Open Quantum Systems/The Lindblad Form

Superoperators and dynamical maps
The Liouville von Neumann equation is given by
 * $$\frac{d}{dt}\rho = -\frac{i}{\hbar}\left[H,\rho\right].$$

We can define a superoperator $$\mathcal{L}$$ such that $$\mathcal{L}\rho = -i/\hbar[H,\rho]$$. It is called a superoperator because it is an object that acts on an operator and results in a new operator. If the Hamiltonian is time-independent, we may formally integrate the Liouville von Neumann equation and obtain
 * $$\rho(t) = \exp\left(\mathcal{L}t\right)\rho(0)\equiv \mathcal{V}(t)\rho(0),$$

where $$\mathcal{V}$$ is another superoperator that maps the density matrix from its initial form to its form at time $$t$$ and therefore is called a dynamical map. It is related to the unitary evolution operator $$U(t)=\exp(-iHt/\hbar)$$ according to
 * $$\mathcal{V}(t)\rho(0) = U(t)\rho(0)U(t)^\dagger.$$

Quantum dynamical semigroups
Now let us include also interaction between the system of interest and its environment. In the following, we will use $$\rho_S$$ for the reduced density operator for the system and $$\mathrm{Tr}_E\{\rho\}$$ for the partial trace over the environment. As the dynamics of the combination of system and environment is fully coherent, we have
 * $$\rho(t) = U(t)\rho(0)U(t)^\dagger,$$

which after taking the trace over the environment on both sides results in
 * $$\rho_S(t) = \mathrm{Tr}_E\{U(t)\rho(0)U(t)^\dagger\}.$$

In many typical situations, the initial state between the system and the environment is a product state of the form
 * $$\rho(0) = \rho_S(0) \otimes \rho_E(0).$$

Then, we may again think of the right hand side of the previous equation to define a superoperator representing a dynamical map $$\mathcal{V}(t)$$, but now for $$S$$ alone! Furthermore, we may use the following decompositions
 * $$\begin{align}\rho_E(0) &= \sum\limits_\alpha \lambda_\alpha |\psi_\alpha\rangle\langle\psi_\alpha|\\

\mathcal{V}(t)\rho_S(0) &= \sum\limits_{\alpha\beta} W_{\alpha\beta}(t)\rho_S(0)W_{\alpha\beta}^\dagger(t),\end{align}$$ where the operators $$W_{\alpha\beta}$$ act only on the Hilbert space of the system and are given by
 * $$W_{\alpha\beta}(t) = \sqrt{\lambda_\beta}\langle \psi_\alpha|U(t)|\psi_\beta\rangle.$$

From the completeness of the states $$|\psi_\alpha\rangle$$ on the Hilbert space of the environment, we may identify the relation
 * $$\sum\limits_{\alpha\beta} W_{\alpha\beta}(t)W_{\alpha\beta}^\dagger(t) = 1_S,$$

from which follows
 * $$\mathrm{Tr}\{\mathcal{V}(t)\rho_S(0)\} = \mathrm{Tr}\{\rho_S\} = 1,$$

i.e., the dynamical map is trace-preserving. Moreover, it is completely positive, mapping a positive density matrix onto another positive density matrix.

In many important cases, we can make one further assumption on the dynamical map $$\mathcal{V}(t)$$. If correlations in the environment decay much faster than the timescale of the evolution in the system of interest, we may neglect memory effects describing how the system has previously interacted with the environment. This is also known as the Markov approximation. For example, consider a thermal state of the environment of the form
 * $$\rho_E = \sum\limits_n\frac{1}{Z}\exp(-\beta E_n)|n\rangle\langle n|,$$

where $$Z = \sum_n \exp(-\beta E_n)$$ is the partition function, $$\beta$$ is the inverse temperature and $$E_n$$ is the energy of the state $$|n\rangle$$. Then, if the environment is large and its dynamics is fast enough, any energy exchanged with the system will quickly dissipate away to form a new thermal state with almost exactly the same temperature. Then, from the viewpoint of the system, the state of the environment will appear to be almost constant all the time.

Formally, we can express the consequences of the Markov approximation on the dynamical map as
 * $$\mathcal{V}(t_1)\mathcal{V}(t_2) = \mathcal{V}(t_1+t_2)\;\;\;t_1,t_2 \geq 0.$$

Note that the constraint on the times being positive means that we can only piecewise propagate the system forward in time, i.e, the inverse of the dynamical map does usually not exist. This is in contrast to coherent dynamics, where there is an inverse operation corresponding to negative time arguments in the unitary evolution operator. Hence, while the dynamical maps of coherent systems form a group, the dynamical maps for open quantum systems only form a semigroup.

The generator of the semigroup is the Liouvillian $$\mathcal{L}$$, which is a generalization of the superoperator appearing on the right hand side of the Liouville von Neumann equation. One important consequence of this generalization is that the von Neumann entropy is no longer a conserved quantity. However, the Liouvillian has to fulfill the property of being the generator of a completely positive and trace-preserving dynamical map. In the following, we will see how the most general form of the Markovian master equation will look like.

Most general form of the dynamics
Similar to the case of a closed quantum system, we can write the dynamical map of an open quantum system as an exponential of the generator of the semigroup,
 * $$\mathcal{V}(t)=\exp\left(\mathcal{L}t\right).$$

The superoperator $$\mathcal{L}$$ reduces to the one of the Liouville von Neumann equation in the case of purely coherent dynamics, but in general will have additional incoherent terms. Expanding the dynamical maps for short times $$\tau$$, we obtain
 * $$\rho(t+\tau)=\mathcal{V}(\tau)\rho(t) = \left(1+\mathcal{L}\tau \right)\rho(t) + O(\tau^2),$$

which in the limit $$\tau \to 0$$ yields a first-order differential equation known as a quantum master equation,
 * $$\frac{d}{dt}\rho(t) = \mathcal{L}\rho(t).$$

Let us now derive an explicit form for the master equation. For this, we need to define an operator basis $$\{F_i\}$$. The inner product for operators is defined as
 * $$\langle F_i, F_j\rangle \equiv \text{Tr}\left\{F_i^\dagger F_j\right\}.$$

A complete orthonormal set consists of $$N^2$$ operators, where $$N$$ is the Hilbert space dimension. It is convenient to choose one of the operators as proportional to the identity, i.e., $$F_{N^2} = 1/\sqrt{N}$$. Then, all other operators are traceless. For example, in a two-level system, the remaining operators are proportional to the Pauli matrices.

We can now express the action of the dynamical map using this operator basis as
 * $$\mathcal{V}(t)\rho = \sum\limits_{i,j=1}^{N^2} c_{ij}(t) F_i\rho F_j^\dagger,$$

where the coefficients $$c_{ij}(t)$$ is given by
 * $$c_{ij}(t) = \sum\limits_{\alpha\beta} \langle F_i, W_{\alpha\beta}(t)\rangle \langle F_j, W_{\alpha\beta}(t)\rangle^*$$

with the operators $$W_{\alpha\beta}$$ defined as above. The coefficients $$c_{ij}(t)$$ form a postive matrix $$C$$, as for any $$N^2$$-dimensional vector $$v$$, we have
 * $$v^\dagger C v = \sum\limits_{\alpha\beta}\left|\left\langle\sum\limits_i v_i F_i, W_{\alpha\beta}(t)\right\rangle\right|^2 \geq 0.$$

Inserting this expansion into the quantum master equation, we obtain
 * $$\begin{align}\mathcal{L}\rho &= \lim\limits_{\tau \to 0} \frac{\mathcal{V}(\tau)\rho-\rho}{\tau}\\

& = \lim\limits_{\tau \to 0}\left[\frac{1}{N}\frac{c_{N^2 N^2}(\tau)-N}{\tau} \rho + \frac{1}{\sqrt{N}}\sum\limits_{i=1}^{N^2-1}\left(\frac{c_{i N^2}(\tau)}{\tau} F_i\rho + \frac{c_{N^2 i}(\tau)}{\tau} \rho F_i^\dagger\right)\right.\\ &\left.+\sum\limits_{i,j=1}^{N^2-1}\frac{c_{ij}(\tau)}{\tau} F_i\rho F_j^\dagger\right],\end{align}$$ where we separated off all terms containing $$F_{N^2N^2} = \frac{1}{\sqrt{N}}$$. As a next step, we define the following quantities
 * $$\begin{align}

a_{ij} & = \lim\limits_{\tau \to 0} \frac{c_{i j}(\tau)}{\tau} \;\;\; i,j=1,\ldots, N^2-1\\ F & = \frac{1}{\sqrt{N}} \sum\limits_{i=1}^{N^2-1} \lim\limits_{\tau \to 0} \frac{c_{i N^2}(\tau)}{\tau} F_i\\ G & = \frac{1}{2 N} \lim\limits_{\tau \to 0} \frac{c_{N^2 N^2}(\tau)-N}{\tau} + \frac{1}{2}(F^\dagger + F)\\ H & = \frac{1}{2i}(F^\dagger-F).\end{align}$$ Note that the operator $$H$$ is Hermitian, although $$F$$ is not because the coefficients $$c_{iN^2}$$ are complex. Using these definitions, we find for the generator
 * $$\mathcal{L}\rho = -i[H,\rho] + \{G,\rho\} + \sum_{ij=1}^{N^2-1} a_{ij} F_i \rho F_j^\dagger.$$

Since the dynamical map is trace-preserving, the trace over quantum master equation has to vanish, i.e.,
 * $$\text{Tr}\left\{\mathcal{L}\rho\right\} = \text{Tr}\left\{\left(2G+\sum_{ij=1}^{N^2-1} a_{ij} F_j^\dagger F_i\right)\rho\right\}=0,$$

from which we can read off that $$G$$ has to be
 * $$G = -\frac{1}{2}\sum_{ij=1}^{N^2-1} a_{ij} F_j^\dagger F_i.$$

Substituting this result back into the quantum master equation, we obtain
 * $$\mathcal{L}\rho = -i[H,\rho] + \sum\limits_{i,j=1}^{N^2-1} a_{ij}\left(F_i\rho F_j^\dagger - \frac{1}{2}\left\{F_j^\dagger F_i, \rho\right\}\right).$$

Finally, the matrix formed by the coefficients $$a_{ij}$$ is again Hermitian and positive, so we can diagonalize it to obtain positive eigenvalues $$\gamma_i$$. Then, we find the most general form of a Markovian quantum master equation to be given by
 * $$\mathcal{L}\rho = -i[H,\rho] + \sum\limits_{i=1}^{N^2-1} \gamma_i\left(A_i\rho A_i^\dagger - \frac{1}{2}\left\{A_i^\dagger A_i, \rho\right\}\right),$$

where the operators $$A_i$$ are appropriate linear combinations of the operators $$F_i$$ obtained from the diagonalization procedure. This form of the quantum master equation is known as the Lindblad form, as Lindblad first showed that the generator of a Markovian master equation has to be of that form.

The system Hamiltonian is contained in the Hermitian operator $$H$$, but the latter can also include additional terms coming from the interaction with the environment. Furthermore, the eigenvalues $$\gamma_i$$ correspond to relaxation rates describing incoherent decay processes in the system. Typically, these decay processes will result in the system eventually reaching a stationary state characterized by $$\frac{d}{dt}\rho = 0$$. However, such a stationary state does not necessarily mean the absence of any dynamics: for example, a single realization of two-level system in a maximally mixed state might still violently jump between both levels! Only when the ensemble average is taken, the dynamics will vanish.