We study the classical limit of quantum mechanics on graphs by introducing a Wigner function for graphs. The classical dynamics is compared to the quantum dynamics obtained from the propagator. In particular, we consider extended open graphs whose classical dynamics generate a diffusion process. The transport properties of the classical system are revealed in the scattering resonances and in the time evolution of the quantum system.
We study the classical limit of quantum mechanics on graphs by introducing a Wigner function for graphs. The classical dynamics is compared to the quantum dynamics obtained from the propagator. In particular, we consider extended open graphs whose classical dynamics generate a diffusion process. The transport properties of the classical system are revealed in the scattering resonances and in the time evolution of the quantum system.
Based on the references [P1–P6], we investigate in this cumulative PhD thesis the dynamics and transport in closed and open interacting spin chain models. We focus on (i) the connection between the open and the closed system dynamics [P1, P2], (ii) the influence of various system parameters on the transport behavior in closed and open systems [P3, P5], and in addtion (iii) the dynamics and transport in closed models [P4, P6]. As for (i), we explore the connection of the dynamics in open systems, modeled by the Lindblad equation, and the correlation functions in the closed system [P1]. Building on this connection for magnetization transport, we study whether the time evolution of the open quantum system can be described in terms of classical correlation functions generated by the Hamiltonian equations of motion for real vectors [P2]. Second, we study in (ii) the influence of different parameters on the diffusion constants in closed and open systems [P3, P5]. In Ref. [P3], we study the impact of integrability-breaking perturbations in both closed and open spin chain systems. While we find a continuous change of the diffusion constant with perturbation strength in the closed system, in the open system, in the weak-perturbation limit, we indicate the possibility of a diverging diffusion constant. Also, we investigate the influence of the system-bath coupling on the transport behavior in boundary-driven open systems [P5]. We observe that the diffusion constants depend on the system-bath coupling and that in the nonequilibrium steady state, they inevitably retain finite-size effects, limiting the computation of long-time transport coefficients. Additional for (iii), in Ref. [P4], we first examine the impact of perturbations on the dynamics in closed spin chain models. We employ projector-operator techniques to understand, on perturbative grounds, the nontrivial damping of the standard correlation function. Second, we also discuss a dynamical quantum typicality approach for studying the energy transport in bipartite systems at low temperatures [P6].
The Wigner formulation of quantum mechanics based on the concept of Wigner function (WF), is particularly suitable for the study of quantum transport in mesoscopic systems since it allows to describe quantum mechanical effects in an (r,p) phase space, in analogy to what is done for classical systems. Recently we have introduced the concept of Wigner paths (WPs) in phase space which both provide a pictorial representation of the quantum evolution of the system of interest and constitute a useful tool for providing numerical solutions of the quantum equation describing the time evolution of the system. The theory has been applied to the following cases: (a) The study of the transport properties of the transient regime for homogeneous systems. (b) The comparison between quantum and classical treatment of the scattering mechanism. (c) The study of the reflection of the WF from an infinite potential barrier. Furthermore, the Wigner approach, originally formulated in terms of single-time WF has been extended by introducing the two-time WF. The Fourier transform of this function contains independent information about both momentum and energy of the electron. The Monte Carlo procedure based on the generation of WPs can be used to evaluate the two-time WF of an electron interacting with the phonon gas.
We employ a recently developed formalism for describing open quantum systems, to device efficient ab initio numerical techniques for the time propagation of the Wigner function. The propagator's complexity is O(NlogN), where N is the length of the array containing the Wigner function. We illustrate the developed method by propagating single particle systems undergoing decoherence and draw a faithful comparison with the classical evolution governed by both the Fokker-Planck and the Koopman-von Neumann equations. These simulations allow us to witness the emergence of the classical world by observing how the Wigner function converges to the solution of the Fokker-Planck equation with sufficient classical diffusion. As a second example we simulate for first time a numerically challenging quantum system made of two coupled particles and observe the loss of decoherence on one of the particles as a result of continuous measurements performed on the other one, therefore obtaining quantitative predictions for the loss of coherence of both particles as a function of time.
The Lindblad quantum master equation is one of the central approaches to the physics of open quantum systems. In particular, boundary driving enables the study of transport, where a steady state emerges in the long-time limit, which features a constant current and a characteristic density profile. While the Lindblad equation complements other approaches to transport in closed quantum systems, it has become clear that a connection between closed and open systems exists in certain cases. Here, we build on this connection for magnetization transport in the spin-1/2 XXZ chain with and without integrability-breaking perturbations. Specifically, we study the question of whether the time evolution of the open quantum system can be described on the basis of classical correlation functions, as generated by the Hamiltonian equations of motion for real vectors. By comparing the results to exact numerical simulations of the Lindblad equation, we find a good accuracy of such a description for a range of model parameters which is consistent with previous studies on closed systems. While this agreement is an interesting physical observation, it also suggests that classical mechanics can be used to solve the Lindblad equation for comparatively large system sizes which lie outside the possibilities of a quantum mechanical treatment. We also point out counterexamples and limitations for the quantitative extraction of transport coefficients. Remarkably, our classical approach to large open systems allows us to detect superdiffusion at the isotropic point.
Quantum state diffusion provides a dynamics for the localization or reduction of a quantum-mechanical wavepacket during a measurement or similar physical process. The essence of this localization dynamics is captured in the classical theory presented here, which applies to the common situation where the localization takes place over macroscopic distances and the wave properties of the system are no longer relevant. It provides a picture of the localization process in classical phase space, and a practical aid for computations on open quantum systems. The theory is developed from classical Hamiltonian dynamics in phase space and the known properties of localization. As an example it is used to illustrate how absorption by a screen leads to quantum jumps. A derivation from quantum state diffusion using the Wigner function is given in the companion paper.
Quantum time correlation functions are often the principal objects of interest in experimental investigations of the dynamics of quantum systems. For instance, transport properties, such as diffusion and reaction rate coefficients, can be obtained by integrating these functions. The evaluation of such correlation functions entails sampling from quantum equilibrium density operators and quantum time evolution of operators. For condensed phase and complex systems, where quantum dynamics is difficult to carry out, approximations must often be made to compute these functions. We present a general scheme for the computation of correlation functions, which preserves the full quantum equilibrium structure of the system and approximates the time evolution with quantum-classical Liouville dynamics. Several aspects of the scheme are discussed, including a practical and general approach to sample the quantum equilibrium density, the properties of the quantum-classical Liouville equation in the context of correlation function computations, simulation schemes for the approximate dynamics and their interpretation and connections to other approximate quantum dynamical methods.