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1
Henkel, C. "Laser theory in manifest Lindblad form." Journal of Physics B: Atomic, Molecular and Optical Physics 40, no. 12 (June 5, 2007): 2359–71. http://dx.doi.org/10.1088/0953-4075/40/12/012.
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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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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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Berrêdo, R. C. de, J. G. P. de Faria, F. Camargo, M. C. Nemes, H. E. Borges, K. M. Fonseca Romero, A. F. R. de Toledo Piza, and A. N. Salgueiro. "On the Physical Content of Lindblad Form Master Equations." Physica Scripta 57, no. 4 (April 1, 1998): 533–34. http://dx.doi.org/10.1088/0031-8949/57/4/010.
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Ginsberg, Daniel. "A priori estimates for a relativistic liquid with free surface boundary." Journal of Hyperbolic Differential Equations 16, no. 03 (September 2019): 401–42. http://dx.doi.org/10.1142/s0219891619500152.
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We prove energy estimates for a relativistic free liquid body with sufficiently small fluid velocity in a general Lorentz spacetime. These estimates control Sobolev norms of the fluid velocity and enthalpy in the interior as well as Sobolev norms of the second fundamental form on the boundary. These estimates are generalizations of the energy estimates of Christodoulou and Lindblad [D. Christodoulou and H. Lindblad, On the motion of the free surface of a liquid, Commun. Pure Appl. Math. 53(12) (2000) 1536–1602] and rely on elliptic estimates which only require bounds for the second fundamental form of the time slices of the free boundary.
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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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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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Fagnola, Franco, and Rolando Rebolledo. "The Approach to Equilibrium of a Class of Quantum Dynamical Semigroups." Infinite Dimensional Analysis, Quantum Probability and Related Topics 01, no. 04 (October 1998): 561–72. http://dx.doi.org/10.1142/s0219025798000302.
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This paper deals with the asymptotic behavior of a quantum dynamical semigroup [Formula: see text] acting on the algebra of all linear bounded operators on a given Hilbert space. In practice, all these semigroups have a generator which can be written in a well-known form named after Lindblad and Davies. If the semigroup has a faithful normal stationary state ρ, necessary and sufficient conditions are derived for the w*-convergence of [Formula: see text] to [Formula: see text], where [Formula: see text] is the conditional expectation of an element X onto the subalgebra of fixed points. Our main results are expressed in terms of the Lindblad–Davies generator .
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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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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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Meyerov, Iosif, Evgeny Kozinov, Alexey Liniov, Valentin Volokitin, Igor Yusipov, Mikhail Ivanchenko, and Sergey Denisov. "Transforming Lindblad Equations into Systems of Real-Valued Linear Equations: Performance Optimization and Parallelization of an Algorithm." Entropy 22, no. 10 (October 6, 2020): 1133. http://dx.doi.org/10.3390/e22101133.
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With their constantly increasing peak performance and memory capacity, modern supercomputers offer new perspectives on numerical studies of open many-body quantum systems. These systems are often modeled by using Markovian quantum master equations describing the evolution of the system density operators. In this paper, we address master equations of the Lindblad form, which are a popular theoretical tools in quantum optics, cavity quantum electrodynamics, and optomechanics. By using the generalized Gell–Mann matrices as a basis, any Lindblad equation can be transformed into a system of ordinary differential equations with real coefficients. Recently, we presented an implementation of the transformation with the computational complexity, scaling as O(N5logN) for dense Lindbaldians and O(N3logN) for sparse ones. However, infeasible memory costs remains a serious obstacle on the way to large models. Here, we present a parallel cluster-based implementation of the algorithm and demonstrate that it allows us to integrate a sparse Lindbladian model of the dimension N=2000 and a dense random Lindbladian model of the dimension N=200 by using 25 nodes with 64 GB RAM per node.
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Ziman, Mário, Peter Štelmachovič, and Vladimír Bužek. "Description of Quantum Dynamics of Open Systems Based on Collision-Like Models." Open Systems & Information Dynamics 12, no. 01 (March 2005): 81–91. http://dx.doi.org/10.1007/s11080-005-0488-0.
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Master equations in the Lindblad form describe evolution of open quantum systems that are completely positive and simultaneously have a semigroup property. We analyze the possibility to derive this type of master equations from an intrinsically discrete dynamics that is modelled as a sequence of collisions between a given quantum system (a qubit) with particles that form the environment. In order to illustrate our approach we analyze in detail how the process of an exponential decay and the process of decoherence can be derived from a collision-like model in which particular collisions are described by SWAP and controlled-NOT interactions, respectively.
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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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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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Siemon, I., A. S. Holevo, and R. F. Werner. "Unbounded Generators of Dynamical Semigroups." Open Systems & Information Dynamics 24, no. 04 (December 2017): 1740015. http://dx.doi.org/10.1142/s1230161217400157.
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Dynamical semigroups have become the key structure for describing open system dynamics in all of physics. Bounded generators are known to be of a standard form, due to Gorini, Kossakowski, Sudarshan and Lindblad. This form is often used also in the unbounded case, but rather little is known about the general form of unbounded generators. In this paper we first give a precise description of the standard form in the unbounded case, emphasizing intuition, and collecting and even proving the basic results around it. We also give a cautionary example showing that the standard form must not be read too naively. Further examples are given of semigroups, which appear to be probability preserving to first order, but are not for finite times. Based on these, we construct examples of generators which are not of standard form.
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PARK, YONG MOON. "REMARKS ON THE STRUCTURE OF DIRICHLET FORMS ON STANDARD FORMS OF VON NEUMANN ALGEBRAS." Infinite Dimensional Analysis, Quantum Probability and Related Topics 08, no. 02 (June 2005): 179–97. http://dx.doi.org/10.1142/s0219025705001925.
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For a von Neumann algebra ࡕ acting on a Hilbert space ℋ with a cyclic and separating vector ξ0, we investigate the structure of Dirichlet forms on the natural standard form associated with the pair (ࡕ, ξ0). For a general bounded Lindblad type generator L of a conservative quantum dynamical semigroup on ࡕ, we give sufficient conditions so that the bounded operator H induced by L via the symmetric embedding of ࡕ into ℋ to be self-adjoint. It turns out that the self-adjoint operator H can be written in the form of a Dirichlet operator associated to a Dirichlet form given in Ref. 23. In order to make the connection possible, we also extend the range of applications of the formula in Ref. 23.
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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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Brüning, E., and F. Petruccione. "Density Matrices and Their Time Evolution." Open Systems & Information Dynamics 15, no. 02 (June 2008): 109–21. http://dx.doi.org/10.1142/s1230161208000109.
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Already in the case of finite dimensional Hilbert spaces [Formula: see text] the general form of density matrices ρ is not known. The main reason for this lack of knowledge is the nonlinear constraint for these matrices. We propose a representation of density matrices on finite dimensional Hilbert spaces in terms of finitely many independent parameters. For dimensions 2, 3, and 4 we write down this representation explicitly. As a further application of this representation we study the time dependence of density matrices ρ(t) which in our case is implemented through time dependence of the independent parameters. Under obvious differentiability assumptions the explicit form of [Formula: see text] is determined. As a special case we recover, for instance, the Lindblad form.
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Schulte-Herbrüggen, Thomas, Gunther Dirr, and Robert Zeier. "Quantum Systems Theory Viewed from Kossakowski-Lindblad Lie Semigroups — and Vice Versa." Open Systems & Information Dynamics 24, no. 04 (December 2017): 1740019. http://dx.doi.org/10.1142/s1230161217400194.
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The solutions to the celebrated Kossakowski-Lindblad equation extended by coherent controls yield Markovian quantum maps. More precisely, the set of all its solutions forms a semigroup of completely positive trace-preserving maps taking the specific form of a Lie semigroup. Non-trivial symmetries of these semigroups are shown to preclude accessibility in Markovian dissipative systems. This is the open-system analogue to closed systems, where triviality of (quadratic) symmetries of the Hamiltonian part suffices to decide that the system is fully controllable. The findings are placed into a unifying Lie frame of quantum systems and control theory alongside with illustrating examples.
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MORA, CARLOS M., and ROLANDO REBOLLEDO. "REGULARITY OF SOLUTIONS TO LINEAR STOCHASTIC SCHRÖDINGER EQUATIONS." Infinite Dimensional Analysis, Quantum Probability and Related Topics 10, no. 02 (June 2007): 237–59. http://dx.doi.org/10.1142/s0219025707002725.
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We develop linear stochastic Schrödinger equations driven by standard cylindrical Brownian motions (LSSs) that unravel quantum master equations in Lindblad form into quantum trajectories. More precisely, this paper establishes the existence and uniqueness of the smooth strong solution Xt to a LSS with regular initial condition. Moreover, we obtain that the mean value of the square norm of Xt is constant. We also treat the approximation of LSSs by ordinary stochastic differential equations. We apply our results to: (i) models of quantum measurements of position and momentum; and (ii) a system formed by fermions.
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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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Strunz, Walter T. "Finite Temperature Dynamics of the Total State in an Open System Model." Open Systems & Information Dynamics 12, no. 01 (March 2005): 65–80. http://dx.doi.org/10.1007/s11080-005-0487-1.
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We determine the dynamics of the total state of a system and environment for an open system model, at finite temperature. Based on a partial Husimi representation, our framework describes the full dynamics very efficiently through equations in the Hilbert space of the open system only. We briefly review the zero-temperature case and present the corresponding new finite temperature theory, within the usual Born-Markov approximation. As we will show, from a reduced point of view, our approach amounts to the derivation of a stochastic Schrödinger equation description of the dynamics. We show how the reduced density operator evolves according to the expected (finite temperature) master equation of Lindblad form.
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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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Dirr, G., U. Helmke, I. Kurniawan, and T. Schulte-Herbrüggen. "Lie-semigroup structures for reachability and control of open quantum systems: kossakowski-lindblad generators form lie wedge to markovian channels." Reports on Mathematical Physics 64, no. 1-2 (August 2009): 93–121. http://dx.doi.org/10.1016/s0034-4877(09)90022-2.
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Kimura, Gen, Shigeru Ajisaka, and Kyouhei Watanabe. "Universal Constraints on Relaxation Times for d-Level GKLS Master Equations." Open Systems & Information Dynamics 24, no. 04 (December 2017): 1740009. http://dx.doi.org/10.1142/s1230161217400091.
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In 1976, Gorini, Kossakowski, Sudarshan and Lindblad independently discovered a general form of master equations for an open quantum Markovian dynamics. In honor of all the authors, the equation is nowadays called the GKLS master equation. In this paper, we show universal constraints on the relaxation times valid for any d-level GKLS master equations, which is a generalization of the well-known constraints for 2-level systems. Specifically, we show that any relaxation rate, the inverse-relaxation time, is not greater than half of the sum of all relaxation rates. Since the relaxation times are measurable in experiments, our constraints provide a direct experimental test for the validity of the GKLS master equations, and hence for the conditions of the complete positivity and Markovianity.
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Tarasov, Vasily E. "General Non-Markovian Quantum Dynamics." Entropy 23, no. 8 (July 31, 2021): 1006. http://dx.doi.org/10.3390/e23081006.
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A general approach to the construction of non-Markovian quantum theory is proposed. Non-Markovian equations for quantum observables and states are suggested by using general fractional calculus. In the proposed approach, the non-locality in time is represented by operator kernels of the Sonin type. A wide class of the exactly solvable models of non-Markovian quantum dynamics is suggested. These models describe open (non-Hamiltonian) quantum systems with general form of nonlocality in time. To describe these systems, the Lindblad equations for quantum observable and states are generalized by taking into account a general form of nonlocality. The non-Markovian quantum dynamics is described by using integro-differential equations with general fractional derivatives and integrals with respect to time. The exact solutions of these equations are derived by using the operational calculus that is proposed by Yu. Luchko for general fractional differential equations. Properties of bi-positivity, complete positivity, dissipativity, and generalized dissipativity in general non-Markovian quantum dynamics are discussed. Examples of a quantum oscillator and two-level quantum system with a general form of nonlocality in time are suggested.
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Campaioli, Francesco, Felix A. Pollock, and Kavan Modi. "Tight, robust, and feasible quantum speed limits for open dynamics." Quantum 3 (August 5, 2019): 168. http://dx.doi.org/10.22331/q-2019-08-05-168.
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Starting from a geometric perspective, we derive a quantum speed limit for arbitrary open quantum evolution, which could be Markovian or non-Markovian, providing a fundamental bound on the time taken for the most general quantum dynamics. Our methods rely on measuring angles and distances between (mixed) states represented as generalized Bloch vectors. We study the properties of our bound and present its form for closed and open evolution, with the latter in both Lindblad form and in terms of a memory kernel. Our speed limit is provably robust under composition and mixing, features that largely improve the effectiveness of quantum speed limits for open evolution of mixed states. We also demonstrate that our bound is easier to compute and measure than other quantum speed limits for open evolution, and that it is tighter than the previous bounds for almost all open processes. Finally, we discuss the usefulness of quantum speed limits and their impact in current research.
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Passarelli, Gianluca, Giulio Filippis, Vittorio Cataudella, and Procolo Lucignano. "May a Dissipative Environment Be Beneficial for Quantum Annealing?" Proceedings 12, no. 1 (July 9, 2019): 26. http://dx.doi.org/10.3390/proceedings2019012026.
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We discuss the quantum annealing of the fully-connected ferromagnetic p-spin model in a dissipative environment at low temperature. This model, in the large p limit, encodes in its ground state the solution to the Grover’s problem of searching in unsorted databases. In the framework of the quantum circuit model, a quantum algorithm is known for this task, providing a quadratic speed-up with respect to its best classical counterpart. This improvement is not recovered in adiabatic quantum computation for an isolated quantum processor. We analyze the same problem in the presence of a low-temperature reservoir, using a Markovian quantum master equation in Lindblad form, and we show that a thermal enhancement is achieved in the presence of a zero temperature environment moderately coupled to the quantum annealer.
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van Driel, Wim, Pieter Mulder, and Françoise Combes. "Modelling the Structure of the Ringed Spiral NGC 4736." International Astronomical Union Colloquium 157 (1996): 253–55. http://dx.doi.org/10.1017/s0252921100049885.
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AbstractWe studied the ringed RSab(r)-type spiral NGC 4736, which has a probably slightly oval disk and a very small bar. We mapped the galaxy in the HI and Hα spectral lines and we obtained long-slit optical spectra. These data were modeled using a 2-D gas dynamical code. The 2-D potential used is axisymmetric in the inner and outer regions and oval (b/a=0.8) at intermediate radii only. The oval component rotates at a pattern speed of 40 km s–1 kpc–1, close to the observed value. Inner and outer rings, like those observed, form at the inner and outer Lindblad resonances, though they co-exist only during a limited time interval in the simulations. The morphology and kinematics of the inner ring and spiral structure as observed in neutral and ionized hydrogen can be well understood in terms of gas dynamical simulations, given the form of the (stellar) potential. What remains to be explained is the origin of the nonaxisymmetric features in the mass distribution defining the potential.
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Szczygielski, Krzysztof, and Robert Alicki. "On Howland time-independent formulation of CP-divisible quantum evolutions." Reviews in Mathematical Physics 32, no. 07 (January 14, 2020): 2050021. http://dx.doi.org/10.1142/s0129055x2050021x.
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We extend Howland time-independent formalism to the case of completely positive and trace preserving dynamics of finite-dimensional open quantum systems governed by periodic, time-dependent Lindbladian in Weak Coupling Limit, expanding our result from previous papers. We propose the Bochner space of periodic, square integrable matrix-valued functions, as well as its tensor product representation, as the generalized space of states within the time-independent formalism. We examine some densely defined operators on this space, together with their Fourier-like expansions and address some problems related to their convergence by employing general results on Banach space-valued Fourier series, such as the generalized Carleson–Hunt theorem. We formulate Markovian dynamics in the generalized space of states by constructing appropriate time-independent Lindbladian in standard (Lindblad–Gorini–Kossakowski–Sudarshan) form, as well as one-parameter semigroup of bounded evolution maps. We show their similarity with Markovian generators and dynamical maps defined on matrix space, i.e. the generator still possesses a standard form (extended by closed perturbation) and the resulting semigroup is also completely positive, trace preserving and a contraction.
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SHAN, CHUAN-JIA, TAO CHEN, JI-BING LIU, WEI-WEN CHENG, TANG-KUN LIU, YAN-XIA HUANG, and HONG LI. "ENTANGLEMENT DYNAMICS OF TWO QUBITS COUPLED TO SQUEEZED DISSIPATIVE ENVIRONMENTS." Modern Physics Letters B 24, no. 26 (October 20, 2010): 2635–45. http://dx.doi.org/10.1142/s0217984910024948.
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By analytically solving the Lindblad form of the master equation, we investigate entanglement dynamics of two qubits coupled via the XY interaction, where each qubit is interacting with an independent reservoir with the squeezing parameters and squeezing angles. In the weak-squeezed reservoir, we show that the entanglement sudden death and entanglement sudden birth will happen for various entangled states. Some initial product states evolve into entangled ones, initially entangled states lose completely or partially their entanglement. The effects of varying the degree of entanglement of the initial states, the spin chain system parameters and different values of the degree of squeezing on the sudden death, revival and birth times are analyzed in detail. We also see that the steady state concurrence appears in the squeezed dissipative environments, which is affected by both the system parameters and the degree of squeezing.
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Combes, Françoise. "Ring and Lens Formation." International Astronomical Union Colloquium 157 (1996): 286–98. http://dx.doi.org/10.1017/s0252921100049927.
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AbstractThe dynamical mechanism to form rings at Lindblad resonances in a barred galaxy is now well-known: due to its dissipative character, the gas is forced in a spiral structure, and experiences torques from the bar potential. Angular momentum is transferred until gas accumulates in the resonant rings. Some problems remain however to account for all observations, such as the very different time-scales for nuclear, inner and outer ring formation, while the three are frequently observed in the same galaxy; the shapes, orientations and thickness of the rings, etc... The adequacy of the present gas dynamical modelizations is discussed.Lenses are secondary components of barred galaxies that could originate from bar evolution. No model until now has met the observational constraints, in particular the sharp edge of the lenses, their strong velocity anisotropy, and their small thickness. We propose here that lenses are the result of partial bar destruction, a necessary step in a feedback cycle of bar formation-destruction, a cycle driven by gas accretion.
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Markmann, Sergej, Martin Franckié, Shovon Pal, David Stark, Mattias Beck, Manfred Fiebig, Giacomo Scalari, and Jérôme Faist. "Two-dimensional spectroscopy on a THz quantum cascade structure." Nanophotonics 10, no. 1 (October 14, 2020): 171–80. http://dx.doi.org/10.1515/nanoph-2020-0369.
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AbstractUnderstanding and controlling the nonlinear optical properties and coherent quantum evolution of complex multilevel systems out of equilibrium is essential for the new semiconductor device generation. In this work, we investigate the nonlinear system properties of an unbiased quantum cascade structure by performing two-dimensional THz spectroscopy. We study the time-resolved coherent quantum evolution after it is driven far from equilibrium by strong THz pulses and demonstrate the existence of multiple nonlinear signals originating from the engineered subbands and find the lifetimes of those states to be in the order of 4–8 ps. Moreover, we observe a coherent population exchange among the first four intersubband levels during the relaxation, which have been confirmed with our simulation. We model the experimental results with a time-resolved density matrix based on the master equation in Lindblad form, including both coherent and incoherent transitions between all density matrix elements. This allows us to replicate qualitatively the experimental observations and provides access to their microscopic origin.
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Lobanov, Y. Y., and E. P. Zhidkov. "Solution of Some Differential Equations of Quantum Physics by the Numerical Functional Integration Method." Computational Methods in Applied Mathematics 3, no. 4 (2003): 560–78. http://dx.doi.org/10.2478/cmam-2003-0035.
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Abstract The application of the numerical functional integration method to the solution of differential equations in quantum physics is discussed. We have developed a method of numerical evaluation of functional integrals in abstract complete separable metric spaces, which proves to have important advantages over the conventional Monte Carlo method of path integration. One of the considered applications is the investigation of open quantum systems (OQS), i.e., systems interacting with their environment. The density operator of OQS satisfies the known Lindblad differential equation. We have obtained the expression for matrix elements of this operator in the form of the double conditional Wiener integral and considered its application to some problems of nuclear physics. Another application is the solution of the Scr¨odinger equation with imaginary time and anticommuting variables for studying many-fermion systems. We have developed a numerical method based on functional integration over ordered subspaces. The binding energies of some nuclei are computed using this method. Comparison of the results with those obtained by other authors and with experimental values is presented.
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Kadir, Hatib Abdul. "Komparasi Munculnya Liberalisme Ekonomi di Indonesia dan Burma." Lembaran Sejarah 13, no. 2 (February 27, 2018): 163. http://dx.doi.org/10.22146/lembaran-sejarah.33541.
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This paper explores the genealogy of the birth of the economic growth system in Indonesia and Burma. First, it is a major transformation that occurs in both countries in the form of commodification of land, labor and money. And the formation of pluralism in the colonial society. Second, the transformation of capitalism in Burma that enters through the system of bureaucratic governance, education and social order in rural communities. Third, the study of comparative application of economic liberalism in Indonesia and Burma and its social effects. And the emergence of middle class society who came from outside of original community. The author uses Karl Polanyi’s approach for looking at the social effects of economic liberalism, based on the transformation of three things: the privatization of the land, the commodification of labor and the emergence of the system of money and debt. This comparison primarily uses extensive data from J.S Furnivall in view of the application of an economic liberalism system which is then enriched with studies from other economic historians, such as Thomas Lindblad, Anne Booth and the study of political economics, Richard Robison.
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Davidović, Dragomir. "Completely Positive, Simple, and Possibly Highly Accurate Approximation of the Redfield Equation." Quantum 4 (September 21, 2020): 326. http://dx.doi.org/10.22331/q-2020-09-21-326.
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Here we present a Lindblad master equation that approximates the Redfield equation, a well known master equation derived from first principles, without significantly compromising the range of applicability of the Redfield equation. Instead of full-scale coarse-graining, this approximation only truncates terms in the Redfield equation that average out over a time-scale typical of the quantum system. The first step in this approximation is to properly renormalize the system Hamiltonian, to symmetrize the gains and losses of the state due to the environmental coupling. In the second step, we swap out an arithmetic mean of the spectral density with a geometric one, in these gains and losses, thereby restoring complete positivity. This completely positive approximation, GAME (geometric-arithmetic master equation), is adaptable between its time-independent, time-dependent, and Floquet form. In the exactly solvable, three-level, Jaynes-Cummings model, we find that the error of the approximate state is almost an order of magnitude lower than that obtained by solving the coarse-grained stochastic master equation. As a test-bed, we use a ferromagnetic Heisenberg spin-chain with long-range dipole-dipole coupling between up to 25-spins, and study the differences between various master equations. We find that GAME has the highest accuracy per computational resource.
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Guo, Hui, Jin-Ming Liu, Cheng-Jie Zhang, and C. H. Oh. "Quantum discord of a three-qubit W-class state in noisy environments." Quantum Information and Computation 12, no. 7&8 (July 2012): 677–92. http://dx.doi.org/10.26421/qic12.7-8-12.
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We study the dynamics of the pairwise quantum discord (QD), classical correlation (CC), and entanglement of formation (EOF) for the three-qubit W-class state |W>_{123}=\frac 12(|100>_{123}+|010>_{123}+\sqrt{2}|001>_{123}) under the influence of various Markovian noises by analytically solving the master equation in the Lindblad form. Through numerical analysis, we find that EOF decreases asymptotically to zero with time for the dephasing noise, but it undergoes sudden death for the bit-flip noise, the isotropic noise, as well as the dissipative and noisy environments. Moreover, QD decays to zero in an asymptotical way for all the noises we investigated. Thus, when the W-class state |W>_{123} is subject to the above Markovian noises, QD is more robust than EOF against decoherence excluding the phase-flip noise, implying that QD is more useful than entanglement to characterize the quantum correlation. We also find a remarkable character for the CC in the presence of the phase-flip noise, i.e., CC displays the behavior of sudden transition and then keeps constant permanently, but the corresponding QD just exhibits a very small sudden change. Furthermore, we verify the monogamic relation between the pairwise QD and EOF of the W-class state.
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Kim, Woong-Tae, Woo-Young Seo, and Yonghwi Kim. "Formation of nuclear rings of barred galaxies and star formation therein." Proceedings of the International Astronomical Union 9, S303 (October 2013): 43–53. http://dx.doi.org/10.1017/s174392131400012x.
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AbstractBarred galaxies contain substructures such as a pair of dust lanes and nuclear rings, with the latter being sites of intense star formation. We study the substructure formation as well as star formation in nuclear rings using numerical simulations. We find that nuclear rings form not by the Lindblad resonances, as previously thought, but by the centrifugal barrier that inflowing gas along dust lanes cannot overcome. This predicts a smaller ring in a more strongly barred galaxy, consistent with observations. Star formation rate (SFR) in a nuclear ring is determined primarily by the mass inflow rate to the ring. In our models, the SFR typically shows a short strong burst associated with the rapid gas infall and stays very small for the rest of the evolution. When the SFR is low, ages of young star clusters exhibit an azimuthal gradient along the ring since star formation takes place mostly near the contact points between the dust lanes and the nuclear ring. When the SFR is large, on the other hand, star formation is widely distributed throughout the whole length of the ring, with no apparent age gradient of star clusters. Since observed ring star formation appears long-lived with episodic bursts, our results suggest that the bar region should be replenished continually with fresh gas from outside.
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Binney, James. "The shearing sheet and swing amplification revisited." Monthly Notices of the Royal Astronomical Society 496, no. 1 (May 30, 2020): 767–83. http://dx.doi.org/10.1093/mnras/staa1485.
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ABSTRACT The principal results of the classic analysis of the shearing sheet and swing amplification by Julian and Toomre (JT) are re-derived in a more accessible way and used to gain a better quantitative understanding of the dynamics of stellar discs. The axisymmetric limit of the shearing sheet is derived and used to re-derive Kalnajs’ 1965 dispersion relation and Toomre’s 1964 stability criterion for axisymmetric disturbances. Using the shearing sheet to revisit Toomre’s important 1969 paper on the group velocity implied by the Lin–Shu–Kalnajs (LSK) dispersion relation, we discover that two wavepackets emerge inside corotation: one each side of the inner Lindblad resonance. An extended form of the JT equation is used to investigate the impact of there being a deficit or surplus of stars in a narrow range of angular momenta. Swing amplification of leading waves introduced by such a groove gives rise to transient trailing spirals that extend further in radius and live longer at smaller azimuthal wavenumbers. Although the LSK dispersion relation provides useful interpretations of wavepackets, the shearing sheet highlights the limitations of the LSK approach to disc dynamics. Disturbances do not avoid an annulus around corotation, as the LSK dispersion relation implies. While disturbances of the shearing sheet have a limited life in real space, they live on much longer in velocity space, which Gaia allows us to probe extensively. c++ code is provided to facilitate applications of winding spiral waves.
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Chia, A., T. Paterek, and L. C. Kwek. "Hitting statistics from quantum jumps." Quantum 1 (July 21, 2017): 19. http://dx.doi.org/10.22331/q-2017-07-21-19.
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We define the hitting time for a model of continuous-time open quantum walks in terms of quantum jumps. Our starting point is a master equation in Lindblad form, which can be taken as the quantum analogue of the rate equation for a classical continuous-time Markov chain. The quantum jump method is well known in the quantum optics community and has also been applied to simulate open quantum walks in discrete time. This method however, is well-suited to continuous-time problems. It is shown here that a continuous-time hitting problem is amenable to analysis via quantum jumps: The hitting time can be defined as the time of the first jump. Using this fact, we derive the distribution of hitting times and explicit exressions for its statistical moments. Simple examples are considered to illustrate the final results. We then show that the hitting statistics obtained via quantum jumps is consistent with a previous definition for a measured walk in discrete time [Phys. Rev. A 73, 032341 (2006)] (when generalised to allow for non-unitary evolution and in the limit of small time steps). A caveat of the quantum-jump approach is that it relies on the final state (the state which we want to hit) to share only incoherent edges with other vertices in the graph. We propose a simple remedy to restore the applicability of quantum jumps when this is not the case and show that the hitting-time statistics will again converge to that obtained from the measured discrete walk in appropriate limits.
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Ramos, P., T. Antoja, and F. Figueras. "Riding the kinematic waves in the Milky Way disk with Gaia." Astronomy & Astrophysics 619 (November 2018): A72. http://dx.doi.org/10.1051/0004-6361/201833494.
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Context. Gaia DR2 has delivered full-sky six-dimensional measurements for millions of stars, and the quest to understand the dynamics of our Galaxy has entered a new phase. Aims. Our aim is to reveal and characterise the kinematic substructure of the different Galactic neighbourhoods, to form a picture of their spatial evolution that can be used to infer the Galactic potential, its evolution, and its components. Methods. We take approximately 5 million stars in the Galactic disk from the Gaia DR2 catalogue and build the velocity distribution in different Galactic neighbourhoods distributed along 5 kpc in Galactic radius and azimuth. We decompose their distribution of stars in the VR–Vϕ plane with the wavelet transformation and asses the statistical significance of the structures found. Results. We detect distinct kinematic substructures (arches and more rounded groups) that diminish their azimuthal velocity as a function of Galactic radius in a continuous way, connecting volumes up to 3 kpc apart in some cases. The average rate of decrease is ∼23 km s−1 kpc−1. In azimuth, the variations are much smaller. We also observe different behaviours: some approximately conserve their vertical angular momentum with radius (e.g. Hercules), while others seem to have nearly constant kinetic energy (e.g. Sirius). These two trends are consistent with the approximate predictions of resonances and phase mixing, respectively. Besides, the overall spatial evolution of Hercules is consistent with being related to the outer Lindblad resonance of the Galactic bar. In addition, we detect new kinematic structures that only appear at either inner or outer Galactic radius, different from the solar neighbourhood. Conclusions. The strong and distinct variation observed for each kinematic substructure with position in the Galaxy, along with the characterisation of extrasolar moving groups, will allow to better model the dynamical processes affecting the velocity distributions.
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Fragkoudi, F., R. J. J. Grand, R. Pakmor, G. Blázquez-Calero, I. Gargiulo, F. Gomez, F. Marinacci, et al. "Chemodynamics of barred galaxies in cosmological simulations: On the Milky Way’s quiescent merger history and in-situ bulge." Monthly Notices of the Royal Astronomical Society 494, no. 4 (April 29, 2020): 5936–60. http://dx.doi.org/10.1093/mnras/staa1104.
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ABSTRACT We explore the chemodynamical properties of a sample of barred galaxies in the Auriga magnetohydrodynamical cosmological zoom-in simulations, which form boxy/peanut (b/p) bulges, and compare these to the Milky Way (MW). We show that the Auriga galaxies which best reproduce the chemodynamical properties of stellar populations in the MW bulge have quiescent merger histories since redshift z ∼ 3.5: their last major merger occurs at $t_{\rm lookback}\gt 12\, \rm Gyr$, while subsequent mergers have a stellar mass ratio of ≤1:20, suggesting an upper limit of a few per cent for the mass ratio of the recently proposed Gaia Sausage/Enceladus merger. These Auriga MW-analogues have a negligible fraction of ex-situ stars in the b/p region ($\lt 1{{\ \rm per\ cent}}$), with flattened, thick disc-like metal-poor stellar populations. The average fraction of ex-situ stars in the central regions of all Auriga galaxies with b/p’s is 3 per cent – significantly lower than in those which do not host a b/p or a bar. While the central regions of these barred galaxies contain the oldest populations, they also have stars younger than 5 Gyr (>30 per cent) and exhibit X-shaped age and abundance distributions. Examining the discs in our sample, we find that in some cases a star-forming ring forms around the bar, which alters the metallicity of the inner regions of the galaxy. Further out in the disc, bar-induced resonances lead to metal-rich ridges in the Vϕ − r plane – the longest of which is due to the Outer Lindblad Resonance. Our results suggest the Milky Way has an uncommonly quiet merger history, which leads to an essentially in-situ bulge, and highlight the significant effects the bar can have on the surrounding disc.
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Jung, E., M. R. Hwang, D. Park, and S. Tamaryan. "Three-party entanglement in tripartite teleportation scheme through noisy channels." Quantum Information and Computation 10, no. 5&6 (May 2010): 377–97. http://dx.doi.org/10.26421/qic10.5-6-2.
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In this paper we have tried to interpret the physical role of three-tangle and $\pi$-tangle in real physical information processes. For the model calculation we adopt the tripartite teleportation scheme through various noisy channels. The three parties consist of sender, accomplice and receiver. It is shown that the $\pi$-tangles for the X- and Z-noisy channels vanish at the limit $\kappa t \rightarrow \infty$, where $\kappa t$ is a decoherence parameter introduced in the master equation in the Lindblad form. At this limit the maximum fidelity of the receiver's state reduces to the classical limit $2/3$. However, this nice feature is not maintained for the Y- and isotropy-noise channels. For the Y-noise channel the $\pi$-tangle vanishes when $0.61 \leq \kappa t$. At $\kappa t = 0.61$ the maximum fidelity becomes $0.57$, which is much less than the classical limit. Similar phenomenon occurs for the isotropic noise channel. We also compute analytically the three-tangles for the X- and Z-noise channels. The remarkable fact is that the three-tangle for the Z-noise channel coincides exactly with the corresponding $\pi$-tangle. In the X-noise channel the three-tangle vanishes when $0.10 \leq \kappa t$. At $\kappa t = 0.10$ the fidelity of the receiver's state can reduce to the classical limit provided that the accomplice performs the measurement appropriately. However, the maximum fidelity becomes $8/9$, which is much larger than the classical limit. Since the Y- and isotropy-noise channels are rank-$8$ mixed states, their three-tangles are not computed explicitly in this paper. Instead, their upper bounds are derived by making use of the analytic formulas of the three-tangle for other noisy channels. Our analysis strongly suggests that different tripartite entanglement measure is needed whose value is between three-tangle and $\pi$-tangle.
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Brož, M., O. Chrenko, D. Nesvorný, and M. Lambrechts. "Dynamics of multiple protoplanets embedded in gas and pebble discs and its dependence on Σ and ν parameters." Astronomy & Astrophysics 620 (December 2018): A157. http://dx.doi.org/10.1051/0004-6361/201833855.
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Protoplanets of super-Earth size may get trapped in convergence zones for planetary migration and form gas giants there. These growing planets undergo accretion heating, which triggers a hot-trail effect that can reverse migration directions, increase planetary eccentricities, and prevent resonant captures of migrating planets. In this work, we study populations of embryos that are accreting pebbles under different conditions, by changing the surface density, viscosity, pebble flux, mass, and the number of protoplanets. For modelling, we used the FARGO-THORIN two-dimensional (2D) hydrocode, which incorporates a pebble disc as a second pressure-less fluid, the coupling between the gas and pebbles, and the flux-limited diffusion approximation for radiative transfer. We find that massive embryos embedded in a disc with high surface density (Σ = 990 g cm−2 at 5.2 au) undergo numerous “unsuccessful” two-body encounters that do not lead to a merger. Only when a third protoplanet arrives in the convergence zone do three-body encounters lead to mergers. For a low-viscosity disc (ν = 5 × 1013 cm2 s−1), a massive co-orbital is a possible outcome, for which a pebble isolation develops and the co-orbital is further stabilised. For more massive protoplanets (5 M⊕), the convergence radius is located further out, in the ice-giant zone. After a series of encounters, there is an evolution driven by a dynamical torque of a tadpole region, which is systematically repeated several times until the co-orbital configuration is disrupted and planets merge. This may be a way to solve the problem that co-orbitals often form in simulations but they are not observed in nature. In contrast, the joint evolution of 120 low-mass protoplanets (0.1 M⊕) reveals completely different dynamics. The evolution is no longer smooth, but rather a random walk. This is because the spiral arms, developed in the gas disc due to Lindblad resonances, overlap with each other and affect not only a single protoplanet but several in the surrounding area. Our hydrodynamical simulations may have important implications for N-body simulations of planetary migration that use simplified torque prescriptions and are thus unable to capture protoplanet dynamics in its full glory.
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Cimerman, Nicolas P., Wilhelm Kley, and Rolf Kuiper. "Formation of a planetary Laplace resonance through migration in an eccentric disk." Astronomy & Astrophysics 618 (October 2018): A169. http://dx.doi.org/10.1051/0004-6361/201833591.
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Context. Orbital mean motion resonances in planetary systems originate from dissipative processes in disk-planet interactions that lead to orbital migration. In multi-planet systems that host giant planets, the perturbation of the protoplanetary disk strongly affects the migration of companion planets. Aims. By studying the well-characterized resonant planetary system around GJ 876 we aim to explore which effects shape disk-driven migration in such a multi-planet system to form resonant chains. Methods. We modelled the orbital migration of three planets embedded in a protoplanetary disk using two-dimensional locally isothermal hydrodynamical simulations. In order to explore the effect of several disk characteristics, we performed a parameter study by varying the disk thickness, α viscosity, mass as well as the initial position of the planets. Moreover, we have carefully analysed and compared simulations with various boundary conditions at the disk’s inner rim. Results. We find that due to the high masses of the giant planets in this system, substantial eccentricity can be excited in the disk. This results in large variations of the torque acting on the outer lower mass planet, which we attribute to a shift of Lindblad and corotation resonances as it approaches the eccentric gap that the giants create. Depending on disk parameters, the migration of the outer planet can be stopped at the gap edge in a non-resonant state. In other models, the outer planet is able to open a partial gap and to circularize the disk again, later entering a 2:1 resonance with the most massive planet in the system to complete the observed 4:2:1 Laplace resonance. Conclusions. Disk-mediated interactions between planets due to spiral waves and excitation of disk eccentricity by massive planets cause deviations from smooth inward migration of exterior lower mass planets. Self-consistent modelling of the disk-driven migration of multi-planet systems is thus mandatory. Constraints can be placed on the properties of the disk during the migration phase, based on the observed resonant state of the system. Our results are compatible with a late migration of the outermost planet into the resonant chain, when the giant planet pair already is in resonance.
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Haywood, M., O. Snaith, M. D. Lehnert, P. Di Matteo, and S. Khoperskov. "Revisiting long-standing puzzles of the Milky Way: the Sun and its vicinity as typical outer disk chemical evolution." Astronomy & Astrophysics 625 (May 2019): A105. http://dx.doi.org/10.1051/0004-6361/201834155.
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We present a scenario of the chemical enrichment of the solar neighborhood that solves the G-dwarf problem by taking into account constraints on a larger scale. We argue that the Milky Way disk within 10 kpc has been enriched to solar metallicity by a massive stellar population: the thick disk, which itself formed from a massive turbulent gaseous disk. While the inner disk, R ≲ 6 kpc, continued this enrichment after a quenching phase (7−10 Gyr), at larger distances radial flows of gas diluted the metals left by the thick disk formation at a time we estimate to be 7−8 Gyr ago, thus partitioning the disk into an inner and outer region characterized by different chemical evolutions. The key new consideration is that the pre-enrichment provided by the thick disk is not related to the mass fraction of this stellar population at the solar radius, as is classically assumed in inside-out scenarios, but is actually related to the formation of the entire massive thick disk, due to the vigorous gas phase mixing that occurred during its formation. Hence, the fact that this population represents only 15−25% of the local stellar surface density today, or 5−10% of the local volume density, is irrelevant for “solving” the G-dwarf problem. The only condition for this scenario to work is that the thick disk was formed from a turbulent gaseous disk that permitted a homogeneous – not radially dependent – distribution of metals, allowing the solar ring to be enriched to solar metallicity. At the solar radius, the gas flowing from the outer disk combined with the solar metallicity gas left over from thick disk formation, providing the fuel necessary to form the thin disk at the correct metallicity to solve the G-dwarf problem. Chemical evolution at R > 6 kpc, and in particular beyond the solar radius, can be reproduced with the same scheme. We suggest that the dilution, occurring at the fringe of the thick disk, was possibly triggered by the formation of the bar and the establishment of the outer Lindblad resonance (OLR), enabling the inflow of metal poorer gas from the outer disk to R ∼ 6 kpc, presumably the position of the OLR at this epoch, and at the same time isolating the inner disk from external influence. These results imply that the local metallicity distribution is not connected to the gas accretion history of the Milky Way. Finally, we argue that the Sun is the result of the evolution typical of stars in the disk beyond ∼6 kpc (i.e., also undergoing dilution), and has none of the characteristics of inner disk stars.
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Torres, Juan Mauricio. "Closed-form solution of Lindblad master equations without gain." Physical Review A 89, no. 5 (May 30, 2014). http://dx.doi.org/10.1103/physreva.89.052133.
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Akamatsu, Yukinao. "Heavy quark master equations in the Lindblad form at high temperatures." Physical Review D 91, no. 5 (March 3, 2015). http://dx.doi.org/10.1103/physrevd.91.056002.
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McCauley, Gavin, Benjamin Cruikshank, Denys I. Bondar, and Kurt Jacobs. "Accurate Lindblad-form master equation for weakly damped quantum systems across all regimes." npj Quantum Information 6, no. 1 (August 27, 2020). http://dx.doi.org/10.1038/s41534-020-00299-6.
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Polyachenko, E. V., P. Berczik, and A. Just. "Bar Formation in Milky Way type Galaxies." Open Astronomy 25, no. 4 (January 1, 2016). http://dx.doi.org/10.1515/astro-2017-0260.
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AbstractMany barred galaxies, possibly including the Milky Way, have cusps in their centers. There is a widespread belief, however, that the usual bar instability, which occurs in bulgeless galaxy models, is impossible for cuspy models because of the presence of the inner Lindblad resonance for any pattern speed. At the same time, there is numerical evidence that the bar instability can form a bar. We analyze this discrepancy by performing accurate and diverse