Weak measurements of trajectories in quantum systems: classical, Bohmian and sum over paths
WWeak measurements of trajectories in quantum systems: classical,Bohmian and sum over paths
A. Matzkin
Laboratoire de Physique Th´eorique et Mod´elisation (CNRS Unit´e 8089),Universit´e de Cergy-Pontoise, 95302 Cergy-Pontoise cedex, France
Abstract
Weak values, obtained from weak measurements, attempt to describe the properties of a quantumsystem as it evolves from an initial to a final state, without practically altering this evolution.Trajectories can be defined from weak measurements of the position, or inferred from weak valuesof the momentum operator. The former can be connected to the asymptotic form of the Feynmanpropagator and the latter to Bohmian trajectories. Employing a time-dependent oscillator as amodel, this work analyzes to what extent weak measurements can shed light on the underlyingdynamics of a quantum system expressed in terms of trajectories, in particular by comparing thetwo approaches.
PACS numbers: 03.65.Ta, 03.65.Ca a r X i v : . [ qu a n t - ph ] J un . INTRODUCTION Contrary to classical physics, the standard formalism of quantum mechanics forbids theuse of space-time trajectories to describe the time evolution of a system. However trajectoriessurreptitiously sneak back into the description, the interpretation and the computation ofquantum phenomena. There are various form of trajectories that have been found useful.Among these the most prominent are the paths of the path integral approach due to Feynman[1] and the trajectories built on the probability flow employed in the de Broglie-Bohm model[2]. Both types of trajectories have been employed to interpret experimental results.The path integral approach has been extremely successful for quantum systems in thesemiclassical regime. Indeed, in this regime the path integral becomes essentially a coherentsum over the classical trajectories of the corresponding classical system [1]. Such classi-cal trajectories have been employed to understand the properties of these systems in theframework of “quantum chaos” [3], and their manifestations have have been experimentallyobserved in many quantum systems (see eg Refs [4]). The trajectories of the Bohmianmodel are essentially obtained by following the probability current density arising from theSchr¨odinger equation. Bohmian trajectories have also been employed to interpret the dy-namics in several systems [5]. They have further been used as a numerical computationmethod, especially in molecular physics or when mixing classical and quantum degrees offreedom in a mean field approximation is required [6].The trajectories of the path integral, generated by the classical Lagrangian, are generi-cally different from the quantum trajectories built on the Schr¨odinger probability flow [7].This is not a problem as long as one sees these trajectories as being computational toolsor mathematical artefacts. However recently the approach of weak measurements, intro-duced some time ago by Aharonov, Albert and Vaidman [8] has been employed to measurenon-perturbatively trajectories in quantum systems. The main idea underlying weak mea-surements [9–11] is to access the properties of a quantum system evolving from a given initialstate towards a final state, practically without disturbing the system evolution. This cannotbe achieved by a standard projective measurement (doing so would irremediably disturb thesystem by projective its premeasurement state to a subspace spanned by the eigenstate ofthe measured observable). Instead, the system observable is coupled unitarily to an ancillaacting as a weak measurement apparatus (WMA). If the coupling is weak, the unitary evo-2ution of the system is practically left unperturbed. The value recorded by a WMA is notan eigenvalue (since there is no state projection) but what is known as a weak value [8] ofthe weakly measured observable.The idea of inferring Bohmian trajectories from weak measurements of the momentum (followed by a projective measurement of the position) has been proposed first by Leavens[12] (see also [13–15] for more recent approaches). This scheme was experimentally imple-mented in a two-slit interferometer [16] allowing to reconstruct Bohmian trajectories fromthe observed data. A method that can in principle allow to observe the Feynman paths withweak measurements of the position (including the coherent paths superposition) was alsosuggested recently [17] (see also [18]). In a first view it is therefore tempting to concludethat the type of trajectory that one sees depends eventually on what is being measured,which in turn calls for a definite experimental setup.The aim of this work is to examine this question in details by displaying expressionsfor the weak measurement of classical and Bohmian trajectories in the same system. Wewill employ a tractable model system – a two dimensional time-dependent linear oscillator(TDLO). While the dynamics of the TDLO is arguably less rich than that of generic systems,its main advantage in the present context is that the Feynman sum over paths can beobtained exactly in closed form (without invoking the semiclassical approximation) whilethe computation of the Bohmian trajectories is numerically tractable. At the same timethe time-dependent aspect allows to ”simulate” dynamical feature that generally appearin systems with more involved dynamics (like recurrences of closed orbits). Moreover theTDLO has often been employed to model quantum systems such as the dynamics of trappedions [19], photon generation in quantum optics [20] or cosmological mini-superspace models[21].The paper is organized as follows. We first introduce the weak measurements frameworkand derive the two types of trajectories that can be inferred from weak measurements. Wethen introduce our model system and detail how the quantum dynamics can be interpretedin terms of classical or Bohmian trajectories (Sec. III), yielding different interpretations ofdynamical phenomena when classical and Bohmian trajectories differ. Sec. IV describesthe trajectories inferred from weak measurements for the TDLO, including derivations andseveral numerical illustrations for specific cases. We discuss our findings and conclude in Sec.V, while an Appendix details how we obtain the closed form solutions of the Schr¨odinger3quation for the TDLO.
II. WEAK MEASUREMENTSA. Weak measurement framework
The underlying idea at the basis of the weak measurement framework is an attempt toanswer the question:” what is the value of a property (represented by an observable ˆ A ) of aquantum system while it is evolving from an initial state | ψ ( t ) (cid:105) to a final state | χ (cid:105) ? ”. Insteadof making a projective measurement the observable ˆ A is coupled unitarily to a dynamicalvariable of an ancilla (the weak measurement apparatus, WMA). There is thus no projectionof the system’s quantum state at this stage. Moreover if the coupling is asymptotically weak,it can be shown that the state of the system is left practically undisturbed. The systemthus continues its evolution until a final projective measurement (of another observableˆ B ) projects its state to | χ (cid:105) , the post-selected state. As a result of the unitary coupling, aprojective measurement on the system also modifies the quantum state of the WMA: thevariable conjugate to the coupled one is shifted by a quantity proportional to Re (cid:104) A w (cid:105) where (cid:104) A w (cid:105) is the weak value of the observable ˆ A when the system is pre-selected in state | ψ ( t ) (cid:105) and post-selected to the state | χ (cid:105) . Letting U ( t (cid:48)(cid:48) , t (cid:48) ) denote the evolution operator of thesystem between times t (cid:48) and t (cid:48)(cid:48) , (cid:68) ˆ A w (cid:69) is given by (cid:68) ˆ A w (cid:69) = (cid:104) χ | U ( t f , t w ) ˆ A [ U ( t w , t ) | ψ ( t ) (cid:105) ] (cid:104) χ | U ( t f , t ) | ψ ( t ) (cid:105) , (1)where t w and t f stand for the times at which the weak measurement and final postselectiontake place respectively. Introducing the notation (cid:104) χ ( t w ) | ≡ (cid:104) χ | U ( t f , t w ) representing thepostselected state evolved backward in time, while | ψ ( t w ) (cid:105) = U ( t w , t ) | ψ ( t ) (cid:105) , the weakvalue (1) is written in terms of quantities taken at the weak measurement time t w as (cid:68) ˆ A w (cid:69) = (cid:104) χ ( t w ) | ˆ A | ψ ( t w ) (cid:105)(cid:104) χ ( t w ) | ψ ( t w ) (cid:105) . (2) Formally, the system and the WMA get entangled, so the coupling interaction changes the dynamics ofthe overall entangled wavefunction. For a small coupling, the postselection probabilities are not modifiedto first order while the WMA’s wavefunction picks up a shift; see eg Secs 2 and 5 of [9], and [22] for aproblem solved with exact solutions of the Schr¨odinger equations. of the weakly measuredobservable. While there has been a controversy on the meaning of weak values from theirinception, in our view the controversy has more to do with the interpretation given to thetheoretical terms of quantum theory in general than to the specificities of weak measurementsas such. There is now ample evidence [9, 10] that the weak values given by Eq. (1) capturea universal effect in which Re (cid:104) A w (cid:105) represents the response on the probe of a minimallydisturbing interaction for the system reflecting the value of the property described by ˆ A relative to the fraction of the initial state that will conditionally end up in the post-selectedstate | χ (cid:105) . The basic property allowing this interpretation follows by writing the expectationvalue of ˆ A when the system is in the state | ψ ( t w ) (cid:105) in terms of the probabilities of reachingeach eigenstate | χ f (cid:105) of a different observable ˆ B, (cid:104) ψ ( t w ) | ˆ A | ψ ( t w ) (cid:105) = (cid:88) f |(cid:104) χ f | ψ ( t w ) (cid:105)| Re (cid:104) χ f | ˆ A | ψ ( t w ) (cid:105)(cid:104) χ f | ψ ( t w ) (cid:105) , (3)where the restriction to the real part comes from the fact that since the left-handside isreal, the sum over the imaginary parts vanishes. The interpretation of this formula in termsof weak measurements assumes impilictly that the probabilities of obtaining the final state | χ f (cid:105) are not modified by the action of ˆ A .Note that in terms of projective measurements, Eq. (3) can also be read similarly to thestandard quantum mechanical expectation value expression (cid:104) ψ ( t w ) | ˆ A | ψ ( t w ) (cid:105) = (cid:88) f |(cid:104) α f | ψ ( t w ) (cid:105)| α f (4)where ˆ A | α f (cid:105) = α f | α f (cid:105) . While Eq. (4) involves a protocol in which the expectation value isobtained by measuring the eigenvalues α f of ˆ A and their relative frequencies, Eq. (3) suggestsa protocol in which the expectation value of ˆ A is obtained by a weak measurement of ˆ A followed by a standard projective measurement of ˆ B, for which only the relative frequenciesfor obtaining the eigenvalues χ f of ˆ B are needed. The pointer can actually register either the real or the imaginary parts of the weak value, which isa complex number. Only the real part is related to the value of the measured observable, while theimaginary part is typically related to the measurement backaction [23]. . Weak measurements of position: ”weak” trajectories The most intuitive way of measuring a trajectory follows from its definition: in a givenframe of reference the position r ( t ) is recorded as a function of time. For an evolving quan-tum system, this involves monitoring the position not only non-destructively, but withoutaffecting the subsequent evolution of the system. A weak measurement of the position isperfectly suited in order to monitor the position. Starting from a localized initial state | ψ ( t ) (cid:105) – this will be the preselected state –, and ending with a projective measurement toa state localized at a given final position at time t f (this will be the postselected state),we can place a series of weak measurement apparata (WMA) that weakly interact with thesystem via a local coupling to the position observable ˆr .Assume first that there is a single WMA, lying at position R and whose wavefunction φ ( R ) is tightly localized around the central position R , eg a Gaussian wavefunction φ ( R ) =(2 /π ∆ ) / e − ( R − R ) / ∆ . The weak interaction is triggered when the system wavefunctionenters the region around R , corresponding to a contact interaction of the form H = γ ( t ) ˆr · R f (cid:16)(cid:12)(cid:12) ˆr − R (cid:12)(cid:12) (cid:17) (5)where γ is a smooth function of t determining the coupling and f is a function sharply peakedat | ˆr − R | = 0 indicating the short-range character of the contact interaction between thesystem and the WMA. Let t w denote the mean time at which the interaction takes place. Ifthe duration τ of the measurement is short relative to the timescale of the system dynamics,then [24] the WMA records the weak value of the position at time t w (cid:104) r ( t w ) (cid:105) W ≡ (cid:104) χ ( t w ) | ˆr f (cid:16) | ˆr − R | (cid:17) | ψ ( t w ) (cid:105)(cid:104) χ ( t w ) | ψ ( t w ) (cid:105) . (6)We will see below that (cid:104) r ( t w ) (cid:105) W can take a simple form for specific choices of the systemwavefunction. Nevertheless one can see qualitatively that the wavefunctions ψ ( t w , r ) and χ ( t w , r ) must overlap significantly around the position of the WMA r ≈ R in order toobtain a non vanishing weak value. If (cid:104) r ( t w ) (cid:105) W = 0 , this essentially means there is nowavefunction exploring the region around R compatible with the postselected state. In the Since the spatial WMA wavefunction picks up an R dependent phase term proportional to the weak value,the WMA pointer is monitored in momentum space. ψ ( t w , r ) (the preselected state propagated forward in time) and χ ( t w , r )(the postselected state propagated backward in time) are identical, then it is easy to seethat under certain conditions (eg, ψ and χ are constant in the region where f (cid:16) | ˆr − R | (cid:17) isnon-zero, or have a maximum at R ) the weak value will be simply given by (cid:104) r ( t w ) (cid:105) W = R ,that is the location of the WMA.Assume now there are several WMA of the same type distributed at positions R k , k =1 , ..., N . It is convenient to label them according to the order in which they interact ( k = 1corresponds to the meter interacting first with the system, k = 2 to the second meter havinginteracted with the system and so on). Each WMA, endowed with its own wavefunction φ ( R k ) localized around R k interacts at time t k with the wavefunction through a contactinteraction of the form (5) and registers a weak value (cid:10) r ( t k ); R k (cid:11) W ≡ (cid:104) χ ( t k ) | ˆr f (cid:16) | ˆr − R k | (cid:17) | ψ ( t k ) (cid:105)(cid:104) χ ( t k ) | ψ ( t k ) (cid:105) (7)where R k , labeling the position of the meter, will be omitted in most of the text. Overall, outof the N WMA that act as meters recording the weak values, only n will display a non-zerovalue, those for which postselection is compatible with the dynamics of the preselected stateat the given WMA positions. Relabeling k in a time-ordered manner reflecting the times atwhich the WMA have interacted, the setWT ψ ( t ) ,χ ( t f ) = { t k , Re [ (cid:104) r ( t k ) (cid:105) W ] } , k = 1 , ...n (8)defines a trajectory in the sense of weak position measurements, that is a “weak trajectory”for given pre and postselected states. Note that the pre and post-selected states | ψ ( t k ) (cid:105) and | χ ( t k ) (cid:105) at times t k cannot be freely chosen but depend on the initial pre-selected state | ψ ( t ) (cid:105) and on the final post-selected state | χ ( t f ) (cid:105) respectively.For an arbitrary quantum system a WT (8) will typically reflect the space-time correlationbetween the forward evolution of the preselected state and the backward evolution of thepostselected state at the positions R k of the weakly interacting meters: only the WMAat positions for which this correlation is non-vanishing will display a non-zero weak value.WTs become particularly interesting in the semiclassical regime, ie when the Feynman pathintegral is approximately given by the semiclassical propagator involving a propagator givenby a coherent sum over the paths of the classical corresponding system. Indeed, as we willsee below (Secs. III and IV) the weak trajectories can in principle be employed to record the7um over paths of the semiclassical propagator. But we will first introduce another type oftrajectories that can be inferred from a different type of weak measurement. C. Weak measurements of momentum: velocity field
The standard textbook form of the quantum mechanical probability current for a systemin state ψ ( r , t ) is given by j ( r , t ) = i (cid:125) m [ ψ ( r , t ) ∇ ψ ∗ ( r , t ) − ψ ∗ ( r , t ) ∇ ψ ( r , t )] . (9)A local velocity field at the space-time point ( r , t ) can be defined from the current densitythrough v ( r , t ) = j ( r , t ) ρ ( r , t ) (10)where ρ ( r , t ) ≡ | ψ ( r , t ) | . Consider now applying the weak value definition (2) to a weakmeasurement of the momentum operator ˆp when the system is in state | ψ ( t ) (cid:105) immediatelyfollowed by a postselection to the position eigenstate | r f (cid:105) . The real and imaginary parts ofthe weak value are given by (cid:104) p ( t ) (cid:105) W = (cid:104) r f | ˆp | ψ ( t ) (cid:105)(cid:104) r f | ψ ( t ) (cid:105) (11)= m v ( r f , t ) − i (cid:125) (cid:79) ρ ( r f , t )2 ρ ( r f , t ) . (12)Hence by performing weak measurements of the momentum at different space-time points( r f , t ), the real part of the weak value (cid:104) p ( t ) (cid:105) W allows to reconstruct the velocity field v ( r , t ).Note that (cid:104) p ( t ) (cid:105) W can also be obtained from the difference of two position measurementsmade in a very small time interval [13]. One first defines a weak position measurement attime t − ε consistent with postselection at position r f at time t : (cid:104) r ( t − ε ) (cid:105) W = (cid:104) r f | U ( t, t − ε ) ˆr | ψ ( t − ε ) (cid:105)(cid:104) r f | U ( t, t − ε ) | ψ ( t − ε ) (cid:105) . (13)To first order in ε → (cid:104) p ( t ) (cid:105) W = m ( r f ( t ) − (cid:104) r ( t − ε ) (cid:105) W ) ε (14)so that the velocity field appears as the real part of the difference ( r f ( t ) − (cid:104) r ( t − ε ) (cid:105) W ) /ε .It turns out [12] that this velocity field matches the local particle velocity in the deBroglie-Bohm interpretation of quantum mechanics (see Sec. III C below). The imaginary8art of the weak value (12) has also very recently [14] been interpreted as the osmoticvelocity of a putative stochastic model underlying the de Broglie-Bohm framework. Therelations (11)-(12) between weak values and the Bohmian momentum are important, notonly because they allow in principle to extract experimentally Bohmian trajectories fromthe weak measurements of the momentum velocity field, but also because these relationsconstitute a link between the momentum operator and the Bohmian particles momentum(the latter having little to do with the eigenvalues of the momentum operator). III. QUANTUM PROPERTIES AND TRAJECTORIESA. Model system and setting
As mentioned in the Introduction, it is often useful to interpret properties of quantumsystems in terms of trajectories. This is particularly the case for quantum systems in thesemiclassical regime for which the underlying classical dynamics drives the quantum evolu-tion operator. Nevertheless a typical system in the semiclassical regime is not easy to handle– the full semiclassical propagation is generally a formidable task, the search for classicalperiodic orbits in general is not trivial and the computation of Bohmian trajectories callsfor a powerful numerical implementation. We will work instead with a simple model system,a two dimensional time-dependent linear oscillator (TDLO). The TDLO allows to simulatethe sum over paths aspect of the semiclassical propagator, given below by Eq. (26) in aneasy and tractable manner. Moreover since the TDLO Hamiltonian (15) is quadratic, thesemiclassical approximation is quantum mechanically exact [1].We shall consider in the following a two-dimensional time-dependent oscillator with time-dependent frequencies V x ( t ) and V y ( t ), whose Hamiltonian is given by H ( t ) = p m + m (cid:0) V x ( t ) x + V y ( t ) y (cid:1) , (15)where for definiteness the time-dependence of the potential will be chosen to take the form V j ( t ) = v j − κ j cos (2 ω j t ) , j = x, y . (16)(see Appendix A for details). Let us take an initial state made up of a single 2D Gaussian ψ ( q , p ) ( r , t ) = (cid:18) mπα (cid:19) / e − m ( r − q ) /α e i p · ( r − q ) / (cid:126) (17)9here r = ( x, y ) , the parameters and q and p are the average values of the position andmomentum operators respectively in that state; α sets the width of the initial Gaussian (forsimplicity, the same initial width is taken along both directions). Eq. (15) is a separableproblem, so the solutions to the time-dependent Schr¨odinger equation are readily obtainedfrom the 1D TDLO described in Appendix A, yielding ψ ( q , p ) ( r , t ) = ψ ( x, t ) ψ ( y, t ) (18)where each 1D wavefunction is given by Eq. (A8). As discussed in Appendix A, ψ ( q , p ) ( r , t )can be written in terms of a wavefunction whose probability density has a maximum alongthe classical trajectory q ( t ) = ( q x ( t ) , q y ( t )) having initial position q ( t ) = q and momentum p ( t ) = p . The phase also depends on the momentum p ( t ) of that classical trajectory. Twoother purely time-dependent functions α ( t ) = ( α x ( t ) , α y ( t )) and φ ( t ) = ( φ x ( t ) , φ y ( t )) linkedto q ( t ) in the framework of Ermakov systems [cf. Eqs (A3) and (A5)] are also necessary todescribe the time-dependent solution, that takes the form ψ ( q , p ) ( r , t ) = (cid:18) mπ det[Re( M ( α ))] (cid:19) / e − [ r − q ( t )] · M ( α ) · [ r − q ( t )] e i p ( t ) · [ x − q ( t )] / (cid:126) e i (cid:126) [ p ( t ) · q ( t ) − p · q ] e − i [ φ ( t ) − φ ] · r /r . (19) M ( α ) is a matrix defined by M ( α ) = m α x ( t ) − iα (cid:48) x ( t )2 (cid:126) α x ( t ) α y ( t ) − iα (cid:48) y ( t )2 (cid:126) α y ( t ) . (20)Note that α ( t ) determines the time-dependent width of the evolving wavefunction; the initialstate (17) corresponds to α ( t ) = ( α , α ) and φ ( t ) = (0 , ψ ( r , t ) = (cid:88) J a J ψ ( q , p J ) ( r , t ) (21)that is a superposition (with normalized real weights a J ) of Gaussians initially localized atthe same position q but with different initial mean momenta p J . The resulting wavefunction ψ ( r , t ) = (cid:88) J a J ψ ( q , p J ) ( r , t ) (22)is a sum of Gaussians (19) each propagating by following the guiding trajectory q J ( t ) . . Path integral and classical trajectories The solutions of the Schr¨odinger equation (A8) and (19) for the TDLO in which thewavefunction amplitude is concentrated along the trajectories of the classical correspondingsystem can best be seen to arise from the Feynman path integral approach. From a qualitativestandpoint, the argument starts from the path integral form of the time evolution operator K ( r , r ; t − t ) = (cid:90) r r D r ( t ) exp i (cid:126) (cid:20)(cid:90) t t L dt (cid:21) , (23)the propagator, that propagates the initial wavefunction along any conceivable path, ac-cording to ψ ( r , t ) = (cid:90) K ( r , r ; t − t ) ψ ( r , t ) dx. (24)The propagator can be expressed in terms of classical trajectories when the action R ( r , r ; t − t ) = (cid:90) t t L dt (25)is huge relative to (cid:126) ( L is the Lagrangian, given here by the 2D extension of Eq. (A2)). Inthat case, the integration in Eq. (23) is handled [1] with the stationary phase approximation,and the stationary points of the action are, by Hamilton’s principle, the classical trajectories. K then takes the generic form K ( r , r ; t − t ) = (2 iπ (cid:126) ) − (cid:88) k (cid:12)(cid:12)(cid:12)(cid:12) det − ∂ R k ∂ r ∂ r (cid:12)(cid:12)(cid:12)(cid:12) / exp i (cid:126) [ R k ( r , r ; t − t ) − µ k ] , (26)where k runs over all the classical trajectories connecting x to x in time t − t . R k isthe classical action and the determinant gives the classical density of paths along the k thclassical trajectory, and µ k are additional phases related to the number of conjugate pointson the trajectory. The sole approximation made in employing the stationary phase impliesneglecting the terms beyond the second order variations along the paths of least action. Butfor quadratic Lagrangians – such as (A2) – the third order and greater order terms vanish,so that the semiclassical propagator (26) is quantum-mechanically exact.The propagator (26) accounts for the fact that the maximum of the initial wavefunc-tion propagates along classical trajectories. The point that remains to be explained is the This generic form actually assumes that the action has isolated non-degenerate stationary points, whichis not the case here (a change of variables is necessary). x and x with the helpof the Ermakov phase and amplitude functions (see Appendix) leads after some tediousmanipulations to R cl ( x , x ; t − t ) = m (cid:26)(cid:18) x α (cid:48) ( t ) α ( t ) − x α (cid:48) ( t ) α ( t ) (cid:19) + (cid:125) cot ( φ ( t ) − φ ( t )) (cid:18) x α ( t ) + x α ( t ) (cid:19) (27) − x x (cid:125) (cid:20) α ( t ) α ( t ) sin ( φ ( t ) − φ ( t )) (cid:21)(cid:27) (28)(see also Ref. [25] describing a method to obtain directly the propagator from the Ermakovsystem solutions). The action is quadratic in x so that with initial wavefunctions of theform (A9) Eq. (24) becomes a Gaussian integral quadratic in x ; finally the classical solutionis identified in the exponent with the help of Eq. (A4). C. De Broglie-Bohm trajectories
According to the Bohmian (or de Broglie-Bohm) model [2, 26], a quantum system canbe seen as the combination of a point-like particle guided by a pilot wave. The wavefunc-tion plays the role of the pilot wave, and through its modulus, it also gives the statisticaldistribution of the particle’s position, thereby recovering by construction the standard (non-relativistic) quantum mechanical probabilities and expectation values. From a dynamicalpoint of view, the resulting Bohmian trajectories are the streamlines of the usual probabilitycurrent density flow derived from the Schr¨odinger equation.If we write the wavefunction in polar form as ψ ( r , t ) = ρ / ( r , t ) exp( iσ ( r , t ) / (cid:126) ) , (29)the current density, Eq. (9) is given as j ( r , t ) = ρ ( r , t ) (cid:53) σ ( r , t ) m . (30)By replacing Eq. (29) in the Schr¨odinger equation it can be seen that ρ and σ obey theequation ∂σ∂t + ( (cid:79) σ ) m + V + Q = 0 . (31)12 is the usual potential (here V = m V x ( t ) x + m V y ( t ) y ) and the term Q ( r , t ) ≡ − (cid:126) m (cid:79) ρρ (32)is known as the quantum potential. The velocity field introduced above [Eq. (10)] gives thevelocity of the Bohmian particle at the space-time point ( r , t ); it can be written in the form v ( r , t ) = (cid:53) σ ( r , t ) m . (33)The momentum field can also be derived [27] without employing the polar form (29) as acomponent of the energy-momentum tensor of the Schr¨odinger field .Applying (cid:79) to Eq. (31) and using Eq. (33) leads to m d v dt = − (cid:79) ( V + Q ) , (34)a Newtonian-like law of motion. This justifies, in the de Broglie-Bohm formulation, theassumption that the streamlines of the probability flow are actually trajectories taken by apoint-like particle governed by Eq. (34), where the dynamics is determined not by the soleusual potential V but by the a total potential function V + Q thus including a wavefunction-dependent ”quantum potential” term.When the wavefunction vanishes the quantum potential becomes singular and dominatesthe dynamics. Therefore generically Bohmian trajectories cannot be classical [7], even insemiclassical systems. Nevertheless the signatures of the underlying classical dynamics inquantum systems, that appear as large scale structures in the quantum properties, arerecovered on a statistical basis. An illustration is given immediately below. The appar-ent conflicting situation between the classical trajectories resulting from the path integralapproach and the de Broglie-Bohm trajectories with regards to the interpretation of thedynamics of a system such as the TDLO becomes particularly acute in the context of weakmeasurements, as illustrated in Sec. IV. This is particularly appealing when generalizations are considered to relativistic quantum fields, in partic-ular for the interpretation of experiments performed in the single photon regime as in the results reportedin Ref. [16], in which the observed Bohmian-like average trajectories can be interpreted as the energyflux of the energy-momentum tensor (just like the Poynting vector appears in the stress-energy tensor forclassical Maxwell fields). IG. 1: (a) The probability density of the initial ( t = 0) wavefunction Eq. (21) with q = 0, J = 1 , a = a is seen to be localized at the origin; the direction of p J is schematicallyindicated by the arrows. The two classical trajectories with initial conditions ( q , p ) and ( q , p )are shown in dashed (purple) and solid (red) lines. Both trajectories are closed at the origin.The probability density is then shown (b): at t > t when the wavepackets start moving; eachwavepacket follows the classical guiding trajectory J = 1 ,
2; (c) at t = t rec (2) , corresponding to thesecond peak of the autocorrelation (see text and Fig. 3); (d) just before the two wavepackets crossand interfere. Atomic units are used throughout, with a TDLO having unit mass. D. Illustration: Recurrence spectrum and returning trajectories in the TDLO
As an illustration of the different dynamical pictures that arise depending on the natureof trajectories that are implemented, let us take our 2D time-dependent linear oscillator(TDLO). Consider an initial wavefunction given by Eq. (21) with q = 0, J = 1 , = a ; the direction of p J is shown schematically in Fig. 1(a) along with the resultingclassical trajectories. Snapshots of the time-dependent wavepackets are shown in Fig. 1(a)-(d). Each wavepacket follows the guiding trajectory q J ( t ) . Both of these guiding trajectoriesare periodic with period 4 π (the plots in Fig. 1 show the trajectories in the interval [0 , π ]).Two typical Bohmian trajectories are shown in Fig. 2. The first (resp. second) Bohmiantrajectory was chosen so that shortly after t = 0 , the Bohmian particle sits at the max-imum of the J = 1(resp. J = 2) wavepacket. The main characteristic of the Bohmiantrajectories is that they seem to ”jump” from one guiding trajectory to the other each timethe wavepackets cross or interfere. This is a simple consequence of Eq. (33): the veloc-ity of the Bohmian ”particle” is proportional to the overall current density resulting fromthe interfering wavepackets, and by definition the current density lines do not cross eachother. Although these two families of Bohmian trajectories are different from the classicalguiding trajectories, on a statistical basis the motion of the wavepackets along the guidingtrajectories is recovered.The Bohmian and classical trajectories can be seen as two alternative manners of definingthe propagation and transport of the probability density in this TDLO. For example ifone focuses on the probability density in a small region V around the origin, the so-calledrecurrence spectrum (Fig. 3), defined by the probability P ( t ) = (cid:82) V | ψ ( x , t ) d x | displays sharppeaks at specific times t rec . These values of t rec correspond to the passage of a wavepacketin the region V and they are obviously given by the times at which one of the classicalguiding trajectories passes through the origin. It can be seen from Figs. 3 and 1, thatthe first recurrence taking place at time t rec (1) is due to the wavepacket propagating alongthe J = 2 (red solid) guiding trajectory. Actually the first return to the origin of the J = 1 (dashed purple) trajectory only happens at t = t rec (7) = 2 π . In terms of the Bohmiantrajectories, the interpretation is more involved: as can be seen from Fig. 2(b) the Bohmiantrajectory plotted in green (light gray) passes through the origin at t = t rec (1) , t rec (3) , t rec (5) and t rec (7) = 2 π and therefore contributes to the relevant peaks in the recurrence spectrumof Fig. 3. On the other hand the purple (dark gray) trajectory is not near the originat those recurrence times, but instead contributes to the peaks at the recurrence times t = t rec (2) , t rec (4) , t rec (6) and t rec (7) (this is not shown in the figures). We thus see that the In the examples given in this work atomic units are used throughout, with a TDLO having unit mass. IG. 2: (a) Two Bohmian trajectories corresponding to the wavefunction shown in Fig. 1 areplotted in light grey (green) and dark grey (purple); their initial conditions are near the maximumof the wavefunction, x dBB (0) = (0 . , .
09) and (0 . , − .
08) respectively. (b): The time evolutionof these 2 Bohmian trajectories are depicted in the time intervals (from top to bottom): (0 , t rec (1) ),( t rec (1) , t rec (3) ) , ( t rec (3) , t rec (5) ) , ( t rec (5) , t rec (7) ) , where t rec ( j ) is the recurrence time defined by thereturn of a wavepacket to the origin (see text for details) and t rec (7) = 2 π . The arrows indicate thedirection of the Bohmian particle motion at the beginning of each time interval. dynamical interpretation of the recurrence spectrum in terms of trajectories is quite differentif couched in terms of classical trajectories or given in the de Broglie-Bohm framework. IV. WEAK MEASUREMENTS AND TRAJECTORIES IN THE TDLOA. General Remarks
We give in this section derivations and specific computations for weak measurementof trajectories for the TDLO. We have seen in Sec. II that weak measurements of theposition and momentum observables lead to different type of trajectories. A crucial differencein the protocols is that the ”weak trajectories” are obtained from the analysis of several weak measurement apparata interacting with the system as it evolves from a given pre-selected state to a unique final post-selected state. Post-selection is not made after eachweak measurement, but only at the end of the evolution. Instead the local average velocity16
IG. 3: The recurrence spectrum defined in the text is given as a function of time. A peak occurswhen part of the wavefunction returns to the region around the origin. 7 recurrence peaks arevisible in the plot, the recurrences taking place at times t rec ( j ) , j = 1 − inferred from the weak measurement of the momentum operator is obtained by performinga single weak measurement immediately followed by post-selection; but in order to obtainthe velocity field, such weak measurements must be repeated by scanning the post-selectedstate over the spatial region of interest.Note also that although the pre-selected state is the initial state of the system in bothcases, the post-selected states will typically be different. For ”weak trajectories”, it is usefulto choose a post-selected state carrying the dynamical information (the mean position andmomentum at the time of post-selection) of the wavepacket. The weak measurement ofBohmian trajectories relies instead on post-selecting ideally to an eigenstate of the positionoperator (see however Ref. [15], in which a protocol allowing to obtain an approximate weakmeasurement of the velocity field in non-ideal conditions is presented).17 . Weak trajectories and sum over paths
1. Weak trajectories and the underlying classical dynamics
Let us start by specializing the weak position weasurements (7) to the case of the TDLOwith some additional assumptions. First let us take the post-selected state to be the Gaussian χ r f , p f ( r , t f ) = (cid:32) πδ f (cid:33) / e − ( r − r f ) /δ f e i p f · ( r − r f ) / (cid:125) . (35)Recall from Eq. (7) that the expression of the weak position at time t k , (cid:104) r ( t k ) (cid:105) W involves thepost-selected state | χ ( t k ) (cid:105) at time t k which is the wavefunction χ r f , p f ( r , t f ) evolved backwardin time. For the TDLO, this means finding the guiding trajectory q f ( t ) having the finalboundary condition q f ( t f ) = r f and p f ( t f ) = p f . At time t k the backward evolved guidingtrajectory will be found at the position q f ( t k ) . Hence a non-vanishing weak value will beregistered by a WMA positioned near R ≈ q f ( t k ) provided the wavefunction ψ ( q , p ) ( r , t = t k ) has a substantial amplitude in the region r ≈ q f ( t k ) . A particular case of practical interest arises when χ r f , p f ( r , t f ) ≈ ψ ( q , p J ) ( r , t f ) , (36)where following the notation of Eqs. (21)-(22) ψ ( q , p J ) ( r , t f ) represents the branch of thepre-selected wavefunction ψ ( r , t ) = (cid:80) J a J ψ ( q , p J ) ( r , t ) [Eq. (21)] propagating along aguiding trajectory q J ( t ) . Then only a WMA positioned along the guiding trajectory q J ( t )will display non-vanishing weak values. Moreover since Eq. (36) then holds for any t, eachweak value (cid:104) r ( t k ) (cid:105) W becomes an expectation value of the position in the state ψ ( q , p J ) ( r , t k ) , which is simply the real term (cid:104) r ( t k ) (cid:105) W = q J ( t k ) . (37)The corresponding weak trajectory (8) is hence the guiding trajectory q J ( t )WT ψ ( t ) ,χ ( t f ) = { t k , Re (cid:104) r ( t k ) (cid:105) W } = (cid:8) t k , q J ( t k ) (cid:9) . (38)Note that in order to obtain (38), Eq. (36) is a sufficient but non-necessary condition. Inparticular, it can be deduced from Eq. (40) given below that any χ r f , p f ( r , t f ) such that q J ( t f ) = q f ( t f ) and p J ( t f ) = p f ( t f ) will also lead to the weak trajectory (38). This is not valid for the WMAs located at a point where guiding trajectories cross. IG. 4: (a) A wavefunction of the form given by Eq. (21) with q = 0, J = 1 , , a ≈ a ≈ a , and with p J schematically indicated by the arrows, initially ( t = 0) compactly localized at theorigin (blue dot) propagates along the the 3 guiding trajectories J (see labels in the plot). Theprobability density is shown at t > t when the wavepackets start moving along the guidingtrajectories. (b) The probability density is shown at t = t f , the time at which the postselection ismade. (c) Postselection, represented by the gray box, is made at point A along the J = 1 trajectorywith a postselection to state (38) with J = 1. A grid of WMAs (weak meaurement apparata, actingas quantum probes) as defined in the text is suggested, some WMAs being explicitly pictured (whiteboxes). Only the WMAs along trajectory A have their quantum state modified (indicated by ablue shading) whereas the quantum states of the other WMAs on the grid are left unchanged.The weak trajectory can be inferred by reading the states of the WMAs. (d) Same as (c) whenpostselection is made at point B along the J = 2 trajectory with a postselected state (38) with J = 2: only the WMAs along trajectory B (shown by a blue shading) have their quantum statemodified, with a phase term proportional to the weak value (37). An illustration is given in Fig. 4. An initial wavefunction ψ ( r , t ) = (cid:88) J =1 a J ψ ( q =0 , p J ) ( r , t ) (39)tightly localized at the origin, subsequently expands as a sum over paths involving 3 guiding19rajectories J = 1 , , t = t f chosen slightlyafter the wavepackets have returned for the first time to the origin [the probability densityat t = t f is plotted in 4(b)]. Let us first choose a post-selected state given by Eq. (36)with r f = r A [the point A is displayed in Fig 4(c)] and J = 1. Assume a set of WMAwas disposed on a grid as indicated schematically in Fig 4(c). Then, only the WMAs placedalong the classical trajectory q J =1 ( t ) record a non-vanishing weak value. This weak valueis given by Eq. (37). Fig 4(d) shows the WMA having a non-vanishing weak-value for thesame preselected state but for a postselected state obeying (36) at r f = r B with J = 2:those WMAs are precisely the ones placed along the classical trajectory q J =2 ( t ).
2. Isolated weak values (cid:104) r ( t k ) (cid:105) W can also be obtained analytically for a post-selected state of the form (35)but not obeying the condition (36). Provided a WMA does not lie at positions wheredifferent branches ψ ( q =0 , p J ) ( r , t ) of the system wavefunction overlap, and further assumingthat the wavepackets are narrower than the range of the system-WMA interaction given by f (cid:16) | ˆr − R k | (cid:17) , the integrals in Eq (7) defining the weak value can be computed analyticallyand where non-vanishing put in the compact form (cid:104) r ( t k ) (cid:105) W = q J k ( t k )+ (cid:2) q J k ( t k ) − q f ( t k ) (cid:3) · m ( t k ) + (cid:2) p J k ( t k ) − p f ( t k ) (cid:3) · m ( t k ) . (40)As above q J k ( t k ) and p J k ( t k ) are the position and momentum of the classical guiding tra-jectory J k at the time the system interacts with the k th WMA. q f ( t k ) and p f ( t k ) are theposition and momentum at time t k of the trajectory having the boundary conditions fixedby the choice of post-selection, q f ( t f ) = r f and p f ( t f ) = p f (so that q f ( t ) is formally thetrajectory with the final boundary conditions determined by postselection evolved backwardin time). m ( t k ) and m ( t k ) are purely time-dependent complex functions.Note that there is now an index k at J k given that Eq. (36) is not obeyed. In that case ifpostselection at t f happens on, a branch, say J f of the wavefunction, a nonzero weak value (cid:104) r ( t k ) (cid:105) W can only be obtained provided the backward evolving q f ( t ) crosses ”accidentally”,at some time t k a wavepacket moving along another branch, denoted J k (thereby evolvingalong the classical trajectory q J k ( t )). At some other time t k (cid:48) , one can imagine that a differenttrajectory q J k (cid:48) ( t ) may be crossed, especially if the inital wavefunction of the form (39)20ontains many branches. In this situation, the weak values will be nonzero for a smallnumber of isolated points. Compared to the previous case, where a weak trajectory can beinferred from (38) involving in principle a dense number of closely positioned WMAs, theset { t k , R k } containing a few points cannot be said to form a trajectory, but rather isolatedpoints belonging to different branches of the wavefunction. An exemple is given in Fig. 5(a).When the conditions stated above Eq. (40) are not fulfilled, then (cid:104) r ( t k ) (cid:105) W must be com-puted numerically, though the compatibility condition for obtaining non-zero weak values(overlap of χ r f , p f ( r , t k ) and ψ ( r , t ) in the neighborhood of R k ) can generally be inferredfrom the classical dynamics that determine the guiding trajectories.
3. Sum over paths
We have just seen that when post-selection takes place along a given branch of thewavefunction, then if the postselected state is dynamically compatible with the guidingclassical trajectory carrying that branch, then only the WMAs placed along that guidingtrajectory will display non-zero weak values. Post-selecting appropriately on a differentbranch will instead yield non-zero weak values along the guiding trajectory associated withthat specific branch. Now since the wavefunction (39), or more generically the semiclassicalpropagator (26) involves a sum over paths, it would be valuable if the corresponding weaktrajectories could be detected simultaneously by the weak measurement apparata.This is possible if post-selection is made at some final position r f where two or more tra-jectories cross and provided the post-selected state can be tailored to take the approximateform χ r f ( r , t f ) ≈ (cid:88) K c K χ r f , p Kf ( r , t f ) , (41)where the c K are arbitrary coefficients and the χ r f , p Kf ( r , t f ) ≈ ψ ( q , p K ) ( r , t f ) as in Eq. (36).Plugging in Eqs. (22) and (41) in the weak value definitions (1) and (7) yields (cid:10) r ( t k ; R k ) (cid:11) W = (cid:10) χ r f ( t f ) (cid:12)(cid:12) U ( t f , t k ) ˆr f (cid:16) | ˆr − R k | (cid:17) [ U ( t k , t ) | ψ ( t ) (cid:105) ] (cid:10) χ r f ( t f ) (cid:12)(cid:12) U ( t f , t ) | ψ ( t ) (cid:105) (42)= (cid:80) K,J c ∗ K a J (cid:68) χ p Kf ( t k ) (cid:12)(cid:12)(cid:12) ˆr f (cid:16) | ˆr − R k | (cid:17) (cid:12)(cid:12)(cid:12) ψ ( q , p J ) ( t k ) (cid:69)(cid:80) K,J c ∗ K a J (cid:68) χ p Kf ( t k ) (cid:12)(cid:12)(cid:12) ψ ( q , p J ) ( t k ) (cid:11) . (43)21 IG. 5: (a)
Isolated weak value : Postselection takes place at point B along the J = 2 trajectory,but unlike the case pictured in Fig. 4(d), the postselected state is of the form (35) but does notobey the condition (36); instead the postselected momentum is “arbitrary”, and corresponds tothe trajectory evolving backward in time shown in red (dashed grey). Then the WMAs placedin a grid are generally not modified (a few are shown in the figure, left unshaded) since the weakvalues vanish and no weak trajectory can be defined. If the post-selected backward-evolved guidingtrajectory crosses accidentally a wavepacket traveling on one of the system’s guiding trajectories,the weak value will be a complex number and the relevant WMA will be modified, as indicated bythe light blue shading of the WMA labeled M . This gives rise to an isolated WMA being modified,and hence no weak trajectory can be defined. (b) Sum over paths : Post-selection takes place at r f = 0 when the wavefunction first returns to the origin (grey box), with a post-selected state ofthe form (41). The WMAs along the three guiding trajectories J = 1 , , q J ( t k ) given by (44) (the state of the other WMAs on the grid– not shown on the figure – is left unchanged). This allows to infer weak trajectories correspondingto the sum over paths appearing in the Feynman propagator. Assuming as we have done up to now that the WMAs set at places where the differentbranches interfere are disregarded, a typical WMA positioned at R k will therefore interactat most with the wavepacket propagating along a given branch, say J . In turn only the samebranch J of the postselected state (41) will overlap with the system wavefunction at R k and22q. (43) becomes (cid:10) r ( t k ); R k ; J (cid:11) W = (cid:68) χ p Jf ( t k ) (cid:12)(cid:12)(cid:12) ˆr f (cid:16) | ˆr − R k | (cid:17) (cid:12)(cid:12)(cid:12) ψ ( q , p J ) ( t k ) (cid:69)(cid:68) χ p Jf ( t k ) (cid:12)(cid:12)(cid:12) ψ ( q , p J ) ( t k ) (cid:11) = q J ( t k ); (44)it is now necessary to explictly state the branch J relevant to the weak value. Indeed,possibly at the same time a different WMA positioned at R k (cid:48) will have interacted withanother branch J (cid:48) of the system wavefunction consistent with the postselection condition(41), recording the weak value (cid:104) r ( t k ); R k (cid:48) ; J (cid:48) (cid:105) W . If there is a sufficient number of WMAsit is then straightforward to arrange the non-zero weak values extracted from the weakmeasurement apparata into time-ordered sets corresponding to different trajectoriesWT ψ ( t ) ,χ ( t f ) ( J ) = { t k , Re (cid:10) r ( t k ); R k ; J (cid:11) W } = (cid:8) t k , q J ( t k ) (cid:9) (45)WT ψ ( t ) ,χ ( t f ) ( J ) = { t k , Re (cid:10) r ( t k ); R k ; J (cid:11) W } = (cid:8) t k , q J ( t k ) (cid:9) ... (46)These trajectories resulting from the weak measurements of the position are a subset of thesum over paths constitutive of the propagating wavefunction compatible with the postse-lected state (41).Consider for example the situation previously shown in Fig. 4 and assume postselectionis made at t f = t r , when the wavepackets return for the first time to the origin r f = 0.The situation is represented in Fig. 5(b): assuming a set of weakly interacting measurementapparata (WMA) laid out in a grid, the WMAs placed along the 3 guiding trajectories J = 1 , q J ( t k ).By retrieving the weak values, a set of weak trajectories corresponding to the three classicalguiding trajectories along which the wavepackets move can be defined. This shows that bypostselecting appropriately at a position where several Feynman paths cross, it is in principlepossible to observe the sum over paths as weak trajectories resulting from the interactionbetween WMAs and the system wavefunction. C. Weak measurement of the velocity field and Bohmian trajectories
As we have seen above [Eqs. (11)-(12)], the weak measurement of the momentum op-erator followed immediately by postselection to an eigenstate | r f (cid:105) of the position operatoryields a velocity field that turns out to correspond to the local velocity at r f of the particle23 IG. 6: (a) A typical Bohmian trajectory with its final ( t = t f )position at the postselectionpoint A (on the J = 1 guiding trajectory) is shown (dark grey [online:red] thick dashed line).The postselection condition is the one illustrated in Fig 4(c), for which the WMAs that haveinteracted (shaded in the figure) are only along the J = 1 trajectory (all the other WMAs, includingthose along the Bohmian trajectory, are left unshaded). The inset details the Bohmian trajectory,starting at the origin, first following the wavepacket travelling along the J = 2 guiding trajectorythen jumping to the wavepacket along the J = 3 guiding trajectory, going back to the origin andleaving along the wavepacket following the J = 1 trajectory. All the Bohmian trajectories with afinal position in the vicinity of A have the form shown in the inset. (b): Same as (a) but for apostselection condition at t = t f at point B, corresponding to the example shown in Fig 4(d). Atypical Bohmian trajectory having its endpoint in the vicinity of B is shown [dark grey (online:red)thick dashed line]. Here the shaded WMAs are those along the J = 2 guiding trajecotry. The insetdetails how the Bohmian trajectory unfolds in time. constitutive of the Bohmian model characterized by the law of motion (34). Here each weakmeasurement, made by a single WMA positioned at R = r f , is followed by a projectivemeasurement ideally at the same position. The projective measurement terminates the sys-tem evolution, so the procedure must be repeated first for the same t throughout space (orat least where the wavefunction amplitude is known to be non-negligible) and this must bedone again for each value of t under consideration. Overall, these weak measurements allowto map unambiguously the velocity field v ( r f , t ) = Re (cid:104) p ( t ) (cid:105) W /m at each space-time point.24ote that strictly speaking, it is not possible to deduce unambiguously particle trajec-tories from a finite sample of velocity field values v ( r f , t k ), without assuming in the firstplace that the streamlines correspond to the actual motion of particles. The additional spe-cific assumption (34) needs to be made. Then the Bohmian trajectories can be integratedfrom the velocity field. The Bohmian trajectories shown in Fig. 6 have been computed bynumerical integration. They correspond to the wavefunction displayed in Fig. 4.Fig. 6(a) shows a typical Bohmian trajectory consistent with postselection at point A [see Fig 4(c)]. In order to infer that Bohmian trajectory from experimental observations,one would need to repeat the weak measurement procedure of Sec. II C by taking differentpostselection points at different times. Instead in Fig. 6(a) we directly show a Bohmiantrajectory arriving at point A in the the postselection region. If we assume we have carriedout the postselection at A as specified in Fig. 4(c),then only the WMAs along trajectory J = 1 will be shaded, the other WMAs on the grid do not have their quantum statemodified. The trajectory shown in Fig. 6(a) is the one that at t = t f has the final position x dBB ( t f ) = q J =1 ( t f ) ie at the maximum of the wavepacket that has followed the guidingtrajectory J = 1. This trajectory is “typical” in the sense that all the other Bohmiantrajectories having the position at t = t f in the vicinity of the postselected point q J =1 ( t f )have the same topology, leaving initially the origin along with the wavepacket travellingalong J = 2 , then jumping to the wavepacket along J = 3 , going back to the origin thenleaving the origin along the wavepacket following the J = 1 trajectory [see the inset in Fig.6(a)]. The important feature is that these Bohmian trajectories are unrelated to the weaktrajectory measurements: for example the WMAs on path J = 1 in the x > A do not reach these WMAs. Conversely as seen in Fig.6(a) there are WMAs that are left unchanged (that are interpreted as not having interactedwith the system given the postselection, represented by the unshaded boxes) on the path ofthe Bohmian trajectories.Fig. 6(b) illustrates the same type of situation shown in Fig. 6(a) but for a weaktrajectory postselected at point B under the conditions depicted in Fig 4(d). The typicalBohmian trajectory displayed in the figure ending at B at the postselection time t = t f , isnot related to the shaded and unshaded WMAs, having interacted (or not) with the systemalong the weak trajectories. 25 . DISCUSSION AND CONCLUSION The starting point of this paper was to note that when trajectories are employed tointerpret the dynamics of quantum systems, the classical trajectories of the semiclassicalpath integral propagator on the one hand and the de Broglie Bohm trajectories on the othergive rise to different accounts of the dynamics taking place in the system. While this is not aproblem if these types of trajectories are envisaged as computational tools or mathematicalartifacts, the main idea developed in this work was to introduce weak measurements in orderto implement a practically non-disturbing observational windows that would allow to followthe evolution of the system. This was done by employing a model, a 2D time-dependentlinear oscillator, introduced as a manner of simulating a more complex (but less tractable)system in the semiclassical regime.We have seen that the classical Feynman trajectories can be observed by weakly measuringthe position of the system by an array of weakly interacting devices, followed by a singlepostselection. Taking an arbitrary postselection state does not yield weak trajectories, giventhat then none of the WMAs will show signs of interaction with the system (their quantumstate has not been modified, except accidentally, giving rise to an isolated weak value). Thedynamical compatibility condition requires that the backward postselected state overlapswith a trajectory of the propagator. Only then will the entire set of WMAs along thattrajectory indicate they have interacted with the system (their quantum state being modifiedby the corresponding weak value). In this sense, the postselected state must contain theinformation on the Feynman path appearing as a weak trajectory. Alternatively, a trial anderror procedure sampling the parameter space for the postselection state can be employed,monitoring the states of the WMAs until they indicate a continuous trajectory. This willprecisely be the weak trajectory of a classical Feynman propagator path.We also saw that Bohmian trajectories can be inferred from the weak measurement of momentum . This is particularly noteworthy, given that the Bohmian momentum is not di-rectly connected to the eigenvalues of the momentum operator. Several weak measurementsmust be done by sampling all the spatial domain, and this must be repeated at each timein order to follow the evolution of the flow. Defining Bohmian trajectories from the velocityrequires an additional assumption, associating the motion of particle with the lines of thecurrent density. In this sense, the weak measurements do not directly yield the observation26f the Bohmian trajectories, but that of the momentum field.As pointed out above [Eq. (14)], the weak measurement of momentum usually relies oninferring a velocity from the difference between two position measurements. In practice itis not possible to do the position measurements with infinite precision, and it has recently[15] been suggested to model the finite resolution using the POVM (Positive Operator val-ued Measure) framework. The authors of Ref. [15] have in particular shown under whichconditions position Gaussian measurement operators allow to recover the Bohmian field ve-locity. In the context of weak trajectories, post-selection to an ideally well-known positioneigenstate is not required, but Gaussian measurement operators could be useful to modelpost-selection to a state having the characteristics of the final wavepacket | χ ( t f ) (cid:105) .So can weak measurements open an observational window that would allow to give a“correct” account of the dynamics of a quantum system in terms of trajectories? Thereare certainly different answers that can be given to this question, ultimately relying oninterpretational commitments (including the meaning of the weak values). A consensualoption (having a Copenhagen interpretation flavour) would state that the type of trajectoriesthat one sees depends on the context – the type of weak measurements that are being madeand hence the entire experimental setup including the WMAs (and in particular in theirinteraction with the system); such an answer would obviously undermine the idea that thereis a meaningful underlying dynamics that can be understood in terms of trajectories, asthese would be relegated to being artifacts or computational tools.Still, the fact that weak trajectories are obtained from a series of weak measurements ofpositions given a final postselected state, as opposed to being inferred from a mean velocityfield at each point, is a natural feature with regard to the definition of a space-time trajectory.Indeed, if we take the weak values in the original [8] sense as referring to a generalized valuefor an observable obtained without appreciably disturbing the system evolution, the weakvalues of the position for a chosen postselection suffice in order to extract a trajectory.On the other hand the observation of the momentum weak value at a postselected space-time point does not as such allow to observe a trajectory, but the local weak value of themomentum field.Moreover, sticking to the situation illustrated in Fig. 6(a), it is noticeable that there isno correlation between the Bohmian trajectories detected at the postselection point A andthe WMAs that interacted with the system (along parts of the J = 1 guiding trajectory in27hich there are no Bohmian trajectories reaching A ) and those that haven’t (the Bohmiantrajectories do not trigger the WMAs along part of their route although the interactioninvolves the position). This fact could apparently be taken as an argument against therelevance of interpreting the dynamics in terms of Bohmian trajectories, on the ground thatthe particle does not comply with its role (which is to make detectors click). This is not thecase, however: one should indeed bear in mind that the weak interactions are unitary and donot translate as clicks until the WMAs themselves are measured. In addition, given the non-local character of the de Broglie Bohm dynamics, the internal states of the WMAs must betaken into account explicitly, and this may affect non-locally some features of the Bohmiandynamics, in particular the ”no-crossing” rule [28]. While it is well-known that Bohmiantrajectories are modified in genuine open systems relative to the Bohmian trajectories of theclosed system (the ones we were interested in throughout this work), the extent to whichthis aspect subsists in the case of weak measurements remains to be investigated. Appendix A: Closed form solutions for the time-dependent harmonic oscillator
The Hamiltonian for a one dimensional linear oscillator of mass m with a time-dependentfrequency V ( t ) is given by H ( t ) = p m + m V ( t ) q . (A1)We will not distinguish in the notation the classical Hamiltonian and phase-space variablesfrom the corresponding quantum Hamiltonian and operators unless required by the context.The solutions of the Schr¨odinger equation can be obtained exactly by employing differentmethods, like algebraic methods [29], path integrals [25] or the more popular procedure basedon solving for the eigenfunctions of dynamical invariants [30]. We will employ a version [31]of the latter method involving solutions of Ermakov systems, which is known to be wellsuited to working with Gaussian wavefunctions.The Lagrangian corresponding to the Hamiltonian (A1) is L = m q − m V ( t ) q (A2)and leads to the classical equation of motion ∂ t q ( t ) + V ( t ) q ( t ) = 0 . (A3)28mploying an amplitude-phase decomposition of the classical solution q ( t ) in the form q ( t ) = α ( t ) ( c cos ( φ ( t ) − φ ( t )) + c sin ( φ ( t ) − φ ( t ))) (A4)where α ( t ) is the amplitude and φ ( t ) the phase leads to an auxiliary nonlinear equation ∂ t α ( t ) α ( t ) + V ( t ) = c α ( t ) (A5)along with the condition ∂ t φ ( t ) = c α ( t ) . (A6)Eqs. (A3)-(A5) form an Ermakov system [32] linking the solutions of a linear and a nonlinearequation. The c i in the equations above denote real constants.The reason for introducing Ermakov systems is that the time-dependent Schr¨odingerequation with the Hamiltonian (A1), i (cid:126) ∂ t ψ ( x, t ) = (cid:20) − (cid:126) m ∂ x + m V ( t ) x (cid:21) ψ ( x, t ) (A7)admits the closed form solution ψ ( q ,p ) ( x, t ) = (cid:18) mπα ( t ) (cid:19) / e − [ x − q ( t )] (cid:16) mα ( t )2 − imα (cid:48) ( t )2 (cid:126) α ( t ) (cid:17) e ip ( t )[ x − q ( t )] / (cid:126) e i (cid:126) [ p ( t ) q ( t ) − p q ] e − i [ φ ( t ) − φ ] . (A8)Hence the exact solutions to the Schr¨odinger equation can be obtained from the knowledgeof the solutions to the Ermakov system, namely q ( t ) which is the solution of the linearequation (that is also the classical equation of motion) and α ( t ) (and its integral φ ( t )) whichis a solution of the nonlinear Ermakov equation. It can be checked explicitly by plugging inEq. (A8) into Eq. (A7) requires setting the constant c of Eq (A5) to c = 2 (cid:126) . Finally, thenotation q etc. indicates the values of the functions at t = t , ie q ≡ q ( t ) and so on. Letus choose specifically the amplitude function such that α (cid:48) ( t ) = 0. Then at t = t the initialwavefunction (A8) is given by ψ ( q ,p ) ( x, t ) = (cid:18) mπα (cid:19) / e − m [ x − q ] /α e ip [ x − q ] / (cid:126) , (A9)that is a standard Gaussian. q and p that appeared as parameters in Eq. (A8) thuscorrespond to the average initial position and momentum of the initial Gaussian and α sets29he initial width. Note that the initial choice of q and p also sets the values of c and c in Eq. (A4), equal respectively to q /α and p α / (cid:126) m .From Eq. (A8) it appears that the exact solution of the Schr¨odinger equation with theinitial condition (A9) has at all times its maximal probability along the curve q ( t ): thisdefines the guiding trajectory. The evolution of the wavefunction ψ ( q ,p ) ( x, t ) depends onthe properties of the guiding trajectory (which as we have seen above, turns out to bethe solution obtained by solving the classical equations with the Hamiltonian (A1)). Theensuing classical correspondence is in line with Ehrenfest’s theorem, though in a strongerform: in case in which the initial wavefunction is a superposition of Gaussians (A9) withdifferent parameters q and p , Ehrenfest’s theorem would then apply for each individualpropagating wavepacket.We will be interested in cases in which the time-dependent part of the potential, V ( t )is periodic. The stability properties of the general solutions of (A3) – Hill’s equation –are well-known [33] by resorting to Floquet theory. For the present purpose of this work,it will suffice to restrict the discussion to the simplest non-trivial case, namely when thetime-dependence takes the form V ( t ) = v − κ cos (2 ωt ) (A10)Then Eq. (A3) becomes a Mathieu equation and the guiding trajectory q ( t ) is given in termsof real even (“cosine”) and odd (“sine”) Mathieu functions. It is well-known [34] that for agiven κ , the solutions are bounded and periodic only for specific “characteristic values” of v ( κ ). [1] L. S. Schulman 1996, Techniques and Applications of Path Integration (Wiley, New York).[2] P. R. Holland, 1995 The Quantum Theory of Motion (Cambridge Univ. Press, Cambridge).[3] S. Wimberger 2014
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