Huygens-Fresnel-Kirchhoff construction for quantum propagators with application to diffraction in space and time

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by  Arseni Goussev

Huygens-Fresnel-Kirchhoff construction for quantum propagators with application to
diffraction in space and time
Arseni Goussev
Max Planck Institute for the Physics of Complex Systems,
No¨thnitzer Straße 38, D-01187 Dresden, Germany
(Dated: January 9, 2012)
Weaddressthephenomenonofdiffractionofnon-relativisticmatterwavesonopeningsinabsorb-
ing screens. To this end, we expand the full quantum propagator, connecting two points on the
oppositesidesofthescreen, intermsofthefreeparticlepropagatorandspatio-temporal properties
of the opening. Our construction, based on the Huygens-Fresnel principle, describes the quantum
phenomena of diffraction in space and diffraction in time, as well as the interplay between the
two. Weillustratethemethodbycalculating diffractionpatternsforlocalized wavepacketspassing
2
through various time-dependentopenings in one and two spatial dimensions.
1
0
PACSnumbers: 03.65.Nk,03.75.-b,42.25.Fx
2

n
a I. INTRODUCTION vatedmethodforevaluatingthewavefunctionofaquan-
J
tum particle passing through an opening in a diffraction

6 Diffraction and interference of matter are among the screenisthe“truncation”approximation,whichisbased

] most fascinating and controversial aspects of quantum onthecompositionpropertyofquantumpropagators. In
h theory. It is not surprising that laboratory exploration this approximation, transmission of a spatially localized
p of these phenomena with subatomic, atomic, and molec- wave packet through the diffraction screen is treated as
t- ular particles has been at the heart of experimental re- a three stage process: (i) the wave packet is propagated
n search since early days of quantum mechanics [1]. As of freely during the time that it takes the corresponding
a today, a wave-like behavior of matter has been success- classical particle to reach the screen, then (ii) the wave
u
q fully demonstratedfor a number of elementaryparticles, function is reshaped (or truncated) in accordance with
[ atoms, simple molecules (see Ref. [1] for a comprehen- the geometry of the aperture, and, finally, (iii) the resul-
sive review), and, most notably, for some heavy organic tantwavefunctionispropagatedfreelyfortheremaining
v1 compounds including C60 [2], C70 [3], and C60F48 [4]. A time interval. The reader is referred to Refs. [11–14] for
9 majority of these experiments involve sending a mono- details and implementation examples of the truncation
8 energetic beam of particles through a screen with open- approximation.
3 ings(apertures)suchasslits,diffractiongratings,orFres- Onedrawbackofthe truncationapproximationis that
1 nel zone plates. Mathematically, the problem of quan- it does not account for diffraction in time. Brukner and
1. tum diffraction on stationary spatial apertures can be Zeilinger [15] proposed another, more versatile method
0 described by the Poisson equation and treated by meth- for solving the problem of quantum diffraction. The
2 ods originallydevelopedinthe contextofdiffractionand central assumption of their method is that the time-
1 interference of light [5, 6]. dependent wave function in question satisfies certain ex-
v: Adiffractionphenomenonofadifferentkind–“diffrac- plicitly known, inhomogeneous time-dependent Dirich-
i tionintime”–wasintroducedbyMoshinskysixdecades let boundary conditions at the surface of the diffraction
X
ago [7] and subsequently studied by many researchers, screen. More specifically, two assumptions are made: (i)
r bothexperimentally[8]andtheoretically(seeRefs.[9,10] the valueofthewavefunctionatapointinsidethe aper-
a
forreviews). Thephenomenonhastodowithtimeevolu- ture is assumed to equal the value that the wave func-
tion (more precisely, with non-uniformspreading) of ini- tion would have at this point if the diffraction screen
tially sharp wave fronts in quantum systems, and mani- was absent, and (ii) the wave function vanishes at every
festsitselfalreadyinonedimension. Thus,intheoriginal point of the diffraction screen outside the aperture. The
set-up proposed by Moshinsky [7], a perfectly absorbing first assumption is commonly referred to as the Kirch-
shutter is placedin the wayof a mono-energeticbeamof hoff approximation, while the second corresponds to the
non-relativistic quantum particles. Then, a sudden re- physical assumption of perfect reflectivity of the screen.
moval of the shutter creates a “chopped” particle beam ThemethodofBruknerandZeilingerhassincebeensuc-
withasharpwavefront. As shownby Moshinsky,sucha cessfully used by several researchers to study quantum
wave front disperses non-uniformly in the course of time diffraction in both space and time [16–18].
and, most interestingly, develops a sequence of diffrac- In this paper, we present another approach to quan-
tion fringes. Mathematically, these fringes appear to be tum diffraction in space and time. Our method is based
analogous to the ones observed in diffraction of light on on the Huygens-Fresnel principle and Kirchhoff theory
the edge of a semi-infinite plane. of diffraction, and allows one to calculate the time-
There are several ways of treating quantum diffrac- dependentquantumpropagatorfortheproblemofparti-
tion theoretically. One commonly used, physically moti- cle diffraction on spatio-temporal openings in otherwise
2
perfectly absorbing screens. Our expressionfor the prop- with the initial condition
agator is especially adaptable for calculating diffraction
′ ′
patterns in situations in which the initial wave function limK(q,q;t)=δ(q q), (3)
t→0 −
is given by a spatially localized wave packet or by a su-
perposition of several localized wave packets. Similar to where Hq denotes the Hamilton operatorin the position
the method of Brukner and Zeilinger, our construction representation. (Thepropagatorisalsosubjecttocertain
provides a unified framework that treats the phenom- absorbingboundaryconditionsthatwedo notdiscussat
ena of diffraction in space and diffraction in time on the this point.) In the case of a free particle of mass m we
soafmBerufokonteargea.ndThZeeimlinaignerdiaffnedrenocuerbmetewtheoedn tihsethmaetthtohde have Hq =−2~m2∇2q and K =K0(q−q′;t) with
fwohrmileerthise dlaetstigerneids otonlytreaaptplpicearfbelcetltyorpeflerefcetcitnlgy sacbrseoernbs-, K0(q−q′;t)= 2πmi~t f2 exp −m|q2i−~tq′|2 . (4)
ing ones. Finally, our approach allows for calculation of (cid:16) (cid:17) (cid:18) (cid:19)
quantum diffraction patterns produced by openings in Hereinafter, the subscript “0” indicates that the corre-
spatially curved screens. sponding quantity refers to the case of a free particle.
The paper is organized as follows. In Sec. II we We also consider the energy-domain Green function
develop an expansion of the propagator for a non- defined as the Laplace transform of the propagator,
relativistic quantum particle passing through a time-
∞
dependent opening in an absorbing screen. In Sec. III G(q,q′;E)= dte−stK(q,q′;t)
wedemonstrateutilityoftheexpansionbyapplyingitto
Z0
someexamplesystemsinoneandtwospatialdimensions. E
[K](q,q′;s) with s= . (5)
InSec.IVwediscussourresultsandmakeconcludingre- ≡L i~
marks. Some technicalities are deferred to an Appendix.
Consequently,thepropagatorisobtainedfromtheGreen
function by means of the inverse Laplace transform,
II. HUYGENS-FRESNEL-KIRCHHOFF K = −1[G]. In the free particle case, Eqs. (4) and (5)
CONSTRUCTION FOR QUANTUM defineLthe free-particle Green function G0(q q′;E) =
PROPAGATORS [K0](q q′;s) that satisfies −
L −
2m 2mE
In this section, we address two quantum-mechanical ∇2qG0+k2G0 =− i~ δ(q−q′) with k2 = ~2 . (6)
phenomena–diffractionintime,pioneeredbyMoshinsky
[7,9,10],anddiffractioninspace,asdescribedbyKirch-
hoff theory [5, 6]. We recast standard descriptions of
B. Diffraction in time
bothphenomenainawayanalogoustoatime-dependent
formulation of the Huygens-Fresnel principle. Expressed
Wenowaddressthephenomenonofdiffractionintime,
in this way, the two diffraction processes appear to be
first considered by Moshinsky [7] and later explored by
closely related and can be straightforwardly unified into
many researchers (see Refs. [9, 10] for reviews). In its
a single model of diffraction in space and time.
one-dimensional formulation, the Moshinsky problem is
concerned with time evolution of a quantum particle,
whose wave function Ψ(ξ;t) is localized to the semi-
A. Notation
infinite interval ( ,x1) at time t = 0, i.e., Ψ(ξ;0)= 0
−∞
for ξ > x1. Over time, an absorbing wall (shutter)
Inordertofacilitatetheclarityofthefollowingpresen-
is switched “on” and “off” at the point x1, according
tation, we begin by introducing central physical quanti-
to a protocol defined by a characteristic function χ(t).
ties and fixing notation.
The latter is allowed to take values between 0 and 1,
The focus of this paper is on the quantum propagator
with 0 representing the case of perfect absorption (shut-
K(q,q′;t) that describes the motion of a quantum par-
ter “on”) and 1 corresponding to perfect transmission
ticle in the f-dimensional coordinate space by relating a
(shutter “off”). The open interval 0 < χ < 1 repre-
particle’s wave function Ψ(q;t) at time t to that at time
sents the case of partially absorbing and partially trans-
t=0 through
mitting (but reflection-free) shutter. One is then inter-
ested in the wave function Ψ(ξ;t) of the particle to the
Ψ(q;t)= dfq′K(q,q′;t)Ψ(q′;0). (1) right of the shutter, ξ > x1, at t > 0. Moshinsky [7]
ZRf analyzed this problem for a “monochromatic” incident
The propagator is the solution of the time-dependent wave Ψ(ξ;0) = Θ(x1 − ξ)eikξ, where k > 0 and Θ is
the Heaviside step function, and a perfectly absorbing
Schr¨odinger equation
shutter that gets suddenly removedat an instantt0, i.e.,
∂K χ(t)=Θ(t t0). Hisanalysisshowedthatattimest>t0,
i~ =HqK (2) thefrontof−theprobabilitydensitywaveexhibitspatterns
∂t
3
identical to those observed in the Fresnel diffraction of ItisinterestingtoobservethatEq.(9)doesnotspecify
light from the edge of a semi-infinite plane, thus giving the function u uniquely. In fact, Eq. (9) turns out to be
rise to the term “diffraction in time”. an identity satisfied exactly by infinitely many different
Our objective is to devise an expressionfor the propa- functions u, some examples being (see App. A)
′
gator K(x,x;t) that describes Moshinsky diffraction by
′
a shutter positioned at a point x1, such that u=ηx−x1 +(1 η)x1−x , (10)
′ t t1 − t1
x <x1 <x, (7) 2−i~ exp( ζ2) m(x x′)2
u= − with ζ2 = − . (11)
and controlled (opened and closed) in accordance with πmt erfc(ζ) 2i~t
r
thecharacteristicfunctionχ(t)ofanarbitraryfunctional
form. Our construction relies on the Huygens-Fresnel Here, η is an arbitrary complex number, and “erfc” de-
principle, which, for the purpose of the current problem, notes the complementary error function.
can be formulated as follows: The disturbance at the
′
point x produced by a source O , located at some other
pointx′,canbeviewedasproducedbyafictitioussource 2. Moshinsky shutter, χ(t)=Θ(t−t0)
′
O1,locatedatapointx1 inbetweenx andx,cf. Eq.(7).
The strengthofthe fictitious sourceO1 is determinedby We now show that the functional form of u can be
the disturbance atthe point x1 produced by the original uniquely determined by comparing Eq. (8) with an ex-
′
sourceO . Whena(partially)absorbingshutterisplaced actexpressionforthequantumpropagatorintheoriginal
at the point x1, the strength of the fictitious source O1 Moshinsky set-up[7], in whichthe absorbingshutter, lo-
is modulated by the characteristic function χ(t) taking catedatx1,iscloseduntilatimet0 andopenafterwards,
values between 0 and 1. Mathematically, this can be i.e., χ(t1)=Θ(t1 t0) for 0 t0,t1 t.
summarized as On one hand, a−direct con≤structio≤n of the propagator
′ t ′ KM(x,x′;t) for the original Moshinsky problem leads to
K(x,x;t)= dt1u(x x1,x1 x;t,t1)
− −
×K0(x−Z0x1;t−t1)χ(t1)K0(x1−x′;t1). (8) KeM(x,x′;t)= −x∞1dx′′K0(x−x′′;t−t0)K0(x′′−x′;t0)
Z
where u, having dimensions of speed, is a yet-to-be-
e 1 mt
′ ′
danetdertmiminesedtfaunndctti1o.n of the distances x−x1 and x1−x =K0(x−x;t)"1− 2erfc (x1−x0)s2i~t0(t−t0)!#
The physical meaning of Eq. (8) is transparent: The (12)
probability amplitude of an event in which the particle
′ with
goes from x to x in time t can be expressed as a sum of
pthreobpaabritliictlyeafimrsptligtouedsefsroovmerxa′ltlocoamnpinotseirtmeeevdeianttesipnowinhtixch1 x0 =xtt0 +x′t−tt0 . (13)
ina time t1 andthenreachesxfromx1 inthe remaining
time t t1. Because of their simple physical interpreta- The first equality in Eq. (12) combines the composition
−
tion, propagator expansions similar to Eq. (8) are often property of quantum propagators and the fact that, at
used for qualitative description of quantum interference time t0, all the probability density to the right of x1 has
phenomena (e.g., see [9] for a qualitative discussion of been absorbed by the shutter. This leads to the trunca-
a two-slit interference experiment), however the explicit tion of the upper limit in the x′′ integral.
functionalformofuisusuallyneitherspecifiednortaken Ontheotherhand,asubstitutionofχ(t1)=Θ(t1 t0)
−
into account. into Eq. (8) yields
t
′ ′
1. Free particle, χ(t)=1 KM(x,x;t)= dt1u(x−x1,x1−x;t,t1)
Zt0
′
K0(x x1;t t1)K0(x1 x;t1). (14)
Interested in determining the functional form of u = × − − −
′
u(x x1,x1 x;t,t1), we first direct our attention to Equations(12)and(14)allowustouniquelydetermine
− −
the simplestpossible scenario,inwhichthe shutter stays
the function form of u by requiring KM = KM. Indeed,
openthroughoutthe entiretimeintervalfrom0tot,i.e., ′
let us for the moment fix the values of x, x, and t, and
χ(t1) = 1 for 0 ≤ t1 ≤ t. In this case, the propagator in treat the propagator KM as a functeion of the shutter
the left-hand side of Eq. (8) is given by the free particle
propagator, K =tK0, requiring u to satisfy othpaetnilnimgtt0i→mte−tK0 Monl=y, i0.ee..,IKndMee=d,KxM0(t→0).xFairsstt,0w→e not−te,
′ ′ and the argument of the ceomplemeentary error function
K0(x−x;t)=Z0 dt1uK0(x−x1;t−t1)K0(x1−x;t1). in Eq. (12) teneds to ei3π/4∞, making the value of the
(9) complementary error function approach2. This limit, of
4
course, corresponds to the trivial case of the absorbing to the Moshinsky shutter problem, for which point-like
shutter being closed throughout the entire time interval absorption can be defined unambiguously.
from 0 to t. Then, in view of this limit, we rewrite the It is interesting to note a formal similarity between
Moshinsky propagator as Eq. (18) and the well-knownLippmann-Schwinger equa-
tion[9,11,19],whichinthecaseofapoint-likeperturba-
t dKM(t1) tion, situated at x1 and described by a spatio-temporal
KM(t0)=−Zt0 dt1 dt1 . (15) potential of the form V(ξ,τ)=δ(ξ−x1)U(τ), reads
e
A straightforweard (but somewhat tedious) calculation Ksc(x,x′;t)=K0(x x′;t)
−
yields i t
′
dKM(t1) 1 x x1 x1 x′ − ~Z0 dt1K0(x−x1;t−t1)U(t1)Ksc(x1,x;t1). (19)
= − + −
dt1 − 2(cid:18) t−t1 t1 (cid:19) As before, K0 is the (free-particle)propagatorin the ab-
e K0(x x1;t t1)K0(x1 x′;t1). (16) senceoftheperturbationpotential,andKsc isthepropa-
× − − − gatorcorrespondingtothefullscatteringproblem. Equa-
ItisnowclearthattheMoshinskypropagatorKM,asex- tion (19) gives rise to a multiple collision representation
pressed by Eqs. (15) and (16), is equivalent to the prop- of the scattering propagator, known as the Dyson series
agator KM, given by Eq. (14) with e [11, 19].
Despite their superficial resemblance, Eqs. (18) and
1 x x1 x1 x′ (19) describe very different physical processes: The
u= − + − . (17)
2 t t1 t1 Lippmann-Schwinger equation pertains to the phe-
(cid:18) − (cid:19) nomenon of quantum scattering, whereas the Huygens-
Note that Eq. (17) is equivalent to Eq. (10) with η = Fresnel propagator expansion represents quantum mo-
1/2. Also, Eq. (17) provides the physicalmeaning of the tion in the presence of obstacles that (partially) absorb
function u: The latter is a characteristic(mean) velocity matter waves without deflecting them. In fact, it is easy
of the particle when it traverses the shutter. to show that the Lippmann-Schwinger equation can not
be used to model perfect absorption, corresponding to
the trivial choice χ = 0 in Eq. (18). Indeed, the case
3. Arbitrary χ(t) of a perfectly absorbing obstacle at x1 would require
′ ′
Ksc(ξ,x;t) = Θ(x1 ξ)K0(ξ x;t), which is clearly
− −
A substitution of Eq. (17) into Eq. (8) yields incompatible with Eq. (19).
In order to avoid possible confusion, we emphasize
′ 1 t x x1 x1 x′ that it is the assumption of a point-like perturbation,
K(x,x;t)= 2Z0 dt1(cid:18) t−−t1 + t−1 ′(cid:19) LVi(pξp,mτ)an=n-Scδh(wξin−gexr1)eUqu(τa)t,ionthtaot pdrooepserlnyotcaapltluowre tthhee
K0(x x1;t t1)χ(t1)K0(x1 x;t1). (18)
× − − − physics of absorption. On the opposite, the Lippmann-
′ Schwinger equation with a smooth, complex-valued po-
Here, the spatial points x, x, and x1 are subject to the
tential function V(ξ,τ), defined over an extended spa-
condition given by Eq. (7), and the characteristic func-
tial interval, is often the method of choice in modeling
tion χ is allowed to take values between 0 (perfect ab-
absorbing boundaries (see Ref. [20] for a comprehensive
sorption) and 1 (perfect transmission).
review).
Equation (18) provides a formulation of the Huygens-
Fresnel principle for a one-dimensional particle in the
presence of a point-like absorbing obstacle, whose ab-
sorbing properties change in the course of time. At this 4. Continuously opening shutter, χ(t)=e−τ/t
point,itisimportanttoemphasizethattheanalysispro-
vided in this section should not be regarded as a rigor- InordertodemonstratetheusefulnessoftheHuygens-
ousmathematicalproofofthepropagatorexpansion(18). Fresnel formulation of diffraction in time we apply the
Unavoidable difficulties in solving the problem from the propagatorexpansion(18)toamodifiedMoshinskyshut-
firstprinciples stemfromthe lackofaproperunambigu- ter problem, in which the absorbing shutter is initially
ous definition of point-like, generally partial and time- closed,χ(t1)=0fort1 0,andthenopenscontinuously
≤
dependent,absorption. Inourapproach,however,weby- in accordance with
pass this issue by modeling the absorptionwith the help
ofatime-dependentcharacteristicfunction, χ(t),andre- χ(t1)=exp( τ/t1) for t1 >0. (20)
−
lyingonthevalidityoftheHuygens-Fresnelconstruction
Here, τ > 0 determines the rate at which the shutter
in its most general form, Eq. (8). Consequently, we re-
opens. Note that the shutter described by Eq. (20) is
move any arbitrariness in the Huygens-Fresnel construc-
completely removed only in the limit of infinitely long
tionbydeterminingthefunctionu,originallyunknownin
Eq.(8),throughanalyzingthecaseofχ(t)corresponding times, t1→lim+∞χ(t1)=1.
5
UsingtheHuygens-Fresnelexpansion(18),wenowde- t>0. Second,itisclearfromEq.(28)thatthe propaga-
′ ′
rive an explicit expression for the propagator K(x,x;t) tor K(x,x;t) is not symmetric in the coordinates x and
′ ′ ′
connecting a point x to the left of the shutter (x <x1) x. Theabsenceofsuchsymmetryistypicalforquantum
attime0withanotherpointxtothe rightofthe shutter motioninthepresenceoftime-dependentobstacles(e.g.,
(x1 < x) at time t. To this end, we rewrite Eq. (18), in see Sec. A.1 in Ref. [9]).
view of Eq. (4), as
i~ ∂ ∂
′
K(x,x;t)= C. Diffraction in space
−2m ∂x − ∂x′
(cid:18) (cid:19)
t
′ WenowaddressKirchhofftheoryofdiffractioninspace
dt1K0(x x1;t t1)χ(t1)K0(x1 x;t1). (21)
0 − − − [5, 6]. In particular, we rewrite the Kirchhoff’s formula-
Z
tion in its time-dependent form, in which it can be di-
Then, taking into account Eq. (20), we write
rectly applied to diffraction of quantum wave packets.
′
χ(t1)K0(x1−x;t1)=K0(x1−x˜;t1), where x˜ satisfies To this end, weconsidera smooth(f 1)-dimensional
−
2i~τ surface S that is defined as the zero set of a real-valued
(x1 x˜)2 =(x1 x′)2+ . (22) function s: Rf R,
− − m →
Solving Eq. (22) for x˜ and choosing the solution that S = q1 Rf : s(q1)=0 . (29)
converges to x′ in the limit τ 0+, we have { ∈ }
→
The surface S is assumed to partition the position space
x˜=x1−ρeiθ, (23) into two disjoint regions, such that the function s takes
different (and constant) signs in the two regions. We
where
also consider two spatial points, q and q′, that lie on
ρ= (x1 x′)4+ 2~τ 2 41 , (24) tthakeeops(pqo)sit>e s0ideasndofst(hqe′)su<rfa0c.e. TFhoer csounrfcarceetenSesgsivwees
" − (cid:18) m (cid:19) # the location of a non-transparent, absorbing screen, in
1 2~τ which some transparent openings (apertures) may be
θ = tan−1 . (25)
2 m(x1 x′)2 “cut out”. These openings and, more generally, the ab-
− sorbing properties of the screen, can be described by a
Equation (21) can now be rewritten as spatially-dependent characteristic function χ(q1) taking
values between zero (perfect absorption) and one (per-
K(x,x′;t)=−2im~ ∂∂x − ∂∂xx˜′∂∂x˜ fdeicfftratrcatinosnm[i5s,si6o]na)llaotwasponoeinttoqe1x∈prSes.sKthirecGhhreoeffntfhuenocrtyioonf
t (cid:18) (cid:19) G(q,q′;E), connecting the points q and q′ at an energy
dt1K0(x x1;t t1)K0(x1 x˜;t1). (26) E, as an integral along the screen:
− − −
Z0
The integral over t1 can now be evaluated explicitly: G(q,q′;E)= i~ dfq1δ s(q1) χ(q1)
t −2mZRf
(cid:0) (cid:1)
Z0 dt1K0(x−x1;t−t1)K0(x1−x˜;t1) ×∇s(q1)· G0(q−q1;E)∇q1G0(q1−q′;E)
(cid:18)
m m
= 2i~erfc(cid:18)r2i~t(x−x˜)(cid:19) . (27) −G0(q1−q′;E)∇q1G0(q−q1;E) . (30)
(cid:19)
This identity is derived in Appendix A for the case of
It is important to emphasize that Kirchhoff method,
real-valued x˜ (see Eqs. (A2) and (A7)); the derivation
Eq. (30), is not applicable to diffraction on apertures
however can be extended to the case of Rex˜ < x1 < x
in transmission screens with reflecting (e.g., Dirichlet or
and Imx˜ 0 (cf. Eqs. (23-25)). Finally, substituting
≤ ′ ′ Neumann) boundary conditions. Instead, Kirchhoff the-
Eq.(27),togetherwith∂x˜/∂x =(x1 x)/(x1 x˜), into
− − ory is known to be a good model for diffraction on per-
Eq. (26) and taking the partial derivatives with respect
fectlyabsorbing,or“black”,infinitelythinobstacles[21–
to x and x˜, we arrive at
23]. In fact, Eq. (30) provides an exact solution to the
′ 1 x1 x′ time-independentdiffractionproblemwiththeboundary
K(x,x;t)= 2 1+ x1− x˜ K0(x−x˜;t). (28) conditions on the screen given by the so-called Kottler
(cid:18) − (cid:19) discontinuity (see Refs. [22, 23] and references within):
Two quick remarks are in order. First, Eq. (28) guar- Theprobabilityamplitudefieldhasadiscontinuityacross
′ ′
anties that, as expected, K(x,x;t) K0(x x;t) as thescreenequaltominusthevalueofthefree-spacefield
τ 0+. Thislimit correspondsto the→standar−dMoshin- atthatpoint. Asimilarconditionisimposedonthenor-
→
sky set-up in whichthe shutter stayscompletely open at mal derivative of the field.
6
For our purposes, it is important to construct a time- the screen (such that s(q′) < 0 and s(q) > 0) in time t,
dependent formulation of Kirchhoff diffraction. This is is given by
straightforwardlyachieved by first rewriting Eq. (30) as
1 t
i~ K(q,q′;t)= dt1 dfq1δ s(q1)
G(q,q′;E)= dfq1δ s(q1) χ(q1) 2Z0 ZRf
×∇s(q1)·(∇−q2−m∇ZqR′f)G0(q−(cid:0) q1;E(cid:1))G0(q1−q′;E()3,1) ×(cid:18)qt−−tq11 + q1t−1 q′(cid:19)·∇s((cid:0)q1) (cid:1) ′
K0(q q1;t t1)χ(q1;t1)K0(q1 q;t1). (33)
× − − −
and then performing the inverse Laplace transform from
Equation (33), which from now on we will refer as to
energy-dependent Green functions to time-dependent
Huygens-Fresnel-Kirchhoff (HFK) construction, consti-
propagators,yielding
tutes the main result of the present paper.
1 t Our arguments supporting the conjecture, given by
K(q,q′;t)= dt1 dfq1δ s(q1) Eq. (33), are as follows. First, the propagatorexpansion
2Z0 ZRf is in accordwith the Huygens-Fresnel principle. Second,
q−q1 + q1−q′ ∇s((cid:0)q1) (cid:1) the HFK construction correctly captures the physics of
× t t1 t1 · diffractionintime: In the specialcase thatχ is indepen-
(cid:18) − (cid:19)
K0(q q1;t t1)χ(q1)K0(q1 q′;t1). (32) dent of q1 and that the surface S is givenby an (f −1)-
× − − − dimensionalplane,s(q1)=n (q1 q0)withaunitvector
· −
Here, an important remark is in order. The energy- n and somefixed vectorq0,Eq.(33)becomes equivalent
′
domain formulation of Kirchhoff diffraction, Eq. (30), to Eq. (18) with x, x, and x1 replaced, respectively, by
only assumes the validity of the Helmholtz equation for n q, n q′, and n q1. Third, in the case that χ is inde-
· · ·
a field in question, and is therefore applicable to a wide pendent of t1, Eq. (33) is nothing but a time-dependent
formulation of Kirchhoff diffraction.
range of wave phenomena encountered, for instance, in
acoustics, optics, and non-relativistic quantum mechan-
ics. Equation (32) however relies on the particular re-
lation, given by Eq. (5), between the energy-dependent III. DIFFRACTION OF SPATIALLY
Green function and time-dependent propagator, and is LOCALIZED WAVE PACKETS
restricted to non-relativistic quantum mechanics only.
Also, we note that the physical picture implied by In this section, we apply the HFK construction,
Eq. (32) is that of a particle traveling freely from q′ Eq. (33), to wave functions that are initially localized in
to a point q1 in the aperture, and then from q1 to q. positionspace. We directoutattentionto some example
It is therefore implicitly assumed that the aperture and systems in one and two spatial dimensions.
pointsqandq′ arechoseninsuchawaythateverypath As before, we consider an absorbing screen that is de-
q′ q1 q′ has no intersection with the screen other fined by a surface S, Eq. (29), and a characteristic func-
→ →
than at q1. In other words, Eq. (32) is only applicable tion χ. The latter, in general, is a function both of the
to configurationsin whichapertures are not “shadowed” position on S and of time. We further consider a quan-
by S and aredirectly “visible”fromthe points q and q′. tum particle described at time t =0 by a wave function
Ψ(q′;0), which is localized in a small neighborhood of a
linear size σ around a spatial point Q, i.e., Ψ(q′;0) 0
D. Diffraction in space and time for q′ Q σ. Here, we consider the case of σ be≃ing
| − | ≫
small compared to the distance L between the point Q
and the surface S. For concreteness, we take s(Q) < 0.
We now note a striking similarity between (i) the
Then, in accordance with Eqs. (1) and (33), the wave
Huygens-Fresnel expansion of the propagator for the
function Ψ at a point q, such that s(q) > 0, and time
problemofdiffractionintime,Eq.(18),and(ii)thetime-
t>0 can be approximately written as
dependent formulation of Kirchhoff theory, Eq. (32).
This leads us to a conjecture that both expressions are
1 t
particular cases of a more general propagator expan- Ψ(q;t) dt1 dfq1δ s(q1)
sion, describing quantum diffraction on apertures which ≃ 2Z0 ZRf
tdheepmensdelevnetsampaeyrtuvraersyainretrheeprceosuernsteedofbtyimaec.haSruacchtetriimsteic- × qt−qt11 + q1t−1 Q ·∇(cid:0)s(q1)(cid:1)
functionχthatdependsonboththepositionq1alongthe (cid:18)K0(q− q1;t t1)χ((cid:19)q1;t1)Ψ0(q1;t1), (34)
dividing surfaceS andthe instantt1 of the time interval × − −
(0,t). As before, χ = χ(q1;t1) is allowed to take values where
between 0 (perfect absorption) and 1 (perfect transmis-
scioonnn)e.ctIinngthtiwsocapsoei,ntwseqcaonndjecqt′uorentthhaetotphpeospirtoepsaidgaestoorf, Ψ0(q1;t1)=ZRf dfq′K0(q1−q′;t1)Ψ(q′;0) (35)
7
is the wave function of the corresponding free particle. 1.5
Equation (34) holds to the leading order in the small
χ=1
parameter σ/L.
χ=Θ(t1−−t0) χ=Θ(t0−−t1)
1
A. One dimension
|t1−−t0|
2 χ=Θ 1−−
As our first example we consider the Moshinksy prob- Ψ| µ ǫ ¶
|
lem in one dimension (f = 1) for a quantum particles
initially described by the Gaussian wave packet 0.5
1 41 i (x′ Q)2
′ ′
Ψ(x;0)= exp P(x Q) − .
πσ2 ~ − − 2σ2
(cid:18) (cid:19) (cid:18) (cid:19)
(36)
0
Here, Q and P represent, respectively, the average posi- 9 9.5 10 10.5 11
x
tion and momentum of the particle, and σ characterizes
the wave packet dispersion in the position space. In our
FIG.1: (Coloronline)Theprobabilitydensity|Ψ(x,t)|2,cal-
set-up, an absorbing shutter is placed at a point d, such
culatedinaccordancewithEq.(37)forvariouscharacteristic
thatQ<dandσ L=d Q. Weareinterestedinthe
≪ − functions χ(t1), as a function of the position x. The initial
wave function Ψ(x;t) at x>d and t>0.
wave packet is characterized by its position Q = 0, average
A substitution of Eq. (36), along with s(x1) = x1 d momentum P = 200, and dispersion σ = 0.1. The position
−
and χ=χ(t1), into Eqs. (34) and (35) yields of the shutter is given by d=8, and the propagation time is
t=0.05. The shutter switching time t0 =0.04 and the time
1 t x d d Q window ǫ=5×10−4. All quantities are given in the atomic
Ψ(x;t)≃ 2Z0 dt1(cid:18)t−−t1 + −t1 (cid:19) units, m=~=1.
K0(x d;t t1)χ(t1)Ψ0(d;t1) (37)
× − −
with χ(t1)=Θ(1−|t1−t0|/ǫ),representingascenarioinwhich
the shutter is open only during a time interval of half-
1 41 width ǫ centered around t0. Here we take t0 = 0.04 and
Ψ0(x;t)=(cid:18)πγt2σ2(cid:19) ǫco=rre5sp×on1d0−in4g. fNreoetecltahssaitcatlhepaprotiscitleioinsaQt +timPet0t/0mof=th8e
i P2 i (x Qt)2 and coincides with the position of the shutter.
exp t+ P(x Q ) − (38)
× (cid:18)~2m ~ − t − 2γtσ2 (cid:19) Oscillations in the spatial dependence of the probabil-
itydensity,seendistinctlyinFig.1,areintroducedbyan
and
instantaneous process of switching the shutter, and are,
P ~t in fact, a manifestation of the diffraction-in-time phe-
Q =Q+ t and γ =1+i . (39)
t m t mσ2 nomenon. Itiswellknownthattheseoscillationsbecome
lesspronouncedandeventuallydisappearasoneswitches
Evaluatingthe integralintheright-handside ofEq.(37)
the shutter “continuously” over longer and longer time
numericallyonobtainsthewavefunctionΨ(x,t)atx>d
intervals [10, 18]. Here we note that the HFK construc-
and t>0.
tion provides a convenient framework for analyzing the
Figure 1 shows the diffraction patterns calculated in
disappearance(alsoknownasapodization)ofthediffrac-
accordance with Eq. (37) for different shutter protocols,
tion pattern for initial states that are localized in the
χ(t1). The initial wave packet, Eq. (36), is centered position space. The phenomenon of the apodization of
around Q = 0 and has the average momentum P = 200
atomicbeams,whichcorrespondtoinitialstateslocalized
and the position dispersion σ =0.1. Hereinafter, we use
in the momentum space, has been previously addressed
the atomic units, m =~ = 1, in all numerical examples.
by different methods [17, 18].
The absorbing shutter is positioned at d = 8. Figure 1
shows the probability density Ψ(x,t)2 as a function of
| |
x for a fixed time, t = 0.05. Note that, for our choice B. Two dimensions
of parameters, the position at the time t of the corre-
sponding free classical particle is Q = Q+Pt/m = 10.
t We now address diffraction of quantum wave packets
The figure shows four probability density distributions
in two dimensions on a (generally curved) shutter whose
for (i) χ(t1) = 1, representing the free-particle case, (ii) absorption properties may both depend on position and
χ(t1) = Θ(t1 − t0), corresponding to the perfectly ab- vary in time. We consider a quantum particle with the
sorbing shutter first being in place and then suddenly
initial state given by
removed at time t0, (iii) χ(t1)=Θ(t0 t1), correspond-
−
ing to the shutter closed instantaneously at t0, and (iv) Ψ(x,y;0)=φ1(x,0)φ2(y,0), (40)
8
where
1
1 4 i (ζ Q )2
j
φ (ζ,0)= exp P (ζ Q ) −
j πσj2! ~ j − j − 2σj2 !
(41)
for j = 1 and 2. Here, Q = (Q1,Q2) is the average po-
sition and P = (P1,P2) average momentum of the par-
ticle, and σ1 and σ2 characterize respectively the x- and
y-component of the wave packet dispersion in the posi-
tion space. As in the one-dimensional case, the initial
distance betweenthe particleandthe shutteris assumed
to be large compared to the spatial extent of the wave
packet.
Analogously to Eqs. (38) and (39), the time evolution
of a free particle is described by the wave function
Ψ0(x,y;t)=φ1(x,t)φ2(y,t), (42)
where
−9 −8 −7 −6 −5 −4 −3 −2 −1
1
1 4 i P2 FIG.2: (Coloronline)Diffractionintimeonacurvedshutter.
j
φj(ζ,t)= πγ2 σ2 exp ~2mt Part (a) of the figure illustrates the set-up. The initial wave
j,t j! " packet is characterized by its average position Q = (0,0),
i (ζ Q )2 average momentum P = (200,0), and dispersion σ1 = σ2 =
+~Pj(ζ −Qj,t)− 2−γj,tσj,j2t # (43) 0d.−1.αTyh12e=sh0utwteitrhisdp=osi8t,ioanneddiatlsontigmtehedecpuernvedesn(xce1,iys1s)p=ecxifi1e−d
byχ(t1)=Θ(1−|t1−t0|/ǫ)witht0 =0.04andǫ=5×10−4.
and Thepropagation time ist=0.05. The (light blue) rectangle,
covering 8.9 < x < 11.1 and −1.6 < y < 1.6, outlines the
P ~t
j spatial region, in which the probability density is computed.
Q =Q + t and γ =1+i , (44)
j,t j m j,t mσ2 Parts (b), (c), and (d) of the figure show ln|Ψ(x,y;t)|2 for
j
α=−0.3, 0, and 0.3 respectively. All quantities are given in
for j = 1 and 2. Then, using this expression for Ψ0 in theatomic units, m=~=1.
Eq.(34)andevaluatingnumericallythetwo-dimensional
integral (over time and along a one-dimensional curve
accommodatingtheshutter),weobtainthewavefunction coincideswiththecenterofthe rectangle. Then,thelog-
Ψ(x,y;t) att>0 andat a point (x,y) onthe side ofthe arithm of the probability density, ln Ψ(x,y;t)2, inside
| |
shutter opposite to Q. the rectangular regionis shownfor three different values
Figure 2 illustrates the effect of shutter curvature on of the parameter α: Fig. 2b corresponds to α = 0.3
−
the diffraction pattern. The set-up is schematically pre- (concave shutter), Fig. 2c to α = 0 (flat shutter), and
sentedinFig.2a. Theinitialstateoftheparticleisgiven Fig.2d to α=0.3 (convexshutter). While the mean po-
by a Gaussianwave packet, Eqs. (40) and (41), with the sition of the quantum particle coincides with that of the
average position Q = (0,0), average momentum P = corresponding classical particle, the diffraction patterns
(200,0), and dispersion σ1 =σ2 =0.1. The wave packet areclearlydifferentforthethreechoicesofα. Compared
is incident upon a shutter, whose spatial geometry is de- to the diffraction pattern produced by the flat shutter,
finedbythecurves(x1,y1)=x1 d αy12 =0withd=8. the concave shutter gives rise to a “divergent” pattern,
− −
Depending onthe signofthe parameterα, the shutter is whiletheconvexshutterproducesa“convergent”diffrac-
concave (α < 0), flat (α = 0), or convex (α > 0). The tion pattern.
timedependenceoftheshutterisgivenbythecharacter- Figure 3 addresses diffraction on a flat screen whose
isticfunctionχ(q1;t1)=Θ(1 t1 t0 /ǫ)witht0 =0.04 absorbing properties depend on time as well as vary in
and ǫ=5 10−4. This corres−po|nd−sto|the case of a per- space. The set-up, schematically illustrated in Fig. 3a.
×
fectly absorbing shutter that is open only during a time As before, the initial wave packet is described by Q =
interval of half-width ǫ centered around t0 (cf. solid red (0,0), P = (200,0), σ1 = σ2 = 0.1, and the propaga-
curveinFig.1). The propagationtime is setto t=0.05, tion time is t = 0.05. Here we consider a flat screen,
and the wave function is investigated inside a spatial re- givenbys(x1,y1)=x1 dwithd=8,andstudydiffrac-
−
gion defined by 8.9 < x < 11.1 and 1.6 < y < 1.6, tion patterns for three different characteristic functions
−
shownschematicallyasa(lightblue)rectangleinFig.2a. χ. As in Fig. 2, we present the logarithm of the prob-
Notethatthepositionofthecorrespondingclassicalpar- ability density, ln Ψ(x,y;t)2, inside a spatial region de-
| |
ticle at time t equals Q = Q + Pt/m = (10,0) and fined by 8.9 < x < 11.1 and 1.6 < y < 1.6, and illus-
t
−
9
χ(q1;t1) = Θ(1 y1 /δ)Θ(1 t1 t0 /ǫ) can be rep-
−| | −| − |
resented as a rectangular opening in the (y1,t1) coordi-
nates, while the aperture corresponding to χ(q1;t1) =
Θ 1 y12/δ2 (t1 t0)2/ǫ2 hasthe shapeofanellipsein
− − −
the same coordinates. This observation suggests that f-
(cid:0) (cid:1)
dimensional quantum dynamics in the presence of time-
dependentdiffractingobstaclesmaybeusedformodeling
diffraction in (f +1)-dimensional systems with station-
ary, time-independent obstacles.

IV. CONCLUSIONS
In this paper we have revisited the phenomenon of
diffraction in non-relativistic quantum mechanics. More
specifically,wehaveconsideredtheproblemofaquantum
particlepassingthroughanopening(orasetofopenings)
in a perfectly absorbing screen. Having assumed the
−10 −9 −8 −7 −6 −5 −4 −3 −2
validity of the Huygens-Fresnel principle, we have con-
FIG. 3: (Color online) Diffraction on a flat absorbing screen structed an expansion of the quantum propagator that
with a time-dependent aperture. Part (a) of the figure il- connectstwospatialpointslyingonthe oppositesidesof
lustrates the set-up. Parameters of the initial wave packet the screen. The expansionrepresentsthe full propagator
are the same as in Fig. 2: Q = (0,0), P = (200,0), and asasumofproductsoftwofreeparticlepropagators,one
σ1 =σ2 =0.1. The propagation time is t=0.05. The plane connectingthe initialpointandthe otherconnectingthe
of the absorbing screen is defined by s(x1,y1)=x1−d with final point to a point in the opening. The construction
d = 8. The (light blue) rectangle, covering 8.9 < x < 11.1
holds in the cases of curved (convex or concave) screens
and −1.6 < y < 1.6, outlines the spatial region, in which
and for openings whose shape changes in time. Conse-
the probability density is computed. Parts (b), (c), and (d)
of the figure show ln|Ψ(x,y;t)|2 for χ(q1;t1) equal, respec- quently,ourapproachallowsonetoanalyzethequantum
phenomenaofdiffractioninspaceanddiffractionintime,
tively, to Θ(1−|y1|/δ), Θ(1−|y1|/δ)Θ(1−|t1−t0|/ǫ), and
(cid:0)1−y12/δ2 −(t1 −t0)2/ǫ2(cid:1) with δ = 0.05, t0 = 0.04, and as well as the interplay between the two.
ǫ = 5×10−4. All quantities are given in the atomic units, In order to illustrate the method, we have used our
m=~=1. propagatorexpansiontocalculatediffractionpatternsfor
an initially localized wave packet passing through vari-
ousspatio-temporaldiffractionscreensinoneandtwodi-
trated by a (light blue) rectangle. Figure 3b shows the mensions. Thus,intwodimensions,wehaveinvestigated
diffraction pattern produced by a slit in the absorbing the effect of screen curvature on the diffraction pattern
screenand correspondsto χ(q1;t1)=Θ(1 y1 /δ) with by analyzing diffraction in time produced by absorbing
−| |
δ = 0.05. Figure 3c represents a scenario, in which the parabolicshuttersanddemonstratinghowaconvex(con-
slit of spatial half-width δ is open only during a time in- cave) shutter gives rise to effective focusing (defocusing)
terval of half-width ǫ = 5 10−4 centered around time of the wavefunction. We havealso studied diffractionof
×
t0 = 0.04. The corresponding characteristic function is spatially localized wave packets passing throughholes of
χ(q1;t1) = Θ(1 y1 /δ)Θ(1 t1 t0 /ǫ). Figure 3d time-dependent size. We have shown that the shape of
− | | −| − |
shows the diffraction pattern obtained for the character- a diffracted wave function is largely determined by the
istic function χ(q1;t1) = Θ 1 y12/δ2 (t1 t0)2/ǫ2 . space-time geometry of the hole, suggesting the use of
− − −
In this case, the width of the slit continuously increases time-dependent obstacles in f-dimensional systems for
(cid:0) (cid:1)
from 0 to 2δ during the time interval t0 ǫ < t1 < t0, modeling quantum diffraction on stationary obstacles in
−
and then shrinks back to 0 during t0 <t1 <t0+ǫ. (f +1)-dimensional systems.
It is interesting to observe some similarity of, on The approach, developed in this paper, is applicable
one hand, the diffraction patterns of Figs. 3c and 3d, to quantum diffraction on perfectly absorbing screens,
and, on the other hand, patterns of Fraunhofer diffrac- and, therefore, complements the method of Brukner and
tion of light on, respectively, rectangular and elliptic Zeilinger [15] valid for perfectly reflecting screens. In
apertures in opaque two-dimensional screens (e.g., see laboratory experiments, however, diffraction screens are
Figs. 8.10 and 8.12 in Ref. [6]). The similarity becomes typicallyneitherperfectlyabsorbingnorperfectlyreflect-
even stronger when one takes into account the space- ing,andthetwomethodshavetobeusedincombination.
time, (y1,t1) representation of the apertures leading to It would interesting to develop such a unified approach
the diffraction patterns shown in Figs. 3c and 3d. In- andtotestitspredictionsagainstresultsofexperimental
deed, the aperture defined by the characteristic function measurements.
10
Acknowledgments where W stands for the Whittaker function. The latter
can be written as (see page 341 in Ref. [25], or section
The author would like to thank Joshua Bodyfelt, 13.18(ii) of Ref. [26])
Andre Eckardt, Orestis Georgiou, Klaus Hornberger,
RAokliarandShKuedtoz,mMeraicrkti,nKSliaeubser,RaicnhdterS,teRveonmaTnomSscohvuicbefrotr, W−14,41(ζ2)=√πz exp(ζ2/2)erfc(ζ) (A6)
helpful discussions and comments at various stages of
this project.
with “erfc” denoting the complementary error function.
Substituting Eqs. (A5) and (A6) into (A4) we obtain
Appendix A: Derivation of Eqs. (10) and (11)
m m
In one dimension the free-particle Green function, = erfc (x x′) . (A7)
I 2i~ 2i~t −
G0(l;E) with l >0, is given by (cid:18)r (cid:19)
∞
G0(l;E)= [K0](l;s)= dte−stK0(l;t) A comparison of the right-hand sides of Eqs. (A2) and
L
Z0 (A7) completes the proof of the identity (9) with the
m ∞ dt ml2 function u chosen in accordance with Eq. (11).
= exp st
r2πi~Z0 √t (cid:18)−2i~t − (cid:19) In order to prove that Eq. (10) offers an alternative
m 2ms choice for the function u we consider
= exp l . (A1)
r2i~s − r i~ !
∂ ∂
′
Here, s = E/(i~) is implied, cf. Eq. (5), and the reader I ≡ η1∂x +η2∂x′ I, (A8)
isreferredtotheformula3.471.15ofRef.[24]forthelast (cid:18) (cid:19)
integral.
We now consider the integral where is given by Eq. (A2), and η1 and η2 are two
I ′
arbitrary, generally complex numbers. On one hand,
t I
′ canbeevaluatedbyadirectsubstitutionofEq.(A2)into
I ≡ dt1K0(x−x1;t−t1)K0(x1−x;t1), (A2) Eq. (A8) which leads to
Z0
′
where x, x, and x1 satisfy Eq. (7). The Laplace trans-
form of is given by ′ im t x x1 x1 x′
I[ ]= [K0](x x1;s) [K0](x1 x′;s) I = ~ Z0 dt1(cid:18)η1 t−−t1 −η2 t−1 (cid:19) ′
LI L − L − K0(x x1;t t1)K0(x1 x;t1). (A9)
× − − −
m 2ms
′
= exp (x x)
2i~s − − r i~ ! On the other hand, using Eqs. (A8), (A3), and (A1), we
m have
′
= 2i~sL[K0](x−x;s). (A3)
r
∂ ∂
′ ′
Then,usingthefactthat1/√s=L[1/√πt]andperform- L[I ]= η1∂x +η2∂x′ L[I](x−x;s)
ing the inverse Laplace transform, we obtain (cid:18) (cid:19)
m ∂ ∂
′
m t dt1 ′ = 2i~s η1∂x +η2∂x′ L[K0](x−x;s)
I =rm2πi~tZ0 √dtt−1t1 K0(x−xm;t(1x) x′)2 =rim~ (η1−(cid:18)η2)L[K0](x−x(cid:19)′;s), (A10)
= exp − . (A4)
2πi~Z0 t1(t−t1) (cid:18)− 2i~t1 (cid:19)
and therefore
The integral inpthe last line of Eq. (A4) is given by the
formula 3.471.2 of Ref. [24],
im
′ ′
t dt1 β I = ~ (η1−η2)K0(x−x;t). (A11)
exp
Z0 t1(t−t1) (cid:18)−t1(cid:19)
1 Finally,comparingEqs.(A9)and(A11),andintroducing
p t 4 β β
=√π β exp −2t W−14,41 t , (A5) a complex number η = η1/(η1 − η2), we arrive at the
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) identity (9) with the function u given by Eq. (10).