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.
The so-called Lindblad equation, a typical master equation describing the dissipative quantum dynamics, is shown to be solvable for finite-level systems in a compact form without resort to writing it down as a set of equations among matrix elements. The solution is then naturally given in an operator form, known as the Kraus representation. Following a few simple examples, the general applicability of the method is clarified.
Using the Bloch representation of an N-dimensional quantum system and immediate results from quantum stochastic calculus, we establish a closed formula for the Bloch vector, hence also for the density operator, of a quantum system following a Lindblad evolution with selfadjoint Lindblad operators.
Markovian dynamics of open quantum systems are described by the L-GKS equation, known also as the Lindblad equation. The equation is expressed by means of left and right matrix multiplications. This formulation hampers numerical implementations. Representing the dynamics by a matrix-vector notation overcomes this problem. We review three approaches to obtain such a representation. The methods are demonstrated for a driven two-level system subject to spontaneous emission.
We present a general scheme to map correlated nonequilibrium quantum impurity problems onto an auxiliary open quantum system of small size. The infinite fermionic reservoirs of the original system are thereby replaced by a small number $N_B$ of noninteracting auxiliary bath sites whose dynamics is described by a Lindblad equation. Due to the presence of the intermediate bath sites, the overall dynamics acting on the impurity site is non-Markovian. With the help of an optimization scheme for the auxiliary Lindblad parameters, an accurate mapping is achieved, which becomes exponentially exact up…
In the framework of the Lindblad theory for open quantum systems, a master equation for the quantum harmonic oscillator interacting with a dissipative environment, in particular with a thermal bath, is derived for the case when the interaction is based on deformed algebra. The equations of motion for observables strongly depend on the deformation function. The expectation values of the number operator and squared number operator are calculated in the limit of a small deformation parameter for the case of zero temperature of the thermal bath. The steady state solution of the equation for the de…
We investigate the critical behavior, both in space and time, of the wetting interface within the coexistence region around the first-order phase transition of a fully-connected quantum Ising model in slab geometry. For that, we employ the Lindblad master equation formalism in which temperature is inherited by the coupling to a dissipative bath, rather than being a functional parameter as in the conventional Cahn's free energy. Lindblad's approach gives not only access to the dissipative dynamics and steady-state configuration of the quantum wetting interface throughout the whole phase diagram…
Given an quantum dynamical semigroup expressed as an exponential superoperator acting on a space of N-dimensional density operators, eigenvalue methods are presented by which canonical Kraus and Lindblad operator sum representations can be computed. These methods provide a mathematical basis on which to develop novel algorithms for quantum process tomography, the statistical estimation of superoperators and their generators, from a wide variety of experimental data. Theoretical arguments and numerical simulations are presented which imply that these algorithms will be quite robust in the prese…
In the framework of the Lindblad theory for open quantum systems the damping of the harmonic oscillator is studied. 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 and Heisenberg 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 the majority of these…
We consider the natural generalization of the Schrödinger equation to Markovian open system dynamics: the so-called the Lindblad equation. We give a quantum algorithm for simulating the evolution of an $n$-qubit system for time $t$ within precision $ε$. If the Lindbladian consists of $\mathrm{poly}(n)$ operators that can each be expressed as a linear combination of $\mathrm{poly}(n)$ tensor products of Pauli operators then the gate cost of our algorithm is $O(t\, \mathrm{polylog}(t/ε)\mathrm{poly}(n))$. We also obtain similar bounds for the cases where the Lindbladian consists of local operato…
The quantum dynamics of a damped and forced harmonic oscillator described by a Lindblad master equation is analyzed. The master equation is converted into a matrix-vector representation and the resulting non-Hermitian Schrödinger equation is solved by Lie-algebraic techniques allowing the construction of the general solution for the density operator.
We perform the real-time lattice simulation of an open quantum system, which is based on the Schwinger-Keldysh path integral representation of the Lindblad formalism. Although the real-time simulation generally suffers from the sign problem, there exist a few exceptional cases. We focus on a sign-problem-free system of a non-relativistic spinless fermion and analyze time evolution under driving and dissipation.
In the preceding paper arXiv:0802.3252 [quant-ph] we treated a model given by a master equation with generalized Lindblad form, and examined the algebraic structure related to some Lie algebras and constructed an approximate solution. In this paper we apply a unitary transformation by the squeezing operator to the master equation. Then the generalized Lindblad form is tranformed to the usual Lindblad one, while the (original) Hamiltonian is tranformed to somewhat complicated one. As a result we have two different representations based on the Lie algebra su(1,1). We examine new algebraic struct…
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.
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 no…
We investigate dissipative extensions of the Su-Schrieffer-Heeger model with regard to different approaches of modeling dissipation. In doing so, we use two distinct frameworks to describe the gain and loss of particles, one uses Lindblad operators within the scope of Lindblad master equations, the other uses complex potentials as an effective description of dissipation. The reservoirs are chosen in such a way that the non-Hermitian complex potentials are $\mathcal{PT}$-symmetric. From the effective theory we extract a state which has similar properties as the non-equilibrium steady state foll…
We derive an approximate Gaussian solution of the Lindblad equation in the semiclassical limit, given a general Hamiltonian and linear coupling with the environment. The theory is carried out in the chord representation and describes the evolved quantum characteristic function, which gives direct access to the Wigner function and the position representation of the density operator by Fourier transforms. The propagation is based on a system of non-linear equations taking place in a double phase space, which coincides with Heller's theory of unitary evolution of Gaussian wave packets when the Li…
By exploiting the peculiarities of a recently introduced formalism for describing open quantum systems (the Parametric Representation with Environmental Coherent States) we derive an equation of motion for the reduced density operator of an open quantum system that has the same structure of the celebrated Gorini-Kossakowski-Sudarshan-Lindblad equation, but holds regardless of markovianity being assumed. The operators in our result have explicit expressions in terms of the Hamiltonian describing the interactions with the environment, and can be computed once a specific model is considered. We f…
The concept of weak invariants is examined in the thermodynamic context. Discussions are made about the temporally-local equilibrium states, corrections to them, and isoenergetic processes based on the quantum master equations of the Lindblad type that admit time-dependent Hamiltonians as weak invariants. The method for determining the correction presented here may be thought of as a quantum-mechanical analog of the Chapman-Enskog expansion in nonequilibrium classical statistical mechanics. Then, the theory is applied to the time-dependent harmonic oscillator as a simple example, and the power…
We study the properties of the Lindbladian quantum mechanical evolution of quarkonia with non-Abelian charges (color-singlet and octet) in the quark-gluon plasma. We confirm that heavy quark recoils in the Lindblad equation correctly thermalize quarkonium colorful states within statistical errors from the simulation method. We also demonstrate that the Lindblad equation in the dipole limit can provide an efficient alternative method, which is applicable to a finite time evolution before thermalization and dramatically reduces the numerical cost. Our findings will serve as a foundation for larg…
The Lindblad equation governs general markovian evolution of the density operator in an open quantum system. An expression for the rate of change of the Wigner function as a sum of integrals is one of the forms of the Weyl representation for this equation. The semiclassical description of the Wigner function in terms of chords, each with its classically defined amplitude and phase, is thus inserted in the integrals, which leads to an explicit differential equation for the Wigner function. All the Lindblad operators are assumed to be represented by smooth phase space functions corresponding to …
Within the framework of the Lindblad master equation, we propose a general methodology to describe the effects of the environment on a system in dilute gas phase. The phenomenological parameters characterizing the transitions between rovibrational states of the system induced by collisions can be extracted from experimental transition kinetic constants, relying on Energy Gap fitting laws. As the availability of this kind of experimental data can be limited, the present work relied on experimental line broadening coefficients, however still using Energy Gap fitting laws. The 3 $μ$m infrared spe…
The operation of autonomous finite-time quantum heat engines rely on the existence of a stable limit cycle in which the dynamics becomes periodic. The two main questions that naturally arise are therefore whether such a limit cycle will eventually be reached and, once it has, what is the state of the system within the limit cycle. In this paper we show that the application of Floquet's theory to Lindblad dynamics offers clear answers to both questions. By moving to a generalized rotating frame, we show that it is possible to identify a single object, the Floquet Liouvillian, which encompasses …
We propose a new representation for several quantum master equations in so-called quasithermodynamic form. This representation (when it exists) let one to write down dynamical equations both for diagonal and non-diagonal elements of density matrix of the quantum system of interest in unified form by means of nonequilibrium potential ("entropy") that is a certain quadratic function depending on all variables describing the state. We prove that above representation exists for the general Pauli master equation and for the Lindblad master equation ( at least in simple cases ) as well. We discuss a…
We propose a complete treatment of a local in time dynamics of open quantum systems. In this approach Markovian evolution turns out to be a special case of a general non-Markovian one. We provide a general representation of the local generator which generalizes well known Lindblad representation for the Markovian dynamics. It shows that the structure of non-Markovian generators is highly intricate and the problem of their classification is still open. Simple examples illustrate our approach.
We derive an effective equation of motion within the steady-state subspace of a large family of Markovian open systems (i.e., Lindbladians) due to perturbations of their Hamiltonians and system-bath couplings. Under mild and realistic conditions, competing dissipative processes destructively interfere without the need for fine-tuning and produce no dissipation within the steady-state subspace. In quantum error-correction, these effects imply that continuously error-correcting Lindbladians are robust to calibration errors, including miscalibrations consisting of operators undetectable by the co…
We develop a Markovian master equation in the Lindblad form that enables the efficient study of a wide range of open quantum many-body systems that would be inaccessible with existing methods. The validity of the master equation is based entirely on properties of the bath and the system-bath coupling, without any requirements on the level structure within the system itself. The master equation is derived using a Markov approximation that is distinct from that used in earlier approaches. We provide a rigorous bound for the error induced by this Markov approximation; the error is controlled by a…
