Lindbladian

Summary

In quantum mechanics, the Gorini–Kossakowski–Sudarshan–Lindblad equation (GKSL equation, named after Vittorio Gorini, Andrzej Kossakowski, George Sudarshan and Göran Lindblad), master equation in Lindblad form, quantum Liouvillian, or Lindbladian is one of the general forms of Markovian master equations describing open quantum systems. It generalizes the Schrödinger equation to open quantum systems; that is, systems in contacts with their surroundings. The resulting dynamics is no longer unitary, but still satisfies the property of being trace-preserving and completely positive for any initial condition. The Schrödinger equation or, actually, the von Neumann equation, is a special case of the GKSL equation, which has led to some speculation that quantum mechanics may be productively extended and expanded through further application and analysis of the Lindblad equation. The Schrödinger equation deals with state vectors, which can only describe pure quantum states and are thus less

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This lecture describes advanced concepts and applications of quantum optics. It emphasizes the connection with ongoing research, and with the fast growing field of quantum technologies. The topics cover some aspects of quantum information processing, quantum sensing and quantum simulation.

This course provides an in-depth treatment of the latest experimental and theoretical topics in quantum sciences and technologies, including for example quantum sensing, quantum optics, cold atoms, theory of quantum measurements and open dissipative quantum systems, etc.

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The characterization of open quantum systems is a central and recurring problem for the development of quantum technologies. For time-independent systems, an (often unique) steady state describes the average physics once all the transient processes have faded out, but interesting quantum properties can emerge at intermediate timescales. Given a Lindblad master equation, these properties are encoded in the spectrum of the Liouvillian whose diagonalization, however, is a challenge even for small-size quantum systems. Here, we propose a new method to efficiently provide the Liouvillian spectral decomposition. We call this method an Arnoldi-Lindblad time evolution, because it exploits the algebraic properties of the Liouvillian superoperator to efficiently construct a basis for the Arnoldi iteration problem. The advantage of our method is double: (i) It provides a faster-than-the-clock method to efficiently obtain the steady state, meaning that it produces the steady state through time evolution shorter than needed for the system to reach stationarity. (ii) It retrieves the low-lying spectral properties of the Liouvillian with a minimal overhead, allowing to determine both which quantum properties emerge and for how long they can be observed in a system. This method is general and model-independent, and lends itself to the study of large systems where the determination of the Liouvillian spectrum can be numerically demanding but the time evolution of the density matrix is still doable. Our results can be extended to time evolution with a time-dependent Liouvillian. In particular, our method works for Floquet (i.e., periodically driven) systems, where it allows not only to construct the Floquet map for the slow-decaying processes, but also to re- trieve the stroboscopic steady state and the eigenspectrum of the Floquet map. Although the method can be applied to any Lindbla- dian evolution (spin, fermions, bosons, ... ), for the sake of simplicity we demonstrate the efficiency of our method on several examples of coupled bosonic resonators (as a particular example). Our method outperforms other di- agonalization techniques and retrieves the Li- ouvillian low-lying spectrum even for system sizes for which it would be impossible to per- form exact diagonalization.

VEREIN FORDERUNG OPEN ACCESS PUBLIZIERENS QUANTENWISSENSCHAF2022

Quantum Monte Carlo approach to the non-equilibrium steady state of open quantum systems

Alexandra Nagy

Many-body open quantum systems are exposed to an essentially uncontrollable environment that acts as a source of decoherence and dissipation. As the exact treatment of such models is generally unfeasible, it is favourable to formulate an approximate description by means of the dynamics of the reduced density operator of the system. When studying models with a weak coupling to a memoryless environment, the existence of such solution is granted and the reduced dynamics is governed by the Lindblad quantum master equation. In recent years, open quantum systems have evolved into a major field of studies. Focus of these studies are the characterization of emergent phenomena and dissipative phase transitions, as well as the ongoing debate about whether quantum computing schemes are still hard to simulate classically --- and thus achieve quantum supremacy --- when in presence of some degree of noise-induced decoherence. However, the challenge posed by this task lies in the complexity of the density matrix that increases exponentially with the system size, and the quest for efficient numerical methods is a research field that is still in its infancy. Here, we first develop a real-time full configuration interaction quantum Monte Carlo technique that stems from a class of methods generally known as projector Monte Carlo. The approach enables the stochastic sampling of the Lindblad time evolution of the density matrix thanks to a massively parallel algorithm, thus providing estimates of observables on the non-equilibrium steady state. We present the underlying theory and introduce an initiator technique and importance sampling to reduce statistical error. We demonstrate the efficiency of the approach by applying it to the dissipative two-dimensional XYZ spin-1/2 model on a lattice. As the importance sampling of projector approaches is often combined with variational results, we then introduce a novel method that is based on the variational Monte Carlo methods and on a neural network representation of the density matrix. Thanks to the stochastic reconfiguration scheme, the application of the variational principle is translated into the actual integration of the Lindblad quantum master equation. We test the effectiveness of the method by modeling the steady state of the dissipative two-dimensional XYZ spin-1/2 model through a dissipative phase transition, and also the real-time dynamics of the dissipative Ising model. In addition, we discuss the application of the developed methods and the open questions of the field.

EPFL2020

Quantum Monte Carlo approach to the non-equilibrium steady state of open quantum systems

Alexandra Nagy

EPFL2020

VEREIN FORDERUNG OPEN ACCESS PUBLIZIERENS QUANTENWISSENSCHAF2022