Extending the quantal adiabatic theorem: Geometry of noncyclic motion

Comment on "Nongeometric Conditional Phase Shift via Adiabatic Evolution of Dark Eigenstates: A New Approach to Quantum Computation"

In [Phys. Rev. Lett. 95, 080502 (2005)], Zheng proposed a scheme for implementing a conditional phase shift via adiabatic passages. The author claims that the gate is "neither of dynamical nor geometric origin" on the grounds that the Hamiltonian does not follow a cyclic change. He further argues that "in comparison with the adiabatic geometric gates, the nontrivial cyclic loop is unnecessary, and thus the errors in obtaining the required solid angle are avoided, which makes this new kind of phase gates superior to the geometric gates." In this Comment, we point out that geometric operations, …

Rydberg-interaction gates via adiabatic passage and phase control of driving fields

In this paperwe propose two theoretical schemes for implementation of quantum phase gates by engineering the phase-sensitive dark state of two atoms subjected to Rydberg-Rydberg interaction. Combining the conventional adiabatic techniques and currently developed approaches of phase control, a feasible proposal for implementation of a geometric phase gate is presented, where the conditional phase shift (Berry phase) is achieved by adiabatically and cyclically changing the parameters of the driving fields. Here we find that the geometric phase acquired is related to the way how the relative phas…

Non-Abelian generalization of off-diagonal geometric phases

If a quantum system evolves in a noncyclic fashion the corresponding geometric phase or holonomy may not be fully defined. Off-diagonal geometric phases have been developed to deal with such cases. Here, we generalize these phases to the non-Abelian case, by introducing off-diagonal holonomies that involve evolution of more than one subspace of the underlying Hilbert space. Physical realizations of the off-diagonal holonomies in adiabatic evolution and interferometry are put forward.

Adiabatically steered open quantum systems: Master equation and optimal phase

We introduce an alternative way to derive the generalized form of the master equation recently presented by J. P. Pekola et al. [Phys. Rev. Lett. 105, 030401 (2010)] for an adiabatically steered two-level quantum system interacting with a Markovian environment. The original derivation employed the effective Hamiltonian in the adiabatic basis with the standard interaction picture approach but without the usual secular approximation. Our approach is based on utilizing a master equation for a non-steered system in the first super-adiabatic basis. It is potentially efficient in obtaining higher-or…

Geometric Phase: a Diagnostic Tool for Entanglement

Using a kinematic approach we show that the non-adiabatic, non-cyclic, geometric phase corresponding to the radiation emitted by a three level cascade system provides a sensitive diagnostic tool for determining the entanglement properties of the two modes of radiation. The nonunitary, noncyclic path in the state space may be realized through the same control parameters which control the purity/mixedness and entanglement. We show analytically that the geometric phase is related to concurrence in certain region of the parameter space. We further show that the rate of change of the geometric phas…

Two geometric phases can dramatically differ from each other even if their evolution paths are sufficiently close in a pointwise manner

One milestone in quantum physics is Berry's seminal work [Proc.~R.~Soc.~Lond.~A \textbf{392}, 45 (1984)], in which a quantal phase factor known as geometric phase was discovered to solely depend on the evolution path in state space. Here, we unveil that even an infinitesimal deviation of the initial state from the eigenstate of the initial Hamiltonian can yield a significant change of the geometric phase accompanying an adiabatic evolution. This leads to the surprising observation that two geometric phases can dramatically differ from each other even if their evolution paths are sufficiently c…

Geometric phases and hidden local gauge symmetry

The analysis of geometric phases associated with level crossing is reduced to the familiar diagonalization of the Hamiltonian in the second quantized formulation. A hidden local gauge symmetry, which is associated with the arbitrariness of the phase choice of a complete orthonormal basis set, becomes explicit in this formulation (in particular, in the adiabatic approximation) and specifies physical observables. The choice of a basis set which specifies the coordinate in the functional space is arbitrary in the second quantization, and a sub-class of coordinate transformations, which keeps the …

Geometric phase effects for wavepacket revivals

The study of wavepacket revivals is extended to the case of Hamiltonians which are made time-dependent through the adiabatic cycling of some parameters. It is shown that the quantal geometric phase (Berry's phase) causes the revived packet to be displaced along the classical trajectory, by an amount equal to the classical geometric phase (Hannay's angle), in one degree of freedom. A physical example illustrating this effect in three degrees of freedom is mentioned.

具有几何结构的结构化近轴电子辐射

Geometry-Enabled Radiation from Structured Paraxial Electrons

本文提出了一种基于几何框架的结构化电子辐射机制，通过结合Foldy-Wouthuysen变换与Lewis-Ermakov不变量，揭示了非循环几何演化与光子辐射之间的直接联系。