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arXiv:2105.01140 · PRX Quantum

Generalized Adiabatic Theorems: Quantum Systems Driven by Modulated Time-Varying Fields

Amro Dodin, Paul Brumer

Published Jul 2, 2021• 11 citations
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Abstract

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.

DOI
10.1103/prxquantum.2.030302

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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 approximation exhibits wide applicability in quantum mechanics, providing a simple approach for nontransitional dynamics in quantum systems governed by slowly varying time-dependent Hamiltonians. However, the standard adiabatic theorem is specifically derived for closed quantum systems. In a realistic open system scenario, the inevitable system-reservoir interaction must be taken into account, which strongly impacts the generalization of the adiabatic behavior. In this paper, we introduce sufficient conditions for the adiabatic approximation in open quantum systems. These conditions are simple yet general, providing a suitable instrument to investigate adiabaticity for arbitrary initial mixed states evolving under time local master equations. We first illustrate our results by showing that the adiabatic approximation for open systems is compatible with the description of quantum thermodynamics at thermal equilibrium, where irreversible entropy production is vanishing. We also apply our sufficient conditions as a tool in quantum control, evaluating the adiabatic behavior for the Hamiltonians of both the Deutsch algorithm and the Landau-Zener model under decoherence.
The adiabatic theorem and shortcuts to adiabaticity for time-dependent open quantum systems are explored in this paper. Starting from the definition of dynamical stable decoherence-free subspace, we show that, under a compact adiabatic condition, the quantum state remains in the time-dependent decoherence-free subspace with an extremely high purity, even though the dynamics of the open quantum system may not be adiabatic. The adiabatic condition mentioned here in the adiabatic theorem for open systems is very similar to that for closed quantum systems, except that the operators required to change slowly are the Lindblad operators. We also show that the adiabatic evolution of decoherence-free subspaces depends on the existence of instantaneous decoherence-free subspaces, which requires that the Hamiltonian of open quantum systems be engineered according to the incoherent control protocol. In addition, shortcuts to adiabaticity for adiabatic decoherence-free subspaces are also presented based on the transitionless quantum driving method. Finally, we provide an example that consists of a two-level system coupled to a broadband squeezed vacuum field to show our theory. Our approach employs Markovian master equations and the theory can apply to finite-dimensional quantum open systems.
The behavior of a quantum-mechanical system with a slowly modulated oscillatory Hamiltonian is characterized by an adiabatic theorem similar to that for a system with a slowly changing ``static'' Hamiltonian. Quasiperiodic states—solutions of the instantaneous Schrödinger equation with an oscillatory Hamiltonian—play the same role as eigenfunctions of the instantaneous Hamiltonian do in an adiabatic theorem for a nearly static Hamiltonian. As an example, the theorem is used to establish the correct wavefunction to be used in computing the refractive index of atomic hydrogen.
The adiabatic theorem is a fundamental result in quantum mechanics, which states that a system can be kept arbitrarily close to the instantaneous ground state of its Hamiltonian if the latter varies in time slowly enough. The theorem has an impressive record of applications ranging from foundations of quantum field theory to computational molecular dynamics. In light of this success it is remarkable that a practicable quantitative understanding of what "slowly enough" means is limited to a modest set of systems mostly having a small Hilbert space. Here we show how this gap can be bridged for a broad natural class of physical systems, namely, many-body systems where a small move in the parameter space induces an orthogonality catastrophe. In this class, the conditions for adiabaticity are derived from the scaling properties of the parameter-dependent ground state without a reference to the excitation spectrum. This finding constitutes a major simplification of a complex problem, which otherwise requires solving nonautonomous time evolution in a large Hilbert space.
For a periodically driven open quantum system, the Floquet theorem states that the time evolution operator Λ(t,0)Λ(t,0)Λ(t,0) of the system can be factorized as Λ(t,0)=D(t)eLefftΛ(t,0)=\mathcal{D}(t)e^{\mathcal{L}_{eff}t}Λ(t,0)=D(t)eLeff​t with micro-motion operator D(t)\mathcal{D}(t)D(t) possessing the same period as the external driving, and time-independent operator Leff\mathcal{L}_{eff}Leff​. In this work, we extend this theorem to open systems that follow a modulated periodic evolution, in which the fast part is periodic while the slow part breaks the periodicity. We derive a factorization for the time evolution operator that separates the long time dynamics and the micro-motion for the open quantum system. High-frequency expansions for the effective evolution operator control the long time dynamics, and the micro-motion operator is also given and discussed. It may find applications in quantum engineering with open systems following modulated periodic evolution.
The evolution of a driven quantum system is said to be adiabatic whenever the state of the system stays close to an instantaneous eigenstate of its time-dependent Hamiltonian. The celebrated quantum adiabatic theorem ensures that such pure state adiabaticity can be maintained with arbitrary accuracy, provided one chooses a small enough driving rate. Here, we extend the notion of quantum adiabaticity to closed quantum systems initially prepared at finite temperature. In this case adiabaticity implies that the (mixed) state of the system stays close to a quasi-Gibbs state diagonal in the basis of the instantaneous eigenstates of the Hamiltonian. We prove a sufficient condition for the finite temperature adiabaticity. Remarkably, it turns out that the finite temperature adiabaticity can be more robust than the pure state adiabaticity with respect to increasing the system size. This can be the case for one-body systems with large Hilbert spaces, such as a particle in a large box, as well as for certain many-body systems. In particular, we present an example of a driven many-body system where, in the thermodynamic limit, the finite temperature adiabaticity is maintained, while the pure state adiabaticity breaks down. On the other hand, for generic many-body systems the scaling of the finite temperature adiabatic condition with the system size is exponential, analogously to pure state adiabatic conditions.