Quantum control theory and applications: A survey <s> Markovian master equations <s> The notion of a quantum dynamical semigroup is defined using the concept of a completely positive map. An explicit form of a bounded generator of such a semigroup onB(ℋ) is derived. This is a quantum analogue of the Levy-Khinchin formula. As a result the general form of a large class of Markovian quantum-mechanical master equations is obtained. <s> BIB001 </s> Quantum control theory and applications: A survey <s> Markovian master equations <s> PREFACE ACKNOWLEDGEMENTS PART 1: PROBABILITY IN CLASSICAL AND QUANTUM MECHANICS 1. Classical probability theory and stochastic processes 2. Quantum Probability PART 2: DENSITY MATRIX THEORY 3. Quantum Master Equations 4. Decoherence PART 3: STOCHASTIC PROCESSES IN HILBERT SPACE 5. Probability distributions on Hilbert space 6. Stochastic dynamics in Hilbert space 7. The stochastic simulation method 8. Applications to quantum optical systems PART 4: NON-MARKOVIAN QUANTUM PROCESSES 9. Projection operator techniques 10. Non-Markovian dynamics in physical systems PART 5: RELATIVISTIC QUANTUM PROCESSES 11. Measurements in relativistic quantum mechanics 12. Open quantum electrodynamics <s> BIB002 </s> Quantum control theory and applications: A survey <s> Markovian master equations <s> In this text the authors develop quantum dynamics of open systems for a wide class of irreversible processes starting from the concept of completely positive semigroups. This unified approach makes the material easily accessible to non-specialists and provides an easy access to practical applications. Written for graduate students, the book presents a wealth of useful examples; in particular, models of unstable and N-level systems are treated systematically and in considerable detail including new types of generated Bloch-equations. The general theory is extensively summarized from abstract dynamical maps to those obtained by a reduction of Hamiltonian dynamics under a Markovian approximation. Various methods of determining semigroup generators and the corresponding master equations are discussed including time-dependent and nonlinear generators. Further topics treated are a generalized H-theorem, quantum detailed balance and return to equilibrium, discrete quantum Boltzmann equation, nonlinear Schrdinger equation, spin relaxation by spin waves, entropy production and its generalization by a measure of irreversibiblity. <s> BIB003
If we use the density matrix ρ(t) to describe the state of a closed quantum system, the evolution equation for ρ(t) can be described by the quantum Liouville equation where [X, ρ] = Xρ − ρX is the commutation operator. In many practical applications, the quantum systems being considered are open quantum systems. In fact, this is the case for most quantum control systems since such systems unavoidably interact with their external environments (including control inputs and measurement devices). For an open quantum system, its evolution cannot generally be described in terms of a unitary transformation. In many situations, a quantum master equation for ρ(t) is a suitable way to describe the dynamics of an open quantum system. One of the simplest cases is when a Markovian approximation can be applied where a short environmental correlation time is supposed and memory effects may be neglected BIB002 . For an Ndimensional open quantum system with Markovian dynamics, its state ρ(t) can be described by the following Markovian master equation (MME) (for details, see, e.g., BIB002 , BIB001 , BIB003 ): Here is a basis for the space of linear bounded operators on H with F 0 = I, the coefficient matrix A = (α jk ) is positive semidefinite and physically specifies the relevant relaxation rates. Markovian master equations have been widely used as models in Markovian quantum feedback (see Section 4.2).