An introduction to Markovian open quantum systems

TL;DR

This work provides a concise, pedagogical overview of Markovian open quantum systems, focusing on the Lindblad master equation, its derivation from CPTP maps via the Choi–Kraus theorem, and the unravelings into stochastic quantum trajectories and continuous measurements. It emphasizes the structure of steady states, the role of weak/strong and dynamical symmetries, and phenomena unique to nonunitary dynamics such as reservoir engineering, the quantum Zeno effect, and both dissipative and measurement-induced phase transitions. By combining formalism with concrete physical realizations—noisy Hamiltonians, continuous monitoring, and reservoir tracing—the notes equip newcomers with a conceptual and practical toolkit for exploring nonequilibrium quantum phenomena. The discussion also covers decoherence-free subspaces and subsystems, as well as the spectral properties of the Liouvillian that govern relaxation and long-time behavior. Overall, the notes establish a foundational perspective on how dissipation and measurements shape quantum evolution and enable novel states and phases of matter.

Abstract

This is a concise, pedagogical introduction to the dynamic field of open quantum systems governed by Markovian master equations. We focus on the mathematical and physical origins of the widely used Lindblad equation, its unraveling in terms of stochastic pure-state trajectories and the corresponding continuous measurement protocols, the structure of steady states with emphasis on the role of symmetry and conservation laws, and a sampling of the novel physical phenomena that arise from nonunitary dynamics (dissipation and measurements). This is far from a comprehensive summary of the field. Rather, the objective is to provide a conceptual foundation and physically illuminating examples that are useful to graduate students and researchers entering this subject. There are exercise problems and references for further reading throughout the notes.

Figures (10)

• Figure 1: (a) Sketch showing possible eigenvalues of a Lindbladian. The steady oscillations are atypical. (b) Spectrum for a damped harmonic oscillator with $\hat{H} = \hat{a}^{\dagger} \hat{a}$ and $\hat{L} = \hat{a}$. (c) Spectrum for a spin-$S$ with $\hat{H} = \hat{S}^x$ and $\hat{L} = \sqrt{0.5/S} \, \hat{S}^-$ for $S=5$.
• Figure 2: (a) Norm of $|\phi_c\rangle$ and (b) $z$-magnetization for a Monte-Carlo trajectory of a spin-$1/2$ with Hamiltonian $\hat{H} = \hat{S}^x$ and jump operator $\hat{L} = \hat{S}^-$. (c) Average over $100$ (blue) and $1000$ (red) trajectories compared to the Lindblad dynamics (gray).
• Figure 3: (a) Expectation (blue) and variance (orange) of the photon number in a given realization of quantum state diffusion for $\hat{H} = {\rm i}(\hat{a}^{\dagger} - \hat{a})$ and $\hat{L} = \hat{a}$. (b) Distance from a coherent state, where $\parallel |\psi\rangle \parallel \coloneqq \sqrt{\langle \psi | \psi \rangle}$. (c) Purity of the ensemble-averaged density matrix.
• Figure 4: (a) Lindblad spectrum for Eq. \ref{['eq:dyn_sym_hubbard']} with $U=1.4$, $B=1.1$, $\gamma=1$, and $\varepsilon_j \in [-0.3, 0.3]$ for $N = 4$, $S=1$, and $4$ sites. (b) Evolution of $\langle \hat{S}^x_i \rangle = \text{Re} \, \langle \hat{c}_{i,\uparrow}^{\dagger} \hat{c}_{i,\downarrow} \rangle$, $i=1,2,3,4$, starting from a random state. Dashed curve shows the (uniform) oscillation resulting from the imaginary eigenvalues (red dots). (c) Evolution for $\gamma = 0$.
• Figure 5: Using drive and loss to create a photonic Mott insulator Ma2019. Each site is a resonator with frequency $\omega_c$ and nonlinearity $U<0$. The end site is driven at two-photon resonance and coupled to a lossy resonator of frequency $\omega_c - |U|$. The box shows energy levels of the last two resonators labeled by their photon numbers.