Quantum-mechanical evolution in a slowly varying double-well potential is analyzed in the semiclassical limit to determine the transition probability due to loss of adiabatic invariance of states having energy close to that of a separatrix in the equivalent classical system. Transformation to the rotating-axis basis reduces the problem to the integration of a set of coupled differential equations. In the adiabatic limit the integration is carried out by the method of successive approximations. For adiabatic parameter \ensuremath{\epsilon} of the order of the inverse, 1/N, of the number of quantum states with energy below the separatrix, the spread of the quantum adiabatic invariant is exponentially small in the inverse of the adiabaticity parameter. Surprisingly, the transition to this quantum adiabatic behavior occurs far below the minimum transition frequency of the system.
Quantum-mechanical evolution in a slowly varying double-well potential is analyzed in the semiclassical limit to determine the transition probability due to loss of adiabatic invariance of states having energy close to that of a separatrix in the equivalent classical system. Transformation to the rotating-axis basis reduces the problem to the integration of a set of coupled differential equations. In the adiabatic limit the integration is carried out by the method of successive approximations. For adiabatic parameter \ensuremath{\epsilon} of the order of the inverse, 1/N, of the number of quantum states with energy below the separatrix, the spread of the quantum adiabatic invariant is exponentially small in the inverse of the adiabaticity parameter. Surprisingly, the transition to this quantum adiabatic behavior occurs far below the minimum transition frequency of the system.
The evolution mod Psi (t)) of a system with slowly-varying Hamiltonian H( Omega t) depends not only on the slowness parameter Omega but also on Planck's constant h(cross). For systems with two (or more) classically separated phase-space regions which are quantally connected by tunnelling, the curves of instantaneous energy levels display hyperbolic near-degeneracies rather than crossings. In such cases the limits Omega to 0 (h(cross) fixed and small) and h(cross) to 0 ( Omega fixed and small) lead to opposite behaviour of mod Psi (t)). As an illustration, the uniform semiclassical adiabatic behaviour ( Omega and h(cross) small, Omega /h(cross) arbitrary) is calculated exactly for a double-well potential for which one well gets shallower as the other gets deeper.
Recently probabilistic hysteresis in isolated Hamiltonian systems of ultracold atoms has been studied in the limit of large particle numbers, where a semiclassical treatment is adequate. The origin of irreversibility in these sweep experiments, where a control parameter is slowly (adiabatically) tuned back and forth, turned out to be a passage back and forth across a separatrix (integrable case) or a passage in and out of a chaotic sea in phase space (chaotic case). Here we focus on the full quantum mechanical description of the integrable system and show how the semiclassical results emerge in the limit of large particle numbers. Instead of the crossing of a separatrix in phase space, where classical adiabaticity fails, the origin of irreversibility in the quantum system is a series of avoided crossings of the adiabatic energy levels: they become so close that already for modest particle numbers the change of the external parameter has to be unrealistically slow to reach the quantum adiabatic limit of perfectly reversible evolution. For a slow but finite sweep rate we find a broad regime where the quantum results agree with the semiclassical results, but only if besides the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$ an initial ensemble of states is considered, with sufficient initial energy width. For a single initial energy eigenstate we find in contrast that the backward sweep reveals strong quantum effects even for very large particle numbers.
We expand upon the standard quantum adiabatic theorem, examining the time-dependence of quantum evolution in the near-adiabatic limit. We examine a Hamiltonian that evolves along some fixed trajectory from H^0 to H^1 in a total evolution-time τ, and our goal is to determine how the final state of the system depends on τ. If the system is initialized in a non-degenerate ground state, the adiabatic theorem says that in the limit of large τ, the system will stay in the ground state. We examine the near-adiabatic limit where the system evolves slowly enough that most but not all of the final state is in the ground state, and we find that the probability of leaving the ground state oscillates in τ with a frequency determined by the integral of the spectral gap along the trajectory of the Hamiltonian, so long as the gap is big. If the gap becomes exceedingly small, the final probability is the sum of oscillatory behavior determined by the integrals of the gap before and after the small gap. We confirm these analytic predictions with numerical evidence from barrier tunneling problems in the context of quantum adiabatic optimization.
Quantum adiabatic evolution is a dynamical evolution of a quantum system under slow external driving. According to the quantum adiabatic theorem, no transitions occur between nondegenerate instantaneous energy eigenstates in such a dynamical evolution. However, this is true only when the driving rate is infinitesimally small. For a small nonzero driving rate, there are generally small transition probabilities between the energy eigenstates. We develop a classical mechanics framework to address the small deviations from the quantum adiabatic theorem order by order. A hierarchy of Hamiltonians is constructed iteratively with the zeroth-order Hamiltonian being determined by the original system Hamiltonian. The k th-order deviations are governed by a k th-order Hamiltonian, which depends on the time derivatives of the adiabatic parameters up to the k th-order. Two simple examples, the Landau–Zener model and a spin-1/2 particle in a rotating magnetic field, are used to illustrate our hierarchical theory. Our analysis also exposes a deep, previously unknown connection between classical adiabatic theory and quantum adiabatic theory.
In a quantum-mechanical problem with a slowly varying Hamiltonian, such as appears in a semiclassical treatment of inelastic atomic collisions, it is common to restrict attention to a small number of states among which transitions are anticipated. This procedure is justified by an extension of the conventional adiabatic theorem to nearly degenerate states. The correct mixing within the restricted basis is determined by a time-dependent variational principle. The contribution of states outside the basis vanishes as the time scale for variation of the Hamiltonian becomes large. A further generalization deals with pseudoeigenfunctions, such as appear in the theory of the Stark effect.
Abstract The postulate of the existence of stationary states in multiply periodic dynamical systems requires that if the condition of such a system, when quantised, is changed in any way by the application of an external field or by the alteration of one of the internal constraints, the new state of the system must also be correctly quantised. It follows that the laws of classical mechanics cannot in general be true, even approximately, during the transition. There is one kind, of change, however, during which one may expect the classical laws to hold, namely, the so-called adiabatic change, which takes place infinitely slowly and regularly, so that the system practically remains multiply periodic all the time. In this case the quantum numbers cannot change, and it should be possible to deduce from the classical laws that the quantum integrals remain invariant. This was attempted by Burgers, who showed that they are invariant provided there are no linear relations of the type Σrmrωr = 0 (1) between the frequencies ωr, of the system, where the mr are integers. In general, however, the frequencies will alter during the adiabatic change, and in so doing will pass through an infinity of values for which relations such as (1) hold. A closer investigation is therefore necessary, as was pointed out by Burgers himself. In the following work, conditions which are rigorously sufficient to ensure the invariance of the quantum integrals, are obtained in such a form that it is possible for one to see whether they are satisfied or not without having to integrate the equations of adiabatic motion.