5.5 Continuum Limit and Master Equation
In this section we want to relate the sequential measurement to continuous measurements and derive a Master equation which approximately describes the state change during an sequen-tial measurement. As already stated in the introduction to this chapter there is an equivalence between selective sequential measurements and sequences of interactions of the qubit with aux-iliary systems. Therefore the continuum limit to be studied here also relates to the physics of permanently open systems.
From a mathematical point of view it is much easier to analyse a continuous evolution than an evolution which proceeds with discrete steps. In the continuum limit we will gain a differential equation for the non-selective motion (Master equation) and a Stochastic differential equation in the non-selective regime (see the following section). We are going to argue that the restriction on weak and frequent measurements within a non-selective sequential measurement is necessary to arrive at a differential equation of motion in the continuum limit.
There are two ways to relate our sequential process to a continuous description:
1. By considering the state motion on a sufficiently large time scale
which includes many elementary measurements. Thus given any sequential measurement with time between consecutive measurements,
has to be chosen such, that
/
as we have already required in Eq. (5.83) in order to derive the Gaussian operation el-ements. On such a time scale the discrete steps of unitary movement followed by a measurement or interaction with an environment are not resolvable and the sequence of measurements or interactions appears as a continuous (in the sense of permanent) mea-surement or interaction.
2. Given a certain time scale
, discrete processes can be selected which look like continu-ous ones. This applies for example if it is desirable to carry out a continucontinu-ous measurement with a certain time resolution . Such a continuous measurement can be approximately realized by a sequential measurement provided that:
is chosen such, that
/
In chapter 3 sequential measurements of Rabi oscillations are studied in the selective regime which are approximately continuous measurements on the time scale of a N-series.
In both cases the continuous process is obtained from the sequential process by a coarse graining with respect to time. If the state changes drastically in the non-selective regime on the coarse grained time scale , it is impossible to resolve its development by an equation of motion adopted to this time scale, i.e. an equation which propagates the state in steps of length
. In general the state will then appear as continuous function of time. If so, the non-selective state evolution cannot be described by a Master equation or by a differential equation
166 Non-selective Discription of Unsharp Measurements with Unitary Backaction at all. On the one hand we demand a coarse grained time scale
, which is much larger than the time distance between two consecutive measurements:
(5.117) (see above), on the other hand thus must not be larger than the time scale on which the state changes considerably. We thus have to demand
(5.118)
where represents the characteristic time scale of the non-selective state evolution. Both re-quirements are met by a sequence of measurements which are weak and follow rapidly upon each other. Such sequential measurements are determined by the conditions (5.83), (5.12),(5.66) and (5.67).
Are these the only sequential measurements12that lead to a sensible description on the time scale , where they appear continuous? No, there are two exceptions: i) (5.12) and (5.67) can be relaxed to incorporate Poissonian processes like spontaneous decay of an excited state which have a very low probability to inflict large state changes. The average state change, i.e. the state change in the non-selective regime will then be small as well. ii) Sequences of measurements with high accuracy
%
and small unitary back-action
. After a short while they lead effectively to events with Poissonian statistics (Quantum jumps). In the continuum limit such sequences can produce the Zeno effect, where const. Hence then is infinitely large and conditions (5.117) and (5.118) are both met.
Let us now formulate the continuum limit. Its purpose is to obtain a differential equation which approximates as good as possible the difference equation for the weak sequential mea-surement. To this end we write down the difference equation (5.116) in a convenient form. The additional terms in Eq. (5.81) will not play a role in the continuum limit.
where we have inserted the definition (5.41) of to see the explicit dependence on the small quantities.
In order to find a differential equation which describes the state propagating with step length
we have to replace the difference ratio by the differential:
(5.120)
Hence an essential ingredient of the continuum limit we seek is
*
. Since
it follows that in the continuum limit also * . The behaviour of does not have to be
12constituted by elementary measurements with two operation elements
5.5 Continuum Limit and Master Equation 167 specified further than that it should be large (cp. (5.118)), because it does not enter (5.119) explicitely. In order to find the differential equation which optimally describes the realistic sequential process all quantities should keep in the continuum limit, if possible, the values they actually posses in the sequential process.
In the case of % this is not possible, because the second term on the r.h.s. of Eq. (5.119) diverges in the limit
*
if
% is kept constant. The divergence would reflect a finite deco-herence on an arbitrary small time scale which results for the time it lasts in the Zeno effect , cp.[MS77]. In order to obtain a finite contribution due to decoherence in the continuum limit, one has to demand
% *
. In the sequential process a finite amount of decoherence occurs in a finite time represented by the decoherence rate , where the subscript “ ” shall remind us here that this is the actual decoherence rate of the sequential process. This decoherence rate has to appear also in the continuum limit and thus in the Master equation:
" % $
% " % $ (5.121)
We refer to this as Barchielli limit , since Barchielli et al. first introduced a similar limit in [BLP82].
Barchielli et al. studied a sequence of unsharp position measurements, each of which has Gaussian operation elements operator and the measurement result. Please note the similarity to the positive part of the back-action (5.103). In order to obtain a continuous measurement they suggested a limit where the
“accuracy of the measurements decreases as the number of measurements increases” [BLP82].
Expressed with time between subsequent measurements this lead to the demand where is constant in the continuum limit * . Since Barchielli et al. postulated Gaussian operation elements they did not obtain a microstructure of as we have.
In fact the Barchielli limit (5.121) can represent our continuum limit completely if is assumed to be finite, because then with also
is going to zero. All other quantities as% ,
and are kept in the continuum limit as they are in the sequential process. The continuum limit of difference equation 5.119 as specified above now reads
(5.122)
Discussion of the Master Equation
Eq. (5.122) is a Master equation of the Lindblad type. It approximately describes the sequential measurement. The only terms that survived in the continuum limit describe the modification of the unitary part of the dynamics by the average Hamiltonian AV from the back-action and the decoherence with respect to the eigenbasis of , i.e. the eigenbasis of the Effects
.
Master equation (5.122) can be employed to approximately describe a sequence of weak and frequent measurements (item 1 in the list above). On the other hand it can serve to find
168 Non-selective Discription of Unsharp Measurements with Unitary Backaction a prescription how to carry out a specific continuous measurement (item 2). Continuous mea-surements are often formulated in an abstract and model-independent way and it is not obvious how to actually perform them. In the restricted path integral formalism for example, a con-tinuous measurement is specified by a certain weight functional. The question how to realize such a measurement can be answered as follows. From the path-integral with the particular weight functional a stochastic master equation and a Master equation can be derived [POU96].
If this stochastic Master equation is of the same form as Eq. (5.127) then also the correspond-ing Master equation will match Eq. (5.122) and the parameters of the sequential measurements which approximately realize the continuous measurement can be read off from (5.122). In the next section weight functionals which correspond to the stochastic Master equation (5.127) are mentioned. How to carry out sequential measurements is discussed at an example in chapter 4.
A third application of Master equation (5.122) lies in the control of quantum systems. The parameters % % as well as the unitary part of the back-action can be thought of as buttons to regulate the dynamics of the system. In praxis however it might be quite difficult “to turn”
these buttons. In the experimental realization of a sequential measurement of photon dynamics for example which is suggested in chapter 4, it turns out to be technically costly to implement a certain feedback
, while it is comparatively easy to tune
% and%
.
In any case one has to take into account that Eq. (5.122) is only an approximation of the dynamics which arise in certain sequential processes. For example let the system be subject randomly to unitary influences , with probabilities% % and% % ( % * ).
In this case Master equation (5.122) suggests that the resulting state evolution is purely unitary, which is of course not possible due to the dephasing caused by such a noisy channel. The error comes from the approximation. The double commutator with in Eq. (5.119) indicates the necessary decoherence but is neglected in the continuum limit.
Last not least we note, that the Barchielli limit of the difference equation for a single ele-mentary cell (5.43) also leads to Master equation (5.122). This match is due to the fact, that terms linear in generate terms linear in when transiting from an elementary cell to a N-series. Nevertheless the concept of coarse graining with repect to time, which stands behind the transition from “sequential” to “continuous” is expressed and becomes visible only by the introduction of a time scale .