Adiabatic evolution of decoherence-free subspaces and its shortcuts

Abstract

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 timedependent decoherence-free subspace with an extremely high purity, even though the dynamics of the open quantum system may be not adiabatic. The adiabatic condition mentioned here in the adiabatic theorem for open systems is very similar with 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 (will be refereed to as adiabatic decoherencefree subspaces(ADFSs) later ) depends on the existence of instantaneous decoherence-free subspaces, which requires that the Hamiltonian of open quantum systems has to be engineered according to the incoherent control protocol. Besides, the shortcuts-to-adiabaticity for the adiabatic decoherence-free subspaces is also presented based on the transitionless quantum driving method. Finally, we provide an example that consists of a two-level system coupled to a broad band squeezed vacuum field to show our theory. Our approach employs Markovian master equations and the theory can apply to finite-dimensional quantum open systems.

Shortcuts to adiabaticity: Concepts, methods, and applications

“…Moreover, STA methods relate synergistically to or overlap partially with other control methods. In particular STA blend well with Optimal Control Theory (OCT), decoherence-free subspaces (DFS) (Wu et al, 2017e), linear response theory (Acconcia et al, 2015), or perturbative and variational schemes (see Sec. II.H and Sec.…”

“…A natural framework to extend the shortcuts to open systems is the explicit use of decoherence-free subspaces (DFSs) (Wu et al, 2017e). Consider that the coupling to the environment is accounted for by a Lindblad form…”

“…, |Φ D } are degenerate eigenstates of the Lindblad operators that obey the relation X k (t)|Φ j (t) = c k (t)|Φ j (t) (Wu et al, 2015). A counterdiabatic driving Hamiltonian in the time-dependent DFS can be shown to be of the form (Wu et al, 2017e)…”

“…The driving based on invariants in the DFS can be equivalently worked out (Ma et al, 2017). An important condition that needs to be fulfilled to control in time such quantum systems is the dynamical stability of the timedependent DFS (Wu et al, 2017e). The application to few-spin systems is explicitly worked out in Ma et al (2017) and Wu et al (2017e).…”

Noise resistant quantum control using dynamical invariants

“…This optimization has been performed so far by minimizing the excitation energy or maximizing the fidelity in perturbative schemes, e.g. in two-level systems (TLSs) [41][42][43][44] or for ion transport [45], using decoherence free subspaces [46,47], super-operator [48] and non-Hermitian invariants [49], and effective Hamiltonians [50].…”