Drake G.W.F. (editor) Handbook of Atomic, Molecular, and Optical Physics

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De Broglie Optics 77.3 Principles of de Broglie Optics 1129

with matrix-valued vector potential

A(x) = i

†

(x)

[

∇S(x)

]

. (77.30)

This matrix is not diagonal; its off-diagonal elements

describe nonadiabatic transitions between the dressed

states. If the detuning is sufﬁciently large, and the atom

moves sufﬁciently slowly, these nonadiabatic transitions

may be neglected in a ﬁrst approximation. In such an

approximation, which is akin to the Born–Oppenheimer

approximation in molecular physics, the dynamics of

the atom is described by two decoupled Hamiltonians of

the generic form (77.3), with scalar potentials U given

U(x) =±

Ω(x)

(77.31)

and vector potentials A(x) given by the diagonal elem-

ents of (77.30). The vector potential is usually neglected.

If included, it describes the Berry phase of the mechan-

ical effects of the radiation interaction of a two-level

atom.

The idea behind the adiabatic approximation is that

the internal state of the atom, which is described by

any of the locally varying dressed states, has enough

time to adjust smoothly to the motion of the atom. For

the important case of strong detuning |∆||Ω

(x)|,

this assumption is usually well justiﬁed. In this case, an

atom in the dressed ground state experiences a potential

which is approximately given by

U(x) =−

|Ω

(x)|

4∆

(77.32)

For red detuning we have ∆>0, in which case the

atom is attracted towards regions of high intensity (high

ﬁeld seeker). For blue detuning, the atom is repelled

from such regions (low ﬁeld seeker). A potential sim-

ilar to (77.32) also applies for complex particles like

molecules whose transitions are far detuned from the

light frequency. The potential is proportional to the

dynamic polarizability α(ω) and the ﬁeld intensity.

Atom Optical Nonlinearity

In very dense atom ensembles at low temperatures, col-

lisions between particles can be described by a contact

interaction whose strength depends, in the simplest case,

on a single parameter, the s-wave scattering length a

(Sect. 75.5.1). In the mean ﬁeld approximation, these

interactions translate into a density-dependent potential

U(x) =

4π

a(N −1)

|ψ(x)|

, (77.33)

where N is the number of particles. This non-

linearity leads to the occurrence of atom soli-

tons [77.13], four-wave mixing [77.11], and parametric

ampliﬁcation [77.12].

77.3 Principles of de Broglie Optics

Since the Schrödinger equation is a linear partial dif-

ferential equation, de Broglie optics shares most of

its principles with principles of other wave phenom-

ena, and in particular with the optical principles of

electromagnetic waves.

77.3.1 Light Optics Analogy

The analogy of de Broglie optics and light optics be-

comes particularly transparent for monoenergetic beams

of scalar particles. Such beams are described by a time

harmonic wave function ψ(x, t) = e

−iEt/

(x),where

(x) obeys the stationary Schrödinger equation

Hψ

= Eψ

. (77.34)

Setting A(x) = 0in(77.3) for simplicity, this equation

assumes the form



∇



1 −

U(x)



(x) = 0 , (77.35)

where the wavenumber

is related to the energy E via

the dispersion relation

E ≡

(77.36)

If U = 0 at the entrance to the interaction region, E is

the kinetic energy of the freely traveling de Broglie wave

and

is the related wavenumber.

Comparing (77.35) with the scalar Helmholtz equa-

tion of electromagnetic theory, and identifying

(x) ≡



1 −U(x)/E



1/2

(77.37)

as an index of refraction for matter waves, one observes

the complete analogy of scalar optics of stationary matter

waves and monochromatic light waves. This analogy can

be generalized for spinor valued wave functions which

would correspond to vector wave optics in anisotropic

index media. However, in contrast to light, spinor-valued

wave functions do not obey a transversality condition.

Part F 77.3

1130 Part F Quantum Optics

In (77.35–77.37), the parameter E describes the kin-

etic energy of the incoming beam. Thus, E is positive,

and therefore

< 1 for positive values of the poten-

tial, while

> 1 for negative values of the potential.

For neutrons, one generally has

< 1. This contrasts

to the index of refraction for light waves which is gener-

ally larger than one. For electrons, ions, and atoms both

< 1and

> 1 may be realized.

77.3.2 WKB Approximation

Waves are described by amplitude and phase. Particles

are described by position and momentum. The link be-

tween these concepts is provided by Hamilton’s ray

optics. For scalar matter waves, a ray follows a clas-

sical trajectory. The optical signature of the ray is

the phase associated with it. The quantum mechan-

ical version of Hamilton’s ray optics is obtained in

the WKB approximation of the stationary Schrödinger

equation (77.34).

Any solution of (77.34) may be written in the form

(x) = A(x)e

iW(x)

(77.38)

with real-valued A(x) and W(x).IntheWKB approxi-

mation A(x) ≈ 1, and

W(x) =



(s)ds+

−1



A(x



) · dx



(77.39)

which is called eikonal in Hamilton’s ray-optics. In

this expression,

(s) ≡

(

x(s)

)

,wherex(s) denotes

the classical trajectory of energy E connecting the

point P

with the point P, x

and x are the coordi-

nates of P

and P, respectively, and ds ≡|dx| is the

element of arc length measured along the classical tra-

jectory. Note that the second contribution in (77.39)is

generally gauge-dependent. However, for closed loops

which are frequently encountered in interferometry, the

gauge-dependence disappears by virtue of Stokes the-

orem which transforms the path integral into an area

integral over the rotor of A.

The eikonal (77.39) may also be written in the form

of a reduced action

W(x) =



p(x



) · dx





k(x



) · dx



, (77.40)

where p(x) is the local value of the canonical momen-

tum of the particle,

k(x) ≡ p(x)/

is the corresponding

wave vector, and the integral is evaluated along the clas-

sical trajectory of the particle. Note that in the presence

of a vector potential, p(x) and dx are no longer parallel

as a result of the difference between canonical momen-

tum p and kinetic momentum M( d/dt) x ≡ p−A.

The WKB approximation becomes invalid in the

vicinity of caustics where neighboring rays intersect.

There, connection formulae are used to ﬁnd the proper

phase factors picked up by the ray in traversing the caus-

tics. Depending on the topology of the intersecting rays,

different classes of diffraction integrals provide uniform

approximations for the wave amplitude near caustics.

For further details see [77.14] and [77.15].

77.3.3 Phase and Group Velocity

The velocity of a particle which traverses a region of

negative potential increases so that p(x)> p

,andthe

phase advances: δW =



[

p(x) − p

]

· dx > 0. In quan-

tum mechanics, the classical velocity corresponds to the

group velocity, while the evolution of the phase is de-

termined by the phase velocity. The phase and group

velocities of de Broglie waves are given by

(x) ≡

p(x)

(x)



(77.41)

(x) ≡

∂E

∂p(x)

(x)



(77.42)

respectively. Note that the product v

= E/M is inde-

pendent of

(x).

77.3.4 Paraxial Approximation

The paraxial approximation is useful in describing the

evolution of wave-like properties and/or distortion of

wavefronts in the immediate neighborhood of an optical

ray.

Let the z-axis be the central optical axis of symmetry

along which the optical elements are aligned. Using the

Ansatz ψ

(x) = e

φ(x, y;z), and dropping ∂



∂z

in a slowly varying envelope approximation, one obtains

∂

∂z

φ(x, y;z)



−

∇

⊥

+U(x, y;z)



φ(x, y;z), (77.43)

where ∇

⊥

= ∂



∂x

+∂



∂y

,andv



M is

the longitudinal velocity of incoming particles. This

equation has exactly the form of a time-dependent

Part F 77.3

De Broglie Optics 77.4 Refraction and Reﬂection 1131

Schrödinger equation in two dimensions, with z/v

play-

ing the role of a ﬁctitious time t. With this interpretation,

the spatial evolution of phase fronts along z can be an-

alyzed in dynamical terms of particles moving in the

xy-plane.

77.3.5 Raman–Nath Approximation

In the Raman–Nath approximation (RNA) (also called

the short-time, thin-hologram, or thin-lens approxima-

tion), the ∇

⊥

term in (77.43) is neglected. The potential

U(x, y;z) then acts as a pure phase structure, and the

solution of (77.43) becomes

φ(x, y;z) =

exp





−





U(x, y;z



)





φ(x, y;z

(77.44)

In terms of a classical particle moving under the in-

ﬂuence of U, the approximation loses validity for

⊥

 U, which is just a quarter cycle for a harmonic

oscillator.

77.4 Refraction and Reﬂection

Consider a particle beam of energy E incident on

a medium with constant index of refraction

.The

boundary plane at z = 0inFig.77.1 divides the vacuum

from the medium. At the boundary, the beam is par-

tially reﬂected and partially transmitted, with the angles

determined by Snell’s law of refraction

sin α =

sin β, (77.45)

and the law of reﬂection

α = α



. (77.46)

The coefﬁcients of reﬂectivity

R, and transmittivity

T = 1 −

R are given by the Fresnel formula

R =



cos α −

cos β

cos α +

cos β



. (77.47)

For

> 1, the interface is “attractive” and

R  1,

with

R → 1 only for glancing incidence α →π/2. For

< 1, the interface is “repulsive” and total reﬂection

α⬘

U ⬎ 0

Fig. 77.1 Reﬂection geometry



R = 1



occurs for α ≥

,where

= sin

−1

is the

critical angle. For α>

, the de Broglie wave becomes

evanescent, with ψ

(z > 0) ∼ e

−

κz

inside the medium

(z > 0), where

κ =



sin

α −sin



1/2

. (77.48)

For thermal neutrons, π/2 −

∼ 3×10

−3

radians.

If E < U,then

is imaginary and total reﬂec-

tion occurs for all α. In neutron optics, this total mirror

reﬂection requires ultracold neutrons (T ≈ 0.5mK). It

has important applications for storage of ultracold neu-

trons in material cavities, and neutron microscopy using

spherical mirrors. For details see [77.2].

77.4.1 Atomic Mirrors

Inelastic processes, such as diffuse scattering and ab-

sorption, inhibit coherent reﬂection of atoms from bare

surfaces. The surface must therefore be coated either

with material of low adsorptivity (noble gas, see [77.16])

or electromagnetic ﬁelds (evanescent light or magnetic

ﬁelds, see below).

Reﬂection of Atoms by Evanescent Laser Light

Evanescent light ﬁelds are produced by total inter-

nal reﬂection of a light beam at a dielectric–vacuum

interface [77.17]. In the vacuum, the ﬁeld decays ex-

ponentially away from the interface on a characteristic

length κ

−1

where

κ = k



sin

−1



1/2

. (77.49)

Here, n is the light index of refraction of the dielectric,

k is the wave number of the light beam in vacuo, and θ

is its angle of incidence.

Part F 77.4

1132 Part F Quantum Optics

If the light is blue-detuned from the atomic res-

onance, an incident beam of ground state atoms

experiences the repulsive potential

U(x) =





Ω

−2κ|z|

+∆



1/2

−|∆|



(77.50)

For α>

, the evanescent ﬁeld acts as a nearly

perfect mirror, the imperfections being due to nonadi-

abatic transitions into the “wrong” dressed state, and

possible spontaneous emission. Reﬂection of atoms

by evanescent laser light was demonstrated by Ba-

lykin et al. [77.18] at grazing incidence and by Kasevich

et al. [77.19] at normal incidence.

Reﬂection of Atoms by Magnetic Near Fields

Magnetic near ﬁelds are produced above substrates with

a spatially modulated permanent magnetization or close

to arrays of stationary currents. In the vacuum above

the substrate, the ﬁeld decays approximately exponen-

tially over a length comparable to the scale of the

magnetic modulation. The motion of atoms that cross

such inhomogeneous magnetic ﬁelds sufﬁciently slowly

is governed by the analog of the adiabatic potential

described in Sect. 77.2.5:

U(x) =−µ







B(x)



, (77.51)

where µ is the magnetic moment and m

is the (con-

served) projection of the atomic spin s onto the local

magnetic ﬁeld direction. A repulsive mirror potential is

achieved for spin states with µm

< 0; these weak ﬁeld

seekers are repelled from the strong ﬁelds close to the

substrate.

Experiments have used magnetic recording media

like magnetic tapes or hard disks [77.20], arrays of

current-carrying wires [77.21], or amorphous magnetic

substrates [77.22].

77.4.2 Atomic Cavities

Atomic reﬂections are used in the two kinds of cavities

proposed so far: the trampoline cavity and the Fabry–

Perot resonator.

The trampoline cavity, also called the gravito-optical

cavity, consists of a single evanescent mirror facing up-

wards, the second mirror being provided by gravitation.

A stable cavity is realized with the evanescent laser ﬁeld

of a parabolically shaped dielectric–vacuum interface,

see [77.23] and the experiment by Aminoff et al. [77.24].

A cavity with transverse conﬁnement provided by a hol-

low blue-detuned laser beam has been demonstrated by

Hammes et al. [77.25].

In the atomic Fabry–Perot resonator both mirrors are

realized by laser light [77.26].

77.4.3 Atomic Lenses

De Broglie waves may be focused by refraction from

a parabolic potential or by diffraction, e.g., by a Fres-

nel zone plate (Sect. 77.5.2). Consider focusing by the

parabolic potential

U(x, y;z) =



Mω

),−w ≤ z ≤ 0

0 , otherwise .



(77.52)

For ground state atoms, such a potential is realized in

the vicinity of the node of a blue detuned standing wave

laser ﬁeld of transverse width w. In this case

= Ω



rec

/|∆|



1/2

, (77.53)

where ω

rec

= k



2M is the recoil frequency.

Comparison with the Raman–Nath approxima-

tion (77.44)atz =0, with the phase fronts of a spherical

wave converging towards a point x

= (0, 0, f ),shows

that U describes a lens of focal length

f =

wω

. (77.54)

The Raman–Nath approximation is only valid for a thin

lens w  f , and breaks down for w>w

=πv

/2ω

In the latter case, oscillations of the particles in the

harmonic potential become relevant, a phenomenon

sometimes called channeling. Channeling may be used

to realize thick lenses with focal length f = w

cor-

responding to a quarter oscillation period.

Focusing of a metastable helium beam using the

anti-node of a large period standing wave laser ﬁeld

has been demonstrated [77.27]. Such a ﬁeld is produced

by reﬂecting a laser beam from a mirror under glancing

incidence. The standing wave forms normal to the mirror

surface. Similar interference patterns provide arrays of

thick lenses that have been exploited in atom lithography

to focus an atomic beam onto a substrate [77.28].

77.4.4 Atomic Waveguides

Atomic waveguides can be realized with potentials that

conﬁne atoms in one or two dimensions [77.29–31].

These devices are key elements for integrated atom

Part F 77.4

De Broglie Optics 77.5 Diffraction 1133

optics, a ﬁeld that has seen a rapid evolution re-

cently [77.32, 33]. A planar waveguide is provided by

the one-dimensional conﬁnement in an optical standing

wave [77.34] or an atomic mirror combined with grav-

ity [77.31]. The discrete nature of the waveguide modes

in that case could be demonstrated by lowering the am-

plitude of the conﬁning potential. Linear waveguides can

be modeled by the parabolic transverse potential (77.52)

that now extends along the waveguide axis (the z-

axis). Physical realizations include hollow, blue-detuned

laser beams [77.30], hollow ﬁbers whose inner wall is

coated with blue-detuned evanescent light [77.35], elon-

gated foci of red-detuned light created by cylindrical

lenses [77.36], and magnetic ﬁeld minima along current-

carrying wires, possibly combined with homogeneous

bias ﬁelds [77.37]. With typical thermal atomic ensem-

bles, these waveguides operate in a multimode regime,

and coherent operation has been demonstrated only with

Bose–Einstein condensates. A strong transverse con-

ﬁnement that facilitates monomode operation, can be

achieved with miniaturized wire networks deposited on

a solid substrate [77.38, 39]. This approach may lead

to the fabrication of atom chips [77.33]. Even mul-

timode waveguides, however, can yield robust atomic

interferometers, as suggested theoretically in [77.40]

and [77.41].

77.5 Diffraction

The diffraction of matter waves is described by the so-

lution of the Schrödinger equation (77.34) subject to

the boundary conditions imposed by the diffracting ob-

ject. For a plane screen Σ made of opaque portions

and apertures, the solution in the source-free region be-

hind the screen is given by the Rayleigh–Sommerfeld

formulation of the Huygens principle

(x) =



dξ dη

2π



1 +



n· R

ψ(ξ) ,

(77.55)

where ξ = (ξ,η,ζ) denotes coordinates of points on Σ,

n is an inwardly directed normal to Σ at a point ξ,and

R= x−ξ.

A diffraction pattern only becomes manifest in the

diffraction limit r  d,wherer is the distance to the

observation point, and dis the length scale of the diffract-

ing system. The two diffraction regimes are then the

Fraunhofer limit r  d

/λ

and the Fresnel regime

r ≈ d

/λ

, also called near-ﬁeld optics.

77.5.1 Fraunhofer Diffraction

In the Fraunhofer limit, the ﬁeld at position (x, y) on

a screen at a distance L downstream from the diffracting

object is given by

ψ(x, y; L) =



dξ dη

2π

×e

−i(

ξ+

η)

ψ(ξ, η;0), (77.56)

where

x /L,

y/L. The ﬁeld at the obser-

vation screen is thus given by the Fourier transform of

the ﬁeld in the object plane; i. e. the momentum repre-

sentation of the diffracted state. Since most diffraction

experiments in atom optics are performed in the Fraun-

hofer limit, most calculations are done in the momentum

representation.

Atomic diffractions from microfabricated transmis-

sion gratings [77.42] and double slits [77.43] have been

observed. Recent experiments have extended de Broglie

wave diffraction to heavier, complex particles like fuller-

ence molecules (C

) [77.44].

77.5.2 Fresnel Diffraction

Typical applications of Fresnel diffraction are Fresnel

zone plates and the effects of Talbot and Lau. Fresnel

zone plates are microfabricated concentric amplitude

structures which act like lenses. They are frequently

employed in optics of α-particles and neutrons. In atom

optics, focusing with a Fresnel zone plate was ﬁrst

demonstrated by Carnal et al. [77.45].

The Talbot effect and the related Lau effect refer

to the self-imaging of a grating of period d,which

appears downstream at distances that are integral multi-

ples of the Talbot length L = 2d

/λ

. For a discrete

set of smaller distances than the Talbot length, im-

ages of the grating appear with smaller periods d/n,

n = 2, 3 ···. For applications in matter wave interfer-

ometry, [77.46]; for applications in atom lithography

see [77.47].

77.5.3 Near-Resonant Kapitza–Dirac Effect

The near resonant Kapitza–Dirac effect refers to the

diffraction of two-level atoms from a standing wave laser

Part F 77.5

1134 Part F Quantum Optics

ﬁeld with a spatially uniform polarization. The dynam-

ics of the effect is described by the Hamiltonian (77.19)

with the mode function of the laser ﬁeld given by (77.16).

Consider atoms traveling predominantly in the z-

direction, i. e., orthogonal to the axis of the laser ﬁeld,

with energy

 Ω

(Fig. 77.2). Kapitza–Dirac

diffraction is then observed in transmission, which in

the theory of diffraction is called Laue geometry. In the

paraxial approximation (Sect. 77.3.4) for motion in the

z-direction, and assuming that the laser proﬁle is homo-

geneous in the y-direction, the description reduces to

an effectively one-dimensional model for the quantum

mechanical motion along the x-axis of the laser ﬁeld.

Due to the periodicity of the standing wave light

ﬁeld, the transverse momenta of the transmitted atom

waves differ by multiples of

k from the transverse

momentum of the undiffracted wave. For the important

case of strong detuning, and assuming that the incoming

atoms are in their electronic ground state moving with

transverse momentum p

=0, the p

-distribution of the

outgoing wave is given in the Raman–Nath approxima-

tion (Sect. 77.3.5)by

Prob(p

= 2n k) ∝





Ω

8|∆|





, (77.57)

where J

is a Bessel function of order n, τ = w/v

is an effective interaction time, w being the width of

the laser ﬁeld, and v

the longitudinal velocity of the

atoms. The distribution (77.57) was observed by Gould

et al. [77.48].

For τ

 τ

,whereτ



rec

Ω

/|∆|



−1/2

,the

Raman–Nath approximation becomes invalid. As a re-

sult of Doppler related phase-mismatch, the momentum

spread saturates and shows a sequence of collapse and

revival as a function of τ [77.49].

If the detuning is too small to allow for a scalar de-

scription, the two-level character of the atoms must be

taken into account. For the particular case of ∆ = 0,

the ground state evolves into an equal superposition

of the two diabatic states 1



√



|e±|g



while enter-

ing the interaction region. Inside the interaction region,

these states experience potentials which differ only

by their sign. For atomic beams with a small spatial

spread δx  2π/k, the diabatic states experience oppo-

site forces, leading to a splitting of the atomic beam

called the atom-optical Stern–Gerlach effect [77.50].

In the general case of arbitrary ∆, the Kapitza–

Dirac Hamiltonian (77.21) is most conveniently

analyzed using band theoretical methods of solid state

theory [77.51].

Fig. 77.2 Geometry of the Kapitza–Dirac effect

77.5.4 Atom Beam Splitters

Beam splitters are optical devices which divide an in-

coming beam into two outgoing beams traveling in

different directions. For thermal neutrons, beam split-

ters may be realized by diffraction from perfect crystals

in Laue geometry. For atoms, they can be realized using

diffraction from crystalline surfaces, microfabricated

structures (see Sect. 77.5.1), or by using diffraction from

an optical standing wave.

The Kapitza–Dirac effect, for example, may be ex-

ploited to split an atomic beam coherently using Bragg

reﬂection at the “lattice planes” provided by the pe-

riodic intensity variations of a standing wave laser

ﬁeld [77.52]. This process is resonant for an incom-

ing atomic beam traveling with transverse momentum

= k because it is energetically degenerate with

the diffracted beam traveling with transverse momen-

tum

=− k. This level degeneracy is lifted while

the atoms enter the interaction region. In the Bragg

regime, the lifting happens slowly enough that only

the momentum states |±

k  participate in the diffrac-

tion (two-beam resonance), and their populations show

Pendellösung type oscillations as a function of the tran-

sit time. The frequency of the oscillations is given

by δE/

,whereδE is the energy splitting of the

two beams inside the interaction region. For a transit

time giv en by a quarter period of the Pendellösung,

a 50% beam splitting is observed [77.52]. In princi-

ple, Bragg resonances may also be realized for higher

diffraction order p

= n k ↔

=−n k.However,

in this case, intermediate momentum states become

populated (multibeam resonance), which makes the

higher-order Bragg resonances less suitable for beam

splitting purposes.

Part F 77.5

De Broglie Optics 77.6 Interference 1135

More promising for the realization of an atomic

beam splitter is the magneto-optical diffraction which

refers to the diffraction of three-level atoms from a laser

ﬁeld with a periodic polarization gradient (lin ⊥ lin

conﬁguration) (Chapt. 75), and a magnetic ﬁeld aligned

parallel to the optical axis of the laser ﬁeld. This

conﬁguration realizes an interaction potential in the

form of a blazed grating, i. e., a phase grating with

an approximately triangular variation of phase. In an

experiment by Pfau et al. [77.53], transverse split-

ting of a beam of metastable helium by 42

k was

observed [77.49].

Diffraction from an evanescent standing wave

involves Bragg reﬂection of atoms under glancing

incidence from the periodic grating of a blue de-

tuned evanescent standing wave laser ﬁeld [77.54–

56]. Diffraction at normal incidence has been

demonstrated with sufﬁciently slow atoms and can

be described by a generalization of the RNA

(Sect. 77.3.5) [77.57, 58].

77.6 Interference

While the overall phase of a wave function ψ is not ob-

servable, interferometry makes detectable the relative

phases of two components ψ

, ψ

in a superposition

ψ = ψ

+ψ

. Two types of interferometers are most

common: the Young double slit as a paradigm for in-

terferometers based on division of wavefront, and the

Mach–Zehnder interferometer as a paradigm for inter-

ferometers based on division of amplitude. In de Broglie

optics, the latter type is realized in the form of a three-

grating interferometer, since division of amplitude is

achieved by diffraction at gratings rather than by semi-

transparent mirrors. Experiments with this geometry

have been reported for atoms [77.59] and recently for

more massive, complex molecules (fullerenes) [77.60].

77.6.1 Interference Phase Shift

From a fundamental point of view, any interferometer

is a ring. At the entrance port of a three-grating inter-

ferometer displayed in Fig. 77.3, for example, the wave

function is split into two coherent parts which spatially

evolve along different paths and subsequently come to-

gether at the exit port where they are superimposed to

produce two outgoing waves

√

(

±ψ

)

(77.58)

where the components from path 1 and 2 are given by

= A

exp

(

)

= A

exp

(

)

(77.59)

For simplicity, assume A

= A



√

2. The rel-

ative ﬂux of the outgoing waves is then



1 ±cos χ



(77.60)

where χ =W

−W

is the relative phase of the two com-

ponents. The sinusoidal variations of I

with varying χ

are called interference fringes, and χ/2π is called the

fringe order number.

In the WKB approximation, the phases W

and W

are given by

= W



k(x) · dx, i = 1, 2 ,

(77.61)

where the integrals extend over the classical paths 1 and

2, respectively, and dx is an element of displacement

along the paths. Using (77.61), the relative phase is

χ =



k(x) · dx ,

(77.62)

where C is the closed interferometer loop. Note, that on

path 2, the path element dx and

k are antiparallel.

Usually, the absolute value of χ is not measured, but

only variations, called phase shifts, which result from

displacements of the diffraction gratings or placement

of optical elements into the beam path. Phase shifts

–

Fig. 77.3 Geometry of the three-grating interferometer

Part F 77.6

1136 Part F Quantum Optics

come in two categories: dispersive and geometric. If

a phase shift χ depends on v

, it is called dispersive,

otherwise it is called geometric. Geometric phases de-

pend only on the geometry of the interferometer loop.

The Sagnac effect (see below), for example, may be

geometric. A phase which depends neither on v

nor

on the geometry of the interferometer loop is called

topological. The Aharonov–Bohm effect (see below) is

topological.

Using (77.39), χ becomes

χ = χ

+χ{U}+χ{A}, (77.63)

where for weak potentials U and A,

(

−L

)

(77.64)

χ{U}=−



U[x(s)]ds ,

(77.65)

χ{A}=



A(x) · dx ,

(77.66)

and L

is the geometric length of the path i. For a constant

potential U

intersecting the interferometer on a length

w, χ{U} is given by

χ{U}=−

. (77.67)

Using Stokes theorem, χ{A} may be written in the

manifestly gauge-invariant form

χ{A}=



[

∇× A(x)

]

· da ,

(77.68)

where the integral extends over the area enclosed by the

interferometer, and da is an inﬁnitesimal area element.

Dispersive Phase Shifts. Atom interferometers have

been able to measure phase shifts of the form (77.65) due

to, for example, the atomic level shift in an electric ﬁeld

(Stark effect) [77.59] or to coherent forward scattering

by background gas atoms, see (77.10) and [77.61].

Sagnac Effect. The Sagnac effect refers to χ{A}in a ro-

tating interferometer. Inserting (77.6)into(77.68), and

assuming that the axis of rotation is oriented perpendic-

ular to the plane of the interferometer, the Sagnac phase

shift is given by

= 4MΩA/ , (77.69)

where A is the geometric area enclosed by the inter-

ferometer loop. χ

may be dispersive or geometric

depending on the type of interferometer. In a Young

double slit, A is independent of energy and χ

is ge-

ometric. In a three-grating interferometer, the area is

A ≈ ϑD

,whereϑ is the splitting angle; see Fig. 77.3.

In this case, χ

is dispersive because of the velocity

dependence of ϑ ≈ 2

k/Mv

. The Sagnac effect for de

Broglie waves was ﬁrst observed by Werner et al. [77.62]

using a neutron interferometer.

Aharonov–Bohm Effect. The Aharonov–Bohm effect

refers to the χ{A} of charged particles encircling

a magnetic ﬂux line [77.63]. Inserting A from (77.7)

into (77.68), and assuming particles of charge q encir-

cling a line of ﬂux Φ once, one ﬁnds

= qΦ/ . (77.70)

A characteristic feature of the Aharonov–Bohm effect is

that the particles actually never “see” the magnetic ﬁeld

of the ﬂux line which is conﬁned to some region inac-

cessible to the particles. χ

is strictly topological, and

only depends on the linking number of the interferome-

ter loop and the ﬂux line. Its appearance is characteristic

for all gauge theories. For further details and a summary

on its experimental veriﬁcation see [77.5].

Aharonov–Casher Effect. The Aharonov–Casher ef-

fect refers to χ{A} of a magnetic spin encircling an

electric line charge [77.64]. Inserting A from (77.11)

into (77.68), one obtains for proper alignment of µ

and E

= 2π

|µ|

(77.71)

where r

is the classical electron radius, µ

is the

Bohr magneton, and ξ = e/ρ

, ρ

being the electric

line charge density. χ

is topological only if the

spin is aligned parallel to the electric line charge and

both are oriented perpendicular to the plane of the

interferometer. χ

for atoms has been observed by

Sangster et al. [77.65].

Electric Dipole Phase. Electric dipole phase refers to the

χ{A} of an electric dipole moment encircling a mag-

netic line charge [77.66]. Inserting A from (77.14)

into (77.68), one obtains for proper alignment of d and

= 2π

|d|

(77.72)

where a

is the Bohr radius, and ξ = Φ

/

, Φ

be-

ing the magnetic ﬂux unit, and 

being the magnetic

line charge density. In analogy to the Aharonov–Casher

Part F 77.6

De Broglie Optics 77.7 Coherence of Scalar Matter Waves 1137

effect, χ

is topological, provided that d is aligned par-

allel to the magnetic line charge, and both are oriented

perpendicular to the interferometer plane.

77.6.2 Internal State Interferometry

Manipulation of the internal state of atoms by means

of electromagnetic ﬁelds makes it possible to realize in-

terferometric setups which involve separation of paths

in internal space rather than in real space. Examples

of such interferometers are the Optical Ramsey inter-

ferometer [77.67], the stimulated Raman interferometer

[77.68], and the interferometers using static electric and

magnetic ﬁelds [77.69, 70].

77.6.3 Manipulation of Cavity Fields

by Atom Interferometry

The entanglement of atomic states and quantized ﬁeld

states opens novel possibilities for manipulating and/or

measuring nonclassical ﬁeld states in a cavity. In the

adiabatic limit, for example, and assuming sufﬁcient

detuning between the atom and the cavity ﬁeld, the in-

teraction and c.m. motion of an atom traversing a cavity

is well described by the potential (77.32)

U(x) =−

∆

f(x)

†

a , (77.73)

where g is the vacuum Rabi frequency, f(x) is a cavity

mode function, and a, a

†

denote cavity photon annihila-

tion and creation operators, respectively.

Because of the presence of the photon-number op-

erator a

†

a in (77.73), the deﬂection and phase shift of

an atom traversing the cavity is quantized, displaying

essentially the photon number statistics in the cavity.

The quantized deﬂection is sometimes called the inverse

Stern–Gerlach effect.

Due to the entanglement of atom and cavity states,

and the position dependence of the interaction strength,

the phase shift induced by U(x) in a standing wave

cavity may be used to measure either the atomic position

via homodyne detection of the cavity ﬁeld [77.71, 72],

or the photon statistics via atom interferometry [77.73,

74]. In a ring cavity, the entanglement of c.m. motion

and cavity ﬁeld may be used to measure the atomic

momentum [77.75] via homodyne detection of the cavity

ﬁeld. For further details see [77.76]andChapt.78.

77.7 Coherence of Scalar Matter Waves

The general solution of the free Schrödinger equa-

tion (77.2) may be written in the form

ψ(x, t) =



ka(

k)e

k·x−ω(

k)t)

, (77.74)

where ω(

k) ≡ E/ =

/2M. If the coefﬁcients a(

are known, the state represented by ψ(x, t) is called

a pure state. Otherwise it is called a mixed state, and

physical quantities are obtained by an ensemble average

over the possible realizations of a(

k).

The degree of coherence of matter waves is described

by the autocorrelation function of Ψ(x, t):

Γ(x, t; x



, t



) ≡ Ψ(x



, t



)

∗

Ψ(x, t), (77.75)

where the overline (···) denotes the ensemble average

over the possible realizations of a(

k). In light optics,

Γ(x, t; x



, t



) is called the mutual coherence function.

In particular, for equal times, Γ(x, t;x



, t) describes the

spatial coherence, and for equal positions, Γ(x, t; x, t



)

describes the temporal coherence.

For a beam of particles, coherence may be either

longitudinal (measured along the beam) or transverse

(measured across the beam). In contrast to light optics,

there is no simple relation between longitudinal coher-

ence and temporal coherence because the dispersion

relation of matter waves is quadratic in the wavenumber.

The spatial coherence function is intimately related

to the quantum mechanical density operator of the par-

ticles (Chapt. 7)

ρ(t) =

|Ψ(t)Ψ(t)| . (77.76)

In the position representation, one has

x|ρ(t)|x



≡ρ(x, x



;t) = Γ(x, t;x



, t). (77.77)

Longitudinal and temporal coherence of a particle

beam is determined mainly by the source of the beam.

The thermal ﬁssion reactors used in neutron optics and

the ovens used in atom optics are analogous to black-

body sources in light optics. In contrast, the transverse

coherence is mainly determined by the way the particles

are extracted from the oven to form a beam.

77.7.1 Atomic Sources

To describe thermal sources, consider a single particle

in an ove n of temperature T and volume V, assuming

Part F 77.7

1138 Part F Quantum Optics

that a(

k) and a(



) are statistically independent:

∗



) = ρ(

k )δ(

k−



), (77.78)

where

ρ(

k ) =

exp



−

2Mk



(77.79)

accounts for the thermal distribution of wavenumbers,

and



2π



1/2

(77.80)

denotes the thermal de Broglie wavelength.

Using (77.78)–(77.80)in(77.75), the mutual coher-

ence function becomes

Γ(x, t; x



, t



) =

{

1 +

[

i(t −t



)/τ

]

}

×exp



−π



x−x





(

1 +

[

i(t −t



)/τ

]

)



(77.81)

where

(77.82)

is the thermal coherence time.

According to (77.81), the spatial coherence of a ther-

mal state falls off in a Gaussian manner on a scale given

by λ

. The temporal coherence, in contrast, falls off al-

gebraically on a time scale given by τ

. Expressed in

physical units, one has

1.74( 5) ×10

−9

√

(M/u)(T/K)

7.63 × 10

−12

(T/K)

(77.83)

where u is the atomic mass unit.

Atomic Beams

Effusive Beams. Effusive beams are produced from ther-

mal sources by a suitable set of collimators placed

in front of the opening of the oven. This produces

a Maxwell–Boltzmann distribution of atomic velocities

in the longitudinal direction. The coherence properties in

the transverse direction are described by the van Cittert–

Zernike theorem [77.77]; for details see any textbook on

classical optics.

Supersonic Beams. Supersonic beams are produced by

supersonic expansion of a high pressure gas which is

forced through an appropriately designed nozzle. The

expansion produces a velocity distribution in the longi-

tudinal direction which is approximately Gaussian with

a velocity ratio v/δv ≈ 10–20.

Pulsed Beams. Pulsed beams are produced by chopping

any of the beams described above. Important applica-

tions for pulsed beams are the resolution of temporal

coherence and the mapping of the relative phases of

the a(

k) in matter wave interferometry.

Laser-like Source of Atoms. In these sources, many

atoms with integral spin (Bosons) occupy one and the

same quantum state of motion [77.78]. Their operational

principle is rooted in the quantum statistical effects of in-

distinguishability. It may be viewed in close analogy to

the operational principle of an ordinary laser (Chapt. 70)

and the mechanism underlying Bose–Einstein Conden-

sation (Sect. 76.1.1). Laser-like sources have indeed

been achieved by letting a small current of atoms leak

out of a trapped Bose–Einstein condensate [77.79].

77.7.2 Atom Decoherence

In any interferometer, the contrast of the interference

fringes quantitatively measures the coherence of the

wave involved. Partially coherent beams show an output

ﬂux given by



1 ±Re



iχ



(77.84)

instead of (77.60), with a complex number C. One has

|C|≤1, with the maximum achieved for a pure state; the

phase of C is measured by scanning the interferometer

phase shift χ.

In de Broglie interferometry, coherence can be lost

when the interfering matter wave gets entangled with

other systems. This happens for atoms, for example,

due to the emission or scattering of photons, as soon

as the detection of these photons permits, in prin-

ciple, the resolution of spatially separated paths in

the interferometer. In fact, the width of Γ(x, t;x



, t)

as a function of x−x



, also called the spatial co-

herence length, is reduced to the photon wavelength

after a single scattering event, see [77.80, 81]. Inter-

ference can be restored when the emitted photons are

detected and correlated with the atom output [77.82,83].

Collisions with background gas atoms between the op-

tical elements of a three-grating interferometer also

Part F 77.7