Open Quantum Systems/The Lindblad Form
Superoperators and dynamical maps
The Liouville von Neumann equation is given by
We can define a superoperator such that . 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
where is another superoperator that maps the density matrix from its initial form to its form at time and therefore is called a dynamical map. It is related to the unitary evolution operator according to
Quantum dynamical semigroups
Now let us include also interaction between the system of interest and its environment. In the following, we will use for the reduced density operator for the system and for the partial trace over the environment. As the dynamics of the combination of system and environment is fully coherent, we have
which after taking the trace over the environment on both sides results in
In many typical situations, the initial state between the system and the environment is a product state of the form
Then, we may again think of the right hand side of the previous equation to define a superoperator representing a dynamical map , but now for alone! Furthermore, we may use the following decompositions 
where the operators act only on the Hilbert space of the system and are given by
From the completeness of the states on the Hilbert space of the environment, we may identify the relation
from which follows
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 . 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
where is the partition function, is the inverse temperature and is the energy of the state . 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 
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 , 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,
The superoperator 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 , we obtain
which in the limit yields a first-order differential equation known as a quantum master equation,
Let us now derive an explicit form for the master equation. For this, we need to define an operator basis . The inner product for operators is defined as
A complete orthonormal set consists of operators, where is the Hilbert space dimension. It is convenient to choose one of the operators as proportional to the identity, i.e., . 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
where the coefficients is given by
with the operators defined as above. The coefficients form a postive matrix , as for any -dimensional vector , we have 
Inserting this expansion into the quantum master equation, we obtain
where we separated off all terms containing . As a next step, we define the following quantities
Note that the operator is Hermitian, although is not because the coefficients are complex. Using these definitions, we find for the generator
Since the dynamical map is trace-preserving, the trace over quantum master equation has to vanish, i.e.,
from which we can read off that has to be
Substituting this result back into the quantum master equation, we obtain
Finally, the matrix formed by the coefficients is again Hermitian and positive, so we can diagonalize it to obtain positive eigenvalues . Then, we find the most general form of a Markovian quantum master equation to be given by
where the operators are appropriate linear combinations of the operators 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 , but the latter can also include additional terms coming from the interaction with the environment. Furthermore, the eigenvalues 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 . 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.
- Breuer, Heinz-Peter; Petruccione, F (2002). The theory of open quantum systems. Oxford; New York: Oxford University Press. ISBN 0198520638 9780198520634 Check
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- Lindblad, G. (1976-06-01). "On the generators of quantum dynamical semigroups". Communications in Mathematical Physics 48 (2): 119–130. doi:10.1007/BF01608499. ISSN 1432-0916 0010-3616, 1432-0916. http://link.springer.com/article/10.1007/BF01608499. Retrieved 2014-05-15.