Quantum measurements in continuous time, non Markovian evolutions and feedback
In this article we reconsider a version of quantum trajectory theory based on the stochastic Schrödinger equation with stochastic coefficients, which was mathematically introduced in the ’90s, and we develop it in order to describe the non Markovian evolution of a quantum system continuously measured and controlled thanks to a measurement based feedback. Indeed, realistic descriptions of a feedback loop have to include delay and thus need a non Markovian theory. The theory allows to put together non Markovian evolutions and measurements in continuous time in agreement with the modern axiomatic formulation of quantum mechanics. To illustrate the possibilities of such a theory, we apply it to a two-level atom stimulated by a laser. We introduce closed loop control too, via the stimulating laser, with the aim to enhance the “squeezing” of the emitted light, or other typical quantum properties. Note that here we change the point of view with respect to the usual applications of control theory. In our model the “system” is the two-level atom, but we do not want to control its state, to bring the atom to a final target state. Our aim is to control the “Mandel -parameter” and the spectrum of the emitted light; in particular the spectrum is not a property at a single time, but involves a long interval of times (a Fourier transform of the autocorrelation function of the observed output is needed).
1 Quantum trajectories and control
Stochastic wave function methods for the description of open quantum systems are now widely used [1, 2, 3, 4, 5] and are often referred to as quantum trajectory theory. These approaches are very important for numerical simulations and allow the continuous measurement description of detection schemes in quantum optics, namely direct, homodyne and heterodyne photo-detection [7, 6, 8, 9]. In the Markovian case, the stochastic differential equations of the quantum trajectory theory can be interpreted in terms of measurements in continuous time because they can be related to positive operator valued measures and instruments [10, 11, 12], which are the objects representing observables and state changes in the modern axiomatic formulation of quantum mechanics. Moreover, these stochastic differential equations can be deduced from purely quantum evolution equations for the measured system coupled with a quantum environment, combined with a continuous monitoring of the environment itself. Such a representation is based on the use of quantum fields and quantum stochastic calculus [13, 14, 15, 16, 17, 8, 18].
The whole quantum trajectory theory is well developed in the Markovian case, but to include memory effects is more and more important. A generalization by Diósi, Gisin and Strunz, based on the introduction of functional derivatives acting on the “past” inside the stochastic Schrödinger equation (SSE), is able to describe dynamical memory effects [19, 20], but fails to have an interpretation in terms of continuous measurements .
Another way to include non Markovian effects and to maintain at the same time the continuous measurement interpretation is to start from the linear SSE and to generalize it by allowing for stochastic coefficients. This can be done without violating the axiomatic formulation of quantum mechanics and a non Markovian quantum trajectory theory can be developed in a mathematically consistent way . The key point to get this result is that the non Markovian character is introduced by using stochastic coefficients in the linear version of the stochastic Schrödinger equation, without other modifications, and this allows to construct completely positive linear dynamics and instruments as required by the principles of a quantum theory.
More recently, motivated by the growing interest for non Markovian evolutions, we begun to analyse possible physical applications of this theory. Some applications to systems affected by coloured noises and continuously monitored have been already developed [23, 24]. This paper, instead, will be focused mainly on feedback control.
In quantum optical systems, even when the Markov approximation for the reduced dynamics is well justified, memory can enter into play when imperfections in the stimulating lasers are taken into account  and when feedback loops are introduced to control the system [26, 27, 28, 29, 30, 31, 4, 32, 33]. The so called closed loop control is based on the continuous monitoring of the system and, so, it fits well in the theory of measurements in continuous time. In some approximations, one can consider an instantaneous and very singular feedback and in this case the usual Markov framework is sufficient; however, more realistic descriptions of the feedback loop, including delay, need a non Markovian theory [27, 28, 29, 9, 30, 31, 32, 33].
In this paper we present the non Markovian version of the theory of quantum measurements in continuous time, based on the SSE and the stochastic master equation (SME) with stochastic coefficients. Then, we develop such a theory, we explain how the stochastic coefficients can describe the most general measurement based feedback and also some imperfections in the apparatus, and we show how to get the physical probabilities for the output of the observation, its moments and other related quantities of physical interest. Indeed, in quantum optical systems the moments of the stochastic output are connected to the Mandel -parameter and to the spectrum of the emitted light (homodyne and heterodyne spectra) and allow for the study of typical quantum properties of the emitted light, such as squeezing [34, 9, 35]. To illustrate these concepts we shall use a prototype model, a two-level atom stimulated by a laser, which is known to have a rich spectrum and to emit squeezed light under particular conditions. Our emphasis will be on the possibility of modelling a non perfectly monochromatic and coherent stimulating laser and of modelling a measurement based feedback. We shall introduce closed loop control, via the stimulating laser, with the aim to control the squeezing in the observed spectrum, not to control the state of the system. Let us stress the change of point of view with respect to the usual applications of control theory [36, 37, 38, 39]. Here the “system” is the two-level atom, but we do not want to control its state, so as to bring the atom to a final target state, which is however possible inside our theory. Our aim is to control the properties of the emitted light; moreover, we want to control the spectrum, which is not a property at a single time, but involves a long interval of times (a Fourier transform in time is needed).
2 The stochastic Schrödinger equation and the stochastic master equation
The best way to introduce memory in quantum evolutions is to start from a dynamical equation in Hilbert space; this approach automatically guarantees the complete positivity of the evolution of the state (statistical operator) of the system. Moreover, considering the linear version of the SSE allows to construct the instruments related to the continuous monitoring even in the non Markov case [22, 23, 24]. We shall introduce first several mathematical objects, from the linear SSE (1) to the instruments (14), and later, thanks to these latter, we shall give a consistent physical interpretation of the whole construction.
Let be the Hilbert space of the quantum system of interest, a separable complex Hilbert space, with inner product . Moreover, we denote by the space of the bounded operators on , by the trace class and by the convex set of the statistical operators.
2.1 The linear SSE and the reference probability
Let us consider a reference probability space with a filtration of -algebras satisfying the usual hypotheses, i.e. with implies , and . As usual, we denote by the generic sample point in the sample space . The driving noises of the key SDE (1) are defined in this filtered probability space; they are continuous standard Wiener processes and Poisson processes . Under the reference probability , all these processes are independent and are adapted, with increments independent from the past, with respect to the given filtration. Every Poisson process is taken with trajectories continuous from the right and with limits from the left (càdlàg); let be the intensity of .
We assume also we have a set of stochastic processes , , with values in , such that and , are adapted processes with trajectories continuous from the left and with limits from the right (càglàd, continuity in the strong operator topology). We assume also
where does not depend on the sample point .
Then, we introduce the linear SSE, for processes with values in ,
with initial condition
Equation (1) is an Itô-type stochastic differential equation admitting a unique strong solution [22, Proposition 2.1]. The solution is continuous from the right and with limits from the left (càdlàg); the symbol means to take the limit from the left.
Here only bounded coefficients are considered, in order not to have mathematical complications, but generalizations to unbounded coefficients are of physical interest. Note that the filtration can be taken bigger than the natural filtration generated by the driving noises and , so that the processes can depend also on some other external noises.
2.2 The linear stochastic master equation
The SSE (1) can be translated into a stochastic differential equation for trace class operators. Indeed, adopting for a moment the Dirac notation, we can consider the rank one selfadjoint operator and compute its stochastic differential. In this way we get a closed linear equation, which can be extended to a process with values in and whose initial condition can be any pure or mixed state. This is the linear stochastic master equation (SME) [22, Propositions 3.2 and 3.4]:
with initial condition
The operator is the stochastic Liouvillian
The SME (3) admits a unique strong solution. Typically, the solution is not Markovian because depends on the past through the random operators , .
2.3 The new probability and the non linear equations
Let us fix a non random state and define the stochastic processes
By taking the trace of (3), we have that satisfies the Doléans equation
with and where the coefficients and depend on the initial condition (4).
The new probability
Due to the martingale property of , the probabilities are consistent, in the sense that for , .
Of course, the new probability modifies the distributions of all the processes appearing in the theory, in particular the distribution of the processes and .
The Girsanov transformation
A very important property is that a Girsanov-type theorem holds [22, Proposition 2.5, Remarks 2.6 and 3.5]. Under , in the time interval , the processes
turn out to be independent Wiener processes. Moreover, become simple regular càdlàg counting process of stochastic intensities ; the meaning of stochastic intensity is given by the heuristic formula
From these results we have immediately
The non linear SSE
In this subsection let us consider again a pure initial condition (2). In this case we have and, so,
where is defined by normalizing the random vector , solution of the linear SSE (1):
The non linear SME
; the operator is the stochastic Liouvillian (5). In  the reader can find a complete discussion of the relations among the four stochastic differential equations (1), (3), (11), (12), in the purely diffusive case with deterministic coefficients.
2.4 The continuous measurement process
Let us introduce now the real processes
where the integral kernels and are deterministic, and where and can be stochastic processes. Even more general expressions than the ones in Eqs. (13) could be considered. These processes will be interpreted as outputs of the continuous measurement, as explained in the next subsection. For this interpretation it is important the natural filtration generated by the processes (13), which we denote by . We assume , .
Instruments and a posteriori states
Now, for , let us define the map-valued measure [22, Remark 4.2]:
Such a measure has the properties:
, is a completely positive linear map on ,
, , is -additive,
Such a map-valued measure is called an instrument with value space [10, 11, 12] and it can be consistently interpreted as a quantum mechanical measurement on the system of the processes and in the time interval ; the instrument gives both the probability distribution of the output and the state changes conditional on the observation.
According to the physical interpretation of the notion of instrument, the probability of the event , when the pre-measurement state is , is given by
and this shows that the physical probability for the observation of the output up to time is indeed the one introduced in Eq. (9) restricted to . When goes from 0 to , the family of instruments gives a consistent description of a continuous measurement performed on the system.
Then, Eq. (13) can be interpreted as the effect of the measuring apparatus which processes the ideal outputs and by the classical response functions and , and which degrades the outputs by adding some more noises and due to the physical realization of the apparatus itself. In quantum optics, the typical output current of an homodyne or heterodyne detector is of the form (13a) with
where is the detector response function. In the final part of this section we shall see how to compute some relevant properties of the outputs under the physical probability.
After all, we can say that is the filtration generated by the outputs, i.e. contains only the events related to the observation of the outputs up to time , while , , is the filtration generated by all the processes involved in the continuous measurement and in the evolution, from the outputs to the unobservable noises.
Now, let us take the conditional expectation on of the random state defined by (6):
The interpretation is that is the conditional state one attributes to the system at time having observed the trajectory of the output up to time . Indeed is -measurable, thus depending only on the trajectories of the output in , and, moreover, one can directly check that [22, Remark 4.4]
The state is the a posteriori state at time .
In the extreme case , which occurs for example when is generated by the processes and and just these processes are the observed output, we get . Therefore the evolution of the a posteriori state is completely defined by the non linear SME (12) satisfied by , or, equivalently, by the SME (3) for , or even, as pure states are mapped to pure states, by the SSE (1) for or the non linear SSE (11) for . Let us stress that the evolution is not Markovian due to the randomness of the operator coefficients , .
When , the a posteriori state has a non Markovian evolution which typically does not even satisfy a differential equation. In this case the SSE (1) and the SME (3) have to be interpreted as an ideal unravelling of the physical evolution of which allows us to consistently define it, by (3) and (17), and allows us to compute, at least numerically, all the quantities of physical interest (that is to define the instruments ).
The randomness of the coefficients and and the distinction between and provide the SME (3) with a great flexibility which allows us to model non Markovian features of the dynamics due both to some environmental noises and to measurement based feedback loops.
A priori states
When the output of the continuous measurement is not taken into account, the state of the system at time is given by the mean state
The state is the a priori state at time . Note that is a completely positive, trace preserving, linear map, i.e. a quantum channel in the terminology of quantum information.
From the SME (3) we get
which is not a closed differential equation when is stochastic, contrary to the Markov case [9, Section 3.4]. By the projection operator technique a closed integro-differential equation for the a priori state could be obtained  (an evolution equation with memory), but this equation is too involved to be of practical use. Again (1) and (3) are an unravelling of a non Markovian evolution. By the non linear SME (12) we have also
Spectra and moments of the diffusive outputs
Let us consider an output current as given by Eq. (13a); is a diffusive stochastic process and its spectrum is given by the classical notion . If is at least asymptotically stationary and the limit in (20) exists in the sense of distributions in , then the spectrum of is defined by
In the case (16) without extra noise, , and for a detector response function going to a Dirac delta, that is in the limit case , the spectrum becomes
The spectrum depends on the distribution of the current , which is the output of the continuous measurement on the system, and, so, gives information on the monitored system. Actually, the spectrum gives information also on the carrier which mediates the measurement. For example, in quantum optics, the system is typically a photoemissive source which is continuously monitored by homodyne or heterodyne detection of its emitted light. In this case, gives information both on the photoemissive source and on the fluorescence light. For example, reveals squeezing of the emitted light.
An expression for the autocorrelation function needed in the computation of the spectrum can be obtained by generalizing the techniques used in the Markovian case [9, Section 4.5]. When the Liouville operator (5) and are independent (which implies that is not used for the feedback), we get
Let us set and assume that the limit exists. Then, we obtain the decomposition of the spectrum in the elastic part and the inelastic one (the spectrum of the fluctuations)
The Mandel -parameter of the counting outputs
When we consider direct detection in quantum optics, in the ideal case of noiseless counter, the output of the measurement is one of the counting processes, say , which means to take and in Eq. (13b). In this case a typical quantity is the Mandel -parameter, defined by
As for a Poisson process this parameter is zero, in quantum optics it is usual to say that in the case of a positive parameter one has super-Poissonian light and sub-Poissonian light in the other case. Sub-Poissonian light is considered an indication of non-classical effects.
In quantum trajectory theory one can find expressions for the moments also in the counting case and we get
When is -independent from , which happens when is not used for feedback, we get
3 A paradigmatic model: the two-level atom
As an application of the theory we consider a two-level atom stimulated by a laser; it is an ideal example, but is sufficiently rich and flexible to illustrate the possibilities of the theory. The Hilbert space is and the Hamiltonian part of the dynamics is given by
where is the resonance frequency of the atom and , are the usual Pauli matrices. The function is the laser wave, which can be noisy and can be controlled by the experimenter; a concrete choice for is given below.
We consider two diffusive channels realized by heterodyne or homodyne detectors of the emitted light with local oscillators represented by the functions :
The observation of the light in channel 1 will be used to control the stimulating laser light. The light in channel 2 is only observed, in order to analyse its properties. We are interested in controlling the properties of this part of the emitted light by controlling the atom via the stimulating laser; in particular we are interested in generating squeezed light.
In the case of homodyne detection the local oscillator is fed by the light produced by the stimulating laser and we have . This is the case illustrated in Figure 1.
In the case of heterodyne detection the local oscillator is fed by an independent laser. For this light we could use for instance the so called phase diffusion model,
where and are extra noises (independent standard Wiener processes). The function represents a nearly monochromatic wave with a Lorentzian spectrum centred on .
We introduce also four jump channels:
The jump channels 3 and 4 are electromagnetic channels: channel 3 represents the emitted light reaching a photo-counter (direct detection), while channel 4 represents the lost light, which is not observed. The output of channel 3 could be used again as a possible signal for closed loop control, but we use it here only to see properties of direct detection. As channel 2 is used to detect properties of the field quadratures (squeezing), the analogous channel 3 is used to explore the counting statistics of the emitted photons.
The counting channels 5 and 6 are used only to introduce dissipation due to a thermal bath; these channels are not connected to observation. So, channels 4, 5, 6 are introduced only to give dissipative effects in the system dynamics; to represent them in the jump form (as done here) or in the diffusive form has no influence on the physical quantities.
The stimulating laser light can be noisy and can be controlled by the output of the diffusive channel 1. In mathematical terms, can be an adapted functional of (extra-noise) and (feedback of the output of the homodyne detector 1).
For this model we need a probability space where are Wiener processes, are Poisson processes and they are all independent. According to the notations of Section 2, the filtration is generated by all these processes, while is generated by , and .
From Eq. (5) and the assumptions of the model we get the random Liouville operator
Note that the whole of the randomness is in the wave and it is due purely to noise in the laser light and to feedback, but this is enough to have a non Markovian evolution of the atom.
3.1 The laser wave
Let us denote by the laser wave without feedback. We assume the laser to be not perfectly coherent and monochromatic and we describe this again by the phase diffusion model, i.e.
is known as Rabi frequency and as detuning. It is easy to compute the spectrum of this wave; we get
So, represents a wave of constant amplitude and Lorentzian spectrum. The effect of such a stimulating laser wave in the heterodyne spectrum of a two-level atom has been studied in Refs. [25, 18]. In that papers, however, the presentation of the theory of continuous measurements was based on the use of Bose fields and quantum stochastic calculus.
The present theory gives also the possibility of describing feedback control, by assuming that is a functional of the output current , . We take to have the form given in Eqs. (13a), (16) with a very simple detector response function; to be concrete we take
The functional dependence of on is determined by the implementation of the feedback mechanism, which should be chosen according to the aims of the controller and to the physical feasibility. Then, in principle, the quantities of interest could be computed by starting from simulations of the non linear SSE of the model. However, in order to have an analytically computable spectrum and to have an idea of the possible behaviours of the model, we consider very simple forms for .
A first choice is to modify the wave (25) by a simple linear amplitude modulation and to take
represents a possible delay. By taking , , one formally obtains , the limiting case of the “Markovian feedback” of Wiseman and Milburn [27, 28]; in  and in [9, Chapt. 10] this feedback scheme has been formulated in terms of SMEs and applied to the homodyne spectrum of a two-level atom. The effect of delay in some models has been studied in [28, 33, 41].
A second choice of the control is to introduce a phase modulation of the wave (25). By taking this phase modulation linear in the output current we get
Our aim is only to show that it is possible to construct in a consistent way a theory with a non Markovian feedback and to demonstrate that, even in simple cases, the feedback can be used to control the squeezing. So, we simplify further on the expression (28) and we take no delay, , and , . Then, we have ; note that the formal instantaneous output is and, thus, is the integrated current: depends on the whole past of the signal detected by the homodyne detector 1. Our final choice for the laser wave is
Whatever form we take for , the feedback acts on the stimulating light and on the local oscillators, as illustrated by Figure 1. As enters the Hamiltonian (24), we can control to some extent the dynamics of the atom. This possibility can be used to control the system state or the time to reach a final state [30, 41]. Instead, what we want to study in this paper is how the feedback can be used to control the output of the quantum system, precisely the squeezing in channel 2 or the Mandel -parameter in channel 3.
3.2 Control of the homodyne spectrum
According to Eqs. (23), to compute the spectrum of the light in channel 2 we need to compute first the quantities and . The best way is to introduce a kind of stochastic rotating frame, that is to make a unitary transformation and to define the quantities