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arXiv:quant-ph/0606203 · Journal of Physics B Atomic Molecular and Optical Physics

Adiabatic approximation in open systems: an alternative approach

X. X. Yi, D. M. Tong, L. C. Kwek, C. H. Oh

Published Jan 3, 2007• 41 citations
arXivHTMLPDF

Abstract

The adiabatic approximation in open systems is formulated through the effective Hamiltonian approach. By introducing an ancilla, we embed the open system dynamics into a non-Hermitian quantum dynamics of a composite system composed of the open system and ancilla, the adiabatic evolution of the open system is then defined as the adiabatic dynamics of the composite system. Validity and invalidity conditions for this approximation as well as the relation to the other definition are established and discussed. A high-order adiabatic approximation for open systems is introduced. As an example, the adiabatic condition for an open spin- particle in time-dependent magnetic fields is analysed.

DOI
10.1088/0953-4075/40/2/004

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The adiabatic approximation in open systems is formulated through the effective Hamiltonian approach. By introducing an ancilla, we embed the open system dynamics into a non-Hermitian quantum dynamics of a composite system composed of the open system and ancilla, the adiabatic evolution of the open system is then defined as the adiabatic dynamics of the composite system. Validity and invalidity conditions for this approximation as well as the relation to the other definition are established and discussed. A high-order adiabatic approximation for open systems is introduced. As an example, the adiabatic condition for an open spin- particle in time-dependent magnetic fields is analysed.
We generalize the standard Born-Oppenheimer approximation to the case of open quantum systems. We define the zeroth-order Born-Oppenheimer approximation for an open quantum system as the regime in which its effective Hamiltonian can be diagonalized with fixed slowly changing variables. We then establish validity and invalidity conditions for this approximation for two types of dissipations---the spin relaxation and the dissipation of center-of-mass motion. As an example, the Born-Oppenheimer approximation for a two-level open system is analyzed.
A precise definition of an adiabaticity parameter \ensuremath{\nu} of a time-dependent Hamiltonian is proposed. A variation of the time-dependent perturbation theory is presented which yields a series expansion of the evolution operator U(\ensuremath{\tau})=${\ensuremath{\sum}}_{\mathrm{\ensuremath{\ell}}}$${\mathrm{U}}^{(\mathrm{\ensuremath{\ell}})}$(\ensuremath{\tau}) with ${\mathrm{U}}^{(\mathrm{\ensuremath{\ell}})}$(\ensuremath{\tau}) being at least of the order ${\ensuremath{\nu}}^{\mathrm{\ensuremath{\ell}}}$. In particular, U(0){\mathrm{U}}^{(0)}U(0)(\ensuremath{\tau}) corresponds to the adiabatic approximation and yields Berry's adiabatic phase. It is shown that this series expansion has nothing to do with the 1/\ensuremath{\tau} expansion of U(\ensuremath{\tau}). It is also shown that the nonadiabatic part of the evolution operator is generated by a transformed Hamiltonian which is off-diagonal in the eigenbasis of the initial Hamiltonian. This suggests the introduction of an adiabatic product expansion for U(\ensuremath{\tau}) which turns out to yield exact expressions for U(\ensuremath{\tau}) for a large number of quantum systems. In particular, a simple application of the adiabatic product expansion is used to show that for the Hamiltonian describing the dynamics of a magnetic dipole in an arbitrarily changing magnetic field, there exists another Hamiltonian with the same eigenvectors for which the Schr\"odinger equation is exactly solvable. Some related issues concerning geometric phases and their physical significance are also discussed.
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.
In this paper the evolution of a quantum system driven by a non-Hermitian Hamiltonian depending on slowly-changing parameters is studied by building an universal high-order adiabatic approximation (HOAA) method with Berry's phase, which is valid for either the Hermitian or the non-Hermitian cases. This method can be regarded as a non-trivial generalization of the HOAA method for closed quantum system presented by this author before. In a general situation, the probabilities of adiabatic decay and non-adiabatic transitions are explicitly obtained for the evolution of the non-Hermitian quantum system. It is also shown that the non-Hermitian analog of the Berry's phase factor for the non-Hermitian case just enjoys the holonomy structure of the dual linear bundle over the parameter manifold. The non-Hermitian evolution of the generalized force harmonic oscillator is discussed as an illustrative example.
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.