A Note on the Adiabatic Theorem Without Gap Condition

We simplify the proof of the adiabatic theorem of quantum mechanics without gap condition of Avron and Elgart by providing an elementary solution of the commutator equation. In addition, a minor modification of their argument allows for more direct treatment of eigenvalue crossings. We also obtain simple, explicit conditions that yield information on the rate of convergence in the adiabatic limit.     

General Adiabatic Evolution with a Gap Condition

We consider the adiabatic regime of two parameters evolution semigroups generated by linear operators that are analytic in time and satisfy the following gap condition for all times: the spectrum of the generator consists in finitely many isolated eigenvalues of finite algebraic multiplicity, away from the rest of the spectrum. The restriction of the generator to the spectral subspace corresponding to the distinguished eigenvalues is not assumed to be diagonalizable. The presence of eigenilpotents in the spectral decomposition of the generator typically leads to solutions which grow exponentially fast in some inverse power of the adiabaticity parameter, even for real spectrum. In turn, this forbids the evolution to follow all instantaneous eigenprojectors of the generator in the adiabatic limit. Making use of superadiabatic renormalization, we construct a different set of time-dependent projectors, close to the instantaneous eigenprojectors of the generator in the adiabatic limit, and an approximation of the evolution semigroup which intertwines exactly between the values of these projectors at the initial and final times. Hence, the evolution semigroup follows the constructed set of projectors in the adiabatic regime, modulo error terms we control.     

Sufficient Condition for Validity of Quantum Adiabatic Theorem

In this paper,we attempt to give a sufficient condition of guaranteeing the validity of the proof of the quantum adiabatic theorem.The new sufficient condition can clearly remove the inconsistency and the counterexample of the quantum adiabatic theorem pointed out by Marzlin and Sanders.     

Design of Selective Adiabatic Inversion Pulses Using the Adiabatic Condition

Adiabatic RF pulses play an important role in spin inversion due to their robust behavior in the presence of inhomogeneous RF fields. These pulses are characterized by the trajectory swept by the tip of theB_effvector and the rate of motion along it. In this paper, we describe a method by which optimized modulation functions can be constructed to render insensitivity toB₁inhomogeneity over a predeterminedB₁range and over a wide band of frequencies. This is accomplished by requiring that the optimized pulse fulfill the adiabatic condition over this range ofB₁inhomogeneity and over the desired frequency band for the complete duration of the pulse. A trajectory similar to the well-known sech/tanh adiabatic pulse, i.e., a half-ellipse, is used. The optimization process improves the slice profile by optimizing the rate of motion along this trajectory. The optimized pulse can be tailored to the specific design requirements; in particular, the transition sharpness may be traded off against the inverted bandwidth. Two design examples, including experimental results, demonstrate the superiority of the optimized pulses over the conventional sech/tanh pulse: in the first example, a large frequency band is to be inverted using a weak RF amplitude in a short time. In the second example, a pulse with a very sharp transition is required.     

Evolutions of Yang Phase Under Cyclic Condition and Adiabatic Condition

There are three non-integrable phases in literatures: Berry phase, Aharonov-Anandan phase, and Yang phase. This article discusses the evolutions of Yang phase under the cyclic condition and the adiabatic condition for the generaltime-dependent harmonic oscillator, thus reveals the intimate relations between these three non-integrable phases.     

Evolutions of Yang Phase Under Cyclic Condition and Adiabatic Condition

There are three non-integrable phases in literatures： Berry phase, Aharonov-Anandan phase, and Yang phase. This article discusses the evolutions of Yang phase under the cyclic condition and the adiabatic condition for the general time-dependent harmonic oscillator, thus reveals the intimate relations between these three non-integrable phases.     

A Solved Model to Show Insufficiency of Quantitative Adiabatic Condition

The adiabatic theorem is a useful tool in processing quantum systems slowly evolving, but its practical application depends on the quantitative condition expressed by Hamiltonian＇s eigenvalues and eigenstates, which is usually taken as a sufficient condition. Recently, the sufficiency of the condition was questioned, and several counterexamples have been reported. Here we present a new solved model to show the insufficiency of the traditional quantitative adiabatic condition.     

光纤布拉格光栅中的隙孤子存在条件

提出光纤布拉格光栅中产生隙孤子的条件和参量制约关系。利用非线性耦合模式方程建立光纤布拉格光栅中孤子的传播方程,通过扰动方法建立了参量的微分方程,计算得到参量近似解。以周期非线性光学介质中隙孤子存在的条件为依据,数学计算分析得到两组参量关系不等式。最终通过数值计算说明了这些参量之间存在制约关系和物理意义。从而理论上说明了在光纤布拉格光栅中隙孤子存在需要选择适当参量。为光纤布拉格光栅中产生隙孤子的实验和进一步的工程应用提供了理论基础。     

A Necessary Condition for Quantum Adiabaticity Applied to the Adiabatic Grover Search

Numerous sufficient conditions for adiabaticity of the evolution of a driven quantum system have been known for quite a long time. In contrast, necessary adiabatic conditions are scarce. Recently a practicable necessary condition well suited for many-body systems has been proved. Here we tailor this condition for estimating run times of adiabatic quantum algorithms. As an illustration, the condition is applied to the adiabatic algorithm for searching in an unstructured database (adiabatic Grover search algorithm). We find that the thus obtained lower bound on the run time of this algorithm reproduces \( \sqrt{N} \) scaling (with N being the number of database entries) of the explicitly known optimum run time. This is in contrast to the poor performance of the known sufficient adiabatic conditions, which guarantee adiabaticity only for a run time on the order of O(N), which does not constitute any speedup over the classical database search. This observation highlights the merits of the new adiabatic condition and its potential relevance to adiabatic quantum computing.     

Adiabatic chaos in a two-dimensional mapping

A close to identity symplectic mapping describing the dynamics of a charged particle in the field of an infinitely wide packet of electrostatic waves is studied. A region of chaotic dynamics, whose width is large for an arbitrarily small deviation of the mapping from the identity, exists on the phase cylinder. This is explained by the quasirandom change occurring in an adiabatic invariant of the problem when the phase trajectory crosses a resonance curve. An asymptotic formula is derived for the jump in the adiabatic invariant. The width of the chaos region and the density of the set of invariant curves near the boundary of the chaos region are estimated. (c) 1996 American Institute of Physics.     

Adiabatic curve of a metastable plasma

Expressions are obtained relating the value of the thermodynamic parameters of a metastable supercooled plasma during adiabatic (isentropic) processes. The dependence of the sound velocity on the plasma parameters is determined.General Physical Institute, Russian Academy of Sciences, Moscow. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 15–20, January, 1994.     

Adiabatic Piston as a Dynamical System

We consider systems of finitely many interacting particles in a cube with a separating wall having a big mass M(adiabatic piston). Assuming that the particles reflect elastically from the ball and the initial velocity of the piston is zero we prove that as Mtends to infinity the dynamics of the piston converges to periodic oscillations.     

On a Nonlinear Model in Adiabatic Evolutions

In this paper, we study a kind of nonlinear model of adiabatic evolution in quantum search problem. As will be seen here, for this problem, there always exists a possibility that this nonlinear model can successfully solve the problem, while the linear model can not. Also in the same setting, when the overlap between the initial state and the final stare is sufficiently large, a simple linear adiabatic evolution can achieve O(1) time efficiency, but infinite time complexity for the nonlinear model of adiabatic evolution is needed. This tells us, it is not always a wise choice to use nonlinear interpolations in adiabatic algorithms. Sometimes, simple linear adiabatic evolutions may be sufficient for using.     

Adiabatic Raman polariton in a Bose condensate

The Raman interaction of optical fields with a Bose condensate is studied in the adiabatic regime. A superposition of operators is found — one annihilating an atom in a metastable state and the other annihilating a photon in resonance with a transition from the ground state to an excited state — which is an adiabatic invariant of the problem (Raman polariton). Possible applications for Bose-condensate diagnostics and development of atomic lasers are proposed. Pis’ma Zh. éksp. Teor. Fiz. 64, No. 7, 473–477 (10 October 1996)     

Adiabatic pumping in a superconductor-normal-superconductor weak link

We present a formalism to study adiabatic pumping through a superconductor-normal-superconductor weak link. At zero temperature, the pumped charge is related to the Berry phase accumulated, in a pumping cycle, by the Andreev bound states. We analyze in detail the case when the normal region is short compared to the superconducting coherence length. The pumped charge turns out to be an even function of the superconducting phase difference. Hence, it can be distinguished from the charge transferred due to the standard Josephson effect.     

Adiabatic compression of matter by a shell

The problem of the spherical compression of condensed matter by a shell was solved in an incompressible medium approximation. Parameter values at the inner boundary of the shell are defined by the solution of a self-modeling problem. At collapse, the velocity and kinetic energy of the shell behave asymptotically.