Figure 4.5: Simulation of a sequence of weak measurements with Poissonian statistics and a detector efficiency of
. The back-action of these measurements has no unitary part. The dashed and the black curve represent G
and " respectively. Parameter values:
,
, ,
and , which lead to an average fuzziness of .
4.2 Simulation of Limited Detector Efficiency in the Selective Regime
A subtle problem arises when the detectors do not always reveal the right energy of the Rydberg atoms. There are several scenarios imaginable.
1. The atoms are transfered by the detectors in one of the energy eigenstates or according to the probabilities for measuring , but the output is flawed. Either there is no output or it does not reflect the state after the measurement.
2. The atom crosses the detectors and the atomic state remains entangled with the photon state. The atoms are not registered by the detector and they do not produce a measurement output.
The first case is much easier to treat than the second since the entanglement between photon and atom is destroyed and the final state of the atom is either or . Both cases are taken
4.2 Simulation of Limited Detector Efficiency in the Selective Regime 129 into account in our Monte Carlo simulation where the energy eigenstate of the atom at the end of each elementary measurement is determined by a random generator according to the right probabilities. However two changes have to be invoked in the simulations: a) the output of each elementary measurement has to be modified in accordance with the probabilities for the detectors to produce a false output or no output at all. b) The detuning of the feedback atoms has to match the output, i.e. false feedback in case of a false output and no feedback for missing output.
What happens if the atoms pass the detector unregistered and without state change? Then decoherence through interaction with the environment will lead to an improper mixture of the energy eigenstates, i.e. the atomic state becomes entangled with the state of the environment and the reduced density operator of the atom appears to represent a classical mixture. Since the atom presumably interacts with many degrees of freedom of the environment the improper mixture is likely to be transfered into a proper mixture by a direct or indirect observation of some of these degrees. If this happens case 2) is reduced to case 1). If nevertheless the improper mixture survives, a correct description of the photonic state can be given in terms of a reduced density matrix. The simulation of the sequential measurement according to this approach would involve the density matrix of the photonic field, which changes due to a correct detection of the energy of the Rydberg atom and subsequent application of the feedback mechanism according to
>" $
(4.22) depending on the outcome , . represents the phase shift due to the feedback. If the atom is not registered the state change amounts to:
5
(4.23) The value of
at time is given by
" $
tr " $ (4.24)
and the probability to measure e, g at time is determined by
%
tr " $ (4.25)
where
are again the effects corresponding to the results . These informa-tions together with the definition (4.20) of the estimation function
" $
and the procedure to reduce the noise of the readout (cf. Appendix C of chapter 3) suffice to carry out the numerical simulation of the sequential measurement.
How big is the mistake if the photonic state is an improper mixture but we treat it as a proper mixture and carry out the simulation described for case 1)? At first glance the answer seems to be: there is no mistake at all, if both kind of mixtures posses the same density operator, they cannot be distinguished by measurements on the system. However, the values of
" $
for both cases are different. Let us shortly discuss this difference.
130 Realisation of Sequential Unsharp Measurements In case of a proper mixture the selective regime shows either the value of
" $
after a projection of the atomic state on or the value of
" $
corresponding to a projection on . Thus
are given by (4.10). Expressing in the second case the entangled state of photon & atom by means of non-orthogonal pointer states (cp. subsection 2.5.3)
it is easy to see that here the value of
Thus in this case the value of
has not changed due to the coupling between photon and atom. Please note, that the value of
" $
in the second case is the average value of the values
" $
assumes in the first case weighted with their probabilities. The difference of the values of
" $
in both cases, however, is not detectable. This is due to the fact that the expectation value
the probabilities for the outcomes of any subsequent measurement or series of measurements is the same in both cases. Therefore its not possible to distinguish both cases by looking at the result of a single measurement or a sequence of consecutive measurements. The different values of
in (4.26) and (4.29)thus describe physically equivalent situations in the context of our setup2. We are thus free to represent the change of the photonic state in case 2) in the same way as in case 1) and carry out the simulations accordingly by assuming that the final state of the atom is either or .
2more precisely, provided measurements of correlations between atom and photon are excluded
Chapter 5
Non-selective Discription of Unsharp Measurements with Unitary Backaction
5.1 Introduction
In this chapter the state change of a qubit subject to a weak sequential measurement is described in the non-selective regime. We thus look at the state change when the result of the measurement is unknown or equivalently at the expected state change when the state after the measurement is averaged over a large ensemble of systems which have been subject to the measurement. In both cases there is no particular result selected, hence the term “non-selective regime”.
The state change due to a single measurement out of the sequence of measurements can be described in the non-selective regime by the trace preserving operation :
":
$ -5
(5.1)
with operation elements (sometimes called Kraus operators)
(5.2) Here we are dealing as in the case of our sequential measurement scheme to monitor Rabi oscillations (cp. chapters 3 and 4) with measurements with two outcomes and . Though a restriction, they correspond to an important class of measurements of a single observable of the qubit as opposed to the joined measurements of several non-commuting observables, which require at least three outcomes1.
We will investigate sequential measurements consisting of a sequence of approximately instantaneous measurements carried out at times , / +) 3 . Between the measurements the system evolves continuously and unitarily (original dynamics). In order to
1The Effects of measurements with only two outcomes always commute since .
132 Non-selective Discription of Unsharp Measurements with Unitary Backaction obtain an expression for the state change during the sequential measurement which can still be analytically handled, it can be advantageous to carry out a “continuum limit” where the time between two consecutive measurements of the sequential measurement is going to zero and to write down the equation of motion for the corresponding continuous measurement. In the non-selective regime of measurement this equation of motion will be a differential equation (master equation) and there are more methods to solve it than to solve the difference equation which describes the original sequential measurement.
In the selective regime, on the other hand, the equation of motion has to comprise the stochastic process of measurement and thus leads in the limit of continuous measurements to a stochastic differential equation2.
We will concentrate here on the non-selective regime. The master equation can be under-stood as a means to compute approximately the dynamics of a system under the influence of a discrete, sequential measurement. A more accurate description taking into account the discrete nature of the measurement is given by a corresponding difference equation. We will derive such a difference equation for elementary measurements which are very weak and follow rapidly upon each other as compared to the timescale of the dynamics of the system without measure-ment. It will reveal properties of the evolution such as decoherence and dissipation. These terms are very similar to terms that have been discovered by Caldera and Leggett [CL83] and Dio´si [Di´o93] in the context of simple but infinite dimensional quantum systems coupled to a heath bath (Quantum Brownian motion).
The emphasis of this investigation lies on the influence of the unitary part
of the back-action of the measurements on the non-selective evolution of the state. Here and in a corre-sponding paper by Audretsch et al. [ADK02b] it is for the first time stated how a very general kind of unitary back-action enters the master equation. The knowledge of the influence of
is important because it can be employed to control the system, since can represent instanta-neous feedback (see below).
The equations of motions to be derived in this chapter can also be used to describe the evolution of a system recurringly interacting with a second system provided that the second system does not “remember” the influence of the first system at former times (Markov Process).
Typically the second system could be an environment or it could consist of a number of systems each of which interacts only once with the first system ( as in a sequence of scattering processes).
We are not dealing with the most general form of such an interaction with a qubit, which would correspond to an operation with four operation elements [NC00], instead of two as in (5.1). But note, that the two operation elements can represent such important operations as amplitude -and phase damping as well as bit flips -and phase flips.
Master equations for special cases of measurements with non-minimal disturbance of the state , i.e., a non-vanishing unitary part of the operation elements have been considered in the literature. A master equation for general feedback was derived by Wiseman [Wis95a]. In the Markovian limit, if the time delay between feedback and measurement vanishes and the
2Stochastic differential equations and methods to solve them are well described in [Gar83], an account on the stochastic differential equation for a continuous (unsharp) position measurement was given by Dio´si [Di´o88]
5.1 Introduction 133 feedback depends only on the outcome of the last measurement (instantaneous feedback), the action of the feedback can be represented in the unitary part of the operation of the measure-ment. However [Wis95a] does not comprise our results since it deals with a special kind of continuous measurements. They have poissonian statistics and allow finite state changes (in the selective regime) during infinitesimal time intervals. More precisely this means that only very seldom a certain measurement result occurs and that this is then connected to a finite state change during an infinitesimal time while for other measurement results the state changes only infinitesimally. Our studies concentrate on moderate probabilities for both measurement results and we require the state change in the continuum limit to be infinitesimal during infinitesimal times. Thus Wiseman’s and our studies do not overlap.
In a later paper Wiseman [Wis95b] employed the operation formalism to analyze a homo-dyne measurement in quantum optics to apply instantaneous feedback in order to minimize disturbance, i.e., compensate the unitary part of the operation. Korotkov investigated a mea-surement with non-minimal disturbance in the context of continuous meamea-surement of a qubit by means of a single electron transistor [Kor00, Kor01b]. He noted that this non-minimal distur-bance acts in the master equation like a change of the distance between the two energy levels of the qubit. We will find this phenomenon as a special case of our studies ( * , where H is the Hamiltonian of the system’s evolution without measuerements). Korotkov also derived a modification of the stochastic master equation (selective regime) due to the non-minimal dis-turbance [Kor00], which is averaged out in the non-selective case.
In the context of continuous position measurement Caves and Milburn [CM87] presented a sequential measurement which posses gaussian shaped Effects. We obtain such Effects for groups of N measurements (N-series). They also employed a special feedback, which served to compensate jumps in the mean position and the mean momentum. In the continuum limit the master equation shows a dissipation term and an additional decoherence term induced by the feedback, similar to the ones found by Caldera and Leggett and the terms occuring in our difference equation.
Many examples of the application of instantaneous feedback in continuous measurements by means of external changes of the Hamiltonian of the system can be found in the literature of quantum control (e.g. [DJJ00, DTPW99]). They correspond to special choices of the unitary part of the measurement operation and obey the equations of motions derived here, provided they are in the limit of continuous measurements connected to Wiener processes [Gar83] with vanishing state changes (in the nonselective - as well as in selective regime) during arbitarary small times.
In the sections of this chapter we proceed as follows. We first consider the non-selective operation of an “elementary cell” (section 5.3), i.e. a period of unitary development followed by a single measurement, which forms the building block of a sequential measurement. We will derive a difference equation for an elementary cell. In this context the influence of a single measurement with two operation elements (5.2) is extensively discussed. We are also interested in bundling the sequence of measurements in subsequences of N measurements (“N-series“) . There are several advantages related to N-series:
134 Non-selective Discription of Unsharp Measurements with Unitary Backaction As we have seen in chapter 3 the motion of the state can be visualized in real time by plotting the sequence of results from the N-series against time. This is because a N-series yields one out of different results thereby encoding the information about the state in vertical hight rather than in the density of points on a horizontal axis as it would be in a plot of the results of the elementary measurements with only two outcomes over time.
The Effects of a N-series have gaussian shape and the sequence of N-series therefore represents a Wiener process. The stochastic master equations of such processes are well known [Di´o88]. The gaussian shape of the Effects indicates a direct relation to restricted path integrals with weighting functionals (see also influence functionals [GJK 96]) of gaussian shape which describe continuous measurements [AM98].
Because of their relation to the well investigated class of continuous measurements with gaussian statistics, sequential measurements employing N-series are good candidates for experimentally feasible realizations of these continuous measurements. Untill now there are not many experimental realizations of continuous measurements known. It can be argued that continuous measurements are in fact sequential measurements which appear only continuous on a coarse grained time scale.
Like the Effects the operation elements of a N-series posses approximately Gaussian shape. Therefore the corresponding nonselective operation can be approximated by an integral with closed form solution.
In section 5.4 we will derive a difference equation for the N-series. This is conveniently done up to second order in the small quantities employing the operation of an elementary cell. We will also pursue a different approach based on the operation elements of a N-series. The latter will enable us to immediately write down a stochastic master equation for the selective regime of measurement. But before we do so in section 5.6, we will discuss how to find the right continuum limit and derive the master equation for the nonselective regime in section 5.5.