Colloquium: Non-Markovian dynamics in open quantum systems

The dynamical behavior of open quantum systems plays a key role in many applications of quantum mechanics, examples ranging from fundamental problems, such as the environment-induced decay of quantum coherence and relaxation in many-body systems, to applications in condensed matter theory, quantum transport, quantum chemistry, and quantum information. In close analogy to a classical Markovian stochastic process, the interaction of an open quantum system with a noisy environment is often modeled phenomenologically by means of a dynamical semigroup with a corresponding time-independent generator in Lindblad form, which describes a memoryless dynamics of the open system typically leading to an irreversible loss of characteristic quantum features. However, in many applications open systems exhibit pronounced memory effects and a revival of genuine quantum properties such as quantum coherence, correlations, and entanglement. Here recent theoretical results on the rich non-Markovian quantum dynamics of open systems are discussed, paying particular attention to the rigorous mathematical definition, to the physical interpretation and classification, as well as to the quantification of quantum memory effects. The general theory is illustrated by a series of physical examples. The analysis reveals that memory effects of the open system dynamics reflect characteristic features of the environment which opens a new perspective for applications, namely, to exploit a small open system as a quantum probe signifying nontrivial features of the environment it is interacting with. This Colloquium further explores the various physical sources of non-Markovian quantum dynamics, such as structured environmental spectral densities, nonlocal correlations between environmental degrees of freedom, and correlations in the initial system-environment state, in addition to developing schemes for their local detection. Recent experiments addressing the detection, quantification, and control of non-Markovian quantum dynamics are also briefly discussed.

Figure 1

An open quantum system described by the Hilbert space ${\mathcal{H}}_S$ and the Hamiltonian $H_S$, which is coupled to an environment with Hilbert space ${\mathcal{H}}_E$ and Hamiltonian $H_E$ through an interaction Hamiltonian $H_I$.

Figure 2

The loss of distinguishability quantified by the trace distance $D$ between density matrices as a consequence of the action of a quantum dynamical map ${\mathrm{\Phi }}_t$. Alice prepares two distinct states, but a dynamical map acts as a noisy channel, thus generally reducing the information available to Bob in order to distinguish among the states by performing measurements on the system only.

Figure 3

The information flow between an open system and its environment according to Eq. (32). Left: The open system loses information to the environment, corresponding to a decrease of ${\mathcal{I}}_{\mathrm{int}}(t)$ and Markovian dynamics. Right: Non-Markovian dynamics is characterized by a backflow of information from the environment to the system and a corresponding increase of ${\mathcal{I}}_{\mathrm{int}}(t)$.

Figure 4

The relations between the concepts of quantum Markovianity and divisibility of the family of quantum dynamical maps ${\mathrm{\Phi }}_t$. The blue (light gray) area to the left of the thick zigzag line represents the set of Markovian quantum processes for which the inverse of the dynamical map exists, which is identical to the set of $P$-divisible processes. This set contains as a subset the set of $CP$-divisible processes (green). The blue (light gray) area to the right represents the non-Markovian processes which are therefore neither $CP$ nor $P$ divisible. The zigzag line dividing the two regions points to the fact that neither of them is convex. The red (dark gray) area marks the Markovian or non-Markovian processes for which the inverse of ${\mathrm{\Phi }}_t$ does not exist.

Figure 5

The non-Markovianity measure $\mathcal{N}(\mathrm{\Phi })$ defined by Eq. (34) for the spin-boson model as a function of temperature $T$ and cutoff frequency $\mathrm{\Omega }$ in units of the transition frequency ${\omega }_0$. The white curve given by Eq. (76) describes the points at which ${\omega }_0$ is exactly in resonance with the maximum of the effective spectral density (75). From [29].

Figure 6

The non-Markovianity measure $\mathcal{N}(\mathrm{\Phi })$ of Eq. (34) for a qubit coupled to a one-dimensional Ising chain in a transverse field as a function of field strength ${\lambda }^{*}$ and particle number $N$. The measure vanishes along the black line and is nonzero everywhere else. From [49].

Figure 7

The behavior of the trace distance with and without initial correlations. If the trace distance between two states, prepared such that ${\rho }_{SE}^2(0)=(\mathrm{\Lambda }\otimes I){\rho }_{SE}^1(0)$, never exceeds its initial value (dashed black line), the dynamics does not witness initial correlations. If, on the other hand, the trace distance for such pair of states increases above the initial value (solid red line), we can conclude the presence of initial correlations.

Figure 8

Scheme for the local detection of system-environment quantum correlations in some state ${\rho }_{SE}^1(0)$, employing a local dephasing operation $\mathrm{\Lambda }$ to generate a second reference state ${\rho }_{SE}^2(0)$.

Figure 9

A system with nonlocal memory effects. The open systems 1 and 2 locally interact with their respective environments. Initial correlations between the local environments cause the occurrence of nonlocal memory effects.

Figure 10

Non-Markovian dynamics arising from an engineered frequency spectrum. (a) The frequency spectrum of the initial state for various values of the tilting angle $\theta$ of the cavity. (b) The change of the trace distance as a function of the tilting angle $\theta$. The transition from the non-Markovian to the Markovian regime occurs at $\theta \approx 4.1°$, and from the Markovian to the non-Markovian regime at $\theta \approx 8.0°$, corresponding to the occurrence of a double peak structure in the frequency spectrum (a). From [73].