Optimization of Short Coherent Control Pulses

The coherent control of small quantum system is considered. For a two-level system coupled to an arbitrary bath we consider a pulse of finite duration. We derive the leading and the next-leading order corrections to the evolution operator due to the non-commutation of the pulse and the bath Hamiltonian. The conditions are computed that make the leading corrections vanish. The pulse shapes optimized in this way are given for $\pi$ and $\frac{\pi}{2}$ pulses.Comment: 9 pages, 6 figures; published versio

Spectrum and transition rates of the XX chain analyzed via Bethe ansatz

As part of a study that investigates the dynamics of the s=1/2 XXZ model in the planar regime |Delta|<1, we discuss the singular nature of the Bethe ansatz equations for the case Delta=0 (XX model). We identify the general structure of the Bethe ansatz solutions for the entire XX spectrum, which include states with real and complex magnon momenta. We discuss the relation between the spinon or magnon quasiparticles (Bethe ansatz) and the lattice fermions (Jordan-Wigner representation). We present determinantal expressions for transition rates of spin fluctuation operators between Bethe wave functions and reduce them to product expressions. We apply the new formulas to two-spinon transition rates for chains with up to N=4096 sites.Comment: 11 pages, 4 figure

Affine pure-jump processes on positive Hilbert–Schmidt operators

We show the existence of a broad class of affine Markov processes on the cone of positive self-adjoint Hilbert–Schmidt operators. Such processes are well-suited as infinite-dimensional stochastic covariance models. The class of processes we consider is an infinite-dimensional analogue of the affine processes on the cone of positive semi-definite and symmetric matrices studied in Cuchiero et al. (2011). As in the finite-dimensional case, the processes we construct allow for a drift depending affine linearly on the state, as well as jumps governed by a jump measure that depends affine linearly on the state. The fact that the cone of positive self-adjoint Hilbert–Schmidt operators has empty interior calls for a new approach to proving existence: instead of using standard localization techniques, we employ the theory on generalized Feller semigroups introduced in Dörsek and Teichmann (2010) and further developed in Cuchiero and Teichmann (2020). Our approach requires a second moment condition on the jump measures involved, consequently, we obtain explicit formulas for the first and second moments of the affine process

Generalization of short coherent control pulses: extension to arbitrary rotations

We generalize the problem of the coherent control of small quantum systems to the case where the quantum bit (qubit) is subject to a fully general rotation. Following the ideas developed in Pasini et al (2008 Phys. Rev. A 77, 032315), the systematic expansion in the shortness of the pulse is extended to the case where the pulse acts on the qubit as a general rotation around an axis of rotation varying in time. The leading and the next-leading corrections are computed. For certain pulses we prove that the general rotation does not improve on the simpler rotation with fixed axis.Comment: 6 pages, no figures; published versio

Lattice vs. continuum theory of the periodic Heisenberg chain

We consider the detailed structure of low energy excitations in the periodic spin-1/2 XXZ Heisenberg chain. By performing a perturbative calculation of the non-linear corrections to the Gaussian model, we determine the exact coefficients of asymptotic expansions in inverse powers of the system length N for a large number of low-lying excited energy levels. This allows us to calculate eigenenergies of the lattice model up to order order N^-4, without having to solve the Bethe Ansatz equations. At the same time, it is possible to express the exact eigenstates of the lattice model in terms of bosonic modes.Comment: 17 pages, 8 Figures. The latest version can be found at http://www.physik.uni-kl.de/eggert/papers/index.htm

Doubly excited ferromagnetic spin-chain as a pair of coupled kicked rotors

We show that the dynamics of a doubly-excited 1D Heisenberg ferromagnetic chain, subject to short pulses from a parabolic magnetic field may be analyzed as a pair of quantum kicked rotors. By focusing on the two-magnon dynamics in the kicked XXZ model we investigate how the anisotropy parameter - which controls the strength of the magnon-magnon interaction - changes the nature of the coupling between the two "image" coupled Kicked Rotors. We investigate quantum state transfer possibilities and show that one may control whether the spin excitations are transmitted together, or separate from each other.Comment: 8 pages, 4 figures; extended appendix and corrected typo