Hamiltonian Control of Quantum Dynamical Semigroups: Stabilization and Convergence Speed

We consider finite-dimensional Markovian open quantum systems, and characterize the extent to which time-independent Hamiltonian control may allow to stabilize a target quantum state or subspace and optimize the resulting convergence speed. For a generic Lindblad master equation, we introduce a dissipation-induced decomposition of the associated Hilbert space, and show how it serves both as a tool to analyze global stability properties for given control resources and as the starting point to synthesize controls that ensure rapid convergence. The resulting design principles are illustrated in realistic Markovian control settings motivated by quantum information processing, including quantum-optical systems and nitrogen-vacancy centers in diamond.

[1]  Herschel Rabitz,et al.  Quantum wavefunction controllability , 2001 .

[2]  Chunfeng Wu,et al.  Driving quantum systems into decoherence-free subspaces by Lyapunov control , 2009, 0908.1048.

[3]  Matthias Steiner,et al.  Single-Shot Readout of a Single Nuclear Spin , 2010, Science.

[4]  J. Wrachtrup,et al.  Universal enhancement of the optical readout fidelity of single electron spins at nitrogen-vacancy c , 2009, 0909.2783.

[5]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[6]  K. Lendi,et al.  Quantum Dynamical Semigroups and Applications , 1987 .

[7]  Matthew Sellars,et al.  Nitrogen-vacancy center in diamond: Model of the electronic structure and associated dynamics , 2006 .

[8]  B. Baumgartner,et al.  Analysis of quantum semigroups with GKS–Lindblad generators: II. General , 2008, 0806.3164.

[9]  Germany,et al.  Quantum states and phases in driven open quantum systems with cold atoms , 2008, 0803.1482.

[10]  G. Lindblad On the generators of quantum dynamical semigroups , 1976 .

[11]  Hans Peter Büchler,et al.  Preparation of Entangled States by Dissipative Quantum Markov Processes , 2008 .

[12]  J. S. Hodges,et al.  Repetitive Readout of a Single Electronic Spin via Quantum Logic with Nuclear Spin Ancillae , 2009, Science.

[13]  U. Helmke,et al.  Lie-Semigroup Structures for Reachability and Control of Open Quantum Systems: Viewing Markovian Quantum Channels as Lie Semigroups and GKS-Lindblad Generators as Lie Wedge , 2008 .

[14]  Uwe Helmke,et al.  Lie-semigroup structures for reachability and control of open quantum systems: kossakowski-lindblad generators form lie wedge to markovian channels , 2009 .

[15]  Claudio Altafini,et al.  Controllability of quantum mechanical systems by root space decomposition of su(N) , 2002 .

[16]  S. Schirmer,et al.  Generating maximal entanglement between non-interacting atoms by collective decay and symmetry breaking , 2010, 1005.2114.

[17]  Lorenza Viola,et al.  Hadamard products of product operators and the design of gradient-diffusion experiments for simulating decoherence by NMR spectroscopy , 2000, quant-ph/0009010.

[18]  Amílcar Sernadas,et al.  Quantum Computation and Information , 2006 .

[19]  J Wrachtrup,et al.  Dynamic polarization of single nuclear spins by optical pumping of nitrogen-vacancy color centers in diamond at room temperature. , 2008, Physical review letters.

[20]  Xiaoting Wang,et al.  Stabilizing Quantum States by Constructive Design of Open Quantum Dynamics , 2010, IEEE Transactions on Automatic Control.

[21]  Claudio Altafini,et al.  Coherent control of open quantum dynamical systems , 2004 .

[22]  S. G. Schirmer,et al.  Stabilizing open quantum systems by Markovian reservoir engineering , 2009, 0909.1596.

[23]  C. Altafini,et al.  QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC) 2357 Controllability properties for finite dimensional quantum Markovian master equations , 2002, quant-ph/0211194.

[24]  Fernando Pastawski,et al.  Quantum memories based on engineered dissipation , 2010, 1010.2901.

[25]  P. Zoller,et al.  Preparation of entangled states by quantum Markov processes , 2008, 0803.1463.

[26]  Lorenza Viola,et al.  Analysis and synthesis of attractive quantum Markovian dynamics , 2008, Autom..

[27]  David J. Tannor,et al.  On the interplay of control fields and spontaneous emission in laser cooling , 1999 .

[28]  E. Sudarshan,et al.  Completely Positive Dynamical Semigroups of N Level Systems , 1976 .

[29]  J. J. Sakurai,et al.  Modern Quantum Mechanics , 1986 .

[30]  Lorenza Viola,et al.  Quantum Markovian Subsystems: Invariance, Attractivity, and Control , 2007, IEEE Transactions on Automatic Control.

[31]  Andrzej Kossakowski,et al.  Properties of Quantum Markovian Master Equations , 1978 .

[32]  Saverio Bolognani,et al.  Engineering Stable Discrete-Time Quantum Dynamics via a Canonical QR Decomposition , 2009, IEEE Transactions on Automatic Control.