Multi‐Slit Diffraction in Scaled Space‐Time

Abstract

ABSTRACT

We revisit multi‐slit diffraction with a scaling of space and time to exploit an equivalence of wave packets describing quantum free and harmonic oscillator (HO) motion. We introduce a co‐moving frame of space and time coordinates to define an effective, time‐independent, HO potential that confines and directs initially displaced wave packets along the classical phase space of the oscillator. $$N$$‐slit diffraction in the lab frame from the slits to a distant detector and the familiar spreading of the propagating wave front is then described as the propagation of HO wave packets over just a quarter cycle of the oscillator and confined to the HO potential well. We show that diffraction features are preserved in the co‐moving frame and well characterized in both the near and far fields by the classical trajectories of the HO phase space.

Introduction

This paper concerns the familiar topic of quantum diffraction of particles with mass. Our emphasis, however, is on the connection between quantum free wave packets in the lab frame and harmonic oscillator (HO) wave packets in a scaled space-time frame and a resulting close correspondence with the underlying classical trajectories which we show to align along key features in the diffraction in both the near and far zones. Our work is a follow-on to two recent papers by Briggs [1, 2] on free-space diffraction and interference in a transformed frame, which also connects with Bohmian trajectories. $$N$$-slit interference in the near field and the connection of Talbot carpets with Bohmian trajectories in the lab frame have been recently studied in detail by Sbitnev [3]. The ordinary classical HO trajectories we identify here are far more regular than the quantum Bohmian trajectories and are shown to link better key features in the diffraction. The scaled space-time frame we introduce facilitates the study of near-field diffraction and the resulting quantum revivals that give rise to Talbot carpets [4, 5]. Our approach here could have application to quantum computing with Fourier optics [6] as well as diffractive neural networks [7]. A recent overview of few-slit diffraction experiments with molecules along with references to conventional experiments with a variety of massive particles has been given by Brand and coworkers [8].

Diffraction experiments begin with a beam of collimated particles. Each particle of mass $$m$$ is released one at a time with well-defined kinetic energy and de Broglie wavelength $$\lambda $$ to pass through a diffraction grating and propagate downstream along the beam axis towards a detector. We envision a grating along the $$x$$ axis with the particle beam propagating along the $$z$$ axis. A small momentum imparted to a particle by the grating will give rise to a small deflection parallel to the grating along $$x$$. Otherwise, the propagation is free and the $$x$$ and $$z$$ motions are independent. A channel-plate detector placed downstream a distance $$z$$ from the grating and along the $$x$$ axis records a particle's arrival at time $$t$$ at position $$x$$ and marks the end of a single diffraction event. The time of flight (TOF) $$t$$ is defined by the grating-detector separation $$z$$ and the fixed downstream velocity $${v}_{z}$$ of the beam as $$t=z/{v}_{z}$$. The co-moving frame introduced in the next section defines a new $$x$$ axis and scaled time co-moving with the particle downstream along the $$z$$ axis.

Summary

• Free-particle wave packet motion is transformed to an effective harmonic oscillator phase space.
• Mult-slit particle diffraction is shown to be fully characterized by the classical trajectories of the oscillator.
• The approach could have application to optical quantum computing and neural networks as well as Bragg scattering from ultracold atoms.

For some events, the diffracted particle will move slowly along $$x$$ with a small transverse momentum $${p}_{x}$$ imparted by the grating and be detected close to the beam axis. For other events, the particle will be fast along $$x$$ and detected far from the beam axis. The TOF $$t$$ is varied by moving the detector along the beam axis. The ensemble of transverse $$x$$-axis detection events as a function of $$t$$ establishes the wave character of the diffraction which we describe with a wave function $$\psi (x,t)$$.

The study of free propagation of Gaussian wave packets in time in one dimension is routine in a wide variety of quantum applications and provides a functional and adaptable model of single- or multi-slit particle diffraction. Because Gaussians can be easily integrated and usually result in Gaussians, we model our grating slit functions with the HO ground state wave function1 $$$\begin{equation} {\psi}_{0}(x)=\frac{{\mathrm{e}}^{-{x}^{2}/(2{\sigma}^{2})}}{{(\pi {\sigma}^{2})}^{1/4}}. \end{equation}$$$Although our approach here can be generalized to other slit functions, the tools we develop with Gaussian slit functions are also directly applicable to classical optics with Gaussian beams. Our emphasis in the following on Gaussian free propagation could also prove useful in the studies of Bragg scattering from ultracold atoms trapped in an optical lattice and then released [9, 10].

Single-slit diffraction is described by the resulting time-dependent free-particle wave function $$\psi (x,t)$$, which is straightforward to calculate as the time-dependent Fourier transform of the HO ground-state momentum wave function $${\tilde{\psi}}_{0}({p}_{x})$$. One obtains 2 $$$\begin{equation} \psi (x,t)={\mathrm{e}}^{-\frac{1}{2}i\hspace*{0.1667em}\arctan \tau}\exp \left[i\frac{{x}^{2}\tau}{2{\sigma}^{2}(1+{\tau}^{2})}\right]{\psi}_{0}\left(\frac{x}{{(1+{\tau}^{2})}^{1/2}}\right), \end{equation}$$$where $$\tau =t/T$$ is a dimensionless time with $$T\equiv m{\sigma}^{2}/\hslash $$ a characteristic timescale. The resulting diffraction density $${|\psi (x,t)|}^{2}$$ spreads in time with a Gaussian width $${\sigma}_{t}=\sigma {(1+{\tau}^{2})}^{1/2}$$ that defines an asymptotic $$t\gg T$$ beam divergence angle [11] 3 $$$\begin{equation} \theta \equiv \frac{{\sigma}_{t}}{z}\simeq \frac{1}{{k}_{z}\sigma}=\frac{\lambda}{2\pi \sigma}, \end{equation}$$$where here the wave number $${k}_{z}$$ is defined by the classical momentum of the particles along the $$z$$ axis, $${p}_{z}=m{v}_{z}=\hslash {k}_{z}$$. This result is familiar from optical diffraction: for a given wavelength, the beam divergence increases as the slit width decreases.

One introduces $$N$$-slit diffraction with a coherent sum over the wave packets emanating from the $$n$$th slit as in Equation (2) with the replacement $$x\to x-\textit{nd}$$, 4 $$$\begin{equation} {\psi}_{\textit{Nd}}(x,t)=\sum _{n=-(N-1)/2}^{(N-1)/2}\psi (x-\textit{nd},t). \end{equation}$$$This expression normalizes to $${N}^{2}$$ if $$d$$ is large enough compared to $$\sigma $$ so that the initial Gaussian overlaps between adjacent slit functions are negligible. The resulting wave function density $$|{\psi}_{\textit{Nd}}{(x,\tau )|}^{2}$$ describes the diffraction pattern as the $$N$$ components expand in time and interfere along the transverse $$x$$ axis with the lowest momentum components requiring $$t\to \infty $$ to reach detector channels even very near the origin $$x=0$$. Figure 1 depicts this diffraction in the lab frame of an $$N=3$$ slit superposition.

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Method

Co-Moving Frame

The wave function $$\psi (x,t)$$ is a solution of the free-particle Schrödinger wave equation. Its form already suggests that a transformation to an alternative space-time frame of reference should begin with the ‘co-moving frame” coordinate $${x}_{c}=x/{(1+{\tau}^{2})}^{1/2}$$. Thus, we introduce the Ansatz 5 $$$\begin{equation} \psi (x,t)=\frac{1}{{(1+{\tau}^{2})}^{1/4}}\exp \left[i\, \frac{{x}_{c}^{2}}{2{\sigma}^{2}}\tau \right]\chi ({x}_{c},{t}_{c}), \end{equation}$$$where $$\chi ({x}_{c},{t}_{c})$$ is to be a solution of the HO wave equation 6 $$$\begin{equation} -\frac{{\hslash}^{2}}{2m}\frac{{\partial}^{2}\chi}{{\partial}{x}_{c}^{2}}+\frac{1}{2}m{\omega}_{c}^{2}{x}_{c}^{2}\, \chi =i\hslash \frac{\partial \chi}{\partial {t}_{c}} \end{equation}$$$with $${t}_{c}={t}_{c}(\tau )$$ the co-moving frame time to be determined as a function of $$\tau $$.

Substituting Equation (5) into the free-particle wave equation, one finds $$\chi ({x}_{c},{t}_{c})$$ satisfies the HO wave equation Equation (6) if the derivative $${t}_{c}^{\prime}(\tau )=T/(1+{\tau}^{2})$$, which integrates to give 7 $$$\begin{align} {t}_{c}(\tau )& =T\arctan\tau \nonumber\\ {\tau}_{c}(\tau )& \equiv {t}_{c}/T. \end{align}$$$Moreover, one finds that the effective HO frequency in the co-moving frame is defined by $${\omega}_{c}=1/T$$ independently of time. The period of the oscillator is then $$2\pi T$$. The resulting HO potential 8 $$$\begin{equation} {V}_{c}=\frac{1}{2}m{\omega}_{c}^{2}{x}_{c}^{2}=\frac{1}{2}m{x}_{c}^{2}/{T}^{2} \end{equation}$$$dictates motion in the co-moving frame in place of the lab-frame free propagation. As time varies $$0\le t<\infty $$ in the lab frame from slit to particle detection at $$x$$, time in the co-moving frame varies over just a quarter cycle of the oscillator $$0\le {\tau}_{c}\le \pi /2$$. Moreover, the resulting diffraction and interference are now constrained by the HO potential to remain near the origin $${x}_{c}=0$$ as opposed to the spreading over all space-time in the lab frame described by $$\psi (x,t)$$ in Equation (2).

Similar results have been obtained by Briggs [1, 2] and compared with lab-frame Bohmian trajectories $$x(t)={x}_{c}{(1+{\tau}^{2})}^{1/2}$$ for Gaussian free propagation defined by Equation (2). Then $${x}_{c}$$ defines the initial position of a Bohm trajectory in the lab, while its asymptotic limit $${x}_{c}\sim \textit{Tx}/t=T{v}_{x}$$ for $$t\gg T$$ defines the lab-frame classical velocity $${v}_{x}$$ of the free particle. In terms of the particle's lab momentum $${p}_{x}=m{v}_{x}$$, one has that $${x}_{c}\sim {p}_{x}{\sigma}^{2}/\hslash ={k}_{x}{\sigma}^{2}$$, where $${k}_{x}={p}_{x}/\hslash $$ is the corresponding wave number. For arbitrary $$\tau $$, $${x}_{c}={k}_{x}{\sigma}^{2}\sin {\tau}_{c}={k}_{x}{\sigma}^{2}\tau /{(1+{\tau}^{2})}^{1/2}$$.

In the co-moving frame, a wave packet $$\chi ({x}_{c},{t}_{c})$$ initially displaced to $${x}_{c}=d$$ in the HO potential well at $${\tau}_{c}=0$$ propagates to the bottom of the well after a quarter cycle at $${\tau}_{c}=\pi /2$$. Since the period of harmonic oscillation is independent of amplitude, wave packets from $$N$$ slits will arrive together at the detector and interfere. The co-moving frame transformation makes available all the resources of HO formalism.

We note in passing that all HO eigenstates have the same dependence on $${x}_{c}=x/{(1+{\tau}^{2})}^{1/2}$$ as the ground state Equation (1) and form a complete set. Hence, our Ansatz Equation (5) extends to arbitrary slit functions and initial states $$\chi ({x}_{c},{t}_{c}=0)$$ beyond the HO ground state but expanded in HO eigenstates.

Equations (5) and (7) are a simplified quantum version of the classical Arnold transformation that maps solutions of certain classical equations of motion to solutions of the classical free particle equations of motion. The quantum Arnold transformation and its connection with Gaussian and free-particle wave packets have been thoroughly reviewed by Guerrero et al. [12, 13]. Our emphasis here is on the close connection for diffraction with the classical HO phase space the transformation affords.

Co-Moving Frame Diffraction

Time development in the co-moving frame is evaluated straightforwardly with the introduction of the HO quantum propagator [14, 15] 9 $$$\begin{equation} \begin{aligned} {K}_{\sigma}({x}_{c},{x}_{c}^{\prime};\hspace*{0.1667em}{t}_{c})& ={\left[\frac{m{\omega}_{c}}{2\pi i\hslash \sin {\tau}_{c}}\right]}^{1/2}\\ &\quad \ensuremath{\times{}}\exp \left[i\frac{m{\omega}_{c}}{2\hslash \sin {\tau}_{c}}\left(({x}_{c}^{2}+{x}_{c}^{\prime 2})\cos {\tau}_{c}-2{x}_{c}{x}_{c}^{\prime}\right)\right]\\ & =\frac{\exp \left[i\left(({x}_{c}^{2}+{x}_{c}^{\prime 2})\cos {\tau}_{c}-2{x}_{c}{x}_{c}^{\prime}\right)/(2{\sigma}^{2}\sin {\tau}_{c}))\right]}{{(2\pi i{\sigma}^{2}\sin {\tau}_{c})}^{1/2}}. \end{aligned} \end{equation}$$$The propagator gives the evolution of an initial wave packet point-wise from a source point $${x}_{c}^{\prime}$$ to a field point $${x}_{c}$$ in time $${t}_{c}$$ and facilitates generalization to non-Gaussian slit functions. In comparing the co-moving- and lab-frame results going forward, it is useful to note that $$\cos {\tau}_{c}=1/\sqrt{1+{\tau}^{2}}$$ while $$\sin {\tau}_{c}=\tau /\sqrt{1+{\tau}^{2}}$$.

We obtain the diffraction in the co-moving frame from a Gaussian slit positioned at $${x}_{c}^{\prime}=d$$ with width $$w$$ narrower (or broader) than the natural width $$\sigma $$ of the oscillator by evaluating 10 $$$\begin{equation} {\chi}_{\textit{dw}}({x}_{c},{t}_{c})=\int {K}_{\sigma}({x}_{c},{x}_{c}^{\prime};{t}_{c})\, {\left.{\psi}_{0}({x}_{c}^{\prime}-d)\right|}_{\sigma \to w}\, d{x}_{c}^{\prime}, \end{equation}$$$where $${\psi}_{0}{|}_{\sigma \to w}$$ is the HO ground-state wave function Equation (1) with the width $$\sigma $$ replaced by $$w$$. The resulting Gaussian integral over $${x}_{c}^{\prime}$$ is easily evaluated, and one obtains (with somewhat convoluted algebra) 11 $$$\begin{equation} {\chi}_{\textit{dw}}({x}_{c},{t}_{c})={\mathrm{e}}^{i{\gamma}_{c}}\hspace*{0.1667em}{\mathrm{e}}^{-\textit{id}{x}_{c}\sin {\tau}_{c}/{\sigma}_{c}^{2}}\hspace*{0.1667em}{\mathrm{e}}^{i{d}^{2}\sin {\tau}_{c}\cos {\tau}_{c}/(2{\sigma}_{c}^{2})}{\left.\hspace*{0.1667em}{\psi}_{0}({x}_{c}-{\bar{x}}_{c})\right|}_{\sigma \to {w}_{c}}. \end{equation}$$$The resulting wave-packet density 12 $$$\begin{equation} |{\chi}_{\textit{dw}}({x}_{c},{t}_{c}){|}^{2}={\left.{\psi}_{0}^{2}({x}_{c}-{\bar{x}}_{c})\right|}_{\sigma \to {w}_{c}}, \end{equation}$$$is form identical with the initial Gaussian slit function except its width now varies with time as 13 $$$\begin{equation} {w}_{c}^{2}=\frac{{w}^{4}{\cos}^{2}{\tau}_{c}+{\sigma}^{4}{\sin}^{2}{\tau}_{c}}{{w}^{2}} \end{equation}$$$while its centroid (and expectation value) moves along the classical HO trajectory 14 $$$\begin{equation} {\bar{x}}_{c}({t}_{c})=d\, \cos {\tau}_{c} \end{equation}$$$from the slit at $${\tau}_{c}=0$$ to the origin $${x}_{c}=0$$ of the potential well Equation (8) at $${\tau}_{c}=\pi /2$$. This is just the scaling trajectory $${x}_{c}=x/{(1+{\tau}^{2})}^{1/2}$$ with $$x=d$$ introduced in Equation (5). If $$w<\sigma $$, the packet width expands somewhat over the quarter cycle as $${w}_{c}=w\to {\sigma}^{2}/w$$.

The width $${\sigma}_{c}$$ in the phase factors in Equation (11) is also time dependent and defined by 15 $$$\begin{equation} {\sigma}_{c}^{2}=\frac{{w}^{4}{\cos}^{2}{\tau}_{c}+{\sigma}^{4}{\sin}^{2}{\tau}_{c}}{{\sigma}^{2}}=\frac{{w}^{2}}{{\sigma}^{2}}{w}_{c}^{2}, \end{equation}$$$while the phase 16 $$$\begin{equation} {\gamma}_{c}=-\frac{1}{2}\arctan \left(\frac{{\sigma}^{2}}{{w}^{2}}\tan {\tau}_{c}\right)+\frac{{x}_{c}^{2}}{2{\sigma}_{c}^{2}}\frac{{\sigma}^{4}-{w}^{4}}{{\sigma}^{4}}\sin {\tau}_{c}\cos {\tau}_{c} \end{equation}$$$is $$d$$ independent and does not contribute to multi-slit diffraction.

Figure 2a illustrates the wave function density in the co-moving frame of an $$N=2$$ superposition as defined in Equation (4) but now based on Equation (11). (See also Equation 23 below.)

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When the initial width $$w$$ is narrower than the natural width $$\sigma $$ of the HO ground state, one refers to a squeezed Gaussian state. If $$w=\sigma $$, then $${w}_{c}=\sigma $$ for all time and one refers to a coherent Gaussian state which evolves without a change of shape. In the coherent-state limit, $${\sigma}_{c}=\sigma $$ for all time and $${\mathrm{e}}^{i{\gamma}_{c}}={\mathrm{e}}^{-i{\tau}_{c}/2}$$ (cf. Equations A10 and A14).

Figure 2b illustrates the coherent-state limit of the squeezed-state density in Figure 2a. One sees that the two contributions from each slit evolve without change of shape along the classical trajectory $${\bar{x}}_{c}=d\, \cos {\tau}_{c}$$ towards the origin of the HO potential well until they begin to overlap and interfere. With a reduced slit width $$w<\sigma $$, the initial squeezed state $${\psi}_{0}{|}_{\sigma \to w}$$ expands with timescale $${T}_{w}=T\, {w}^{2}/{\sigma}^{2}$$ more quickly than the HO ground state Equation (1) with timescale $$T$$. Thus, the wave front contributions from each slit overlap and interfere much earlier than their coherent state counterparts in Figure 2b. As we shall see, $$\sigma $$ then takes on the status of a free parameter that can be chosen to highlight various diffraction features.

One evaluates the squeezed-state momentum wave function by direct Fourier transform of Equation (11) and finds it is closely related in form to the coordinate wave function by a simple HO phase-space reciprocity 17 $$$\begin{equation} {\tilde{\chi}}_{\textit{dw}}({k}_{c},{t}_{c})={\mathrm{e}}^{+i\pi /4}\sigma \, {\chi}_{\textit{dw}}({x}_{c}\to {\sigma}^{2}{k}_{c},{\tau}_{c}\to {\tau}_{c}+\pi /2), \end{equation}$$$where $${k}_{c}={p}_{c}/\hslash $$ is the momentum-space wave number. (Here the extra factor $$\sigma $$ on the right-hand side multiplying $${\chi}_{\textit{dw}}$$ is a renormalization for the change of variables $$d{x}_{c}={\sigma}^{2}d{k}_{c}$$.) The momentum wave packet density is an anti-squeezed version of Equation (12), 18 $$$\begin{equation} |{\tilde{\chi}}_{\textit{dw}}({k}_{c},{t}_{c}){|}^{2}=\frac{1}{{\tilde{w}}_{c}\sqrt{\pi}}\exp \left[-\frac{1}{{\tilde{w}}_{c}^{2}}{\left({k}_{c}-{\bar{k}}_{c}\right)}^{2}\right] \end{equation}$$$with the time-dependent Gaussian width 19 $$$\begin{equation} {\tilde{w}}_{c}^{2}=\frac{{w}^{4}{\sin}^{2}{\tau}_{c}+{\sigma}^{4}{\cos}^{2}{\tau}_{c}}{{w}^{2}{\sigma}^{4}} \end{equation}$$$and with a centroid (and expectation value) that follows the classical momentum trajectory $${\bar{p}}_{c}=m\, d{\bar{x}}_{c}/d{t}_{c}=-m{\omega}_{c}d\sin {\tau}_{c}$$, that is, along 20 $$$\begin{equation} {\bar{k}}_{c}({t}_{c})=-\frac{d\sin {\tau}_{c}}{{\sigma}^{2}}. \end{equation}$$$At $${\tau}_{c}=0$$, $${\tilde{\chi}}_{\textit{dw}}$$ is just the Fourier transform of the displaced slit function, $${\tilde{\chi}}_{\textit{dw}}({k}_{c},0)={\mathrm{e}}^{-\textit{id}{k}_{c}}{\tilde{\psi}}_{0}({k}_{c}){|}_{\sigma \to w}$$ (note $${\tilde{w}}_{c}(0)=1/w$$), and for all times 21 $$$\begin{equation} |{\tilde{\chi}}_{\textit{dw}}({k}_{c},{t}_{c}){|}^{2}={|{\tilde{\psi}}_{0}({k}_{c}-{\bar{k}}_{c})|}_{\sigma \to 1/{\tilde{w}}_{c}}^{2}. \end{equation}$$$

Introducing the lab-frame wave number with the replacement $${x}_{c}\to {k}_{x}{\sigma}^{2}\sim x/\tau $$ for $${\tau}_{c}\to \pi /2$$ (see Equation 8), Equation (17) gives 22 $$$\begin{equation} {\chi}_{\textit{dw}}({x}_{c}\sim x/\tau ,{\tau}_{c}\to \pi /2)=\frac{{e}^{-i\pi /4}}{\sigma}{\tilde{\chi}}_{\textit{dw}}({k}_{x},0)\propto {\tilde{\psi}}_{0}({k}_{x}){|}_{\sigma \to w} \end{equation}$$$describing the familiar single-slit Fraunhofer diffraction proportional to the Fourier transform of the slit function.

The wave packets Equations (11) and (17) start out at $${x}_{c}=d$$ at $${\tau}_{c}=0$$ with zero momentum but then accelerate towards the origin along $${\bar{x}}_{c}({t}_{c})$$ with momentum $${\bar{p}}_{c}({t}_{c})$$ reaching maximum momentum $${\bar{p}}_{c}=-\hslash d/{\sigma}^{2}$$ at the origin at $${\tau}_{c}=\pi /2$$. These trajectories account for the phases in Equation (11) and in particular their $${d}^{2}$$ dependence, which makes the sum over $$N$$ slits nontrivial.

The momentum Gaussian width $${\tilde{w}}_{c}({\tau}_{c})$$ in Equation (19) is one-quarter cycle out of phase with the coordinate-space width $${w}_{c}({\tau}_{c})$$ in Equation (13). One sees that generally $$\hslash \hspace*{0.1667em}{\tilde{w}}_{c}({\tau}_{c})\hspace*{0.1667em}{w}_{c}({\tau}_{c})\ge \hslash $$ so that squeezed Gaussian wave packets are not minimum uncertainty states except at the endpoints $${\tau}_{c}=0$$ and $$\pi /2$$. However, in the limit $$w=\sigma $$, one has $${w}_{c}=\sigma $$, $${\tilde{w}}_{c}=1/\sigma $$ for all time, so that coherent Gaussian wave packets are minimum uncertainty states $$\hslash \hspace*{0.1667em}{\tilde{w}}_{c}({\tau}_{c})\hspace*{0.1667em}{w}_{c}({\tau}_{c})=\hslash $$ for all $${\tau}_{c}$$.

All aspects of squeezed and coherent states and especially their connection with the classical HO phase space can be established with raising/lowering operator algebra, as is well demonstrated in many quantum texts. However, the time-dependent coordinate wave functions of squeezed states are cumbersome to derive with operator formalism and are not usually presented (see however [12, 16] and references therein). For completeness, we outline the approach in the appendices and an alternative derivation of the squeezed-state wave function Equation (11).

$$N$$-Slit Details

Collecting the $$d$$ and $${d}^{2}$$ dependence in the phases in Equation (11) and introducing $$d\to \textit{nd}$$ for the $$n$$th slit, one describes $$N$$-slit squeezed-state diffraction with 23 $$$\begin{eqnarray} {\chi}_{\textit{Ndw}}({x}_{c},{t}_{c})&& =\sum _{n=-(N-1)/2}^{(N-1)/2}{\chi}_{\textit{ndw}}({x}_{c},{t}_{c})\nonumber\\ && ={\mathrm{e}}^{i{\gamma}_{c}}{\left.\hspace*{0.1667em}{\psi}_{0}({x}_{c})\right|}_{\sigma \to {w}_{c}}\nonumber\\ && \ensuremath{\times{}}\sum _{n}\exp \left[\frac{\textit{nd}{x}_{c}{\mathrm{e}}^{-i{\phi}_{c}/2}}{\sigma {\sigma}_{c}}\right]\exp \left[-\frac{{n}^{2}{d}^{2}{\mathrm{e}}^{-i{\phi}_{c}/2}\cos {\tau}_{c}}{2\sigma {\sigma}_{c}}\right],\nonumber\\ \end{eqnarray}$$$where 24 $$$\begin{equation} {\phi}_{c}=\arctan({\sigma}^{2}\tan {\tau}_{c}/{w}^{2}) \end{equation}$$$from the phase $${\gamma}_{c}$$ in Equation (16). In the coherent-state limit $$w=\sigma $$, this expression reduces to 25 $$$\begin{eqnarray} {\chi}_{\textit{Nd}\sigma}({x}_{c},{t}_{c})&& ={\mathrm{e}}^{-i{\tau}_{c}/2}{\left.\hspace*{0.1667em}{\psi}_{0}({x}_{c})\right|}_{\sigma}\nonumber\\ && \ensuremath{\times{}}\sum _{n}\exp \left[\frac{\textit{nd}{x}_{c}{\mathrm{e}}^{-i{\tau}_{c}/2}}{{\sigma}^{2}}\right]\hspace*{0.1667em}\exp \left[-\frac{{n}^{2}{d}^{2}{\mathrm{e}}^{-i{\tau}_{c}/2}\cos {\tau}_{c}}{2{\sigma}^{2}}\right].\nonumber\\ \end{eqnarray}$$$The contributions here from the $$n$$th slit retain their form for all $${\tau}_{c}$$ and are a key motivation for introducing scaled coordinates and time. These expressions normalize to $${N}^{2}$$ if $$d$$ is large enough compared to $$\sigma $$ so that the initial Gaussian overlaps between adjacent slit functions are negligible.

The $$N$$-slit squeezed and coherent states are remarkably similar in form to the lab-frame $$N$$-slit description, perhaps predictable with the Gaussian connection but nevertheless surprising given the algebra required to achieve these forms. Collecting the $$d$$ and $${d}^{2}$$ dependence in the phases in Equation (4), one has that 26 $$$\begin{eqnarray} {\psi}_{\textit{Ndw}}(x,t)&& ={\mathrm{e}}^{i{\gamma}_{w}}{\left.\hspace*{0.1667em}{\psi}_{0}\left(\frac{x}{{(1+{\tau}_{w}^{2})}^{1/2}}\right)\right|}_{\sigma \to w}\nonumber\\ && \ensuremath{\times{}}\sum _{n}\exp \left[\frac{\textit{ndx}\hspace*{0.1667em}{\mathrm{e}}^{-i{\phi}_{w}/2}}{{w}^{2}{(1+{\tau}_{w}^{2})}^{1/2}}\right]\hspace*{0.1667em}\exp \left[-\frac{{n}^{2}{d}^{2}\hspace*{0.1667em}{\mathrm{e}}^{-i{\phi}_{w}/2}}{2{w}^{2}{(1+{\tau}_{w}^{2})}^{1/2}}\right],\nonumber\\ \end{eqnarray}$$$assuming now an initial Gaussian with width $$w$$ as in Equation (23), so that here $${\tau}_{w}=t/{T}_{w}$$ with scale $${T}_{w}=m{w}^{2}/\hslash $$. The phase $${\phi}_{w}=\arctan{\tau}_{w}$$ while $${\gamma}_{w}$$ is $$d$$ independent. Unlike the squeezed and coherent state descriptions in the co-moving frame, the slit contributions to the wave function in the lab frame expand in time.

In the limit $${\tau}_{c}\to \pi /2$$ in Equations (23) and (25) and $$t\to \infty $$ in Equation (26), the sums over $$n$$ become geometric series. In the co-moving frame, Equation (23) gives 27 $$$\begin{equation} {\chi}_{\textit{Ndw}}({x}_{c},{\tau}_{c}\to \pi /2)={e}^{-i\pi /4}\hspace*{0.1667em}{\psi}_{0}({x}_{c}){|}_{\sigma \to {\sigma}^{2}/w}\hspace*{0.1667em}\frac{\sin [N{x}_{c}d/(2{\sigma}^{2})]}{\sin [{x}_{c}d/(2{\sigma}^{2})]}, \end{equation}$$$a form familiar to $$N$$-slit diffraction.

For $$N$$ large, this function shows strong peaks around $${x}_{c}\approx 2\pi \nu {\sigma}^{2}/d$$ for integer diffraction-order $$\nu $$. Introducing the lab-frame wave number $${k}_{x}$$ as in Equation (22) with $${x}_{c}\to {k}_{x}{\sigma}^{2}\sim x/\tau $$ for $${\tau}_{c}\to \pi /2$$, one has the familiar result of diffraction peaks in the lab frame for $${k}_{x}\approx \nu {k}_{g}$$, where $${k}_{g}=2\pi /d$$ is the wave number of the small momentum transfer along the grating, $${k}_{g}/{k}_{z}=\lambda /d\ll 1$$. Noting that $${{\psi}_{0}({k}_{x}{\sigma}^{2})|}_{\sigma \to {\sigma}^{2}/w}=\sigma {[{\tilde{\psi}}_{0}({k}_{x})]}_{\sigma \to w}$$ in Equation (27), one has 28 $$$\begin{eqnarray} {\chi}_{\textit{Ndw}}({x}_{c}\sim x/\tau ,{\tau}_{c}\to \pi /2)={\mathrm{e}}^{-i\pi /4}\sigma \hspace*{0.1667em}{\tilde{\psi}}_{0}({k}_{x}){|}_{\sigma \to w}\hspace*{0.1667em}\frac{\sin [N\pi {k}_{x}/{k}_{g}]}{\sin [\pi {k}_{x}/{k}_{g}]},\nonumber\\ \end{eqnarray}$$$describing the familiar Fraunhofer diffraction pattern of an $$N$$-slit grating masked by the single-slit diffraction of Equation (22). In the limit $$N\to \infty $$ the peaks become Dirac $$\delta $$ functions giving 29 $$$\begin{align} {\chi}_{\infty \textit{dw}}({k}_{x}{\sigma}^{2},{\tau}_{c}=\pi /2)\propto \sigma \, {\tilde{\psi}}_{0}({k}_{x}){|}_{\sigma \to w}\sum _{\nu}\delta ({k}_{x}-\nu {k}_{g}) \end{align}$$$summed over all diffraction orders $$\nu $$. As in Equation (17), the extra factor $$\sigma $$, here multiplying $${\tilde{\psi}}_{0}$$, is a renormalization factor for the change of variables $$d{x}_{c}={\sigma}^{2}d{k}_{c}$$. (Also, Equation (22) can be derived in the same way as Equation (28) directly from $${\chi}_{\textit{dw}}({x}_{c}={\sigma}^{2}{k}_{x},{\tau}_{c}=\pi /2)$$.)

Given the phase-space reciprocity Equation (17), the result Equation (28) is also proportional to the momentum Fourier transform $${\tilde{\chi}}_{\textit{Ndw}}({k}_{x},{\tau}_{c}=0)$$ of the grating transmission function $${\chi}_{\textit{Ndw}}(x,t=0)$$. (Note at $$t={\tau}_{c}=0$$, $${x}_{c}=x$$ and $${k}_{c}={k}_{x}$$). One sees explicitly that the grating delivers momentum kicks $$\nu \hslash {k}_{g}$$ depending on the order $$\nu $$ of the diffraction.

Our focus all along has been on the connection of the quantum diffraction in the co-moving frame with the underlying classical HO phase space. To that end, we compare the $$N$$-slit density $$|{\chi}_{\textit{Ndw}}({x}_{c},{t}_{c}){|}^{2}$$ from grating to a detector with sets of classical HO trajectories defined by say the slit positions $${x}_{c}=\textit{nd}$$ at $${\tau}_{c}=0$$ and the diffraction-peak positions $${x}_{c}=\nu {k}_{g}{\sigma}^{2}$$ at $${\tau}_{c}=\pi /2$$. One finds that 30 $$$\begin{equation} {\bar{x}}_{c}({t}_{c},n,\nu )=\textit{nd}\cos {\tau}_{c}+\nu {k}_{g}{\sigma}^{2}\sin {\tau}_{c}. \end{equation}$$$The classical momentum $${\bar{p}}_{c}=m\, d{\bar{x}}_{c}/d{t}_{c}=\hslash {\bar{k}}_{c}$$ defines a corresponding wave number in the co-moving frame, 31 $$$\begin{equation} {\bar{k}}_{c}({t}_{c},n,\nu )=\nu {k}_{g}\cos {\tau}_{c}-\frac{\textit{nd}}{{\sigma}^{2}}\sin {\tau}_{c} \end{equation}$$$with the same phase-space reciprocity $${\bar{x}}_{c}({t}_{c}+\pi /2,n,\nu )={\sigma}^{2}{\bar{k}}_{c}({t}_{c},n,\nu )$$ as in Equation (17). These trajectories are perpendicular to the wave fronts emanating from the slits and propagating to points on the detector. In order for particles to reach the diffraction peaks at $${x}_{c}=\nu {k}_{g}{\sigma}^{2}$$ as $${\tau}_{c}\to \pi /2$$, the grating must impart momentum kicks $${p}_{c}=\nu \hslash {k}_{g}$$ as the particles depart the slits at $${\tau}_{c}=0$$. There is of course a continuum of such trajectories connecting an arbitrary initial point on the grating with any final point on the detector. The wave function density gives the probability of finding a particle on any given trajectory.

Transforming Equation (30) back to the lab frame with $$x={(1+{\tau}^{2})}^{1/2}{x}_{c}$$, one obtains 32 $$$\begin{equation} \bar{x}(t,n,\nu )=\textit{nd}+\nu \, {k}_{g}{\sigma}^{2}\tau =\textit{nd}+\nu \, \frac{\hslash {k}_{g}}{m}t \end{equation}$$$and $$\bar{p}(t,n,\nu )=\nu \hslash {k}_{g}$$, the classical straight-line trajectories of the lab free-particle motion.

Figure 3 shows the squeezed-state density of an $$N=10$$ superposition $$|{\chi}_{\textit{Ndw}}({x}_{c},{t}_{c}){|}^{2}$$ from Equation (23). Here $$\sigma =2$$ and $$d=10$$ so the spacing from Equation (27) between asymptotic diffraction peaks is $$\mathrm{\Delta}{x}_{c}\approx 2\pi \, \mathrm{\Delta}\nu \, {\sigma}^{2}/d\simeq 2.51$$, as is evident in the plot. Also, $$N$$ is large enough to illustrate revivals (repeating patterns) of peaks and nodes, all familiar fractal-like characteristics of Talbot carpets [3–5]. The plot has been overlaid with a set of classical HO trajectories from Equation (30). The crisscross pattern of classical trajectories clearly matches the nodal patterns seen in the plot. Also, the trajectories cross where the quantum density shows peaks and revivals for all $${\tau}_{c}$$.

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The momentum kicks $${p}_{c}=\nu \hslash {k}_{g}$$ along trajectories exiting the grating, as described by Equation (31), give rise to wave fronts colliding between and near any given pair of slits with equal and opposite momenta creating diffraction patterns that tend to imitate an additional ghost slit between actual slits. Thus, in addition to the trajectories described by Equation (30), we have included in Figure 3 trajectories starting from the halfway points between the slits and ending at the halfway points between the diffraction peaks.

Figure 4 shows for comparison an $$N=50$$ density zoomed in as in Figure 3. Here one sees the diffraction peaks at $${\tau}_{c}=\pi /2$$ have become sharply peaked as described by the $$\delta $$-function limit Equation (29). The overall structure is identical with Figure 3. However, there is now considerably more fine detail among the peaks and nodes of Figure 3 with the appearance of endless revivals in the Talbot carpet as one looks back in time towards $${\tau}_{c}=0$$.

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The Infinite ‘Comb’ Grating

Because of the $${d}^{2}\to {n}^{2}{d}^{2}$$ dependence in the exponent in either Equation (23) or Equation (26), the sum over $$n$$ for $$N$$ finite cannot be done in closed form and generally must be computed numerically. However, for an infinite comb grating defined in the limit $$N\to \infty $$ the sums can be expressed in terms of an elliptic (Jacobi) theta function defined for complex-valued $$\zeta $$ and $$\eta $$ by [17]33 $$$\begin{equation} {\vartheta}_{3}(\zeta ,{e}^{\eta})=\sum _{n=-\infty}^{\infty}{\mathrm{e}}^{2i\zeta n}{\mathrm{e}}^{\eta {n}^{2}}\hspace*{14.45377pt}|{\mathrm{e}}^{\eta}|<1. \end{equation}$$$Comparing with Equation (23), one obtains by inspection 34 $$$\begin{equation} \begin{aligned} {\chi}_{\infty \textit{dw}}({x}_{c},{t}_{c}) & = {\mathrm{e}}^{i{\gamma}_{c}}\hspace*{0.1667em}{\psi}_{0}({x}_{c}){|}_{\sigma \to {w}_{c}}\\ &\quad \ensuremath{\times{}}{\vartheta}_{3}\left(\frac{d{x}_{c}\hspace*{0.1667em}{\mathrm{e}}^{-i{\phi}_{c}/2}}{2i\sigma {\sigma}_{c}},\exp \left[-\frac{{d}^{2}{\mathrm{e}}^{-i{\phi}_{c}/2}\cos {\tau}_{c}}{2\sigma {\sigma}_{c}}\right]\right).\\ \end{aligned} \end{equation}$$$Given the restriction $${|{\mathrm{e}}^{\eta}|}<1$$ in Equation (33), this expression is not defined for $${\tau}_{c}=\pi /2$$ where the δ-function limit Equation (29) holds instead. For $${\tau}_{c}<\pi /2,$$ Equation (34) is valid (though not normalized) as long as $$\exp [-{d}^{2}/(2{\sigma}^{2})]<1$$, which is satisfied since $$d>\sigma $$ is required to render the initial Gaussian overlap between adjacent slits negligible. Nevertheless, because of the δ-function limit Equation (29), this expression becomes increasingly ‘noisy’ and difficult to evaluate numerically as $${\tau}_{c}\to \pi /2$$.

Figure 5 shows the squeezed-state density $$|{\chi}_{\infty \textit{dw}}({x}_{c},{t}_{c}){|}^{2}$$ from Equation (34). Here $$\sigma =0.2$$ and $$d=2$$ so the approximate spacing from Equation (27) between asymptotic diffraction peaks is $$\mathrm{\Delta}{x}_{c}=2\pi \, \mathrm{\Delta}\nu \, {\sigma}^{2}/d\simeq 0.13$$, roughly where the three large peaks in the plot have begun to develop as $${\tau}_{c}\to \pi /2$$.

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Although Equation (34) simulates an infinite number of slit excitations which all propagate together in the co-moving frame towards the origin $${x}_{c}=0$$ as $${\tau}_{c}\to \pi /2$$, a realistic beam incident on the infinite comb will have a finite width and excite only a finite number of slits, albeit possibly a large number. Nevertheless, we generally find good convergence of $${\chi}_{\textit{Ndw}}({x}_{c},{t}_{c})$$ with $$N=50$$ to the $$N\to \infty $$ limit $${\chi}_{\infty \textit{dw}}({x}_{c},{t}_{c})$$.

Discussion

With Equation (5), one connects the co-moving frame $$N$$ -slit density from Equation (23) with the lab frame density from Equation (26) according to35 $$$\begin{equation} |{\chi}_{\textit{Ndw}}({x}_{c},{t}_{c}){|}^{2}={(1+{\tau}^{2})}^{1/2}{|{\psi}_{\textit{Ndw}}(x,t)|}^{2} \end{equation}$$$with the lab frame coordinates $$x,t$$ connected to the co-moving frame space-time according to $${x}_{c}=x/{(1+{\tau}^{2})}^{-1/2}=x\cos ({t}_{c}/T)$$ and $${t}_{c}=T\arctan (t/T)$$. At $$t=0$$, $${x}_{c}=x$$ while $${x}_{c}\sim x/\tau $$ as $$t\to \infty$$. However, in the co-moving frame the HO phase space shrinks as well towards the origin as $${\tau}_{c}\to \pi /2$$ along with the HO potential Equation (8). Unlike the diverging straight-line classical trajectories Equation (32) in the lab frame, the classical HO trajectories in the co-moving frame essentially follow the scaling trajectory $${x}_{c}=x\cos {\tau}_{c}$$ from Equation (5), as is evident in the figures presented here. As we see from Equation (30), HO trajectories and their wave fronts emerging from the $$N$$ slits at $${\tau}_{c}=0$$ over the large initial range $$\mathrm{\Delta}{x}_{c}=\textit{Nd}$$ converge as $${\tau}_{c}\to \pi /2$$ on the small intervals $$\mathrm{\Delta}{x}_{c}={k}_{g}{\sigma}^{2}=2\pi {\sigma}^{2}/d$$ between diffraction peaks.

Quantum connections with underlying classical phase spaces are familiar from collision physics where one uses classical trajectories to connect detector arrival-time distributions with the momentum distributions of scattered particles emerging near the microscopic collision volume. One introduces a quantum propagator to image the momentum wave function describing the collision with the macroscopic coordinate wave function at the detector. Stationary phase arguments connect the two wave function extremes via classical trajectories [18, 19]. Here we have seen how the classical HO trajectories closely follow and overlap with the quantum diffraction peaks in both the near and far fields.

A re-mapping of the detection data in the co-moving frame could facilitate a straightforward which-slit determination, that is, a means to establish for each event through which slit $$n$$ the particle passed. As is evident in Figure 1, which-slit information in the lab frame is lost in the multi-slit interference except very close to the slits. However, if one re-maps the co-moving-frame data in the coherent-state mode by taking $$\sigma \equiv w$$, as illustrated in Figure 2b, the wave packets from each slit evolve without expanding or overlapping and remain distinguishable except for times nearer to $${\tau}_{c}=\pi /2.$$

For a narrow-slit grating with $$w\ll d$$, the wave packets at any given time are sharply peaked along the classical trajectories $${\bar{x}}_{c}({t}_{c},n,\nu =0)=\textit{nd}\cos {\tau}_{c}$$ from Equation (30) connecting the $$n$$th slit with the $$\nu =0$$ central diffraction order at the origin. A particle detection at $$x$$ and $$t$$ which maps onto $${x}_{c}=x\cos {\tau}_{c}$$ will then pinpoint, as long as $$t$$ is not too large, the $$n$$th classical trajectory with $$x\approx \textit{nd}$$, a virtual co-moving frame detection comparable to a lab frame detection at the slit. Of course, which-path information at 100% certainty fully destroys all interference. At longer times nearer to $${\tau}_{c}=\pi /2$$ when the coherent-state components begin to overlap and interfere, one would obtain only partial which-slit information and as is well-known interference with reduced visibility [22].

Conclusion

We have examined in the context of quantum diffraction the equivalence of free-particle and HO motion in a scaled space-time. The scaling introduces an effective HO potential Equation (8) that confines and connects the diffraction closely to the classical HO phase space. The relative strength and position of diffraction features are transformed to remain within the bounds of the potential and the quarter cycle $$0\le {\tau}_{c}\le \pi /2$$ of HO propagation it defines. The resulting ‘time dilation’ $$\tau =\tan {\tau}_{c}$$ in the lab frame compared to the co-moving frame means that diffraction features in the co-moving frame evolve more slowly especially in the near field where the Talbot carpets show the most detail. The transformation neither creates nor destroys diffraction features, rather it aligns features in a uniform way along the classical phase space of the oscillator. Of course, there is a vast literature available for the analysis of Gaussian wave packet propagation in an HO potential. Only Gaussian integrals are required, and the analytic details are a familiar component of quantum and optics lectures.

Our presentation has focussed on demonstrating the close relation of the propagated Gaussian wave packets in the co-moving frame as a translating and mostly shape-invariant HO ground-state wave function. We connect directly with the classical HO phase space to characterize details of the wave packet motion and the resulting time development of the $$N$$-slit diffraction as packets from multiple slits overlap. Figure 3 is a punchline of sorts for this paper. Compared to the Bohmian trajectories predicted for $$N$$ slits in the lab frame [3], the diffraction features observed in the co-moving frame and highlighted instead by the classical trajectories of the HO phase space are orderly and uniform.

The arrival-time distribution of detection events maps across the HO phase space of the co-moving frame. The ensemble of TOF detection events transformed to the co-moving frame is naturally collected as histograms that define approximately the density $$|{\chi}_{\textit{Ndw}}({x}_{c},{t}_{c}){|}^{2}$$ from Equation (23). The total number of events detected along with the detector resolution $$\delta x$$, $$\delta t$$ determine how faithfully the resulting histogram will represent the diffraction. A re-mapping of detection events in the co-moving frame in the coherent-state limit $$\sigma \equiv w$$ suggests a straightforward which-slit determination for each detection event, which could have application to optical quantum computing [6] and diffractive neural networks [7].

The co-moving frame analysis and our squeezed state description could also prove useful in the studies of experiments with Bragg scattering from ultra-cold atoms trapped in an optical lattice [9, 10]. When the trap is suddenly turned off, the atoms are left to expand freely as Gaussian wave packets defined by the HO ground state. The light scattering imparts a momentum kick to each atom that is readily included in the coherent state formalism with a simple complex-valued shift of the coherent-state eigenvalue $$\alpha $$ (cf. Equation A4 in Appendix A) [21–23].

Our approach based on the HO propagator to calculate the time evolution of wave packets emanating from a diffraction grating is applicable to any slit function. However, our emphasis on Gaussian slit functions is also relevant to optical diffraction. In the paraxial approximation with light beams, one demonstrates that a two-dimensional time-dependent Schrödinger wave equation approximates the classical time-independent Helmholtz equation with the beam axis $$z$$ defining time according to $$t=z/c$$ with $$c$$ the speed of light [11]. Light beams with Gaussian profiles along a transverse $$x$$ axis are widely studied and implemented. The resulting Gaussian wave $$\psi (x,z)$$ in the paraxial approximation is form identical with Equation (2) [1]. One can therefore transform to a scaled space-time to define an effective HO potential as in Equation (8) and thereby introduce the functionality of HO wave packet propagation [12] we have considered here [24].

Author Contributions

J. M. Feagin is the sole author of this paper.

Acknowledgements

I much appreciate the ongoing collaboration and correspondence on all aspects of this research with Prof. John S. Briggs. I am grateful to Prof. Jan-Michael Rost for the hospitality and stimulation of his group at MPIPKS Dresden where the threads of this work originated.

Ethical Statement

This research did not involve studies on human subjects, human data or tissue, or animals.

Conflicts of Interest

The author declares no conflicts of interest.

Data Availability Statement

Data sharing is not applicable. The article describes entirely theoretical research.

Peer Review

The peer review history for this article is available at .

A

Appendix

Coherent State Algebra

Coherent/squeezed states are characterized by the HO raising and lowering operators defined by (we set $$m=\hslash =1$$ going forward) A1 $$$\begin{align} a& =\frac{1}{\sqrt{2}}\left(\frac{x}{\sigma}+i\sigma p\right)\nonumber\\ {a}^{\ensuremath{\dag}}& =\frac{1}{\sqrt{2}}\left(\frac{x}{\sigma}-i\sigma p\right), \end{align}$$$where $$x$$ and $$p$$ are the coordinate and momentum operators. Coherent states are eigenstates of the lowering operator $$a|\alpha \rangle =\alpha |\alpha \rangle $$ with eigenvalue $$\alpha $$, and the HO ground state gives a trivial solution $$a|0 \rangle =0$$. When a displaced ground state $${\psi}_{0}({x}_{c}-d)$$ is propagated in the co-moving frame in the HO potential well Equation (8), it gains momentum and its coherent-state eigenvalue $$\alpha $$ becomes complex valued.

A general eigenstate is constructed from the unitary displacement operator A2 $$$\begin{equation} {D}_{\alpha}=\exp (\alpha {a}^{\ensuremath{\dag}}-{\alpha}^{\ast}a) \end{equation}$$$with the defining property $${D}_{\alpha}^{\ensuremath{\dag}}a{D}_{\alpha}=a+\alpha $$ [14]. Therefore, displaced HO ground states $$|\alpha \rangle ={D}_{\alpha}|0 \rangle $$ are general solutions of $$a|\alpha \rangle =\alpha |\alpha \rangle $$ (i.e., $$a{D}_{\alpha}|0 \rangle ={D}_{\alpha}(a+\alpha )|0 \rangle =\alpha {D}_{\alpha}|0 \rangle $$).

The HO time development operator in the co-moving frame is $$\exp (-i{H}_{0}{t}_{c})={\mathrm{e}}^{-i{\tau}_{c}({a}^{\ensuremath{\dag}}a+1/2)}$$, so a coherent state describing wave propagation from a Gaussian slit width $$\sigma $$ evolves in time according to [14, 15] A3 $$$\begin{equation} \exp (-i{H}_{0}{t}_{c})|\alpha \rangle ={\mathrm{e}}^{-i{\tau}_{c}}|\alpha {\mathrm{e}}^{-i{\tau}_{c}} \rangle . \end{equation}$$$At any given time, one defines complex $$\alpha $$ in terms of the classical phase-space positions $${x}_{\alpha},{p}_{\alpha}$$ of the coherent state wave packet according to A4 $$$\begin{align} \text{Re}\, \alpha & =\frac{1}{2}({\alpha}^{\ast}+\alpha )\equiv \frac{1}{\sigma \sqrt{2}}{x}_{\alpha}\nonumber\\ \text{Im}\, \alpha & =\frac{i}{2}({\alpha}^{\ast}-\alpha )\equiv \frac{\sigma}{\sqrt{2}}{p}_{\alpha}. \end{align}$$$One thereby transforms the displacement operator to the phase-space Weyl operator A5 $$$\begin{equation} {D}_{\alpha}=\exp (\alpha {a}^{\ensuremath{\dag}}-{\alpha}^{\ast}a)=\exp (i({p}_{\alpha}x-{x}_{\alpha}p)), \end{equation}$$$with the properties A6 $$$\begin{equation} {D}_{\alpha}^{\ensuremath{\dag}}x{D}_{\alpha}=x+{x}_{\alpha},\hspace*{14.45377pt}{D}_{\alpha}^{\ensuremath{\dag}}p{D}_{\alpha}=p+{p}_{\alpha}. \end{equation}$$$The Weyl operator performs this shift in any system and is not specific to HO Gaussians [14], which facilitates generalization to non-Gaussian slit functions. Disentangling Equation (A5) gives [14, 16, 20] A7 $$$\begin{equation} {D}_{\alpha}={\mathrm{e}}^{-\frac{1}{2}i{p}_{\alpha}{x}_{\alpha}}{\mathrm{e}}^{+i{p}_{\alpha}x}{\mathrm{e}}^{-i{x}_{\alpha}p}, \end{equation}$$$where the last factor effects a familiar shift $${\mathrm{e}}^{-i{x}_{\alpha}p}\, \psi ({x}_{c})=\psi ({x}_{c}-{x}_{\alpha})$$ in the coordinate representation with $$p=-i\partial /\partial {x}_{c}$$ while $${\mathrm{e}}^{i{p}_{\alpha}{x}_{c}}$$ gives a corresponding momentum boost.

In the co-moving frame, the coherent state $$|\alpha \rangle ={D}_{\alpha}|0 \rangle $$ defines an HO ground-state wave function shifted by $${x}_{\alpha}$$ with momentum boost $${p}_{\alpha}$$, that is, A8 $$$\begin{equation} {\chi}_{\alpha}({x}_{c})= \langle{x}_{c}|\alpha \rangle ={\mathrm{e}}^{-\frac{1}{2}i{p}_{\alpha}{x}_{\alpha}}\hspace*{0.1667em}{\mathrm{e}}^{+i{p}_{\alpha}{x}_{c}}\hspace*{0.1667em}{\psi}_{0}({x}_{c}-{x}_{\alpha}). \end{equation}$$$This expression (and Equation A10 below) can also be derived by summing an expansion in HO eigenfunctions $${\psi}_{n}({x}_{c})$$ expressed in terms of Hermite polynomials [15].

Since for $$N$$-slit diffraction, we wish to consider only displaced Gaussians initially with zero initial momentum kick, it is tempting at this point to set $${p}_{\alpha}=0$$ and simplify $${D}_{\alpha}$$ to a single factor defined by a real-valued $$\alpha $$ with slit displacement $${x}_{\alpha}=d$$. However, this fails to recognize the HO dynamics of the co-moving frame. The reason is simple: when a classical pendulum is displaced and released from rest, it nevertheless gains momentum with time as it falls. This classical dynamic is built into coherent states by allowing $$\alpha $$ to be complex valued.

Equation (A8) develops in time with the replacement $$\alpha \to \alpha {\mathrm{e}}^{-i{\tau}_{c}}$$ so that its phase-space trajectory from Equation (A4) evolves in time as A9 $$$\begin{align} {x}_{\alpha}& \to {\bar{x}}_{c}=\sigma \sqrt{2}\, (\text{Re}\, \alpha \, \cos {\tau}_{c}+\text{Im}\, \alpha \, \sin {\tau}_{c}),\nonumber\\ {p}_{\alpha}& \to {\bar{p}}_{c}=\frac{\sqrt{2}}{\sigma}\, (\text{Im}\, \alpha \, \cos {\tau}_{c}-\text{Re}\, \alpha \, \sin {\tau}_{c}). \end{align}$$$These results are general solutions of the classical HO equations of motion. One obtains from Equation (A8) including the phase $${\mathrm{e}}^{-i{\tau}_{c}/2}$$ from Equation (A3) the time-dependent coherent-state wave function [15] A10 $$$\begin{equation} \begin{aligned} {\chi}_{\alpha}({x}_{c},{t}_{c})& ={\mathrm{e}}^{-i{\tau}_{c}/2}\hspace*{0.1667em}{\mathrm{e}}^{-\frac{1}{2}i{\bar{p}}_{c}{\bar{x}}_{c}}\hspace*{0.1667em}{\mathrm{e}}^{+i{\bar{p}}_{c}{x}_{c}}\hspace*{0.1667em}{\psi}_{0}({x}_{c}-{\bar{x}}_{c})\\ & ={\mathrm{e}}^{-i{\tau}_{c}/2}\hspace*{0.1667em}{\mathrm{e}}^{-\frac{1}{2}i{\bar{p}}_{c}{\bar{x}}_{c}}\frac{{\mathrm{e}}^{+i{\bar{p}}_{c}{x}_{c}}}{{(\pi {\sigma}^{2})}^{1/4}}\exp \left[-\frac{{({x}_{c}-{\bar{x}}_{c})2}}{2{\sigma}^{2}}\right]. \end{aligned} \end{equation}$$$

One just as easily obtains the momentum Fourier transform of Equation (A10) by reverse disentangling $${D}_{\alpha}$$ in Equation (A7) according to A11 $$$\begin{equation} {D}_{\alpha}={\mathrm{e}}^{+\frac{1}{2}i{p}_{\alpha}{x}_{\alpha}}{\mathrm{e}}^{-i{x}_{\alpha}p}{\mathrm{e}}^{+i{p}_{\alpha}x}, \end{equation}$$$where now the last factor gives a boost in momentum space as $${\mathrm{e}}^{+i{p}_{\alpha}x}\, \tilde{\psi}({p}_{c})=\tilde{\psi}({p}_{c}-{p}_{\alpha})$$ since $$x=+i\partial /\partial {p}_{c}$$ while $${\mathrm{e}}^{-i{x}_{\alpha}{p}_{c}}$$ gives a corresponding coordinate shift. Then A12 $$$\begin{equation} \begin{aligned} {\tilde{\chi}}_{\alpha}({p}_{c},{\tau}_{c})& ={\mathrm{e}}^{-i{\tau}_{c}/2}\hspace*{0.1667em} \langle{p}_{c}|\alpha {\mathrm{e}}^{-i{\tau}_{c}} \rangle \\ & ={\mathrm{e}}^{-i{\tau}_{c}/2}\hspace*{0.1667em}{\mathrm{e}}^{+\frac{1}{2}i{\bar{p}}_{c}{\bar{x}}_{c}}\hspace*{0.1667em}{\mathrm{e}}^{-i{\bar{x}}_{c}{p}_{c}}\hspace*{0.1667em}{\tilde{\psi}}_{0}({p}_{c}-{\bar{p}}_{c})\\ & ={\mathrm{e}}^{-i{\tau}_{c}/2}\hspace*{0.1667em}{\mathrm{e}}^{+\frac{1}{2}i{\bar{p}}_{c}{\bar{x}}_{c}}\hspace*{0.1667em}\frac{{\mathrm{e}}^{-i{\bar{x}}_{c}{p}_{c}}\sqrt{\sigma}}{{\pi}^{1/4}}\hspace*{0.1667em}\exp \left[-\frac{{\sigma}^{2}}{2}{({p}_{c}-{\bar{p}}_{c})2}\right], \end{aligned} \end{equation}$$$which can be verified by direct integration.

The corresponding probability distributions $$|{\chi}_{\alpha}({x}_{c},{t}_{c}){|}^{2}$$ and $$|{\tilde{\chi}}_{\alpha}({p}_{c},{\tau}_{c}){|}^{2}$$ evolve in time along the phase space trajectory $${\bar{x}}_{c},{\bar{p}}_{c}$$ with no change in shape, which justifies the designation coherent state.

One simulates diffraction from a Gaussian slit width $$\sigma $$ and initial displacement $${x}_{\alpha}=d$$ (with zero initial momentum kick $${p}_{\alpha}=0$$) by defining $$\alpha \equiv d/(\sigma \sqrt{2})$$ real-valued. The classical phase space Equation (A9) is then described by A13 $$$\begin{equation} {\bar{x}}_{c}=d\, \cos {\tau}_{c},\hspace*{14.45377pt}{\bar{p}}_{c}=-\frac{d}{{\sigma}^{2}}\, \sin {\tau}_{c}, \end{equation}$$$which transforms Equation (A10) as A14 $$$\begin{equation} \begin{aligned} {\chi}_{d}({x}_{c},{\tau}_{c})& ={\mathrm{e}}^{-i{\tau}_{c}/2}\hspace*{0.1667em}{\mathrm{e}}^{i{d}^{2}\sin {\tau}_{c}\cos {\tau}_{c}/(2{\sigma}^{2})}\\ &\quad \ensuremath{\times{}}{\mathrm{e}}^{-\textit{id}{x}_{c}\sin {\tau}_{c}/{\sigma}^{2}}\hspace*{0.1667em}{\psi}_{0}({x}_{c}-d\cos {\tau}_{c}). \end{aligned} \end{equation}$$$This is just Equation (11) in the limit $$w=\sigma $$ so that $${w}_{c}={\sigma}_{c}=\sigma $$ and $${\mathrm{e}}^{i{\gamma}_{c}}={\mathrm{e}}^{-i{\tau}_{c}/2}$$.

B

Appendix

Squeezed State Algebra

The displaced HO ground state wave function in Equation (10) defines a coherent state with $${\alpha}_{d}\equiv d/(\sigma \sqrt{2})$$ and width $$\sigma \to w$$, B1 $$$\begin{equation} {\left.{\psi}_{0}({x}_{c}-d)\right|}_{\sigma \to w}={\left. \langle {x}_{c}|{\alpha}_{d} \rangle \right|}_{\sigma \to w}. \end{equation}$$$We obtain the time evolution of this state in the co-moving frame by applying the time-development operator as in Equation (A3) to define B2 $$$\begin{equation} {\chi}_{\textit{dw}}({x}_{c},{t}_{c})= \langle {x}_{c}|\ {\mathrm{e}}^{-i{\tau}_{c}({a}^{\ensuremath{\dag}}a+1/2)}{|{\alpha}_{d} \rangle}_{w}. \end{equation}$$$

Unfortunately, the slit states $${|{\alpha}_{d} \rangle}_{w}$$ are defined by a different set of raising and lowering operators form identical with $$a,{a}^{\ensuremath{\dag}}$$ in Equation (A1) but with $$w$$ in place of $$\sigma $$. However, if we insert a complete set of coherent states width $$\sigma $$ defined by Equation (A2), $${\pi}^{-1}\int {d}^{2}\alpha \, |\alpha \rangle \langle \alpha |=1$$, we obtainB3 $$$\begin{eqnarray} {\chi}_{\textit{dw}}({x}_{c},{t}_{c})& =&{\mathrm{e}}^{-i{\tau}_{c}/2}\int \frac{{d}^{2}\alpha}{\pi} \langle{x}_{c}|\alpha {\mathrm{e}}^{-i{\tau}_{c}} \rangle \langle\alpha {|{\alpha}_{d} \rangle}_{w}\nonumber\\ & =&{\mathrm{e}}^{-i{\tau}_{c}/2}{\pi}^{-1}\int d\text{Re}\alpha \hspace*{0.1667em}d\text{Im}\alpha \hspace*{0.1667em}{\chi}_{\alpha}({x}_{c},{t}_{c})\hspace*{0.1667em} \langle\alpha {|{\alpha}_{d}\rangle}_{w}, \end{eqnarray}$$$and we can use the results of the previous section. With Equation (A8), the time-independent expansion coefficient in Equation (B3) is easily evaluated as a Gaussian overlap integral,B4 $$$\begin{equation} \begin{aligned} {\langle\alpha |{\alpha}_{d} \rangle}_{w}& =\int d{x}_{c}\hspace*{0.1667em}{\chi}_{\alpha}^{\ast}({x}_{c}){\left.\hspace*{0.1667em}{\psi}_{c}({x}_{c}-d)\right|}_{\sigma \to w}\\ & =\sqrt{\frac{2w\sigma}{{w}^{2}+{\sigma}^{2}}}\exp \left[-\frac{{d}^{2}}{2({w}^{2}+{\sigma}^{2})}\right]\\ &\quad \ensuremath{\times{}}\exp \left[-\frac{({x}_{\alpha}-i{\sigma}^{2}{p}_{\alpha})({x}_{\alpha}+i{w}^{2}{p}_{\alpha}-2d)}{2({w}^{2}+{\sigma}^{2})}\right]. \end{aligned} \end{equation}$$$With this result and with $$ \langle{x}_{c}|\alpha {\mathrm{e}}^{-i{\tau}_{c}} \rangle ={\chi}_{\alpha}({x}_{c},{t}_{c})$$ from Equation (A10), Equation (B3) can be integrated once $${x}_{\alpha},{p}_{\alpha}$$ and $${x}_{\bar{\tau}},{p}_{\bar{\tau}}$$ have been expressed with Equation (A9) in terms of $$\text{Re}\, \alpha ,\text{Im}\, \alpha $$. The two integrations over $$\alpha $$ are again Gaussian overlap integrals form identical with the previous one. One thus obtains (after rather protracted algebra) precisely $${\chi}_{\textit{dw}}({x}_{c},{t}_{c})$$ in Equation (11).

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