Geometrical and mathematical aspects of the adiabatic theorem of quantum mechanics

This work is devoted to rigorous results about the adiabatic theorem of quantum mechanics. This theorem deals with the time dependent Schrödinger equation when the hamiltonian is a slowly varying function of time, characterizing the so-called adiabatic regime. Mathematically, the adiabatic theorem describes the solutions ψε(t) in an Hilbert space H of the rescaled Schrödinger equation in the limit ε→ 0. Suppose the hamiltonian possesses for any time t two spectral projectors, P1(t) and P2(t), which are spectrally isolated. Let us consider a normalized solution which belongs at time t = - ∞ to the spectral subspace P1(-∞)H, i.e. which satisfies the boundary condition Then the transition probability P21(ε) from P1(-∞)H to P2(+∞)H between the times -∞ and +∞ is defined by The adiabatic theorem states that P21(ε) tends to zero in the limit ε→ 0. Our main concern is the study of the decay of P21(ε) as ε→ 0. We fist show that if H(t) is an analytic unbounded operator then P21(ε) decays exponentially fast to zero in the adiabaticity parameter ε: for some positive constant τ. Then we turn to two-level systems for which we have a finer control on the behaviour of P21(ε) as ε→ 0. Indeed, in the generic case we give an explicit asymptotic formula for the transition probability P21(ε) which reads The prefactor exp {2Imθ1} is of geometrical nature and the exponential decay rate 21m ∫γ e1(z)dz is computed by means of the integral of the analytic continuation of the eigenvalue e1 (t) along a suitable path γ in the complex plane. This expression constitutes a generalization of the so-called Dykhne formula which does not contain the geometric prefactor. Moreover, we improve this result and compute the leading term of P21(ε) up to a correction of order Ο(εq) for any q, instead of Ο(ε). This result shows as well that the logarithm of P21(ε) admits an asymptotic power series in ε up to any order. Finally we push the estimates to get the leading term of P21(ε) up to a correction of order Ο(e-τ/c). We consider also cases where the 2 X 2 hamiltonian possesses some symmetry, as the time reversal symmetry for example. In these situations, the leading term of P21(ε) changes qualitatively since it is given by a decreasing exponential multiplying an oscillatory function of 1 /ε. Then we come back to general systems driven by unbounded hamiltonians and study the case where P1(t) and P2(t) are both one-dimensional. These projectors are thus associated with non-degenerate instantaneous eigenvalues el(t) and e2(t) of the hamiltonian H(t). We prove that, in this case too, an asymptotic formula for P21(ε) exists, provided the two levels el(t) and e2(t) are sufficiently isolated in the spectrum of H(t). This formula turns out to be the same as the formula valid for two-level systems. Finally, we consider the situation frequently encountered in applications where the two levels el(t) and e2(t) display an avoided crossing during the evolution. For an avoided crossing located at time t = 0, this means that the levels behave as where δ ≪ 1. As a consequence, the gap between el(t) and e2(t) is minimum for where its value is In this case, we show that for ε and δ small enough, the above formula for P21(ε) reduces to the well-known Landau-Zener formula When c = 0 we recover the familiar Landau-Zener formula. This gives a rigorous mathematical status to a formula which has been widely applied for years in a variety of circumstances.