Lindblad superoperator and generated dynamics

Question

In quantum mechanics, in order to evolve the state of an open system, I can use an equation like this $\dot\rho(t)=\mathcal{L}\rho(t)$, where $\mathcal{L}$ is the Lindblad superoperator. In general, $\mathcal{L}$ satisfies $$\frac{\partial V(t,t_0)}{\partial t} = \mathcal{L}V(t,t_0) \, .$$ On Breuer-Petruccione's book about the theory of open quantum systems, it seems that it is remarked a difference between the dynamics generated by $\mathcal{L}$, depending on whether $\mathcal{L}$ itself is time dependent or not. I don't understand in particular in which case the dynamics is Markovian and why. I am also a bit confused about the semigroup structure followed by this dynamics, in particular I am not sure if the semigroup property vanishes when $\mathcal{L}$ is time-dependent. Thank you in advance

Answer 1

I try to unravel the question as far as I know about the topic.

I try to answer the question of when evolution is markovian or not. We recall that in general a quantum evolution is described by a one-parameter family of dynamical maps $\Phi_t$ which are CPT (completely positive and trace-preserving) maps from the set of states.

At this point, to define what is markovian and what is not, we must deal with the property of these dynamical maps. Several definitions of markovianity can be found in the literature (if you are interested, just ask me). In particular, we must consider two-parameter family $\Phi_{t,s} = \Phi_{t} \Phi_s^{-1}$. We assume the existence of the inverse, but, pay attention, we can not assure that the inverse is CP and neither is Positive, thus $\Phi_{t,s}$ may not be dynamical map even if $\Phi_t$ and $\Phi_s$ are so. This is a further request one has to take into account. If the map is CPT as well, then the map is called divisible. Then, one defines a markovian evolution as the set of CPT-divisible map.

Other definitions are possible, in terms of trace distance or of flow of information, and so on (I can indicate several reviews if you are interested in the topic of non-markovianity).

Anyway, the point I'm trying to make clear is the following: a markovian evolution is not necessarily described by a Lindblad evolution. More specifically: if the CPT-divisible map is also differentiable, that is, the following limit exists (in the norm topology, and with other mathematical assumptions) $$ \lim_{\epsilon \to 0^+} \left[\frac{\Vert \Phi_{t+\epsilon,t} - \mathbb{I} \Vert}{\epsilon}\right] := \mathcal{L}_t $$ then we obtain a Quantum Markovian Semigroup whose generator is the operator obtained from the latter limits. In this sense, this is a subclass of Markovian processes, which are homogeneous in time, namely, we can write the two-parameter family as a one-parameter family since $$ \Phi_{t,s} = e^{\mathcal{L}(t-s)} \Longrightarrow \Phi_{t} = e^{\mathcal{L}t}. $$ However, as previously stated, these are not all the markovian evolution possible, that is, that are not homogeneous in the time parameter.

After this brief review on the definition of markovianity (I repeat there: markovianity $\neq$ semigroup), I move more precisely to your question, about semigroup property. The GKLS theorem stated the following: to have a semigroup property you need that the Lindbladian operator and the coefficient $\gamma_i$ are all time-independent. This is not sufficient to have dynamical maps: you need also CP, and this is possible if all the coefficients $\gamma_i > 0$ are positives. So the answer is: yes if the coefficient are time-dependent, the semigroup property does not hold anymore. However, if the inverse of the one-parameter family exists, you can still write a sort of Lindbladian equation, that is, a time-local quantum master equation, but you have to keep attention to many properties that do not hold anymore.

In order to make all the things clear, I also answer directly a question you made in a comment to another answer, which should be clear after all the discussion above. What you said is

Instead of semigroup property, we now have $V(t,t_1)V(t_1,t_0)=V(t,t_0)$". This actually seems to me a semigroup structure, but I think I am missing something, maybe on the mathematical side

Ok, that's actually true, this is not the semigroup property stated in 3.45 in Breuer Petruccione, which I report here

$$ V(t_1) V(t_2) = V(t_1 + t_2), $$

since it is not homogeneuous in time, but it is still markovian, since it represent a divisible map. Pay attention also to the fact that $V(t,t_1)V(t_1,t_0)=V(t,t_0)$ is two-parameter family, while $V(t_1) V(t_2) = V(t_1 + t_2)$ is a one-parameter family.

Answer 2

In general, it helps to unravel the details of the superoperator $\mathcal{L}$, which helps to adress your questions. Let us write the open-system Liouville von Neumann equation in Lindblad form or quantum master equation (QME) as

$$\partial_t\hat{\rho}(t)=\left(\mathcal{L}_0+\mathcal{L}_D\right)\hat{\rho}(t)$$

where $\mathcal{L}_0\hat{\rho}(t)=-\frac{\text{i}}{\hbar}[\hat{H},\hat{\rho}(t)]$ is the generator of the unitary time-evolution of the density operator $\hat{\rho}(t)$ and usually refers to the system part of the open problem. The second term $\mathcal{L}_D$ is regularly denoted as dissipator and describes the implicit interaction with the environment or bath. If $\mathcal{L}_D$ takes a Lindblad form, we have

$$\mathcal{L}_D\hat{\rho}(t)=\sum_k\gamma_k\left(C_k\hat{\rho}(t)C^\dagger_k-\frac{1}{2}\{C^\dagger_kC_k,\hat{\rho}(t)\}\right)$$

with Lindblad operators $C^\dagger_k,C_k$ and decay rates $\gamma_k$ of the individual dissipation channels. In this case, we have by definition a Markovian QME and the superoperator $\mathcal{L}=\mathcal{L}_0+\mathcal{L}_D$ is the generator of a semigroup. If you study the book by Breuer and Petruccione, I highly recommend to read both chapters on the derivation of markovian QME: the very formal one, going back to work done by Lindblad and Kossakowski, Gorini and Sudarshan, as well as the physically motivated microscopic derivation tracing out the bath DoF. Especially in the latter, the nature and consequences of the Markov approximation in open quantum systems becomes clear.

Now, consider time-dependence. In the above equation, the time-dependence might either attributed to the unitary term, e.g., by coupling an external time-dependent perturbation to the system, or to the non-unitary term. The former is quite commonly the case, when studying driven dissipative dynamics of quantum systems. The solution of the Lindblad type QME becomes definitely more involved, but it is still a Markovian QME and the semigroup-structure does not vanish. Time-dependence of the dissipator is a more subtle topic, since the Lindblad operators usually take the form of projectors in the system eigenstate basis, e.g. $|1\rangle\langle 2|$, and the time-dependence might occur in the rates $\gamma_k$. At this stage, it might be helpful to refer to the physically motivated derivation of the QME. The $\gamma_k$ are derived from the bath correlation function, and become explicitly time-independent after applying the Markov approximation. At this stage it would be helpful, if you specify your question at this point and maybe outline the statement of confusion in Breuer and Petruccione's book in some detail.