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arXiv:quant-ph/0412198

Berry Phase and Adiabatic Breakdown in Optical Modulator

Eyal Buks

Published Dec 26, 2004• 0 citations
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Abstract

We consider an all in-fiber optical modulator based on a ring resonator configuration. The case of adiabatic to nonadiabatic transition is considered, where the geometrical (Berry) phase acquired in a round trip along the ring changes abruptly by pi. Degradation of the responsivity of the modulator due to finite linewidth of the optical input is discussed. We show that the responsivity of the proposed modulator can be significantly enhanced with optimum design and compare with other configurations.

DOI
10.48550/arxiv.quant-ph/0412198

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We consider an all in-fiber optical modulator based on a ring resonator configuration. The case of adiabatic to nonadiabatic transition is considered, where the geometrical (Berry) phase acquired in a round trip along the ring changes abruptly by pi. Degradation of the responsivity of the modulator due to finite linewidth of the optical input is discussed. We show that the responsivity of the proposed modulator can be significantly enhanced with optimum design and compare with other configurations.
It is shown that, similar to resonant enhancement of linear phase, birefringent phase, and chirality/optical activity by a ring resonator (RR), Berry's phase, or the geometrical phase, can be amplified by repeatedly passing light through a Berry's phase-generating element placed in a RR. Small-signal amplification of Berry's phase is equal to the resonator finesse F. This amplification can be exploited when Berry's phase is used for performing quantum computation or in fiber-optic and other sensors.
The Berry phase acquired by an eigenstate that experienced a nonlinear adiabatic evolution is investigated thoroughly. The circuit integral of the Berry connection of the instantaneous eigenstate cannot account for the adiabatic geometric phase, while the Bogoliubov excitations around the eigenstates are found to be accumulated during the nonlinear adiabatic evolution and contribute a finite phase of geometric nature. A two-mode model is used to illustrate our theory. Our theory is applicable to Bose-Einstein condensate, nonlinear light propagation, and Ginzburg-Landau equations for complex order parameters in condensed-matter physics.
We present a simple and intuitive model based on the impulse response of linear electrical systems for describing the propagation of optical pulses through a dynamic Fabry–Perot resonator whose refractive index changes with time. Our model shows that the adiabatic wavelength conversion process in resonators results from a scaling of the round-trip time with index changes. For pulses longer than the cavity round-trip time, we find that more energy can be transferred to the new wavelength when the input pulses are slightly detuned from the cavity resonance and the refractive index does not change too rapidly. In fact, the optimum duration of index changes scales with the photon lifetime of the resonator. We describe the evolution of the shape and spectrum of picosecond pulses inside a resonator under a variety of input conditions and with the magnitude and duration of index variations. We also apply our general theory to the case of pulses whose widths are shorter than the round-trip time and derive an analytical expression for the output field under quite general conditions. This analysis reveals a shifting of the spectral comb as well as compression of the temporal pulse train that depends on the both the magnitude and sign of the index change. Our results should find applications in the area of optical signal processing using resonant photonic structures.
If the time evolution of a quantum state leads back to the initial state, a geometric phase is accumulated that is known as the Berry phase for adiabatic evolution or as the Aharonov-Anandan (AA) phase for nonadiabatic evolution. We evaluate these geometric phases using Floquet theory for systems in time-dependent external fields with a focus on paths leading through a degeneracy of the eigenenergies. Contrary to expectations, the low-frequency limits of the two phases do not always coincide. This happens as the degeneracy leads to a slow convergence of the quantum states to adiabaticity, resulting in a nonzero finite or divergent contribution to the AA phase. Steering the system adiabatically through a degeneracy provides control over the geometric phase as it can cause a π shift of the Berry phase. On the other hand, we revisit an example of degeneracy crossing proposed by AA. We find that, at suitable driving frequencies, both geometric-phase definitions give the same result and the dynamical phase is zero due to the symmetry of time evolution about the point of degeneracy, providing an advantageous setup for manipulation of quantum states.
A formalism for calculating Berry phases for nonadiabatic systems containing an additional adiabatic time dependence is developed. This is then applied to a two-level atom in a strong resonant polarized laser, where the laser polarization provides the adiabatic parameter. It is found that, when the laser polarization is kept constant, judicious polarization choices allow one to eliminate either the rotating-wave or non-rotating-wave part of the interaction. Furthermore, the adiabatic Berry phasor obtained when the laser polarization is varied is found to be complex. This can be compared to the case of purely adiabatic time-reversal even systems, which can only have real Berry phasors.
In this work, we propose an experimentally feasible nonlinear optical platform to realize nonintegrable geometric phases that emerge in interacting quantum systems undergoing quantum phase transitions. Specifically, we demonstrate that an exotic nonlinear term in the dynamical equations governs the polarization evolution of the optical field propagating along an anisotropic low-birefringence fiber with tetragonal symmetry. Intriguingly, by adiabatically tuning the nonlinear susceptibilities along the fiber, the Stokes vector on the Poincar\'e sphere accumulates a nonintegrable phase known as the Hannay angle---the classical counterpart to the Berry phase---which exhibits the same geometric gauge structure, including conic singularities in parameter space, as those associated with quantum phase transitions in systems like Bose-Einstein condensates. We further discuss a practical experimental implementation through the adiabatic deposition of nanocrystals along the fiber to control these susceptibilities, highlighting assumptions such as negligible losses and low birefringence for realistic fiber lengths.