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Generalized stochastic processes and continual observations in quantum mechanics
Journal of Mathematical Physics
Abstract
We give here a mathematically rigorous form to an earlier work by Barchielli, Lanz, and Prosperi, in which it was found that a generalized stochastic process describes the results of continual observations of the position of a quantum particle. With the help of Albeverio and Ho&slash;egh-Krohn’s theory of Feynman path integrals, we define the characteristic functional of this process and demonstrate that it possesses the necessary properties of normalization, continuity, and positive definiteness. An explicit calculation of the Feynman path integral which defines the functional allows an analysis of the process to be made.
... By this we mean the situation in which one or more quantities are followed in their dynamical evolution and probabilities on their "trajectory space" are extracted from quantum mechanics. A consistent theory of measurements continuous in time has been developed (Davies, 1969(Davies, , 1970(Davies, , 1971(Davies, , 1976Barchielli et al., 1982Barchielli et al., , 1983Barchielli et al., , 1985Lupieri, 1983;Barchielli and Lupieri, 1985a,b;Barchielli, 1986a,b;Holevo, 1988Holevo, , 1989) and applications worked out Davies, 1981, 1982;Holevo, 1982;Barchietli, 1983Barchietli, , 1985Barchietli, , 1987Barchietli, , 1988Barchietli, , 1990Barchietli, , 1991Barchietli, , 1993 Now a natural question is: if during a continuous measurement a certain trajectory of the measured observable is registered, what is the state of the system soon after, conditioned upon this information (the a posteriori state)? By using ideas from the classical filtering theory for stochastic processes and the formulation of continuous measurements in terms of quantum stochastic differential equations (Barchielli and Lupieri, 1985a,b;Barchielli, 1986a) an It6 stochastic differential equation for the a posteriori states has been obtained and solved in some significant cases (Belavkin, 1988(Belavkin, , 1989a(Belavkin, ,b, 1990a(Belavkin, ,b,c, 1992Staszewski, 1989, 1991;Chru~cifiski and Staszewski, 1992;Holevo, 1991;Staszewski and Staszewska, 1992). ...
... The theory of continuous measurements in the case of quantum point processes (typically, counting of particles) was initiated by Davies (1969Davies ( , 1970Davies ( , 1970, while the general formulation of continuous measurements of any kind of observables is due to the Milan group (Barchielli et al., 1982Lupieri, 1983). The idea is to use families of instruments to represent measurements continuous in time. ...
... A large class of such families of instruments has been constructed by using Fourier transform techniques (Barchielli et al., 1982Lupieri, 1983;Barchielli and Lupieri, 1985a,b;Holevo, 1988Holevo, , 1989). Another way to obtain continuous measurements is by the indirect measurement scheme described by equation (1.13) (Barchielli and Lupieri, 1985a,b;Barchielli, 1986a); the mathematical techniques one needs in this case are based on "quantum stochastic calculus" (Hudson and Parthasarathy, 1984). ...
In recent years a consistent theory describing measurements continuous in time in quantum mechanics has been developed. The result of such a measurement is atrajectoryfor one or more quantities observed with continuity in time. Applications are connected especially with detection theory in quantum optics. In such a theory of continuous measurements one can ask what is the state of the system given that a certain trajectory up to timet has been observed. The response to this question is the notion ofa posteriori states and afilteringequation governing the evolution of such states: this turns out to be a nonlinear stochastic differential equation for density matrices or for pure vectors. The driving noise appearing in such an equation is not an external one, but its probability law is determined by the system itself (it is the probability measure on the trajectory space given by the theory of continuous measurements).
... The theory of measurements continuous in time in quantum mechanics (quantum continual measurements) has been formulated by using the notions of instrument and positive operator valued measure [1]- [6] arisen inside the operational approach [1,7] to quantum mechanics, by using functional integrals [8,9,10], by using quantum stochastic differential equations [11]- [23] and by using classical stochastic differential equations (SDE's) [12]- [6]. Various types of SDE's are involved, and precisely linear and non linear equations for vectors in Hilbert spaces and for trace-class operators. ...
... Let us end by discussing the connections between the invariant measure for eq. (12) and the equilibrium state of the master equation (9). The following considerations apply both to the diffusive and to the jump case; we assume that H is finite dimensional and that there exists a unique invariant probability measure µ and that Supp µ =: M 0 ⊂ M . ...
The theory of measurements continuous in time in quantum mechanics (quantum continual measurements) has been formulated by using the notions of instrument and positive operator valued measure, functional integrals, quantum stochastic differential equations and classical stochastic differential equations (SDE's). Various types of SDE's are involved, and precisely linear and non linear equations for vectors in Hilbert spaces and for trace-class operators. All such equations contain either a diffusive part, or a jump one, or both. In this paper we introduce a class of linear SDE's for trace-class operators, relevant to the theory of continual measurements, and we recall how such SDE's are related to instruments and master equations and, so, to the general formulation of quantum mechanics. We do not present the Hilbert space formulation of such SDE's and we make some mathematical simplifications: no time dependence is introduced into the coefficients and only bounded operators on the Hilbert space of the quantum system are considered. Then we introduce the notion of a posteriori state and the non linear SDE satisfied by such states and we give conditions from which such equation is assured to preserve pure states and to send any mixed state into a pure one for large times. Finally we review the known results about the existence and uniqueness of invariant measures in the purely diffusive case and we give some concrete examples of physical systems.
... That a continuous measurement of a physical observable can be described in this way has been demonstrated by , and also by Ueda et al. [15] using a quite different approach. For the theory of continuous measurement the reader is referred to these works and references [16,17,18,19]. We refer to this measurement process as a continuous projection measurement because in the absence of any system evolution, the sole effect of this term is to reduce the off-diagonal elements of the density matrix to zero in the eigenbasis of that observable. ...
We present a method for obtaining evolution operators for linear quantum trajectories. We apply this to a number of physical examples of varying mathematical complexity, in which the quantum trajectories describe the continuous projection measurement of physical observables. Using this method we calculate the average conditional uncertainty for the measured observables, being a central quantity of interest in these measurement processes. Comment: Revtex, 10 pages, 1 eps figure. v2: corrections to the operator disentangling relation in appendix B
Quantum mechanics started as a theory of closed systems: the state of the system is a vector of norm one in a Hilbert space and it evolves in time according to the Schrödinger equation (B.11). In order to describe also a possible uncertainty on the initial state, a “statistical” formulation of quantum mechanics has been developed: the states are represented by statistical operators (Sect. B.3.1), also called density matrices, and their evolution is given by the von Neumann equation (B.18). This statistical formulation revealed to be well suited also for open systems. General evolution equations for density operators appeared under the names of master equations and quantum dynamical semigroups [1–5] (Sect. B.3.3); these concepts were generalised and gave rise to the theory of quantum Markov semigroups [6,7].
Ordinary Quantum Mechanics has to do with an idealized situation in which a system is prepared in a given state at an initial time t
0
, it is left to evolve freely for some time and it is submitted to some kind of measurement at a single final time t
1 or at certain well separated subsequent times t
1, t
2,... In any case the single process is considered as pratically istantaneous.
The irreversible and stochastic behaviour of the continuously (in time) observed quantum system expressed by the reduction of the wave function cannot be described within the standard formulation of quantum mechanics. The state of a quantum system under continuous nondemolition observation evolves according to weakly-nonlinear filtering equation announced by Belavkin in 1988. This quantum stochastic differential equation of Ito type can be obtained from the unitary evolution of the compound system — “system plus measuring device” by conditioning with respect to the measuring trajectories. The equation has been applied to some significant physical problems. These include: stochastic resolution of quantum Zeno paradox for a free particle, watchdog effects for quantum particle for various cases of the diffusion nondemolition measurement and relaxation without mixing of an atom under a counting observation.
Operation‐valued stochastic processes give a formalization of the concept of continuous (in time) measurements in quantum mechanics. In this article, a first stage M of a measuring apparatus coupled to the system S is explicitly introduced, and continuous measurement of some observables of M is considered (one can speak of an indirect continuous measurement on S). When the degrees of freedom of the measuring apparatus M are eliminated and the weak coupling limit is taken, it is shown that an operation‐valued stochastic process describing a direct continuous observation of the system S is obtained.
Limiting formulas are deduced for the state of a quantum system which is subjected to a rapid sequence of many measurements of the same two-valued observable. In the case of a semibounded Hamiltonian, the dynamics usually gets reduced to the subspace spanned by the eigenstates of the observable. In particular, the values of the observable "freeze." Exceptions to freezing occur for states which cannot be approximated by eigenstates of finite energy.
In Quantum Mechanics, as well known, due to the occurrence of the so called interference terms, any statement has to refer to a definite experiment performed on an object, i.e. to the response of a specific apparatus. No unambiguous meaning can be attached to statements on the value of any quantity independently of an explicit measurement of it.
This course-based monograph introduces the reader to the theory of continuous measurements in quantum mechanics and provides some benchmark applications.
The approach chosen, quantum trajectory theory, is based on the stochastic Schrödinger and master equations, which determine the evolution of the a-posteriori state of a continuously observed quantum system and give the distribution of the measurement output. The present introduction is restricted to finite-dimensional quantum systems and diffusive outputs. Two appendices introduce the tools of probability theory and quantum measurement theory which are needed for the theoretical developments in the first part of the book.
First, the basic equations of quantum trajectory theory are introduced, with all their mathematical properties, starting from the existence and uniqueness of their solutions. This makes the text also suitable for other applications of the same stochastic differential equations in different fields such as simulations of master equations or dynamical reduction theories.
In the next step the equivalence between the stochastic approach and the theory of continuous measurements is demonstrated.
To conclude the theoretical exposition, the properties of the output of the continuous measurement are analyzed in detail. This is a stochastic process with its own distribution, and the reader will learn how to compute physical quantities such as its moments and its spectrum. In particular this last concept is introduced with clear and explicit reference to the measurement process.
The two-level atom is used as the basic prototype to illustrate the theory in a concrete application. Quantum phenomena appearing in the spectrum of the fluorescence light, such as Mollow’s triplet structure, squeezing of the fluorescence light, and the linewidth narrowing, are presented.
Last but not least, the theory of quantum continuous measurements is the natural starting point to develop a feedback control theory in continuous time for quantum systems. The two-level atom is again used to introduce and study an example of feedback based on the observed output.
In this chapter we complete the task of showing that the SDE approach can be reduced to the usual formulation of quantum mechanics.
The last notion which we need is the one of “instrument”; its definition, meaning and properties are presented in Sect. B.4.
We give also the important concept of characteristic operator, a kind of Fourier transform of the instruments, we show that it satisfies an evolution equation, in some sense similar to a master equation, and we show how to obtain explicit formulae for the moments of the output by means of this characteristic operator.
A generalised stochastic process is obtained for the Boltzmann distribution function in quantum mechanics: this is an example of an objective state-space description of a quantum system.
A two-stage measuring apparatus (detector and meter) for detecting gravitational waves is analyzed using a recently developed formalism for continuous observations in quantum mechanics. Detector and meter are modeled as quantum-mechanical dissipative linear systems; a large class of couplings between them is considered, so obtaining a unified treatment of various models which have appeared in the literature. In the simplest cases, explicit expressions for uncertainties in the estimate of the gravitational force are obtained. In the general case it is shown how the problem of the estimation of the force can be treated and general results are given.
The physical idea of a continual observation on a quantum system has been recently formalized by means of the concept of operation valued stochastic process (OVSP). In this article, it is shown how the formalism of quantum stochastic calculus of Hudson and Parthasarathy allows, in a simple way, for constructing a large class of OVSP’s that in particular contains the quantum counting processes of Davies and Srinivas and continual ‘‘Gaussian’’ measurements. This result is obtained by means of a stochastic dilation of the OVSP’s: at the level of the enlarged system probabilities turn out to be expressed in terms of projection valued measures associated with certain time-dependent, commuting, self-adjoint operators.
Transient phenomena that occur when a free quantum particle undergoes a continuous nondemolition observation of its position are discussed. Independently of a particular case of observation (specified by particle mass, the strength of coupling, and the value of initial dispersion of the Gaussian wave packet) the dispersion of the posterior Guassian wave packet always decreases in the beginning of observation, then (after some time) takes the form of oscillations decaying to the asymptotic value (/2m)1/2. We say that the quantum particle is frightened when observation begins, then it is trembling, and finally relaxes. The discussion is preceded by a brief presentation of basic ideas of the theory of continuous in time quantum measurements.
A structure of generator of a quantum dynamical semigroup for the dynamics of a test particle interacting through collisions with the environment is considered, which has been obtained from a microphysical model. The related master-equation is shown to go over to a Fokker–Planck equation for the description of Brownian motion at quantum level in the long wavelength limit. The structure of this Fokker–Planck equation is expressed in this paper in terms of superoperators, giving explicit expressions for the coefficient of diffusion in momentum in correspondence with two cases of interest for the interaction potential. This Fokker–Planck equation gives an example of a physically motivated generator of quantum dynamical semigroup, which serves as a starting point for the theory of measurement continuous in time, allowing for the introduction of trajectories in quantum mechanics. This theory has in fact already been applied to the problem of Brownian motion referring to similar phenomenological structures obtained only on the basis of mathematical requirements.
In the more refined description of concrete measuring procedures that is allowed by modern quantum theory, objectivity elements can be recovered that are usually deemed to be forbidden in quantum mechanics; actually, in the case of amacrosystem, a way towards an objective physical description appears.
A formalism developed in previous papers for the description of continual observations of some quantities in the framework of quantum mechanics is reobtained and generalized, starting from a more axiomatic point of view. The statistics of the observations of continuous state trajectories is treated from the beginning as a generalized stochastic process in the sense of Gel'fand. An effect-valued measure and an operation-valued measure on the -algebra generated by the cylinder sets in the space of trajectories are introduced. The properties of the characteristic functional for the operation-valued stochastic process are discussed and, through a suitableansatz, a significant class of such processes is explicitly constructed, which contains the examples of the preceding papers as particular cases.
An alarming result has been derived in the theory of unstable systems by considering a sequence of instantaneous observations.
According to it, continued observation prevents decay. We show that this paradox is avoided if the fiction of the instantaneous
observation is replaced with the realistic hypothesis that a measurement (hence the corresponding reduction of the wave packet)
takes a nonvanishing time interval. We derive the corresponding formulae, first for a single measurement on a single unstable
system, then for a sequence of measurements. Actually the results hold not only for measurements, but for interactions of
all kinds. The upshot is that the lifetime of an unstable system, while depending on its interactions with a second system,
cannot be prolonged arbitrarily by mere continued observation. And in any event such a life span is an objective property
of the system under investigation.
Un risultato allarmante è stato derivato nella teoria dei sistemi instabili considerando una sequenza di osservazioni istantanee.
In accordo con questa l'osservazione continuata impedisce il decadimento. Si mostra che si evita questo paradosso se la finzione
dell'osservazione istantanea è sostituita con l'ipotesi realistica che la misura (e quindi la riduzione corrispondente del
pacchetto d'onde) prende un'intervallo di tempo che non tende a zero. Si derivano le formule corrispondenti prima per una
misura singola su un sistema instabile singolo, quindi per una sequenza di misurazioni. Effettivamente, i risultati valgono
non solo per le misurazioni ma anche per interazioni di tutti i tipi. Il risultato è che la vita media di un sistema instabile,
mentre dipende dalle sue interazioni con un secondo sistema, non può essere prolungata arbitrariamente da semplici osservazioni
continuate. E in qualsiasi evento tale durata di vita è una proprietà oggettiva del sistema in esame.
Рассматривая последовательность мтновенных наблюдений, получается «тревожный» результат в теории нестабильных систем. Согласно
этому результату непрерывное наблюдение препятствует распаду. Мы показываем, что этот парадокс устраняется, если абстракция
мгновенного наблюдения заменяется реальной гипотезой, что измерение (следовательно, соответствующее преобразование волнового
пакета) занимает конечный временной интервал. Мы выводим соответствующие формулы, сначала для одиночного измерения на отдельной
нестабильной системе, а затем для последовательности измерений. Естественно, полученные результаты справедливы не только для
измерений, но и для взаимодействий всех типов. Мы получаем, что время жизни нестабильной системы, хотя и зависит от взаимодействия
со второй системой, не может быть увеличено произвольно за счет непрерывного наблюдения. Протяженность времени жизни любого
события представляет объективное свойство рассматриваемой системы.
Starting from the recently introduced formalism of continual measurements in quantum mechanics, it is shown that, for the
quantum open systems, it is possible to construct probability distributions for the values at all times of certaion observables,
without the continual measurement of such observables perturbing the dynamics of the system. More precisely, starting from
the quantum description of an open system, a generalized stochastic process for certain observables is constructed, which
is independent of the fact that these observables are actually measured or not. The example of the quantum Brownian motion
is developed in detail. In such an example it is shown how thea priori arbitrary elements of the formalism are in reality determined by the dynamics of the system.
Partendo dal formalismo delle misurazioni continue in meccanica quantistica recentemente introdotto, si mostra che per i sistemi
aperti quantistici è possibile costruire delle distribuzioni di probabilità per i valori a tutti i tempi di certe osservabili,
senza che la misurazione continua di tali osservabili perturbi la dinamica del sistema. Piú precisamente, partendo dalla descrizione
quantistica di un sistema aperto, si costruisce un processo stocastico generalizzato per certe osservabili, che è indipendente
dal fatto che queste osservabili siano effettivamente misurate o no. L'esempio del moto browniano quantistico è sviluppato
in dettaglio. In tale esempio si mostra come gli elementia priori arbitrari del formalismo siano in realtà determinati dalla dinamica del sistema.
A quantum stochastic model of continuous nondemolition observation of photons emitted by an atom is given. The statistics of photon counts is found. The posterior stochastic Schrödinger equation for the atom is solved. A limit theorem for this equation showing the relaxation of the atom to its ground state without mixing is proved.
Continuous (in time) measurements can be introduced in quantum mechanics by using operation-valued measures and quantum stochastic calculus. In this paper quantum stochastic calculus is used for showing the connections between measurement theory and open-system theory. In particular, it is shown how continuous measurements are strictly related to the concept of output channels, introduced in the framework of quantum stochastic differential equations by Gardiner and Collet.
Starting from the results obtained by Caves, a certain class of continuous measurements is constructed. These measurements describe a situation in which the output is some continuous function of time and in which the apparatus needs some finite-time interval to produce any output.
Starting from the idea of generalized observables, related to effect-valued measures, as introduced by Ludwig, some examples oi continual observations in quantum mechanics are discussed. A functional probability distribution, on the set of the trajectories which are obtained as output of the continual observation, is constructed in the form of a Feynman integral. Interesting connections with the theory of dynamical semi-groups are pointed out. The examples refer to small systems, but they are interesting for the light they may shed on the problem of the connections between the quantum and the macroscopic levels of description for a large body; the idea of continuous trajectories indeed seems to be essential for the macroscopic level of description.
We generalize Umegaki's definition of an expectation on the algebra of all bounded operators on a Hilbert space, and classify certain classes of expectations subject to a covariance condition with respect to a unitary representation of a given locally compact group. While Umegaki's expectations only exist for discrete observables, we show that interesting classes of our expectations exist in the general case of continuous observables, and discuss the implications of our work in the theory of quantum measurements.
By introducing the family of Feynman maps Fs, we show that our earlier definition of the Feynman path integral F=F1 can be obtained as the analytic continuation of the Wiener integral E =F−i. This leads to some new results for the Wiener and Feynman integrals. We establish a translation and Cameron–Martin formula for the Feynman maps Fs, having applications to nonrelativistic quantum mechanics. We also estalish a (weak) dominated convergence theorem for F1=F.
An attempt is made to interpret quantum mechanics as a statistical theory, or more exactly as a form of non-deterministic statistical dynamics. The paper falls into three parts. In the first, the distribution functions of the complete set of dynamical variables specifying a mechanical system (phase-space distributions), which are fundamental in any form of statistical dynamics, are expressed in terms of the wave vectors of quantum theory. This is shown to be equivalent to specifying a theory of functions of non-commuting operators, and may hence be considered as an interpretation of quantum kinematics. In the second part, the laws governing the transformation with time of these phase-space distributions are derived from the equations of motion of quantum dynamics and found to be of the required form for a dynamical stochastic process. It is shown that these phase-space transformation equations can be used as an alternative to the Schrödinger equation in the solution of quantum mechanical problems, such as the evolution with time of wave packets, collision problems and the calculation of transition probabilities in perturbed systems; an approximation method is derived for this purpose. The third part, quantum statistics, deals with the phase-space distribution of members of large assemblies, with a view to applications of quantum mechanics to kinetic theories of matter. Finally, the limitations of the theory, its uniqueness and the possibilities of experimental verification are discussed.(Received November 12 1947)
Some criticisms to von Neumann's treatment of the measuring process are recalled and discussed.The problem is reconsidered and reinvestigated, in the spirit of the “philosophy” of Jordan and Ludwig, on the basis of an ergodic theorem recently given.The measuring apparatus is schematized as a macroscopic system which possesses, besides the energy, at least another macroscopic constant of the motion. The value of this constant characterizes an invariant manifold (“channel”). In each manifold certain ergodicity conditions hold and there exists an equilibrium macro-state towards which the system evolves spontaneously. The apparatus is assumed to be initially in the equilibrium state belonging to a given channel and the interaction with the observed system determines a transition of the apparatus towards a state belonging to another channel, which depends on the initial state of the observed system. Then the apparatus evolves towards a new equilibrium state.The ergodicity conditions employed are sufficiently realistic, since it has been proved that they are in particular satisfied by that class of Hamiltonians for which Van Hove succeeded in deriving a master equation.
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