Next: Example: spontaneous emission.
Up: The Markovian limit
Previous: The Markovian limit
The theory of quantum operations supposes that things just `happen' to the system's density matrixwe don't ask why, or how fast. Now let's start looking at the dynamics, but let's do so on a timescale that has to satisfy two condiitons.
 should be small compared with the characteristic timescale of the systemso the system density matrix only evolves `a little bit' in this time interval (i.e.
).
 But should also be long compared with the time over which the environment `forgets' its information about the system (i.e.
).
Since we are beyond the timescale , we might hope that the evolution of the system will depend only on the present system density matrix, and not on anything that has happened in the past. In that case the evolution through time should be described by a quantum operation on the current system density matrix.
The idea is to look for a suitable quantum operation such that should be altered only to order :

(96) 
Thus it follows that one of the Kraus operators, say, must be
, and the others must be
. So, let's write
Here and are Hermitian operators, but are otherwise arbitrary at this stage; the operators are also arbitrary and are known as Lindblad operators (note that they need be neither unitary nor Hermitian). However, the normalization condition on the Kraus operators requires

(98) 
Hence

(99) 
and therefore
where
represents the anticommutator
.
Taking the limit
we obtain the Lindblad master equation:

(102) 
Note that:
 If there were no Lindblad operators (i.e., if there were only one Kraus operator in the decomposition (96), this formula would reduce to equation (12). We would then identify as the Hamiltonian of the (closed) system.
 However, there is in general to reason to suppose that the operator appearing in equation(103) is the Hamiltonian of the isolated system. Indeed, we shall see later that there are (potentially important) corrections to it that come from the interaction with the environment.
 Indeed, is not even unique; the equation of motion remains invariant under the changes

(103) 
where and are arbitrary scalars. The equation of motion also remains invariant under an arbitrary unitary transformation of the Lindblad operators:

(104) 
 The righthand side of equation (103) is a linear functional of ; it defines the Lindbladian superoperator through

(105) 
The formal solution to this can be written in the form of a timeevolution superoperator:

(106) 
Here
is the same entity we previously called : the timeordering operator that puts earliest times to the right and latest times to the left.
Provided the Lindbladian is timeindependent, this can be simplified to

(107) 
Note however that this is not a recipe for efficient practical calculations; if the dimension of the system's Hilbert space is , a matrix representation for would contain elements; directly exponentiating it would therefore require
operations.
 The term involving the Lindblad operators on the RHS of equation (103) is known as the dissipator, written
; thus we have

(108) 
 This is all in the Schrödinger representation, where the wavefunction (or density matrix) is timedependent but operators are not. An alternative way of representing the information is to transfer the timedependence to the operators: we then require that the expectation value of any (system) operator be the same in either picture.

(109) 
where
, and the operator
orders in the opposite sense to normal (i.e. earliest times to the left).
Note that obeys the equation of motion

(110) 
In the case of a timeindependent Lindbladian things simplify once again, and

(111) 
Next: Example: spontaneous emission.
Up: The Markovian limit
Previous: The Markovian limit
Andrew Fisher
20040714