Adiabatic theorem

Tracing the arctan over a full revolution by the method described in Section rV and noting the factor 1 /2 in Eq. (171) establishes our result. (The case that E — m needs more careful consideration, since it leads to a breakdown of the adiabatic theorem. However, this case will be of no consequence for the results.)... 

The limit equation governing limj -,o qc can be motivated by referring to the quantum adiabatic theorem which originates from work of Born and FOCK [4, 20] The classical position g influences the Hamiltonian very slowly compared to the time scale of oscillations of in fact, infinitely slowly in the limit e — 0. Thus, in analogy to the quantum adiabatic theorem, one would expect that the population of the energy levels remain invariant during the evolution ... 

To understand many NMR phenomena we must recognize that the rate at which resonance is approached can be quite important. A very slow change of magnetic field or frequency is called adiabatic, and the adiabatic theorem tells us that if the rate of change is slow enough that... 

According to the adiabatic theorem [19,20], if the change of direction of the magnetic field is sufficiently slow, the angle between the magnetization M and the instantaneous direction of the effective field (ooeff) is a constant of motion. The adiabatic condition for spin inversion is given by equations (6) and (7) ... 

An avoided crossing will mainly limit the application of the adiabatic theorem If the dynamics is not slow enough, dynamical transitions, so-called nonadiabatic transitions, will be induced between the dressed states forming the avoided crossing. A local Landau-Zener analysis can be invoked to determine... 

The adiabatic theorem is valid in two quite different situations, which can be illustrated with a two-level example ... 

Thus, in both cases in the adiabatic limit the population is carried at all times by a single branch of instantaneous eigenstates. A quite general formulation of the adiabatic theorem, which imposes only smoothness conditions on the instantaneous eigenprojections, has been presented recently by Avron and Elgart [45,46],... 

A sketch of an argument that leads to the adiabatic theorem for the Floquet Hamiltonian of an iV-level system is given in Appendix C. 

We remark that in this formulation the choice of the reference basis is fixed but arbitrary. We use this formulation in the discussion of the adiabatic theorem in Appendix C. 

This leads to the Dirac variation-of-constants method [10]. Although generall> successful, an unsatisfying feature of this method can be seen when we consider an adiabatically switched-on static perturbation. By the adiabatic theorem [11] the perturbed wave function as t — -)-< has the form... 

Goldstone s approach exploited a time-dependent PT in the interaction picture and was based on the Gell-Mann and Low adiabatic theorem [42], as was the work of Hubbard... 

B. Simon, Holonomy, the quantum adiabatic theorem, and Berry s phase, Phys. Rev. Lett. 51 2167 (1983). 

Adiabatic switching is based on the Ehrenfest adiabatic theorem [65-67], which states that classical actions and quantum numbers are preserved in adiabatic, slow processes. It is assumed that the actual Hamiltonian H may be written as a sum of a separable zero-order Hamiltonian H0 and a nonseparable part AH ... 

Quantum adiabatic theorem (QAT) has been extensively explored since the birth of QM (Born and Fock 1928 Messiah 1962). The essence of QAT is as follows. Let Hi be the initial Hamiltonian at time t = 0, and let Hf be the final Hamiltonian after the lapse of a long time t = T. We can construct a time-dependent Hamiltonian H(t) that continuously and infinitesimally slowly interpolates between and Hf in the time interval 0<,t T ... 

Kato, T. 1950. On the adiabatic theorem of quantum mechanics. Journal of the Physical Society of Japan 5 435. 

In quantum computing there are various computational models such as the standard model with quantum circuits, the geometrical model with quantum gates derived from holonomy groups and the adiabatic model with its application of the adiabatic theorem of the quantum mechanics [30]. All these models are very well known in quantum computing and computer science and it is not necessary to describe them here. 

Our algorithm uses the well known adiabatic theorem of the quantum mechanics. 

If for the considered value of T, some component of I P(T)> has a probability more greater than 0.5, then the algorithm is finished and the dominant component of I T (T)> is justly the Kauffman bracket for given link L. This is due to the adiabatic theorem and the equation (19). But if I P(T)> has no a dominant component then a new more greater value of T is assigned and the algorithm is executed again. 

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