Continuous simultaneous measurement of position and momentum of a particle

For more than a century the principles of quantum mechanics are employed successfully in the study of an immensely broad range of phenomena and it forms a basis of the modern physical worldview. The theory is developed from a set of axioms including, at least in the conventional Copenhagen formulation [1, 2], the postulate of wavefunction collapse: upon observation, the pre-measurement state of the system immediately changes to the projection onto the eigensubspace associated with the measurement’s outcome [3, 4, 5, 6]. From its outset this rule sparked an ongoing controversy, and as part of the measurement problem it remains a subject of debate to the present day. Measurement grows into the central issue in most attempts to understand the emergence of the classical from the quantum [7]. It is often treated as a destructive process performed repeatedly on an ensemble of identically prepared quantum states. The problem of continuous monitoring of a single quantum system was considered merely of academic interest by the founding fathers of quantum mechanics since in these days it was inconceivable to experimentally study an isolated quantum system in a single (non-averaged) realization.

Single quantum systems suitable for controlled long-time observations appeared with the first trapped ions [8]. This experimental achievement triggered theoretical advances in the field of repeated measurement of optical transitions in a single quantum system [9]. Advances in techniques of manipulating and observing atoms in periodic lattices, in particular the construction of atomic microscopes, allowed for direct observation of the motion of individual quantum objects in present-day experiments with ultracold atoms [10, 11, 12, 13, 14, 15, 16]. One can directly monitor atoms illuminated by light and jumping between adjacent sites in optical lattices. These experiments raised again some interest in the effect of continuous observation of quantum systems [17, 18, 19, 20].

One of the motivations for the present study stems from the work on single-shot measurements of many-body quantum systems [21, 22] where high-order spatial correlations may reveal distinct geometric structures. Similarly, simultaneous detection of many particles unveils the solitonic nature of type-II excitation in the Lieb-Linniger model on a ring [23, 24]. In both these cases the one-particle density does not indicate any of said structures. They emerge only in the course of multipoint measurement on the same system. A similar effect, but in the time domain, is responsible for the phenomenon of time crystals as envisioned by Wilczek [25] (see also [26, 27, 28]). Breaking the time-translational symmetry of stationary states initiates a periodic motion of the system. Repeated measurement of the same system is necessary to prove this periodic motion. If the sequence of measurements was performed each time on a newly prepared system some chaotic rather than periodic motion would be observed. Every measurement on a newly prepared system breaks the symmetry differently, so the initial position of the system will vary randomly from one realization to the other.

To account for the effects mentioned above a back-action of the meters must be incorporated into the description. Canonical theory of quantum measurement [4, 30, 29, 31, 32, 33] assumes a quantum system coupled to a meter. The interaction between the two introduces correlations, the meter wavefunction becomes entangled with the system; then a projection operator corresponding to the observable of interest is applied to readout the meter. If the measurement is described in terms of mapping on a complete set of self-adjoint projectors then it constitutes a projection-valued measure (PVM).

Historically, continuous measurements of quantum objects were first considered in 1969 by Davies as a quantum stochastic process in the context of photon counting [34]. Later, description of continuous measurement in the framework of path integrals formalism was suggested by Mensky [35]. This approach was further developed by Caves [32]. Observables with continous spectra, like the position operator, were considered by Barchielli, Lanz and Prosperi [36, 37] (see also [38, 39, 40, 41]) who instead of projection operators introduced the more general concepts of ”effects” and ”operations”. ”Effect” (positive operator valued measure) is a generalization of the notion of observable, typically viewed as a projective-valued measure. The term ”operation” generalizes the reduction of the wavepacket [38, 39, 40, 41, 42]. Most approaches are based on the stochastic Schrödinger equation or on stochastic quantum differential equations for the density operator, [32, 33, 43, 44, 45, 46] describing a system which experiences some external noise.

Investigations of measurement of observables with continous spectra lead to the formulation of a theory where the effect of measurement of a particle is accounted for by multiplying its wavefunction by a meter-dependent envelope - a filter localized at some spatial position [18, 36, 43, 32]. This kind of description departs from the orthodox postulate of the collapse of the wavefunction onto the meter state. The localization scheme does not affect significantly the wavefunction locally at the position of the meter $x_{0}$, but filters the wavefunction by diminishing distant amplitudes. In particular the filtering does not change substantially the local phase $\varphi(x_{0})$ of the wavefunction i.e. the local velocity $v(x_{0})\propto\partial_{x}\varphi(x_{0})$ is ”almost” not modified. The post-measurement state of the system is not completely determined by the meter, it preserves some information of its pre-measurement history. Nevertheless, in the words of Caves [32], this filtering-type of localization ”nowhere invokes wave-function collapse, yet hidden within it must be a way of thinking in terms of collapse”. The projective measurement is performed on the system’s environment.

In this paper we want to present an approach rooted in standard quantum mechanics which is capable of tracing the dynamics of a quantum system at the level of a single realization of a sequential measurement – a trajectory of meter’s clicks – where each readout is related to the position and momentum of the observed quantum particle at the given time. Although position and momentum are not commuting variables, their simultaneous measurement is possible, though limits on precision are set by the Heisenberg uncertainty relation, as discussed by Arthurs and Kelly [31], and later by Scott and Milburn [33]. We assume both for Born’s rule, stating that the probability of transitioning to a (detector) state is given by its squared overlap with the wavefunction, as well as von Neumann’s postulate that after the measurement the system collapses to said ”detector state”. After such a measurement, the wavefunction does not preserve any information of its pre-measurement state.

Position and momentum are canonical variables of unique importance. They define a phase-space – the stage for the theory of classical physics with trajectories as the fundamental object. The problem of observing unique trajectories of quantum particles was first raised by Darwin [47], who in 1929 noticed that though an $\alpha$-particle emitted by a nucleus should be described by a spherical wave, it leaves a straight continuous track in a cloud chamber. The issue was taken by Mott [48] who formulated the problem as follows: ” … the $\alpha$-particle, once emerged, has particle-like properties, the most striking being the ray tracks that it forms in a Wilson cloud chamber. It is a little difficult to picture how it is that an outgoing spherical wave can produce a straight track …”.

He resolved the trajectory issue by noticing a proper setting for the problem: accounting not only for the $\alpha$-particle but also for every atom in the Wilson chamber – the detectors. Only those located at a straight line passing through the nucleus have a significant probability of joint ionization. This marks the first description of emergence of classical trajectories from the wavefunction formalism. In this case, ignoring the back-action of detectors on the highly-energetic $\alpha$-particle was reasonable. At low energies this approximation is not valid - a quantum particle is disturbed by an observation.

One peculiarity of standard quantum mechanics is that a probabilistic quantity - the wavefunction - is at the very heart of its formalism. Comparing theoretical predictions with experimental observations in quantum theory inheres the necessity of repeated measurements. In a vast majority of experiments single-particle observables are measured on an ensemble of identically prepared samples. Results are then averaged over the ensemble - a probabilistic description fits perfectly to such an arrangement. The situation changed dramatically with advent of ion trapping [8].

In 1985 Cook and Kimble, based on the idea of Dehmelt [49], proposed to monitor in situ quantum jumps - transitions of an electron between levels of a single trapped atom (ion) [50]. The quantum jumps to or from the metastable state would be marked by a sudden interruption or resumption of the fluorescence of an atom (ion) resonantly driven at the optical transition. The picture of a jumping electron is present in our thinking about emission or absorption of radiation since Niels Bohr introduced his model of the atom more than a century ago [51]. Nevertheless the relation of such discontinuous processes to standard quantum mechanics is a matter of debate (see [9] for discussion).

Rather than with the Schrödinger equation, the rapid switching between bright and dark periods in the fluorescence signal as observed in 1986 [52, 53, 54] seem to be understood in terms of a stochastic process. However, the discontinuities in fluorescence do not necessarily imply classical stochastic electron jumps, as noted in [55]. The stochastic jumps might simply be induced by the measurement of the fluorescence giving a classical random sequence of photon counts. Measurement links the quantum and classical world and it is not clear at which stage, on the part of the atom or the photons, the transition between the two worlds takes place.

The continuous quantum mechanical formalism was merged with a stochastic approach in the Monte-Carlo Wave-Function (MCWF) method introduced by Dalibard, Castin and Mølmer [56]. The method is designed to describe the dynamics of a small system coupled to a Markovian environment. The standard treatment of such a problem involves solving the master equation, which yields the evolution of the density matrix. However, the master equation can be substituted by an ensemble of pure state vectors of the small subsystem where each vector undergoes some random evolution [56, 57]. The smooth (non-unitary) evolution of the wavefunction describing the system is interrupted by random quantum jumps due to the interaction with the environment. Each ”quantum trajectory” obtained this way can be interpreted as a single realization of the systems’ dynamics. Averaging over many such realizations gives a smooth evolution equivalent to the solution of the Gorini-Kossakowski-Sudarshan-Lindblad equation for the density matrix in the Markovian approximation [58, 59]. In the following we will use this approach to describe the collapse of the system upon a measurement. A very similar quantum trajectory method was developed by Carmichael [60, 61]. Similar approach based also on a Monte Carlo method was introduced by Dum, Gardiner and Zoller [62, 63, 64].

Below we define a theoretical scheme of repeated measurements on the ”same system” in the language of quantum jumps. We use formalism allowing for tracing of a single quantum realization of the process. It is based on the Monte-Carlo Wave-Function approach. We define the model of detectors to monitor the particles and discuss its implementation. Our formalism accounts directly for the von Neumann (Lüders) postulate [4, 5] that upon measurement the system (not the environment) is found in the state the apparatus is ”projecting on”, $\phi(x)\rightarrow f(x-x_{0})$, where $\phi(x)$ describes a state of the system and $f(x-x_{0})$ a meter centered at position $x_{0}$. By this means we incorporate into the scheme the postulate of wavepacket reduction. Our model of detection conceptually differs from approaches where detectors act as ”filters”. [36, 43, 32, 18], $\phi(x)\rightarrow\,\propto\phi(x)f(x-x_{0})$. The ”measurement operators” we introduce do not correspond to a set of orthogonal states. Therefore our approach is a particular realization of the Positive Operator-Valued Measure (POVM). We will refer to the scheme we introduce as the ”collapse approach”.

First we illustrate the method in action by showing trajectories of a particle appearing in course of a continued measurement. Then we study in detail a single detector and show how the Zeno effect emerges in the limit of very intense measurement. Next we consider spatially sparse detectors and show that the observed motion of a free particle varies depending on the frequency of observation; from stroboscopic jumps for rarely clicking meters, up to the Zeno-kind delayed motion for a very frequent interrogation of the system. We analyse the particle’s motion in terms of a stochastic renewal process. Finally we study the statistical properties of the observed trajectories, i.e. mean values of position as well as dispersion in position and momentum. We show how these quantities scale with time for a dense and a sparse spatial grid of detectors. These results are compared to another model of continuous measurement of position which is based on filtering of the particle’s wavefunction in position space [18]. We summarize the results in the final section.

Our exemplary object of studies is a particle ”observed” by a grid of detectors. The approach can be generalized to a larger number of particles, though this is not the subject of the present study. Each of the detectors is able to measure simultaneously position and momentum of a quantum particle. The detectors are located at discrete points in phase space on a rectangular grid of spacing $D_{x}\times D_{p}$. This way positions and momenta measured can take discrete values of $(x_{m},k_{n})$, and the spacing is $D_{x}=x_{m}-x_{m-1}$, $D_{p}=k_{n}-k_{n-1}$. As we will elaborate at a further point, each possible measurement is associated with a projection on a single detector wavefunction. A convenient choice for the detector wavefunction (and thus of the post-measurement state of the particle) in one dimension (and with trivial generalization to higher dimensions) is the Gaussian wavepacket centered at $(x_{m},k_{n})$ having a width in position space equal to $\sigma$:

Every measurement in the phase space with readout values $x=x_{m}$ and $p=k_{n}$ is the result of a collapse of the wavefunction on the detector state $|\alpha_{mn}\rangle$. In the limit of infinitely small detector spacing, the functions $\alpha_{mn}$ become coherent (squeezed coherent) states of an overcomplete, non-orthogonal basis of the particles’ Hilbert space. The detector wavefunction in momentum space is the Fourier transform of (1), i.e.

Evidently, the resolution of measurement in position space is equal to $\sigma_{x}=\sigma$ while in momentum space to $\sigma_{p}=1/2\sigma$. This way the simultaneous measurement of the noncommuting observables minimizes the Heisenberg uncertainty principle [31],

Here we work in dimensionless units. In particular both the reduced Planck’s constant and the mass of the particle are set to one, $\hbar=m=1$. We also specify $a_{0}=\sqrt{2}\sigma$ as our unit of length. In these units the detector wavefunctions have the same width in position and momentum space meaning that the detectors have equal ”precision” in measuring both conjugate variables. Similarly, the spacings between detectors in position and momentum are assumed to be equal, $D_{x}=D_{p}=D$. The unit of time is $\tau_{0}=m\sigma^{2}/\hbar$. Since every detector wavefunction is associated with a measurement outcome $(x_{m},k_{n})$ our approach differs from the method introduced by Scott and Milburn [33], where position and momentum are measured by two principally independent detectors.

The crucial issue is to define the effect of the detectors on the particle under observation. To this end we denote by $C_{\alpha}$ the jump operators acting in the space of the small system, where, to simplify notation, by $\alpha$ we denote the detector state $\alpha=|\alpha_{m,n}\rangle$. The jump operators describe the effect of the environment on the particle. Their choice is crucial as it defines the physical processes of interest. Here we want to associate the jump with an act of measurement, [60, 61]. Therefore we define $C_{\alpha}$ to be proportional to be projectors onto the $|\alpha\rangle$ state:

Here $\gamma$ is a coupling strength determining the characteristic detection frequency. We assume that it has constant value, but in general it can depend both on position and momentum of the meter $\alpha$. This way, we incorporate Born’s rule and the von Neumann postulate of wavefunction collapse upon measurement into our approach, [1, 4, 5, 6]. To generate individual sequence of detectors’ clicks we make use of the Monte-Carlo Wave-Function (MCWF) method [56, 57]. The following paragraphs give a summary of the formalism.

In detail, the procedure consists of one of two options for each small time step $\delta t$ of the evolution. First the wavefunction $|\phi(t)\rangle$ is evolved using the non-hermitian Hamiltonian:

where $H_{S}$ is the Hamiltonian of the small system, $H_{S}=\frac{k^{2}}{2}+V(x)$. For an infinitely small time increment $\delta t$ we can write:

The above evolution is non-unitary so the norm is not preserved

with

The $\delta p_{\alpha}$ are interpreted as probabilities of quantum jumps ’in direction’ of the respective states $C_{\alpha}|\phi(t)\rangle$. Thus, the wavefunction at $t+\delta t$ is selected according to the values of probabilities $\delta p_{\alpha}$, i.e.:

1. 1.

   with probability $1-\sum_{\alpha}\delta p_{\alpha}$ the wavefunction is the one obtained from nonunitary evolution (with necessary normalization)

   $$|\phi(t+\delta t)\rangle=\frac{|\phi^{(1)}(t+\delta t)\rangle}{||\,|\phi^{(1)}(t+\delta t)\rangle\,||},$$

   (9)

2. 2.

   one of the meters clicks with probability $\delta p_{\alpha}/\delta p$ and the particle jumps to the measured state

   $$|\phi(t+\delta t)\rangle=\frac{C_{\alpha}|\phi(t)\rangle}{||C_{\alpha}|\phi(t)\rangle||}=|\alpha\rangle.$$

   (10)

Evidently $\delta p=\sum_{\alpha}\delta p_{\alpha}$ needs to be smaller than 1. This is usually achieved by tuning the timestep parameter $\delta t$ to be sufficiently small. Note also that although we study the case of a single particle possibly in an external potential, the evolution is far from trivial due to dissipation, i.e. since the particle permanently ”feels” all the detectors in its proximity.

Assuming that the particle wavefunction itself is roughly Gaussian one can get from Eq. (8) an estimate for the total jump probability per time step, $\delta p\approx\gamma\delta t\left(\frac{4\pi^{2}}{D}+1\right)$. This formula is particularly useful when choosing appropriate parameters for numerical calculations. It provides a good estimate for both limiting cases of dense (small $D$) and sparse (large $D$) detector grids.

The evolution of the initial coherent state $\alpha_{0}$ which is generated by the free Hamiltonian $H_{S}$ and ”dissipation” implied from presence of the meters is illustrated in Fig. 1 for a motionless particle located at the central detector. In the consecutive panels it is shown that as the time grows (left to right, top to bottom), in addition to quantum diffusion, the modulus (squared) of the wavefunction tends to increase around the positions of the meters – as if the detectors ”attracted” the particle. Note that we presented here a scenario in which no jumps occurred within the given time interval.

In MCWF, single ”trajectories” $|\phi(t)\rangle$ of the system are generated via a stochastic method. Given the same initial conditions, the average $\overline{|\phi(t)\rangle\langle\phi(t)|}$ over multiple trajectories is equal to $\rho_{S}$, the density operator of the particle, in the limit of many realizations.

In the quantum statistical description we treat the particle as a (small) open system coupled to the ”reservoir” of detectors. Averaging the individual trajectories one show that the statistical operator satisfies the Gorini-Kossakowski-Sudarshan-Lindblad equation, [58, 59] for the (sub)system’s density operator $\rho_{S}$:

and the relaxation operator $\mathcal{L}_{relax}$ describes the interaction with the environment. For general Markovian and time-homogeneous processes $\mathcal{L}_{relax}$ is of the Lindblad form, which for the jump operators defined by Eq. (4), is:

and summation is extended over the set of all detectors’ states, $\sum_{\alpha}=\sum_{\{\alpha_{mn}\}}$. Solving (11) leads to statistical predictions such as one-time operator average values $\langle A\rangle(t)=Tr(A\rho_{S}(t))$ within the conventional probabilistic interpretation of quantum mechanics, but it leaves open the question of how the evolution of the system in a single realization might look like. The MCWF method is thus fully equivalent to the master equation treatment when it comes to predicting statistical quantities. On the other hand in practice it may provide a computational advantage as well as possible additional physical insight from studying the pre-averaged single trajectories. Such observations are well within the reach of modern experimental techniques and we want to include them into our model.

We now briefly discuss the relation of this procedure with the stochastic Schrödinger equation approach, which is often found in related studies, [37, 45, 46, 41]. These approaches usually postulate a stochastic differential equation as a generalization of the standard Schroedinger/von Neumann equation, supplied with some dissipative term, accounting for the continuous observation. Such an equation is mathematically interpreted within the Ito formalism or an alternative quantum stochastic calculus. Indeed, there is a correspondence between the master equation Eq. (11) and various approaches yielding individual realizations, whose expectation values comply with the solution of Eq. (11) – the Monte Carlo Wave-Function method being one of them. In our case, the corresponding stochastic equationF would be given as

where $dN_{\alpha}(t)$ is a marked point process [41] and $dN_{\alpha}(t)dN_{\beta}(t)=\delta(\alpha-\beta)dN_{\alpha}(t)$ and $\langle dN_{\alpha}(t)\rangle=\langle C^{\dagger}_{\alpha}C_{\alpha}\rangle dt$.

We have shown that repeated measurements reveal classical dynamics perturbed by the back-action of the meters, as visible in the exemplary trajectories Figs. (2,3). Due to the probabilistic nature of ”detection” the trajectories form a random sequence of clicks. In this section we present some statistical characteristics of an ensemble of such trajectories.

We will compare them with statistics obtained from an alternative model of continouos observation based on measurement described as a spatial filtering of the particle’s wavefunction by the filter function centered at $X$ [36, 32, 43, 18]. In these approaches the assumed form of the measurement (Kraus) operator is:

Where $f$ is a suitable localized function. The action of $K(X)$ effectively reduces to multiplying the wavefunction in position space by an envelope centered around the measurement outcome $X$. We are going to refer to the measurement described by the operator $K(X)$ as ”filtering”, as opposed to the ”projection” on coherent states related to our jump operators $C_{\alpha}$. The filtering approach can be easily incorporated into the scheme of the MCWF method we use here. It amounts to a substitution of the jump operators $C_{\alpha}$ by the filters $K(X)$. To have a direct correspondence between the two methods we assume that in both cases the meters are located on the same spatial grid of spacing $D$. Moreover, we choose the filtering functions $f(x_{m}-x)$, centered at $x_{m}$ to be identical (up to the phase factor $e^{-ixk_{n}}$) to the Gaussian function of the jump operators $C_{\alpha}$, i.e. we choose $f(x_{m}-x)=\frac{1}{(2\pi\sigma^{2})^{\frac{1}{4}}}e^{-\frac{(x_{m}-x)^{2}}{4\sigma^{2}}}$ with $\sigma=1/\sqrt{2}$. An important difference between the two approaches is that the effect of filtering results only in a measurement of the particle’s position, and not its momentum.

Note also that repeated applying the operator $K(X)$ differs from a single measurement. This is in contrast to our method in which the jump operator $C_{\alpha}$ is proportional to a projection operator. Hence immediately after detection, the particle is in an eigenstate of the measurement operation, with maximum probability of being captured again at this position and momentum (indeed, for an orthogonal set of detector functions this would be the only possible measurement outcome at this point in time).

We limit our study to the case of a free particle launched with initial momentum $k_{0}=5$ and meters characterized by measurement intensity $\gamma=1$. We assume that at $t=0$ the particle’s wavefunction is described by a Gaussian wavepacket located at $x_{0}=0$ and identical to the spatial profile of the detector $\langle x|\alpha\rangle$ which we set at the origin of the coordinate system, $\alpha=(x_{0}=0,k_{0}=5)$.

As discussed earlier, the dynamics crucially depend on the detector spacing $D$. According to our choice the separation between different momentum states is also equal to $D$. We shall discuss the case when the spatial distance between the detectors is larger then the range of their sensitivity, $D=5.1>\sigma=\frac{1}{\sqrt{2}}$, and the opposite regime when the detectors cover ”almost continuously” the entire space, $D=0.73\simeq\sigma=\frac{1}{\sqrt{2}}$. In the first case the classical time to travel the distance $D$ is of the order of $t=1$, i.e. it is of the order of the average time between two clicks of a single detector. For the dense grid, the particle arrives at a neighbouring meter in a time significantly smaller than the ”reaction” time $1/\gamma=1$ of the meters.

To compare the statistics of trajectories provided by the two models we simulate a number of individual realizations using the MCWF method. In the case of our jump model, each trajectory is a set of measurement events $\{x_{i},k_{i},t_{i}\}$. In the case of filtering measurement operators (35), the results only comprise position measurements $x_{i}$ at times $t_{i}$, so we infer the particle’s velocity at these times as $k_{i}=\frac{x_{i}-x_{i-1}}{t_{i}-t_{i-1}}$, where $x_{0}$ is the particle’s initial position. We collect these results for both a very dense and very sparse detector grid. As before, the detection events of the trajectory ensemble are grouped into small time intervals, for which we calculate the mean value and standard deviation of $x$ and $k$.

The time dependence of the mean position of the particle is shown in the upper left panel of Fig. 9. In all figures the orange lines correspond to our quantum jump approach while blue lines indicate a filtering measurement. As we have already shown, the initial evolution of the particle localized initially at one of the meters has a stroboscopic (step-like) character if detectors are very sparse. The particle spends some substantial time at one of the detectors and then rapidly jumps to the neighbouring one. As soon as the characteristic width of the wavepacket is similar to the detectors spacing, the observed motion of the particle becomes classical (top right panel). Both approaches give here the same results. The dashed black line shows the classical trajectory of a particle moving with initial velocity $k_{0}=5$.

In the middle panel we present the dispersion of the position of the particle as a function of time. For sparse detectors (left middle panels) both methods predict large oscillations. They are directly related to the ”stroboscopic motion” of the particle predicted at the early times of the evolution. The dispersion has peaks at times classically corresponding to half the distance between the detectors. This effect is analogous to shot noise found in many physical settings. At larger times, the dispersion grows linearly, though the slope is larger in the filtering approach. For dense detector spacing the dispersion of the particle’s position scales as $\sim t^{3/2}$ for the quantum jump approach while scaling as $\sim t^{2}$ for the filtering method (right middle panel). The growth of the dispersion is again faster in the filtering scenario.

Significant differences may be observed in the case of the dispersion of the particle’s momentum. For sparse detectors, the jump measurement leaves the particle’s average momentum constant and its dispersion is negligible (left low panel). This is because for the chosen spacing, it is extremely unlikely to jump to a neighbouring momentum state. In the case of filtering meters the dispersion of the momentum exhibits oscillations at initial time mimicking the change of the particle’s average position, and stays constant later on.

In the case of dense detector spacing the filtering method gives linear growth of the momentum dispersion, $\sigma_{k}\sim t$, while the dispersion obtained from the jump measurements grows as $\sigma_{k}\sim\sqrt{t}$. This dependence revokes the characteristic of a random walk in momentum space. This is supported by the fact that the momenta maintain a Gaussian distribution around their mean (i.e. initial) value over time.

From a theoretical point of view this scaling of the dispersion in the quantum jump scheme is plausible since indeed, in the absence of an external potential the particle always has the same probability of jumping to $k+dk$ as to $k-dk$. In the limit of dense detectors and frequent measurement we thus expect that the random walk will pass into a Wiener process, and the momentum distribution will be governed by a standard diffusion equation, which is physically equivalent to constantly ”heating” the system.

If we do not wish the system to ”heat up” indefinitely, the approach must be generalized. The simplest way is to introduce a dependency of the coupling strength $\gamma$ on the absolute momentum measured by the respective detector $\alpha_{m,n}=(x_{m},k_{n})$. One particularly simple approach is to set $C_{\alpha_{m,n}}=\sqrt{\gamma_{n}}|\alpha_{m,n}\rangle\langle\alpha_{m,n}|$ where $\gamma_{n}=\gamma_{0}e^{-k_{n}^{2}/k_{cut}^{2}}$, where $k_{cut}$ is a characteristic cut-off momentum above which the detectors are supposed to become less sensible. One might also consider different modifications of the scheme going beyond the Markov approximation. This is however not the issue of the present study.

In this paper we discussed the dynamics of a particle whose position and momentum are continuously measured. The detectors, regularly distributed in phase-space, act as a permanent perturbation on the particle. Their effect on the measured system is to collapse the particle wavefunction $\psi(t)$ to a coherent state (the detector ”eigenstate”) centered at the phase-space point $\alpha=(x_{m},k_{n})$. The probability per unit time of a detector click is proportional to the overlap (squared) $|\langle\alpha|\psi(t)\rangle|^{2}$ of the particle wavefunction and the coherent eigenstate of the detector, in direct analogy to Born’s rule. Likewise, the post-measurement state of the particle is fully determined by the detector-eigenstate.

After introducing the model, we presented a number of (semi-)analytical results explaining several features of the dynamics, coming from different choices of the parameters $\gamma$ and $D$. In particular, we identify the occurrence of a Zeno effect in the limit of a sparse spatial grid of meters and ”frequent clicking”, $\gamma>v/D$. In this case, the particle’s motion (found initially at one of the detectors) is delayed as compared to its free evolution. We showed that the delay time can be explained in terms of renewal theory. Our estimation of the inter-arrival time matches numerical results. Moreover, we showed that if the ”measuring intensity” is low, $\gamma<v/D$, the average position of the particle is initially a steplike function of time, in contrast to a dense detector grid, where classical trajectories emerge independent of $\gamma$.

Finally we compared our method to an alternative model of continouos measurement, in which the post-measurement state is obtained by applying a Gaussian filter centered around a given position on the wavefunction. For better comparison we assume that these filters are localized at the same discrete points in space as our projective meters. We show that both approaches provide similar results when it comes to the mean position of the particle, however they differ with respect to how the dispersion of position and momentum scale with time.

In this study, we assumed that the detectors project on Gaussian states with equivalent width in position and momentum space. One could instead study detectors characterized by squeezed coherent states $\langle x|\alpha_{m,n}\rangle\propto e^{(x-x_{m})^{2}/(2\sigma^{2})+ik_{n}x}$ – where $\sigma\neq 1/2$ determines the level of squeezing. By varying $\sigma$ one can smoothly change the relative precision of position and momentum measurement. Likewise, varying the relative spacing in position and momentum of the detectors is another way of further exploring the model. As mentioned at the end of last section, introducing a $\gamma$ parameter which varies between the detectors could also lead to new dynamical features. However, probably the most rewarding extension of this work is to explore an equivalent model in a many-body setting.

Commenting on what sets apart the model presented here from the various other continuous measurement schemes studied so far – and summarized in the introduction of this article – we would like to point out that we consider a simultaneous measurement of non-commuting observables, that is position and momentum. The result of this measurement operation is the projection on a pre-defined state, in the spirit of Born’s rule. This simplicity introduces a potential advantage when it comes to theoretical study of the model: a dynamics which consists of returning every so often to the same Gaussian state (eventually translated in position and/or momentum) may be understood in the context of extended renewal theory. By this we mean that the evolution of the system is characterized completely by functions $\lambda_{u}(t)$ which denote the overlap squared of a Gaussian detector with a neighbor at a distance $u=(\Delta x,\Delta p)$ in phase space as a function of time. The computational advantage coming from this fact was not explored fully in the present work, since the systems under study were simple enough to be treated within MCWF. Nevertheless we believe that further investigation in this direction could lead to future valuable insights.