AArrival Time Distributions of Spin-1/2 Particles
Siddhant Das and Detlef D ¨urr
Mathematisches Institut, Ludwig-Maximilians-Universitat M ¨unchen, Theresienstr. 39, D-80333 M ¨unchen, [email protected]@mathematik.uni-muenchen.de
ABSTRACT
The arrival time statistics of spin-1/2 particles governed by Pauli’s equation, and defined by their Bohmian trajectories, showunexpected and very well articulated features. Comparison with other proposed statistics of arrival times that arise from eitherthe usual (convective) quantum flux or from semiclassical considerations suggest testing the notable deviations in an arrivaltime experiment, thereby probing the predictive power of Bohmian trajectories. The suggested experiment, including thepreparation of the wave functions, could be done with present-day experimental technology.
Introduction
In non-relativistic quantum mechanics, the probability of finding a particle in a small spatial volume d r around position rrr at a fixed time t is given by Born’s rule | Ψ ( rrr , t ) | d r , where Ψ ( rrr , t ) is the wave function of the particle. This formula isexperimentally well established. However, a formula for the probability of finding the particle at a fixed point rrr between times t and t + d t is the matter of an ongoing debate . Let us consider a typical time of arrival experiment, in which a particleis initially trapped in a region Σ ⊂ R , e.g., the interior of a potential well. The trap is released at, say, t =
0, allowing theparticle to propagate freely in space. If the trapping potential is deep enough, the wave function of the particle at this instant, Ψ ( rrr ) ≡ Ψ ( rrr , t = ) , practically vanishes outside the region Σ . Particle detectors placed on the boundary ∂ Σ measure the timeof arrival of the particle, denoted by τ . If the experiment is repeated many times, the recorded arrival times are random, evenif the initial wave function of the particle ( Ψ ) is kept unchanged in each experiment. What is the probability distribution ofarrival times Π Ψ ( τ ) as a functional depending on Ψ and on ∂ Σ ?Measurement in quantum mechanics has become in recent decades a tricky notion. Traditionally, measurement outcomeswere solely associated with observables, represented by self-adjoint operators on the Hilbert space of the measured system.However, it has long been known that for time measurements, such as arrival times, no such observable exists [2, § 8.5]. In fact,the notion of self-adjoint operators defining quantum observables does not apply to many other experiments as well. To remedythis situation, the notion of an observable was generalized to positive operator valued measures (POVMs), and there are varioussuggestions for arrival time POVMs (see for a discussion). In Dürr et al., all measurements describable by POVMs werecalled linear measurements. Another class of measurements have also been performed, the so-called weak measurements, which are nonlinear in the sense of . So far, however, no theoretical predictions for the arrival time distribution Π Ψ ( τ ) have been backed up by experiments (see for various proposals). On the other hand, recent ‘attoclock’ experiments(claimed to be measuring the tunnel delay time of ionized electrons) have shown some of the theoretical ideas to be empiricallyinadequate .One problem with arrival time measurements is that detection events are based on interactions of the detectors with thedetected particle, which may disturb its wave function in an uncontrollable way, leading to backscattering and in extreme casesto the quantum Zeno effect. While this is a valid concern, we note that the double-slit experiment (mentioned also below) is anexample where the distribution of arrival positions of the particle on the detector screen (the ubiquitous interference picture) isanalyzed without any reference to the presence of the detector. Note well that the particles strike the detector surface at randomtimes , a fact blissfully ignored in the usual discussions of the double-slit experiment–and there are good reasons why that isjustified. We expect that in the experiment proposed in this paper the same will be true, i.e., the detection event should notbe drastically disturbed by the presence of the detector. Our expectation is based on our results, namely, on the very strikingarticulated features of the computed arrival time distributions, which should survive mild disturbances.A further problem is that the notion of arrival time is most naturally connected with that of particle trajectories, an ideawhich is hard to concretize in the orthodox interpretation of quantum mechanics. Bohmian mechanics (or de Broglie-Bohm pilotwave theory) is a quantum theory (and not simply an alternative interpretation of quantum mechanics) where particles move onwell defined smooth trajectories, hence it is naturally suited for computing arrival times of a particle. See Kocsis et al. for a a r X i v : . [ qu a n t - ph ] N ov eak measurement of average quantum trajectories, which can indeed be seen as Bohmian trajectories. Bohmian mechanics hasbeen proven to be empirically equivalent to standard quantum mechanics, wherever the latter is unambiguous (e.g., in positionand momentum measurements) . It has been shown that Bohmian mechanics provides in (far field) scattering situations ideal arrival time statistics for spin-0 particles , via the quantum flux (or the probability current) JJJ: Π Ψ ( τ ) = (cid:90) ∂ Σ JJJ ( rrr , τ ) · dsss (= : Π qf ( τ )) . (1)Only in scattering situations (i.e., when the detector surface ∂ Σ is far away from the support of the initial wave function Ψ ),is the surface integral in (1) demonstrably positive , otherwise it cannot be interpreted as a probability density (see for adiscussion of POVMs versus flux statistics). Although often not recognized or emphasized in textbooks, it is the quantum flux JJJ,integrated over time, that yields the double-slit interference pattern of arrival positions of particles on the screen––as there is no given time at which the particles arrive at the screen.Here, we must mention as well another flux based (Bohmian) arrival time distribution derived by C. R. Leavens (Eq. (9)of ), which is valid only in one space dimension (as discussed in [1, Ch. 5]). Therefore, it is not applicable for particles withspin-1/2, except perhaps in some idealized situations. In this paper, we propose a Bohmian formula for the distribution of first arrival times of a spin-1/2 particle (Eq. (9) below), which may be referred to as an ideal or intrinsic distribution, since it isformulated without referring to any particular measurement device, just as equation (1). In fact, the spin-1/2 analogue of (1)becomes a special case of (9), whenever the so-called current positivity condition is met. Evaluating our formula numericallyfor a specific, carefully chosen experiment, we find that the resulting arrival time distributions show drastic and unexpectedchanges when control parameters are varied. Since the predicted distributions show such interesting and significant behavior,we suggest that the proposed experiment be performed to test the predictive power of Bohmian mechanics for spin-1/2 particles.One may legitimately ask, how can the idea of an ideal first arrival time distribution be entertained at all? Our standpoint onthis is as follows: Given the relative simplicity of making the Bohmian prediction for the ideal arrival time distribution, andgiven the ambiguity of quantum mechanical proposals mentioned above, why not just do the experiment to check it? Howeverthe experimental results turn out, they provide in any case valuable experimental data that would enrich our understanding ofquantum mechanics and of Bohmian mechanics as well. A word of clarification is, however, in order: we do not construe in ourwork any contradiction to quantum mechanics, since an unambiguous answer to the question of arrival times has not been givenwithin this framework. The main advantage of Bohmian mechanics is the clear picture of reality, independent of observation, itprovides and which in the problem of arrival times allows an unambiguous answer in contrast to the answers given in quantummechanics, so far. y x z ≈ Ld ≈ Figure 1.
Schematic drawing of the experimental setup. The barrier at d is switched off at t = z = L .We proceed now to a description of the proposed experiment. A spin-1/2 particle of mass m is constrained to movewithin a semi-infinite cylindrical waveguide (Fig. 1). Initially, it is trapped between the end face of the waveguide and animpenetrable potential barrier placed at a distance d . At the start of the experiment, the particle is prepared in a ground state Ψ of this cylindrical box, then the barrier at d is suddenly switched off, allowing the particle to propagate freely within thewaveguide. The arrival surface ∂ Σ is the plane situated at distance L ( > d ) from the end face of the waveguide. We thencompute numerically from the Bohmian equations of motion how long it takes for the particle to arrive at ∂ Σ , and determinethe empirical distribution Π Ψ Bohm ( τ ) from typical trajectories (trajectories whose initial points are randomly drawn from theBorn | Ψ | distribution) for different initial ground state wave functions.The Bohmian equations for spin-1/2 particles are as follows: The wave function Ψ ( rrr , t ) is a two-component complex-valued pinor solution of the Pauli equation i ¯ h ∂∂ t Ψ ( rrr , t ) = − ¯ h m ( σσσ · ∇∇∇ ) Ψ ( rrr , t ) + V ( rrr , t ) Ψ ( rrr , t ) , (2)with given initial condition Ψ ( rrr ) . Here, V ( rrr , t ) is an external potential, and σσσ = σ x ˆ xxx + σ y ˆ yyy + σ z ˆ zzz is a 3-vector of Pauli spinmatrices. The quantum continuity equation for the Pauli equation reads ∂ | Ψ | ∂ t = ¯ hm ∇∇∇ · (cid:16) Im [ Ψ † ∇∇∇Ψ ] + ∇∇∇ × ( Ψ † σσσ Ψ ) (cid:17) = ∇∇∇ · JJJ
Pauli = : ∇∇∇ · (cid:16) vvv Ψ Bohm | Ψ | (cid:17) , (3)where Ψ † is the adjoint of Ψ , and | Ψ | = Ψ † Ψ . The rightmost equality of (3) defines the Bohmian velocity field vvv Ψ Bohm = JJJ
Pauli / | Ψ | . The term ¯ hm Im [ Ψ † ∇∇∇Ψ ] is the so-called convective flux, while ¯ h m ∇∇∇ × ( Ψ † σσσ Ψ ) is the spin flux. The alert readerwill recognize that the spin flux is divergenceless, hence one may argue that neither the flux JJJ Pauli nor the Bohmian velocityare uniquely defined. However, observing that the Pauli equation and its flux emerge as non-relativistic limits of the Diracequation and the Dirac flux, respectively, which are unique , one is led directly to the current and Bohmian velocity givenhere.The Bohmian trajectories are integral curves of the velocity field vvv Ψ Bohm , hence the Bohmian guidance law readsdd t RRR ( t ) = vvv Ψ Bohm ( RRR ( t ) , t ) = ¯ hm Im (cid:20) Ψ † ∇∇∇ΨΨ † Ψ (cid:21) ( RRR ( t ) , t ) + ¯ h m (cid:34) ∇∇∇ × (cid:0) Ψ † σσσ Ψ (cid:1) Ψ † Ψ (cid:35) ( RRR ( t ) , t ) . (4)Here, RRR ( t ) is the position of the particle at time t . In Bohm’s theory, the spin-1/2 particle has no degrees of freedom other thanthose specifying its position in space, so spin is not an extra degree of freedom. Thus, all quantum mechanical phenomenaattributed to spin (such as the deflection of particles in the Stern-Gerlach experiment) arise solely from the non-linear equationof motion . The guidance law (4) is time reversal invariant and its right-hand side transforms as a velocity under Galileantransformations (see for further discussion). We integrate Eq. (4) for a statistical ensemble of | Ψ | -distributed initialparticle positions RRR ( ) (see for a justification). Methods
Our computations and results are for the following setup: Let the cylindrical waveguide be mounted on the xy − plane ofa right-handed orthogonal coordinate system, the axis of the cylinder defining the z − axis (Fig.1). Employing cylindricalcoordinates rrr ≡ ( ρ , φ , z ) , we model the potential field of the waveguide as V ( rrr , t ) = V ⊥ ( ρ ) + V (cid:107) ( z , t ) , where V ⊥ ( ρ ) = m ω ρ is a transverse confining potential, and V (cid:107) ( z , t ) = v ( z ) + θ ( − t ) v ( d − z ) is a time dependent axial potential comprised of animpenetrable hard-wall at z =
0, viz., v ( z ) = (cid:40) ∞ z ≤ z > , (5)and another impenetrable potential barrier v ( d − z ) that is switched off at t =
0. The wall at z = θ ( x ) is Heaviside’s step function. Fortunately, near perfect harmonic confinements can be realized in conventional (ultrahighvacuum) Penning traps, which can trap single electrons and protons over a wide range of trapping frequencies. Forelectrons, typical waveguide parameters read: L ≈ − mm , ω ≈ − rad/s . In we give a detailed analysis ofthe proposed experiment for a quadrupole ion trap (Paul trap) waveguide.The particle is prepared in a ground state of the cylindrical box at t =
0, which can be written as Ψ ( rrr ) = ψ ( rrr ) χ , where(setting ¯ h = m = d = ψ ( rrr ) = (cid:114) ωπ θ ( z ) θ ( − z ) sin ( π z ) exp (cid:16) − ω ρ (cid:17) (6)is the spatial part of the wave function, and χ = (cid:18) cos ( α / ) sin ( α / ) e i β (cid:19) , ≤ α ≤ π , ≤ β < π , (7)is a normalized Bloch spinor ( χ † χ = α and β , we obtain different ground state wave functions. For instance, α = ( π ) gives the spin-up (spin-down) ground state wave function, usually denoted by Ψ ↑ ( Ψ ↓ ) , while α = π and β = ields the so-called up-down ground state wave function Ψ (cid:108) = √ ( Ψ ↑ + Ψ ↓ ) . We refer to α and β as spin orientation angles ,because they specify the orientation of the “spin vector” sss : = ( Ψ †0 σσσ Ψ †0 ) / | Ψ | , given bysss = (cid:0) sin α cos β ˆ xxx + sin α sin β ˆ yyy + cos α ˆ zzz (cid:1) . The instant the barrier is switched off, the wave function spreads dispersively, filling the volume of the waveguide. Theparticle moves according to (4) on the Bohmian trajectory
RRR ( t ) = R ( t ) [ cos Φ ( t ) ˆ xxx + sin Φ ( t ) ˆ yyy ] + Z ( t ) ˆ zzz . In this choice ofcoordinates the first arrival time of a trajectory starting at RRR ( ) and arriving at z = L is τ ( RRR ( )) = inf { t | Z ( t , RRR ( )) = L , RRR ( ) ∈ supp ( Ψ ) } , (8)where Z ( t , RRR ( )) ≡ Z ( t ) is the z − coordinate of the particle at time t , and supp ( Ψ ) denotes the support of the initial wavefunction (the interior of the cylindrical box). Since the initial position RRR ( ) is | Ψ | -distributed , the distribution of τ ( RRR ( )) isgiven by Π Ψ Bohm ( τ ) = (cid:90) supp ( Ψ ) d RRR ( ) δ (cid:0) τ ( RRR ( )) − τ (cid:1) | Ψ | ( RRR ( )) . (9)In general, there is no closed form expression for (8), hence the integral in (9) cannot be evaluated analytically. However, if theBohmian trajectories cross ∂ Σ at most once, or in other words if the quantum flux JJJ Pauli is outward directed at every point of ∂ Σ , at all times (also referred to as the current positivity condition), then Π Ψ Bohm ( τ ) reduces to the integrated quantum flux Π qf ( τ ) , Eq. (1) with the Pauli current JJJ Pauli replacing JJJ.
Note well that by the very meaning of the quantum flux, (1) is a natural guess for the arrival time distribution from the pointof view of standard quantum mechanics as well . However, (1) makes sense only if the left-hand side is positive, which neednot be the case. Of course, if the current positivity condition holds, the left-hand side of (1) ≥
0, and Π qf becomes a specialcase of (9). Generally, this condition does not hold, in which case one computes Π Ψ Bohm ( τ ) numerically from a large number ofBohmian trajectories. Results and Discussion (i) For the spin-up ( α =
0) and spin-down ( α = π ) wave functions the arrival time distribution Π Ψ Bohm ( τ ) coincides withthe quantum flux expression (1), since in these cases the current positivity condition is satisfied. Moreover, in these casesJJJ Pauli ( rrr , τ ) can be replaced by the convective flux ¯ hm Im [ Ψ † ∇∇∇Ψ ] in (1). The resulting distribution has a heavy tail ∼ τ − as τ → ∞ . (ii) For other initial wave functions Π Ψ Bohm ( τ ) differs from (1) and falls off faster than ∼ τ − . For any initial groundstate wave function, the arrival time distribution displays an infinite sequence of self-similar lobes below τ = mdL π ¯ h (see Fig. 2below), which diminish in size as τ →
0. These lobes mirror typical wave function evolution when suddenly released to spreadfreely into the volume of the waveguide (see also ). (iii) If the initial wave function is an equal superposition of the spin-upand spin-down wave functions ( α = π ), the arrival time distribution pinches off at a maximum arrival time τ max , i.e., no particlearrivals occur for τ > τ max . Moreover an even more striking manifestation of the lobes can be seen: characteristic “no-arrival windows ” appear between the smaller lobes, inside which the arrival time distribution is zero . (iv) Time of flight measurementsrefer in general to semiclassical expressions based on the momentum distribution. Our distributions deviate significantly fromthis alleged semiclassical formula.A few details concerning the computation of our results are in order. First note that the Pauli equation (2) with initialcondition Ψ ( rrr ) can be solved in closed form, facilitating very fast numerical computation of Bohmian trajectories. The timedependent wave function takes the form Ψ ( rrr , t ) = ψ ( rrr , t ) χ , where the spin part is given by (7), while ψ ( rrr , t ) = (cid:114) ωπ exp (cid:16) − ω ρ − i ω t (cid:17) W ( z , t ) , (10)where W ( z , t ) = θ ( z ) (cid:2) D ( z − , t ) + D ( − z , t ) − D ( + z , t ) − D ( − − z , t ) (cid:3) , (11)which we call the ‘time evolution integral’. In (11): D ( x , t ) : = e − i π t i (cid:40) e i π x erfc (cid:34) i / √ (cid:18) x √ t − π √ t (cid:19)(cid:35) − e − i π x erfc (cid:34) i / √ (cid:18) x √ t + π √ t (cid:19)(cid:35)(cid:41) , (12) here erfc ( x ) is the complementary error function. A detailed derivation of this result will be given elsewhere . Substitutingour solution for the time dependent wave function in the guidance law (4), we obtain coupled non-linear equations of motionfor the spin-1/2 particle:˙ R ( t ) = sin α sin ( Φ ( t ) − β ) Re (cid:20) W (cid:48) W (cid:21) ( Z ( t ) , t ) , (13a)˙ Φ ( t ) = sin α R ( t ) cos ( Φ ( t ) − β ) Re (cid:20) W (cid:48) W (cid:21) ( Z ( t ) , t ) + ω cos α , (13b)˙ Z ( t ) = Im (cid:20) W (cid:48) W (cid:21) ( Z ( t ) , t ) + ω sin α sin ( Φ ( t ) − β ) R ( t ) , (13c)where W (cid:48) = ∂ W / ∂ z . The parameters in the initial wave function are, in view of (7), α and β , so we denote Π Ψ Bohm ( τ ) ≡ Π α | β Bohm ( τ ) . In fact, it is enough to consider β = Π α | β Bohm ( τ ) = Π α | Bohm ( τ ) , (14a) Π α | β Bohm ( τ ) = Π π − α | β Bohm ( τ ) . (14b)These results are proven in . We sample N ≈ initial positions from the | Ψ | distribution, solve Eq. (13) numerically foreach point in this ensemble, continuing until the trajectory hits z = L , then record the arrival time and plot the histogram for Π α | Bohm ( τ ) .For the spin-up and spin-down wave functions, Eq. (13c) reduces to ˙ Z = Im [ W (cid:48) / W ]( Z ( t ) , t ) and numerically it turns out thatIm [ W (cid:48) / W ]( L , t ) >
0. Hence the spin-up and spin-down trajectories cross ∂ Σ at most once, and the first arrival time distribution(or simply the arrival time distribution) in these cases equals (cf Eq. (1)) Π qf ( τ ) = [ W ∗ ( L , τ ) W (cid:48) ( L , τ )] . (15)As noted, (15) is non-negative for all values of τ , and features prominently a large main lobe for τ > L π . The main lobe fallsoff as (cid:0) L π (cid:1) τ , as τ → ∞ . An infinite train of smaller lobes permeates the interval 0 < τ < L π , which are well approximated bythe formula π L sinc (cid:0) L τ (cid:1) , whenever τ (cid:28) L . Apart from that we find that Π qf ( τ ) is a function only of the arrival distance L . It isalso independent of the trapping frequency ω , which is rather surprising. In fact, the spin-up (down) arrival time distributionsare independent of the exact shape of the transverse confining potential V ⊥ ( ρ ) of the waveguide as well . Figure 2 depicts ourresults for L =
100 ( ≈ mm for a d = μ m trap) and ω = ( ≈ . × rad/s ). Note: We have expressed L , ω and τ inunits of d , ¯ hmd and md ¯ h , respectively. For a d = μ m trap, known electron mass m ≈ . × − kg and reduced Planck’sconstant ¯ h ≈ . × − Js , the frequency and time units are ≈ . × rad/s and ≈ . μ s , respectively.For wave functions corresponding to 0 < α ≤ π (cf. (14b)), the first arrival time distribution is not given by the integratedflux (1). This is because the Bohmian trajectories in these cases cross ∂ Σ more than once, hence the aforementioned currentpositivity condition is not met. As α approaches π , the tail of Π α | Bohm ( τ ) thins gradually, pinching off completely at acharacteristic maximum arrival time τ max for α = π (i.e. all Bohmian trajectories with wave function Ψ (cid:108) strike the detectorsurface z = L before t = τ max ). In Fig. 2 τ max ≈ .
9, which corresponds to ≈ ms . This behavior results in a sharp drop in themean first arrival time (cid:104) τ (cid:105) in the vicinity of α = π , as shown in Fig. 3 below.Unlike the spin-up (down) case, the up-down arrival time statistics are influenced by the trapping frequency ω . Keeping L fixed, we find that the maximum arrival time τ max , mean first arrival time (cid:104) τ (cid:105) , and the standard deviation σ corresponding to Π π | Bohm ( τ ) decrease with increasing ω , each approaching a constant for ω (cid:29)
1. Conversely, when these quantities are graphedas functions of L with ω fixed, we see a clear linear growth in each (see Fig. 4). Remarkably, these effects persist even forlarge L , provided ω is also made suitably large (a welcome feature for Penning traps). Increasing L causes the arrival timedistributions to shift to larger values of τ , thus the smaller lobes can be easily seen, especially in experiments incapable ofresolving very small arrival times.Since the smaller (self-similar) lobes become progressively smaller, resolving the n th lobe (denoting the main lobe by n = δ t ) of the measuring apparatus. Roughly, δ t < width of n th lobe should suffice to resolve thefirst n lobes. This translates into δ t < (cid:18) md π ¯ h (cid:19) Ln . (16) (cid:8)(cid:8)(cid:8)(cid:25) (cid:0) L π (cid:1) τ @@R τ max τ @@R Π | Bohm @@R Π π | Bohm @@R Π sc Figure 2.
Arrival time histograms for spin-up (cid:0) Π | Bohm ( τ ) (cid:1) and up-down (cid:0) Π π | Bohm ( τ ) (cid:1) wave functions, L =
100 and ω = graphed along with the semiclassical arrival time distribution Π sc ( τ ) (dashed line) and the quantum (convective) fluxdistribution Π qf ( τ ) (solid line). We see agreement between Π | Bohm ( τ ) and Π qf ( τ ) . For the up-down case, no arrivals arerecorded for τ > . = τ max ). Note the disagreement of all distributions with Π sc ( τ ) . Each histogram in this figure has beengenerated with 10 Bohmian trajectories. The time scale on the horizontal axis is ≈ . μ s , assuming d = μ m . Inset:Magnified view of the self-similar smaller lobes of the up-down histogram, separated by distinct no-arrival windows. π π π π π π π
16 5 π π π π π . . . . . . . h τ i qf Ψ ↓ Ψ ↑ Ψ l α h τ i ω = 50 ω = 100 Figure 3.
Mean first arrival time (cid:104) τ (cid:105) vs. spin orientation angle α for L =
10 and β =
0. The symmetry of the curves about α = π is a consequence of property (14b).For d = μ m and L = mm (Fig. 2), a modest δ t ≈ μ s will successfully resolve 8 lobes (main + 7 smaller lobes), while δ t ≈ . μ s will resolve as many as 83 lobes (main + 82 smaller lobes). However, we must also understand that only a few datapoints (about (cid:16) π (cid:17) Nn in N experiments) contribute to the n th lobe, especially when n (cid:29)
1. This number, being independent ofany tunable parameters like L , ω , etc., sets an intrinsic limit on the experimenter’s ability to resolve the distant lobes.
20 40 60 80 1000204060 L h τ i στ max h τ i qf (a) ω = . . . . . τ max . . . h τ i . . . . . . ω/ σ (b) L = Figure 4. (a) Graphs of mean first arrival time (cid:104) τ (cid:105) , standard deviation σ and maximum arrival time τ max for the up-downwave function vs. L , keeping ω fixed. The mean arrival time of Π qf ( τ ) is also shown here. (b) Graphs of mean, standarddeviation and maximum arrival time vs. ω , keeping L fixed.Finally, we come to the semiclassical arrival time distribution (dashed line in Fig. 2), Π sc ( τ ) = (cid:90) ∂ Σ rrr · dsss τ (cid:12)(cid:12)(cid:12) ˜ ψ (cid:16) rrr τ (cid:17)(cid:12)(cid:12)(cid:12) , (17)routinely used in the interpretation of time-of-flight experiments. Here, ˜ ψ denotes the Fourier transform of the initial wavefunction ψ . Although (17) is based on the tacit assumption that the particle moves classically between preparation andmeasurement stages (see § 5.3.1 of ), it can also be motivated from the scattering formalism [6, pg 971], provided the detectorsurface ( ∂ Σ ) is placed far away from the support of the initial wave function ψ (far field regime). In typical cold atomexperiments these conditions are met, hence the semiclassical formula (17) is empirically adequate. Therefore, solicitingdeviations from (17), theorists have recommended “moving the detectors closer to the region of coherent wave packet production,or closer to the interaction region” [2, 33, pg 419] (i.e. L ≈ d ). However, such a relocation may disturb the wave function of theparticle in an undesirable way.For a meaningful comparison with our results, Eq. (17) (which is only applicable for free propagation) must be generalizedto account for the presence of the waveguide. A careful calculation yields Π sc ( τ ) = π L τ cos ( L / τ )(( L / τ ) − π ) , (18)which falls off as ( π ) L τ (compare this with Eq. (15) and its τ − fall-off). In Fig. 2, we see that the semiclassical formula (18)for L =
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The authors would like to thank J. M. Wilkes for critically reviewing the manuscript. He and Markus Nöth verified our analyticalcalculations meticulously, which led to the correction of a mistake in the early stages of the work. The assistance of GrzesioGradziuk and Leopold Kellers with the numerical simulations was essential, and greatly appreciated. Thanks are also dueto Serj Aristarkhov, Lukas Nickel, Ward Struyve, Nikolai Leopold, Roderich Tumulka and Nicola Vona for inspiration andenriching discussions on the subject of this letter.
Author contributions statement
S.D. conceived the experiment and conducted the numerical computations. The paper was written equally by S.D. and D.D.
Additional information
Competing interests : The authors declare no competing interests.
Data availability statement : The datasets generated and analyzed during the current study are available from the correspondingauthor on reasonable request.: The datasets generated and analyzed during the current study are available from the correspondingauthor on reasonable request.