5.4 N-series
5.4.2 Gaussian Operation Elements
with a component perpendicular to
. By an appropriate choice of
the expectation value of any observable can be shifted in a N-series.
The rest of the terms are proportional to double commutators of applied to single commutators containing a Hamiltonian or vice versa. They can be interpreted to generate decoherence of the unitary evolved state or unitary evolution of the partly decohered state, respectively.
5.4.2 Gaussian Operation Elements
In this subsection the Effects and operation elements of a N-series are derived. The Effects are approximately of Gaussian shape as well as the corresponding operation elements. The latter lead to the same difference equation up to first order in the small parameters %
and as the one we have derived in the previous subsection based on the composition of operations of an elementary cell. In order to generate a Gaussian probability distribution for the outcomes of a N-series we require the number to be very large:
/ (5.83)
Moreover we assume conditions (5.66)-(5.68) to hold. One more assumptions has to be made in order to obtain approximately Gaussian operation elements:
%
%
?
*
(5.84) The last assumption explicitely excludes processes with Poissonian statistics (see discussion in the beginning of section 5.3.
The change of the density operator during a N-series of measurements with results
3
, each of which can take the values “
”and “ ” takes modulo normalisation the form
"
where the original evolution between the measurements is represented by
;
(5.86)
The order of the influences of the original dynamics and the measurement can in general not be permuted. Now conditions (5.66)-(5.68) allow certain permutations within a N-series. First of all it is possible to separate the influences as follows (cf. Appendix A of this chapter)
3
" $ (5.87)
160 Non-selective Discription of Unsharp Measurements with Unitary Backaction where the order of magnitude of the correcting term
Also the order of results
is in general important. In Appendix A of this chapter it is shown that
of the assumptions (5.66)-(5.68) we may also neglect . To ignore corrections
and simplifies the calculations reasonably. The influence (5.85) of the N-series becomes a function of , independent of the
different orderings of the “ ” and “ ” results. If is kept as the only measurement result (readout) irrespective of the above detailed order then the total N-series of duration possesses operation elements
" The factor in front of
ensures the normalisation of the operation elements:
5
" $ " $
/ (5.94)
A derivation of this factor is given in chapter 3. The unitary part may be written as
" $ ;
5.4 N-series 161 We now make use of the fact that for large a binomial distribution can be approximated by a Gaussian distribution10
% " % $
[Kre70]. With (5.96) we can express " $ by means of square roots of Gaussians:
" $
or written in a more concise form:
" $ Because we assumed according to (5.68) that the measurements are unsharp, the ’spread ’of the Gaussian becomes in lowest order in
% a c-number
% " %
$ " % $
% % + % " % $ (5.99)
where we have ignored terms of order % and higher. The error thus committed in the Gaussian (5.98) is of order
%
, which can be seen by inserting from (5.10) and from (5.100) and expanding the Gaussian (5.98) in powers of % . We introduce the new variable to replace the readout by defining
; % + % / (5.100)
Because is large we may approximately regard to be continuous. Its range is limited:
* % + % / (5.101)
.
Employing the approximation in (5.99) in (5.98), then inserting (5.100) and expressing the remaining by means of as in equation (5.10), the exponential of the operation element reads
" $ %
% " % $ (5.102)
10This is the content of the Local Limit theorem of de Moivre and Laplace, which can be found for example in [Kre70].
162 Non-selective Discription of Unsharp Measurements with Unitary Backaction Expressing the exponential by means of the decoherence rate ,
" % $
%
" %
$
one obtains the ultimate form of the Gaussian operation elements:
It was first shown by Barchielli et al. [BLP82], that operation elements of the form of (5.103) lead in a certain continuum limit (cp. Barchielli limit in section 5.5) to continuous measure-ments. Barchielli et al. found a description of the thus obtained continuous measurements in the Restricted Path integral formalism. Later Di´osi [Di´o88] was able to derive stochastic differ-ential equations of the Ito-type to represent these continuous measurements. The links between the different representations of continuous measurements are reviewed in [POU96]. Here and in [ADK02b] the unitary back-action of measurements is taken into account. A representation of the corresponding continuous measurements in form of a Master equation and a stochastic differential equation is given in sections 5.5 and 5.6.
The operation element could also be induced by a single measurement with continu-ously many outcomes. A model for such a measurement (although for a system with infinite-dimensional Hilbert space) has been presented by Caves and Milburn [CM87]. Also for qubits, realizations of measurements with uncountably many outcomes exist. One just has to think of the Stern Gerlach experiment, where the spin of an silver atom (spin 1/2) couples to its position and afterwards a position measurement is made. If the latter is an unsharp measurement of the kind suggested in [CM87]11the operation elements are realized.
The operation including its unitary part is given by
"
$ (5.104)
where, using (5.95) and (5.40), we obtain
The N-series operations above correspond for large approximately to a continuous set of Effects:
11The position of the atom couples to the momentum of a meter prepared in a Gaussian wave function, after a certain time the position of the meter is detected.
5.4 N-series 163 Normalisation is formally satisfied if we extend the range of to the whole real axis:
(5.108) The statistical weight of the unphysical values of will be negligible, provided% @% ? * as required in (5.84) and this justifies the formal extension of the values of beyond the physical range (5.101).
Non-Selective Evolution with Gaussian Operation Elements
The last subsection served to bring the operation elements in Gaussian form, which makes it easy to calculate the state change in the non-selective regime. This state change is given in the operator-sum representation by
We are going to expand the r.h.s. up to the linear term in . The expansion of the unitary part of the operation generated by and leads to
Introducing operators acting from the left and from the right (see subsection 5.3.1, denoted by superscript "
$
and "
$
, we can write the integrand of ":
$
In order to restore the original operator ordering after the integration
and
should be understood to act in products with from the left and from the right of , respectively, for example
. Because of the predetermined order all operators carrying a super index commute and the product of exponentials can be written as exponential of the sum of exponents. The integral is Gaussian in :
"'
164 Non-selective Discription of Unsharp Measurements with Unitary Backaction Therefore the quadrature has a closed form solution:
"'
Now we can expand it to the leading linear term in and we can restore the usual operator formalism:
Together with the contribution from and in equation (5.111) we obtain as our final result
The result is in complete agreement with our previous calculation based on carrying out times the operation of an elementary cell, cp. Eq. (5.81). The first term on the r.h.s. represents the unitary dynamical evolution given by the Hamiltonians , and . The second term represents decoherence with decoherence rate from (5.41). The third term corresponds to fur-ther decoherence induced by the unitary part of the back-action. The rightmost term describes generalized friction. The physical meaning of the terms is discussed in detail in subsections 5.3.3 and 5.3.4.
Difference equation (5.116) describes a discrete sequence of measurements subject to con-ditions (5.66)-(5.68). Based on these concon-ditions we have derived (5.116) employing certain approximations – we have changed the order of , and in the operation of the N-series, neglecting the commutators between them cp. (5.87) and (5.89), the estimated order of magnitude of the error is smaller than " % $ "
. We also have approximated the q-number denominator of the Gaussian operation elements by a c-q-number, cp. (5.99), which leads to an error of order " % $ . We further expanded the operation in powers of up to the first order, cf. (5.110) and (5.112), which results in errors of order "
. In the continuum limit the errors all vanish but they can play a role for discrete sequences of measurements. Subsection 5.3.2 contains a more accurate calculation, which confirms the order of magnitude of the errors estimated here, in case of the neglected commutators in (5.87) and (5.89) the errors turn out to be actually smaller: there are no terms proportional to % in difference equation (5.81). These terms cancel when summing over different outcomes of a N-series cp. [ADK02b].