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2013 Rep. Prog. Phys. 76 086402
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IOP PUBLISHING
REPORTS ON PROGRESS IN PHYSICS
Rep. Prog. Phys. 76 (2013) 086402 (28pp)
doi:10.1088/0034-4885/76/8/086402
Experimental methods of molecular
matter-wave optics
Thomas Juffmann1,2 , Hendrik Ulbricht3 and Markus Arndt1
1
2
3
Faculty of Physics, University of Vienna, VCQ, QuNaBioS, Boltzmanngasse 5, A-1090 Vienna, Austria
Department of Physics, Stanford University, Stanford, CA 94305, USA
School of Physics and Astronomy, University of Southampton, Highfield Southampton, SO17 1BJ, UK
E-mail: [email protected]
Received 12 July 2010, in final form 3 June 2013
Published 2 August 2013
Online at stacks.iop.org/RoPP/76/086402
Abstract
We describe the state of the art in preparing, manipulating and detecting coherent molecular matter. We focus
on experimental methods for handling the quantum motion of compound systems from diatomic molecules to
clusters or biomolecules.
Molecular quantum optics offers many challenges and innovative prospects: already the combination of two
atoms into one molecule takes several well-established methods from atomic physics, such as for instance laser
cooling, to their limits. The enormous internal complexity that arises when hundreds or thousands of atoms are
bound in a single organic molecule, cluster or nanocrystal provides a richness that can only be tackled by
combining methods from atomic physics, chemistry, cluster physics, nanotechnology and the life sciences.
We review various molecular beam sources and their suitability for matter-wave experiments. We discuss
numerous molecular detection schemes and give an overview over diffraction and interference experiments that
have already been performed with molecules or clusters.
Applications of de Broglie studies with composite systems range from fundamental tests of physics up to
quantum-enhanced metrology in physical chemistry, biophysics and the surface sciences.
Nanoparticle quantum optics is a growing field, which will intrigue researchers still for many years to
come. This review can, therefore, only be a snapshot of a very dynamical process.
(Some figures may appear in colour only in the online journal)
Contents
1. Introduction: Why matter-wave optics with
clusters and molecules?
2. A brief history of matter-wave physics with
molecules, clusters and nanoparticles
3. Molecular beam sources and coherence
3.1. Molecular beam sources for quantum optics
experiments
3.2. Preparation of coherence: transverse and
longitudinal selection and cooling
4. Molecular beam detectors
4.1. Ionizing detectors
4.2. Surface-based detection schemes
5. Quantum diffraction of molecules
5.1. Molecular speckle patterns: diffraction at
arbitrarily shaped pinholes
5.2. Poisson’s spot: diffraction at an opaque obstacle
5.3. Diffraction of fast molecules at crystal surfaces
5.4. Reflective quantum diffraction at
micromechanical gratings
0034-4885/13/086402+28$88.00
5.5. Transmission line gratings
5.6. Optical manipulation of molecular matter waves
6. Advanced molecule interferometry with multiple
optical elements
6.1. Ramsey–Bordé interferometry measures
molecular spectra and transition moments
6.2. Mach–Zehnder experiments
6.3. Near-field (Talbot–Lau) interferometry
6.4. TLI for molecule metrology
7. Prospects of molecular and nanoparticle
interferometry
7.1. Decoherence, phase averaging and
fundamental limitations to nanoparticle
interferometry
8. Conclusion
Acknowledgments
References
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© 2013 IOP Publishing Ltd
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Printed in the UK & the USA
Rep. Prog. Phys. 76 (2013) 086402
T Juffmann et al
Table of symbols
b
C4
d
d
|ep0 + k
Ei
E
F
G1 , G2 ,
G3
|g, p0 h
Jn (ϕ)
k
kB
lc
L
LT
m
n
ne
N
po
pin
p
PL
r
T
Ttrans
Trot
Tvib
v
v0
vlong
v
vtrans
VCP
wx
wy
x
xc
x
x3
Xc
α
αopt
βn
ηion
ηse
λdB
λdB
λL
ψ
σion (E)
τ
grating thickness
Casimir–Polder interaction constant
grating period
variation of grating period
two-level atom in its excited state, after
photon absorption
ionization energy
electric field
force
gratings 1, 2 and 3
phase shift of the matter wave
de Broglie wavelength
spread of de Broglie wavelength distribution
wavelength of light
wave function
molecular ionization cross section
transit time of particle through laser waist
Table of acronyms, alphabetically sorted
2PI
AFM
BEC
EMCCD
two-level atom in its ground state moving
with momentum p0
Planck’s constant
h/2π
Bessel function of nth order with modulation
index ϕ
photon wavenumber, 2π/λ
Boltzmann’s constant
longitudinal (spectral) coherent length
length
Talbot length
mass
index, n ∈ N
electron number density
number of grating openings
linear forward-directed momentum of a
particle
initial momentum of a particle
momentum change
laser power
interaction distance molecule/grating wall
temperature
translational temperature
rotational temperature
vibrational temperature
modulus of the velocity v = |
v|
most probable velocity
longitudinal velocity
spread of longitudinal velocity
transverse velocity
Casimir–Polder interaction potential
waist of laser beam in the x-direction
waist of laser beam in the y-direction
position
Casimir cut-off distance
source width
increment of position of G3
transverse (spatial) coherence width
static (frequency-independent) polarizability
optical (frequency-dependent, dynamic)
polarizability
diffraction angle of nth diffraction order
ionization efficiency
secondary electron yield
phase of the matter wave
ESI
FWHM
IOP
KDTLI
LD
LIAD
LIFT
MALDI
MCP
MUPI
MZI
NSOM
OTIMA
PSF
QMS
R2PI
REMPI
SPI
SQUID
SSPD
STM
TLI
TOF-MS or TOF
UV/VUV
two-photon ionization
atomic force microscope
Bose–Einstein condensate
electron multiplying charge coupled
device
electrospray ionization
full-width at half-maximum
Institute of Physics, UK
Kapitza–Dirac–Talbot–Lau
interferometer/interferometry
laser desorption
laser-induced acoustic desorption
laser-induced forward transfer
matrix-assisted laser desorption
ionization
multi-channel plate
multi-photon ionization
Mach–Zehnder interferometer
near-field scanning optical microscope
optical time-domain ionizing
matter-wave interferometer
point spread function
quadrupole mass
spectrometer/spectroscopy
resonant two-photon ionization
resonantly enhanced multi-photon
ionization
single-photon ionization
superconducting quantum interference
device
originally: superconducting
single-photon detectors;
now and here also: superconducting
single-particle detectors
scanning tunnelling microscope
Talbot–Lau
interferometer/interferometry
time-of-flight-mass spectroscopy
ultraviolet (10–400 nm)/vacuum
ultraviolet (10–200 nm)
1. Introduction: Why matter-wave optics with
clusters and molecules?
The physics of matter waves has intrigued a large scientific
community over the past few decades [1–4]. It is both
at the core of many non-intuitive but fundamental quantum
2
Rep. Prog. Phys. 76 (2013) 086402
T Juffmann et al
phenomena and it has become a central element of modern
devices, which use tiny shifts of the quantum waves to measure
material structures [6, 7], properties [8–10] or external forces
[11]. In this report, we focus specifically on the coherent optics
of objects in the range between dimers and nanoparticles.
Coherence may exist both in the internal and translational
degrees of freedom. Internally, it may be preserved in the
dynamics of electronic, vibrational, rotational and excitonic
states [12–14]. In contrast to that, our present review
discusses the external translational (motional) evolution, that
is pure de Broglie wave mechanics, which may be enriched
by the particles’ size, temperature and internal complexity:
internal state properties can eventually modify the motional
dynamics in the presence of external fields [15] or in scattering
processes [16].
We aim to understand the evolution of many-body
systems, which may comprise hundreds of atoms, and we
are interested in effects that can be associated with the de
Broglie wavelength λdB = h/mv, where m and v = |
v | are the
mass and centre-of-mass velocity of the composite particle,
respectively, and h is Planck’s constant. The following key
questions are in the current focus of research:
If even very massive objects are allowed to propagate
in compliance with the predictions of quantum wave
dynamics, why don’t we observe quantum delocalization
in our macroscopic world? Will the quantum-classical
‘boundary’ simply shift with future advances of experimental
refinements or is it reasonable to expect a fundamental limit,
for instance imposed by a mass-scaling non-linearity in the
underlying quantum equations [17–19]? How well can we
control decoherence [20–22] to advance experimental quantum
physics? Can we test new physics by exploring ‘quantum
macroscopicity’ [23–28]? Can gravity play a role at some
point? Such questions are also relevant in the context of
counter-propagating ring currents in superconducting quantum
devices (SQUIDs [29]), in the coherent splitting of atomic
Bose–Einstein condensates (BECs) [30] or in the superposition
of micromechanical oscillator states [31]. Here we study the
experimental basis for pursuing them in de Broglie experiments
[32, 33], where we ask whether nanoparticle interference may
allow us to search for modifications of established quantum
mechanics [17, 34, 35].
Most of them rely on a non-linear extension of the
Schrödinger equation and their spirit is always very similar:
develop a master equation for the time evolution of the system’s
density matrix ρ, which will no longer be described by the
conservative von Neumann term [Hρ]/i alone. Instead, it
is supposed to be enriched by a Lindblad superoperator L:
∂ρ/∂t = [H, ρ]/i + Lρ, where H is the Hamiltonian of the
quantum system and the Lindblad operator may encompass
standard decoherence [21], a spontaneous collapse of the wave
function [17, 19], interactions with the gravitational field [36]
or modifications of space–time [18, 37]. The principal goal of
many interference experiments with massive particles is then
to explore and quantify the relevance of this extension.
The devices that are being built to test such fundamental
questions are also relevant for practical applications, such as
quantum-assisted metrology [8, 15, 38]. Many matter-wave
interferometers generate a free-flying periodic particle density
pattern, which can be used as a ruler to measure tiny forces
or molecular properties through interference fringe shifts in
the presence of external fields [9, 10, 15, 39, 40]. Forces on
the level of y N are easily measurable when molecules interact
with external electric, magnetic or optical fields through their
polarizability, magnetic susceptibility, absorption cross section
or other internal properties.
The key idea behind these experiments is very analogous
to that of atom interferometry: if we can divide a particle’s
wave function to travel along at least two spatially distinct
paths and if we expose their distinct amplitudes to different
force fields they will pick up a phase difference, which finally
shows up as a modulation of the particle density and count rate
behind the interferometer. In practice, the accumulated phase
commonly grows with the enclosed interferometer area, both
in space or in a space–time picture, and it often also depends on
the mass of the particle. Since the interferometer area usually
grows with decreasing forward velocity, many experiments
depend on the availability of slow particle beams. Their generation will be an important aspect of the following discussion.
Finally, we foresee a new avenue in the interferenceassisted preparation and probing of nanostructures [41, 42,
194]. Molecules exhibit tiny de Broglie wavelengths (1–
10 pm) at low kinetic energies (0.1–10 eV). Coherent matterwave optics might thus allow us to deposit molecular
nanostructures onto a surface [42] or to probe surfaces with
molecules.
In section 2, we briefly review the history of molecular
matter-wave experiments. Section 3 outlines methods to
prepare neutral matter beams. We briefly discuss the concept
of coherence and discuss techniques to increase the degree of
coherence of molecular beams and we analyse different particle
detection schemes in section 4. While section 5 is dedicated
to the diffraction of molecules, section 6 discusses molecule
interferometers that combine several optical elements to more
refined instruments linked to the names of Ramsey and
Bordé interferometer, Mach–Zehnder interferometer (MZI) or
variants of Talbot–Lau interferometer. We finally distinguish
molecular phase averaging and decoherence in section 7 and
we give a short outlook in section 8.
In order to increase the legibility of the paper we have
introduced an icon system (see table 1), which represents the
building blocks of each experiment: some experiment can be
conceptually derived from another by exchanging one or a
few components. We also note that throughout the text we
occasionally use the word ‘molecule’ as a placeholder for either
‘cluster’, ‘molecule’ or ‘nanoparticle’ when the particular
character of the internal atomic bonds or arrangement is of
no further relevance.
2. A brief history of matter-wave physics with
molecules, clusters and nanoparticles
The origins of molecule interferometry can be traced back
to as early as the year 1930 when Estermann and Stern
demonstrated coherent scattering of H2 molecules at a clean
LiF and NaCl surface [43] (see figure 1). The de Broglie
3
Rep. Prog. Phys. 76 (2013) 086402
T Juffmann et al
Table 1. Icons to guide the reader through the sections and experimental building blocks of molecular matter-wave interferometry.
Icon
Description
Icon
Description
Thermal sublimation, section 3.1.1
Fluorescence detection in free flight, section 4.1.7
Supersonic beam, section 3.1.2
Scanning tunnelling microscopy section 4.2.1
Laser evaporation, p 6
Fluorescence microscopy, section 4.2.2
Neutralization of molecular ions, p 6
Two-dimensional diffraction mask, section 5.1
Slotted disc velocity selector, section 3.2.1
Diffraction at a disc/sphere, section 5.2
Helical velocity selector, section 3.2.1
Diffraction at a crystal, section 5.3
Gravitational velocity selection, section 3.2.2
Blazed material grating, section 5.4
Electron impact ionization, section 4.1.1
Material grating, section 5.5
Langmuir–Taylor ionization, section 4.1.2
Optical grating, section 5.6
Thermal photoionization, section 4.1.4
Ramsey–Bordé interferometer, section 6.1
Field ionization, section 4.1.5
Mach–Zehnder interferometer, section 6.2
Magnetic sector field, p 8
Three-grating Talbot–Lau interferometer, section 6.3
Two-dimensional multi-channel plate array, p 8
Two-grating Talbot–Lau interferometer, section 6.3.2
wavelength of supersonically expanded hydrogen dimers at
a velocity of v = 2000 m s−1 amounts to λdB = 1 Å. This
is well compatible with the interatomic spacing of a clean
crystal surface and the H2 molecules scatter with an angular
distribution as explained by quantum diffraction.
Coherent surface scattering has recently been revived
with dimers at kinetic energies up to 10 keV and de Broglie
wavelengths down to 100 fm [16, 47].
Even earlier than that, a number of molecule
interferometers have been developed: Already in 1981,
Christian Bordé and colleagues performed optical Ramsey
spectroscopy on SF6 [48] but it took the scientific community
about a decade to realize [49] that the internal state
labelling provided by photon absorption is also accompanied
by coherent momentum transfer. Various diffraction and
interference experiments were performed with nanofabricated
gratings: the weakly bound helium dimer He2 was discovered
because of its diffraction pattern [50], Mach–Zehnder
interferometry allowed us to explore collisional properties of
Na2 [51] and diffraction of internally hot many-body systems
was first realized with the fullerenes C60 and C70 [32, 52].
Dimers formed from Bose–Einstein condensed (BEC) are
the most spatially extended molecules, by far. They may
extend to dozens of micrometres close to their dissociation
threshold [53]. Nowadays, BECs provide the coldest possible
sources for quantum experiments with ground-state dimers
[54] or with ultracold cesium tetramers [55].
A number of de Broglie experiments relied on supersonic
molecular beams: magic clusters were revealed in the
diffraction of helium clusters up to He100 [56, 57] and D2 was
quantum imaged into the shadow behind a microdisc [58], thus
implementing the idea of a Poisson spot with molecules [59].
Macromolecule interferometry has recently been extended
in various directions: diffraction at a grating [41, 60] is
4
Rep. Prog. Phys. 76 (2013) 086402
T Juffmann et al
In order to even advance the field of nanoparticle quantum
optics, a number of challenges have to be overcome: new
sources shall prepare cold and coherent molecular beams.
New diffraction elements must be compatible with the rich
internal structure of complex molecules. Decoherence has to
be avoided and new detectors should combine high efficiency
with position accuracy and selectivity to mass, chemical
composition or internal states. In the following we document
progress along the way and future perspectives.
3. Molecular beam sources and coherence
Figure 1. Experiments by Estermann and Stern in 1930
demonstrated quantum diffraction of helium atoms and molecular
hydrogen at clean LiF and NaCl (shown here) surfaces. Following
the pioneering experiments by Davisson and Germer [44] and
Thomson [45], they were the first to corroborate de Broglie’s
quantum wave hypothesis [46] for composite and significantly more
massive systems than electrons. The line is drawn as a guide to
the eye.
An analogy between electromagnetic and quantum wave
physics can be established based on the fact that the stationary
Schrödinger equation is formally equivalent to a Helmholtz
equation, as it is found in the classical electrodynamics
of scalar fields [77].
Also the concept of coherence
can therefore be transferred to matter waves. We here
divide the source requirements into those referring to the
longitudinal and transverse coherence, respectively. This
division may be questioned for ultracold particles with
an isotropic velocity distribution, but it is justified in de
Broglie experiments with collimated molecular beams, with
vtrans /vlong = 10−3 –10−6 .
The degree of coherence [4, 77] quantifies the correlations
and phase relations in a wave field. Two wavelets may interfere
if they originate from the same cell in phase space and if they
are fundamentally indistinguishable. For massive particles this
includes their internal and external dynamics—from the source
to the detector.
All molecular de Broglie experiments to date have realized
single-particle interference, where partial wave functions of
one and the same particle interfere and all molecules are
prepared in similar—but not necessarily identical—states to
add to the same fringe pattern. Two-particle interference would
require the wave functions of two independent molecules to
overlap in space while the molecules are equal in their atomic,
isotopic and conformational composition. They should occupy
the same electronic, rotational, vibrational and spin states.
At present this can only be done for diatomic ground-state
molecules in BECs. But experiments with these ensembles
have remained in the regime of single-particle interference,
see also [53, 78].
Longitudinal and transverse coherence are measures for
the width of the Fourier transform both of the de Broglie wave
spectrum and of the effective source emission function [79].
Their numerical value varies in the literature by factors of π ,
depending on whether the initial state is better described, for
instance, by a Gaussian wave packet or a plane wave with finite
support [77, 80].
As a rule of thumb, longitudinal coherence can be
estimated by lc λ2dB /λdB ∝ 1/vlong . It measures
the spectral purity of the source. A measure of the lateral
correlations is given by the transverse coherence, which is
estimated by Xc λdB L/x, where L is the distance to the
source and λdB is the spread in the distribution of de Broglie
wavelengths and x is the size of the source.
conceptually close to the textbook example of a double
slit, but it also requires a molecular beam, which is
sufficiently coherent to cover at least two neighbouring
slits by wavelets with a well-defined phase relation. This
triggered efforts in Vienna and Southampton to implement a
multiplexing near-field interferometer, which was first realized
for atoms by Clauser [61, 62] and then implemented with
molecules in Vienna—meanwhile in three different variants,
namely Talbot–Lau interferometry (TLI [63]), Kapitza–Dirac–
Talbot–Lau interferometry (KDTLI [64, 65]) and the optical
time-domain ionizing matter-wave interferometer (OTIMA
[66–68]).
The TLI design requires typically three gratings for
coherence preparation, for diffraction and to scan the
interference fringe pattern. It was used for the first de Broglie
experiments with biomolecules [69], to study collisional [70]
and thermal [22] decoherence and to establish quantumenhanced molecule metrology [71] and lithography [42].
The idea of a KDTL interferometer builds on the
Talbot–Lau concept but the central grating is replaced by an
optical phase mask to eliminate the attractive van der Waals
force that arises during the interaction between matter and
material diffraction gratings. The KDTLI concept enabled
a series of interference experiments with the most massive
particles to date [72, 73] and with new applications in quantumassisted metrology [1, 8, 15].
The OTIMA concept [66, 67] extends this idea by
replacing all mechanical gratings by optical standing light
waves whose photon energy suffices to ionize or ‘deplete’ the
incident molecular beam in the antinodes of the standing wave.
Because of the tiny immaterial grating structure, this concept
is believed to be particularly versatile and useful for future
high-mass interference studies.
Several alternative schemes have been suggested for
coherence experiments with nanoparticles [74–76]. They all
assume the availability of cooling schemes, which would allow
one to trap and confine a particle’s wave function so tightly that
its high momentum uncertainty ensures a fast expansion of the
wave function, once the particle is released.
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Rep. Prog. Phys. 76 (2013) 086402
T Juffmann et al
A rather comprehensive treatment of various theoretical
methods in matter-wave physics is provided by recent reviews
on atom interferometry [2], molecule interferometry [1], as
well as treatments on relativistic phases [81], Feynman path
integrals [82] and a phase-space approach based on Wigner
functions [83].
expansion provides translational cooling and improves the
longitudinal coherence as well as the forward-directed particle
flux. This process affects the temperature of all degrees of
freedom including translation, rotation and vibration with a
temperature ranking Ttrans < Trot < Tvib [89]. The finite energy
level spacing in molecules is the reason why vibrational modes
may be completely frozen in small molecules [85], while in
massive particles cooling to the rotational ground state is a big
challenge. Jet cooling of internal states is only effective for
molecules up to a few dozen atoms. The final temperature
is determined by the molecular heat capacity as well as the
number of thermalizing collisions with the carrier gas.
Supersonic beams have a wide range of applications since
they can be loaded with free gases [43, 49, 50, 58] or particles
that are volatilized in laser desorption (LD) [90–92] or thermal
sources [51].
Adiabatic expansion comes, however, at the price of an
overall acceleration to velocities in the range of 300 m s−1
for cryogenic nozzles and more than 1000 m s−1 for helium
at room temperature [51]. This high velocity can be reduced
in the lab-frame when the nozzle is placed on a backwardrotating wheel [93]. At velocities below 100 m s−1 [94]
small molecules would be compatible with matter-wave
interferometry.
3.1. Molecular beam sources for quantum optics experiments
The goal of this section is to describe methods that aim at
generating neutral molecular beams of sufficient flux and
coherence to be compatible with molecule interferometry.
We will discuss both established techniques and conceivable
alternatives.
3.1.1. Effusive beam sources. Thermal effusive sources have
served as workhorses in various interference experiments with
complex molecules [84, 85]. Stable particles can be heated up
to the point where they sublimate or evaporate in a free effusive
regime, where their velocity distribution is determined by a
Maxwell–Boltzmann law. This is strictly only the case when
the collisional mean free path of the molecules is larger than
the smallest dimension of the source opening [85] and for that
reason, one finds a supersonic contribution in many practical
realizations, also.
The volatilization temperature has to stay low enough to
avoid particle fragmentation. This is a challenge for complex
particles with large polarizabilities and therefore strong van
der Waals interactions to neighbouring molecules—which
increase the enthalpy of evaporation. Based on this insight it
is now possible to alleviate this problem by chemical tailoring
the molecular bonds, appropriately [84]. The attachment of
perfluoroalkyl chains to organic molecules by Mayor and coworkers in Basel has proven to be an important and very
successful method that paved the path to the realization of
quantum interference of particles in excess of 10 000 amu
[72, 73, 84, 86].
The
√ most probable velocity in a thermal beam is
v = 2kB T /m. This corresponds to a de Broglie wavelength
of λdB 5 × 10−12 m in the case of C60 at 900 K
[32]. Current state-of-the-art matter-wave interferometers
can cope with λdB 200 fm [72, 87], corresponding to a
particle of m = 105 amu thermalized at room temperature or
m 106 amu at 10 K.
The potentially detrimental effect of heating on the
integrity of fragile molecules is a particular problem for
biomolecules. The thermal load may, however, be reduced
by limiting the heating power to a local spot, from which
the molecules are rapidly laser evaporated. This method
induces substantially less thermal damage than a Knudsen
cell and it proved useful in far-field diffraction of tailor-made
phthalocyanine derivatives [41]. A laser beam can be focused
to less than a micrometre, which provides the transverse
coherence that is required for molecule interferometry.
3.1.3. Future alternatives. Beams of large neutral clusters
may be formed in aggregation sources, based on magnetron
sputtering [95], laser ablation or thermal evaporation [96].
They generate neutral or singly charged particles of either sign
and are prime candidates for quantum interference experiments
with metal or semiconductor clusters. Aggregation sources
may produce beams of internally cold particles but still require
further slowing or cooling for high-mass interferometry.
Charged particles may be acceptable precursors for
molecular interference experiments. Charge may provide a
handle for beam preparation and cooling. It seems, however,
advised to add a neutralization scheme and to perform all
quantum delocalization studies with neutral particles to avoid
decoherence and phase averaging in stray electromagnetic
fields. Matrix-assisted laser desorption ionization [91, 97]
(MALDI) and electrospray ionization [92] (ESI) could serve
as promising beam sources for such experiments. This is
particularly true for thermolabile biomolecules of almost any
size [98].
In MALDI, molecules are initially isolated in and
desorbed from an organic matrix. The matrix molecules
act both as absorbers for the desorbing laser light pulse
and as collision partners in the following expansion into the
vacuum. They are responsible for cooling as well as for
charge transfer with the analyte particles. The result is a
mixed molecular beam composed of matrix molecules and
isolated analyte particles travelling at 300–800 m s−1 . The
method produces singly charged molecules and a substantial
fraction of neutrals [99]. MALDI has been shown to volatilize
objects with masses in excess of 1010 amu [99, 100], even
functional biomaterials, which may become relevant for farfuture quantum experiments.
3.1.2. Supersonic sources. Molecule interferometry has also
been performed with particles entrained in a supersonically
expanding carrier gas [43, 49, 50, 58, 87, 88]. The adiabatic
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Rep. Prog. Phys. 76 (2013) 086402
T Juffmann et al
A number of matrix-free optical methods are conceivable
and currently being explored in several laboratories. This
includes laser-induced acoustic desorption, LIAD [101], laserinduced forward transfer, LIFT [102], or matrix-free LD [103].
They are all compatible with a high-vacuum environment.
In contrast to that, ESI experiments start at atmospheric
pressure and require the subsequent transfer of the isolated
molecules through differential pumping stages into the ultrahigh-vacuum science chamber. The molecules are first
dissolved in a carrier liquid and filled into a thin capillary. High
voltage (1–2 kV) between the capillary and the vacuum inlet
leads to the formation of a liquid cone which fractions when
the electrical forces overcome the surface tension (Coulomb
limit). Release of small droplets and a repetitive cycle of
solvent evaporation and charge instabilities finally lead to the
isolation of individual solvent-free molecules. This method
is again particularly useful for labile biomolecules. However,
since electrosprays are intrinsically based on Coulomb fission,
even a single protein of m 105 amu ends up loaded with
dozens of elementary charges. Charge reduction, efficient
transfer into the vacuum chamber and slowing of the beam
remain the key challenges for quantum experiments based on
this source [66].
many slits into this disc. The resulting convolution of
molecular packages travelling at different velocities and
partially overtaking each other can complicate the velocity
analysis. The signal can, however, be deconvoluted a
posteriori if a known pseudo-random sequence is modulated
onto the molecular beam [107]. Many different velocity
classes can then be used in a single experiment.
The particle velocity can also be selected by a rotating
turbine with helical grooves [42, 108]. Depending on the
groove geometry and the rotational speed, only molecules
within a certain velocity band can pass. This method has
been widely used in neutron interferometry, but the vibrational
noise of a mechanical device spinning at 100 Hz may also
scramble quantum fringes: while far-field diffraction is rather
inert against such perturbations [32, 51, 52] because the fringe
spacing exceeds the grating period by orders of magnitude,
vibration amplitudes as tiny as 10 nm may be detrimental for
near-field interferometry [109]. It is, therefore, often advised
to choose passive selection methods.
3.2.2. Vibration-free velocity selection by Earth’s gravity. In
the early days of molecular beam physics, Otto Stern selected
the molecular beam velocity by rotating the entire vacuum
apparatus [110]. Nowadays, it appears rather favourable to
employ a static scheme that exploits the free fall in the Earth’s
gravitational field [111]. Three horizontal slits placed between
the source and the detector can define a free-flight parabola
and select molecules that travel at a desired velocity [41, 111].
The selectivity of this scheme depends on the height and
separation of all slits. It improves with decreasing velocity and
enables the selection of velocity bands with a typical width of
v/v ∼ 5 − 30%. In practice, the first and the third slit can be
realized by the source and the detection aperture, respectively.
3.2. Preparation of coherence: transverse and longitudinal
selection and cooling
With no molecular analogues to atom lasers at hand [104, 105]
and transverse laser cooling still being demanding [86], the
preparation of transverse coherence in molecular beams often
relies on restricting the size of the effective source. This is
either done by reducing the width of the emitter or by placing
mechanical collimators to define the beam.
The longitudinal coherence length lc ∝ λ2dB /|λdB | =
λdB · vlong /|vlong | increases with a narrowing of the velocity
spread. A typical thermal beam source emits particles with
a velocity distribution whose full-width at half-maximum
vlong (FWHM) reaches up to 60% of the forward velocity
v0 [32, 52]. Since the de Broglie wavelength depends on
the particle velocity the diffraction fringe pattern itself can
be used to determine or select a certain velocity, if the mass
is already known [41, 106]. More commonly, the timeof-flight (TOF) between two well-defined points provides
velocity information. If the particle beam emerges from a
pulsed adiabatic expansion [68] a time-resolving detector is
the appropriate instrument.
3.2.3. Slowing and cooling. Slowing and cooling of
complex molecules is still a grand challenge, but a number
of sophisticated techniques have been demonstrated in recent
years: laser cooling, which is tremendously successful for
atoms [112], has recently been extended to diatomic molecules,
such as SrF [113] and current efforts explore the possibility of
going even further. It has also been shown that cold molecules
could be generated by joining cold atoms [114, 115].
Since these techniques require cycling optical transitions
and the exchange of many photons it is difficult to extrapolate
them to compounds with dense and open energy level systems.
A number of research teams have started sympathetic cooling
of molecules co-trapped with laser-cooled atomic ions. This
is promising but in order to trap two species in the same Paul
trap it seems that the atom and the molecule should have the
same mass-to-charge ratio [116–120].
When optical cycling is no option, one may also exploit
selective removal from the ensemble following the idea of
Maxwell’s demon: open and close a door between two
chambers when you ‘know’ a slow particle wants to pass.
An atom or molecule oscillating in any kind of trap reaches
its lowest velocity at its turning point. When it is extracted
from the ensemble at this very point, its motional energy is
3.2.1. Rotating mechanical masks. A continuous molecular
beam can be made compatible with TOF selection by encoding
a time sequence on it. The beam can be chopped using a single
rotating disc with a slit. Two discs are needed to define a travel
time when both the source and the detector are operating in
continuous mode. In practice several, often up to six, discs
are used to suppress higher order velocity bands [85]. The
transmitted velocity range can typically be reduced down to
vlong /v0 = 1%.
When a single disc is used in combination with a timeresolving detector, the signal can be enhanced by etching
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minimized. This trick has been implemented with atoms and it
will be interesting to see how to extend it to larger objects [121].
It has been predicted that laser cooling of complex
dielectric nanoparticles could also be realized using offresonant optical forces in a cavity-assisted Sisyphus cooling
scheme [122, 123]. Although several groups succeeded in
proving the working principle for atoms [124, 125], substantial
cooling of mesoscopic particles is still a challenge [126–128].
Another concept relies on non-radiative but modulated
electric or magnetic fields [129]. Since neutral molecules
couple only weakly to external fields, many interaction
steps are required to achieve sizeable deceleration. Electric
[130–132] and optical Stark deceleration [133], reverse
magnetic coil guns [134, 135] and Zeeman slowers [136] have
been used to manipulate molecules up to about 100 amu.
Extrapolation of these methods to more complex objects is
a subject for future research.
For gaseous polar molecules, Stark selection [137] rather
than deceleration was shown to provide high phase-space
densities and a good starting point for further storage and
advanced laser cooling schemes [138–140]. This technique
works best when it can select from an existing dense cold
and gaseous ensemble—which makes it less compatible with
massive objects such as biomolecules or metal clusters.
Finally, cryogenic buffer gas cooling [141] and the
implantation of molecules into liquid helium droplets
[142, 143] are techniques that allow one to cool both the
internal and the translational degrees of freedom. As many
other schemes, they are still waiting to be integrated into
preparatory stages for future matter-wave experiments with
nanoparticles.
in biomolecular ionization mostly because of competing
processes such as electron recapture, re-neutralization and
fragmentation processes.
This may be circumvented by functionalizing (tagging)
large molecules with smaller ones of low ionization potential
[149] or by forming large clusters from well-ionizing smaller
organic compounds [150]. A general method for post-ionizing
or detecting complex bionanomatter has still to be developed.
This is also the reason why it has been possible to demonstrate
quantum delocalization for functionalized large molecules [72]
or complex clusters of small organic dyes [87] in the mass range
of small proteins, while quantum experiments with genuine
proteins, DNA or RNA are still at the core of an intense
open research programme. In principle, ion detection can
be position sensitive with micrometre resolution, for instance
when using multi-channel plates (MCPs).
4.1.1. Electron impact ionization. When a molecular beam is
bombarded with electrons of energy E the ionization efficiency
ηion ∝ ne ·σ ion (E) · l depends on the electron density ne , the
molecular ionization cross section σion (E) and the interaction
length l. For many molecules σion (E) is maximized for
electron energies between 40 and 100 eV [85, 86]. The
maximal electron density is limited by the energy-dependent
space charge, i.e. by Coulomb repulsion between the electrons.
With ionization cross sections ranging between 10−16 and
10−14 cm2 , a total ionization yield of 10−3 –10−5 can typically
be achieved. For complex molecules, fragmentation is an
important issue, since a large manifold of internal states
enables fast conversion between electronic, vibrational and
rotational energies.
In spite of these difficulties, electron impact was the
method of choice for a large number of quantum diffraction and
interference experiments with diatomic and organic molecules
up to 104 amu [8, 15, 16, 39, 50, 58, 65, 69, 73].
4. Molecular beam detectors
Molecular beam detectors serve various purposes [85, 144].
Apart from measuring molecular flux or density they may
be sensitive to the particle’s mass, internal state, position or
velocity.
4.1.2. Hot wires and Langmuir–Taylor ionization. A neutral
particle may also ionize when it hits a surface whose work
function is higher than its own ionization potential. If the
surface is sufficiently hot, the ionized particle may re-desorb
and be collected by an ion counter [151]. That is the basic
idea of Langmuir–Taylor or hot-wire detectors that have been
widely used in alkali atom interferometers [51, 152].
For certain atom/surface combinations this process is
highly efficient [153]. It is, however, less compatible with
TOF mass spectrometry, since time constants between 10−6
and 10−2 s spoil the mass resolution [85]. On materials with
a low work function and particles with a high electron affinity
the reverse process, electron attachment, can also occur [154].
Some molecule/surface combinations, in particular those
involving molecules, may favour molecular dissociation into
products with lower ionization potentials [144]. In selected
cases, up to the size of insulin, intact surface ionization has
been reported in interaction with hot rhenium oxide, which
exhibits a particularly high work function. Even then it was
necessary to compensate for the missing potential energy by
the kinetic energy of the incident hyperthermal molecular beam
[155]. Since ‘hyperthermal’ is tantamount to saying ‘very fast’
4.1. Ionizing detectors
Since ions can be manipulated in electromagnetic fields
and counted and mass selected with high efficiency many
matter-wave detectors comprise an ionization, mass selection
and ion counting unit. We first focus on a discussion of
different ionization strategies. Mass selection is then typically
implemented through one of several well-established schemes,
for instance using radiofrequency quadrupole guides (QMS),
TOF spectrometers or magnetic sector field mass spectroscopy
(SF-MS) [145].
While post-ionization of metals is generally efficient
even at low energies, the ionization of noble gas clusters or
large organic molecules turns out to be notoriously difficult
[146–148]: the ionization energies of most biomolecules range
between 8 and 12 eV, i.e. a regime that is not accessible to
commercial lasers. Interestingly, even dedicated extreme UV
and x-ray light sources realized by plasma lamps or electron
synchrotrons have not yet produced the expected breakthrough
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this technique is not well adapted to macromolecule quantum
experiments with long de Broglie wavelengths.
Field ionization may also be interesting for studying
Rydberg molecules, since the required electric fields can
be very low for such highly excited molecules [136]. The
combination of molecular Rydberg physics with matter-wave
interferometry is still open for future research.
4.1.3. Single-, two- and multi-photon ionization. The typical
photoionization cross-sections of clusters and molecules up
to 10 000 amu vary between 10−18 and 10−16 cm2 [149],
i.e. about one hundred times smaller than that for electron
impact ionization. In spite of that, photoionization is usually
substantially more efficient. The reduced cross section
can be overcompensated by the available photon flux, and
photoionization is generally softer since it deposits less excess
energy in the ionized particles.
Single-photon ionization (SPI) can be applied to organic
molecules and clusters even above 10 000 amu [103, 150] and
it has also been proposed [66, 67] and implemented [87] as the
basis for a universal measurement-induced diffraction grating
for matter waves.
If the energy of a single photon is not enough to overcome
the ionization potential, a resonant or non-resonant two- (2PI)
or multi-photon (MUPI) process may be used. Resonant MUPI
can succeed if subsequent absorption events occur faster than
the competing internal relaxation processes. In that case, the
absorbed energy just barely exceeds the ionization potential
and MUPI can also be a soft ionization scheme [156–158].
4.1.6. Ion counting techniques. For particles with masses up
to about 104 amu the overall detection efficiency is dominated
by the ionization process. At higher masses the subsequent
secondary electron yield ηSE becomes an additional factor.
Depending on the particle species and the overall energy,
ηSE ∝ v n grows with a power law in the ion velocity v, where
n = 4 is typically valid for velocities up to 104 m s−1 [166].
Efficient detection, therefore, requires acceleration voltages of
several 10 kV for masses beyond 105 amu.
While commercial mass spectrometers are equipped with
secondary electron multipliers to register energetic ions, highmass ions may also be counted using image charge detectors.
Their current sensitivity of 1–10 fA sets a detection limit of
about 104 elementary charges per detector bin and second.
4.1.7.
Optical detection in free flight.
Small and
cold molecules with narrow optical resonances and strong
absorption lines are routinely detected via optical ionization,
absorption or fluorescence in free flight. The access to specific
internal states allows one to couple and even entangle internal
and external degrees of freedom. This has been exploited in
molecular Ramsey Bordé interferometry [48].
Free-flight optical detection is, however, hampered
for complex molecules since they lack closed fluorescent
transitions. Only in selected cases the short light–matter
interaction time will be sufficient to capture sufficient
fluorescence to detect a single particle. Also, the repetitive
deposition of Stokes energy in the particle can lead to its rapid
heating and damage. Such limitations are overcome when the
molecule is bound to a solid substrate. Fluorescence can be
collected over extended times and the substrate stabilizes the
molecule by dissipating the Stokes energy.
4.1.4. Delayed ionization of laser-heated molecules. Multiphoton thermal ionization has been studied for highly stable
molecules, such as the fullerenes C60 and C70 [159, 160].
When they are internally heated to energies between 50
and 200 eV several competing processes may be observed
[161, 162]: the emission of thermal radiation, the emission
of electrons and finally fragmentation.
The rotational and vibrational modes may couple energy
to the electronic states and, given the thermo-statistical nature
of the energy randomization, ionization may be delayed by
many microseconds after the exciting laser pulse.
Delayed ionization was an enabling technique for the
first fullerene diffraction experiments [32] where the total
ionization efficiency was estimated to be above 10% and
the spatial resolution was fixed to a few micrometres, as
determined by the waist of the laser beam [163].
4.2. Surface-based detection schemes
Surface-deposited clusters and molecules may be visualized by
several modern microscopy techniques, ranging from scanning
tunnelling (STM) and atomic microscopy (AFM) to nearfield scanning optical microscopy (NSOM) or fluorescence
microscopy in one of its many modern high-resolution
implementations.
Depending on the method, the molecules can be identified
by their size, shape and tunnelling spectra in tunnelling
microscopy or by their absorption and emission bands in
fluorescence microscopy. For this to be useful, it is important
that the particles of interest are sufficiently distinct from all
other species in the beam and on the substrate. Furthermore, it
is crucial that surface diffusion processes [167] are negligible
to prevent the deposited molecular interferograms from
washing out.
Probing a surface-deposited quantum interferogram
combines high detection efficiency with high spatial resolution.
4.1.5. Field ionization. When a positive electric potential
is applied to a tip as sharp as 10 nm, it is possible to create
a local electric field of up to 1010 V m−1 . A molecule in the
vicinity of the needle will be polarized and attracted to the
tip. The internal states are then shifted to the point that an
electron can tunnel between the molecule and the surface.
The molecular ions can again be counted [164]. Since field
ionization is a soft process it is also applicable to complex
molecules [144]. However, the effective interaction region
around the tip is limited to 200 nm [85], which is both an
asset and a drawback: it can be used to image a molecular
beam distribution with high resolution, but it also takes a long
time to scan a two-dimensional (2D) distribution. When the
interaction zone is reduced to the size of a single atom [165]
the emitter also becomes promising as a coherent source for
ion interferometry [3].
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Figure 3. Single-molecule fluorescence detection of
surface-deposited phthalocyanines. The graph shows the time
evolution of the fluorescence intensity of two molecules attached to
the quartz surface with a separation of about 5 µm. This distance
can be easily resolved, optically. Constant emission followed by an
abrupt disappearance of the entire signal is indicative of true
single-molecule fluorescence. Each molecular peak can be fitted by
an Airy function whose centre can be determined to much better
than Abbé’s resolution limit, here to about 10 nm. Picture
reproduced from [41].
Figure 2. Isolated C60 molecules deposited and immobilized on an
atomically resolved Si (1 1 1) 7 × 7 surface. Even some
intra-molecular features can be identified (see the inset) [42].
Copyright 2013 by the American Physical Society.
Moreover, intensity fluctuations of the molecular beam do
not affect the measurement outcome since the entire fringe
system is recorded simultaneously. A number of systematic
uncertainties are therefore eliminated. This is to be balanced
against the difficulty to monitor grating drifts or interferometer
phase shifts in real time.
absorption and emission bands from their surroundings.
Single-molecule fluorescence imaging was introduced about
20 years ago [174, 175] and it has found numerous applications
in physics, chemistry and biology ever since. Recently, it has
been used to record the build-up of a quantum interference
pattern from individually arriving phthalocyanine molecules
in real time (see figure 3) [41].
Fluorescence microscopy [176, 177] can be directly
interfaced to matter-wave interferometry when the molecular
fringe pattern is deposited on a quartz window that closes the
vacuum chamber. All optics can be mounted in air.
One may object that the spatial resolution of tunnelling
microscopy exceeds that of optical microscopy by two or three
orders of magnitude, because of Abbé’s diffraction limit for
optical imaging devices. In recent years, however, several
methods have been shown to overcome these limits in optical
microscopy [178–183]. They rely on the selective isolation
of molecules by bleaching or on non-linear optical processes
and they might eventually enable the detection of molecular
near-field interference patterns with features much smaller
than the wavelength of light, down to the level of a few
nanometres.
With a fixed optical resolution, the position accuracy in the
measurement of each individual molecule can be enormously
increased when the molecular surface density is low—which
is even desired in single-particle interferometry [184]. If two
neighbouring molecules are further apart than the width of
the point spread function (PSF) of the imaging system, one
can fit the PSF to the detected signal for each molecule to
determine its centre. The ultimate accuracy is determined
by the signal-to-noise ratio in the image. Values as tiny as
1.5 nm [184] and 10 nm [41] have been achieved in recent
experiments.
4.2.1. Scanning tunnelling microscopy. Scanning tunnelling
microscopy is known to excel in imaging single atoms and
molecules [168–171]. Given its high spatial resolution and the
possibility of post-processing surface-deposited interference
patterns, this technique is also suitable for exploring quantumassisted nanolithography.
In molecule interferometry it has been applied as a
fullerene detector, specifically for C60 molecules. Figure 2
shows an atomically resolved, flat silicon 7 × 7 (1 1 1) surface
reconstruction, which provides 19 dangling bonds per unit
cell and therefore plenty of possibilities for each fullerene to
bind in the vicinity of where it lands on the surface, even at
room temperature [172]. The surface physics involved in this
process has been studied in detail by others with an interest
in doped fullerenes as quantum logic elements [173]. The
specific preparation of a reconstructed silicon surface requires
ultra-high vacuum conditions around 10−10 mbar. This is a
challenge in combination with molecular beams that may even
be initially created at the mbar pressure level.
STM imaging offers the highest possible spatial resolution
that is physically meaningful for the purpose of macromolecule
interferometry, but a single image scan may take 45 min for an
area as small as 2 µm2 .
4.2.2. Fluorescence microscopy. Fluorescence microscopy
provides an excellent tool to discriminate molecules of known
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Optical microscopy hinges on molecular parameters as
well as the experimental design. Dye molecules, such as
for instance phthalocyanines and rhodamines are particularly
well suited for that purpose, since they combine high optical
absorption cross-sections and a high fluorescence quantum
yield with long bleaching times. Larger molecules can be
tailor-made and this enabled far-field diffraction with the most
massive molecules to date.
Similarly, almost all proteins and other biomolecules can
be labelled with a wide range of dye markers, if they are not
already fluorescent, such as for instance the green fluorescent
protein GFP [185]. Many other nanoparticles, in particular
quantum dots, also show strong fluorescence.
Recent studies have shown that even non-fluorescent
nanoparticles can be optically detected either by their
stimulated emission [186], their absorption [187–189] or
a change in the refractive index [191].
In order to
reach a sufficient signal-to-noise ratio, these methods
need a tightly focused and scanning probe beam, which
increases the recording time when a wide area needs to be
imaged.
4.2.3. Cryogenic and superconducting bolometers. Neutral
molecules could also be detected by cryogenic bolometers.
While these are not sensitive to the molecular mass, they are
sensitive to energy. This can be used in experiments, where
the final internal energy is correlated with the de Broglie phase
accumulated throughout the interferometer. The detection
principle would be based on the transfer of energy from internal
excitations to the detector. When the detector is cooled to
its superconducting state the transferred energy can trigger
the phase transition to the normal conducting state, which is
detected as an increase in the electronic resistance. Already
in the early days of molecular coherence, with SF6 molecules
in a Ramsey–Bordé interferometer, the output was monitored
by recording the energy increase that resulted from a resonant
laser excitation of a specific vibrational state in one arm of the
interferometer [48]. In future experiments, this idea may be
extended to a large class of molecules when it becomes possible
to refine the detector sensitivity to the level of single particles.
This is expected to be achievable with superconducting singleparticle detectors (SSPD) that have emerged over recent
years [190].
A key challenge here is to keep the detector sufficiently
clean throughout every experiment to avoid energy dissipation
in a cushion of surface adsorbates [192].
Figure 4. Diffraction of H2 at an arbitrarily shaped aperture.
(a) Scanning electron micrograph of the hole, and (c) the observed
interference pattern. (b) Theoretically expected speckle pattern. The
figure is adapted from [193]. Copyright 2006 by the American
Physical Society. The pictograms introduced here on the right-hand
side summarize in a single view the source (top), diffraction method
(middle) and detection mechanisms (lower panel/s).
5.1. Molecular speckle patterns: diffraction at arbitrarily
shaped pinholes
5. Quantum diffraction of molecules
An arbitrarily shaped pinhole is probably the simplest and most
generic of all possible diffractive objects. It has been studied
using a supersonic beam of neutral H2 .
With a bundle of molecular beam generation and detection
methods now at hand, the following section is dedicated
to coherent beam splitting methods that have already been
realized with molecules. More complex compositions of these
matter-wave optical elements to full-fledged interferometers
and their applications will then be discussed in section 6.
For navigation purposes, the figures in this section are
complemented with icons that link to the sections that describe
the relevant experimental techniques.
The Fourier image of a micrometre-sized arbitrarily
shaped mechanical aperture leads to a speckle pattern (see
figure 4) [193], which can be regarded as a first step towards
diffractive coherent imaging of microscopic objects [194].
Field ionization at a tungsten tip served the detection of H2
very well. The limited detection area required scanning of the
tip but 2D field ionization arrays may become part of future
wide-area detectors [164].
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Figure 6. 2D intensity distribution for scattering of H2 at
E = 0.6 keV under grazing incidence at a sulfur superstructure on
Fe(1 1 0). Molecular intensity increases from blue to red. Reprinted
from [201], with permission from IOP Publishing.
Figure 5. Poisson spot seen with D2 diffracted at an opaque circular
disc [58]. This is the molecular analogue to an earlier experiment
that showed diffraction of atoms around a wire [197]. Picture
reproduced from [58]. Copyright 2009 by the American
Physical Society.
Recent studies have extended this theme to fast molecular
beams with energies up to 25 keV and de Broglie wavelengths
down to λdB 100 fm [199, 200]. The molecular beam
was directed at a crystal surface under grazing incidence to
observe quantum diffraction and channelling within the surface
potential. A typical interference pattern is shown in figure 6.
5.2. Poisson’s spot: diffraction at an opaque obstacle
An opaque object of cylindrical symmetry is the inverse
structure to the pinhole discussed before. Wave theory predicts
a bright spot at the centre of the shadow region behind the
object, when it is illuminated coherently. Historically, an
experiment performed by Arago and named after Poisson was
crucial to defend Fresnel’s wave theory of light. Recently, such
an experiment has also been successfully performed with D2
matter waves [58] (see figure 5).
At first sight, the Poisson experiment appears very
promising for probing the wave nature of large molecules as
well: due to the symmetry of the arrangement, alignment
is easy and the Poisson spot is ‘white’, i.e. all wavelengths
interfere constructively on the optical axis behind the obstacle.
Since the bright spot appears at the centre of the classical
shadow region, one might even think that it is backgroundfree.
It is, however, important to see that the van der Waals
force between the molecule and the obstacle also classically
attracts all particles towards the shadow centre [195]. Adding
to that a realistic velocity distribution, finite edge effects, the
finite source size and a limited detector resolution one finds
that the distinction of classical and quantum shadows becomes
increasingly difficult for increasingly massive objects. For
molecules in the size range of fullerenes the experimental
prospects are still promising [196].
5.4. Reflective quantum diffraction at micromechanical
gratings
Molecule diffraction can act as a surface probe as before,
but surface scattering can also reveal specific characteristics
of molecules and textbook examples of quantum physics.
Quantum mechanics teaches us that, contrary to the situation
in classical physics, a particle may be reflected by an attractive
potential well. This has recently been shown for He2 and He3
reflected by a blazed diffraction grating [202]. When these
weakly bound molecules approach the grating under grazing
incidence they approach a van der Waals potential that is four
orders of magnitude bigger than their internal binding energy
(∼10−7 eV). The fact that molecular dimers with a mean bond
length of 5.2 nm are not torn apart by the external potential
shows that quantum reflection must take place several tens of
nanometres above the surface. Under grazing incidence the
molecular wave vector normal to the surface is sufficiently
small to realize diffraction even at a grating with a period as
large as d = 20 µm, which is orders of magnitude wider than
the size of the molecular de Broglie wavelength (see figure 7).
5.5. Transmission line gratings
5.3. Diffraction of fast molecules at crystal surfaces
5.5.1. Manufacturing of transmission gratings. Since
quantum reflection is a peculiarity of small and less-polarizable
particles, transmission line gratings have been regarded as
more generic and essential in a large number of studies with
molecules. SiNx gratings are technologically particularly
interesting since they remain pre-stressed and stretched even
when they are pierced with holes of 50–100 nm width in
periods of 100–300 nm [32, 42, 50, 51]. Far-field diffraction
at a single grating is tolerant to small deviations from
perfect periodicity and the required nanomask can readily
In contrast to the two previous experiments, which
probed microscopic structures in transmission, a number of
experiments focused on probing nanostructures in molecular
reflection. Coherent scattering of molecular beams can thus
be used as an analytical tool to study the potential energy
surface of crystals [198]. These experiments are preferentially
performed with light molecules, such as H2 or D2 , at energies of
about 10–100 meV, corresponding to de Broglie wavelengths
between 50 and 200 pm [16].
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Figure 7. Quantum reflection of He2 and He3 at a blazed diffraction grating under grazing incidence. The angles, at which the nth order
diffraction peaks of He2 and He3 are expected, are indicated by the red and dashed blue lines, respectively. Picture adapted from [202].
Reprinted with permission from AAAS.
be written using focused ion beam machines [41]. The
concatenation of three gratings to a full interferometer requires
additionally extraordinary reproducibility and precision
between independent masks. This is especially important
for wide-beam Talbot–Lau interferometers [51], where the
average periodicity error needs to be smaller than 1 Å [65, 83].
Photolithography can provide large masks with a precision of
d/d 10−5 in the grating period. This feat can actually
even be achieved for gratings with a period down to 100 nm,
using achromatic interference lithography [203].
5.5.2. General aspects of grating diffraction: near-field versus
far-field. Textbooks teach us that far-field interference peaks
appear at sin βn = nλdB /d, where β is the angle to the optical
axis. Near-field physics is usually less well covered in books
but equally important for a number of coherence experiments.
In order to visualize the transition between both regimes
we plot the evolution of a plane wave diffracted at a grating in
figure 8. The vertical axis represents the decadic logarithm
of the scaled distance behind the gating, where L/LT is
measured in units of the Talbot length LT ≡ d2 /λdB . The
horizontal axis is linear in the scaled position x/x̄, where
x̄ = λdB L/d + NλdB LT /6d. In the far-field (L LT ) the
scaled position corresponds to the diffraction angle in units
of β1 , i.e. the angle to the first diffraction maximum. In the
near-field, i.e. for L < LT , x̄ reduces to N d/6L, which is
proportional to the width of the grating.
The scaling in figure 8 allows us to follow the wave
evolution from the emergence of the Talbot carpets in the nearfield (top) [204, 205] to the domain of far-field diffraction with
well-resolved side peaks (bottom).
Figure 8. Evolution of the wave intensity behind a diffraction
grating, from the near-field (top) to the far-field (bottom): The
grating is coherently illuminated by a plane wave. Close to the
grating, in the near-field, shifted self-images of the grating occur at
multiples of the Talbot distance. The logarithmic distance scale
allows us to see the full evolution in one picture. The lateral position
scaling is described in the text.
5.5.3. Applications of transmission line gratings: quantumassisted mass spectrometry. An interesting application of
de Broglie interference in physical chemistry is to prove the
existence of the weakly bound He2 through a mass assignment
of its molecular diffraction peaks [50].
The helium dimer usually escapes mass spectrometry as
it is bound with an energy of about 1.1 mK (100 neV) [206],
which is too small to survive impact ionization with typical
electron energies in the range 60–100 eV. The de Broglie
relation, however, allows one to deduce the mass of a particle
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Figure 10. Coherent diffraction of C60 molecules at a
nanomechanical grating. Experiments with fullerenes C60 and C70
were the first to demonstrate quantum interferometry with thermally
excited complex compounds. In spite of internal temperatures of the
order of 900 K they exhibited high-contrast interference in their
centre-of-mass motion. Data are fitted with a Fraunhofer diffraction
equation taking into account an effective narrowing of the slits due
to van der Waals forces between the molecules and the grating
walls [32, 52]. Reprinted with permission from [52]. Copyright
2003, American Association of Physics Teachers.
Figure 9. The analysis of interference patterns allowed the
detection of weakly bound clusters, up to He100 atoms. The different
lines represent the cluster diffraction curves that appear for different
source pressures P0 . Larger clusters are formed at a higher
stagnation pressure and show up under a smaller diffraction angle.
Reprinted with permission from [57]. Copyright 2006, American
Institute of Physics.
in our daily lives. Many reasons can be invoked for this
phenomenon: the kinematic explanation of non-interference
is inherent to quantum physics and points to the fact that the
de Broglie wavelength of macroscopic objects is just way too
small for us to have hope to ever resolve it. Equally well rooted
in quantum mechanics are the predictions of decoherence
theory [20, 21, 209–212], which points to the fact that quantum
observations are usually made on isolated systems. Once
the individual system is coupled to a complex environment,
quantum information diffuses into a large ensemble and
becomes essentially unobservable in the partial subsystem.
Genuine decoherence is based on quantum entanglement
between systems. Quantum physics itself thus ensures
that certain quantum phenomena become unobservable on a
large scale. Also experiments with molecular matter waves
corroborate the presence and significance of this effect [22, 70].
This also motivates a growing set of quantum interference
experiments with massive and complex molecules, clusters and
nanoparticles [1, 33], which was experimentally initiated with
a demonstration of quantum interference with C60 and C70 [32]
(see figure 10).
When teaching quantum interference to students we
usually discuss the deterministic fringe pattern that is formed
from individually and stochastically arriving quanta on a screen
at some distance behind a diffraction grating. This aspect of
the wave–particle duality has recently been visualized with
molecules in two experiments, where an interference pattern
was first deposited onto a substrate and then imaged using
either fluorescence [41] or tunnelling [42] microscopy. They
both offer single-molecule sensitivity and a spatial resolution
down to the molecular level in two dimensions.
Figure 11 shows the quantum fringe pattern that was
recorded with phthalocyanine molecules after diffraction at a
SiNx mask with a period of 100 nm and a membrane thickness
from its diffraction angle at a given nanostructure when its
velocity is known (figure 9). Supersonic expansion of helium
from a small nozzle leads to a fast and well-collimated beam,
which is sufficiently cold and dense to form helium dimers
and clusters. Transverse coherence can be established by tight
collimation, longitudinal coherence by the intrinsic velocity
compression during the adiabatic expansion of the noble gas
jet. It is interesting to see that diffraction at a mechanical
mask is sufficiently non-invasive to leave the dimers intact
during the scattering process, in spite of the fact that the
kinetic energy exceeds the intra-molecular binding energy
by orders of magnitude. The particles are thus first sorted
by quantum interference and then detected using electron
ionization quadrupole mass spectrometry.
Far-field diffraction was also used to observe the quantum
behaviour of supersonically expanded Na2 [51] and later
also of clusters up to about He100 . This also facilitated, in
particular, the investigation of ‘magic clusters’, i.e. clusters of
particular structural stability [56, 207] and bound HeH2 halo
molecules [208].
5.5.4. Applications of transmission line gratings: establishing
quantum superposition at high mass, temperature and
complexity. All experiments described in this report are
compatible with the rules of quantum mechanics. While this
is a rather satisfying fact it is still in marked discrepancy with
the observation that we do not see quantum superposition
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T Juffmann et al
dispersive. Slow molecules acquire a stronger dephasing than
fast ones. While a correct description of the van der Waals
forces requires a detailed knowledge of the frequency spectrum
both of the molecule and the surface, we estimate the cut-off
distance based on the computationally simpler Casimir–Polder
potential, VCP = −C4 /r 4 . One finds xc = (18C4 b2 /mv 2 )1/6
to be of the order of 20 nm for gratings with a thickness of
b = 100 nm and molecules with a static polarizability of the
order of α 100 Å3 × 4π ε0 , a typical mass of 500–700 amu
and a velocity around 150 m s−1 [83]. The potential VCP
generally scales with the fourth inverse power of the distance
or length parameter r, while the interaction strength is defined
by the constant C4 . More information about the construction
of the interaction constant can be found elsewhere [213].
These effects have been investigated in great detail in
diffraction [214, 215] and Mach–Zehnder interference [216]
of atoms and small molecules and they turned out to reduce
the effective open slit width in the diffraction of fullerenes by
50%, for grating slits with an initial width of 50 nm [32, 52].
This indicates that the Casimir–Polder approximation
overestimates the cut-off xc .
To minimize these effects, recent quantum diffraction
experiments with phthalocyanines were performed with
gratings of only 10 nm thickness. And yet the van der
Waals forces are still substantial, as also seen in figure 11.
Further experiments will, therefore, be needed to explore the
extreme limit of material gratings, which shall finally even
enable quantum diffraction at graphene, i.e. a membrane whose
thickness measures only a single atom.
Figure 11. Single-molecule imaging in real time allows visualizing
the wave–particle duality of fluorescent dyes; here of
phthalocyanine [41]. Each dot represents a single molecule after its
passage through a nanomechanical diffraction grating. The
ensemble shows that the propagation of each particle must be
described by quantum mechanics. The scale bar is 20 µm wide.
5.6. Optical manipulation of molecular matter waves
as small as 10 nm. Due to the gravitational velocity selection
applied in these experiments, slower molecules are deposited
further down on the detection screen. Their longer de Broglie
wavelength implies larger diffraction angles and therefore
larger separations between the interference orders than those
observed for the fast molecules, which arrive further up on the
screen. An area of 400 × 400 µm2 can be imaged within a
few seconds, orders of magnitude faster than using STM or
AFM techniques. A sequence of images recorded at different
deposition times can show how the deterministic fringe pattern
is built-up from individual and randomly arriving molecules.
Optical gratings are very appealing for matter-wave
experiments for various reasons. They can be realized with the
high periodicity that is determined by the quality of modern
lasers and mirrors. They can be timed, their open fraction
can be changed both in situ and in real time, and optical
phase gratings are also perfectly transparent for the particle
beam. Optical absorption gratings can be realized without the
influence of van der Waals dephasing.
5.6.1. Optical absorption and ionization gratings. An
optical absorption grating is based on the spatially periodic
depletion of the molecular beam. It thus represents the idea of
a measurement-induced grating [217] based on the selection
of positions at which the molecules are removed from the
beam—either by extracting them or by pumping them into
non-detectable states [218].
Various molecules undergo SPI at photon energies beyond
7.9 eV, the energy of a VUV photon at 157 nm. One can achieve
high ionization probability at the antinodes of a standing light
wave and full transmission of neutral particles at its nodes.
Since ions can be easily extracted by electric fields an optical
ionization grating is effectively absorptive [66, 68].
Molecular absorption cross-sections between 10−18 and
−16
10 cm2 necessitate the use of pulsed lasers with energies
in the range of 1 mJ or very high-power (cavity-enhanced)
continuous lasers. Pulsed lasers are advantageous as they also
5.5.5. Applications of transmission line gratings: molecular
diffraction assesses van der Waals forces. Material gratings
are often regarded as binary transmission masks, i.e. as purely
absorptive structures. However, due to the van der Waals
interaction of each polarizable molecule with the grating bars
we also have to consider an additional phase contribution
[41, 83]. Quantum physics causes that every polarizable
particle undergoes stochastic charge fluctuations, typically
most pronounced on the femtosecond time scale. They induce
image charges in near-by surfaces such as the side wall of
a diffraction slit. The resulting attraction to the wall has
three consequences: first, it removes particles if they get too
close to the wall within a cut-off distance xc . Second, it
dephases the matter wave, the stronger the closer the molecule
approaches its image charge. Third, this phenomenon is also
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T Juffmann et al
enable the realization of interferometers in the time domain
[67, 68]. In practice, the molecules will not only experience a
spatially modulated transmission but also a position-dependent
phase shift, which is determined by the optical dipole force.
many molecules and many recoils then results in a smearing
of the interference pattern.
Third, it is interesting to realize that absorption from a
coherent laser wave does not induce decoherence, by itself.
Removal of a photon from a coherent beam does not leave any
information behind and all phases are conserved. Pictorially,
the interferogram is only laterally shifted by a fixed amount,
which is related to the recoil of a single photon. Since each
photon is in a superposition of two momentum states—being
incident onto the mirror and being reflected—the absorption
of a single photon is by itself a coherent beam splitting process
with momentum transfer p = ±k (see [220] for singlephoton emission close to a mirror).
The internal heating associated with the absorption and
internal storage of a photon does not deteriorate the de Broglie
interference pattern as long as it does not invoke any photoactivated or thermal emission process, which could release
‘which-path’ information to the environment. Even in spite
of its coherent nature absorption will reduce the interference
contrast, since the diffraction peaks related to the dipole phase
grating are separated by p = 2n · k and will be interlaced
with those related to the single-photon absorption peaks at
p = k or even higher (odd) orders at high laser intensities.
This relates molecular diffraction at optical gratings to studies
of quantum random walks as well [221].
In [60] a green laser beam was reflected from a mirror to
form a standing light wave with a period of 257 nm at which
a coherent beam of fullerenes C60 was successfully diffracted,
as shown in figure 12.
The interaction time of molecules with a cw optical grating
is velocity-dependent (dispersive) and the phase shift reads
∝ αopt E 2 (xyz)/v. Even though the spatial phase
gradient is much weaker than that in the case of van der Waals
interactions it is still necessary to pre-select a velocity class to
maintain a high interference contrast.
The advantage of light gratings is particularly important
in quantum interference experiments with highly polarizable
molecules. Light gratings were therefore also integrated into
KDTL [65, 222] and OTIMA [68] interferometry.
5.6.2. Dipole force and phase gratings. The success of
atom optics throughout the last decades was strongly driven
by methods to manipulate the internal and external degrees
of freedom with tailored laser light [219]. In contrast to
atoms, with resonant absorption cross-sections as high as
λ2dB /2π 10−9 cm2 and strong optical polarizabilities close
to narrow resonance lines, molecular line widths can extend
to dozens of nanometres. Molecular absorption cross sections
therefore typically range between 10−18 and 10−15 cm2 .
Even through there is no substantial resonance enhancement of dipole forces in warm and complex molecules, recent
experiments have successfully utilized the coherent interaction
between a non-resonant standing light field with a molecular
optical polarizability αopt : the optical field E induces an oscillating dipole moment, which interacts again with the optical
field. The resulting force F = −αopt ∇E 2 /2 depends on the
intensity gradient.
In atomic physics this force can be attractive or repulsive,
depending on the relative detuning between the laser frequency
and the internal resonance line. While this choice is still
possible in diatomic molecules, most complex compounds
studied so far were effective high-field seekers, i.e. attracted
towards the field maximum.
When laser light is retro-reflected at a smooth mirror
surface it generates a standing wave with a periodicity of
λL /2 whose periodicity and contrast depends on about half
the coherence length of the incident laser beam. This may
extend to thousands of kilometres for narrow-band continuous
wave (cw) lasers and it is limited to several millimetres for
pulsed excimer lasers, in practice.
When a spatially extended molecular matter wave falls
onto a dipole phase grating, the incident molecules will leave
the interaction zone in a superposition of transverse momenta
with po = pin + p and p = 2n · k, where n ∈ N .
The relative weight of the nth momentum states is determined
by the Bessel function Jn (ϕ), where the modulation index
ϕ ∝ PL τ/wx wy is proportional to the laser powerPL , its beam
waists wx and wy and the molecular transit time τ ∝ 1/vlong .
Since the molecular lines are broad and the interaction
cross-sections are small, intense and well-focused laser beams
are required to induce a significant dipole force. At the
same time, it is important to avoid real absorption processes,
which would be accompanied by three effects that are usually
undesirable.
First, photon absorption can be followed by partial nonradiative internal conversion of its energy into excitations of
vibrational and rotational states. A repeating sequence of many
such processes may eventually heat the molecule to a degree
that it fragments.
Second, photon absorption may be followed by
spontaneous reemission and momentum recoil whose direction
varies stochastically from event to event. Integration over
6. Advanced molecule interferometry with multiple
optical elements
Far-field diffraction of massive matter is probably the
most intuitive and pictorial realization of a quantum wave
phenomenon.
Yet, although it is still conceivable to
extrapolate this method to nanoparticles in the mass range
of 106 amu or more in a fountain configuration [195], more
sophisticated experimental arrangements are desirable for
certain applications, such as molecular precision metrology
and nanoparticle interferometry.
6.1. Ramsey–Bordé interferometry measures molecular
spectra and transition moments
The first molecule interferometer composed of several beam
splitters was built with the purpose of high-resolution
spectroscopy in SF6 [48, 49]. The introduction of separated
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T Juffmann et al
Figure 13. Ramsey–Bordé interferometry with molecules:
Ramsey’s method of separated oscillatory fields was originally
implemented for molecular beams with the goal to achieve a new
level of high-resolution spectroscopy. Already in 1981 this
technique was employed on molecules as complex as SF6 [48]. The
beam splitting was, however, not yet spatially resolved because of
the width of the incident molecular beam and the small diffraction
angles due to the high forward velocity of the molecules. This
method was also used for the analysis of physical properties of
iodine I2 [49] and the potassium dimer K2 [88]. Picture reprinted
from [228] with kind permission from Springer Science + Business
Media.
Figure 12. Diffraction of C60 molecules at an optical phase grating.
A collimated and coarsely velocity-selected (v/v 1/6) fullerene
beam is phase-modulated by the interaction with a standing laser
light wave. The far-field diffraction pattern is described by the
Fourier transform of the optical potential. (a) Free molecular beam,
(b)–(d) far-field diffraction pattern for increasing laser power. In all
cases, the probability for a molecule to absorb a photon from the
standing light wave is less than one. Diffraction at phase gratings
allows the suppression of the forward-directed zeroth-order beam.
Reprinted with permission from [60]. Copyright 2001 by the
American Physical Society.
laser removes this momentum again and takes the molecule
back into the ground state. A similar sequence of two laser
pulses is then repeated to close the molecule interferometer,
i.e. to recombine the two possible beam paths.
Because of the finite lifetime of the excited state only
the black trajectory—with the molecular ground state in
both arms—contributes to the final interference signal. The
same short lifetime is also the reason why the spatial
splitting between the two interferometer arms remains smaller
than the width of the molecular beam and why it took
the community many years to realize that this advanced
spectroscopy scheme can also be seen as a genuine molecular
matter-wave interferometer. Its key is nowadays used in
metrology: applied to atoms with long-lived excited states
this scheme implements the prototype of an optical atom
clock [226]. Recent studies with potassium dimers 39 K2 were
successfully employed to demonstrate the measurement of a
molecular transition dipole moment [227].
oscillatory fields by Ramsey [223] had led to a boost in
spectroscopic resolution and it was therefore translated into
molecular physics. Such experiments start by creating a
superposition of two internal states whose different phase
evolution in free flight is finally converted into a detectable
population difference.
Figure 13 shows how this is
implemented in practice: when a laser beam interacts with an
effective atomic or molecular two-level system, it induces Rabi
oscillations [224], i.e. periodic population oscillations between
the two coupled states. The collimated molecular beam crosses
the first running laser wave with an interaction time and
laser intensity adjusted to realize a ‘π/2-pulse’ yielding a
coherent and equal superposition of both states. In quantum
information language this beam splitting corresponds to a
Hadamard gate [225]. At the same time, this process generates
entanglement as it couples the internal and external molecular
degrees of freedom such that neither of them is known before
a measurement, but both are inseparably coupled. If the
atom stays in the ground state |g, the initial momentum
also remains unchanged. If it gets excited to |e it receives
ψ ∝
a recoil by the momentum of one laser photon k:
iφ
|g, p0 + e |ep0 + k. Stimulated emission by the second
6.2. Mach–Zehnder experiments
One of the first closed interferometer for atoms (dashed line in
figure 14) [229] was also used on molecular sodium (solid
line). It served for the first demonstration of molecular
interference with macroscopically split trajectories (several
micrometres) and allowed the MIT team to measure the
molecular polarizability both in an external electric field as
well as in collision experiments [51]. The thermal particle
beam was seeded into a supersonic source, sent through three
nanomechanical gratings in Mach–Zehnder (MZI) geometry
and detected by a hot-wire detector.
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T Juffmann et al
Figure 14. MZI for Na2 [51]. One of the first interferometers for
sodium atoms (dashed line) [229] was also used on sodium
molecules (solid line) [51]. The two lasers shown in this scheme can
be used to induce absorption, deflection and decoherence.
Copyright 2013 by the American Physical Society.
Figure 15. The concept of molecular matter-wave interferometry in
a TLI near-field configuration [63, 64]. The KDTLI and OTIMA
configurations are derived from this concept. The TLI is transformed
into a KDTLI, when G2 is replaced by an optical phase grating, i.e. a
non-resonant standing light wave interacting with the molecular
optical polarizability. The TLI can be converted into an OTIMA by
replacing every nanomechanical mask G1 –G3 by a pulsed and
absorptive optical standing light wave, as for instance implemented
by three vacuum ultraviolet laser pulses of several nanosecond
duration and with equal time intervals between subsequent pulses.
Copyright 2002 by the American Physical Society.
In an MZI configuration the gratings are positioned in the
molecular far-field where the beam paths become physically
separated. It is important that the initial collimation angle
of the particle beam is smaller than the first-order diffraction
angle at the nanogratings. The accessibility of the individual
interferometer arms to lasers, static electric fields or even
separate gas cells was used for metrology on atoms and
molecules [51].
The key advantage of TLI over MZI is its high
multiplexing capability. Without the need of collimation one
can use broader molecular beams. The signal enhancement of
three to five orders of magnitude was the enabling step for a
multitude of novel experiments. At the same time, the scaling
behaviour of near-field optics is favourable for the very short
de Broglie wavelengths of very massive particles.
The Talbot–Lau setup is based on the idea that coherent
self-imaging may occur when the diffraction orders of
neighbouring slits overlap at the same positions on the
screen. While the diffraction pattern originating from each
individual starting point in G1 is a pure wave phenomenon, the
mutual overlap between neighbouring interference patterns is
a consequence of the experimental geometry.
This resonance condition is fulfilled if the gratings are
positioned at the Talbot distance LT = d 2 /λdB , with d the
grating period and λdB the de Broglie wavelength. It is also
fulfilled at multiples of the Talbot length and it even leads
to (weaker) fringes with a smaller spatial period at rational
fractions of LT [204, 232].
The Talbot criterion implies that for a given experimental
dimension—for instance limited
by the lab size—the grating
√
period only shrinks as d ∝ λdB . Two molecular species that
travel at the same speed and differ in mass by a factor of 100
require only a ratio of 10 in their mask periods. For high-mass
interferometry this is a key√
advantage compared with far-field
interferometry where d ∝ λdB .
Over the years, different requirements led to a number
of variations of this basic layout: first, it turns out that
massive things are usually also highly polarizable. The
dephasing van der Waals force that influenced earlier molecular
far-field experiments is already detrimental for near-field
interferometry with molecules in the range of 10 000 amu and
grating openings of several dozen nanometres [65]. This is
the reason why it proved necessary and successful to replace
the central grating G2 in figure 15 by an optical phase grating.
6.3. Near-field (Talbot–Lau) interferometry
6.3.1. The concept. Mach–Zehnder interferometry with
atoms and molecules was pioneering in matter-wave physics
but also showed the challenges for using rare materials such as
macromolecules. In the absence of coherent sources, far-field
experiments require the preparation of spatial coherence, and
thus in practice tight collimation. This is acceptable for intense
atomic beams but it represents an almost insurmountable
obstacle for macromolecular quantum optics—until new
cooling schemes are developed. In the meantime, molecular
quantum delocalization experiments can also be pursued in a
near-field matter-wave interferometer.
The Talbot–Lau concept was already known from optics
[204, 230] and it has been realized for the first time with atoms
by Clauser [61] who also suggested that this idea would be a
key to super-massive interferometry [62].
The implementation shown in figure 15 refers to a series of
experiments with macromolecules performed in Vienna [63].
Molecules may arrive at the first transmission mask G1 without
any initial spatial coherence. The comb of narrow slits preselects the possible starting points for the subsequent matter
waves to evolve. All slits act individually as coherencepreparing elements. The tight localization in each slit of
G1 imposes sufficient quantum uncertainty in the momentum
of the particles to establish the transverse delocalization
and coherence that is required to illuminate more than two
neighbouring slits in the second grating. Quantum near-field
interference then generates a particle density pattern at the
position of G3 , which is a self-image of G2 [62, 64, 231], as
also seen in figure 8. If G3 is scanned across the interference
pattern, the totally transmitted molecular flux will give a sinelike signal.
The idea of TLI is nowadays used for verifying the wave–
particle duality of massive molecules, for quantum-enhanced
lithography and quantum-enhanced molecule metrology.
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T Juffmann et al
experiment with C60 molecules, where STM imaging was
used to visualize the resulting interference pattern after its
deposition on a reconstructed Si (1 1 1) 7 × 7 surface. Such
images also permit us to visualize the quantum wave–particle
duality. In figure 16, every single spot represents a detected
C60 molecule and although there is no way to predict a
priori the position that any of them will have, the vertical
ensemble average (bottom) shows the expected highly regular
fringe pattern. And even though one would also expect a
classical fringe shadow behind a two-grating arrangement
[232], the expected and observed quantum fringe contrast are
significantly higher—even in the presence of van der Waals
forces. Quantum-enhanced molecule lithography may become
relevant in nanotechnology as a positive and soft non-contact
deposition technique for creating surface modifications on the
nanoscale and with deposition energies as small as 0.1 eV.
6.3.3.
Comoving fluorescence detector for near-field
experiments. Spatially highly resolving imaging methods,
such as scanning tunnelling or single-molecule fluorescence
microscopy require the suppression of molecular surface
diffusion to less than a few ten nanometres. This is not
granted for all molecule–surface combinations. But even
then, position-encoded fluorescence detection can still be
implemented using a mechanical scheme that magnifies the
quantum fringe pattern to the extent that even sizeable surface
diffusion can be completely neglected [237].
In a three-grating interferometer of the Talbot–Lau type
(figure 17), the third grating, G3 , can be regarded as part of
the detector. It is scanned in the direction of the grating vector
k = ex · 2π /d, i.e. normal to the molecular beam and normal
to the extension of the diffraction slits. For each position, the
transmitted molecular flux is then recorded.
In a recent experiment this method was combined
with fluorescence imaging [237] of molecules that were
accumulated on a quartz substrate mounted onto a translation
stage behind G3 . For each position shift of the third grating
x3 , the quartz plate was shifted by ∼4300x3 . This trick
magnifies the molecular interference pattern mechanically to
an extent that it can be easily analysed and that molecular
diffusion on the detection plate can be completely neglected.
When using gravitational velocity selection, a 2D image, such
as shown in figure 17 allows visualizing the velocity-dependent
near-field matter-wave interferogram.
Figure 16. Scan of the surface-deposited molecular interference
pattern behind a Talbot–Lau configuration with two mechanical SiN
gratings and one Si-detection screen. Every single C60 molecule is
depicted as a white dot. A vertical sum over the entire image yields
the blue circles in the bottom part. The interference pattern is
expected to be described by a sine curve, which was also used to fit
the data [83]. Reprinted with permission from [42]. Copyright 2009
by the American Physical Society.
While the coherence preparation and fringe selection still have
to be performed by absorptive gratings, here realized by the
material masks G1 and G3 , the diffraction is then caused by
the Kapitza–Dirac effect, i.e. the interaction of delocalized
matter with a non-resonant standing light wave. Originally
proposed by Kapitza and Dirac for electrons [233] the effect
of phase gratings was first seen for atoms [234], then with
electrons [235] and molecules [60] in far-field diffraction. In
combination with the near-field idea of the TLI it becomes the
Kapitza–Dirac–Talbot–Lau interferometer (KDTLI) [65].
While the KDTLI interferometer still integrates the
position-selective nanomechanical gratings G1 and G3 one can
replace them also by absorptive structures of light. They can,
for instance, be achieved when the photon energy exceeds the
molecular ionization energy [66]. An interferometer based
on three (typically ultraviolet) ionizing laser gratings can
be operated both in space and in the time domain [67, 87].
An optical time-domain ionizing (OTIMA) interferometer is
universal in the sense that it can be applied to a large variety of
different particles, from single atoms to clusters of molecules.
The quantum state in such interferometers may in the future
also be retrieved by using state tomography [234].
6.4. TLI for molecule metrology
All variants of TLI can be applied as tools for molecule
metrology, i.e. the measurement of internal molecular
properties [1]. The basic concept is to use the accurate spatial
modulation of the molecular beam to observe the coupling
of external fields on internal properties, which will cause a
force that shifts the interference fringe pattern. The range of
properties to be investigated is wide. Some experiments have
already been performed to measure the static and dynamic
polarizability [71] as well as the permanent or thermally
activated electric dipole moment of molecules in the range of
1000 amu [39]. For that the shift of the interference pattern was
6.3.2. Interferometric deposition of surface-bound molecular
nanostructures. Molecular TLI as shown in figure 15
intrinsically generates a nanosized molecular density pattern
at a well-defined position behind the second grating. While
in many experiments this pattern is sampled and visualized
by scanning the third grating G3 across the molecular beam,
it is also possible to capture the interference fringes on a
screen to generate a surface-bound molecular nanostructure
[42]. This idea has been pursued in a recent matter-wave
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T Juffmann et al
Figure 17. Quantum interference of tetraphenylporphyrin (TPP) detected in a comoving and mechanically magnifying fluorescence
detection scheme. A local vertical average over the 2D image (red rectangles) allows one to retrieve the quantum fringes (right-hand graph,
black dotted lines). We expect and find a sinusoidal shape of the fringe whose contrast varies with the particles’ height on the screen, i.e.
their most probable velocity. A = 33 µm (v ≈ 300 m s−1 ), B = 371 µm (v ≈ 270 m s−1 ), C = 938 µm (v ≈ 160 m s−1 ) and D = 1234 µm
(v ≈ 140 m s−1 ). Reprinted from [237], with permission from IOP Publishing.
monitored in the presence of an inhomogeneous electrostatic
field (E∇)E. The deflection can, in principle, be used to
spatially separate and sort pure molecular samples starting
from a mixtures of species [38]. The conformational state
dynamics of diazobenzene molecules could be mapped to the
centre-of-mass motion by comparing the observed deflection
with ab initio simulations [15]. Quantum deflection may
thus provide benchmark data for increasingly sophisticated
molecular simulations, both with regard to static properties
or for instance optical absorption cross sections [40].
In the following, we briefly discuss a number of practical
and possibly fundamental limitations for quantum coherence
experiments with massive bodies. We distinguish phase
averaging mechanisms, genuine quantum decoherence and the
possibility of subtle deviations from quantum linearity.
7.1.1. Decoherence. A key distinguishing element in this
comparison is the role of quantum information in the transition
between observable quantum coherences and classical
probabilities. Decoherence theory [21, 211, 212, 239–242]
is applied quantum mechanics and it describes the nonobservability of coherence in a system as a result of the
coupling to its complex physical environment: The exchange
of quantum information, the diffusion of coherence into
the environment by virtue of entangling interactions or
‘measurements’—even without the need for invoking any
human observer—is fully described by established quantum
theory. It reduces quantum coherence effectively if we neglect
to include all partners of the extended many-body system.
Understanding quantum decoherence is also central to the
overall enterprise that is currently being pursued under the
heading of quantum information technologies. Only if we
know and master the effects that destroy quantum phenomena
we can also take measures to extend coherence to an extent
that enables us to build quantum-based sensors or information
processing devices on the mesoscopic scale.
A number of matter-wave experiments were able to
corroborate the ideas of decoherence theory: when particles
are excited, such that each of them spontaneously emits a
7. Prospects of molecular and nanoparticle
interferometry
The present review focused mainly on experimental techniques
that have been used with small molecules and methods that are
required to establish quantum phenomena with large molecules
and nanoparticles. In this outlook, we focus on the interest
and possibilities in extending this approach to high mass and
complexity.
7.1. Decoherence, phase averaging and fundamental
limitations to nanoparticle interferometry
Molecule and nanoparticle interferometry is currently believed
to be a valid and actually even one of the best possible
approaches for testing the quantum superposition principle
in a domain where a number of still speculative models
have emerged over recent years with predictions of non-linear
additions or other modification of quantum physics [35, 238].
20
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T Juffmann et al
photon while they are delocalized inside a MZI (see figure 14),
the overall quantum fringe contrast is reduced, if the emitted
photon wavelength suffices to yield which-path information,
i.e. information about where the particle was [243].
The ‘Copenhagen complementarity’ interpretation, which
states that path information and quantum interference exclude
each other, can nowadays be rephrased in terms of quantum
information theory: information can only be extracted by
a physical particle. If this scattering partner becomes
quantum entangled with the originally delocalized nanoobject, quantum coherence is diluted into the more complex
environment.
In pure de Broglie interference, i.e. experiments which
only consider the centre-of-mass motion, this information
theoretical view can finally be complemented by a third
perspective, namely the analogy to ‘Heisenberg’s microscope’:
when there is no coupling to the internal states, the only way
to extract position information is momentum exchange. In
that case, random kicks by the interacting environment can
dephase each individual delocalized molecule differently to a
degree that the coherence of the ensemble is finally lost. This
effect depends on the separation of the wave packets as well as
on the momentum exchange with the probing radiation [244].
In TLI with C60 decoherence was observed for the first
time both caused by collisions with residual gases in the
vacuum chamber [70, 245] and by thermal radiation emitted
by the molecules themselves [22, 246]. Decoherence is also
believed to be an important reason why molecules appear with a
certain chirality in nature, instead of the energetically favoured
superposition of left- and right-handed configurations [247].
The only way to circumvent decoherence is to isolate
a particle from all external agents.
In practice, this
imposes rather stringent requirements on the background
vacuum or temperature [35, 74–76]. Extrapolations of particle
interferometry with masses around 1010 amu are conceptually
still compatible with existing technologies, demanding as they
may be in practice [35, 74]. This will require cooling of the
entire experiment to the temperature of liquid helium in order
to reduce the base pressure and the level of black body radiation
to a level acceptable for high-mass interference [248].
as they shift a classical particle beam. Also different modes
of interferometer vibrations will reduce the quantum contrast,
without extracting quantum information [109].
The OTIMA concept [67, 68] eliminates a large number
of dispersive effects and is therefore particularly promising
for this kind of macro-interference. In addition, it starts
from spatially and spectrally incoherent sources, which is a
tremendous benefit, given the difficulties in preparing large
ensembles of ultracold and similar nanoparticles. Also singlegrating experiments become feasible again, once ultracold
nanoparticle sources become available.
7.1.3. Is quantum mechanics all there is? In neither one of
the two above arguments the superposition principle is ever
broken. Conceptually, this leads either to an ever-growing
entanglement in the universe or a many-world interpretation
of quantum mechanics [251].
This is the reason why a number of proposals have been
made in the attempt to establish ‘reality’, i.e. physical states
without the superposition of classically mutually exclusive
phenomena. This has for instance been done by adding
non-linearities to the Schrödinger equation [17, 19, 252] or by
exploring the relation of quantum mechanics in combination
with gravity theory [18, 36, 37].
This summarizes to a picture where quantum mechanics,
at present, is the best-confirmed theory of non-relativistic
physics—and even better tested in its relativistic form of
quantum electrodynamics—but where a growing number of
models emerge to explore potential limits and modifications
of the superposition principle for very massive bodies.
8. Conclusion
This report has been written to sketch the present state of the
art in matter-wave interferometry with particles more complex
than single atoms. We have discussed several experimental
techniques to prepare, manipulate and detect molecular
beams. Applications in mass spectroscopy, in surface studies,
explorations of van der Waals forces, characterization of
molecular bonds and energies as well as demonstrations
of fundamental quantum phenomena with complex particles
show the wide applicability of mesoscopic matter waves.
Coherent optics with large molecules is still a young
field and we expect major progress in future years:
more sophisticated cooling techniques and coherent beam
sources will open new possibilities in manipulating complex
nanoparticles. Coupling internal with external degrees of
freedom is expected to lead to new forms of spectroscopy.
Improved coherent control of motional states shall take us
to coherent molecular microscopy and lithography. Modern
experiments in molecule interferometry cover de Broglie
wavelengths between 100 fm and 100 µm with energies
ranging between 100 neV and 10 keV.
Advances in the manipulation of single molecules open
a new reductionist way to molecular beam physics with full
quantum control at the level of individual molecular systems.
The advent of measurement-induced gratings made of
light has led to new experiments that will allow testing quantum
7.1.2.
Phase averaging. In contrast to decoherence,
phase averaging may appear trivial: any advanced quantum
experiment may suffer from many external perturbations,
which do not invoke quantum information arguments and
yet they impose very relevant boundary conditions to highmass interference experiments. Some of the phase averaging
mechanisms may even be related to fundamental physics from
outside of quantum mechanics, such as gravitational wave
background noise [249] or space–time fluctuations [250].
However, most causes for phase averaging are related
to a much more basic and lab-oriented level. This includes
dispersive effects in interferometry with molecular beams of
finite velocity spread: the Coriolis-force on the Earth, the
particle’s free fall in the Earth’s gravitational field, the van
der Waals interaction with solid surfaces or the deflection in
an inhomogeneous electric or magnetic field may all influence
the phase of the quantum fringe very much in the same way
21
Rep. Prog. Phys. 76 (2013) 086402
T Juffmann et al
theory in an unprecedented mass range. Several breakthroughs
throughout recent years have given credibility to nanoparticle
interferometry and the possibility to quantitatively test the
linearity of quantum physics in a new domain.
A fascinating aspect of quantum physics with complex
compounds is the seemingly unlimited number of test objects,
ranging from tailored inorganic materials, over metals and
semiconductor clusters or nanospheres up to biomolecules and
nanomaterials at the interface to life. This opens the access to
an unprecedented wealth of interaction possibilities to tailor
and tune the particles to the needs of the experiments.
Quantum-interference-assisted metrology, based on the
coupling of internal and external degrees of freedom can
be exploited for new measurements of electronic, magnetic,
optical or structural properties, and the exchange of quantum
information between internal and external states may open a
new avenue to fundamental quantum studies.
We conclude by expressing the desire that many partners
will join us in our research. The complexity of the subject
is enormous and many hands and brains will be needed to
harvest the fruits that grow on the wide field of molecular and
nanoparticle quantum optics.
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Acknowledgments
MA owes special thanks to Anton Zeilinger with whom we
started the first quantum experiments with macromolecules
and who has remained a driving force in questioning the
status quo of knowledge. We thank our co-workers at the
University of Vienna and the University of Southampton as
well as our collaboration partners, in particular the groups
around Klaus Hornberger, Marcel Mayor, Ori Cheshnovsky,
Uzi Even, Angelo Bassi and Markus Aspelmeyer, for their
work and inspiration that make matter-wave optics with
clusters, molecules and nanoparticles an exciting adventure.
We thank the Austrian Science Fund, FWF for financial
support in the projects Z149-N16 (Wittgenstein) as well as
the European Research Council in project (ERC AdG 320694
PROBIOTIQUS) and the European Commission in project
(304886 NANOQUESTFIT). HU thanks the UK funding
agency EPSRC (EP/J014664/1), the Foundational Questions
Institute (FQXi-RFP3-1021) and the John F Templeton
foundation (ID 39530) for support. TJ thanks the Gordon and
Betty Moore Foundation for support.
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