We present a derivation of the Lindblad equation -an important tool for the treatment of nonunitary evolutions -that is accessible to undergraduate students in physics or mathematics with a basic background on quantum mechanics. We consider a specific case, corresponding to a very simple situation, where a primary system interacts with a bath of harmonic oscillators at zero temperature, with an interaction Hamiltonian that resembles the Jaynes-Cummings formato We start with the Born-Markov equation and, tracing out the bath degrees of freedom, we obtain an equation in the Lindblad formo The specific situation is very instructive, for it makes it easy to realize that the Lindblads represent the effect on the main system caused by the interaction with the bath, and that the Markov approximation is a fundamental condition for the emergence of the Lindbladian operator. The formal derivation of the Lindblad equation for a more general case requires the use of quantum dynamical semi-groups and broader considerations regarding the environment and temperature than we have considered in the particular case treated here.
We present a derivation of the Lindblad equation -an important tool for the treatment of nonunitary evolutions -that is accessible to undergraduate students in physics or mathematics with a basic background on quantum mechanics. We consider a specific case, corresponding to a very simple situation, where a primary system interacts with a bath of harmonic oscillators at zero temperature, with an interaction Hamiltonian that resembles the Jaynes-Cummings formato We start with the Born-Markov equation and, tracing out the bath degrees of freedom, we obtain an equation in the Lindblad formo The specific situation is very instructive, for it makes it easy to realize that the Lindblads represent the effect on the main system caused by the interaction with the bath, and that the Markov approximation is a fundamental condition for the emergence of the Lindbladian operator. The formal derivation of the Lindblad equation for a more general case requires the use of quantum dynamical semi-groups and broader considerations regarding the environment and temperature than we have considered in the particular case treated here.
The Lindblad theory of open quantum systems has been successfully applied in various fields ranging from quantum optics, condensed matter physics, and quantum thermodynamics to quantum chemistry. Nevertheless, there are situations where we can compare its predictions to those of exact methods or numerically exact results and find discrepancies between them. One example is the case where a single harmonic oscillator is coupled to a thermal bath of independent harmonic oscillators. In this case, for certain parameters, the exact master equation describes equilibration of the system to a thermal state with a temperature different from the one predicted by the Lindblad master equation. The Lindblad theory, based on a weak coupling between the system and environment as well as the Born-Markov approximation, predicts that the system will evolve to a thermal state whose temperature is equal to the initial temperature of the bath. This result is a consequence of the Born approximation, which supposes that throughout the evolution, the bath only fluctuates around its initial equilibrium state. In this master thesis we study the validity of the Born approximation as well as its consistency with other approximations. By starting from a Hamiltonian for the total system we show that, if the the Born approximation is not invoked, the dynamics of the system is determined by a hierarchy of equations for matrices which can be used to reconstruct the reduced density matrix of the system. This hierarchy approach is fundamentally different from the standard approach, which describes the system dynamics with a weak-coupling master equation, because the hierarchy does not restrict the amount of entanglement possible between the system and the environment. Furthermore, we show how invoking the Born approximation reduces to the usual result and how a generalised ansatz, which we refer to as a generalised Born approximation, reduces the hierarchy to a master equation where the bath temperature is an additional degree of freedom. Finally, we analyse the conservation of the energy of the total system and its relationship to the Markov and secular approximations, and discuss whether imposing it as a condition can fix the time-dependent temperature.
The effective dynamics of a system interacting with a bath or environment is presented in two ways, (1) the (LGKS) replacement of the von Neuman equation for the density matrix and (2) the Feynman-Vernon path-integral derivation, by integrating out the bath degree of freedom, to arrive at a system's density matrix. In this paper, I connect the two methods by deriving a Lindbladian in a mechanical example, a point particle interacting with a bath of harmonic oscillators, previously considered by Feynman and Vernon (FV) and expounded on later by Caldeira and Leggett (CL). But the (FV)/(CL) results only in non-Markov effect, memory terms from the bath interaction. To derive a Lindbladian, I changed the interaction term they considered to take into account the point particle interacting with the bath harmonic oscillators to something more realistic. From the resulting path-integral expression of the system's propagator for the density matrix, the Lindbladian and non-Markov terms are read for this simple problem. I also point out the causes of these terms, the Markov Lindbladian from the very local interaction of the point particle with the classical solutions of the harmonic oscillator and the non-Markov term from the global interaction of the point particle with the fluctuation of the classical solutions.
The celebrated Lindblad equation governs the nonunitary time evolution of density operators used in the description of open quantum systems. It is usually derived from the von Neumann equation for a large system, at given physical conditions, when a small subsystem is explicitly singled out and the rest of the system acts as an environment whose degrees of freedom are traced out. In the specific case of a subsystem with variable particle number, the equilibrium density operator is given by the well-known grand canonical Gibbs state. Consequently, solving the Lindblad equation in this case should automatically yield, without any additional assumptions, the corresponding density operator in the limiting case of statistical equilibrium. Current studies of the Lindblad equation with varying particle number assume, however, the grand canonical Gibbs state a priori: the chemical potential is externally imposed rather than derived from first principles, and hence the corresponding density operator is not obtained as a natural solution of the equation. In this work, we investigate the compatibility of grand canonical statistical mechanics with the derivation of the Lindblad equation. We propose an alternative and complementary approach to the current literature that consists in using a generalized system Hamiltonian which includes a term μN. In a previous paper, this empirically well-known term has been formally derived from the von Neumann equation for the specific case of equilibrium. Including μN in the system Hamiltonian leads to a modified Lindblad equation which yields the grand canonical state as a natural solution, meaning that all the quantities involved are obtained from the physics of the system without any external assumptions.
Away from equilibrium, the properties of open quantum systems depend on the details of their environment. A microscopic derivation of a master equation (ME) is therefore crucial. Of particular interest are Lindblad-type equations, not only because they provide the most general class of Markovian MEs, but also since they are the starting point for efficient quantum trajectory simulations. Lindblad-type MEs are commonly derived from the Born-Markov-Redfield equation via a rotating-wave approximation (RWA). However the RWA is valid only for ultraweak system-bath coupling and often fails to accurately describe nonequilibrium processes. Here we derive an alternative Lindbladian approximation to the Redfield equation, which does not rely on ultraweak system-bath coupling. Applying it to an extended Hubbard model coupled to Ohmic baths, we show that, especially away from equilibrium, it provides a good approximation in large parameter regimes where the RWA fails.
In this paper we derive an extra class of non-Markovian master equations where the system state is written as a sum of auxiliary matrixes whose evolution involve Lindblad contributions with local coupling between all of them, resembling the structure of a classical rate equation. The system dynamics may develop strong nonlocal effects such as the dependence of the stationary properties with the system initialization. These equations are derived from alternative microscopic interactions, such as complex environments described in a generalized Born-Markov approximation and tripartite system-environment interactions, where extra unobserved degrees of freedom mediates the entanglement between the system and a Markovian reservoir. Conditions that guarantee the completely positive condition of the solution map are found. Quantum stochastic processes that recover the system dynamics in average are formulated. We exemplify our results by analyzing the dynamical action of nontrivial structured dephasing and depolarizing reservoirs over a single qubit.
The time evolution of an open quantum system is governed by the Gorini-Kossakowski-Sudarshan-Lindlad equation for the reduced density operator of the system. This operator is obtained from the full density operator of the composite system involving the system itself, the bath, and the interactions between them, by performing a partial trace over the bath degrees of freedom. The entanglement between the system and the bath leads to a generalized Liouville evolution that involves, amongst other things, dissipation and decoherence of the system. In a similar fashion, the time evolution of a physical observable in a classically constrained dynamical system is governed by a generalization of the Liouville equation, in which the usual Poisson bracket is replaced by the so-called Dirac bracket. The generalization takes into account the reduction in the phase space of the system because of constraints which arise either because they are introduced by hand, or because of singular Lagrangians. We derive an intriguing, but precise classical-quantum correspondence between the aforementioned situations which connects the Lindblad operators to the constraints. The correspondence is illustrated in the ubiquitous example of coupled simple harmonic oscillators studied earlier in various contexts like quantum optics, dissipative quantum systems, Brownian motion, and the area law for the entropy of black holes.