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We extend the concept of superadiabatic dynamics, or transitionless quantum driving, to quantum open systems whose evolution is governed by a master equation in the Lindblad form. We provide the general framework needed to determine the control strategy required to achieve superadiabaticity. We apply our formalism to two examples consisting of a two-level system coupled to environments with time-dependent bath operators.
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We analyze a class of dynamics of open quantum systems which is governed by the dynamical map mutually commuting at different times. Such evolution may be effectively described via spectral analysis of the corresponding time dependent generators. We consider both Markovian and non-Markovian cases.
Simple examples are used to introduce and examine symmetries of open quantum dynamics that can be described by unitary operators. For the Hamiltonian dynamics of an entire closed system, the symmetry takes the expected form which, when the Hamiltonian has a lower bound, says that the unitary symmetry operator commutes with the Hamiltonian operator. There are many more symmetries that are only for the open dynamics of a subsystem. Examples show how these symmetries alone can reveal properties of the dynamics and reduce what needs to be done to work out the dynamics. A symmetry of the open dynamics of a subsystem can even imply properties of the dynamics for the entire system that are not implied by the symmetries of the dynamics of the entire system. The symmetries are generally not related to constants of the motion for the open dynamics of the subsystem. There are many symmetries that cannot be seen in the Schrodinger picture as symmetries of dynamical maps of density matrices for the subsystem. There are symmetries of the open dynamics of a subsystem that depend only on the dynamics. In the simplest examples, these are also symmetries of the dynamics of the entire system. There are many more symmetries, of a new kind, that also depend on correlations, or absence of correlations, between the subsystem and the rest of the entire system, or on the state of the rest of the entire system.
We present an object-oriented open-source framework for solving the dynamics of open quantum systems written in Python. Arbitrary Hamiltonians, including time-dependent systems, may be built up from operators and states defined by a quantum object class, and then passed on to a choice of master equation or Monte-Carlo solvers. We give an overview of the basic structure for the framework before detailing the numerical simulation of open system dynamics. Several examples are given to illustrate the build up to a complete calculation. Finally, we measure the performance of our library against that of current implementations. The framework described here is particularly well-suited to the fields of quantum optics, superconducting circuit devices, nanomechanics, and trapped ions, while also being ideal for use in classroom instruction.
The Hamilton operator of an open quantum system is non-Hermitian. Its eigenvalues are, generally, complex and provide not only the energies but also the lifetimes of the states of the system. The states may couple via the common environment of scattering wavefunctions into which the system is embedded. This causes an {it external mixing} (EM) of the states. Mathematically, EM is related to the existence of singular (the so-called exceptional) points (EPs). The eigenfunctions of a non-Hermitian operator are biorthogonal, in contrast to the orthogonal eigenfunctions of a Hermitian operator. A quantitative measure for the ratio between biorthogonality and orthogonality is the phase rigidity of the wavefunctions. At and near an EP, the phase rigidity takes its minimum value. The lifetimes of two nearby eigenstates of a quantum system bifurcate under the influence of an EP. At the parameter value of maximum width bifurcation, the phase rigidity approaches the value one, meaning that the two eigenfunctions become orthogonal. However, the eigenfunctions are externally mixed at this parameter value. The S-matrix and therewith the cross section do contain, in the one-channel case, almost no information on the EM of the states. The situation is completely different in the case with two (or more) channels where the resonance structure is strongly influenced by the EM of the states and interesting features of non-Hermitian quantum physics are revealed. We provide numerical results for two and three nearby eigenstates of a non-Hermitian Hamilton operator which are embedded in one common continuum and influenced by two adjoining EPs. The results are discussed. They are of interest for an experimental test of the non-Hermitian quantum physics as well as for applications.
The Stochastic Liouville-von Neumann (SLN) equation describes the dynamics of an open quantum system reduced density matrix coupled to a non-Markovian harmonic environment. The interaction with the environment is represented by complex coloured noises which drive the system, and whose correlation functions are set by the properties of the environment. We present a number of schemes capable of generating coloured noises of this kind that are built on a noise amplitude reduction procedure [Imai et al, Chem. Phys. 446, 134 (2015)], including two analytically optimised schemes. In doing so, we pay close attention to the properties of the correlation functions in Fourier space, which we derive in full. For some schemes the method of Wiener filtering for deconvolutions leads to the realisation that weakening causality in one of the noise correlation functions improves numerical convergence considerably, allowing us to introduce a well controlled method for doing so. We compare the ability of these schemes, along with an alternative optimised scheme [Schmitz and Stockburger, Eur. Phys. J.: Spec. Top. 227, 1929 (2019)], to reduce the growth in the mean and variance of the trace of the reduced density matrix, and their ability to extend the region in which the dynamics is stable and well converged for a range of temperatures. By numerically optimising an additional noise scaling freedom, we identify the scheme which performs best for the parameters used, improving convergence by orders of magnitude and increasing the time accessible by simulation.
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