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Journal articles on the topic 'Equation maitresse de Lindblad'

Author: Grafiati

Published: 4 June 2021

Last updated: 11 February 2022

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1

Tarasov, Vasily E. "Quantum Maps with Memory from Generalized Lindblad Equation." Entropy 23, no. 5 (April 28, 2021): 544. http://dx.doi.org/10.3390/e23050544.

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In this paper, we proposed the exactly solvable model of non-Markovian dynamics of open quantum systems. This model describes open quantum systems with memory and periodic sequence of kicks by environment. To describe these systems, the Lindblad equation for quantum observable is generalized by taking into account power-law fading memory. Dynamics of open quantum systems with power-law memory are considered. The proposed generalized Lindblad equations describe non-Markovian quantum dynamics. The quantum dynamics with power-law memory are described by using integrations and differentiation of non-integer orders, as well as fractional calculus. An example of a quantum oscillator with linear friction and power-law memory is considered. In this paper, discrete-time quantum maps with memory, which are derived from generalized Lindblad equations without any approximations, are suggested. These maps exactly correspond to the generalized Lindblad equations, which are fractional differential equations with the Caputo derivatives of non-integer orders and periodic sequence of kicks that are represented by the Dirac delta-functions. The solution of these equations for coordinates and momenta are derived. The solutions of the generalized Lindblad equations for coordinate and momentum operators are obtained for open quantum systems with memory and kicks. Using these solutions, linear and nonlinear quantum discrete-time maps are derived.

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2

Pearle, Philip. "Simple derivation of the Lindblad equation." European Journal of Physics 33, no. 4 (April 27, 2012): 805–22. http://dx.doi.org/10.1088/0143-0807/33/4/805.

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3

Ou, Congjie, Yuho Yokoi, and Sumiyoshi Abe. "Spin Isoenergetic Process and the Lindblad Equation." Entropy 21, no. 5 (May 17, 2019): 503. http://dx.doi.org/10.3390/e21050503.

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A general comment is made on the existence of various baths in quantum thermodynamics, and a brief explanation is presented about the concept of weak invariants. Then, the isoenergetic process is studied for a spin in a magnetic field that slowly varies in time. In the Markovian approximation, the corresponding Lindbladian operators are constructed without recourse to detailed information about the coupling of the subsystem with the environment called the energy bath. The entropy production rate under the resulting Lindblad equation is shown to be positive. The leading-order expressions of the power output and work done along the isoenergetic process are obtained.

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4

Chruściński, Dariusz, and Saverio Pascazio. "A Brief History of the GKLS Equation." Open Systems & Information Dynamics 24, no. 03 (September 2017): 1740001. http://dx.doi.org/10.1142/s1230161217400017.

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5

FUJII, KAZUYUKI. "ALGEBRAIC STRUCTURE OF A MASTER EQUATION WITH GENERALIZED LINDBLAD FORM." International Journal of Geometric Methods in Modern Physics 05, no. 07 (November 2008): 1033–40. http://dx.doi.org/10.1142/s0219887808003168.

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The quantum damped harmonic oscillator is described by the master equation with usual Lindblad form. The equation has been solved completely by us in arXiv: 0710.2724 [quant-ph]. To construct the general solution a few facts of representation theory based on the Lie algebra su(1,1) were used. In this paper we treat a general model described by a master equation with generalized Lindblad form. Then we examine the algebraic structure related to some Lie algebras and construct the interesting approximate solution.

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6

Binney, James. "Angle-action variables for orbits trapped at a Lindblad resonance." Monthly Notices of the Royal Astronomical Society 495, no. 1 (May 19, 2020): 886–94. http://dx.doi.org/10.1093/mnras/staa092.

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ABSTRACT The conventional approach to orbit trapping at Lindblad resonances via a pendulum equation fails when the parent of the trapped orbits is too circular. The problem is explained and resolved in the context of the Torus Mapper and a realistic Galaxy model. Tori are computed for orbits trapped at both the inner and outer Lindblad resonances of our Galaxy. At the outer Lindblad resonance, orbits are quasi-periodic and can be accurately fitted by torus mapping. At the inner Lindblad resonance, orbits are significantly chaotic although far from ergodic, and each orbit explores a small range of tori obtained by torus mapping.

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7

Hod, Oded, César A. Rodríguez-Rosario, Tamar Zelovich, and Thomas Frauenheim. "Driven Liouville von Neumann Equation in Lindblad Form." Journal of Physical Chemistry A 120, no. 19 (February 16, 2016): 3278–85. http://dx.doi.org/10.1021/acs.jpca.5b12212.

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8

Manzano, Daniel. "A short introduction to the Lindblad master equation." AIP Advances 10, no. 2 (February 1, 2020): 025106. http://dx.doi.org/10.1063/1.5115323.

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9

Chetrite, R., and K. Mallick. "Quantum Fluctuation Relations for the Lindblad Master Equation." Journal of Statistical Physics 148, no. 3 (August 2012): 480–501. http://dx.doi.org/10.1007/s10955-012-0557-z.

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10

Dubois, Jonathan, Ulf Saalmann, and Jan M. Rost. "Semi-classical Lindblad master equation for spin dynamics." Journal of Physics A: Mathematical and Theoretical 54, no. 23 (May 7, 2021): 235201. http://dx.doi.org/10.1088/1751-8121/abf79b.

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11

Lange, Stefan, and Carsten Timm. "Random-matrix theory for the Lindblad master equation." Chaos: An Interdisciplinary Journal of Nonlinear Science 31, no. 2 (February 2021): 023101. http://dx.doi.org/10.1063/5.0033486.

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12

Fagnola, Franco, and Carlos M. Mora. "Basic Properties of a Mean Field Laser Equation." Open Systems & Information Dynamics 26, no. 03 (September 2019): 1950015. http://dx.doi.org/10.1142/s123016121950015x.

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We study the nonlinear quantum master equation describing a laser under the mean field approximation. The quantum system is formed by a single mode optical cavity and two level atoms, which interact with reservoirs. Namely, we establish the existence and uniqueness of the regular solution to the nonlinear operator equation under consideration, as well as we get a probabilistic representation for this solution in terms of a mean field stochastic Schrödinger equation. To this end, we find a regular solution for the nonautonomous linear quantum master equation in Gorini–Kossakowski–Sudarshan–Lindblad form, and we prove the uniqueness of the solution to the nonautonomous linear adjoint quantum master equation in Gorini–Kossakowski–Sudarshan–Lindblad form. Moreover, we obtain rigorously the Maxwell–Bloch equations from the mean field laser equation.

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13

Kirchanov, V. S. "The Lindblad equation for a quantum dissipative harmonic oscillator." ВЕСТНИК ПЕРМСКОГО УНИВЕРСИТЕТА. ФИЗИКА, no. 2 (2018): 5–12. http://dx.doi.org/10.17072/1994-3598-2018-2-05-12.

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14

Andrianov, A. A., M. V. Ioffe, and O. O. Novikov. "Supersymmetrization of the Franke–Gorini–Kossakowski–Lindblad–Sudarshan equation." Journal of Physics A: Mathematical and Theoretical 52, no. 42 (September 23, 2019): 425301. http://dx.doi.org/10.1088/1751-8121/ab4338.

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15

Chebotarev, A. M., J. C. Garcia, and R. B. Quezada. "On the lindblad equation with unbounded time-dependent coefficients." Mathematical Notes 61, no. 1 (January 1997): 105–17. http://dx.doi.org/10.1007/bf02355012.

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16

Ozorio de Almeida, A. M., and O. Brodier. "Nonlinear semiclassical dynamics of open systems." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 369, no. 1935 (January 28, 2011): 260–77. http://dx.doi.org/10.1098/rsta.2010.0261.

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A semiclassical approximation for an evolving density operator, driven by a ‘closed’ Hamiltonian and ‘open’ Markovian Lindblad operators, is reviewed. The theory is based on the chord function, i.e. the Fourier transform of the Wigner function. It reduces to an exact solution of the Lindblad master equation if the Hamiltonian is a quadratic function and the Lindblad operators are linear functions of positions and momenta. The semiclassical formulae are interpreted within a (real) double phase space, generated by an appropriate classical double Hamiltonian. An extra ‘open’ term in the double Hamiltonian is generated by the non-Hermitian part of the Lindblad operators in the general case of dissipative Markovian evolution. Decoherence narrows the relevant region of double phase space to the neighbourhood of a caustic for both the Wigner function and the chord function. This difficulty is avoided by the definition of a propagator, here developed in both representations. Generalized asymptotic equilibrium solutions are thus presented for the first time.

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17

ISAR, A., A. SANDULESCU, H. SCUTARU, E. STEFANESCU, and W. SCHEID. "OPEN QUANTUM SYSTEMS." International Journal of Modern Physics E 03, no. 02 (June 1994): 635–714. http://dx.doi.org/10.1142/s0218301394000164.

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The damping of the harmonic oscillator is studied in the framework of the Lindblad theory for open quantum systems. A generalization of the fundamental constraints on quantum mechanical diffusion coefficients which appear in the master equation for the damped quantum oscillator is presented; the Schrödinger, Heisenberg and Weyl-Wigner-Moyal representations of the Lindblad equation are given explicitly. On the basis of these representations it is shown that various master equations for the damped quantum oscillator used in the literature are particular cases of the Lindblad equation and that not all of these equations are satisfying the constraints on quantum mechanical diffusion coefficients. Analytical expressions for the first two moments of coordinate and momentum are obtained by using the characteristic function of the Lindblad master equation. The master equation is transformed into Fokker-Planck equations for quasiprobability distributions and a comparative study is made for the Glauber P representation, the antinormal ordering Q representation, and the Wigner W representation. The density matrix is represented via a generating function, which is obtained by solving a timedependent linear partial differential equation derived from the master equation. Illustrative examples for specific initial conditions of the density matrix are provided. The solution of the master equation in the Weyl-Wigner-Moyal representation is of Gaussian type if the initial form of the Wigner function is taken to be a Gaussian corresponding (for example) to a coherent wavefunction. The damped harmonic oscillator is applied for the description of the charge equilibration mode observed in deep inelastic reactions. For a system consisting of two harmonic oscillators the time dependence of expectation values, Wigner function and Weyl operator, are obtained and discussed. In addition models for the damping of the angular momentum are studied. Using this theory to the quantum tunneling through the nuclear barrier, besides Gamow’s transitions with energy conservation, additional transitions with energy loss are found. The tunneling spectrum is obtained as a function of the barrier characteristics. When this theory is used to the resonant atom-field interaction, new optical equations describing the coupling through the environment of the atomic observables are obtained. With these equations, some characteristics of the laser radiation absorption spectrum and optical bistability are described.

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18

Bravyi, Sergey, and Robert Konig. "Classical simulation of dissipative fermionic linear optics." Quantum Information and Computation 12, no. 11&12 (November 2012): 925–43. http://dx.doi.org/10.26421/qic12.11-12-2.

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Fermionic linear optics is a limited form of quantum computation which is known to be efficiently simulable on a classical computer. We revisit and extend this result by enlarging the set of available computational gates: in addition to unitaries and measurements, we allow dissipative evolution governed by a Markovian master equation with linear Lindblad operators. We show that this more general form of fermionic computation is also simulable efficiently by classical means. Given a system of $N$~fermionic modes, our algorithm simulates any such gate in time $O(N^3)$ while a single-mode measurement is simulated in time $O(N^2)$. The steady state of the Lindblad equation can be computed in time $O(N^3)$.

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19

Bengs, Christian. "Markovian exchange phenomena in magnetic resonance and the Lindblad equation." Journal of Magnetic Resonance 322 (January 2021): 106868. http://dx.doi.org/10.1016/j.jmr.2020.106868.

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20

Bondarev, Boris V. "Lindblad Equation for Harmonic Oscillator: Uncertainty Relation Depending on Temperature." Applied Mathematics 08, no. 11 (2017): 1529–38. http://dx.doi.org/10.4236/am.2017.811111.

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21

Davidsson, Eric, and Markus Kowalewski. "Simulating photodissociation reactions in bad cavities with the Lindblad equation." Journal of Chemical Physics 153, no. 23 (December 21, 2020): 234304. http://dx.doi.org/10.1063/5.0033773.

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22

Kosov, Daniel S., Tomaž Prosen, and Bojan Žunkovič. "Lindblad master equation approach to superconductivity in open quantum systems." Journal of Physics A: Mathematical and Theoretical 44, no. 46 (October 21, 2011): 462001. http://dx.doi.org/10.1088/1751-8113/44/46/462001.

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23

Vacchini, Bassano. "General structure of quantum collisional models." International Journal of Quantum Information 12, no. 02 (March 2014): 1461011. http://dx.doi.org/10.1142/s0219749914610115.

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We point to the connection between a recently introduced class of non-Markovian master equations and the general structure of quantum collisional models. The basic construction relies on three basic ingredients: a collection of time dependent completely positive maps, a completely positive trace preserving transformation and a waiting time distribution characterizing a renewal process. The relationship between this construction and a Lindblad dynamics is clarified by expressing the solution of a Lindblad master equation in terms of demixtures over different stochastic trajectories for the statistical operator weighted by suitable probabilities on the trajectory space.

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24

Villegas-Martínez, B. M., F. Soto-Eguibar, and H. M. Moya-Cessa. "Application of Perturbation Theory to a Master Equation." Advances in Mathematical Physics 2016 (2016): 1–7. http://dx.doi.org/10.1155/2016/9265039.

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We develop a matrix perturbation method for the Lindblad master equation. The first- and second-order corrections are obtained and the method is generalized for higher orders. The perturbation method developed is applied to the problem of a lossy cavity filled with a Kerr medium; the second-order corrections are estimated and compared with the known exact analytic solution. The comparison is done by calculating theQ-function, the average number of photons, and the distance between density matrices.

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25

Isar, A., A. Sandulescu, and W. Scheid. "Lindblad master equation for the damped harmonic oscillator with deformed dissipation." Physica A: Statistical Mechanics and its Applications 322 (May 2003): 233–46. http://dx.doi.org/10.1016/s0378-4371(02)01828-9.

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26

Selstø, Sølve. "Non-Hermitian quantum mechanics in the context of the Lindblad equation." Journal of Physics: Conference Series 388, no. 15 (November 5, 2012): 152016. http://dx.doi.org/10.1088/1742-6596/388/15/152016.

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27

Asano, Masanari, Masanori Ohya, Yoshiharu Tanaka, Andrei Khrennikov, and Irina Basieva. "On Application of Gorini-Kossakowski-Sudarshan-Lindblad Equation in Cognitive Psychology." Open Systems & Information Dynamics 18, no. 01 (March 2011): 55–69. http://dx.doi.org/10.1142/s1230161211000042.

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We proceed towards an application of the mathematical formalism of quantum mechanics to cognitive psychology — the problem of decision-making in games of the Prisoners Dilemma type. These games were used as tests of rationality of players. Experiments performed in cognitive psychology by Shafir and Tversky [1, 2], Croson [3], Hofstader [4, 5] demonstrated that in general real players do not use "rational strategy" provided by classical game theory; this psychological phenomenon was called the disjunction effect. We elaborate a model of quantum-like decision making which can explain this effect ("irrationality" of plays). Our model is based on quantum information theory. The main result of this paper is the derivation of Gorini-Kossakowski-Sudarshan-Lindblad equation whose equilibrium solution gives the quantum state used for decision making. It is the first application of this equation in cognitive psychology.

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28

Barchielli, A., and C. Pellegrini. "Jump-diffusion unravelling of a non-Markovian generalized Lindblad master equation." Journal of Mathematical Physics 51, no. 11 (November 2010): 112104. http://dx.doi.org/10.1063/1.3514539.

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29

Cao, Yu, and Jianfeng Lu. "Lindblad equation and its semiclassical limit of the Anderson-Holstein model." Journal of Mathematical Physics 58, no. 12 (December 2017): 122105. http://dx.doi.org/10.1063/1.4993431.

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30

Prosen, Tomaž. "Spectral theorem for the Lindblad equation for quadratic open fermionic systems." Journal of Statistical Mechanics: Theory and Experiment 2010, no. 07 (July 23, 2010): P07020. http://dx.doi.org/10.1088/1742-5468/2010/07/p07020.

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31

Kirchanov, V. S. "Quantum Damped Fock Oscillator with Linear Dissipation and the Lindblad Equation." Russian Physics Journal 62, no. 2 (June 2019): 382–85. http://dx.doi.org/10.1007/s11182-019-01723-x.

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32

Antão, T. V. C., and N. M. R. Peres. "Two-level systems coupled to Graphene plasmons: A Lindblad equation approach." International Journal of Modern Physics B 35, no. 20 (August 10, 2021): 2130007. http://dx.doi.org/10.1142/s0217979221300073.

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In this paper, we review the theory of open quantum systems and macroscopic quantum electrodynamics, providing a self-contained account of many aspects of these two theories. The former is presented in the context of a qubit coupled to a electromagnetic thermal bath, the latter is presented in the context of a quantization scheme for surface-plasmon polaritons (SPPs) in graphene based on Langevin noise currents. This includes a calculation of the dyadic Green’s function (in the electrostatic limit) for a Graphene sheet between two semi-infinite linear dielectric media, and its subsequent application to the construction of SPP creation and annihilation operators. We then bring the two fields together and discuss the entanglement of two qubits in the vicinity of a graphene sheet which supports SPPs. The two qubits communicate with each other via the emission and absorption of SPPs. We find that a Schrödinger cat state involving the two qubits can be partially protected from decoherence by taking advantage of the dissipative dynamics in graphene. A comparison is also drawn between the dynamics at zero temperature, obtained via Schrödinger’s equation, and at finite temperature, obtained using the Lindblad equation.

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33

ISAR, A., A. SANDULESCU, and W. SCHEID. "PHASE SPACE REPRESENTATION FOR OPEN QUANTUM SYSTEMS WITHIN THE LINDBLAD THEORY." International Journal of Modern Physics B 10, no. 22 (October 10, 1996): 2767–79. http://dx.doi.org/10.1142/s0217979296001240.

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The Lindblad master equation for an open quantum system with a Hamiltonian containing an arbitrary potential is written as an equation for the Wigner distribution function in the phase space representation. The time derivative of this function is given by a sum of three parts: the classical one, the quantum corrections and the contribution due to the opening of the system. In the particular case of a harmonic oscillator, quantum corrections do not exist.

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34

Schaller, Gernot, and Julian Ablaßmayer. "Thermodynamics of the Coarse-Graining Master Equation." Entropy 22, no. 5 (May 5, 2020): 525. http://dx.doi.org/10.3390/e22050525.

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We study the coarse-graining approach to derive a generator for the evolution of an open quantum system over a finite time interval. The approach does not require a secular approximation but nevertheless generally leads to a Lindblad–Gorini–Kossakowski–Sudarshan generator. By combining the formalism with full counting statistics, we can demonstrate a consistent thermodynamic framework, once the switching work required for the coupling and decoupling with the reservoir is included. Particularly, we can write the second law in standard form, with the only difference that heat currents must be defined with respect to the reservoir. We exemplify our findings with simple but pedagogical examples.

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35

Furusho, K., S. Iriyama, and M. Ohya. "Chaos Amplification Process Can Be Described by the GKSL Master Equation." Open Systems & Information Dynamics 24, no. 02 (June 2017): 1750008. http://dx.doi.org/10.1142/s1230161217500081.

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In 2000, Ohya et al. proposed a quantum algorithm with the amplification process of success probability, so-called chaos amplifier. They defined the process based on the logistic map, and its chaos behaviour amplifies the probability very fast. In this paper, we construct the chaos amplifier using a lifting map, a master equation and a partial trace. We also calculate the condition on the Lindblad operators in the GKSL master equation to achieve an effective amplification.

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36

ABE, SUMIYOSHI. "FRACTIONAL DIFFUSION EQUATION, QUANTUM SUBDYNAMICS AND EINSTEIN'S THEORY OF BROWNIAN MOTION." International Journal of Modern Physics B 21, no. 23n24 (September 30, 2007): 3993–99. http://dx.doi.org/10.1142/s0217979207045086.

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The fractional diffusion equation for describing the anomalous diffusion phenomenon is derived in the spirit of Einstein's 1905 theory of Brownian motion. It is shown how naturally fractional calculus appears in the theory. Then, Einstein's theory is examined in view of quantum theory. An isolated quantum system composed of the objective system and the environment is considered, and then subdynamics of the objective system is formulated. The resulting quantum master equation is found to be of the Lindblad type.

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37

Li, Ming. "Simulation of Quantum Dynamics Based on the Quantum Stochastic Differential Equation." Scientific World Journal 2013 (2013): 1–6. http://dx.doi.org/10.1155/2013/424137.

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The quantum stochastic differential equation derived from the Lindblad form quantum master equation is investigated. The general formulation in terms of environment operators representing the quantum state diffusion is given. The numerical simulation algorithm of stochastic process of direct photodetection of a driven two-level system for the predictions of the dynamical behavior is proposed. The effectiveness and superiority of the algorithm are verified by the performance analysis of the accuracy and the computational cost in comparison with the classical Runge-Kutta algorithm.

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38

Espoukeh, Pakhshan, and Pouria Pedram. "The lower bound to the concurrence for four-qubit W state under noisy channels." International Journal of Quantum Information 13, no. 01 (February 2015): 1550004. http://dx.doi.org/10.1142/s0219749915500045.

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We study the dynamics of four-qubit W state under various noisy environments by solving analytically the master equation in the Lindblad form in which the Lindblad operators correspond to the Pauli matrices and describe the decoherence of states. Also, we investigate the dynamics of the entanglement using the lower bound to the concurrence. It is found that while the entanglement decreases monotonically for Pauli-Z noise, it decays suddenly for other three noises. Moreover, by studying the time evolution of entanglement of various maximally entangled four-qubit states, we indicate that the four-qubit W state is more robust under same-axis Pauli channels. Furthermore, three-qubit W state preserves more entanglement with respect to the four-qubit W state, except for the Pauli-Z noise.

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39

Dorda, A., M. Sorantin, W. von der Linden, and E. Arrigoni. "Optimized auxiliary representation of non-Markovian impurity problems by a Lindblad equation." New Journal of Physics 19, no. 6 (June 2, 2017): 063005. http://dx.doi.org/10.1088/1367-2630/aa6ccc.

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40

Moodley, Mervlyn, S. Paul, and T. Nsio Nzundu. "Stochastic wave-function unravelling of the generalized Lindblad equation using correlated states." Physica Scripta 85, no. 4 (March 7, 2012): 045002. http://dx.doi.org/10.1088/0031-8949/85/04/045002.

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41

Roux, Filippus S. "The Lindblad equation for the decay of entanglement due to atmospheric scintillation." Journal of Physics A: Mathematical and Theoretical 47, no. 19 (April 23, 2014): 195302. http://dx.doi.org/10.1088/1751-8113/47/19/195302.

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42

Wang, Yi-Siang, Parmeet Nijjar, Xin Zhou, Denys I. Bondar, and Oleg V. Prezhdo. "Combining Lindblad Master Equation and Surface Hopping to Evolve Distributions of Quantum Particles." Journal of Physical Chemistry B 124, no. 21 (May 4, 2020): 4326–37. http://dx.doi.org/10.1021/acs.jpcb.0c03030.

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43

Ou, Congjie, and Sumiyoshi Abe. "Weak invariants, temporally local equilibria and isoenergetic processes described by the Lindblad equation." EPL (Europhysics Letters) 125, no. 6 (May 2, 2019): 60004. http://dx.doi.org/10.1209/0295-5075/125/60004.

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44

Trushechkin, A. S. "Finding Stationary Solutions of the Lindblad Equation by Analyzing the Entropy Production Functional." Proceedings of the Steklov Institute of Mathematics 301, no. 1 (May 2018): 262–71. http://dx.doi.org/10.1134/s008154381804020x.

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45

ACCARDI, L., A. N. PECHEN, and I. V. VOLOVICH. "A STOCHASTIC GOLDEN RULE AND QUANTUM LANGEVIN EQUATION FOR THE LOW DENSITY LIMIT." Infinite Dimensional Analysis, Quantum Probability and Related Topics 06, no. 03 (September 2003): 431–53. http://dx.doi.org/10.1142/s0219025703001304.

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A rigorous derivation of quantum Langevin equation from microscopic dynamics in the low density limit is given. We consider a quantum model of a microscopic system (test particle) coupled with a reservoir (gas of light Bose particles) via interaction of scattering type. We formulate a mathematical procedure (the so-called stochastic golden rule) which allows us to determine the quantum Langevin equation in the limit of large time and small density of particles of the reservoir. The quantum Langevin equation describes not only dynamics of the system but also the reservoir. We show that the generator of the corresponding master equation has the Lindblad form of most general generators of completely positive semigroups.

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46

Marmo, Giuseppe, and Saverio Pascazio. "The Legacy of George Sudarshan." Open Systems & Information Dynamics 26, no. 03 (September 2019): 1950011. http://dx.doi.org/10.1142/s1230161219500112.

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George Sudarshan was an eclectic thinker, and one of the most profound physicists of the last century. We review here a small part of his oeuvre, focusing on his pioneering contributions to the quantum Zeno effect, quantum channels (the Kraus–Sudarshan representation) and quantum semigroups (the Gorini–Kossakowski–Lindblad–Sudarshan equation). These topics are of crucial importance in the booming field of quantum mechanics and applications.

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47

Chebotarev, Alexander M. "Quantum Stochastic Differential Equation is Unitarily Equivalent to a Symmetric Boundary Value Problem in Fock Space." Infinite Dimensional Analysis, Quantum Probability and Related Topics 01, no. 02 (April 1998): 175–99. http://dx.doi.org/10.1142/s0219025798000120.

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We show a new remarkable connection between the symmetric form of a quantum stochastic differential equation (QSDE) and the strong resolvent limit of the Schrödinger equations in Fock space: the strong resolvent limit is unitarily equivalent to QSDE in the adapted (or Ito) form, and the weak limit is unitarily equivalent to the symmetric (or Stratonovich) form of QSDE. We also prove that QSDE is unitarily equivalent to a symmetric boundary value problem for the Schrödinger equation in Fock space. The boundary condition describes standard jumps in phase and amplitude of components of Fock vectors belonging to the range of the resolvent. The corresponding Markov evolution equation (the Lindblad or Markov master equation) is derived from the boundary value problem for the Schrödinger equation.

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48

vom Ende, Frederik, Gunther Dirr, Michael Keyl, and Thomas Schulte-Herbrüggen. "Reachability in Infinite-Dimensional Unital Open Quantum Systems with Switchable GKS–Lindblad Generators." Open Systems & Information Dynamics 26, no. 03 (September 2019): 1950014. http://dx.doi.org/10.1142/s1230161219500148.

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In quantum systems theory one of the fundamental problems boils down to: given an initial state, which final states can be reached by the dynamic system in question. Here we consider infinite-dimensional open quantum dynamical systems following a unital Kossakowski–Lindblad master equation extended by controls. More precisely, their time evolution shall be governed by an inevitable potentially unbounded Hamiltonian drift term H0, finitely many bounded control Hamiltonians Hj allowing for (at least) piecewise constant control amplitudes [Formula: see text] plus a bang-bang (i.e., on-off) switchable noise term ГV in Kossakowski–Lindblad form. Generalizing standard majorization results from finite to infinite dimensions, we show that such bilinear quantum control systems allow to approximately reach any target state majorized by the initial one as up to now it only has been known in finite dimensional analogues. The proof of the result is currently limited to the bounded control Hamiltonians Hj and for noise terms ГV with compact normal V.

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49

SINGH, NAVINDER, and N. KUMAR. "QUANTUM DIFFUSION ON A DYNAMICALLY DISORDERED AND HARMONICALLY DRIVEN LATTICE WITH STATIC BIAS: DECOHERENCE." Modern Physics Letters B 19, no. 07n08 (April 10, 2005): 379–87. http://dx.doi.org/10.1142/s0217984905008426.

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We revisit the problem of quantum diffusion of a particle moving on a lattice with dynamical disorder. Decoherence, essential for the diffusive motion, is introduced via a set of Lindblad operators, known to guarantee per se the positivity, Hermiticity and the trace-class nature of the reduced density matrix, are derived and solved analytically for several transport quantities of interest. For the special Hermitian choice of the Lindblad operators projected onto the lattice sites, we recover several known results, obtained by others, e.g. through the stochastic Liouville equation using phenomenological damping terms for the off-diagonal density-matrix elements. An interesting result that we obtained is for the case of a 1D lattice with static potential bias and a time-harmonic modulation (ac drive) of its transition-matrix element, where the diffusion coefficient shows an oscillatory behavior as function of the drive amplitude and frequency — clearly, a Wannier–Stark ladder signature. The question of dissipation is also briefly discussed.

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50

Stratonovich, Rouslan L. "The quantum langevin forces for dynamical systems with linear dissipation and the Lindblad equation." Physica A: Statistical Mechanics and its Applications 236, no. 3-4 (March 1997): 335–52. http://dx.doi.org/10.1016/s0378-4371(96)00261-0.

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