We discuss adiabatic quantum phenomena at a level crossing. Given a path in the parameter space which passes through a degeneracy point, we find a criterion which determines whether the adiabaticity condition can be satisfied. For paths that can be traversed adiabatically we also derive a differential equation which specifies the time dependence of the system parameters, for which transitions between distinct energy levels can be neglected. We also generalize the well-known geometric connections to the case of adiabatic paths containing arbitrarily many level-crossing points and degenerate levels.
We discuss adiabatic quantum phenomena at a level crossing. Given a path in the parameter space which passes through a degeneracy point, we find a criterion which determines whether the adiabaticity condition can be satisfied. For paths that can be traversed adiabatically we also derive a differential equation which specifies the time dependence of the system parameters, for which transitions between distinct energy levels can be neglected. We also generalize the well-known geometric connections to the case of adiabatic paths containing arbitrarily many level-crossing points and degenerate levels.
We elucidate the geometry of quantum adiabatic evolution. By minimizing the deviation from adiabaticity, we find a Riemannian metric tensor underlying adiabatic evolution. Equipped with this tensor, we identify a unified geometric description of quantum adiabatic evolution and quantum phase transitions that generalizes previous treatments to allow for degeneracy. The same structure is relevant for applications in quantum information processing, including adiabatic and holonomic quantum computing, where geodesics over the manifold of control parameters correspond to paths which minimize errors. We illustrate this geometric structure with examples, for which we explicitly find adiabatic geodesics. By solving the geodesic equations in the vicinity of a quantum critical point, we identify universal characteristics of optimal adiabatic passage through a quantum phase transition. In particular, we show that in the vicinity of a critical point describing a second-order quantum phase transition, the geodesic exhibits power-law scaling with an exponent given by twice the inverse of the product of the spatial and scaling dimensions.
This thesis discusses different aspects of time-dependent two-level quantum systems and their coherent dynamics. In the introductory chapters, we present some basic models used to study them and some basic results related to the level-crossing problems generally. As these systems are only rarely analytically solvable, some emphasis is put on the existing approximative theories. In particular, the generalization of the method of Dykhne, Davis and Pechukas (DDP), originally developed for problems in near-adiabatic region, is studied further. In the DDP theory, the final populations of different states are determined by the complex zero points of the eigenenergies. Superparabolic level-crossing models are introduced in this thesis and they prove to be simple but versatile test models for these studies. It is shown that, by considering all of the zero points, one can obtain accurate approximations also for highly non-adiabatic regions. In Chapter 4, a general differential geometric framework for time-dependent level-crossing models is developed and its basic character is discussed. A natural way of associating a plane curve to a time-dependent two-level model is introduced. This association allows one to use all of the mathematical results concerning plane curves to study these models. As an example, we use the Four-vertex theorem to discuss the adiabatic limit of certain type of systems. We also consider the prospect of enhancing the population transfer by transforming the time-dependent coupling pulse into a zero-area coupling. This is done either smoothly or by an abrupt jump in the phase of the coupling. It has been shown earlier, that one can obtain a complete population inversion (CPI) in a robust way with strong, non-resonant zero-area pulses. We study this in the more general time-dependent setting in Chapter 5, where also the energy levels are driven in a time-dependent fashion. This allows to transport the CPI phenomenon from the strong coupling region towards more moderate couplings. As an example of the smooth case, we consider the SechTanh-model, and show that even the nearest-zero DDP approximation can be used, despite the highly non-adiabatic origin of the phenomenon. We develop an approximative scheme also for the phase-jump case, using the parabolic model as an example. As a limiting case, we derive an approximative formula for the transition probability, which has some universal character and which proves to be useful in the most interesting case where the robust CPI is obtained. It is shown that even the highly suppressed transitions in tunnelling cases can be strongly enhanced in the phase-jump scenario.
In an adiabatic evolution, the wave function follows the path of ground states ∣Φ0(s)⟩ of a Hamiltonian H(s) as s is tuned from 0 to 1. The geometry of this adiabatic path is described by quantities such as its length L=∫01∥∂s∣Φ0(s)⟩∥ds and its total curvature K. This geometry affects the evolution time T. For instance, T is at least of order L/Δ∗, where Δ∗ is the minimum energy gap over s∈[0,1]. The traditional approach gives L=O(Δ∗−1/2) and K=O(Δ∗−1), which are loose in many cases. We improve these bounds by using μ<(γ), the width of the region in s on which the gap is below γ. The gap profile is in width class p when μ<(γ) falls at least as fast as γ1/p. A typical avoided crossing is in width class p=1, which gives L=O(logΔ∗−1) and K=O(logΔ∗−1). A profile in width class p>1 gives power laws with exponents (p−1)/(2p) for L and (p−1)/p for K. The width class also bounds the evolution time. At p=1 we have T=O(Δ∗−2) with a schedule that advances s at a constant rate, and T=O(Δ∗−1) up to a polylogarithmic correction with a schedule that traverses the path at a constant geometric speed. The bounds on L hold for any twice continuously differentiable H(s), and the bounds on K for affine H(s). We also prove the tightness of scaling of L in Δ∗ for every integer p≥1, and the scaling of K at p=1. We demonstrate the bounds on the adiabatic Grover search, the XXZ spin chain, and molecular electronic Hamiltonians.
ADIABATIC passage through an almost degenerate intermediate state is a common experimental technique.It is ba sed on the Ehrenfest principle which states that if the external conditions for a system are changed slowly enough, the change of the system can be predicted by computing qu asi statically the modification of its initial quantum state.The Ehrenfest principle is not a law of na ture beca use the system has an alternate path it can follow, which might appropriately be termed 'inertial*.This path is favored by a rapid change of the external variables.Among the two paths, the adiabatic one has a privileged position.Thus, it has been shown re cently that if the external change is measured by a "slowness parameter" inversely proportional to the time ta ken to effect the change then the Ehrenfest principle is valid to all orders in th is para meter [l].The following lines are to derive a rigorous formula for the probability that the adiabatic path is not followed in the simple case of the near crossing of two quantum levels.