We present generalized adiabatic theorems for closed and open quantum systems that can be applied to slow modulations of rapidly varying fields, such as oscillatory fields that occur in optical experiments and light-induced processes.The generalized adiabatic theorems show that a sufficiently slow modulation conserves the dynamical modes of time-dependent reference Hamiltonians.In the limiting case of modulations of static fields, the standard adiabatic theorems are recovered.Applying these results to periodic fields shows that they remain in Floquet states rather than in energy eigenstates.More generally, these adiabatic theorems can be applied to transformations of arbitrary time-dependent fields, by accounting for the rapidly varying part of the field through the dynamical normal modes, and treating the slow modulation adiabatically.As examples, we apply the generalized theorem to (a) predict the dynamics of a two-level system driven by a frequency-modulated resonant oscillation, a pathological situation beyond the applicability of traditional adiabatic theorems, and (b) to show that open quantum systems driven by slowly turned-on incoherent light, such as biomolecules under natural illumination conditions, can display only coherences that survive in the steady state.
The adiabatic theorem states that an initial eigenstate of a slowly varying Hamiltonian remains close to an instantaneous eigenstate of the Hamiltonian at a later time. We show that a perfunctory application of this statement is problematic if the change in eigenstate is significant, regardless of how closely the evolution satisfies the requirements of the adiabatic theorem. We also introduce an example of a two-level system with an exactly solvable evolution to demonstrate the inapplicability of the adiabatic approximation for a particular slowly varying Hamiltonian.
The adiabatic theorem is a powerful result concerning the evolution of quantum mechanical systems. Intuitively, it states the following: suppose we want to study the evolution of a quantum mechanical system governed by a given Hamiltonian
H (Def. 6.1), and the initial state of the system is the eigenstate of
H corresponding to its lowest eigenvalue. The laws of quantum physics tell us that the evolution of the system over time follows the Schrüdinger equation, Eq. (6.1). Suppose now that the Hamiltonian is not fixed, but rather, it changes very slowly over time, so that the evolution of the system is described by the differential equation
idtd∣ψ(t)⟩=H(t)∣ψ(t)⟩, and
H(t) is slowly changing. If, for any value of the time parameter
t, there is always a gap between the lowest eigenvalue of
H(t) and all other eigenvalues, then the adiabatic theorem asserts that the state of the system always remains in an eigenstate with the lowest eigenvalue throughout the evolution. One specific case of interest in this chapter occurs when the time-dependent Hamiltonian
H(t) varies from an initial (time-independent) Hamiltonian
HI to a final Hamiltonian
HF. If the change happens slowly, and the initial state of the system is an eigenstate of
HI with minimum eigenvalue, the adiabatic theorem asserts that the evolution eventually leads to the eigenstate with minimum eigenvalue of the final Hamiltonian
HF. This result can be used for optimization: imagine that the initial Hamiltonian encodes an easy problem for which we can construct the eigenstate with minimum eigenvalue, and the final Hamiltonian encodes a difficult optimization problem whose solution is given by the eigenstate with minimum eigenvalue. By the adiabatic theorem, evolving the eigenstate of the initial Hamiltonian while slowly changing the Hamiltonian into the final one leads to the solution of the difficult optimization problem. Of course we need to know how slowly
HI should be changed into
HF. The adiabatic theorem is at the heart of the quantum approximate optimization algorithm (QAOA), a framework that has been widely used to implement optimization algorithms on existing quantum hardware, because it has low requirements of quantum resources–although it does not guarantee improvements over classical algorithms in general. In this chapter we first discuss the adiabatic theorem, and then give an overview of QAOA.
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.
The consistency of the quantum adiabatic theorem has been doubted recently. It is shown in the present paper that the difference between the adiabatic solution and the exact solution to the Schr\"odinger equation with a slowly changing driving Hamiltonian is small; while the difference between their time derivatives is not small. This explains why substituting the adiabatic solution back into the Schr\"odinger equation leads to ``inconsistency'' of the adiabatic theorem. Physics is determined completely by the state vector, and not by its time derivative. Therefore the quantum adiabatic theorem is physically correct.