of magnitude. Considering a system composed of two two-level systems, it is proven that for a class of nonlocal interactions of the two subsystems including (with ) and the Heisenberg interaction, the geometric description of the motion is particularly simple: each of the two Bloch vectors follows an elliptical orbit within the Bloch sphere. The utility of this result is demonstrated in two applications, the first of which bears on quantum control via quantum interfaces. By employing nonunitary control operations, we extend the idea of controllability to a set of points which are not necessarily connected by unitary transformations. The second application shows how the orbit of the coherence vector can be used to assess the entangling power of Heisenberg exchange interaction.
The quantum states of cold ensemble of two-level atoms were rotated around axis 2 on the Bloch sphere geometrically using a laser-controlled resonant pulse sequence with pulse areas of $\ensuremath{\pi}∕2$, $\ensuremath{\pi}$, and $\ensuremath{\pi}∕2$, whose central pulse has a relative phase difference of $\ensuremath{\pi}+\ensuremath{\delta}∕2$. It was ascertained that the population probability of the excited state varies according to a sinusoidal function of $\ensuremath{\delta}$ within an uncertainty of 1%. Hybrid operations of two-level atoms were realized combining the geometrical operation with the Rabi excitation around axis 2 or the time evolution around axis 3.